messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Determine the correct solution for the given 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": 2100
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Compute the determinant: $A=\\begin{pmatrix}7&4\\\\5&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}7&4\\\\\\5&5\\end{pmatrix} $\n\nUse $\\d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2101
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. -22 + 193x + 118 = -194x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -22 + 193x + 118 = -194x^2 $\n\nMove all terms to the left (standard form):\n$ 194x^2 + 193x + 96 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2102
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Prove that $f(x)=8x^2+1x+5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+1x+5$.\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": 2103
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Find the modular inverse of 9 modulo 10, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $9\\cdot a^{-1} \\equiv 1 \\pmod{10}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2104
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Prove that $f(x,y)=\\frac12(3x^2+3y^2)+3x+-5y+5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+3y^2)+3x+-5y+5 $\n\n$ \\frac{\\partial^2 f... | {
"type": "optimization",
"seed": 1337,
"index": 2105
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. -39 + 126x^2 + 86x = -68"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -39 + 126x^2 + 86x = -68 $\n\nMove all terms to the left (standard form):\n$ 126x^2 + 86x + 29 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2106
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 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\nT... | {
"type": "trigonometry",
"seed": 1337,
"index": 2107
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Differentiate with respect to $x$: $f(x) = -3x^3 + 9x^2 + -19x + -16$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -3x^3 + 9x^2 + -19x + -16 $\n\nApply the standard differentiation rul... | {
"type": "calculus",
"seed": 1337,
"index": 2108
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Evaluate: ((112 + -130/13) * 17) + 6^5 - log_7(2401) + (232 mod 18) + ceil(-104/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((112 + -130/13) * 17) + 6^5 - log_7(2401) + (232 mod 18) + ceil(-104/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 2109
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}4&3&4\\\\\\-4&5&2\\\\\\3&3&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}-22\\\\\\6\\\\\\-18\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2110
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. -180 - 732x = -742"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -180 - 732x = -742 $\n\nMove all terms to the left (standard form):\n$ -732x + 562 = 0 $\n\nIsolate the $x$ term:\n$ -732x = -562 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 2111
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. -4x - 138x^2 - 68 = 197"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -4x - 138x^2 - 68 = 197 $\n\nMove all terms to the left (standard form):\n$ -138x^2 + -4x + -265 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2112
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Prove that $(\\mathbb{Z}_27, +)$ is a group under addition modulo 27."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_27=\\{0,1,\\dots,26\\}$ with operation $+$ modulo $27$.\n\n**Closure:** If $a,b\\in\\math... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2113
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Compute the determinant: $A=\\begin{pmatrix}-3&9\\\\-2&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-3&9\\\\\\-2&6\\end{pmatrix}... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2114
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. -306 - 631 = -68x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -306 - 631 = -68x $\n\nMove all terms to the left (standard form):\n$ 68x + -937 = 0 $\n\nIsolate the $x$ term:\n$ 68x = 937 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2115
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Compute $(78+177) \\bmod 13$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (78+177) \\bmod 13 $\n\nUse the rule that addition is compatible with reduction modul... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2116
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. -8 + 15x - 178x^2 = -190"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -8 + 15x - 178x^2 = -190 $\n\nMove all terms to the left (standard form):\n$ -178x^2 + 15x + 182 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2117
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Differentiate with respect to $x$: $f(x) = \\ln(6x + -9)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(6x + -9) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{... | {
"type": "calculus",
"seed": 1337,
"index": 2118
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}6&0\\\\\\-8&-1\\end{pmatrix},\\quad b=\\begin{pmatrix}54\\\\\\-80\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}6&0\\\\\\-8&-1\\end{... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2119
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Evaluate: ((129 + 234/13) * 2) + 6^4 - log_2(64) + (52 mod 26) + ceil(154/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((129 + 234/13) * 2) + 6^4 - log_2(64) + (52 mod 26) + ceil(154/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 2120
