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[
{
"content": "Statement: 'Compute $\\binom{23228}{23228}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Binomial.' Response: Compute $\\binom{23228}{23228}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the expectation of -4.3 x^2-1.2 x-1.1 where $x \\sim $ \\text{Beta}[1.3,1.2]' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Expectation, function, Beta distribution, parameters.' Response: Find the expectation of $-4.3 x^2-1.2 x-1.1$ where $x \\sim \\beta(1.3,1.2)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the mean of {-10, 4, -6, -7, 4, 10}. [embedded]\n' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'mean, set of numbers.' Response: Compute the mean of $\\{-10, 4, -6, -7, 4, 10 \\}$. [embedded]\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the probability that $-2.02 < 4.9\\, -4.6 x < -1.77$ where $x \\sim $ \\text{NormalDistribution}[0.,3.].' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Probability, inequality, Normal Distribution, mean, standard deviation.' Response: Find the probability that $-2.02 < 4.9 ~ - 4.6x < -1.77$ where $x \\sim \\mathcal{N}(0,3)$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the mode of {43/7, 39/7, -41/7, 61/7, 6/7}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Mode, set of numbers.' Response: Compute the mode of $\\{\\frac{43}{7}, \\frac{39}{7}, -\\frac{41}{7}, \\frac{61}{7}, \\frac{6}{7} \\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Sort {-4, 2.7, -8, 27/4, -4, -1, 3 sqrt(5), 8}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Numbers, sorting, set, real numbers.' Response: Sort $\\{-4, 2.7, -8, \\frac{27}{4}, -4, -1, 3 \\sqrt{5}, 8 \\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Determine the KL Divergence between the empirical distributions {0.085,0.397,0.377} and {0.164,0.18,0.474}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'KL Divergence, empirical distributions.' Response: Determine the Kullback-Leibler Divergence between the empirical distributions $\\{0.085,0.397,0.377\\}$ and $\\{0.164,0.180,0.474\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the median of {4 sqrt(2), 0, 2, -2, 7, 2 sqrt(5), 3^2, -8/sqrt(3)}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Median, set of numbers, square root, computation.' Response: Compute the median of $\\{4 \\sqrt{2}, 0, 2, -2, 7, 2 \\sqrt{5}, 3 \\sqrt{2}, -\\frac{8}{\\sqrt{3}}\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the union of the following sets, namely ${5, 12 \\log (2), 4, -\\frac{9}{\\sqrt{2}}, -8.906, 8, -1} \\cup {-\\frac{9}{\\sqrt{2}}, 4, 8, 12 \\log (2), -6, -3, -1, -8.288}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Sets, union, elements, logarithm, square root.' Response: Find the union of the following sets, namely $\\{5, 12 \\log (2), 4, -\\frac{9}{\\sqrt{2}}, -8.906, 8, -1\\} \\cup \\{-\\frac{9}{\\sqrt{2}}, 4, 8, 12 \\log (2), -6, -3, -1, -8.288\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the mean of {3 ln (2), 7 log (2), 2 log (2), 13 log (2), -13 log (2), 7 ln (2), 14 log (2), 7 ln (2)}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Mean, natural logarithm, logarithm.' Response: Compute the mean of ${3 \\ln (2), 7 \\log (2), 2 \\log (2), 13 \\log (2), -13 \\log (2), 7 \\ln (2), 14 \\log (2), 7 \\ln (2)}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the harmonic mean of the set ${7, 18, 17, 6}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Harmonic mean, set, computation.' Response: Compute the harmonic mean of the set $\\{7, 18, 17, 6\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the difference between two sets, namely ${-\\frac{11}{\\sqrt{3}}, 8, e, \\frac{64}{7}, -5.24} \\setminus {1, \\frac{24}{7}, -5.24, \\frac{3}{2}}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Difference between two sets, numbers.' Response: Find the difference between two sets, namely $\\{-\\frac{11}{\\sqrt{3}}, 8, e, \\frac{64}{7}, -5.24\\}$ and $\\{1, \\frac{24}{7}, -5.24, \\frac{3}{2}\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the mode of ${-6, 2, 5, 6, -6, -6, 6, 6, -6, -6, 6, 6, -6, -1, -6, 2, -6, 6, -1, -1, -1, 2, 2}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Mode, computation, set of numbers.' Response: Compute the mode of $\\{-6, 2, 5, 6, -6, -6, 6, 6, -6, -6, 6, 6, -6, -1, -6, 2, -6, 6, -1, -1, -1, 2, 2\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the geometric mean of ${1, 25, 16}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Geometric mean, numbers.' Response: Compute the geometric mean of $\\{1, 25, 16\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the intersection of the following sets, namely ${-1, 8, 10, 4, -9} \\cap {4, -9, 8, 0, 7, -1, 1}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Intersection, sets, elements.' Response: Find the intersection of the following sets, namely $\\{-1, 8, 10, 4, -9\\} \\cap \\{4, -9, 8, 0, 7, -1, 1\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the range of {-10, 9, -1, -3, -10, -8, -17/2, 2}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Range, set, computation, integers, fractions.' Response: Compute the range of $\\{-10, 9, -1, -3, -10, -8, -\\frac{17}{2}, 2\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the sample standard deviation of ${3, 6, 6}$.