diff options
| author | zitian-gao <zitian.gao@outlook.com> | 2025-05-27 16:45:31 +0800 |
|---|---|---|
| committer | zitian-gao <zitian.gao@outlook.com> | 2025-05-27 16:45:31 +0800 |
| commit | 7c792461c8e4e4f1f8734fed143630c74e76b27f (patch) | |
| tree | cf6341ff9f2727424751da7a11a3bea6c39015bb /Qwen2.5-Eval/evaluation/data/college_math/test.jsonl | |
| parent | 16815c8c5ec263c4bd1a0af60030c1c0efa1421e (diff) | |
init eval
Diffstat (limited to 'Qwen2.5-Eval/evaluation/data/college_math/test.jsonl')
| -rwxr-xr-x | Qwen2.5-Eval/evaluation/data/college_math/test.jsonl | 2818 |
1 files changed, 2818 insertions, 0 deletions
diff --git a/Qwen2.5-Eval/evaluation/data/college_math/test.jsonl b/Qwen2.5-Eval/evaluation/data/college_math/test.jsonl new file mode 100755 index 0000000..1f88418 --- /dev/null +++ b/Qwen2.5-Eval/evaluation/data/college_math/test.jsonl @@ -0,0 +1,2818 @@ +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.61", "question": "Simplify: $-10-4(n-5)$", "answer": "$10-4 n$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.9.7", "question": "Solve the following systems of equation: \n$x y=90$ \n$(x-5)(y+1)=120$", "answer": "$(45,2),(-10,-9)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.42", "question": "Evaluate the expression: $\\frac{8 i}{6-7 i}$", "answer": "$\\frac{48 i-56}{85}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.9", "question": "Find three consecutive integers such that their sum is 108.", "answer": "$35,36,37$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.31", "question": "Find a quadratic equation with the solutions $2 \\pm \\sqrt{6}$.", "answer": "$x^{2}-4 x-2=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.5", "question": "Specify the domain of the function: $f(x)=x^{2}-3 x-4$", "answer": "all real numbers", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.21", "question": "Find the square: $(p+7)^{2}$", "answer": "$p^{2}+14 p+49$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.36", "question": "Simplify: $\\frac{k^{2}-12 k+32}{k^{2}-64}$", "answer": "$\\frac{k-8}{k+4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.47", "question": "Distribute: $-9 x(4-x)$", "answer": "$-36 x+9 x^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.4", "question": "Write the number in scientific notation: 1.09", "answer": "$1.09 \\times 10^{0}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.15", "question": "Solve the proportion: $\\frac{3}{10}=\\frac{a}{a+2}$", "answer": "$a=\\frac{6}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.31", "question": "Solve the compound inequality and give interval notation: $2 x+9 \\geqslant 10 x+1$ and $3 x-2<7 x+2$", "answer": "$-1<x \\leqslant 1:(-1,1]$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.31", "question": "Simplify: $\\frac{\\sqrt{2 p^{2}}}{\\sqrt{3 p}}$", "answer": "$\\frac{\\sqrt{6 p}}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.28", "question": "Evaluate $\\left(7+4 m+8 m^{4}\\right)-\\left(5 m^{4}+1+6 m\\right)$.", "answer": "$3 m^{4}-2 m+6$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.1", "question": "A merchant bought some pieces of silk for $\\$ 900$. Had he bought 3 pieces more for the same money, he would have paid $\\$ 15$ less for each piece. Find the number of pieces purchased.", "answer": "12", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.19", "question": "Solve the equation: $9^{2 n+3}=243$", "answer": "$-\\frac{1}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.5.11", "question": "Find the slope of a line perpendicular to the given line: $y=-\\frac{1}{3} x$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.19", "question": "Evaluate the expression: $(-8)-(-5)$", "answer": "-3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.11", "question": "Solve the equation: $\\frac{3}{2}\\left(\\frac{7}{3} n+1\\right)=\\frac{3}{2}$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.28", "question": "Evaluate the expression: $7-3$", "answer": "4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.4", "question": "Solve: $4 x^{3}-2=106$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.35", "question": "Solve the equation: $13 a=-143$", "answer": "-11", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.77", "question": "Evaluate the expression: $\\left(-\\frac{15}{8}\\right)+\\frac{5}{3}$", "answer": "$-\\frac{5}{24}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.34", "question": "Simplify the expression: $\\frac{n-7}{n^{2}-2 n-35} \\div \\frac{9 n+54}{10 n+50}$", "answer": "$\\frac{10}{9(n+6)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.43", "question": "An electrician cuts a $30$ ft piece of wire into two pieces. One piece is $2$ ft longer than the other. Find the lengths of the two pieces.", "answer": "14,16", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.29", "question": "Build up each fraction by finding the Least Common Denominator: $\\frac{4 x}{x^{2}-x-6}, \\frac{x+2}{x-3}$", "answer": "$\\frac{4 x}{(x-3)(x+2)}, \\frac{x^{2}+4 x+4}{(x-3)(x+2)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.17", "question": "Evaluate using the given values: $k \\times 3^{2}-(j+k)-5$; $j=4, k=5$", "answer": "36", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.37", "question": "Simplify: $-8 \\sqrt{180 x^{4} y^{2} z^{4}}$", "answer": "$-48 x^{2} z^{2} y \\sqrt{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.18", "question": "Solve the following systems of equation: \n$x+2 y-3 z=9$\n$2 x-y+2 z=-8$\n$3 x-y-4 z=3$", "answer": "$(-1,2,-2)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.3", "question": "Find the product: $(1+3 p)(1-3 p)$", "answer": "$1-9 p^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.13", "question": "Multiply and simplify: $(2+2 \\sqrt{2})(-3+\\sqrt{2})$", "answer": "$-2-4 \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.3", "question": "Solve the equation $\\frac{f}{g} x=b$ for $\\mathrm{x}$.", "answer": "$x=\\frac{\\mathrm{gb}}{f}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.20", "question": "Simplify and write the answer in scientific notation: $\\frac{7.2 \\times 10^{-1}}{7.32 \\times 10^{-1}}$", "answer": "$9.836 \\times 10^{-1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.19", "question": "Divide: $\\frac{10 x^{2}-32 x+9}{10 x-2}$", "answer": "$x-3+\\frac{3}{10 x-2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.12", "question": "Solve the absolute value equation: $|9 p+6|=3$", "answer": "$-\\frac{1}{3},-1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.32", "question": "Add or subtract the rational expressions and simplify: $\\frac{4 x}{x^{2}-2 x-3}-\\frac{3}{x^{2}-5 x+6}$", "answer": "$\\frac{4 x+1}{(x+1)(x-2)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.5", "question": "Find the principal that will amount to $\\$ 2500$ if invested at $5 \\%$ interest compounded semiannually for 7.5 years.", "answer": "1726.16", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.18", "question": "Solve the compound inequality and give interval notation: $1 \\leqslant \\frac{p}{8} \\leqslant 0$", "answer": "No solution : $\\oslash$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.25", "question": "Simplify and express with positive exponents: $\\frac{3 y^{-\\frac{5}{4}}}{y^{-1} \\cdot 2 y^{-\\frac{1}{3}}}$", "answer": "$\\frac{3 y^{\\frac{1}{12}}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.25", "question": "Solve the proportion: $\\frac{2}{p+4}=\\frac{p+5}{3}$", "answer": "$p=-7,-2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.23", "question": "Solve the equation for $x$: $\\frac{3}{x+2}+\\frac{x-1}{x+5}=\\frac{5 x+20}{6 x+24}$", "answer": "4,7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.27", "question": "Combine Like Terms: $-7 x-2 x$", "answer": "$-9 x$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.6", "question": "State if the given functions are inverses:\n$f(x)=\\frac{x-5}{10}$\n$h(x)=10 x+5$", "answer": "Yes", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.24", "question": "Solve the equation $\\mathrm{q}=6(L-p)$ for $\\mathrm{L}$.", "answer": "$L=\\frac{q+6 p}{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.6", "question": "Divide: $\\frac{5 p^{4}+16 p^{3}+16 p^{2}}{4 p}$", "answer": "$\\frac{5 p^{3}}{4}+4 p^{2}+4 p$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.9", "question": "Find the product: $(8 b+3)(7 b-5)$", "answer": "$56 b^{2}-19 b-15$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.26", "question": "Solve the following systems of equation: \n$2 x+y=z$\n$4 x+z=4 y$\n$y=x+1$", "answer": "$(1,2,4)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.40", "question": "Solve the equation: $4+\\frac{a}{3}=1$", "answer": "-9", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.27", "question": "Solve: $n^{2}=-21+10 n$", "answer": "7,3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.34", "question": "Solve the equation: $\\log _{7}-3 n=4$", "answer": "$-\\frac{2401}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.27", "question": "Solve the following equation: $x^{8}-17 x^{4}+16=0$", "answer": "$\\pm 1, \\pm i, \\pm 2, \\pm 2 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.17", "question": "The rate of a stream is 3 miles an hour. If a crew rows downstream for a distance of 8 miles and then back again, the round trip occupying 5 hours, what is the rate of the crew in still water?", "answer": "$\\mathrm{r}=5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.15", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{u^{2}}{4 u^{0} v^{3} \\cdot 3 v^{2}}$", "answer": "$\\frac{u^{2}}{12 v^{5}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.40", "question": "Simplify: $\\frac{2 x^{4} y^{5} \\cdot 2 z^{10} x^{2} y^{7}}{\\left(x y^{2} z^{2}\\right)^{4}}$", "answer": "$4 x^{2} y^{4} z^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.26", "question": "Solve the equation: $625^{2 x}=25$", "answer": "$\\frac{1}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.22", "question": "Simplify the expression: $\\frac{2 r}{r+6} \\div \\frac{2 r}{7 r+42}$", "answer": "7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.3", "question": "State if the given functions are inverses:\n$f(x)=\\frac{-x-1}{x-2}$\n$g(x)=\\frac{-2 x+1}{-x-1}$", "answer": "Yes", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.26", "question": "Solve the proportion: $\\frac{5}{n+1}=\\frac{n-4}{10}$", "answer": "$n=-6,9$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.22", "question": "Solve the absolute value equation: $3-|6 n+7|=-40$", "answer": "$6,-\\frac{25}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.12", "question": "Add or subtract the rational expressions and simplify: $\\frac{2 a-1}{3 a^{2}}+\\frac{5 a+1}{9 a}$", "answer": "$\\frac{5 a^{2}+7 a-3}{9 a^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.11", "question": "The sum of the ages of a china plate and a glass plate is 16 years. Four years ago the china plate was three times the age of the glass plate. Find the present age of each plate.", "answer": "10,6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.14", "question": "Find the slope of the line through the points $(13,-2)$ and $(7,7)$.", "answer": "$-\\frac{3}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.11", "question": "Solve the equation for $x$: $\\frac{4-x}{1-x}=\\frac{12}{3-x}$", "answer": "$-5,0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.71", "question": "Simplify: $5(1-6 k)+10(k-8)$", "answer": "$-75-20 k$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.12", "question": "Simplify: $\\left(3^{2}\\right)^{3}$", "answer": "$3^{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.46", "question": "Evaluate the expression: $\\sqrt{-12} \\sqrt{-2}$", "answer": "$-2 \\sqrt{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.24", "question": "Solve the equation: $-\\frac{149}{16}-\\frac{11}{3} r=-\\frac{7}{4} r-\\frac{5}{4}\\left(-\\frac{4}{3} r+1\\right)$", "answer": "$-\\frac{9}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.1", "question": "Reduce the following radical: $\\sqrt[8]{16 x^{4} y^{6}}$", "answer": "$\\sqrt[4]{4 x^{2} y^{3}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.3", "question": "Solve the equation for $x$: $x+\\frac{20}{x-4}=\\frac{5 x}{x-4}-2$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.7.15", "question": "A page is to have a margin of 1 inch and is to contain $35 \\mathrm{in}^{2}$ of painting. How large must the page be if the length is to exceed the width by 2 inches?", "answer": "$7 \\times 9$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.39", "question": "Solve the equation: $5^{-3 n-3} \\cdot 5^{2 n}=1$", "answer": "-3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.33", "question": "Find a quadratic equation with the solutions $1 \\pm 3 i$.", "answer": "$x^{2}-2 x+10=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.11", "question": "State the excluded values for $\\frac{10 m^{2}+8 m}{10 m}$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.31", "question": "Combine Like Terms: $x-10-6 x+1$", "answer": "$-5 x-9$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.14", "question": "Solve the equation: $\\frac{5}{9}=\\frac{b}{9}$", "answer": "5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.7", "question": "Evaluate using the given values: $5 j+\\frac{k h}{2}$; $h=5, j=4, k=2$", "answer": "25", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.12", "question": "Simplify: $\\frac{5}{2 \\sqrt{3}-\\sqrt{2}}$", "answer": "$\\frac{2 \\sqrt{3}+\\sqrt{2}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.12", "question": "Find the Least Common Denominator: $5 x^{2} y, 25 x^{3} y^{5} z$", "answer": "$25 x^{3} y^{5} z$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.7", "question": "Solve: $0=-8(p-5)$", "answer": "5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.29", "question": "Solve the absolute value equation: $-7+8|-7 x-3|=73$", "answer": "$-\\frac{13}{7}, 1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.30", "question": "Find the inverse of the function: $f(x)=\\frac{5 x-5}{4}$", "answer": "$f^{-1}(x)=\\frac{5+4 x}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.33", "question": "Divide: $\\frac{3 n^{3}+9 n^{2}-64 n-68}{n+6}$", "answer": "$3 n^{2}-9 n-10-\\frac{8}{n+6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.43", "question": "Add or subtract the rational expressions and simplify: $\\frac{2 x+7}{x^{2}-2 x-3}-\\frac{3 x-2}{x^{2}+6 x+5}$", "answer": "$\\frac{-(x-29)}{(x-3)(x+5)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.26", "question": "Solve the following equation: $8 x^{6}+7 x^{3}-1=0$", "answer": "$\\frac{1}{2},-1, \\frac{-1 \\pm i \\sqrt{3}}{4}, \\frac{1 \\pm i \\sqrt{3}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.9", "question": "Simplify: $2 \\sqrt{2}-3 \\sqrt{18}-\\sqrt{2}$", "answer": "$-8 \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.40", "question": "Find the product: $3(2 x+3)(6 x+9)$", "answer": "$36 x^{2}+108 x+81$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.26", "question": "Solve the equation: $3 n=24$", "answer": "8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.12", "question": "Tim can finish a certain job in 10 hours. It takes his wife JoAnn only 8 hours to do the same job. If they work together, how long will it take them to complete the job?", "answer": "$4 \\frac{4}{9}$ days", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.25", "question": "Simplify: $\\frac{a+\\sqrt{\\mathrm{ab}}}{\\sqrt{a}+\\sqrt{b}}$", "answer": "$\\sqrt{a}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.15", "question": "A coin purse contains 18 coins in nickels and dimes. The coins have a total value of $\\$ 1.15$. Find the number of nickels and dimes in the coin purse.", "answer": "$13 \\mathrm{n}, 5 \\mathrm{~d}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.36", "question": "Simplify: $\\frac{x^{-3}+y^{-3}}{x^{-2}-x^{-1} y^{-1}+y^{-2}}$", "answer": "$\\frac{x+y}{x y}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.3.22", "question": "Solve: $\\frac{-9 \\cdot 2-(3-6)}{1-(-2+1)-(-3)}$", "answer": "-3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.1.3", "question": "Solve: $\\sqrt{6 x-5}-x=0$", "answer": "1,5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.10", "question": "Multiply and simplify: $5 \\sqrt{15}(3 \\sqrt{3}+2)$", "answer": "$45 \\sqrt{5}+10 \\sqrt{15}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.9", "question": "Find a quadratic equation with the solutions $-4$ and 11.", "answer": "$x^{2}-7 x-44=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.16", "question": "Simplify: $\\sqrt{100 n^{3}}$", "answer": "$10 n \\sqrt{n}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.23", "question": "Simplify and express with positive exponents: $\\frac{a^{\\frac{3}{4}} b^{-1} \\cdot b^{\\frac{7}{4}}}{3 b^{-1}}$", "answer": "$\\frac{b^{\\frac{7}{4}} a^{\\frac{3}{4}}}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.8", "question": "Find a quadratic equation with the solutions $-2$ and $-5$.", "answer": "$x^{2}+7 x+10=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.1", "question": "Find the product: $6(p-7)$", "answer": "$6 p-42$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.6", "question": "Solve the equation: $-4-b=8$", "answer": "-12", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.50", "question": "Find quotient: $\\frac{30}{6}$", "answer": "5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.12", "question": "Solve the equation: $4 r=-28$", "answer": "-7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.39", "question": "Simplify: $\\frac{q^{3} r^{2} \\cdot\\left(2 p^{2} q^{2} r^{3}\\right)^{2}}{2 p^{3}}$", "answer": "$2 q^{7} r^{8} p$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.9.5", "question": "Solve the following systems of equation: \n$x y=45$ \n$(x+2)(y+1)=70$", "answer": "$(5,9),(18,2.5)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.15", "question": "Simplify: $3 \\sqrt{18}-\\sqrt{2}-3 \\sqrt{2}$", "answer": "$5 \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.38", "question": "The sum of the ages of two children is 16 years. Four years ago, the age of the older child was three times the age of the younger child. Find the present age of each child.", "answer": "10,6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.3", "question": "Solve the absolute value equation: $|b|=1$", "answer": "$1,-1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.32", "question": "Divide: $\\frac{x^{3}-16 x^{2}+71 x-56}{x-8}$", "answer": "$x^{2}-8 x+7$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.13", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{u^{2} v^{-1}}{2 u^{0} v^{4} \\cdot 2 u v}$", "answer": "$\\frac{u}{4 v^{6}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.29", "question": "Simplify: $\\frac{2 x^{7} y^{5}}{3 x^{3} y \\cdot 4 x^{2} y^{3}}$", "answer": "$\\frac{x^{2} y}{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.25", "question": "Solve the following equation: $8 x^{6}-9 x^{3}+1=0$", "answer": "$1, \\frac{1}{2}, \\frac{-1 \\pm i \\sqrt{3}}{4}, \\frac{-1 \\pm i \\sqrt{3}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.29", "question": "Simplify: $\\sqrt{320 x^{4} y^{4}}$", "answer": "$8 x^{2} y^{2} \\sqrt{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.25", "question": "Combine the following radicals: $\\sqrt{a} \\sqrt[4]{a^{3}}$", "answer": "$a \\sqrt[4]{a}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.39", "question": "Divide: $\\frac{r^{3}-r^{2}-16 r+8}{r-4}$", "answer": "$r^{2}+3 r-4-\\frac{8}{r-4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.24", "question": "Find the product: $(4 x+8)\\left(4 x^{2}+3 x+5\\right)$", "answer": "$16 x^{3}+44 x^{2}+44 x+40$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.9", "question": "Solve the absolute value equation: $|9+7 x|=30$", "answer": "$3,-\\frac{39}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.7", "question": "Solve: $(x+2)^{5}=-243$", "answer": "-5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.25", "question": "Evaluate the expression: $(-6)+3$", "answer": "-3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.9", "question": "Multiply and simplify: $-5 \\sqrt{15}(3 \\sqrt{3}+2)$", "answer": "$-45 \\sqrt{5}-10 \\sqrt{15}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.15", "question": "Given $f(n)=-2|-n-2|+1$, find $f(-6)$", "answer": "-7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.33", "question": "Simplify: $\\left(\\frac{2 y^{17}}{\\left(2 x^{2} y^{4}\\right)^{4}}\\right)^{3}$", "answer": "$\\frac{y^{3}}{512 x^{24}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.33", "question": "Find product: $(10)(-8)$", "answer": "-80", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.14", "question": "Find the product: $(6 a+4)(a-8)$", "answer": "$6 a^{2}-44 a-32$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.10", "question": "Simplify and leave your answer as an improper fraction: $\\frac{48}{18}$", "answer": "$\\frac{8}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.3", "question": "Simplify: $\\frac{4+2 \\sqrt{3}}{5 \\sqrt{4}}$", "answer": "$\\frac{2+\\sqrt{3}}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.25", "question": "Let the first angle of a triangle be $x$ degrees. Find the measures of the second and third angles.", "answer": "$25,100,55$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.40", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{2 m p n^{-3}}{\\left(m^{0} n^{-4} p^{2}\\right)^{3} \\cdot 2 n^{2} p^{0}}$", "answer": "$\\frac{m n^{7}}{p^{5}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.3", "question": "Solve the equation: $102=-7 r+4$", "answer": "-14", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.4", "question": "How much alcohol must be added to 24 gallons of a $14 \\%$ solution of alcohol in order to produce a $20 \\%$ solution?", "answer": "1,8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.21", "question": "Solve the absolute value equation: $8|x+7|-3=5$", "answer": "$-6,-8$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.28", "question": "The time required to drive a fixed distance varies inversely as the speed. It takes $5 \\mathrm{hr}$ at a speed of $80 \\mathrm{~km} / \\mathrm{h}$ to drive a fixed distance. How long will it take to drive the same distance at a speed of $70 \\mathrm{~km} / \\mathrm{h}$ ?", "answer": "$5.7 \\mathrm{hr}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.38", "question": "Find the inverse of the function: $f(x)=-\\frac{3 x}{4}$", "answer": "$f^{-1}(x)=-\\frac{4 x}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.35", "question": "Solve: $7 n^{2}-n+7=7 n+6 n^{2}$", "answer": "7,1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.40", "question": "Divide: $\\frac{2 x^{3}+12 x^{2}+4 x-37}{2 x+6}$", "answer": "$x^{2}+3 x-7+\\frac{5}{2 x+6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.34", "question": "Solve the equation: $32^{2 p-2} \\cdot 8^{p}=\\left(\\frac{1}{2}\\right)^{2 p}$", "answer": "$\\frac{2}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.25", "question": "Solve the equation: $180=12 x$", "answer": "15", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.1", "question": "Simplify: $\\frac{4+2 \\sqrt{3}}{\\sqrt{9}}$", "answer": "$\\frac{4+2 \\sqrt{3}}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.32", "question": "A family drove to a resort at an average speed of $25 \\mathrm{mph}$ and later returned over the same road at an average speed of $40 \\mathrm{mph}$. Find the distance to the resort if the total driving time was $13 \\mathrm{~h}$.", "answer": "200", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.8", "question": "A cistern can be filled by one pipe in 20 minutes and by another in 30 minutes. How long will it take both pipes together to fill the tank?", "answer": "$12 \\mathrm{~min}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.3", "question": "Evaluate $\\frac{x-3}{x^{2}-4 x+3}$ when $x=-4$", "answer": "$-\\frac{1}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.13", "question": "Find the slope of the line through the points $(-15,10)$ and $(16,-7)$.", "answer": "$-\\frac{17}{31}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.18", "question": "Solve: $\\frac{\\frac{15}{x^{2}}-\\frac{2}{x}-1}{\\frac{4}{x^{2}}-\\frac{5}{x}+4}$", "answer": "$-\\frac{(x-3)(x+5)}{4 x^{2}-5 x+4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.6", "question": "Solve the following system of equations by elimination:\n$5 x-5 y=-15$\n$5 x-5 y=-15$", "answer": "Infinite number of solutions", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.37", "question": "Solve: $-8(n-7)+3(3 n-3)=41$", "answer": "-6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.16", "question": "Solve: $4+3 x=-12 x+4$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.16", "question": "Solve the compound inequality and give interval notation: $-11 \\leqslant n-9 \\leqslant-5$", "answer": "$-2 \\leqslant n \\leqslant 4:[-2,4]$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.19", "question": "Simplify: $\\frac{1}{1+\\sqrt{2}}$", "answer": "$\\sqrt{2}-1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.60", "question": "Simplify: $-10(x-2)-3$", "answer": "$-10 x+17$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.27", "question": "Solve the equation: $20 b=-200$", "answer": "-10", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.9", "question": "Solve: $-2+2(8 x-7)=-16$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.24", "question": "Solve: $-7(1+b)=-5-5 b$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.38", "question": "Solve the equation: $3 x-3=-3$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.32", "question": "Find product: $\\left(-\\frac{17}{9}\\right)\\left(-\\frac{3}{5}\\right)$", "answer": "$-\\frac{19}{20}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.28", "question": "Find the square: $(4 m-n)^{2}$", "answer": "$16 m^{2}-8 m n+n^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.11", "question": "Simplify: $-\\sqrt[4]{112}$", "answer": "$-2 \\sqrt[4]{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.11", "question": "Multiply and simplify: $5 \\sqrt{10}(5 n+\\sqrt{2})$", "answer": "$25 n \\sqrt{10}+10 \\sqrt{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.7.4", "question": "The length of a room is $8 \\mathrm{ft}$ greater than its width. If each dimension is increased by $2 \\mathrm{ft}$, the area will be increased by $60 \\mathrm{sq}$. ft. Find the dimensions of the room.", "answer": "$10 \\mathrm{ft} x 18 \\mathrm{ft}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.4", "question": "Solve the equation for $x$: $\\frac{x^{2}+6}{x-1}+\\frac{x-2}{x-1}=2 x$", "answer": "$-1,4$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.79", "question": "Simplify: $\\left(4-2 k^{2}\\right)+\\left(8-2 k^{2}\\right)$", "answer": "$-4 k^{2}+12$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.20", "question": "Simplify: $\\frac{21 k}{24 k^{2}}$", "answer": "$\\frac{7}{8 k}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.37", "question": "Evaluate $\\left(2 n^{2}+7 n^{4}-2\\right)+\\left(2+2 n^{3}+4 n^{2}+2 n^{4}\\right)$.", "answer": "$9 n^{4}+2 n^{3}+6 n^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.18", "question": "Simplify the expression: $\\frac{p-8}{p^{2}-12 p+32} \\div \\frac{1}{p-10}$", "answer": "$\\frac{p-10}{p-4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.38", "question": "Find a quadratic equation with the solutions $\\frac{2 \\pm 5 i}{3}$.", "answer": "$9 x^{2}-12 x+29=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.40", "question": "Solve the equation: $4^{3 r} \\cdot 4^{-3 r}=\\frac{1}{64}$", "answer": "No solution", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.33", "question": "Solve the equation: $\\log _{11}(x+5)=-1$", "answer": "$-\\frac{54}{11}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.29", "question": "Given $k(n)=|n-1|$, find $k(3)$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.18", "question": "Evaluate the expression: $5-7$", "answer": "-2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.31", "question": "Solve the absolute value equation: $|5 x+3|=|2 x-1|$", "answer": "$-\\frac{4}{3},-\\frac{2}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.16", "question": "Solve the equation: $4^{-3 v}=64$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.42", "question": "Solve: $-4(1+a)=2 a-8(5+3 a)$", "answer": "-2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.11", "question": "How many kilograms of hard candy that cost $\\$ 7.50$ per kilogram must be mixed with $24 \\mathrm{~kg}$ of jelly beans that cost $\\$ 3.25$ per kilogram to make a mixture that sells for $\\$ 4.50$ per kilogram?", "answer": "10", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.27", "question": "Divide: $\\frac{4 x^{2}-33 x+28}{4 x-5}$", "answer": "$x-7-\\frac{7}{4 x-5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.7", "question": "Solve the equation: $\\frac{635}{72}=-\\frac{5}{2}\\left(-\\frac{11}{4}+x\\right)$", "answer": "$-\\frac{7}{9}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.11", "question": "A $\\$ 10,000$ Treasury Bill earned $16 \\%$ compounded monthly. If the bill matured in 2 years, what was it worth at maturity?", "answer": "13742.19", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.5", "question": "Multiply and simplify: $\\sqrt[3]{4 x^{3}} \\cdot \\sqrt[3]{2 x^{4}}$", "answer": "$2 x^{2} \\sqrt[3]{x}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.12", "question": "Simplify the expression: $\\frac{9}{b^{2}-b-12} \\div \\frac{b-5}{b^{2}-b-12}$", "answer": "$\\frac{9}{b-5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.24", "question": "You are buying carpet to cover a room that measures $38 \\mathrm{ft}$ by $40 \\mathrm{ft}$. The carpet cost $\\$ 18$ per square yard. How much will the carpet cost?", "answer": "$\\$ 3040$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.2", "question": "A number of men subscribed a certain amount to make up a deficit of $\\$ 100$ but 5 men failed to pay and thus increased the share of the others by $\\$ 1$ each. Find the amount that each man paid.", "answer": "$\\$ 4$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.45", "question": "Solve the equation $3 \\mathrm{x}+2 \\mathrm{y}=7$ for $\\mathrm{y}$.", "answer": "$y=\\frac{7-3 x}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.13", "question": "State the excluded values for $\\frac{r^{2}+3 r+2}{5 r+10}$", "answer": "-2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.37", "question": "Solve the following equation: $3(y+1)^{2}-14(y+1)=5$", "answer": "$4,-\\frac{4}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.3.6", "question": "Solve: $7-5+6$", "answer": "8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.17", "question": "Evaluate the expression: $\\log _{4} 16$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.36", "question": "Find the product: $(b+4)(b-4)$", "answer": "$b^{2}-16$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.13", "question": "Simplify: $\\frac{3}{4-3 \\sqrt{3}}$", "answer": "$\\frac{-12-9 \\sqrt{3}}{11}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.78", "question": "Evaluate the expression: $\\frac{3}{2}+\\frac{9}{7}$", "answer": "$\\frac{39}{14}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.3.12", "question": "Solve: $\\frac{-10-6}{(-2)^{2}}-5$", "answer": "-9", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.15", "question": "Solve: $\\frac{\\frac{a^{2}-b^{2}}{4 a^{2} b}}{\\frac{a+b}{16 \\mathrm{ab}^{2}}}$", "answer": "$\\frac{4 b(a-b)}{a}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.1.16", "question": "Solve: $\\sqrt{2-3 x}-\\sqrt{3 x+7}=3$", "answer": "$-\\frac{7}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.7.6", "question": "The length of a rectangle is $15 \\mathrm{ft}$ greater than its width. If each dimension is decreased by $2 \\mathrm{ft}$, the area will be decreased by $106 \\mathrm{ft}^{2}$. Find the dimensions.", "answer": "$20 \\mathrm{ft} x 35 \\mathrm{ft}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.17", "question": "The age of the older of two boys is twice that of the younger; 5 years ago it was three times that of the younger. Find the age of each.", "answer": "10,20", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.4", "question": "Multiply and simplify: $\\sqrt{5 r^{3}} \\cdot-5 \\sqrt{10 r^{2}}$", "answer": "$-25 r^{2} \\sqrt{2 r}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.3.23", "question": "Solve: $\\frac{2^{3}+4}{-18-6+(-4)-[-5(-1)(-5)]}$", "answer": "-4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.48", "question": "Find quotient: $\\frac{-13}{8} \\div \\frac{-15}{8}$", "answer": "$\\frac{13}{15}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.27", "question": "A local zoning ordinance says that a house's \"footprint\" (area of its ground floor) cannot occupy more than $\\frac{1}{4}$ of the lot it is built on. Suppose you own a $\\frac{1}{3}$ acre lot, what is the maximum allowed footprint for your house in square feet? in square inches? $\\left(1\\right.$ acre $\\left.=43560 \\mathrm{ft}^{2}\\right)$", "answer": "$3630 \\mathrm{ft}^{2}, 522,720 \\mathrm{in}^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.5", "question": "Solve: $\\frac{\\frac{1}{a^{2}}-\\frac{1}{a}}{\\frac{1}{a^{2}}+\\frac{1}{a}}$", "answer": "$-\\frac{a-1}{a+1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.21", "question": "Find a quadratic equation with the solutions $\\pm 5$.", "answer": "$x^{2}-25=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.30", "question": "At a constant temperature, the volume of a gas varies inversely as the pres- sure. If the pressure of a certain gas is 40 newtons per square meter when the volume is 600 cubic meters what will the pressure be when the volume is reduced by 240 cubic meters?", "answer": "$100 \\mathrm{~N}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.39", "question": "Given the points $(x, 5)$ and $(8,0)$ and a slope of $-\\frac{5}{6}$, find the value of $x$.", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.21", "question": "Find product: $(-4+5 i)(2-7 i)$", "answer": "$27+38 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.1", "question": "Simplify: $\\sqrt{245}$", "answer": "$7 \\sqrt{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.27", "question": "Solve the equation: $\\frac{3}{2}\\left(v+\\frac{3}{2}\\right)=-\\frac{7}{4} v-\\frac{19}{6}$", "answer": "$-\\frac{5}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.25", "question": "It takes Sally $10 \\frac{1}{2}$ minutes longer than Patricia to clean up their dorm room. If they work together, they can clean it in 5 minutes. How long will it take each of them if they work alone?", "answer": "$\\mathrm{P}=7, \\mathrm{~S}=17 \\frac{1}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.2", "question": "Write the number in scientific notation: 0.000744", "answer": "$7.44 \\times 10^{-4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.7.5", "question": "The length of a rectangular lot is 4 rods greater than its width, and its area is 60 square rods. Find the dimensions of the lot.", "answer": "$6 \\times 10$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.1", "question": "Solve the equation $a b=c$ for $b$.", "answer": "$b=\\frac{c}{a}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.2", "question": "State if the given functions are inverses:\n$g(x)=\\frac{4-x}{x}$\n$f(x)=\\frac{4}{x}$", "answer": "No", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.23", "question": "Solve: $\\frac{x-4+\\frac{9}{2 x+3}}{x+3-\\frac{5}{2 x+3}}$", "answer": "$\\frac{x-3}{x+4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.10", "question": "Solve: $\\frac{\\frac{-5}{b-5}-3}{\\frac{10}{b-5}+6}$", "answer": "$-\\frac{1}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.5", "question": "Solve: $66=6(6+5 x)$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.25", "question": "Find product: $(1+5 i)(2+i)$", "answer": "$-3+11 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.7", "question": "Find the product: $(5 n-8)(5 n+8)$", "answer": "$25 n^{2}-64$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.2", "question": "Evaluate the expression: $4-(-1)$", "answer": "5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.3.17", "question": "Solve: $[6 \\cdot 2+2-(-6)]\\left(-5+\\left|\\frac{-18}{6}\\right|\\right)$", "answer": "-40", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.55", "question": "Evaluate the expression: $\\frac{3}{7}-\\frac{1}{7}$", "answer": "$\\frac{2}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.54", "question": "Evaluate the expression: $i^{68}$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.20", "question": "A chair lift at the Divide ski resort in Cold Springs, WY is 4806 feet long and takes 9 minutes. What is the average speed in miles per hour? How many feet per second does the lift travel?", "answer": "$6.608 \\mathrm{mi} / \\mathrm{hr}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.26", "question": "Build up each fraction by finding the Least Common Denominator: $\\frac{5 x+1}{x^{2}-3 x-10}, \\frac{4}{x-5}$", "answer": "$\\frac{5 x+1}{(x-5)(x+2)}, \\frac{4 x+8}{(x-5)(x+2)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.27", "question": "Add or subtract the rational expressions and simplify: $\\frac{x}{x^{2}+5 x+6}-\\frac{2}{x^{2}+3 x+2}$", "answer": "$\\frac{x-3}{(x+3)(x+1)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.37", "question": "Solve the equation: $\\log _{5}(-10 x+4)=4$", "answer": "$-\\frac{621}{10}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.37", "question": "Simplify: $6 \\sqrt[3]{-54 m^{8} n^{3} p^{7}}$", "answer": "$-18 m^{2} n p^{2} \\sqrt[3]{2 m^{2} p}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.34", "question": "Add or subtract the rational expressions and simplify: $\\frac{x-1}{x^{2}+3 x+2}+\\frac{x+5}{x^{2}+4 x+3}$", "answer": "$\\frac{2 x+7}{x^{2}+5 x+6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.33", "question": "Solve the equation: $4^{3 k-3} \\cdot 4^{2-2 k}=16^{-k}$", "answer": "$\\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.32", "question": "Simplify: $\\frac{\\sqrt{8 n^{2}}}{\\sqrt{10 n}}$", "answer": "$\\frac{2 \\sqrt{5 n}}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.10", "question": "Simplify: $(5-4 i)+(8-4 i)$", "answer": "$13-8 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.16", "question": "Solve the equation for $x$: $\\frac{x+2}{3 x-1}-\\frac{1}{x}=\\frac{3 x-3}{3 x^{2}-x}$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.37", "question": "Divide: $\\frac{9 p^{3}+45 p^{2}+27 p-5}{9 p+9}$", "answer": "$p^{2}+4 p-1+\\frac{4}{9 p+9}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.28", "question": "Solve: $\\frac{\\frac{x-1}{x+1}-\\frac{x+1}{x-1}}{\\frac{x-1}{x+1}+\\frac{x+1}{x-1}}$", "answer": "$-\\frac{2 x}{x^{2}+1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.6", "question": "Multiply and simplify: $3 \\sqrt[3]{4 a^{4}} \\cdot \\sqrt[3]{10 a^{3}}$", "answer": "$6 a^{2} \\sqrt[3]{5 a}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.56", "question": "Solve: $10 v^{2}-15 v=27+4 v^{2}-6 v$", "answer": "$3,-\\frac{3}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.29", "question": "Combine Like Terms: $k-2+7$", "answer": "$k+5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.11", "question": "Write the number in standard notation: $2 \\times 10^{0}$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.35", "question": "Given the points $(-8, y)$ and $(-1,1)$ and a slope of $\\frac{6}{7}$, find the value of $y$.", "answer": "-5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.30", "question": "Chelsea's age is double Daniel's age. Eight years ago the sum of their ages was 32 . How old are they now?", "answer": "16,32", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.15", "question": "Find three consecutive odd integers such that the sum of the first, twice the second, and three times the third is 70.", "answer": "$9,11,13$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.21", "question": "Solve the equation for $x$: $\\frac{x-2}{x+3}-\\frac{1}{x-2}=\\frac{1}{x^{2}+x-6}$", "answer": "0,5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.1", "question": "A is 60 miles from B. An automobile at A starts for B at the rate of 20 miles an hour at the same time that an automobile at B starts for A at the rate of 25 miles an hour. How long will it be before the automobiles meet?", "answer": "$1 \\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.23", "question": "Solve: $-4 n+11=2(1-8 n)+3 n$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.20", "question": "Combine the following radicals: $\\sqrt{a b} \\sqrt[5]{2 a^{2} b^{2}}$", "answer": "$\\sqrt[10]{4 a^{9} b^{9}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.1", "question": "Simplify and leave your answer as an improper fraction: $\\frac{42}{12}$", "answer": "$\\frac{7}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.27", "question": "Find the slope of the line through the points $(7,-14)$ and $(-8,-9)$.", "answer": "$-\\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.14", "question": "Solve the equation: $27^{-2 n-1}=9$", "answer": "$-\\frac{5}{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.34", "question": "Simplify and write the answer in scientific notation: $\\frac{9.58 \\times 10^{-2}}{1.14 \\times 10^{-3}}$", "answer": "$8.404 \\times 10^{1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.27", "question": "Find a quadratic equation with the solutions $\\pm \\frac{\\sqrt{3}}{4}$.", "answer": "$16 x^{2}-3=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.29", "question": "The sum of the ages of two ships is 12 years. Two years ago, the age of the older ship was three times the age of the newer ship. Find the present age of each ship.", "answer": "8,4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.50", "question": "Evaluate the expression: $\\frac{6+\\sqrt{-32}}{4}$", "answer": "$\\frac{3+2 i \\sqrt{2}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.27", "question": "Simplify: $\\sqrt[3]{-32 x^{4} y^{4}}$", "answer": "$-2 x y \\sqrt[3]{4 x y}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.9", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{2 x^{-3} y^{2}}{3 x^{-3} y^{3} \\cdot 3 x^{0}}$", "answer": "$\\frac{2}{9 y}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.10", "question": "Solve the equation: $\\frac{3}{2}-\\frac{7}{4} v=-\\frac{9}{8}$", "answer": "$\\frac{3}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.32", "question": "Simplify: $-4 \\sqrt[3]{56 x^{2} y^{8}}$", "answer": "$-8 y^{2} \\sqrt[3]{7 x^{2} y^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.42", "question": "Divide: $\\frac{24 b^{3}-38 b^{2}+29 b-60}{4 b-7}$", "answer": "$6 b^{2}+b+9+\\frac{3}{4 b-7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.18", "question": "Simplify: $\\sqrt[6]{256 x^{6}}$", "answer": "$2 x \\sqrt[6]{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.2", "question": "Solve the following system of equations by elimination:\n$-7 x+y=-10$\n$-9 x-y=-22$", "answer": "$(2,4)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.21", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{2 n m^{4}}{\\left(2 m^{2} n^{2}\\right)^{4}}$", "answer": "$\\frac{1}{8 m^{4} n^{7}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.24", "question": "Evaluate the expression: $8-(-1)$", "answer": "9", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.45", "question": "Solve: $k^{2}-7 k+50=3$", "answer": "$\\frac{7+i \\sqrt{139}}{2}, \\frac{7-i \\sqrt{139}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.2", "question": "Solve the equation $\\mathrm{g}=\\frac{h}{i}$ for $\\mathrm{h}$.", "answer": "$\\mathrm{h}=\\mathrm{gi}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.6", "question": "Simplify: $\\sqrt{72}$", "answer": "$6 \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.30", "question": "Combine the following radicals: $\\sqrt[5]{a^{3} b} \\sqrt{a b}$", "answer": "$a \\sqrt[10]{a b^{7}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.33", "question": "Solve the equation: $10=x-4$", "answer": "14", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.26", "question": "Evaluate $\\left(6 v+8 v^{3}\\right)+\\left(3+4 v^{3}-3 v\\right)$.", "answer": "$12 v^{3}+3 v+3$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.17", "question": "Simplify: $-3 \\sqrt{6}-3 \\sqrt{6}-\\sqrt{3}+3 \\sqrt{6}$", "answer": "$-3 \\sqrt{6}-\\sqrt{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.18", "question": "Find the Least Common Denominator: $x^{2}-9, x^{2}-6 x+9$", "answer": "$(x-3)^{2}(x+3)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.1.1", "question": "Solve: $\\sqrt{2 x+3}-3=0$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.20", "question": "The sum of Jason and Mandy's age is 35. Ten years ago Jason was double Mandy's age. How old are they now?", "answer": "15,20", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.47", "question": "State the excluded values for $\\frac{n^{2}-2 n+1}{6 n+6}$", "answer": "$\\frac{(n-1)^{2}}{6(n+1)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.6", "question": "Solve: $32=2-5(-4 n+6)$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.38", "question": "Solve the equation: $3^{2-x} \\cdot 3^{3 m}=1$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.7.24", "question": "A third farmer has a field that is $500 \\mathrm{ft}$ by $550 \\mathrm{ft}$. He wants to increase his field by $20 \\%$. How wide a ring should he cultivate around the outside of his field?", "answer": "$25 \\mathrm{ft}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.31", "question": "A caterer made an ice cream punch by combining fruit juice that cost $\\$ 2.25$ per gallon with ice cream that costs $\\$ 3.25$ per gallon. How many gallons of each were used to make 100 gal of punch costing $\\$ 2.50$ per pound?", "answer": "75,25", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.26", "question": "Simplify: $\\left(u^{2} v^{2} \\cdot 2 u^{4}\\right)^{3}$", "answer": "$8 u^{18} v^{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.9.8", "question": "Solve the following systems of equation: \n$x y=48$ \n$(x-6)(y+3)=60$", "answer": "$(16,3),(-6,-8)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.30", "question": "Solve the following system of equations by elimination:\n$-7 x+10 y=13$\n$4 x+9 y=22$", "answer": "$(1,2)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.44", "question": "Solve: $4 x^{2}+4 x+25=0$", "answer": "$\\frac{-1+2 i \\sqrt{6}}{2}, \\frac{-1-2 i \\sqrt{6}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.39", "question": "Distribute: $8 n(n+9)$", "answer": "$8 n^{2}+72 n$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.12", "question": "Solve: $-3 n-27=-27-3 n$", "answer": "All real numbers", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.1", "question": "A tank contains 8000 liters of a solution that is $40 \\%$ acid. How much water should be added to make a solution that is $30 \\%$ acid?", "answer": "2666.7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.2", "question": "Simplify: $\\sqrt[3]{375}$", "answer": "$5 \\sqrt[3]{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.9", "question": "Solve the equation: $2 b+\\frac{9}{5}=-\\frac{11}{5}$", "answer": "-2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.18", "question": "Solve the equation $S=L+2 B$ for $\\mathrm{L}$.", "answer": "$L=S-2 B$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.25", "question": "Simplify and write the answer in scientific notation: $\\left(7.8 \\times 10^{-2}\\right)^{5}$", "answer": "$2.887 \\times 10^{-6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.40", "question": "An investment portfolio earned $\\$ 2010$ in interest last year. If $\\$ 3000$ was invested at a certain rate of return and $\\$ 24000$ was invested in a fund with a rate that was $4 \\%$ lower, find the two rates of interest.", "answer": "$\\$ 3000 @ 11 \\%$; $\\$ 24000 @ 7 \\%$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.34", "question": "Simplify: $\\frac{4-4 x^{-1}+x^{-2}}{4-x^{-2}}$", "answer": "$\\frac{2 x-1}{2 x+1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.19", "question": "Find product: $(-7-4 i)(-8+6 i)$", "answer": "$80-10 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.7.16", "question": "A picture 10 inches long by 8 inches wide has a frame whose area is one half the area of the picture. What are the outside dimensions of the frame?", "answer": "1 in", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.5", "question": "Solve the equation $3 x=\\frac{a}{b}$ for $x$.", "answer": "$x=\\frac{a}{3 b}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.34", "question": "Solve the equation: $-80=4 x-28$", "answer": "-13", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.58", "question": "Simplify: $-9-10(1+9 a)$", "answer": "$-19-90 a$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.62", "question": "Evaluate the expression: $\\frac{12}{7}-\\frac{9}{7}$", "answer": "$\\frac{3}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.26", "question": "Simplify: $\\frac{n-9}{9 n-81}$", "answer": "$\\frac{1}{9}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.13", "question": "Two drivers are testing the same model car at speeds that differ by $20 \\mathrm{~km} / \\mathrm{hr}$. The one driving at the slower rate drives 70 kilometers down a speedway and returns by the same route. The one driving at the faster rate drives 76 kilometers down the speedway and returns by the same route. Both drivers leave at the same time, and the faster car returns $\\frac{1}{2}$ hour earlier than the slower car. At what rates were the cars driven?", "answer": "56,76", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.8", "question": "Find the value that completes the square and then rewrite as a perfect square: $p^{2}-17 p+$", "answer": "$\\frac{289}{4} ;\\left(p-\\frac{17}{2}\\right)^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.19", "question": "Solve: $-32-24 v=34-2 v$", "answer": "-3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.13", "question": "Find product: $(-5 i)(8 i)$", "answer": "40", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.34", "question": "How many pounds of coffee that is $40 \\%$ java beans must be mixed with $80 \\mathrm{lb}$ of coffee that is $30 \\%$ java beans to make a coffee blend that is $32 \\%$ java beans?", "answer": "20", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.9", "question": "Simplify: $\\left(3^{3}\\right)^{4}$", "answer": "$3^{12}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.26", "question": "Solve the following system of equations by elimination:\n$-4 x-5 y=12$\n$-10 x+6 y=30$", "answer": "$(-3,0)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.18", "question": "Let the first angle of a triangle be $x$ degrees. Find the measures of the second and third angles.", "answer": "$36,36,108$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.37", "question": "Simplify: $\\frac{5 \\sqrt[4]{5 r^{4}}}{\\sqrt[4]{8 r^{2}}}$", "answer": "$\\frac{5 \\sqrt[4]{10 r^{2}}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.13", "question": "Evaluate using the given values: $q-p-(q-1-3)$; $p=3, q=6$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.13", "question": "Solve the equation: $-a-\\frac{5}{4}\\left(-\\frac{8}{3} a+1\\right)=-\\frac{19}{4}$", "answer": "$-\\frac{3}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.38", "question": "Simplify the following expression: $\\frac{\\sqrt[3]{x^{2}}}{\\sqrt[5]{x}}$", "answer": "$\\sqrt[15]{x^{7}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.37", "question": "Solve the equation: $\\frac{p}{20}=-12$", "answer": "-240", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.1.4", "question": "Solve: $\\sqrt{x+2}-\\sqrt{x}=2$", "answer": "no solution", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.13", "question": "Evaluate the expression: $\\log _{125} 5$", "answer": "$\\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.38", "question": "Simplify and write the answer in scientific notation: $\\frac{5 \\times 10^{6}}{6.69 \\times 10^{2}}$", "answer": "$7.474 \\times 10^{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.38", "question": "Simplify: $-6 \\sqrt[4]{80 m^{4} p^{7} q^{4}}$", "answer": "$-12 m p q \\sqrt[4]{5 p^{3}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.7", "question": "Simplify: $3 \\sqrt{6}+3 \\sqrt{5}+2 \\sqrt{5}$", "answer": "$3 \\sqrt{6}+5 \\sqrt{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.28", "question": "Simplify: $-3 \\sqrt[4]{4}+3 \\sqrt[4]{324}+2 \\sqrt[4]{64}$", "answer": "$10 \\sqrt[4]{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.56", "question": "Evaluate the expression: $\\frac{1}{3}+\\frac{5}{3}$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.7", "question": "Solve the following systems of equation: \n$x+y+z=6$\n$2 x-y-z=-3$\n$x-2 y+3 z=6$", "answer": "$(1,2,3)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.19", "question": "Simplify and write the answer in scientific notation: $\\frac{4.9 \\times 10^{1}}{2.7 \\times 10^{-3}}$", "answer": "$1.815 \\times 10^{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.6", "question": "How many grams of pure acid must be added to 40 grams of a $20 \\%$ acid solution to make a solution which is $36 \\%$ acid?", "answer": "10", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.27", "question": "Solve the following systems of equation: \n$m+6 n+3 p=8$\n$3 m+4 n=-3$\n$5 m+7 n=1$", "answer": "$(-25,18,-25)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.5.14", "question": "Find the slope of a line perpendicular to the given line: $3 x-y=-3$", "answer": "$-\\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.15", "question": "Simplify and express with positive exponents: $\\left(a^{\\frac{1}{2}} b^{\\frac{1}{2}}\\right)^{-1}$", "answer": "$\\frac{1}{a^{\\frac{1}{2}} b^{\\frac{1}{2}}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.35", "question": "Simplify: $\\frac{2-\\sqrt{5}}{-3+\\sqrt{5}}$", "answer": "$\\frac{-1+\\sqrt{5}}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.18", "question": "A drawer contains $15 \\mathbb{C}$ stamps and $18 \\mathbb{C}$ stamps. The number of $15 \\mathbb{C}$ stamps is four less than three times the number of $18 \\mathbb{C}$ stamps. The total value of all the stamps is $\\$ 1.29$. How many $15 \\mathbb{C}$ stamps are in the drawer?", "answer": "5 4.6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.15", "question": "Simplify: $\\frac{4}{3+\\sqrt{5}}$", "answer": "$3-\\sqrt{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.22", "question": "A total of 26 bills are in a cash box. Some of the bills are one-dollar bills, and the rest are five-dollar bills. The total amount of cash in the box is $\\$ 550$. Find the number of each type of bill in the cash box.", "answer": "$20 \\$ 1,6 \\$ 5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.52", "question": "Solve: $3 x^{2}-11 x=-18$", "answer": "$\\frac{11+i \\sqrt{95}}{6}, \\frac{11-i \\sqrt{95}}{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.40", "question": "Solve the following equation: $\\left(x^{2}+x+3\\right)^{2}+15=8\\left(x^{2}+x+3\\right)$", "answer": "$0, \\pm 1,-2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.30", "question": "Solve the equation: $n+8=10$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.32", "question": "Given the points $(8, y)$ and $(-2,4)$ and a slope of $-\\frac{1}{5}$, find the value of $y$.", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.13", "question": "Solve the compound inequality and give interval notation: $3 \\leqslant 9+x \\leqslant 7$", "answer": "$-6 \\leqslant x \\leqslant-2:[-6,-2]$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.11", "question": "Solve the following equation: $z^{6}-216=19 z^{3}$", "answer": "$-2,3,1 \\pm i \\sqrt{3}, \\frac{-3 \\pm i \\sqrt{3}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.35", "question": "Simplify: $-3 \\sqrt[3]{192 a b^{2}}$", "answer": "$-12 \\sqrt[3]{3 a b^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.30", "question": "The perimeter of a college basketball court is $96$ meters and the length is $14$ meters more than the width. Find the dimensions of the basketball court.", "answer": "17,31", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.33", "question": "Find the inverse of the function: $h(x)=\\frac{4-\\sqrt[3]{4 x}}{2}$", "answer": "$h^{-1}(x)=\\frac{(-2 x+4)^{3}}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.31", "question": "Divide: $\\frac{x^{3}-26 x-41}{x+4}$", "answer": "$x^{2}-4 x-10-\\frac{1}{x+4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.23", "question": "Find the product: $(r-7)\\left(6 r^{2}-r+5\\right)$", "answer": "$6 r^{3}-43 r^{2}+12 r-35$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.47", "question": "Evaluate the expression: $47 \\frac{3+\\sqrt{-27}}{6}$", "answer": "$\\frac{1+i \\sqrt{3}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.4", "question": "Solve the following systems of equation: \n$x+y+z=2$\n$6 x-4 y+5 z=31$\n$5 x+2 y+2 z=13$", "answer": "$(3,-2,1)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.35", "question": "Solve the equation: $\\log _{4}(6 b+4)=0$", "answer": "$-\\frac{1}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.13", "question": "Simplify and leave your answer as an improper fraction: $\\frac{63}{18}$", "answer": "$\\frac{7}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.31", "question": "A truck leaves a depot at 11 A.M. and travels at a speed of $45 \\mathrm{mph}$. At noon, a van leaves the same place and travels the same route at a speed of $65 \\mathrm{mph}$. At what time does the van overtake the truck?", "answer": "$2: 15$ PM", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.34", "question": "Simplify: $\\frac{4 \\sqrt{2}+3}{3 \\sqrt{2}+\\sqrt{3}}$", "answer": "$\\frac{24-4 \\sqrt{6}+9 \\sqrt{2}-3 \\sqrt{3}}{15}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.32", "question": "Simplify and express with positive exponents: $\\frac{\\left(x^{\\frac{1}{2}} y^{0}\\right)^{-\\frac{4}{3}}}{y^{4} \\cdot x^{-2} y^{-\\frac{2}{3}}}$", "answer": "$\\frac{x^{\\frac{4}{3}}}{y^{\\frac{10}{3}}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.17", "question": "Simplify and leave your answer as an improper fraction: $\\frac{72}{60}$", "answer": "$\\frac{6}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.30", "question": "The largest single rough diamond ever found, the Cullinan diamond, weighed 3106 carats; how much does the diamond weigh in miligrams? in pounds?", "answer": "$621,200 \\mathrm{mg} ; 1.42 \\mathrm{lb}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.29", "question": "Solve the equation: $6^{2-2 x}=6^{2}$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.28", "question": "Simplify: $\\sqrt{512 a^{4} b^{2}}$", "answer": "$16 a^{2} b \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.16", "question": "Find product: $(-i)(7 i)(4-3 i)$", "answer": "$28-21 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.43", "question": "Distribute: $-6(1+6 x)$", "answer": "$-6-36 x$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.12", "question": "Solve the equation: $216^{-3 v}=36^{3 v}$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.36", "question": "Simplify: $3 \\sqrt[3]{135 x y^{3}}$", "answer": "$9 y \\sqrt[3]{5 x}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.6", "question": "Simplify the expression $x^{4}-5 x^{3}-x+13$ when $x=5$.", "answer": "8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.40", "question": "Solve the equation: $\\log _{8}(3 k-1)=1$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.33", "question": "Add or subtract the rational expressions and simplify: $\\frac{2 x}{x^{2}-1}-\\frac{4}{x^{2}+2 x-3}$", "answer": "$\\frac{2 x+4}{x^{2}+4 x+3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.51", "question": "Solve: $8 r^{2}+10 r=-55$", "answer": "$\\frac{-5+i \\sqrt{415}}{8}, \\frac{-5-i \\sqrt{415}}{8}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.8", "question": "Solve the equation: $-\\frac{16}{9}=-\\frac{4}{3}\\left(\\frac{5}{3}+n\\right)$", "answer": "$-\\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.22", "question": "Solve the following systems of equation: \n$3 x+y-z=10$\n$8 x-y-6 z=-3$\n$5 x-2 y-5 z=1$", "answer": "no solution", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.5", "question": "A boy has $\\$ 2.25$ in nickels and dimes. If there are twice as many dimes as nickels, how many of each kind has he?", "answer": "9,18", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.4", "question": "Solve the proportion: $\\frac{8}{x}=\\frac{4}{8}$", "answer": "$x=16$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.1.11", "question": "Solve: $\\sqrt{2 x+4}-\\sqrt{x+3}=1$", "answer": "6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.19", "question": "Add or subtract the rational expressions and simplify: $\\frac{8}{x^{2}-4}-\\frac{3}{x+2}$", "answer": "$\\frac{14-3 x}{x^{2}-4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.16", "question": "An automobile made a trip of 120 miles and then returned, the round trip occupying 7 hours. Returning, the rate was increased 10 miles an hour. Find the rate of each.", "answer": "$30 \\mathrm{mph}, 40 \\mathrm{mph}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.1", "question": "Convert $7 \\mathrm{mi}$ to yards.", "answer": "$12320 \\mathrm{yd}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.34", "question": "Divide: $\\frac{k^{3}-4 k^{2}-6 k+4}{k-1}$", "answer": "$k^{2}-3 k-9-\\frac{5}{k-1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.15", "question": "Evaluate $\\left(8 n+n^{4}\\right)-\\left(3 n-4 n^{4}\\right)$.", "answer": "$5 n^{4}+5 n$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.33", "question": "Evaluate $\\left(n-5 n^{4}+7\\right)+\\left(n^{2}-7 n^{4}-n\\right)$.", "answer": "$-12 n^{4}+n^{2}+7$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.13", "question": "Find the Least Common Denominator: $x^{2}-3 x, x-3, x$", "answer": "$x(x-3)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.30", "question": "Simplify: $\\frac{4}{5 \\sqrt{3 x y^{4}}}$", "answer": "$\\frac{4 \\sqrt{3 x}}{16 x y^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.24", "question": "Find the slope of the line through the points $(8,11)$ and $(-3,-13)$.", "answer": "$\\frac{24}{11}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.13", "question": "Simplify the expression: $\\frac{x-10}{35 x+21} \\div \\frac{7}{35 x+21}$", "answer": "$\\frac{x-10}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.27", "question": "Simplify: $\\sqrt{16 x^{3} y^{3}}$", "answer": "$4 x y \\sqrt{x y}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.12", "question": "Find the product: $(7 n-6)(n+7)$", "answer": "$7 n^{2}+43 n-42$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.33", "question": "Simplify: $2 \\sqrt[3]{375 u^{2} v^{8}}$", "answer": "$10 v^{2} \\sqrt[3]{3 u^{2} v^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.17", "question": "Convert $7.50 \\mathrm{~T} / \\mathrm{yd}^{2}$ (tons per square yard) to pounds per square inch.", "answer": "$11.6 \\mathrm{lb} / \\mathrm{in}^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.39", "question": "How many ounces of pure bran flakes must be added to 50 oz of cereal that is $40 \\%$ bran flakes to produce a mixture that is $50 \\%$ bran flakes?", "answer": "10", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.16", "question": "Solve the equation $\\mathrm{F}=\\mathrm{k}(R-L)$ for $\\mathrm{k}$.", "answer": "$k=\\frac{F}{R-L}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.6", "question": "Convert $4.5 \\mathrm{ft}^{2}$ to square yards.", "answer": "$0.5 \\mathrm{yd}^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.21", "question": "Solve the compound inequality and give interval notation: $-16 \\leqslant 2 n-10 \\leqslant-22$", "answer": "No solution : $\\oslash$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.17", "question": "Simplify: $\\sqrt[5]{224 p^{5}}$", "answer": "$2 p \\sqrt[5]{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.17", "question": "Simplify: $-\\frac{4}{4-4 \\sqrt{2}}$", "answer": "$1+\\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.1", "question": "Solve the equation for $x$: $3 x-\\frac{1}{2}-\\frac{1}{x}=0$", "answer": "$-\\frac{1}{2}, \\frac{2}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.24", "question": "Solve the following equation: $x^{6}-10 x^{3}+16=0$", "answer": "$2, \\sqrt[3]{2},-1 \\pm i \\sqrt{3}, \\frac{-\\sqrt[3]{2} \\pm i \\sqrt[6]{108}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.39", "question": "State the excluded values for $\\frac{2 n^{2}+19 n-10}{9 n+90}$", "answer": "$\\frac{2 n-1}{9}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.17", "question": "Solve the equation for $x$: $\\frac{x+1}{x-1}-\\frac{x-1}{x+1}=\\frac{5}{6}$", "answer": "$-\\frac{1}{5}, 5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.24", "question": "Find the square: $(4 x-5)^{2}$", "answer": "$16 x^{2}-40 x+25$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.31", "question": "Solve the equation: $\\log _{5}(-3 m)=3$", "answer": "$-\\frac{125}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.34", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{2 b^{4} c^{-2} \\cdot\\left(2 b^{3} c^{2}\\right)^{-4}}{a^{-2} b^{4}}$", "answer": "$\\frac{a^{2}}{8 c^{10} b^{12}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.34", "question": "Evaluate $\\left(8 x^{2}+2 x^{4}+7 x^{3}\\right)+\\left(7 x^{4}-7 x^{3}+2 x^{2}\\right)$.", "answer": "$9 x^{2}+10 x^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.37", "question": "Find quotient: $-2 \\div \\frac{7}{4}$", "answer": "$-\\frac{8}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.16", "question": "Solve the equation: $-37=8+3 x$", "answer": "-15", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.34", "question": "Solve the following equation: $(a+1)^{2}+2(a-1)=15$", "answer": "$2,-6$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.64", "question": "Simplify: $-2 r(1+4 r)+8 r(-r+4)$", "answer": "$30 r-16 r^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.24", "question": "Find a quadratic equation with the solutions $\\pm \\sqrt{7}$.", "answer": "$x^{2}-7=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.5", "question": "Solve the proportion: $\\frac{6}{x}=\\frac{8}{2}$", "answer": "$x=\\frac{3}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.17", "question": "Simplify and express with positive exponents: $\\frac{a^{2} b^{0}}{3 a^{4}}$", "answer": "$\\frac{1}{3 a^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.29", "question": "A bus traveling at a rate of $60 \\mathrm{mph}$ overtakes a car traveling at a rate of 45 mph. If the car had a $1 \\mathrm{~h}$ head start, how far from the starting point does the bus overtake the car?", "answer": "180", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.5", "question": "Solve the equation: $x + x + 2x + 4x = -104$", "answer": "-13", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.41", "question": "Distribute: $7 k(-k+6)$", "answer": "$-7 k^{2}+42 k$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.4", "question": "Solve the equation: $\\frac{3}{2} n-\\frac{8}{3}=-\\frac{29}{12}$", "answer": "$\\frac{1}{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.22", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{2 y^{2}}{\\left(x^{4} y^{0}\\right)^{-4}}$", "answer": "$2 x^{16} y^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.17", "question": "Solve the equation: $2=-12+2 r$", "answer": "7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.7", "question": "Solve the equation: $7x - 11 = 6x + 5$", "answer": "16", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.22", "question": "Solve each equation by completing the square: $6 r^{2}+12 r-24=-6$", "answer": "$1,-3$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.31", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{y\\left(2 x^{4} y^{2}\\right)^{2}}{2 x^{4} y^{0}}$", "answer": "$2 y^{5} x^{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.18", "question": "Simplify: $\\frac{4}{4 \\sqrt{3}-\\sqrt{5}}$", "answer": "$\\frac{16 \\sqrt{3}+4 \\sqrt{5}}{43}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.6", "question": "Evaluate using the given values: $x+6 z-4 y$; $x=6, y=4, z=4$", "answer": "14", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.55", "question": "Evaluate the expression: $i^{62}$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.34", "question": "Simplify: $2 \\sqrt[4]{48}-3 \\sqrt[4]{405}-3 \\sqrt[4]{48}-\\sqrt[4]{162}$", "answer": "$-2 \\sqrt[4]{3}-9 \\sqrt[4]{5}-3 \\sqrt[4]{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.25", "question": "Divide: $\\frac{27 b^{2}+87 b+35}{3 b+8}$", "answer": "$9 b+5-\\frac{5}{3 b+8}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.3", "question": "Simplify: $\\sqrt[3]{750}$", "answer": "$5 \\sqrt[3]{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.31", "question": "The time required to empty a tank varies inversely as the rate of pumping. If a pump can empty a tank in $45 \\mathrm{~min}$ at the rate of $600 \\mathrm{~kL} / \\mathrm{min}$, how long will it take the pump to empty the same tank at the rate of $1000 \\mathrm{~kL} / \\mathrm{min}$ ?", "answer": "$27 \\mathrm{~min}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.38", "question": "Find product: $(6)(-1)$", "answer": "-6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.5", "question": "Divide: $\\frac{12 x^{4}+24 x^{3}+3 x^{2}}{6 x}$", "answer": "$2 x^{3}+4 x^{2}+\\frac{x}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.15", "question": "Multiply and simplify: $(\\sqrt{5}-5)(2 \\sqrt{5}-1)$", "answer": "$15-11 \\sqrt{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.14", "question": "Simplify and write the answer in scientific notation: $\\left(2 \\times 10^{-6}\\right)\\left(8.8 \\times 10^{-5}\\right)$", "answer": "$1.76 \\times 10^{-10}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.35", "question": "Simplify: $\\left(\\frac{2 m n^{4} \\cdot 2 m^{4} n^{4}}{m n^{4}}\\right)^{3}$", "answer": "$64 m^{12} n^{12}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.25", "question": "Simplify: $\\frac{20}{4 p+2}$", "answer": "$\\frac{10}{2 p+1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.25", "question": "Solve the equation: $\\log _{2} x=-2$", "answer": "$\\frac{1}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.32", "question": "Simplify the expression: $\\frac{9 x^{3}+54 x^{2}}{x^{2}+5 x-14} \\cdot \\frac{x^{2}+5 x-14}{10 x^{2}}$", "answer": "$\\frac{9(x+6)}{10}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.12", "question": "Evaluate: $100^{-\\frac{3}{2}}$", "answer": "$\\frac{1}{1000}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.32", "question": "Simplify: $\\frac{a b}{a \\sqrt{b}-b \\sqrt{a}}$", "answer": "$\\frac{a \\sqrt{b}+b \\sqrt{a}}{a-b}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.13", "question": "Find the product: $(4 m-8 n)(4 m+8 n)$", "answer": "$16 m^{2}-64 n^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.11", "question": "Solve the following system of equations by elimination:\n$2 x-y=5$\n$5 x+2 y=-28$", "answer": "$(-2,-9)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.38", "question": "Evaluate the expression: $\\frac{4}{4+6 i}$", "answer": "$\\frac{4-6 i}{13}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.19", "question": "By going 15 miles per hour faster, a train would have required 1 hour less to travel 180 miles. How fast did it travel?", "answer": "$45 \\mathrm{mph}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.35", "question": "Simplify: $\\frac{x^{-2}-6 x^{-1}+9}{x^{2}-9}$", "answer": "$\\frac{(1-3 x)^{2}}{x^{2}(x+3)(x-3)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.23", "question": "Find a quadratic equation with the solutions $\\pm \\frac{1}{5}$.", "answer": "$25 x^{2}-1=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.31", "question": "Simplify: $\\frac{x^{-2}-y^{-2}}{x^{-1}+y^{-1}}$", "answer": "$\\frac{y-x}{x y}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.5", "question": "Solve the following equation: $a^{4}-50 a^{2}+49=0$", "answer": "$\\pm 1, \\pm 7$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.16", "question": "Solve the proportion: $\\frac{x+1}{9}=\\frac{x+2}{2}$", "answer": "$v=-\\frac{16}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.21", "question": "Find the slope of the line through the points $(12,-19)$ and $(6,14)$.", "answer": "$-\\frac{11}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.16", "question": "A long-distance runner started on a course running at an average speed of 6 mph. One hour later, a second runner began the same course at an average speed of $8 \\mathrm{mph}$. How long after the second runner started will the second runner overtake the first runner?", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.3", "question": "Solve: $-5(-4+2 v)=-50$", "answer": "7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.32", "question": "Millicent earned $\\$ 435$ last year in interest. If $\\$ 3000$ was invested at a certain rate of return and $\\$ 4500$ was invested in a fund with a rate that was $2 \\%$ lower, find the two rates of interest.", "answer": "$\\$ 3000 @ 4.6 \\%$ $\\$ 4500$ @ 6.6\\%", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.35", "question": "The intensity of a light from a light bulb varies inversely as the square of the distance from the bulb. Suppose intensity is $90 \\mathrm{~W} / \\mathrm{m}^{2}$ (watts per square meter) when the distance is $5 \\mathrm{~m}$. How much further would it be to a point where the intesity is $40 \\mathrm{~W} / \\mathrm{m}^{2}$ ?", "answer": "$2.5 \\mathrm{~m}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.13", "question": "Solve each equation by completing the square: $6 x^{2}+12 x+63=0$", "answer": "$\\frac{-2+i \\sqrt{38}}{2}, \\frac{-2-i \\sqrt{38}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.15", "question": "A motorboat leaves a harbor and travels at an average speed of $8 \\mathrm{mph}$ toward a small island. Two hours later a cabin cruiser leaves the same harbor and travels at an average speed of $16 \\mathrm{mph}$ toward the same island. In how many hours after the cabin cruiser leaves will the cabin cruiser be alongside the motorboat?", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.14", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{2 x y^{2} \\cdot 4 x^{3} y^{-4}}{4 x^{-4} y^{-4} \\cdot 4 x}$", "answer": "$\\frac{x^{7} y^{2}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.57", "question": "Evaluate the expression: $i^{154}$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.13", "question": "Solve the equation: $-9=\\frac{n}{12}$", "answer": "-108", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.34", "question": "Find the product: $5(x-4)(2 x-3)$", "answer": "$10 x^{2}-55 x+60$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.34", "question": "Simplify: $8 \\sqrt[3]{-750 x y}$", "answer": "$-40 \\sqrt[3]{6 x y}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.40", "question": "Simplify: $-\\sqrt{32 x y^{2} z^{3}}$", "answer": "$-4 y z \\sqrt{2 x z}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.4", "question": "Simplify: $\\sqrt[3]{250}$", "answer": "$5 \\sqrt[3]{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.34", "question": "The drag force on a boat varies jointly as the wetted surface area and the square of the velocity of a boat. If a boat going $6.5 \\mathrm{mph}$ experiences a drag force of $86 \\mathrm{~N}$ when the wetted surface area is $41.2 \\mathrm{ft}^{2}$, how fast must a boat with $28.5 \\mathrm{ft}^{2}$ of wetted surface area go in order to experience a drag force of $94 \\mathrm{~N} ?$", "answer": "$8.2 \\mathrm{mph}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.9", "question": "Simplify: $(i)-(2+3 i)-6$", "answer": "$-8-2 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.28", "question": "Solve the proportion: $\\frac{1}{n+3}=\\frac{n+2}{2}$", "answer": "$n=-4,-1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.8", "question": "Solve the following equation: $y^{4}-40 y^{2}+144=0$", "answer": "$\\pm 6, \\pm 2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.1", "question": "Solve the equation: $x + 3 = 19$", "answer": "11", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.29", "question": "Find product: $(8)\\left(\\frac{1}{2}\\right)$", "answer": "4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.35", "question": "Given $h(t)=2|-3 t-1|+2$, find $h\\left(n^{2}\\right)$", "answer": "$2\\left|-3 n^{2}-1\\right|+2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.13", "question": "Solve the equation for $y$: $\\frac{7}{y-3}-\\frac{1}{2}=\\frac{y-2}{y-4}$", "answer": "$\\frac{16}{3}, 5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.15", "question": "Simplify and leave your answer as an improper fraction: $\\frac{80}{60}$", "answer": "$\\frac{4}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.20", "question": "Simplify the expression: $\\frac{x^{2}-7 x+10}{x-2} \\cdot \\frac{x+10}{x^{2}-x-20}$", "answer": "$\\frac{x+10}{x+4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.28", "question": "Evaluate the expression: $\\frac{-3+2 i}{-3 i}$", "answer": "$\\frac{-3 i-2}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.44", "question": "Solve the following equation: $\\left(2 x^{2}+3 x\\right)^{2}=8\\left(2 x^{2}+3 x\\right)+9$", "answer": "$-3,-1, \\frac{3}{2},-\\frac{1}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.3", "question": "Find the principal that will amount to $\\$ 3500$ if invested at $4 \\%$ interest compounded quarterly for 5 years.", "answer": "2868.41", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.27", "question": "Simplify: $\\frac{2 \\sqrt{4}}{3 \\sqrt{3}}$", "answer": "$\\frac{4 \\sqrt{3}}{9}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.9.11", "question": "Solve the following systems of equation: \n$x y=45$ \n$(x-5)(y+3)=160$", "answer": "$(45,1),\\left(-\\frac{5}{3},-27\\right)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.29", "question": "Simplify: $\\frac{32 x^{2}}{28 x^{2}+28 x}$", "answer": "$\\frac{8 x}{7(x+1)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.39", "question": "Simplify: $6 \\sqrt[4]{648 x^{5} y^{7} z^{2}}$", "answer": "$18 x y \\sqrt[4]{8 x y^{3} z^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.20", "question": "A water tank is being filled by two inlet pipes. Pipe A can fill the tank in $4 \\frac{1}{2}$ hrs, while both pipes together can fill the tank in 2 hours. How long does it take to fill the tank using only pipe B?", "answer": "3.6 hours", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.31", "question": "Add or subtract the rational expressions and simplify: $\\frac{5 x}{x^{2}-x-6}-\\frac{18}{x^{2}-9}$", "answer": "$\\frac{5 x+12}{x^{2}+5 x+6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.6", "question": "Simplify. Your answer should contain only positive expontents: $\\left(m^{0} n^{3} \\cdot 2 m^{-3} n^{-3}\\right)^{0}$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.28", "question": "Solve the equation $S=\\pi \\mathrm{rh}+\\pi r^{2}$ for $\\mathrm{h}$.", "answer": "$h=\\frac{s-\\pi r^{2}}{\\pi r}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.8", "question": "State the excluded values for $\\frac{27 p}{18 p^{2}-36 p}$", "answer": "0,2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.37", "question": "How many ounces of dried apricots must be added to 18 oz of a snack mix that contains $20 \\%$ dried apricots to make a mixture that is $25 \\%$ dried apricots?", "answer": "1.2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.16", "question": "Simplify: $(2 x y)^{4}$", "answer": "$16 x^{4} y^{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.36", "question": "The volume of a cone varies jointly as its height, and the square of its radius. If a cone with a height of 8 centimeters and a radius of 2 centimeters has a volume of $33.5 \\mathrm{~cm}^{3}$, what is the volume of a cone with a height of 6 centimeters and a radius of 4 centimeters?", "answer": "$\\mathrm{V}=100.5 \\mathrm{~cm}^{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.7", "question": "Find the value that completes the square and then rewrite as a perfect square: $y^{2}-y+$", "answer": "$\\frac{1}{4} ;\\left(y-\\frac{1}{2}\\right)^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.7.9", "question": "The area of a rectangle is $48 \\mathrm{ft}^{2}$ and its perimeter is $32 \\mathrm{ft}$. Find its length and width.", "answer": "$4 \\mathrm{ft} \\times 12 \\mathrm{ft}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.31", "question": "Solve the equation for $x$: $\\frac{x-3}{x-6}+\\frac{x+5}{x+3}=\\frac{-2 x^{2}}{x^{2}-3 x-18}$", "answer": "$\\frac{13}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.29", "question": "The manager of a garden shop mixes grass seed that is $60 \\%$ rye grass with 70 lb of grass seed that is $80 \\%$ rye grass to make a mixture that is $74 \\%$ rye grass. How much of the $60 \\%$ mixture is used?", "answer": "30", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.30", "question": "Find the square: $(8 x+5 y)^{2}$", "answer": "$64 x^{2}+80 x y+25 y^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.26", "question": "A takes $7 \\frac{1}{2}$ minutes longer than $\\mathrm{B}$ to do a job. Working together, they can do the job in 9 minutes. How long does it take each working alone?", "answer": "15 and $22.5 \\mathrm{~min}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.31", "question": "Simplify and express with positive exponents: $\\frac{\\left(x^{-\\frac{4}{3}} y^{-\\frac{1}{3}} \\cdot y\\right)^{-1}}{x^{\\frac{1}{3}} y^{-2}}$", "answer": "$x y^{\\frac{4}{3}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.11", "question": "Find the inverse of the function: $f(x)=(x-2)^{5}+3$", "answer": "$f^{-1}(x)=\\sqrt[5]{x-3}+2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.7", "question": "Simplify: $(3-3 i)+(-7-8 i)$", "answer": "$-4-11 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.53", "question": "Simplify: $9(b+10)+5 b$", "answer": "$14 b+90$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.18", "question": "Find product: $(8 i)(-2 i)(-2-8 i)$", "answer": "$-32-128 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.3", "question": "Reduce the following radical: $\\sqrt[12]{64 x^{4} y^{6} z^{8}}$", "answer": "$\\sqrt[6]{8 x^{2} y^{3} z^{4}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.16", "question": "Solve: $\\frac{1-\\frac{1}{x}-\\frac{6}{x^{2}}}{1-\\frac{4}{x}+\\frac{3}{x^{2}}}$", "answer": "$\\frac{x+2}{x-1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.25", "question": "Find the slope of the line through the points $(-17,19)$ and $(10,-7)$.", "answer": "$-\\frac{26}{27}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.37", "question": "Given the points $(x,-7)$ and $(-9,-9)$ and a slope of $\\frac{2}{5}$, find the value of $x$.", "answer": "-4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.2", "question": "Simplify: $(3 i)-(7 i)$", "answer": "$-4 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.55", "question": "Find quotient: $\\frac{-8}{-2}$", "answer": "4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.24", "question": "Solve each equation by completing the square: $6 n^{2}-12 n-14=4$", "answer": "$3,-1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.1.9", "question": "Solve: $\\sqrt{4 x+5}-\\sqrt{x+4}=2$", "answer": "5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.46", "question": "Find quotient: $\\frac{30}{-10}$", "answer": "-3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.1", "question": "A boy is 10 years older than his brother. In 4 years he will be twice as old as his brother. Find the present age of each.", "answer": "6,16", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.28", "question": "Add or subtract the rational expressions and simplify: $\\frac{2 x}{x^{2}-1}-\\frac{3}{x^{2}+5 x+4}$", "answer": "$\\frac{2 x+3}{(x-1)(x+4)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.19", "question": "Marge is twice as old as Consuelo. The sum of their ages seven years ago was 13. How old are they now?", "answer": "9,18", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.33", "question": "Given the points $(-3,-2)$ and $(x, 6)$ and a slope of $-\\frac{8}{5}$, find the value of $x$.", "answer": "-8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.26", "question": "Simplify: $\\frac{\\sqrt{2}}{3 \\sqrt{5}}$", "answer": "$\\frac{\\sqrt{10}}{15}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.5", "question": "Fred is 4 years older than Barney. Five years ago the sum of their ages was 48 . How old are they now?", "answer": "27,31", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.25", "question": "Find a quadratic equation with the solutions $\\pm \\sqrt{11}$.", "answer": "$x^{2}-11=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.4", "question": "Simplify the expression $n^{3}-9 n^{2}+23 n-21$ when $n=5$.", "answer": "-6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.10", "question": "Specify the domain of the function: $y(x)=\\frac{x}{x^{2}-25}$", "answer": "$x \\neq \\pm 5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.35", "question": "Find the product: $3(2 x+1)(4 x-5)$", "answer": "$24 x^{2}-18 x-15$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.29", "question": "Solve the following systems of equation: \n$-2 w+2 x+2 y-2 z=-10$\n$w+x+y+z=-5$\n$3 w+2 x+2 y+4 z=-11$\n$w+3 x-2 y+2 z=-6$", "answer": "$(1,-3,-2,-1)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.11", "question": "Evaluate the expression: $3-(-1)$", "answer": "4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.13", "question": "Simplify: $\\sqrt{192 n}$", "answer": "$8 \\sqrt{3 n}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.32", "question": "Simplify: $\\frac{b^{2}+14 b+48}{b^{2}+15 b+56}$", "answer": "$\\frac{b+6}{b+7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.7.1", "question": "Find the value: $\\cos 71^{\\circ}$", "answer": "0.3256", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.75", "question": "Simplify: $(5 p-6)+(1-p)$", "answer": "$4 p-5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.4", "question": "Solve the equation: $27=21-3 x$", "answer": "-2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.18", "question": "Solve: $-16 n+12=39-7 n$", "answer": "-3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.63", "question": "Simplify: $4(x+7)+8(x+4)$", "answer": "$12 x+60$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.34", "question": "Solve the following system of equations by elimination:\n$-6-42 y=-12 x$\n$x-\\frac{1}{2}-\\frac{7}{2} y=0$", "answer": "Infinite number of solutions", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.6", "question": "A clothier bought a lot of suits for $\\$ 750$. He sold all but 3 of them for $\\mathbb{\\$} 864$ making a profit of $\\$ 7$ on each suit sold. How many suits did he buy?", "answer": "30", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.20", "question": "Solve the equation for $x$: $\\frac{3 x-5}{5 x-5}+\\frac{5 x-1}{7 x-7}-\\frac{x-4}{1-x}=2$", "answer": "10", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.3.20", "question": "Solve: $\\frac{-5^{2}+(-5)^{2}}{\\left|4^{2}-2^{5}\\right|-2 \\cdot 3}$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.38", "question": "Divide: $\\frac{8 m^{3}-57 m^{2}+42}{8 m+7}$", "answer": "$m^{2}-8 m+7-\\frac{7}{8 m+7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.27", "question": "Build up each fraction by finding the Least Common Denominator: $\\frac{x+1}{x^{2}-36}, \\frac{2 x+3}{x^{2}+12 x+36}$", "answer": "$\\frac{x^{2}+7 x+6}{(x-6)(x+6)^{2}}, \\frac{2 x^{2}-9 x-18}{(x-6)(x+6)^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.6", "question": "Find the value that completes the square and then rewrite as a perfect square: $r^{2}-\\frac{1}{9} r+$", "answer": "$\\frac{1}{324} ;\\left(r-\\frac{1}{18}\\right)^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.26", "question": "The wavelength of a radio wave varies inversely as its frequency. A wave with a frequency of 1200 kilohertz has a length of 300 meters. What is the length of a wave with a frequency of 800 kilohertz?", "answer": "$450 \\mathrm{~m}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.20", "question": "Multiply and simplify: $(5 \\sqrt{2}-1)(-\\sqrt{2 m}+5)$", "answer": "$-10 \\sqrt{m}+25 \\sqrt{2}+\\sqrt{2 m}-5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.29", "question": "Combine the following radicals: $\\sqrt{x y^{3}} \\sqrt[3]{x^{2} y}$", "answer": "$x y \\sqrt[6]{x y^{5}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.9", "question": "Solve each equation by completing the square: $x^{2}-16 x+55=0$", "answer": "11,5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.26", "question": "Combine the following radicals: $\\sqrt[3]{x^{2}} \\sqrt[6]{x^{5}}$", "answer": "$x \\sqrt{x}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.14", "question": "Multiply and simplify: $(-2+\\sqrt{3})(-5+2 \\sqrt{3})$", "answer": "$16-9 \\sqrt{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.35", "question": "A wool tapestry is 32 years older than a linen tapestry. Twenty years ago, the wool tapestry was twice as old as the linen tapestry. Find the present age of each.", "answer": "84,52", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.30", "question": "Evaluate $\\left(7 n+1-8 n^{4}\\right)-\\left(3 n+7 n^{4}+7\\right)$.", "answer": "$-15 n^{4}+4 n-6$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.10", "question": "Simplify the expression: $\\frac{6 x(x+4)}{x-3} \\cdot \\frac{(x-3)(x-6)}{6 x(x-6)}$", "answer": "$x+4$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.29", "question": "Solve the following equation: $(y+b)^{2}-4(y+b)=21$", "answer": "$-(b+3), 7-b$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.73", "question": "Evaluate the expression: $6-\\frac{8}{7}$", "answer": "$\\frac{34}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.7", "question": "A group of boys bought a boat for $\\$ 450$. Five boys failed to pay their share, hence each remaining boys were compelled to pay $\\$ 4.50$ more. How many boys were in the original group and how much had each agreed to pay?", "answer": "$25 @ \\$ 18$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.31", "question": "Simplify: $\\left(\\frac{(2 x)^{3}}{x^{3}}\\right)^{2}$", "answer": "64", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.37", "question": "Simplify: $\\frac{6 a-10}{10 a+4}$", "answer": "$\\frac{3 a-5}{5 a+2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.19", "question": "Evaluate the expression: $\\log _{6} 36$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.28", "question": "Computer memory is measured in units of bytes, where one byte is enough memory to store one character (a letter in the alphabet or a number). How many typical pages of text can be stored on a 700-megabyte compact disc? Assume that one typical page of text contains 2000 characters. (1 megabyte $=$ $1,000,000$ bytes)", "answer": "350,000 pages", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.39", "question": "Simplify: $2 \\sqrt{80 h j^{4} k}$", "answer": "$8 j^{2} \\sqrt{5 h k}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.37", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{\\left(c b^{3}\\right)^{2} \\cdot 2 a^{-3} b^{2}}{\\left(a^{3} b^{-2} c^{3}\\right)^{3}}$", "answer": "$\\frac{2 b^{14}}{a^{12} c^{7}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.22", "question": "Find the inverse of the function: $g(x)=\\sqrt[5]{\\frac{-x+2}{2}}$", "answer": "$g^{-1}(x)=-2 x^{5}+2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.8", "question": "Simplify: $-8 \\sqrt[4]{48}$", "answer": "$-16 \\sqrt[4]{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.6", "question": "Solve the equation for $x$: $\\frac{x-4}{x-1}=\\frac{12}{3-x}+1$", "answer": "$\\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.6", "question": "Add or subtract the rational expressions and simplify: $\\frac{3}{x}+\\frac{4}{x^{2}}$", "answer": "$\\frac{3 x+4}{x^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.67", "question": "Simplify: $7(7+3 v)+10(3-10 v)$", "answer": "$79-79 v$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.34", "question": "Combine the following radicals: $\\sqrt{a^{4} b^{3} c^{4}} \\sqrt[3]{a b^{2} c}$", "answer": "$a^{2} b^{2} c^{2} \\sqrt[6]{a^{2} b c^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.20", "question": "Simplify: $-8 \\sqrt[7]{384 b^{8}}$", "answer": "$-16 b \\sqrt[7]{3 b}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.3", "question": "Simplify the expression $n^{3}-7 n^{2}+15 n-20$ when $n=2$.", "answer": "-10", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.54", "question": "Simplify: $4 v-7(1-8 v)$", "answer": "$60 v-7$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.5", "question": "Simplify: $\\frac{2-5 \\sqrt{5}}{4 \\sqrt{13}}$", "answer": "$\\frac{2 \\sqrt{13}-5 \\sqrt{65}}{52}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.7", "question": "Write the number in standard notation: $8.7 \\times 10^{5}$", "answer": "870000", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.30", "question": "Simplify the expression: $\\frac{k-7}{k^{2}-k-12} \\cdot \\frac{7 k^{2}-28 k}{8 k^{2}-56 k}$", "answer": "$\\frac{7}{8(k+3)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.17", "question": "A car traveling at $48 \\mathrm{mph}$ overtakes a cyclist who, riding at $12 \\mathrm{mph}$, has had a 3-hour head start. How far from the starting point does the car overtake the cyclist?", "answer": "48", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.54", "question": "Evaluate the expression: $\\frac{1}{7}+\\left(-\\frac{11}{7}\\right)$", "answer": "$-\\frac{10}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.29", "question": "Solve the equation for $x$: $\\frac{x-5}{x-9}+\\frac{x+3}{x-3}=\\frac{-4 x^{2}}{x^{2}-12 x+27}$", "answer": "$-\\frac{2}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.5", "question": "Solve: $3 x^{2}+1=73$", "answer": "$\\pm 2 \\sqrt{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.5", "question": "Solve the compound inequality and give interval notation: $x-6<-13$ or $6 x \\leqslant-60$", "answer": "$x<-7:(-\\infty,-7)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.4", "question": "Find the missing numerator to build up the denominator: $\\frac{5}{2 x^{2}}=\\frac{?