messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Find the solution to this math problem. 9 - 124x^2 + 122x = 157"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 9 - 124x^2 + 122x = 157 $\n\nMove all terms to the left (standard form):\n$ -124x^2 + 122x + -148 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 900
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. -849 - 769 = 942x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -849 - 769 = 942x $\n\nMove all terms to the left (standard form):\n$ -942x + -1618 = 0 $\n\nIsolate the $x$ term:\n$ -942x = 1618 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 901
} |
[
{
"role": "user",
"content": "Please determine the value of 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 $f'(x)=0$, henc... | {
"type": "optimization",
"seed": 1337,
"index": 902
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. 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\\in\\mathb... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 903
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n Compute the indefinite integral: $\\int 1x^3 + -9x^2 + -15x + -14\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1x^3 + -9x^2 + -15x + -14\\,dx $\n\nUse standard antiderivatives ... | {
"type": "calculus",
"seed": 1337,
"index": 904
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Differentiate with respect to $x$: $f(x) = 7\\sin(1x) + 8\\cos(9x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 7\\sin(1x) + 8\\cos(9x) $\n\nApply the standard differentiation rules:\n$ f... | {
"type": "calculus",
"seed": 1337,
"index": 905
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. -358 + 929 = -251x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -358 + 929 = -251x $\n\nMove all terms to the left (standard form):\n$ 251x + 571 = 0 $\n\nIsolate the $x$ term:\n$ 251x = -571 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 906
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Evaluate: ((-105 + 65/13) * 14) + 4^3 - log_2(4) + (418 mod 24) + floor(197/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-105 + 65/13) * 14) + 4^3 - log_2(4) + (418 mod 24) + floor(197/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 907
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Compute $(200+-120) \\bmod 21$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (200+-120) \\bmod 21 $\n\nUse the rule that addition is compatible with reduction modulo ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 908
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Evaluate: ((-4 + -750/15) * 17) + 4^2 - log_3(243) + (-246 mod 3) + floor(-13/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-4 + -750/15) * 17) + 4^2 - log_3(243) + (-246 mod 3) + floor(-13/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 909
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Differentiate with respect to $x$: $f(x) = -3\\sin(4x) + 8\\cos(2x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -3\\sin(4x) + 8\\cos(2x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\b... | {
"type": "calculus",
"seed": 1337,
"index": 910
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&3&-3\\\\\\2&-3&-2\\\\\\5&0&2\\end{pmatrix},\\quad b=\\begin{pmatrix}13\\\\\\-10\\\\\\-18\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 911
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Find the modular inverse of 16 modulo 25, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $16\\cdot a^{-1} \\equiv 1 \\pmod{25}$.\n\nSince $\\gcd(a,m)=1$, Bézout's iden... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 912
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Solve for $x$ (in radians): tan(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -1 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(-1) = -0.785398 $\n\nThe... | {
"type": "trigonometry",
"seed": 1337,
"index": 913
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Evaluate: ((141 + 968/22) * 2) + 9^5 - log_10(10000) + (127 mod 3) + floor(-45/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((141 + 968/22) * 2) + 9^5 - log_10(10000) + (127 mod 3) + floor(-45/... | {
"type": "arithmetic",
"seed": 1337,
"index": 914
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. -922 - 722 = 807x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -922 - 722 = 807x $\n\nMove all terms to the left (standard form):\n$ -807x + -1644 = 0 $\n\nIsolate the $x$ term:\n$ -807x = 1644 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 915
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. 585 - 38 = 197x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 585 - 38 = 197x $\n\nMove all terms to the left (standard form):\n$ -197x + 547 = 0 $\n\nIsolate the $x$ term:\n$ -197x = -547 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 916
} |
[
{
"role": "user",
"content": "Please determine the outcome of 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": 917
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. -43 + 37 + 30x^2 = -49x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -43 + 37 + 30x^2 = -49x $\n\nMove all terms to the left (standard form):\n$ 30x^2 + 49x + -6 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 918
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 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": 919
