messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Compute the final answer to this equation. -876 + 867x = 54"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -876 + 867x = 54 $\n\nMove all terms to the left (standard form):\n$ 867x + -930 = 0 $\n\nIsolate the $x$ term:\n$ 867x = 930 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 800
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-4&5\\\\\\1&5\\end{pmatrix},\\quad b=\\begin{pmatrix}4\\\\\\24\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-4&5\\\\\\1&5\\end{p... | {
"type": "linear_algebra",
"seed": 1337,
"index": 801
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Evaluate: ((-61 + -440/20) * 2) + 6^4 - log_9(729) + (321 mod 5) + ceil(-64/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-61 + -440/20) * 2) + 6^4 - log_9(729) + (321 mod 5) + ceil(-64/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 802
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{... | {
"type": "optimization",
"seed": 1337,
"index": 803
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Solve for $x$ (in radians): sin(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 804
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. 82 - 537 = -721x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 82 - 537 = -721x $\n\nMove all terms to the left (standard form):\n$ 721x + -455 = 0 $\n\nIsolate the $x$ term:\n$ 721x = 455 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 805
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Compute the determinant: $A=\\begin{pmatrix}3&9\\\\7&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}3&9\\\\\\7&-7\\end{pmatrix} $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 806
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Prove that $f(x)=2x^2+-10x+3$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+-10x+3$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = ... | {
"type": "optimization",
"seed": 1337,
"index": 807
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. -261x + 259 = 268"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -261x + 259 = 268 $\n\nMove all terms to the left (standard form):\n$ -261x + -9 = 0 $\n\nIsolate the $x$ term:\n$ -261x = 9 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 808
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. -796x + 969 = -313"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -796x + 969 = -313 $\n\nMove all terms to the left (standard form):\n$ -796x + 1282 = 0 $\n\nIsolate the $x$ term:\n$ -796x = -1282 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 809
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\... | {
"type": "optimization",
"seed": 1337,
"index": 810
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. 628 - 828x = -171"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 628 - 828x = -171 $\n\nMove all terms to the left (standard form):\n$ -828x + 799 = 0 $\n\nIsolate the $x$ term:\n$ -828x = -799 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 811
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 812
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Find the modular inverse of 31 modulo 40, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $31\\cdot a^{-1} \\equiv 1 \\pmod{40}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees in... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 813
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Compute the indefinite integral: $\\int (1x + 9)e^{5x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (1x + 9)e^{5x}\\,dx $\n\nUse standard antiderivatives (and integration by part... | {
"type": "calculus",
"seed": 1337,
"index": 814
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}1&2\\\\9&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 1 & 2 \\\\ 9 & 9 \\end{pmatrix} $\n\nCompute the dete... | {
"type": "linear_algebra",
"seed": 1337,
"index": 815
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-2&3&-4\\\\\\-5&2&0\\\\\\3&-4&1\\end{pmatrix},\\quad b=\\begin{pmatrix}-18\\\\\\-24\\\\\\21\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $A... | {
"type": "linear_algebra",
"seed": 1337,
"index": 816
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Solve for $x$ (in radians): cos(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\... | {
"type": "trigonometry",
"seed": 1337,
"index": 817
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. -462 + 131x = 462"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -462 + 131x = 462 $\n\nMove all terms to the left (standard form):\n$ 131x + -924 = 0 $\n\nIsolate the $x$ term:\n$ 131x = 924 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 818
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. -31 + 115x^2 - 98 = -194x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -31 + 115x^2 - 98 = -194x $\n\nMove all terms to the left (standard form):\n$ 115x^2 + 194x + -129 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 819
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-5&3\\\\\\4&-9\\end{pmatrix},\\quad b=\\begin{pmatrix}54\\\\\\-96\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-5&3\\\\\\4&... | {
"type": "linear_algebra",
"seed": 1337,
"index": 820
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Solve for $x$ (in radians): cos(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\ar... | {
"type": "trigonometry",
"seed": 1337,
"index": 821
} |
[
{
"role": "user",