}{8 x^{3} y}$", "answer": "$20 x y$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.38", "question": "A man bought a cow and a calf for $\\$990$. He paid $8$ times as much for the cow as for the calf. What was the cost of each?", "answer": "110,880", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.4", "question": "Convert $1.35 \\mathrm{~km}$ to centimeters.", "answer": "$135,000 \\mathrm{~cm}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.14", "question": "Solve the compound inequality and give interval notation: $0 \\geqslant \\frac{x}{9} \\geqslant-1$", "answer": "$-9 \\leqslant x \\leqslant 0:[-9,0]$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.8", "question": "$\\$ 3.25$ in dimes and nickels were distributed among 45 boys. If each received one coin, how many received dimes and how many received nickels?", "answer": "25,20", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.5", "question": "Simplify: $\\sqrt{12}$", "answer": "$2 \\sqrt{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.12", "question": "Find the product: $(7 a+7 b)(7 a-7 b)$", "answer": "$49 a^{2}-49 b^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.4", "question": "Evaluate $\\frac{a+2}{a^{2}+3 a+2}$ when $a=-1$", "answer": "undefined", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.38", "question": "Solve the equation: $-15=\\frac{x}{9}$", "answer": "-135", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.24", "question": "Find product: $(-2)\\left(\\frac{1}{3}\\right)$", "answer": "$-\\frac{2}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.28", "question": "It takes John 24 minutes longer than Sally to mow the lawn. If they work together, they can mow the lawn in 9 minutes. How long will it take each to mow the lawn if they work alone?", "answer": "12 and $36 \\mathrm{~min}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.33", "question": "A bicycle and a bicycle helmet cost $\\$240$. If the bicycle cost $5$ times as much as the helmet, what was the cost of each?", "answer": "40, 200", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.3.19", "question": "Solve: $\\frac{-13-2}{2-(-1)^{3}+(-6)-[-1-(-3)]}$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.36", "question": "Simplify: $-5 \\sqrt{72 x^{3} y^{4}}$", "answer": "$-30 y^{2} x \\sqrt{2 x}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.27", "question": "Find product: $\\left(-\\frac{6}{5}\\right)\\left(-\\frac{11}{8}\\right)$", "answer": "$\\frac{33}{20}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.16", "question": "Evaluate the expression: $\\log _{7} 1$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.26", "question": "Simplify: $3 \\sqrt[3]{135}-\\sqrt[3]{81}-\\sqrt[3]{135}$", "answer": "$6 \\sqrt[3]{5}-3 \\sqrt[3]{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.33", "question": "Solve: $-2(m-2)+7(m-8)=-67$", "answer": "-3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.16", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{2 x^{-2} y^{2}}{4 y x^{2}}$", "answer": "$\\frac{y}{2 x^{4}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.4", "question": "Solve: $2-8(-4+3 x)=34$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.46", "question": "Solve: $a^{2}-5 a+25=3$", "answer": "$\\frac{5+3 i \\sqrt{7}}{2}, \\frac{5-3 i \\sqrt{7}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.14", "question": "An $8.5 \\%$ account earns continuous interest. If $\\$ 2500$ is deposited for 5 years, what is the total accumulated?", "answer": "3823.98", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.40", "question": "Solve: $-6(x-8)-4(x-2)=-4$", "answer": "6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.23", "question": "Solve the following system of equations by elimination:\n$9 x-2 y=-18$\n$5 x-7 y=-10$", "answer": "$(-2,0)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.14", "question": "An athlete plans to row upstream a distance of 2 kilometers and then return to his starting point in a total time of 2 hours and 20 minutes. If the rate of the current is $2 \\mathrm{~km} / \\mathrm{hr}$, how fast should he row?", "answer": "$3.033 \\mathrm{~km} / \\mathrm{hr}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.2", "question": "Solve the proportion: $\\frac{7}{9}=\\frac{n}{6}$", "answer": "$n=\\frac{14}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.42", "question": "Simplify: $\\left(\\frac{2 q^{3} p^{3} r^{4} \\cdot 2 p^{3}}{\\left(q r p^{3}\\right)^{2}}\\right)^{4}$", "answer": "$256 q^{4} r^{8}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.18", "question": "Find the product: $(2 u+3 v)(8 u-7 v)$", "answer": "$16 u^{2}+10 u v-21 v^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.12", "question": "Simplify and leave your answer as an improper fraction: $\\frac{48}{42}$", "answer": "$\\frac{8}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.29", "question": "Solve the equation: $\\frac{47}{9}+\\frac{3}{2} x=\\frac{5}{3}\\left(\\frac{5}{2} x+1\\right)$", "answer": "$\\frac{4}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.31", "question": "Jason earned $\\$ 256$ interest last year on his investments. If $\\$ 1600$ was invested at a certain rate of return and $\\$ 2400$ was invested in a fund with a rate that was double the rate of the first fund, find the two rates of interest.", "answer": "$\\$ 1600 @ 4 \\%$; $\\$ 2400$ @ 8\\%", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.35", "question": "Combine Like Terms: $9 n-1+n+4$", "answer": "$10 n+3$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.7", "question": "Evaluate the expression: $3-(-5)$", "answer": "8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.14", "question": "Solve: $56 p-48=6 p+2$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.30", "question": "Simplify: $\\sqrt{512 m^{4} n^{3}}$", "answer": "$16 m^{2} n \\sqrt{2 n}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.14", "question": "Solve the proportion: $\\frac{n}{8}=\\frac{n-4}{3}$", "answer": "$n=\\frac{32}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.4", "question": "Simplify: $-2 \\sqrt{6}-\\sqrt{3}-3 \\sqrt{6}$", "answer": "$-5 \\sqrt{6}-\\sqrt{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.12", "question": "Convert $3.5 \\mathrm{mph}$ (miles per hour) to feet per second.", "answer": "$5.13 \\mathrm{ft} / \\mathrm{sec}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.17", "question": "Simplify: $\\frac{4^{5}}{4^{3}}$", "answer": "$4^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.30", "question": "Solve the proportion: $\\frac{x-5}{4}=\\frac{-3}{x+3}$", "answer": "$x=-1,3$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.12", "question": "Multiply and simplify: $\\sqrt{15}(\\sqrt{5}-3 \\sqrt{3 v})$", "answer": "$5 \\sqrt{3}-9 \\sqrt{5 v}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.13", "question": "Combine the following radicals: $\\sqrt[3]{5} \\sqrt{6}$", "answer": "$\\sqrt[6]{5400}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.3", "question": "Jack can wash and wax the family car in one hour less than Bob can. The two working together can complete the job in $1 \\frac{1}{5}$ hours. How much time would each require if they worked alone?", "answer": "2 and 3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.27", "question": "Solve the equation $h=\\mathrm{vt}-16 t^{2}$ for $\\mathrm{v}$.", "answer": "$v=\\frac{16 t^{2}+h}{t}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.6", "question": "Solve the compound inequality and give interval notation: $9+n<2$ or $5 n>40$", "answer": "$n<-7$ or $n>8:(-\\infty-7), \\bigcup(8, \\infty)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.22", "question": "An average human heart beats 60 times per minute. If an average person lives to the age of 75 , how many times does the average heart beat in a lifetime?", "answer": "$2,365,200,000$ beats/lifetime", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.11", "question": "A family drove to a resort at an average speed of $30 \\mathrm{mph}$ and later returned over the same road at an average speed of $50 \\mathrm{mph}$. Find the distance to the resort if the total driving time was 8 hours.", "answer": "150", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.40", "question": "Simplify and write the answer in scientific notation: $\\left(9 \\times 10^{-2}\\right)^{-3}$", "answer": "$1.372 \\times 10^{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.30", "question": "Solve the equation: $\\log _{11}(x-4)=-1$", "answer": "$\\frac{45}{11}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.2", "question": "Convert $234 \\mathrm{oz}$ to tons.", "answer": "$0.0073125 \\mathrm{~T}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.23", "question": "Evaluate $\\left(8 x^{3}+1\\right)-\\left(5 x^{4}-6 x^{3}+2\\right)$.", "answer": "$-5 x^{4}+14 x^{3}-1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.9", "question": "Add or subtract the rational expressions and simplify: $\\frac{8}{9 t^{3}}+\\frac{5}{6 t^{2}}$", "answer": "$\\frac{15 t+16}{18 t^{3}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.3", "question": "Solve the following equation: $m^{4}-7 m^{2}-8=0$", "answer": "$\\pm i, \\pm 2 \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.10", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{3 y^{3}}{3 y x^{3} \\cdot 2 x^{4} y^{-3}}$", "answer": "$\\frac{y^{5}}{2 x^{7}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.30", "question": "Solve the compound inequality and give interval notation: $8-10 r \\leqslant 8+4 r$ or $-6+8 r<2+8 r$", "answer": "$\\{$ All real numbers. $\\}: \\mathbb{R}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.14", "question": "Simplify: $(x y)^{3}$", "answer": "$x^{3} y^{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.27", "question": "The number of kilograms of water in a human body varies directly as the mass of the body. A 96-kg person contains $64 \\mathrm{~kg}$ of water. How many kilo grams of water are in a $60-\\mathrm{kg}$ person?", "answer": "$40 \\mathrm{~kg}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.3", "question": "Find the product: $2(6 x+3)$", "answer": "$12 x+6$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.25", "question": "Build up each fraction by finding the Least Common Denominator: $\\frac{x}{x^{2}-16}, \\frac{3 x}{x^{2}-8 x+16}$", "answer": "$\\frac{x^{2}-4 x}{(x-4)^{2}(x+4)}, \\frac{3 x^{2}+12 x}{(x-4)^{2}(x+4)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.32", "question": "Evaluate $\\left(7 x^{2}+2 x^{4}+7 x^{3}\\right)+\\left(6 x^{3}-8 x^{4}-7 x^{2}\\right)$.", "answer": "$-6 x^{4}+13 x^{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.27", "question": "Solve the equation: $-2 x+4=22$", "answer": "-9", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.38", "question": "Simplify: $\\frac{\\left(2 y^{3} x^{2}\\right)^{2}}{2 x^{2} y^{4} \\cdot x^{2}}$", "answer": "$2 y^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.7", "question": "Find a quadratic equation with the solutions 0 and 0.", "answer": "$x^{2}=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.13", "question": "Add or subtract the rational expressions and simplify: $\\frac{x-1}{4 x}-\\frac{2 x+3}{x}$", "answer": "$\\frac{-7 x-13}{4 x}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.6", "question": "Simplify: $-3 \\sqrt{3}+2 \\sqrt{3}-2 \\sqrt{3}$", "answer": "$-3 \\sqrt{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.20", "question": "Solve the equation: $\\frac{1}{12}=\\frac{4}{3} x+\\frac{5}{3}\\left(x-\\frac{7}{4}\\right)$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.35", "question": "Simplify: $\\frac{12 x^{2}-42 x}{30 x^{2}-42 x}$", "answer": "$\\frac{2 x-7}{5 x-7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.27", "question": "Evaluate the expression: $\\frac{-9+5 i}{i}$", "answer": "$9 i+5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.33", "question": "Simplify and express with positive exponents: $\\frac{\\left(u v^{2}\\right)^{\\frac{1}{2}}}{v^{-\\frac{1}{4}} v^{2}}$", "answer": "$\\frac{u^{\\frac{1}{2}}}{v^{\\frac{3}{4}}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.45", "question": "The cost of a private pilot course is $\\$1,275$. The flight portion costs $\\$625$ more than the ground school portion. What is the cost of each?", "answer": "325,950", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.32", "question": "Solve the equation: $\\frac{216}{6^{-2 a}}=6^{3 a}$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.2", "question": "Simplify and leave your answer as an improper fraction: $\\frac{25}{20}$", "answer": "$\\frac{5}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.19", "question": "Evaluate $\\left(5 n^{4}+6 n^{3}\\right)+\\left(8-3 n^{3}-5 n^{4}\\right)$.", "answer": "$3 n^{3}+8$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.31", "question": "Find the product: $\\left(5 k^{2}+3 k+3\\right)\\left(3 k^{2}+3 k+6\\right)$", "answer": "$15 k^{4}+24 k^{3}+48 k^{2}+27 k+18$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.36", "question": "Solve: $-8(6+6 x)+4(-3+6 x)=-12$", "answer": "-2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.39", "question": "Find quotient: $\\frac{-1}{9} \\div \\frac{-1}{2}$", "answer": "$\\frac{2}{9}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.35", "question": "Solve the equation lwh $=\\mathrm{V}$ for $\\mathrm{w}$.", "answer": "$w=\\frac{V}{\\ell h}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.36", "question": "Find product: $\\left(\\frac{1}{2}\\right)\\left(\\frac{5}{7}\\right)$", "answer": "$\\frac{5}{14}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.28", "question": "Find product: $\\left(-\\frac{3}{7}\\right)\\left(-\\frac{11}{8}\\right)$", "answer": "$\\frac{33}{56}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.20", "question": "Evaluate the expression: $\\log _{36} 6$", "answer": "$\\frac{1}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.9", "question": "Simplify and leave your answer as an improper fraction: $\\frac{27}{18}$", "answer": "$\\frac{3}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.11", "question": "Solve the absolute value equation: $|8+6 m|=50$", "answer": "$7,-\\frac{29}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.3.16", "question": "Solve: $-4-\\left[2+4(-6)-4-\\left|2^{2}-5 \\cdot 2\\right|\\right]$", "answer": "28", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.17", "question": "A farmer has some cream which is $21 \\%$ butterfat and some which is $15 \\%$ butter fat. How many gallons of each must be mixed to produce 60 gallons of cream which is $19 \\%$ butterfat?", "answer": "40,20", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.50", "question": "Solve: $n^{2}-n=-41$", "answer": "$\\frac{1+i \\sqrt{163}}{2}, \\frac{1-i \\sqrt{163}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.23", "question": "Simplify: $\\frac{\\sqrt{5}}{4 \\sqrt{125}}$", "answer": "$\\frac{1}{20}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.11", "question": "Convert $5,500 \\mathrm{~cm}^{3}$ to cubic yards.", "answer": "$0.0072 \\mathrm{yd}^{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.29", "question": "Find the product: $\\left(8 n^{2}+4 n+6\\right)\\left(6 n^{2}-5 n+6\\right)$", "answer": "$48 n^{4}-16 n^{3}+64 n^{2}-6 n+36$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.38", "question": "Find quotient: $\\frac{-12}{7} \\div \\frac{-9}{5}$", "answer": "$\\frac{20}{21}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.29", "question": "Solve the compound inequality and give interval notation: $1+5 k \\leqslant 7 k-3$ or $k-10>2 k+10$", "answer": "$k \\geqslant 2$ or $k<-20:(-\\infty,-20) \\cup[2, \\infty)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.16", "question": "Find the Least Common Denominator: $x, x-7, x+1$", "answer": "$x(x-7)(x+1)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.31", "question": "Simplify: $6 \\sqrt{80 x y^{2}}$", "answer": "$24 y \\sqrt{5 x}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.7", "question": "Convert $435,000 \\mathrm{~m}^{2}$ to sqaure kilometers.", "answer": "$0.435 \\mathrm{~km}^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.14", "question": "Solve: $\\frac{\\frac{x}{3 x-2}}{\\frac{x}{9 x^{2}-4}}$", "answer": "$3 x+2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.10", "question": "Find the product: $(b-7)(b+7)$", "answer": "$b^{2}-49$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.28", "question": "Find the inverse of the function: $g(x)=\\frac{-x+2}{3}$", "answer": "$g^{-1}(x)=-3 x+2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.11", "question": "Find product: $(6 i)(-8 i)$", "answer": "48", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.44", "question": "Simplify: $-2(n+1)$", "answer": "$-2 n-2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.34", "question": "Harry and his sister collected $240$ stamps in total. Harry collected $3$ times as many stamps as his sister. How many stamps did each of them collect?", "answer": "60,180", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.13", "question": "Simplify and write the answer in scientific notation: $\\left(7 \\times 10^{-1}\\right)\\left(2 \\times 10^{-3}\\right)$", "answer": "$1.4 \\times 10^{-3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.5", "question": "Solve the following system of equations by elimination:\n$-6 x+9 y=3$\n$6 x-9 y=-9$", "answer": "No solution", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.29", "question": "Solve the equation $Q_{1}=P\\left(Q_{2}-Q_{1}\\right)$ for $Q_{2}$.", "answer": "$Q_{2}=\\frac{Q_{1}+\\mathrm{PQ}_{1}}{P}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.8", "question": "Divide: $\\frac{3 m^{4}+18 m^{3}+27 m^{2}}{9 m^{2}}$", "answer": "$\\frac{m^{2}}{3}+2 m+3$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.16", "question": "Find a quadratic equation with the solutions 2 and $\\frac{2}{9}$.", "answer": "$9 x^{2}-20 x+4=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.8.1", "question": "Find the angle measure to the nearest degree given $\\sin Z=0.4848$", "answer": "$29^{\\circ}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.25", "question": "Solve the equation for $x$: $\\frac{x}{x-1}-\\frac{2}{x+1}=\\frac{4 x^{2}}{x^{2}-1}$", "answer": "$\\frac{2}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.33", "question": "Simplify and write the answer in scientific notation: $\\left(1.8 \\times 10^{-5}\\right)^{-3}$", "answer": "$1.715 \\times 10^{14}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.1", "question": "Multiply and simplify: $3 \\sqrt{5} \\cdot-4 \\sqrt{16}$", "answer": "$-48 \\sqrt{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.40", "question": "Given the points $(6,2)$ and $(x, 6)$ and a slope of $-\\frac{4}{5}$, find the value of $x$.", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.24", "question": "Solve: $(3-2 x)^{\\frac{4}{3}}=-81$", "answer": "No Solution", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.44", "question": "Simplify the expression: $\\frac{x^{2}+3 x-10}{x^{2}+6 x+5} \\cdot \\frac{2 x^{2}-x-3}{2 x^{2}+x-6} \\div \\frac{8 x+20}{6 x+15}$", "answer": "$\\frac{3(x-2)}{4(x+2)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.25", "question": "Brandon is 9 years older than Ronda. In four years the sum of their ages will be 91 . How old are they now?", "answer": "37,46", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.36", "question": "A tea that is $20 \\%$ jasmine is blended with a tea that is $15 \\%$ jasmine. How many pounds of each tea are used to make $5 \\mathrm{lb}$ of tea that is $18 \\%$ jasmine?", "answer": "3,2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.28", "question": "Given $k(x)=-2 \\cdot 4^{2 x-2}$, find $k(2)$", "answer": "-32", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.22", "question": "Find the square: $(7 k-7)^{2}$", "answer": "$49 k^{2}-98 k+49$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.6", "question": "Solve the equation: $625^{-n-2}=\\frac{1}{125}$", "answer": "$-\\frac{5}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.8", "question": "Simplify and leave your answer as an improper fraction: $\\frac{36}{27}$", "answer": "$\\frac{4}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.23", "question": "Find the slope of the line through the points $(-5,-10)$ and $(-5,20)$.", "answer": "Undefined", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.41", "question": "Evaluate $\\left(8 x^{4}+2 x^{3}+2 x\\right)+\\left(2 x+2-2 x^{3}-x^{4}\\right)-\\left(x^{3}+5 x^{4}+8 x\\right)$.", "answer": "$2 x^{4}-x^{3}-4 x+2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.39", "question": "Solve: $5 x^{2}+5 x=-31-5 x$", "answer": "$\\frac{5+i \\sqrt{130}}{5}, \\frac{5-i \\sqrt{130}}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.25", "question": "Simplify: $\\left(x^{3} y^{4} \\cdot 2 x^{2} y^{3}\\right)^{2}$", "answer": "$4 x^{10} y^{14}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.42", "question": "Add or subtract the rational expressions and simplify: $\\frac{x+2}{x^{2}-4 x+3}+\\frac{4 x+5}{x^{2}+4 x-5}$", "answer": "$\\frac{5 x+5}{x^{2}+2 x-15}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.36", "question": "Solve the equation: $\\log _{11}(10 v+1)=-1$", "answer": "$-\\frac{1}{11}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.42", "question": "State the excluded values for $\\frac{56 x-48}{24 x^{2}+56 x+32}$", "answer": "$\\frac{9 r}{5(r+1)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.22", "question": "Build up each fraction by finding the Least Common Denominator: $\\frac{3 x}{x-4}, \\frac{2}{x+2}$", "answer": "$\\frac{3 x^{2}+6 x}{(x-4)(x+2)}, \\frac{2 x-8}{(x-4)(x+2)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.1", "question": "Find a quadratic equation with the solutions 2 and 5.", "answer": "$x^{2}-7 x+10=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.25", "question": "Solve: $x^{2}=-10 x-29$", "answer": "$-5+2 i,-5-2 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.34", "question": "Find a quadratic equation with the solutions $-2 \\pm 4 i$.", "answer": "$x^{2}+4 x+20=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.10", "question": "Find a quadratic equation with the solutions 3 and $-1$.", "answer": "$x^{2}-2 x-3=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.41", "question": "Simplify the following expression: $\\frac{\\sqrt{a b^{3} c}}{\\sqrt[5]{a^{2} b^{3} c^{-1}}}$", "answer": "$\\sqrt[10]{a b^{9} c^{7}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.15", "question": "Find the inverse of the function: $f(x)=\\frac{-2 x-2}{x+2}$", "answer": "$f^{-1}(x)=\\frac{-2 x-2}{x+2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.35", "question": "Find a quadratic equation with the solutions $6 \\pm i \\sqrt{3}$.", "answer": "$x^{2}-12 x+39=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.29", "question": "Solve: $\\frac{\\frac{y}{y+2}-\\frac{y}{y-2}}{\\frac{y}{y+2}+\\frac{y}{y-2}}$", "answer": "$-\\frac{2}{y}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.3", "question": "Solve the equation: $6x - 18 = -42$", "answer": "-4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.14", "question": "Convert $153 \\mathrm{ft} / \\mathrm{s}$ (feet per second) to miles per hour.", "answer": "$104.32 \\mathrm{mi} / \\mathrm{hr}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.45", "question": "Find product: $(4)(-6)$", "answer": "-24", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.4", "question": "Find the product: $3 n^{2}(6 n+7)$", "answer": "$18 n^{3}+21 n^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.6", "question": "Solve the following equation: $b^{4}-10 b^{2}+9=0$", "answer": "$\\pm 3, \\pm 1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.23", "question": "The number of aluminum cans used each year varies directly as the number of people using the cans. If 250 people use 60,000 cans in one year, how many cans are used each year in Dallas, which has a population of $1,008,000$ ?", "answer": "$241,920,000$ cans", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.17", "question": "Find the square: $(a+5)^{2}$", "answer": "$a^{2}+10 a+25$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.44", "question": "Evaluate the square root of $-45$", "answer": "$3 i \\sqrt{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.8", "question": "Simplify: $(-4-i)+(1-5 i)$", "answer": "$-3-6 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.33", "question": "The stopping distance of a car after the brakes have been applied varies directly as the square of the speed $\\mathrm{r}$. If a car, traveling $60 \\mathrm{mph}$ can stop in 200 $\\mathrm{ft}$, how fast can a car go and still stop in $72 \\mathrm{ft}$ ?", "answer": "$\\mathrm{r}=36$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.25", "question": "Solve: $-6 v-29=-4 v-5(v+1)$", "answer": "8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.23", "question": "Build up each fraction by finding the Least Common Denominator: $\\frac{x+2}{x-3}, \\frac{x-3}{x+2}$", "answer": "$\\frac{x^{2}+4 x+4}{(x-3)(x+2)}, \\frac{x^{2}-6 x+9}{(x-3)(x+2)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.14", "question": "Solve: $(2 x+3)^{\\frac{4}{3}}=16$", "answer": "$-\\frac{11}{2}, \\frac{5}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.6", "question": "Simplify and leave your answer as an improper fraction: $\\frac{30}{24}$", "answer": "$\\frac{5}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.29", "question": "Evaluate the expression: $\\frac{-10-9 i}{6 i}$", "answer": "$\\frac{10 i-9}{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.8", "question": "Solve the equation: $8x - 14 = 4x + 3$", "answer": "$\\frac{17}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.27", "question": "Simplify: $\\frac{2+\\sqrt{6}}{2+\\sqrt{3}}$", "answer": "$4-2 \\sqrt{3}+2 \\sqrt{6}-3 \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.70", "question": "Evaluate the expression: $\\frac{6}{5}-\\frac{8}{5}$", "answer": "$-\\frac{2}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.54", "question": "Find quotient: $\\frac{80}{-8}$", "answer": "-10", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.18", "question": "Simplify and express with positive exponents: $\\frac{2 x^{\\frac{1}{2}} y^{\\frac{1}{3}}}{2 x^{\\frac{4}{3}} y^{-\\frac{7}{4}}}$", "answer": "$\\frac{y^{\\frac{25}{12}}}{x^{\\frac{5}{6}}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.6", "question": "Write the number in scientific notation: 15000", "answer": "$1.5 \\times 10^{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.26", "question": "Simplify: $\\frac{a+\\sqrt{\\mathrm{ab}}}{\\sqrt{a}+\\sqrt{b}}$", "answer": "$\\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.12", "question": "Solve the equation: $-6=15+3 p$", "answer": "-7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.5", "question": "Evaluate $\\frac{b+2}{b^{2}+4 b+4}$ when $b=0$", "answer": "$\\frac{1}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.4", "question": "Solve the equation: $-14=x-18$", "answer": "4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.21", "question": "Solve: $-2-5(2-4 m)=33+5 m$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.16", "question": "Evaluate the expression: $(-1)-8$", "answer": "-9", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.15", "question": "Find the square of $-7 i$", "answer": "-49", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.19", "question": "A sink has two faucets, one for hot water and one for cold water. The sink can be filled by a cold-water faucet in 3.5 minutes. If both faucets are open, the sink is filled in 2.1 minutes. How long does it take to fill the sink with just the hot-water faucet open?", "answer": "$5 \\frac{1}{4} \\mathrm{~min}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.24", "question": "Simplify: $8 \\sqrt{112 p^{2}}$", "answer": "$32 p \\sqrt{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.53", "question": "Evaluate the expression: $\\frac{1}{3}+\\left(-\\frac{4}{3}\\right)$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.21", "question": "Combine Like Terms: $r-9+10$", "answer": "$r+1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.26", "question": "Simplify: $\\sqrt[3]{64 u^{5} v^{3}}$", "answer": "$4 u v \\sqrt[3]{u^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.15", "question": "You lend $\\$ 100$ at $10 \\%$ continuous interest. If you are repaid 2 months later, what is owed?", "answer": "101.68", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.8", "question": "Find the product: $(2 r+3)(2 r-3)$", "answer": "$4 r^{2}-9$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.29", "question": "Simplify and express with positive exponents: $\\frac{\\left(m^{2} n^{\\frac{1}{2}}\\right)^{0}}{n^{\\frac{3}{4}}}$", "answer": "$\\frac{1}{n^{\\frac{3}{4}}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.30", "question": "Solve: $\\frac{\\frac{x+1}{x-1}-\\frac{1-x}{1+x}}{\\frac{1}{(x+1)^{2}}+\\frac{1}{(x-1)^{2}}}$", "answer": "$x^{2}-1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.3", "question": "Simplify: $(7 i)-(3-2 i)$", "answer": "$-3+9 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.36", "question": "Carolyn's age is triple her daughter's age. In eight years the sum of their ages will be 72 . How old are they now?", "answer": "14,42", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.29", "question": "Simplify: $\\frac{5 x^{2}}{4 \\sqrt{3 x^{3} y^{3}}}$", "answer": "$\\frac{5 \\sqrt{3 x y}}{12 y^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.23", "question": "Simplify: $\\frac{4 x^{3} y^{4}}{3 x y^{3}}$", "answer": "$\\frac{4 x^{2} y}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.80", "question": "Simplify: $\\left(7 a^{2}+7 a\\right)-\\left(6 a^{2}+4 a\\right)$", "answer": "$a^{2}+3 a$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.40", "question": "Simplify the expression: $\\frac{12 x+24}{10 x^{2}+34 x+28} \\cdot \\frac{15 x+21}{5}$", "answer": "$\\frac{18}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.26", "question": "A kerosene lamp is 95 years old, and an electric lamp is 55 years old. How many years ago was the kerosene lamp twice the age of the electric lamp?", "answer": "15", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.14", "question": "Simplify: $\\frac{4}{\\sqrt{2}-2}$", "answer": "$-2 \\sqrt{2}-4$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.10", "question": "Solve the proportion: $\\frac{9}{n+2}=\\frac{3}{9}$", "answer": "$n=25$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.28", "question": "Simplify: $\\frac{28 m+12}{36}$", "answer": "$\\frac{7 m+3}{9}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.34", "question": "Find the product: $(8 n+7)(8 n-7)$", "answer": "$64 n^{2}-49$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.3", "question": "Pat is 20 years older than his son James. In two years Pat will be twice as old as James. How old are they now?", "answer": "18,38", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.69", "question": "Evaluate the expression: $\\frac{1}{5}+\\frac{3}{4}$", "answer": "$\\frac{19}{20}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.10", "question": "Solve the equation: $22=16+m$", "answer": "6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.32", "question": "Solve the equation: $\\frac{m}{4}-1=-2$", "answer": "-4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.52", "question": "Find quotient: $\\frac{27}{3}$", "answer": "9", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.22", "question": "Evaluate the expression: $1+(-1)$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.16", "question": "Solve each equation by completing the square: $8 a^{2}+16 a-1=0$", "answer": "$\\frac{-4+3 \\sqrt{2}}{4}, \\frac{-4-3 \\sqrt{2}}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.4", "question": "A purse contains $\\$ 3.90$ made up of dimes and quarters. If there are 21 coins in all, how many dimes and how many quarters were there?", "answer": "$9 \\mathrm{~d}, 12 \\mathrm{q}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.28", "question": "Divide: $\\frac{4 n^{2}-23 n-38}{4 n+5}$", "answer": "$n-7-\\frac{3}{4 n+5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.38", "question": "Distribute: $3(8 v+9)$", "answer": "$24 v+27$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.21", "question": "Simplify: $-2 \\sqrt[3]{-48 v^{7}}$", "answer": "$4 v^{2} \\sqrt[3]{6 v}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.40", "question": "Find a quadratic equation with the solutions $\\frac{-2 \\pm i \\sqrt{15}}{2}$.", "answer": "$4 x^{2}+8 x+19=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.22", "question": "A candy mix sells for $\\$ 2.20$ per kilogram. It contains chocolates worth $\\$ 1.80$ per kilogram and other candy worth $\\$ 3.00$ per kilogram. How much of each are in 15 kilograms of the mixture?", "answer": "10,5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.66", "question": "Evaluate the expression: $\\frac{1}{2}-\\frac{11}{6}$", "answer": "$-\\frac{4}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.13", "question": "Two people working together can complete a job in 6 hours. If one of them works twice as fast as the other, how long would it take the faster person, working alone, to do the job?", "answer": "9 hours", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.8", "question": "A man walks at the rate of 4 miles per hour. How far can he walk into the country and ride back on a trolley that travels at the rate of 20 miles per hour, if he must be back home 3 hours from the time he started?", "answer": "10", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.27", "question": "Simplify: $2 x\\left(x^{4} y^{4}\\right)^{4}$", "answer": "$2 x^{17} y^{16}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.3", "question": "Simplify and leave your answer as an improper fraction: $\\frac{35}{25}$", "answer": "$\\frac{7}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.20", "question": "Solve the following systems of equation: \n$4 x-7 y+3 z=1$\n$3 x+y-2 z=4$\n$4 x-7 y+3 z=6$", "answer": "no solution", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.17", "question": "It takes 10 hours to fill a pool with the inlet pipe. It can be emptied in $15 \\mathrm{hrs}$ with the outlet pipe. If the pool is half full to begin with, how long will it take to fill it from there if both pipes are open?", "answer": "15 hours", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.39", "question": "Evaluate $\\left(8-b+7 b^{3}\\right)-\\left(3 b^{4}+7 b-8+7 b^{2}\\right)+\\left(3-3 b+6 b^{3}\\right)$.", "answer": "$-3 b^{4}+13 b^{3}-7 b^{2}-$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.31", "question": "Solve: $-57=-(-p+1)+2(6+8 p)$", "answer": "-4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.19", "question": "Solve the following equation: $x^{4}-2 x^{2}-3=0$", "answer": "$\\pm i, \\pm \\sqrt{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.31", "question": "Find the square: $(5+2 r)^{2}$", "answer": "$25+20 r+4 r^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.9", "question": "Simplify: $6 \\sqrt{128}$", "answer": "$48 \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.2", "question": "Solve the equation: $14=b+3$", "answer": "11", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.22", "question": "Simplify: $\\frac{2+\\sqrt{10}}{\\sqrt{2}+\\sqrt{5}}$", "answer": "$\\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.30", "question": "How many ounces of water evaporated from 50 oz of a $12 \\%$ salt solution to produce a $15 \\%$ salt solution?", "answer": "10", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.29", "question": "Simplify: $3 \\sqrt[4]{2}-2 \\sqrt[4]{2}-\\sqrt[4]{243}$", "answer": "$\\sqrt[4]{2}-3 \\sqrt[4]{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.7.4", "question": "Find the value: $\\sin 50^{\\circ}$", "answer": "0.7660", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.15", "question": "Solve each equation by completing the square: $5 k^{2}-10 k+48=0$", "answer": "$\\frac{5+i \\sqrt{215}}{5}, \\frac{5-i \\sqrt{215}}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.34", "question": "Find product: $(-7)(-2)$", "answer": "14", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.13", "question": "A, who travels 4 miles an hour starts from a certain place 2 hours in advance of B, who travels 5 miles an hour in the same direction. How many hours must B travel to overtake A?", "answer": "8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.14", "question": "Solve the equation: $\\frac{1}{3}\\left(-\\frac{7}{4} k+1\\right)-\\frac{10}{3} k=-\\frac{13}{8}$", "answer": "$\\frac{1}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.17", "question": "Given $f(t)=3^{t}-2$, find $f(-2)$", "answer": "$-\\frac{17}{9}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.48", "question": "Solve: $2 p^{2}-p+56=-8$", "answer": "$\\frac{1+i \\sqrt{511}}{4}, \\frac{1-i \\sqrt{511}}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.42", "question": "Find quotient: $\\frac{5}{3} \\div \\frac{7}{5}$", "answer": "$\\frac{25}{21}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.15", "question": "Add or subtract the rational expressions and simplify: $\\frac{5 x+3 y}{2 x^{2} y}-\\frac{3 x+4 y}{x y^{2}}$", "answer": "$\\frac{3 y^{2}-3 x y-6 x^{2}}{2 x^{2} y^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.28", "question": "Solve the equation: $-\\frac{8}{3}-\\frac{1}{2} x=-\\frac{4}{3} x-\\frac{2}{3}\\left(-\\frac{13}{4} x+1\\right)$", "answer": "$-\\frac{3}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.3", "question": "Solve the proportion: $\\frac{7}{6}=\\frac{2}{k}$", "answer": "$k=\\frac{12}{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.27", "question": "Solve the equation: $\\log _{11} k=2$", "answer": "121", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.26", "question": "A cargo container is $50 \\mathrm{ft}$ long, $10 \\mathrm{ft}$ wide, and $8 \\mathrm{ft}$ tall. Find its volume in cubic yards and cubic meters.", "answer": "$148.15 \\mathrm{yd}^{3} ; 113 \\mathrm{~m}^{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.6", "question": "$\\$ 3.75$ is made up of quarters and half dollars. If the number of quarters exceeds the number of half dollars by 3, how many coins of each denomination are there?", "answer": "$7 \\mathrm{q}, 4 \\mathrm{~h}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.7", "question": "Solve the equation $\\mathrm{E}=\\mathrm{mc}^{2}$ for $\\mathrm{m}$.", "answer": "$m=\\frac{E}{c^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.1", "question": "Simplify the expression: $\\frac{8 x^{2}}{9} \\cdot \\frac{9}{2}$", "answer": "$4 x^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.24", "question": "Solve the equation: $4^{2 n}=4^{2-3 n}$", "answer": "$\\frac{2}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.4", "question": "Find the principal that will amount to $\\$ 3000$ if invested at $3 \\%$ interest compounded semiannually for 10 years.", "answer": "2227.41", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.15", "question": "Solve the following equation: $x^{\\frac{2}{3}}-35=2 x^{\\frac{1}{3}}$", "answer": "$-125,343$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.23", "question": "Divide: $\\frac{n^{2}-4}{n-2}$", "answer": "$n+2$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.30", "question": "Solve the equation $\\mathrm{L}=\\pi\\left(r_{1}+r_{2}\\right)+2 d$ for $r_{1}$.", "answer": "$r_{1}=\\frac{L-2 d-\\pi r^{2}}{\\pi}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.5.36", "question": "Simplify: $\\frac{-1+\\sqrt{5}}{2 \\sqrt{5}+5 \\sqrt{2}}$", "answer": "$\\frac{2 \\sqrt{5}-5 \\sqrt{2}-10+5 \\sqrt{10}}{30}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.21", "question": "Solve: $(3 x-2)^{\\frac{4}{5}}=16$", "answer": "$-\\frac{34}{3},-10$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.20", "question": "Let the first angle of a triangle be $x$ degrees. Find the measures of the second and third angles.", "answer": "$30,90,60$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.5", "question": "Simplify: $3 m \\cdot 4 m n$", "answer": "$12 m^{2} n$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.25", "question": "Solve the following system of equations by elimination:\n$9 x+6 y=-21$\n$-10 x-9 y=28$", "answer": "$(-1,-2)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.3", "question": "Find a quadratic equation with the solutions 20 and 2.", "answer": "$x^{2}-22 x+40=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.22", "question": "Combine the following radicals: $\\sqrt[5]{a^{2} b^{3}} \\sqrt[4]{a^{2} b}$", "answer": "$\\sqrt[20]{a^{18} b^{17}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.28", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{2 y x^{2} \\cdot x^{-2}}{\\left(2 x^{0} y^{4}\\right)^{-1}}$", "answer": "$4 y^{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.39", "question": "Find product: $(9)(-4)$", "answer": "-36", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.47", "question": "Find quotient: $\\frac{-49}{-7}$", "answer": "7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.26", "question": "Simplify: $\\sqrt{72 a^{3} b^{4}}$", "answer": "$6 b^{2} a \\sqrt{2 a}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.6", "question": "Simplify: $(-8 i)-(7 i)-(5-3 i)$", "answer": "$5-12 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.9", "question": "Reduce the following radical: $\\sqrt[8]{x^{6} y^{4} z^{2}}$", "answer": "$\\sqrt[4]{x^{3} y^{2} z}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.27", "question": "Find the inverse of the function: $g(x)=\\frac{8-5 x}{4}$", "answer": "$g^{-1}(x)=\\frac{-4 x+8}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.17", "question": "A postal clerk sold some $15 \\mathbb{C}$ stamps and some $25 \\mathbb{C}$ stamps. Altogether, 15 stamps were sold for a total cost of $\\$ 3.15$. How many of each type of stamps were sold?", "answer": "$615 \\mathbb{C}, 925 \\mathbb{C}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.28", "question": "Solve the equation: $216^{2 n}=36$", "answer": "$\\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.1", "question": "Solve the absolute value equation: $|x|=8$", "answer": "$8,-8$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.40", "question": "Given $h(n)=5^{n-1}+1$, find $h\\left(\\frac{n}{2}\\right)$", "answer": "$5^{\\frac{-2+n}{2}}+1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.19", "question": "Solve: $(x-1)^{-\\frac{5}{2}}=32$", "answer": "$\\frac{5}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.15", "question": "Find a quadratic equation with the solutions $\\frac{3}{7}$ and 4.", "answer": "$7 x^{2}-31 x+12=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.56", "question": "Simplify: $-8 x+9(-9 x+9)$", "answer": "$-89 x+81$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.10", "question": "Add or subtract the rational expressions and simplify: $\\frac{x+5}{8}+\\frac{x-3}{12}$", "answer": "$\\frac{5 x+9}{24}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.42", "question": "Simplify the expression: $\\frac{a^{3}+b^{3}}{a^{2}+3 \\mathrm{ab}+2 b^{2}} \\cdot \\frac{3 a-6 b}{3 a^{2}-3 \\mathrm{ab}+3 b^{2}} \\div \\frac{a^{2}-4 b^{2}}{a+2 b}$", "answer": "$\\frac{1}{a+2 b}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.44", "question": "The total cost for tuition plus room and board at State University is $\\$2,584$. Tuition costs $\\$704$ more than room and board. What is the tuition fee?", "answer": "1644", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.7", "question": "Reduce the following radical: $\\sqrt[12]{x^{6} y^{9}}$", "answer": "$\\sqrt[4]{x^{2} y^{3}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.40", "question": "Add or subtract the rational expressions and simplify: $\\frac{2 r}{r^{2}-s^{2}}+\\frac{1}{r+s}-\\frac{1}{r-s}$", "answer": "$\\frac{2}{r+s}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.8", "question": "Solve the absolute value equation: $|3-x|=6$", "answer": "$-3,9$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.82", "question": "Simplify: $\\left(3-7 n^{2}\\right)+\\left(6 n^{2}+3\\right)$", "answer": "$-n^{2}+6$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.7", "question": "Add or subtract the rational expressions and simplify: $\\frac{5}{6 r}-\\frac{5}{8 r}$", "answer": "$\\frac{5}{24 r}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.10.4", "question": "A dealer bought a number of sheep for $\\$ 440$. After 5 had died he sold the remainder at a profit of $\\mathbb{\\$} 2$ each making a profit of $\\mathbb{\\$} 60$ for the sheep. How many sheep did he originally purchase?", "answer": "55", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.33", "question": "Combine the following radicals: $\\sqrt[3]{3 x y^{2} z} \\sqrt[4]{9 x^{3} y z^{2}}$", "answer": "$x \\sqrt[12]{59049 x y^{11} z^{10}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.33", "question": "Solve the following equation: $(x-3)^{2}-2(x-3)=35$", "answer": "$-2,10$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.17", "question": "Solve the proportion: $\\frac{v-5}{v+6}=\\frac{4}{9}$", "answer": "$v=\\frac{69}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.40", "question": "Find the product: $(3 a-8)(3 a+8)$", "answer": "$9 a^{2}-64$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.22", "question": "Simplify: $4 \\sqrt[3]{250 a^{6}}$", "answer": "$20 a^{2} \\sqrt[3]{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.5.5", "question": "Find the slope of a line parallel to the given line: $x-y=4$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.10", "question": "You lend out $\\$ 5500$ at $10 \\%$ compounded monthly. If the debt is repaid in 18 months, what is the total owed at the time of repayment?", "answer": "6386.12", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.33", "question": "Simplify: $\\frac{9 v+54}{v^{2}-4 v-60}$", "answer": "$\\frac{9}{v-10}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.18", "question": "Find the square: $(v+4)^{2}$", "answer": "$v^{2}+8 v+16$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.11", "question": "Divide: $\\frac{n^{2}+13 n+32}{n+5}$", "answer": "$n+8-\\frac{8}{n+5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.67", "question": "Evaluate the expression: $\\left(-\\frac{1}{2}\\right)+\\frac{3}{2}$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.16", "question": "Simplify the expression: $\\frac{1}{a-6} \\cdot \\frac{8 a+80}{8}$", "answer": "$\\frac{a+10}{a-6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.8", "question": "Solve: $(5 x+1)^{4}=16$", "answer": "$\\frac{1}{5},-\\frac{3}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.3", "question": "Find the value that completes the square and then rewrite as a perfect square: $m^{2}-36 m+$", "answer": "$324 ;(m-18)^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.17", "question": "Solve the equation: $\\frac{16}{9}=-\\frac{4}{3}\\left(-\\frac{4}{3} n-\\frac{4}{3}\\right)$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.5.1", "question": "Find the slope of a line parallel to the given line: $y=2 x+4$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.1", "question": "Solve the following system of equations by elimination:\n$4 x+2 y=0$\n$-4 x-9 y=-28$", "answer": "$(-2,4)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.23", "question": "Solve the compound inequality and give interval notation: $-5 b+10 \\leqslant 30$ and $7 b+2 \\leqslant-40$", "answer": "No solution : $\\oslash$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.23", "question": "Simplify: $3 \\sqrt{24}-3 \\sqrt{27}+2 \\sqrt{6}+2 \\sqrt{8}$", "answer": "$8 \\sqrt{6}-9 \\sqrt{3}+4 \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.24", "question": "Simplify: $\\frac{\\sqrt{12}}{\\sqrt{3}}$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.33", "question": "Simplify: $2 \\sqrt[4]{2}+2 \\sqrt[4]{3}+3 \\sqrt[4]{64}-\\sqrt[4]{3}$", "answer": "$2 \\sqrt[4]{2}+\\sqrt[4]{3}+6 \\sqrt[4]{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.21", "question": "Simplify: $\\frac{3 n m^{2}}{3 n}$", "answer": "$m^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.9", "question": "Simplify the expression: $\\frac{7 r}{7 r(r+10)} \\div \\frac{r-6}{(r-6)^{2}}$", "answer": "$\\frac{r-6}{r+10}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.10", "question": "Solve each equation by completing the square: $n^{2}-8 n-12=0$", "answer": "$4+2 \\sqrt{7}, 4-2 \\sqrt{7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.66", "question": "Simplify: $9(6 b+5)-4 b(b+3)$", "answer": "$-42 b-45-4 b^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.15", "question": "Solve: $(2 x-3)^{\\frac{2}{3}}=4$", "answer": "$\\frac{11}{2},-\\frac{5}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.3.3", "question": "Solve: $3+(8) \\div|4|$", "answer": "5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.18", "question": "Simplify and leave your answer as an improper fraction: $\\frac{126}{108}$", "answer": "$\\frac{7}{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.5", "question": "Solve the absolute value equation: $|5+8 a|=53$", "answer": "$6,-\\frac{29}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.45", "question": "Simplify the following expression: $\\frac{\\sqrt[3]{(2 x+1)^{2}}}{\\sqrt[5]{(2 x+1)^{2}}}$", "answer": "$\\sqrt[15]{(2 x+1)^{4}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.27", "question": "Simplify and express with positive exponents: $\\left(\\frac{m^{\\frac{3}{2}} n^{-2}}{\\left(m n^{\\frac{4}{3}}\\right)^{-1}}\\right)^{\\frac{7}{4}}$", "answer": "$\\frac{m^{\\frac{35}{8}}}{n^{\\frac{7}{6}}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.30", "question": "Simplify: $\\frac{2 b a^{7} \\cdot 2 b^{4}}{b a^{2} \\cdot 3 a^{3} b^{4}}$", "answer": "$\\frac{4 a^{2}}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.8.3", "question": "Convert $11.2 \\mathrm{mg}$ to grams.", "answer": "$0.0112 \\mathrm{~g}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.32", "question": "Solve the compound inequality and give interval notation: $-9 m+2<-10-6 m$ or $-m+5 \\geqslant 10+4 m$", "answer": "$m>4$ or $m \\leqslant-1:(-\\infty,-1] \\cup(4, \\infty)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.3.2.1", "question": "Solve the compound inequality and give interval notation: $\\frac{n}{3} \\leqslant-3$ or $-5 n \\leqslant-10$", "answer": "$n \\leqslant-9$ or $n \\geqslant 2:(-\\infty,-9] \\cup[2, \\infty)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.2", "question": "Find a quadratic equation with the solutions 3 and 6.", "answer": "$x^{2}-9 x+18=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.24", "question": "Solve the equation: $m-4=-13$", "answer": "-9", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.5", "question": "Evaluate using the given values: $c^{2}-(a-1)$; $a=3$ and $c=5$", "answer": "23", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.30", "question": "Solve: $5 x^{2}=-26+10 x$", "answer": "$\\frac{5+i \\sqrt{105}}{5}, \\frac{5-i \\sqrt{105}}{5}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.7.21", "question": "Hooke's law states that the distance that a spring is stretched by hanging object varies directly as the mass of the object. If the distance is $20 \\mathrm{~cm}$ when the mass is $3 \\mathrm{~kg}$, what is the distance when the mass is $5 \\mathrm{~kg}$ ?", "answer": "$33.3 \\mathrm{~cm}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.12", "question": "How many kilograms of soil supplement that costs $\\$ 7.00$ per kilogram must be mixed with $20 \\mathrm{~kg}$ of aluminum nitrate that costs $\\$ 3.50$ per kilogram to make a fertilizer that costs $\\$ 4.50$ per kilogram?", "answer": "8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.29", "question": "Solve: $-a-5(8 a-1)=39-7 a$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.8.22", "question": "Let the first angle of a triangle be $x$ degrees. Find the measures of the second and third angles.", "answer": "$28,84,68$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.34", "question": "Given the points $(-2,y)$ and $(2,4)$ and a slope of $\\frac{1}{4}$, find the value of $y$.", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.76", "question": "Evaluate the expression: $(-1)-\\left(-\\frac{1}{3}\\right)$", "answer": "$-\\frac{2}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.25", "question": "Simplify: $-2 \\sqrt[3]{16}+2 \\sqrt[3]{16}+2 \\sqrt[3]{2}$", "answer": "$2 \\sqrt[3]{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.24", "question": "Simplify the expression: $\\frac{2 n^{2}-12 n-54}{n+7} \\div(2 n+6)$", "answer": "$\\frac{n-9}{n+7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.38", "question": "Find the product: $5(2 x-1)(4 x+1)$", "answer": "$40 x^{2}-10 x-5$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.9", "question": "$\\$ 1750$ is invested in an account earning $13.5 \\%$ interest compounded monthly for a 2 year period. What is the balance at the end of 9 years?", "answer": "2288.98", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.18", "question": "Solve the equation: $-8+\\frac{n}{12}=-7$", "answer": "12", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.52", "question": "Evaluate the expression: $i^{251}$", "answer": "$-i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.20", "question": "Solve each equation by completing the square: $m^{2}-8 m-3=6$", "answer": "$9,-1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.75", "question": "Evaluate the expression: $\\frac{3}{2}-\\frac{15}{8}$", "answer": "$-\\frac{3}{8}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.32", "question": "Solve the equation: $\\log _{2}-8 r=1$", "answer": "$-\\frac{1}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.28", "question": "Solve: $-8 n-19=-2(8 n-3)+3 n$", "answer": "5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.18", "question": "A pitcher is 30 years old, and a vase is 22 years old. How many years ago was the pitcher twice as old as the vase?", "answer": "14", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.35", "question": "Find product: $(-4)(-2)$", "answer": "8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.4", "question": "Simplify: $5+(-6-6 i)$", "answer": "$-1-6 i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.38", "question": "Given the points $(2,-5)$ and $(3, y)$ and a slope of 6, find the value of $y$.", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.34", "question": "Simplify: $\\frac{y x^{2} \\cdot\\left(y^{4}\\right)^{2}}{2 y^{4}}$", "answer": "$\\frac{y^{5} x^{2}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.9", "question": "Solve: $\\frac{\\frac{3}{2 a-3}+2}{\\frac{-6}{2 a-3}-4}$", "answer": "$-\\frac{1}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.18", "question": "Find a quadratic equation with the solutions $\\frac{5}{3}$ and $-\\frac{1}{2}$.", "answer": "$6 x^{2}-7 x-5=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.22", "question": "Simplify: $-2 \\sqrt{128 n}$", "answer": "$-16 \\sqrt{2 n}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.40", "question": "Find quotient: $-2 \\div \\frac{-3}{2}$", "answer": "$\\frac{4}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.6", "question": "Reduce the following radical: $\\sqrt[15]{x^{9} y^{12} z^{6}}$", "answer": "$\\sqrt[5]{x^{3} y^{4} z^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.21", "question": "Build up each fraction by finding the Least Common Denominator: $\\frac{3 a}{5 b^{2}}, \\frac{2}{10 a^{3} b}$", "answer": "$\\frac{6 a^{4}}{10 a^{3} b^{2}}, \\frac{2 b}{10 a^{3} b^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.5.1", "question": "Solve: $\\frac{1+\\frac{1}{x}}{1-\\frac{1}{x^{2}}}$", "answer": "$\\frac{x}{x-1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.21", "question": "Simplify: $-7 \\sqrt{64 x^{4}}$", "answer": "$-56 x^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.7", "question": "Simplify the expression $x^{2}+9 x+23$ when $x=-3$.", "answer": "5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.36", "question": "Simplify. Your answer should contain only positive expontents: $\\left(\\frac{\\left(2 x^{-3} y^{0} z^{-1}\\right)^{3} \\cdot x^{-3} y^{2}}{2 x^{3}}\\right)^{-2}$", "answer": "$\\frac{x^{30} z^{6}}{16 y^{4}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.5.21", "question": "Evaluate the expression: $\\log _{2} 64$", "answer": "6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.7.3", "question": "Find the value: $\\sin 75^{\\circ}$", "answer": "0.9659", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.1.12", "question": "Solve: $\\sqrt{7 x+2}-\\sqrt{3 x+6}=6$", "answer": "46", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.6.34", "question": "Simplify and express with positive exponents: $\\left(\\frac{y^{\\frac{1}{3}} y^{-2}}{\\left(x^{\\frac{5}{3}} y^{3}\\right)^{-\\frac{3}{2}}}\\right)^{\\frac{3}{2}}$", "answer": "$x^{\\frac{15}{4}} y^{\\frac{17}{4}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.28", "question": "Solve the equation for $x$: $\\frac{x}{x+1}-\\frac{3}{x+3}=\\frac{-2 x^{2}}{x^{2}+4 x+3}$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.37", "question": "Distribute: $-8(x-4)$", "answer": "$-8 x+32$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.51", "question": "Evaluate the expression: $i^{73}$", "answer": "$i$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.52", "question": "Distribute: $2 x(8 x-10)$", "answer": "$16 x^{2}-20 x$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.4", "question": "Simplify the expression: $\\frac{9 m}{5 m^{2}} \\cdot \\frac{7}{2}$", "answer": "$\\frac{63}{10 m}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.30", "question": "Evaluate the expression: $\\frac{-4+2 i}{3 i}$", "answer": "$\\frac{4 i+2}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.8", "question": "Specify the domain of the function: $f(x)=\\frac{-2}{x^{2}-3 x-4}$", "answer": "$x \\neq-1,4$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.13", "question": "Find the product: $(3 v-4)(5 v-2)$", "answer": "$15 v^{2}-26 v+8$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.9", "question": "Solve the equation for $m$: $\\frac{3 m}{2 m-5}-\\frac{7}{3 m+1}=\\frac{3}{2}$", "answer": "-5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.2.26", "question": "Find the slope of the line through the points $(11,-2)$ and $(1,17)$.", "answer": "$-\\frac{19}{10}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.21", "question": "A motorboat leaves a harbor and travels at an average speed of $18 \\mathrm{mph}$ to an island. The average speed on the return trip was $12 \\mathrm{mph}$. How far was the island from the harbor if the total trip took $5 \\mathrm{~h}$?", "answer": "36", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.19", "question": "Find the square: $(x-8)^{2}$", "answer": "$x^{2}-16 x+64$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.31", "question": "Simplify the expression: $\\frac{x^{2}-12 x+32}{x^{2}-6 x-16} \\cdot \\frac{7 x^{2}+14 x}{7 x^{2}+21 x}$", "answer": "$\\frac{x-4}{x+3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.7", "question": "A thousand dollars is left in a bank savings account drawing $7 \\%$ interest, compounded quarterly for 10 years. What is the balance at the end of that time?", "answer": "2001.60", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.7", "question": "Specify the domain of the function: $f(x)=\\sqrt{x-16}$", "answer": "$x \\geqslant 16$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.24", "question": "Solve the equation for $x$: $\\frac{x}{x+3}-\\frac{4}{x-2}=\\frac{-5 x^{2}}{x^{2}+x-6}$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.6.6", "question": "Find the principal that will amount to $\\$ 1750$ if invested at $3 \\%$ interest compounded quarterly for 5 years.", "answer": "1507.08", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.24", "question": "Add or subtract the rational expressions and simplify: $\\frac{3 a}{4 a-20}+\\frac{9 a}{6 a-30}$", "answer": "$\\frac{9 a}{4(a-5)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.2.18", "question": "Solve: $(x-1)^{-\\frac{5}{3}}=32$", "answer": "$\\frac{9}{8}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.24", "question": "Solve the following system of equations by elimination:\n$3 x+7 y=-8$\n$4 x+6 y=-4$", "answer": "$(2,-2)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.10", "question": "How many liters of a solvent that costs $\\$ 80$ per liter must be mixed with $6 \\mathrm{~L}$ of a solvent that costs $\\$ 25$ per liter to make a solvent that costs $\\$ 36$ per liter?", "answer": "1.5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.9", "question": "Evaluate the expression: $(-7)-(-5)$", "answer": "-2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.7", "question": "A man having ten hours at his disposal made an excursion, riding out at the rate of 10 miles an hour and returning on foot, at the rate of 3 miles an hour. Find the distance he rode.", "answer": "$\\frac{300}{13}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.15", "question": "Find the product: $(6 x-7)(4 x+1)$", "answer": "$24 x^{2}-22 x-7$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.3.8", "question": "State if the given functions are inverses:\n$f(x)=\\sqrt[5]{\\frac{x+1}{2}}$\n$g(x)=2 x^{5}-1$", "answer": "Yes", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.30", "question": "Simplify. Your answer should contain only positive expontents: $\\frac{u^{-3} v^{-4}}{2 v\\left(2 u^{-3} v^{4}\\right)^{0}}$", "answer": "$\\frac{1}{2 u^{3} v^{5}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.23", "question": "Evaluate the expression: $5-(-6)$", "answer": "11", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.16", "question": "Solve the following systems of equation: \n$p+q+r=1$\n$p+2 q+3 r=4$\n$4 p+5 q+6 r=7$", "answer": "$\\propto$ solutions", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.24", "question": "Build up each fraction by finding the Least Common Denominator: $\\frac{5}{x^{2}-6 x}, \\frac{2}{x}, \\frac{-3}{x-6}$", "answer": "$\\frac{5}{x(x-6)}, \\frac{2 x-12}{x(x-6)}, \\frac{-3 x}{x(x-6)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.28", "question": "Find a quadratic equation with the solutions $\\pm 11 i$.", "answer": "$x^{2}+121=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.13", "question": "A is now 34 years old, and $\\mathrm{B}$ is 4 years old. In how many years will A be twice as old as B?", "answer": "26", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.19", "question": "A chemist wants to make $50 \\mathrm{ml}$ of a $16 \\%$ acid solution by mixing a $13 \\%$ acid solution and an $18 \\%$ acid solution. How many milliliters of each solution should the chemist use?", "answer": "20,30", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.1", "question": "Solve the proportion: $\\frac{10}{a}=\\frac{6}{8}$", "answer": "$\\frac{40}{3}=a$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.18", "question": "Divide: $\\frac{48 k^{2}-70 k+16}{6 k-2}$", "answer": "$8 k-9-\\frac{1}{3 k-1}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.6.23", "question": "Find the square: $(7-5 n)^{2}$", "answer": "$49-70 n+25 n^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.17", "question": "Solve each equation by completing the square: $x^{2}+10 x-57=4$", "answer": "$-5+\\sqrt{86},-5-\\sqrt{86}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.15", "question": "State the excluded values for $\\frac{b^{2}+12 b+32}{b^{2}+4 b-32}$", "answer": "$-8,4$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.19", "question": "Solve the proportion: $\\frac{7}{x-1}=\\frac{4}{x-6}$", "answer": "$x=\\frac{38}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.48", "question": "Solve: $-5(x+7)=4(-8 x-2)$", "answer": "1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.5.39", "question": "Find the product: $6(4 x-1)(4 x+1)$", "answer": "$96 x^{2}-6$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.7", "question": "Solve the equation: $0=-6 v$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.3.17", "question": "Solve the following system of equations by elimination:\n$-7 x+4 y=-4$\n$10 x-8 y=-8$", "answer": "$(4,6)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.36", "question": "Solve the equation $\\mathrm{V}=\\frac{\\pi r^{2} h}{3}$ for $\\mathrm{h}$.", "answer": "$h=\\frac{3 v}{\\pi r^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.17", "question": "Solve the equation: $340=20 n$", "answer": "17", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.19", "question": "Simplify. Your answer should contain only positive expontents: $\\left(\\frac{2 a^{2} b^{3}}{a^{-1}}\\right)^{4}$", "answer": "$16 a^{12} b^{12}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.27", "question": "Simplify: $\\frac{x+1}{x^{2}+8 x+7}$", "answer": "$\\frac{1}{x+7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.21", "question": "Solve the following equation: $2 x^{4}-5 x^{2}+2=0$", "answer": "$\\pm \\sqrt{2}, \\pm \\frac{\\sqrt{2}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.6.27", "question": "A goldsmith combined an alloy that costs $\\$ 4.30$ per ounce with an alloy that costs $\\$ 1.80$ per ounce. How many ounces of each were used to make a mixture of 200 oz costing $\\$ 2.50$ per ounce?", "answer": "56,144", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.9.27", "question": "A father is three times as old as his son, and his daughter is 3 years younger than the son. If the sum of their ages 3 years ago was 63 years, find the present age of the father.", "answer": "45", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.3.8", "question": "Solve: $-55=8+7(k-5)$", "answer": "-4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.6", "question": "Solve the equation $\\frac{\\mathrm{ym}}{b}=\\frac{c}{d}$ for $\\mathrm{y}$.", "answer": "$y=\\frac{\\mathrm{cb}}{\\mathrm{dm}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.2.2", "question": "Simplify. Your answer should contain only positive expontents: $2 a^{-2} b^{-3} \\cdot\\left(2 a^{0} b^{4}\\right)^{4}$", "answer": "$\\frac{32 b^{13}}{a^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.54", "question": "Solve: $4 b^{2}-15 b+56=3 b^{2}$", "answer": "8,7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.26", "question": "Find a quadratic equation with the solutions $\\pm 2 \\sqrt{3}$.", "answer": "$x^{2}-12=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.4", "question": "Solve the equation: $16^{-3 p}=64^{-3 p}$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.1.1", "question": "Simplify: $4 \\cdot 4^{4} \\cdot 4^{4}$", "answer": "$4^{9}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.3.9", "question": "Find the missing numerator to build up the denominator: $\\frac{x-4}{x+2}=\\frac{?}{x^{2}+5 x+6}$", "answer": "$x^{2}-x-12$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.6.2", "question": "Solve the absolute value equation: $|n|=7$", "answer": "$7,-7$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.5.15", "question": "Find the slope of a line perpendicular to the given line: $x+2 y=8$", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.4.28", "question": "Solve the following systems of equation: \n$3 x+2 y=z+2$\n$y=1-2 x$\n$3 z=-2 y$", "answer": "$\\left(\\frac{2}{7}, \\frac{3}{7},-\\frac{2}{7}\\right)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.10", "question": "Evaluate the expression: $(-4)+(-1)$", "answer": "-5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.4.11", "question": "Evaluate $\\left(5 p-5 p^{4}\\right)-\\left(8 p-8 p^{4}\\right)$.", "answer": "$11 b+19$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.8", "question": "Add or subtract the rational expressions and simplify: $\\frac{7}{x y^{2}}+\\frac{3}{x^{2} y}$", "answer": "$\\frac{7 x+3 y}{x^{2} y^{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.42", "question": "Solve the equation $\\mathrm{x}+5 \\mathrm{y}=3$ for $\\mathrm{x}$.", "answer": "$x=3-5 y$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.1.8", "question": "Simplify: $5 \\sqrt{32}$", "answer": "$20 \\sqrt{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.8.18", "question": "A sink is $\\frac{1}{4}$ full when both the faucet and the drain are opened. The faucet alone can fill the sink in 6 minutes, while it takes 8 minutes to empty it with the drain. How long will it take to fill the remaining $\\frac{3}{4}$ of the sink?", "answer": "$18 \\mathrm{~min}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.7.34", "question": "Solve the equation for $x$: $\\frac{3 x-1}{x+6}-\\frac{2 x-3}{x-3}=\\frac{-3 x^{2}}{x^{2}+3 x-18}$", "answer": "$\\frac{7}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.39", "question": "Simplify and write the answer in scientific notation: $\\frac{2.4 \\times 10^{-6}}{6.5 \\times 10^{0}}$", "answer": "$3.692 \\times 10^{-7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.7.23", "question": "Combine the following radicals: $\\sqrt[4]{a^{2} b c^{2}} \\sqrt[5]{a^{2} b^{3} c}$", "answer": "$\\sqrt[20]{a^{18} b^{17} c^{14}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.36", "question": "Solve: $n^{2}+4 n=12$", "answer": "$2,-6$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.42", "question": "Solve the following equation: $\\left(x^{2}+x\\right)^{2}-8\\left(x^{2}+x\\right)+12=0$", "answer": "$-3, \\pm 2,1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.1.8", "question": "Solve: $\\sqrt{2 x+2}=3+\\sqrt{2 x-1}$", "answer": "no solution", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.7", "question": "Solve the equation: $x-7=-26$", "answer": "-19", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.10", "question": "Solve the equation: $5^{2 n}=5^{-n}$", "answer": "0", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.22", "question": "Given $w(x)=-4 x+3$, find $w(6)$", "answer": "-21", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.4.38", "question": "Simplify: $\\frac{4}{\\sqrt[4]{64 m^{4} n^{2}}}$", "answer": "$\\frac{\\sqrt[4]{4 n^{2}}}{m n}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.5", "question": "Simplify the expression: $\\frac{5 x^{2}}{4} \\cdot \\frac{6}{5}$", "answer": "$\\frac{3 x^{2}}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.11", "question": "Given $g(x)=4 x-4$, find $g(0)$", "answer": "-4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.21", "question": "Find product: $(9)\\left(\\frac{8}{9}\\right)$", "answer": "8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.23", "question": "Combine Like Terms: $n+n$", "answer": "$2 n$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.45", "question": "State the excluded values for $\\frac{7 n^{2}-32 n+16}{4 n-16}$", "answer": "$\\frac{7 n-4}{4}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.33", "question": "Combine Like Terms: $m-2 m$", "answer": "$-m$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.5.3", "question": "Find the slope of a line parallel to the given line: $y=4 x-5$", "answer": "4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.2.15", "question": "Simplify the expression: $\\frac{x^{2}-6 x-7}{x+5} \\cdot \\frac{x+5}{x-7}$", "answer": "$x+1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.6.38", "question": "Sarah worked 10 more hours than Josh. If Sarah worked $7 \\mathrm{hr}$ for every $2 \\mathrm{hr}$ Josh worked, how long did they each work?", "answer": "J: $4 \\mathrm{hr}, \\mathrm{S}: 14 \\mathrm{hr}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.4.22", "question": "Solve the equation: $\\frac{7}{6}-\\frac{4}{3} n=-\\frac{3}{2} n+2\\left(n+\\frac{3}{2}\\right)$", "answer": "-1", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.2.15", "question": "Simplify: $\\sqrt[5]{224 n^{3}}$", "answer": "$2 \\sqrt[5]{7 n^{3}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.1.11", "question": "Solve the equation: $340=-17 x$", "answer": "-20", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.8.39", "question": "Evaluate the expression: $\\frac{7}{10-7 i}$", "answer": "$\\frac{70+49 i}{149}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.33", "question": "Solve the equation $a x+b=c$ for $a$.", "answer": "$a=\\frac{c-b}{x}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.9.1", "question": "Solve the following systems of equation: \n$x y=72$\n$(x+2)(y-4)=128$", "answer": "$(2,36),(-18,-4)$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.9", "question": "Divide: $\\frac{x^{2}-2 x-71}{x+8}$", "answer": "$x-10+\\frac{9}{x+8}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.33", "question": "Three campers left their campsite by canoe and paddled downstream at an average rate of $10 \\mathrm{mph}$. They then turned around and paddled back upstream at an average rate of $5 \\mathrm{mph}$ to return to their campsite. How long did it take the campers to canoe downstream if the total trip took $1 \\mathrm{hr}$?", "answer": "$\\frac{1}{3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.58", "question": "Find quotient: $\\frac{48}{8}$", "answer": "6", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.2.5.7", "question": "Find the slope of a line parallel to the given line: $7 x+y=-2$", "answer": "-7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.8.3.12", "question": "Simplify: $-\\sqrt{5}-\\sqrt{5}-2 \\sqrt{54}$", "answer": "$-2 \\sqrt{5}-6 \\sqrt{6}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.36", "question": "Find a quadratic equation with the solutions $-9 \\pm i \\sqrt{5}$.", "answer": "$x^{2}+18 x+86=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.1.30", "question": "Evaluate the expression: $(-3)+(-5)$", "answer": "-8", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.3", "question": "The attendance at a school concert was 578. Admission was $\\$ 2.00$ for adults and $\\$ 1.50$ for children. The total receipts were $\\$ 985.00$. How many adults and how many children attended?", "answer": "236 adult, 342 child", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.2.1", "question": "Solve the equation: $5+\\frac{n}{4}=4$", "answer": "-4", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.4.21", "question": "Solve the equation: $3^{3 x-2}=3^{3 x+1}$", "answer": "No solution", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.4.9", "question": "Evaluate using the given values: $\\frac{4-(p-m)}{2}+q$; $m=4, p=6, q=6$", "answer": "7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.10", "question": "Solve the equation $\\mathrm{E}=\\frac{\\mathrm{mv}^{2}}{2}$ for $\\mathrm{m}$.", "answer": "$m=\\frac{2 E}{v_{2}}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.4.21", "question": "Add or subtract the rational expressions and simplify: $\\frac{t}{t-3}-\\frac{5}{4 t-12}$", "answer": "$\\frac{4 t-5}{4(t-3)}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.7.1.18", "question": "Simplify: $\\frac{12 n}{4 n^{2}}$", "answer": "$\\frac{3}{n}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.24", "question": "Divide: $\\frac{2 x^{2}-5 x-8}{2 x+3}$", "answer": "$x-4+\\frac{4}{2 x+3}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.38", "question": "Given $h(t)=t^{2}+t$, find $h\\left(t^{2}\\right)$", "answer": "$t^{4}+t^{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.8.4", "question": "Find the angle measure to the nearest degree given $\\cos \\mathrm{Y}=0.6157$", "answer": "$52^{\\circ}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.41", "question": "Find quotient: $\\frac{-3}{2} \\div \\frac{13}{7}$", "answer": "$-\\frac{21}{26}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.5.39", "question": "Solve the equation $\\mathrm{at}-\\mathrm{bw}=s$ for $\\mathrm{t}$.", "answer": "$t=\\frac{5+\\mathrm{bw}}{a}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.33", "question": "Find product: $33(2)\\left(\\frac{3}{2}\\right)$", "answer": "3", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.3.19", "question": "Solve each equation by completing the square: $n^{2}-16 n+67=4$", "answer": "9,7", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.3.10", "question": "Write the number in standard notation: $5 \\times 10^{4}$", "answer": "50000", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.1.10.22", "question": "A motorboat leaves a harbor and travels at an average speed of $9 \\mathrm{mph}$ toward a small island. Two hours later a cabin cruiser leaves the same harbor and travels at an average speed of $18 \\mathrm{mph}$ toward the same island. In how many hours after the cabin cruiser leaves will the cabin cruiser be alongside the motorboat?", "answer": "2", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.4.5.21", "question": "A coin bank contains nickels and dimes. The number of dimes is 10 less than twice the number of nickels. The total value of all the coins is $\\$ 2.75$. Find the number of each type of coin in the bank.", "answer": "$15 \\mathrm{n}, 20 \\mathrm{~d}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.6.32", "question": "Solve the following equation: $(m-1)^{2}-5(m-1)=14$", "answer": "$8,-1$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.5.7.29", "question": "Divide: $\\frac{a^{3}+15 a^{2}+49 a-55}{a+7}$", "answer": "$a^{2}+8 a-7-\\frac{6}{a+7}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.9.5.30", "question": "Find a quadratic equation with the solutions $\\pm 5 i \\sqrt{2}$.", "answer": "$x^{2}+50=0$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.0.2.11", "question": "Simplify and leave your answer as an improper fraction: $\\frac{40}{16}$", "answer": "$\\frac{5}{2}$", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.Beginning_and_Intermediate_Algebra", "question_number": "exercise.10.1.20", "question": "Given $w(x)=x^{2}+4 x$, find $w(-5)$", "answer": "5", "license": "Creative Commons Attribution 3.0 Unported License (CC BY 3.0)", "data_topic": "college_math.algebra"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.2.32", "question": "Express the repeating decimal as a fraction of integers: $-5.8 \\overline{67}$", "answer": "$-\\frac{5809}{990}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.91", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \\pi \\leq x \\leq 2 \\pi$: $\\sin (2 x) \\geq \\sin (x)$", "answer": "$\\left[-2 \\pi,-\\frac{5 \\pi}{3}\\right] \\cup\\left[-\\pi,-\\frac{\\pi}{3}\\right] \\cup\\left[0, \\frac{\\pi}{3}\\right] \\cup\\left[\\pi, \\frac{5 \\pi}{3}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.84", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\\pi \\leq x \\leq \\pi$: $\\sin ^{2}(x)<\\frac{3}{4}$", "answer": "$\\left[-\\pi,-\\frac{\\pi}{4}\\right] \\cup\\left(0, \\frac{3 \\pi}{4}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.35", "question": "Evaluate the expression: $\\ln \\left(e^{5}\\right)$", "answer": "$\\ln \\left(e^{5}\\right)=5$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.10", "question": "Solve the equation analytically: $5^{-x}=2$", "answer": "$x=-\\frac{\\ln (2)}{\\ln (5)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.23", "question": "Simplify the given power of $i$: $i^{15}$", "answer": "$i^{15}=\\left(i^{4}\\right)^{3} \\cdot i^{3}=1 \\cdot(-i)=-i$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.57", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sin (6 x)+\\sin (x)=0$", "answer": "$x=0, \\frac{2 \\pi}{7}, \\frac{4 \\pi}{7}, \\frac{6 \\pi}{7}, \\frac{8 \\pi}{7}, \\frac{10 \\pi}{7}, \\frac{12 \\pi}{7}, \\frac{\\pi}{5}, \\frac{3 \\pi}{5}, \\pi, \\frac{7 \\pi}{5}, \\frac{9 \\pi}{5}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.17", "question": "Solve the equation analytically: $\\log _{169}(3 x+7)-\\log _{169}(5 x-9)=\\frac{1}{2}$", "answer": "$x=2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.40", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\csc ^{3}(x)+\\csc ^{2}(x)=4 \\csc (x)+4$", "answer": "$x=\\frac{\\pi}{6}, \\frac{5 \\pi}{6}, \\frac{7 \\pi}{6}, \\frac{3 \\pi}{2}, \\frac{11 \\pi}{6}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.13", "question": "Simplify the quantity $\\sqrt{-25}\\sqrt{-4}$", "answer": "-10", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.4", "question": "Solve the rational equation: $\\frac{2 x+17}{x+1}=x+5$", "answer": "$x=-6, x=2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.23", "question": "The height $h$ in feet of a model rocket above the ground $t$ seconds after lift-off is given by $h(t)=-5 t^{2}+100 t$, for $0 \\leq t \\leq 20$. When does the rocket reach its maximum height above the ground? What is its maximum height?", "answer": "The rocket reaches its maximum height of 500 feet 10 seconds after lift-off.