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. -48x + 28 - 198x^2 = -41"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -48x + 28 - 198x^2 = -41 $\n\nMove all terms to the left (standard form):\n$ -198x^2 + -48x + 69 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2121
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((-170 + -374/17) * 4) + 10^3 - log_7(7) + (-349 mod 25) + round(-38/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-170 + -374/17) * 4) + 10^3 - log_7(7) + (-349 mod 25) + round(-38/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 2122
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. 22 + 183x - 60x^2 = 197"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 22 + 183x - 60x^2 = 197 $\n\nMove all terms to the left (standard form):\n$ -60x^2 + 183x + -175 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2123
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}4&-9\\\\\\9&1\\end{pmatrix},\\quad b=\\begin{pmatrix}45\\\\\\-5\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}4&-9\\\\\\9&1\\end{pmatrix} ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2124
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Compute the indefinite integral: $\\int (-4x + -6)\\sin(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-4x + -6)\\sin(4x)\\,dx $\n\nUse standard antiderivatives (and integration by part... | {
"type": "calculus",
"seed": 1337,
"index": 2125
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. 598 + 970x = 683"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 598 + 970x = 683 $\n\nMove all terms to the left (standard form):\n$ 970x + -85 = 0 $\n\nIsolate the $x$ term:\n$ 970x = 85 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 2126
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Compute the indefinite integral: $\\int 8e^{3x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 8e^{3x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if ne... | {
"type": "calculus",
"seed": 1337,
"index": 2127
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Solve for $x$ (in radians): tan(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ ... | {
"type": "trigonometry",
"seed": 1337,
"index": 2128
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Evaluate: ((-159 + 648/24) * 8) + 2^3 - log_8(4096) + (-362 mod 7) + ceil(-113/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-159 + 648/24) * 8) + 2^3 - log_8(4096) + (-362 mod 7) + ceil(-113/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 2129
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. 246x - 37 = 727"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 246x - 37 = 727 $\n\nMove all terms to the left (standard form):\n$ 246x + -764 = 0 $\n\nIsolate the $x$ term:\n$ 246x = 764 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 2130
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&-2&-5\\\\\\4&5&2\\\\\\4&1&-4\\end{pmatrix},\\quad b=\\begin{pmatrix}2\\\\\\1\\\\\\-3\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2131
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Evaluate: ((-177 + -230/5) * 10) + 5^2 - log_8(64) + (246 mod 11) + ceil(38/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-177 + -230/5) * 10) + 5^2 - log_8(64) + (246 mod 11) + ceil(38/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 2132
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Evaluate: ((130 + 946/22) * 2) + 12^2 - log_2(64) + (20 mod 9) + round(77/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((130 + 946/22) * 2) + 12^2 - log_2(64) + (20 mod 9) + round(77/2) $\n\nCompute each co... | {
"type": "arithmetic",
"seed": 1337,
"index": 2133
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. -252 + 90 = -97x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -252 + 90 = -97x $\n\nMove all terms to the left (standard form):\n$ 97x + -162 = 0 $\n\nIsolate the $x$ term:\n$ 97x = 162 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 2134
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Evaluate: ((-72 + 0/6) * 16) + 8^4 - log_5(15625) + (-300 mod 25) + floor(16/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-72 + 0/6) * 16) + 8^4 - log_5(15625) + (-300 mod 25) + floor(16/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 2135
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. 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(\\fr... | {
"type": "trigonometry",
"seed": 1337,
"index": 2136
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Evaluate: ((-114 + -162/6) * 4) + 4^2 - log_4(1024) + (-210 mod 23) + round(-27/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-114 + -162/6) * 4) + 4^2 - log_4(1024) + (-210 mod 23) + round(... | {
"type": "arithmetic",
"seed": 1337,
"index": 2137
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Prove that $f(x)=1x^2+-4x+-5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+-4x+-5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\... | {
"type": "optimization",
"seed": 1337,
"index": 2138
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. -21 + 80x + 153x^2 = -72"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -21 + 80x + 153x^2 = -72 $\n\nMove all terms to the left (standard form):\n$ 153x^2 + 80x + 51 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2139
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Find $A^{-1}$ for $A=\\begin{pmatrix}-8&-1\\\\4&4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -8 & -1 \\\\ 4 & 4 \\end{pmatrix} $\n\nC... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2140