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Prove that $f(x,y)=\\frac12(6x^2+1y^2)+4x+-3y+3$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(6x^2+1y^2)+4x+-3y+3 $\n\n$ \\frac{\\partial^... | {
"type": "optimization",
"seed": 1337,
"index": 920
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Compute the indefinite integral: $\\int -2\\sin(7x) + 3\\cos(9x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -2\\sin(7x) + 3\\cos(9x)\\,dx $\n\nUse standard antiderivatives (and... | {
"type": "calculus",
"seed": 1337,
"index": 921
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Compute the determinant: $A=\\begin{pmatrix}6&3\\\\-9&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}6&3\\\\\\-9&-4\\end{pmatrix} $\n\nUse $\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 922
} |
[
{
"role": "user",
"content": "Please calculate this equation. Solve for $x$ (in radians): cos(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = 1 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(1) = 0 $\n\nThe general solutions... | {
"type": "trigonometry",
"seed": 1337,
"index": 923
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Evaluate: ((78 + 828/23) * 8) + 4^3 - log_2(4) + (132 mod 17) + ceil(-1/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((78 + 828/23) * 8) + 4^3 - log_2(4) + (132 mod 17) + ceil(-1/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 924
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Solve for $x$ (in radians): sin(x) = -\\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(... | {
"type": "trigonometry",
"seed": 1337,
"index": 925
} |
[
{
"role": "user",
"content": "Find the solution to the following 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 $f'(x)=0$, henc... | {
"type": "optimization",
"seed": 1337,
"index": 926
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Prove that $f(x,y)=\\frac12(5x^2+6y^2)+-2x+5y+5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(5x^2+6y^2)+-2x+5y+5 $\n\n$ \\frac{\\partial^2 f}{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 927
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute the indefinite integral: $\\int 3\\sin(2x) + -5\\cos(8x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 3\\sin(2x) + -5\\cos(8x)\\,dx $\n\nUse standard antiderivatives (and int... | {
"type": "calculus",
"seed": 1337,
"index": 928
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Evaluate: ((-107 + -1012/22) * 17) + 7^4 - log_4(4) + (-369 mod 3) + floor(73/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-107 + -1012/22) * 17) + 7^4 - log_4(4) + (-369 mod 3) + floor(73/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 929
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Evaluate: ((-52 + 481/13) * 14) + 9^5 - log_10(1000) + (334 mod 12) + ceil(-179/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-52 + 481/13) * 14) + 9^5 - log_10(1000) + (334 mod 12) + ceil(-179/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 930
} |
[
{
"role": "user",
"content": "Please work out this equation. Compute the indefinite integral: $\\int 5e^{-4x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 5e^{-4x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \\int 5e^... | {
"type": "calculus",
"seed": 1337,
"index": 931
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Evaluate: ((5 + 0/14) * 9) + 8^3 - log_10(1000) + (311 mod 27) + floor(-144/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((5 + 0/14) * 9) + 8^3 - log_10(1000) + (311 mod 27) + floor(-144/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 932
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Find the modular inverse of 25 modulo 29, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $25\\cdot a^{-1} \\equiv 1 \\pmod{29}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 933
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Compute $(-142+67) \\bmod 22$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-142+67) \\bmod 22 $\n\nUse the rule that addition is compatible with reduction ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 934
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Differentiate with respect to $x$: $f(x) = 1\\sin(7x) + -6\\cos(2x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 1\\sin(7x) + -6\\cos(2x) $\n\nApply the standard differentiation rules:\n$ f... | {
"type": "calculus",
"seed": 1337,
"index": 935
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Compute the indefinite integral: $\\int 6\\sin(7x) + 0\\cos(7x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 6\\sin(7x) + 0\\cos(7x)\\,dx $\n\nUse standard antiderivatives (and int... | {
"type": "calculus",
"seed": 1337,
"index": 936
} |
[
{
"role": "user",
"content": "Can you determine the solution of 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} & \\fra... | {
"type": "optimization",
"seed": 1337,
"index": 937
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Solve for $x$ (in radians): tan(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha =... | {
"type": "trigonometry",
"seed": 1337,
"index": 938
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Prove that $f(x)=3x^2+10x+-3$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=3x^2+10x+-3$.\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": 939