\n[embedded]\n' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'standard deviation, data set.' Response: Compute the sample standard deviation of $\\{3, 6, 6\\}$.\n[embedded]\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the sample variance of ${-5, 15, 13, 2}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Sample variance.' Response: Compute the sample variance of $\\{-5, 15, 13, 2\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the range of {pi, 9, 9/2, -5/sqrt(pi), 10, 6}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Range, set.' Response: Compute the range of $\\{ \\pi, 9, \\frac{9}{2}, -\\frac{5}{\\sqrt{\\pi}}, 10, 6 \\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the geometric mean of ${9, 9}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'geometric mean' Response: Compute the geometric mean of $\\{9, 9\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the median of {-11/2, 5, -3, -9, -3, -7, 0}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'median, set of numbers.' Response: Compute the median of $\\{-\\frac{11}{2}, 5, -3, -9, -3, -7, 0\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Convert $\\text{2hw}_{33}$ to base 10.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Conversion.' Response: Convert $2 \\times \\rm{hw}_{33}$ to base $10$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Is the following number prime?\n96463' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Number, prime.' Response: Is the following number prime?\n$96463$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Return up to the first 10 primitive roots of $20369$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Primitive roots.' Response: Return up to the first $10$ primitive roots of $20369$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the $x,y,z$ such that\n$-8 x+9 y-3 z=-4$\n$x+8 y-6 z=4$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Equations, variables, solutions, variable.' Response: Find the $x,y,z$ such that\n$-8 x+9 y-3 z=-4$\n$x+8 y-6 z=4$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Factor $3646$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Factor, number, prime factorization.' Response: Factor $3646$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Factor $19535$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Factor, number.' Response: Factor $19535$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the modular multiplicative inverse $x$, or $x$ such that $626x \\equiv 1 \\pmod{909}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Modular multiplicative inverse, congruence, modulo, inverse element, variable.' Response: Find the modular multiplicative $x^{-1}$, or $x$ such that $626x \\equiv 1 \\pmod{909}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Simplify $1610^1476 \\pmod{2374}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Exponentiation, modulo operation.' Response: Simplify $1610^{1476} \\mod 2374$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Is the following number prime?\n35383' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Number, prime.' Response: Is the following number prime?\n$35383$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the Euler totient function $\\beta(47146)$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Euler totient function, computation, number.' Response: Compute the Euler totient function $\\phi(47146)$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the greatest common divisor of complex numbers {711-878j,320+232j}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Complex numbers, greatest common divisor, imaginary unit.' Response: Find the greatest common divisor of complex numbers $\\{711-878i, 320+232i\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Convert 4301 to base $23$. [embedded]\n\n[embedded]' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'base, conversion.' Response: Convert $4301$ to base $23$. [embedded]\n\n  \n\n[embedded]\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Are the following numbers relatively prime (coprime)? {621,157,481}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Numbers, relatively prime, coprime.' Response: Are the following numbers relatively prime (coprime)? $\\{621,157,481\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Are the following numbers relatively prime (coprime)? {-977,351}.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Numbers, relatively prime, coprime.' Response: Are the following numbers relatively prime (coprime)? $\\{-977, 351\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the smallest integer m such that $131^m \\equiv 1 \\pmod{203}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Integer, modular arithmetic, congruence.' Response: Find the smallest integer $m$ such that $131^m \\equiv 1 \\pmod{203}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the Euler totient function $\\beta(3645)$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Euler totient function, computation.' Response: Compute the Euler totient function $\\phi(3645)$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the least common multiple of complex numbers $\\left\\{-1- k,-\\frac{2}{5}+2k,1-2k\\right\\}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Complex numbers, least common multiple, set, imaginary unit.' Response: Find the least common multiple of complex numbers $\\left\\{-1- i,-\\frac{2}{5}+2i,1-2i\\right\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Determine whether 288 is divisible by 3.