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 298 - 499 = -997x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 298 - 499 = -997x $\n\nMove all terms to the left (standard form):\n$ 997x + -201 = 0 $\n\nIsolate the $x$ term:\n$ 997x = 201 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2141
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n -846 - 702x = 680"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -846 - 702x = 680 $\n\nMove all terms to the left (standard form):\n$ -702x + -1526 = 0 $\n\nIsolate the $x$ term:\n$ -702x = 1526 ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2142
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? 190x + 485 = 899"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 190x + 485 = 899 $\n\nMove all terms to the left (standard form):\n$ 190x + -414 = 0 $\n\nIsolate the $x$ term:\n$ 190x = 414 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2143
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Find $A^{-1}$ for $A=\\begin{pmatrix}-5&3\\\\-3&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -5 & 3 \\\\ -3 & 3 \\end{pmatrix} $\n\nCompute the determi... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2144
} |
[
{
"role": "user",
"content": "Please evaluate and compute 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 genera... | {
"type": "trigonometry",
"seed": 1337,
"index": 2145
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Compute the indefinite integral: $\\int (-3x + 3)\\sin(1x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-3x + 3)\\sin(1x)\\,dx $\n\nUse standard antiderivatives (and inte... | {
"type": "calculus",
"seed": 1337,
"index": 2146
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. -14 + 65x - 129x^2 = 14"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -14 + 65x - 129x^2 = 14 $\n\nMove all terms to the left (standard form):\n$ -129x^2 + 65x + -28 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2147
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Evaluate: ((-188 + -560/20) * 18) + 9^2 - log_3(3) + (-490 mod 10) + ceil(-198/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-188 + -560/20) * 18) + 9^2 - log_3(3) + (-490 mod 10) + ceil(-198/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 2148
} |
[
{
"role": "user",
"content": "Please solve this math problem. 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 general solutions... | {
"type": "trigonometry",
"seed": 1337,
"index": 2149
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Differentiate with respect to $x$: $f(x) = 9\\sin(8x) + -6\\cos(8x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 9\\sin(8x) + -6\\cos(8x) $\n\nApply the standard differentiation rules:\n$ f'... | {
"type": "calculus",
"seed": 1337,
"index": 2150
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Differentiate with respect to $x$: $f(x) = \\ln(3x + 6)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(3x + 6) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{3}{3x + 6}}... | {
"type": "calculus",
"seed": 1337,
"index": 2151
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Prove that $(\\mathbb{Z}_2, +)$ is a group under addition modulo 2."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_2=\\{0,1,\\dots,1\\}$ with operation $+$ modulo $2$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, then $(a... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2152
} |
[
{
"role": "user",
"content": "Solve the following math equation. Evaluate: ((159 + -154/11) * 5) + 7^2 - log_6(36) + (-201 mod 25) + floor(-173/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((159 + -154/11) * 5) + 7^2 - log_6(36) + (-201 mod 25) + floor(-173/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 2153
} |
[
{
"role": "user",
"content": "Solve this equation accurately. -172 + 830 = -212x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -172 + 830 = -212x $\n\nMove all terms to the left (standard form):\n$ 212x + 658 = 0 $\n\nIsolate the $x$ term:\n$ 212x = -658 $\n\nDivide both sides by... | {
"type": "linear_equation",
"seed": 1337,
"index": 2154
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Differentiate with respect to $x$: $f(x) = (1x + -6)\\sin(1x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (1x + -6)\\sin(1x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{... | {
"type": "calculus",
"seed": 1337,
"index": 2155
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Compute the determinant: $A=\\begin{pmatrix}2&-3\\\\8&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}2&-3\\\\\\8&6\\end{pmatrix} $\n\nUse... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2156
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Prove that $(\\mathbb{Z}_16, +)$ is a group under addition modulo 16."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_16=\\{0,1,\\dots,15\\}$ with operation $+$ modulo $16$.\n\n**Closure:** If $a,b\\in\\mat... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2157
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. -650x - 566 = 447"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -650x - 566 = 447 $\n\nMove all terms to the left (standard form):\n$ -650x + -1013 = 0 $\n\nIsolate the $x$ term:\n$ -650x = 1013 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 2158
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Evaluate: ((-30 + -176/22) * 7) + 2^2 - log_4(4) + (-444 mod 7) + ceil(-88/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-30 + -176/22) * 7) + 2^2 - log_4(4) + (-444 mod 7) + ceil(-88/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 2159