"content": "Solve the following math equation. Evaluate: ((-34 + 135/9) * 20) + 2^3 - log_8(32768) + (-437 mod 23) + ceil(12/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-34 + 135/9) * 20) + 2^3 - log_8(32768) + (-437 mod 23) + ceil(12/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 822
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}4&-5&1\\\\\\5&-2&5\\\\\\5&-5&0\\end{pmatrix},\\quad b=\\begin{pmatrix}19\\\\\\11\\\\\\20\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax... | {
"type": "linear_algebra",
"seed": 1337,
"index": 823
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Prove that $f(x,y)=\\frac12(10x^2+2y^2)+-6x+5y+3$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(10x^2+2y^2)+-6x+5y+3 $\n\n$ \\frac{\\partial^2... | {
"type": "optimization",
"seed": 1337,
"index": 824
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. 19x + 8 - 85x^2 = -9"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 19x + 8 - 85x^2 = -9 $\n\nMove all terms to the left (standard form):\n$ -85x^2 + 19x + 17 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 825
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Prove that $f(x)=7x^2+9x+10$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+9x+10$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x... | {
"type": "optimization",
"seed": 1337,
"index": 826
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Prove that $f(x)=10x^2+-9x+6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=10x^2+-9x+6$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ ... | {
"type": "optimization",
"seed": 1337,
"index": 827
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Compute $(77+-87) \\bmod 9$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (77+-87) \\bmod 9 $\n\nUse the rule that addition is compatible with reduction m... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 828
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. -40 - 154x + 39 = 34x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -40 - 154x + 39 = 34x^2 $\n\nMove all terms to the left (standard form):\n$ -34x^2 + -154x + -1 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 829
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}0&-2\\\\6&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 0 & -2 \\\\ 6 & 6 \\end{pmatrix} $\n\nCompute the ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 830
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Solve for $x$ (in radians): cos(x) = -\\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -\\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\... | {
"type": "trigonometry",
"seed": 1337,
"index": 831
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Evaluate: ((196 + 945/21) * 13) + 8^4 - log_7(7) + (252 mod 4) + ceil(-137/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((196 + 945/21) * 13) + 8^4 - log_7(7) + (252 mod 4) + ceil(-137/2) $\n\nCompute eac... | {
"type": "arithmetic",
"seed": 1337,
"index": 832
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}1&-4&-4\\\\\\1&5&-1\\\\\\5&-1&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}16\\\\\\-8\\\\\\8\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $A... | {
"type": "linear_algebra",
"seed": 1337,
"index": 833
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Prove that $f(x)=2x^2+7x+5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+7x+5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 4x... | {
"type": "optimization",
"seed": 1337,
"index": 834
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. -85 - 952x = -7"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -85 - 952x = -7 $\n\nMove all terms to the left (standard form):\n$ -952x + -78 = 0 $\n\nIsolate the $x$ term:\n$ -952x = 78 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 835
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Compute $(-160+195) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-160+195) \\bmod 8 $\n\nUse the rule that addition is compatible with reduction modul... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 836
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Compute the determinant: $A=\\begin{pmatrix}4&6\\\\7&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}4&6\\\\\\7&5\\end{pmatrix} $\n\nUse $\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 837
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Prove that $f(x,y)=\\frac12(5x^2+1y^2)+2x+5y+6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(5x^2+1y^2)+2x+5y+6 $\n\n$ \\frac{\\partial^2 f}{\\partia... | {
"type": "optimization",
"seed": 1337,
"index": 838
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. 38 + 64 + 53x^2 = -141x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 38 + 64 + 53x^2 = -141x $\n\nMove all terms to the left (standard form):\n$ 53x^2 + 141x + 102 = 0 $\n\nApply the quadratic ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 839
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Evaluate: ((-175 + 494/19) * 11) + 11^5 - log_6(1296) + (133 mod 2) + floor(-171/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-175 + 494/19) * 11) + 11^5 - log_6(1296) + (133 mod 2) + floor(-171/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 840
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Solve for $x$ (in radians): sin(x) = -\\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\al... | {
"type": "trigonometry",
"seed": 1337,
"index": 841
