messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Please work through this equation and solve it. Prove that $(\\mathbb{Z}_15, +)$ is a group under addition modulo 15."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_15=\\{0,1,\\dots,14\\}$ with operation $+$ modulo $15$.\n\n**Closure:** If $a,b\\in\\ma... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4900
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. 360 - 15 = -215x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 360 - 15 = -215x $\n\nMove all terms to the left (standard form):\n$ 215x + 345 = 0 $\n\nIsolate the $x$ term:\n$ 215x = -345 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 4901
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. 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) = 3.14159 $\n\nThe ... | {
"type": "trigonometry",
"seed": 1337,
"index": 4902
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. -6 + 7 - 41x^2 = 8x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -6 + 7 - 41x^2 = 8x $\n\nMove all terms to the left (standard form):\n$ -41x^2 + -8x + 1 = 0 $\n\nApply the quadratic formula:\n$ x = \\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4903
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Evaluate: ((58 + 238/14) * 13) + 5^3 - log_10(1000000) + (273 mod 13) + floor(-112/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((58 + 238/14) * 13) + 5^3 - log_10(1000000) + (273 mod 13) + floor(-112/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 4904
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. 46x^2 + 97 + 176 = -193x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 46x^2 + 97 + 176 = -193x $\n\nMove all terms to the left (standard form):\n$ 46x^2 + 193x + 273 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4905
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Find the modular inverse of 5 modulo 6, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{6}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantee... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4906
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Find $A^{-1}$ for $A=\\begin{pmatrix}9&5\\\\8&-6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 9 & 5 \\\\ 8 & -6 \\end{pmatrix} $\n\nCompu... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4907
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? 12 + 67x - 33x^2 = 74"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 12 + 67x - 33x^2 = 74 $\n\nMove all terms to the left (standard form):\n$ -33x^2 + 67x + -62 = 0 $\n\nApply the quadratic formula:\n$ x = \\fr... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4908
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Find the modular inverse of 11 modulo 30, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $11\\cdot a^{-1} \\equiv 1 \\pmod{30}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guar... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4909
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. 958x + 389 = 767"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 958x + 389 = 767 $\n\nMove all terms to the left (standard form):\n$ 958x + -378 = 0 $\n\nIsolate the $x$ term:\n$ 958x = 378 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 4910
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Prove that $f(x)=7x^2+3x+6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+3x+6$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(... | {
"type": "optimization",
"seed": 1337,
"index": 4911
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Prove that $f(x)=9x^2+-2x+4$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=9x^2+-2x+4$.\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": 4912
} |
[
{
"role": "user",
"content": "Compute the value of 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": 4913
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. - + 161x^2 + 102x = -153"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ - + 161x^2 + 102x = -153 $\n\nMove all terms to the left (standard form):\n$ 161x^2 + 102x + 153 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4914
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Compute $(-173+197) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-173+197) \\bmod 8 $\n\nUse the rule that addition is compatible with reduction m... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4915
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Compute the indefinite integral: $\\int 9e^{-1x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 9e^{-1x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ ... | {
"type": "calculus",
"seed": 1337,
"index": 4916
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}2&-4&0\\\\\\-2&1&0\\\\\\-4&0&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\-4\\\\\\-13\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "S... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4917
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 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": 4918
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. -343 - 484x = -270"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -343 - 484x = -270 $\n\nMove all terms to the left (standard form):\n$ -484x + -73 = 0 $\n\nIsolate the $x$ term:\n$ -484x = 73 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 4919