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.13", "question": "Solve the equation: $|x|=x^{2}$", "answer": "$x=-1, x=0$ or $x=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.7.3.19", "question": "The mirror in Carl's flashlight is a paraboloid of revolution. If the mirror is 5 centimeters in diameter and 2.5 centimeters deep, where should the light bulb be placed so it is at the focus of the mirror?", "answer": "The bulb should be placed 0.625 centimeters above the vertex of the mirror. (As verified by Carl himself!)", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.13", "question": "Solve the equation analytically: $6-3 \\log _{5}(2 x)=0$", "answer": "$x=\\frac{25}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.55", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sin (5 x)=\\sin (3 x)$", "answer": "$x=0, \\frac{\\pi}{8}, \\frac{3 \\pi}{8}, \\frac{5 \\pi}{8}, \\frac{7 \\pi}{8}, \\pi, \\frac{9 \\pi}{8}, \\frac{11 \\pi}{8}, \\frac{13 \\pi}{8}, \\frac{15 \\pi}{8}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.5", "question": "Write out the first four terms of the given sequence: $\\left\\{\\frac{x^{n}}{n^{2}}\\right\\}_{n=1}^{\\infty}$", "answer": "$x, \\frac{x^{2}}{4}, \\frac{x^{3}}{9}, \\frac{x^{4}}{16}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.36", "question": "Solve the inequality analytically: $2^{\\left(x^{3}-x\\right)}<1$", "answer": "$(-\\infty,-1) \\cup(0,1)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.28", "question": "Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(2 \\sqrt{45}, \\sqrt{12}),(\\sqrt{20}, \\sqrt{27})$.", "answer": "$d=\\sqrt{83}, M=\\left(4 \\sqrt{5}, \\frac{5 \\sqrt{3}}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.19", "question": "Evaluate the expression: $\\log _{6}\\left(\\frac{1}{36}\\right)$", "answer": "$\\log _{6}\\left(\\frac{1}{36}\\right)=-2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.33", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sin (2 x)=\\tan (x)$", "answer": "$x=0, \\pi, \\frac{\\pi}{4}, \\frac{3 \\pi}{4}, \\frac{5 \\pi}{4}, \\frac{7 \\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.41", "question": "Convert the angle from radian measure into degree measure: $\\frac{\\pi}{3}$", "answer": "$60^{\\circ}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.52", "question": "Create a polynomial $f$ that is degree 5 and has the following characteristics:\n- $x=6$, $x=i$, and $x=1-3i$ are zeros of $f$\n- As $x \\rightarrow -\\infty$, $f(x) \\rightarrow \\infty$", "answer": "$f(x)=a(x-6)(x-i)(x+i)(x-(1-3 i))(x-(1+3 i))$ where $a$ is any real number, $a<0$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.23", "question": "Convert the point from polar coordinates into rectangular coordinates: $\\left(9, \\frac{7 \\pi}{2}\\right)$", "answer": "$(0,-9)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.39", "question": "Find all of the angles which satisfy the given equation: $\\cos (\\theta)=-1.001$", "answer": "$\\cos (\\theta)=-1.001$ never happens", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.43", "question": "Use your calculator to help you solve the inequality: $e^{-x}-x e^{-x} \\geq 0$", "answer": "$(-\\infty, 1]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.5.3.20", "question": "Solve the equation or inequality: $3 x+\\sqrt{6-9 x}=2$", "answer": "$x=-\\frac{1}{3}, \\frac{2}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.79", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $\\sec (x) \\leq \\sqrt{2}$", "answer": "$\\left[0, \\frac{\\pi}{4}\\right] \\cup\\left(\\frac{\\pi}{2}, \\frac{3 \\pi}{2}\\right) \\cup\\left[\\frac{7 \\pi}{4}, 2 \\pi\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.41", "question": "Find the real solutions of the polynomial equation $x^{3}+x^{2}=\\frac{11 x+10}{3}$.", "answer": "$x= \\pm \\sqrt{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.7", "question": "Expand the given logarithm and simplify: $\\log _{\\sqrt{2}}\\left(4 x^{3}\\right)$", "answer": "$3 \\log _{\\sqrt{2}}(x)+4$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.32", "question": "Solve the quadratic equation $y^{2}-4 y=x^{2}-4$ for $x$.", "answer": "$x= \\pm(y-2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.50", "question": "Approximate the given value to three decimal places: $\\cos (-2.01)$", "answer": "$\\cos (-2.01) \\approx-0.425$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.32", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (x) \\csc (x) \\cot (x)=6-\\cot ^{2}(x)$", "answer": "$x=\\frac{\\pi}{6}, \\frac{7 \\pi}{6}, \\frac{5 \\pi}{6}, \\frac{11 \\pi}{6}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.19", "question": "Solve the equation analytically: $2 \\log _{7}(x)=\\log _{7}(2)+\\log _{7}(x+12)$", "answer": "$x=6$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.63", "question": "If $\\theta=5^{\\circ}$ and the hypotenuse has length 10 , how long is the side adjacent to $\\theta$ ?", "answer": "The side adjacent to $\\theta$ has length $10 \\cos \\left(5^{\\circ}\\right) \\approx 9.962$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.17", "question": "Solve the equation analytically: $70+90 e^{-0.1 t}=75$", "answer": "$t=\\frac{\\ln \\left(\\frac{1}{18}\\right)}{-0.1}=10 \\ln (18)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.7.13", "question": "Solve the system of nonlinear equations: $\\left\\{\\begin{array}{rr}y & =x^{3}+8 \\\\ y & =10 x-x^{2}\\end{array}\\right.$", "answer": "$(-4,-56),(1,9),(2,16)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.17", "question": "Evaluate the expression: $\\log _{6}(216)$", "answer": "$\\log _{6}(216)=3$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.35", "question": "Find the real solutions of the polynomial equation $9 x^{3}=5 x^{2}+x$.", "answer": "$x=0, \\frac{5 \\pm \\sqrt{61}}{18}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.14", "question": "Expand the given logarithm and simplify: $\\log _{\\frac{1}{2}}\\left(\\frac{4 \\sqrt[3]{x^{2}}}{y \\sqrt{z}}\\right)$", "answer": "$-2+\\frac{2}{3} \\log _{\\frac{1}{2}}(x)-\\log _{\\frac{1}{2}}(y)-\\frac{1}{2} \\log _{\\frac{1}{2}}(z)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.44", "question": "Find the real solutions of the polynomial equation $2 x^{5}+3 x^{4}=18 x+27$.", "answer": "$\\left\\{-\\frac{1}{2}\\right\\} \\cup[1, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.22", "question": "Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(1,2),(-3,5)$", "answer": "$d=5, M=\\left(-1, \\frac{7}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.1", "question": "Write out the first four terms of the given sequence: $a_{n}=2^{n}-1, n \\geq 0$", "answer": "$0,1,3,7$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.10", "question": "Write out the first four terms of the given sequence: $c_{0}=-2, c_{j}=\\frac{c_{j-1}}{(j+1)(j+2)}, j \\geq 1$", "answer": "$-2,-\\frac{1}{3},-\\frac{1}{36},-\\frac{1}{720}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.52", "question": "Convert the equation from rectangular coordinates into polar coordinates: $x^{2}+y^{2}=x$", "answer": "$\\left(\\frac{1}{3}, \\pi+\\arctan (2)\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.2.29", "question": "Express the repeating decimal as a fraction of integers: $0 . \\overline{7}$", "answer": "$\\frac{7}{9}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.66", "question": "Solve the equation: $9 \\arccos ^{2}(x)-\\pi^{2}=0$", "answer": "$x=-1,0$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.21", "question": "Solve the equation analytically: $\\frac{150}{1+29 e^{-0.8 t}}=75$", "answer": "$t=\\frac{\\ln \\left(\\frac{1}{29}\\right)}{-0.8}=\\frac{5}{4} \\ln (29)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.86", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\\pi \\leq x \\leq \\pi$: $\\cos (x) \\geq \\sin (x)$", "answer": "$\\left[-\\frac{3 \\pi}{4}, \\frac{\\pi}{4}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.27", "question": "Solve the inequality analytically: $10 \\log \\left(\\frac{x}{10^{-12}}\\right) \\geq 90$", "answer": "$\\left[10^{-3}, \\infty\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.4", "question": "Solve the equation analytically: $4^{2 x}=\\frac{1}{2}$", "answer": "$x=-\\frac{1}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.34", "question": "Solve the inequality analytically: $e^{x}>53$", "answer": "$(\\ln (53), \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.103", "question": "Express the domain of the function using the extended interval notation: $f(x)=\\csc (2 x)$", "answer": "$\\bigcup_{k=-\\infty}^{\\infty}\\left(\\frac{k \\pi}{2}, \\frac{(k+1) \\pi}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.4.80", "question": "Write the given sum as a product: $\\cos (3 \\theta)+\\cos (5 \\theta)$", "answer": "$2 \\cos (4 \\theta) \\cos (\\theta)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.46", "question": "Solve the polynomial inequality $x^{4}-9 x^{2} \\leq 4 x-12$ and state your answer using interval notation.", "answer": "$\\{2\\} \\cup[4, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.47", "question": "Convert the equation from rectangular coordinates into polar coordinates: $x=3 y+1$", "answer": "$\\left(10, \\arctan \\left(\\frac{4}{3}\\right)\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.7.15", "question": "Solve the system of nonlinear equations: $\\left\\{\\begin{aligned} x^{2}+y^{2} & =25 \\\\ 4 x^{2}-9 y & =0 \\\\ 3 y^{2}-16 x & =0\\end{aligned}\\right.$", "answer": "$(3,4)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.78", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $\\cos (3 x) \\leq 1$", "answer": "$[0,2 \\pi]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.9", "question": "Solve the equation analytically: $3^{2 x}=5$", "answer": "$x=\\frac{\\ln (5)}{2 \\ln (3)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.32", "question": "Find all of the points on the $x$-axis which are 2 units from the point $(-1,1)$.", "answer": "$(-1+\\sqrt{3}, 0),(-1-\\sqrt{3}, 0)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.74", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $\\sin \\left(x+\\frac{\\pi}{3}\\right)>\\frac{1}{2}$", "answer": "$\\left(0, \\frac{\\pi}{3}\\right] \\cup\\left[\\frac{2 \\pi}{3}, \\pi\\right) \\cup\\left(\\pi, \\frac{4 \\pi}{3}\\right] \\cup\\left[\\frac{5 \\pi}{3}, 2 \\pi\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.18", "question": "Simplify the quantity $-\\sqrt{(-9)}$", "answer": "$-3 i$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.62", "question": "Convert the equation from polar coordinates into rectangular coordinates: $\\theta=\\pi$", "answer": "$\\theta=\\frac{\\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.2", "question": "Solve the equation analytically: $3^{(x-1)}=27$", "answer": "$x=4$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.88", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \\pi \\leq x \\leq 2 \\pi$: $\\cos (x) \\leq \\frac{5}{3}$", "answer": "$[-2 \\pi, 2 \\pi]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.4", "question": "Solve the equation analytically: $\\log _{5}\\left(18-x^{2}\\right)=\\log _{5}(6-x)$", "answer": "$x=-3,4$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.34", "question": "Evaluate the expression: $\\log _{36}\\left(36^{216}\\right)$", "answer": "$\\log _{36}\\left(36^{216}\\right)=216$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.5.17", "question": "Find the inverse of the given matrix: $B=\\left[\\begin{array}{rr}12 & -7 \\\\ -5 & 3\\end{array}\\right]$", "answer": "$B^{-1}=\\left[\\begin{array}{rr}3 & 7 \\\\ 5 & 12\\end{array}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.56", "question": "Find the domain of the function: $f(x)=\\frac{\\sqrt{-1-x}}{\\log _{\\frac{1}{2}}(x)}$", "answer": "No domain", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.10", "question": "Solve the equation: $|2 x-1|=x+1$", "answer": "$x=0$ or $x=2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.29", "question": "Solve the equation analytically: $e^{2 x}=e^{x}+6$", "answer": "$x=\\ln (3)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.49", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sqrt{2} \\cos (x)-\\sqrt{2} \\sin (x)=1$", "answer": "$x=\\frac{\\pi}{12}, \\frac{17 \\pi}{12}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.17", "question": "Simplify the power of a complex number: $\\left(\\frac{\\sqrt{2}}{2}-\\frac{\\sqrt{2}}{2} i\\right)^{4}$", "answer": "-1", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.23", "question": "Solve the equation analytically: $(\\log (x))^{2}=2 \\log (x)+15$", "answer": "$x=10^{-3}, 10^{5}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.11", "question": "Solve the rational inequality and express your answer using interval notation: $\\frac{x^{2}-x-12}{x^{2}+x-6}>0$", "answer": "$(-\\infty,-3) \\cup(-3,2) \\cup(4, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.22", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} x-3 y-4 z & =3 \\\\ 3 x+4 y-z & =13 \\\\ 2 x-19 y-19 z & =2\\end{aligned}\\right.$", "answer": "$\\left(\\frac{19}{13} t+\\frac{51}{13},-\\frac{11}{13} t+\\frac{4}{13}, t\\right)$ for all real numbers $t$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.1", "question": "Solve the equation analytically: $2^{4 x}=8$", "answer": "$x=\\frac{3}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.28", "question": "Solve the inequality analytically: $5.6 \\leq \\log \\left(\\frac{x}{10^{-3}}\\right) \\leq 7.1$", "answer": "$\\left[10^{2.6}, 10^{4.1}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.42", "question": "Use your calculator to help you solve the equation: $e^{\\sqrt{x}}=x+1$", "answer": "$x=0$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.29", "question": "Solve the inequality analytically: $2.3<-\\log (x)<5.4$", "answer": "$\\left(10^{-5.4}, 10^{-2.3}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.4.2", "question": "Find the inverse of the matrix or state that the matrix is not invertible: $B=\\left[\\begin{array}{rr}12 & -7 \\\\ -5 & 3\\end{array}\\right]$", "answer": "$B^{-1}=\\left[\\begin{array}{rr}3 & 7 \\\\ 5 & 12\\end{array}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.29", "question": "Use the properties of logarithms to write the expression as a single logarithm: $\\log _{2}(x)+\\log _{\\frac{1}{2}}(x-1)$", "answer": "$\\log _{2}\\left(\\frac{x}{x-1}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.35", "question": "Solve the inequality analytically: $1000(1.005)^{12 t} \\geq 3000$", "answer": "$\\left[\\frac{\\ln (3)}{12 \\ln (1.005)}, \\infty\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.25", "question": "Use the properties of logarithms to write the expression as a single logarithm: $\\log _{7}(x)+\\log _{7}(x-3)-2$", "answer": "$\\log _{7}\\left(\\frac{x(x-3)}{49}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.21", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sin (2 x)=\\cos (x)$", "answer": "$x=\\frac{\\pi}{6}, \\frac{\\pi}{2}, \\frac{5 \\pi}{6}, \\frac{3 \\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.59", "question": "If $\\theta=12^{\\circ}$ and the side adjacent to $\\theta$ has length 4 , how long is the hypotenuse?", "answer": "The hypotenuse has length $\\frac{4}{\\cos \\left(12^{\\circ}\\right)} \\approx 4.089$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.19", "question": "Find the exact value of the cosine and sine of the given angle: $\\theta=\\frac{10 \\pi}{3}$", "answer": "$\\cos \\left(\\frac{10 \\pi}{3}\\right)=-\\frac{1}{2}, \\sin \\left(\\frac{10 \\pi}{3}\\right)=-\\frac{\\sqrt{3}}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.34", "question": "Let's assume for a moment that we are standing at the origin and the positive $y$-axis points due North while the positive $x$-axis points due East. Our Sasquatch-o-meter tells us that Sasquatch is 3 miles West and 4 miles South of our current position. What are the coordinates of his position? How far away is he from us? If he runs 7 miles due East what would his new position be?", "answer": "(-3, -4), 5 miles, $(4,-4)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.8", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} x+y+z & =3 \\\\ 2 x-y+z & =0 \\\\ -3 x+5 y+7 z & =7\\end{aligned}\\right.$", "answer": "$(-3,20,19)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.26", "question": "Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $\\left(\\frac{24}{5}, \\frac{6}{5}\\right),\\left(-\\frac{11}{5},-\\frac{19}{5}\\right)$.", "answer": "$d=\\sqrt{74}, M=\\left(\\frac{13}{10},-\\frac{13}{10}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.8", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\cos (9 x)=9$", "answer": "No solution", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.33", "question": "Solve the equation or inequality using your calculator: $\\ln \\left(x^{2}+1\\right) \\geq 5$", "answer": "$\\approx(-\\infty,-12.1414) \\cup(12.1414, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.7", "question": "Find the exact value of the cosine and sine of the given angle: $\\theta=\\pi$", "answer": "$\\cos (\\pi)=-1, \\sin (\\pi)=0$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.53", "question": "Approximate the given value to three decimal places: $\\sin \\left(\\pi^{\\circ}\\right)$", "answer": "$\\sin \\left(\\pi^{\\circ}\\right) \\approx 0.055$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.4.11", "question": "Use one matrix inverse to solve the following system of linear equations:\n$\\left\\{\\begin{aligned} 3 x+7 y & =-7 \\\\ 5 x+12 y & =5\\end{aligned}\\right.$", "answer": "$\\left[\\begin{array}{rr}12 & -7 \\\\ -5 & 3\\end{array}\\right]\\left[\\begin{array}{r}-7 \\\\ 5\\end{array}\\right]=\\left[\\begin{array}{r}-119 \\\\ 50\\end{array}\\right]$ So $x=-119$ and $y=50$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.4.5", "question": "Find the inverse of the matrix or state that the matrix is not invertible: $E=\\left[\\begin{array}{rrr}3 & 0 & 4 \\\\ 2 & -1 & 3 \\\\ -3 & 2 & -5\\end{array}\\right]$", "answer": "$E^{-1}=\\left[\\begin{array}{rrr}-1 & 8 & 4 \\\\ 1 & -3 & -1 \\\\ 1 & -6 & -3\\end{array}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.24", "question": "Use the properties of logarithms to write the expression as a single logarithm: $3-\\log (x)$", "answer": "$\\log \\left(\\frac{1000}{x}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.23", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} x+y+z & =4 \\\\ 2 x-4 y-z & =-1 \\\\ x-y & =2\\end{aligned}\\right.$", "answer": "Inconsistent", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.20", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sin (2 x)=\\sin (x)$", "answer": "$x=0, \\frac{\\pi}{3}, \\pi, \\frac{5 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.25", "question": "Simplify the given power of $i$: $i^{117}$", "answer": "$i^{117}=\\left(i^{4}\\right)^{29} \\cdot i=1 \\cdot i=i$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.16", "question": "Write the set using interval notation: $\\{x \\mid x \\leq-3$ or $x>0\\}$", "answer": "$(-\\infty,-3] \\cup(0, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.20", "question": "Solve the equation: $|2-5 x|=5|x+1|$", "answer": "$x=-\\frac{3}{10}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.3", "question": "Convert the angle into the DMS system and round the answer to the nearest second: $-317.06^{\\circ}$", "answer": "$-317^{\\circ} 3^{\\prime} 36^{\\prime \\prime} \\quad", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.33", "question": "Convert the angle from degree measure into radian measure, giving the exact value in terms of $\\pi$: $-315^{\\circ}$", "answer": "$-\\frac{7 \\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.2", "question": "Solve the equation analytically: $\\log _{2}\\left(x^{3}\\right)=\\log _{2}(x)$", "answer": "$x=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.9", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} 4 x-y+z & =5 \\\\ 2 y+6 z & =30 \\\\ x+z & =5\\end{aligned}\\right.$", "answer": "$(-3 t+4,-6 t-6,2, t)$ for all real numbers $t$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.25", "question": "Evaluate the expression: $\\log \\left(\\frac{1}{1000000}\\right)$", "answer": "$\\log \\frac{1}{1000000}=-6$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.18", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\sin ^{2}(x)=\\frac{3}{4}$", "answer": "$x=\\frac{\\pi}{3}+\\pi k$ or $x=\\frac{2 \\pi}{3}+\\pi k ; x=\\frac{\\pi}{3}, \\frac{2 \\pi}{3}, \\frac{4 \\pi}{3}, \\frac{5 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.92", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \\pi \\leq x \\leq 2 \\pi$: $\\cos (2 x) \\leq \\sin (x)$", "answer": "$\\left[-\\frac{11 \\pi}{6},-\\frac{7 \\pi}{6}\\right] \\cup\\left[\\frac{\\pi}{6}, \\frac{5 \\pi}{6}\\right] \\cup,\\left\\{-\\frac{\\pi}{2}, \\frac{3 \\pi}{2}\\right\\}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.37", "question": "Evaluate the expression: $\\log \\left(\\sqrt[3]{10^{5}}\\right)$", "answer": "$\\log \\left(\\sqrt[3]{10^{5}}\\right)=\\frac{5}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.31", "question": "Solve the equation analytically: $e^{x}-3 e^{-x}=2$", "answer": "$x=\\ln (3)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.2.30", "question": "Express the repeating decimal as a fraction of integers: $0 . \\overline{13}$", "answer": "$\\frac{13}{99}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.6", "question": "Write out the first four terms of the given sequence: $\\left\\{\\frac{\\ln (n)}{n}\\right\\}_{n=1}^{\\infty}$", "answer": "$0, \\frac{\\ln (2)}{2}, \\frac{\\ln (3)}{3}, \\frac{\\ln (4)}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.26", "question": "Solve the equation analytically: $3^{(x-1)}=\\left(\\frac{1}{2}\\right)^{(x+5)}$", "answer": "$x=\\frac{\\ln (3)+5 \\ln \\left(\\frac{1}{2}\\right)}{\\ln (3)-\\ln \\left(\\frac{1}{2}\\right)}=\\frac{\\ln (3)-5 \\ln (2)}{\\ln (3)+\\ln (2)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.47", "question": "Solve the polynomial inequality $(x-1)^{2} \\geq 4$ and state your answer using interval notation.", "answer": "$(-\\infty,-\\sqrt[3]{3}) \\cup(\\sqrt[3]{2}, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.54", "question": "Convert the equation from rectangular coordinates into polar coordinates: $(x+2)^{2}+y^{2}=4$", "answer": "$\\left(15,2 \\pi-\\arctan \\left(\\frac{3}{4}\\right)\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.13", "question": "Expand the given logarithm and simplify: $\\log \\left(\\frac{100 x \\sqrt{y}}{\\sqrt[3]{10}}\\right)$", "answer": "$\\frac{5}{3}+\\log (x)+\\frac{1}{2} \\log (y)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.1.30", "question": "Suppose the cost, in thousands of dollars, to produce $x$ hundred LCD TVs is given by $C(x)=200 x+25$ for $x \\geq 0$. Find and simplify an expression for the profit function $P(x)$. (Remember: Profit $=$ Revenue - Cost.)", "answer": "$p(t)=-t^{2}(3-5 t)\\left(t^{2}+t+4\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.31", "question": "Solve the quadratic equation $x^{2}-10 y^{2}=0$ for $x$.", "answer": "$x= \\pm y \\sqrt{10}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.12", "question": "Write the set using interval notation: $\\{x \\mid x \\neq 2,-2\\}$", "answer": "$(-\\infty,-2) \\cup(-2,2) \\cup(2, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.21", "question": "Use the properties of logarithms to write the expression as a single logarithm: $\\log (x)-\\frac{1}{3} \\log (z)+\\frac{1}{2} \\log (y)$", "answer": "$\\log \\left(\\frac{x \\sqrt{y}}{\\sqrt[3]{z}}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.19", "question": "Determine if the given sequence is arithmetic, geometric or neither. If it is arithmetic, find the common difference $d$; if it is geometric, find the common ratio $r$: $2,22,222,2222, \\ldots$", "answer": "neither", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.19", "question": "Solve the equation analytically: $\\frac{100 e^{x}}{e^{x}+2}=50$", "answer": "$x=\\ln (2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.31", "question": "Convert the point from polar coordinates into rectangular coordinates: $\\left(2, \\pi-\\arctan \\left(\\frac{1}{2}\\right)\\right)$", "answer": "$\\left(-\\frac{4 \\sqrt{5}}{5}, \\frac{2 \\sqrt{5}}{5}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.2", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\cos (3 x)=\\frac{1}{2}$", "answer": "$x=\\frac{\\pi}{9}+\\frac{2 \\pi k}{3}$ or $x=\\frac{5 \\pi}{9}+\\frac{2 \\pi k}{3} ; x=\\frac{\\pi}{9}, \\frac{5 \\pi}{9}, \\frac{7 \\pi}{9}, \\frac{11 \\pi}{9}, \\frac{13 \\pi}{9}, \\frac{17 \\pi}{9}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.4", "question": "Solve the equation: $4-|x|=3$", "answer": "$x=-1$ or $x=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.39", "question": "Evaluate the expression: $\\log _{5}\\left(3^{\\log _{3}(5)}\\right)$", "answer": "$\\log _{5}\\left(3^{\\log _{3} 5}\\right)=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.42", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\tan (x)=\\sec (x)$", "answer": "No solution", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.34", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cot ^{4}(x)=4 \\csc ^{2}(x)-7$", "answer": "$x=\\frac{\\pi}{6}, \\frac{\\pi}{4}, \\frac{3 \\pi}{4}, \\frac{5 \\pi}{6}, \\frac{7 \\pi}{6}, \\frac{5 \\pi}{4}, \\frac{7 \\pi}{4}, \\frac{11 \\pi}{6}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.102", "question": "Express the domain of the function using the extended interval notation: $f(x)=\\sqrt{2-\\sec (x)}$", "answer": "$\\bigcup_{k=-\\infty}^{\\infty}\\left\\{\\left[\\frac{(6 k-1) \\pi}{3}, \\frac{(6 k+1) \\pi}{3}\\right] \\cup\\left(\\frac{(4 k+1) \\pi}{2}, \\frac{(4 k+3) \\pi}{2}\\right)\\right\\}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.1.39", "question": "A local pizza store offers medium two-topping pizzas delivered for $\\$ 6.00$ per pizza plus a $\\$ 1.50$ delivery charge per order. On weekends, the store runs a 'game day' special: if six or more medium two-topping pizzas are ordered, they are $\\$ 5.50$ each with no delivery charge. Write a piecewise-defined linear function which calculates the $\\operatorname{cost} C$ (in dollars) of $p$ medium two-topping pizzas delivered during a weekend.", "answer": "$C(p)=\\left\\{\\begin{array}{rll}6 p+1.5 & \\text { if } & 1 \\leq p \\leq 5 \\\\ 5.5 p & \\text { if } & p \\geq 6\\end{array}\\right.$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.2.14", "question": "Rewrite the sum using summation notation: $-\\ln (3)+\\ln (4)-\\ln (5)+\\cdots+\\ln (20)$", "answer": "$\\sum_{k=3}^{20}(-1)^{k} \\ln (k)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.7", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\cot (2 x)=-\\frac{\\sqrt{3}}{3}$", "answer": "$x=\\frac{\\pi}{3}+\\frac{\\pi k}{2} ; x=\\frac{\\pi}{3}, \\frac{5 \\pi}{6}, \\frac{4 \\pi}{3}, \\frac{11 \\pi}{6}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.7", "question": "Convert the angle into decimal degrees and round the answer to three decimal places: $502^{\\circ} 35^{\\prime}$", "answer": "$502.583^{\\circ}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.13", "question": "Expand the binomial: $\\left(x-x^{-1}\\right)^{4}$", "answer": "$\\left(x-x^{-1}\\right)^{4}=x^{4}-4 x^{2}+6-4 x^{-2}+x^{-4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.42", "question": "Convert the angle from radian measure into degree measure: $\\frac{5 \\pi}{3}$", "answer": "$300^{\\circ}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.89", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \\pi \\leq x \\leq 2 \\pi$: $\\cot (x) \\geq 5$", "answer": "$(-2 \\pi, \\operatorname{arccot}(5)-2 \\pi] \\cup(-\\pi, \\operatorname{arccot}(5)-\\pi] \\cup(0, \\operatorname{arccot}(5)] \\cup(\\pi, \\pi+\\operatorname{arccot}(5)]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.13", "question": "Write out the first four terms of the given sequence: $F_{0}=1, F_{1}=1, F_{n}=F_{n-1}+F_{n-2}, n \\geq 2$ (This is the famous Fibonacci Sequence)", "answer": "$1,1,2,3$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.61", "question": "If $\\theta=59^{\\circ}$ and the side opposite $\\theta$ has length 117.42 , how long is the hypotenuse?", "answer": "The hypotenuse has length $\\frac{117.42}{\\sin \\left(59^{\\circ}\\right)} \\approx 136.99$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.95", "question": "Solve the given inequality: $6 \\operatorname{arccot}(7 x) \\geq \\pi$", "answer": "$\\left(-\\infty, \\frac{\\sqrt{3}}{7}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.17", "question": "Use the properties of logarithms to write the expression as a single logarithm: $\\log _{2}(x)+\\log _{2}(y)-\\log _{2}(z)$", "answer": "$\\log _{2}\\left(\\frac{x y}{z}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.1.29", "question": "Carl can stuff 6 envelopes per minute. Find a linear function $E$ that represents the total number of envelopes Carl can stuff after $t$ hours, assuming he doesn't take any breaks.", "answer": "$E(t)=360 t, t \\geq 0$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.7", "question": "Solve the equation analytically: $3^{7 x}=81^{4-2 x}$", "answer": "$x=\\frac{16}{15}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.4.23", "question": "The height $h$ in feet of a model rocket above the ground $t$ seconds after lift-off is given by $h(t)=-5 t^{2}+100 t$, for $0 \\leq t \\leq 20$. When does the rocket reach its maximum height above the ground? What is its maximum height?", "answer": "$\\left[-\\frac{1}{3}, 4\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.19", "question": "Simplify the given power of $i$: $i^5$", "answer": "$i^{5}=i^{4} \\cdot i=1 \\cdot i=i$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.34", "question": "Solve the equation or inequality using your calculator: $\\ln \\left(-2 x^{3}-x^{2}+13 x-6\\right)<0$", "answer": "$\\approx(-3.0281,-3) \\cup(0.5,0.5991) \\cup(1.9299,2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.6", "question": "Solve the equation: $|7 x-1|+2=0$", "answer": "no solution", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.33", "question": "Evaluate the expression: $36^{\\log _{36}(216)}$", "answer": "$36^{\\log _{36}(216)}=216$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.12", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $2 \\cos \\left(x+\\frac{7 \\pi}{4}\\right)=\\sqrt{3}$", "answer": "$x=-\\frac{19 \\pi}{12}+2 \\pi k$ or $x=\\frac{\\pi}{12}+2 \\pi k ; x=\\frac{\\pi}{12}, \\frac{5 \\pi}{12}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.44", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $x^{3}+y^{3}=4$", "answer": "Function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.32", "question": "Solve the equation or inequality using your calculator: $\\ln (x)=\\sqrt[4]{x}$", "answer": "$x \\approx 4.177, x \\approx 5503.665$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.19", "question": "Solve the rational inequality and express your answer using interval notation: $\\frac{x^{4}-4 x^{3}+x^{2}-2 x-15}{x^{3}-4 x^{2}} \\geq x$", "answer": "$[-3,0) \\cup(0,4) \\cup[5, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.42", "question": "Find the real solutions of the polynomial equation $x^{4}+2 x^{2}=15$.", "answer": "$x=-\\frac{3}{2}, \\pm \\sqrt{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.16", "question": "Use the properties of logarithms to write the expression as a single logarithm: $4 \\ln (x)+2 \\ln (y)$", "answer": "$\\ln \\left(x^{4} y^{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.43", "question": "Find the domain of the function: $f(x)=\\ln \\left(x^{2}+1\\right)$", "answer": "$(-\\infty, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.14", "question": "Solve the equation analytically: $3 \\ln (x)-2=1-\\ln (x)$", "answer": "$x=e^{3 / 4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.29", "question": "Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(0,0),(x, y)$", "answer": "$d=\\sqrt{x^{2}+y^{2}}, M=\\left(\\frac{x}{2}, \\frac{y}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.7.3.16", "question": "Find an equation for the parabola which fits the given criteria: Focus $(10,1)$, directrix $x=5$", "answer": "$(y-1)^{2}=10\\left(x-\\frac{15}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.100", "question": "Express the domain of the function using the extended interval notation: $f(x)=\\frac{\\cos (x)}{\\sin (x)+1}$", "answer": "$\\bigcup_{k=-\\infty}^{\\infty}\\left(\\frac{(4 k-1) \\pi}{2}, \\frac{(4 k+3) \\pi}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.45", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (2 x) \\cos (x)+\\sin (2 x) \\sin (x)=1$", "answer": "$x=\\frac{\\pi}{48}, \\frac{11 \\pi}{48}, \\frac{13 \\pi}{48}, \\frac{23 \\pi}{48}, \\frac{25 \\pi}{48}, \\frac{35 \\pi}{48}, \\frac{37 \\pi}{48}, \\frac{47 \\pi}{48}, \\frac{49 \\pi}{48}, \\frac{59 \\pi}{48}, \\frac{61 \\pi}{48}, \\frac{71 \\pi}{48}, \\frac{73 \\pi}{48}, \\frac{83 \\pi}{48}, \\frac{85 \\pi}{48}, \\frac{95 \\pi}{48}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.39", "question": "Find the real solutions of the polynomial equation $x^{3}-7 x^{2}=7-x$.", "answer": "$x=7$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.90", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \\pi \\leq x \\leq 2 \\pi$: $\\tan ^{2}(x) \\geq 1$", "answer": "$\\left[-\\frac{7 \\pi}{4},-\\frac{3 \\pi}{2}\\right) \\cup\\left(-\\frac{3 \\pi}{2},-\\frac{5 \\pi}{4}\\right] \\cup\\left[-\\frac{3 \\pi}{4},-\\frac{\\pi}{2}\\right) \\cup\\left(-\\frac{\\pi}{2},-\\frac{\\pi}{4}\\right] \\cup\\left[\\frac{\\pi}{4}, \\frac{\\pi}{2}\\right) \\cup\\left(\\frac{\\pi}{2}, \\frac{3 \\pi}{4}\\right] \\cup\\left[\\frac{5 \\pi}{4}, \\frac{3 \\pi}{2}\\right) \\cup\\left(\\frac{3 \\pi}{2}, \\frac{7 \\pi}{4}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.15", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\tan ^{2}(x)=3$", "answer": "$x=\\frac{\\pi}{3}+\\pi k$ or $x=\\frac{2 \\pi}{3}+\\pi k ; x=\\frac{\\pi}{3}, \\frac{2 \\pi}{3}, \\frac{4 \\pi}{3}, \\frac{5 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.17", "question": "The temperature $T$, in degrees Fahrenheit, $t$ hours after $6 \\mathrm{AM}$ is given by:\n$T(t)=-\\frac{1}{2} t^{2}+8 t+32, \\quad 0 \\leq t \\leq 12$\nWhat is the warmest temperature of the day? When does this happen?", "answer": "$64^{\\circ}$ at 2 PM (8 hours after 6 AM.)", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.80", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $\\cot (x) \\leq 4$", "answer": "$\\left(-\\frac{\\pi}{6}, \\frac{\\pi}{6}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.2.13", "question": "Rewrite the sum using summation notation: $2+\\frac{3}{2}+\\frac{4}{3}+\\frac{5}{4}+\\frac{6}{5}$", "answer": "$\\sum_{k=1}^{5} \\frac{k+1}{k}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.35", "question": "Convert the point from polar coordinates into rectangular coordinates: $(\\pi, \\arctan (\\pi))$", "answer": "$\\left(\\frac{\\pi}{\\sqrt{1+\\pi^{2}}}, \\frac{\\pi^{2}}{\\sqrt{1+\\pi^{2}}}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.3", "question": "Write out the first four terms of the given sequence: $\\{5 k-2\\}_{k=1}^{\\infty}$", "answer": "$3,8,13,18$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.22", "question": "Use the properties of logarithms to write the expression as a single logarithm: $-\\frac{1}{3} \\ln (x)-\\frac{1}{3} \\ln (y)+\\frac{1}{3} \\ln (z)$", "answer": "$\\ln \\left(\\sqrt[3]{\\frac{z}{x y}}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.19", "question": "Write the set using interval notation: $\\{x \\mid-3<x<3$ or $x=4\\}$", "answer": "$(-3,3) \\cup\\{4\\}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.4.8", "question": "Find the inverse of the matrix or state that the matrix is not invertible: $H=\\left[\\begin{array}{rrrr}1 & 0 & -3 & 0 \\\\ 2 & -2 & 8 & 7 \\\\ -5 & 0 & 16 & 0 \\\\ 1 & 0 & 4 & 1\\end{array}\\right]$", "answer": "$H^{-1}=\\left[\\begin{array}{rrrr}16 & 0 & 3 & 0 \\\\ -90 & -\\frac{1}{2} & -\\frac{35}{2} & \\frac{7}{2} \\\\ 5 & 0 & 1 & 0 \\\\ -36 & 0 & -7 & 1\\end{array}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.7.27", "question": "Solve the system of nonlinear equations: $\\left\\{\\begin{aligned} 2 y+2 z & =\\lambda y z \\\\ 2 x+2 z & =\\lambda x z \\\\ 2 y+2 x & =\\lambda x y \\\\ x y z & =1000\\end{aligned}\\right.$", "answer": "$x=10, y=10, z=10, \\lambda=\\frac{2}{5}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.81", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\\pi \\leq x \\leq \\pi$: $\\cos (x)>\\frac{\\sqrt{3}}{2}$", "answer": "$\\left[-\\pi,-\\frac{\\pi}{2}\\right) \\cup\\left[-\\frac{\\pi}{3}, \\frac{\\pi}{3}\\right] \\cup\\left(\\frac{\\pi}{2}, \\pi\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.2", "question": "Determine whether or not the relation represents $y$ as a function of $x$ and find the domain and range of those relations which are functions:\n$\\{(-3,0),(1,6),(2,-3),(4,2),(-5,6),(4,-9),(6,2)\\}$", "answer": "Not a function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.24", "question": "Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $\\left(\\frac{1}{2}, 4\\right),\\left(\\frac{3}{2},-1\\right)$", "answer": "$d=\\sqrt{26}, M=\\left(1, \\frac{3}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.2", "question": "Expand the given logarithm and simplify: $\\log _{2}\\left(\\frac{128}{x^{2}+4}\\right)$", "answer": "$7-\\log _{2}\\left(x^{2}+4\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.27", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $3 \\cos (2 x)=\\sin (x)+2$", "answer": "$x=\\arctan (2), \\pi+\\arctan (2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.4.3", "question": "Find the inverse of the matrix or state that the matrix is not invertible: $C=\\left[\\begin{array}{rr}6 & 15 \\\\ 14 & 35\\end{array}\\right]$", "answer": "$C$ is not invertible", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.40", "question": "Evaluate the expression: $\\log \\left(e^{\\ln (100)}\\right)$", "answer": "$\\log \\left(e^{\\ln (100)}\\right)=2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.64", "question": "Solve the equation: $12 \\operatorname{arccsc}\\left(\\frac{x}{3}\\right)=2 \\pi$", "answer": "$x=\\frac{1}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.29", "question": "Find the quadratic function passing through the points $(-2,1),(1,4),(3,-2)$", "answer": "$f(x)=-\\frac{4}{5} x^{2}+\\frac{1}{5} x+\\frac{23}{5}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.1.32", "question": "A salesperson is paid $\\$ 200$ per week plus $5 \\%$ commission on her weekly sales of $x$ dollars. Find a linear function that represents her total weekly pay, $W$ (in dollars) in terms of $x$. What must her weekly sales be in order for her to earn $\\$ 475.00$ for the week?", "answer": "$W(x)=200+.05 x, x \\geq 0$ She must make $\\$ 5500$ in weekly sales.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.54", "question": "Approximate the given value to three decimal places: $\\cos (e)$", "answer": "$\\cos (e) \\approx-0.912$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.35", "question": "Convert the angle from degree measure into radian measure, giving the exact value in terms of $\\pi$: $45^{\\circ}$", "answer": "$\\frac{\\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.56", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (5 x)=-\\cos (2 x)$", "answer": "$x=\\frac{\\pi}{7}, \\frac{\\pi}{3}, \\frac{3 \\pi}{7}, \\frac{5 \\pi}{7}, \\pi, \\frac{9 \\pi}{7}, \\frac{11 \\pi}{7}, \\frac{5 \\pi}{3}, \\frac{13 \\pi}{7}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.19", "question": "Convert the point from polar coordinates into rectangular coordinates: $\\left(11,-\\frac{7 \\pi}{6}\\right)$", "answer": "$\\left(-\\frac{11 \\sqrt{3}}{2}, \\frac{11}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.9", "question": "Find the exact value of the cosine and sine of the given angle: $\\theta=\\frac{5 \\pi}{4}$", "answer": "$\\cos \\left(\\frac{5 \\pi}{4}\\right)=-\\frac{\\sqrt{2}}{2}, \\sin \\left(\\frac{5 \\pi}{4}\\right)=-\\frac{\\sqrt{2}}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.46", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $2 x y=4$", "answer": "Function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.7.20", "question": "Solve the following system: $\\left\\{\\begin{aligned} x^{2}+\\sqrt{y}+\\log _{2}(z) & =6 \\\\ 3 x^{2}-2 \\sqrt{y}+2 \\log _{2}(z) & =5 \\\\ -5 x^{2}+3 \\sqrt{y}+4 \\log _{2}(z) & =13\\end{aligned}\\right.$", "answer": "$(1,4,8),(-1,4,8)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.5", "question": "Solve the rational equation: $\\frac{x^{2}-2 x+1}{x^{3}+x^{2}-2 x}=1$", "answer": "No solution", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.18", "question": "Suppose $C(x)=x^{2}-10 x+27$ represents the costs, in hundreds, to produce $x$ thousand pens. How many pens should be produced to minimize the cost? What is this minimum cost?", "answer": "5000 pens should be produced for a cost of $\\$ 200$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.39", "question": "Convert the equation from rectangular coordinates into polar coordinates: $y=7$", "answer": "$\\left(7 \\sqrt{2}, \\frac{7 \\pi}{4}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.38", "question": "Evaluate the expression: $\\ln \\left(\\frac{1}{\\sqrt{e}}\\right)$", "answer": "$\\ln \\left(\\frac{1}{\\sqrt{e}}\\right)=-\\frac{1}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.33", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $y=x^{3}-x$", "answer": "Function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.18", "question": "Evaluate the expression: $\\log _{2}(32)$", "answer": "$\\log _{2}(32)=5$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.5", "question": "Solve the equation analytically: $\\log _{3}(7-2 x)=2$", "answer": "$x=-1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.43", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sin (6 x) \\cos (x)=-\\cos (6 x) \\sin (x)$", "answer": "$x=0, \\frac{\\pi}{7}, \\frac{2 \\pi}{7}, \\frac{3 \\pi}{7}, \\frac{4 \\pi}{7}, \\frac{5 \\pi}{7}, \\frac{6 \\pi}{7}, \\pi, \\frac{8 \\pi}{7}, \\frac{9 \\pi}{7}, \\frac{10 \\pi}{7}, \\frac{11 \\pi}{7}, \\frac{12 \\pi}{7}, \\frac{13 \\pi}{7}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.72", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $\\cos ^{2}(x)>\\frac{1}{2}$", "answer": "$\\left[0, \\frac{\\pi}{4}\\right) \\cup\\left(\\frac{3 \\pi}{4}, \\frac{5 \\pi}{4}\\right) \\cup\\left(\\frac{7 \\pi}{4}, 2 \\pi\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.16", "question": "Solve the equation analytically: $500\\left(1-e^{2 x}\\right)=250$", "answer": "$x=\\frac{1}{2} \\ln \\left(\\frac{1}{2}\\right)=-\\frac{1}{2} \\ln (2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.4.33", "question": "Solve the quadratic equation $x^{2}-m x=1$ for $x$.", "answer": "$(-\\infty, 1) \\cup\\left(2, \\frac{3+\\sqrt{17}}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.26", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} x_{1}-x_{3} & =-2 \\\\ 2 x_{2}-x_{4} & =0 \\\\ x_{1}-2 x_{2}+x_{3} & =0 \\\\ -x_{3}+x_{4} & =1\\end{aligned}\\right.$", "answer": "$(1,2,3,4)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.58", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\tan (x)=\\cos (x)$", "answer": "$x=\\arcsin \\left(\\frac{-1+\\sqrt{5}}{2}\\right) \\approx 0.6662, \\pi-\\arcsin \\left(\\frac{-1+\\sqrt{5}}{2}\\right) \\approx 2.4754$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.41", "question": "Solve the equation for $t$: $\\sin (t)=-\\frac{\\sqrt{2}}{2}$", "answer": "$\\sin (t)=-\\frac{\\sqrt{2}}{2}$ when $t=\\frac{5 \\pi}{4}+2 \\pi k$ or $t=\\frac{7 \\pi}{4}+2 \\pi k$ for any integer $k$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.82", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\\pi \\leq x \\leq \\pi$: $\\sin (x)>\\frac{1}{3}$", "answer": "$\\left(\\arcsin \\left(\\frac{1}{3}\\right), \\pi-\\arcsin \\left(\\frac{1}{3}\\right)\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.41", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $2 \\tan (x)=1-\\tan ^{2}(x)$", "answer": "$x=\\frac{\\pi}{8}, \\frac{5 \\pi}{8}, \\frac{9 \\pi}{8}, \\frac{13 \\pi}{8}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.26", "question": "Evaluate the expression: $\\log (0.01)$", "answer": "$\\log (0.01)=-2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.3", "question": "Solve the equation analytically: $\\ln \\left(8-x^{2}\\right)=\\ln (2-x)$", "answer": "$x=-2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.15", "question": "Find the exact value of the cosine and sine of the given angle: $\\theta=-\\frac{13 \\pi}{2}$", "answer": "$\\cos \\left(-\\frac{13 \\pi}{2}\\right)=0, \\sin \\left(-\\frac{13 \\pi}{2}\\right)=-1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.12", "question": "Write out the first four terms of the given sequence: $s_{0}=1, s_{n+1}=x^{n+1}+s_{n}, n \\geq 0$", "answer": "$1, x+1, x^{2}+x+1, x^{3}+x^{2}+x+1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.44", "question": "Convert the angle from radian measure into degree measure: $\\frac{\\pi}{2}$", "answer": "$90^{\\circ}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.25", "question": "Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $\\left(-\\frac{2}{3}, \\frac{3}{2}\\right),\\left(\\frac{7}{3}, 2\\right)$", "answer": "$d=\\frac{\\sqrt{37}}{2}, M=\\left(\\frac{5}{6}, \\frac{7}{4}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.9", "question": "Evaluate: $\\left(\\begin{array}{c}n \\\\ n-2\\end{array}\\right), n \\geq 2$", "answer": "$\\frac{n(n-1)}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.8.53", "question": "A small boat leaves the dock at Camp DuNuthin and heads across the Nessie River at 17 miles per hour (that is, with respect to the water) at a bearing of $S 68^{\\circ} \\mathrm{W}$. The river is flowing due east at 8 miles per hour. What is the boat's true speed and heading? Round the speed to the nearest mile per hour and express the heading as a bearing, rounded to the nearest tenth of a degree.", "answer": "The current is moving at about 10 miles per hour bearing $\\mathrm{N} 54.6^{\\circ} \\mathrm{W}$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.4.29", "question": "Let $L$ be the line $y=2 x+1$. Find a function $D(x)$ which measures the distance squared from a point on $L$ to $(0,0)$. Use this to find the point on $L$ closest to $(0,0)$.", "answer": "$(-\\infty, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.31", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sec (x)=2 \\csc (x)$", "answer": "$x=\\frac{\\pi}{6}, \\frac{\\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.7.2.14", "question": "Find the standard equation of the circle which satisfies the given criteria: center $(3,6)$, passes through $(-1,4)$", "answer": "$(x-3)^{2}+(y-6)^{2}=20$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.5.29", "question": "Carbon-14 cannot be used to date inorganic material such as rocks, but there are many other methods of radiometric dating which estimate the age of rocks. One of them, RubidiumStrontium dating, uses Rubidium-87 which decays to Strontium-87 with a half-life of 50 billion years. Use Equation 6.5 to express the amount of Rubidium-87 left from an initial 2.3 micrograms as a function of time $t$ in billions of years. Research this and other radiometric techniques and discuss the margins of error for various methods with your classmates.", "answer": "$A(t)=2.3 e^{-0.0138629 t}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.4", "question": "Simplify the expression: $\\frac{9 !}{4 ! 3 ! 2 !}$", "answer": "1260", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.28", "question": "Find all of the points on the line $y=1-x$ which are 2 units from $(1,-1)$.", "answer": "$\\left(\\frac{3-\\sqrt{7}}{2}, \\frac{-1+\\sqrt{7}}{2}\\right),\\left(\\frac{3+\\sqrt{7}}{2}, \\frac{-1-\\sqrt{7}}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.12", "question": "Solve the equation: $|x-4|=x-5$", "answer": "no solution", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.45", "question": "Find the domain of the function: $f(x)=\\ln (4 x-20)$", "answer": "$(5, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.11", "question": "Solve the equation analytically: $-\\log (x)=5.4$", "answer": "$x=10^{-5.4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.10", "question": "Write the set using interval notation: $\\{x \\mid x \\neq-3,4\\}$", "answer": "$(-\\infty,-3) \\cup(-3,4) \\cup(4, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.27", "question": "Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(\\sqrt{2}, \\sqrt{3}),(-\\sqrt{8},-\\sqrt{12})$", "answer": "$d=3 \\sqrt{5}, M=\\left(-\\frac{\\sqrt{2}}{2},-\\frac{\\sqrt{3}}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.94", "question": "Solve the given inequality: $3 \\arccos (x) \\leq \\pi$", "answer": "$\\left[\\frac{1}{2}, 1\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.24", "question": "Solve the equation analytically: $\\ln \\left(x^{2}\\right)=(\\ln (x))^{2}$", "answer": "$x=1, x=e^{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.21", "question": "Evaluate the expression: $\\log _{36}(216)$", "answer": "$\\log _{36}(216)=\\frac{3}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.40", "question": "Solve the equation for $t$: $\\cos (t)=0$", "answer": "$\\cos (t)=0$ when $t=\\frac{\\pi}{2}+\\pi k$ for any integer $k$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.18", "question": "Use the properties of logarithms to write the expression as a single logarithm: $\\log _{3}(x)-2 \\log _{3}(y)$", "answer": "$\\log _{3}\\left(\\frac{x}{y^{2}}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.28", "question": "Solve the equation analytically: $e^{2 x}-3 e^{x}-10=0$", "answer": "$x=\\ln (5)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.5.14", "question": "For each isotope:\n- Find the decay constant $k$. Round your answer to four decimal places.\n- Find a function which gives the amount of isotope $A$ which remains after time $t$. (Keep the units of $A$ and $t$ the same as the given data.)\n- Determine how long it takes for $90 \\%$ of the material to decay. Round your answer to two decimal places. (HINT: If $90 \\%$ of the material decays, how much is left?)\n14. Cobalt 60, used in food irradiation, initial amount 50 grams, half-life of 5.27 years.", "answer": "$\\bullet k=\\frac{\\ln (1 / 2)}{5.27} \\approx-0.1315$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.28", "question": "Use the properties of logarithms to write the expression as a single logarithm: $\\log _{2}(x)+\\log _{4}(x-1)$", "answer": "$\\log _{2}(x \\sqrt{x-1})$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.5.3.23", "question": "Solve the equation or inequality: $x^{\\frac{2}{3}}=4$", "answer": "$x= \\pm 8$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.36", "question": "Evaluate the expression: $\\log \\left(\\sqrt[9]{10^{11}}\\right)$", "answer": "$\\log \\left(\\sqrt[9]{10^{11}}\\right)=\\frac{11}{9}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.2", "question": "Solve the rational equation: $\\frac{3 x-1}{x^{2}+1}=1$", "answer": "$x=1, x=2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.85", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\\pi \\leq x \\leq \\pi$: $\\cot (x) \\geq-1$", "answer": "$\\left(-\\pi,-\\frac{\\pi}{4}\\right] \\cup\\left(0, \\frac{3 \\pi}{4}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.73", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $\\cos (2 x) \\leq 0$", "answer": "$\\left[\\frac{\\pi}{4}, \\frac{3 \\pi}{4}\\right] \\cup\\left[\\frac{5 \\pi}{4}, \\frac{7 \\pi}{4}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.34", "question": "Convert the angle from degree measure into radian measure, giving the exact value in terms of $\\pi$: $150^{\\circ}$", "answer": "$\\frac{5 \\pi}{6}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.3", "question": "Find the exact value of the cosine and sine of the given angle: $\\theta=\\frac{\\pi}{3}$", "answer": "$\\cos \\left(\\frac{\\pi}{3}\\right)=\\frac{1}{2}, \\sin \\left(\\frac{\\pi}{3}\\right)=\\frac{\\sqrt{3}}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.57", "question": "Find the domain of the function: $f(x)=\\ln \\left(-2 x^{3}-x^{2}+13 x-6\\right)$", "answer": "$(-\\infty,-3) \\cup\\left(\\frac{1}{2}, 2\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.35", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (2 x)+\\csc ^{2}(x)=0$", "answer": "$x=0, \\frac{\\pi}{3}, \\frac{2 \\pi}{3}, \\pi, \\frac{4 \\pi}{3}, \\frac{5 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.46", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (5 x) \\cos (3 x)-\\sin (5 x) \\sin (3 x)=\\frac{\\sqrt{3}}{2}$", "answer": "$x=0, \\frac{\\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.51", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (2 x)-\\sqrt{3} \\sin (2 x)=\\sqrt{2}$", "answer": "$x=0, \\pi, \\frac{\\pi}{3}, \\frac{4 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.70", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $\\tan (x) \\geq \\sqrt{3}$", "answer": "$x=\\left[\\frac{\\pi}{3}, \\frac{\\pi}{2}\\right) \\cup\\left[\\frac{4 \\pi}{3}, \\frac{3 \\pi}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.9", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\sin \\left(\\frac{x}{3}\\right)=\\frac{\\sqrt{2}}{2}$", "answer": "$x=\\frac{3 \\pi}{4}+6 \\pi k$ or $x=\\frac{9 \\pi}{4}+6 \\pi k ; x=\\frac{3 \\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.31", "question": "Find all of the points on the $y$-axis which are 5 units from the point $(-5,3)$.", "answer": "$(0,3)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.60", "question": "Solve the equation: $\\pi-2 \\arcsin (x)=2 \\pi$", "answer": "$x=-1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.19", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} x-y+z & =-4 \\\\ -3 x+2 y+4 z & =-5 \\\\ x-5 y+2 z & =-18\\end{aligned}\\right.$", "answer": "$(1,3,-2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.3", "question": "Solve the equation analytically: $5^{2 x-1}=125$", "answer": "$x=2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.50", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sqrt{3} \\sin (2 x)+\\cos (2 x)=1$", "answer": "$x=\\frac{17 \\pi}{24}, \\frac{41 \\pi}{24}, \\frac{23 \\pi}{24}, \\frac{47 \\pi}{24}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.50", "question": "A yo-yo which is 2.25 inches in diameter spins at a rate of 4500 revolutions per minute. How fast is the edge of the yo-yo spinning in miles per hour? Round your answer to two decimal places.", "answer": "About 30.12 miles per hour", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.47", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sin (x)+\\cos (x)=1$", "answer": "$x=0, \\frac{\\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.76", "question": "Convert the equation from polar coordinates into rectangular coordinates: $r=1+\\sin (\\theta)$", "answer": "$r=\\sin (\\theta)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.1.41", "question": "A mobile plan charges a base monthly rate of $\\$ 10$ for the first 500 minutes of air time plus a charge of $15 \\notin$ for each additional minute. Write a piecewise-defined linear function which calculates the monthly cost $C$ (in dollars) for using $m$ minutes of air time.", "answer": "$C(m)=\\left\\{\\begin{array}{rll}10 & \\text { if } & 0 \\leq m \\leq 500 \\\\ 10+0.15(m-500) & \\text { if } & m>500\\end{array}\\right.$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.4.9", "question": "Use one matrix inverse to solve the following system of linear equations:\n$\\left\\{\\begin{array}{r}3 x+7 y=26 \\\\ 5 x+12 y=39\\end{array}\\right.$", "answer": "$\\left[\\begin{array}{rr}12 & -7 \\\\ -5 & 3\\end{array}\\right]\\left[\\begin{array}{l}26 \\\\ 39\\end{array}\\right]=\\left[\\begin{array}{r}39 \\\\ -13\\end{array}\\right]$ So $x=39$ and $y=-13$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.53", "question": "Create a polynomial $f$ with the following characteristics:\n- The leading term of $f(x)$ is $-2x^3$\n- $c=2i$ is a zero\n- $f(0)=-16$", "answer": "$f(x)=-2(x-2 i)(x+2 i)(x+2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.22", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (2 x)=\\sin (x)$", "answer": "$x=\\frac{\\pi}{6}, \\frac{5 \\pi}{6}, \\frac{3 \\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.20", "question": "Evaluate the expression: $\\log _{8}(4)$", "answer": "$\\log _{8}(4)=\\frac{2}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.5.9", "question": "How much money needs to be invested now to obtain $\\$ 5000$ in 10 years if the interest rate in a CD is $2.25 \\%$, compounded monthly? Round your answer to the nearest cent.", "answer": "$P=\\frac{5000}{\\left(1+\\frac{0.025}{12}\\right)^{12 \\cdot 10}} \\approx \\$ 3993.42$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.5", "question": "Simplify the expression: $\\frac{(n+1) !}{n !}, n \\geq 0$.", "answer": "$n+1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.1", "question": "Convert the angle into the DMS system and round the answer to the nearest second: $63.75^{\\circ}$", "answer": "$63^{\\circ} 45^{\\prime}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.6", "question": "Solve the equation analytically: $\\log _{\\frac{1}{2}}(2 x-1)=-3$", "answer": "$x=\\frac{9}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.29", "question": "Let $L$ be the line $y=2 x+1$. Find a function $D(x)$ which measures the distance squared from a point on $L$ to $(0,0)$. Use this to find the point on $L$ closest to $(0,0)$.", "answer": "$D(x)=x^{2}+(2 x+1)^{2}=5 x^{2}+4 x+1, D$ is minimized when $x=-\\frac{2}{5}$, so the point on $y=2 x+1$ closest to $(0,0)$ is $\\left(-\\frac{2}{5}, \\frac{1}{5}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.15", "question": "Solve the equation analytically: $2000 e^{0.1 t}=4000$", "answer": "$t=\\frac{\\ln (2)}{0.1}=10 \\ln (2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.19", "question": "Solve the equation: $|4-x|-|x+2|=0$", "answer": "$x=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.3", "question": "Solve the equation: $|4-x|=7$", "answer": "$x=-3$ or $x=11$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.25", "question": "Solve the inequality analytically: $\\frac{1-\\ln (x)}{x^{2}}<0$", "answer": "$(e, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.4.7", "question": "Find the inverse of the matrix or state that the matrix is not invertible: $G=\\left[\\begin{array}{rrr}1 & 2 & 3 \\\\ 2 & 3 & 11 \\\\ 3 & 4 & 19\\end{array}\\right]$", "answer": "$G$ is not invertible", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.62", "question": "Solve the equation: $6 \\operatorname{arccot}(2 x)-5 \\pi=0$", "answer": "$x=-\\frac{\\sqrt{3}}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.1", "question": "Solve the rational equation: $\\frac{x}{5 x+4}=3$", "answer": "$x=-\\frac{6}{7}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.4.10", "question": "Use one matrix inverse to solve the following system of linear equations:\n$\\left\\{\\begin{aligned} 3 x+7 y & =0 \\\\ 5 x+12 y & =-1\\end{aligned}\\right.$", "answer": "$\\left[\\begin{array}{rr}12 & -7 \\\\ -5 & 3\\end{array}\\right]\\left[\\begin{array}{r}0 \\\\ -1\\end{array}\\right]=\\left[\\begin{array}{r}7 \\\\ -3\\end{array}\\right] \\quad$ So $x=7$ and $y=-3$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.13", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\csc (x)=0$", "answer": "No solution", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.53", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (3 x)=\\cos (5 x)$", "answer": "$x=0, \\frac{\\pi}{4}, \\frac{\\pi}{2}, \\frac{3 \\pi}{4}, \\pi, \\frac{5 \\pi}{4}, \\frac{3 \\pi}{2}, \\frac{7 \\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.24", "question": "Simplify the given power of $i$: $i^{26}$", "answer": "$i^{26}=\\left(i^{4}\\right)^{6} \\cdot i^{2}=1 \\cdot(-1)=-1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.44", "question": "Solve the equation for $t$: $\\cos (t)=\\frac{1}{2}$", "answer": "$\\cos (t)=\\frac{1}{2}$ when $t=\\frac{\\pi}{3}+2 \\pi k$ or $t=\\frac{5 \\pi}{3}+2 \\pi k$ for any integer $k$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.8", "question": "Solve the equation: $\\frac{2}{3}|5-2 x|-\\frac{1}{2}=5$", "answer": "$x=-\\frac{13}{8}$ or $x=\\frac{53}{8}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.20", "question": "Simplify the given power of $i$: $i^6$", "answer": "$i^{6}=i^{4} \\cdot i^{2}=1 \\cdot(-1)=-1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.48", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sin (x)+\\sqrt{3} \\cos (x)=1$", "answer": "$x=\\frac{\\pi}{2}, \\frac{11 \\pi}{6}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.14", "question": "Solve the equation analytically: $e^{-5730 k}=\\frac{1}{2}$", "answer": "$k=\\frac{\\ln \\left(\\frac{1}{2}\\right)}{-5730}=\\frac{\\ln (2)}{5730}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.23", "question": "Solve the equation analytically: $e^{2 x}=2 e^{x}$", "answer": "$x=\\ln (2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.101", "question": "Express the domain of the function using the extended interval notation: $f(x)=\\sqrt{\\tan ^{2}(x)-1}$", "answer": "$\\bigcup_{k=-\\infty}^{\\infty}\\left\\{\\left[\\frac{(4 k+1) \\pi}{4}, \\frac{(2 k+1) \\pi}{2}\\right) \\cup\\left(\\frac{(2 k+1) \\pi}{2}, \\frac{(4 k+3) \\pi}{4}\\right]\\right\\}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.7.2.16", "question": "Find the standard equation of the circle which satisfies the given criteria: endpoints of a diameter: $\\left(\\frac{1}{2}, 4\\right),\\left(\\frac{3}{2},-1\\right)$", "answer": "$(x-1)^{2}+\\left(y-\\frac{3}{2}\\right)^{2}=\\frac{13}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.96", "question": "Solve the given inequality: $\\pi>2 \\arctan (x)$", "answer": "$(-\\infty, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.52", "question": "Find the domain of the function: $f(x)=\\sqrt[4]{\\log _{4}(x)}$", "answer": "$[1, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.4.4", "question": "Find the inverse of the matrix or state that the matrix is not invertible: $D=\\left[\\begin{array}{rr}2 & -1 \\\\ 16 & -9\\end{array}\\right]$", "answer": "$D^{-1}=\\left[\\begin{array}{cc}\\frac{9}{2} & -\\frac{1}{2} \\\\ 8 & -1\\end{array}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.5.33", "question": "The current $i$ measured in amps in a certain electronic circuit with a constant impressed voltage of 120 volts is given by $i(t)=2-2 e^{-10 t}$ where $t \\geq 0$ is the number of seconds after the circuit is switched on. Determine the value of $i$ as $t \\rightarrow \\infty$. (This is called the steady state current.)", "answer": "The steady state current is 2 amps.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.5.3.29", "question": "Solve the equation or inequality: $2(x-2)^{-\\frac{1}{3}}-\\frac{2}{3} x(x-2)^{-\\frac{4}{3}} \\leq 0$", "answer": "$(-\\infty, 2) \\cup(2,3]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.5", "question": "Solve the equation: $2|5 x+1|-3=0$", "answer": "$x=-\\frac{1}{2}$ or $x=\\frac{1}{10}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.5.3.35", "question": "Solve the equation or inequality: $\\frac{2}{3}(x+4)^{\\frac{3}{5}}(x-2)^{-\\frac{1}{3}}+\\frac{3}{5}(x+4)^{-\\frac{2}{5}}(x-2)^{\\frac{2}{3}} \\geq 0$", "answer": "$(-\\infty,-4) \\cup\\left(-4,-\\frac{22}{19}\\right] \\cup(2, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.17", "question": "Solve the rational inequality and express your answer using interval notation: $\\frac{-x^{3}+4 x}{x^{2}-9} \\geq 4 x$", "answer": "$(-\\infty,-3) \\cup[-2 \\sqrt{2}, 0] \\cup[2 \\sqrt{2}, 3)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.52", "question": "Approximate the given value to three decimal places: $\\cos \\left(207^{\\circ}\\right)$", "answer": "$\\cos \\left(207^{\\circ}\\right) \\approx-0.891$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.5.3.43", "question": "Find the inverse of $k(x)=\\frac{2 x}{\\sqrt{x^{2}-1}}$.", "answer": "$k^{-1}(x)=\\frac{x}{\\sqrt{x^{2}-4}}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.29", "question": "Convert the angle from degree measure into radian measure, giving the exact value in terms of $\\pi$: $0^{\\circ}$", "answer": "0", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.15", "question": "Write the set using interval notation: $\\{x \\mid x<3$ or $x \\geq 2\\}$", "answer": "$(-\\infty, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.41", "question": "Evaluate the expression: $\\log _{2}\\left(3^{-\\log _{3}(2)}\\right)$", "answer": "$\\log _{2}\\left(3^{-\\log _{3}(2)}\\right)=-1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.16", "question": "Simplify the power of a complex number: $\\left(\\frac{\\sqrt{3}}{2}+\\frac{1}{2} i\\right)^{3}$", "answer": "$i$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.3", "question": "Solve the rational equation: $\\frac{1}{x+3}+\\frac{1}{x-3}=\\frac{x^{2}-3}{x^{2}-9}$", "answer": "$x=-1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.54", "question": "Find the domain of the function: $f(x)=\\ln (\\sqrt{x-4}-3)$", "answer": "$(13, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.75", "question": "Convert the equation from polar coordinates into rectangular coordinates: $r=1-2 \\cos (\\theta)$", "answer": "$r=6 \\sin (\\theta)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.45", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $2 x+3 y=4$", "answer": "Function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.55", "question": "Find the domain of the function: $f(x)=\\frac{1}{3-\\log _{5}(x)}$", "answer": "$(0,125) \\cup(125, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.1.31", "question": "A plumber charges $\\$ 50$ for a service call plus $\\$ 80$ per hour. If she spends no longer than 8 hours a day at any one site, find a linear function that represents her total daily charges $C$ (in dollars) as a function of time $t$ (in hours) spent at any one given location.", "answer": "$C(t)=80 t+50,0 \\leq t \\leq 8$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.5.8", "question": "How much money needs to be invested now to obtain $\\$ 2000$ in 3 years if the interest rate in a savings account is $0.25 \\%$, compounded continuously? Round your answer to the nearest cent.", "answer": "$P=\\frac{2000}{e^{0.0025 \\cdot 3}} \\approx \\$ 1985.06$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.64", "question": "If $\\theta=37.5^{\\circ}$ and the side opposite $\\theta$ has length 306 , how long is the side adjacent to $\\theta$ ?", "answer": "The hypotenuse has length $c=\\frac{306}{\\sin \\left(37.5^{\\circ}\\right)} \\approx 502.660$, so the side adjacent to $\\theta$ has length $\\sqrt{c^{2}-306^{2}} \\approx 398.797$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.6", "question": "Solve the rational equation: $\\frac{-x^{3}+4 x}{x^{2}-9}=4 x$", "answer": "$x=0, x= \\pm 2 \\sqrt{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.21", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} 2 x-y+z & =1 \\\\ 2 x+2 y-z & =1 \\\\ 3 x+6 y+4 z & =9\\end{aligned}\\right.$", "answer": "$\\left(\\frac{1}{3}, \\frac{2}{3}, 1\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.7.78", "question": "Use a calculator to approximate the five fifth roots of 1.", "answer": "$w_{0}=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.7", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned}-5 x+y & =17 \\\\ x+y & =5\\end{aligned}\\right.$", "answer": "$(-2,7)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.15", "question": "Solve the rational inequality and express your answer using interval notation: $\\frac{3 x-1}{x^{2}+1} \\leq 1$", "answer": "$(-\\infty, 1] \\cup[2, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.4.91", "question": "If $\\tan (\\theta)=\\frac{x}{7}$ for $-\\frac{\\pi}{2}<\\theta<\\frac{\\pi}{2}$, find an expression for $\\sin (2 \\theta)$ in terms of $x$.", "answer": "$\\frac{14 x}{x^{2}+49}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.4", "question": "Write out the first four terms of the given sequence: $\\left\\{\\frac{n^{2}+1}{n+1}\\right\\}_{n=0}^{\\infty}$", "answer": "$1,1, \\frac{5}{3}, \\frac{5}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.37", "question": "Convert the angle from radian measure into degree measure: $\\pi$", "answer": "$180^{\\circ}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.11", "question": "Solve the equation analytically: $5^{x}=-2$", "answer": "No solution.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.1.1", "question": "The sounds we hear are made up of mechanical waves. The note ' $A$ ' above the note 'middle $\\mathrm{C}^{\\prime}$ is a sound wave with ordinary frequency $f=440 \\mathrm{Hertz}=440 \\frac{\\mathrm{cycles}}{\\text { second }}$. Find a sinusoid which models this note, assuming that the amplitude is 1 and the phase shift is 0 .", "answer": "$S(t)=\\sin (880 \\pi t)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.38", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $x=-6$", "answer": "Not a function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.32", "question": "Convert the angle from degree measure into radian measure, giving the exact value in terms of $\\pi$: $-270^{\\circ}$", "answer": "$-\\frac{3 \\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.1", "question": "Simplify the expression: $(3 !)^{2}$", "answer": "36", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.1", "question": "Solve the equation analytically: $\\log (3 x-1)=\\log (4-x)$", "answer": "$x=\\frac{5}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.27", "question": "Convert the point from polar coordinates into rectangular coordinates: $(6, \\arctan (2))$", "answer": "$\\left(\\frac{6 \\sqrt{5}}{5}, \\frac{12 \\sqrt{5}}{5}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.53", "question": "Find the domain of the function: $f(x)=\\log _{9}(|x+3|-4)$", "answer": "$(-\\infty,-7) \\cup(1, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.83", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\\pi \\leq x \\leq \\pi$: $\\sec (x) \\leq 2$", "answer": "$\\left[-\\frac{2 \\pi}{3},-\\frac{\\pi}{3}\\right) \\cup\\left(\\frac{\\pi}{3}, \\frac{2 \\pi}{3}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.21", "question": "Determine if the given sequence is arithmetic, geometric or neither. If it is arithmetic, find the common difference $d$; if it is geometric, find the common ratio $r$: $a_{n}=\\frac{n !}{2}, n \\geq 0$", "answer": "neither", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.31", "question": "Convert the angle from degree measure into radian measure, giving the exact value in terms of $\\pi$: $135^{\\circ}$", "answer": "$\\frac{3 \\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.30", "question": "Convert the angle from degree measure into radian measure, giving the exact value in terms of $\\pi$: $240^{\\circ}$", "answer": "$\\frac{4 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.4.90", "question": "If $\\sin (\\theta)=\\frac{x}{2}$ for $-\\frac{\\pi}{2}<\\theta<\\frac{\\pi}{2}$, find an expression for $\\cos (2 \\theta)$ in terms of $x$.", "answer": "$1-\\frac{x^{2}}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.20", "question": "Convert the point from polar coordinates into rectangular coordinates: $(-20,3 \\pi)$", "answer": "$(20,0)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.20", "question": "Solve the equation analytically: $\\frac{5000}{1+2 e^{-3 t}}=2500$", "answer": "$t=\\frac{1}{3} \\ln (2)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.26", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (2 x)=5 \\sin (x)-2$", "answer": "$x=0, \\frac{2 \\pi}{3}, \\frac{4 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.1", "question": "Solve the equation: $|x|=6$", "answer": "$x=-6$ or $x=6$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.11", "question": "Simplify the quantity $\\sqrt{-49}$", "answer": "$7 i$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.45", "question": "Solve the polynomial inequality $-2 x^{3}+19 x^{2}-49 x+20>0$ and state your answer using interval notation.", "answer": "$(-\\infty,-1) \\cup(-1,0) \\cup(2, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.5", "question": "Find the exact value of the cosine and sine of the given angle: $\\theta=\\frac{2 \\pi}{3}$", "answer": "$\\cos \\left(\\frac{2 \\pi}{3}\\right)=-\\frac{1}{2}, \\sin \\left(\\frac{2 \\pi}{3}\\right)=\\frac{\\sqrt{3}}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.12", "question": "Solve the equation analytically: $3^{(x-1)}=29$", "answer": "$x=\\frac{\\ln (29)+\\ln (3)}{\\ln (3)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.11", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\sin \\left(2 x-\\frac{\\pi}{3}\\right)=-\\frac{1}{2}$", "answer": "$x=\\frac{3 \\pi}{4}+\\pi k$ or $x=\\frac{13 \\pi}{12}+\\pi k ; x=\\frac{\\pi}{12}, \\frac{3 \\pi}{4}, \\frac{13 \\pi}{12}, \\frac{7 \\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.5", "question": "Expand the given logarithm and simplify: $\\ln \\left(\\frac{\\sqrt{z}}{x y}\\right)$", "answer": "$\\frac{1}{2} \\ln (z)-\\ln (x)-\\ln (y)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.12", "question": "Expand the binomial: $\\left(\\frac{1}{3} x+y^{2}\\right)^{3}$", "answer": "$\\left(\\frac{1}{3} x+y^{2}\\right)^{3}=\\frac{1}{27} x^{3}+\\frac{1}{3} x^{2} y^{2}+x y^{4}+y^{6}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.23", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (2 x)=\\cos (x)$", "answer": "$x=0, \\frac{2 \\pi}{3}, \\frac{4 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.21", "question": "What is the largest rectangular area one can enclose with 14 inches of string?", "answer": "The largest rectangle has area 12.25 square inches.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.24", "question": "Evaluate the expression: $\\log _{36}(36)$", "answer": "$\\log _{36}(36)=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.71", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $\\sec ^{2}(x) \\leq 4$", "answer": "$\\left(-\\infty, \\frac{\\pi}{3}\\right] \\cup\\left[\\frac{2 \\pi}{3}, \\frac{4 \\pi}{3}\\right] \\cup\\left[\\frac{5 \\pi}{3}, 2 \\pi\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.7.2.15", "question": "Find the standard equation of the circle which satisfies the given criteria: endpoints of a diameter: $(3,6)$ and $(-1,4)$", "answer": "$(x-1)^{2}+(y-5)^{2}=5$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.41", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}+y^{2}=4$", "answer": "Not a function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.2.12", "question": "Rewrite the sum using summation notation: $1+2+4+\\cdots+2^{29}$", "answer": "$\\sum_{k=1}^{30} 2^{k-1}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.6", "question": "Simplify the expression: $\\frac{(k-1) !}{(k+2) !}, k \\geq 1$.", "answer": "$\\frac{1}{k(k+1)(k+2)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.13", "question": "Solve the rational inequality and express your answer using interval notation: $\\frac{x^{3}+2 x^{2}+x}{x^{2}-x-2} \\geq 0$", "answer": "$(-1,0] \\cup(2, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.43", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}-y^{2}=4$", "answer": "Not a function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.11", "question": "Write the set using interval notation: $\\{x \\mid x \\neq 0,2\\}$", "answer": "$(-\\infty, 0) \\cup(0,2) \\cup(2, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.8", "question": "Solve the equation analytically: $\\log \\left(x^{2}-3 x\\right)=1$", "answer": "$x=-2,5$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.30", "question": "Evaluate the expression: $\\log _{13}(\\sqrt{13})$", "answer": "$\\log _{13}(\\sqrt{13})=\\frac{1}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.7.3.15", "question": "Find an equation for the parabola which fits the given criteria: Vertex $(7,0)$, focus $(0,0)$", "answer": "$y^{2}=-28(x-7)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.30", "question": "Solve the equation analytically: $4^{x}+2^{x}=12$", "answer": "$x=\\frac{\\ln (3)}{\\ln (2)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.17", "question": "Simplify the quantity $\\sqrt{-(-9)}$", "answer": "3", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.18", "question": "Write the set using interval notation: $\\{x \\mid x>2$ or $x= \\pm 1\\}$", "answer": "$\\{-1\\} \\cup\\{1\\} \\cup(2, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.75", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $\\cot ^{2}(x) \\geq \\frac{1}{3}$", "answer": "$\\left(0, \\frac{\\pi}{3}\\right] \\cup\\left[\\frac{2 \\pi}{3}, \\pi\\right) \\cup\\left(\\pi, \\frac{4 \\pi}{3}\\right] \\cup\\left[\\frac{5 \\pi}{3}, 2 \\pi\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.37", "question": "Solve the inequality analytically: $25\\left(\\frac{4}{5}\\right)^{x} \\geq 10$", "answer": "$\\left(-\\infty, \\frac{\\ln \\left(\\frac{2}{5}\\right)}{\\ln \\left(\\frac{4}{5}\\right)}\\right]=\\left(-\\infty, \\frac{\\ln (2)-\\ln (5)}{\\ln (4)-\\ln (5)}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.17", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\cos ^{2}(x)=\\frac{1}{2}$", "answer": "$x=\\frac{\\pi}{4}+\\frac{\\pi k}{2} ; x=\\frac{\\pi}{4}, \\frac{3 \\pi}{4}, \\frac{5 \\pi}{4}, \\frac{7 \\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.20", "question": "Solve the equation analytically: $\\log (x)-\\log (2)=\\log (x+8)-\\log (x+2)$", "answer": "$x=4$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.43", "question": "Convert the angle from radian measure into degree measure: $-\\frac{\\pi}{6}$", "answer": "$-30^{\\circ}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.8.58", "question": "A 600 pound Sasquatch statue is suspended by two cables from a gymnasium ceiling. If each cable makes a $60^{\\circ}$ angle with the ceiling, find the tension on each cable. Round your answer to the nearest pound.", "answer": "The resultant force is only about 296 pounds so the couch doesn't budge. Even if it did move, the stronger force on the third rope would have made the couch drift slightly to the south as it traveled down the street.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.22", "question": "The height of an object dropped from the roof of an eight story building is modeled by $h(t)=-16 t^{2}+64,0 \\leq t \\leq 2$. Here, $h$ is the height of the object off the ground, in feet, $t$ seconds after the object is dropped. How long before the object hits the ground?", "answer": "2 seconds.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.73", "question": "Convert the equation from polar coordinates into rectangular coordinates: $r=-\\csc (\\theta) \\cot (\\theta)$", "answer": "$r=7 \\sin (\\theta)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.26", "question": "Use the properties of logarithms to write the expression as a single logarithm: $\\ln (x)+\\frac{1}{2}$", "answer": "$\\ln (x \\sqrt{e})$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.5.3.26", "question": "Solve the equation or inequality: $5-(4-2 x)^{\\frac{2}{3}}=1$", "answer": "$x=-2,6$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.22", "question": "Evaluate the expression: $\\log _{\\frac{1}{5}}(625)$", "answer": "$\\log _{\\frac{1}{5}}(625)=-4$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.10", "question": "Solve the equation analytically: $\\log \\left(\\frac{x}{10^{-3}}\\right)=4.7$", "answer": "$x=10^{1.7}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.2.9", "question": "Rewrite the sum using summation notation: $8+11+14+17+20$", "answer": "$\\sum_{k=1}^{5}(3 k+5)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.44", "question": "Find the domain of the function: $f(x)=\\log _{7}(4 x+8)$", "answer": "$(-2, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.4", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\tan (6 x)=1$", "answer": "$x=\\frac{\\pi}{24}+\\frac{\\pi k}{6} ; x=\\frac{\\pi}{24}, \\frac{5 \\pi}{24}, \\frac{3 \\pi}{8}, \\frac{13 \\pi}{24}, \\frac{17 \\pi}{24}, \\frac{7 \\pi}{8}, \\frac{25 \\pi}{24}, \\frac{29 \\pi}{24}, \\frac{11 \\pi}{8}, \\frac{37 \\pi}{24}, \\frac{41 \\pi}{24}, \\frac{15 \\pi}{8}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.32", "question": "Evaluate the expression: $7^{\\log _{7}(3)}$", "answer": "$7^{\\log _{7}(3)}=3$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.23", "question": "Use the properties of logarithms to write the expression as a single logarithm: $\\log _{5}(x)-3$", "answer": "$\\log _{5}\\left(\\frac{x}{125}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.31", "question": "Solve the equation or inequality using your calculator: $\\ln (x)=e^{-x}$", "answer": "$x \\approx 1.3098$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.39", "question": "Solve the inequality analytically: $70+90 e^{-0.1 t} \\leq 75$", "answer": "$\\left[\\frac{\\ln \\left(\\frac{1}{18}\\right)}{-0.1}, \\infty\\right)=[10 \\ln (18), \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.26", "question": "Simplify the given power of $i$: $i^{304}$", "answer": "$i^{304}=\\left(i^{4}\\right)^{76}=1^{76}=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.32", "question": "Solve the equation analytically: $e^{x}+15 e^{-x}=8$", "answer": "$x=\\ln (3), \\ln (5)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.22", "question": "Solve the equation analytically: $\\ln (\\ln (x))=3$", "answer": "$x=e^{e^{3}}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.51", "question": "Approximate the given value to three decimal places: $\\sin (392.994)$", "answer": "$\\sin (392.994) \\approx-0.291$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.15", "question": "Simplify the power of a complex number: $(-1+i \\sqrt{3})^{3}$", "answer": "8", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.2", "question": "Solve the equation: $|3 x-1|=10$", "answer": "$x=-3$ or $x=\\frac{11}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.16", "question": "Evaluate the expression: $\\log _{3}(27)$", "answer": "$\\log _{3}(27)=3$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.1", "question": "Find the exact value of the cosine and sine of the given angle: $\\theta=0$", "answer": "$\\cos (0)=1, \\sin (0)=0$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.54", "question": "A rock got stuck in the tread of my tire and when I was driving 70 miles per hour, the rock came loose and hit the inside of the wheel well of the car. How fast, in miles per hour, was the rock traveling when it came out of the tread? (The tire has a diameter of 23 inches.)", "answer": "70 miles per hour", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.35", "question": "Solve the quadratic equation $y^{2}-4 y=x^{2}-4$ for $y$.", "answer": "$y=2 \\pm x$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.42", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $y=\\sqrt{4-x^{2}}$", "answer": "Function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.7", "question": "Solve the equation: $\\frac{5-|x|}{2}=1$", "answer": "$x=-3$ or $x=3$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.50", "question": "Find the domain of the function: $f(x)=\\ln (4 x-20)+\\ln \\left(x^{2}+9 x+18\\right)$", "answer": "$(5, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.4", "question": "Convert the angle into the DMS system and round the answer to the nearest second: $179.999^{\\circ}$", "answer": "$179^{\\circ} 59^{\\prime} 56^{\\prime \\prime}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.47", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}=y^{2}$", "answer": "Not a function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.36", "question": "Solve the equation analytically: $\\ln (3-y)-\\ln (y)=2 x+\\ln (5)$ for $y$.", "answer": "$y=\\frac{3}{5 e^{2 x}+1}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.25", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $3 \\cos (2 x)+\\cos (x)+2=0$", "answer": "$x=\\frac{7 \\pi}{6}, \\frac{11 \\pi}{6}, \\arcsin \\left(\\frac{1}{3}\\right), \\pi-\\arcsin \\left(\\frac{1}{3}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.36", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}-y^{2}=1$", "answer": "Not a function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.24", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (2 x)=2-5 \\cos (x)$", "answer": "$x=\\frac{2 \\pi}{3}, \\frac{4 \\pi}{3}, \\arccos \\left(\\frac{1}{3}\\right), 2 \\pi-\\arccos \\left(\\frac{1}{3}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.59", "question": "Solve the equation: $\\arccos (2 x)=\\pi$", "answer": "$x=-\\frac{1}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.9", "question": "Solve the equation analytically: $\\log _{125}\\left(\\frac{3 x-2}{2 x+3}\\right)=\\frac{1}{3}$", "answer": "$x=-\\frac{17}{7}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.18", "question": "Solve the equation analytically: $30-6 e^{-0.1 x}=20$", "answer": "$x=-10 \\ln \\left(\\frac{5}{3}\\right)=10 \\ln \\left(\\frac{3}{5}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.17", "question": "Write the set using interval notation: $\\{x \\mid x \\leq 5$ or $x=6\\}$", "answer": "$(-\\infty, 5] \\cup\\{6\\}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.27", "question": "At a local buffet, 22 diners (5 of whom were children) feasted for $\\$162.25$, before taxes. If the kids buffet is $\\$4.50$, the basic buffet is $\\$7.50$, and the deluxe buffet (with crab legs) is $\\$9.25$, find out how many diners chose the deluxe buffet.", "answer": "This time, 7 diners chose the deluxe buffet.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.67", "question": "Solve the equation: $8 \\operatorname{arccot}^{2}(x)+3 \\pi^{2}=10 \\pi \\operatorname{arccot}(x)$", "answer": "$x=-1,0$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.22", "question": "Simplify the given power of $i$: $i^8$", "answer": "$i^{8}=i^{4} \\cdot i^{4}=\\left(i^{4}\\right)^{2}=(1)^{2}=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.8", "question": "Write the set using interval notation: $\\{x \\mid x \\neq 5\\}$", "answer": "$(-\\infty, 5) \\cup(5, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.34", "question": "Solve the quadratic equation $y^{2}-3 y=4 x$ for $y$.", "answer": "$y=\\frac{3 \\pm \\sqrt{16 x+9}}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.20", "question": "Use the properties of logarithms to write the expression as a single logarithm: $2 \\ln (x)-3 \\ln (y)-4 \\ln (z)$", "answer": "$\\ln \\left(\\frac{x^{2}}{y^{3} z^{4}}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.25", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} 2 x-3 y+z & =-1 \\\\ 4 x-4 y+4 z & =-13 \\\\ 6 x-5 y+7 z & =-25\\end{aligned}\\right.$", "answer": "$\\left(-2 t-\\frac{35}{4},-t-\\frac{11}{2}, t\\right)$ for all real numbers $t$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.49", "question": "Approximate the given value to three decimal places: $\\sin \\left(78.95^{\\circ}\\right)$", "answer": "$\\sin \\left(78.95^{\\circ}\\right) \\approx 0.981$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.8", "question": "Solve the equation analytically: $9 \\cdot 3^{7 x}=\\left(\\frac{1}{9}\\right)^{2 x}$", "answer": "$x=-\\frac{2}{11}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.93", "question": "Solve the given inequality: $\\arcsin (2 x)>0$", "answer": "$\\left(0, \\frac{1}{2}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.44", "question": "Use your calculator to help you solve the inequality: $3^{(x-1)}<2^{x}$", "answer": "$\\approx(-\\infty, 2.7095)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.3", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\sin (-2 x)=\\frac{\\sqrt{3}}{2}$", "answer": "$x=\\frac{2 \\pi}{3}+\\pi k$ or $x=\\frac{5 \\pi}{6}+\\pi k ; x=\\frac{2 \\pi}{3}, \\frac{5 \\pi}{6}, \\frac{5 \\pi}{3}, \\frac{11 \\pi}{6}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.49", "question": "Create a polynomial $f$ with real number coefficients that has the following characteristics:\n- The zeros of $f$ are $c=\\pm 1$ and $c=\\pm i$\n- The leading term of $f(x)$ is $42x^4$", "answer": "$f(x)=42(x-1)(x+1)(x-i)(x+i)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.6", "question": "Solve the equation analytically: $2^{\\left(x^{3}-x\\right)}=1$", "answer": "$x=-1,0,1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.20", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} 2 x-4 y+z & =-7 \\\\ x-2 y+2 z & =-2 \\\\ -x+4 y-2 z & =3\\end{aligned}\\right.