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Evaluate: ((-32 + -18/3) * 3) + 6^5 - log_5(625) + (-12 mod 12) + round(-137/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-32 + -18/3) * 3) + 6^5 - log_5(625) + (-12 mod 12) + round(-137/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 842
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. 12 + 191 + 125x^2 = 20x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 12 + 191 + 125x^2 = 20x $\n\nMove all terms to the left (standard form):\n$ 125x^2 + -20x + 203 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 843
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. -16 - 113 + 128x^2 = -49x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -16 - 113 + 128x^2 = -49x $\n\nMove all terms to the left (standard form):\n$ 128x^2 + 49x + -129 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 844
} |
[
{
"role": "user",
"content": "solve: \n\n Evaluate: ((-77 + -594/22) * 15) + 12^4 - log_10(10000) + (-57 mod 24) + floor(-139/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-77 + -594/22) * 15) + 12^4 - log_10(10000) + (-57 mod 24) + floor(-139/2) $\n\nCompute each compo... | {
"type": "arithmetic",
"seed": 1337,
"index": 845
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Prove that $f(x)=5x^2+9x+0$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+9x+0$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n... | {
"type": "optimization",
"seed": 1337,
"index": 846
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. -5x^2 + 36x + 22 = -74"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -5x^2 + 36x + 22 = -74 $\n\nMove all terms to the left (standard form):\n$ -5x^2 + 36x + 96 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 847
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Compute the determinant: $A=\\begin{pmatrix}4&0\\\\-5&4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}4&0\\\\\\-5&4\\end{pmatrix} $\n\nUse $\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 848
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. 962x + 290 = 79"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 962x + 290 = 79 $\n\nMove all terms to the left (standard form):\n$ 962x + 211 = 0 $\n\nIsolate the $x$ term:\n$ 962x = -211 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 849
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Evaluate: ((142 + 36/6) * 16) + 12^5 - log_7(117649) + (342 mod 9) + ceil(-116/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((142 + 36/6) * 16) + 12^5 - log_7(117649) + (342 mod 9) + ceil(-116/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 850
} |
[
{
"role": "user",
"content": "Please work out this equation. Prove that $f(x,y)=\\frac12(3x^2+7y^2)+2x+2y+6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+7y^2)+2x+2y+6 $\n\n$ \\frac{\\partial^2 f}{\\partial x^2} = ... | {
"type": "optimization",
"seed": 1337,
"index": 851
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Evaluate: ((10 + -828/23) * 10) + 6^4 - log_5(625) + (250 mod 18) + round(-198/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((10 + -828/23) * 10) + 6^4 - log_5(625) + (250 mod 18) + round(-198/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 852
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Differentiate with respect to $x$: $f(x) = 5x^3 + 2x^2 + 20x + -17$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 5x^3 + 2x^2 + 20x + -17 $\n\nApply the standard differentiation rules:\n$... | {
"type": "calculus",
"seed": 1337,
"index": 853
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Prove that $f(x,y)=\\frac12(2x^2+10y^2)+-1x+3y+5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(2x^2+10y^2)+-1x+3y+5 $\n\n$ \\frac{\\partial^2 f}{... | {
"type": "optimization",
"seed": 1337,
"index": 854
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Differentiate with respect to $x$: $f(x) = \\ln(5x + 6)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(5x + 6) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{5}{5... | {
"type": "calculus",
"seed": 1337,
"index": 855
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Compute the indefinite integral: $\\int (4x + -3)\\sin(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (4x + -3)\\sin(4x)\\,dx $\n\nUse standard antiderivatives (and inte... | {
"type": "calculus",
"seed": 1337,
"index": 856
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-2&4&1\\\\\\2&5&0\\\\\\1&1&4\\end{pmatrix},\\quad b=\\begin{pmatrix}13\\\\\\-1\\\\\\10\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 857
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Find the modular inverse of 13 modulo 24, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $13\\cdot a^{-1} \\equiv 1 \\pmod{24}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identit... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 858
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((-35 + -416/13) * 13) + 7^5 - log_3(243) + (-204 mod 19) + floor(-85/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-35 + -416/13) * 13) + 7^5 - log_3(243) + (-204 mod 19) + floo... | {
"type": "arithmetic",
"seed": 1337,
"index": 859
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Prove that $(\\mathbb{Z}_26, +)$ is a group under addition modulo 26."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_26=\\{0,1,\\dots,25\\}$ with operation $+$ modulo $26$.\n\n**Closure:** If $a,b\\in\\mathbb{... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 860