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. 293x - 151 = 190"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 293x - 151 = 190 $\n\nMove all terms to the left (standard form):\n$ 293x + -341 = 0 $\n\nIsolate the $x$ term:\n$ 293x = 341 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 940
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((180 + 646/19) * 16) + 6^5 - log_5(15625) + (366 mod 16) + round(-184/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((180 + 646/19) * 16) + 6^5 - log_5(15625) + (366 mod 16) + rou... | {
"type": "arithmetic",
"seed": 1337,
"index": 941
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Solve for $x$ (in radians): sin(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 0 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(0) = 0 $\n\nThe gen... | {
"type": "trigonometry",
"seed": 1337,
"index": 942
} |
[
{
"role": "user",
"content": "solve: 17x^2 - 150 - 188 = -115x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 17x^2 - 150 - 188 = -115x $\n\nMove all terms to the left (standard form):\n$ 17x^2 + 115x + -338 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\pm \\sqrt{b^... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 943
} |
[
{
"role": "user",
"content": "Solve this equation step by step. -45 - 61 - 51x^2 = 81x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -45 - 61 - 51x^2 = 81x $\n\nMove all terms to the left (standard form):\n$ -51x^2 + -81x + -106 = 0 $\n\nApply the quadratic formula:\n$ x = \\f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 944
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Compute the indefinite integral: $\\int -6x^3 + -6x^2 + -6x + 8\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -6x^3 + -6x^2 + -6x + 8\\,dx $\n\nUse standard antiderivatives (and integration ... | {
"type": "calculus",
"seed": 1337,
"index": 945
} |
[
{
"role": "user",
"content": "Please evaluate and compute 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{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 946
} |
[
{
"role": "user",
"content": "Solve the equation shown below. -546 + 637 = 835x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -546 + 637 = 835x $\n\nMove all terms to the left (standard form):\n$ -835x + 91 = 0 $\n\nIsolate the $x$ term:\n$ -835x = -91 $\n\nDivide both sides by t... | {
"type": "linear_equation",
"seed": 1337,
"index": 947
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-2&-4&-3\\\\\\-2&-4&-5\\\\\\5&4&2\\end{pmatrix},\\quad b=\\begin{pmatrix}10\\\\\\14\\\\\\-8\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 948
} |
[
{
"role": "user",
"content": "Find the numerical solution to 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": 949
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Prove that $f(x)=1x^2+1x+-1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+1x+-1$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\... | {
"type": "optimization",
"seed": 1337,
"index": 950
} |
[
{
"role": "user",
"content": "Calculate the result of the given 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).\n$ \\... | {
"type": "trigonometry",
"seed": 1337,
"index": 951
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Solve for $x$ (in radians): sin(x) = -\\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(-... | {
"type": "trigonometry",
"seed": 1337,
"index": 952
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Compute $(-90+-147) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-90+-147) \\bmod 8 $\n\nUse the rule that addition is compatible with reduction... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 953
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Prove that $f(x,y)=\\frac12(1x^2+6y^2)+3x+-2y+-5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+6y^2)+3x+-2y+-5 $\n\n$ \\frac{\\partial^2 f}{\\... | {
"type": "optimization",
"seed": 1337,
"index": 954
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. 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)... | {
"type": "trigonometry",
"seed": 1337,
"index": 955
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. -674x - 853 = 24"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -674x - 853 = 24 $\n\nMove all terms to the left (standard form):\n$ -674x + -877 = 0 $\n\nIsolate the $x$ term:\n$ -674x = 877 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 956
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Evaluate: ((8 + -108/9) * 14) + 6^3 - log_7(343) + (-60 mod 29) + ceil(45/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((8 + -108/9) * 14) + 6^3 - log_7(343) + (-60 mod 29) + ceil(45/2) $\n\nCompute e... | {
"type": "arithmetic",
"seed": 1337,
"index": 957
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? -654 + 864x = 110"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -654 + 864x = 110 $\n\nMove all terms to the left (standard form):\n$ 864x + -764 = 0 $\n\nIsolate the $x$ term:\n$ 864x = 764 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 958
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Prove that $(\\mathbb{Z}_22, +)$ is a group under addition modulo 22."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_22=\\{0,1,\\dots,21\\}$ with operation $+$ modulo $22$.\n\n**Closure:** If $a,b\\in\\math... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 959