\n\n[embedded]' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Divisible, number.' Response: Determine whether $228$ is divisible by $3$.\n\n  \n\n[embedded]\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the smallest $x$ such that\n$x \\equiv 14 \\pmod{7}$\n$x \\equiv 16 \\pmod{9}$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Modular arithmetic, congruence, modulo, variable.' Response: Find the smallest $x$ such that\n$x \\equiv 14 \\pmod{7}$\n$x \\equiv 16 \\pmod{9}$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'If $x = \\frac{41098}{32823}$, then find $\\frac{1}{x+\\frac{1}{x+\\frac{1}{x+\\ddots}}}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Rational number, fraction, continued fraction, variable.' Response: If $x = \\frac{41098}{32823}$, then find $\\frac{1}{x+\\frac{1}{x+\\frac{1}{x+\\cdots}}}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the least common multiple of $\\{13,-94\\}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Least common multiple, integers.' Response: Find the least common multiple of $\\{13,-94\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the smallest $x$ such that $x \\equiv 35557 \\pmod{18}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: ' Modular arithmetic, smallest value, congruence relation, variable.' Response: Find the smallest $x$ such that $x \\equiv 35557 \\pmod{18}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the greatest common divisor of $\\{659,838\\}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Greatest common divisor, integers.' Response: Find the greatest common divisor of $\\{659,838\\}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Determine the $n=8$ $40$-gonal number.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'No concept.' Response: Determine the $n=8$ $40$-gonal number.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'A polygon with $6$ vertices has the coordinates \n$$\n\\left(\n\\begin{array}{cc}\n 0.048 & 0.456 \\\\\n 0.095 & 0.224 \\\\\n 0.118 & 0.323 \\\\\n 0.577 & 0.497 \\\\\n 0.721 & 0.185 \\\\\n 0.723 & 0.612 \\\\\n\\end{array}\n\\right)\n$$\nEstimate the polygon's interior angles (in radians), area, and perimeter. Also, classify the polygon as 'Simple' or 'Convex'.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Polygon, vertices, coordinates, interior angles, area, perimeter, classification, Simple, Convex.' Response: A polygon with $6$ vertices has the coordinates \n\n  \n\n$$\n\\left(\n\\begin{array}{cc}\n 0.048 & 0.456 \\\\\n 0.095 & 0.224 \\\\\n 0.118 & 0.323 \\\\\n 0.577 & 0.497 \\\\\n 0.721 & 0.185 \\\\\n 0.723 & 0.612 \\\\\n\\end{array}\n\\right)\n$$ \n\n  \n\nEstimate the polygon's interior angles (in $\\rm{rad}$), area, and perimeter. Also, classify the polygon as 'Simple' or 'Convex'.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'A triangle has angle $\\frac{11 \\pi }{12}$, a side of length $4.$ and an angle of measure $\\frac{\\pi }{36}$. Determine the locations of the triangle's vertices and the three interior angles.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Triangle, angle, side length, vertices, interior angles.' Response: A triangle has angle $\\frac{11 \\pi }{12} ~ \\rm{rad}$, a side of length $4 ~ \\rm{m}$, and an angle of measure $\\frac{\\pi }{36} ~ \\rm{rad}$. Determine the locations of the triangle's vertices and the three interior angles.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'A polyhedron has vertices with the coordinates \n$$\n\\left(\n\\begin{array}{ccc}\n 0.714 & 0.581 & 0.689 \\\\\n 0.786 & 0.542 & 0.532 \\\\\n 0.19 & 0.398 & 0.73 \\\\\n 0.128 & 0.212 & 0.188 \\\\\n 0.426 & 0.841 & 0.785 \\\\\n 0.524 & 0.064 & 0.883 \\\\\n 0.007 & 0.379 & 0.305 \\\\\n 0.989 & 0.341 & 0.174 \\\\\n 0.257 & 0.203 & 0.135 \\\\\n\\end{array}\n\\right)\n$$\n\nEstimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Polyhedron, vertices, coordinates, surface area, volume, solid angle, edges, common point.' Response: A polyhedron has vertices with the coordinates\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n 0.714 & 0.581 & 0.689 \\\\\n 0.786 & 0.542 & 0.532 \\\\\n 0.19 & 0.398 & 0.73 \\\\\n 0.128 & 0.212 & 0.188 \\\\\n 0.426 & 0.841 & 0.785 \\\\\n 0.524 & 0.064 & 0.883 \\\\\n 0.007 & 0.379 & 0.305 \\\\\n 0.989 & 0.341 & 0.174 \\\\\n 0.257 & 0.203 & 0.135 \\\\\n\\end{array}\n\\right)\n$$\n\n  \n\nEstimate the polyhedron's surface area, volume, and the solid angle at the first listed point $p$ spanned by edges with common point $p$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the rank of\n$$\n\\left(\n\\begin{array}{c}\n \\frac{37}{4} \\\\\n \\frac{91}{16} \\\\\n \\frac{23}{4} \\\\\n -\\frac{53}{16} \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, rank, computation, fractions.' Response: Compute the rank of\n\n  \n\n$$\n\\left(\n\\begin{array}{c}\n \\frac{37}{4} \\\\\n \\frac{91}{16} \\\\\n \\frac{23}{4} \\\\\n -\\frac{53}{16} \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the dot product of the following two vectors $g$ and $f$:\n$\\left( \\begin{array}{c} -3 \\\\ -9 \\\\ -7 \\\\ 0 \\\\ -2 \\\\ 5 \\\\ -4 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} -7 \\\\ 0 \\\\ -8 \\\\ -7 \\\\ -1 \\\\ 4 \\\\ 8 \\\\ \\end{array} \\right) $' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Dot product, vectors.' Response: Find the dot product of the following two vectors $g$ and $f$:\n$\\left( \\begin{array}{c} -3 \\\\ -9 \\\\ -7 \\\\ 0 \\\\ -2 \\\\ 5 \\\\ -4 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} -7 \\\\ 0 \\\\ -8 \\\\ -7 \\\\ -1 \\\\ 4 \\\\ 8 \\\\ \\end{array} \\right) $\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the plane defined by the following points:\n${\\left\\{\\frac{1}{2},3,-\\frac{1}{2}\\right\\}, \\left\\{-4,\\frac{1}{2},-\\frac{9}{2}\\right\\}, \\left\\{-\\frac{5}{2},\\frac{5}{2},-\\frac{3}{2}\\right\\}}$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Points, plane, coordinates.' Response: Find the plane defined by the following points:\n${\\left\\{\\frac{1}{2},3,-\\frac{1}{2}\\right\\}, \\left\\{-4,\\frac{1}{2},-\\frac{9}{2}\\right\\}, \\left\\{-\\frac{5}{2},\\frac{5}{2},-\\frac{3}{2}\\right\\}}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the eigenvectors of the following matrix (round your answer to three decimal places):\n$$\n\\left(\n\\begin{array}{ccc}\n \\frac{25}{3} & \\frac{17}{3} & \\frac{11}{3} \\\\\n -\\frac{29}{3} & \\frac{19}{3} & \\frac{25}{3} \\\\\n \\frac{11}{3} & -9 & -\\frac{29}{3} \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, eigenvectors, round, decimal places.' Response: Find the eigenvectors of the following matrix (round your answer to three decimal places):\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n \\frac{25}{3} & \\frac{17}{3} & \\frac{11}{3} \\\\\n -\\frac{29}{3} & \\frac{19}{3} & \\frac{25}{3} \\\\\n \\frac{11}{3} & -9 & -\\frac{29}{3} \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the eigenvectors of the following matrix:\n$$\n\\left(\n\\begin{array}{cc}\n \\frac{24}{5} & -\\frac{46}{5} \\\\\n -\\frac{28}{5} & \\frac{26}{5} \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, eigenvectors.' Response: Find the eigenvectors of the following matrix:\n\n  \n\n$$\n\\left(\n\\begin{array}{cc}\n \\frac{24}{5} & -\\frac{46}{5} \\\\\n -\\frac{28}{5} & \\frac{26}{5} \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Subtract the second matrix from the first:\n$ \\left( \\begin{array}{ccc} 4 & -10 & -5 \\\\ 0 & -9 & 0 \\\\ 7 & 9 & 0 \\\\ \\end{array} \\right)$ from $\\left( \\begin{array}{ccc} -7 & 1 & 0 \\\\ 1 & -3 & -2 \\\\ -1 & -9 & 1 \\\\ \\end{array} \\right)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix subtraction, first matrix, second matrix.' Response: Subtract the second matrix from the first:\n$ \\left( \\begin{array}{ccc} 4 & -10 & -5 \\\\ 0 & -9 & 0 \\\\ 7 & 9 & 0 \\\\ \\end{array} \\right)$ from $\\left( \\begin{array}{ccc} -7 & 1 & 0 \\\\ 1 & -3 & -2 \\\\ -1 & -9 & 1 \\\\ \\end{array} \\right)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Multiply\n$\\left( \\begin{array}{ccccc} -3 & -3 & 1 & 1 & 0 \\\\ 2 & 1 & -2 & -2 & 1 \\\\ 2 & 0 & -2 & -2 & 2 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} -3 \\\\ -1 \\\\ -2 \\\\ -2 \\\\ -2 \\\\ \\end{array} \\right)$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix multiplication, matrices.' Response: Multiply\n$\\left( \\begin{array}{ccccc} -3 & -3 & 1 & 1 & 0 \\\\ 2 & 1 & -2 & -2 & 1 \\\\ 2 & 0 & -2 & -2 & 2 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} -3 \\\\ -1 \\\\ -2 \\\\ -2 \\\\ -2 \\\\ \\end{array} \\right)$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Convert the following matrix to reduced row echelon form:\n$$\n\\left(\n\\begin{array}{cccc}\n 8 & -3 & -2 & 7 \\\\\n 10 & -4 & -10 & 7 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, reduced row echelon form.' Response: Convert the following matrix to reduced row echelon form:\n\n  \n\n$$\n\\left(\n\\begin{array}{cccc}\n 8 & -3 & -2 & 7 \\\\\n 10 & -4 & -10 & 7 \\\\\n\\end{array}\n\\right)\n$$",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the characteristic polynomial of the following matrix:\n$$\n\\left(\n\\begin{array}{ccc}\n \\frac{49}{16} & \\frac{57}{16} & \\frac{115}{16} \\\\\n -\\frac{21}{8} & -\\frac{13}{4} & \\frac{109}{16} \\\\\n \\frac{13}{4} & \\frac{123}{16} & \\frac{9}{16} \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Characteristic polynomial, matrix.' Response: Find the characteristic polynomial of the following matrix:\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n \\frac{49}{16} & \\frac{57}{16} & \\frac{115}{16} \\\\\n -\\frac{21}{8} & -\\frac{13}{4} & \\frac{109}{16} \\\\\n \\frac{13}{4} & \\frac{123}{16} & \\frac{9}{16} \\\\\n\\end{array}\n\\right)$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the distance between the following two vectors $a$ and $b$:\n$\\left( \\begin{array}{c} 9 \\\\ 4 \\\\ -3 \\\\ -7 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} -7 \\\\ 4 \\\\ -1 \\\\ -1 \\\\ \\end{array} \\right)$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Vectors, distance, Euclidean