$", "answer": "$\\left(-3, \\frac{1}{2}, 1\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.54", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos (4 x)=\\cos (2 x)$", "answer": "$x=0, \\frac{\\pi}{3}, \\frac{2 \\pi}{3}, \\pi, \\frac{4 \\pi}{3}, \\frac{5 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.7", "question": "Solve the rational inequality and express your answer using interval notation: $\\frac{1}{x+2} \\geq 0$", "answer": "$(-2, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.36", "question": "Convert the angle from degree measure into radian measure, giving the exact value in terms of $\\pi$: $-225^{\\circ}$", "answer": "$-\\frac{5 \\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.21", "question": "Solve the equation: $3|x-1|=2|x+1|$", "answer": "$x=\\frac{1}{5}$ or $x=5$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.62", "question": "If $\\theta=5^{\\circ}$ and the hypotenuse has length 10 , how long is the side opposite $\\theta$ ?", "answer": "The side opposite $\\theta$ has length $10 \\sin \\left(5^{\\circ}\\right) \\approx 0.872$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.12", "question": "Expand the given logarithm and simplify: $\\log _{6}\\left(\\frac{216}{x^{3} y}\\right)^{4}$", "answer": "$12-12 \\log _{6}(x)-4 \\log _{6}(y)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.15", "question": "Solve the equation: $\\left|x^{2}-1\\right|=3$", "answer": "$x=-2$ or $x=2$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.26", "question": "Solve the inequality analytically: $x \\ln (x)-x>0$", "answer": "$(e, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.47", "question": "Find the domain of the function: $f(x)=\\log \\left(\\frac{x+2}{x^{2}-1}\\right)$", "answer": "$(-2,-1) \\cup(1, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.27", "question": "Solve the equation analytically: $7^{3+7 x}=3^{4-2 x}$", "answer": "$x=\\frac{4 \\ln (3)-3 \\ln (7)}{7 \\ln (7)+2 \\ln (3)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.9", "question": "Write the set using interval notation: $\\{x \\mid x \\neq-1\\}$", "answer": "$(-\\infty,-1) \\cup(-1, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.9.21", "question": "Find the work done pushing a 200 pound barrel 10 feet up a $12.5^{\\circ}$ incline. Ignore all forces acting on the barrel except gravity, which acts downwards. Round your answer to two decimal places.\n\nHINT: Since you are working to overcome gravity only, the force being applied acts directly upwards. This means that the angle between the applied force in this case and the motion of the object is not the $12.5^{\\circ}$ of the incline!", "answer": "(1500 pounds) $\\left(300\\right.$ feet) $\\cos \\left(0^{\\circ}\\right)=450,000$ foot-pounds", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.2", "question": "Write out the first four terms of the given sequence: $d_{j}=(-1)^{\\frac{j(j+1)}{2}}, j \\geq 1$", "answer": "$-1,-1,1,1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.66", "question": "Convert the equation from polar coordinates into rectangular coordinates: $r=3 \\sin (\\theta)$", "answer": "$r=\\frac{19}{4 \\cos (\\theta)-\\sin (\\theta)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.52", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $3 \\sqrt{3} \\sin (3 x)-3 \\cos (3 x)=3 \\sqrt{3}$", "answer": "$x=\\frac{\\pi}{6}, \\frac{5 \\pi}{18}, \\frac{5 \\pi}{6}, \\frac{17 \\pi}{18}, \\frac{3 \\pi}{2}, \\frac{29 \\pi}{18}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.39", "question": "Convert the angle from radian measure into degree measure: $\\frac{7 \\pi}{6}$", "answer": "$210^{\\circ}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.76", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \\leq x \\leq 2 \\pi$: $2 \\cos (x) \\geq 1$", "answer": "$\\left[0, \\frac{\\pi}{2}\\right) \\cup\\left(\\frac{11 \\pi}{6}, 2 \\pi\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.24", "question": "A faucet can fill a sink in 5 minutes while a drain will empty the same sink in 8 minutes. If the faucet is turned on and the drain is left open, how long will it take to fill the sink?", "answer": "$\\frac{40}{3} \\approx 13.33$ minutes", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.7.3.21", "question": "A parabolic arch is constructed which is 6 feet wide at the base and 9 feet tall in the middle. Find the height of the arch exactly 1 foot in from the base of the arch.", "answer": "The arch can be modeled by $x^{2}=-(y-9)$ or $y=9-x^{2}$. One foot in from the base of the arch corresponds to either $x= \\pm 2$, so the height is $y=9-( \\pm 2)^{2}=5$ feet.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.38", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cos ^{3}(x)=-\\cos (x)$", "answer": "$x=\\frac{\\pi}{2}, \\frac{3 \\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.6", "question": "Expand the given logarithm and simplify: $\\log _{5}\\left(x^{2}-25\\right)$", "answer": "$\\log _{5}(x-5)+\\log _{5}(x+5)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.4.15", "question": "The International Silver Strings Submarine Band holds a bake sale each year to fund their trip to the National Sasquatch Convention. It has been determined that the cost in dollars of baking $x$ cookies is $C(x)=0.1 x+25$ and that the demand function for their cookies is $p=10-.01 x$. How many cookies should they bake in order to maximize their profit?", "answer": "$\\left(1, \\frac{5}{3}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.28", "question": "Evaluate the expression: $\\log _{4}(8)$", "answer": "$\\log _{4}(8)=\\frac{3}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.40", "question": "Convert the angle from radian measure into degree measure: $\\frac{11 \\pi}{6}$", "answer": "$330^{\\circ}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.1.49", "question": "Find the domain of the function: $f(x)=\\ln (7-x)+\\ln (x-4)$", "answer": "$(4,7)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.40", "question": "Use your calculator to help you solve the equation: $2^{x}=x^{2}$", "answer": "$x \\approx-0.76666, x=2, x=4$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.13", "question": "Write the set using interval notation: $\\{x \\mid x \\neq 0, \\pm 4\\}$", "answer": "$(-\\infty,-4) \\cup(-4,0) \\cup(0,4) \\cup(4, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.36", "question": "Find the real solutions of the polynomial equation $9 x^{2}+5 x^{3}=6 x^{4}$.", "answer": "$x=0, \\frac{5 \\pm \\sqrt{241}}{12}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.30", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\cot ^{2}(x)=3 \\csc (x)-3$", "answer": "$x=\\frac{\\pi}{6}, \\frac{7 \\pi}{6}, \\frac{5 \\pi}{6}, \\frac{11 \\pi}{6}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.7", "question": "Write out the first four terms of the given sequence: $a_{1}=3, a_{n+1}=a_{n}-1, n \\geq 1$", "answer": "$3,2,1,0$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.1.34", "question": "The Topology Taxi Company charges $\\$ 2.50$ for the first fifth of a mile and $\\$ 0.45$ for each additional fifth of a mile. Find a linear function which models the taxi fare $F$ as a function of the number of miles driven, $m$. Interpret the slope of the linear function and find and interpret $F(0)$.", "answer": "$F(m)=2.25 m+2.05$ The slope 2.25 means it costs an additional $\\$ 2.25$ for each mile beyond the first 0.2 miles. $F(0)=2.05$, so according to the model, it would cost $\\$ 2.05$ for a trip of 0 miles. Would this ever really happen? Depends on the driver and the passenger, we suppose.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.38", "question": "Solve the inequality analytically: $\\frac{150}{1+29 e^{-0.8 t}} \\leq 130$", "answer": "$\\left(-\\infty, \\frac{\\ln \\left(\\frac{2}{377}\\right)}{-0.8}\\right]=\\left(-\\infty, \\frac{5}{4} \\ln \\left(\\frac{377}{2}\\right)\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.97", "question": "Solve the given inequality: $2 \\arcsin (x)^{2}>\\pi \\arcsin (x)$", "answer": "$[-1,0)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.99", "question": "Express the domain of the function using the extended interval notation: $f(x)=\\frac{1}{\\cos (x)-1}$", "answer": "$\\bigcup_{k=-\\infty}^{\\infty}(2 k \\pi,(2 k+2) \\pi)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.7", "question": "Evaluate: $\\left(\\begin{array}{l}8 \\\\ 3\\end{array}\\right)$", "answer": "56", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.21", "question": "Simplify the given power of $i$: $i^7$", "answer": "$i^{7}=i^{4} \\cdot i^{3}=1 \\cdot(-i)=-i$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.50", "question": "Convert the equation from rectangular coordinates into polar coordinates: $x^{2}+y^{2}-2 y=0$", "answer": "$(20, \\pi-\\arctan (3))$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.2.11", "question": "Rewrite the sum using summation notation: $x-\\frac{x^{3}}{3}+\\frac{x^{5}}{5}-\\frac{x^{7}}{7}$", "answer": "$\\sum_{k=1}^{4}(-1)^{k-1} \\frac{x^{2 k-1}}{2 k-1}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.27", "question": "Use the properties of logarithms to write the expression as a single logarithm: $\\log _{2}(x)+\\log _{4}(x)$", "answer": "$\\log _{2}\\left(x^{3 / 2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.5.18", "question": "Find the inverse of the given matrix: $F=\\left[\\begin{array}{rrr}4 & 6 & -3 \\\\ 3 & 4 & -3 \\\\ 1 & 2 & 6\\end{array}\\right]$", "answer": "$F^{-1}=\\left[\\begin{array}{rrr}-\\frac{5}{2} & \\frac{7}{2} & \\frac{1}{2} \\\\ \\frac{7}{4} & -\\frac{9}{4} & -\\frac{1}{4} \\\\ -\\frac{1}{6} & \\frac{1}{6} & \\frac{1}{6}\\end{array}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.4.3.9", "question": "Solve the rational inequality and express your answer using interval notation: $\\frac{x}{x^{2}-1}>0$", "answer": "$(-1,0) \\cup(1, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.43", "question": "Solve the equation for $t$: $\\sin (t)=-\\frac{1}{2}$", "answer": "$\\sin (t)=-\\frac{1}{2}$ when $t=\\frac{7 \\pi}{6}+2 \\pi k$ or $t=\\frac{11 \\pi}{6}+2 \\pi k$ for any integer $k$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.1.38", "question": "Convert the angle from radian measure into degree measure: $-\\frac{2 \\pi}{3}$", "answer": "$-120^{\\circ}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.33", "question": "Solve the equation analytically: $3^{x}+25 \\cdot 3^{-x}=10$", "answer": "$x=\\frac{\\ln (5)}{\\ln (3)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.5", "question": "Determine whether or not the relation represents $y$ as a function of $x$ and find the domain and range of those relations which are functions:\n$\\{(x, y) \\mid x$ is an odd integer, and $y$ is an even integer $\\}$", "answer": "Not a function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.8", "question": "Expand the given logarithm and simplify: $\\log _{\\frac{1}{3}}\\left(9 x\\left(y^{3}-8\\right)\\right)$", "answer": "$-2+\\log _{\\frac{1}{3}}(x)+\\log _{\\frac{1}{3}}(y-2)+\\log _{\\frac{1}{3}}\\left(y^{2}+2 y+4\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.7.16", "question": "A certain bacteria culture follows the Law of Uninbited Growth, Equation 6.4. After 10 minutes, there are 10,000 bacteria. Five minutes later, there are 14,000 bacteria. How many bacteria were present initially? How long before there are 50,000 bacteria?", "answer": "Initially, there are $\\frac{250000}{49} \\approx 5102$ bacteria. It will take $\\frac{5 \\ln (49 / 5)}{\\ln (7 / 5)} \\approx 33.92$ minutes for the colony to grow to 50,000 bacteria.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.11", "question": "Solve the equation: $4-|x|=2 x+1$", "answer": "$x=1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.3.15", "question": "The International Silver Strings Submarine Band holds a bake sale each year to fund their trip to the National Sasquatch Convention. It has been determined that the cost in dollars of baking $x$ cookies is $C(x)=0.1 x+25$ and that the demand function for their cookies is $p=10-.01 x$. How many cookies should they bake in order to maximize their profit?", "answer": "495 cookies", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.2.11", "question": "Find the exact value of the cosine and sine of the given angle: $\\theta=\\frac{3 \\pi}{2}$", "answer": "$\\cos \\left(\\frac{3 \\pi}{2}\\right)=0, \\sin \\left(\\frac{3 \\pi}{2}\\right)=-1$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.2.10", "question": "Solve the following system of linear equations: $\\left\\{\\begin{aligned} x-2 y+3 z & =7 \\\\ -3 x+y+2 z & =-5 \\\\ 2 x+2 y+z & =3\\end{aligned}\\right.$", "answer": "Inconsistent", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.87", "question": "Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \\pi \\leq x \\leq 2 \\pi$: $\\csc (x)>1$", "answer": "$\\left(-2 \\pi,-\\frac{3 \\pi}{2}\\right) \\cup\\left(-\\frac{3 \\pi}{2},-\\pi\\right) \\cup\\left(0, \\frac{\\pi}{2}\\right) \\cup\\left(\\frac{\\pi}{2}, \\pi\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.106", "question": "Express the domain of the function using the extended interval notation: $f(x)=\\ln (|\\cos (x)|)$", "answer": "$\\bigcup_{k=-\\infty}^{\\infty}\\left(\\frac{(2 k-1) \\pi}{2}, \\frac{(2 k+1) \\pi}{2}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.14", "question": "Simplify the quantity $\\sqrt{(-25)(-4)}$", "answer": "10", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.2.31", "question": "Express the repeating decimal as a fraction of integers: $10 . \\overline{159}$", "answer": "$\\frac{3383}{333}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.11.4.56", "question": "Convert the equation from rectangular coordinates into polar coordinates: $4 x^{2}+4\\left(y-\\frac{1}{2}\\right)^{2}=1$", "answer": "$(\\sqrt{13}, \\pi-\\arctan (2))$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.4.15", "question": "Simplify the quantity $\\sqrt{-9}\\sqrt{-16}$", "answer": "-12", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.4", "question": "Expand the given logarithm and simplify: $\\log \\left(1.23 \\times 10^{37}\\right)$", "answer": "$\\log (1.23)+37$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.36", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\tan ^{3}(x)=3 \\tan (x)$", "answer": "$x=0, \\frac{\\pi}{3}, \\frac{2 \\pi}{3}, \\pi, \\frac{4 \\pi}{3}, \\frac{5 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.4.17", "question": "The temperature $T$, in degrees Fahrenheit, $t$ hours after $6 \\mathrm{AM}$ is given by $T(t)=-\\frac{1}{2} t^{2}+8 t+32, \\quad 0 \\leq t \\leq 12$. What is the warmest temperature of the day? When does this happen?", "answer": "$(-\\infty,-3] \\cup[1, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.40", "question": "Find the real solutions of the polynomial equation $2 x^{3}=19 x^{2}-49 x+20$.", "answer": "$x=\\frac{1}{2}, 4,5$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.2.3", "question": "Expand the given logarithm and simplify: $\\log _{5}\\left(\\frac{z}{25}\\right)^{3}$", "answer": "$3 \\log _{5}(z)-6$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.28", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $2 \\sec ^{2}(x)=3-\\tan (x)$", "answer": "$x=\\frac{\\pi}{6}, \\frac{5 \\pi}{6}, \\frac{\\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.8.7.19", "question": "Solve the system of nonlinear equations after making the appropriate substitutions: $\\left\\{\\begin{array}{l}4 \\ln (x)+3 y^{2}=1 \\\\ 3 \\ln (x)+2 y^{2}=-1\\end{array}\\right.$", "answer": "$\\left(e^{-5}, \\pm \\sqrt{7}\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.37", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\tan ^{2}(x)=\\frac{3}{2} \\sec (x)$", "answer": "$x=\\frac{\\pi}{2}, \\frac{3 \\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.4.2", "question": "Simplify the expression: $\\frac{10 !}{7 !}$", "answer": "720", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.1.28", "question": "Jeff can walk comfortably at 3 miles per hour. Find a linear function $d$ that represents the total distance Jeff can walk in $t$ hours, assuming he doesn't take any breaks.", "answer": "$d(t)=3 t, t \\geq 0$.", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.5.3.17", "question": "Solve the equation or inequality: $x+1=\\sqrt{3 x+7}$", "answer": "$x=3$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.16", "question": "Solve the equation analytically: $\\log _{5}(2 x+1)+\\log _{5}(x+2)=1$", "answer": "$x=\\frac{1}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.45", "question": "Use your calculator to help you solve the inequality: $e^{x}<x^{3}-x$", "answer": "$\\approx(2.3217,4.3717)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.104", "question": "Express the domain of the function using the extended interval notation: $f(x)=\\frac{\\sin (x)}{2+\\cos (x)}$", "answer": "$(-\\infty, \\infty)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.2.2.14", "question": "Solve the equation: $|x|=12-x^{2}$", "answer": "$x=-3$ or $x=3$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.5.3.40", "question": "The period of a pendulum in seconds is given by\n$$\nT=2 \\pi \\sqrt{\\frac{L}{g}}\n$$\n(for small displacements) where $L$ is the length of the pendulum in meters and $g=9.8$ meters per second per second is the acceleration due to gravity. My Seth-Thomas antique schoolhouse clock needs $T=\\frac{1}{2}$ second and I can adjust the length of the pendulum via a small dial on the bottom of the bob. At what length should I set the pendulum?", "answer": "$9.8\\left(\\frac{1}{4 \\pi}\\right)^{2} \\approx 0.062$ meters or 6.2 centimeters", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.5.12", "question": "A finance company offers a promotion on $\\$ 5000$ loans. The borrower does not have to make any payments for the first three years, however interest will continue to be charged to the loan at $29.9 \\%$ compounded continuously. What amount will be due at the end of the three-year period, assuming no payments are made? If the promotion is extended an additional three years, and no payments are made, what amount would be due?", "answer": "$A(3)=5000 e^{0.299 \\cdot 3} \\approx \\$ 12,226.18, A(6)=5000 e^{0.299 \\cdot 6} \\approx \\$ 30,067.29$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.7.2.13", "question": "Find the standard equation of the circle which satisfies the given criteria: center $(3,5)$, passes through $(-1,-2)$", "answer": "$(x-3)^{2}+(y-5)^{2}=65$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.9.1.9", "question": "Write out the first four terms of the given sequence: $b_{1}=2, b_{k+1}=3 b_{k}+1, k \\geq 1$", "answer": "$2,7,22,67$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.41", "question": "Use your calculator to help you solve the equation: $e^{x}=\\ln (x)+5$", "answer": "$x \\approx 0.01866, x \\approx 1.7115$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.3.3.43", "question": "Find the real solutions of the polynomial equation $14 x^{2}+5=3 x^{4}$.", "answer": "$\\{-2\\} \\cup[1,3]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.30", "question": "Solve the inequality analytically: $\\ln \\left(x^{2}\\right) \\leq(\\ln (x))^{2}$", "answer": "$(0,1] \\cup\\left[e^{2}, \\infty\\right)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.4.15", "question": "Solve the equation analytically: $\\log _{3}(x-4)+\\log _{3}(x+4)=2$", "answer": "$x=5$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.10", "question": "Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \\pi)$: $\\cos \\left(x+\\frac{5 \\pi}{6}\\right)=0$", "answer": "$x=-\\frac{\\pi}{3}+\\pi k ; x=\\frac{2 \\pi}{3}, \\frac{5 \\pi}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.40", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $y=x^{2}+4$", "answer": "Function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.39", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\tan (2 x)-2 \\cos (x)=0$", "answer": "$x=\\frac{\\pi}{6}, \\frac{\\pi}{2}, \\frac{5 \\pi}{6}, \\frac{3 \\pi}{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.107", "question": "Express the domain of the function using the extended interval notation: $f(x)=\\arcsin (\\tan (x))$", "answer": "$\\bigcup_{k=-\\infty}^{\\infty}\\left[\\frac{(4 k-1) \\pi}{4}, \\frac{(4 k+1) \\pi}{4}\\right]$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.63", "question": "Solve the equation: $4 \\operatorname{arcsec}\\left(\\frac{x}{2}\\right)=\\pi$", "answer": "$x=2 \\sqrt{2}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.5", "question": "Solve the equation analytically: $8^{x}=\\frac{1}{128}$", "answer": "$x=-\\frac{7}{3}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.1.23", "question": "Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(3,-10),(-1,2)$", "answer": "$d=4 \\sqrt{10}, M=(1,-4)$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.6.3.13", "question": "Solve the equation analytically: $(1.005)^{12 x}=3$", "answer": "$x=\\frac{\\ln (3)}{12 \\ln (1.005)}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.10.7.19", "question": "Solve the equation, giving the exact solutions which lie in $[0,2 \\pi)$: $\\sin (x)=\\cos (x)$", "answer": "$x=\\frac{\\pi}{4}, \\frac{5 \\pi}{4}$", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.PRECALCULUS", "question_number": "exercise.1.3.35", "question": "Determine whether or not the equation represents $y$ as a function of $x$: $x^{3} y=-4$", "answer": "Function", "license": "Creative Commons License", "data_topic": "college_math.precalculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.6.2", "question": "Determine whether the series $\\sum_{n=1}^{\\infty}(-1)^{n-1} \\frac{3 n^{2}+4}{2 n^{2}+3 n+5} $ converges absolutely, converges conditionally, or diverges.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.1", "question": "Find the antiderivative: $\\int \\sin ^{2} x d x $", "answer": "$x / 2-\\sin (2 x) / 4+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.8", "question": "An object moves along a straight line with acceleration given by $a(t)=-\\cos (t)$, and $s(0)=1$ and $v(0)=0$. Find the maximum distance the object travels from zero, and find its maximum speed. Describe the motion of the object.", "answer": "$s(t)=\\cos t, v(t)=-\\sin t$, maximum distance is 1 , maximum speed is 1", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.10", "question": "Find all local maximum and minimum points of the function: $y=(x+1) / \\sqrt{5 x^{2}+35} $", "answer": "$\\max$ at $x=7$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.5", "question": "Find the domain of the function: $y=f(x)=\\sqrt[3]{x} $", "answer": "$\\{x \\mid x \\in \\mathbb{R}\\}$, i.e., all $x$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.8.1", "question": "Compute the limit of $\\lim _{x \\rightarrow 0} \\frac{\\cos x-1}{\\sin x} $.", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.10", "question": "Find the antiderivative: $\\int x \\sin x \\cos x d x $", "answer": "$x / 4-\\left(x \\cos ^{2} x\\right) / 2+(\\cos x \\sin x) / 4+$ C", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.9", "question": "Find the derivative of the function: $(1+3 x)^{2} $", "answer": "$6+18 x$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.10.3", "question": "Find the derivative of $\\operatorname{arccot} x$, the inverse cotangent.", "answer": "$-1 /\\left(1+x^{2}\\right)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.2.1", "question": "Find the derivative of the function: $5 x^{3}+12 x^{2}-15 $", "answer": "$15 x^{2}+24 x$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.8", "question": "A thin plate lies in the region contained by $y=4-x^{2}$ and the $x$-axis. Find the centroid.", "answer": "$\\bar{x}=0, \\bar{y}=8 / 5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.6", "question": "Determine whether the series converges: $\\sum_{n=0}^{\\infty} \\frac{1}{\\sqrt{n^{2}+4}} $", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.3.11", "question": "A hemispheric bowl of radius $r$ contains water to a depth $h$. Find the volume of water in the bowl.", "answer": "$\\pi h^{2}(3 r-h) / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.5.6", "question": "Describe all functions with derivative $x^{2}+47 x-5$.", "answer": "$x^{3} / 3+47 x^{2} / 2-5 x+k$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.2", "question": "Find the derivative of the function: $x^{3}-2 x^{2}+4 \\sqrt{x} $", "answer": "$3 x^{2}-4 x+2 / \\sqrt{x}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.6.5", "question": "Determine whether the series $\\sum_{n=2}^{\\infty}(-1)^{n} \\frac{1}{\\ln n} $ converges absolutely, converges conditionally, or diverges.", "answer": "converges conditionally", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.9", "question": "Determine whether the series converges: $\\sum_{n=0}^{\\infty} \\frac{n !}{1 \\cdot 3 \\cdot 5 \\cdots(2 n-1)} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.5", "question": "Determine whether the series converges: $1-\\frac{3}{4}+\\frac{5}{8}-\\frac{7}{12}+\\frac{9}{16}+\\cdots $", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.2", "question": "Evaluate the integral: $\\int t\\left(t^{2}-9\\right)^{3 / 2} d t $", "answer": "$\\frac{\\left(t^{2}-9\\right)^{5 / 2}}{5}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.7.6", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{n !}{n^{n}} $ converges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.5", "question": "Find the antiderivative: $\\int x \\sqrt{1-x^{2}} d x $", "answer": "$-\\left(1-x^{2}\\right)^{3 / 2} / 3+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.6", "question": "Find the domain of the function: $y=f(x)=\\sqrt[4]{x} $", "answer": "$\\{x \\mid x \\geq 0\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.3.7", "question": "Find the volume of the solid obtained by revolving the region bounded by $y=\\sqrt{\\sin x}$, the $y$-axis, and the lines $y=1$ and $x=\\pi / 2$ around the $x$-axis.", "answer": "$\\pi(\\pi / 2-1)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.1", "question": "Find the derivative of the function: $3^{x^{2}} $", "answer": "$2 \\ln (3) x 3^{x^{2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.18", "question": "Find the derivative of the function: $G(x)=\\int_{1}^{x^{2}} t^{2}-3 t d t $", "answer": "$2 x\\left(x^{4}-3 x^{2}\\right)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.1.5", "question": "Find the derivative of the function: $x^{3 / 4} $", "answer": "$(3 / 4) x^{-1 / 4}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.6", "question": "A box with square base and no top is to hold a volume $V$. Find (in terms of $V$ ) the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. (This ratio will not involve $V$.)", "answer": "$w=l=2^{1 / 3} V^{1 / 3}, h=V^{1 / 3} / 2^{2 / 3}$, $h / w=1 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.9", "question": "Does the improper integral $\\int_{-\\infty}^{\\infty} \\frac{x^{2}}{4+x^{6}} d x$ converge or diverge? If it converges, find the value.", "answer": "$\\pi / 6$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.5.5", "question": "A water tank has the shape of the bottom half of a sphere with radius $r=1$ meter. If the tank is full, how much work is required to pump all the water out the top of the tank?", "answer": "$2450 \\pi \\mathrm{N}-\\mathrm{m}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.15", "question": "Find the derivative of the function: $\\sqrt[3]{x+x^{3}} $", "answer": "$\\frac{1+3 x^{2}}{3\\left(x+x^{3}\\right)^{2 / 3}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.9.4", "question": "Find the arc length of $f(x)=\\ln (\\sin x)$ on the interval $[\\pi / 4, \\pi / 3] . $", "answer": "$\\ln ((\\sqrt{2}+1) / \\sqrt{3})$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.10", "question": "Find the derivative of the function: $e^{4 x} / x $", "answer": "$e^{4 x}(4 x-1) / x^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.5.3", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{1}{2 n^{2}-3 n-5} $ converges or diverges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.24", "question": "Find a series representation for the function: $\\sum_{n=0}^{\\infty} \\frac{(x-1)^{n}}{n !} $", "answer": "$(-\\infty, \\infty)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.27", "question": "Find the derivative of the function: $\\left(3 x^{2}+1\\right)(2 x-4)^{3} $", "answer": "$120 x^{4}-576 x^{3}+888 x^{2}-480 x+96$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.1", "question": "A beam 10 meters long has density $\\sigma(x)=x^{2}$ at distance $x$ from the left end of the beam. Find the center of mass $\\bar{x}$.", "answer": "$15 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.6", "question": "Find the area bounded by the curves: $y=\\sin (\\pi x / 3)$ and $y=x$ (in the first quadrant)", "answer": "$3 / \\pi-3 \\sqrt{3} /(2 \\pi)-1 / 8$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.1", "question": "Find the antiderivative: $\\int \\csc x d x $", "answer": "$-\\ln |\\csc x+\\cot x|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.3.6", "question": "For all $x \\geq 0,4 x-9 \\leq f(x) \\leq x^{2}-4 x+7$. Find $\\lim _{x \\rightarrow 4} f(x)$.", "answer": "7", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.4.2", "question": "Let $f(x)=\\sqrt{x}$. If $a=1$ and $d x=\\Delta x=1 / 10$, what are $\\Delta y$ and $d y$ ?", "answer": "$\\Delta y=\\sqrt{11 / 10}-1, d y=0.05$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.7", "question": "Determine whether the series converges or diverges: $\\sum_{n=2}^{\\infty} \\frac{1}{n \\ln n} $", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.1", "question": "A cylindrical tank standing upright (with one circular base on the ground) has radius 20 $\\mathrm{cm}$. Find the rate at which the water level in the tank drops when the water is being drained at 25 $\\mathrm{cm}^{3} / \\mathrm{sec}$.", "answer": "$1 /(16 \\pi) \\mathrm{cm} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.2.4", "question": "Find the derivative of the function: $f(x)+g(x)$, where $f(x)=x^{2}-3 x+2$ and $g(x)=2 x^{3}-5 x $", "answer": "$6 x^{2}+2 x-8$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.1", "question": "Determine whether the area under the curve $y=1 / x$ from 1 to infinity is finite or infinite. If it is finite, compute the area.", "answer": "$\\infty$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.10", "question": "Find the antiderivative of the function: $|2 t-4| $", "answer": "$4 t-t^{2}+C, t<2 ; t^{2}-4 t+8+C$, $t \\geq 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.4.3", "question": "Find the derivative of the function: $\\frac{1}{\\sin x} $", "answer": "$-\\frac{\\cos x}{\\sin ^{2} x}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.7", "question": "Find the antiderivative: $\\int x \\arctan x d x $", "answer": "$\\left(x^{2} \\arctan x+\\arctan x-x\\right) / 2+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.11", "question": "Find the antiderivative: $\\int \\arctan (\\sqrt{x}) d x $", "answer": "$x \\arctan (\\sqrt{x})+\\arctan (\\sqrt{x})-\\sqrt{x}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.9.3", "question": "Find a power series representation for $2 /(1-x)^{3}$.", "answer": "$\\sum_{n=0}^{\\infty}(n+1)(n+2) x^{n}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.12", "question": "Determine whether the series converges: $\\sum_{n=1}^{\\infty} \\frac{1 \\cdot 3 \\cdot 5 \\cdots(2 n-1)}{(2 n) !} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.11", "question": "Find all critical points of the function $f(x)=x^{3} /(x+1) $. Identify them as local maximum points, local minimum points, or neither.", "answer": "$\\min$ at $x=-3 / 2$, neither at $x=0$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.2", "question": "Find all local maximum and minimum points of the function: $y=2+3 x-x^{3} $", "answer": "$\\min$ at $x=-1, \\max$ at $x=1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.1.11", "question": "Find all of the solutions of $2 \\sin (t)-1-\\sin ^{2}(t)=0$ in the interval $[0,2 \\pi]$.", "answer": "$t=\\pi / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.6.7", "question": "Determine whether the series $\\sum_{n=0}^{\\infty}(-1)^{n} \\frac{3^{n}}{2^{n}+3^{n}} $ converges absolutely, converges conditionally, or diverges.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.7", "question": "Find the antiderivative: $\\int \\frac{1}{\\sqrt{1+x^{2}}} d x $", "answer": "$\\ln \\left|x+\\sqrt{1+x^{2}}\\right|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.26", "question": "Find the derivative of the function: $\\left(x^{2}+1\\right)(5-2 x) / 2 $", "answer": "$-3 x^{2}+5 x-1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.23", "question": "Evaluate the integral: $\\int \\sin ^{3} t \\cos ^{4} t d t $", "answer": "$\\frac{\\cos ^{7} t}{7}-\\frac{\\cos ^{5} t}{5}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.6", "question": "Find the derivative of the function: $\\csc x $", "answer": "$-\\csc x \\cot x$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.9", "question": "Marketing tells you that if you set the price of an item at $\\$ 10$ then you will be unable to sell it, but that you can sell 500 items for each dollar below $\\$ 10$ that you set the price. Suppose your fixed costs total $\\$ 3000$, and your marginal cost is $\\$ 2$ per item. What is the most profit you can make?", "answer": "$\\$ 5000$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.1.6", "question": "Determine whether the sequence $\\left\\{\\frac{2^{n}}{n !}\\right\\}_{n=0}^{\\infty}$ converges or diverges.", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.2.1", "question": "Explain why the series $\\sum_{n=1}^{\\infty} \\frac{n^{2}}{2 n^{2}+1}$ diverges.", "answer": "$\\lim _{n \\rightarrow \\infty} n^{2} /\\left(2 n^{2}+1\\right)=1 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.25", "question": "What fraction of the volume of a sphere is taken up by the largest cylinder that can be fit inside the sphere?", "answer": "$1 / \\sqrt{3} \\approx 58 \\%$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.5.3", "question": "A water tank has the shape of a cylinder with radius $r=1$ meter and height 10 meters. If the depth of the water is 5 meters, how much work is required to pump all the water out the top of the tank?", "answer": "$367,500 \\pi \\mathrm{N}-\\mathrm{m}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.10", "question": "Find the area bounded by the curves: $y=\\sin x \\cos x$ and $y=\\sin x, 0 \\leq x \\leq \\pi $", "answer": "2", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.5.8", "question": "Determine whether the series $\\sum_{n=2}^{\\infty} \\frac{1}{\\ln n} $ converges or diverges.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.10", "question": "An object moves along a straight line with acceleration given by $a(t)=1+\\sin (\\pi t)$. Assume that when $t=0, s(t)=v(t)=0$. Find $s(t)$ and $v(t)$.", "answer": "$s(t)=t^{2} / 2-\\sin (\\pi t) / \\pi^{2}+t / \\pi$, $v(t)=t-\\cos (\\pi t) / \\pi+1 / \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.12", "question": "A thin plate lies in the region between the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}=1$ in the first quadrant. Find the centroid.", "answer": "$\\bar{x}=\\bar{y}=28 /(9 \\pi)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.4.5", "question": "Find the derivative of the function: $\\sqrt{1-\\sin ^{2} x} $", "answer": "$\\frac{-\\sin x \\cos x}{\\sqrt{1-\\sin ^{2} x}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.24", "question": "Evaluate the integral: $\\int \\frac{1}{t^{2}-6 t+9} d t $", "answer": "$\\frac{-1}{t-3}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.14", "question": "Find the derivative of the function: $100 /\\left(100-x^{2}\\right)^{3 / 2} $", "answer": "$\\frac{300 x}{\\left(100-x^{2}\\right)^{5 / 2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.5.5", "question": "Find the antiderivative: $\\int \\frac{x^{4}}{4+x^{2}} d x $", "answer": "$-4 x+x^{3} / 3+8 \\arctan (x / 2)+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.11", "question": "Find the domain of the function: $y=f(x)=1 /(\\sqrt{x}-1) $", "answer": "$\\{x \\mid x \\geq 0$ and $x \\neq 1\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.10", "question": "Describe the concavity of the function: $y=(x+1) / \\sqrt{5 x^{2}+35} $", "answer": "concave up on $(-\\infty,(21-\\sqrt{497}) / 4)$ and $(21+\\sqrt{497}) / 4, \\infty)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.3", "question": "Compute the limit: $\\lim _{x \\rightarrow-4} \\frac{x^{2}+x-12}{x-3} $. If a limit does not exist, explain why.", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.13", "question": "Determine whether the series converges: $\\sum_{n=0}^{\\infty} \\frac{6^{n}}{n !} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.5.8", "question": "Describe all functions with derivative $x^{3}-\\frac{1}{x}$.", "answer": "$x^{4} / 4-\\ln x+k$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.9.14", "question": "Find an equation for the tangent line to $x^{4}=y^{2}+x^{2}$ at $(2, \\sqrt{12})$. (This curve is the kampyle of Eudoxus.)", "answer": "$y=7 x / \\sqrt{3}-8 / \\sqrt{3}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.1.2", "question": "Suppose an object moves in a straight line so that its speed at time $t$ is given by $v(t)=t^{2}+2$, and that at $t=0$ the object is at position 5 . Find the position of the object at $t=2 . $", "answer": "$35 / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.8.2", "question": "Find the radius and interval of convergence for the series: $\\sum_{n=0}^{\\infty} \\frac{x^{n}}{n !} $", "answer": "$R=\\infty, I=(-\\infty, \\infty)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.13", "question": "For a cylinder with given surface area $S$, including the top and the bottom, find the ratio of height to base radius that maximizes the volume.", "answer": "$h / r=2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.2", "question": "Determine whether the area under the curve $y=1 / x^{3}$ from 1 to infinity is finite or infinite. If it is finite, compute the area.", "answer": "$1 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.4.1", "question": "Find the derivative of the function: $\\sin ^{2}(\\sqrt{x}) $", "answer": "$\\sin (\\sqrt{x}) \\cos (\\sqrt{x}) / \\sqrt{x}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.6", "question": "Express the improper integral $\\int_{0}^{1 / 2}(2 x-1)^{-3} d x$ as a limit and determine whether it converges or diverges. If it converges, find the value.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.12", "question": "Find an $N$ such that $\\sum_{n=2}^{\\infty} \\frac{1}{n(\\ln n)^{2}}=\\sum_{n=2}^{N} \\frac{1}{n(\\ln n)^{2}} \\pm 0.005$.", "answer": "any integer greater than $e^{200}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.16", "question": "A police helicopter is flying at a speed of $150 \\mathrm{~mph}$ at a constant altitude of 0.5 mile above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 1 mile from the helicopter, and that this distance is decreasing at a rate of $190 \\mathrm{~mph}$. Find the speed of the car.", "answer": "$380 / \\sqrt{3}-150 \\approx 69.4 \\mathrm{mph}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.2", "question": "Find the domain of the function: $y=f(x)=1 /(x+1) $", "answer": "$\\{x \\mid x \\neq-1\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.4", "question": "Determine whether the series converges or diverges: $\\sum_{n=1}^{\\infty} \\frac{1}{n^{2}+1} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.8.2", "question": "Compute the limit of $\\lim _{x \\rightarrow \\infty} \\frac{e^{x}}{x^{3}} $.", "answer": "$\\infty$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.5.9", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{3^{n}}{2^{n}+5^{n}} $ converges or diverges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.24", "question": "A light shines from the top of a pole $20 \\mathrm{~m}$ high. An object is dropped from the same height from a point $10 \\mathrm{~m}$ away, so that its height at time $t$ seconds is $h(t)=20-9.8 t^{2} / 2$. Find the rate at which the object's shadow is moving on the ground one second later.", "answer": "$4000 / 49 \\mathrm{~m} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.4.4", "question": "Find the derivative of the function: $\\frac{x^{2}+x}{\\sin x} $", "answer": "$\\frac{(2 x+1) \\sin x-\\left(x^{2}+x\\right) \\cos x}{\\sin ^{2} x}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.4", "question": "Find the antiderivative of the function: $2 / z^{2} $", "answer": "$-2 / z+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.9", "question": "Find the antiderivative of the function: $\\int \\frac{\\sin x}{\\cos ^{3} x} d x $", "answer": "$1 /\\left(2 \\cos ^{2} x\\right)=(1 / 2) \\sec ^{2} x+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.22", "question": "Find the derivative of the function: $5(x+1-1 / x) $", "answer": "$5+5 / x^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.5.10", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{3^{n}}{2^{n}+3^{n}} $ converges or diverges.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.15", "question": "Evaluate the definite integral: $\\int_{3}^{4} \\frac{1}{(3 x-7)^{2}} d x $", "answer": "$1 / 10$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.4", "question": "Find the antiderivative: $\\int x e^{x^{2}} d x $", "answer": "$(1 / 2) e^{x^{2}}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.13", "question": "Evaluate the integral: $\\int \\frac{1}{t^{2}+3 t} d t $", "answer": "$\\frac{\\ln |t|}{3}-\\frac{\\ln |t+3|}{3}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.20", "question": "Find the derivative of the function: $\\left(6-2 x^{2}\\right)^{3} $", "answer": "$-12 x\\left(6-2 x^{2}\\right)^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.11", "question": "Find all local maximum and minimum points of the function: $y=x^{5}-x $", "answer": "$\\max$ at $-5^{-1 / 4}$, min at $5^{-1 / 4}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.8", "question": "Find the area bounded by the curves: $y=\\sqrt{x}$ and $y=\\sqrt{x+1}, 0 \\leq x \\leq 4 $", "answer": "$10 \\sqrt{5} / 3-6$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.3", "question": "Find the antiderivative: $\\int \\sin ^{4} x d x $", "answer": "$3 x / 8-(\\sin 2 x) / 4+(\\sin 4 x) / 32+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.3", "question": "Find all critical points of the function $y=x^{3}-9 x^{2}+24 x $. Identify them as local maximum points, local minimum points, or neither.", "answer": "$\\max$ at $x=2$, min at $x=4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.8.4", "question": "Compute the limit of $\\lim _{x \\rightarrow \\infty} \\frac{\\ln x}{x} $.", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.15", "question": "Find an equation for the tangent line to $\\sin ^{2}(x)$ at $x=\\pi / 3$.", "answer": "$\\sqrt{3} x / 2+3 / 4-\\sqrt{3} \\pi / 6$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.20", "question": "Find the derivative of the function: $G(x)=\\int_{1}^{x^{2}} e^{t^{2}} d t $", "answer": "$2 x e^{x^{4}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.14", "question": "Find the antiderivative of the function: $\\int \\frac{\\sin (\\tan x)}{\\cos ^{2} x} d x $", "answer": "$-\\cos (\\tan x)+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.8.5", "question": "Find the radius and interval of convergence for the series: $\\sum_{n=1}^{\\infty} \\frac{(n !)^{2}}{n^{n}}(x-2)^{n} $", "answer": "$R=0$, converges only when $x=2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.2.5", "question": "Find the derivative of the function: $(x+1)\\left(x^{2}+2 x-3\\right) $", "answer": "$3 x^{2}+6 x-1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.8.7", "question": "The function $f(x)=\\frac{x}{\\sqrt{x^{2}+1}}$ has two horizontal asymptotes. Find them and give a rough sketch of $f$ with its horizontal asymptotes.", "answer": "$y=1$ and $y=-1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.9", "question": "Find the domain of the function: $y=f(x)=1 / \\sqrt{1-(3 x)^{2}} $", "answer": "$\\{x \\mid-1 / 3<x<1 / 3\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.4", "question": "Find the domain of the function: $y=f(x)=\\sqrt{-1 / x} $", "answer": "$\\{x \\mid x<0\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.5", "question": "Find the derivative of the function: $e^{\\sin x} $", "answer": "$\\cos (x) e^{\\sin x}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.5.1", "question": "How much work is done in lifting a 100 kilogram weight from the surface of the earth to an orbit 35,786 kilometers above the surface of the earth?", "answer": "$\\approx 5,305,028,517 \\mathrm{~N}-\\mathrm{m}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.1.6", "question": "Find the derivative of the function: $x^{-9 / 7} $", "answer": "$-(9 / 7) x^{-16 / 7}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.2", "question": "A cylindrical tank standing upright (with one circular base on the ground) has radius 1 meter. Find the rate at which the water level in the tank drops when the water is being drained at 3 liters per second.", "answer": "$3 /(1000 \\pi)$ meters $/$ second", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.4.5", "question": "Find an equation for the tangent line to $f(x)=\\left(x^{2}-4\\right) /(5-x)$ at $x=3$.", "answer": "$y=17 x / 4-41 / 4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.8", "question": "Describe the concavity of the function: $y=\\sin x+\\cos x $", "answer": "concave down on $((8 n-1) \\pi / 4,(8 n+$ $3) \\pi / 4)$, concave up on $((8 n+$ $3) \\pi / 4,(8 n+7) \\pi / 4)$, for integer $n$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.4", "question": "Find the antiderivative of the function: $\\int \\frac{1}{\\sqrt[3]{1-5 t}} d t $", "answer": "$-3(1-5 t)^{2 / 3} / 10+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.5.7", "question": "Find the antiderivative: $\\int \\frac{x^{3}}{4+x^{2}} d x $", "answer": "$x^{2} / 2-2 \\ln \\left(4+x^{2}\\right)+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.13", "question": "Determine whether the volume of the solid obtained by rotating the curve $y=1 / x$ around the x-axis, from $x=1$ to infinity, is finite or infinite. If it is finite, compute the volume.", "answer": "$\\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.6", "question": "A baseball diamond is a square $90 \\mathrm{ft}$ on a side. A player runs from first base to second base at a speed of $15 \\mathrm{ft} / \\mathrm{sec}$. At what rate is the player's distance from third base decreasing when she is halfway from first to second base?", "answer": "$3 \\sqrt{5} \\mathrm{ft} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.12", "question": "Find the antiderivative: $\\int \\sin (\\sqrt{x}) d x $", "answer": "$2 \\sin (\\sqrt{x})-2 \\sqrt{x} \\cos (\\sqrt{x})+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.4.2", "question": "Find the derivative of the function: $y=f(t)=80-4.9 t^{2}$.", "answer": "$-9.8 t$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.13", "question": "Compute the value of the integral: $\\int_{1}^{10} \\frac{1}{x} d x $", "answer": "$\\ln (10)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.13", "question": "Find the antiderivative: $\\int \\sec ^{2} x \\csc ^{2} x d x $", "answer": "$\\sec x \\csc x-2 \\cot x+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.4.1", "question": "Find the derivative of the function: $y=f(x)=\\sqrt{169-x^{2}}$.", "answer": "$-x / \\sqrt{169-x^{2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.20", "question": "Find the value of $a$ so that the tangent line to $y=\\ln (x)$ at $x=a$ is a line through the origin. Sketch the resulting situation.", "answer": "$e$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.1", "question": "Find all local maximum and minimum points of the function: $y=x^{2}-x $", "answer": "$\\min$ at $x=1 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.1.7", "question": "Let $f(x)=x^{2}+3 x+2$. Approximate the area under the curve between $x=0$ and $x=2$ using 4 rectangles and also using 8 rectangles.", "answer": "4 rectangles: $41 / 4=10.25$, 8 rectangles: $183 / 16=11.4375$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.24", "question": "Find the derivative of the function: $\\frac{1}{1+1 / x} $", "answer": "$1 /(x+1)^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.12", "question": "Describe the concavity of the function: $y=6 x+\\sin 3 x $", "answer": "concave down on $(2 n \\pi / 3,(2 n+$ 1) $\\pi / 3)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.10.6", "question": "Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $1 / x^{2}, a=1 $", "answer": "$\\sum_{n=0}^{\\infty}(-1)^{n}(n+1)(x-1)^{n}, R=1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.2", "question": "Find the antiderivative: $\\int \\csc ^{3} x d x $", "answer": "$-\\csc x \\cot x / 2-(1 / 2) \\ln \\mid \\csc x+$ $\\cot x \\mid+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.4.9", "question": "If $f^{\\prime}(4)=5, g^{\\prime}(4)=12,(f g)(4)=f(4) g(4)=2$, and $g(4)=6$, compute $f(4)$ and $\\frac{d}{d x} \\frac{f}{g}$ at 4 .", "answer": "$13 / 18$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.37", "question": "Find an equation for the tangent line to $y=9 x^{-2}$ at $(3,1)$.", "answer": "$y=3-2 x / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.4.2", "question": "Determine whether the series $\\sum_{n=4}^{\\infty} \\frac{(-1)^{n-1}}{\\sqrt{n-3}} $ converges or diverges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.15", "question": "Describe the concavity of the function: $y=(x+5)^{1 / 4} $", "answer": "concave down everywhere", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.19", "question": "Evaluate the integral: $\\int \\frac{t^{3}}{\\left(2-t^{2}\\right)^{5 / 2}} d t $", "answer": "$\\frac{2}{3\\left(2-t^{2}\\right)^{3 / 2}}-\\frac{1}{\\left(2-t^{2}\\right)^{1 / 2}}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.10", "question": "Find an $N$ such that $\\sum_{n=0}^{\\infty} \\frac{1}{e^{n}}=\\sum_{n=0}^{N} \\frac{1}{e^{n}} \\pm 10^{-4}$.", "answer": "$N=10$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.6.6", "question": "Determine whether the series $\\sum_{n=0}^{\\infty}(-1)^{n} \\frac{3^{n}}{2^{n}+5^{n}} $ converges absolutely, converges conditionally, or diverges.", "answer": "converges absolutely", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.2.5", "question": "Compute the value of the series $\\sum_{n=0}^{\\infty} \\frac{3}{2^{n}}+\\frac{4}{5^{n}}$.", "answer": "11", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.29", "question": "Find the derivative of the function: $\\frac{x^{2}-1}{x^{2}+1} $", "answer": "$4 x /\\left(x^{2}+1\\right)^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.10", "question": "Does the improper integral $\\int_{-\\infty}^{\\infty} x d x$ converge or diverge? If it converges, find the value. Also, find the Cauchy Principal Value, if it exists.", "answer": "diverges, 0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.3", "question": "Find all local maximum and minimum points of the function: $y=x^{3}-9 x^{2}+24 x $", "answer": "$\\max$ at $x=2$, min at $x=4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.12", "question": "Find the derivative of the function: $\\sqrt{\\frac{169}{x}-x} $", "answer": "$\\frac{1}{2}\\left(\\frac{-169}{x^{2}}-1\\right) / \\sqrt{\\frac{169}{x}-x}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.8", "question": "Compute the limit: $\\lim _{x \\rightarrow 4} 3 x^{3}-5 x $.", "answer": "172", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.16", "question": "Find an equation for the tangent line to $\\sec ^{2} x$ at $x=\\pi / 3$.", "answer": "$8 \\sqrt{3} x+4-8 \\sqrt{3} \\pi / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.11", "question": "An object moves along a straight line with acceleration given by $a(t)=1-\\sin (\\pi t)$. Assume that when $t=0, s(t)=v(t)=0$. Find $s(t)$ and $v(t)$.", "answer": "$s(t)=t^{2} / 2+\\sin (\\pi t) / \\pi^{2}-t / \\pi$, $v(t)=t+\\cos (\\pi t) / \\pi-1 / \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.1", "question": "Find the derivative of the function: $x^{4}-3 x^{3}+(1 / 2) x^{2}+7 x-\\pi $", "answer": "$4 x^{3}-9 x^{2}+x+7$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.7", "question": "Find the derivative of the function: $x^{3} e^{x} $", "answer": "$3 x^{2} e^{x}+x^{3} e^{x}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.17", "question": "Find the derivative of the function: $(x+8)^{5} $", "answer": "$5(x+8)^{4}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.3", "question": "Find the derivative of the function: $\\left(x^{2}+1\\right)^{3} $", "answer": "$6\\left(x^{2}+1\\right)^{2} x$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.2.6", "question": "Find the standard equation of the circle passing through $(-2,1)$ and tangent to the line $3 x-2 y=6$ at the point $(4,3)$. Sketch. (Hint: The line through the center of the circle and the point of tangency is perpendicular to the tangent line.)", "answer": "$(x+2 / 7)^{2}+(y-41 / 7)^{2}=1300 / 49$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.1.3", "question": "Determine whether the sequence $\\{\\sqrt{n+47}-\\sqrt{n}\\}_{n=0}^{\\infty}$ converges or diverges. If it converges, compute the limit.", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.13", "question": "Find the derivative of the function: $\\sqrt{x^{3}-x^{2}-(1 / x)} $", "answer": "$\\frac{3 x^{2}-2 x+1 / x^{2}}{2 \\sqrt{x^{3}-x^{2}-(1 / x)}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.6", "question": "Describe the concavity of the function: $y=\\left(x^{2}-1\\right) / x $", "answer": "concave up when $x<0$, concave down when $x>0$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.4.3", "question": "Determine whether the series $\\sum_{n=1}^{\\infty}(-1)^{n-1} \\frac{n}{3 n-2} $ converges or diverges.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.2", "question": "Find the antiderivative: $\\int x^{2} \\cos x d x $", "answer": "$x^{2} \\sin x-2 \\sin x+2 x \\cos x+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.5.9", "question": "Describe all functions with derivative $\\sin (2 x)$.", "answer": "$-\\cos (2 x) / 2+k$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.30", "question": "If you fit the cone with the largest possible surface area (lateral area plus area of base) into a sphere, what percent of the volume of the sphere is occupied by the cone?", "answer": "The ratio of the volume of the sphere to the volume of the cone is $1033 / 4096+33 / 4096 \\sqrt{17} \\approx 0.2854$, so the cone occupies approximately $28.54 \\%$ of the sphere.", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.10", "question": "A thin plate lies in the region contained by $\\sqrt{x}+\\sqrt{y}=1$ and the axes in the first quadrant. Find the centroid.", "answer": "$\\bar{x}=\\bar{y}=1 / 5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.3.2", "question": "Use Newton's Method to approximate the cube root of 10 to two decimal places.", "answer": "2.15", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.4.5", "question": "An object moves with velocity $v(t)=-t^{2}+1$ feet per second between $t=0$ and $t=2$. Find the average velocity and the average speed of the object between $t=0$ and $t=2 . $", "answer": "$-1 / 3,1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.6", "question": "Determine whether the series converges or diverges: $\\sum_{n=1}^{\\infty} \\frac{n}{e^{n}} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.1.3", "question": "Use an angle sum identity to compute $\\cos (\\pi / 12)$.", "answer": "$(\\sqrt{2}+\\sqrt{6}) / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.4.3", "question": "Find the derivative of the function: $y=f(x)=x^{2}-(1 / x)$.", "answer": "$2 x+1 / x^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.2.2", "question": "Find the derivative of the function: $-4 x^{5}+3 x^{2}-5 / x^{2} $", "answer": "$-20 x^{4}+6 x+10 / x^{3}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.4", "question": "For the velocity function $v=\\sin (\\pi t / 3)-t$, find both the net distance and the total distance traveled during the time interval $0 \\leq t \\leq 1$.", "answer": "$(3-\\pi) /(2 \\pi),(18-12 \\sqrt{3}+\\pi) /(4 \\pi)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.18", "question": "Find the derivative of the function: $(4-x)^{3} $", "answer": "$-3(4-x)^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.18", "question": "Evaluate the integral: $\\int\\left(t^{3 / 2}+47\\right)^{3} \\sqrt{t} d t $", "answer": "$\\frac{\\left(t^{3 / 2}+47\\right)^{4}}{6}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.7", "question": "Evaluate the integral: $\\int \\frac{1}{t\\left(t^{2}-4\\right)} d t $", "answer": "$\\frac{1}{8} \\ln \\left|1-4 / t^{2}\\right|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.5", "question": "Find the derivative of the function: $\\left(x^{2}-4 x+5\\right) \\sqrt{25-x^{2}} $", "answer": "$(2 x-4) \\sqrt{25-x^{2}}-$ $\\left(x^{2}-4 x+5\\right) x / \\sqrt{25-x^{2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.18", "question": "Describe the concavity of the function: $y=\\sin ^{3} x $", "answer": "inflection points at $n \\pi$, $\\pm \\arcsin (\\sqrt{2 / 3})+n \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.16", "question": "Compute the value of the integral: $\\int_{1}^{2} x^{5} d x $", "answer": "$2^{6} / 6-1 / 6$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.10.5", "question": "Find the derivative of $\\arctan \\left(e^{x}\\right)$.", "answer": "$\\frac{e^{x}}{1+e^{2 x}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.3.1", "question": "Approximate the fifth root of 7 , using $x_{0}=1.5$ as a first guess. Use Newton's method to find $x_{3}$ as your approximation.", "answer": "$x_{3}=1.475773162$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.2.3", "question": "Find the derivative of the function: $5\\left(-3 x^{2}+5 x+1\\right) $", "answer": "$-30 x+25$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.5.10", "question": "Find the antiderivative: $\\int \\frac{1}{x^{2}+3 x} d x $", "answer": "$(1 / 3) \\ln |x|-(1 / 3) \\ln |x+3|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.8.1", "question": "Find the radius and interval of convergence for the series: $\\sum_{n=0}^{\\infty} n x^{n} $", "answer": "$R=1, I=(-1,1)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.5", "question": "Find the antiderivative of the function: $7 s^{-1} $", "answer": "$7 \\ln s+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.4.6", "question": "The observation deck on the 102nd floor of the Empire State Building is 1,224 feet above the ground. If a steel ball is dropped from the observation deck its velocity at time $t$ is approximately $v(t)=-32 t$ feet per second. Find the average speed between the time it is dropped and the time it hits the ground, and find its speed when it hits the ground.", "answer": "$-4 \\sqrt{1224} \\mathrm{ft} / \\mathrm{s} ;-8 \\sqrt{1224} \\mathrm{ft} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.31", "question": "Find the derivative of the function: $\\frac{2 x^{-1}-x^{-2}}{3 x^{-1}-4 x^{-2}} $", "answer": "$-5 /(3 x-4)^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.1.8", "question": "Let $f(x)=x^{2}-2 x+3$. Approximate the area under the curve between $x=1$ and $x=3$ using 4 rectangles.", "answer": "$23 / 4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.1.1", "question": "Compute the limit: $\\lim _{x \\rightarrow \\infty} x^{1 / x} . $", "answer": "1", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.4.4", "question": "Find the average height of $\\sqrt{1-x^{2}}$ over the interval $[-1,1] . $", "answer": "$\\pi / 4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.1.5", "question": "Determine whether the sequence $\\left\\{\\frac{n+47}{\\sqrt{n^{2}+3 n}}\\right\\}_{n=1}^{\\infty}$ converges or diverges. If it converges, compute the limit.", "answer": "1", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.15", "question": "Compute the limit: $\\lim _{x \\rightarrow 1}\\left\\{\\begin{array}{ll}x-5 & x \\neq 1, \\\\ 7 & x=1 .\\end{array} \\right.$. If a limit does not exist, explain why.", "answer": "-4", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.1", "question": "Find the antiderivative of the function: $\\int(1-t)^{9} d t $", "answer": "$-(1-t)^{10 / 10+C}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.8", "question": "A boat is pulled in to a dock by a rope with one end attached to the front of the boat and the other end passing through a ring attached to the dock at a point $5 \\mathrm{ft}$ higher than the front of the boat. The rope is being pulled through the ring at a rate of $0.6 \\mathrm{ft} / \\mathrm{sec}$. Find the rate at which the boat is approaching the dock when $13 \\mathrm{ft}$ of rope are out.", "answer": "$13 / 20 \\mathrm{ft} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.23", "question": "You are designing a poster to contain a fixed amount $A$ of printing (measured in square centimeters) and have margins of $a$ centimeters at the top and bottom and $b$ centimeters at the sides. Find the ratio of vertical dimension to horizontal dimension of the printed area on the poster if you want to minimize the amount of posterboard needed.", "answer": "$a / b$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.3.3", "question": "An object moves so that its velocity at time $t$ is $v(t)=1+2 \\sin t \\mathrm{~m} / \\mathrm{s}$. Find the net distance traveled by the object between $t=0$ and $t=2 \\pi$, and find the total distance traveled during the same period.", "answer": "net: $2 \\pi$, total: $2 \\pi / 3+4 \\sqrt{3}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.4", "question": "Find the area bounded by the curves: $x=3 y-y^{2}$ and $x+y=3 $", "answer": "$4 / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.2.8", "question": "Find an equation for the tangent line to $f(x)=3 x^{2}-\\pi^{3}$ at $x=4$.", "answer": "$y=24 x-48-\\pi^{3}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.15", "question": "Find the derivative of the function: $x^{\\sin (x)} $", "answer": "$x^{\\sin (x)}(\\cos (x) \\ln (x)+\\sin (x) / x)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.3", "question": "Find the antiderivative: $\\int \\sqrt{x^{2}-1} d x $", "answer": "$x \\sqrt{x^{2}-1} / 2-\\ln \\left|x+\\sqrt{x^{2}-1}\\right| / 2+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.9", "question": "Find the antiderivative of the function: $\\frac{2}{x \\sqrt{x}} $", "answer": "$-4 / \\sqrt{x}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.3.6", "question": "Evaluate the three integrals:\n$$\nA=\\int_{0}^{3}-x^{2}+9 d x \\quad B=\\int_{0}^{4}-x^{2}+9 d x \\quad C=\\int_{4}^{3}-x^{2}+9 d x,\n$$\nand verify that $A=B+C$. $$", "answer": "$A=18, B=44 / 3, C=10 / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.10", "question": "Find the area of the largest rectangle that fits inside a semicircle of radius 10 (one side of the rectangle is along the diameter of the semicircle).", "answer": "100", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.4", "question": "Does the improper integral $\\int_{1}^{\\infty} 1 / \\sqrt{x} d x$ converge or diverge? If it converges, find the value.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.35", "question": "Find the derivative of the function: $(2 x+1)^{3}\\left(x^{2}+1\\right)^{2} $", "answer": "$56 x^{6}+72 x^{5}+110 x^{4}+100 x^{3}+$ $60 x^{2}+28 x+6$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.11", "question": "A thin plate lies in the region between the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}=1$, above the $x$-axis. Find the centroid.", "answer": "$\\bar{x}=0, \\bar{y}=28 /(9 \\pi)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.12", "question": "Find the domain of the function: $h(x)=\\left\\{\\begin{array}{ll}\\left(x^{2}-9\\right) /(x-3) & x \\neq 3 \\\\ 6 & \\text { if } x=3 .\\end{array} \\right.$", "answer": "$\\mathbb{R}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.10", "question": "Find the antiderivative of the function: $\\int \\tan x d x $", "answer": "$-\\ln |\\cos x|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.4.1", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{(-1)^{n-1}}{2 n+5} $ converges or diverges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.1", "question": "Find the antiderivative of the function: $8 \\sqrt{x} $", "answer": "$(16 / 3) x^{3 / 2}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.7", "question": "Determine whether the series converges: $\\sum_{n=0}^{\\infty} \\frac{\\sin ^{3}(n)}{n^{2}} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.3", "question": "Find the antiderivative of the function: $4 / \\sqrt{x} $", "answer": "$8 \\sqrt{x}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.4.1", "question": "Find the average height of $\\cos x$ over the intervals $[0, \\pi / 2],[-\\pi / 2, \\pi / 2]$, and $[0,2 \\pi] . $", "answer": "$2 / \\pi ; 2 / \\pi ; 0$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.11", "question": "Describe the concavity of the function: $y=x^{5}-x $", "answer": "concave up on $(0, \\infty)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.2", "question": "For the velocity function $v=-9.8 t+49$, find both the net distance and the total distance traveled during the time interval $0 \\leq t \\leq 10$.", "answer": "0,245", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.12", "question": "Find the derivative of the function: $\\ln (\\cos (x)) $", "answer": "$-\\tan (x)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.34", "question": "Find the derivative of the function: $\\left((2 x+1)^{-1}+3\\right)^{-1} $", "answer": "$1 /\\left(2(2+3 x)^{2}\\right)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.25", "question": "Find the derivative of the function: $\\frac{-3}{4 x^{2}-2 x+1} $", "answer": "$3(8 x-2) /\\left(4 x^{2}-2 x+1\\right)^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.9.15", "question": "Find an equation for the tangent line to $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$ at a point $\\left(x_{1}, y_{1}\\right)$ on the curve, with $x_{1} \\neq 0$ and $y_{1} \\neq 0$. (This curve is an astroid.)", "answer": "$y=\\left(-y_{1}^{1 / 3} x+y_{1}^{1 / 3} x_{1}+x_{1}^{1 / 3} y_{1}\\right) / x_{1}^{1 / 3}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.10", "question": "Find the antiderivative: $\\int \\tan ^{3} x \\sec x d x $", "answer": "$\\left(\\sec ^{3} x\\right) / 3-\\sec x+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.27", "question": "Evaluate the integral: $\\int t^{3} e^{t} d t $", "answer": "$\\left(t^{3}-3 t^{2}+6 t-6\\right) e^{t}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.3", "question": "Find the derivative of the function: $\\sqrt{x \\tan x} $", "answer": "$\\frac{\\tan x+x \\sec ^{2} x}{2 \\sqrt{x \\tan x}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.10.4", "question": "Find the derivative of $\\arcsin \\left(x^{2}\\right) . $", "answer": "$\\frac{2 x}{\\sqrt{1-x^{4}}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.18", "question": "Find all local maximum and minimum points of the function: $y=\\sin ^{3} x $", "answer": "$\\max$ at $\\pi / 2+2 n \\pi$, min at $3 \\pi / 2+2 n \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.7.5", "question": "Determine whether the series $\\sum_{n=0}^{\\infty}(-1)^{n} \\frac{3^{n}}{5^{n}} $ converges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.4.1", "question": "Let $f(x)=x^{4}$. If $a=1$ and $d x=\\Delta x=1 / 2$, what are $\\Delta y$ and $d y$ ?", "answer": "$\\Delta y=65 / 16, d y=2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.2", "question": "Find all critical points of the function $y=2+3 x-x^{3} $. Identify them as local maximum points, local minimum points, or neither.", "answer": "$\\min$ at $x=-1, \\max$ at $x=1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.8", "question": "You have $l$ feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area?", "answer": "$l^{2} / 8$ square feet", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.13", "question": "Determine the domain of the composition $(g \\circ f)(x)$ if $f(x)=3 x-9$ and $g(x)=\\sqrt{x}$. What is the domain of $(f \\circ g)(x) ? $", "answer": "$\\{x \\mid x \\geq 3\\},\\{x \\mid x \\geq 0\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.14", "question": "You want to make cylindrical containers to hold 1 liter using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side $2 r$, so that $2(2 r)^{2}=8 r^{2}$ of material is needed (rather than $2 \\pi r^{2}$, which is the total area of the top and bottom). Find the dimensions of the container using the least amount of material, and also find the ratio of height to radius for this container.", "answer": "$r=5, h=40 / \\pi, h / r=8 / \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.9", "question": "Find the derivative of the function: $\\sin (\\cos (6 x)) $", "answer": "$-6 \\cos (\\cos (6 x)) \\sin (6 x)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.19", "question": "Find the derivative of the function: $G(x)=\\int_{1}^{x} e^{t^{2}} d t $", "answer": "$e^{x^{2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.11", "question": "The sun is rising at a rate of $1 / 4 \\mathrm{deg} / \\mathrm{min}$ and appears to be climbing into the sky perpendicular to the horizon. Find the rate at which the shadow of a 200 meter building is shrinking at the moment when the shadow is 500 meters long.", "answer": "$145 \\pi / 72 \\mathrm{~m} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.1", "question": "Describe the concavity of the function: $y=x^{2}-x $", "answer": "concave up everywhere", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.11", "question": "Does the improper integral $\\int_{-\\infty}^{\\infty} \\sin x d x$ converge or diverge? If it converges, find the value. Also, find the Cauchy Principal Value, if it exists.", "answer": "diverges, 0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.10", "question": "Determine whether the series converges: $\\sum_{n=1}^{\\infty} \\frac{1}{n \\sqrt{n}} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.2.2", "question": "Explain why the series $\\sum_{n=1}^{\\infty} \\frac{5}{2^{1 / n}+14}$ diverges.", "answer": "$\\lim _{n \\rightarrow \\infty} 5 /\\left(2^{1 / n}+14\\right)=1 / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.7", "question": "Find the derivative of the function: $x^{3} \\sin \\left(23 x^{2}\\right) $", "answer": "$3 x^{2} \\sin \\left(23 x^{2}\\right)+46 x^{4} \\cos \\left(23 x^{2}\\right)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.39", "question": "Find an equation for the tangent line to $\\frac{\\left(x^{2}+x+1\\right)}{(1-x)}$ at $(2,-7)$.", "answer": "$y=2 x-11$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.2", "question": "Find the area bounded by the curves: $x=y^{3}$ and $x=y^{2} $", "answer": "$1 / 12$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.14", "question": "Describe the concavity of the function: $y=x^{2}+1 / x $", "answer": "concave up on $(-\\infty,-1)$ and $(0, \\infty)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.1", "question": "Find the derivative of the function: $\\sin x \\cos x $", "answer": "$\\cos ^{2} x-\\sin ^{2} x$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.5.3", "question": "Find the antiderivative: $\\int \\frac{1}{x^{2}+10 x+25} d x $", "answer": "$-1 /(x+5)+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.5", "question": "Find the antiderivative: $\\int \\sin ^{2} x d x $", "answer": "$(x / 2)-\\sin (2 x) / 4+C=$ $(x / 2)-(\\sin x \\cos x) / 2+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.10", "question": "Find the domain of the function: $y=f(x)=\\sqrt{x}+1 /(x-1) $", "answer": "$\\{x \\mid x \\geq 0$ and $x \\neq 1\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.9.3", "question": "Find the arc length of $f(x)=(1 / 3)\\left(x^{2}+2\\right)^{3 / 2}$ on the interval $[0, a]$.", "answer": "$a+a^{3} / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.1.3", "question": "Find the derivative of the function: $\\frac{1}{x^{5}} $", "answer": "$-5 x^{-6}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.20", "question": "Find the antiderivative of the function: $\\int f(x) f^{\\prime}(x) d x $", "answer": "$f(x)^{2} / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.22", "question": "A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top. Suppose that the colored glass transmits only $k$ times as much light per unit area as the clear glass ( $k$ is between 0 and 1). If the distance from top to bottom (across both the rectangle and the semicircle) is a fixed distance $H$, find (in terms of $k$ ) the ratio of vertical side to horizontal side of the rectangle for which the window lets through the most light.", "answer": "If $k \\leq 2 / \\pi$ the ratio is $(2-k \\pi) / 4$; if $k \\geq 2 / \\pi$, the ratio is zero: the window should be semicircular with no rectangular part.", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.5", "question": "Find all critical points of the function $y=3 x^{4}-4 x^{3} $. Identify them as local maximum points, local minimum points, or neither.", "answer": "$\\min$ at $x=1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.9.5", "question": "Find a power series representation for $\\int \\ln (1-x) d x$.", "answer": "$C+\\sum_{n=0}^{\\infty} \\frac{-1}{(n+1)(n+2)} x^{n+2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.1.4", "question": "Determine whether the sequence $\\left\\{\\frac{n^{2}+1}{(n+1)^{2}}\\right\\}_{n=0}^{\\infty}$ converges or diverges. If it converges, compute the limit.", "answer": "1", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.5.4", "question": "Find the antiderivative: $\\int \\frac{x^{2}}{4-x^{2}} d x $", "answer": "$-x-\\ln |x-2|+\\ln |x+2|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.5", "question": "Does the improper integral $\\int_{0}^{\\infty} e^{-x} d x$ converge or diverge? If it converges, find the value.", "answer": "1", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.6", "question": "An object is shot upwards from ground level with an initial velocity of 3 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground.", "answer": "$45 / 98$ meters, $30 / 49$ seconds", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.2", "question": "Compute the limit: $\\lim _{x \\rightarrow 1} \\frac{x^{2}+x-12}{x-3} $. If a limit does not exist, explain why.", "answer": "5", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.4", "question": "Find the antiderivative: $\\int \\sqrt{9+4 x^{2}} d x $", "answer": "$x \\sqrt{9+4 x^{2}} / 2+$ $(9 / 4) \\ln \\left|2 x+\\sqrt{9+4 x^{2}}\\right|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.8", "question": "Find the antiderivative of the function: $\\int \\cos (\\pi t) \\cos (\\sin (\\pi t)) d t $", "answer": "$\\sin (\\sin \\pi t) / \\pi+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.21", "question": "A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top; the colored glass transmits only $1 / 2$ as much light per unit area as the the clear glass. If the distance from top to bottom (across both the rectangle and the semicircle) is 2 meters and the window may be no more than 1.5 meters wide, find the dimensions of the rectangular portion of the window that lets through the most light.", "answer": "1.5 meters wide by 1.25 meters tall", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.9", "question": "Find the antiderivative: $\\int \\sec ^{2} x \\csc ^{2} x d x $", "answer": "$\\tan x-\\cot x+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.18", "question": "Find the points on the curve $y=x+2 \\cos x$ that have a horizontal tangent line.", "answer": "$\\pi / 6+2 n \\pi, 5 \\pi / 6+2 n \\pi$, any integer $n$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.5.9", "question": "Find the antiderivative: $\\int \\frac{1}{2 x^{2}-x-3} d x $", "answer": "$(1 / 5) \\ln |2 x-3|-(1 / 5) \\ln |1+x|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.5", "question": "Find the area bounded by the curves: $y=\\cos (\\pi x / 2)$ and $y=1-x^{2}$ (in the first quadrant)", "answer": "$2 / 3-2 / \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.7", "question": "Compute the limit: $\\lim _{x \\rightarrow 2} 3 $.", "answer": "3", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.26", "question": "Find a series representation for the function: $\\ln (1+x) $", "answer": "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{n+1} x^{n+1}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.2", "question": "Find the dimensions of the rectangle of largest area having fixed perimeter $100 . $", "answer": "$25 \\times 25$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.3.5", "question": "Compute the limit: $\\lim _{x \\rightarrow \\pi / 4} \\frac{\\sin x-\\cos x}{\\cos (2 x)} $", "answer": "$-\\sqrt{2} / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.11.1", "question": "Find a polynomial approximation for $\\cos x$ on $[0, \\pi]$, accurate to $\\pm 10^{-3} $", "answer": "$1-\\frac{x^{2}}{2}+\\frac{x^{4}}{24}-\\frac{x^{6}}{720}+\\cdots+\\frac{x^{12}}{12 !}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.20", "question": "Find a series representation for the function: $\\sum_{n=1}^{\\infty} \\frac{x^{n}}{n 3^{n}} $", "answer": "$(-3,3)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.3", "question": "Does the improper integral $\\int_{0}^{\\infty} x^{2}+2 x-1 d x$ converge or diverge? If it converges, find the value.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.3.5", "question": "Consider the function $f(x)=x^{2}-3 x+2$ on $[0,4]$. Find the total area between the curve and the $x$-axis (measuring all area as positive).", "answer": "$17 / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.4.2", "question": "Find the derivative of the function: $\\sqrt{x} \\sin x $", "answer": "$\\frac{\\sin x}{2 \\sqrt{x}}+\\sqrt{x} \\cos x$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.11", "question": "Evaluate the integral: $\\int \\frac{e^{t}}{\\sqrt{e^{t}+1}} d t $", "answer": "$2 \\sqrt{e^{t}+1}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.7", "question": "A thin plate lies in the region contained by $y=x$ and $y=x^{2}$. Find the centroid.", "answer": "$\\bar{x}=1 / 2, \\bar{y}=2 / 5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.4", "question": "Find the derivative of the function: $\\tan x /(1+\\sin x) $", "answer": "$\\frac{\\sec ^{2} x(1+\\sin x)-\\tan x \\cos x}{(1+\\sin x)^{2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.5.2", "question": "How much work is done in lifting a 100 kilogram weight from an orbit 1000 kilometers above the surface of the earth to an orbit 35,786 kilometers above the surface of the earth?", "answer": "$\\approx 4,457,854,041 \\mathrm{~N}-\\mathrm{m}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.18", "question": "Find a series representation for the function: $\\sum_{n=0}^{\\infty} \\frac{2^{n}}{n !} x^{n} $", "answer": "$(-\\infty, \\infty)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.6.4", "question": "Determine whether the series $\\sum_{n=1}^{\\infty}(-1)^{n-1} \\frac{\\ln n}{n^{3}} $ converges absolutely, converges conditionally, or diverges.", "answer": "converges absolutely", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.3.7", "question": "For all $x, 2 x \\leq g(x) \\leq x^{4}-x^{2}+2$. Find $\\lim _{x \\rightarrow 1} g(x)$.", "answer": "2", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.9", "question": "Find all local maximum and minimum points of the function: $y=4 x+\\sqrt{1-x} $", "answer": "$\\max$ at $x=63 / 64$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.5", "question": "Compute the limit: $\\lim _{x \\rightarrow 1} \\frac{\\sqrt{x+8}-3}{x-1} $. If a limit does not exist, explain why.", "answer": "$1 / 6$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.8", "question": "Find the derivative of the function: $x+2^{x} $", "answer": "$1+2^{x} \\ln (2)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.3", "question": "Determine whether the series converges or diverges: $\\sum_{n=1}^{\\infty} \\frac{\\ln n}{n^{2}} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.3.6", "question": "Find the volume of the solid obtained by revolving the region bounded by $y=x-x^{2}$ and the $x$-axis around the $x$-axis.", "answer": "$\\pi / 30$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.9.1", "question": "Find a series representation for $\\ln 2$.", "answer": "the alternating harmonic series", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.5", "question": "A rotating beacon is located 2 miles out in the water. Let $A$ be the point on the shore that is closest to the beacon. As the beacon rotates at $10 \\mathrm{rev} / \\mathrm{min}$, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. Find the speed at which the point where the beam hits the shore is moving when the beam is lighting up a point 2 miles along the shore from the point $A$ .", "answer": "$80 \\pi \\mathrm{mi} / \\mathrm{min}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.10", "question": "Evaluate the integral: $\\int t \\sec ^{2} t d t $", "answer": "$t \\tan t+\\ln |\\cos t|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.11", "question": "Compute the limit: $\\lim _{x \\rightarrow 0^{+}} \\frac{\\sqrt{2-x^{2}}}{x} $. If a limit does not exist, explain why.", "answer": "does not exist", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.13", "question": "Find all critical points of the function $f(x)=\\sin ^{2} x $. Identify them as local maximum points, local minimum points, or neither.", "answer": "$\\min$ at $n \\pi, \\max$ at $\\pi / 2+n \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.7", "question": "Describe the concavity of the function: $y=3 x^{2}-\\left(1 / x^{2}\\right) $", "answer": "concave up when $x<-1$ or $x>1$, concave down when $-1<x<0$ or $0<x<1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.19", "question": "Find the derivative of the function: $\\left(x^{2}+5\\right)^{3} $", "answer": "$6 x\\left(x^{2}+5\\right)^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.9", "question": "Find the antiderivative: $\\int x \\sin ^{2} x d x $", "answer": "$x^{2} / 4-\\left(\\cos ^{2} x\\right) / 4-(x \\sin x \\cos x) / 2+$ C", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.2", "question": "Determine whether the series converges: $\\frac{1}{1 \\cdot 2}+\\frac{1}{3 \\cdot 4}+\\frac{1}{5 \\cdot 6}+\\frac{1}{7 \\cdot 8}+\\cdots $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.16", "question": "Describe the concavity of the function: $y=\\tan ^{2} x $", "answer": "concave up everywhere", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.7", "question": "Does the improper integral $\\int_{0}^{1} 1 / \\sqrt{x} d x$ converge or diverge? If it converges, find the value.", "answer": "2", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.15", "question": "Evaluate the integral: $\\int \\frac{\\sec ^{2} t}{(1+\\tan t)^{3}} d t $", "answer": "$\\frac{-1}{2(1+\\tan t)^{2}}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.6", "question": "A thin plate fills the upper half of the unit circle $x^{2}+y^{2}=1$. Find the centroid.", "answer": "$\\bar{x}=0, \\bar{y}=4 /(3 \\pi)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.16", "question": "Evaluate the definite integral: $\\int_{0}^{\\pi / 6}\\left(\\cos ^{2} x-\\sin ^{2} x\\right) d x $", "answer": "$\\sqrt{3} / 4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.3", "question": "Find the antiderivative: $\\int x e^{x} d x $", "answer": "$(x-1) e^{x}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.8", "question": "Find the antiderivative: $\\int \\sqrt{x^{2}+2 x} d x $", "answer": "$(x+1) \\sqrt{x^{2}+2 x} / 2-$ $\\ln \\left|x+1+\\sqrt{x^{2}+2 x}\\right| / 2+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.8", "question": "Find the domain of the function: $y=f(x)=\\sqrt{1-(1 / x)} $", "answer": "$\\{x \\mid x \\geq 1\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.21", "question": "Find the derivative of the function: $\\left(1-4 x^{3}\\right)^{-2} $", "answer": "$24 x^{2}\\left(1-4 x^{3}\\right)^{-3}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.2", "question": "Find the antiderivative of the function: $\\int\\left(x^{2}+1\\right)^{2} d x $", "answer": "$x^{5} / 5+2 x^{3} / 3+x+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.1.6", "question": "Find an algebraic expression for the difference quotient $(f(x+\\Delta x)-f(x)) / \\Delta x$ when $f(x)=$ $m x+b$. Simplify the expression as much as possible. Then determine what happens as $\\Delta x$ approaches 0 . That value is $f^{\\prime}(x)$.", "answer": "$m$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.4", "question": "Find the antiderivative: $\\int \\cos ^{2} x \\sin ^{3} x d x $", "answer": "$\\left(\\cos ^{5} x\\right) / 5-\\left(\\cos ^{3} x\\right) / 3+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.2.4", "question": "Compute the value of the series $\\sum_{n=0}^{\\infty} \\frac{4}{(-3)^{n}}-\\frac{3}{3^{n}}$.", "answer": "$-3 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.11", "question": "Find the derivative of the function: $\\ln \\left(x^{3}+3 x\\right) $", "answer": "$\\left(3 x^{2}+3\\right) /\\left(x^{3}+3 x\\right)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.14", "question": "Find the maxima and minima of the function $f(x)=\\sec x $.", "answer": "$\\min$ at $2 n \\pi, \\max$ at $(2 n+1) \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.11", "question": "Compute $\\frac{d}{d t} t^{5} \\cos (6 t)$.", "answer": "$5 t^{4} \\cos (6 t)-6 t^{5} \\sin (6 t)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.5.7", "question": "Describe all functions with derivative $\\frac{1}{1+x^{2}}$.", "answer": "$\\arctan x+k$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.18", "question": "Evaluate the definite integral: $\\int_{-1}^{1}\\left(2 x^{3}-1\\right)\\left(x^{4}-2 x\\right)^{6} d x $", "answer": "$-\\left(3^{7}+1\\right) / 14$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.6", "question": "Find the derivative of the function: $\\sqrt{r^{2}-x^{2}}, r$ is a constant", "answer": "$-x / \\sqrt{r^{2}-x^{2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.7.8", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{(n !)^{2}}{n^{n}} $ converges.