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Compute the indefinite integral: $\\int 6e^{-4x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 6e^{-4x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if nee... | {
"type": "calculus",
"seed": 1337,
"index": 960
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Differentiate with respect to $x$: $f(x) = -1e^{4x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -1e^{4x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-4e^{4... | {
"type": "calculus",
"seed": 1337,
"index": 961
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Evaluate: ((100 + 329/7) * 17) + 12^2 - log_8(32768) + (-461 mod 12) + round(148/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((100 + 329/7) * 17) + 12^2 - log_8(32768) + (-461 mod 12) + round(148/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 962
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Prove that $f(x)=8x^2+10x+10$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+10x+10$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 16x +... | {
"type": "optimization",
"seed": 1337,
"index": 963
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Differentiate with respect to $x$: $f(x) = \\ln(9x + -6)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(9x + -6) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{9}... | {
"type": "calculus",
"seed": 1337,
"index": 964
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. Evaluate: ((-34 + 60/10) * 18) + 9^3 - log_8(64) + (46 mod 17) + ceil(162/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-34 + 60/10) * 18) + 9^3 - log_8(64) + (46 mod 17) + ceil(162/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 965
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Compute $(200+42) \\bmod 10$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (200+42) \\bmod 10 $\n\nUse the rule that addition is compatible with reductio... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 966
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Evaluate: ((139 + 184/23) * 14) + 11^5 - log_4(4) + (293 mod 25) + round(15/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((139 + 184/23) * 14) + 11^5 - log_4(4) + (293 mod 25) + round(15/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 967
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 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": 968
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Find the modular inverse of 3 modulo 5, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $3\\cdot a^{-1} \\equiv 1 \\pmod{5}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarant... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 969
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Evaluate: ((-66 + -578/17) * 1) + 8^2 - log_3(3) + (209 mod 4) + ceil(-18/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-66 + -578/17) * 1) + 8^2 - log_3(3) + (209 mod 4) + ceil(-18/2) $\n\nCompute each compo... | {
"type": "arithmetic",
"seed": 1337,
"index": 970
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. 33x + 25 - 95x^2 = -171"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 33x + 25 - 95x^2 = -171 $\n\nMove all terms to the left (standard form):\n$ -95x^2 + 33x + 196 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 971
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Solve for $x$ (in radians): sin(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 1 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(1) = 1.5708 $\n\nThe gene... | {
"type": "trigonometry",
"seed": 1337,
"index": 972
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Differentiate with respect to $x$: $f(x) = \\ln(8x + -6)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(8x + -6) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\f... | {
"type": "calculus",
"seed": 1337,
"index": 973
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Compute $(-112+-121) \\bmod 9$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-112+-121) \\bmod 9 $\n\nUse the rule that addition is compatible with reductio... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 974
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Differentiate with respect to $x$: $f(x) = \\ln(5x + 7)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(5x + 7) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{... | {
"type": "calculus",
"seed": 1337,
"index": 975
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}1&-2&-2\\\\\\-5&0&-1\\\\\\-5&2&5\\end{pmatrix},\\quad b=\\begin{pmatrix}-1\\\\\\-6\\\\\\0\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., by... | {
"type": "linear_algebra",
"seed": 1337,
"index": 976
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. Prove that $f(x)=2x^2+-5x+6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+-5x+6$.\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": 977
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Prove that $f(x,y)=\\frac12(2x^2+4y^2)+-6x+-1y+-6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(2x^2+4y^2)+-6x+-1y+-6 $\n\n$ \\frac{\\partial^... | {
"type": "optimization",
"seed": 1337,
"index": 978