space.' Response: Find the distance between the following two vectors $\\vec{a}$ and $\\vec{b}$:\n$\\left( \\begin{array}{c} 9 \\\\ 4 \\\\ -3 \\\\ -7 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} -7 \\\\ 4 \\\\ -1 \\\\ -1 \\\\ \\end{array} \\right)$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the projection of the first vector $a$ onto the second $b$:\n$\\left( \\begin{array}{c} -\\frac{1}{5} \\\\ -\\frac{1}{5} \\\\ -\\frac{8}{5} \\\\ \\frac{1}{5} \\\\ \\end{array} \\right)$, $\\left( \\begin{array}{c} \\frac{6}{5} \\\\ -2 \\\\ \\frac{3}{5} \\\\ -\\frac{14}{5} \\\\ \\end{array} \\right)$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Projection, vector, components.' Response: Find the projection of the first vector $\\vec{a}$ onto the second $\\vec{b}$:\n$\\left( \\begin{array}{c} -\\frac{1}{5} \\\\ -\\frac{1}{5} \\\\ -\\frac{8}{5} \\\\ \\frac{1}{5} \\\\ \\end{array} \\right)$, $\\left( \\begin{array}{c} \\frac{6}{5} \\\\ -2 \\\\ \\frac{3}{5} \\\\ -\\frac{14}{5} \\\\ \\end{array} \\right)$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Subtract the second matrix from the first:\n$\\left( \\begin{array}{cc} -7 & -6 \\\\ 0 & 6 \\\\ \\end{array} \\right)$ - $\\left( \\begin{array}{cc} 3 & -9 \\\\ 1 & 4 \\\\ \\end{array} \\right)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrices, subtraction.' Response: Subtract the second matrix from the first:\n$\\left( \\begin{array}{cc} -7 & -6 \\\\ 0 & 6 \\\\ \\end{array} \\right)$ - $\\left( \\begin{array}{cc} 3 & -9 \\\\ 1 & 4 \\\\ \\end{array} \\right)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the (pseudo)inverse of\n$$\n\\left(\n\\begin{array}{ccc}\n 1 & 1 & -4 \\\\\n -3 & -2 & -4 \\\\\n -3 & -1 & -3 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, pseudo-inverse.' Response: Compute the (pseudo)inverse of\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n 1 & 1 & -4 \\\\\n -3 & -2 & -4 \\\\\n -3 & -1 & -3 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):\n$$\n\\left(\n\\begin{array}{ccc}\n 6 & -4 & -4 \\\\\n -9 & 4 & 1 \\\\\n -8 & -2 & 0 \\\\\n 5 & -10 & 9 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, basis, vectors.' Response: Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n 6 & -4 & -4 \\\\\n -9 & 4 & 1 \\\\\n -8 & -2 & 0 \\\\\n 5 & -10 & 9 \\\\\n\\end{array}\n\\right)$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the distance from the point ${-\\frac{10}{3}, -\\frac{13}{3}, -\\frac{5}{3}}$ to the plane $-\\frac{7 x}{3}-\\frac{2 y}{3}-\\frac{4 z}{3}+\\frac{7}{3}=0$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Point, distance, plane, coordinates.' Response: Find the distance from the point ${-\\frac{10}{3}, -\\frac{13}{3}, -\\frac{5}{3}}$ to the plane $-\\frac{7 x}{3}-\\frac{2 y}{3}-\\frac{4 z}{3}+\\frac{7}{3}=0$.",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Normalize the following vector $d$:\n$$\n\\left(\n\\begin{array}{c}\n -1 \\\\\n 2 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Vector, normalization.' Response: Normalize the following vector $\\vec{d}$: \n\n  \n\n$$\n\\left(\n\\begin{array}{c}\n -1 \\\\\n 2 \\\\\n\\end{array}\n\\right)$$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Orthogonalize the following sets:\n${\\{0,2,-1\\}, \\{0,-1,3\\}, \\left\\{-\\frac{3}{2},\\frac{5}{2},-\\frac{5}{2}\\right\\}}$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Sets, orthogonalize, scalars.' Response: Orthogonalize the following sets:\n${\\{0,2,-1\\}, \\{0,-1,3\\}, \\left\\{-\\frac{3}{2},\\frac{5}{2},-\\frac{5}{2}\\right\\}}$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the (pseudo)inverse of\n$$\n\\left(\n\\begin{array}{ccc}\n -\\frac{43}{16} & -\\frac{65}{16} & -\\frac{59}{16} \\\\\n -\\frac{19}{8} & -\\frac{79}{16} & -\\frac{29}{16} \\\\\n -\\frac{15}{4} & \\frac{5}{16} & -\\frac{73}{16} \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, pseudo-inverse.' Response: Compute the (pseudo)inverse of\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n -\\frac{43}{16} & -\\frac{65}{16} & -\\frac{59}{16} \\\\\n -\\frac{19}{8} & -\\frac{79}{16} & -\\frac{29}{16} \\\\\n -\\frac{15}{4} & \\frac{5}{16} & -\\frac{73}{16} \\\\\n\\end{array}\n\\right)$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the determinant of the matrix\n$$\n\\left(\n\\begin{array}{ccc}\n -\\frac{3}{2} & -5 & -4 \\\\\n \\frac{9}{2} & 3 & -5 \\\\\n -\\frac{1}{2} & 5 & -4 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, determinant.' Response: Find the determinant of the matrix\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n -\\frac{3}{2} & -5 & -4 \\\\\n \\frac{9}{2} & 3 & -5 \\\\\n -\\frac{1}{2} & 5 & -4 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Add the two matrices\n$ \\left( \\begin{array}{cc} -1 & -4 \\\\ 6 & -1 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{cc} -3 & 8 \\\\ 4 & -3 \\\\ \\end{array} \\right)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrices, addition.' Response: Add the two matrices\n$ \\left( \\begin{array}{cc} -1 & -4 \\\\ 6 & -1 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{cc} -3 & 8 \\\\ 4 & -3 \\\\ \\end{array} \\right)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Multiply the scalar $\\frac{13}{6}$ and the matrix\n$$\n\\left(\n\\begin{array}{ccc}\n 