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Evaluate: ((53 + -700/20) * 14) + 11^3 - log_9(9) + (-39 mod 10) + ceil(166/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((53 + -700/20) * 14) + 11^3 - log_9(9) + (-39 mod 10) + ceil(166/2) $\n\nCompute each co... | {
"type": "arithmetic",
"seed": 1337,
"index": 4920
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Evaluate: ((200 + -117/13) * 6) + 6^3 - log_6(36) + (-147 mod 24) + ceil(27/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((200 + -117/13) * 6) + 6^3 - log_6(36) + (-147 mod 24) + ceil(27/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 4921
} |
[
{
"role": "user",
"content": "solve: \n\n Evaluate: ((159 + 99/9) * 16) + 9^3 - log_7(49) + (27 mod 21) + round(165/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((159 + 99/9) * 16) + 9^3 - log_7(49) + (27 mod 21) + round(165/2) $\n\nCompute each component:\n$ 99/9 = 11 $... | {
"type": "arithmetic",
"seed": 1337,
"index": 4922
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. 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$ \\alph... | {
"type": "trigonometry",
"seed": 1337,
"index": 4923
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. -454 - 175 = -453x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -454 - 175 = -453x $\n\nMove all terms to the left (standard form):\n$ 453x + -629 = 0 $\n\nIsolate the $x$ term:\n$ 453x = 629 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 4924
} |
[
{
"role": "user",
"content": "Please calculate and resolve 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": 4925
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Compute the determinant: $A=\\begin{pmatrix}-7&3\\\\8&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-7&3\\\\\\8&3\\end{pmatrix} $\n\nUse ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4926
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Compute the determinant: $A=\\begin{pmatrix}-3&-8\\\\8&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-3&-8\\\\\\8&3\\end{pmatrix} $\n\nUse ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4927
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Differentiate with respect to $x$: $f(x) = -6e^{2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -6e^{2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-12e^{2x}} $"
}... | {
"type": "calculus",
"seed": 1337,
"index": 4928
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Find $A^{-1}$ for $A=\\begin{pmatrix}-9&5\\\\-5&-9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -9 & 5 \\\\ -5 & -9 \\end{pmatrix} $\n\nCompute t... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4929
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Compute the determinant: $A=\\begin{pmatrix}5&-2\\\\-9&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}5&-2\\\\\\-9&5\\end{pmatrix} $\n\nUse $\\det(A)=ad-bc$: \n$ \\det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4930
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. -42x - 135 + 131x^2 = 39"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -42x - 135 + 131x^2 = 39 $\n\nMove all terms to the left (standard form):\n$ 131x^2 + -42x + -174 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4931
} |
[
{
"role": "user",
"content": "Please evaluate and compute this 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": 4932
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Solve for $x$ (in radians): tan(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = 0 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(0) = 0 $\n\nThe general s... | {
"type": "trigonometry",
"seed": 1337,
"index": 4933
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&-4&5\\\\\\4&-2&1\\\\\\-2&1&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}20\\\\\\16\\\\\\-18\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4934
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n -112x + 148 = -66"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -112x + 148 = -66 $\n\nMove all terms to the left (standard form):\n$ -112x + 214 = 0 $\n\nIsolate the $x$ term:\n$ -112x = -214 $\... | {
"type": "linear_equation",
"seed": 1337,
"index": 4935
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Evaluate: ((-63 + -207/23) * 11) + 4^2 - log_2(4) + (40 mod 27) + round(-198/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-63 + -207/23) * 11) + 4^2 - log_2(4) + (40 mod 27) + round(-198/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 4936
} |
[
{
"role": "user",
"content": "Can you find the solution to 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": 4937
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Evaluate: ((-113 + 30/5) * 6) + 12^3 - log_6(6) + (20 mod 29) + round(-5/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-113 + 30/5) * 6) + 12^3 - log_6(6) + (20 mod 29) + round(-5/2) $\n\nCompute e... | {
"type": "arithmetic",
"seed": 1337,
"index": 4938
} |
[
{
"role": "user",
"content": "solve: Evaluate: ((-43 + -516/12) * 20) + 10^4 - log_10(1000) + (-258 mod 26) + floor(112/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-43 + -516/12) * 20) + 10^4 - log_10(1000) + (-258 mod 26) + floor(112/2) $\n\nCompute each component:\... | {
"type": "arithmetic",
"seed": 1337,