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: 21 + 193x + 146x^2 = -118"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 21 + 193x + 146x^2 = -118 $\n\nMove all terms to the left (standard form):\n$ 146x^2 + 193x + 139 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\pm \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2160
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Find the modular inverse of 5 modulo 37, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{37}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2161
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Evaluate: ((198 + 14/14) * 7) + 10^3 - log_5(625) + (-54 mod 17) + ceil(-17/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((198 + 14/14) * 7) + 10^3 - log_5(625) + (-54 mod 17) + ceil(-17/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 2162
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. -26x^2 + 55 - 137 = 45x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -26x^2 + 55 - 137 = 45x $\n\nMove all terms to the left (standard form):\n$ -26x^2 + -45x + -82 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2163
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? 12 - 148 - 154x = -20x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 12 - 148 - 154x = -20x^2 $\n\nMove all terms to the left (standard form):\n$ 20x^2 + -154x + -136 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2164
} |
[
{
"role": "user",
"content": "solve: Find $A^{-1}$ for $A=\\begin{pmatrix}9&-2\\\\0&-3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 9 & -2 \\\\ 0 & -3 \\end{pmatrix} $\n\nCompute the determinant: $\\det(A)=ad-bc$.\n$ \... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2165
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. -40x - 14 - 10 = -146x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -40x - 14 - 10 = -146x^2 $\n\nMove all terms to the left (standard form):\n$ 146x^2 + -40x + -24 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2166
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. 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": 2167
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Differentiate with respect to $x$: $f(x) = 3x^3 + 3x^2 + -1x + -11$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 3x^3 + 3x^2 + -1x + -11 $\n\nApply the standard differentiation rules:\n$ f'(x) = ... | {
"type": "calculus",
"seed": 1337,
"index": 2168
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Compute the indefinite integral: $\\int \\ln(9x + -3)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(9x + -3)\\,dx $\n\nUse standard antiderivatives (and integration by parts... | {
"type": "calculus",
"seed": 1337,
"index": 2169
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Find $A^{-1}$ for $A=\\begin{pmatrix}8&-6\\\\2&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 8 & -6 \\\\ 2 & 9 \\end{pmatrix} $\n\nCo... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2170
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. -11x - 85x^2 + 43 = -73"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -11x - 85x^2 + 43 = -73 $\n\nMove all terms to the left (standard form):\n$ -85x^2 + -11x + 116 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2171
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Compute the determinant: $A=\\begin{pmatrix}3&-3\\\\-8&4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}3&-3\\\\\\-8&4\\end{pmatri... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2172
} |
[
{
"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": 2173
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Compute the determinant: $A=\\begin{pmatrix}-8&0\\\\-5&-3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-8&0\\\\\\-5&-3\\end{pmatrix} $\n\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2174
} |
[
{
"role": "user",
"content": "Solve the following math equation. Differentiate with respect to $x$: $f(x) = -3\\sin(3x) + -8\\cos(6x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -3\\sin(3x) + -8\\cos(6x) $\n\nApply the standard differentiation rules:\n$ f'(x) =... | {
"type": "calculus",
"seed": 1337,
"index": 2175
} |
[
{
"role": "user",
"content": "solve: Find $A^{-1}$ for $A=\\begin{pmatrix}5&-8\\\\-7&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 5 & -8 \\\\ -7 & 1 \\end{pmatrix} $\n\nCompute the determinant: $\\det(A)=ad-bc$.\n$ \... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2176
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Solve for $x$ (in radians): tan(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal va... | {
"type": "trigonometry",
"seed": 1337,
"index": 2177
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Prove that $f(x,y)=\\frac12(3x^2+8y^2)+-2x+-3y+9$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+8y^2)+-2x+-3y+9 $\n\n$ \\frac{\\partial^2... | {
"type": "optimization",
"seed": 1337,
"index": 2178
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Evaluate: ((84 + -456/12) * 8) + 8^4 - log_5(25) + (-207 mod 8) + ceil(107/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((84 + -456/12) * 8) + 8^4 - log_5(25) + (-207 mod 8) + ceil(107/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 2179
} |
[
{
"role": "user",
"content": "Find the answer to the following 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": 2180