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Evaluate: ((-193 + -112/8) * 16) + 9^4 - log_10(10) + (141 mod 17) + floor(-33/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-193 + -112/8) * 16) + 9^4 - log_10(10) + (141 mod 17) + floor(-3... | {
"type": "arithmetic",
"seed": 1337,
"index": 979
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Solve for $x$ (in radians): tan(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arct... | {
"type": "trigonometry",
"seed": 1337,
"index": 980
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Evaluate: ((-107 + -250/10) * 4) + 9^2 - log_2(4) + (70 mod 22) + floor(82/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-107 + -250/10) * 4) + 9^2 - log_2(4) + (70 mod 22) + floor(82/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 981
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. 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\\in\\math... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 982
} |
[
{
"role": "user",
"content": "Please find the solution of 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{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 983
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. -49x - 103 + 193 = -14x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -49x - 103 + 193 = -14x^2 $\n\nMove all terms to the left (standard form):\n$ 14x^2 + -49x + 90 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 984
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Evaluate: ((-52 + -84/4) * 16) + 3^3 - log_2(32) + (393 mod 7) + floor(-102/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-52 + -84/4) * 16) + 3^3 - log_2(32) + (393 mod 7) + floor(-102/2) $\n\nCompute each component:\n$ -84... | {
"type": "arithmetic",
"seed": 1337,
"index": 985
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Compute the determinant: $A=\\begin{pmatrix}-7&-7\\\\6&-9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-7&-7\\\\\\6&-9\\end{pmatrix} $\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 986
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. -791x + 623 = -478"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -791x + 623 = -478 $\n\nMove all terms to the left (standard form):\n$ -791x + 1101 = 0 $\n\nIsolate the $x$ term:\n$ -791x = -1101 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 987
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 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 = \\arccos(... | {
"type": "trigonometry",
"seed": 1337,
"index": 988
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Prove that $f(x)=1x^2+-5x+8$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+-5x+8$.\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": 989
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Find the modular inverse of 31 modulo 32, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $31\\cdot a^{-1} \\equiv 1 \\pmod{32}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarant... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 990
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. 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{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 991
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Compute the indefinite integral: $\\int 5\\sin(4x) + -5\\cos(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 5\\sin(4x) + -5\\cos(4x)\\,dx $\n\nUse standard antiderivatives (a... | {
"type": "calculus",
"seed": 1337,
"index": 992
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Differentiate with respect to $x$: $f(x) = -3\\sin(7x) + -4\\cos(7x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -3\\sin(7x) + -4\\cos(7x) $\n\nApply the standard differentiation rules:\n$ f'... | {
"type": "calculus",
"seed": 1337,
"index": 993
} |
[
{
"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": 994
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Compute the indefinite integral: $\\int -1x^3 + -3x^2 + -2x + 1\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -1x^3 + -3x^2 + -2x + 1\\,dx $\n\nUse standard antiderivatives (and i... | {
"type": "calculus",
"seed": 1337,
"index": 995
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Find the modular inverse of 21 modulo 25, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $21\\cdot a^{-1} \\equiv 1 \\pmod{25}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guara... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 996
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Compute the determinant: $A=\\begin{pmatrix}2&3\\\\8&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}2&3\\\\\\8&1\\end{pmatrix} $\n\nUse $\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 997
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Prove that $f(x)=1x^2+6x+10$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+6x+10$.\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": 998
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Evaluate: ((75 + 60/4) * 7) + 10^2 - log_3(81) + (-26 mod 2) + round(9/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((75 + 60/4) * 7) + 10^2 - log_3(81) + (-26 mod 2) + round(9/2) $\n\nCompute each compone... | {
"type": "arithmetic",
"seed": 1337,
"index": 999
} |
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