"index": 4939
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. 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$ \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 4940
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Differentiate with respect to $x$: $f(x) = 1\\sin(2x) + 4\\cos(2x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 1\\sin(2x) + 4\\cos(2x) $\n\nApply the standard differentiation rules:\n$ f'... | {
"type": "calculus",
"seed": 1337,
"index": 4941
} |
[
{
"role": "user",
"content": "solve: \n\n -619x + 488 = -590"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -619x + 488 = -590 $\n\nMove all terms to the left (standard form):\n$ -619x + 1078 = 0 $\n\nIsolate the $x$ term:\n$ -619x = -1078 $\n\nDivide both sides by the coefficient... | {
"type": "linear_equation",
"seed": 1337,
"index": 4942
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Prove that $f(x,y)=\\frac12(4x^2+10y^2)+3x+3y+1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(4x^2+10y^2)+3x+3y+1 $\n\n$ \\frac{\\partial^2 f}{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 4943
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. 29 - 102x - 68 = 157x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 29 - 102x - 68 = 157x^2 $\n\nMove all terms to the left (standard form):\n$ -157x^2 + -102x + -39 = 0 $\n\nApply the quadrat... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4944
} |
[
{
"role": "user",
"content": "Solve this equation and return 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).\n$... | {
"type": "trigonometry",
"seed": 1337,
"index": 4945
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 26x - 112x^2 - 55 = -153"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 26x - 112x^2 - 55 = -153 $\n\nMove all terms to the left (standard form):\n$ -112x^2 + 26x + 98 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4946
} |
[
{
"role": "user",
"content": "Compute the value of 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": 4947
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. -768x + 996 = 178"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -768x + 996 = 178 $\n\nMove all terms to the left (standard form):\n$ -768x + 818 = 0 $\n\nIsolate the $x$ term:\n$ -768x = -818 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 4948
} |
[
{
"role": "user",
"content": "Please calculate this equation. Prove that $f(x)=2x^2+-6x+-3$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+-6x+-3$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 4x + -6... | {
"type": "optimization",
"seed": 1337,
"index": 4949
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Evaluate: ((-150 + 108/4) * 18) + 12^3 - log_3(9) + (424 mod 9) + ceil(187/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-150 + 108/4) * 18) + 12^3 - log_3(9) + (424 mod 9) + ceil(187/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 4950
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Find $A^{-1}$ for $A=\\begin{pmatrix}-6&-3\\\\-9&-1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & -3 \\\\ -9 & -1 \\end{pmatrix} $\n\nCompute th... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4951
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Prove that $(\\mathbb{Z}_15, +)$ is a group under addition modulo 15."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_15=\\{0,1,\\dots,14\\}$ with operation $+$ modulo $15$.\n\n**Closure:** If $a,b\\in\\m... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4952
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Compute $(-67+85) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-67+85) \\bmod 8 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ (... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4953
} |
[
{
"role": "user",
"content": "Can you compute this equation? Prove that $(\\mathbb{Z}_23, +)$ is a group under addition modulo 23."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_23=\\{0,1,\\dots,22\\}$ with operation $+$ modulo $23$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, then ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4954
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Compute the indefinite integral: $\\int 2x^3 + 2x^2 + 1x + -9\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + 2x^2 + 1x + -9\\,dx $\n\nUse standard antiderivatives (and in... | {
"type": "calculus",
"seed": 1337,
"index": 4955
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Evaluate: ((-90 + -252/9) * 4) + 6^2 - log_9(729) + (-335 mod 29) + floor(-190/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-90 + -252/9) * 4) + 6^2 - log_9(729) + (-335 mod 29) + floor... | {
"type": "arithmetic",
"seed": 1337,
"index": 4956
} |
[
{
"role": "user",
"content": "Please solve 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(-\\frac{1... | {
"type": "trigonometry",
"seed": 1337,
"index": 4957
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. 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 $\n\nThe... | {
"type": "trigonometry",
"seed": 1337,
"index": 4958
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 48 - 929 = 176x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 48 - 929 = 176x $\n\nMove all terms to the left (standard form):\n$ -176x + -881 = 0 $\n\nIsolate the $x$ term:\n$ -176x = 881 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 4959