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.14", "question": "Evaluate the integral: $\\int \\frac{1}{t^{2} \\sqrt{1+t^{2}}} d t $", "answer": "$\\frac{-1}{\\sin \\arctan t}+C=-\\sqrt{1+t^{2}} / t+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.6", "question": "Find the antiderivative: $\\int x^{2} \\sqrt{1-x^{2}} d x $", "answer": "$\\arcsin (x) / 8-\\sin (4 \\arcsin x) / 32+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.3", "question": "Find the antiderivative of the function: $\\int x\\left(x^{2}+1\\right)^{100} d x $", "answer": "$\\left(x^{2}+1\\right)^{101} / 202+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.16", "question": "Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the center of the base of the larger cone and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let $H$ and $R$ be the height and base radius of the larger cone, and let $h$ and $r$ be the height and base radius of the smaller cone. Hint: Use similar triangles to get an equation relating $h$ and $r$.)", "answer": "$4 / 27$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.3.4", "question": "Consider the function $f(x)=(x+2)(x+1)(x-1)(x-2)$ on $[-2,2]$. Find the total area between the curve and the $x$-axis (measuring all area as positive).", "answer": "8", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.5", "question": "Find the antiderivative: $\\int \\cos ^{3} x d x $", "answer": "$\\sin x-\\left(\\sin ^{3} x\\right) / 3+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.5.7", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{\\ln n}{n^{3}} $ converges or diverges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.9.2", "question": "Find the arc length of $f(x)=x^{2} / 8-\\ln x$ on the interval $[1,2]$.", "answer": "$\\ln (2)+3 / 8$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.9", "question": "Find the derivative of the function: $(1 / 3)^{x^{2}} $", "answer": "$-2 x \\ln (3)(1 / 3)^{x^{2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.2", "question": "Determine whether the series converges or diverges: $\\sum_{n=1}^{\\infty} \\frac{n}{n^{2}+1} $", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.5", "question": "Determine whether the series converges or diverges: $\\sum_{n=1}^{\\infty} \\frac{1}{e^{n}} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.5", "question": "Find all local maximum and minimum points of the function: $y=3 x^{4}-4 x^{3} $", "answer": "$\\min$ at $x=1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.3", "question": "Find the area bounded by the curves: $x=1-y^{2}$ and $y=-x-1 $", "answer": "$9 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.8.3", "question": "Compute the limit of $\\lim _{x \\rightarrow \\infty} \\sqrt{x^{2}+x}-\\sqrt{x^{2}-x} $.", "answer": "1", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.11", "question": "Determine whether the series converges: $\\frac{1}{2 \\cdot 3 \\cdot 4}+\\frac{2}{3 \\cdot 4 \\cdot 5}+\\frac{3}{4 \\cdot 5 \\cdot 6}+\\frac{4}{5 \\cdot 6 \\cdot 7}+\\cdots $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.12", "question": "Evaluate the integral: $\\int \\cos ^{4} t d t $", "answer": "$\\frac{3 t}{8}+\\frac{\\sin 2 t}{4}+\\frac{\\sin 4 t}{32}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.9", "question": "Find the antiderivative: $\\int \\frac{1}{x^{2}\\left(1+x^{2}\\right)} d x $", "answer": "$-\\arctan x-1 / x+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.12", "question": "Compute the limit: $\\lim _{x \\rightarrow 0^{+}} \\frac{\\sqrt{2-x^{2}}}{x+1} $. If a limit does not exist, explain why.", "answer": "$\\sqrt{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.10.9", "question": "Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for $x \\cos \\left(x^{2}\\right)$.", "answer": "$\\sum_{n=0}^{\\infty}(-1)^{n} x^{4 n+1} /(2 n)$ !", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.4", "question": "Compute the limit: $\\lim _{x \\rightarrow 2} \\frac{x^{2}+x-12}{x-2} $. If a limit does not exist, explain why.", "answer": "undefined", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.4", "question": "Find the derivative of the function: $x \\sqrt{169-x^{2}} $", "answer": "$\\sqrt{169-x^{2}}-x^{2} / \\sqrt{169-x^{2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.5", "question": "A thin plate lies in the region between $y=x^{2}$ and the $x$-axis between $x=1$ and $x=2$. Find the centroid.", "answer": "$\\bar{x}=45 / 28, \\bar{y}=93 / 70$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.14", "question": "Compute the value of the integral: $\\int_{0}^{5} e^{x} d x $", "answer": "$e^{5}-1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.7", "question": "Find the domain of the function: $y=f(x)=\\sqrt{r^{2}-(x-h)^{2}}$, where $r$ and $h$ are positive constants.", "answer": "$\\{x \\mid h-r \\leq x \\leq h+r\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.9.1", "question": "Find the arc length of $f(x)=x^{3 / 2}$ on the interval $[0,2]$.", "answer": "$(22 \\sqrt{22}-8) / 27$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.8", "question": "Find the derivative of the function: $\\frac{1}{\\sqrt{5-\\sqrt{x}}} \\cdot $", "answer": "$\\frac{1}{4 \\sqrt{x}(5-\\sqrt{x})^{3 / 2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.12", "question": "For a cylinder with surface area 50, including the top and the bottom, find the ratio of height to base radius that maximizes the volume.", "answer": "$h / r=2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.1.9", "question": "Determine whether the lines $3 x+6 y=7$ and $2 x+4 y=5$ are parallel.", "answer": "yes", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.13", "question": "Find all local maximum and minimum points of the function: $y=x+1 / x $", "answer": "$\\max$ at $-1, \\min$ at 1", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.17", "question": "Find all local maximum and minimum points of the function: $y=\\cos ^{2} x-\\sin ^{2} x $", "answer": "$\\max$ at $n \\pi$, min at $\\pi / 2+n \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.5.6", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{\\ln n}{n} $ converges or diverges.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.38", "question": "Find an equation for the tangent line to $\\left(x^{2}-4 x+5\\right) \\sqrt{25-x^{2}}$ at $(3,8)$.", "answer": "$y=13 x / 2-23 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.5", "question": "Find the antiderivative of the function: $\\int \\sin ^{3} x \\cos x d x $", "answer": "$\\left(\\sin ^{4} x\\right) / 4+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.7.7", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{n^{5}}{n^{n}} $ converges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.25", "question": "Evaluate the integral: $\\int \\frac{1}{t(\\ln t)^{2}} d t $", "answer": "$\\frac{-1}{\\ln t}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.17", "question": "Find the derivative of the function: $G(x)=\\int_{1}^{x} t^{2}-3 t d t $", "answer": "$x^{2}-3 x$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.20", "question": "Evaluate the integral: $\\int \\frac{1}{t\\left(9+4 t^{2}\\right)} d t $", "answer": "$\\frac{\\ln |\\sin (\\arctan (2 t / 3))|}{9}+C=$ $\\left(\\ln \\left(4 t^{2}\\right)-\\ln \\left(9+4 t^{2}\\right)\\right) / 18+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.4.6", "question": "Find an equation for the tangent line to $f(x)=(x-2) /\\left(x^{3}+4 x-1\\right)$ at $x=1$.", "answer": "$y=11 x / 16-15 / 16$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.3", "question": "A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The foot of the ladder is pulled away from the wall at a rate of $0.6 \\mathrm{~m} / \\mathrm{sec}$. Find the rate at which the top of the ladder is sliding down the wall when the foot of the ladder is $5 \\mathrm{~m}$ from the wall.", "answer": "$1 / 4 \\mathrm{~m} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.25", "question": "Find a series representation for the function: $2^{x} $", "answer": "$\\sum_{n=0}^{\\infty} \\frac{(\\ln (2))^{n}}{n !} x^{n}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.10", "question": "Find all critical points of the function $f(x)=\\left|x^{2}-121\\right| $. Identify them as local maximum points, local minimum points, or neither.", "answer": "$\\max$ at $x=0$, min at $x= \\pm 11$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.6.1", "question": "Determine whether the series $\\sum_{n=1}^{\\infty}(-1)^{n-1} \\frac{1}{2 n^{2}+3 n+5} $ converges absolutely, converges conditionally, or diverges.", "answer": "converges absolutely", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.8", "question": "Find all critical points of the function $y=\\cos (2 x)-x $. Identify them as local maximum points, local minimum points, or neither.", "answer": "$\\min$ at $x=7 \\pi / 12+k \\pi, \\max$ at $x=-\\pi / 12+k \\pi$, for integer $k$.", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.2", "question": "Find the antiderivative: $\\int \\sin ^{3} x d x $", "answer": "$-\\cos x+\\left(\\cos ^{3} x\\right) / 3+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.1.13", "question": "A car rental firm has the following charges for a certain type of car: $\\$ 25$ per day with 100 free miles included, $\\$ 0.15$ per mile for more than 100 miles. Suppose you want to rent a car for one day, and you know you'll use it for more than 100 miles. What is the equation relating the cost $y$ to the number of miles $x$ that you drive the car?", "answer": "$y=0.15 x+10$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.9", "question": "Evaluate the integral: $\\int \\frac{\\cos 3 t}{\\sqrt{\\sin 3 t}} d t $", "answer": "$\\frac{2}{3} \\sqrt{\\sin 3 t}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.4.6", "question": "Approximate the value of the series $\\sum_{n=1}^{\\infty}(-1)^{n-1} \\frac{1}{n^{4}}$ to two decimal places.", "answer": "0.95", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.5", "question": "Find the derivative of the function: $\\cot x $", "answer": "$-\\csc ^{2} x$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.1", "question": "For the velocity function $v=\\cos (\\pi t)$, find both the net distance and the total distance traveled during the time interval $0 \\leq t \\leq 2.5$.", "answer": "$1 / \\pi, 5 / \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.29", "question": "Find a series representation for the function: $\\frac{1}{1+x^{2}} $", "answer": "$\\sum_{n=0}^{\\infty}(-1)^{n} x^{2 n}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.5.7", "question": "A force of 2 Newtons will compress a spring from 1 meter (its natural length) to 0.8 meters. How much work is required to stretch the spring from 1.1 meters to 1.5 meters?", "answer": "$6 / 5 \\mathrm{~N}-\\mathrm{m}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.14", "question": "A woman $5 \\mathrm{~ft}$ tall walks at a speed of $3.5 \\mathrm{~ft} / \\mathrm{sec}$ away from a streetlight that is $12 \\mathrm{~ft}$ above the ground. Find the rate at which the tip of her shadow is moving. Find the rate at which her shadow is lengthening.", "answer": "tip: $6 \\mathrm{ft} / \\mathrm{s}$, length: $5 / 2 \\mathrm{ft} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.17", "question": "Evaluate the integral: $\\int e^{t} \\sin t d t $", "answer": "$\\frac{e^{t} \\sin t-e^{t} \\cos t}{2}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.7", "question": "An object is shot upwards from ground level with an initial velocity of 100 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground.", "answer": "25000/49 meters, 1000/49 seconds", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.16", "question": "Evaluate the integral: $\\int t^{3} \\sqrt{t^{2}+1} d t $", "answer": "$\\frac{\\left(t^{2}+1\\right)^{5 / 2}}{5}-\\frac{\\left(t^{2}+1\\right)^{3 / 2}}{3}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.30", "question": "Find a series representation for the function: $\\arctan (x) $", "answer": "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{2 n+1} x^{2 n+1}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.13", "question": "Find the antiderivative of the function: $\\int_{0}^{\\sqrt{\\pi} / 2} x \\sec ^{2}\\left(x^{2}\\right) \\tan \\left(x^{2}\\right) d x $", "answer": "$1 / 4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.3", "question": "Find the domain of the function: $y=f(x)=1 /\\left(x^{2}-1\\right) $", "answer": "$\\{x \\mid x \\neq 1$ and $x \\neq-1\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.3", "question": "Describe the concavity of the function: $y=x^{3}-9 x^{2}+24 x $", "answer": "concave down when $x<3$, concave up when $x>3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.14", "question": "Compute the limit: $\\lim _{x \\rightarrow 2}\\left(x^{2}+4\\right)^{3} $.", "answer": "512", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.4", "question": "Find all local maximum and minimum points of the function: $y=x^{4}-2 x^{2}+3 $", "answer": "$\\min$ at $x= \\pm 1, \\max$ at $x=0$.", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.6", "question": "Find the derivative of the function: $x^{\\sin x} $", "answer": "$x^{\\sin x}\\left(\\cos x \\ln x+\\frac{\\sin x}{x}\\right)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.12", "question": "Find the antiderivative of the function: $\\int \\sec ^{2} x \\tan x d x $", "answer": "$\\tan ^{2}(x) / 2+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.1.2", "question": "Find all values of $\\theta$ such that $\\cos (2 \\theta)=1 / 2$; give your answer in radians.", "answer": "$n \\pi \\pm \\pi / 6$, any integer $n$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.1.4", "question": "Use an angle sum identity to compute $\\tan (5 \\pi / 12)$.", "answer": "$-(1+\\sqrt{3}) /(1-\\sqrt{3})$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.7.8", "question": "Does the improper integral $\\int_{0}^{\\pi / 2} \\sec ^{2} x d x$ converge or diverge? If it converges, find the value.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.11", "question": "Find the area of the largest rectangle that fits inside a semicircle of radius $r$ (one side of the rectangle is along the diameter of the semicircle).", "answer": "$r^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.5.8", "question": "A 20 meter long steel cable has density 2 kilograms per meter, and is hanging straight down. How much work is required to lift the entire cable to the height of its top end?", "answer": "$3920 \\mathrm{~N}-\\mathrm{m}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.6", "question": "Find the antiderivative of the function: $(5 x+1)^{2} $", "answer": "$(5 x+1)^{3} / 15+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.2", "question": "A beam 10 meters long has density $\\sigma(x)=\\sin (\\pi x / 10)$ at distance $x$ from the left end of the beam. Find the center of mass $\\bar{x}$.", "answer": "5", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.5", "question": "An object is shot upwards from ground level with an initial velocity of 2 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground.", "answer": "$10 / 49$ meters, $20 / 49$ seconds", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.4.4", "question": "Determine whether the series $\\sum_{n=1}^{\\infty}(-1)^{n-1} \\frac{\\ln n}{n} $ converges or diverges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.27", "question": "Find a series representation for the function: $\\ln \\left(\\frac{1+x}{1-x}\\right) $", "answer": "$\\sum_{n=0}^{\\infty} \\frac{2}{2 n+1} x^{2 n+1}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.10.3", "question": "Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $1 / x, a=5 $", "answer": "$\\sum_{n=0}^{\\infty}(-1)^{n} \\frac{(x-5)^{n}}{5^{n+1}}, R=5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.16", "question": "Determine whether the series converges: $1+\\frac{5^{2}}{2^{2}}+\\frac{5^{4}}{(2 \\cdot 4)^{2}}+\\frac{5^{6}}{(2 \\cdot 4 \\cdot 6)^{2}}+\\frac{5^{8}}{(2 \\cdot 4 \\cdot 6 \\cdot 8)^{2}}+\\cdots $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.9.9", "question": "A hyperbola passing through $(8,6)$ consists of all points whose distance from the origin is a constant more than its distance from the point $(5,2)$. Find the slope of the tangent line to the hyperbola at $(8,6) . $", "answer": "1", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.4", "question": "Find all critical points of the function $y=x^{4}-2 x^{2}+3 $. Identify them as local maximum points, local minimum points, or neither.", "answer": "$\\min$ at $x= \\pm 1, \\max$ at $x=0$.", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.8", "question": "Determine whether the series converges or diverges: $\\sum_{n=2}^{\\infty} \\frac{1}{n(\\ln n)^{2}} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.12", "question": "Find the area bounded by the curves: $y=x^{2}-2 x$ and $y=x-2 $", "answer": "$1 / 6$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.7", "question": "Find the area bounded by the curves: $y=\\sqrt{x}$ and $y=x^{2} $", "answer": "$1 / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.3.3", "question": "Compute the limit: $\\lim _{x \\rightarrow 0} \\frac{\\cot (4 x)}{\\csc (3 x)} $", "answer": "$3 / 4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.3", "question": "Find the dimensions of the rectangle of largest area having fixed perimeter $P$.", "answer": "$P / 4 \\times P / 4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.5.8", "question": "Find the derivative of the function: $\\sin ^{2} x+\\cos ^{2} x $", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.4", "question": "A box with square base and no top is to hold a volume 100. Find the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base.", "answer": "$w=l=2 \\cdot 5^{2 / 3}, h=5^{2 / 3}, h / w=$ $1 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.7", "question": "Find the antiderivative of the function: $\\int \\frac{x^{2}}{\\sqrt{1-x^{3}}} d x $", "answer": "$-2 \\sqrt{1-x^{3}} / 3+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.4.4", "question": "Find the derivative of the function: $y=f(x)=a x^{2}+b x+c$ (where $a, b$, and $c$ are constants).", "answer": "$2 a x+b$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.8", "question": "Find all local maximum and minimum points of the function: $y=\\cos (2 x)-x $", "answer": "$\\min$ at $x=7 \\pi / 12+n \\pi$, $\\max$ at $x=-\\pi / 12+n \\pi$, for integer $n$.", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.5", "question": "Describe the concavity of the function: $y=3 x^{4}-4 x^{3} $", "answer": "concave up when $x<0$ or $x>2 / 3$, concave down when $0<x<2 / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.6", "question": "Find the antiderivative of the function: $\\int x \\sqrt{100-x^{2}} d x $", "answer": "$-\\left(100-x^{2}\\right)^{3 / 2} / 3+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.6", "question": "Compute the limit: $\\lim _{x \\rightarrow 0} \\sqrt{\\frac{1}{x}+2}-\\sqrt{\\frac{1}{x}} $.", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.9", "question": "Find an $N$ such that $\\sum_{n=1}^{\\infty} \\frac{1}{n^{4}}=\\sum_{n=1}^{N} \\frac{1}{n^{4}} \\pm 0.005$.", "answer": "$N=5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.3.5", "question": "Use integration to find the volume of the solid obtained by revolving the region bounded by $x+y=2$ and the $x$ and $y$ axes around the $x$-axis.", "answer": "$8 \\pi / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.11", "question": "Find an $N$ such that $\\sum_{n=1}^{\\infty} \\frac{\\ln n}{n^{2}}=\\sum_{n=1}^{N} \\frac{\\ln n}{n^{2}} \\pm 0.005$.", "answer": "$N=1687$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.3", "question": "Find the derivative of the function: $\\left(e^{x}\\right)^{2} $", "answer": "$2 e^{2 x}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.15", "question": "A man 1.8 meters tall walks at a speed of 1 meter per second toward a streetlight that is 4 meters above the ground. Find the rate at which the tip of his shadow is moving. Find the rate at which his shadow is shortening.", "answer": "tip: $20 / 11 \\mathrm{~m} / \\mathrm{s}$, length: $9 / 11 \\mathrm{~m} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.3.1", "question": "Compute the limit: $\\lim _{x \\rightarrow 0} \\frac{\\sin (5 x)}{x} $", "answer": "5", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.2.1", "question": "Find all critical points of the function $y=x^{2}-x $. Identify them as local maximum points, local minimum points, or neither.", "answer": "$\\min$ at $x=1 / 2$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.3.3", "question": "The function $f(x)=x^{3}-3 x^{2}-3 x+6$ has a root between 3 and 4 , because $f(3)=-3$ and $f(4)=10$. Approximate the root to two decimal places.", "answer": "3.36", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.10", "question": "Find the antiderivative: $\\int \\frac{x^{2}}{\\sqrt{4-x^{2}}} d x $", "answer": "$2 \\arcsin (x / 2)-x \\sqrt{4-x^{2}} / 2+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.8", "question": "Find the antiderivative: $\\int x^{2} \\sin x d x $", "answer": "$-x^{2} \\cos x+2 x \\sin x+2 \\cos x+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.3.2", "question": "An object moves so that its velocity at time $t$ is $v(t)=\\sin t$. Set up and evaluate a single definite integral to compute the net distance traveled between $t=0$ and $t=2 \\pi$.", "answer": "$\\int_{0}^{2 \\pi} \\sin t d t=0$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.13", "question": "Describe the concavity of the function: $y=x+1 / x $", "answer": "concave up on $(0, \\infty)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.8.6", "question": "Find the radius and interval of convergence for the series: $\\sum_{n=1}^{\\infty} \\frac{(x+5)^{n}}{n(n+1)} $", "answer": "$R=1, I=(-6,-4)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.1", "question": "Determine whether the series converges: $\\sum_{n=0}^{\\infty} \\frac{n}{n^{2}+4} $", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.1.4", "question": "Find the derivative of the function: $x^{\\pi} $", "answer": "$\\pi x^{\\pi-1}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.3.4", "question": "Compute the limit: $\\lim _{x \\rightarrow 0} \\frac{\\tan x}{x} $", "answer": "1", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.7", "question": "Find the derivative of the function: $\\sqrt{1+x^{4}} $", "answer": "$2 x^{3} / \\sqrt{1+x^{4}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.3", "question": "A beam 4 meters long has density $\\sigma(x)=x^{3}$ at distance $x$ from the left end of the beam. Find the center of mass $\\bar{x}$.", "answer": "$16 / 5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.1.3.14", "question": "A farmer wants to build a fence along a river. He has 500 feet of fencing and wants to enclose a rectangular pen on three sides (with the river providing the fourth side). If $x$ is the length of the side perpendicular to the river, determine the area of the pen as a function of $x$. What is the domain of this function?", "answer": "$A=x(500-2 x),\\{x \\mid 0 \\leq x \\leq 250\\}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.9", "question": "Compute the limit: $\\lim _{x \\rightarrow 0} \\frac{4 x-5 x^{2}}{x-1} $. If a limit does not exist, explain why.", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.10.8", "question": "Find the first four terms of the Maclaurin series for $\\tan x$ (up to and including the $x^{3}$ term).", "answer": "$x+x^{3} / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.8", "question": "Find the antiderivative: $\\int \\sin x(\\cos x)^{3 / 2} d x $", "answer": "$-2(\\cos x)^{5 / 2} / 5+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.6", "question": "Find the antiderivative: $\\int \\sin ^{2} x \\cos ^{2} x d x $", "answer": "$x / 8-(\\sin 4 x) / 32+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.16", "question": "Find the derivative of the function: $\\sqrt{\\left(x^{2}+1\\right)^{2}+\\sqrt{1+\\left(x^{2}+1\\right)^{2}}} $", "answer": "$\\left(4 x\\left(x^{2}+1\\right)+\\frac{4 x^{3}+4 x}{2 \\sqrt{1+\\left(x^{2}+1\\right)^{2}}}\\right) /$ $2 \\sqrt{\\left(x^{2}+1\\right)^{2}+\\sqrt{1+\\left(x^{2}+1\\right)^{2}}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.7", "question": "Find the antiderivative of the function: $(x-6)^{2} $", "answer": "$(x-6)^{3} / 3+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.9.4", "question": "Find a power series representation for $1 /(1-x)^{3}$. What is the radius of convergence?", "answer": "$\\sum_{n=0}^{\\infty} \\frac{(n+1)(n+2)}{2} x^{n}, R=1$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.4", "question": "Describe the concavity of the function: $y=x^{4}-2 x^{2}+3 $", "answer": "concave up when $x<-1 / \\sqrt{3}$ or $x>1 / \\sqrt{3}$, concave down when $-1 / \\sqrt{3}<x<1 / \\sqrt{3}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.8", "question": "Find the antiderivative of the function: $x^{3 / 2} $", "answer": "$2 x^{5 / 2} / 5+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.12", "question": "The sun is setting at a rate of $1 / 4 \\mathrm{deg} / \\mathrm{min}$ and appears to be climbing into the sky perpendicular to the horizon. Find the rate at which the shadow of a 25 meter wall is lengthening at the moment when the shadow is 50 meters.", "answer": "$25 \\pi / 144 \\mathrm{~m} / \\mathrm{min}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.1.2", "question": "Find the derivative of the function: $x^{-100} $", "answer": "$-100 x^{-101}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.9", "question": "Find the area bounded by the curves: $x=0$ and $x=25-y^{2} $", "answer": "$500 / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.4", "question": "Find the derivative of the function: $\\sin \\left(e^{x}\\right) $", "answer": "$e^{x} \\cos \\left(e^{x}\\right)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.6", "question": "Evaluate the integral: $\\int \\frac{2 t+1}{t^{2}+t+3} d t $", "answer": "$\\ln \\left|t^{2}+t+3\\right|+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.13", "question": "Compute the limit: $\\lim _{x \\rightarrow a} \\frac{x^{3}-a^{3}}{x-a} $.", "answer": "$3 a^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.9", "question": "An object moves along a straight line with acceleration given by $a(t)=\\sin (\\pi t)$. Assume that when $t=0, s(t)=v(t)=0$. Find $s(t), v(t)$, and the maximum speed of the object. Describe the motion of the object.", "answer": "$s(t)=-\\sin (\\pi t) / \\pi^{2}+t / \\pi$, $v(t)=-\\cos (\\pi t) / \\pi+1 / \\pi$, maximum speed is $2 / \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.14", "question": "Find all local maximum and minimum points of the function: $y=x^{2}+1 / x $", "answer": "$\\min$ at $2^{-1 / 3}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.5.4", "question": "Determine whether the series $\\sum_{n=1}^{\\infty} \\frac{3 n+4}{2 n^{2}+3 n+5} $ converges or diverges.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.21", "question": "Evaluate the integral: $\\int \\frac{\\arctan 2 t}{1+4 t^{2}} d t $", "answer": "$\\frac{(\\arctan (2 t))^{2}}{4}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.19", "question": "A piece of carboard is 1 meter by $1 / 2$ meter. A square is to be cut from each corner and the sides folded up to make an open-top box. What are the dimensions of the box with maximum possible volume?", "answer": "$\\frac{\\sqrt{3}}{6} \\times \\frac{\\sqrt{3}}{6}+\\frac{1}{2} \\times \\frac{1}{4}-\\frac{\\sqrt{3}}{12}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.4", "question": "Determine whether the series converges: $\\sum_{n=0}^{\\infty} \\frac{n !}{8^{n}} $", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.17", "question": "A police helicopter is flying at a speed of 200 kilometers per hour at a constant altitude of $1 \\mathrm{~km}$ above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 2 kilometers from the helicopter, and that this distance is decreasing at a rate of $250 \\mathrm{~kph}$. Find the speed of the car.", "answer": "$81 \\mathrm{~km} / \\mathrm{hr}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.2", "question": "Find the antiderivative of the function: $3 t^{2}+1 $", "answer": "$t^{3}+t+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.12", "question": "Compute the value of the integral: $\\int_{0}^{\\pi} \\sin t d t $", "answer": "2", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.1.1", "question": "Find the derivative of the function: $x^{100} $", "answer": "$100 x^{99}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.1.1", "question": "Suppose an object moves in a straight line so that its speed at time $t$ is given by $v(t)=2 t+2$, and that at $t=1$ the object is at position 5 . Find the position of the object at $t=2 . $", "answer": "10", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.8.3", "question": "Find the radius and interval of convergence for the series: $\\sum_{n=1}^{\\infty} \\frac{n !}{n^{n}} x^{n} $", "answer": "$R=e, I=(-e, e)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.28", "question": "Find the derivative of the function: $\\frac{x+1}{x-1} $", "answer": "$-2 /(x-1)^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.36", "question": "Find an equation for the tangent line to $f(x)=(x-2)^{1 / 3} /\\left(x^{3}+4 x-1\\right)^{2}$ at $x=1$.", "answer": "$y=23 x / 96-29 / 96$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.5.2", "question": "Determine whether the series $\\sum_{n=2}^{\\infty} \\frac{1}{2 n^{2}+3 n-5} $ converges or diverges.", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.33", "question": "Find the derivative of the function: $\\frac{1}{(2 x+1)(x-3)} $", "answer": "$(5-4 x) /\\left((2 x+1)^{2}(x-3)^{2}\\right)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.5.1", "question": "Find the antiderivative: $\\int \\frac{1}{4-x^{2}} d x $", "answer": "$-\\ln |x-2| / 4+\\ln |x+2| / 4+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.9", "question": "A balloon is at a height of 50 meters and is rising at a constant rate of $5 \\mathrm{~m} / \\mathrm{sec}$. A bicyclist passes beneath it, traveling in a straight line at a constant speed of $10 \\mathrm{~m} / \\mathrm{sec}$. Find the rate at which the distance between the bicyclist and the balloon is increasing 2 seconds later.", "answer": "$5 \\sqrt{10} / 2 \\mathrm{~m} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.2", "question": "Find the derivative of the function: $\\frac{\\sin x}{e^{x}} $", "answer": "$\\frac{\\cos x-\\sin x}{e^{x}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.5.6", "question": "A spring has constant $k=10 \\mathrm{~kg} / \\mathrm{s}^{2}$. How much work is done in compressing it $1 / 10$ meter from its natural length?", "answer": "$0.05 \\mathrm{~N}-\\mathrm{m}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.4.3", "question": "Find the average height of $1 / x^{2}$ over the interval $[1, A]$.", "answer": "$1 / A$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.21", "question": "Find a series representation for the function: $x+\\frac{1}{2} \\frac{x^{3}}{3}+\\frac{1 \\cdot 3}{2 \\cdot 4} \\frac{x^{5}}{5}+\\frac{1 \\cdot 3 \\cdot 5}{2 \\cdot 4 \\cdot 6} \\frac{x^{7}}{7}+\\cdots $", "answer": "$(-1,1)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.4", "question": "Evaluate the integral: $\\int \\sin t \\cos 2 t d t $", "answer": "$\\cos t-\\frac{2}{3} \\cos ^{3} t+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.10", "question": "Compute the limit: $\\lim _{x \\rightarrow 1} \\frac{x^{2}-1}{x-1} $. If a limit does not exist, explain why.", "answer": "2", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.1", "question": "Evaluate the integral: $\\int(t+4)^{3} d t $", "answer": "$\\frac{(t+4)^{4}}{4}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.2.3", "question": "For the velocity function $v=3(t-3)(t-1)$, find both the net distance and the total distance traveled during the time interval $0 \\leq t \\leq 5$.", "answer": "20,28", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.2.7", "question": "Find an equation for the tangent line to $f(x)=x^{3} / 4-1 / x$ at $x=-2$.", "answer": "$y=13 x / 4+5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.2.7", "question": "Find the antiderivative: $\\int \\cos ^{3} x \\sin ^{2} x d x $", "answer": "$\\left(\\sin ^{3} x\\right) / 3-\\left(\\sin ^{5} x\\right) / 5+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.3.1", "question": "Compute the limit: $\\lim _{x \\rightarrow 3} \\frac{x^{2}+x-12}{x-3} $. If a limit does not exist, explain why.", "answer": "7", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.4.4", "question": "Use differentials to estimate the amount of paint needed to apply a coat of paint $0.02 \\mathrm{~cm}$ thick to a sphere with diameter 40 meters. (Recall that the volume of a sphere of radius $r$ is $V=(4 / 3) \\pi r^{3}$. Notice that you are given that $d r=0.02$.)", "answer": "$d V=32 \\pi / 25$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.10.2", "question": "Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $e^{x} $", "answer": "$\\sum_{n=0}^{\\infty} x^{n} / n !, R=\\infty$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.3", "question": "Determine whether the series converges: $\\sum_{n=0}^{\\infty} \\frac{n}{\\left(n^{2}+4\\right)^{2}} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.7.2.15", "question": "Compute the value of the integral: $\\int_{0}^{3} x^{3} d x $", "answer": "$3^{4} / 4$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.6.8", "question": "Determine whether the series $\\sum_{n=1}^{\\infty}(-1)^{n-1} \\frac{\\arctan n}{n} $ converges absolutely, converges conditionally, or diverges.", "answer": "converges conditionally", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.17", "question": "Describe the concavity of the function: $y=\\cos ^{2} x-\\sin ^{2} x $", "answer": "concave up on $(\\pi / 4+n \\pi, 3 \\pi / 4+n \\pi)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.32", "question": "Find the derivative of the function: $3\\left(x^{2}+1\\right)\\left(2 x^{2}-1\\right)(2 x+3) $", "answer": "$60 x^{4}+72 x^{3}+18 x^{2}+18 x-6$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.3", "question": "Evaluate the integral: $\\int\\left(e^{t^{2}}+16\\right) t e^{t^{2}} d t $", "answer": "$\\frac{\\left(e^{t^{2}}+16\\right)^{2}}{4}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.6.22", "question": "Evaluate the integral: $\\int \\frac{t}{t^{2}+2 t-3} d t $", "answer": "$\\frac{3 \\ln |t+3|}{4}+\\frac{\\ln |t-1|}{4}+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.27", "question": "Find the dimensions of the lightest cylindrical can containing 0.25 liter $\\left(=250 \\mathrm{~cm}^{3}\\right)$ if the top and bottom are made of a material that is twice as heavy (per unit area) as the material used for the side.", "answer": "$r=5 /(2 \\pi)^{1 / 3} \\approx 2.7 \\mathrm{~cm}$, $h=5 \\cdot 2^{5 / 3} / \\pi^{1 / 3}=4 r \\approx 10.8 \\mathrm{~cm}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.2.4", "question": "A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The top of the ladder is being pulled up the wall at a rate of 0.1 meters per second. Find the rate at which the foot of the ladder is approaching the wall when the foot of the ladder is $5 \\mathrm{~m}$ from the wall.", "answer": "$-6 / 25 \\mathrm{~m} / \\mathrm{s}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.4.6", "question": "Find the antiderivative: $\\int \\ln x d x $", "answer": "$x \\ln x-x+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.2.9", "question": "Suppose the position of an object at time $t$ is given by $f(t)=-49 t^{2} / 10+5 t+10$. Find a function giving the speed of the object at time $t$. The acceleration of an object is the rate at which its speed is changing, which means it is given by the derivative of the speed function. Find the acceleration of the object at time $t . $", "answer": "$-49 t / 5+5,-49 / 5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.4.9", "question": "Describe the concavity of the function: $y=4 x+\\sqrt{1-x} $", "answer": "concave down everywhere", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.8", "question": "Determine whether the series converges: $\\sum_{n=0}^{\\infty} \\frac{n}{e^{n}} $", "answer": "converges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.40", "question": "Find an equation for the tangent line to $\\sqrt{\\left(x^{2}+1\\right)^{2}+\\sqrt{1+\\left(x^{2}+1\\right)^{2}}}$ at $(1, \\sqrt{4+\\sqrt{5}})$.", "answer": "$y=\\frac{20+2 \\sqrt{5}}{5 \\sqrt{4+\\sqrt{5}}} x+\\frac{3 \\sqrt{5}}{5 \\sqrt{4+\\sqrt{5}}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.1.11", "question": "Find the area bounded by the curves: $y=x^{3 / 2}$ and $y=x^{2 / 3} $", "answer": "$1 / 5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.3.1", "question": "Determine whether the series converges or diverges: $\\sum_{n=1}^{\\infty} \\frac{1}{n^{\\pi / 4}} $", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.5.3.16", "question": "Find all local maximum and minimum points of the function: $y=\\tan ^{2} x $", "answer": "$\\min$ at $n \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.7.13", "question": "Find the derivative of the function: $\\sqrt{\\ln \\left(x^{2}\\right)} / x $", "answer": "$\\left(1-\\ln \\left(x^{2}\\right)\\right) /\\left(x^{2} \\sqrt{\\ln \\left(x^{2}\\right)}\\right)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.11", "question": "Find the antiderivative of the function: $\\int_{0}^{\\pi} \\sin ^{5}(3 x) \\cos (3 x) d x $", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.10.1", "question": "Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $\\cos x $", "answer": "$\\sum_{n=0}^{\\infty}(-1)^{n} x^{2 n} /(2 n) !, R=\\infty$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.3.12", "question": "Find the antiderivative: $\\int \\frac{x^{3}}{\\sqrt{4 x^{2}-1}} d x $", "answer": "$\\left(2 x^{2}+1\\right) \\sqrt{4 x^{2}-1} / 24+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.2.4.5", "question": "Find the derivative of the function: $y=f(x)=x^{3}$.", "answer": "$3 x^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.6.1.15", "question": "You want to make cylindrical containers of a given volume $V$ using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side $2 r$, so that $2(2 r)^{2}=8 r^{2}$ of material is needed (rather than $2 \\pi r^{2}$, which is the total area of the top and bottom). Find the optimal ratio of height to radius.", "answer": "$8 / \\pi$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.23", "question": "Find the derivative of the function: $4\\left(2 x^{2}-x+3\\right)^{-2} $", "answer": "$-8(4 x-1)\\left(2 x^{2}-x+3\\right)^{-3}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.5.6", "question": "Find the antiderivative: $\\int \\frac{1}{x^{2}+10 x+29} d x $", "answer": "$(1 / 2) \\arctan (x / 2+5 / 2)+C$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.30", "question": "Find the derivative of the function: $\\frac{(x-1)(x-2)}{x-3} $", "answer": "$\\left(x^{2}-6 x+7\\right) /(x-3)^{2}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.19", "question": "Find a series representation for the function: $\\sum_{n=0}^{\\infty} \\frac{x^{n}}{1+3^{n}} $", "answer": "$(-3,3)$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.4.1.1", "question": "Find all values of $\\theta$ such that $\\sin (\\theta)=-1$; give your answer in radians.", "answer": "$2 n \\pi-\\pi / 2$, any integer $n$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.17", "question": "Determine whether the series converges: $\\sum_{n=1}^{\\infty} \\sin (1 / n) $. Find the interval and radius of convergence; you need not check the endpoints of the intervals.", "answer": "diverges", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.12.23", "question": "Find a series representation for the function: $\\sum_{n=1}^{\\infty} \\frac{(-1)^{n}}{n^{2} 3^{n}} x^{2 n} $", "answer": "$(-\\sqrt{3}, \\sqrt{3})$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.3.5.10", "question": "Find the derivative of the function: $\\frac{\\left(x^{2}+x+1\\right)}{(1-x)} $", "answer": "$\\frac{2 x+1}{1-x}+\\frac{x^{2}+x+1}{(1-x)^{2}}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.8.1.19", "question": "Evaluate the definite integral: $\\int_{-1}^{1} \\sin ^{7} x d x $", "answer": "0", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.4.2", "question": "Find the average height of $x^{2}$ over the interval $[-2,2] . $", "answer": "$4 / 3$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.9.2", "question": "Find a power series representation for $1 /(1-x)^{2}$.", "answer": "$\\sum_{n=0}^{\\infty}(n+1) x^{n}$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.10.4.5", "question": "Approximate the value of the series $\\sum_{n=1}^{\\infty}(-1)^{n-1} \\frac{1}{n^{3}}$ to two decimal places.", "answer": "0.90", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Calculus", "question_number": "exercise.9.6.9", "question": "A thin plate lies in the region contained by $y=x^{1 / 3}$ and the $x$-axis between $x=0$ and $x=1$. Find the centroid.", "answer": "$\\bar{x}=4 / 7, \\bar{y}=2 / 5$", "license": "Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)", "data_topic": "college_math.calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.2.1", "question": "Evaluate the given double integral: $\\int_{0}^{1} \\int_{\\sqrt{x}}^{1} 24 x^{2} y d y d x$", "answer": "$1 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.4.5", "question": "Calculate $\\mathbf{v} \\times \\mathbf{w}$:\n$\\mathbf{v}=-\\mathbf{i}+2 \\mathbf{j}+\\mathbf{k}, \\mathbf{w}=-3 \\mathbf{i}+6 \\mathbf{j}+3 \\mathbf{k}$", "answer": "0", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.1.3", "question": "For the points $P=(0,0,0), Q=(1,3,2), R=(1,0,1), S=(2,3,4)$, does $\\overrightarrow{P Q}=\\overrightarrow{R S}$ ?", "answer": "No", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.5.9", "question": "Find all local maxima and minima of the function $f(x, y) = 4x^{2} - 4xy + 2y^{2} + 10x - 6y$.", "answer": "local min. $(-1,1 / 2) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.6.3", "question": "Find the Laplacian of the function $f(x, y, z)=\\left(x^{2}+y^{2}+z^{2}\\right)^{3 / 2}$.", "answer": "$12 \\sqrt{x^{2}+y^{2}+z^{2}} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.4.31", "question": "Describe geometrically the set of points with position vector $\\mathbf{x}$ satisfying the equation\n$$(\\mathbf{v} \\times \\mathbf{x}) \\times \\mathbf{x}=\\mathbf{v}$$\nfor given vector $\\mathbf{v} \\neq \\mathbf{0}$", "answer": "A circle of radius $\\frac{1}{\\|\\mathbf{v}\\|}$ centered at the origin in the normal plane to $\\mathbf{v}$.", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.1.3", "question": "Find the volume under the surface $z=f(x, y)$ over the rectangle $R$: $f(x, y)=x^{3}+y^{2}, R=[0,1] \\times[0,1]$", "answer": "$\\frac{7}{12} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.3.3", "question": "Evaluate the given triple integral: $\\int_{0}^{\\pi} \\int_{0}^{x} \\int_{0}^{x y} x^{2} \\sin z d z d y d x$", "answer": "$\\left(2 \\cos \\left(\\pi^{2}\\right)+\\pi^{4}-2\\right) / 4 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.6.1", "question": "Find the center of mass of the region $R$ with the given density function $\\delta(x, y)$:\n$R=\\{(x, y): 0 \\leq x \\leq 2,0 \\leq y \\leq 4\\}, \\delta(x, y)=2 y$", "answer": "$(1,8 / 3)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.1.5", "question": "State the domain and range of the given function: $f(x, y, z)=\\sin (x y z)$", "answer": "domain: $\\mathbb{R}^{3}$, range: $[-1,1] $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.5.7", "question": "Calculate $\\int_{C} \\mathbf{f} \\cdot d \\mathbf{r}$ for the given vector field $\\mathbf{f}(x, y, z)$ and curve $C$: $\\mathbf{f}(x, y, z)=(y-2 z) \\mathbf{i}+x y \\mathbf{j}+(2 x z+y) \\mathbf{k} ; \\quad C: x=t, y=2 t, z=t^{2}-1,0 \\leq t \\leq 1$", "answer": "$67 / 15 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.3.7", "question": "Evaluate the given triple integral: $\\int_{1}^{2} \\int_{2}^{4} \\int_{0}^{3} 1 d x d y d z$", "answer": "6", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.1.13", "question": "Evaluate the limit: $\\lim _{(x, y) \\rightarrow(1,1)} \\frac{x^{2}-y^{2}}{x-y}$", "answer": "$2 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.1.17", "question": "Evaluate the limit: $\\lim _{(x, y) \\rightarrow(0,0)} \\frac{x}{y}$", "answer": "does not exist", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.2.5", "question": "Evaluate the given double integral: $\\int_{0}^{\\pi / 2} \\int_{0}^{y} \\cos x \\sin y d x d y$", "answer": "$\\frac{\\pi}{4} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.1.7", "question": "Evaluate the double integral: $\\int_{0}^{2} \\int_{0}^{1}(x+2) d x d y$", "answer": "$5 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.2.3", "question": "Evaluate the given double integral: $\\int_{1}^{2} \\int_{0}^{\\ln x} 4 x d y d x$", "answer": "$8 \\ln 2-3$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.1.3", "question": "Calculate the line integral $\\int_{C} f(x, y) d s$ for the given function $f(x, y)$ and curve $C$.\n$f(x, y)=2 x+y ; \\quad C$ : polygonal path from $(0,0)$ to $(3,0)$ to $(3,2)$", "answer": "23", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.1.1", "question": "Find the volume under the surface $z=f(x, y)$ over the rectangle $R$: $f(x, y)=4 x y, R=[0,1] \\times[0,1]$", "answer": "$1 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.5.13", "question": "Find three positive numbers $x, y, z$ whose sum is 10 such that $x^{2}y^{2}z$ is a maximum.", "answer": "$x=y=4, z=2$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.5.7", "question": "Find all local maxima and minima of the function $f(x, y) = \\sqrt{x^{2} + y^{2}}$.", "answer": "local min. $(0,0) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.5.3", "question": "Find all local maxima and minima of the function $f(x, y) = x^{3} - 3x + y^{3} - 3y$.", "answer": "local min. $(1,1)$; local max. $(-1,-1)$; saddle pts. $(1,-1),(-1,1) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.1.7", "question": "Calculate the line integral $\\int_{C} \\mathbf{f} \\cdot d \\mathbf{r}$ for the given vector field $\\mathbf{f}(x, y)$ and curve $C$.\n$\\mathbf{f}(x, y)=y \\mathbf{i}-x \\mathbf{j} ; \\quad C: x=\\cos t, y=\\sin t, 0 \\leq t \\leq 2 \\pi$", "answer": "$-2 \\pi$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.7.2", "question": "For $\\sigma>0$ and $\\mu>0$, evaluate\n\n$$\n\\int_{-\\infty}^{\\infty} \\frac{1}{\\sigma \\sqrt{2 \\pi}} e^{-(x-\\mu)^{2} / 2 \\sigma^{2}} d x\n$$", "answer": "1", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.6.5", "question": "Find the center of mass of the region $R$ with the given density function $\\delta(x, y)$:\n$R=\\left\\{(x, y): y \\geq 0, x^{2}+y^{2} \\leq 1\\right\\}, \\delta(x, y)=y$", "answer": "$(0,3 \\pi / 16) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.6.9", "question": "Find the center of mass of the solid $S$ with the given density function $\\delta(x, y, z)$:\n$S=\\{(x, y, z): 0 \\leq x \\leq 1,0 \\leq y \\leq 1,0 \\leq z \\leq 1\\}, \\delta(x, y, z)=x^{2}+y^{2}+z^{2}$", "answer": "$(7 / 12,7 / 12,7 / 12)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.2.3.1", "question": "Find the tangent line, the osculating plane, and the curvature at each point of the curve $\\mathbf{f}(t)= (\\cos t, \\sin t, t)$.", "answer": "$\\frac{3 \\pi \\sqrt{5}}{2} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.4.3", "question": "Calculate $\\mathbf{v} \\times \\mathbf{w}$:\n$\\mathbf{v}=(2,1,4), \\mathbf{w}=(1,-2,0)$", "answer": "$(8,4,-5) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.2.1.5", "question": "Find the velocity $\\mathbf{v}(t)$ and acceleration $\\mathbf{a}(t)$ of an object with the given position vector $\\mathbf{r}(t)$: $\\mathbf{r}(t)=(t, t-\\sin t, 1-\\cos t)$", "answer": "$\\mathbf{v}(t)=(1,1-\\cos t, \\sin t)$, $\\mathbf{a}(t)=(0, \\sin t, \\cos t) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.3.9", "question": "Find the equation of the tangent plane to the given surface at the point $P$: $x^{2}+y^{2}-z^{2}=0$, $P=(3,4,5)$.", "answer": "$3 x+4 y-5 z=0$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.3.1", "question": "Evaluate the given triple integral: $\\int_{0}^{3} \\int_{0}^{2} \\int_{0}^{1} x y z d x d y d z$", "answer": "$\\frac{9}{2} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.3.1", "question": "Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x^{2}+y^{3}$, $P=(1,1,2)$.", "answer": "$2 x+3 y-z-3=0 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.4.7", "question": "Calculate the area of the triangle $\\triangle P Q R$:\n$P=(5,1,-2), Q=(4,-4,3), R=(2,4,0)$", "answer": "$16.72 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.2.10", "question": "Evaluate the double integral: $\\iint_{R} f(x, y) d A$, where $f(x, y)=x^{2}+y$ and $R$ is the triangle with vertices $(0,0),(2,0)$ and $(0,1)$.", "answer": "$\\frac{6}{5} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.6.7", "question": "Find the center of mass of the solid $S$ with the given density function $\\delta(x, y, z)$:\n$S=\\left\\{(x, y, z): z \\geq 0, x^{2}+y^{2}+z^{2} \\leq a^{2}\\right\\}, \\delta(x, y, z)=x^{2}+y^{2}+z^{2}$", "answer": "$(0,0,5 a / 12) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.5.13", "question": "State whether or not the vector field $\\mathbf{f}(x, y, z)$ has a potential in $\\mathbb{R}^{3}$ (you do not need to find the potential itself): $\\mathbf{f}(x, y, z)=x y \\mathbf{i}-\\left(x-y z^{2}\\right) \\mathbf{j}+y^{2} z \\mathbf{k}$", "answer": "No", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.5.7", "question": "Evaluate $\\iint_{R} \\sin \\left(\\frac{x+y}{2}\\right) \\cos \\left(\\frac{x-y}{2}\\right) d A$, where $R$ is the triangle with vertices $(0,0),(2,0)$ and $(1,1)$. (Hint: Use the change of variables $u=(x+y) / 2, v=(x-y) / 2$.)", "answer": "$1-\\frac{\\sin 2}{2}$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.3.9", "question": "Let $\\mathbf{v}=(8,4,3)$ and $\\mathbf{w}=(-2,1,4)$. Is $\\mathbf{v} \\perp \\mathbf{w}$ ? Justify your answer.", "answer": "Yes, since $\\mathbf{v} \\cdot \\mathbf{w}=0", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.4.7", "question": "Compute the gradient $\\nabla f$ for the function $f(x, y, z)=\\sin (x y z)$.", "answer": "$\\quad(y z \\cos (x y z), x z \\cos (x y z), x y \\cos (x y z))$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.7.5", "question": "For $\\mathbf{f}(\\rho, \\theta, \\phi)=\\mathbf{e}_{\\rho}+\\rho \\cos \\theta \\mathbf{e}_{\\theta}+\\rho \\mathbf{e}_{\\phi}$ in spherical coordinates, find $\\operatorname{div} \\mathbf{f}$ and curlf.", "answer": "$\\operatorname{div} \\mathbf{f}=\\frac{2}{\\rho}-\\frac{\\sin \\theta}{\\sin \\phi}+\\cot \\phi, \\operatorname{curl} \\mathbf{f}=\\cot \\phi \\cos \\theta \\mathbf{e}_{\\rho}+$ $2 \\mathbf{e}_{\\theta}-2 \\cos \\theta \\mathbf{e}_{\\phi} \\mathbf{6}$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.1.11", "question": "Evaluate the double integral: $\\int_{0}^{2} \\int_{1}^{4} x y d x d y$", "answer": "$15 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.7.3", "question": "Let $f(x, y, z)=\\frac{z}{x^{2}+y^{2}}$ in Cartesian coordinates. Find $\\nabla f$ in cylindrical coordinates.", "answer": "$-\\frac{2 z}{r^{3}} \\mathbf{e}_{r}+\\frac{1}{r^{2}} \\mathbf{e}_{z}$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.4.9", "question": "Compute the gradient $\\nabla f$ for the function $f(x, y, z)=x^{2}+y^{2}+z^{2}$.", "answer": "$\\quad(2 x, 2 y, 2 z) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.1.9", "question": "Evaluate the double integral: $\\int_{0}^{\\pi / 2} \\int_{0}^{1} x y \\cos \\left(x^{2} y\\right) d x d y$", "answer": "$\\frac{1}{2} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.3.7", "question": "Is there a potential $F(x, y)$ for $\\mathbf{f}(x, y)=(8 x y+3) \\mathbf{i}+4\\left(x^{2}+y\\right) \\mathbf{j}$ ? If so, find one.", "answer": "Yes. $F(x, y)=4 x^{2} y+2 y^{2}+3 x$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.2.1", "question": "Evaluate $\\oint_{C}\\left(x^{2}+y^{2}\\right) d x+2 x y d y$ for $C: x=\\cos t, y=\\sin t, 0 \\leq t \\leq 2 \\pi$.", "answer": "$0 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.2.7", "question": "Evaluate the given double integral: $\\int_{0}^{2} \\int_{0}^{y} 1 d x d y$", "answer": "$2 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.3.7", "question": "Find the equation of the tangent plane to the given surface at the point $P$: $\\frac{x^{2}}{4}+\\frac{y^{2}}{9}+\\frac{z^{2}}{16}=1$, $P=\\left(1,2, \\frac{2 \\sqrt{11}}{3}\\right)$.", "answer": "$\\frac{1}{2}(x-1)+\\frac{4}{9}(y-2)+\\frac{\\sqrt{11}}{12}(z-\\frac{2 \\sqrt{11}}{3})=0 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.2.3.3", "question": "Find the tangent line, the osculating plane, and the curvature at each point of the curve $\\mathbf{f}(t)= (t \\sin t, t \\cos t)$.", "answer": "$2\\left(5^{3 / 2}-8\\right) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.4.3", "question": "Compute the gradient $\\nabla f$ for the function $f(x, y)=\\sqrt{x^{2}+y^{2}+4}$.", "answer": "$\\left(\\frac{x}{\\sqrt{x^{2}+y^{2}+4}}, \\frac{y}{\\sqrt{x^{2}+y^{2}+4}}\\right) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.4.5", "question": "Compute the gradient $\\nabla f$ for the function $f(x, y)=\\ln (x y)$.", "answer": "$\\left(\\frac{1}{x}, \\frac{1}{y}\\right)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.6.9", "question": "Find the trace of the hyperbolic paraboloid $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=\\frac{z}{c}$ in the $x y$-plane", "answer": "lines $\\frac{x}{a}=\\frac{y}{b}, z=0$ and $\\frac{x}{a}=-\\frac{y}{b}, z=0$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.5.1", "question": "Find the volume $V$ inside the paraboloid $z=x^{2}+y^{2}$ for $0 \\leq z \\leq 4$.", "answer": "$8 \\pi$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.3.7", "question": "Find the angle $\\theta$ between the vectors $\\mathbf{v}=-\\mathbf{i}+2 \\mathbf{j}+\\mathbf{k}$ and $\\mathbf{w}=-3 \\mathbf{i}+6 \\mathbf{j}+3 \\mathbf{k}$.", "answer": "$0^{\\circ} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.1.1", "question": "State the domain and range of the given function: $f(x, y)=x^{2}+y^{2}-1$", "answer": "domain: $\\mathbb{R}^{2}$, range: $[-1, \\infty) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.4.1", "question": "Calculate $\\mathbf{v} \\times \\mathbf{w}$:\n$\\mathbf{v}=(5,1,-2), \\mathbf{w}=(4,-4,3)$", "answer": "$(-5,-23,-24) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.3.3", "question": "Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x^{2} y$, $P=(-1,1,1)$.", "answer": "$-2 x+y-z-2=0$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.2.3", "question": "Is there a potential $F(x, y)$ for $\\mathbf{f}(x, y)=y \\mathbf{i}-x \\mathbf{j}$ ? If so, find one.", "answer": "No", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.5.1", "question": "Find all local maxima and minima of the function $f(x, y) = x^{3} - 3x + y^{2}$.", "answer": "local min. $(1,0)$; saddle pt. $(-1,0)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.3.1", "question": "Let $\\mathbf{v}=(5,1,-2)$ and $\\mathbf{w}=(4,-4,3)$. Calculate $\\mathbf{v} \\cdot \\mathbf{w}$.", "answer": "$10 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.5.11", "question": "For a rectangular solid of volume 1000 cubic meters, find the dimensions that will minimize the surface area. (Hint: Use the volume condition to write the surface area as a function of just two variables.)", "answer": "width $=$ height $=\\operatorname{depth}=10$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.6.1", "question": "Determine if the given equation describes a sphere. If so, find its radius and center: $x^{2}+y^{2}+z^{2}-4 x-6 y-10 z+37=0$", "answer": "radius: 1 , center: $(2,3,5)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.4.1", "question": "Compute the gradient $\\nabla f$ for the function $f(x, y)=x^{2}+y^{2}-1$.", "answer": "$(2 x, 2 y)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.4.13", "question": "Find the directional derivative of $f(x, y)=\\sqrt{x^{2}+y^{2}+4}$ at the point $P=(1,1)$ in the direction of $\\mathbf{v}=\\left(\\frac{1}{\\sqrt{2}}, \\frac{1}{\\sqrt{2}}\\right)$.", "answer": "$\\quad \\frac{1}{\\sqrt{3}}$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.4.15", "question": "Find the directional derivative of $f(x, y, z)=\\sin (x y z)$ at the point $P=(1,1,1)$ in the direction of $\\mathbf{v}=\\left(\\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}\\right)$.", "answer": "$\\sqrt{3} \\cos (1) ; 1$. increase: $(45,20)$, decrease: $(-45,-20)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.4.11", "question": "Find the volume of the parallelepiped with adjacent sides $\\mathbf{u}, \\mathbf{v}, \\mathbf{w}$:\n$\\mathbf{u}=(1,1,3), \\mathbf{v}=(2,1,4), \\mathbf{w}=(5,1,-2)$", "answer": "$9 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.5.5", "question": "Find all local maxima and minima of the function $f(x, y) = 2x^{3} + 6xy + 3y^{2}$.", "answer": "local min. $(1,-1)$, saddle pt. $(0,0) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.2.5", "question": "Is there a potential $F(x, y)$ for $\\mathbf{f}(x, y)=x y^{2} \\mathbf{i}+x^{3} y \\mathbf{j}$ ? If so, find one.", "answer": "No", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.5.9", "question": "Calculate $\\int_{C} \\mathbf{f} \\cdot d \\mathbf{r}$ for the given vector field $\\mathbf{f}(x, y, z)$ and curve $C$: $\\mathbf{f}(x, y, z)=x y \\mathbf{i}+(z-x) \\mathbf{j}+2 y z \\mathbf{k} ; \\quad C$ : the polygonal path from $(0,0,0)$ to $(1,0,0)$ to $(1,2,0)$ to $(1,2,-2)$", "answer": "6", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.5.3", "question": "Find the volume $V$ of the solid inside both $x^{2}+y^{2}+z^{2}=4$ and $x^{2}+y^{2}=1$.", "answer": "$\\frac{4 \\pi}{3}\\left(8-3^{3 / 2}\\right)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.2.4", "question": "Is there a potential $F(x, y)$ for $\\mathbf{f}(x, y)=x \\mathbf{i}-y \\mathbf{j}$ ? If so, find one.", "answer": "Yes. $F(x, y)=\\frac{x^{2}}{2}-\\frac{y^{2}}{2} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.2.1.1", "question": "Calculate $\\mathbf{f}^{\\prime}(t)$ and find the tangent line at $\\mathbf{f}(0)$ for the following function: $\\mathbf{f}(t)=\\left(t+1, t^{2}+1, t^{3}+1\\right)$", "answer": "$\\mathbf{f}^{\\prime}(t)=\\left(1,2 t, 3 t^{2}\\right), \\quad x=1+t, \\quad y=z=$ $1 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.1.11", "question": "Calculate the line integral $\\int_{C} \\mathbf{f} \\cdot d \\mathbf{r}$ for the given vector field $\\mathbf{f}(x, y)$ and curve $C$.\n$\\mathbf{f}(x, y)=\\left(x^{2}+y^{2}\\right) \\mathbf{i} ; \\quad C: x=2+\\cos t, y=\\sin t, 0 \\leq t \\leq 2 \\pi$", "answer": "0", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.4.11", "question": "Find the directional derivative of $f(x, y)=x^{2}+y^{2}-1$ at the point $P=(1,1)$ in the direction of $\\mathbf{v}=\\left(\\frac{1}{\\sqrt{2}}, \\frac{1}{\\sqrt{2}}\\right)$.", "answer": "$\\quad 2 \\sqrt{2} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.5.5", "question": "Calculate $\\int_{C} \\mathbf{f} \\cdot d \\mathbf{r}$ for the given vector field $\\mathbf{f}(x, y, z)$ and curve $C$: $\\mathbf{f}(x, y, z)=y \\mathbf{i}-x \\mathbf{j}+z \\mathbf{k} ; \\quad C: x=\\cos t, y=\\sin t, z=t, 0 \\leq t \\leq 2 \\pi$", "answer": "$2 \\pi(\\pi-1)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.2.1.3", "question": "Calculate $\\mathbf{f}^{\\prime}(t)$ and find the tangent line at $\\mathbf{f}(0)$ for the following function: $\\mathbf{f}(t)=(\\cos 2 t, \\sin 2 t, t)$", "answer": "$\\mathbf{f}^{\\prime}(t)=(-2 \\sin 2 t, 2 \\cos 2 t, 1) ; \\quad x=1$, $y=2 t, \\quad z=t $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.1.2", "question": "For the points $P=(1,-1,1), Q=(2,-2,2), R=(2,0,1), S=(3,-1,2)$, does $\\overrightarrow{P Q}=\\overrightarrow{R S}$ ?", "answer": "Yes", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.6.5", "question": "Find the Laplacian of the function $f(x, y, z)=x^{3}+y^{3}+z^{3}$.", "answer": "$6(x+y+z) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.5.2", "question": "Calculate $\\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=\\frac{x}{y}+y+2 y z ; \\quad C: x=t^{2}, y=t, z=1,1 \\leq t \\leq 2$", "answer": "$(17 \\sqrt{17}-5 \\sqrt{5}) / 3 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.1.5", "question": "Evaluate the double integral: $\\int_{0}^{1} \\int_{1}^{2}(1-y) x^{2} d x d y$", "answer": "$\\frac{7}{6} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.6.1", "question": "Find the Laplacian of the function $f(x, y, z)=x+y+z$.", "answer": "$0 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.7.1", "question": "Find the constrained maxima and minima of $f(x, y)=2 x+y$ given that $x^{2}+y^{2}=4$.", "answer": "$\\max .\\left(\\frac{4}{\\sqrt{5}}, \\frac{2}{\\sqrt{5}}\\right) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.5.4", "question": "Calculate $\\int_{C} \\mathbf{f} \\cdot d \\mathbf{r}$ for the given vector field $\\mathbf{f}(x, y, z)$ and curve $C$: $\\mathbf{f}(x, y, z)=\\mathbf{i}-\\mathbf{j}+\\mathbf{k} ; \\quad C: x=3 t, y=2 t, z=t, 0 \\leq t \\leq 1$", "answer": "2", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.1.1", "question": "Calculate the line integral $\\int_{C} f(x, y) d s$ for the given function $f(x, y)$ and curve $C$.\n$f(x, y)=x y ; \\quad C: x=\\cos t, y=\\sin t, 0 \\leq t \\leq \\pi / 2$", "answer": "$1 / 2$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.1.3", "question": "State the domain and range of the given function: $f(x, y)=\\sqrt{x^{2}+y^{2}-4}$", "answer": "domain: $\\left\\{(x, y): x^{2}+y^{2} \\geq 4\\right\\}$, range: $[0, \\infty) $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.3.10", "question": "Find the volume $V$ of the solid $S$ bounded by the three coordinate planes, bounded above by the plane $x+y+z=2$, and bounded below by the plane $z=x+y$.", "answer": "$\\frac{1}{3}$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.3.5", "question": "Is there a potential $F(x, y)$ for $\\mathbf{f}(x, y)=\\left(y^{2}+3 x^{2}\\right) \\mathbf{i}+2 x y \\mathbf{j}$ ? If so, find one.", "answer": "Yes. $F(x, y)=x y^{2}+x^{3}$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.6.3", "question": "Find the center of mass of the region $R$ with the given density function $\\delta(x, y)$:\n$R=\\left\\{(x, y): y \\geq 0, x^{2}+y^{2} \\leq a^{2}\\right\\}, \\delta(x, y)=1$", "answer": "$\\left(0, \\frac{4 a}{3 \\pi}\\right) \\quad$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.2.6", "question": "Evaluate the given double integral: $\\int_{0}^{\\infty} \\int_{0}^{\\infty} x y e^{-\\left(x^{2}+y^{2}\\right)} d x d y$", "answer": "$\\frac{1}{4} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.1.11", "question": "Evaluate the limit: $\\lim _{(x, y) \\rightarrow(1,-1)} \\frac{x^{2}-2 x y+y^{2}}{x-y}$", "answer": "$2 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.1.7", "question": "Evaluate the limit: $\\lim _{(x, y) \\rightarrow(0,0)} \\cos (x y)$", "answer": "$1 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.5.3", "question": "Calculate $\\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=z^{2} ; \\quad C: x=t \\sin t, y=t \\cos t, z=\\frac{2 \\sqrt{2}}{3} t^{3 / 2}, 0 \\leq t \\leq 1$", "answer": "$2 / 5 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.7.2", "question": "Let $f(x, y, z)=e^{-x^{2}-y^{2}-z^{2}}$ in Cartesian coordinates. Find the Laplacian of the function in spherical coordinates.", "answer": "$\\left(4 \\rho^{2}-6\\right) e^{-\\rho^{2}} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.3.5", "question": "Find the angle $\\theta$ between the vectors $\\mathbf{v}=(2,1,4)$ and $\\mathbf{w}=(1,-2,0)$.", "answer": "$90^{\\circ} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.7.5", "question": "Find the constrained maxima and minima of $f(x, y, z)=x+y^{2}+2 z$ given that $4 x^{2}+9 y^{2}-36 z^{2}=36$.", "answer": "$\\frac{8 a b c}{3 \\sqrt{3}}$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.5.9", "question": "Find the volume inside the elliptic cylinder $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}} \\leq 1$ for $0 \\leq z \\leq 2$.", "answer": "$2 \\pi a b$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.4.9", "question": "Calculate the area of the parallelogram $P Q R S$:\n$P=(2,1,3), Q=(1,4,5), R=(2,5,3), S=(3,2,1)$", "answer": "4 \\sqrt{5}", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.3.5", "question": "Evaluate the given triple integral: $\\int_{1}^{e} \\int_{0}^{y} \\int_{0}^{1 / y} x^{2} z d x d z d y$", "answer": "$\\frac{1}{6} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.5.11", "question": "State whether or not the vector field $\\mathbf{f}(x, y, z)$ has a potential in $\\mathbb{R}^{3}$ (you do not need to find the potential itself): $\\mathbf{f}(x, y, z)=a \\mathbf{i}+b \\mathbf{j}+c \\mathbf{k}(a, b, c$ constant $)$", "answer": "Yes", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.1.9", "question": "Calculate the line integral $\\int_{C} \\mathbf{f} \\cdot d \\mathbf{r}$ for the given vector field $\\mathbf{f}(x, y)$ and curve $C$.\n$\\mathbf{f}(x, y)=\\left(x^{2}-y\\right) \\mathbf{i}+\\left(x-y^{2}\\right) \\mathbf{j} ; \\quad C: x=\\cos t, y=\\sin t, 0 \\leq t \\leq 2 \\pi$", "answer": "$2 \\pi$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.4.13", "question": "Calculate $\\mathbf{u} \\cdot(\\mathbf{v} \\times \\mathbf{w})$ and $\\mathbf{u} \\times(\\mathbf{v} \\times \\mathbf{w})$:\n$\\mathbf{u}=(1,1,1), \\mathbf{v}=(3,0,2), \\mathbf{w}=(2,2,2)$", "answer": "0 and $(8,-10,2)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.7.1", "question": "Let $f(x, y, z)=\\left(x^{2}+y^{2}+z^{2}\\right)^{3 / 2}$ in Cartesian coordinates. Find the Laplacian of $f$ in spherical coordinates.", "answer": "$12 \\rho $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.6.3", "question": "Determine if the given equation describes a sphere. If so, find its radius and center: $2 x^{2}+2 y^{2}+2 z^{2}+4 x+4 y+4 z-44=0$", "answer": "radius: 5, center: $(-1,-1,-1)$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.1.9", "question": "Evaluate the limit: $\\lim _{(x, y) \\rightarrow(0,0)} \\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$", "answer": "does not exist", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.1.15", "question": "Evaluate the limit: $\\lim _{(x, y) \\rightarrow(0,0)} \\frac{y^{4} \\sin (x y)}{x^{2}+y^{2}}$", "answer": "$0 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.3.3.5", "question": "Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x+2 y$, $P=(2,1,4)$.", "answer": "$x+2 y=z $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.5.5.1", "question": "Calculate $\\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=z ; \\quad C: x=\\cos t, y=\\sin t, z=t, 0 \\leq t \\leq 2 \\pi$", "answer": "$2 \\sqrt{2} \\pi^{2} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.3.3", "question": "Find the angle $\\theta$ between the vectors $\\mathbf{v}=(5,1,-2)$ and $\\mathbf{w}=(4,-4,3)$.", "answer": "$73.4^{\\circ} $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.1.4.15", "question": "Calculate $(\\mathbf{u} \\times \\mathbf{v}) \\cdot(\\mathbf{w} \\times \\mathbf{z})$:\n$\\mathbf{u}=(1,1,1), \\mathbf{v}=(3,0,2), \\mathbf{w}=(2,2,2), \\mathbf{z}=(2,1,4)$", "answer": "$14 $", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Corrals_Vector_Calculus", "question_number": "exercise.4.7.1", "question": "Evaluate the integral\n\n$$\n\\int_{-\\infty}^{\\infty} e^{-x^{2}} d x\n$$\n\nusing anything you have learned so far.", "answer": "$\\sqrt{\\pi}$", "license": "GNU Free Documentation License", "data_topic": "college_math.vector_calculus"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.2.3", "question": "How many seven-element subsets are there in a set of nine elements?", "answer": "$\\left(\\begin{array}{l}9 \\\\ 7\\end{array}\\right)=36$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.6.2.9", "question": "A die is loaded so that the probability of a face coming up is proportional to the number on that face. The die is rolled with outcome $X$. Find $V(X)$ and $D(X)$.", "answer": "$V(X)=\\frac{20}{9}, \\quad D(X)=\\frac{2 \\sqrt{5}}{3}$.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.1.21", "question": "Modify the program AllPermutations to count the number of permutations of $n$ objects that have exactly $j$ fixed points for $j=0,1,2, \\ldots, n$. Run your program for $n=2$ to 6 . Make a conjecture for the relation between the number that have 0 fixed points and the number that have exactly 1 fixed point. A proof of the correct conjecture can be found in Wilf. ${ }^{12}$", "answer": "They are the same.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.6.2.7", "question": "A coin is tossed three times. Let $X$ be the number of heads that turn up. Find $V(X)$ and $D(X)$.", "answer": "$V(X)=\\frac{3}{4}, \\quad D(X)=\\frac{\\sqrt{3}}{2}$.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.6.3.3", "question": "The lifetime, measure in hours, of the ACME super light bulb is a random variable $T$ with density function $f_{T}(t)=\\lambda^{2} t e^{-\\lambda t}$, where $\\lambda=.05$. What is the expected lifetime of this light bulb? What is its variance?", "answer": "$\\mu=40, \\sigma^{2}=800$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.6.1.5", "question": "In a second version of roulette in Las Vegas, a player bets on red or black. Half of the numbers from 1 to 36 are red, and half are black. If a player bets a dollar on black, and if the ball stops on a black number, he gets his dollar back and another dollar. If the ball stops on a red number or on 0 or 00 he loses his dollar. Find the expected winnings for this bet.", "answer": "$-1 / 19$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.2.5", "question": "Use the program BinomialProbabilities to find the probability that, in 100 tosses of a fair coin, the number of heads that turns up lies between 35 and 65 , between 40 and 60 , and between 45 and 55 .", "answer": ".998,.965,.729", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.1.1", "question": "Four people are to be arranged in a row to have their picture taken. In how many ways can this be done?", "answer": "24", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.5.1.27", "question": "Assume that the probability that there is a significant accident in a nuclear power plant during one year's time is .001. If a country has 100 nuclear plants, estimate the probability that there is at least one such accident during a given year.", "answer": "$m=100 \\times(.001)=.1$. Thus $P$ (at least one accident $)=1-e^{-.1}=.0952$.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.6.1.13", "question": "You have 80 dollars and play the following game. An urn contains two white balls and two black balls. You draw the balls out one at a time without replacement until all the balls are gone. On each draw, you bet half of your present fortune that you will draw a white ball. What is your expected final fortune?", "answer": "45", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.6.1.1", "question": "A card is drawn at random from a deck consisting of cards numbered 2 through 10. A player wins 1 dollar if the number on the card is odd and loses 1 dollar if the number if even. What is the expected value of his winnings?", "answer": "$-1 / 9$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.2.2.15", "question": "At the Tunbridge World's Fair, a coin toss game works as follows. Quarters are tossed onto a checkerboard. The management keeps all the quarters, but for each quarter landing entirely within one square of the checkerboard the management pays a dollar. Assume that the edge of each square is twice the diameter of a quarter, and that the outcomes are described by coordinates chosen at random. Is this a fair game?", "answer": "Yes.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.1.15", "question": "A computing center has 3 processors that receive $n$ jobs, with the jobs assigned to the processors purely at random so that all of the $3^{n}$ possible assignments are equally likely. Find the probability that exactly one processor has no jobs.", "answer": "$\\frac{\\left(\\begin{array}{l}3 \\\\ 1\\end{array}\\right) \\times\\left(2^{n}-2\\right)}{3^{n}}$.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.4.1.19", "question": "In a poker hand, John has a very strong hand and bets 5 dollars. The probability that Mary has a better hand is .04. If Mary had a better hand she would raise with probability .9 , but with a poorer hand she would only raise with probability .1. If Mary raises, what is the probability that she has a better hand than John does?", "answer": ".273.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.11.2.9", "question": "A process moves on the integers 1, 2, 3, 4, and 5. It starts at 1 and, on each successive step, moves to an integer greater than its present position, moving with equal probability to each of the remaining larger integers. State five is an absorbing state. Find the expected number of steps to reach state five.", "answer": "2.08", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.5.1.17", "question": "The probability of a royal flush in a poker hand is $p=1 / 649,740$. How large must $n$ be to render the probability of having no royal flush in $n$ hands smaller than $1 / e$ ?", "answer": "649741", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.4.1.21", "question": "It is desired to find the probability that in a bridge deal each player receives an ace. A student argues as follows. It does not matter where the first ace goes. The second ace must go to one of the other three players and this occurs with probability $3 / 4$. Then the next must go to one of two, an event of probability $1 / 2$, and finally the last ace must go to the player who does not have an ace. This occurs with probability $1 / 4$. The probability that all these events occur is the product $(3 / 4)(1 / 2)(1 / 4)=3 / 32$. Is this argument correct?", "answer": "No.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.4.3.1", "question": "One of the first conditional probability paradoxes was provided by Bertrand. ${ }^{23}$ It is called the Box Paradox. A cabinet has three drawers. In the first drawer there are two gold balls, in the second drawer there are two silver balls, and in the third drawer there is one silver and one gold ball. A drawer is picked at random and a ball chosen at random from the two balls in the drawer. Given that a gold ball was drawn, what is the probability that the drawer with the two gold balls was chosen?", "answer": "$2 / 3$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.2.21", "question": "A lady wishes to color her fingernails on one hand using at most two of the colors red, yellow, and blue. How many ways can she do this?", "answer": "$3\\left(2^{5}\\right)-3=93$ (we subtract 3 because the three pure colors are each counted twice).", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.8.1.5", "question": "Let $X$ be a random variable with $E(X)=0$ and $V(X)=1$. What integer value $k$ will assure us that $P(|X| \\geq k) \\leq .01$ ?", "answer": "$k=10$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.6.2.1", "question": "A number is chosen at random from the set $S=\\{-1,0,1\\}$. Let $X$ be the number chosen. Find the expected value, variance, and standard deviation of $X$.", "answer": "$E(X)=0, V(X)=\\frac{2}{3}, \\quad \\sigma=D(X)=\\sqrt{\\frac{2}{3}}$.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.5.2.37", "question": "Let $X$ be a random variable having a normal density and consider the random variable $Y=e^{X}$. Then $Y$ has a $\\log$ normal density. Find this density of $Y$.", "answer": "$F_{Y}(y)=\\frac{1}{\\sqrt{2 \\pi y}} e^{-\\frac{\\log ^{2}(y)}{2}}$, for $y>0$.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.1.2.9", "question": "A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability $5 / 8$, French with probability $5 / 8$, and art and French together with probability $1 / 4$. What is the probability that he chooses mathematics? What is the probability that he chooses either art or French?", "answer": "$3 / 4,1$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.8.1.1", "question": "A fair coin is tossed 100 times. The expected number of heads is 50, and the standard deviation for the number of heads is $(100 \\cdot 1 / 2 \\cdot 1 / 2)^{1 / 2}=5$. What does Chebyshev's Inequality tell you about the probability that the number of heads that turn up deviates from the expected number 50 by three or more standard deviations (i.e., by at least 15 )?", "answer": "$1 / 9$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.2.9", "question": "Find integers $n$ and $r$ such that the following equation is true:\n$$\n\\left(\\begin{array}{c}\n13 \\\\\n5\n\\end{array}\\right)+2\\left(\\begin{array}{c}\n13 \\\\\n6\n\\end{array}\\right)+\\left(\\begin{array}{c}\n13 \\\\\n7\n\\end{array}\\right)=\\left(\\begin{array}{l}\nn \\\\\nr\n\\end{array}\\right)\n$$", "answer": "$n=15, r=7$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.4.1.13", "question": "Two cards are drawn from a bridge deck. What is the probability that the second card drawn is red?", "answer": "$1 / 2$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.11.1.1", "question": "It is raining in the Land of Oz. Determine a tree and a tree measure for the next three days' weather. Find $\\mathbf{w}^{(1)}, \\mathbf{w}^{(2)}$, and $\\mathbf{w}^{(3)}$ and compare with the results obtained from $\\mathbf{P}, \\mathbf{P}^{2}$, and $\\mathbf{P}^{3}$.", "answer": "$\\mathbf{w}(1)=(.5, .25, .25)$\n\n$\\mathbf{w}(2)=(.4375, .1875, .375)$\n\n$\\mathbf{w}(3)=(.40625, .203125, .390625)$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.7.1.3", "question": "Let $X_{1}$ and $X_{2}$ be independent random variables with common distribution\n$$\np_{X}=\\left(\\begin{array}{ccc}\n0 & 1 & 2 \\\\\n1 / 8 & 3 / 8 & 1 / 2\n\\end{array}\\right) .\n$$\nFind the distribution of the sum $X_{1}+X_{2}$.", "answer": "$\\quad\\left(\\begin{array}{ccccc}0 & 1 & 2 & 3 & 4 \\\\ \\frac{1}{64} & \\frac{3}{32} & \\frac{17}{64} & \\frac{3}{8} & \\frac{1}{4}\\end{array}\\right)$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.6.1.3", "question": "In a class there are 20 students: 3 are 5' 6\u201d, 5 are 5'8\u201d, 4 are 5'10\", 4 are 6 ', and 4 are 6' 2\". A student is chosen at random. What is the student's expected height?", "answer": "$5^{\\prime} 10.1^{\\prime \\prime}$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.5.1.13", "question": "The Poisson distribution with parameter $\\lambda=.3$ has been assigned for the outcome of an experiment. Let $X$ be the outcome function. Find $P(X=0)$, $P(X=1)$, and $P(X>1)$.", "answer": ".7408,.2222, .0370", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.4.1.43", "question": "The Yankees are playing the Dodgers in a world series. The Yankees win each game with probability .6. What is the probability that the Yankees win the series? (The series is won by the first team to win four games.)", "answer": ".710.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.1.3", "question": "In a digital computer, a bit is one of the integers $\\{0,1\\}$, and a word is any string of 32 bits. How many different words are possible?", "answer": "$2^{32}$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.1.7", "question": "Five people get on an elevator that stops at five floors. Assuming that each has an equal probability of going to any one floor, find the probability that they all get off at different floors.", "answer": "$\\frac{5 !}{5^{5}}$.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.2.11", "question": "A restaurant offers apple and blueberry pies and stocks an equal number of each kind of pie. Each day ten customers request pie. They choose, with equal probabilities, one of the two kinds of pie. How many pieces of each kind of pie should the owner provide so that the probability is about .95 that each customer gets the pie of his or her own choice?", "answer": "Eight pieces of each kind of pie.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.2.15", "question": "A baseball player, Smith, has a batting average of .300 and in a typical game comes to bat three times. Assume that Smith's hits in a game can be considered to be a Bernoulli trials process with probability .3 for success. Find the probability that Smith gets $0,1,2$, and 3 hits.", "answer": ".343,.441, .189, .027.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.2.31", "question": "Each of the four engines on an airplane functions correctly on a given flight with probability .99 , and the engines function independently of each other. Assume that the plane can make a safe landing if at least two of its engines are functioning correctly. What is the probability that the engines will allow for a safe landing?", "answer": ".999996.", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.1.2.13", "question": "In a horse race, the odds that Romance will win are listed as $2: 3$ and that Downhill will win are $1: 2$. What odds should be given for the event that either Romance or Downhill wins?", "answer": "$11: 4$", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Grinstead_and_Snells_Introduction_to_Probability", "question_number": "exercise.3.1.5", "question": "There are three different routes connecting city A to city B. How many ways can a round trip be made from A to B and back? How many ways if it is desired to take a different route on the way back?", "answer": "9,6 .", "license": "GNU Free Documentation License", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.1.9", "question": "Suppose you start with eight pennies and flip one fair coin. If the coin comes up heads, you get to keep all your pennies; if the coin comes up tails, you have to give half of them back. Let $X$ be the total number of pennies you have at the end. Compute $E(X)$.", "answer": "$E(X)=6$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.3.9", "question": "Let $Z \\sim$ Negative-Binomial $(3,1 / 4)$. Compute $P(Z \\leq 2)$.", "answer": "53/512", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.5.7", "question": "Suppose we repeat a certain experiment 2000 times and obtain a sample average of -5 and a standard error of 17 . In terms of this, specify an interval that is virtually certain to contain the experiment's (unknown) true mean $\\mu$.", "answer": "$(-6.1404, -3.8596)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.6.5", "question": "Suppose $P([0,1])=1$, but $P([1 / n, 1])=0$ for all $n=1,2,3, \\ldots$. What must $P(\\{0\\})$ be?", "answer": "1", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.6.7", "question": "Let $X_{1}, X_{2}, \\ldots, X_{n+1}$ be i.i.d. with distribution $N(0,1)$. Find a value of $C$ such that\n$$\nC \\frac{X_{1}}{\\sqrt{X_{2}^{2}+\\cdots+X_{n}^{2}+X_{n+1}^{2}}} \\sim t(n) .\n$$", "answer": "$C=\\sqrt{n}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.3.3", "question": "Consider flipping two fair coins. Let $X=1$ if the first coin is heads, and $X=0$ if the first coin is tails. Let $Y=1$ if the second coin is heads, and $Y=5$ if the second coin is tails. Let $Z=X Y$. What is the probability function of $Z$ ?", "answer": "$p_{Z}(1)=p_{Z}(5)=1 / 4, p_{Z}(0)=1 / 2$, and $p_{Z}(z)=0$ otherwise", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.2.9", "question": "Suppose $S=\\{1,2,3,4\\}$, and $P(\\{1\\})=1 / 12$, and $P(\\{1,2\\})=1 / 6$, and $P(\\{1,2,3\\})=1 / 3$. Compute $P(\\{1\\}), P(\\{2\\}), P(\\{3\\})$, and $P(\\{4\\})$.", "answer": "$P(\\{1\\})=1 / 12, P(\\{2\\})=1 / 12, P(\\{3\\})=1 / 6, P(\\{4\\})=2 / 3$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.1.3", "question": "Suppose that the lifelengths (in thousands of hours) of light bulbs are distributed $\\operatorname{Exponential}(\\theta)$, where $\\theta>0$ is unknown. If we observe $\\bar{x}=5.2$ for a sample of 20 light bulbs, record a representative likelihood function. Why is it that we only need to observe the sample average to obtain a representative likelihood?", "answer": "$L\\left(\\theta \\mid x_{1}, \\ldots, x_{20}\\right)=\\theta^{20} \\exp (-(20 \\bar{x}) \\theta)$ and $\\bar{x}$ is a sufficient statistic.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.2.9", "question": "Suppose you simultaneously roll two dice $n$ times and record the outcomes. Based on these values, how would you assess the null hypothesis that the outcome on each die is independent of the outcome on the other?", "answer": "Then there are 36 possible pairs $(i, j)$ for $i, j=1, \\ldots, 6$. Let $f_{i j}$ denote the frequency for $(i, j)$ and compute chi-squared statistic, $X^{2}=\\sum_{i=1}^{6} \\sum_{j=1}^{6}\\left(f_{i j}-\\right.$ $\\left.f_{i} \\cdot f_{\\cdot j} / n\\right)^{2} /\\left(f_{i \\cdot} \\cdot f_{\\cdot j} / n\\right)$. Compute the P-value $P\\left(\\chi^{2}(25)>X^{2}\\right)$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.1.9", "question": "Suppose a statistical model is given by $\\left\\{f_{1}, f_{2}\\right\\}$, where $f_{i}$ is an $N(i, 1)$ distribution. Compute the likelihood ratio $L(1 \\mid 0) / L(2 \\mid 0)$ and explain how you interpret this number.", "answer": "$L(1 \\mid 0) / L(2 \\mid 0)=4.4817$, the distribution $f_{1}$ is 4.4817 times more likely than $f_{2}$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.3.9", "question": "Suppose you know that the probability distribution of a variable $X$ is either $P_{1}$ or $P_{2}$. If you observe $X=1$ and $P_{1}(X=1)=0.75$ while $P_{2}(X=1)=0.001$, then what would you guess as the true distribution of $X$ ? Give your reasoning for this conclusion.", "answer": "$P_{1}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.6.9", "question": "Suppose $P([0,1 / 2])=1 / 3$. Must there be some $n$ such that $P([1 / n, 1 / 2])>$ $1 / 4$ ?", "answer": "No", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.8.2.7", "question": "Suppose you want to test the null hypothesis $H_{0}: \\mu=0$ based on a sample of $n$ from an $N(\\mu, 1)$ distribution, where $\\mu \\in\\{0,2\\}$. How large does $n$ have to be so that the power at $\\mu=2$, of the optimal size 0.05 test, is equal to 0.99 ?", "answer": "$n \\geq 4$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.1.5", "question": "Let $X \\sim \\operatorname{Geometric}(\\theta)$ and $Y \\sim \\operatorname{Poisson}(\\lambda)$. Compute $E(8 X-Y+12)$.", "answer": "$E(8 X-Y+12)=8((1-p) / p)-\\lambda+12$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.7.5", "question": "Suppose we are told only that $P(X>x)=1 / x^{2}$ for $x \\geq 1$, and $P(X>x)=1$ for $x<1$, but we are not told if $X$ is discrete or continuous or neither. Compute $E(X)$.", "answer": "$E(X)=2$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.7.1", "question": "Let $X \\sim \\operatorname{Bernoulli}(1 / 3)$, and let $Y=4 X-2$. Compute the joint $\\operatorname{cdf} F_{X, Y}$.", "answer": "$$\nF_{X, Y}(x, y)= \\begin{cases}0 & \\min [x,(y+2) / 4]<0 \\\\ 1 / 3 & 0 \\leq \\min [x,(y+2) / 4]<1 \\\\ 1 & \\min [x,(y+2) / 4] \\geq 1\\end{cases}\n$$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.3.13", "question": "Let $X$ and $Y$ be independent, with $X \\sim \\operatorname{Bernoulli}(1 / 2)$ and $Y \\sim \\operatorname{Bernoulli}(1 / 3)$. Let $Z=X+Y$ and $W=X-Y$. Compute $\\operatorname{Cov}(Z, W)$ and $\\operatorname{Corr}(Z, W)$.", "answer": "$\\operatorname{Cov}(Z, W)=0, \\operatorname{Corr}(Z, W)=0$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.7.4.7", "question": "Determine Jeffreys' prior for the Bernoulli $(\\theta)$ model and determine the posterior distribution of $\\theta$ based on this prior.", "answer": "Jeffreys' prior is $\\sqrt{n} \\theta^{-1 / 2}(1-\\theta)^{-1 / 2}$. The posterior distribution of $\\theta$ is $\\operatorname{Beta}(n \\bar{x}$ $+1 / 2, n(1-\\bar{x})+1 / 2)$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.5.7", "question": "Suppose that a statistical model is given by the family of $N\\left(\\mu, \\sigma_{0}^{2}\\right)$ distributions where $\\theta=\\mu \\in R^{1}$ is unknown, while $\\sigma_{0}^{2}$ is known. If our interest is in making inferences about the first quartile of the true distribution, then determine $\\psi(\\mu)$.", "answer": "$\\psi(\\mu)=\\mu+\\sigma_{0} z_{0.25}$, where $z_{0.25}$ satisfies $\\Phi\\left(z_{0.25}\\right)=0.25$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.7.7", "question": "Suppose $P(W>w)=e^{-5 w}$ for $w \\geq 0$ and $P(W>w)=1$ for $w<0$. Compute $E(W)$.", "answer": "$E(W)=1 / 5$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.1.3", "question": "Suppose that an urn contains a proportion $p$ of chips labelled 0 and proportion $1-p$ of chips labelled 1. For a sample of $n=2$, drawn with replacement, determine the distribution of the sample mean.", "answer": "If $Z$ is the sample mean, then $P(Z=0)=p^{2}, P(Z=0.5)=2 p(1-p)$, and $P(Z=1)=(1-p)^{2}$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.5.7", "question": "Suppose a baseball pitcher throws fastballs $80 \\%$ of the time and curveballs $20 \\%$ of the time. Suppose a batter hits a home run on $8 \\%$ of all fastball pitches, and on $5 \\%$ of all curveball pitches. What is the probability that this batter will hit a home run on this pitcher's next pitch?", "answer": "0.074", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.11.4.1", "question": "Suppose we define a process $\\left\\{X_{n}\\right\\}$ as follows. Given $X_{n}$, with probability $3 / 8$ we let $X_{n+1}=X_{n}-4$, while with probability $5 / 8$ we let $X_{n+1}=X_{n}+C$. What value of $C$ will make $\\left\\{X_{n}\\right\\}$ be a martingale?", "answer": "$C=12 / 5$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.3.11", "question": "Let $Y \\sim \\operatorname{Binomial}(10, \\theta)$. Compute $P(Y=10)$.", "answer": "$\\theta^{10}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.1.11", "question": "Suppose we have a statistical model $\\left\\{f_{\\theta}: \\theta \\in[0,1]\\right\\}$ and we observe $x_{0}$. Is it true that $\\int_{0}^{1} L\\left(\\theta \\mid x_{0}\\right) d \\theta=1$ ? Explain why or why not.", "answer": "No", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.6.3", "question": "Let $X \\sim N(3,5)$ and $Y \\sim N(-7,2)$ be independent. Find values of $C_{1} \\neq$ $0, C_{2}, C_{3} \\neq 0, C_{4}, C_{5}$ so that $C_{1}\\left(X+C_{2}\\right)^{2}+C_{3}\\left(Y+C_{4}\\right)^{2} \\sim \\chi^{2}\\left(C_{5}\\right)$.", "answer": "$C_{1}=1 / 5, C_{2}=-3, C_{3}=1 / 2, C_{4}=7, C_{5}=2$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.6.1", "question": "Let $Z \\sim$ Poisson(3). Use Markov's inequality to get an upper bound on $P(Z \\geq$ $7)$.", "answer": "$3 / 7$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.2.13", "question": "Explain why it is not possible that the function $\\theta^{3} \\exp \\left(-(\\theta-5.3)^{2}\\right)$ for $\\theta \\in R^{1}$ is a likelihood function.", "answer": "A likelihood function cannot take negative values.