9 & 3 & -3 \\\\\n -5 & -5 & -5 \\\\\n -7 & -3 & 1 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Scalar, matrix, multiplication.' Response: Multiply the scalar $\\frac{13}{6}$ and the matrix\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n 9 & 3 & -3 \\\\\n -5 & -5 & -5 \\\\\n -7 & -3 & 1 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute\n$$\ne^\\left(\n\\begin{array}{cc}\n 1 & -1 \\\\\n 1 & -1 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, exponential function.' Response: Compute\n\n  \n\n$$\ne^\\left(\n\\begin{array}{cc}\n 1 & -1 \\\\\n 1 & -1 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Give a list of vectors that forms a basis for the null space of the following matrix:\n$$\n\\left(\n\\begin{array}{ccc}\n 4 & 7 & 3 \\\\\n -4 & 5 & -7 \\\\\n 9 & 8 & -4 \\\\\n 4 & 4 & -1 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, basis, null space, vectors.' Response: Give a list of vectors that forms a basis for the null space of the following matrix:\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n 4 & 7 & 3 \\\\\n -4 & 5 & -7 \\\\\n 9 & 8 & -4 \\\\\n 4 & 4 & -1 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the cross product of the following vectors $a$ and $g$:\n$\\left( \\begin{array}{c} -9 \\\\ 3 \\\\ 7 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} 6 \\\\ 0 \\\\ 3 \\\\ \\end{array} \\right)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Cross product, vectors, components.' Response: Find the cross product of the following vectors $\\vec{a}$ and $\\vec{g}$:\n\n  \n\n$\\left( \\begin{array}{c} -9 \\\\ 3 \\\\ 7 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} 6 \\\\ 0 \\\\ 3 \\\\ \\end{array} \\right)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Multiply the scalar $-\\frac{7}{4}$ and the matrix\n$$\n\\left( \\begin{array}{cccc}\n -8 & 3 & -1 & 0 \\\\\n 8 & -6 & 8 & -6 \\\\\n -6 & -6 & 1 & -6 \\\\\n\\end{array}\n\\right)\n$$\n\n[embedded]' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Scalar, matrix, multiplication.' Response: Multiply the scalar $-\\frac{7}{4}$ and the matrix\n\n  \n\n$$\n\\left( \\begin{array}{cccc}\n -8 & 3 & -1 & 0 \\\\\n 8 & -6 & 8 & -6 \\\\\n -6 & -6 & 1 & -6 \\\\\n\\end{array}\n\\right)\n$$\n\n  \n\n[embedded]\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Add the two matrices\n$\\left( \\begin{array}{c} -3 \\\\ \\frac{8}{5} \\\\ 9 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} \\frac{22}{5} \\\\ \\frac{19}{5} \\\\ \\frac{18}{5} \\\\ \\end{array} \\right)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrices, addition.' Response: Add the two matrices\n$\\left( \\begin{array}{c} -3 \\\\ \\frac{8}{5} \\\\ 9 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} \\frac{22}{5} \\\\ \\frac{19}{5} \\\\ \\frac{18}{5} \\\\ \\end{array} \\right)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Compute the trace of\n$$\n\\left(\n\\begin{array}{cccc}\n 6 & 1 & -10 & -7 \\\\\n -10 & 4 & -8 & 1 \\\\\n -8 & -9 & 1 & 9 \\\\\n -10 & 8 & -7 & 3 \\\\\n\\end{array}\n\\right)\n$$\n\n[embedded]' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, trace.' Response: Compute the trace of\n\n  \n\n$$\n\\left(\n\\begin{array}{cccc}\n 6 & 1 & -10 & -7 \\\\\n -10 & 4 & -8 & 1 \\\\\n -8 & -9 & 1 & 9 \\\\\n -10 & 8 & -7 & 3 \\\\\n\\end{array}\n\\right)\n$$\n\n  \n\n[embedded]\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the characteristic polynomial of the following matrix:\n$$\n\\left(\n\\begin{array}{cc}\n -3 & -\\frac{4}{5} \\\\\n -\\frac{3}{5} & \\frac{18}{5} \\\\\n\\end{array}\n\\right)\n$$\n[embedded]' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Characteristic polynomial, matrix.' Response: Find the characteristic polynomial of the following matrix:\n\n  \n\n$$\n\\left(\n\\begin{array}{cc}\n -3 & -\\frac{4}{5} \\\\\n -\\frac{3}{5} & \\frac{18}{5} \\\\\n\\end{array}\n\\right)\n$$\n\n  \n\n[embedded]\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the least squares vector given the matrix\n$\\left( \\begin{array}{cc} 1 & 1 \\\\ 0 & -2 \\\\ 1 & -2 \\\\ -3 & 0 \\\\ \\end{array} \\right)$ and the vector $\\left( \\begin{array}{c} -2.65 \\\\ 2.69 \\\\ -2.86 \\\\ -0.45 \\\\ \\end{array} \\right)$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Least squares, vector, matrix, solution.' Response: Find the least squares vector given the matrix\n\n  \n\n$\\left( \\begin{array}{cc} 1 & 1 \\\\ 0 & -2 \\\\ 1 & -2 \\\\ -3 & 0 \\\\ \\end{array} \\right)$ and the vector $\\left( \\begin{array}{c} -2.65 \\\\ 2.69 \\\\ -2.86 \\\\ -0.45 \\\\ \\end{array} \\right)$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Convert the following matrix to reduced row echelon form:\n$$\n\\left(\n\\begin{array}{ccccc}\n 10 & -8 & 6 & 0 & -1 \\\\\n -1 & 1 & -1 & -1 & -2 \\\\\n -4 & -1 & -5 & 0 & -10 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, reduced row echelon form.' Response: Convert the following matrix to reduced row echelon form:\n\n  \n\n$$\n\\left(\n\\begin{array}{ccccc}\n 10 & -8 & 6 & 0 & -1 \\\\\n -1 & 1 & -1 & -1 & -2 \\\\\n -4 & -1 & -5 & 0 & -10 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the dot product of the following two vectors m and n:\n$\\left( \\begin{array}{c} 3 \\sqrt{2} \\\\ -6 \\sqrt{2} \\\\ -\\sqrt{2} \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} -\\sqrt{2} \\\\ 0 \\\\ -\\frac{9}{\\sqrt{2}} \\\\ \\end{array} \\right)$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Dot product, vectors, components, calculation.' Response: Find the dot product of the following two vectors $\\vec{m}$ and $\\vec{n}$:\n\n  \n\n$\\left( \\begin{array}{c} 3 \\sqrt{2} \\\\ -6 \\sqrt{2} \\\\ -\\sqrt{2} \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{c} -\\sqrt{2} \\\\ 0 \\\\ -\\frac{9}{\\sqrt{2}} \\\\ \\end{array} \\right)$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Multiply\n$\\left( \\begin{array}{ccc} 1 & 2 & 0 \\\\ -2 & 2 & -2 \\\\ -2 & -1 & 0 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{ccc} -3 & 2 & -1 \\\\ -1 & 2 & 2 \\\\ -1 & -1 & 2 \\\\ \\end{array} \\right)$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix multiplication, matrices.' Response: Multiply\n$\\left( \\begin{array}{ccc} 1 & 2 & 0 \\\\ -2 & 2 & -2 \\\\ -2 & -1 & 0 \\\\ \\end{array} \\right)$ and $\\left( \\begin{array}{ccc} -3 & 2 & -1 \\\\ -1 & 2 & 2 \\\\ -1 & -1 & 2 \\\\ \\end{array} \\right)$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the determinant of the matrix\n$$\n\\left(\n\\begin{array}{ccc}\n \\frac{7}{6} & \\frac{23}{6} & -\\frac{23}{6} \\\\\n -2 & -\\frac{1}{3} & -\\frac{11}{6} \\\\\n \\frac{5}{3} & \\frac{14}{3} & \\frac{3}{2} \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, determinant, fractions.' Response: Find the determinant of the matrix\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n \\frac{7}{6} & \\frac{23}{6} & -\\frac{23}{6} \\\\\n -2 & -\\frac{1}{3} & -\\frac{11}{6} \\\\\n \\frac{5}{3} & \\frac{14}{3} & \\frac{3}{2} \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the $\\ell_\\infty$ norm of the following vector $y$:\n$$\n\\left(\n\\begin{array}{c}\n \\frac{40}{7} \\\\\n -5 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'norm, vector, absolute value, maximum value.' Response: Find the $\\ell_\\infty$ norm of the following vector $\\vec{y}$:\n\n  \n\n$$\n\\left(\n\\begin{array}{c}\n \\frac{40}{7} \\\\\n -5 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the eigenvalues of the following matrix:\n$$\n\\left(\n\\begin{array}{cc}\n -\\frac{20}{3} & \\frac{11}{3} \\\\\n -7 & 8 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, eigenvalues.' Response: Find the eigenvalues of the following matrix:\n\n  \n\n$$\n\\left(\n\\begin{array}{cc}\n -\\frac{20}{3} & \\frac{11}{3} \\\\\n -7 & 8 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the eigenvalues of the following matrix (round your answer to three decimal places):\n$$\n\\left(\n\\begin{array}{ccc}\n -1 & 9 & 8 \\\\\n 1 & 10 & 2 \\\\\n 8 & -8 & -9 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Matrix, eigenvalues, round, three decimal places.' Response: Find the eigenvalues of the following matrix (round your answer to three decimal places):\n\n  \n\n$$\n\\left(\n\\begin{array}{ccc}\n -1 & 9 & 8 \\\\\n 1 & 10 & 2 \\\\\n 8 & -8 & -9 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the $\\ell_2$ norm of the following vector $x$:\n$$\n\\left(\n\\begin{array}{c}\n 1 \\\\\n -8 \\\\\n -7 \\\\\n -6 \\\\\n -4 \\\\\n 0 \\\\\n -6 \\\\\n\\end{array}\n\\right)\n$$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Euclidean norm, Norm, vector.' Response: Find the $\\ell_2$ norm of the following vector $\\vec{x}$:\n\n  \n\n$$\n\\left(\n\\begin{array}{c}\n 1 \\\\\n -8 \\\\\n -7 \\\\\n -6 \\\\\n -4 \\\\\n 0 \\\\\n -6 \\\\\n\\end{array}\n\\right)\n$$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the cross product of the following vectors $p$ and $q$:\n$\\left( \\begin{array}{c} -\\frac{39}{5} \\\\ \\frac{31}{10} \\\\ -\\frac{67}{10} \\\\ \\end{array}\n\\right)$ and $\\left( \\begin{array}{c} \\frac{9}{10} \\\\ \\frac{43}{5} \\\\ 3 \\\\ \\end{array} \\right)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Vectors, cross product, components.' Response: Find the cross product of the following vectors $\\vec{p}$ and $\\vec{q}$:\n$\\left( \\begin{array}{c} -\\frac{39}{5} \\\\ \\frac{31}{10} \\\\ -\\frac{67}{10} \\\\ \\end{array}\n\\right)$ and $\\left( \\begin{array}{c} \\frac{9}{10} \\\\ \\frac{43}{5} \\\\ 3 \\\\ \\end{array} \\right)$",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the gradient of the function f(x,y,z) where $f(x,y,z) = x^3$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Function, gradient, variables, cubic function.' Response: Find the gradient of the function $f(x,y,z)$ where $f(x,y,z) = x^3$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the laplacian of the function $f(x,y,z)$ where $f(x,y,z) = \\sqrt{\\frac{x}{y z}}$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Laplacian, function, square root, division, variables.' Response: Find the Laplacian of the function $f(x,y,z)$ where $f(x,y,z) = \\sqrt{\\frac{x}{yz}}$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the arclength of the function $f(x) = \\cosh (5x)$ on the interval $x = 4$ to $x = 4$.' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Function, arclength, interval, hyperbolic cosine, variable.' Response: Find the arclength of the function $f(x) = \\cosh (5x)$ on the interval $x = 4$ to $x = 4$.