} |
[
{
"role": "user",
"content": "Evaluate the following equation. -665x - 482 = -864"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -665x - 482 = -864 $\n\nMove all terms to the left (standard form):\n$ -665x + 382 = 0 $\n\nIsolate the $x$ term:\n$ -665x = -382 $\n\nDivide both sides... | {
"type": "linear_equation",
"seed": 1337,
"index": 4960
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Prove that $f(x)=9x^2+-4x+-4$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=9x^2+-4x+-4$.\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": 4961
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. -12 - 38 - 115x = 105x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -12 - 38 - 115x = 105x^2 $\n\nMove all terms to the left (standard form):\n$ -105x^2 + -115x + -50 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4962
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-1&5&5\\\\\\5&5&4\\\\\\5&-4&1\\end{pmatrix},\\quad b=\\begin{pmatrix}-1\\\\\\6\\\\\\0\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4963
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Find the modular inverse of 25 modulo 28, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $25\\cdot a^{-1} \\equiv 1 \\pmod{28}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarante... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4964
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. -11x + 20 - 38 = -19x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -11x + 20 - 38 = -19x^2 $\n\nMove all terms to the left (standard form):\n$ 19x^2 + -11x + -18 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4965
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Evaluate: ((-170 + 170/5) * 17) + 2^2 - log_6(46656) + (239 mod 8) + round(144/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-170 + 170/5) * 17) + 2^2 - log_6(46656) + (239 mod 8) + round(144/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 4966
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Compute the indefinite integral: $\\int \\ln(9x + -2)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(9x + -2)\\,dx $\n\nUse standard antiderivatives (and integration by parts ... | {
"type": "calculus",
"seed": 1337,
"index": 4967
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Evaluate: ((187 + 336/24) * 13) + 10^4 - log_6(7776) + (313 mod 28) + ceil(-159/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((187 + 336/24) * 13) + 10^4 - log_6(7776) + (313 mod 28) + ceil(-159/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 4968
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}0&-8\\\\5&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 0 & -8 \\\\ 5 & -4 \\end{pmatrix} $\n\nCompute the det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4969
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. 46 - 19 - 74x^2 = 25x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 46 - 19 - 74x^2 = 25x $\n\nMove all terms to the left (standard form):\n$ -74x^2 + -25x + 27 = 0 $\n\nApply the quadratic formula:\n$ x = \\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4970
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Evaluate: ((-93 + -182/7) * 18) + 9^3 - log_2(8) + (-405 mod 23) + ceil(-112/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-93 + -182/7) * 18) + 9^3 - log_2(8) + (-405 mod 23) + ceil(-112/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 4971
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. 738 + 715x = 845"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 738 + 715x = 845 $\n\nMove all terms to the left (standard form):\n$ 715x + -107 = 0 $\n\nIsolate the $x$ term:\n$ 715x = 107 $\n\nDi... | {
"type": "linear_equation",
"seed": 1337,
"index": 4972
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. 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).... | {
"type": "trigonometry",
"seed": 1337,
"index": 4973
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&-5&-3\\\\\\-2&3&2\\\\\\5&-5&5\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\4\\\\\\-40\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4974
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. 726x + 120 = -762"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 726x + 120 = -762 $\n\nMove all terms to the left (standard form):\n$ 726x + 882 = 0 $\n\nIsolate the $x$ term:\n$ 726x = -882 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 4975
} |
[
{
"role": "user",
"content": "solve Find the modular inverse of 11 modulo 26, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $11\\cdot a^{-1} \\equiv 1 \\pmod{26}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integers $x,y$ with:\n$ 11x + 26... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4976
} |
[
{
"role": "user",
"content": "Determine the solution of the given 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 $... | {
"type": "trigonometry",
"seed": 1337,
"index": 4977
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Find the modular inverse of 4 modulo 33, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $4\\cdot a^{-1} \\equiv 1 \\pmod{33}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integers ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4978