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.11.6.5", "question": "Let $\\{N(t)\\}_{t \\geq 0}$ be a Poisson process with intensity $a>0$. Compute (with explanation) the conditional probability $P\\left(N_{2.6}=2 \\mid N_{2.9}=2\\right)$.", "answer": "$P\\left(N_{2.6}=2 \\mid N_{2.9}=2\\right)=(2.6 / 2.9)^{2}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.2.7", "question": "Suppose that $X \\sim \\operatorname{Gamma}(3,6)$. What value would you record as a prediction of a future value of $X$ ? How would you justify your choice?", "answer": "The mode is $1 / 3$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.4.5", "question": "Let $Y=3 X+4$. Compute $m_{Y}(s)$ in terms of $m_{X}$.", "answer": "$m_{Y}(s)=e^{4 s} m_{X}(3 s)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.4.7", "question": "Determine the empirical distribution function based on the sample given below.\n\\begin{tabular}{|rrrrr|}\n\\hline 1.06 & -1.28 & 0.40 & 1.36 & -0.35 \\\\\n-1.42 & 0.44 & -0.58 & -0.24 & -1.34 \\\\\n0.00 & -1.02 & -1.35 & 2.05 & 1.06 \\\\\n0.98 & 0.38 & 2.13 & -0.03 & -1.29 \\\\\n\\hline\n\\end{tabular}\n\nUsing the empirical cdf, determine the sample median, the first and third quartiles, and the interquartile range. What is your estimate of $F(2)$ ?", "answer": "The sample median is estimated by -0.03 and the estimate of the first quartile is -1.28 , and for the third quartile is 0.98 . Also $\\hat{F}(2)=\\hat{F}(1.36)=0.90$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.2.15", "question": "Suppose basketball teams $A$ and $B$ each have five players and that each member of team A is being \"guarded\" by a unique member of team B. Suppose it is noticed that each member of team A is taller than the corresponding guard from team B. Does it necessarily follow that the mean height of team $\\mathrm{A}$ is larger than the mean height of team B? Why or why not?", "answer": "Yes", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.1.9", "question": "Suppose four fair coins are flipped, and let $Y$ be the number of pairs of coins which land the same way (i.e., the number of pairs that are either both heads or both tails). Compute the exact distribution of $Y$.", "answer": "$p_{Y}(y)=1 / 2$ for $y=1$, 2; otherwise, $p_{Y}(y)=0$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.4.11", "question": "Suppose $X$ has density $f$ and $f(x)>f(y)$ whenever $0<x<1<y<2$. Does it follow that $P(0<X<1)>P(1<X<2)$ ? Explain.", "answer": "Yes", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.2.13", "question": "Suppose darts are randomly thrown at a wall. Let $X$ be the distance (in centimeters) from the left edge of the dart's point to the left end of the wall, and let $Y$ be the distance from the right edge of the dart's point to the left end of the wall. Assume the dart's point is 0.1 centimeters thick, and that $E(X)=214$. Compute $E(Y)$.", "answer": "$E(Y)=214.1$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.2.9", "question": "Suppose $X$ has density function $f(x)=3 / 20\\left(x^{2}+x^{3}\\right)$ for $0<x<2$, otherwise $f(x)=0$. Compute each of $E(X), E\\left(X^{2}\\right)$, and $E\\left(X^{3}\\right)$, and rank them from largest to smallest.", "answer": "Let $\\mu_{k}=E\\left(X^{k}\\right)$, then $\\mu_{1}=39 / 25, \\mu_{2}=64 / 25, \\mu_{3}=152 / 35$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.8.1.9", "question": "Suppose a statistical model comprises all continuous distributions on $R^{1}$. Based on a sample of $n$, determine a UMVU estimator of $P((-1,1))$, where $P$ is the true probability measure. Justify your answer.", "answer": "$n^{-1} \\sum_{i=1}^{n} I_{(-1,1)}\\left(X_{i}\\right)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.3.9", "question": "A coin was tossed $n=1000$ times, and the proportion of heads observed was 0.51. Do we have evidence to conclude that the coin is unfair?", "answer": "$\\mathrm{P}$-value $=0.527$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.4.1", "question": "Suppose we obtained the following sample from a distribution that we know has its first six moments. Determine an approximate 0.95-confidence interval for $\\mu_{3}$.\n\\begin{tabular}{|rrrrrrrrrr|}\n3.27 & -1.24 & 3.97 & 2.25 & 3.47 & -0.09 & 7.45 & 6.20 & 3.74 & 4.12 \\\\\n1.42 & 2.75 & -1.48 & 4.97 & 8.00 & 3.26 & 0.15 & -3.64 & 4.88 & 4.55 \\\\\n\\hline\n\\end{tabular}", "answer": "$m_{3} \\pm z_{(1+\\gamma) / 2} s_{3} / \\sqrt{n}=(26.027,151.373)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.1.15", "question": "Suppose we have a quantitative response variable $Y$ and two categorical predictor variables $W$ and $X$, both taking values in $\\{0,1\\}$. Suppose the conditional distributions of $Y$ are given by\n$$\n\\begin{aligned}\n& Y \\mid W=0, X=0 \\sim N(2,5) \\\\\n& Y \\mid W=1, X=0 \\sim N(3,5) \\\\\n& Y \\mid W=0, X=1 \\sim N(4,5) \\\\\n& Y \\mid W=1, X=1 \\sim N(4,5) .\n\\end{aligned}\n$$\nDoes $W$ have a relationship with $Y$ ? Does $X$ have a relationship with $Y$ ? Explain your answers.", "answer": "$W$ has a relationship with $Y$ and $X$ has a relationship with $Y$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.4.7", "question": "Let $M>0$, and suppose $f(x)=c x^{2}$ for $0<x<M$, otherwise $f(x)=0$. For what value of $c$ (depending on $M$ ) is $f$ a density?", "answer": "$c=3 / M^{3}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.11.4.5", "question": "Let $\\left\\{X_{n}\\right\\}$ be a martingale, with initial value $X_{0}=5$. Suppose we know that $P\\left(X_{8}=3\\right)+P\\left(X_{8}=4\\right)+P\\left(X_{8}=6\\right)=1$, i.e., $X_{8}$ is always either 3 , 4, or 6 . Suppose further that $P\\left(X_{8}=3\\right)=2 P\\left(X_{8}=6\\right)$. Compute $P\\left(X_{8}=4\\right)$.", "answer": "$P\\left(X_{n}=4\\right)=5 / 8$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.6.11", "question": "Let $X$ have density function $f_{X}(x)=(1 / 2) \\sin (x)$ for $0<x<\\pi$, otherwise $f_{X}(x)=0$. Let $Y=X^{2}$. Compute the density function $f_{Y}(y)$ for $Y$.", "answer": "$f_{Y}(y)=y^{-1 / 2} \\sin \\left(y^{1 / 2}\\right) / 4$ for $0<y<\\pi^{2}$ and 0 otherwise", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.6.13", "question": "Let $X \\sim \\operatorname{Normal}(0,1)$. Let $Y=X^{3}$. Compute the density function $f_{Y}(y)$ for $Y$.", "answer": "$f_{Y}(y)=(2 \\pi)^{-1 / 2}\\left(3|y|^{2 / 3}\\right)^{-1} \\exp \\left(-|y|^{2 / 3} / 2\\right)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.3.11", "question": "Suppose that $X$ and $Y$ are random variables such that a regression model describes the relationship between $Y$ and $X$. If $E(Y \\mid X)=\\beta_{1}+\\beta_{2} X^{2}$, then discuss whether or not this is a simple linear regression model (perhaps involving a predictor other than $X$ ).", "answer": "We can write $E(Y \\mid X)=E\\left(Y \\mid X^{2}\\right)$ in this case and $E\\left(Y \\mid X^{2}\\right)=\\beta_{1}+\\beta_{2} X^{2}$, so this is a simple linear regression model but the predictor is $X^{2}$ not $X$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.6.9", "question": "Let $X \\sim N(3,5)$ and $Y \\sim N(-7,2)$ be independent. Find values of $C_{1}, C_{2}, C_{3}$, $C_{4}, C_{5}, C_{6}, C_{7}$ so that\n$$\n\\frac{C_{1}\\left(X+C_{2}\\right)^{C_{3}}}{\\left(Y+C_{4}\\right)^{C_{5}}} \\sim F\\left(C_{6}, C_{7}\\right) .\n$$", "answer": "$C_{1}=2 / 5, C_{2}=-3, C_{3}=2, C_{4}=7, C_{5}=2, C_{6}=1, C_{7}=1$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.5.3", "question": "Describe a Monte Carlo approximation of $\\int_{0}^{\\infty} e^{-5 x-14 x^{2}} d x$. (Hint: Remember the Exponential(5) distribution.)", "answer": "This integral equals $(1 / 5) E\\left(e^{-14 Z^{2}}\\right)$, where $Z \\sim$ Exponential(5).", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.5.1", "question": "Describe a Monte Carlo approximation of $\\int_{-\\infty}^{\\infty} \\cos ^{2}(x) e^{-x^{2} / 2} d x$.", "answer": "The integral equals $\\sqrt{2 \\pi} E\\left(\\cos ^{2}(Z)\\right)$, where $Z \\sim N(0,1)$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.5.5", "question": "Suppose we deal five cards from an ordinary 52-card deck. What is the conditional probability that the hand contains all four aces, given that the hand contains at least four aces?", "answer": "1", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.3.19", "question": "Suppose an urn contains 1000 balls - one of these is black, and the other 999 are white. Suppose that 100 balls are randomly drawn from the urn with replacement. Use the appropriate Poisson distribution to approximate the probability that five black balls are observed.", "answer": "$P(X=5) \\approx\\left((100 / 1000)^{5} / 5\\right.$ ! $) \\exp \\{-100 / 1000\\}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.2.5", "question": "If $\\left(x_{1}, \\ldots, x_{n}\\right)$ is a sample from a $\\operatorname{Gamma}\\left(\\alpha_{0}, \\theta\\right)$ distribution, where $\\alpha_{0}>0$ and $\\theta \\in(0, \\infty)$ is unknown, then determine the MLE of $\\theta$.", "answer": "$\\hat{\\theta}=\\alpha_{0} / \\bar{x}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.2.11", "question": "Suppose men's heights (in centimeters) follow the distribution $N\\left(174,20^{2}\\right)$, while those of women follow the distribution $N\\left(160,15^{2}\\right)$. Compute the mean total height of a man-woman married couple.", "answer": "334", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.2.11", "question": "Suppose $S=\\{1,2,3\\}$, and $P(\\{1\\})=P(\\{2\\})+1 / 6$, and $P(\\{3\\})=2 P(\\{2\\})$. Compute $P(\\{1\\}), P(\\{2\\})$, and $P(\\{3\\})$.", "answer": "$P(\\{2\\})=5 / 24, P(\\{1\\})=3 / 8, P(\\{3\\})=5 / 12$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.4.7", "question": "Let $Y \\sim$ Poisson $(\\lambda)$. Compute $E\\left(Y^{3}\\right)$, the third moment of $Y$.", "answer": "$m_{Y}^{\\prime \\prime \\prime}(s)=e^{\\lambda\\left(e^{s}-1\\right)} e^{s} \\lambda\\left(1+3 e^{s} \\lambda+e^{2 s} \\lambda^{2}\\right), E\\left(Y^{3}\\right)=m_{Y}^{\\prime \\prime \\prime}(0)=\\lambda\\left(1+3 \\lambda+\\lambda^{2}\\right)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.2.11", "question": "Suppose that a chi-squared test is carried out, based on a random sample of $n$ from a population, to assess whether or not two categorical variables $X$ and $Y$ are independent. Suppose the P-value equals 0.001 and the investigator concludes that there is evidence against independence. Discuss how you would check to see if the deviation from independence was of practical significance.", "answer": "We look at the differences $\\left|f_{i j}-f i \\cdot f_{\\cdot j} / n\\right|$ to see how big these are.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.6.5", "question": "Let $X \\sim \\operatorname{Exponential}(\\lambda)$. Let $Y=X^{3}$. Compute the density $f_{Y}$ of $Y$.", "answer": "$f_{Y}(y)$ equals $(\\lambda / 3) y^{-2 / 3} e^{-\\lambda y^{1 / 3}}$ for $y>0$ and otherwise equals 0 .", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.1.5", "question": "Suppose that a symmetrical die is tossed $n=20$ independent times. Work out the exact sampling distribution of the maximum of this sample.", "answer": "For $1 \\leq j \\leq 6, P(\\max =j)=(j / 6)^{20}-((j-1) / 6)^{20}$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.5.11", "question": "Suppose that a statistical model is given by the family of $N\\left(\\mu, \\sigma^{2}\\right)$ distributions where $\\theta=\\left(\\mu, \\sigma^{2}\\right) \\in R^{1} \\times R^{+}$is unknown. If our interest is in making inferences about the distribution function evaluated at 3 , then determine $\\psi\\left(\\mu, \\sigma^{2}\\right)$.", "answer": "$\\psi\\left(\\mu, \\sigma^{2}\\right)=\\Phi((3-\\mu) / \\sigma)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.11.4.3", "question": "Suppose we define a process $\\left\\{X_{n}\\right\\}$ as follows. Given $X_{n}$, with probability $p$ we let $X_{n+1}=2 X_{n}$, while with probability $1-p$ we let $X_{n+1}=X_{n} / 2$. What value of $p$ will make $\\left\\{X_{n}\\right\\}$ be a martingale?", "answer": "$p=1 / 3$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.4.7", "question": "Suppose we keep dealing cards from an ordinary 52-card deck until the first jack appears. What is the probability that at least 10 cards go by before the first jack?", "answer": "$\\left(\\begin{array}{c}48 \\\\ 10\\end{array}\\right) /\\left(\\begin{array}{c}52 \\\\ 10\\end{array}\\right)=246 / 595=0.4134$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.11.5.9", "question": "Let $X_{t}=10-1.5 t+4 B_{t}$. Compute $E\\left(X_{3} X_{5}\\right)$.", "answer": "$E\\left(X_{3} X_{5}\\right)=61.75$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.4.3", "question": "Suppose we flip 100 fair independent coins. What is the probability that at least three of them are heads? (Hint: You may wish to use (1.3.1).)", "answer": "$1-5051 / 2^{100}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.4.5", "question": "Verify that the third moment of an $N\\left(\\mu, \\sigma^{2}\\right)$ distribution is given by $\\mu_{3}=$ $\\mu^{3}+3 \\mu \\sigma^{2}$. Because the normal distribution is specified by its first two moments, any characteristic of the normal distribution can be estimated by simply plugging in the MLE estimates of $\\mu$ and $\\sigma^{2}$. Compare the method of moments estimator of $\\mu_{3}$ with this plug-in MLE estimator, i.e., determine whether they are the same or not.", "answer": "From the mgf, $m_{X}^{\\prime \\prime \\prime}(0)=3 \\sigma^{2} \\mu+\\mu^{3}$. The plug-in estimator is $\\hat{\\mu}_{3}=3\\left(m_{2}-m_{1}^{2}\\right) \\times$ $m_{1}+m_{1}^{3}$, while the method of moments estimator of $\\mu_{3}$ is $m_{3}=\\frac{1}{n} \\sum x_{i}^{3}$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.1.5", "question": "The following data were generated from an $N(\\mu, 1)$ distribution by a student. Unfortunately, the student forgot which value of $\\mu$ was used, so we are uncertain about the correct probability distribution to use to describe the variation in the data.\n\n\\begin{tabular}{rrrrrrrr|}\n\\hline 0.2 & -0.7 & 0.0 & -1.9 & 0.7 & -0.3 & 0.3 & 0.4 \\\\\n0.3 & -0.8 & 1.5 & 0.1 & 0.3 & -0.7 & -1.8 & 0.2 \\\\\n\\hline\n\\end{tabular}\n\nCan you suggest a plausible value for $\\mu$ ? Explain your reasoning.", "answer": "$\\bar{x}=-0.1375$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.2.11", "question": "Suppose you are measuring the volume of a cubic box in centimeters by taking repeated independent measurements of one of the sides. Suppose it is reasonable to assume that a single measurement follows an $N\\left(\\mu, \\sigma_{0}^{2}\\right)$ distribution, where $\\mu$ is unknown and $\\sigma_{0}^{2}$ is known. Based on a sample of measurements, you obtain the MLE of $\\mu$ as 3.2 $\\mathrm{cm}$. What is your estimate of the volume of the box? How do you justify this in terms of the likelihood function?", "answer": "$\\hat{\\mu}^{3}=32.768 \\mathrm{~cm}^{3}$ is the MLE", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.3.13", "question": "Suppose that a simple linear model is fit to data. An analysis of the residuals indicates that there is no reason to doubt that the model is correct; the ANOVA test indicates that there is substantial evidence against the null hypothesis of no relationship between the response and predictor. The value of $R^{2}$ is found to be 0.05 . What is the interpretation of this number and what are the practical consequences?", "answer": "$R^{2}=0.05$ indicates that the linear model explains only $5 \\%$ of the variation in the response, so the model will not have much predictive power.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.3.1", "question": "Suppose there are three coins - one is known to be fair, one has probability $1 / 3$ of yielding a head on a single toss, and one has probability $2 / 3$ for head on a single toss. A coin is selected (not randomly) and then tossed five times. The goal is to make an inference about which of the coins is being tossed, based on the sample. Fully describe a statistical model for a single response and for the sample.", "answer": "The statistical model for a single response consists of three probability functions \\{Bernoulli(1/2), Bernoulli(1/3), Bernoulli(2/3)\\}.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.1.7", "question": "Let $X \\sim \\operatorname{Binomial}(80,1 / 4)$, and let $Y \\sim \\operatorname{Poisson}(3 / 2)$. Assume $X$ and $Y$ are independent. Compute $E(X Y)$.", "answer": "$E(X Y)=30$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.3.3", "question": "Marks on an exam in a statistics course are assumed to be normally distributed with unknown mean but with variance equal to 5 . A sample of four students is selected, and their marks are $52,63,64,84$. Assess the hypothesis $H_{0}: \\mu=60$ by computing the relevant $\\mathrm{P}$-value and compute a 0.95 -confidence interval for the unknown $\\mu$.", "answer": "P-value $=0.000$ and 0.95 -confidence interval is $(63.56,67.94)$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.4.9", "question": "Suppose two measurements, $Y_{1}$ and $Y_{2}$, corresponding to different treatments, are taken on the same individual who has been randomly sampled from a population $\\Pi$. Suppose that $Y_{1}$ and $Y_{2}$ have the same variance and are negatively correlated. Our goal is to compare the treatment means. Explain why it would have been better to have randomly sampled two individuals from $\\Pi$ and applied the treatments to these individuals separately. (Hint: Consider $\\operatorname{Var}\\left(Y_{1}-Y_{2}\\right)$ in these two sampling situations.)", "answer": "When $Y_{1}$ and $Y_{2}$ are measured on the same individual, we have that $\\operatorname{Var}\\left(Y_{1}-\\right.$ $\\left.Y_{2}\\right)=2\\left(\\operatorname{Var}\\left(Y_{1}\\right)-\\operatorname{Cov}\\left(Y_{1}, Y_{2}\\right)\\right)>2 \\operatorname{Var}\\left(Y_{1}\\right)$ since $\\operatorname{Cov}\\left(Y_{1}, Y_{2}\\right)<0$. If we had measured $Y_{1}$ and $Y_{2}$ on independently randomly selected individuals, then we would have that $\\operatorname{Var}\\left(Y_{1}-Y_{2}\\right)=2 \\operatorname{Var}\\left(Y_{1}\\right)$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.2.3", "question": "Suppose $S=\\{1,2,3\\}$, with $P(\\{1\\})=1 / 2$ and $P(\\{1,2\\})=2 / 3$. What must $P(\\{2\\})$ be?", "answer": "$P(\\{2\\})=1 / 6$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.7.4.9", "question": "Suppose a student wants to put a prior on the mean grade out of 100 that their class will obtain on the next statistics exam. The student feels that a normal prior centered at 66 is appropriate and that the interval $(40,92)$ should contain $99 \\%$ of the marks. Fully identify the prior.", "answer": "The prior distribution is $\\theta \\sim N\\left(66, \\sigma^{2}\\right)$ with $\\sigma^{2}=101.86$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.7.9", "question": "Suppose the cdf of $W$ is given by $F_{W}(w)=0$ for $w<10, F_{W}(w)=w-10$ for $10 \\leq w \\leq 11$, and by $F_{W}(w)=1$ for $w>11$. Compute $E(W)$. (Hint: Remember that $F_{W}(w)=P(W \\leq w)=1-P(W>w)$.)", "answer": "$E(W)=21 / 2$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.8.1.3", "question": "Suppose that $\\left(x_{1}, \\ldots, x_{n}\\right)$ is a sample from an $N\\left(\\mu, \\sigma_{0}^{2}\\right)$ distribution, where $\\mu \\in$ $R^{1}$ is unknown and $\\sigma_{0}^{2}$ is known. Determine a UMVU estimator of the second moment $\\mu^{2}+\\sigma_{0}^{2}$.", "answer": "$\\bar{x}^{2}+(1-1 / n) \\sigma_{0}^{2}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.6.5", "question": "Let $W \\sim \\operatorname{Binomial}(100,1 / 2)$, as in the number of heads when flipping 100 fair coins. Use Chebychev's inequality to get an upper bound on $P(|W-50| \\geq 10)$.", "answer": "$1 / 4$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.3.17", "question": "A P-value was computed to assess the hypothesis $H_{0}: \\psi(\\theta)=0$ and the value 0.22 was obtained. The investigator says this is strong evidence that the hypothesis is correct. How do you respond?", "answer": "The P-value 0.22 does not imply the null hypothesis is correct. It may be that we have just not taken a large enough sample size to detect a difference.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.11.6.3", "question": "Let $\\{N(t)\\}_{t \\geq 0}$ be a Poisson process with intensity $a=1 / 3$. Compute $P\\left(N_{2}=\\right.$ 6) and $P\\left(N_{3}=5\\right)$.", "answer": "$P\\left(N_{2}=6\\right)=e^{-2 / 3}(2 / 3)^{6} / 6$ !, $P\\left(N_{3}=5\\right)=e^{-3 / 3}(3 / 3)^{5} / 5$ !", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.2.9", "question": "Suppose that $X \\sim \\operatorname{Geometric}(1 / 3)$. What value would you record as a prediction of a future value of $X$ ?", "answer": "The mode is $x=0$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.2.15", "question": "If two functions of $\\theta$ are equivalent versions of the likelihood when one is a positive multiple of the other, then when are two log-likelihood functions equivalent?", "answer": "Equivalent log-likelihood functions differ by an additive constant.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.5.13", "question": "Suppose that a statistical model is given by the family of $\\operatorname{Bernoulli}(\\theta)$ distributions where $\\theta \\in \\Omega=[0,1]$. If our interest is in making inferences about the probability that in two independent observations from this model we obtain a 0 and a 1 , then determine $\\psi(\\theta)$.", "answer": "$\\psi(\\theta)=2 \\theta(1-\\theta)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.4.5.9", "question": "Suppose a certain experiment has probability $\\theta$ of success, where $0<\\theta<1$ but $\\theta$ is unknown. Suppose we repeat the experiment 1000 times, of which 400 are successes and 600 are failures. Compute an interval of values that are virtually certain to contain $\\theta$.", "answer": "$(0.354,0.447)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.2.7", "question": "Suppose a university is composed of $55 \\%$ female students and $45 \\%$ male students. A student is selected to complete a questionnaire. There are 25 questions on the questionnaire administered to a male student and 30 questions on the questionnaire administered to a female student. If $X$ denotes the number of questions answered by a randomly selected student, then compute $P(X=x)$ for every real number $x$.", "answer": "$P(X=25)=0.45, P(X=30)=0.55$, and $P(X=x)=0$ otherwise", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.1.5", "question": "Let $A$ and $B$ be events, and let $X=I_{A} \\cdot I_{B}$. Is $X$ an indicator function? If yes, then of what event?", "answer": "Yes, for $A \\cap B$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.3.11", "question": "Suppose a possibly biased die is rolled 30 times and that the face containing two pips comes up 10 times. Do we have evidence to conclude that the die is biased?", "answer": "P-value $=0.014$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.8.1.5", "question": "Suppose that $\\left(x_{1}, \\ldots, x_{n}\\right)$ is a sample from an $N\\left(\\mu, \\sigma_{0}^{2}\\right)$ distribution, where $\\mu \\in$ $R^{1}$ is unknown and $\\sigma_{0}^{2}$ is known. Is $2 \\bar{x}+3$ a UMVU estimator of anything? If so, what is it UMVU for? Justify your answer.", "answer": "UMVU for $5+2 \\mu$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.3.7", "question": "Suppose your team has a $40 \\%$ chance of winning or tying today's game and has a $30 \\%$ chance of winning today's game. What is the probability that today's game will be a tie?", "answer": "$10 \\%$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.3.6.13", "question": "Suppose a species of beetle has length 35 millimeters on average. Find an upper bound on the probability that a randomly chosen beetle of this species will be over 80 millimeters long.", "answer": "$7 / 16$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.5.9", "question": "Suppose that a statistical model is given by the family of $N\\left(\\mu, \\sigma_{0}^{2}\\right)$ distributions where $\\theta=\\mu \\in R^{1}$ is unknown, while $\\sigma_{0}^{2}$ is known. If our interest is in making inferences about the distribution function evaluated at 3, then determine $\\psi(\\mu)$.", "answer": "$\\psi(\\mu)=\\Phi\\left((3-\\mu) / \\sigma_{0}\\right)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.1.5", "question": "Suppose that $X$ is a random variable and $Y=X^{2}$. Determine whether or not $X$ and $Y$ are related. What happens when $X$ has a degenerate distribution?", "answer": "The conditional distributions $P(Y=y \\mid X=x)$ will change with $x$ whenever $X$ is not degenerate.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.5.2.5", "question": "Suppose that $X \\sim N(10,2)$. What value would you record as a prediction of a future value of $X$ ? How would you justify your choice?", "answer": "$x=10$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.3.1", "question": "Suppose measurements (in centimeters) are taken using an instrument. There is error in the measuring process and a measurement is assumed to be distributed $N\\left(\\mu, \\sigma_{0}^{2}\\right)$, where $\\mu$ is the exact measurement and $\\sigma_{0}^{2}=0.5$. If the $(n=10)$ measurements 4.7, 5.5, 4.4, 3.3, 4.6, 5.3, 5.2, 4.8, 5.7, 5.3 were obtained, assess the hypothesis $H_{0}: \\mu=5$ by computing the relevant P-value. Also compute a 0.95 -confidence interval for the unknown $\\mu$.", "answer": "$\\mathrm{P}$-value $=0.592$ and 0.95 -confidence interval is $(4.442,5.318)$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.1.19", "question": "Suppose a variable $X$ takes the values 1 and 2 on a population and the conditional distributions of $Y$ given $X$ are $N(0,5)$ when $X=1$, and $N(0,7)$ when $X=2$. Determine whether $X$ and $Y$ are related and if so, describe their relationship.", "answer": "$X$ and $Y$ are related. We see that only the variance of the conditional distribution changes as we change $X$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.3.3", "question": "Suppose that $\\left(x_{1}, \\ldots, x_{n}\\right)$ is a sample from the Exponential $(\\theta)$, where $\\theta>0$ is unknown. What is the least-squares estimate of the mean of this distribution?", "answer": "$\\bar{x}$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.2.1", "question": "Consider flipping two independent fair coins. Let $X$ be the number of heads that appear. Compute $P(X=x)$ for all real numbers $x$.", "answer": "$P(X=0)=P(X=2)=1 / 4, P(X=1)=1 / 2, P(X=x)=0$ for $x \\neq 0,1,2$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.3.7", "question": "Let $X \\sim \\operatorname{Binomial}(12, \\theta)$. For what value of $\\theta$ is $P(X=11)$ maximized?", "answer": "$\\theta=11 / 12$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.2.4.5", "question": "Is the function defined by $f(x)=x / 3$ for $-1<x<2$ and 0 otherwise, a density? Why or why not?", "answer": "No", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.7.1.7", "question": "Suppose we have a sample\n\n\\begin{tabular}{|llllllllll|}\n\\hline 6.56 & 6.39 & 3.30 & 3.03 & 5.31 & 5.62 & 5.10 & 2.45 & 8.24 & 3.71 \\\\\n4.14 & 2.80 & 7.43 & 6.82 & 4.75 & 4.09 & 7.95 & 5.84 & 8.44 & 9.36 \\\\\n\\hline\n\\end{tabular}\n\nfrom an $N\\left(\\mu, \\sigma^{2}\\right)$ distribution and we determine that a prior specified by $\\mu \\mid \\sigma^{2} \\sim$ $N\\left(3,4 \\sigma^{2}\\right), \\sigma^{-2} \\sim \\operatorname{Gamma}(1,1)$ is appropriate. Determine the posterior distribution of $\\left(\\mu, 1 / \\sigma^{2}\\right)$.", "answer": "$\\mu\\left|\\sigma^{2}, x_{1}, \\ldots, x_{n} \\sim N\\left(5.5353, \\frac{4}{81} \\sigma^{2}\\right), 1 / \\sigma^{2}\\right| x_{1}, \\ldots, x_{n} \\sim \\operatorname{Gamma}(11,41.737)$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.1.3.3", "question": "Suppose that an employee arrives late $10 \\%$ of the time, leaves early $20 \\%$ of the time, and both arrives late and leaves early $5 \\%$ of the time. What is the probability that on a given day that employee will either arrive late or leave early (or both)?", "answer": "$P$ (late or early or both) $=25 \\%$", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.10.2.7", "question": "Write out in full how you would generate a value from a Dirichlet $(1,1,1,1)$ distribution.", "answer": "We should first generate a value for $X_{1} \\sim \\operatorname{Dirichlet}(1,3)$. Then generate $U_{2}$ from the $\\operatorname{Beta}(1,2)$ distribution and set $X_{2}=\\left(1-X_{1}\\right) U_{2}$. Next generate $U_{3}$ from the $\\operatorname{Beta}(1,1)$ distribution and set $X_{3}=\\left(1-X_{1}-X_{2}\\right) U_{3}$. Finally, set $X_{4}=1-X_{1}-$ $X_{2}-X_{3}$.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Probability_and_Statistics-The_Science_of_Uncertainty", "question_number": "exercise.6.2.3", "question": "If $\\left(x_{1}, \\ldots, x_{n}\\right)$ is a sample from a $\\operatorname{Bernoulli}(\\theta)$ distribution, where $\\theta \\in[0,1]$ is unknown, then determine the MLE of $\\theta^{2}$.", "answer": "$\\psi(\\theta)=\\theta^{2}$ is $1-1$, and so $\\psi\\left(\\hat{\\theta}\\left(x_{1}, \\ldots, x_{n}\\right)\\right)=\\bar{x}^{2}$ is the MLE.", "license": "The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.", "data_topic": "college_math.probability"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.12", "question": "Determine if the system is consistent. If so, is the solution unique?\n$$\n\\begin{gathered}\nx+2 y+z-w=2 \\\\\nx-y+z+w=1 \\\\\n2 x+y-z=1 \\\\\n4 x+2 y+z=5\n\\end{gathered}\n$$", "answer": "There is no solution.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.5.19", "question": "Suppose that $A, B, C, D$ are $n \\times n$-matrices, and that all relevant matrices are invertible. Further, suppose that $(A+B)^{-1}=C B^{-1}$. Solve this equation for $A$ (in terms of $B$ and $C$ ), $B$ (in terms of $A$ and $C$ ), and $C$ (in terms of $A$ and $B$ ).", "answer": "To solve for $A$, we invert both sides of the equation $(A+B)^{-1}=C B^{-1}$ and use matrix algebra to get $A+B=\\left(C B^{-1}\\right)^{-1}=\\left(B^{-1}\\right)^{-1} C^{-1}=B C^{-1}$. Therefore, $A=B C^{-1}-B$.\n\nTo solve for $B$, we note that $A=B C^{-1}-B=B\\left(C^{-1}-I\\right)$. Multiplying both sides of the equation on the right by the inverse of $C^{-1}-I$, we get $B=A\\left(C^{-1}-I\\right)^{-1}$.\n\nTo solve for $C$, we take the original equation $(A+B)^{-1}=C B^{-1}$ and right-multiply both sides of the equation $B$. This yields $C=(A+B)^{-1} B$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.5.2.5", "question": "Are the following vectors linearly independent? If not, write one of them as a linear combination of the others.\n\n$$\n\\mathbf{u}=\\left[\\begin{array}{l}\n1 \\\\\n3 \\\\\n1\n\\end{array}\\right], \\quad \\mathbf{v}=\\left[\\begin{array}{l}\n1 \\\\\n4 \\\\\n2\n\\end{array}\\right], \\quad \\mathbf{w}=\\left[\\begin{array}{r}\n1 \\\\\n1 \\\\\n-1\n\\end{array}\\right]\n$$", "answer": "The vectors are linearly dependent. We have $\\mathbf{w}=3 \\mathbf{u}-2 \\mathbf{v}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.18", "question": "Simplify $\\|\\mathbf{u} \\times \\mathbf{v}\\|^{2}+(\\mathbf{u} \\cdot \\mathbf{v})^{2}-\\|\\mathbf{u}\\|^{2}\\|\\mathbf{v}\\|^{2}$.", "answer": "We have\n\n$$\n\\begin{aligned}\n\\|\\mathbf{u} \\times \\mathbf{v}\\|^{2} & =\\|\\mathbf{u}\\|^{2}\\|\\mathbf{v}\\|^{2} \\sin ^{2} \\theta \\\\\n& =\\|\\mathbf{u}\\|^{2}\\|\\mathbf{v}\\|^{2}\\left(1-\\cos ^{2} \\theta\\right) \\\\\n& =\\|\\mathbf{u}\\|^{2}\\|\\mathbf{v}\\|^{2}-\\|\\mathbf{u}\\|^{2}\\|\\mathbf{v}\\|^{2} \\cos ^{2} \\theta \\\\\n& =\\|\\mathbf{u}\\|^{2}\\|\\mathbf{v}\\|^{2}-(\\mathbf{u} \\cdot \\mathbf{v})^{2}\n\\end{aligned}\n$$\n\nwhich implies the expression equals 0 .", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.4", "question": "Find the area of the parallelogram determined by the vectors $\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 3\\end{array}\\right],\\left[\\begin{array}{r}4 \\\\ -2 \\\\ 1\\end{array}\\right]$.", "answer": "$\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 3\\end{array}\\right] \\times\\left[\\begin{array}{r}4 \\\\ -2 \\\\ 1\\end{array}\\right]=\\left[\\begin{array}{r}6 \\\\ 11 \\\\ -2\\end{array}\\right]$. The area is of the parallelogram is $\\sqrt{36+121+4}=\\sqrt{161}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.22", "question": "Suppose a system of equations has fewer equations than variables and you have found a solution to this system of equations. Is it possible that your solution is the only one? Explain.", "answer": "No. Consider $x+y+z=2$ and $x+y+z=1$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.7.3", "question": "Which of the following matrices are symmetric, antisymmetric, both, or neither?\n$$\nA=\\left[\\begin{array}{rr}\n0 & 1 \\\\\n-1 & 0\n\\end{array}\\right], \\quad B=\\left[\\begin{array}{ll}\n2 & 1 \\\\\n1 & 3\n\\end{array}\\right], \\quad C=\\left[\\begin{array}{rr}\n1 & 2 \\\\\n-2 & 0\n\\end{array}\\right], \\quad D=\\left[\\begin{array}{ll}\n0 & 0 \\\\\n0 & 0\n\\end{array}\\right] .\n\n$$", "answer": "$A$ is antisymmetric, $B$ is symmetric, $C$ is neither, and $D$ is both.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.8.4.2", "question": "Find the eigenvalues and eigenvectors of the matrix\n$$\n\\left[\\begin{array}{rrr}\n-13 & -28 & 28 \\\\\n4 & 9 & -8 \\\\\n-4 & -8 & 9\n\\end{array}\\right]\n$$\nOne eigenvalue is 3. Diagonalize if possible.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{l}\n2 \\\\\n0 \\\\\n1\n\\end{array}\\right],\\left[\\begin{array}{c}\n-2 \\\\\n1 \\\\\n0\n\\end{array}\\right]\\right\\} \\text { for eigenvalue } 1, \\quad\\left\\{\\left[\\begin{array}{c}\n7 \\\\\n-2 \\\\\n2\n\\end{array}\\right]\\right\\} \\text { for eigenvalue } 3 .\n$$\n\nThe matrix $P$ needed to diagonalize the above matrix is\n\n$$\n\\left[\\begin{array}{rrr}\n2 & -2 & 7 \\\\\n0 & 1 & -2 \\\\\n1 & 0 & 2\n\\end{array}\\right]\n$$\n\nand the diagonal matrix $D$ is\n\n$$\n\\left[\\begin{array}{lll}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 3\n\\end{array}\\right]\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.5.2.10", "question": "Here are some vectors.\n\n$$\n\\mathbf{u}_{1}=\\left[\\begin{array}{r}\n1 \\\\\n2 \\\\\n-2\n\\end{array}\\right], \\quad \\mathbf{u}_{2}=\\left[\\begin{array}{r}\n2 \\\\\n2 \\\\\n-4\n\\end{array}\\right], \\quad \\mathbf{u}_{3}=\\left[\\begin{array}{r}\n2 \\\\\n7 \\\\\n-4\n\\end{array}\\right], \\quad \\mathbf{u}_{4}=\\left[\\begin{array}{r}\n5 \\\\\n7 \\\\\n-10\n\\end{array}\\right], \\quad \\mathbf{u}_{5}=\\left[\\begin{array}{r}\n12 \\\\\n17 \\\\\n-24\n\\end{array}\\right] .\n$$\n\nDescribe the span of these vectors as the span of as few vectors as possible.", "answer": "$$\n\\left[\\begin{array}{rrrrr}\n1 & 2 & 2 & 5 & 12 \\\\\n1 & 2 & 7 & 7 & 17 \\\\\n-2 & -4 & -4 & -10 & -24\n\\end{array}\\right] \\simeq\\left[\\begin{array}{llllr}\n1 & 1 & 2 & 5 & 12 \\\\\n0 & 0 & 5 & 2 & 5 \\\\\n0 & 0 & 0 & 0 & 0\n\\end{array}\\right]\n$$\nLinearly independent subset: $\\left\\{\\mathbf{u}_{1}, \\mathbf{u}_{3}\\right\\}$. Since the rank is 2 , this is the smallest possible.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.1.3", "question": "What are the dimensions of the following matrices?\n$$\nA=\\left[\\begin{array}{rrr}\n1 & -2 & 0 \\\\\n4 & 3 & 2\n\\end{array}\\right], \\quad B=\\left[\\begin{array}{lll}\n3 & 4 & 1 \\\\\n1 & 3 & 1 \\\\\n6 & 2 & 2\n\\end{array}\\right], \\quad C=\\left[\\begin{array}{ll}\n1 & 0 \\\\\n4 & 0 \\\\\n2 & 0 \\\\\n0 & 0\n\\end{array}\\right]\n$$", "answer": "$2 \\times 3,3 \\times 3,4 \\times 2$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.3.1.6", "question": "Consider the following vector equation for a line in $\\mathbb{R}^{3}$ :\n$$\n\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz\n\\end{array}\\right]=\\left[\\begin{array}{l}\n1 \\\\\n2 \\\\\n0\n\\end{array}\\right]+t\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n1\n\\end{array}\\right]\n$$\nFind a new vector equation for the same line by doing the change of parameter $t=2-s$.", "answer": "We have\n\n$$\n\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz\n\\end{array}\\right]=\\left[\\begin{array}{l}\n1 \\\\\n2 \\\\\n0\n\\end{array}\\right]+(2-s)\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n1\n\\end{array}\\right]=\\left[\\begin{array}{l}\n3 \\\\\n2 \\\\\n2\n\\end{array}\\right]+s\\left[\\begin{array}{r}\n-1 \\\\\n0 \\\\\n-1\n\\end{array}\\right] .\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.5.3", "question": "Use Gauss-Jordan elimination to solve the system of equations $-8 x+2 y+5 z=18,-8 x+$ $3 y+5 z=13$, and $-4 x+y+5 z=19$.", "answer": "Solution is: $[x=-1, y=-5, z=4]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.10.7", "question": "Which of the following vectors, if any, are orthogonal?\n$\\mathbf{u}_{1}=\\left[\\begin{array}{c}1 \\\\ i \\\\ 0\\end{array}\\right]$,\n$\\mathbf{u}_{2}=\\left[\\begin{array}{c}1 \\\\ -i \\\\ 1\\end{array}\\right]$,\n$\\mathbf{u}_{3}=\\left[\\begin{array}{c}0 \\\\ 1 \\\\ -i\\end{array}\\right]$,\n$\\mathbf{u}_{4}=\\left[\\begin{array}{c}0 \\\\ 0 \\\\ 1\\end{array}\\right]$.", "answer": "$\\mathbf{u}_{1} \\perp \\mathbf{u}_{2}, \\mathbf{u}_{1} \\perp \\mathbf{u}_{4}$, and $\\mathbf{u}_{2} \\perp \\mathbf{u}_{3}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.4.12", "question": "Compute $A^{4}$, where\n$$\nA=\\left[\\begin{array}{ll}\n1 & -3 \\\\\n2 & -5\n\\end{array}\\right]\n$$\nHint: you can save some work by calculating $A^{2}$ times $A^{2}$.", "answer": "$\\left[\\begin{array}{rr}-71 & 168 \\\\ -112 & 265\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.11.9", "question": "Unitarily diagonalize the hermitian matrix $A=\\left[\\begin{array}{cc}2 & 1+i \\\\ 1-i & 3\\end{array}\\right]$.", "answer": "Eigenvalues: $\\lambda_{1}=1$ and $\\lambda_{2}=4$. Normalized eigenvectors: $\\mathbf{v}_{1}=\\frac{1}{\\sqrt{3}}\\left[\\begin{array}{c}1+i \\\\ -1\\end{array}\\right], \\mathbf{v}_{2}=\\frac{1}{\\sqrt{3}}\\left[\\begin{array}{c}1 \\\\ 1-i\\end{array}\\right]$.\n\n$$\nD=\\left[\\begin{array}{ll}\n1 & 0 \\\\\n0 & 4\n\\end{array}\\right], \\quad P=\\frac{1}{\\sqrt{3}}\\left[\\begin{array}{cc}\n1+i & 1 \\\\\n-1 & 1-i\n\\end{array}\\right] \\text {. }\n$$`", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.5.4", "question": "Consider the points $\\left(x_{1}, y_{1}\\right)=(-1,4),\\left(x_{2}, y_{2}\\right)=(0,-2),\\left(x_{3}, y_{3}\\right)=(1,4),\\left(x_{4}, y_{4}\\right)=$ $(2,2)$. Find the least squares parabola for these points.", "answer": "$y=1-x+x^{2}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.5", "question": "Find the area of the parallelogram with vertices $(-2,3,1),(2,1,1),(1,2,-1)$, and $(5,0,-1)$.", "answer": "Let $P=(-2,3,1), Q=(2,1,1), R=(1,2,-1)$, and $S=(5,0,-1)$. We have $\\overrightarrow{P Q} \\times \\overrightarrow{P R}=\\left[\\begin{array}{r}4 \\\\ -2 \\\\ 0\\end{array}\\right] \\times$ $\\left[\\begin{array}{r}3 \\\\ -1 \\\\ -2\\end{array}\\right]=\\left[\\begin{array}{l}5 \\\\ 8 \\\\ 2\\end{array}\\right]$. The area of the parallelogram is $\\|\\overrightarrow{P Q} \\times \\overrightarrow{P R}\\|=\\sqrt{5^{2}+8^{2}+2^{2}}=\\sqrt{93}$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.9.2.9", "question": "Assume $\\mathbf{u}, \\mathbf{v}, \\mathbf{w}$ are linearly independent elements of some vector space $V$. Consider the set of vectors\n$$\nR=\\left\\{2 \\mathbf{u}-\\mathbf{w}, \\mathbf{w}+\\mathbf{v}, 3 \\mathbf{v}+\\frac{1}{2} \\mathbf{u}\\right\\}\n$$\nDetermine whether $R$ is linearly independent.", "answer": "Therefore, the set $R$ is linearly independent.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.5.2.1", "question": "Which of the following vectors are redundant? If there are redundant vectors, write each of them as a linear combination of previous vectors.\n\n$$\n\\mathbf{u}_{1}=\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n1\n\\end{array}\\right], \\quad \\mathbf{u}_{2}=\\left[\\begin{array}{l}\n2 \\\\\n0 \\\\\n2\n\\end{array}\\right], \\quad \\mathbf{u}_{3}=\\left[\\begin{array}{l}\n1 \\\\\n2 \\\\\n1\n\\end{array}\\right], \\quad \\mathbf{u}_{4}=\\left[\\begin{array}{l}\n1 \\\\\n6 \\\\\n1\n\\end{array}\\right]\n$$", "answer": "$\\mathbf{u}_{2}$ and $\\mathbf{u}_{4}$ are redundant. We have $\\mathbf{u}_{2}=2 \\mathbf{u}_{1}$ and $\\mathbf{u}_{4}=3 \\mathbf{u}_{3}-2 \\mathbf{u}_{1}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.5.2.6", "question": "Find a linearly independent set of vectors that has the same span as the given vectors.\n\n$$\n\\mathbf{u}_{1}=\\left[\\begin{array}{l}\n2 \\\\\n0 \\\\\n3\n\\end{array}\\right], \\quad \\mathbf{u}_{2}=\\left[\\begin{array}{l}\n1 \\\\\n3 \\\\\n5\n\\end{array}\\right], \\quad \\mathbf{u}_{3}=\\left[\\begin{array}{l}\n3 \\\\\n3 \\\\\n8\n\\end{array}\\right], \\quad \\mathbf{u}_{4}=\\left[\\begin{array}{r}\n3 \\\\\n-3 \\\\\n1\n\\end{array}\\right]\n$$", "answer": "$\\left[\\begin{array}{rrrr}2 & 1 & 3 & 3 \\\\ 0 & 3 & 3 & -3 \\\\ 3 & 5 & 8 & 1\\end{array}\\right] \\simeq\\left[\\begin{array}{rrrr}1 & 4 & 5 & -2 \\\\ 0 & 1 & 1 & -1 \\\\ 0 & 0 & 0 & 0\\end{array}\\right]$. Linearly independent subset: $\\left\\{\\mathbf{u}_{1}, \\mathbf{u}_{2}\\right\\}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.5.8", "question": "Which of the following are unit vectors?\n$$\n\\mathbf{u}=\\frac{1}{2}\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right], \\quad \\mathbf{v}=\\frac{1}{3}\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right], \\quad \\mathbf{w}=\\frac{1}{2}\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right]\n$$", "answer": "Only $\\mathbf{w}$ is a unit vector.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.6.9", "question": "Find $\\operatorname{proj}_{\\mathbf{v}}(\\mathbf{w})$ where $\\mathbf{w}=\\left[\\begin{array}{r}1 \\\\ 2 \\\\ -2\\end{array}\\right]$ and $\\mathbf{v}=\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 3\\end{array}\\right]$.", "answer": "$\\frac{\\mathbf{v} \\cdot \\mathbf{w}}{\\mathbf{v} \\cdot \\mathbf{v}} \\mathbf{v}=\\frac{-5}{10}\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 3\\end{array}\\right]=\\left[\\begin{array}{c}-\\frac{1}{2} \\\\ 0 \\\\ -\\frac{3}{2}\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.6.14", "question": "Decompose the vector $\\mathbf{v}$ into $\\mathbf{v}=\\mathbf{a}+\\mathbf{b}$ where $\\mathbf{a}$ is parallel to $\\mathbf{u}$ and $\\mathbf{b}$ is orthogonal to $\\mathbf{u}$.\n$$\n\\mathbf{v}=\\left[\\begin{array}{r}\n3 \\\\\n2 \\\\\n-5\n\\end{array}\\right], \\quad \\mathbf{u}=\\left[\\begin{array}{r}\n1 \\\\\n-1 \\\\\n2\n\\end{array}\\right]\n$$", "answer": "$$\n\\mathbf{a}=\\left[\\begin{array}{c}\n-3 / 2 \\\\\n3 / 2 \\\\\n-3\n\\end{array}\\right], \\quad \\mathbf{b}=\\left[\\begin{array}{c}\n9 / 2 \\\\\n1 / 2 \\\\\n-2\n\\end{array}\\right]\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.1.1", "question": "Can a column vector ever be equal to a row vector?", "answer": "Yes, a $1 \\times 1$-matrix is both a row vector and a column vector. However, a column vector of dimension 2 or greater can never be equal to a row vector, because one is an $n \\times 1$-matrix and the other is a $1 \\times n$ matrix.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.3.1", "question": "In $\\mathbb{R}^{3}$ with the usual dot product, find an orthogonal basis for\n$$\n\\operatorname{span}\\left\\{\\left[\\begin{array}{l}\n1 \\\\\n2 \\\\\n3\n\\end{array}\\right],\\left[\\begin{array}{l}\n2 \\\\\n6 \\\\\n0\n\\end{array}\\right]\\right\\}\n$$", "answer": "$\\mathbf{u}_{1}=\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right], \\mathbf{u}_{2}=\\left[\\begin{array}{c}1 \\\\ 4 \\\\ -3\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.A.1.1", "question": "Let $z=2+7 i$ and let $w=3-8 i$. Compute $z+w, z-2 w$, $z w$, and $\\frac{w}{z}$.", "answer": "$z+w=5-i, z-2 w=-4+23 i, z w=62+5 i$, and $\\frac{w}{z}=-\\frac{50}{53}-\\frac{37}{53} i$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.8", "question": "Find $h$ such that\n$$\n\\left[\\begin{array}{ll|l}\n1 & h & 3 \\\\\n2 & 4 & 6\n\\end{array}\\right]\n$$\nis the augmented matrix of a consistent system.", "answer": "Any $h$ will work.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.3.2", "question": "Use elementary operations to find the point $(x, y)$ that lies on both lines $x+3 y=1$ and $4 x-y=3$.", "answer": "Solution is: $(x, y)=\\left(\\frac{10}{13}, \\frac{1}{13}\\right)$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.5.1.3", "question": "Describe the span of the following vectors in $\\mathbb{R}^{4}$:\n$$\n\\mathbf{u}_{1}=\\left[\\begin{array}{r}\n1 \\\\\n1 \\\\\n-1 \\\\\n-1\n\\end{array}\\right], \\quad \\mathbf{u}_{2}=\\left[\\begin{array}{l}\n5 \\\\\n1 \\\\\n1 \\\\\n0\n\\end{array}\\right], \\quad \\mathbf{u}_{3}=\\left[\\begin{array}{r}\n0 \\\\\n-2 \\\\\n2 \\\\\n1\n\\end{array}\\right], \\quad \\mathbf{u}_{4}=\\left[\\begin{array}{l}\n3 \\\\\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right]\n$$", "answer": "It is the hyperplane given by the equation $x-2 y-3 z+2 w=0$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.2.1", "question": "Find $\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right]+\\left[\\begin{array}{l}1 \\\\ 5 \\\\ 1\\end{array}\\right]+\\left[\\begin{array}{c}-1 \\\\ 2 \\\\ -4\\end{array}\\right]$.", "answer": "$\\left[\\begin{array}{l}1 \\\\ 9 \\\\ 0\\end{array}\\right]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.6.3", "question": "My system of equations has a solution $(x, y, z)=(1,2,4)$. The associated homogeneous system has basic solutions $(x, y, z)=(1,0,1)$ and $(x, y, z)=(0,1,-1)$. What is the general solution of my system of equations?", "answer": "The general solution is $(x, y, z)=(1,2,4)+s(1,0,1)+t(0,1,-1)$, or equivalently, $(x, y, z)=(1+$ $s, 2+t, 4+s-t)$, where $s$ and $t$ are parameters.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.8.4.3", "question": "Find the eigenvalues and eigenvectors of the matrix\n$$\n\\left[\\begin{array}{rrr}\n5 & -18 & -32 \\\\\n0 & 5 & 4 \\\\\n2 & -5 & -11\n\\end{array}\\right]\n$$\nOne eigenvalue is 1. Diagonalize if possible.", "answer": "The eigenvalues are -1 and 1. The eigenvectors corresponding to the eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n10 \\\\\n-2 \\\\\n3\n\\end{array}\\right]\\right\\} \\text { for eigenvalue }-1, \\quad\\left\\{\\left[\\begin{array}{c}\n7 \\\\\n-2 \\\\\n2\n\\end{array}\\right]\\right\\} \\text { for eigenvalue } 1 .\n$$\n\nSince there are only 2 linearly independent eigenvectors, this matrix is not diagonalizable.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.5.5", "question": "Find the length of each of the following vectors.\n$$\n\\mathbf{u}=\\left[\\begin{array}{c}\n-3 \\\\\n2\n\\end{array}\\right], \\quad \\mathbf{v}=\\left[\\begin{array}{c}\n1 \\\\\n-2 \\\\\n5\n\\end{array}\\right], \\quad \\mathbf{w}=\\left[\\begin{array}{c}\n1 \\\\\n4 \\\\\n-2 \\\\\n1\n\\end{array}\\right]\n$$", "answer": "$\\|\\mathbf{u}\\|=\\sqrt{13},\\|\\mathbf{v}\\|=\\sqrt{30},\\|\\mathbf{w}\\|=\\sqrt{22}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.4.9", "question": "Let $A=\\left[\\begin{array}{ll}1 & 2 \\\\ 3 & 4\\end{array}\\right]$ and $B=\\left[\\begin{array}{ll}1 & 2 \\\\ 3 & k\\end{array}\\right]$. Is it possible to find $k$ such that $A B=B A$? If so, what should $k$ equal?", "answer": "Solution is: $k=4$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.9.2", "question": "Decrypt the message \"ERM DXYBJUWW. JWQLD, HL\" using the Hill cipher with block size 3 and encryption matrix\n$$\nA=\\left[\\begin{array}{lll}\n2 & 1 & 1 \\\\\n1 & 3 & 1 \\\\\n1 & 1 & 4\n\\end{array}\\right]\n$$", "answer": "The decryption matrix is\n\n$$\nA^{-1}=\\left[\\begin{array}{ccc}\n16 & 22 & 5 \\\\\n22 & 26 & 17 \\\\\n5 & 17 & 2\n\\end{array}\\right]\n$$\n\nPlaintext: \"Spies are at the gate\".", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.3.2", "question": "Find $-3\\left[\\begin{array}{r}5 \\\\ -1 \\\\ 2 \\\\ -3\\end{array}\\right]+5\\left[\\begin{array}{r}-8 \\\\ 2 \\\\ -3 \\\\ 6\\end{array}\\right]$.", "answer": "$\\left[\\begin{array}{r}-55 \\\\ 13 \\\\ -21 \\\\ 39\\end{array}\\right]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.3.4", "question": "Let\n$$\nA=\\left[\\begin{array}{ccc}\n3 & -1 & 0 \\\\\n-1 & 5 & 2 \\\\\n0 & 2 & 3\n\\end{array}\\right]\n$$\nand consider the vector space $\\mathbb{R}^{3}$ with the inner product given by $\\langle\\mathbf{v}, \\mathbf{w}\\rangle=\\mathbf{v}^{T}$ Aw. Let\n$$\n\\mathbf{v}_{1}=\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n2\n\\end{array}\\right], \\quad \\mathbf{v}_{2}=\\left[\\begin{array}{c}\n-1 \\\\\n1 \\\\\n-5\n\\end{array}\\right], \\quad \\text { and } \\quad \\mathbf{v}_{3}=\\left[\\begin{array}{l}\n2 \\\\\n2 \\\\\n3\n\\end{array}\\right]\n$$\nApply the Gram-Schmidt procedure to $\\mathbf{v}_{1}, \\mathbf{v}_{2}, \\mathbf{v}_{3}$ to find an orthogonal basis $\\left\\{\\mathbf{u}_{1}, \\mathbf{u}_{2}, \\mathbf{u}_{3}\\right\\}$ for $\\mathbb{R}^{3}$ with respect to the above inner product.", "answer": "$\\mathbf{u}_{1}=\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 2\\end{array}\\right], \\mathbf{u}_{2}=\\left[\\begin{array}{c}1 \\\\ 1 \\\\ -1\\end{array}\\right], \\mathbf{u}_{3}=\\left[\\begin{array}{c}-1 \\\\ 1 \\\\ 0\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.9.3", "question": "Eve intercepts the following encrypted message sent by Bob:\n\u201cTGVXKHGSW, JU, JHY JSCDSBQIRPEV\u201d\nEve knows that Alice uses a Hill cipher with block length 2, but she does not know the secret encryption matrix. Eve also knows that Bob begins all of his letters with \"Hello\". Decrypt the message.", "answer": "The first two plaintext blocks are $(8,5),(12,12)$ and the first two ciphertext blocks are $(20,7),(22,24)$. Eve solves the equation\n\n$$\nA^{-1}\\left[\\begin{array}{cc}\n20 & 22 \\\\\n7 & 24\n\\end{array}\\right]=\\left[\\begin{array}{ll}\n8 & 12 \\\\\n5 & 12\n\\end{array}\\right]\n$$\n\nto find the secret decryption matrix\n\n$$\nA^{-1}=\\left[\\begin{array}{ll}\n3 & 5 \\\\\n1 & 2\n\\end{array}\\right] .\n$$\n\nThe plaintext is \"Hello, password is kiwifruit\".", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.2.5", "question": "Suppose $B=\\left\\{\\mathbf{u}_{1}, \\mathbf{u}_{2}, \\mathbf{u}_{3}\\right\\}$ is an orthogonal basis for an inner product space $V$, such that $\\left\\|\\mathbf{u}_{1}\\right\\|=2,\\left\\|\\mathbf{u}_{2}\\right\\|=\\sqrt{3}$, and $\\left\\|\\mathbf{u}_{3}\\right\\|=\\sqrt{5}$. Moreover, suppose that $\\mathbf{v} \\in V$ is a vector such that $\\left\\langle\\mathbf{v}, \\mathbf{u}_{1}\\right\\rangle=1$, $\\left\\langle\\mathbf{v}, \\mathbf{u}_{2}\\right\\rangle=2$, and $\\left\\langle\\mathbf{v}, \\mathbf{u}_{3}\\right\\rangle=-4$. Find the coordinates of $\\mathbf{v}$ with respect to $B$.", "answer": "$\\mathbf{v}=\\frac{1}{4} \\mathbf{u}_{1}+\\frac{2}{3} \\mathbf{u}_{2}-\\frac{4}{5} \\mathbf{u}_{3}$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.12", "question": "Find the volume of the parallelepiped determined by the vectors $\\left[\\begin{array}{r}1 \\\\ -7 \\\\ -5\\end{array}\\right],\\left[\\begin{array}{r}1 \\\\ -2 \\\\ -6\\end{array}\\right]$, and $\\left[\\begin{array}{l}3 \\\\ 2 \\\\ 3\\end{array}\\right]$", "answer": "\\left(\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 3\\end{array}\\right] \\times\\left[\\begin{array}{r}-7 \\\\ -2 \\\\ 2\\end{array}\\right]\\right) \\cdot\\left[\\begin{array}{r}-5 \\\\ -6 \\\\ 3\\end{array}\\right]=\\left[\\begin{array}{r}8 \\\\ -23 \\\\ 5\\end{array}\\right] \\cdot\\left[\\begin{array}{r}-5 \\\\ -6 \\\\ 3\\end{array}\\right]=113$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.3.3", "question": "In $\\mathbb{R}^{4}$ with the usual dot product, find an orthogonal basis for\n$$\n\\operatorname{span}\\left\\{\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n1 \\\\\n0\n\\end{array}\\right],\\left[\\begin{array}{c}\n1 \\\\\n3 \\\\\n1 \\\\\n-1\n\\end{array}\\right],\\left[\\begin{array}{l}\n2 \\\\\n4 \\\\\n2 \\\\\n2\n\\end{array}\\right]\\right\\}\n$$", "answer": "$\\mathbf{u}_{1}=\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 1 \\\\ 0\\end{array}\\right], \\mathbf{u}_{2}=\\left[\\begin{array}{c}0 \\\\ 3 \\\\ 0 \\\\ -1\\end{array}\\right], \\mathbf{u}_{3}=\\left[\\begin{array}{l}0 \\\\ 1 \\\\ 0 \\\\ 3\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.7.6.8", "question": "Consider the matrix\n$$\nA=\\left[\\begin{array}{ccc}\ne^{t} & \\cosh t & \\sinh t \\\\\ne^{t} & \\sinh t & \\cosh t \\\\\ne^{t} & \\cosh t & \\sinh t\n\\end{array}\\right]\n$$\nDoes there exist a value of t for which this matrix fails to be invertible? Explain.", "answer": "Since the matrix $A$ has two identical rows, we have $\\operatorname{det}(A)=0$ for all $t$. So this matrix is noninvertible for all $t$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.3.6", "question": "Consider the inner product space $C[0,2]$, with the inner product given by\n$$\n\\langle p, q\\rangle=\\int_{0}^{2} f(x) g(x) d x .\n$$\nUse the Gram-Schmidt procedure to find an orthogonal basis for $\\operatorname{span}\\left\\{1, x, x^{2}\\right\\}$.", "answer": "$\\mathbf{u}_{1}=1, \\mathbf{u}_{2}=x-1, x^{2}-2 x+\\frac{2}{3}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.4.2", "question": "Decide whether\n$$\n\\mathbf{v}=\\left[\\begin{array}{r}\n4 \\\\\n4 \\\\\n-3\n\\end{array}\\right]\n$$\nis a linear combination of the vectors\n$$\n\\mathbf{u}_{1}=\\left[\\begin{array}{r}\n3 \\\\\n1 \\\\\n-1\n\\end{array}\\right] \\quad \\text { and } \\quad \\mathbf{u}_{2}=\\left[\\begin{array}{r}\n2 \\\\\n-2 \\\\\n1\n\\end{array}\\right]\n$$\nIf yes, find the coefficients.", "answer": "$$\n\\left[\\begin{array}{r}\n4 \\\\\n4 \\\\\n-3\n\\end{array}\\right]=2\\left[\\begin{array}{r}\n3 \\\\\n1 \\\\\n-1\n\\end{array}\\right]-\\left[\\begin{array}{r}\n2 \\\\\n-2 \\\\\n1\n\\end{array}\\right]\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.6.8", "question": "Find $\\operatorname{proj}_{\\mathbf{v}}(\\mathbf{w})$ where $\\mathbf{w}=\\left[\\begin{array}{r}1 \\\\ 0 \\\\ -2\\end{array}\\right]$ and $\\mathbf{v}=\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right]$.", "answer": "$\\frac{\\mathbf{v} \\cdot \\mathbf{w}}{\\mathbf{v} \\cdot \\mathbf{v}} \\mathbf{v}=\\frac{-5}{14}\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right]=\\left[\\begin{array}{c}-\\frac{5}{14} \\\\ -\\frac{5}{7} \\\\ -\\frac{15}{14}\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.2.3", "question": "Let $A=\\left[\\begin{array}{rrr}1 & 2 & -1 \\\\ -1 & 4 & 0\\end{array}\\right]$ and $B=\\left[\\begin{array}{rrr}0 & 3 & 0 \\\\ 1 & -1 & 1\\end{array}\\right]$.\n\nFind a matrix $X$ such that $(A+X)-(B+0)=B+A$. Hint: first use the properties of matrix addition to simplify the equation and solve for $X$.", "answer": "The equation simplifies to $X=B+B$, so $X=\\left[\\begin{array}{rrr}0 & 6 & 0 \\\\ 2 & -2 & 2\\end{array}\\right]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.5.4", "question": "Use Gauss-Jordan elimination to solve the system of equations $3 x-y-2 z=3, y-4 z=0$, and $-2 x+y=-2$.", "answer": "Solution is: $[x=2 t+1, y=4 t, z=t]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.6.4", "question": "Find $\\cos \\theta$ where $\\theta$ is the angle between the vectors\n$$\n\\mathbf{u}=\\left[\\begin{array}{r}\n3 \\\\\n-1 \\\\\n-1\n\\end{array}\\right], \\mathbf{v}=\\left[\\begin{array}{l}\n1 \\\\\n4 \\\\\n2\n\\end{array}\\right]\n$$", "answer": "$\\cos \\theta=\\frac{\\left[\\begin{array}{ccc}3 & -1 & -1\\end{array}\\right]^{T} \\cdot\\left[\\begin{array}{lll}1 & 4 & 2\\end{array}\\right]^{T}}{\\sqrt{9+1+1} \\sqrt{1+16+4}}=\\frac{-3}{\\sqrt{11} \\sqrt{21}}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.6", "question": "Find the area of the triangle determined by the three points, $(1,0,3),(4,1,0)$ and $(-3,1,1)$.", "answer": "Let $P=(1,0,3), Q=(4,1,0)$, and $R=(-3,1,1) . \\overrightarrow{P Q} \\times \\overrightarrow{P R}=\\left[\\begin{array}{r}3 \\\\ 1 \\\\ -3\\end{array}\\right] \\times\\left[\\begin{array}{r}-4 \\\\ 1 \\\\ -2\\end{array}\\right]=\\left[\\begin{array}{c}1 \\\\ 18 \\\\ 7\\end{array}\\right]$. The area of the triangle is $\\frac{1}{2}\\|\\overrightarrow{P Q} \\times \\overrightarrow{P R}\\|=\\frac{1}{2} \\sqrt{1+18^{2}+7^{2}}=\\frac{1}{2} \\sqrt{374}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.5.5", "question": "Use Gauss-Jordan elimination to solve the system of equations $-9 x+15 y=66,-11 x+$ $18 y=79,-x+y=4$, and $z=3$.", "answer": "Solution is: $[x=1, y=5, z=3]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.3.5", "question": "Let\n$$\nA=\\left[\\begin{array}{llll}\n3 & 2 & 0 & 0 \\\\\n2 & 5 & 1 & 0 \\\\\n0 & 1 & 3 & 1 \\\\\n0 & 0 & 1 & 3\n\\end{array}\\right]\n$$\nand consider the vector space $\\mathbb{R}^{4}$ with the inner product given by $\\langle\\mathbf{v}, \\mathbf{w}\\rangle=\\mathbf{v}^{T}$ Aw. Let\n$$\n\\mathbf{v}_{1}=\\left[\\begin{array}{c}\n1 \\\\\n-1 \\\\\n1 \\\\\n1\n\\end{array}\\right], \\quad \\mathbf{v}_{2}=\\left[\\begin{array}{l}\n2 \\\\\n0 \\\\\n0 \\\\\n2\n\\end{array}\\right], \\quad \\text { and } \\quad \\mathbf{v}_{3}=\\left[\\begin{array}{l}\n2 \\\\\n0 \\\\\n2 \\\\\n3\n\\end{array}\\right]\n$$\nand let $W=\\operatorname{span}\\left\\{\\mathbf{v}_{1}, \\mathbf{v}_{2}, \\mathbf{v}_{3}\\right\\}$. Apply the Gram-Schmidt procedure to $\\mathbf{v}_{1}, \\mathbf{v}_{2}, \\mathbf{v}_{3}$ to find an orthogonal basis $\\left\\{\\mathbf{u}_{1}, \\mathbf{u}_{2}, \\mathbf{u}_{3}\\right\\}$ for $W$ with respect to the above inner product.", "answer": "$\\mathbf{u}_{1}=\\left[\\begin{array}{c}1 \\\\ -1 \\\\ 1 \\\\ 1\\end{array}\\right], \\mathbf{u}_{2}=\\left[\\begin{array}{c}1 \\\\ 1 \\\\ -1 \\\\ 1\\end{array}\\right], \\mathbf{u}_{3}=\\left[\\begin{array}{c}-1 \\\\ 1 \\\\ 1 \\\\ 0\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.6.2.6", "question": "Find the matrix for $T(\\mathbf{w})=\\operatorname{proj}_{\\mathbf{v}}(\\mathbf{w})$, where $\\mathbf{v}=[1,-2,3]^{T}$.", "answer": "Recall that the desired matrix has $i^{\\text {th }}$ column equal to $\\operatorname{proj}_{\\mathbf{u}}\\left(\\mathbf{e}_{i}\\right)=\\frac{\\mathbf{u} \\cdot \\mathbf{e}_{i}}{\\|\\mathbf{u}\\|^{2}} \\mathbf{u}$. Therefore, the matrix is\n\n$$\n\\frac{1}{14}\\left[\\begin{array}{rrr}\n1 & -2 & 3 \\\\\n-2 & 4 & -6 \\\\\n3 & -6 & 9\n\\end{array}\\right]\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.8.5.5", "question": "Let $A=\\left[\\begin{array}{lll}-2 & 0 & 6 \\\\ -3 & 1 & 6 \\\\ -3 & 0 & 7\\end{array}\\right]$. Find a square root of $A$.", "answer": "$A=\\left[\\begin{array}{rrr}0 & 0 & 2 \\\\ -1 & 1 & 2 \\\\ -1 & 0 & 3\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.5.3.3", "question": "Let $\\mathbf{w}, \\mathbf{v}$ be given vectors in $\\mathbb{R}^{4}$ and define\n$$\nM=\\left\\{\\mathbf{u}=\\left[\\begin{array}{l}\nu_{1} \\\\\nu_{2} \\\\\nu_{3} \\\\\nu_{4}\n\\end{array}\\right] \\in \\mathbb{R}^{4} \\mid \\mathbf{w} \\cdot \\mathbf{u}=0 \\text { and } \\mathbf{v} \\cdot \\mathbf{u}=0\\right\\}\n$$\nIs $M$ a subspace of $\\mathbb{R}^{4}$ ? Explain.", "answer": "Yes, this is a subspace.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.7", "question": "Find $h$ such that\n$$\n\\left[\\begin{array}{ll|l}\n2 & h & 4 \\\\\n3 & 6 & 7\n\\end{array}\\right]\n$$\nis the augmented matrix of an inconsistent system.", "answer": "$h=4$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.6.5", "question": "Find $\\cos \\theta$ where $\\theta$ is the angle between the vectors\n$$\n\\mathbf{u}=\\left[\\begin{array}{r}\n1 \\\\\n-2 \\\\\n1\n\\end{array}\\right], \\mathbf{v}=\\left[\\begin{array}{r}\n1 \\\\\n2 \\\\\n-7\n\\end{array}\\right]\n$$", "answer": "$\\cos \\theta=\\frac{-10}{\\sqrt{1+4+1} \\sqrt{1+4+49}}$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.4.15", "question": "Find $3 \\times 3$-matrices $A$ and $B$ such that $A B \\neq B A$.", "answer": "$A=\\left[\\begin{array}{lll}0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0\\end{array}\\right], B=\\left[\\begin{array}{lll}1 & 2 & 0 \\\\ 3 & 4 & 0 \\\\ 0 & 0 & 0\\end{array}\\right]$. Then $A B=\\left[\\begin{array}{lll}3 & 4 & 0 \\\\ 1 & 2 & 0 \\\\ 0 & 0 & 0\\end{array}\\right]$ and $B A=\\left[\\begin{array}{lll}2 & 1 & 0 \\\\ 4 & 3 & 0 \\\\ 0 & 0 & 0\\end{array}\\right]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.15", "question": "What does it mean geometrically if the box product of three vectors equals zero?", "answer": "It means that if you place them so that they all have their tails at the same point, the three will lie in the same plane.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.24", "question": "Suppose the coefficient matrix of a system of $n$ equations with $n$ variables has the property that every column is a pivot column. Does it follow that the system of equations must have a solution? If so, must the solution be unique? Explain.", "answer": "Yes. It has a unique solution.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.1.3", "question": "Find $x$ and $y$ so that $\\mathbf{u}=[5 x-3 y, 4]^{T}$ and $\\mathbf{v}=[2 x-2 y, 2 y]^{T}$ are equal in $\\mathbb{R}^{2}$.", "answer": "We need $5 x-3 y=2 x-2 y$ and $4=2 y$. The unique solution is $x=\\frac{2}{3}$ and $y=2$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.9.4.6", "question": "Let $V$ be a 5-dimensional vector space. If you have 5 linearly independent vectors in $V$, can you conclude that the vectors span $V$ ?", "answer": "Yes, because the set of 5 linearly independent vectors can be extended to a basis $B$ of $V$. But since $V$ is 5-dimensional, $B$ has only 5 elements, which must be the original 5 vectors.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.4.1", "question": "Consider $\\mathbb{R}^{3}$ with the usual dot product. Let\n$$\n\\mathbf{u}_{1}=\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n1\n\\end{array}\\right], \\quad \\mathbf{u}_{2}=\\left[\\begin{array}{r}\n-1 \\\\\n-2 \\\\\n1\n\\end{array}\\right], \\quad \\text { and } \\quad \\mathbf{v}=\\left[\\begin{array}{r}\n-1 \\\\\n5 \\\\\n3\n\\end{array}\\right]\n$$\nNote that $\\mathbf{u}_{1}$ and $\\mathbf{u}_{2}$ are orthogonal. Find the best approximation of $\\mathbf{v}$ in $\\operatorname{span}\\left\\{\\mathbf{u}_{1}, \\mathbf{u}_{2}\\right\\}$.", "answer": "The best approximation is $\\mathbf{v}^{\\prime}=3 \\mathbf{u}_{1}-\\mathbf{u}_{2}=\\left[\\begin{array}{r}-2 \\\\ 5 \\\\ 2\\end{array}\\right]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.3.5", "question": "Solve the following system of equations by back substitution.\n$$\n\\begin{array}{r}\nx+3 y-2 z=5 \\\\\ny+3 z=4 \\\\\nz=1 .\n\\end{array}\n$$", "answer": "$(x, y, z)=(4,1,1)$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.9", "question": "Find $h$ such that\n$$\n\\left[\\begin{array}{ll|r}\n1 & 1 & 4 \\\\\n3 & h & 12\n\\end{array}\\right]\n$$\nis the augmented matrix of a consistent system.", "answer": "Any $h$ will work.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.5.6", "question": "Use Gauss-Jordan elimination to solve the system of equations $-19 x+8 y=-108,-71 x+$ $30 y=-404,-2 x+y=-12,4 x+z=14$.", "answer": "Solution is: $[x=4, y=-4, z=-2]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.5.3", "question": "Consider the points $\\left(x_{1}, y_{1}\\right)=(-1,0),\\left(x_{2}, y_{2}\\right)=(0,3),\\left(x_{3}, y_{3}\\right)=(1,3),\\left(x_{4}, y_{4}\\right)=(2,5)$, $\\left(x_{5}, y_{5}\\right)=(3,9)$. Find the least squares line for these points.", "answer": "$y=2+2 x$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.3.2", "question": "In $\\mathbb{R}^{4}$ with the usual dot product, find an orthogonal basis for\n$$\n\\operatorname{span}\\left\\{\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n1 \\\\\n0\n\\end{array}\\right],\\left[\\begin{array}{c}\n3 \\\\\n0 \\\\\n2 \\\\\n-1\n\\end{array}\\right]\\right\\}\n$$", "answer": "$\\mathbf{u}_{1}=\\left[\\begin{array}{l}0 \\\\ 1 \\\\ 1 \\\\ 0\\end{array}\\right], \\mathbf{u}_{2}=\\left[\\begin{array}{c}3 \\\\ -1 \\\\ 1 \\\\ -1\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.13", "question": "Determine if the system is consistent. If so, is the solution unique?\n$$\n\\begin{gathered}\nx+2 y+z-w=2 \\\\\nx-y+z+w=0 \\\\\n2 x+y-z=1 \\\\\n4 x+2 y+z=3\n\\end{gathered}\n$$", "answer": "The solution is: $x=\\frac{1}{3}-\\frac{1}{3} t, y=\\frac{2}{3}+\\frac{2}{3} t, z=\\frac{1}{3}, w=t$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.5.2.14", "question": "Let $\\mathbf{u}, \\mathbf{v}, \\mathbf{w}$ be linearly independent vectors in $\\mathbb{R}^{n}$. Are the vectors $\\mathbf{u}+\\mathbf{v}, \\mathbf{u}+\\mathbf{w}$, and $\\mathbf{w}+\\mathbf{v}$ linearly independent?", "answer": "From $a(\\mathbf{u}+\\mathbf{v})+b(\\mathbf{u}+\\mathbf{w})+c(\\mathbf{w}+\\mathbf{v})=\\mathbf{0}$ we get $(a+b) \\mathbf{u}+(a+c) \\mathbf{v}+(b+c) \\mathbf{w}=\\mathbf{0}$. Since $\\mathbf{u}, \\mathbf{v}, \\mathbf{w}$ are linearly independent, this last system has only the trivial solution, so $a+b=0, a+c=0$, and $b+c=0$. Solving, we find the unique solution $(a, b, c)=(0,0,0)$. So the vectors $\\mathbf{u}+\\mathbf{v}, \\mathbf{u}+\\mathbf{w}$, and $\\mathbf{w}+\\mathbf{v}$ are linearly independent.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.21", "question": "Solve the system of equations $8 x+2 y+3 z=-3,8 x+3 y+3 z=-1$, and $4 x+y+3 z=-9$.", "answer": "Solution is: $[x=1, y=2, z=-5]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.7", "question": "Find the area of the triangle determined by the three points, $(1,2,3),(2,3,4)$ and $(3,4,5)$. Did something interesting happen here? What does it mean geometrically?", "answer": "Let $P=(1,2,3), Q=(2,3,4)$, and $R=(3,4,5) . \\overrightarrow{P Q} \\times \\overrightarrow{P R}=\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 1\\end{array}\\right] \\times\\left[\\begin{array}{l}2 \\\\ 2 \\\\ 2\\end{array}\\right]=\\left[\\begin{array}{l}0 \\\\ 0 \\\\ 0\\end{array}\\right]$. The area of the triangle is 0. It means the three points are on a line.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.6.1", "question": "Find $\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3 \\\\ 4\\end{array}\\right] \\cdot\\left[\\begin{array}{l}2 \\\\ 0 \\\\ 1 \\\\ 3\\end{array}\\right]$.", "answer": "$\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3 \\\\ 4\\end{array}\\right] \\cdot\\left[\\begin{array}{l}2 \\\\ 0 \\\\ 1 \\\\ 3\\end{array}\\right]=17$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.4", "question": "Consider the following augmented matrix in which $*$ denotes an arbitrary number and $\\mathbf{I}$ denotes a non-zero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?\n$$\n\\left[\\begin{array}{ccccc|c}\n\\mathbf{\\square} & * & * & * & * & * \\\\\n0 & \\mathbf{\\square} & * & * & 0 & * \\\\\n0 & 0 & 0 & 0 & \\mathbf{\\square} & 0 \\\\\n0 & 0 & 0 & 0 & * & \\mathbf{\\square}\n\\end{array}\\right]\n$$", "answer": "There might be a solution. If so, there are infinitely many.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.2.2", "question": "On $C[-1,1]$, which of the following functions are orthogonal to each other?\n$$\nf_{1}(x)=x, \\quad f_{2}(x)=x^{2}, \\quad f_{3}(x)=x^{3}-x, \\quad f_{4}(x)=1-x^{4} .\n$$", "answer": "$f_{1} \\perp f_{2}, f_{1} \\perp f_{4}, f_{2} \\perp f_{3}$, and $f_{3} \\perp f_{4}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.5.1", "question": "Find the least squares approximation for the system of equations\n$$\n\\begin{array}{rr}\nx+2 y+2 z= & 5, \\\\\nx+y-z= & 11, \\\\\nx+2 y-z= & -18 \\\\\n2 x-y+2 z= & 0 .\n\\end{array}\n$$", "answer": "$(x, y, z)=(-1,-1,2)$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.5.2", "question": "Find the least squares approximation for the system of equations\n$$\n\\left[\\begin{array}{rrr}\n-1 & 2 & 1 \\\\\n-1 & 0 & -1 \\\\\n2 & 0 & 2 \\\\\n0 & 0 & 2 \\\\\n1 & 2 & 2\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz\n\\end{array}\\right]=\\left[\\begin{array}{r}\n1 \\\\\n1 \\\\\n3 \\\\\n-2 \\\\\n4\n\\end{array}\\right]\n$$", "answer": "$(x, y, z)=(2,2,-1)$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.18", "question": "Solve the system of equations $3 x-y+4 z=6, y+8 z=0$, and $-2 x+y=-4$.", "answer": "Solution is: $[x=2-4 t, y=-8 t, z=t]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.9.8", "question": "Find the principal axes of the ellipsoid $2 x^{2}+2 y^{2}+3 z^{2}+2 x z-2 y z$.", "answer": "The principal axes are $\\frac{1}{\\sqrt{3}}\\left[\\begin{array}{r}1 \\\\ -1 \\\\ -1\\end{array}\\right], \\frac{1}{\\sqrt{2}}\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 0\\end{array}\\right], \\frac{1}{\\sqrt{6}}\\left[\\begin{array}{r}1 \\\\ -1 \\\\ 2\\end{array}\\right]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.7.6.6", "question": "Consider the matrix\n$$\nA=\\left[\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & \\cos t & -\\sin t \\\\\n0 & \\sin t & \\cos t\n\\end{array}\\right]\n$$\nDoes there exist a value of t for which this matrix fails to be invertible? Explain.", "answer": "No. It has non-zero determinant $\\operatorname{det}(A)=\\cos ^{2} t+\\sin ^{2} t=1$ for all $t$, so it is invertible for all $t$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.8.4.4", "question": "Find the eigenvalues and eigenvectors of the matrix\n$$\n\\left[\\begin{array}{rrr}\n8 & 0 & 10 \\\\\n-6 & -3 & -6 \\\\\n-5 & 0 & -7\n\\end{array}\\right] \\text {. }\n$$\nOne eigenvalue is -3. Diagonalize if possible.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n0\n\\end{array}\\right]\\right\\} \\text { for eigenvalue }-3, \\quad\\left\\{\\left[\\begin{array}{c}\n-2 \\\\\n1 \\\\\n1\n\\end{array}\\right]\\right\\} \\text { for eigenvalue } 3, \\quad\\left\\{\\left[\\begin{array}{c}\n-1 \\\\\n0 \\\\\n1\n\\end{array}\\right]\\right\\} \\text { for eigenvalue }-2\n$$\n\nThe matrix $P$ needed to diagonalize the above matrix is\n\n$$\n\\left[\\begin{array}{rrr}\n0 & -2 & -1 \\\\\n1 & 1 & 0 \\\\\n0 & 1 & 1\n\\end{array}\\right]\n$$\n\nand the diagonal matrix $D$ is\n\n$$\n\\left[\\begin{array}{rrr}\n-3 & 0 & 0 \\\\\n0 & 3 & 0 \\\\\n0 & 0 & -2\n\\end{array}\\right]\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.4.14", "question": "Find $2 \\times 2$-matrices $A$ and $B$ such that $A \\neq 0$ and $B \\neq 0$, but $A B=0$.", "answer": "$A=\\left[\\begin{array}{rr}1 & -1 \\\\ -1 & 1\\end{array}\\right], B=\\left[\\begin{array}{ll}1 & 1 \\\\ 1 & 1\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.6.2.7", "question": "Find the matrix for $T(\\mathbf{w})=\\operatorname{proj}_{\\mathbf{v}}(\\mathbf{w})$, where $\\mathbf{v}=[1,5,3]^{T}$.", "answer": "$$\n\\frac{1}{35}\\left[\\begin{array}{rrr}\n1 & 5 & 3 \\\\\n5 & 25 & 15 \\\\\n3 & 15 & 9\n\\end{array}\\right]\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.25", "question": "Suppose there is a unique solution to a system of linear equations. What must be true of the pivot columns in the augmented matrix?", "answer": "The last column must not be a pivot column. The remaining columns must each be pivot columns.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.5.20", "question": "Which of the following matrices is right invertible? Find a right inverse if one exists. If possible, find two different right inverses.\n\n$$\nA=\\left[\\begin{array}{lll}\n1 & 2 & 3 \\\\\n0 & 1 & 0\n\\end{array}\\right] \\quad B=\\left[\\begin{array}{lll}\n1 & 1 & 2 \\\\\n2 & 2 & 4\n\\end{array}\\right] \\quad C=\\left[\\begin{array}{ll}\n1 & 2 \\\\\n1 & 0 \\\\\n0 & 1\n\\end{array}\\right] \\quad D=\\left[\\begin{array}{ll}\n1 & 2 \\\\\n2 & 3\n\\end{array}\\right]\n$$", "answer": "The matrix $A$ is right invertible. Two possible right inverses are\n\n$$\n\\left[\\begin{array}{cc}\n1 & -2 \\\\\n0 & 1 \\\\\n0 & 0\n\\end{array}\\right] \\text { and }\\left[\\begin{array}{cc}\n-2 & 1 \\\\\n0 & 1 \\\\\n1 & -1\n\\end{array}\\right]\n$$\n\nThe matrices $B$ and $C$ are not right invertible. The matrix $D$ is right invertible with inverse\n\n$$\n\\left[\\begin{array}{cc}\n-3 & 2 \\\\\n2 & -1\n\\end{array}\\right]\n$$\n\nSince $D$ is square, its right inverse is actually an inverse, and therefore unique.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.4.2", "question": "Consider $\\mathbb{R}^{4}$ with the usual dot product. Let\n$$\n\\mathbf{u}_{1}=\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n1 \\\\\n3\n\\end{array}\\right], \\quad \\mathbf{u}_{2}=\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n0 \\\\\n0\n\\end{array}\\right], \\quad \\mathbf{u}_{3}=\\left[\\begin{array}{r}\n1 \\\\\n-1 \\\\\n3 \\\\\n-1\n\\end{array}\\right], \\quad \\text { and } \\quad \\mathbf{v}=\\left[\\begin{array}{r}\n6 \\\\\n-2 \\\\\n-5 \\\\\n5\n\\end{array}\\right]\n$$\nNote that $\\mathbf{u}_{1}, \\mathbf{u}_{2}$, and $\\mathbf{u}_{3}$ are orthogonal. Find the best approximation of $\\mathbf{v}$ in $\\operatorname{span}\\left\\{\\mathbf{u}_{1}, \\mathbf{u}_{2}, \\mathbf{u}_{3}\\right\\}$.", "answer": "The best approximation is $\\mathbf{v}^{\\prime}=\\mathbf{u}_{1}+2 \\mathbf{u}_{2}-\\mathbf{u}_{3}=\\left[\\begin{array}{r}1 \\\\ 3 \\\\ -2 \\\\ 4\\end{array}\\right]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.7.6.9", "question": "Consider the matrix\n$$\nA=\\left[\\begin{array}{ccc}\ne^{t} & e^{-t} \\cos t & e^{-t} \\sin t \\\\\ne^{t} & -e^{-t} \\cos t-e^{-t} \\sin t & -e^{-t} \\sin t+e^{-t} \\cos t \\\\\ne^{t} & 2 e^{-t} \\sin t & -2 e^{-t} \\cos t\n\\end{array}\\right]\n$$\nDoes there exist a value of t for which this matrix fails to be invertible? Explain.", "answer": "$$\n\\operatorname{det}\\left[\\begin{array}{ccc}\ne^{t} & e^{-t} \\cos t & e^{-t} \\sin t \\\\\ne^{t} & -e^{-t} \\cos t-e^{-t} \\sin t & -e^{-t} \\sin t+e^{-t} \\cos t \\\\\ne^{t} & 2 e^{-t} \\sin t & -2 e^{-t} \\cos t\n\\end{array}\\right]=5 e^{-t} \\neq 0\n$$\nand so this matrix is always invertible.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.30", "question": "Consider the system $-5 x+2 y-z=0$ and $-5 x-2 y-z=0$. Both equations equal zero and so $-5 x+2 y-z=-5 x-2 y-z$ which is equivalent to $y=0$. Does it follow that $x$ and $z$ can equal anything? Notice that when $x=1, z=-4$, and $y=0$ are plugged in to the equations, the equations do not equal 0 . Why?", "answer": "These are not legitimate row operations. They do not preserve the solution set of the system.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.3.1", "question": "Use elementary operations to solve the system of equations\n$$\n\\begin{aligned}\n& 3 x+y=3 \\\\\n& x+2 y=1\n\\end{aligned}\n$$", "answer": "$(x, y)=(1,0)$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.9.3.2", "question": "Consider the set of all vectors $\\left[\\begin{array}{l}x \\\\ y\\end{array}\\right] \\in \\mathbb{R}^{2}$ such that $x+y \\geq 0$. Is this a subspace of $\\mathbb{R}^{2}$ ?", "answer": "No. It is not closed under scalar multiplication.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.7.6.11", "question": "Find the inverse, if it exists, of the matrix\n$$\nA=\\left[\\begin{array}{ccc}\ne^{t} & \\cos t & \\sin t \\\\\ne^{t} & -\\sin t & \\cos t \\\\\ne^{t} & -\\cos t & -\\sin t\n\\end{array}\\right]\n$$", "answer": "$$\n\\left[\\begin{array}{ccc}\ne^{t} & \\cos t & \\sin t \\\\\ne^{t} & -\\sin t & \\cos t \\\\\ne^{t} & -\\cos t & -\\sin t\n\\end{array}\\right]^{-1}=\\left[\\begin{array}{ccc}\n\\frac{1}{2} e^{-t} & 0 & \\frac{1}{2} e^{-t} \\\\\n\\frac{1}{2} \\cos t+\\frac{1}{2} \\sin t & -\\sin t & \\frac{1}{2} \\sin t-\\frac{1}{2} \\cos t \\\\\n\\frac{1}{2} \\sin t-\\frac{1}{2} \\cos t & \\cos t & -\\frac{1}{2} \\cos t-\\frac{1}{2} \\sin t\n\\end{array}\\right]\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.5.9", "question": "Normalize the following vectors.\n$$\n\\mathbf{u}=\\left[\\begin{array}{l}\n1 \\\\\n2\n\\end{array}\\right], \\quad \\mathbf{v}=\\left[\\begin{array}{r}\n-2 \\\\\n3 \\\\\n2\n\\end{array}\\right], \\quad \\mathbf{w}=\\left[\\begin{array}{r}\n5 \\\\\n-3 \\\\\n1 \\\\\n-1\n\\end{array}\\right]\n$$", "answer": "$$\n\\frac{1}{\\|\\mathbf{u}\\|} \\mathbf{u}=\\frac{1}{\\sqrt{5}}\\left[\\begin{array}{l}\n1 \\\\\n2\n\\end{array}\\right], \\quad \\frac{1}{\\|\\mathbf{v}\\|} \\mathbf{v}=\\frac{1}{\\sqrt{17}}\\left[\\begin{array}{r}\n-2 \\\\\n3 \\\\\n2\n\\end{array}\\right], \\quad \\frac{1}{\\|\\mathbf{w}\\|} \\mathbf{w}=\\frac{1}{6}\\left[\\begin{array}{r}\n5 \\\\\n-3 \\\\\n1 \\\\\n-1\n\\end{array}\\right]\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.9.1.13", "question": "Let $X=\\{1,2, \\ldots, n\\}$, and consider the space Func $_{X, \\mathbb{R}}$ of real-valued functions defined on $X$. Explain how Func $_{X, \\mathbb{R}}$ can be considered as $\\mathbb{R}^{n}$.", "answer": "Let $f(i)$ be the $i^{\\text {th }}$ component of a vector $\\mathbf{x} \\in \\mathbb{R}^{n}$. Thus a typical element in $\\mathbb{R}^{n}$ is $(f(1), \\ldots, f(n))$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.4.3", "question": "Decide whether\n$$\n\\mathbf{v}=\\left[\\begin{array}{l}\n4 \\\\\n4 \\\\\n4\n\\end{array}\\right]\n$$\nis a linear combination of the vectors\n$$\n\\mathbf{u}_{1}=\\left[\\begin{array}{r}\n3 \\\\\n1 \\\\\n-1\n\\end{array}\\right], \\quad \\mathbf{u}_{2}=\\left[\\begin{array}{r}\n8 \\\\\n0 \\\\\n-1\n\\end{array}\\right] \\quad \\text { and } \\quad \\mathbf{u}_{3}=\\left[\\begin{array}{r}\n2 \\\\\n-2 \\\\\n1\n\\end{array}\\right]\n$$\nIf yes, find the coefficients.", "answer": "The system\n\n$$\n\\left[\\begin{array}{l}\n4 \\\\\n4 \\\\\n4\n\\end{array}\\right]=a_{1}\\left[\\begin{array}{r}\n3 \\\\\n1 \\\\\n-1\n\\end{array}\\right]+a_{2}\\left[\\begin{array}{r}\n8 \\\\\n0 \\\\\n-1\n\\end{array}\\right]+a_{3}\\left[\\begin{array}{r}\n2 \\\\\n-2 \\\\\n1\n\\end{array}\\right]\n$$\n\nhas no solution.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.8.2.8", "question": "Is it possible for a non-zero matrix to have only 0 as an eigenvalue?", "answer": "Yes. $\\left[\\begin{array}{ll}0 & 1 \\\\ 0 & 0\\end{array}\\right]$ works.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.7.1", "question": "Let $X=\\left[\\begin{array}{lll}-1 & -1 & 1\\end{array}\\right]$ and $Y=\\left[\\begin{array}{lll}0 & 1 & 2\\end{array}\\right]$. Find $X^{T} Y$ and $X Y^{T}$ if possible.", "answer": "$X^{T} Y=\\left[\\begin{array}{rrr}0 & -1 & -2 \\\\ 0 & -1 & -2 \\\\ 0 & 1 & 2\\end{array}\\right], X Y^{T}=1$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.4.13", "question": "Find $2 \\times 2$-matrices $A, B$, and $C$ such that $A \\neq 0, C \\neq B$, but $A C=A B$.", "answer": "$A=\\left[\\begin{array}{rr}1 & -1 \\\\ -1 & 1\\end{array}\\right], B=\\left[\\begin{array}{ll}1 & 1 \\\\ 1 & 1\\end{array}\\right], C=\\left[\\begin{array}{ll}2 & 2 \\\\ 2 & 2\\end{array}\\right]$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.9.1", "question": "Encrypt the message \"Rendezvous at dawn\" using the Hill cipher with block size 3 and encryption matrix\n$$\nA=\\left[\\begin{array}{lll}\n2 & 1 & 1 \\\\\n1 & 3 & 1 \\\\\n1 & 1 & 4\n\\end{array}\\right]\n$$", "answer": "Ciphertext: \"ZRUJPZVAEJTWOXGJZV\".", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.5.7", "question": "Solve the following two systems of equations simultaneously, by using a single augmented matrix with two constant vectors.\n$$\n\\begin{array}{rrr}\nx+2 y-z=0 & x+2 y-z=1 \\\\\n2 x+3 y+z=3 & 2 x+3 y+z=7 \\\\\nx-y+2 z=3 & x-y+2 z=4\n\\end{array}\n$$", "answer": "The rank of the coefficient matrix is 3 , so both systems have a unique solution. The solution of the first system is $(x, y, z)=(1,0,1)$, and the solution of the second system is $(x, y, z)=(1,1,2)$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.14", "question": "Suppose $\\mathbf{u}, \\mathbf{v}$, and $\\mathbf{w}$ are three vectors whose components are all integers. Can you conclude the volume of the parallelepiped determined from these three vectors will always be an integer?", "answer": "Yes. It will involve the sum of a product of integers and so it will be an integer.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.7.6.7", "question": "Consider the matrix\n$$\nA=\\left[\\begin{array}{rrr}\n1 & t & t^{2} \\\\\n0 & 1 & 2 t \\\\\nt & 0 & 2\n\\end{array}\\right]\n$$\nDoes there exist a value of t for which this matrix fails to be invertible? Explain.", "answer": "$\\operatorname{det}(A)=\\left|\\begin{array}{ccc}1 & t & t^{2} \\\\ 0 & 1 & 2 t \\\\ t & 0 & 2\\end{array}\\right|=t^{3}+2$, and so $A$ has no inverse when $t=-\\sqrt[3]{2}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.1.2", "question": "Find scalars $x, y, z$ such that the following two matrices are equal.\n$$\n\\left[\\begin{array}{rr}\nx & -1 \\\\\n2 & 4\n\\end{array}\\right] \\text { and }\\left[\\begin{array}{ll}\n2 & y \\\\\nz & 4\n\\end{array}\\right]\n$$", "answer": "$x=2, y=-1, z=2$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.3", "question": "Find the area of the parallelogram determined by the vectors $\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right],\\left[\\begin{array}{r}3 \\\\ -2 \\\\ 1\\end{array}\\right]$.", "answer": "$\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 3\\end{array}\\right] \\times\\left[\\begin{array}{r}-7 \\\\ -2 \\\\ 2\\end{array}\\right]=\\left[\\begin{array}{r}8 \\\\ -23 \\\\ 5\\end{array}\\right]$. The area of the parallelogram is $8 \\sqrt{3}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.10.4.8", "question": "Let $\\mathbf{v}=\\left[\\begin{array}{r}1 \\\\ -2 \\\\ 3\\end{array}\\right]$ and consider the linear function $T(\\mathbf{w})=\\operatorname{proj}_{\\mathbf{v}}(\\mathbf{w})$. Find the matrix of $T$ with respect to the basis\n\n$$\nB=\\left\\{\\mathbf{v}_{1}, \\mathbf{v}_{2}, \\mathbf{v}_{3}\\right\\}=\\left\\{\\left[\\begin{array}{r}\n1 \\\\\n-2 \\\\\n3\n\\end{array}\\right],\\left[\\begin{array}{l}\n2 \\\\\n1 \\\\\n0\n\\end{array}\\right],\\left[\\begin{array}{l}\n3 \\\\\n0 \\\\\n1\n\\end{array}\\right]\\right\\}\n$$", "answer": "We have $T\\left(\\mathbf{v}_{1}\\right)=\\mathbf{v}_{1}, T\\left(\\mathbf{v}_{2}\\right)=\\mathbf{0}$, and $T\\left(\\mathbf{v}_{3}\\right)=\\mathbf{0}$. Therefore\n\n$$\n[T]_{B, B}=\\left[\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{array}\\right] .\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.10.4.2", "question": "Let $B=\\left\\{\\left[\\begin{array}{r}1 \\\\ -1 \\\\ 2\\end{array}\\right],\\left[\\begin{array}{l}2 \\\\ 1 \\\\ 2\\end{array}\\right],\\left[\\begin{array}{r}-1 \\\\ 0 \\\\ 2\\end{array}\\right]\\right\\}$ be a basis of $\\mathbb{R}^{3}$ and let $\\mathbf{x}=\\left[\\begin{array}{r}5 \\\\ -1 \\\\ 4\\end{array}\\right]$ be a vector in $\\mathbb{R}^{2}$. Find $[\\mathbf{x}]_{B}$.", "answer": "$[\\mathbf{x}]_{B}=\\left[\\begin{array}{r}2 \\\\ 1 \\\\ -1\\end{array}\\right]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.7.7.5", "question": "Find the value of $y$ in the following system of equations:\n$$\n\\left[\\begin{array}{ccc}\n1 & 1 & 1 \\\\\n1 & t & t^{2} \\\\\n1 & s & s^{2}\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz\n\\end{array}\\right]=\\left[\\begin{array}{l}\nt \\\\\ns \\\\\n1\n\\end{array}\\right] .\n$$", "answer": "By Cramer's rule, we have\n$$\ny=\\frac{\\left|\\begin{array}{ccc}\n1 & t & 1 \\\\\n1 & s & t^{2} \\\\\n1 & 1 & s^{2}\n\\end{array}\\right|}{\\left|\\begin{array}{ccc}\n1 & 1 & 1 \\\\\n1 & t & t^{2} \\\\\n1 & s & s^{2}\n\\end{array}\\right|}=\\frac{s^{3}+t^{3}+1-2-t s^{2}-t^{2}}{t s^{2}+t^{2}+s-t-s t^{2}-s^{2}} .\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.9.3.8", "question": "Let $U=\\left\\{[x, y, z]^{T} \\in \\mathbb{R}^{3}|| x \\mid \\leq 4\\right\\}$. Is $U$ a subspace of $\\mathbb{R}^{3}$ ?", "answer": "This is not a subspace. $\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 1\\end{array}\\right]$ is in it, but $5\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 1\\end{array}\\right]$ is not.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.2", "question": "Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a non-zero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?\n$$\n\\left[\\begin{array}{ccc|c}\n\\mathbf{\\square} & * & * & * \\\\\n0 & \\mathbf{0} & * & * \\\\\n0 & 0 & \\mathbf{\\square} & *\n\\end{array}\\right]\n$$", "answer": "A solution exists and is unique.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.5.2", "question": "Find the distance between the points $P=(1,3,-1,0)$ and $Q=(2,2,3,3)$ in $\\mathbb{R}^{4}$.", "answer": "$\\sqrt{27}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.3.2.2", "question": "Consider the following vector equation for a plane in $\\mathbb{R}^{4}$ :\n$$\n\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz \\\\\nw\n\\end{array}\\right]=\\left[\\begin{array}{l}\n1 \\\\\n2 \\\\\n0 \\\\\n0\n\\end{array}\\right]+t\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n0 \\\\\n1\n\\end{array}\\right]+s\\left[\\begin{array}{r}\n-1 \\\\\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right] .\n$$\nFind a new vector equation for the same plane by doing the change of parameters $t=1-r_{1}, s=r_{1}+r_{2}$.", "answer": "We have\n\n$$\n\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz \\\\\nw\n\\end{array}\\right]=\\left[\\begin{array}{l}\n1 \\\\\n2 \\\\\n0 \\\\\n0\n\\end{array}\\right]+\\left(1-r_{1}\\right)\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n0 \\\\\n1\n\\end{array}\\right]+\\left(r_{1}+r_{2}\\right)\\left[\\begin{array}{r}\n-1 \\\\\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right]=\\left[\\begin{array}{l}\n2 \\\\\n2 \\\\\n0 \\\\\n1\n\\end{array}\\right]+r_{1}\\left[\\begin{array}{r}\n-2 \\\\\n-1 \\\\\n1 \\\\\n-1\n\\end{array}\\right]+r_{2}\\left[\\begin{array}{r}\n-1 \\\\\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right] .\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.17", "question": "Solve the system of equations $7 x+14 y+15 z=22,2 x+4 y+3 z=5$, and $3 x+6 y+10 z=$ 13.", "answer": "Solution is: $[x=1-2 t, z=1, y=t]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.2.7.8", "question": "Is $\\mathbf{u} \\times(\\mathbf{v} \\times \\mathbf{w})=(\\mathbf{u} \\times \\mathbf{v}) \\times \\mathbf{w}$ ? What is the meaning of $\\mathbf{u} \\times \\mathbf{v} \\times \\mathbf{w}$ ? Explain. Hint: Try $(\\mathbf{i} \\times \\mathbf{j}) \\times \\mathbf{j}$.", "answer": "$(\\mathbf{i} \\times \\mathbf{j}) \\times \\mathbf{j}=\\mathbf{k} \\times \\mathbf{j}=-\\mathbf{i}$. However, $\\mathbf{i} \\times(\\mathbf{j} \\times \\mathbf{j})=\\mathbf{0}$ and so the cross product is not associative. The expression $\\mathbf{u} \\times \\mathbf{v} \\times \\mathbf{w}$ has no meaning.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.8.5.4", "question": "Let $A=\\left[\\begin{array}{rr}-5 & -6 \\\\ 9 & 10\\end{array}\\right]$. Find a square root of $A$, i.e., find a matrix $B$ such that $B^{2}=A$.", "answer": "$B=\\left[\\begin{array}{rr}-1 & -2 \\\\ 3 & 4\\end{array}\\right]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.5.2", "question": "Suppose $A B=A C$ and $A$ is an invertible $n \\times n$-matrix. Does it follow that $B=C$ ? Explain why or why not.", "answer": "Yes $B=C$. Multiply $A B=A C$ on the left by $A^{-1}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.19", "question": "Solve the system of equations $9 x-2 y+4 z=-17,13 x-3 y+6 z=-25$, and $-2 x-z=3$.", "answer": "Solution is: $[x=-1, y=2, z=-1]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.9.3.3", "question": "Consider the set of all vectors $\\left[\\begin{array}{l}x \\\\ y\\end{array}\\right] \\in \\mathbb{R}^{2}$ such that $x y=0$. Is this a subspace of $\\mathbb{R}^{2}$ ?", "answer": "No. It is not closed under addition.