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"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: If $e^{i \\alpha} = \\frac{3}{5} +\\frac{4}{5} i$ and $e^{i \\beta} = -\\frac{12}{13} + \\frac{5}{13} i,$ then find $\\sin (\\alpha + \\beta).$",
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[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: The matrices\n\\[\\begin{pmatrix} 3 & -8 \\\\ a & 11 \\end{pmatrix} \\quad \\text{and} \\quad \\begin{pmatrix} 11 & b \\\\ 4 & 3 \\end{pmatrix}\\]are inverses. Enter the ordered pair $(a,b).$",
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[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: Given that $\\mathbf{a}$ and $\\mathbf{b}$ are nonzero vectors such that $\\|\\mathbf{a} + \\mathbf{b}\\| = \\|\\mathbf{a} - \\mathbf{b}\\|,$ find the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees.",
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[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: Two lines are perpendicular. One line has a direction vector of $\\begin{pmatrix} 3 \\\\ -7 \\end{pmatrix}.$ The other line has a direction vector of $\\begin{pmatrix} a \\\\ 2 \\end{pmatrix}.$ Find $a.$",
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[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be unit vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{b} \\times \\mathbf{c}) = \\frac{\\mathbf{b} + \\mathbf{c}}{\\sqrt{2}},\\]and such that $\\{\\mathbf{a}, \\mathbf{b}, \\mathbf{c}\\}$ is a linearly independent set.\n\nFind the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees.",
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[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: If $\\det \\mathbf{M} = -2,$ then find $ \\det (\\mathbf{M}^4).$",
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[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: Find the smallest positive integer $k$ such that $\nz^{10} + z^9 + z^6+z^5+z^4+z+1\n$ divides $z^k-1$.",
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[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: Define the sequence $a_1, a_2, a_3, \\ldots$ by $a_n = \\sum\\limits_{k=1}^n \\sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$.",
"role": "user"
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] |
[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: Find the number of real solutions of the equation\n\\[\\frac{x}{100} = \\sin x.\\]",
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[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate\n\\[\\begin{vmatrix} \\sin^2 A & \\cot A & 1 \\\\ \\sin^2 B & \\cot B & 1 \\\\ \\sin^2 C & \\cot C & 1 \\end{vmatrix}.\\]",
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[
{
"content": "Answer the following question. Output a solution and then your answer with no units (just a numerical value) on the last line. Use LaTeX to format your answer. Ouput in the following format: \nSolution: <solution> \nAnswer: <answer>. MAKE SURE TO FOLLOW THE FORMAT EXACTLY. \nQuestion: If angle $A$ lies in the second quadrant and $\\sin A = \\frac{3}{4},$ find $\\cos A.$",
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] |
Subsets and Splits