\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Differentiate the following function:\n$\\cos (4 x+9)-\\cos (8-2 x)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Function, differentiation, cosine, algebra, trigonometry, variables.' Response: Differentiate the following function:\n$\\cos (4 x+9)-\\cos (8-2 x)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the jacobian of the vector valued function $\\begin{pmatrix} f(x,y,z)\\\\ g(x,y,z) \\\\ h(x,y,z)\\end{pmatrix}$ where $f(x,y,z) = \\sin (x)$, $g(x,y,z) = \\tan (y+z)$, and $h(x,y,z) = \\frac{1}{z^2}$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Jacobian, vector-valued function, sine function, tangent function, reciprocal function, variables.' Response: Find the Jacobian of the vector-valued function $\\begin{pmatrix} f(x,y,z)\\\\ g(x,y,z) \\\\ h(x,y,z)\\end{pmatrix}$ where $f(x,y,z) = \\sin(x)$, $g(x,y,z) = \\tan(y+z)$, and $h(x,y,z) = \\frac{1}{z^2}$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Differentiate the following function:\n$e^{2 x^2+6}+\\log (4-2 x)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Function, differentiation, exponential function, logarithmic function, variables.' Response: Differentiate the following function:\n$e^{2 x^2+6}+\\log (4-2 x)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the divergence of the vector field $l = f(x,y,z) = \\uvec{h} + g(x,y,z)\\uvec{j} + h(x,y,z)\\uvec{k}$ where $f(x,y,z) = x$, $g(x,y,z) = \\cos ^{-1}\\left(y^2\\right)$, and $h(x,y,z) = \\cos (z)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Vector field, divergence, function, unit vectors, components, cosine function.' Response: Find the divergence of the vector field $\\vec{l} = f(x,y,z) = \\hat{\\bf{i}} + g(x,y,z)\\hat{\\bf{j}} + h(x,y,z)\\hat{\\bf{k}}$ where $f(x,y,z) = x$, $g(x,y,z) = \\cos^{-1}(y^2)$, and $h(x,y,z) = \\cos(z)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the second derivative of the following function:\n$e^{8 x-7}$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Function, second derivative, exponential function, variables.' Response: Find the second derivative of the following function:\n$e^{8x-7}$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the curl of the vector field $p = f(x,y,z) = \\uvec{i} + g(x,y,z)\\uvec{j} + h(x,y,z)\\uvec{j}$ where $f(x,y,z) = \\tanh ^{-1}(x y)$, $g(x,y,z) = \\frac{x}{z}$, and $h(x,y,z) = \\frac{1}{z}$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Vector field, curl, function, inverse hyperbolic tangent, division, reciprocal, unit vector, variables.' Response: Find the curl of the vector field $\\vec{p} = f(x,y,z) = \\hat{\\bf{i}} + g(x,y,z)\\hat{\\bf{j}} + h(x,y,z)\\hat{\\bf{k}}$ where $f(x,y,z) =\\tanh ^{-1}(xy)$, $g(x,y,z) = \\frac{x}{z}$, and $h(x,y,z) = \\frac{1}{z}$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find an indefinite integral (which can vary by a constant) of the following function:\n$\\frac{x^2 \\left(64 x \\left(x^3+2\\right) \\cot \\left(1-8 x^4\\right)+3\\right) \\csc \\left(1-8 x^4\\right)}{\\sqrt{2} \\sqrt{-x^3-2}}$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Indefinite integral, function, variable, constant, cotangent, cosecant, square root.' Response: Find an indefinite integral (which can vary by a constant) of the following function:\n$\\frac{x^2 \\left(64 x \\left(x^3+2\\right) \\cot \\left(1-8 x^4\\right)+3\\right) \\csc \\left(1-8 x^4\\right)}{\\sqrt{2} \\sqrt{-x^3-2}}$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: '\n[embedded]\n\nFind the first order expansion about $x = -1$ of the composition $f(g(x))$ for $f(x) = e^{x^2}$ and $g(t) = $e^x$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'First order expansion, composition, function, exponential function, variable.' Response: [embedded]\n\n  \n\nFind the first order expansion about $x = -1$ of the composition$f(g(x))$ for $f(x) = e^{x^2}$ and $g(x) = e^x$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the second order series of the inverse of the following function around 2:\n$\\frac{9 x^2}{4}-e^{-7 x/2}$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Function, second order series, inverse function, point, expansion point, variable.' Response: Find the second order series of the inverse of the following function around $2$:\n$\\frac{9x^2}{4}-e^{-\\frac{7x}{2}}$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Find the range of the following function:\n$\\cos (5 x+6)$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Function, range, cosine function, argument, variable.' Response: Find the range of the following function:\n$\\cos(5x + 6)$\n",
"role": "assistant"
}
] |
[
{
"content": "Statement: 'Multiply and expand $p(j) = 7 x^2-15 x+8$ and $q(t) = 6 x^2+8 x+11$' List the concepts in the statement. With the concepts identified in mind, rewrite the statement, replacing existing notations, variables and units with the most conventional notations, most common variables and correct units in LaTex without paraphrasing the statement.",
"role": "user"
},
{
"content": "Related concepts: 'Polynomial, multiplication, expansion, variable.' Response: Multiply and expand $p(x) = 7x^2 - 15x + 8$ and $q(x) = 6x^2 + 8x + 11$.\n",
"role": "assistant"
}
] |
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