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. 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 = \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 4979
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. -245 + 477 = -280x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -245 + 477 = -280x $\n\nMove all terms to the left (standard form):\n$ 280x + 232 = 0 $\n\nIsolate the $x$ term:\n$ 280x = -232 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 4980
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Compute $(141+147) \\bmod 15$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (141+147) \\bmod 15 $\n\nUse the rule that addition is compatible with reductio... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4981
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. 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$... | {
"type": "trigonometry",
"seed": 1337,
"index": 4982
} |
[
{
"role": "user",
"content": "solve Evaluate: ((105 + -144/12) * 11) + 8^3 - log_7(343) + (20 mod 28) + round(4/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((105 + -144/12) * 11) + 8^3 - log_7(343) + (20 mod 28) + round(4/2) $\n\nCompute each component:\n$ -144/12 = -1... | {
"type": "arithmetic",
"seed": 1337,
"index": 4983
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. -12x^2 + 85x + 100 = 161"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -12x^2 + 85x + 100 = 161 $\n\nMove all terms to the left (standard form):\n$ -12x^2 + 85x + -61 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4984
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Compute the indefinite integral: $\\int -7e^{-1x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -7e^{-1x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):... | {
"type": "calculus",
"seed": 1337,
"index": 4985
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical 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} & \\... | {
"type": "optimization",
"seed": 1337,
"index": 4986
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. -3x^2 - 104 + 200x = -95"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -3x^2 - 104 + 200x = -95 $\n\nMove all terms to the left (standard form):\n$ -3x^2 + 200x + -9 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4987
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. 17x^2 + 126x - 124 = -7"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 17x^2 + 126x - 124 = -7 $\n\nMove all terms to the left (standard form):\n$ 17x^2 + 126x + -117 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4988
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. 18x + 3 + 163 = 110x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 18x + 3 + 163 = 110x^2 $\n\nMove all terms to the left (standard form):\n$ -110x^2 + 18x + 166 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4989
} |
[
{
"role": "user",
"content": "Please evaluate and solve 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:\n... | {
"type": "optimization",
"seed": 1337,
"index": 4990
} |
[
{
"role": "user",
"content": "Find the answer 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 $f'(x)=0$, hence:... | {
"type": "optimization",
"seed": 1337,
"index": 4991
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Prove that $(\\mathbb{Z}_29, +)$ is a group under addition modulo 29."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_29=\\{0,1,\\dots,28\\}$ with operation $+$ modulo $29$.\n\n**Closure:** If $a,b\\in\\mathbb{... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4992
} |
[
{
"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": 4993
} |
[
{
"role": "user",
"content": "solve: -39x^2 - 43 + 163x = -41"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -39x^2 - 43 + 163x = -41 $\n\nMove all terms to the left (standard form):\n$ -39x^2 + 163x + -2 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\pm \\sqrt{b^2 -... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4994
} |
[
{
"role": "user",
"content": "Can you compute 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 $a,b\\in\\mathbb{Z}_n$, then ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4995
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. -45x - 167 + 46 = -167x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -45x - 167 + 46 = -167x^2 $\n\nMove all terms to the left (standard form):\n$ 167x^2 + -45x + -121 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4996
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Compute the indefinite integral: $\\int 2x^3 + -1x^2 + -7x + -11\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + -1x^2 + -7x + -11\\,dx $\n\nUse standard antiderivatives (and... | {
"type": "calculus",
"seed": 1337,
"index": 4997
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute the indefinite integral: $\\int -1\\sin(9x) + 5\\cos(3x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -1\\sin(9x) + 5\\cos(3x)\\,dx $\n\nUse standard antiderivatives (and int... | {
"type": "calculus",
"seed": 1337,
"index": 4998
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Find $A^{-1}$ for $A=\\begin{pmatrix}-4&-5\\\\3&4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -4 & -5 \\\\ 3 & 4 \\end{pmatrix} $\n\nCompute the d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4999
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.