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Prove that $(\\mathbb{Z}_20, +)$ is a group under addition modulo 20."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_20=\\{0,1,\\dots,19\\}$ with operation $+$ modulo $20$.\n\n**Closure:** If $a,b\... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 861
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Find $A^{-1}$ for $A=\\begin{pmatrix}3&-3\\\\-5&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 3 & -3 \\\\ -5 & 6 \\end{pmatrix} $\n\nCompute the determi... | {
"type": "linear_algebra",
"seed": 1337,
"index": 862
} |
[
{
"role": "user",
"content": "Please work out this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}5&1\\\\\\-8&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}-21\\\\\\3\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}5&1\\\\\\-8&-5\\end{pmatrix}... | {
"type": "linear_algebra",
"seed": 1337,
"index": 863
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Prove that $(\\mathbb{Z}_28, +)$ is a group under addition modulo 28."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_28=\\{0,1,\\dots,27\\}$ with operation $+$ modulo $28$.\n\n**Closure:** If $a,b\... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 864
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}1&3&1\\\\\\3&-3&-1\\\\\\-2&4&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\-8\\\\\\6\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Sol... | {
"type": "linear_algebra",
"seed": 1337,
"index": 865
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Compute the indefinite integral: $\\int (1x + -3)\\sin(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (1x + -3)\\sin(4x)\\,dx $\n\nUse standard antiderivatives (and integration... | {
"type": "calculus",
"seed": 1337,
"index": 866
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. 34 - 10x = -725"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 34 - 10x = -725 $\n\nMove all terms to the left (standard form):\n$ -10x + 759 = 0 $\n\nIsolate the $x$ term:\n$ -10x = -759 $\n\nDivide both sid... | {
"type": "linear_equation",
"seed": 1337,
"index": 867
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. -16x - 29 + 22x^2 = 90"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -16x - 29 + 22x^2 = 90 $\n\nMove all terms to the left (standard form):\n$ 22x^2 + -16x + -119 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 868
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Compute the indefinite integral: $\\int \\ln(4x + 9)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(4x + 9)\\,dx $\n\nUse standard antiderivatives (and integration by parts if n... | {
"type": "calculus",
"seed": 1337,
"index": 869
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. 331x + 925 = 321"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 331x + 925 = 321 $\n\nMove all terms to the left (standard form):\n$ 331x + 604 = 0 $\n\nIsolate the $x$ term:\n$ 331x = -604 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 870
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Evaluate: ((-20 + -84/6) * 6) + 2^3 - log_8(32768) + (-400 mod 12) + floor(110/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-20 + -84/6) * 6) + 2^3 - log_8(32768) + (-400 mod 12) + floor(110/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 871
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Prove that $(\\mathbb{Z}_3, +)$ is a group under addition modulo 3."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_3=\\{0,1,\\dots,2\\}$ with operation $+$ modulo $3$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 872
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Prove that $f(x,y)=\\frac12(6x^2+8y^2)+6x+-6y+-4$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(6x^2+8y^2)+6x+-6y+-4 $\n\n$ \\frac{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 873
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Differentiate with respect to $x$: $f(x) = -7e^{-5x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -7e^{-5x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{35e^{-5x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 874
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Compute the determinant: $A=\\begin{pmatrix}1&1\\\\-5&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}1&1\\\\\\-5&6\\end{pmatrix} $\n\nUse $\\det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 875
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Evaluate: ((-164 + 48/12) * 5) + 4^3 - log_4(4) + (282 mod 24) + ceil(38/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-164 + 48/12) * 5) + 4^3 - log_4(4) + (282 mod 24) + ceil(38/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 876
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. Prove that $f(x)=6x^2+-2x+8$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=6x^2+-2x+8$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x... | {
"type": "optimization",
"seed": 1337,
"index": 877
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Find $A^{-1}$ for $A=\\begin{pmatrix}-4&0\\\\2&-3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -4 & 0 \\\\ 2 & -3 \\end{pmatrix} $\n\nCompute the determina... | {
"type": "linear_algebra",
"seed": 1337,
"index": 878
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Prove that $(\\mathbb{Z}_20, +)$ is a group under addition modulo 20."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_20=\\{0,1,\\dots,19\\}$ with operation $+$ modulo $20$.\n\n**Closure:** If $a,b\\in\\math... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 879