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.23", "question": "Suppose a system of linear equations has an augmented matrix with 2 rows and 4 columns and the last column is a pivot column. Could the system of linear equations be consistent? Explain.", "answer": "No. This would lead to $0=1$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.10.4.7", "question": "Let $\\mathbf{v}=\\left[\\begin{array}{r}1 \\\\ -2 \\\\ 3\\end{array}\\right]$ and consider the linear function $T(\\mathbf{w})=\\operatorname{proj}_{\\mathbf{v}}(\\mathbf{w})$. Find the matrix of $T$ with respect to the standard basis of $\\mathbb{R}^{3}$.", "answer": "Recall that $\\operatorname{proj}_{\\mathbf{v}}(\\mathbf{w})=\\frac{\\mathbf{v}^{\\bullet} \\mathbf{w}}{\\|\\mathbf{v}\\|^{2}} \\mathbf{v}$. The desired matrix has $i^{\\text {th }}$ column equal to $\\operatorname{proj}_{\\mathbf{v}}\\left(\\mathbf{e}_{i}\\right)$. Therefore, the desired matrix is\n\n$$\n\\frac{1}{14}\\left[\\begin{array}{rrr}\n1 & -2 & 3 \\\\\n-2 & 4 & -6 \\\\\n3 & -6 & 9\n\\end{array}\\right] .\n$$", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.10.9", "question": "Consider $\\mathbb{C}^{3}$ with the complex dot product. Let $\\mathbf{v}_{1}=\\left[\\begin{array}{c}0 \\\\ i \\\\ 2\\end{array}\\right]$ and $\\mathbf{v}_{2}=\\left[\\begin{array}{c}1 \\\\ 1+i \\\\ 3 i+2\\end{array}\\right]$. Use the Gram-Schmidt procedure to find an orthogonal basis for $\\operatorname{span}\\left\\{\\mathbf{v}_{1}, \\mathbf{v}_{2}\\right\\}$. Then find an orthonormal basis.", "answer": "Orthogonal: $[0, i, 2]^{T},[1,2, i]^{T}$. Orthonormal: $\\frac{1}{\\sqrt{5}}[0, i, 2]^{T}, \\frac{1}{\\sqrt{6}}[1,2, i]^{T}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.11.2.6", "question": "Suppose $B=\\left\\{\\mathbf{u}_{1}, \\mathbf{u}_{2}, \\mathbf{u}_{3}\\right\\}$ is an orthogonal basis of $\\mathbb{R}^{3}$. We have been told that\n$$\n\\mathbf{u}_{1}=\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n0\n\\end{array}\\right],\n$$\nbut it is not known what $\\mathbf{u}_{2}$ and $\\mathbf{u}_{3}$ are. Find the first coordinate of the vector\n$$\n\\mathbf{v}=\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n2\n\\end{array}\\right]\n$$\nwith respect to the basis $B$.", "answer": "We have $\\mathbf{v}=a_{1} \\mathbf{u}_{1}+a_{2} \\mathbf{u}_{2}+a_{3} \\mathbf{u}_{3}$ where $a_{1}=\\frac{\\left\\langle\\mathbf{u}_{1}, \\mathbf{v}\\right\\rangle}{\\left\\langle\\mathbf{u}_{1}, \\mathbf{u}_{1}\\right\\rangle}=\\frac{1}{2}$. So the first coordinate is $\\frac{1}{2}$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.5.1.2", "question": "Describe the span of the vectors $\\mathbf{u}=\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 2\\end{array}\\right]$ and $\\mathbf{v}=\\left[\\begin{array}{r}2 \\\\ -1 \\\\ 1\\end{array}\\right]$ in $\\mathbb{R}^{3}$.", "answer": "It is the plane $2 x+3 y-z=0$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.1", "question": "Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a non-zero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?\n$$\n\\left[\\begin{array}{lllll|l}\n\\mathbf{\\square} & * & * & * & * & * \\\\\n0 & \\mathbf{\\square} & * & * & 0 & * \\\\\n0 & 0 & \\mathbf{\\square} & * & * & * \\\\\n0 & 0 & 0 & 0 & \\mathbf{\\square} & *\n\\end{array}\\right]\n$$", "answer": "The solution exists but is not unique.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.1.4.20", "question": "Solve the system of equations $65 x+84 y+16 z=546,81 x+105 y+20 z=682$, and $84 x+110 y+21 z=713$.", "answer": "Solution is: $[x=2, y=4, z=5]$.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.Matrix_Theory_and_Linear_Algebra", "question_number": "exercise.4.4.10", "question": "Let $A=\\left[\\begin{array}{ll}1 & 2 \\\\ 3 & 4\\end{array}\\right]$ and $B=\\left[\\begin{array}{ll}1 & 2 \\\\ 1 & k\\end{array}\\right]$. Is it possible to choose $k$ such that $A B=B A$? If so, what should $k$ equal?", "answer": "There is no possible choice of $k$ which will make these matrices commute.", "license": "Creative Commons Attribution License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.44", "question": "Suppose there is a unique solution to a system of linear equations. What must be true of the pivot columns in the augmented matrix?", "answer": "The last column must not be a pivot column. The remaining columns must each be pivot columns.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.23", "question": "An object moves 10 meters in the direction of $\\vec{j}+\\vec{i}$. There are two forces acting on this object, $\\vec{F}_{1}=\\vec{i}+2 \\vec{j}+2 \\vec{k}$, and $\\vec{F}_{2}=5 \\vec{i}+2 \\vec{j}-6 \\vec{k}$. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. Why?", "answer": "$$\n\\begin{aligned}\n\\vec{F}_{1} \\bullet\\left[\\begin{array}{c}\n\\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} \\\\\n0\n\\end{array}\\right] 10+\\vec{F}_{2} \\bullet\\left[\\begin{array}{c}\n\\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} \\\\\n0\n\\end{array}\\right] 10 & =\\left(\\vec{F}_{1}+\\vec{F}_{2}\\right) \\bullet\\left[\\begin{array}{c}\n\\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} \\\\\n0\n\\end{array}\\right] 10 \\\\\n& =\\left[\\begin{array}{r}\n6 \\\\\n4 \\\\\n-4\n\\end{array}\\right] \\bullet\\left[\\begin{array}{c}\n\\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} \\\\\n0\n\\end{array}\\right] 10 \\\\\n& =50 \\sqrt{2}\n\\end{aligned}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.9.13", "question": "Find the matrix for $T(\\vec{w})=\\operatorname{proj}_{\\vec{v}}(\\vec{w})$ where $\\vec{v}=\\left[\\begin{array}{lll}1 & 0 & 3\\end{array}\\right]^{T}$.", "answer": "$$\n\\frac{1}{10}\\left[\\begin{array}{lll}\n1 & 0 & 3 \\\\\n0 & 0 & 0 \\\\\n3 & 0 & 9\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.18", "question": "Determine if the system is consistent. If so, is the solution unique?\n$$\n\\begin{gathered}\nx+2 y+z-w=2 \\\\\nx-y+z+w=0 \\\\\n2 x+y-z=1 \\\\\n4 x+2 y+z=3\n\\end{gathered}\n$$", "answer": "Solution is: $\\left[w=\\frac{3}{2} y-1, x=\\frac{2}{3}-\\frac{1}{2} y, z=\\frac{1}{3}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.4.1", "question": "Let $M=\\left\\{\\vec{u}=\\left(u_{1}, u_{2}, u_{3}, u_{4}\\right) \\in \\mathbb{R}^{4}:\\left|u_{1}\\right| \\leq 4\\right\\}$. Is $M$ a subspace of $\\mathbb{R}^{4}$ ?", "answer": "This is not a subspace. $\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 1 \\\\ 1\\end{array}\\right]$ is in it, but $20\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 1 \\\\ 1\\end{array}\\right]$ is not.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.37", "question": "Find the solution to the system of equations, $-8 x+2 y+5 z=18,-8 x+3 y+5 z=13$, and $-4 x+y+5 z=19$.", "answer": "Solution is: $[x=-1, y=-5, z=4]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.2", "question": "Find the area of the triangle determined by the three points, $(1,2,3),(4,2,0)$ and $(-3,2,1)$.", "answer": "$\\left[\\begin{array}{r}3 \\\\ 0 \\\\ -3\\end{array}\\right] \\times\\left[\\begin{array}{r}-4 \\\\ 0 \\\\ -2\\end{array}\\right]=\\left[\\begin{array}{r}0 \\\\ 18 \\\\ 0\\end{array}\\right]$. So the area is 9.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.9.13", "question": "Write the solution set of the following system as a linear combination of vectors\n$$\n\\left[\\begin{array}{rrrr}\n1 & 1 & 0 & 1 \\\\\n1 & -1 & 1 & 0 \\\\\n3 & 1 & 1 & 2 \\\\\n3 & 3 & 0 & 3\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz \\\\\nw\n\\end{array}\\right]=\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{array}\\right]\n$$", "answer": "Solution is:\n\n$$\n\\left[\\begin{array}{c}\n-\\frac{1}{2} s-\\frac{1}{2} t \\\\\n\\frac{1}{2} s-\\frac{1}{2} t \\\\\ns \\\\\nt\n\\end{array}\\right]\n$$\n\nfor $s, t \\in \\mathbb{R}$. A basis is\n\n$$\n\\left\\{\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n2 \\\\\n0\n\\end{array}\\right],\\left[\\begin{array}{c}\n-1 \\\\\n1 \\\\\n0 \\\\\n1\n\\end{array}\\right]\\right\\}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.2.11", "question": "Find the matrix for $T(\\vec{w})=\\operatorname{proj}_{\\vec{v}}(\\vec{w})$ where $\\vec{v}=\\left[\\begin{array}{lll}1 & -2 & 3\\end{array}\\right]^{T}$.", "answer": "Recall that $\\operatorname{proj}_{\\vec{u}}(\\vec{v})=\\frac{\\vec{v} \\vec{u}}{\\|\\vec{u}\\|^{2}} \\vec{u}$ and so the desired matrix has $i^{t h}$ column equal to $\\operatorname{proj}_{\\vec{u}}\\left(\\vec{e}_{i}\\right)$. Therefore, the matrix desired is\n\n$$\n\\frac{1}{14}\\left[\\begin{array}{rrr}\n1 & -2 & 3 \\\\\n-2 & 4 & -6 \\\\\n3 & -6 & 9\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.1.14", "question": "Is it possible for a nonzero matrix to have only 0 as an eigenvalue?", "answer": "Yes. $\\left[\\begin{array}{ll}0 & 1 \\\\ 0 & 0\\end{array}\\right]$ works.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.5", "question": "Four times the weight of Gaston is 150 pounds more than the weight of Ichabod. Four times the weight of Ichabod is 660 pounds less than seventeen times the weight of Gaston. Four times the weight of Gaston plus the weight of Siegfried equals 290 pounds. Brunhilde would balance all three of the others. Find the weights of the four people.", "answer": "Solution is : $\\{g=60, I=90, b=200, s=50\\}$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.9.11", "question": "Find the matrix for $T(\\vec{w})=\\operatorname{proj}_{\\vec{v}}(\\vec{w})$ where $\\vec{v}=\\left[\\begin{array}{lll}1 & -2 & 3\\end{array}\\right]^{T}$.", "answer": "Recall that $\\operatorname{proj}_{\\vec{u}}(\\vec{v})=\\frac{\\vec{v} \\bullet \\vec{u}}{\\|\\vec{u}\\|^{2}} \\vec{u}$ and so the desired matrix has $i^{t h}$ column equal to $\\operatorname{proj}_{\\vec{u}}\\left(\\vec{e}_{i}\\right)$. Therefore, the matrix desired is\n\n$$\n\\frac{1}{14}\\left[\\begin{array}{rrr}\n1 & -2 & 3 \\\\\n-2 & 4 & -6 \\\\\n3 & -6 & 9\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.9.15", "question": "Write the solution set of the following system as a linear combination of vectors\n$$\n\\left[\\begin{array}{rrrr}\n1 & 1 & 0 & 1 \\\\\n2 & 1 & 1 & 2 \\\\\n1 & 0 & 1 & 1 \\\\\n0 & -1 & 1 & 1\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz \\\\\nw\n\\end{array}\\right]=\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{array}\\right]\n$$", "answer": "Solution is: $\\left[\\begin{array}{r}-\\hat{t} \\\\ \\hat{t} \\\\ \\hat{t} \\\\ 0\\end{array}\\right]$, a basis is $\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 1 \\\\ 0\\end{array}\\right]$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.6.3.2", "question": "Find the complex cube roots of 8.", "answer": "Solution is: $i \\sqrt{3}+1,1-i \\sqrt{3},-2$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.2.3", "question": "Decide whether $\\vec{v}=\\left[\\begin{array}{r}4 \\\\ 4 \\\\ -3\\end{array}\\right]$ is a linear combination of the vectors $\\vec{u}_{1}=\\left[\\begin{array}{r}3 \\\\ 1 \\\\ -1\\end{array}\\right]$ and $\\vec{u}_{2}=\\left[\\begin{array}{r}2 \\\\ -2 \\\\ 1\\end{array}\\right]$.", "answer": "$$\n\\left[\\begin{array}{r}\n4 \\\\\n4 \\\\\n-3\n\\end{array}\\right]=2\\left[\\begin{array}{r}\n3 \\\\\n1 \\\\\n-1\n\\end{array}\\right]-\\left[\\begin{array}{r}\n2 \\\\\n-2 \\\\\n1\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.9.5", "question": "Write the solution set of the following system as a linear combination of vectors.\n$$\n\\left[\\begin{array}{lll}\n1 & -1 & 2 \\\\\n1 & -2 & 0 \\\\\n3 & -4 & 4\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz\n\\end{array}\\right]=\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n0\n\\end{array}\\right]\n$$", "answer": "Solution is: $\\left[\\begin{array}{r}-4 \\hat{t} \\\\ -2 \\hat{t} \\\\ \\hat{t}\\end{array}\\right]$. A basis is $\\left[\\begin{array}{r}-4 \\\\ -2 \\\\ 1\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.3.1", "question": "Let $A=\\left[\\begin{array}{ll}1 & 2 \\\\ 2 & 1\\end{array}\\right]$. Diagonalize A to find $A^{10}$.", "answer": "First we write $A=P D P^{-1}$.\n\n$$\n\\left[\\begin{array}{ll}\n1 & 2 \\\\\n2 & 1\n\\end{array}\\right]=\\left[\\begin{array}{rr}\n-1 & 1 \\\\\n1 & 1\n\\end{array}\\right]\\left[\\begin{array}{rr}\n-1 & 0 \\\\\n0 & 3\n\\end{array}\\right]\\left[\\begin{array}{rr}\n-\\frac{1}{2} & \\frac{1}{2} \\\\\n\\frac{1}{2} & \\frac{1}{2}\n\\end{array}\\right]\n$$\n\nTherefore $A^{10}=P D^{10} P^{-1}$.\n\n$$\n\\begin{aligned}\n{\\left[\\begin{array}{ll}\n1 & 2 \\\\\n2 & 1\n\\end{array}\\right]^{10} } & =\\left[\\begin{array}{rr}\n-1 & 1 \\\\\n1 & 1\n\\end{array}\\right]\\left[\\begin{array}{rr}\n-1 & 0 \\\\\n0 & 3^{10}\n\\end{array}\\right]\\left[\\begin{array}{rr}\n-\\frac{1}{2} & \\frac{1}{2} \\\\\n\\frac{1}{2} & \\frac{1}{2}\n\\end{array}\\right] \\\\\n& =\\left[\\begin{array}{ll}\n29525 & 29524 \\\\\n29524 & 29525\n\\end{array}\\right]\n\\end{aligned}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.36", "question": "Let\n$$\nA=\\left[\\begin{array}{ll}\n0 & 1 \\\\\n5 & 3\n\\end{array}\\right]\n$$\nFind $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.", "answer": "$\\left[\\begin{array}{ll}0 & 1 \\\\ 5 & 3\\end{array}\\right]^{-1}=\\left[\\begin{array}{cc}-\\frac{3}{5} & \\frac{1}{5} \\\\ 1 & 0\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.39", "question": "Find the solution to the system of equations, $-9 x+15 y=66,-11 x+18 y=79,-x+y=4$, and $z=3$.", "answer": "Solution is: $[x=1, y=5, z=3]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.2.9", "question": "Consider the matrix\n$$\nA=\\left[\\begin{array}{ccc}\ne^{t} & \\cosh t & \\sinh t \\\\\ne^{t} & \\sinh t & \\cosh t \\\\\ne^{t} & \\cosh t & \\sinh t\n\\end{array}\\right]\n$$\nDoes there exist a value of $t$ for which this matrix fails to have an inverse? Explain.", "answer": "$$\n\\operatorname{det}\\left[\\begin{array}{ccc}\ne^{t} & \\cosh t & \\sinh t \\\\\ne^{t} & \\sinh t & \\cosh t \\\\\ne^{t} & \\cosh t & \\sinh t\n\\end{array}\\right]=0\n$$\n\nand so this matrix fails to have a nonzero determinant at any value of $t$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.42", "question": "Suppose a system of linear equations has a $2 \\times 4$ augmented matrix and the last column is a pivot column. Could the system of linear equations be consistent? Explain.", "answer": "No. This would lead to $0=1$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.2.8", "question": "Suppose the characteristic polynomial of an $n \\times n$ matrix $A$ is $1-X^{n}$. Find $A^{m n}$ where $m$ is an integer.", "answer": "The eigenvalues are distinct because they are the $n^{\\text {th }}$ roots of 1 . Hence if $X$ is a given vector with\n\n$$\nX=\\sum_{j=1}^{n} a_{j} V_{j}\n$$\n\nthen\n\n$$\nA^{n m} X=A^{n m} \\sum_{j=1}^{n} a_{j} V_{j}=\\sum_{j=1}^{n} a_{j} A^{n m} V_{j}=\\sum_{j=1}^{n} a_{j} V_{j}=X\n$$\n\nso $A^{n m}=I$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.14", "question": "Find $h$ such that\n$$\n\\left[\\begin{array}{rr|r}\n1 & 1 & 4 \\\\\n3 & h & 12\n\\end{array}\\right]\n$$\nis the augmented matrix of a consistent system.", "answer": "Any $h$ will work.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.4", "question": "Find the following determinant by expanding along the first row and second column.\n\n$$\n\\left|\\begin{array}{lll}\n1 & 2 & 1 \\\\\n2 & 1 & 3 \\\\\n2 & 1 & 1\n\\end{array}\\right|\n$$", "answer": "$$\n\\left|\\begin{array}{lll}\n1 & 2 & 1 \\\\\n2 & 1 & 3 \\\\\n2 & 1 & 1\n\\end{array}\\right|=6\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.2.3", "question": "Find the eigenvalues and eigenvectors of the matrix\n$$\n\\left[\\begin{array}{rrr}\n89 & 38 & 268 \\\\\n14 & 2 & 40 \\\\\n-30 & -12 & -90\n\\end{array}\\right]\n$$\nOne eigenvalue is -3. Diagonalize if possible.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{l}\n-6 \\\\\n-1 \\\\\n-2\n\\end{array}\\right]\\right\\} \\leftrightarrow 6,\\left\\{\\left[\\begin{array}{c}\n-5 \\\\\n-2 \\\\\n2\n\\end{array}\\right]\\right\\} \\leftrightarrow-3,\\left\\{\\left[\\begin{array}{c}\n-8 \\\\\n-2 \\\\\n3\n\\end{array}\\right]\\right\\} \\leftrightarrow-2\n$$\n\nThe matrix $P$ needed to diagonalize the above matrix is\n\n$$\n\\left[\\begin{array}{rrr}\n-6 & -5 & -8 \\\\\n-1 & -2 & -2 \\\\\n2 & 2 & 3\n\\end{array}\\right]\n$$\n\nand the diagonal matrix $D$ is\n\n$$\n\\left[\\begin{array}{rrr}\n6 & 0 & 0 \\\\\n0 & -3 & 0 \\\\\n0 & 0 & -2\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.2.10", "question": "Consider the matrix\n$$\nA=\\left[\\begin{array}{ccc}\ne^{t} & e^{-t} \\cos t & e^{-t} \\sin t \\\\\ne^{t} & -e^{-t} \\cos t-e^{-t} \\sin t & -e^{-t} \\sin t+e^{-t} \\cos t \\\\\ne^{t} & 2 e^{-t} \\sin t & -2 e^{-t} \\cos t\n\\end{array}\\right]\n$$\nDoes there exist a value of $t$ for which this matrix fails to have an inverse? Explain.", "answer": "$$\n\\operatorname{det}\\left[\\begin{array}{ccc}\ne^{t} & e^{-t} \\cos t & e^{-t} \\sin t \\\\\ne^{t} & -e^{-t} \\cos t-e^{-t} \\sin t & -e^{-t} \\sin t+e^{-t} \\cos t \\\\\ne^{t} & 2 e^{-t} \\sin t & -2 e^{-t} \\cos t\n\\end{array}\\right]=5 e^{-t} \\neq 0\n$$\n\nand so this matrix is always invertible.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.9", "question": "An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.\n\n$$\n\\left[\\begin{array}{ll}\na & b \\\\\nc & d\n\\end{array}\\right] \\rightarrow \\cdots \\rightarrow\\left[\\begin{array}{ll}\na & c \\\\\nb & d\n\\end{array}\\right]\n$$", "answer": "It does not change the determinant. This was just taking the transpose.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.35", "question": "Let\n$$\nA=\\left[\\begin{array}{rr}\n2 & 1 \\\\\n-1 & 3\n\\end{array}\\right]\n$$\nFind $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.", "answer": "$\\left[\\begin{array}{rr}2 & 1 \\\\ -1 & 3\\end{array}\\right]^{-1}=\\left[\\begin{array}{rr}\\frac{3}{7} & -\\frac{1}{7} \\\\ \\frac{1}{7} & \\frac{2}{7}\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.3.10", "question": "A person sets off on a random walk with three possible locations. The Markov matrix of probabilities $A=\\left[a_{i j}\\right]$ is given by\n$$\n\\left[\\begin{array}{rrr}\n0.5 & 0.1 & 0.6 \\\\\n0.2 & 0.9 & 0.2 \\\\\n0.3 & 0 & 0.2\n\\end{array}\\right]\n$$\nIt is unknown where the walker starts, but the probability of starting in each location is given by\n$$\nX_{0}=\\left[\\begin{array}{r}\n0.2 \\\\\n0.25 \\\\\n0.55\n\\end{array}\\right]\n$$\nWhat is the probability of the walker being in location 1 at time $n=2$ ?", "answer": "$$\nX_{2}=\\left[\\begin{array}{l}\n0.367 \\\\\n0.4625 \\\\\n0.1705\n\\end{array}\\right]\n$$\n\nTherefore the probability of ending up in location 1 is 0.367 .", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.2.8", "question": "Consider the matrix\n$$\nA=\\left[\\begin{array}{rrr}\n1 & t & t^{2} \\\\\n0 & 1 & 2 t \\\\\nt & 0 & 2\n\\end{array}\\right]\n$$\nDoes there exist a value of t for which this matrix fails to have an inverse? Explain.", "answer": "$$\n\\operatorname{det}\\left[\\begin{array}{ccc}\n1 & t & t^{2} \\\\\n0 & 1 & 2 t \\\\\nt & 0 & 2\n\\end{array}\\right]=t^{3}+2\n$$\n\nand so it has no inverse when $t=-\\sqrt[3]{2}$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.37", "question": "Let\n$$\nA=\\left[\\begin{array}{ll}\n2 & 1 \\\\\n3 & 0\n\\end{array}\\right]\n$$\nFind $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.", "answer": "$\\left[\\begin{array}{ll}2 & 1 \\\\ 3 & 0\\end{array}\\right]^{-1}=\\left[\\begin{array}{cc}0 & \\frac{1}{3} \\\\ 1 & -\\frac{2}{3}\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.5.8", "question": "Explain why an $n \\times n$ matrix $A$ is both one to one and onto if and only if its rank is $n$.", "answer": "The rank is $n$ is the same as saying the columns are independent which is the same as saying $A$ is one to one which is the same as saying the columns are a basis. Thus the span of the columns of $A$ is all of $\\mathbb{R}^{n}$ and so $A$ is onto. If $A$ is onto, then the columns must be linearly independent since otherwise the span of these columns would have dimension less than $n$ and so the dimension of $\\mathbb{R}^{n}$ would be less than $n$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.7.11", "question": "Find $\\operatorname{proj}_{\\vec{v}}(\\vec{w})$ where $\\vec{w}=\\left[\\begin{array}{r}1 \\\\ 2 \\\\ -2\\end{array}\\right]$ and $\\vec{v}=\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 3\\end{array}\\right]$.", "answer": "$\\overrightarrow{\\vec{u} \\bullet \\vec{v}} \\overrightarrow{\\vec{u}} \\vec{u}=\\frac{-5}{10}\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 3\\end{array}\\right]=\\left[\\begin{array}{r}-\\frac{1}{2} \\\\ 0 \\\\ -\\frac{3}{2}\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.21", "question": "How much work does it take to slide a crate 20 meters along a loading dock by pulling on it with a 200 Newton force at an angle of $30^{\\circ}$ from the horizontal? Express your answer in Newton meters.", "answer": "$200\\left(\\cos \\left(\\frac{\\pi}{6}\\right)\\right) 20=3464.1$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.10", "question": "An airplane is flying due north at 150.0 miles per hour but it is not actually going due North because there is a wind which is pushing the airplane due east at 40.0 miles per hour. After one hour, the plane starts flying $30^{\\circ}$ East of North. Assuming the plane starts at $(0,0)$, where is it after 2 hours? Let North be the direction of the positive y axis and let East be the direction of the positive $x$ axis.", "answer": "After two hours it is then at $(40,150)+150\\left[\\begin{array}{ll}\\frac{1}{2} & \\frac{\\sqrt{3}}{2}\\end{array}\\right]+\\left[\\begin{array}{ll}40 & 0\\end{array}\\right]=\\left[\\begin{array}{ll}155 & 75 \\sqrt{3}+150\\end{array}\\right]=\\left[\\begin{array}{ll}155.0 & 279.9\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.3", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.\n$$\nA=\\left[\\begin{array}{rrr}\n-1 & 1 & 1 \\\\\n1 & -1 & 1 \\\\\n1 & 1 & -1\n\\end{array}\\right]\n$$\nHint: One eigenvalue is -2.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{3} \\sqrt{3}\n\\end{array}\\right]\\right\\} \\leftrightarrow 1,\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{2} \\sqrt{2} \\\\\n\\frac{1}{2} \\sqrt{2} \\\\\n0\n\\end{array}\\right],\\left[\\begin{array}{c}\n-\\frac{1}{6} \\sqrt{6} \\\\\n-\\frac{1}{6} \\sqrt{6} \\\\\n\\frac{1}{3} \\sqrt{2} \\sqrt{3}\n\\end{array}\\right]\\right\\} \\leftrightarrow-2 \\\\\n{\\left[\\begin{array}{ccc}\n\\sqrt{3} / 3 & -\\sqrt{2} / 2 & -\\sqrt{6} / 6 \\\\\n\\sqrt{3} / 3 & \\sqrt{2} / 2 & -\\sqrt{6} / 6 \\\\\n\\sqrt{3} / 3 & 0 & \\frac{1}{3} \\sqrt{2} \\sqrt{3}\n\\end{array}\\right]^{T}\\left[\\begin{array}{rrr}\n-1 & 1 & 1 \\\\\n1 & -1 & 1 \\\\\n1 & 1 & -1\n\\end{array}\\right]} \\\\\n.\\left[\\begin{array}{ccc}\n\\sqrt{3} / 3 & -\\sqrt{2} / 2 & -\\sqrt{6} / 6 \\\\\n\\sqrt{3} / 3 & \\sqrt{2} / 2 & -\\sqrt{6} / 6 \\\\\n\\sqrt{3} / 3 & 0 & \\frac{1}{3} \\sqrt{2} \\sqrt{3}\n\\end{array}\\right] \\\\\n=\\left[\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & -2 & 0 \\\\\n0 & 0 & -2\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.4", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.\n$$\nA=\\left[\\begin{array}{rrr}\n17 & -7 & -4 \\\\\n-7 & 17 & -4 \\\\\n-4 & -4 & 14\n\\end{array}\\right]\n$$\nHint: Two eigenvalues are 18 and 24.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{3} \\sqrt{3}\n\\end{array}\\right]\\right\\} \\leftrightarrow 6,\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{6} \\sqrt{6} \\\\\n-\\frac{1}{6} \\sqrt{6} \\\\\n\\frac{1}{3} \\sqrt{2} \\sqrt{3}\n\\end{array}\\right]\\right\\} \\leftrightarrow 18,\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{2} \\sqrt{2} \\\\\n\\frac{1}{2} \\sqrt{2} \\\\\n0\n\\end{array}\\right]\\right\\} \\leftrightarrow 24\n$$\n\nThe matrix $U$ has these as its columns.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.11", "question": "City $A$ is located at the origin $(0,0)$ while city $B$ is located at $(300,500)$ where distances are in miles. An airplane flies at 250 miles per hour in still air. This airplane wants to fly from city A to city $B$ but the wind is blowing in the direction of the positive y axis at a speed of 50 miles per hour. Find a unit vector such that if the plane heads in this direction, it will end up at city B having flown the shortest possible distance. How long will it take to get there?", "answer": "Therefore, it takes $\\frac{583.1}{291.55}=2$ hours.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.6", "question": "Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?\n$$\n\\left[\\begin{array}{ccccc|c}\n\\mathbf{\\square} & * & * & * & * & * \\\\\n0 & \\mathbf{\\square} & * & * & 0 & * \\\\\n0 & 0 & \\mathbf{\\square} & * & * & * \\\\\n0 & 0 & 0 & 0 & \\mathbf{\\square} & *\n\\end{array}\\right]\n$$", "answer": "The solution exists but is not unique.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.20", "question": "A large dog drags a sled for 300 feet along the ground by pulling on a rope which is 45 degrees from the horizontal with a force of 20 pounds. How much work does this force do?", "answer": "$20\\left(\\cos \\frac{\\pi}{4}\\right) 300=4242.6$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.6", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.\n$$\nA=\\left[\\begin{array}{ccc}\n-\\frac{5}{3} & \\frac{1}{15} \\sqrt{6} \\sqrt{5} & \\frac{8}{15} \\sqrt{5} \\\\\n\\frac{1}{15} \\sqrt{6} \\sqrt{5} & -\\frac{14}{5} & -\\frac{1}{15} \\sqrt{6} \\\\\n\\frac{8}{15} \\sqrt{5} & -\\frac{1}{15} \\sqrt{6} & \\frac{7}{15}\n\\end{array}\\right]\n$$\nHint: The eigenvalues are $-3,-2,1$.", "answer": "eigenvectors:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{6} \\sqrt{6} \\\\\n0 \\\\\n\\frac{1}{6} \\sqrt{5} \\sqrt{6}\n\\end{array}\\right]\\right\\} \\leftrightarrow 1,\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{3} \\sqrt{2} \\sqrt{3} \\\\\n-\\frac{1}{5} \\sqrt{5} \\\\\n\\frac{1}{15} \\sqrt{2} \\sqrt{15}\n\\end{array}\\right]\\right\\} \\leftrightarrow-2,\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{6} \\sqrt{6} \\\\\n\\frac{2}{5} \\sqrt{5} \\\\\n\\frac{1}{30} \\sqrt{30}\n\\end{array}\\right]\\right\\} \\leftrightarrow-3\n$$\n\nThese vectors are the columns of $U$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.8", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.\n$$\nA=\\left[\\begin{array}{lll}\n2 & 0 & 0 \\\\\n0 & 5 & 1 \\\\\n0 & 1 & 5\n\\end{array}\\right]\n$$", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n1 \\\\\n0 \\\\\n0\n\\end{array}\\right]\\right\\} \\leftrightarrow 2,\\left\\{\\left[\\begin{array}{c}\n0 \\\\\n-\\frac{1}{2} \\sqrt{2} \\\\\n\\frac{1}{2} \\sqrt{2}\n\\end{array}\\right]\\right\\} \\leftrightarrow 4,\\left\\{\\left[\\begin{array}{c}\n0 \\\\\n\\frac{1}{2} \\sqrt{2} \\\\\n\\frac{1}{2} \\sqrt{2}\n\\end{array}\\right]\\right\\} \\leftrightarrow 6\n$$\n\nThese vectors are the columns of $U$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.35", "question": "Find the solution to the system of equations, $65 x+84 y+16 z=546,81 x+105 y+20 z=682$, and $84 x+110 y+21 z=713$.", "answer": "Solution is: $[x=2, y=4, z=5]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.2.4", "question": "Decide whether $\\vec{v}=\\left[\\begin{array}{l}4 \\\\ 4 \\\\ 4\\end{array}\\right]$ is a linear combination of the vectors $\\vec{u}_{1}=\\left[\\begin{array}{r}3 \\\\ 1 \\\\ -1\\end{array}\\right]$ and $\\vec{u}_{2}=\\left[\\begin{array}{r}2 \\\\ -2 \\\\ 1\\end{array}\\right]$.", "answer": "The system\n$$\n\\left[\\begin{array}{l}\n4 \\\\\n4 \\\\\n4\n\\end{array}\\right]=a_{1}\\left[\\begin{array}{r}\n3 \\\\\n1 \\\\\n-1\n\\end{array}\\right]+a_{2}\\left[\\begin{array}{r}\n2 \\\\\n-2 \\\\\n1\n\\end{array}\\right]\n$$\nhas no solution.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.3.9", "question": "A person sets off on a random walk with three possible locations. The Markov matrix of probabilities $A=\\left[a_{i j}\\right]$ is given by\n$$\n\\left[\\begin{array}{lll}\n0.1 & 0.3 & 0.7 \\\\\n0.1 & 0.3 & 0.2 \\\\\n0.8 & 0.4 & 0.1\n\\end{array}\\right]\n$$\nIf the walker starts in location 2 , what is the probability of ending back in location 2 at time $n=3$ ?", "answer": "$$\nX_{3}=\\left[\\begin{array}{l}\n0.38 \\\\\n0.18 \\\\\n0.44\n\\end{array}\\right]\n$$\n\nTherefore the probability of ending up back in location 2 is 0.18 .", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.19", "question": "A girl drags a sled for 200 feet along the ground by pulling on a rope which is 30 degrees from the horizontal with a force of 20 pounds. How much work does this force do?", "answer": "$20 \\cos \\left(\\frac{\\pi}{6}\\right) 200=3464.1$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.4", "question": "Find the area of the triangle determined by the three points, $(1,2,3),(2,3,4)$ and $(3,4,5)$. Did something interesting happen here? What does it mean geometrically?", "answer": "$\\left[\\begin{array}{lll}1 & 1 & 1\\end{array}\\right] \\times\\left[\\begin{array}{lll}2 & 2 & 2\\end{array}\\right]=\\left[\\begin{array}{lll}0 & 0 & 0\\end{array}\\right]$. The area is 0. It means the three points are on the same line.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.11", "question": "Suppose $\\vec{u}, \\vec{v}$, and $\\vec{w}$ are three vectors whose components are all integers. Can you conclude the volume of the parallelepiped determined from these three vectors will always be an integer?", "answer": "Yes. It will involve the sum of product of integers and so it will be an integer.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.2.14", "question": "Find the inverse, if it exists, of the matrix\n$$\nA=\\left[\\begin{array}{ccc}\ne^{t} & \\cos t & \\sin t \\\\\ne^{t} & -\\sin t & \\cos t \\\\\ne^{t} & -\\cos t & -\\sin t\n\\end{array}\\right]\n$$", "answer": "$$\n\\begin{aligned}\n& {\\left[\\begin{array}{ccc}\ne^{t} & \\cos t & \\sin t \\\\\ne^{t} & -\\sin t & \\cos t \\\\\ne^{t} & -\\cos t & -\\sin t\n\\end{array}\\right]^{-1} } \\\\\n= & {\\left[\\begin{array}{ccc}\n\\frac{1}{2} e^{-t} & 0 & \\frac{1}{2} e^{-t} \\\\\n\\frac{1}{2} \\cos t+\\frac{1}{2} \\sin t & -\\sin t & \\frac{1}{2} \\sin t-\\frac{1}{2} \\cos t \\\\\n\\frac{1}{2} \\sin t-\\frac{1}{2} \\cos t & \\cos t & -\\frac{1}{2} \\cos t-\\frac{1}{2} \\sin t\n\\end{array}\\right] }\n\\end{aligned}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.16", "question": "Simplify $\\|\\vec{u} \\times \\vec{v}\\|^{2}+(\\vec{u} \\bullet \\vec{v})^{2}-\\|\\vec{u}\\|^{2}\\|\\vec{v}\\|^{2}$.", "answer": "$$\n\\begin{aligned}\n\\|\\vec{u} \\times \\vec{v}\\|^{2} & =\\varepsilon_{i j k} u_{j} v_{k} \\varepsilon_{i r s} u_{r} v_{s}=\\left(\\delta_{j r} \\delta_{k s}-\\delta_{k r} \\delta_{j s}\\right) u_{r} v_{s} u_{j} v_{k} \\\\\n& =u_{j} v_{k} u_{j} v_{k}-u_{k} v_{j} u_{j} v_{k}=\\|\\vec{u}\\|^{2}\\|\\vec{v}\\|^{2}-(\\vec{u} \\bullet \\vec{v})^{2}\n\\end{aligned}\n$$\nIt follows that the expression reduces to 0. You can also do the following.\n$$\n\\begin{aligned}\n\\|\\vec{u} \\times \\vec{v}\\|^{2} & =\\|\\vec{u}\\|^{2}\\|\\vec{v}\\|^{2} \\sin ^{2} \\theta \\\\\n& =\\|\\vec{u}\\|^{2}\\|\\vec{v}\\|^{2}\\left(1-\\cos ^{2} \\theta\\right) \\\\\n& =\\|\\vec{u}\\|^{2}\\|\\vec{v}\\|^{2}-\\|\\vec{u}\\|^{2}\\|\\vec{v}\\|^{2} \\cos ^{2} \\theta \\\\\n& =\\|\\vec{u}\\|^{2}\\|\\vec{v}\\|^{2}-(\\vec{u} \\bullet \\vec{v})^{2}\n\\end{aligned}\n$$\nwhich implies the expression equals 0.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.2.1", "question": "Find $-3\\left[\\begin{array}{r}5 \\\\ -1 \\\\ 2 \\\\ -3\\end{array}\\right]+5\\left[\\begin{array}{r}-8 \\\\ 2 \\\\ -3 \\\\ 6\\end{array}\\right]$.", "answer": "$\\left[\\begin{array}{r}-55 \\\\ 13 \\\\ -21 \\\\ 39\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.41", "question": "Let\n$$\nA=\\left[\\begin{array}{lll}\n1 & 0 & 3 \\\\\n2 & 3 & 4 \\\\\n1 & 0 & 2\n\\end{array}\\right]\n$$\nFind $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.", "answer": "$\\left[\\begin{array}{lll}1 & 0 & 3 \\\\ 2 & 3 & 4 \\\\ 1 & 0 & 2\\end{array}\\right]^{-1}=\\left[\\begin{array}{rrr}-2 & 0 & 3 \\\\ 0 & \\frac{1}{3} & -\\frac{2}{3} \\\\ 1 & 0 & -1\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.5", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.\n$$\nA=\\left[\\begin{array}{rrr}\n13 & 1 & 4 \\\\\n1 & 13 & 4 \\\\\n4 & 4 & 10\n\\end{array}\\right]\n$$\nHint: Two eigenvalues are 12 and 18.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{6} \\sqrt{6} \\\\\n-\\frac{1}{6} \\sqrt{6} \\\\\n\\frac{1}{3} \\sqrt{2} \\sqrt{3}\n\\end{array}\\right]\\right\\} \\leftrightarrow 6,\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{2} \\sqrt{2} \\\\\n\\frac{1}{2} \\sqrt{2} \\\\\n0\n\\end{array}\\right]\\right\\} \\leftrightarrow 12,\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{3} \\sqrt{3}\n\\end{array}\\right]\\right\\} \\leftrightarrow 18 .\n$$\n\nThe matrix $U$ has these as its columns.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.10", "question": "An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.\n\n$$\n\\left[\\begin{array}{ll}\na & b \\\\\nc & d\n\\end{array}\\right] \\rightarrow \\cdots \\rightarrow\\left[\\begin{array}{ll}\nc & d \\\\\na & b\n\\end{array}\\right]\n$$", "answer": "In this case two rows were switched and so the resulting determinant is -1 times the first.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.6", "question": "Find the following determinant by expanding along the second row and first column.\n\n$$\n\\left|\\begin{array}{lll}\n1 & 2 & 1 \\\\\n2 & 1 & 3 \\\\\n2 & 1 & 1\n\\end{array}\\right|\n$$", "answer": "$$\n\\left|\\begin{array}{lll}\n1 & 2 & 1 \\\\\n2 & 1 & 3 \\\\\n2 & 1 & 1\n\\end{array}\\right|=6\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.9", "question": "Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?\n$$\n\\left[\\begin{array}{lllll|l}\n\\mathbf{\\square} & * & * & * & * & * \\\\\n0 & \\mathbf{\\square} & * & * & 0 & * \\\\\n0 & 0 & 0 & 0 & \\mathbf{\\square} & 0 \\\\\n0 & 0 & 0 & 0 & * & \\mathbf{\\square}\n\\end{array}\\right]\n$$", "answer": "There might be a solution. If so, there are infinitely many.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.11", "question": "An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.\n\n$$\n\\left[\\begin{array}{ll}\na & b \\\\\nc & d\n\\end{array}\\right] \\rightarrow \\cdots \\rightarrow\\left[\\begin{array}{cc}\na & b \\\\\na+c & b+d\n\\end{array}\\right]\n$$", "answer": "The determinant is unchanged. It was just the first row added to the second.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.9.9", "question": "Write the solution set of the following system as a linear combination of vectors\n$$\n\\left[\\begin{array}{rrrr}\n1 & 0 & 1 & 1 \\\\\n1 & -1 & 1 & 0 \\\\\n3 & -1 & 3 & 2 \\\\\n3 & 3 & 0 & 3\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz \\\\\nw\n\\end{array}\\right]=\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{array}\\right]\n$$", "answer": "Solution is: $\\left[\\begin{array}{c}0 \\\\ -\\hat{t} \\\\ -\\hat{t} \\\\ \\hat{t}\\end{array}\\right], \\hat{t} \\in \\mathbb{R}$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.7.4", "question": "Let $V=\\mathbb{R}^{3}$ and let\n\n$$\nW=\\operatorname{span}\\left\\{\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right],\\left[\\begin{array}{r}\n-1 \\\\\n2 \\\\\n-1\n\\end{array}\\right]\\right\\}\n$$\n\nExtend this basis of $W$ to a basis of $V$.", "answer": "There are many possible such extensions, one is (how do we know?):\n\n$$\n\\left\\{\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right],\\left[\\begin{array}{r}\n-1 \\\\\n2 \\\\\n-1\n\\end{array}\\right],\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n1\n\\end{array}\\right]\\right\\}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.38", "question": "Find the solution to the system of equations, $3 x-y-2 z=3, y-4 z=0$, and $-2 x+y=-2$.", "answer": "Solution is: $[x=2 t+1, y=4 t, z=t]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.12", "question": "An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.\n\n$$\n\\left[\\begin{array}{ll}\na & b \\\\\nc & d\n\\end{array}\\right] \\rightarrow \\cdots \\rightarrow\\left[\\begin{array}{cc}\na & b \\\\\n2 c & 2 d\n\\end{array}\\right]\n$$", "answer": "The second row was multiplied by 2 so the determinant of the result is 2 times the original determinant.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.9.7", "question": "Write the solution set of the following system as a linear combination of vectors\n$$\n\\left[\\begin{array}{rrr}\n0 & -1 & 2 \\\\\n1 & 0 & 1 \\\\\n1 & -2 & 5\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz\n\\end{array}\\right]=\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n0\n\\end{array}\\right]\n$$", "answer": "Solution is: $\\left[\\begin{array}{c}-\\hat{t} \\\\ 2 \\hat{t} \\\\ \\hat{t}\\end{array}\\right], \\hat{t} \\in \\mathbb{R}$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.3.30", "question": "If you have 6 vectors in $\\mathbb{R}^{5}$, is it possible they are linearly independent? Explain.", "answer": "No. They can't be.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.25", "question": "Find the determinant using row operations to first simplify.\n\n$$\n\\left|\\begin{array}{rrrr}\n1 & 4 & 1 & 2 \\\\\n3 & 2 & -2 & 3 \\\\\n-1 & 0 & 3 & 3 \\\\\n2 & 1 & 2 & -2\n\\end{array}\\right|\n$$", "answer": "One can row reduce this using only row operation 3 to\n\n$$\n\\left[\\begin{array}{rrrr}\n1 & 4 & 1 & 2 \\\\\n0 & -10 & -5 & -3 \\\\\n0 & 0 & 2 & \\frac{19}{5} \\\\\n0 & 0 & 0 & -\\frac{211}{20}\n\\end{array}\\right]\n$$\n\nThus the determinant is given by\n\n$$\n\\left|\\begin{array}{rrrr}\n1 & 4 & 1 & 2 \\\\\n3 & 2 & -2 & 3 \\\\\n-1 & 0 & 3 & 3 \\\\\n2 & 1 & 2 & -2\n\\end{array}\\right|=211\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.6.3.9", "question": "If $n$ is an integer, is it always true that $(\\cos \\theta-i \\sin \\theta)^{n}=\\cos (n \\theta)-i \\sin (n \\theta)$ ? Explain.", "answer": "Yes, this is true.\n\n$$\n\\begin{aligned}\n(\\cos \\theta-i \\sin \\theta)^{n} & =(\\cos (-\\theta)+i \\sin (-\\theta))^{n} \\\\\n& =\\cos (-n \\theta)+i \\sin (-n \\theta) \\\\\n& =\\cos (n \\theta)-i \\sin (n \\theta)\n\\end{aligned}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.6", "question": "Find the area of the parallelogram determined by the vectors $\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 3\\end{array}\\right],\\left[\\begin{array}{r}4 \\\\ -2 \\\\ 1\\end{array}\\right]$.", "answer": "$\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 3\\end{array}\\right] \\times\\left[\\begin{array}{r}4 \\\\ -2 \\\\ 1\\end{array}\\right]=\\left[\\begin{array}{r}6 \\\\ 11 \\\\ -2\\end{array}\\right]$. The area is $\\sqrt{36+121+4}=\\sqrt{161}$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.2.12", "question": "Find the matrix for $T(\\vec{w})=\\operatorname{proj}_{\\vec{v}}(\\vec{w})$ where $\\vec{v}=\\left[\\begin{array}{lll}1 & 5 & 3\\end{array}\\right]^{T}$.", "answer": "$$\n\\frac{1}{35}\\left[\\begin{array}{rrr}\n1 & 5 & 3 \\\\\n5 & 25 & 15 \\\\\n3 & 15 & 9\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.9.17", "question": "Suppose $A \\vec{x}=\\vec{b}$ has a solution. Explain why the solution is unique precisely when $A \\vec{x}=\\overrightarrow{0}$ has only the trivial solution.", "answer": "If not, then there would be a infintely many solutions to $A \\vec{x}=\\overrightarrow{0}$ and each of these added to a solution to $A \\vec{x}=\\vec{b}$ would be a solution to $A \\vec{x}=\\vec{b}$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.2", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$.\n$$\nA=\\left[\\begin{array}{rrr}\n4 & 1 & -2 \\\\\n1 & 4 & -2 \\\\\n-2 & -2 & 7\n\\end{array}\\right]\n$$\nHint: One eigenvalue is 3.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right], \\frac{1}{\\sqrt{3}}\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right]\\right\\} \\leftrightarrow 3,\\left\\{\\frac{1}{\\sqrt{6}}\\left[\\begin{array}{r}\n-1 \\\\\n-1 \\\\\n2\n\\end{array}\\right]\\right\\} \\leftrightarrow 9\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.9.1", "question": "Write the solution set of the following system as a linear combination of vectors\n$$\n\\left[\\begin{array}{lll}\n1 & -1 & 2 \\\\\n1 & -2 & 1 \\\\\n3 & -4 & 5\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz\n\\end{array}\\right]=\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n0\n\\end{array}\\right]\n$$", "answer": "Solution is: $\\left[\\begin{array}{r}-3 \\hat{t} \\\\ -\\hat{t} \\\\ \\hat{t}\\end{array}\\right], \\hat{t}_{3} \\in \\mathbb{R}$. A basis for the solution space is $\\left[\\begin{array}{r}-3 \\\\ -1 \\\\ 1\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.1.1", "question": "If $A$ is an invertible $n \\times n$ matrix, compare the eigenvalues of $A$ and $A^{-1}$. More generally, for $m$ an arbitrary integer, compare the eigenvalues of $A$ and $A^{m}$.", "answer": "$A^{m} X=\\lambda^{m} X$ for any integer. In the case of $-1, A^{-1} \\lambda X=A A^{-1} X=X$ so $A^{-1} X=\\lambda^{-1} X$. Thus the eigenvalues of $A^{-1}$ are just $\\lambda^{-1}$ where $\\lambda$ is an eigenvalue of $A$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.12", "question": "What does it mean geometrically if the box product of three vectors gives zero?", "answer": "It means that if you place them so that they all have their tails at the same point, the three will lie in the same plane.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.10", "question": "Find the volume of the parallelepiped determined by the vectors $\\left[\\begin{array}{r}1 \\\\ -7 \\\\ -5\\end{array}\\right]$,\n$\\left[\\begin{array}{r}1 \\\\ -2 \\\\ -6\\end{array}\\right]$, and $\\left[\\begin{array}{l}3 \\\\ 2 \\\\ 3\\end{array}\\right]$", "answer": "$\\left|\\begin{array}{rrr}1 & -7 & -5\\end{array}\\right|=113$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.9", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.\n$$\nA=\\left[\\begin{array}{ccc}\n\\frac{4}{3} & \\frac{1}{3} \\sqrt{3} \\sqrt{2} & \\frac{1}{3} \\sqrt{2} \\\\\n\\frac{1}{3} \\sqrt{3} \\sqrt{2} & 1 & -\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{3} \\sqrt{2} & -\\frac{1}{3} \\sqrt{3} & \\frac{5}{3}\n\\end{array}\\right]\n$$\nHint: The eigenvalues are 0,2,2 where 2 is listed twice because it is a root of multiplicity 2.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{5} \\sqrt{2} \\sqrt{5} \\\\\n\\frac{1}{5} \\sqrt{3} \\sqrt{5} \\\\\n\\frac{1}{5} \\sqrt{5}\n\\end{array}\\right]\\right\\} \\leftrightarrow 0,\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{3} \\sqrt{3} \\\\\n0 \\\\\n\\frac{1}{3} \\sqrt{2} \\sqrt{3}\n\\end{array}\\right],\\left[\\begin{array}{c}\n\\frac{1}{5} \\sqrt{2} \\sqrt{5} \\\\\n\\frac{1}{5} \\sqrt{3} \\sqrt{5} \\\\\n-\\frac{1}{5} \\sqrt{5}\n\\end{array}\\right]\\right\\} \\leftrightarrow 2\n$$\n\nThe columns are these vectors.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.2.13", "question": "Find the matrix for $T(\\vec{w})=\\operatorname{proj}_{\\vec{v}}(\\vec{w})$ where $\\vec{v}=\\left[\\begin{array}{lll}1 & 0 & 3\\end{array}\\right]^{T}$.", "answer": "$$\n\\frac{1}{10}\\left[\\begin{array}{lll}\n1 & 0 & 3 \\\\\n0 & 0 & 0 \\\\\n3 & 0 & 9\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.1.21", "question": "Consider functions defined on $\\{1,2, \\cdots, n\\}$ having values in $\\mathbb{R}$. Explain how, if $V$ is the set of all such functions, $V$ can be considered as $\\mathbb{R}^{n}$.", "answer": "Let $f(i)$ be the $i^{t h}$ component of a vector $\\vec{x} \\in \\mathbb{R}^{n}$. Thus a typical element in $\\mathbb{R}^{n}$ is $(f(1), \\cdots, f(n))$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.45", "question": "The steady state temperature, $u$, of a plate solves Laplace's equation, $\\Delta u=0$. One way to approximate the solution is to divide the plate into a square mesh and require the temperature at each node to equal the average of the temperature at the four adjacent nodes. In the following picture, the numbers represent the observed temperature at the indicated nodes. Find the temperature at the interior nodes, indicated by $x, y, z$, and $w$. One of the equations is $z=\\frac{1}{4}(10+0+w+x)$.", "answer": "Solution is: $[w=15, x=15, y=20, z=10]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.7.12", "question": "Find $\\operatorname{proj}_{\\vec{v}}(\\vec{w})$ where $\\vec{w}=\\left[\\begin{array}{r}1 \\\\ 2 \\\\ -2 \\\\ 1\\end{array}\\right]$ and $\\vec{v}=\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3 \\\\ 0\\end{array}\\right]$.", "answer": "$\\overrightarrow{\\vec{u} \\bullet \\vec{v}} \\overrightarrow{\\vec{u}} \\vec{u}=\\frac{\\left[\\begin{array}{llll}1 & 2 & -2 & 1\\end{array}\\right]^{T} \\bullet\\left[\\begin{array}{llll}1 & 2 & 3 & 0\\end{array}\\right]^{T}}{1+4+9}\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3 \\\\ 0\\end{array}\\right]=\\left[\\begin{array}{r}-\\frac{1}{14} \\\\ -\\frac{1}{7} \\\\ -\\frac{3}{14} \\\\ 0\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.24", "question": "Find the determinant using row operations to first simplify.\n\n$$\n\\left|\\begin{array}{rrrr}\n1 & 2 & 1 & 2 \\\\\n3 & 1 & -2 & 3 \\\\\n-1 & 0 & 3 & 1 \\\\\n2 & 3 & 2 & -2\n\\end{array}\\right|\n$$", "answer": "One can row reduce this using only row operation 3 to\n\n$$\n\\left[\\begin{array}{rrrr}\n1 & 2 & 1 & 2 \\\\\n0 & -5 & -5 & -3 \\\\\n0 & 0 & 2 & \\frac{9}{5} \\\\\n0 & 0 & 0 & -\\frac{63}{10}\n\\end{array}\\right]\n$$\n\nand therefore, the determinant is -63.\n\n$$\n\\left|\\begin{array}{rrrr}\n1 & 2 & 1 & 2 \\\\\n3 & 1 & -2 & 3 \\\\\n-1 & 0 & 3 & 1 \\\\\n2 & 3 & 2 & -2\n\\end{array}\\right|=63\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.7", "question": "Is $\\vec{u} \\times(\\vec{v} \\times \\vec{w})=(\\vec{u} \\times \\vec{v}) \\times \\vec{w}$ ? What is the meaning of $\\vec{u} \\times \\vec{v} \\times \\vec{w}$ ? Explain. Hint: Try $(\\vec{i} \\times \\vec{j}) \\times \\vec{k}$", "answer": "$(\\vec{i} \\times \\vec{j}) \\times \\vec{j}=\\vec{k} \\times \\vec{j}=-\\vec{i}$. However, $\\vec{i} \\times(\\vec{j} \\times \\vec{j})=\\overrightarrow{0}$ and so the cross product is not associative.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.5", "question": "Find the following determinant by expanding along the first column and third row.\n\n$$\n\\left|\\begin{array}{lll}\n1 & 2 & 1 \\\\\n1 & 0 & 1 \\\\\n2 & 1 & 1\n\\end{array}\\right|\n$$", "answer": "$$\n\\left|\\begin{array}{lll}\n1 & 2 & 1 \\\\\n1 & 0 & 1 \\\\\n2 & 1 & 1\n\\end{array}\\right|=2\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.9.12", "question": "Find the matrix for $T(\\vec{w})=\\operatorname{proj}_{\\vec{v}}(\\vec{w})$ where $\\vec{v}=\\left[\\begin{array}{lll}1 & 5 & 3\\end{array}\\right]^{T}$.", "answer": "$$\n\\frac{1}{35}\\left[\\begin{array}{rrr}\n1 & 5 & 3 \\\\\n5 & 25 & 15 \\\\\n3 & 15 & 9\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.29", "question": "Find the solution of the system whose augmented matrix is\n$$\n\\left[\\begin{array}{lll|l}\n1 & 1 & 0 & 1 \\\\\n1 & 0 & 4 & 2\n\\end{array}\\right]\n$$", "answer": "The solution is $z=t, y=4 t, x=2-4 t$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.9.3", "question": "Write the solution set of the following system as a linear combination of vectors\n$$\n\\left[\\begin{array}{lll}\n0 & -1 & 2 \\\\\n1 & -2 & 1 \\\\\n1 & -4 & 5\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz\n\\end{array}\\right]=\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n0\n\\end{array}\\right]\n$$", "answer": "Solution is: $\\left[\\begin{array}{c}3 \\hat{t} \\\\ 2 \\hat{t} \\\\ \\hat{t}\\end{array}\\right]$, A basis is $\\left[\\begin{array}{l}3 \\\\ 2 \\\\ 1\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.36", "question": "Find the solution to the system of equations, $8 x+2 y+3 z=-3,8 x+3 y+3 z=-1$, and $4 x+y+3 z=-9$.", "answer": "Solution is: $[x=1, y=2, z=-5]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.7.1", "question": "Find $\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3 \\\\ 4\\end{array}\\right] \\bullet\\left[\\begin{array}{l}2 \\\\ 0 \\\\ 1 \\\\ 3\\end{array}\\right]$.", "answer": "$\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3 \\\\ 4\\end{array}\\right] \\bullet\\left[\\begin{array}{l}2 \\\\ 0 \\\\ 1 \\\\ 3\\end{array}\\right]=17$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.18", "question": "A boy drags a sled for 100 feet along the ground by pulling on a rope which is 20 degrees from the horizontal with a force of 40 pounds. How much work does this force do?", "answer": "$40 \\cos \\left(\\frac{20}{180} \\pi\\right) 100=3758.8$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.12", "question": "Find $h$ such that\n$$\n\\left[\\begin{array}{ll|l}\n2 & h & 4 \\\\\n3 & 6 & 7\n\\end{array}\\right]\n$$\nis the augmented matrix of an inconsistent system.", "answer": "$h=4$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.40", "question": "Let\n$$\nA=\\left[\\begin{array}{lll}\n1 & 2 & 3 \\\\\n2 & 1 & 4 \\\\\n1 & 0 & 2\n\\end{array}\\right]\n$$\nFind $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.", "answer": "$\\left[\\begin{array}{lll}1 & 2 & 3 \\\\ 2 & 1 & 4 \\\\ 1 & 0 & 2\\end{array}\\right]^{-1}=\\left[\\begin{array}{rrr}-2 & 4 & -5 \\\\ 0 & 1 & -2 \\\\ 1 & -2 & 3\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.7", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.\n$$\nA=\\left[\\begin{array}{ccc}\n3 & 0 & 0 \\\\\n0 & \\frac{3}{2} & \\frac{1}{2} \\\\\n0 & \\frac{1}{2} & \\frac{3}{2}\n\\end{array}\\right]\n$$", "answer": "The eigenvectors and eigenvalues are: $\\left\\{\\left[\\begin{array}{c}0 \\\\ -\\frac{1}{2} \\sqrt{2} \\\\ \\frac{1}{2} \\sqrt{2}\\end{array}\\right]\\right\\} \\leftrightarrow 1,\\left\\{\\left[\\begin{array}{c}0 \\\\ \\frac{1}{2} \\sqrt{2} \\\\ \\frac{1}{2} \\sqrt{2}\\end{array}\\right]\\right\\} \\leftrightarrow 2,\\left\\{\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 0\\end{array}\\right]\\right\\} \\leftrightarrow 3$.\n\nThese vectors are the columns of the matrix $U$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.22", "question": "Find the determinant using row operations to first simplify.\n\n$$\n\\left|\\begin{array}{rrr}\n1 & 2 & 1 \\\\\n2 & 3 & 2 \\\\\n-4 & 1 & 2\n\\end{array}\\right|\n$$", "answer": "$$\n\\left|\\begin{array}{rrr}\n1 & 2 & 1 \\\\\n2 & 3 & 2 \\\\\n-4 & 1 & 2\n\\end{array}\\right|=-6\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.23", "question": "Find the determinant using row operations to first simplify.\n\n$$\n\\left|\\begin{array}{rrr}\n2 & 1 & 3 \\\\\n2 & 4 & 2 \\\\\n1 & 4 & -5\n\\end{array}\\right|\n$$", "answer": "$$\n\\left|\\begin{array}{rrr}\n2 & 1 & 3 \\\\\n2 & 4 & 2 \\\\\n1 & 4 & -5\n\\end{array}\\right|=-32\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.33", "question": "Find the solution to the system of equations, $3 x-y+4 z=6, y+8 z=0$, and $-2 x+y=-4$.", "answer": "Solution is: $[x=2-4 t, y=-8 t, z=t]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.2.7", "question": "Consider the matrix\n$$\nA=\\left[\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & \\cos t & -\\sin t \\\\\n0 & \\sin t & \\cos t\n\\end{array}\\right]\n$$\nDoes there exist a value of t for which this matrix fails to have an inverse? Explain.", "answer": "No. It has a nonzero determinant for all $t$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.11", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix\n$$\nA=\\left[\\begin{array}{ccc}\n\\frac{1}{3} & \\frac{1}{6} \\sqrt{3} \\sqrt{2} & \\frac{1}{6} \\sqrt{3} \\sqrt{6} \\\\\n\\frac{1}{6} \\sqrt{3} \\sqrt{2} & \\frac{3}{2} & \\frac{1}{12} \\sqrt{2} \\sqrt{6} \\\\\n\\frac{1}{6} \\sqrt{3} \\sqrt{6} & \\frac{1}{12} \\sqrt{2} \\sqrt{6} & \\frac{7}{6}\n\\end{array}\\right]\n$$\nHint: The eigenvalues are 1,2,-2.", "answer": "eigenvectors:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{2} \\sqrt{2} \\\\\n\\frac{1}{6} \\sqrt{6}\n\\end{array}\\right]\\right\\} \\leftrightarrow 1,\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{3} \\sqrt{3} \\\\\n0 \\\\\n\\frac{1}{3} \\sqrt{2} \\sqrt{3}\n\\end{array}\\right]\\right\\} \\leftrightarrow-2,\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{2} \\sqrt{2} \\\\\n-\\frac{1}{6} \\sqrt{6}\n\\end{array}\\right]\\right\\} \\leftrightarrow 2 .\n$$\n\nThen the columns of $U$ are these vectors", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.34", "question": "Find the solution to the system of equations, $9 x-2 y+4 z=-17,13 x-3 y+6 z=-25$, and $-2 x-z=3$.", "answer": "Solution is: $[x=-1, y=2, z=-1]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.6.3.1", "question": "Give the complete solution to $x^{4}+16=0$.", "answer": "Solution is:\n\n$$\n(1-i) \\sqrt{2},-(1+i) \\sqrt{2},-(1-i) \\sqrt{2},(1+i) \\sqrt{2}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.1.2", "question": "If $A$ is an $n \\times n$ matrix and $c$ is a nonzero constant, compare the eigenvalues of $A$ and $c A$.", "answer": "Say $A X=\\lambda X$. Then $c A X=c \\lambda X$ and so the eigenvalues of $c A$ are just $c \\lambda$ where $\\lambda$ is an eigenvalue of $A$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.10", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$. Diagonalize $A$ by finding an orthogonal matrix $U$ and a diagonal matrix $D$ such that $U^{T} A U=D$.\n$$\nA=\\left[\\begin{array}{ccc}\n1 & \\frac{1}{6} \\sqrt{3} \\sqrt{2} & \\frac{1}{6} \\sqrt{3} \\sqrt{6} \\\\\n\\frac{1}{6} \\sqrt{3} \\sqrt{2} & \\frac{3}{2} & \\frac{1}{12} \\sqrt{2} \\sqrt{6} \\\\\n\\frac{1}{6} \\sqrt{3} \\sqrt{6} & \\frac{1}{12} \\sqrt{2} \\sqrt{6} & \\frac{1}{2}\n\\end{array}\\right]\n$$\nHint: The eigenvalues are 2, 1,0.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{3} \\sqrt{3} \\\\\n0 \\\\\n\\frac{1}{3} \\sqrt{2} \\sqrt{3}\n\\end{array}\\right]\\right\\} \\leftrightarrow 0,\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{3} \\sqrt{3} \\\\\n-\\frac{1}{2} \\sqrt{2} \\\\\n\\frac{1}{6} \\sqrt{6}\n\\end{array}\\right]\\right\\} \\leftrightarrow 1,\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{3} \\sqrt{3} \\\\\n\\frac{1}{2} \\sqrt{2} \\\\\n\\frac{1}{6} \\sqrt{6}\n\\end{array}\\right]\\right\\} \\leftrightarrow 2\n$$\n\nThe columns are these vectors.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.40", "question": "Find the solution to the system of equations, $-19 x+8 y=-108,-71 x+30 y=-404$, $-2 x+y=-12,4 x+z=14$.", "answer": "Solution is: $[x=4, y=-4, z=-2]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.2.2", "question": "Find the eigenvalues and eigenvectors of the matrix\n$$\n\\left[\\begin{array}{rrr}\n-13 & -28 & 28 \\\\\n4 & 9 & -8 \\\\\n-4 & -8 & 9\n\\end{array}\\right]\n$$\nOne eigenvalue is 3. Diagonalize if possible.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{l}\n2 \\\\\n0 \\\\\n1\n\\end{array}\\right]\\right\\} \\leftrightarrow 1,\\left\\{\\left[\\begin{array}{c}\n-2 \\\\\n1 \\\\\n0\n\\end{array}\\right]\\right\\} \\leftrightarrow 1,\\left\\{\\left[\\begin{array}{c}\n7 \\\\\n-2 \\\\\n2\n\\end{array}\\right]\\right\\} \\leftrightarrow 3\n$$\n\nThe matrix $P$ needed to diagonalize the above matrix is\n\n$$\n\\left[\\begin{array}{rrr}\n2 & -2 & 7 \\\\\n0 & 1 & -2 \\\\\n1 & 0 & 2\n\\end{array}\\right]\n$$\n\nand the diagonal matrix $D$ is\n\n$$\n\\left[\\begin{array}{lll}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 3\n\\end{array}\\right]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.3.1.13", "question": "An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant.\n\n$$\n\\left[\\begin{array}{ll}\na & b \\\\\nc & d\n\\end{array}\\right] \\rightarrow \\cdots \\rightarrow\\left[\\begin{array}{ll}\nb & a \\\\\nd & c\n\\end{array}\\right]\n$$", "answer": "In this case the two columns were switched so the determinant of the second is -1 times the determinant of the first.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.7", "question": "The wind blows from West to East at a speed of 50 miles per hour and an airplane which travels at 400 miles per hour in still air is heading North West. What is the velocity of the airplane relative to the ground? What is the component of this velocity in the direction North?", "answer": "The velocity is the sum of two vectors. $50 \\vec{i}+\\frac{300}{\\sqrt{2}}(\\vec{i}+\\vec{j})=\\left(50+\\frac{300}{\\sqrt{2}}\\right) \\vec{i}+\\frac{300}{\\sqrt{2}} \\vec{j}$. The component in the direction of North is then $\\frac{300}{\\sqrt{2}}=150 \\sqrt{2}$ and the velocity relative to the ground is $\\left(50+\\frac{300}{\\sqrt{2}}\\right) \\vec{i}+\\frac{300}{\\sqrt{2}} \\vec{j}$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.1", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for $A$.\n$$\nA=\\left[\\begin{array}{rrr}\n11 & -1 & -4 \\\\\n-1 & 11 & -4 \\\\\n-4 & -4 & 14\n\\end{array}\\right]\n$$\nHint: Two eigenvalues are 12 and 18.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\frac{1}{\\sqrt{3}}\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right]\\right\\} \\leftrightarrow 6,\\left\\{\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right]\\right\\} \\leftrightarrow 12,\\left\\{\\frac{1}{\\sqrt{6}}\\left[\\begin{array}{r}\n-1 \\\\\n-1 \\\\\n2\n\\end{array}\\right]\\right\\} \\leftrightarrow 18\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.41", "question": "Suppose a system of equations has fewer equations than variables and you have found a solution to this system of equations. Is it possible that your solution is the only one? Explain.", "answer": "No. Consider $x+y+z=2$ and $x+y+z=1$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.3.29", "question": "If you have 5 vectors in $\\mathbb{R}^{5}$ and the vectors are linearly independent, can it always be concluded they span $\\mathbb{R}^{5}$ ?", "answer": "Yes. If not, there would exist a vector not in the span. But then you could add in this vector and obtain a linearly independent set of vectors with more vectors than a basis.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.9.11", "question": "Write the solution set of the following system as a linear combination of vectors\n$$\n\\left[\\begin{array}{llll}\n1 & 1 & 0 & 1 \\\\\n2 & 1 & 1 & 2 \\\\\n1 & 0 & 1 & 1 \\\\\n0 & 0 & 0 & 0\n\\end{array}\\right]\\left[\\begin{array}{l}\nx \\\\\ny \\\\\nz \\\\\nw\n\\end{array}\\right]=\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{array}\\right]\n$$", "answer": "Solution is: $\\left[\\begin{array}{c}-s-t \\\\ s \\\\ s \\\\ t\\end{array}\\right], s, t \\in \\mathbb{R}$. A basis is\n\n$$\n\\left\\{\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n1 \\\\\n0\n\\end{array}\\right],\\left[\\begin{array}{r}\n-1 \\\\\n0 \\\\\n0 \\\\\n1\n\\end{array}\\right]\\right\\}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.1", "question": "Find the point $\\left(x_{1}, y_{1}\\right)$ which lies on both lines, $x+3 y=1$ and $4 x-y=3$.", "answer": "Solution is: $\\left[x=\\frac{10}{13}, y=\\frac{1}{13}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.6.3.3", "question": "Find the four fourth roots of 16.", "answer": "Solution is:\n\n$$\n(1-i) \\sqrt{2},-(1+i) \\sqrt{2},-(1-i) \\sqrt{2},(1+i) \\sqrt{2}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.4.2", "question": "Let $M=\\left\\{\\vec{u}=\\left(u_{1}, u_{2}, u_{3}, u_{4}\\right) \\in \\mathbb{R}^{4}: \\sin \\left(u_{1}\\right)=1\\right\\}$. Is $M$ a subspace of $\\mathbb{R}^{4}$ ?", "answer": "This is not a subspace.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.3", "question": "Find the area of the triangle determined by the three points, $(1,0,3),(4,1,0)$ and $(-3,1,1)$.", "answer": "$\\left[\\begin{array}{r}3 \\\\ 1 \\\\ -3\\end{array}\\right] \\times\\left[\\begin{array}{r}-4 \\\\ 1 \\\\ -2\\end{array}\\right]=\\left[\\begin{array}{c}1 \\\\ 18 \\\\ 7\\end{array}\\right]$. The area is given by\n$$\n\\frac{1}{2} \\sqrt{1+(18)^{2}+49}=\\frac{1}{2} \\sqrt{374}\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.8.1", "question": "Let $V=\\mathbb{R}^{3}$ and let\n$$\nW=\\operatorname{span}(S), \\text { where } S=\\left\\{\\left[\\begin{array}{r}\n1 \\\\\n-1 \\\\\n1\n\\end{array}\\right],\\left[\\begin{array}{r}\n-2 \\\\\n2 \\\\\n-2\n\\end{array}\\right],\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n1\n\\end{array}\\right],\\left[\\begin{array}{r}\n1 \\\\\n-1 \\\\\n3\n\\end{array}\\right]\\right\\}\n$$\nFind a basis of $W$ consisting of vectors in $S$.", "answer": "In this case $\\operatorname{dim}(W)=1$ and a basis for $W$ consisting of vectors in $S$ can be obtained by taking any (nonzero) vector from $S$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.9.9.15", "question": "Let $B=\\left\\{\\left[\\begin{array}{r}1 \\\\ -1 \\\\ 2\\end{array}\\right],\\left[\\begin{array}{l}2 \\\\ 1 \\\\ 2\\end{array}\\right],\\left[\\begin{array}{r}-1 \\\\ 0 \\\\ 2\\end{array}\\right]\\right\\}$ be a basis of $\\mathbb{R}^{3}$ and let $\\vec{x}=\\left[\\begin{array}{r}5 \\\\ -1 \\\\ 4\\end{array}\\right]$ be a vector in $\\mathbb{R}^{2}$. Find $C_{B}(\\vec{x})$.", "answer": "$C_{B}(\\vec{x})=\\left[\\begin{array}{r}2 \\\\ 1 \\\\ -1\\end{array}\\right]$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.11.21", "question": "Find the least squares solution to the following system.\n$$\n\\begin{gathered}\nx+2 y=1 \\\\\n2 x+3 y=2 \\\\\n3 x+5 y=4\n\\end{gathered}\n$$", "answer": "Solution is: $\\left[\\begin{array}{c}\\frac{2}{3} \\\\ \\frac{1}{3}\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.7.8", "question": "Find the angle between the vectors\n$$\n\\vec{u}=\\left[\\begin{array}{r}\n3 \\\\\n-1 \\\\\n-1\n\\end{array}\\right], \\vec{v}=\\left[\\begin{array}{l}\n1 \\\\\n4 \\\\\n2\n\\end{array}\\right]\n$$", "answer": "$\\frac{\\left[\\begin{array}{lll}3 & -1 & -1\\end{array}\\right]^{T} \\cdot\\left[\\begin{array}{lll}1 & 4 & 2\\end{array}\\right]^{T}}{\\sqrt{9+1+1} \\sqrt{1+16+4}}=\\frac{-3}{\\sqrt{11} \\sqrt{21}}=-0.19739=\\cos \\theta$\nTherefore we need to solve\n$$\n-0.19739=\\cos \\theta\n$$\nThus $\\theta=1.7695$ radians.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.7.1", "question": "Let $V=\\mathbb{R}^{3}$ and let\n\n$$\nW=\\operatorname{span}(S), \\text { where } S=\\left\\{\\left[\\begin{array}{r}\n1 \\\\\n-1 \\\\\n1\n\\end{array}\\right],\\left[\\begin{array}{r}\n-2 \\\\\n2 \\\\\n-2\n\\end{array}\\right],\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n1\n\\end{array}\\right],\\left[\\begin{array}{r}\n1 \\\\\n-1 \\\\\n3\n\\end{array}\\right]\\right\\}\n$$\n\nFind a basis of $W$ consisting of vectors in $S$.", "answer": "In this case $\\operatorname{dim}(W)=1$ and a basis for $W$ consisting of vectors in $S$ can be obtained by taking any (nonzero) vector from $S$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.15", "question": "Simplify $(\\vec{u} \\times \\vec{v}) \\bullet(\\vec{v} \\times \\vec{w}) \\times(\\vec{w} \\times \\vec{z})$.", "answer": "Here $[\\vec{v}, \\vec{w}, \\vec{z}]$ denotes the box product. Consider the cross product term. From the above,\n$$\n\\begin{aligned}\n(\\vec{v} \\times \\vec{w}) \\times(\\vec{w} \\times \\vec{z}) & =[\\vec{v}, \\vec{w}, \\vec{z}] \\vec{w}-[\\vec{w}, \\vec{w}, \\vec{z}] \\vec{v} \\\\\n& =[\\vec{v}, \\vec{w}, \\vec{z}] \\vec{w}\n\\end{aligned}\n$$\nThus it reduces to\n$$\n(\\vec{u} \\times \\vec{v}) \\bullet[\\vec{v}, \\vec{w}, \\vec{z}] \\vec{w}=[\\vec{v}, \\vec{w}, \\vec{z}][\\vec{u}, \\vec{v}, \\vec{w}]\n$$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.5.8.2", "question": "Let $B=\\left\\{\\left[\\begin{array}{r}1 \\\\ -1 \\\\ 2\\end{array}\\right],\\left[\\begin{array}{l}2 \\\\ 1 \\\\ 2\\end{array}\\right],\\left[\\begin{array}{r}-1 \\\\ 0 \\\\ 2\\end{array}\\right]\\right\\}$ be a basis of $\\mathbb{R}^{3}$ and let $\\vec{x}=\\left[\\begin{array}{r}5 \\\\ -1 \\\\ 4\\end{array}\\right]$ be a vector in $\\mathbb{R}^{2}$. Find $C_{B}(\\vec{x})$.", "answer": "$C_{B}(\\vec{x})=\\left[\\begin{array}{r}2 \\\\ 1 \\\\ -1\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.59", "question": "Consider the system $-5 x+2 y-z=0$ and $-5 x-2 y-z=0$. Both equations equal zero and so $-5 x+2 y-z=-5 x-2 y-z$ which is equivalent to $y=0$. Does it follow that $x$ and $z$ can equal anything? Notice that when $x=1, z=-4$, and $y=0$ are plugged in to the equations, the equations do not equal 0 . Why?", "answer": "These are not legitimate row operations. They do not preserve the solution set of the system.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.7.10", "question": "Find $\\operatorname{proj}_{\\vec{v}}(\\vec{w})$ where $\\vec{w}=\\left[\\begin{array}{r}1 \\\\ 0 \\\\ -2\\end{array}\\right]$ and $\\vec{v}=\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right]$.", "answer": "$\\frac{\\vec{u} \\bullet \\vec{v}}{\\vec{u} \\bullet \\vec{u}} \\vec{u}=\\frac{-5}{14}\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right]=\\left[\\begin{array}{r}-\\frac{5}{14} \\\\ -\\frac{5}{7} \\\\ -\\frac{15}{14}\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.4.12", "question": "Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix\n$$\nA=\\left[\\begin{array}{ccc}\n-\\frac{1}{2} & -\\frac{1}{5} \\sqrt{6} \\sqrt{5} & \\frac{1}{10} \\sqrt{5} \\\\\n-\\frac{1}{5} \\sqrt{6} \\sqrt{5} & \\frac{7}{5} & -\\frac{1}{5} \\sqrt{6} \\\\\n\\frac{1}{10} \\sqrt{5} & -\\frac{1}{5} \\sqrt{6} & -\\frac{9}{10}\n\\end{array}\\right]\n$$\nHint: The eigenvalues are $-1,2,-1$ where -1 is listed twice because it has multiplicity 2 as a zero of the characteristic equation.", "answer": "The eigenvectors and eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n-\\frac{1}{6} \\sqrt{6} \\\\\n0 \\\\\n\\frac{1}{6} \\sqrt{5} \\sqrt{6}\n\\end{array}\\right],\\left[\\begin{array}{c}\n\\frac{1}{3} \\sqrt{2} \\sqrt{3} \\\\\n\\frac{1}{5} \\sqrt{5} \\\\\n\\frac{1}{15} \\sqrt{2} \\sqrt{15}\n\\end{array}\\right]\\right\\} \\leftrightarrow-1,\\left\\{\\left[\\begin{array}{c}\n\\frac{1}{6} \\sqrt{6} \\\\\n-\\frac{2}{5} \\sqrt{5} \\\\\n\\frac{1}{30} \\sqrt{30}\n\\end{array}\\right]\\right\\} \\leftrightarrow 2 .\n$$\n\nThe columns of $U$ are these vectors.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.9.5", "question": "Find the area of the parallelogram determined by the vectors $\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right],\\left[\\begin{array}{r}3 \\\\ -2 \\\\ 1\\end{array}\\right]$.", "answer": "$\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right] \\times\\left[\\begin{array}{r}3 \\\\ -2 \\\\ 1\\end{array}\\right]=\\left[\\begin{array}{r}8 \\\\ 8 \\\\ -8\\end{array}\\right]$. The area is $8 \\sqrt{3}$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.24", "question": "An object moves 20 meters in the direction of $\\vec{k}+\\vec{j}$. There are two forces acting on this object, $\\vec{F}_{1}=\\vec{i}+\\vec{j}+2 \\vec{k}$, and $\\vec{F}_{2}=\\vec{i}+2 \\vec{j}-6 \\vec{k}$. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. Why?", "answer": "$\\left[\\begin{array}{r}2 \\\\ 3 \\\\ -4\\end{array}\\right] \\bullet\\left[\\begin{array}{c}0 \\\\ \\frac{1}{\\sqrt{2}} \\\\ \\frac{1}{\\sqrt{2}}\\end{array}\\right] 20=-10 \\sqrt{2}$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.32", "question": "Suppose $A B=A C$ and $A$ is an invertible $n \\times n$ matrix. Does it follow that $B=C$ ? Explain why or why not.", "answer": "Yes $B=C$. Multiply $A B=A C$ on the left by $A^{-1}$.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.14", "question": "Let $A=\\left[\\begin{array}{ll}1 & 2 \\\\ 3 & 4\\end{array}\\right], B=\\left[\\begin{array}{ll}1 & 2 \\\\ 3 & k\\end{array}\\right]$. Is it possible to choose $k$ such that $A B=B A$ ? If so, what should $k$ equal?", "answer": "Solution is: $[k=4]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.32", "question": "Find the solution to the system of equations, $7 x+14 y+15 z=22,2 x+4 y+3 z=5$, and $3 x+6 y+10 z=13$.", "answer": "Solution is: $[x=1-2 t, z=1, y=t]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.7", "question": "Consider the following augmented matrix in which $*$ denotes an arbitrary number and denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?\n$$\n\\left[\\begin{array}{ccc|c}\n\\mathbf{0} & * & * & * \\\\\n0 & \\mathbf{\\square} & * & * \\\\\n0 & 0 & \\mathbf{\\square} & *\n\\end{array}\\right]\n$$", "answer": "A solution exists and is unique.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.17", "question": "Determine if the system is consistent. If so, is the solution unique?\n$$\n\\begin{gathered}\nx+2 y+z-w=2 \\\\\nx-y+z+w=1 \\\\\n2 x+y-z=1 \\\\\n4 x+2 y+z=5\n\\end{gathered}\n$$", "answer": "There is no solution.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.7.2.1", "question": "Find the eigenvalues and eigenvectors of the matrix\n$$\n\\left[\\begin{array}{rrr}\n5 & -18 & -32 \\\\\n0 & 5 & 4 \\\\\n2 & -5 & -11\n\\end{array}\\right]\n$$\nOne eigenvalue is 1. Diagonalize if possible.", "answer": "The eigenvalues are $-1,-1,1$. The eigenvectors corresponding to the eigenvalues are:\n\n$$\n\\left\\{\\left[\\begin{array}{c}\n10 \\\\\n-2 \\\\\n3\n\\end{array}\\right]\\right\\} \\leftrightarrow-1,\\left\\{\\left[\\begin{array}{c}\n7 \\\\\n-2 \\\\\n2\n\\end{array}\\right]\\right\\} \\leftrightarrow 1\n$$\n\nTherefore this matrix is not diagonalizable.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.43", "question": "Suppose the coefficient matrix of a system of $n$ equations with $n$ variables has the property that every column is a pivot column. Does it follow that the system of equations must have a solution? If so, must the solution be unique? Explain.", "answer": "Yes. It has a unique solution.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.13", "question": "Let $X=\\left[\\begin{array}{lll}-1 & -1 & 1\\end{array}\\right]$ and $Y=\\left[\\begin{array}{lll}0 & 1 & 2\\end{array}\\right]$. Find $X^{T} Y$ and $X Y^{T}$ if possible.", "answer": "$X^{T} Y=\\left[\\begin{array}{rrr}0 & -1 & -2 \\\\ 0 & -1 & -2 \\\\ 0 & 1 & 2\\end{array}\\right], X Y^{T}=1$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.42", "question": "Let\n$$\nA=\\left[\\begin{array}{lll}\n1 & 2 & 1 \\\\\n2 & 1 & 4 \\\\\n4 & 5 & 10\n\\end{array}\\right]\n$$\nFind $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.", "answer": "$\\left[\\begin{array}{rrrr}1 & 2 & 0 & 2 \\\\ 1 & 1 & 2 & 0 \\\\ 2 & 1 & -3 & 2 \\\\ 1 & 2 & 1 & 2\\end{array}\\right]^{-1}=\\left[\\begin{array}{rrrr}-1 & \\frac{1}{2} & \\frac{1}{2} & \\frac{1}{2} \\\\ 3 & \\frac{1}{2} & -\\frac{1}{2} & -\\frac{5}{2} \\\\ -1 & 0 & 0 & 1 \\\\ -2 & -\\frac{3}{4} & \\frac{1}{4} & \\frac{9}{4}\\end{array}\\right]$", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.12", "question": "A certain river is one half mile wide with a current flowing at 2 miles per hour from East to West. A man swims directly toward the opposite shore from the South bank of the river at a speed of 3 miles per hour. How far down the river does he find himself when he has swam across? How far does he end up traveling?", "answer": "Water: $\\left[\\begin{array}{ll}-2 & 0\\end{array}\\right]$ Swimmer: $\\left[\\begin{array}{ll}0 & 3\\end{array}\\right]$ Speed relative to earth: $\\left[\\begin{array}{ll}-2 & 3\\end{array}\\right]$. It takes him $1 / 6$ of an hour to get across. Therefore, he ends up travelling $\\frac{1}{6} \\sqrt{4+9}=\\frac{1}{6} \\sqrt{13}$ miles. He ends up $1 / 3$ mile down stream.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.12.22", "question": "An object moves 10 meters in the direction of $\\vec{j}$. There are two forces acting on this object, $\\vec{F}_{1}=\\vec{i}+\\vec{j}+2 \\vec{k}$, and $\\vec{F}_{2}=-5 \\vec{i}+2 \\vec{j}-6 \\vec{k}$. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. Why?", "answer": "$\\left[\\begin{array}{r}-4 \\\\ 3 \\\\ -4\\end{array}\\right] \\bullet\\left[\\begin{array}{l}0 \\\\ 1 \\\\ 0\\end{array}\\right] \\times 10=30$\nYou can consider the resultant of the two forces because of the properties of the dot product.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.4.7.9", "question": "Find the angle between the vectors\n$$\n\\vec{u}=\\left[\\begin{array}{r}\n1 \\\\\n-2 \\\\\n1\n\\end{array}\\right], \\vec{v}=\\left[\\begin{array}{r}\n1 \\\\\n2 \\\\\n-7\n\\end{array}\\right]\n$$", "answer": "$\\frac{-10}{\\sqrt{1+4+1} \\sqrt{1+4+49}}=-0.55555=\\cos \\theta$\nTherefore we need to solve $-0.55555=\\cos \\theta$, which gives $\\theta=2.0313$ radians.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.1.2.13", "question": "Find $h$ such that\n$$\n\\left[\\begin{array}{ll|l}\n1 & h & 3 \\\\\n2 & 4 & 6\n\\end{array}\\right]\n$$\nis the augmented matrix of a consistent system.", "answer": "Any $h$ will work.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.A_First_Course_in_Linear_Algebra", "question_number": "exercise.2.1.15", "question": "Let $A=\\left[\\begin{array}{ll}1 & 2 \\\\ 3 & 4\\end{array}\\right], B=\\left[\\begin{array}{ll}1 & 2 \\\\ 1 & k\\end{array}\\right]$. Is it possible to choose $k$ such that $A B=B A$ ? If so, what should $k$ equal?", "answer": "There is no possible choice of $k$ which will make these matrices commute.", "license": "Creative Commons License (CC BY)", "data_topic": "college_math.linear_algebra"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.1.4", "question": "Find solutions $y_{1}, y_{2}, \\ldots, y_{n}$ of the equation $y^{(n)}=0$ that satisfy the initial conditions $y_{i}^{(j)}\\left(x_{0}\\right)=\\left\\{\\begin{array}{ll}0, & j \\neq i-1, \\\\1, & j=i-1,\\end{array} \\quad 1 \\leq i \\leq n .$", "answer": "$y_{i}=\\frac{\\left(x-x_{0}\\right)^{i-1}}{(i-1) !}, 1 \\leq i \\leq n$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.17", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $(x-1) y^{\\prime \\prime}-x y^{\\prime}+y=0$, given that $y_{1}=e^{x}$.", "answer": "$y_{2}=x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.24", "question": "Find a curve $y=y(x)$ through $(2,1)$ such that the normal to the curve at any point $\\left(x_{0}, y\\left(x_{0}\\right)\\right)$ intersects the $y$ axis at $y_{I}=2 y\\left(x_{0}\\right)$.", "answer": "$y=\\sqrt{x^{2}-3}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.30", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}-4 & 0 & -1 \\\\ -1 & -3 & -1 \\\\ 1 & 0 & -2\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right] e^{-2 t}+c_{2}\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n1\n\\end{array}\\right] e^{-2 t}+c_{3}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n0 \\\\\n0\n\\end{array}\\right] e^{-2 t}+\\left[\\begin{array}{r}\n1 \\\\\n-1 \\\\\n1\n\\end{array}\\right] t e^{-2 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.26", "question": "Solve the equation $y^{\\prime}=\\frac{\\cos x}{\\sin y}, \\quad y(\\pi)=\\frac{\\pi}{2}$ explicitly. HINT: Use the identity $\\cos (x+\\pi / 2)=-\\sin x$ and the periodicity of the cosine.", "answer": "$y=-x+3 \\pi / 2$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.28", "question": "Find the orthogonal trajectories of the given family of curves: $x y e^{x^{2}}=c$", "answer": "$y^{2}=-\\frac{1}{2} \\ln \\left(1+2 x^{2}\\right)+k$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.6", "question": "A 3200-lb car is moving at $64 \\mathrm{ft} / \\mathrm{s}$ down a 30-degree grade when it runs out of fuel. Find its velocity after that if friction exerts a resistive force with magnitude proportional to the square of the speed, with $k=1 \\mathrm{lb}-\\mathrm{s}^{2} / \\mathrm{ft}^{2}$. Also find its terminal velocity.", "answer": "$v=-\\frac{40\\left(13+3 e^{-4 t / 5}\\right)}{13-3 e^{-4 t / 5}} ;-40 \\mathrm{ft} / \\mathrm{s}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.10", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}-1 & 1 & -1 \\\\ -2 & 0 & 2 \\\\ -1 & 3 & -1\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n1\n\\end{array}\\right] e^{2 t}+c_{2}\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n1\n\\end{array}\\right] e^{-2 t}+c_{3}\\left(\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{e^{-2 t}}{2}+\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n1\n\\end{array}\\right] t e^{-2 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.11", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}3 & 2 \\\\ -5 & 1\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1} e^{2 t}\\left[\\begin{array}{c}3 \\sin 3 t-\\cos 3 t \\\\ 5 \\cos 3 t\\end{array}\\right]+c_{2} e^{2 t}\\left[\\begin{array}{c}-3 \\cos 3 t-\\sin 3 t \\\\ 5 \\sin 3 t\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.16", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}1 & 2 & -2 \\\\ 0 & 2 & -1 \\\\ 1 & 0 & 0\\end{array}\\right] \\mathbf{y}^{\\prime}$", "answer": "$\\mathbf{y}=c_{1}\\left[\\begin{array}{r}6 \\\\ -3 \\\\ 3\\end{array}\\right] e^{8 t}+\\left[\\begin{array}{c}10 \\cos 4 t-4 \\sin 4 t \\\\ 17 \\cos 4 t-\\sin 4 t \\\\ 3 \\cos 4 t-7 \\sin 4 t\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.19", "question": "Two objects suspended from identical springs are set into motion. The period of one object is twice the period of the other. How are the weights of the two objects related?", "answer": "The object with the longer period weighs four times as much as the other.", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.10", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $3 x^{2} y^{\\prime \\prime}-x y^{\\prime}+y=0$", "answer": "$y=c_{1} x+c_{2} x^{1 / 3}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.3.30", "question": "Find the general solution of $y^{\\prime \\prime}+\\omega_{0}^{2} y=M \\cos \\omega x+N \\sin \\omega x$, where $M$ and $N$ are constants and $\\omega$ and $\\omega_{0}$ are distinct positive numbers.", "answer": "$y=\\frac{1}{\\omega_{0}^{2}-\\omega^{2}}(M \\cos \\omega x+N \\sin \\omega x)+c_{1} \\cos \\omega_{0} x+c_{2} \\sin \\omega_{0} x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.1.1", "question": "Find the general solution: $y^{\\prime}+a y=0(a=$ constant $)$", "answer": "$y=e^{-a x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.11", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $2 x^{2} y^{\\prime \\prime}-3 x y^{\\prime}+2 y=0$", "answer": "$y=c_{1} x^{2}+c_{2} x^{1 / 2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.19", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $4 x^{2}(\\sin x) y^{\\prime \\prime}-4 x(x \\cos x+\\sin x) y^{\\prime}+(2 x \\cos x+3 \\sin x) y=0$, given that $y_{1}=x^{1 / 2}$.", "answer": "$y_{2}=x^{1 / 2} \\cos x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.13", "question": "A $2 \\mathrm{lb}$ weight stretches a spring 6 inches in equilibrium. An external force $F(t)=\\sin 8 t \\mathrm{lb}$ is applied to the weight, which is released from rest 2 inches below equilibrium. Find its displacement for $t>0$.", "answer": "$y=-t \\cos 8 t-\\frac{1}{6} \\cos 8 t+\\frac{1}{8} \\sin 8 t \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.3", "question": "At $12: 00 \\mathrm{PM}$ a thermometer reading $10^{\\circ} \\mathrm{F}$ is placed in a room where the temperature is $70^{\\circ} \\mathrm{F}$. It reads $56^{\\circ}$ when it's placed outside, where the temperature is $5^{\\circ} \\mathrm{F}$, at $12: 03$. What does it read at 12:05 PM?", "answer": "$\\approx 24.33^{\\circ} \\mathrm{F}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.3", "question": "A spring with natural length $.5 \\mathrm{~m}$ has length $50.5 \\mathrm{~cm}$ with a mass of $2 \\mathrm{gm}$ suspended from it. The mass is initially displaced $1.5 \\mathrm{~cm}$ below equilibrium and released with zero velocity. Find its displacement for $t>0$.", "answer": "$y=1.5 \\cos 14 \\sqrt{10} t \\mathrm{~cm}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.6", "question": "An object is placed in a room where the temperature is $20^{\\circ} \\mathrm{C}$. The temperature of the object drops by $5^{\\circ} \\mathrm{C}$ in 4 minutes and by $7^{\\circ} \\mathrm{C}$ in 8 minutes. What was the temperature of the object when it was initially placed in the room?", "answer": "$(85 / 3)^{\\circ} \\mathrm{C}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.27", "question": "Find the orthogonal trajectories of the given family of curves: $y=c e^{2 x}$", "answer": "$y^{2}=-x+k$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.31", "question": "Find a fundamental set of solutions: $\\left(D^{2}+9\\right)^{3} D^{2} y=0$", "answer": "$\\left\\{\\cos 3 x, x \\cos 3 x, x^{2} \\cos 3 x, \\sin 3 x, x \\sin 3 x, x^{2} \\sin 3 x, 1, x\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.5", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}4 & 12 \\\\ -3 & -8\\end{array}\\right] \\mathbf{y}$", "answer": "c_{1}\\left[\\begin{array}{r}\n-2 \\\\\n1\n\\end{array}\\right]+c_{2}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n0\n\\end{array}\\right] \\frac{e^{-2 t}}{3}+\\left[\\begin{array}{r}\n-2 \\\\\n1\n\\end{array}\\right] t e^{-2 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.2", "question": "Find the general solution: $x^{2} y^{\\prime \\prime}+x y^{\\prime}-y=\\frac{4}{x^{2}} ; \\quad y_{1}=x$", "answer": "$y=\\frac{4}{3 x^{2}}+c_{1} x+\\frac{c_{2}}{x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.32", "question": "Find a fundamental set of solutions: $(D-2)^{3}(D+1)^{2} D y=0$", "answer": "$\\left\\{e^{2 x}, x e^{2 x}, x^{2} e^{2 x}, e^{-x}, x e^{-x}, 1\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.29", "question": "Find the orthogonal trajectories of the given family of curves: $y=\\frac{c e^{x}}{x}$", "answer": "$y^{2}=-2 x-\\ln (x-1)^{2}+k$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.6", "question": "An $8 \\mathrm{lb}$ weight stretches a spring $.32 \\mathrm{ft}$. The weight is initially displaced 6 inches above equilibrium and given an upward velocity of $4 \\mathrm{ft} / \\mathrm{sec}$. Find its displacement for $t>0$ if the medium exerts a damping force of $1.5 \\mathrm{lb}$ for each $\\mathrm{ft} / \\mathrm{sec}$ of velocity.", "answer": "$y=\\frac{1}{2} e^{-3 t}\\left(\\cos \\sqrt{91} t+\\frac{11}{\\sqrt{91}} \\sin \\sqrt{91} t\\right) \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.1.15", "question": "Find a power series solution $y(x)=\\sum_{n=0}^{\\infty} a_{n} x^{n}$ for $\\left(1+3 x^{2}\\right) y^{\\prime \\prime}-2 x y^{\\prime}+4 y$.", "answer": "$b_{n}=(n+2)(n+1) a_{n+2}+\\left(3 n^{2}-5 n+4\\right) a_{n}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.20", "question": "A mass of one $\\mathrm{kg}$ stretches a spring $49 \\mathrm{~cm}$ in equilibrium. It is attached to a dashpot that supplies a damping force of $4 \\mathrm{~N}$ for each $\\mathrm{m} / \\mathrm{sec}$ of speed. Find the steady state component of its displacement if it's subjected to an external force $F(t)=8 \\sin 2 t-6 \\cos 2 t \\mathrm{~N}$.", "answer": "$y=-\\frac{1}{2} \\cos 2 t+\\frac{1}{4} \\sin 2 t \\mathrm{~m}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.7", "question": "Find the steady state current in the circuit described by the equation.\n$\\frac{1}{20} Q^{\\prime \\prime}+2 Q^{\\prime}+100 Q=10 \\cos 25 t-5 \\sin 25 t$", "answer": "$I_{p}=\\frac{20}{37}(\\cos 25 t-6 \\sin 25 t)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.1", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}3 & 4 \\\\ -1 & 7\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{l}\n2 \\\\\n1\n\\end{array}\\right] e^{5 t}+c_{2}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n0\n\\end{array}\\right] e^{5 t}+\\left[\\begin{array}{l}\n2 \\\\\n1\n\\end{array}\\right] t e^{5 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.14", "question": "Find the general solution: $2 x y^{\\prime \\prime}+(4 x+1) y^{\\prime}+(2 x+1) y=3 x^{1 / 2} e^{-x} ; \\quad y_{1}=e^{-x}$", "answer": "$y=e^{-x}\\left(x^{3 / 2}+c_{1}+c_{2} x^{1 / 2}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.19", "question": "Find a fundamental set of solutions: $x^{2} y^{\\prime \\prime}-4 x y^{\\prime}+6 y=0 ; \\quad y_{1}=x^{2}$", "answer": "$\\{x^{2}, x^{3}\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.14", "question": "Find the general solution: $y^{(4)}-4 y^{\\prime \\prime \\prime}+7 y^{\\prime \\prime}-6 y^{\\prime}+2 y=0$", "answer": "$y=e^{x}\\left(c_{1}+c_{2} x+c_{3} \\cos x+c_{4} \\sin x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.3.60", "question": "Find the general solution: $y^{\\prime \\prime \\prime}-y^{\\prime \\prime}-y^{\\prime}+y=e^{2 x}(10+3 x)$", "answer": "$y=e^{2 x}(1+x)+c_{1} e^{-x}+e^{x}\\left(c_{2}+c_{3} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.20", "question": "Two objects suspended from identical springs are set into motion. The weight of one object is twice the weight of the other. How are the periods of the resulting motions related?", "answer": "$T_{2}=\\sqrt{2} T_{1}$, where $T_{1}$ is the period of the smaller object.", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.3", "question": "Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.\n$R=2$ ohms; $L=.1$ henrys; $C=.01$ farads; $Q_{0}=2$ coulombs; $I_{0}=0$ amperes.", "answer": "$I=-\\frac{200}{3} e^{-10 t} \\sin 30 t$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.11", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}4 & -2 & -2 \\\\ -2 & 3 & -1 \\\\ 2 & -1 & 3\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-2 \\\\\n-3 \\\\\n1\n\\end{array}\\right] e^{2 t}+c_{2}\\left[\\begin{array}{r}\n0 \\\\\n-1 \\\\\n1\n\\end{array}\\right] e^{4 t}+c_{3}\\left(\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n0\n\\end{array}\\right] \\frac{e^{4 t}}{2}+\\left[\\begin{array}{r}\n0 \\\\\n-1 \\\\\n1\n\\end{array}\\right] t e^{4 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.16", "question": "Solve the equation explicitly: $y^{\\prime}=\\frac{y^{2}+2 x y}{x^{2}}$", "answer": "$y=\\frac{c x^{2}}{1-c x} \\quad y=-x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.2", "question": "Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.