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Differentiate with respect to $x$: $f(x) = -5e^{-5x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -5e^{-5x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{25e^{-5x}} $"
}
... | {
"type": "calculus",
"seed": 1337,
"index": 2181
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. 43x^2 - 200 - 37x = 115"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 43x^2 - 200 - 37x = 115 $\n\nMove all terms to the left (standard form):\n$ 43x^2 + -37x + -315 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2182
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. 45x - 47x^2 + 108 = -80"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 45x - 47x^2 + 108 = -80 $\n\nMove all terms to the left (standard form):\n$ -47x^2 + 45x + 188 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2183
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Prove that $f(x,y)=\\frac12(4x^2+6y^2)+1x+2y+-6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(4x^2+6y^2)+1x+2y+-6 $\n\n$ \\frac{\\partial^2 f}{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 2184
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. -13 + 219 = -454x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -13 + 219 = -454x $\n\nMove all terms to the left (standard form):\n$ 454x + 206 = 0 $\n\nIsolate the $x$ term:\n$ 454x = -206 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2185
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Evaluate: ((-8 + -84/2) * 9) + 3^2 - log_8(512) + (233 mod 4) + ceil(2/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-8 + -84/2) * 9) + 3^2 - log_8(512) + (233 mod 4) + ceil(2/2) $\n\nCompute eac... | {
"type": "arithmetic",
"seed": 1337,
"index": 2186
} |
[
{
"role": "user",
"content": "Provide 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 $f'(x)=0$, h... | {
"type": "optimization",
"seed": 1337,
"index": 2187
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((-33 + -416/13) * 4) + 12^5 - log_8(4096) + (-58 mod 8) + ceil(47/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-33 + -416/13) * 4) + 12^5 - log_8(4096) + (-58 mod 8) + ceil(47/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 2188
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. -47 + 66x - 98 = 111x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -47 + 66x - 98 = 111x^2 $\n\nMove all terms to the left (standard form):\n$ -111x^2 + 66x + -145 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2189
} |
[
{
"role": "user",
"content": "Can you compute this equation? 33 - 160x^2 + 110x = 179"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 33 - 160x^2 + 110x = 179 $\n\nMove all terms to the left (standard form):\n$ -160x^2 + 110x + -146 = 0 $\n\nApply the quadratic formula:\n$ x = \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2190
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. -658 - 678 = -733x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -658 - 678 = -733x $\n\nMove all terms to the left (standard form):\n$ 733x + -1336 = 0 $\n\nIsolate the $x$ term:\n$ 733x = 1336 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 2191
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Evaluate: ((171 + 490/10) * 6) + 3^4 - log_5(3125) + (25 mod 3) + ceil(-171/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((171 + 490/10) * 6) + 3^4 - log_5(3125) + (25 mod 3) + ceil(-171/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 2192
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Compute $(-113+181) \\bmod 3$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-113+181) \\bmod 3 $\n\nUse the rule that addition is compatible with red... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2193
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Find $A^{-1}$ for $A=\\begin{pmatrix}-8&1\\\\2&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -8 & 1 \\\\ 2 & -7 \\end{pmatrix} $\n\nC... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2194
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: -33 + 80 - 35x^2 = 40x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -33 + 80 - 35x^2 = 40x $\n\nMove all terms to the left (standard form):\n$ -35x^2 + -40x + 47 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\pm \\sqrt{b... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2195
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Differentiate with respect to $x$: $f(x) = -4e^{-3x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -4e^{-3x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{12e^{-3x}} $"
... | {
"type": "calculus",
"seed": 1337,
"index": 2196
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. -44 + 131x^2 - 57 = -79x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -44 + 131x^2 - 57 = -79x $\n\nMove all terms to the left (standard form):\n$ 131x^2 + 79x + -101 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2197
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 3 - 81x + 29x^2 = 99"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 3 - 81x + 29x^2 = 99 $\n\nMove all terms to the left (standard form):\n$ 29x^2 + -81x + -96 = 0 $\n\nApply the quadratic formula:\n$ x = \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2198
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Compute the determinant: $A=\\begin{pmatrix}-8&2\\\\-9&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-8&2\\\\\\-9&9\\end{pmatrix} $\n\nUse $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2199
} |
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