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Prove that $(\\mathbb{Z}_24, +)$ is a group under addition modulo 24."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_24=\\{0,1,\\dots,23\\}$ with operation $+$ modulo $24$.\n\n**Closure:** If $a,b\\i... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 880
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Solve for $x$ (in radians): cos(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\... | {
"type": "trigonometry",
"seed": 1337,
"index": 881
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. -7x - 113 - 181 = 108x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -7x - 113 - 181 = 108x^2 $\n\nMove all terms to the left (standard form):\n$ -108x^2 + -7x + -294 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 882
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Find $A^{-1}$ for $A=\\begin{pmatrix}-9&0\\\\6&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -9 & 0 \\\\ 6 & -5 \\end{pmatrix} $\n\nCompute the determina... | {
"type": "linear_algebra",
"seed": 1337,
"index": 883
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Evaluate: ((175 + 672/16) * 1) + 4^3 - log_2(8) + (-359 mod 12) + floor(-62/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((175 + 672/16) * 1) + 4^3 - log_2(8) + (-359 mod 12) + floor(-62/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 884
} |
[
{
"role": "user",
"content": "solve Differentiate with respect to $x$: $f(x) = (2x + -9)e^{5x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (2x + -9)e^{5x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{2e^{5x} + (2x + -9)\\cdot 5e^{5x}} $"
... | {
"type": "calculus",
"seed": 1337,
"index": 885
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Find $A^{-1}$ for $A=\\begin{pmatrix}-6&7\\\\8&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & 7 \\\\ 8 & 5 \\end{pmatrix} $\n\nCompute t... | {
"type": "linear_algebra",
"seed": 1337,
"index": 886
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Differentiate with respect to $x$: $f(x) = \\ln(4x + 9)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(4x + 9) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{4}... | {
"type": "calculus",
"seed": 1337,
"index": 887
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-1&-2&-1\\\\\\-1&3&3\\\\\\1&-5&1\\end{pmatrix},\\quad b=\\begin{pmatrix}0\\\\\\5\\\\\\-7\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 888
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence... | {
"type": "optimization",
"seed": 1337,
"index": 889
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{... | {
"type": "optimization",
"seed": 1337,
"index": 890
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Evaluate: ((-10 + -338/13) * 17) + 6^2 - log_6(46656) + (317 mod 3) + round(17/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-10 + -338/13) * 17) + 6^2 - log_6(46656) + (317 mod 3) + round(17... | {
"type": "arithmetic",
"seed": 1337,
"index": 891
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Prove that $(\\mathbb{Z}_30, +)$ is a group under addition modulo 30."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_30=\\{0,1,\\dots,29\\}$ with operation $+$ modulo $30$.\n\n**Closure:** If $a,b\\in\\mat... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 892
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Evaluate: ((-6 + -616/22) * 9) + 12^5 - log_10(10000) + (-246 mod 13) + floor(99/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-6 + -616/22) * 9) + 12^5 - log_10(10000) + (-246 mod 13) + floor(9... | {
"type": "arithmetic",
"seed": 1337,
"index": 893
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Evaluate: ((91 + 644/23) * 1) + 10^5 - log_5(625) + (417 mod 2) + ceil(-59/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((91 + 644/23) * 1) + 10^5 - log_5(625) + (417 mod 2) + ceil(-59/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 894
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Differentiate with respect to $x$: $f(x) = (1x + -1)e^{3x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (1x + -1)e^{3x} $\n\nApply the standard differentiation rules:\n$ f'(x) =... | {
"type": "calculus",
"seed": 1337,
"index": 895
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Compute $(-12+-200) \\bmod 10$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-12+-200) \\bmod 10 $\n\nUse the rule that addition is compatible with ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 896
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. 327 - 705x = -764"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 327 - 705x = -764 $\n\nMove all terms to the left (standard form):\n$ -705x + 1091 = 0 $\n\nIsolate the $x$ term:\n$ -705x = -1091 $\n\nDiv... | {
"type": "linear_equation",
"seed": 1337,
"index": 897
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. 41x - 125 + 95x^2 = -33"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 41x - 125 + 95x^2 = -33 $\n\nMove all terms to the left (standard form):\n$ 95x^2 + 41x + -92 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 898
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Prove that $f(x,y)=\\frac12(1x^2+10y^2)+3x+-2y+1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+10y^2)+3x+-2y+1 $\n\n$ \\frac{\\partial^2 f}{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 899
} |
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