\n$R=2$ ohms; $L=.05$ henrys; $C=.01$ farads'; $Q_{0}=2$ coulombs; $I_{0}=-2$ amperes.", "answer": "$I=e^{-20 t}(2 \\cos 40 t-101 \\sin 40 t)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.3", "question": "A boat weighs $64,000 \\mathrm{lb}$. Its propellor produces a constant thrust of $50,000 \\mathrm{lb}$ and the water exerts a resistive force with magnitude proportional to the speed, with $k=2000 \\mathrm{lb}-\\mathrm{s} / \\mathrm{ft}$. Assuming that the boat starts from rest, find its velocity as a function of time, and find its terminal velocity.", "answer": "$v=25\\left(1-e^{-t}\\right) ; 25 \\mathrm{ft} / \\mathrm{s}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.13", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}1 & 1 & 2 \\\\ 1 & 0 & -1 \\\\ -1 & -2 & -1\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1}\\left[\\begin{array}{r}-1 \\\\ 1 \\\\ 1\\end{array}\\right] e^{-2 t}+c_{2} e^{t}\\left[\\begin{array}{r}\\sin t \\\\ -\\cos t \\\\ \\cos t\\end{array}\\right]+c_{3} e^{t}\\left[\\begin{array}{r}-\\cos t \\\\ -\\sin t \\\\ \\sin t\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.18", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $2 x^{2} y^{\\prime \\prime}+10 x y^{\\prime}+9 y=0$", "answer": "$y=\\frac{1}{x^{2}}\\left[c_{1} \\cos \\left(\\frac{1}{\\sqrt{2}} \\ln x\\right)+c_{2} \\sin \\left(\\frac{1}{\\sqrt{2}} \\ln x\\right)\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.2.1", "question": "Find the power series in $x$ for the general solution: $\\left(1+x^{2}\\right) y^{\\prime \\prime}+6 x y^{\\prime}+6 y=0$", "answer": "$y=a_{0} \\sum_{m=0}^{\\infty}(-1)^{m}(2 m+1) x^{2 m}+a_{1} \\sum_{m=0}^{\\infty}(-1)^{m}(m+1) x^{2 m+1}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.8", "question": "A mass of $20 \\mathrm{gm}$ stretches a spring $5 \\mathrm{~cm}$. The spring is attached to a dashpot with damping constant 400 dyne sec/cm. Determine the displacement for $t>0$ if the mass is initially displaced $9 \\mathrm{~cm}$ above equilibrium and released from rest.", "answer": "$y=e^{-10 t}\\left(9 \\cos 4 \\sqrt{6} t+\\frac{45}{2 \\sqrt{6}} \\sin 4 \\sqrt{6} t\\right) \\mathrm{cm}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.9", "question": "Find the general solution: $x^{2} y^{\\prime \\prime}+x y^{\\prime}-4 y=-6 x-4 ; \\quad y_{1}=x^{2}$", "answer": "$y=2 x+1+c_{1} x^{2}+\\frac{c_{2}}{x^{2}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.21", "question": "A mass $m$ is suspended from a spring with constant $k$ and subjected to an external force $F(t)=$ $\\alpha \\cos \\omega_{0} t+\\beta \\sin \\omega_{0} t$, where $\\omega_{0}$ is the natural frequency of the spring-mass system without damping. Find the steady state component of the displacement if a dashpot with constant $c$ supplies damping.", "answer": "$y_{p}=\\frac{1}{c \\omega_{0}}\\left(-\\beta \\cos \\omega_{0} t+\\alpha \\sin \\omega_{0} t\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.23", "question": "Find a curve $y=y(x)$ through $(0,2)$ such that the normal to the curve at any point $\\left(x_{0}, y\\left(x_{0}\\right)\\right)$ intersects the $x$ axis at $x_{I}=x_{0}+1$.", "answer": "$y=\\sqrt{2 x+4}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.6", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $y^{\\prime \\prime}+3\\left(x^{2}+1\\right) y^{\\prime}-2 y=0$, given that $W(\\pi)=0$.", "answer": "0", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.30", "question": "Find a fundamental set of solutions: $(D-1)^{2}(2 D-1)^{3}\\left(D^{2}+1\\right) y=0$", "answer": "$\\left\\{e^{x}, x e^{x}, e^{x / 2}, x e^{x / 2}, x^{2} e^{x / 2}, \\cos x, \\sin x\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.6", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}-10 & 9 \\\\ -4 & 2\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{l}\n3 \\\\\n2\n\\end{array}\\right] e^{-4 t}+c_{2}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n0\n\\end{array}\\right] \\frac{e^{-4 t}}{2}+\\left[\\begin{array}{l}\n3 \\\\\n2\n\\end{array}\\right] t e^{-4 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.4", "question": "Find the general solution: $y^{\\prime \\prime}-4 y^{\\prime}+4 y=0$", "answer": "$y=e^{2 x}\\left(c_{1}+c_{2} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.1.7", "question": "Find the Wronskian $W$ of a set of three solutions of $y^{\\prime \\prime \\prime}+2 x y^{\\prime \\prime}+e^{x} y^{\\prime}-y=0$, given that $W(0)=2$.", "answer": "$2 e^{-x^{2}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.14", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}-x y^{\\prime}+10 y=0$", "answer": "$y=x\\left[c_{1} \\cos (3 \\ln x)+c_{2} \\sin (3 \\ln x)\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.9", "question": "Find the general solution: $y^{(4)}-16 y=0$", "answer": "$y=c_{1} e^{2 x}+c_{2} e^{-2 x}+c_{3} \\cos 2 x+c_{4} \\sin 2 x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.24", "question": "Solve the equation $y^{\\prime}=\\frac{\\left(1+y^{2}\\right)}{\\left(1+x^{2}\\right)}$ explicitly. Hint: Use the identity $\\tan (A+B)=\\frac{\\tan A+\\tan B}{1-\\tan A \\tan B}$.", "answer": "$y=\\frac{x+c}{1-c x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.2", "question": "Solve the given Bernoulli equation: $7 x y^{\\prime}-2 y=-\\frac{x^{2}}{y^{6}}$", "answer": "$y=x^{2 / 7}(c-\\ln |x|)^{1 / 7}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.8", "question": "Find the general solution: $y^{(4)}+y^{\\prime \\prime}=0$", "answer": "$y=c_{1}+c_{2} x+c_{3} \\cos x+c_{4} \\sin x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.5", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}3 & -3 & 1 \\\\ 0 & 2 & 2 \\\\ 5 & 1 & 1\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1}\\left[\\begin{array}{c}\n-1 \\\\\n-1 \\\\\n2\n\\end{array}\\right] e^{-2 t}+c_{2} e^{4 t}\\left[\\begin{array}{c}\n\\cos 2 t-\\sin 2 t \\\\\n\\cos 2 t+\\sin 2 t \\\\\n2 \\cos 2 t\n\\end{array}\\right]+c_{3} e^{4 t}\\left[\\begin{array}{c}\n\\sin 2 t+\\cos 2 t \\\\\n\\sin 2 t-\\cos 2 t \\\\\n2 \\sin 2 t\n\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.12", "question": "Find the general solution: $10 y^{\\prime \\prime}-3 y^{\\prime}-y=0$", "answer": "$y=c_{1} e^{-x / 5}+c_{2} e^{x / 2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.52", "question": "Find two linearly independent Frobenius solutions of the equation: $4 x^{2} y^{\\prime \\prime}+2 x\\left(4-x^{2}\\right) y^{\\prime}+\\left(1+7 x^{2}\\right) y=0$", "answer": "$y_{1}=x^{-1 / 2}\\left(1-\\frac{1}{2} x^{2}+\\frac{1}{32} x^{4}\\right)$\n$y_{2}=y_{1} \\ln x+x^{3 / 2}\\left(\\frac{5}{8}-\\frac{9}{128} x^{2}+\\sum_{m=2}^{\\infty} \\frac{1}{4^{m+1}(m-1) m(m+1)(m+1) !} x^{2 m}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.2", "question": "A fluid initially at $100^{\\circ} \\mathrm{C}$ is placed outside on a day when the temperature is $-10^{\\circ} \\mathrm{C}$, and the temperature of the fluid drops $20^{\\circ} \\mathrm{C}$ in one minute. Find the temperature $T(t)$ of the fluid for $t>0$.", "answer": "$T=-10+110 e^{-t \\ln \\frac{11}{9}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.4.16", "question": "Find the general solution for the equation: $y^{\\prime \\prime}-6 y^{\\prime}+8 y=e^{x}(11-6 x)$", "answer": "$y=e^{x}(1-2 x)+c_{1} e^{2 x}+c_{2} e^{4 x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.29", "question": "Find a fundamental set of solutions: $\\left(x^{2}-2 x\\right) y^{\\prime \\prime}+\\left(2-x^{2}\\right) y^{\\prime}+(2 x-2) y=0 ; \\quad y_{1}=e^{x}$", "answer": "$\\left\\{e^{x}, x^{2}\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.17", "question": "A mass of one $\\mathrm{kg}$ is attached to a spring with constant $k=4 \\mathrm{~N} / \\mathrm{m}$. An external force $F(t)=$ $-\\cos \\omega t-2 \\sin \\omega t \\mathrm{n}$ is applied to the mass. Find the displacement $y$ for $t>0$ if $\\omega$ equals the natural frequency of the spring-mass system. Assume that the mass is initially displaced $3 \\mathrm{~m}$ above equilibrium and given an upward velocity of $450 \\mathrm{~cm} / \\mathrm{s}$.", "answer": "$y=\\frac{t}{2} \\cos 2 t-\\frac{t}{4} \\sin 2 t+3 \\cos 2 t+2 \\sin 2 t \\mathrm{~m}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.3", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}1 & 2 \\\\ -4 & 5\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1} e^{3 t}\\left[\\begin{array}{c}\n\\cos 2 t+\\sin 2 t \\\\\n2 \\cos 2 t\n\\end{array}\\right]+c_{2} e^{3 t}\\left[\\begin{array}{c}\n\\sin 2 t-\\cos 2 t \\\\\n2 \\sin 2 t\n\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.1", "question": "Find the general solution: $y^{\\prime \\prime \\prime}-3 y^{\\prime \\prime}+3 y^{\\prime}-y=0$", "answer": "$y=e^{x}\\left(c_{1}+c_{2} x+c_{3} x^{2}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.8", "question": "Find a fundamental set of Frobenius solutions for the equation: $18 x^{2}(1+x) y^{\\prime \\prime}+3 x\\left(5+11 x+x^{2}\\right) y^{\\prime}-\\left(1-2 x-5 x^{2}\\right) y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=x^{1 / 3}\\left(1-\\frac{1}{3} x+\\frac{2}{15} x^{2}-\\frac{5}{63} x^{3}+\\cdots\\right)$\n$y_{2}=x^{-1 / 6}\\left(1-\\frac{1}{12} x^{2}+\\frac{1}{18} x^{3}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.7", "question": "Find the general solution: $y^{\\prime \\prime}-2 y^{\\prime}+2 y=e^{x} \\sec x ; \\quad y_{1}=e^{x} \\cos x$", "answer": "$y=e^{x}\\left(x \\sin x+\\cos x \\ln |\\cos x|+c_{1} \\cos x+c_{2} \\sin x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.2", "question": "An object stretches a string 1.2 inches in equilibrium. Find its displacement for $t>0$ if it's initially displaced 3 inches below equilibrium and given a downward velocity of $2 \\mathrm{ft} / \\mathrm{s}$.", "answer": "$y=-\\frac{1}{4} \\cos 8 \\sqrt{5} t-\\frac{1}{4 \\sqrt{5}} \\sin 8 \\sqrt{5} t \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.2", "question": "Find the general solution: $y^{\\prime \\prime}-4 y^{\\prime}+5 y=0$", "answer": "$y=e^{2 x}\\left(c_{1} \\cos x+c_{2} \\sin x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.3", "question": "Find the general solution: $y^{\\prime \\prime \\prime}-y^{\\prime \\prime}+16 y^{\\prime}-16 y=0$", "answer": "$y=c_{1} e^{x}+c_{2} \\cos 4 x+c_{3} \\sin 4 x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.3.64", "question": "Find the general solution: $y^{\\prime \\prime \\prime}-3 y^{\\prime \\prime}+3 y^{\\prime}-y=e^{x}(1+x)$", "answer": "$y=\\frac{x^{3} e^{x}}{24}(4+x)+e^{x}\\left(c_{1}+c_{2} x+c_{3} x^{2}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.5", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}+x y^{\\prime}+y=0$", "answer": "$y=c_{1} \\cos (\\ln x)+c_{2} \\sin (\\ln x)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.1", "question": "A $64 \\mathrm{lb}$ object stretches a spring $4 \\mathrm{ft}$ in equilibrium. It is attached to a dashpot with damping constant $c=8 \\mathrm{lb}-\\mathrm{sec} / \\mathrm{ft}$. The object is initially displaced 18 inches above equilibrium and given a downward velocity of $4 \\mathrm{ft} / \\mathrm{sec}$. Find its displacement and time-varying amplitude for $t>0$.", "answer": "$y=\\frac{e^{-2 t}}{2}(3 \\cos 2 t-\\sin 2 t) \\mathrm{ft} ; \\sqrt{\\frac{5}{2}} e^{-2 t} \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.9", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}5 & -4 \\\\ 10 & 1\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1} e^{3 t}\\left[\\begin{array}{c}\\cos 6 t-3 \\sin 6 t \\\\ 5 \\cos 6 t\\end{array}\\right]+c_{2} e^{3 t}\\left[\\begin{array}{c}\\sin 6 t+3 \\cos 6 t \\\\ 5 \\sin 6 t\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.32", "question": "Solve the given homogeneous equation implicitly: $y^{\\prime}=\\frac{y}{y-2 x}$", "answer": "$y^{2}(y-3 x)=c ; \\quad y \\equiv 0 ; y=3 x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.1", "question": "Find the general solution: $y^{\\prime \\prime}+5 y^{\\prime}-6 y=0$", "answer": "$y=c_{1} e^{-6 x}+c_{2} e^{x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.1.5", "question": "Find the general solution: $x^{2} y^{\\prime}+y=0$", "answer": "$y=c e^{1 / x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.9", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $4 x^{2} y^{\\prime \\prime}+8 x y^{\\prime}+y=0$", "answer": "$y=x^{-1 / 2}\\left(c_{1}+c_{2} \\ln x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.2.4", "question": "Find the power series in $x$ for the general solution: $\\left(1-x^{2}\\right) y^{\\prime \\prime}-8 x y^{\\prime}-12 y=0$", "answer": "$y=a_{0} \\sum_{m=0}^{\\infty}(m+1)(2 m+1) x^{2 m}+\\frac{a_{1}}{3} \\sum_{m=0}^{\\infty}(m+1)(2 m+3) x^{2 m+1}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.11", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $y^{\\prime \\prime}-6 y^{\\prime}+9 y=0$, given that $y_{1}=e^{3 x}$.", "answer": "$y_{2}=x e^{3 x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.7", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}-13 & 16 \\\\ -9 & 11\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{l}\n4 \\\\\n3\n\\end{array}\\right] e^{-t}+c_{2}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n0\n\\end{array}\\right] \\frac{e^{-t}}{3}+\\left[\\begin{array}{l}\n4 \\\\\n3\n\\end{array}\\right] t e^{-t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.13", "question": "Find the general solution: $4 y^{(4)}+12 y^{\\prime \\prime \\prime}+3 y^{\\prime \\prime}-13 y^{\\prime}-6 y=0$", "answer": "$y=c_{1} e^{x}+c_{2} e^{-2 x}+c_{3} e^{-x / 2}+c_{4} e^{-3 x / 2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.9", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\frac{1}{3}\\left[\\begin{array}{rrr}1 & 1 & -3 \\\\ -4 & -4 & 3 \\\\ -2 & 1 & 0\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n1\n\\end{array}\\right] e^{t}+c_{2}\\left[\\begin{array}{r}\n1 \\\\\n-1 \\\\\n1\n\\end{array}\\right] e^{-t}+c_{3}\\left(\\left[\\begin{array}{l}\n0 \\\\\n3 \\\\\n0\n\\end{array}\\right] e^{-t}+\\left[\\begin{array}{r}\n1 \\\\\n-1 \\\\\n1\n\\end{array}\\right] t e^{-t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.3.66", "question": "Find the general solution: $y^{\\prime \\prime \\prime}+2 y^{\\prime \\prime}-y^{\\prime}-2 y=e^{-2 x}[(23-2 x) \\cos x+(8-9 x) \\sin x]$", "answer": "$y=e^{-2 x}\\left[\\left(1+\\frac{x}{2}\\right) \\cos x+\\left(\\frac{3}{2}-2 x\\right) \\sin x\\right]+c_{1} e^{x}+c_{2} e^{-x}+c_{3} e^{-2 x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.2", "question": "Find a fundamental set of Frobenius solutions for the equation: $3 x^{2} y^{\\prime \\prime}+2 x\\left(1+x-2 x^{2}\\right) y^{\\prime}+\\left(2 x-8 x^{2}\\right) y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=x^{1 / 3}\\left(1-\\frac{2}{3} x+\\frac{8}{9} x^{2}-\\frac{40}{81} x^{3}+\\cdots\\right)$\n$y_{2}=1-x+\\frac{6}{5} x^{2}-\\frac{4}{5} x^{3}+\\cdots$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.1", "question": "Find a fundamental set of Frobenius solutions for the equation: $2 x^{2}\\left(1+x+x^{2}\\right) y^{\\prime \\prime}+x\\left(3+3 x+5 x^{2}\\right) y^{\\prime}-y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=x^{1 / 2}\\left(1-\\frac{1}{5} x-\\frac{2}{35} x^{2}+\\frac{31}{315} x^{3}+\\cdots\\right)$\n$y_{2}=x^{-1}\\left(1+x+\\frac{1}{2} x^{2}-\\frac{1}{6} x^{3}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.2.8", "question": "Find the power series in $x$ for the general solution: $\\left(1+x^{2}\\right) y^{\\prime \\prime}-10 x y^{\\prime}+28 y=0$", "answer": "$y=a_{0}\\left(1-14 x^{2}+\\frac{35}{3} x^{4}\\right)+a_{1}\\left(x-3 x^{3}+\\frac{3}{5} x^{5}+\\frac{1}{35} x^{7}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.3", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}-x y^{\\prime}+y=0$", "answer": "$y=x\\left(c_{1}+c_{2} \\ln x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.21", "question": "Find a fundamental set of solutions: $4 x y^{\\prime \\prime}+2 y^{\\prime}+y=0 ; \\quad y_{1}=\\sin \\sqrt{x}$", "answer": "$\\{\\sin \\sqrt{x}, \\cos \\sqrt{x}\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.1", "question": "Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.\n$R=3$ ohms; $L=.1$ henrys; $C=.01$ farads; $Q_{0}=0$ coulombs; $I_{0}=2$ amperes.", "answer": "$I=e^{-15 t}\\left(2 \\cos 5 \\sqrt{15} t-\\frac{6}{\\sqrt{31}} \\sin 5 \\sqrt{31} t\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.51", "question": "Find two linearly independent Frobenius solutions of the equation: $x\\left(1+x^{2}\\right) y^{\\prime \\prime}+\\left(1-x^{2}\\right) y^{\\prime}-8 x y=0$", "answer": "$y_{1}=\\left(1+x^{2}\\right)^{2}$\n$y_{2}=y_{1} \\ln x-\\frac{3}{2} x^{2}-\\frac{3}{2} x^{4}+\\sum_{m=3}^{\\infty} \\frac{(-1)^{m}}{m(m-1)(m-2)} x^{2 m}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.2.2", "question": "Find the power series in $x$ for the general solution: $\\left(1+x^{2}\\right) y^{\\prime \\prime}+2 x y^{\\prime}-2 y=0$", "answer": "$y=a_{0} \\sum_{m=0}^{\\infty}(-1)^{m+1} \\frac{x^{2 m}}{2 m-1}+a_{1} x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.7", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}+3 x y^{\\prime}-3 y=0$", "answer": "$y=c_{1} x+\\frac{c_{2}}{x^{3}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.10", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\frac{1}{3}\\left[\\begin{array}{rr}7 & -5 \\\\ 2 & 5\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1} e^{2 t}\\left[\\begin{array}{c}\\cos t-3 \\sin t \\\\ 2 \\cos t\\end{array}\\right]+c_{2} e^{2 t}\\left[\\begin{array}{c}\\sin t+3 \\cos t \\\\ 2 \\sin t\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.36", "question": "Find a fundamental set of solutions: $D^{3}(D-2)^{2}\\left(D^{2}+4\\right)^{2} y=0$", "answer": "$\\left\\{1, x, x^{2}, e^{2 x}, x e^{2 x}, \\cos 2 x, x \\cos 2 x, \\sin 2 x, x \\sin 2 x\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.3.68", "question": "Find the general solution: $y^{\\prime \\prime \\prime}-4 y^{\\prime \\prime}+14 y^{\\prime \\prime}-20 y^{\\prime}+25 y=e^{x}[(2+6 x) \\cos 2 x+3 \\sin 2 x]$", "answer": "$y=-\\frac{x^{2} e^{x}}{16}(1+x) \\cos 2 x+e^{x}\\left[\\left(c_{1}+c_{2} x\\right) \\cos 2 x+\\left(c_{3}+c_{4} x\\right) \\sin 2 x\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.33", "question": "Solve the given homogeneous equation implicitly: $y^{\\prime}=\\frac{x y^{2}+2 y^{3}}{x^{3}+x^{2} y+x y^{2}}$", "answer": "$(x-y)^{3}(x+y)=c y^{2} x^{4} ; \\quad y=0 ; y=x ; y=-x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.46", "question": "Find two linearly independent Frobenius solutions of the equation: $x^{2}(1-x) y^{\\prime \\prime}+x(3-2 x) y^{\\prime}+(1+2 x) y=0$", "answer": "$y_{1}=\\frac{(x-1)^{2}}{x}$\n$y_{2}=y_{1} \\ln x+3-3 x+2 \\sum_{n=2}^{\\infty} \\frac{1}{n\\left(n^{2}-1\\right)} x^{n}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.1.8", "question": "Find the Wronskian $W$ of a set of four solutions of $y^{(4)}+(\\tan x) y^{\\prime \\prime \\prime}+x^{2} y^{\\prime \\prime}+2 x y=0$, given that $W(\\pi / 4)=K$.", "answer": "$\\sqrt{2} K \\cos x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.2.3", "question": "Find the power series in $x$ for the general solution: $\\left(1+x^{2}\\right) y^{\\prime \\prime}-8 x y^{\\prime}+20 y=0$", "answer": "$y=a_{0}\\left(1-10 x^{2}+5 x^{4}\\right)+a_{1}\\left(x-2 x^{3}+\\frac{1}{5} x^{5}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.2", "question": "A firefighter who weighs $192 \\mathrm{lb}$ slides down an infinitely long fire pole that exerts a frictional resistive force with magnitude proportional to her speed, with constant of proportionality $k$. Find $k$, given that her terminal velocity is $-16 \\mathrm{ft} / \\mathrm{s}$, and then find her velocity $v$ as a function of $t$. Assume that she starts from rest.", "answer": "$k=12 ; \\quad v=-16\\left(1-e^{-2 t}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.10", "question": "Find a fundamental set of Frobenius solutions for the equation: $10 x^{2}\\left(1+x+2 x^{2}\\right) y^{\\prime \\prime}+x\\left(13+13 x+66 x^{2}\\right) y^{\\prime}-\\left(1+4 x+10 x^{2}\\right) y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=x^{1 / 2}\\left(1+\\frac{3}{17} x-\\frac{7}{153} x^{2}-\\frac{547}{5661} x^{3}+\\cdots\\right)$\n$y_{2}=x^{-1 / 2}\\left(1+x+\\frac{14}{13} x^{2}-\\frac{556}{897} x^{3}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.8", "question": "Find the steady state current in the circuit described by the equation.\n$\\frac{1}{10} Q^{\\prime \\prime}+2 Q^{\\prime}+100 Q=3 \\cos 50 t-6 \\sin 50 t$", "answer": "$I_{p}=\\frac{3}{13}(8 \\cos 50 t-\\sin 50 t)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.10", "question": "An object weighing $256 \\mathrm{lb}$ is dropped from rest in a medium that exerts a resistive force with magnitude proportional to the square of the speed. The magnitude of the resisting force is $1 \\mathrm{lb}$ when $|v|=4 \\mathrm{ft} / \\mathrm{s}$. Find $v$ for $t>0$, and find its terminal velocity.", "answer": "$v=-\\frac{64\\left(1-e^{-t}\\right)}{1+e^{-t}} ;-64 \\mathrm{ft} / \\mathrm{s}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.4", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{ll}5 & -6 \\\\ 3 & -1\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1} e^{2 t}\\left[\\begin{array}{c}\n\\cos 3 t-\\sin 3 t \\\\\n\\cos 3 t\n\\end{array}\\right]+c_{2} e^{2 t}\\left[\\begin{array}{c}\n\\sin 3 t+\\cos 3 t \\\\\n\\sin 3 t\n\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.5", "question": "A stone weighing $1 / 2 \\mathrm{lb}$ is thrown upward from an initial height of $5 \\mathrm{ft}$ with an initial speed of 32 $\\mathrm{ft} / \\mathrm{s}$. Air resistance is proportional to speed, with $k=1 / 128 \\mathrm{lb}-\\mathrm{s} / \\mathrm{ft}$. Find the maximum height attained by the stone.", "answer": "$\\approx 17.10 \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.4", "question": "Find a fundamental set of Frobenius solutions for the equation: $4 x^{2} y^{\\prime \\prime}+x\\left(7+2 x+4 x^{2}\\right) y^{\\prime}-\\left(1-4 x-7 x^{2}\\right) y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=x^{1 / 4}\\left(1-\\frac{1}{2} x-\\frac{19}{104} x^{2}+\\frac{1571}{10608} x^{3}+\\cdots\\right)$\n$y_{2}=x^{-1}\\left(1+2 x-\\frac{11}{6} x^{2}-\\frac{1}{7} x^{3}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.20", "question": "Find all curves $y=y(x)$ such that the tangent to the curve at any point passes through a given point $\\left(x_{1}, y_{1}\\right)$.", "answer": "$y=y_{1}+c\\left(x-x_{1}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.1", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}+7 x y^{\\prime}+8 y=0$", "answer": "$y=c_{1} x^{-4}+c_{2} x^{-2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.7", "question": "A $32 \\mathrm{lb}$ weight stretches a spring $2 \\mathrm{ft}$ in equilibrium. It is attached to a dashpot with constant $c=8$ $\\mathrm{lb}-\\mathrm{sec} / \\mathrm{ft}$. The weight is initially displaced 8 inches below equilibrium and released from rest. Find its displacement for $t>0$.", "answer": "$y=-\\frac{e^{-4 t}}{3}(2+8 t)$ ft", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.5", "question": "Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.\n$R=4$ ohms; $L=.05$ henrys; $C=.008$ farads; $Q_{0}=-1$ coulombs; $I_{0}=2$ amperes.", "answer": "$I=-e^{-40 t}(2 \\cos 30 t-86 \\sin 30 t)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.9", "question": "A tank initially contains a solution of 10 pounds of salt in 60 gallons of water. Water with $1 / 2$ pound of salt per gallon is added to the tank at $6 \\mathrm{gal} / \\mathrm{min}$, and the resulting solution leaves at the same rate. Find the quantity $Q(t)$ of salt in the tank at time $t>0$.", "answer": "$Q(t)=30-20 e^{-t / 10}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.20", "question": "Find a fundamental set of solutions: $x^{2}(\\ln |x|)^{2} y^{\\prime \\prime}-(2 x \\ln |x|) y^{\\prime}+(2+\\ln |x|) y=0 ; \\quad y_{1}=\\ln |x|$", "answer": "$\\{\\ln |x|, x \\ln |x|\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.16", "question": "Find the general solution: $4 x^{2} y^{\\prime \\prime}-4 x(x+1) y^{\\prime}+(2 x+3) y=4 x^{5 / 2} e^{2 x} ; \\quad y_{1}=x^{1 / 2}$", "answer": "$y=x^{1 / 2}\\left(\\frac{e^{2 x}}{2}+c_{1}+c_{2} e^{x}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.12", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}34 & 52 \\\\ -20 & -30\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1} e^{2 t}\\left[\\begin{array}{c}\\sin 4 t-8 \\cos 4 t \\\\ 5 \\cos 4 t\\end{array}\\right]+c_{2} e^{2 t}\\left[\\begin{array}{c}-\\cos 4 t-8 \\sin 4 t \\\\ 5 \\sin 4 t\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.10", "question": "Find the general solution: $x^{2} y^{\\prime \\prime}+2 x(x-1) y^{\\prime}+\\left(x^{2}-2 x+2\\right) y=x^{3} e^{2 x} ; \\quad y_{1}=x e^{-x}$", "answer": "$y=\\frac{x e^{2 x}}{9}+x e^{-x}\\left(c_{1}+c_{2} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.5.34", "question": "Find conditions on the constants $A, B, C, D, E$, and $F$ such that the equation\n$\\left(A x^{2}+B x y+C y^{2}\\right) d x+\\left(D x^{2}+E x y+F y^{2}\\right) d y=0$\nis exact.", "answer": "$B=2 D, \\quad E=2 C$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.17", "question": "Solve the equation explicitly: $x y^{3} y^{\\prime}=y^{4}+x^{4}$", "answer": "$y= \\pm x(4 \\ln |x|+c)^{1 / 4}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.1", "question": "A firefighter who weighs $192 \\mathrm{lb}$ slides down an infinitely long fire pole that exerts a frictional resistive force with magnitude proportional to his speed, with $k=2.5 \\mathrm{lb}-\\mathrm{s} / \\mathrm{ft}$. Assuming that he starts from rest, find his velocity as a function of time and find his terminal velocity.", "answer": "$v=-\\frac{384}{5}\\left(1-e^{-5 t / 12}\\right) ;-\\frac{384}{5} \\mathrm{ft} / \\mathrm{s}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.3", "question": "Find the general solution: $y^{\\prime \\prime}+8 y^{\\prime}+7 y=0$", "answer": "$y=c_{1} e^{-7 x}+c_{2} e^{-x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.2", "question": "Find the general solution: $y^{(4)}+8 y^{\\prime \\prime}-9 y=0$", "answer": "$y=c_{1} e^{x}+c_{2} e^{-x}+c_{3} \\cos 3 x+c_{4} \\sin 3 x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.17", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}-3 x y^{\\prime}+4 y=0$", "answer": "$y=x^{2}\\left(c_{1}+c_{2} \\ln x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.24", "question": "Find a fundamental set of solutions: $x^{2} y^{\\prime \\prime}-2 x y^{\\prime}+\\left(x^{2}+2\\right) y=0 ; \\quad y_{1}=x \\sin x$", "answer": "$\\{x \\sin x, x \\cos x\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.4", "question": "An object stretches a spring 6 inches in equilibrium. Find its displacement for $t>0$ if it's initially displaced 3 inches above equilibrium and given a downward velocity of 6 inches/s. Find the frequency, period, amplitude and phase angle of the motion.", "answer": "$y=\\frac{1}{4} \\cos 8 t-\\frac{1}{16} \\sin 8 t \\mathrm{ft} ; R=\\frac{\\sqrt{17}}{16} \\mathrm{ft} ; \\omega_{0}=8 \\mathrm{rad} / \\mathrm{s} ; T=\\pi / 4 \\mathrm{~s}$; $\\phi \\approx-.245 \\mathrm{rad} \\approx-14.04^{\\circ}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.2.6", "question": "Find the power series in $x$ for the general solution: $\\left(1+x^{2}\\right) y^{\\prime \\prime}+2 x y^{\\prime}+\\frac{1}{4} y=0$", "answer": "$y=a_{0} \\sum_{m=0}^{\\infty}(-1)^{m}\\left[\\prod_{j=0}^{m-1} \\frac{(4 j+1)^{2}}{2 j+1}\\right] \\frac{x^{2 m}}{8^{m} m !}+a_{1} \\sum_{m=0}^{\\infty}(-1)^{m}\\left[\\prod_{j=0}^{m-1} \\frac{(4 j+3)^{2}}{2 j+3}\\right] \\frac{x^{2 m+1}}{8^{m} m !}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.28", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}-2 & -12 & 10 \\\\ 2 & -24 & 11 \\\\ 2 & -24 & 8\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-2 \\\\\n1 \\\\\n2\n\\end{array}\\right] e^{-6 t}+c_{2}\\left(-\\left[\\begin{array}{l}\n6 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{e^{-6 t}}{6}+\\left[\\begin{array}{r}\n-2 \\\\\n1 \\\\\n2\n\\end{array}\\right] t e^{-6 t}\\right)+c_{3}\\left(-\\left[\\begin{array}{c}\n12 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{e^{-6 t}}{36}-\\left[\\begin{array}{l}\n6 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{t e^{-6 t}}{6}+\\left[\\begin{array}{r}\n-2 \\\\\n1 \\\\\n2\n\\end{array}\\right] \\frac{t^{2} e^{-6 t}}{2}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.5", "question": "Find the general solution: $y^{\\prime \\prime}+2 y^{\\prime}+10 y=0$", "answer": "$y=e^{-x}\\left(c_{1} \\cos 3 x+c_{2} \\sin 3 x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.32", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}-3 & -1 & 0 \\\\ 1 & -1 & 0 \\\\ -1 & -1 & -2\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-1 \\\\\n0 \\\\\n1\n\\end{array}\\right] e^{-3 t}+c_{2}\\left[\\begin{array}{l}\n0 \\\\\n0 \\\\\n1\n\\end{array}\\right] e^{-3 t}+c_{3}\\left(\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n0\n\\end{array}\\right] e^{-3 t}+\\left[\\begin{array}{l}\n-1 \\\\\n-1 \\\\\n1\n\\end{array}\\right] t e^{-3 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.31", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}-3 & -3 & 4 \\\\ 4 & 5 & -8 \\\\ 2 & 3 & -5\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{l}\n2 \\\\\n0 \\\\\n1\n\\end{array}\\right] e^{-t}+c_{2}\\left[\\begin{array}{r}\n-3 \\\\\n2 \\\\\n0\n\\end{array}\\right] e^{-t}+c_{3}\\left(\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n0\n\\end{array}\\right] \\frac{e^{-t}}{2}+\\left[\\begin{array}{r}\n-1 \\\\\n2 \\\\\n1\n\\end{array}\\right] t e^{-t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.4.15", "question": "Find the general solution for the equation: $y^{\\prime \\prime}-3 y^{\\prime}+2 y=e^{3 x}(1+x)$", "answer": "$y=\\frac{e^{3 x}}{4}(-1+2 x)+c_{1} e^{x}+c_{2} e^{2 x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.1.16", "question": "Suppose $y(x)=\\sum_{n=0}^{\\infty} a_{n}(x+1)^{n}$ on an open interval that contains $x_{0}=-1$. Find a power series in $x+1$ for $x y^{\\prime \\prime}+(4+2 x) y^{\\prime}+(2+x) y$.", "answer": "$b_{0}=-2 a_{2}+2 a_{1}+a_{0}$,\n\n$b_{n}=-(n+2)(n+1) a_{n+2}+(n+1)(n+2) a_{n+1}+(2 n+1) a_{n}+a_{n-1}, n \\geq 2$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.18", "question": "Solve the equation explicitly: $y^{\\prime}=\\frac{y}{x}+\\sec \\frac{y}{x}$", "answer": "$y=x \\sin ^{-1}(\\ln |x|+c)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.10", "question": "A tank initially contains 100 liters of a salt solution with a concentration of $.1 \\mathrm{~g} / \\mathrm{liter}$. A solution with a salt concentration of $.3 \\mathrm{~g} / \\mathrm{liter}$ is added to the tank at 5 liters $/ \\mathrm{min}$, and the resulting mixture is drained out at the same rate. Find the concentration $K(t)$ of salt in the tank as a function of $t$.", "answer": "$K(t)=.3-.2 e^{-t / 20}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.6", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}-3 x y^{\\prime}+13 y=0$", "answer": "$y=x^{2}\\left[c_{1} \\cos (3 \\ln x)+c_{2} \\sin (3 \\ln x)\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.22", "question": "Find a fundamental set of solutions: $x y^{\\prime \\prime}-(2 x+2) y^{\\prime}+(x+2) y=0 ; \\quad y_{1}=e^{x}$", "answer": "$\\{e^{x}, x^{3} e^{x}\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.4", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}3 & 1 \\\\ -1 & 1\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-1 \\\\\n1\n\\end{array}\\right] e^{2 t}+c_{2}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n0\n\\end{array}\\right] e^{2 t}+\\left[\\begin{array}{r}\n-1 \\\\\n1\n\\end{array}\\right] t e^{2 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.3.65", "question": "Find the general solution: $y^{(4)}-2 y^{\\prime \\prime}+y=-e^{-x}\\left(4-9 x+3 x^{2}\\right)$", "answer": "$y=\\frac{x^{2} e^{-x}}{16}\\left(1+2 x-x^{2}\\right)+e^{x}\\left(c_{1}+c_{2} x\\right)+e^{-x}\\left(c_{3}+c_{4} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.18", "question": "Control mechanisms allow fluid to flow into a tank at a rate proportional to the volume $V$ of fluid in the tank, and to flow out at a rate proportional to $V^{2}$. Suppose $V(0)=V_{0}$ and the constants of proportionality are $a$ and $b$, respectively. Find $V(t)$ for $t>0$ and find $\\lim _{t \\rightarrow \\infty} V(t)$.", "answer": "$V=\\frac{a}{b} \\frac{V_{0}}{V_{0}-\\left(V_{0}-a / b\\right) e^{-a t}}, \\quad \\lim _{t \\rightarrow \\infty} V(t)=a / b$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.17", "question": "Find the general solution: $x^{2} y^{\\prime \\prime}-5 x y^{\\prime}+8 y=4 x^{2} ; \\quad y_{1}=x^{2}$", "answer": "$y=-2 x^{2} \\ln x+c_{1} x^{2}+c_{2} x^{4}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.4", "question": "Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.\n$R=6$ ohms; $L=.1$ henrys; $C=.004$ farads'; $Q_{0}=3$ coulombs; $I_{0}=-10$ amperes.", "answer": "$I=-10 e^{-30 t}(\\cos 40 t+18 \\sin 40 t)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.1", "question": "Find the general solution: $(2 x+1) y^{\\prime \\prime}-2 y^{\\prime}-(2 x+3) y=(2 x+1)^{2} ; \\quad y_{1}=e^{-x}$", "answer": "$y=1-2 x+c_{1} e^{-x}+c_{2} x e^{x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.4", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}+5 x y^{\\prime}+4 y=0$", "answer": "$y=x^{-2}\\left(c_{1}+c_{2} \\ln x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.19", "question": "Find all curves $y=y(x)$ such that the tangent to the curve at any point $\\left(x_{0}, y\\left(x_{0}\\right)\\right)$ intersects the $x$ axis at $x_{I}=x_{0}^{3}$.", "answer": "$y=\\frac{c x}{\\sqrt{\\left|x^{2}-1\\right|}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.14", "question": "A $10 \\mathrm{gm}$ mass suspended on a spring moves in simple harmonic motion with period $4 \\mathrm{~s}$. Find the period of the simple harmonic motion of a 20 gm mass suspended from the same spring.", "answer": "$T=4 \\sqrt{2} \\mathrm{~s}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.35", "question": "Solve the equation using variation of parameters followed by separation of variables: $y^{\\prime}+y=\\frac{2 x e^{-x}}{1+y e^{x}}$", "answer": "$y=e^{-x}\\left(-1 \\pm \\sqrt{2 x^{2}+c}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.47", "question": "Find two linearly independent Frobenius solutions of the equation: $4 x^{2}(1+x) y^{\\prime \\prime}-4 x^{2} y^{\\prime}+(1-5 x) y=0$", "answer": "$y_{1}=x^{1 / 2}(x+1)^{2}$\n$y_{2}=y_{1} \\ln x-x^{3 / 2}\\left(3+3 x+2 \\sum_{n=2}^{\\infty} \\frac{(-1)^{n}}{n\\left(n^{2}-1\\right)} x^{n}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.8", "question": "A tank initially contains 40 gallons of pure water. A solution with 1 gram of salt per gallon of water is added to the tank at $3 \\mathrm{gal} / \\mathrm{min}$, and the resulting solution drains out at the same rate. Find the quantity $Q(t)$ of salt in the tank at time $t>0$.", "answer": "$Q(t)=40\\left(1-e^{-3 t / 40}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.6", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}-3 & 3 & 1 \\\\ 1 & -5 & -3 \\\\ -3 & 7 & 3\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1}\\left[\\begin{array}{c}\n-1 \\\\\n-1 \\\\\n1\n\\end{array}\\right] e^{-t}+c_{2} e^{-2 t}\\left[\\begin{array}{c}\n\\cos 2 t-\\sin 2 t \\\\\n-\\cos 2 t-\\sin 2 t \\\\\n2 \\cos 2 t\n\\end{array}\\right]+c_{3} e^{-2 t}\\left[\\begin{array}{c}\n\\sin 2 t+\\cos 2 t \\\\\n-\\sin 2 t+\\cos 2 t \\\\\n2 \\sin 2 t\n\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.10", "question": "Find the general solution: $y^{\\prime \\prime}+6 y^{\\prime}+13 y=0$", "answer": "$y=e^{-3 x}\\left(c_{1} \\cos 2 x+c_{2} \\sin 2 x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.4", "question": "Find all solutions: $x^{2} y y^{\\prime}=\\left(y^{2}-1\\right)^{3 / 2}$", "answer": "$\\frac{(\\ln y)^{2}}{2}=-\\frac{x^{3}}{3}+c$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.9", "question": "Find the general solution: $y^{\\prime \\prime}-2 y^{\\prime}+3 y=0$", "answer": "$y=e^{x}\\left(c_{1} \\cos \\sqrt{2} x+c_{2} \\sin \\sqrt{2} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.11", "question": "An $8 \\mathrm{lb}$ weight stretches a spring 2 inches. It is attached to a dashpot with damping constant $c=4 \\mathrm{lb}-\\mathrm{sec} / \\mathrm{ft}$. The weight is initially displaced 3 inches above equilibrium and given a downward velocity of $4 \\mathrm{ft} / \\mathrm{sec}$. Find its displacement for $t>0$.", "answer": "$y=e^{-8 t}\\left(\\frac{1}{4} \\cos 8 \\sqrt{2} t-\\frac{1}{4 \\sqrt{2}} \\sin 8 \\sqrt{2} t\\right) \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.1.3", "question": "A mass $m_{1}$ is suspended from a rigid support on a spring $S_{1}$ with spring constant $k_{1}$ and damping constant $c_{1}$. A second mass $m_{2}$ is suspended from the first on a spring $S_{2}$ with spring constant $k_{2}$ and damping constant $c_{2}$, and a third mass $m_{3}$ is suspended from the second on a spring $S_{3}$ with spring constant $k_{3}$ and damping constant $c_{3}$. Let $y_{1}=y_{1}(t), y_{2}=y_{2}(t)$, and $y_{3}=y_{3}(t)$ be the displacements of the three masses from their equilibrium positions at time $t$, measured positive upward. Derive a system of differential equations for $y_{1}, y_{2}$ and $y_{3}$, assuming that the masses of the springs are negligible and that vertical external forces $F_{1}, F_{2}$, and $F_{3}$ also act on the masses.", "answer": "\\begin{align*}\nm_{1} y_{1}^{\\prime \\prime}&=-\\left(c_{1}+c_{2}\\right) y_{1}^{\\prime}+c_{2} y_{2}^{\\prime}-\\left(k_{1}+k_{2}\\right) y_{1}+k_{2} y_{2}+F_{1} \\\\\nm_{2} y_{2}^{\\prime \\prime}&=\\left(c_{2}-c_{3}\\right) y_{1}^{\\prime}-\\left(c_{2}+c_{3}\\right) y_{2}^{\\prime}+c_{3} y_{3}^{\\prime}+\\left(k_{2}-k_{3}\\right) y_{1}-\\left(k_{2}+k_{3}\\right) y_{2}+k_{3} y_{3}+F_{2} \\\\\nm_{3} y_{3}^{\\prime \\prime}&=c_{3} y_{1}^{\\prime}+c_{3} y_{2}^{\\prime}-c_{3} y_{3}^{\\prime}+k_{3} y_{1}+k_{3} y_{2}-k_{3} y_{3}+F_{3}\n\\end{align*}", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.7", "question": "Find a fundamental set of Frobenius solutions for the equation: $8 x^{2} y^{\\prime \\prime}-2 x\\left(3-4 x-x^{2}\\right) y^{\\prime}+\\left(3+6 x+x^{2}\\right) y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=x^{3 / 2}\\left(1-x+\\frac{11}{26} x^{2}-\\frac{109}{1326} x^{3}+\\cdots\\right)$\n$y_{2}=x^{1 / 4}\\left(1+4 x-\\frac{131}{24} x^{2}+\\frac{39}{14} x^{3}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.2.7", "question": "Find the power series in $x$ for the general solution: $\\left(1-x^{2}\\right) y^{\\prime \\prime}-5 x y^{\\prime}-4 y=0$", "answer": "$y=a_{0} \\sum_{m=0}^{\\infty} \\frac{2^{m} m !}{\\prod_{j=0}^{m-1}(2 j+1)} x^{2 m}+a_{1} \\sum_{m=0}^{\\infty} \\frac{\\prod_{j=0}^{m-1}(2 j+3)}{2^{m} m !} x^{2 m+1}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.34", "question": "Find a fundamental set of solutions: $\\left(D^{4}-16\\right)^{2} y=0$", "answer": "$\\left\\{e^{2 x}, x e^{2 x}, e^{-2 x}, x e^{-2 x}, \\cos 2 x, x \\cos 2 x, \\sin 2 x, x \\sin 2 x\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.7", "question": "A cup of boiling water is placed outside at 1:00 PM. One minute later the temperature of the water is $152^{\\circ} \\mathrm{F}$. After another minute its temperature is $112^{\\circ} \\mathrm{F}$. What is the outside temperature?", "answer": "$32^{\\circ} \\mathrm{F}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.15", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}-6 y=0$", "answer": "$y=c_{1} x^{3}+\\frac{c_{2}}{x^{2}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.4", "question": "A constant horizontal force of $10 \\mathrm{~N}$ pushes a $20 \\mathrm{~kg}$-mass through a medium that resists its motion with $.5 \\mathrm{~N}$ for every $\\mathrm{m} / \\mathrm{s}$ of speed. The initial velocity of the mass is $7 \\mathrm{~m} / \\mathrm{s}$ in the direction opposite to the direction of the applied force. Find the velocity of the mass for $t>0$.", "answer": "$v=20-27 e^{-t / 40}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.4.4", "question": "An object with mass $m$ moves in a spiral orbit $r=c \\theta^{2}$ under a central force\n\n$$\n\\mathbf{F}(r, \\theta)=f(r)(\\cos \\theta \\mathbf{i}+\\sin \\theta \\mathbf{j}) .\n$$\n\nFind $f$.", "answer": "$f(r)=-m h^{2}\\left(\\frac{6 c}{r^{4}}+\\frac{1}{r^{3}}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.49", "question": "Find two linearly independent Frobenius solutions of the equation: $x^{2}\\left(1+x^{2}\\right) y^{\\prime \\prime}-x\\left(1-9 x^{2}\\right) y^{\\prime}+\\left(1+25 x^{2}\\right) y=0$", "answer": "$y_{1}=x-4 x^{3}+x^{5}$\n$y_{2}=y_{1} \\ln x+6 x^{3}-3 x^{5}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.8", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}-3 & 1 & -3 \\\\ 4 & -1 & 2 \\\\ 4 & -2 & 3\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1}\\left[\\begin{array}{r}-1 \\\\ 1 \\\\ 1\\end{array}\\right] e^{t}+c_{2} e^{-t}\\left[\\begin{array}{c}-\\sin 2 t-\\cos 2 t \\\\ 2 \\cos 2 t \\\\ 2 \\cos 2 t\\end{array}\\right]+c_{3} e^{-t}\\left[\\begin{array}{c}\\cos 2 t-\\sin 2 t \\\\ 2 \\sin 2 t \\\\ 2 \\sin 2 t\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.2", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{ll}0 & -1 \\\\ 1 & -2\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{l}\n1 \\\\\n1\n\\end{array}\\right] e^{-t}+c_{2}\\left(\\left[\\begin{array}{l}\n1 \\\\\n0\n\\end{array}\\right] e^{-t}+\\left[\\begin{array}{l}\n1 \\\\\n1\n\\end{array}\\right] t e^{-t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.4", "question": "Find the general solution: $y^{\\prime \\prime}-3 y^{\\prime}+2 y=\\frac{1}{1+e^{-x}} ; \\quad y_{1}=e^{2 x}$", "answer": "$y=\\left(e^{2 x}+e^{x}\\right) \\ln \\left(1+e^{-x}\\right)+c_{1} e^{2 x}+c_{2} e^{x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.31", "question": "Solve the given homogeneous equation implicitly: $y^{\\prime}=\\frac{x+2 y}{2 x+y}$", "answer": "$(y+x)=c(y-x)^{3} ; \\quad y=x ; \\quad y=-x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.7.42", "question": "Find a fundamental set of Frobenius solutions of Bessel's equation:\n\n$$\nx^{2} y^{\\prime \\prime}+x y^{\\prime}+\\left(x^{2}-v^{2}\\right) y=0\n$$\n\nin the case where $v$ is a positive integer.", "answer": "$y_{1}=x^{\\nu} \\sum_{m=0}^{\\infty} \\frac{(-1)^{m}}{4^{m} m ! \\prod_{j=1}^{m}(j+v)} x^{2 m}$;\n$y_{2}=x^{-v} \\sum_{m=0}^{v-1} \\frac{(-1)^{m}}{4^{m} m ! \\prod_{j=1}^{m}(j-v)} x^{2 m}-\\frac{2}{4^{v} v !(v-1) !}\\left(y_{1} \\ln x-\\frac{x^{v}}{2} \\sum_{m=1}^{\\infty} \\frac{(-1)^{m}}{4^{m} m ! \\prod_{j=1}^{m}(j+v)}\\left(\\sum_{j=1}^{m} \\frac{2 j+v}{j(j+v)}\\right) x^{2 m}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.1", "question": "Find the general solution: $y^{\\prime}+a y=0(a=$ constant $)$", "answer": "$y_{1}=x\\left(1-x+\\frac{3}{4} x^{2}-\\frac{13}{36} x^{3}+\\cdots\\right)$\n$y_{2}=y_{1} \\ln x+x^{2}\\left(1-x+\\frac{65}{108} x^{2}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.8", "question": "A weight stretches a spring 6 inches in equilibrium. The weight is initially displaced 6 inches above equilibrium and given a downward velocity of $3 \\mathrm{ft} / \\mathrm{s}$. Find its displacement for $t>0$.", "answer": "$y=\\frac{1}{2} \\cos 8 t-\\frac{3}{8} \\sin 8 t \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.11", "question": "Find the general solution: $16 y^{(4)}-72 y^{\\prime \\prime}+81 y=0$", "answer": "$y=e^{3 x / 2}\\left(c_{1}+c_{2} x\\right)+e^{-3 x / 2}\\left(c_{3}+c_{4} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.45", "question": "Find two linearly independent Frobenius solutions of the equation: $x(1+x) y^{\\prime \\prime}+(1-x) y^{\\prime}+y=0$", "answer": "$y_{1}=1-x$\n$y_{2}=y_{1} \\ln x+4 x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.7", "question": "Find the general solution: $27 y^{\\prime \\prime \\prime}+27 y^{\\prime \\prime}+9 y^{\\prime}+y=0$", "answer": "$y=e^{-x / 3}\\left(c_{1}+c_{2} x+c_{3} x^{2}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.15", "question": "Find the general solution: $x y^{\\prime \\prime}-(2 x+1) y^{\\prime}+(x+1) y=-e^{x} ; \\quad y_{1}=e^{x}$", "answer": "$y=e^{x}\\left(x+c_{1}+c_{2} x^{2}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.12", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $y^{\\prime \\prime}-2 a y^{\\prime}+a^{2} y=0$ $(a=$ constant $)$, given that $y_{1}=e^{a x}$.", "answer": "$y_{2}=x e^{a x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.13", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $9 x^{2} y^{\\prime \\prime}+15 x y^{\\prime}+y=0$", "answer": "$y=x^{-1 / 3}\\left(c_{1}+c_{2} \\ln x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.2", "question": "Find the general solution: $y^{\\prime}+3 x^{2} y=0$", "answer": "$y_{1}=x^{-1}\\left(1-2 x+\\frac{9}{2} x^{2}-\\frac{20}{3} x^{3}+\\cdots\\right)$\n$y_{2}=y_{1} \\ln x+1-\\frac{15}{4} x+\\frac{133}{18} x^{2}+\\cdots$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.23", "question": "Find a fundamental set of solutions: $x^{2} y^{\\prime \\prime}-(2 a-1) x y^{\\prime}+a^{2} y=0 ; \\quad y_{1}=x^{a}$", "answer": "$\\left\\{x^{a}, x^{a} \\ln x\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.1.2", "question": "Find the general solution: $y^{\\prime}+3 x^{2} y=0$", "answer": "$y=c e^{-x^{3}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.7", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}2 & 1 & -1 \\\\ 0 & 1 & 1 \\\\ 1 & 0 & 1\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1}\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right] e^{2 t}+c_{2} e^{t}\\left[\\begin{array}{r}\n-\\sin t \\\\\n\\sin t \\\\\n\\cos t\n\\end{array}\\right]+c_{3} e^{t}\\left[\\begin{array}{r}\n\\cos t \\\\\n-\\cos t \\\\\n\\sin t\n\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.3.67", "question": "Find the general solution: $y^{\\prime \\prime \\prime}-3 y^{\\prime \\prime}+4 y^{\\prime \\prime}-2 y^{\\prime}=e^{x}[(28+6 x) \\cos 2 x+(11-12 x) \\sin 2 x]$", "answer": "$y=-x e^{x} \\sin 2 x+c_{1}+c_{2} e^{x}+e^{x}\\left(c_{3} \\cos x+c_{4} \\sin x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.5", "question": "Find the general solution: $x^{2} y^{\\prime}+y=0$", "answer": "$y_{1}=x\\left(1-4 x+\\frac{19}{2} x^{2}-\\frac{49}{3} x^{3}+\\cdots\\right)$\n$y_{2}=y_{1} \\ln x+x^{2}\\left(3-\\frac{43}{4} x+\\frac{208}{9} x^{2}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.7", "question": "Find the general solution: $y^{\\prime \\prime}-8 y^{\\prime}+16 y=0$", "answer": "$y=e^{4 x}\\left(c_{1}+c_{2} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.11", "question": "Find the general solution: $4 y^{\\prime \\prime}+4 y^{\\prime}+10 y=0$", "answer": "$y=e^{-x / 2}\\left(c_{1} \\cos \\frac{3 x}{2}+c_{2} \\sin \\frac{3 x}{2}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.17", "question": "A $192 \\mathrm{lb}$ weight is suspended from a spring with constant $k=6 \\mathrm{lb} / \\mathrm{ft}$ and subjected to an external force $F(t)=8 \\cos 3 t \\mathrm{lb}$. Find the steady state component of the displacement for $t>0$ if the medium resists the motion with a force equal to 8 times the speed in $\\mathrm{ft} / \\mathrm{sec}$.", "answer": "$y_{p}=-\\frac{2}{15} \\cos 3 t+\\frac{1}{15} \\sin 3 t \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.2.5", "question": "Find the power series in $x$ for the general solution: $\\left(1+2 x^{2}\\right) y^{\\prime \\prime}+7 x y^{\\prime}+2 y=0$", "answer": "$y=a_{0} \\sum_{m=0}^{\\infty}(-1)^{m}\\left[\\prod_{j=0}^{m-1} \\frac{4 j+1}{2 j+1}\\right] x^{2 m}+a_{1} \\sum_{m=0}^{\\infty}(-1)^{m}\\left[\\prod_{j=0}^{m-1}(4 j+3)\\right] \\frac{x^{2 m+1}}{2^{m} m !}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.11", "question": "An object with mass $m$ is given an initial velocity $v_{0} \\leq 0$ in a medium that exerts a resistive force with magnitude proportional to the square of the speed. Find the velocity of the object for $t>0$, and find its terminal velocity.", "answer": "$v=\\alpha \\frac{v_{0}\\left(1+e^{-\\beta t}\\right)-\\alpha\\left(1-e^{-\\beta t}\\right)}{\\alpha\\left(1+e^{-\\beta t}\\right)-v_{0}\\left(1-e^{-\\beta t}\\right)} ; \\quad-\\alpha$, where $\\alpha=\\sqrt{\\frac{m g}{k}}$ and $\\beta=2 \\sqrt{\\frac{k g}{m}}$.", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.4.5", "question": "An object with mass $m$ moves in the orbit $r=r_{0} e^{\\gamma \\theta}$ under a central force\n\n$$\n\\mathbf{F}(r, \\theta)=f(r)(\\cos \\theta \\mathbf{i}+\\sin \\theta \\mathbf{j}) .\n$$\n\nFind $f$.", "answer": "$f(r)=-\\frac{m h^{2}\\left(\\gamma^{2}+1\\right)}{r^{3}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.15", "question": "A mass of one $\\mathrm{kg}$ stretches a spring $49 \\mathrm{~cm}$ in equilibrium. A dashpot attached to the spring supplies a damping force of $4 \\mathrm{~N}$ for each $\\mathrm{m} / \\mathrm{sec}$ of speed. The mass is initially displaced $10 \\mathrm{~cm}$ above equilibrium and given a downward velocity of $1 \\mathrm{~m} / \\mathrm{sec}$. Find its displacement for $t>0$.", "answer": "$y=e^{-2 t}\\left(\\frac{1}{10} \\cos 4 t-\\frac{1}{5} \\sin 4 t\\right) \\mathrm{m}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.13", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $x^{2} y^{\\prime \\prime}+x y^{\\prime}-y=0$, given that $y_{1}=x$.", "answer": "$y_{2}=\\frac{1}{x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.12", "question": "Find the general solution: $(1-2 x) y^{\\prime \\prime}+2 y^{\\prime}+(2 x-3) y=\\left(1-4 x+4 x^{2}\\right) e^{x} ; \\quad y_{1}=e^{x}$", "answer": "$y=-\\frac{(2 x-1)^{2} e^{x}}{8}+c_{1} e^{x}+c_{2} x e^{-x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.10", "question": "A $32 \\mathrm{lb}$ weight stretches a spring $1 \\mathrm{ft}$ in equilibrium. The weight is initially displaced 6 inches above equilibrium and given a downward velocity of $3 \\mathrm{ft} / \\mathrm{sec}$. Find its displacement for $t>0$ if the medium resists the motion with a force equal to 3 times the speed in $\\mathrm{ft} / \\mathrm{sec}$.", "answer": "$y=e^{-\\frac{3}{2} t}\\left(\\frac{1}{2} \\cos \\frac{\\sqrt{119}}{2} t-\\frac{9}{2 \\sqrt{119}} \\sin \\frac{\\sqrt{119}}{2} t\\right) \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.1.17", "question": "Suppose $y(x)=\\sum_{n=0}^{\\infty} a_{n}(x-2)^{n}$ on an open interval that contains $x_{0}=2$. Find a power series in $x-2$ for $x^{2} y^{\\prime \\prime}+2 x y^{\\prime}-3 x y$.", "answer": "$b_{0}=8 a_{2}+4 a_{1}-6 a_{0}$,\n\n$b_{n}=4(n+2)(n+1) a_{n+2}+4(n+1)^{2} a_{n+1}+\\left(n^{2}+n-6\\right) a_{n}-3 a_{n-1}, n \\geq 1$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.4.19", "question": "Find the general solution for the equation: $y^{\\prime \\prime}-2 y^{\\prime}+y=e^{x}(2-12 x)$", "answer": "$y=e^{x}\\left[x^{2}(1-2 x)+c_{1}+c_{2} x\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.6", "question": "Find the general solution: $4 x^{2} y^{\\prime \\prime}+\\left(4 x-8 x^{2}\\right) y^{\\prime}+\\left(4 x^{2}-4 x-1\\right) y=4 x^{1 / 2} e^{x}(1+4 x) ; \\quad y_{1}=x^{1 / 2} e^{x}$", "answer": "$y=e^{x}\\left(2 x^{3 / 2}+x^{1 / 2} \\ln x+c_{1} x^{1 / 2}+c_{2} x^{-1 / 2}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.9", "question": "A $64 \\mathrm{lb}$ weight is suspended from a spring with constant $k=25 \\mathrm{lb} / \\mathrm{ft}$. It is initially displaced 18 inches above equilibrium and released from rest. Find its displacement for $t>0$ if the medium resists the motion with $6 \\mathrm{lb}$ of force for each $\\mathrm{ft} / \\mathrm{sec}$ of velocity.", "answer": "$y=e^{-3 t / 2}\\left(\\frac{3}{2} \\cos \\frac{\\sqrt{41}}{2} t+\\frac{9}{2 \\sqrt{41}} \\sin \\frac{\\sqrt{41}}{2} t\\right) \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.50", "question": "Find two linearly independent Frobenius solutions of the equation: $9 x^{2} y^{\\prime \\prime}+3 x\\left(1-x^{2}\\right) y^{\\prime}+\\left(1+7 x^{2}\\right) y=0$", "answer": "$y_{1}=x^{1 / 3}\\left(1-\\frac{1}{6} x^{2}\\right)$\n$y_{2}=y_{1} \\ln x+x^{7 / 3}\\left(\\frac{1}{4}-\\frac{1}{12} \\sum_{m=1}^{\\infty} \\frac{1}{6^{m} m(m+1)(m+1) !} x^{2 m}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.44", "question": "Find two linearly independent Frobenius solutions of the equation: $x^{2}(1-2 x) y^{\\prime \\prime}+3 x y^{\\prime}+(1+4 x) y=0$", "answer": "$y_{1}=\\frac{1}{x}$\n$y_{2}=y_{1} \\ln x-6+6 x-\\frac{8}{3} x^{2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.34", "question": "Solve the given homogeneous equation implicitly: $y^{\\prime}=\\frac{x^{3}+x^{2} y+3 y^{3}}{x^{3}+3 x y^{2}}$", "answer": "$\\frac{y}{x}+\\frac{y^{3}}{x^{3}}=\\ln |x|+c$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.15", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}6 & 0 & -3 \\\\ -3 & 3 & 3 \\\\ 1 & -2 & 6\\end{array}\\right] \\mathbf{y}^{\\prime}$", "answer": "$\\mathbf{y}=c_{1}\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 1\\end{array}\\right] e^{3 t}+c_{2} e^{6 t}\\left[\\begin{array}{r}-\\sin 3 t \\\\ \\sin 3 t \\\\ \\cos 3 t\\end{array}\\right]+c_{3} e^{6 t}\\left[\\begin{array}{r}\\cos 3 t \\\\ -\\cos 3 t \\\\ \\sin 3 t\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.21", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $\\left(x^{2}-4\\right) y^{\\prime \\prime}+4 x y^{\\prime}+2 y=0$, given that $y_{1}=\\frac{1}{x-2}$.", "answer": "$y_{2}=\\frac{1}{x^{2}-4}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.27", "question": "Find a fundamental set of solutions: $4 x^{2} y^{\\prime \\prime}-4 x y^{\\prime}+\\left(3-16 x^{2}\\right) y=0 ; \\quad y_{1}=x^{1 / 2} e^{2 x}$", "answer": "$\\left\\{x^{1 / 2} e^{2 x}, x^{1 / 2} e^{-2 x}\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.29", "question": "Find a fundamental set of solutions: $\\left(D^{2}+6 D+13\\right)(D-2)^{2} D^{3} y=0$", "answer": "$\\left\\{e^{-3 x} \\cos 2 x, e^{-3 x} \\sin 2 x, e^{2 x}, x e^{2 x}, 1, x, x^{2}\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.8", "question": "Find the general solution: $y^{\\prime \\prime}+y^{\\prime}=0$", "answer": "$y=c_{1}+c_{2} e^{-x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.30", "question": "Find a fundamental set of solutions: $x y^{\\prime \\prime}-(4 x+1) y^{\\prime}+(4 x+2) y=0 ; \\quad y_{1}=e^{2 x}$", "answer": "$\\left\\{e^{2 x}, x^{2} e^{2 x}\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.9", "question": "Find the steady state current in the circuit described by the equation.\n$\\frac{1}{10} Q^{\\prime \\prime}+6 Q^{\\prime}+250 Q=10 \\cos 100 t+30 \\sin 100 t$", "answer": "$I_{p}=\\frac{20}{123}(17 \\sin 100 t-11 \\cos 100 t)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.5", "question": "Find a fundamental set of Frobenius solutions for the equation: $12 x^{2}(1+x) y^{\\prime \\prime}+x\\left(11+35 x+3 x^{2}\\right) y^{\\prime}-\\left(1-10 x-5 x^{2}\\right) y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=x^{1 / 3}\\left(1-x+\\frac{28}{31} x^{2}-\\frac{1111}{1333} x^{3}+\\cdots\\right)$\n$y_{2}=x^{1 / 4}\\left(1-x+\\frac{7}{8} x^{2}-\\frac{19}{24} x^{3}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.8.7.29", "question": "Given the equation $m y^{\\prime \\prime}+c y^{\\prime}+k y=0, \\quad y(0)=y_{0}, \\quad y^{\\prime}(0)=v_{0}$, find the impulse that would have to be applied to the object at $t=\\tau$ to put it in equilibrium if $y(\\tau)=0$.", "answer": "$y=(-1)^{k} m \\omega_{1} R e^{-c \\tau / 2 m} \\delta(t-\\tau)$ if $\\omega_{1} \\tau-\\phi=(2 k+1) \\pi / 2(k=$ integer)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.8", "question": "Find the general solution: $y^{\\prime \\prime}+4 x y^{\\prime}+\\left(4 x^{2}+2\\right) y=8 e^{-x(x+2)} ; \\quad y_{1}=e^{-x^{2}}$", "answer": "$y=e^{-x^{2}}\\left(2 e^{-2 x}+c_{1}+c_{2} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.1", "question": "Solve the given Bernoulli equation: $y^{\\prime}+y=y^{2}$", "answer": "$y=\\frac{1}{1-c e^{x}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.37", "question": "Find a fundamental set of solutions: $\\left(4 D^{2}+1\\right)^{2}\\left(9 D^{2}+4\\right)^{3} y=0$", "answer": "$\\{\\cos (x / 2), x \\cos (x / 2), \\sin (x / 2), x \\sin (x / 2), \\cos 2 x / 3 x \\cos (2 x / 3)$, $\\left.x^{2} \\cos (2 x / 3), \\sin (2 x / 3), x \\sin (2 x / 3), x^{2} \\sin (2 x / 3)\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.16", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $2 x^{2} y^{\\prime \\prime}+3 x y^{\\prime}-y=0$", "answer": "$y=\\frac{c_{1}}{x}+c_{2} x^{1 / 2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.26", "question": "Find the orthogonal trajectories of the given family of curves: $x^{2}+4 x y+y^{2}=c$", "answer": "$(y-x)^{3}(y+x)=k$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.12", "question": "Suppose water is added to a tank at $10 \\mathrm{gal} / \\mathrm{min}$, but leaks out at the rate of $1 / 5 \\mathrm{gal} / \\mathrm{min}$ for each gallon in the tank. What is the smallest capacity the tank can have if the process is to continue indefinitely?", "answer": "50 gallons", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.6", "question": "Find the steady state current in the circuit described by the equation.\n$\\frac{1}{10} Q^{\\prime \\prime}+3 Q^{\\prime}+100 Q=5 \\cos 10 t-5 \\sin 10 t$", "answer": "$I_{p}=-\\frac{1}{3}(\\cos 10 t+2 \\sin 10 t)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.8", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}0 & 2 & 1 \\\\ -4 & 6 & 1 \\\\ 0 & 4 & 2\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-1 \\\\\n-1 \\\\\n2\n\\end{array}\\right]+c_{2}\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n2\n\\end{array}\\right] e^{4 t}+c_{3}\\left(\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{e^{4 t}}{2}+\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n2\n\\end{array}\\right] t e^{4 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.3.10", "question": "Find the steady state current in the circuit described by the equation.\n$\\frac{1}{20} Q^{\\prime \\prime}+4 Q^{\\prime}+125 Q=15 \\cos 30 t-30 \\sin 30 t$", "answer": "$I_{p}=-\\frac{45}{52}(\\cos 30 t+8 \\sin 30 t)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.15", "question": "Solve the equation explicitly: $y^{\\prime}=\\frac{y+x}{x}$", "answer": "$y=x(\\ln |x|+c)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.25", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}1 & 10 & -12 \\\\ 2 & 2 & 3 \\\\ 2 & -1 & 6\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n1\n\\end{array}\\right] e^{3 t}+c_{2}\\left(\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n0\n\\end{array}\\right] \\frac{e^{3 t}}{2}+\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n1\n\\end{array}\\right] t e^{3 t}\\right)+c_{3}\\left(\\left[\\begin{array}{l}\n1 \\\\\n2 \\\\\n0\n\\end{array}\\right] \\frac{e^{3 t}}{36}+\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n0\n\\end{array}\\right] \\frac{t e^{3 t}}{2}+\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n1\n\\end{array}\\right] \\frac{t^{2} e^{3 t}}{2}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.18", "question": "A 2 gm mass is attached to a spring with constant 20 dyne/cm. Find the steady state component of the displacement if the mass is subjected to an external force $F(t)=3 \\cos 4 t-5 \\sin 4 t$ dynes and a dashpot supplies 4 dynes of damping for each $\\mathrm{cm} / \\mathrm{sec}$ of velocity.", "answer": "$y_{p}=\\frac{11}{100} \\cos 4 t+\\frac{27}{100} \\sin 4 t \\mathrm{~cm}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.16", "question": "A mass of 100 grams stretches a spring $98 \\mathrm{~cm}$ in equilibrium. A dashpot attached to the spring supplies a damping force of 600 dynes for each $\\mathrm{cm} / \\mathrm{sec}$ of speed. The mass is initially displaced 10 $\\mathrm{cm}$ above equilibrium and given a downward velocity of $1 \\mathrm{~m} / \\mathrm{sec}$. Find its displacement for $t>0$.", "answer": "$y=e^{-3 t}(10 \\cos t-70 \\sin t) \\mathrm{cm}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.1.13", "question": "Find a power series solution $y(x)=\\sum_{n=0}^{\\infty} a_{n} x^{n}$ for $\\left(1+2 x^{2}\\right) y^{\\prime \\prime}+(2-3 x) y^{\\prime}+4 y$.", "answer": "$b_{n}=(n+2)(n+1) a_{n+2}+2(n+1) a_{n+1}+\\left(2 n^{2}-5 n+4\\right) a_{n}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.19", "question": "Identical tanks $T_{1}$ and $T_{2}$ initially contain $W$ gallons each of pure water. Starting at $t_{0}=0$, a salt solution with constant concentration $c$ is pumped into $T_{1}$ at $r \\mathrm{gal} / \\mathrm{min}$ and drained from $T_{1}$ into $T_{2}$ at the same rate. The resulting mixture in $T_{2}$ is also drained at the same rate. Find the concentrations $c_{1}(t)$ and $c_{2}(t)$ in tanks $T_{1}$ and $T_{2}$ for $t>0$.", "answer": "$c_{1}=c\\left(1-e^{-r t / W}\\right), c_{2}=c\\left(1-e^{-r t / W}-\\frac{r}{W} t e^{-r t / W}\\right)$.", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.10", "question": "A $64 \\mathrm{lb}$ weight is attached to a spring with constant $k=8 \\mathrm{lb} / \\mathrm{ft}$ and subjected to an external force $F(t)=2 \\sin t$. The weight is initially displaced 6 inches above equilibrium and given an upward velocity of $2 \\mathrm{ft} / \\mathrm{s}$. Find its displacement for $t>0$.", "answer": "$y=\\frac{1}{3} \\sin t+\\frac{1}{2} \\cos 2 t+\\frac{5}{6} \\sin 2 t$ ft", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.3", "question": "Solve the given Bernoulli equation: $x^{2} y^{\\prime}+2 y=2 e^{1 / x} y^{1 / 2}$", "answer": "$y=e^{2 / x}(c-1 / x)^{2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.5", "question": "Find the general solution: $y^{\\prime \\prime \\prime}+5 y^{\\prime \\prime}+9 y^{\\prime}+5 y=0$", "answer": "$y=c_{1} e^{-x}+e^{-2 x}\\left(c_{1} \\cos x+c_{2} \\sin x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.8.4.29", "question": "Find $L(u(t-\\tau))$.", "answer": "$\\frac{e^{-\\tau s}}{s}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.26", "question": "Find a fundamental set of solutions: $4 x^{2}(\\sin x) y^{\\prime \\prime}-4 x(x \\cos x+\\sin x) y^{\\prime}+(2 x \\cos x+3 \\sin x) y=0 ; \\quad y_{1}=x^{1 / 2}$", "answer": "$\\left\\{x^{1 / 2}, x^{1 / 2} \\cos x\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.5.33", "question": "Find conditions on the constants $A, B, C$, and $D$ such that the equation\n$(A x+B y) d x+(C x+D y) d y=0$\nis exact.", "answer": "$B=C$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.9", "question": "Find a fundamental set of Frobenius solutions for the equation: $x\\left(3+x+x^{2}\\right) y^{\\prime \\prime}+\\left(4+x-x^{2}\\right) y^{\\prime}+x y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=1-\\frac{1}{14} x^{2}+\\frac{1}{105} x^{3}+\\cdots$\n$y_{2}=x^{-1 / 3}\\left(1-\\frac{1}{18} x-\\frac{71}{405} x^{2}+\\frac{719}{34992} x^{3}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.22", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $(2 x+1) x y^{\\prime \\prime}-2\\left(2 x^{2}-1\\right) y^{\\prime}-4(x+1) y=0$, given that $y_{1}=\\frac{1}{x}$.", "answer": "$y_{2}=e^{2 x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.3", "question": "Find a fundamental set of Frobenius solutions for the equation: $x^{2}\\left(3+3 x+x^{2}\\right) y^{\\prime \\prime}+x\\left(5+8 x+7 x^{2}\\right) y^{\\prime}-\\left(1-2 x-9 x^{2}\\right) y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=x^{1 / 3}\\left(1-\\frac{4}{7} x-\\frac{7}{45} x^{2}+\\frac{970}{2457} x^{3}+\\cdots\\right)$\n$y_{2}=x^{-1}\\left(1-x^{2}+\\frac{2}{3} x^{3}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.29", "question": "Solve the given homogeneous equation implicitly: $\\left(y^{\\prime} x-y\\right)(\\ln |y|-\\ln |x|)=x$", "answer": "$(x+y) \\ln |x|+y(1-\\ln |y|)+c x=0$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.5", "question": "A $16 \\mathrm{lb}$ weight stretches a spring 6 inches in equilibrium. It is attached to a damping mechanism with constant $c$. Find all values of $c$ such that the free vibration of the weight has infinitely many oscillations.", "answer": "$0 \\leq c<8 \\mathrm{lb}-\\mathrm{sec} / \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.4.18", "question": "Find the general solution for the equation: $y^{\\prime \\prime}+2 y^{\\prime}-3 y=-16 x e^{x}$", "answer": "$y=x e^{x}(1-2 x)+c_{1} e^{x}+c_{2} e^{-3 x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.1.43", "question": "Experiments indicate that glucose is absorbed by the body at a rate proportional to the amount of glucose present in the bloodstream. Let $\\lambda$ denote the (positive) constant of proportionality. Now suppose glucose is injected into a patient's bloodstream at a constant rate of $r$ units per unit of time. Let $G=G(t)$ be the number of units in the patient's bloodstream at time $t>0$. Then\n\n$$\nG^{\\prime}=-\\lambda G+r\n$$\n\nwhere the first term on the right is due to the absorption of the glucose by the patient's body and the second term is due to the injection. Determine $G$ for $t>0$, given that $G(0)=G_{0}$. Also, find $\\lim _{t \\rightarrow \\infty} G(t)$.", "answer": "$G=\\frac{r}{\\lambda}+\\left(G_{0}-\\frac{r}{\\lambda}\\right) e^{-\\lambda t}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.5.6", "question": "Find a fundamental set of Frobenius solutions for the equation: $x^{2}\\left(5+x+10 x^{2}\\right) y^{\\prime \\prime}+x\\left(4+3 x+48 x^{2}\\right) y^{\\prime}+\\left(x+36 x^{2}\\right) y=0$. Compute $a_{0}, a_{1} \\ldots, a_{N}$ for $N$ at least 7 in each solution.", "answer": "$y_{1}=x^{1 / 5}\\left(1-\\frac{6}{25} x-\\frac{1217}{625} x^{2}+\\frac{41972}{46875} x^{3}+\\cdots\\right)$\n$y_{2}=x-\\frac{1}{4} x^{2}-\\frac{35}{18} x^{3}+\\frac{11}{12} x^{4}+\\cdots$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.18", "question": "Find a fundamental set of solutions: $x y^{\\prime \\prime}+(2-2 x) y^{\\prime}+(x-2) y=0 ; \\quad y_{1}=e^{x}$", "answer": "$\\{e^{x}, e^{x} / x\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.29", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}-1 & -12 & 8 \\\\ 1 & -9 & 4 \\\\ 1 & -6 & 1\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-4 \\\\\n0 \\\\\n1\n\\end{array}\\right] e^{-3 t}+c_{2}\\left[\\begin{array}{l}\n6 \\\\\n1 \\\\\n0\n\\end{array}\\right] e^{-3 t}+c_{3}\\left(\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n0\n\\end{array}\\right] e^{-3 t}+\\left[\\begin{array}{l}\n2 \\\\\n1 \\\\\n1\n\\end{array}\\right] t e^{-3 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.28", "question": "Find a fundamental set of solutions: $(2 x+1) x y^{\\prime \\prime}-2\\left(2 x^{2}-1\\right) y^{\\prime}-4(x+1) y=0 ; \\quad y_{1}=1 / x$", "answer": "$\\left\\{1 / x, e^{2 x}\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.6", "question": "Find the general solution: $4 y^{\\prime \\prime \\prime}-8 y^{\\prime \\prime}+5 y^{\\prime}-y=0$", "answer": "$y=c_{1} e^{x}+e^{x / 2}\\left(c_{2}+c_{3} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.12", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}+3 x y^{\\prime}+5 y=0$", "answer": "$y=\\frac{1}{x}\\left[c_{1} \\cos (2 \\ln x)+c_{2} \\sin (2 \\ln x]\\right.$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.15", "question": "A $6 \\mathrm{lb}$ weight stretches a spring 6 inches in equilibrium. Suppose an external force $F(t)=$ $\\frac{3}{16} \\sin \\omega t+\\frac{3}{8} \\cos \\omega t \\mathrm{lb}$ is applied to the weight. For what value of $\\omega$ will the displacement be unbounded? Find the displacement if $\\omega$ has this value. Assume that the motion starts from equilibrium with zero initial velocity.", "answer": "$\\omega=8 \\mathrm{rad} / \\mathrm{s} y=-\\frac{t}{16}(-\\cos 8 t+2 \\sin 8 t)+\\frac{1}{128} \\sin 8 t \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.5.1", "question": "Determine which equations are exact and solve them:\n1. $6 x^{2} y^{2} d x+4 x^{3} y d y=0$\n2. $\\left(3 y \\cos x+4 x e^{x}+2 x^{2} e^{x}\\right) d x+(3 \\sin x+3) d y=0$\n3. $14 x^{2} y^{3} d x+21 x^{2} y^{2} d y=0$\n4. $\\left(2 x-2 y^{2}\\right) d x+\\left(12 y^{2}-4 x y\\right) d y=0$\n5. $(x+y)^{2} d x+(x+y)^{2} d y=0$\n6. $(4 x+7 y) d x+(3 x+4 y) d y=0$\n7. $\\left(-2 y^{2} \\sin x+3 y^{3}-2 x\\right) d x+\\left(4 y \\cos x+9 x y^{2}\\right) d y=0$\n8. $(2 x+y) d x+(2 y+2 x) d y=0$\n9. $\\left(3 x^{2}+2 x y+4 y^{2}\\right) d x+\\left(x^{2}+8 x y+18 y\\right) d y=0$\n10. $\\left(2 x^{2}+8 x y+y^{2}\\right) d x+\\left(2 x^{2}+x y^{3} / 3\\right) d y=0$\n11. $\\left(\\frac{1}{x}+2 x\\right) d x+\\left(\\frac{1}{y}+2 y\\right) d y=0$\n12. $\\left(y \\sin x y+x y^{2} \\cos x y\\right) d x+\\left(x \\sin x y+x y^{2} \\cos x y\\right) d y=0$\n13. $\\frac{x d x}{\\left(x^{2}+y^{2}\\right)^{3 / 2}}+\\frac{y d y}{\\left(x^{2}+y^{2}\\right)^{3 / 2}}=0$\n14. $\\left(e^{x}\\left(x^{2} y^{2}+2 x y^{2}\\right)+6 x\\right) d x+\\left(2 x^{2} y e^{x}+2\\right) d y=0$\n15. $\\left(x^{2} e^{x^{2}+y}\\left(2 x^{2}+3\\right)+4 x\\right) d x+\\left(x^{3} e^{x^{2}+y}-12 y^{2}\\right) d y=0$\n16. $\\left(e^{x y}\\left(x^{4} y+4 x^{3}\\right)+3 y\\right) d x+\\left(x^{5} e^{x y}+3 x\\right) d y=0$\n17. $\\left(3 x^{2} \\cos x y-x^{3} y \\sin x y+4 x\\right) d x+\\left(8 y-x^{4} \\sin x y\\right) d y=0$", "answer": "$2 x^{3} y^{2}=c$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.6", "question": "A $10 \\mathrm{~kg}$ mass stretches a spring $70 \\mathrm{~cm}$ in equilibrium. Suppose a $2 \\mathrm{~kg}$ mass is attached to the spring, initially displaced $25 \\mathrm{~cm}$ below equilibrium, and given an upward velocity of $2 \\mathrm{~m} / \\mathrm{s}$. Find its displacement for $t>0$. Find the frequency, period, amplitude, and phase angle of the motion.", "answer": "$y=-\\frac{1}{4} \\cos \\sqrt{70} t+\\frac{2}{\\sqrt{70}} \\sin \\sqrt{70} t \\mathrm{~m} ; \\quad R=\\frac{1}{4} \\sqrt{\\frac{67}{35}} \\mathrm{~m} \\omega_{0}=\\sqrt{70} \\mathrm{rad} / \\mathrm{s}$;\n\n$T=2 \\pi / \\sqrt{70} \\mathrm{~s} ; \\phi \\approx 2.38 \\mathrm{rad} \\approx 136.28^{\\circ}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.38", "question": "Solve the equation using variation of parameters followed by separation of variables: $y^{\\prime}-2 y=\\frac{x e^{2 x}}{1-y e^{-2 x}}$", "answer": "$y=e^{2 x}\\left(1 \\pm \\sqrt{c-x^{2}}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.7", "question": "A $96 \\mathrm{lb}$ weight is dropped from rest in a medium that exerts a resistive force with magnitude proportional to the speed. Find its velocity as a function of time if its terminal velocity is $-128 \\mathrm{ft} / \\mathrm{s}$.", "answer": "$v=-128\\left(1-e^{-t / 4}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.18", "question": "A space vehicle is to be launched from the moon, which has a radius of about 1080 miles. The acceleration due to gravity at the surface of the moon is about $5.31 \\mathrm{ft} / \\mathrm{s}^{2}$. Find the escape velocity in miles/s.", "answer": "$\\approx 1.47$ miles/s", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.1.4", "question": "Find the general solution: $x y^{\\prime}+3 y=0$", "answer": "$y=\\frac{c}{x^{3}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.3.17", "question": "A space probe is to be launched from a space station 200 miles above Earth. Determine its escape velocity in miles/s. Take Earth's radius to be 3960 miles.", "answer": "$\\approx 6.76$ miles/s", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.1.11", "question": "Find a power series solution $y(x)=\\sum_{n=0}^{\\infty} a_{n} x^{n}$ for $(2+x) y^{\\prime \\prime}+x y^{\\prime}+3 y$.", "answer": "$b_{n}=2(n+2)(n+1) a_{n+2}+(n+1) n a_{n+1}+(n+3) a_{n}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.25", "question": "Find a fundamental set of solutions: $x y^{\\prime \\prime}-(4 x+1) y^{\\prime}+(4 x+2) y=0 ; \\quad y_{1}=e^{2 x}$", "answer": "$\\left\\{e^{2 x}, x^{2} e^{2 x}\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.4", "question": "Find the general solution: $x y^{\\prime}+3 y=0$", "answer": "$y_{1}=x^{1 / 2}\\left(1-2 x+\\frac{5}{2} x^{2}-2 x^{3}+\\cdots\\right)$\n$y_{2}=y_{1} \\ln x+x^{3 / 2}\\left(1-\\frac{9}{4} x+\\frac{17}{6} x^{2}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.3.63", "question": "Find the general solution: $y^{\\prime \\prime \\prime}+2 y^{\\prime \\prime}+y^{\\prime}=-2 e^{-x}\\left(7-18 x+6 x^{2}\\right)$", "answer": "$y=x^{2} e^{-x}(1-x)^{2}+c_{1}+e^{-x}\\left(c_{2}+c_{3} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.21", "question": "Find a curve $y=y(x)$ through $(1,-1)$ such that the tangent to the curve at any point $\\left(x_{0}, y\\left(x_{0}\\right)\\right)$ intersects the $y$ axis at $y_{I}=x_{0}^{3}$.", "answer": "$y=-\\frac{x^{3}}{2}-\\frac{x}{2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.36", "question": "Solve the equation using variation of parameters followed by separation of variables: $x y^{\\prime}-2 y=\\frac{x^{6}}{y+x^{2}}$", "answer": "$y=x^{2}\\left(-1+\\sqrt{x^{2}+c}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.33", "question": "Find a fundamental set of solutions: $\\left(D^{2}+1\\right)\\left(D^{2}+9\\right)^{2}(D-2) y=0$", "answer": "$\\left\\{\\cos x, \\sin x, \\cos 3 x, x \\cos 3 x, \\sin 3 x, x \\sin 3 x, e^{2 x}\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.48", "question": "Find two linearly independent Frobenius solutions of the equation: $x^{2}(1-x) y^{\\prime \\prime}-x(3-5 x) y^{\\prime}+(4-5 x) y=0$", "answer": "$y_{1}=x^{2}(1-x)^{3}$\n$y_{2}=y_{1} \\ln x+x^{3}\\left(4-7 x+\\frac{11}{3} x^{2}-6 \\sum_{n=3}^{\\infty} \\frac{1}{n(n-2)\\left(n^{2}-1\\right)} x^{n}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.11", "question": "Find the general solution: $x^{2} y^{\\prime \\prime}-x(2 x-1) y^{\\prime}+\\left(x^{2}-x-1\\right) y=x^{2} e^{x} ; \\quad y_{1}=x e^{x}$", "answer": "$y=x e^{x}\\left(\\frac{x}{3}+c_{1}+\\frac{c_{2}}{x^{2}}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.20", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $(3 x-1) y^{\\prime \\prime}-(3 x+2) y^{\\prime}-(6 x-8) y=0$, given that $y_{1}=e^{2 x}$.", "answer": "$y_{2}=x e^{-x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.2", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{ll}-11 & 4 \\\\ -26 & 9\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1} e^{-t}\\left[\\begin{array}{c}\n5 \\cos 2 t+\\sin 2 t \\\\\n13 \\cos 2 t\n\\end{array}\\right]+c_{2} e^{-t}\\left[\\begin{array}{c}\n5 \\sin 2 t-\\cos 2 t \\\\\n13 \\sin 2 t\n\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.14", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}3 & -4 & -2 \\\\ -5 & 7 & -8 \\\\ -10 & 13 & -8\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1}\\left[\\begin{array}{l}2 \\\\ 2 \\\\ 1\\end{array}\\right] e^{-2 t}+c_{2} e^{2 t}\\left[\\begin{array}{c}-\\cos 3 t-\\sin 3 t \\\\ -\\sin 3 t \\\\ \\cos 3 t\\end{array}\\right]+c_{3} e^{2 t}\\left[\\begin{array}{c}-\\sin 3 t+\\cos 3 t \\\\ \\cos 3 t \\\\ \\sin 3 t\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.22", "question": "Find all curves $y=y(x)$ such that the tangent to the curve at any point $\\left(x_{0}, y\\left(x_{0}\\right)\\right)$ intersects the $y$ axis at $y_{I}=x_{0}$.", "answer": "$y=-x \\ln |x|+c x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.4", "question": "A $96 \\mathrm{lb}$ weight stretches a spring $3.2 \\mathrm{ft}$ in equilibrium. It is attached to a dashpot with damping constant $c=18 \\mathrm{lb}-\\mathrm{sec} / \\mathrm{ft}$. The weight is initially displaced 15 inches below equilibrium and given a downward velocity of $12 \\mathrm{ft} / \\mathrm{sec}$. Find its displacement for $t>0$.", "answer": "$y=-\\frac{e^{-3 t}}{4}(5 \\cos t+63 \\sin t) \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.13", "question": "Find the general solution: $x^{2} y^{\\prime \\prime}-3 x y^{\\prime}+4 y=4 x^{4} ; \\quad y_{1}=x^{2}$", "answer": "$y=x^{4}+c_{1} x^{2}+c_{2} x^{2} \\ln |x|$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.1.3", "question": "Find the general solution: $x y^{\\prime}+(\\ln x) y=0$", "answer": "$y=c e^{-(\\ln x)^{2} / 2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.13", "question": "An $8 \\mathrm{lb}$ weight stretches a spring 8 inches in equilibrium. It is attached to a dashpot with damping constant $c=.5 \\mathrm{lb}-\\mathrm{sec} / \\mathrm{ft}$ and subjected to an external force $F(t)=4 \\cos 2 t \\mathrm{lb}$. Determine the steady state component of the displacement for $t>0$.", "answer": "$y_{p}=\\frac{22}{61} \\cos 2 t+\\frac{2}{61} \\sin 2 t \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.30", "question": "Solve the given homogeneous equation implicitly: $y^{\\prime}=\\frac{y^{3}+2 x y^{2}+x^{2} y+x^{3}}{x(y+x)^{2}}$", "answer": "$(y+x)^{3}=3 x^{3}(\\ln |x|+c)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.8", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $12 x^{2} y^{\\prime \\prime}-5 x y^{\\prime \\prime}+6 y=0$", "answer": "$y=c_{1} x^{2 / 3}+c_{2} x^{3 / 4}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.6.1", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{ll}-1 & 2 \\\\ -5 & 5\\end{array}\\right] \\mathbf{y}$", "answer": "$\\mathbf{y}=c_{1} e^{2 t}\\left[\\begin{array}{c}\n3 \\cos t+\\sin t \\\\\n5 \\cos t\n\\end{array}\\right]+c_{2} e^{2 t}\\left[\\begin{array}{c}\n3 \\sin t-\\cos t \\\\\n5 \\sin t\n\\end{array}\\right]$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.4.17", "question": "Find the general solution for the equation: $y^{\\prime \\prime}+6 y^{\\prime}+9 y=e^{2 x}(3-5 x)$", "answer": "$y=\\frac{e^{2 x}}{5}(1-x)+e^{-3 x}\\left(c_{1}+c_{2} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.11", "question": "A unit mass hangs in equilibrium from a spring with constant $k=1 / 16$. Starting at $t=0$, a force $F(t)=3 \\sin t$ is applied to the mass. Find its displacement for $t>0$.", "answer": "$y=\\frac{16}{5}\\left(4 \\sin \\frac{t}{4}-\\sin t\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.21", "question": "Two identical objects suspended from different springs are set into motion. The period of one motion is 3 times the period of the other. How are the two spring constants related?", "answer": "$k_{1}=9 k_{2}$, where $k_{1}$ is the spring constant of the system with the shorter period.", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.28", "question": "The population $P=P(t)$ of a species satisfies the logistic equation: $P^{\\prime}=a P(1-\\alpha P)$ and $P(0)=P_{0}>0$. Find $P$ for $t>0$, and find $\\lim _{t \\rightarrow \\infty} P(t)$.", "answer": "$P=\\frac{P_{0}}{\\alpha P_{0}+\\left(1-\\alpha P_{0}\\right) e^{-a t}} ; \\lim _{t \\rightarrow \\infty} P(t)=1 / \\alpha$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.2.14", "question": "A $32 \\mathrm{lb}$ weight stretches a spring $1 \\mathrm{ft}$ in equilibrium. It is attached to a dashpot with constant $c=12 \\mathrm{lb}-\\mathrm{sec} / \\mathrm{ft}$. The weight is initially displaced 8 inches above equilibrium and released from rest. Find its displacement for $t>0$.", "answer": "$y=-\\frac{2}{3}\\left(e^{-8 t}-2 e^{-4 t}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.35", "question": "Find a fundamental set of solutions: $\\left(4 D^{2}+4 D+9\\right)^{3} y=0$", "answer": "$\\left\\{e^{-x / 2} \\cos 2 x, x e^{-x / 2} \\cos 2 x, x^{2} e^{-x / 2} \\cos 2 x, e^{-x / 2} \\sin 2 x, x e^{-x / 2} \\sin 2 x\\right.$, $\\left.x^{2} e^{-x / 2} \\sin 2 x\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.3.62", "question": "Find the general solution: $y^{\\prime \\prime \\prime}-6 y^{\\prime \\prime}+11 y^{\\prime}-6 y=e^{2 x}\\left(5-4 x-3 x^{2}\\right)$", "answer": "$y=x e^{2 x}(1+x)^{2}+c_{1} e^{x}+c_{2} e^{2 x}+c_{3} e^{3 x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.7", "question": "A weight stretches a spring 1.5 inches in equilibrium. The weight is initially displaced 8 inches above equilibrium and given a downward velocity of $4 \\mathrm{ft} / \\mathrm{s}$. Find its displacement for $t>0$.", "answer": "$y=\\frac{2}{3} \\cos 16 t-\\frac{1}{4} \\sin 16 t \\mathrm{ft}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.14", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $x^{2} y^{\\prime \\prime}-x y^{\\prime}+y=0$, given that $y_{1}=x$.", "answer": "$y_{2}=x \\ln x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.11", "question": "A 200 gallon tank initially contains 100 gallons of water with 20 pounds of salt. A salt solution with $1 / 4$ pound of salt per gallon is added to the tank at $4 \\mathrm{gal} / \\mathrm{min}$, and the resulting mixture is drained out at $2 \\mathrm{gal} / \\mathrm{min}$. Find the quantity of salt in the tank as it's about to overflow.", "answer": "$Q(50)=47.5$ (pounds)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.10", "question": "Find the general solution: $y^{(4)}+12 y^{\\prime \\prime}+36 y=0$", "answer": "$y=\\left(c_{1}+c_{2} x\\right) \\cos \\sqrt{6} x+\\left(c_{3}+c_{4} x\\right) \\sin \\sqrt{6} x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.6.1.9", "question": "A spring-mass system has natural frequency $7 \\sqrt{10} \\mathrm{rad} / \\mathrm{s}$. The natural length of the spring is $.7 \\mathrm{~m}$. What is the length of the spring when the mass is in equilibrium?", "answer": "$.72 \\mathrm{~m}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.38", "question": "Find a fundamental set of solutions: $\\left[(D-1)^{4}-16\\right] y=0$", "answer": "$\\left\\{e^{-x}, e^{3 x}, e^{x} \\cos 2 x, e^{x} \\sin 2 x\\right\\}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.14", "question": "A 1200-gallon tank initially contains 40 pounds of salt dissolved in 600 gallons of water. Starting at $t_{0}=0$, water that contains $1 / 2$ pound of salt per gallon is added to the tank at the rate of 6 $\\mathrm{gal} / \\mathrm{min}$ and the resulting mixture is drained from the tank at $4 \\mathrm{gal} / \\mathrm{min}$. Find the quantity $Q(t)$ of salt in the tank at any time $t>0$ prior to overflow.", "answer": "$Q=t+300-\\frac{234 \\times 10^{5}}{(t+300)^{2}}, \\quad 0 \\leq t \\leq 300$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.3", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rr}-7 & 4 \\\\ -1 & -11\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-2 \\\\\n1\n\\end{array}\\right] e^{-9 t}+c_{2}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n0\n\\end{array}\\right] e^{-9 t}+\\left[\\begin{array}{r}\n-2 \\\\\n1\n\\end{array}\\right] t e^{-9 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.5", "question": "Find the general solution: $y^{\\prime \\prime}-2 y^{\\prime}+y=7 x^{3 / 2} e^{x} ; \\quad y_{1}=e^{x}$", "answer": "$y=e^{x}\\left(\\frac{4}{5} x^{7 / 2}+c_{1}+c_{2} x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.4.2", "question": "Find the general solution of the given Euler equation on $(0, \\infty)$: $x^{2} y^{\\prime \\prime}-7 x y^{\\prime}+7 y=0$", "answer": "$y=c_{1} x+c_{2} x^{7}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.16", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $4 x^{2} y^{\\prime \\prime}-4 x y^{\\prime}+\\left(3-16 x^{2}\\right) y=0$, given that $y_{1}=x^{1 / 2} e^{2 x}$.", "answer": "$y_{2}=x^{1 / 2} e^{-2 x}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.1.12", "question": "Find a power series solution $y(x)=\\sum_{n=0}^{\\infty} a_{n} x^{n}$ for $\\left(1+3 x^{2}\\right) y^{\\prime \\prime}+3 x^{2} y^{\\prime}-2 y$.", "answer": "$b_{0}=2 a_{2}-2 a_{0} b_{n}=(n+2)(n+1) a_{n+2}+[3 n(n-1)-2] a_{n}+3(n-1) a_{n-1}, n \\geq 1$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.3.61", "question": "Find the general solution: $y^{\\prime \\prime \\prime}+y^{\\prime \\prime}-2 y=-e^{3 x}\\left(9+67 x+17 x^{2}\\right)$", "answer": "$y=e^{3 x}\\left(1-x-\\frac{x^{2}}{2}\\right)+c_{1} e^{x}+e^{-x}\\left(c_{2} \\cos x+c_{3} \\sin x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.5.25", "question": "Find the orthogonal trajectories of the given family of curves: $x^{2}+2 y^{2}=c^{2}$", "answer": "$y=k x^{2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.12", "question": "Find the general solution: $6 y^{(4)}+5 y^{\\prime \\prime \\prime}+7 y^{\\prime \\prime}+5 y^{\\prime}+y=0$", "answer": "$y=c_{1} e^{-x / 2}+c_{2} e^{-x / 3}+c_{3} \\cos x+c_{4} \\sin x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.24", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}5 & -1 & 1 \\\\ -1 & 9 & -3 \\\\ -2 & 2 & 4\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n1\n\\end{array}\\right] e^{6 t}+c_{2}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{e^{6 t}}{4}+\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n1\n\\end{array}\\right] t e^{6 t}\\right)+c_{3}\\left(\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{e^{6 t}}{8}+\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{t e^{6 t}}{4}+\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n1\n\\end{array}\\right] \\frac{t^{2} e^{6 t}}{2}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.6.3", "question": "Find the general solution: $x y^{\\prime}+(\\ln x) y=0$", "answer": "$y_{1}=1+x-x^{2}+\\frac{1}{3} x^{3}+\\cdots$\n$y_{2}=y_{1} \\ln x-x\\left(3-\\frac{1}{2} x-\\frac{31}{18} x^{2}+\\cdots\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.23", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $\\left(x^{2}-2 x\\right) y^{\\prime \\prime}+\\left(2-x^{2}\\right) y^{\\prime}+(2 x-2) y=0$, given that $y_{1}=e^{x}$.", "answer": "$y_{2}=x^{2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.15", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $x^{2} y^{\\prime \\prime}-(2 a-1) x y^{\\prime}+a^{2} y=0$ $(a=$ nonzero constant $)$, $x>0$, given that $y_{1}=x^{a}$.", "answer": "$y_{2}=x^{a} \\ln x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.25", "question": "Solve the equation $y^{\\prime} \\sqrt{1-x^{2}}+\\sqrt{1-y^{2}}=0$ explicitly. Hint: Use the identity $\\sin (A-B)=\\sin A \\cos B-$ $\\cos A \\sin B$.", "answer": "$y=-x \\cos c+\\sqrt{1-x^{2}} \\sin c ; \\quad y \\equiv 1 ; y \\equiv-1$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.26", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}-6 & -4 & -4 \\\\ 2 & -1 & 1 \\\\ 2 & 3 & 1\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n0 \\\\\n-1 \\\\\n1\n\\end{array}\\right] e^{-2 t}+c_{2}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right] e^{-2 t}+\\left[\\begin{array}{r}\n0 \\\\\n-1 \\\\\n1\n\\end{array}\\right] t e^{-2 t}\\right)+c_{3}\\left(\\left[\\begin{array}{r}\n3 \\\\\n-2 \\\\\n0\n\\end{array}\\right] \\frac{e^{-2 t}}{4}+\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right] t e^{-2 t}+\\left[\\begin{array}{r}\n0 \\\\\n-1 \\\\\n1\n\\end{array}\\right] \\frac{t^{2} e^{-2 t}}{2}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.27", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}0 & 2 & -2 \\\\ -1 & 5 & -3 \\\\ 1 & 1 & 1\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n1\n\\end{array}\\right] e^{2 t}+c_{2}\\left(\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{e^{2 t}}{2}+\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n1\n\\end{array}\\right] t e^{2 t}\\right)+c_{3}\\left(\\left[\\begin{array}{r}\n-1 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{e^{2 t}}{8}+\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n0\n\\end{array}\\right] \\frac{t e^{2 t}}{2}+\\left[\\begin{array}{l}\n0 \\\\\n1 \\\\\n1\n\\end{array}\\right] \\frac{t^{2} e^{2 t}}{2}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.7.1.14", "question": "Find a power series solution $y(x)=\\sum_{n=0}^{\\infty} a_{n} x^{n}$ for $\\left(1+x^{2}\\right) y^{\\prime \\prime}+(2-x) y^{\\prime}+3 y$.", "answer": "$b_{n}=(n+2)(n+1) a_{n+2}+2(n+1) a_{n+1}+\\left(n^{2}-2 n+3\\right) a_{n}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.3", "question": "Find all solutions: $\\left(3 y^{3}+3 y \\cos y+1\\right) y^{\\prime}+\\frac{(2 x+1) y}{1+x^{2}}=0$", "answer": "$y=\\frac{c}{x-c} \\quad y \\equiv-1$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.9.2.4", "question": "Find the general solution: $2 y^{\\prime \\prime \\prime}+3 y^{\\prime \\prime}-2 y^{\\prime}-3 y=0$", "answer": "$y=c_{1} e^{x}+c_{2} e^{-x}+c_{3} e^{-3 x / 2}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.2.37", "question": "Solve the equation using variation of parameters followed by separation of variables: $y^{\\prime}-y=\\frac{(x+1) e^{4 x}}{\\left(y+e^{x}\\right)^{2}}$", "answer": "$y=e^{x}\\left(-1+\\left(3 x e^{x}+c\\right)^{1 / 3}\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.4.2.1", "question": "A thermometer is moved from a room where the temperature is $70^{\\circ} \\mathrm{F}$ to a freezer where the temperature is $12^{\\circ} \\mathrm{F}$. After 30 seconds the thermometer reads $40^{\\circ} \\mathrm{F}$. What does it read after 2 minutes?", "answer": "$\\approx 15.15^{\\circ} \\mathrm{F}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.1.18", "question": "Find the Wronskian of a given set $\\left\\{y_{1}, y_{2}\\right\\}$ of solutions of $x^{2} y^{\\prime \\prime}-2 x y^{\\prime}+\\left(x^{2}+2\\right) y=0$, given that $y_{1}=x \\cos x$.", "answer": "$y_{2}=x \\sin x$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.2.6", "question": "Find the general solution: $y^{\\prime \\prime}+6 y^{\\prime}+10 y=0$", "answer": "$y=e^{-3 x}\\left(c_{1} \\cos x+c_{2} \\sin x\\right)$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.5.6.3", "question": "Find the general solution: $x^{2} y^{\\prime \\prime}-x y^{\\prime}+y=x ; \\quad y_{1}=x$", "answer": "$y=\\frac{x(\\ln |x|)^{2}}{2}+c_{1} x+c_{2} x \\ln |x|$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.10.5.12", "question": "Find the general solution: $\\mathbf{y}^{\\prime}=\\left[\\begin{array}{rrr}6 & -5 & 3 \\\\ 2 & -1 & 3 \\\\ 2 & 1 & 1\\end{array}\\right] \\mathbf{y}$", "answer": "\\mathbf{y}=c_{1}\\left[\\begin{array}{r}\n-1 \\\\\n-1 \\\\\n1\n\\end{array}\\right] e^{-2 t}+c_{2}\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right] e^{4 t}+c_{3}\\left(\\left[\\begin{array}{l}\n1 \\\\\n0 \\\\\n0\n\\end{array}\\right] \\frac{e^{4 t}}{2}+\\left[\\begin{array}{l}\n1 \\\\\n1 \\\\\n1\n\\end{array}\\right] t e^{4 t}\\right)", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} +{"data_source": "college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS", "question_number": "exercise.2.4.4", "question": "Solve the given Bernoulli equation: $\\left(1+x^{2}\\right) y^{\\prime}+2 x y=\\frac{1}{\\left(1+x^{2}\\right) y}$", "answer": "$y= \\pm \\frac{\\sqrt{2 x+c}}{1+x^{2}}$", "license": "Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)", "data_topic": "college_math.differential_equation"} |