messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Find the final answer to the given equation. Prove that $f(x)=7x^2+2x+-6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+2x+-6$.\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": 6200
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. -10 + 132x^2 - 110x = 196"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -10 + 132x^2 - 110x = 196 $\n\nMove all terms to the left (standard form):\n$ 132x^2 + -110x + -206 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6201
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Prove that $f(x)=9x^2+3x+-1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=9x^2+3x+-1$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 1... | {
"type": "optimization",
"seed": 1337,
"index": 6202
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. 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{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 6203
} |
[
{
"role": "user",
"content": "Find the value that satisfies 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 = \\arcco... | {
"type": "trigonometry",
"seed": 1337,
"index": 6204
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Compute $(-74+6) \\bmod 20$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-74+6) \\bmod 20 $\n\nUse the rule that addition is compatible with reduction modu... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6205
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Evaluate: ((-174 + -820/20) * 16) + 4^3 - log_3(81) + (338 mod 10) + round(-69/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-174 + -820/20) * 16) + 4^3 - log_3(81) + (338 mod 10) + round(-69/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6206
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. 10x^2 - 109x - 120 = -157"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 10x^2 - 109x - 120 = -157 $\n\nMove all terms to the left (standard form):\n$ 10x^2 + -109x + 37 = 0 $\n\nApply the quadrat... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6207
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Evaluate: ((-109 + 539/11) * 2) + 5^4 - log_7(7) + (-279 mod 9) + floor(180/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-109 + 539/11) * 2) + 5^4 - log_7(7) + (-279 mod 9) + floor(180/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 6208
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Evaluate: ((-67 + -225/25) * 13) + 3^2 - log_3(3) + (376 mod 20) + floor(195/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-67 + -225/25) * 13) + 3^2 - log_3(3) + (376 mod 20) + floor(195/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6209
} |
[
{
"role": "user",
"content": "Solve the equation and calculate 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{\\... | {
"type": "optimization",
"seed": 1337,
"index": 6210
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Evaluate: ((111 + 44/2) * 18) + 11^5 - log_6(36) + (231 mod 16) + ceil(73/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((111 + 44/2) * 18) + 11^5 - log_6(36) + (231 mod 16) + ceil(73/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 6211
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Compute the determinant: $A=\\begin{pmatrix}7&5\\\\-5&4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}7&5\\\\\\-5&4\\end{pmatrix} $\n\n... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6212
} |
[
{
"role": "user",
"content": "Please work out 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$ \\alpha = \\arc... | {
"type": "trigonometry",
"seed": 1337,
"index": 6213
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 6214
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-4&9\\\\-6&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -4 & 9 \\\\ -6 & 5 \\end{pmatrix} $\n\nCompute th... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6215
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Find $A^{-1}$ for $A=\\begin{pmatrix}0&-6\\\\-7&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 0 & -6 \\\\ -7 & -5 \\end{pmatrix} $\n\nCompute the determin... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6216
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Evaluate: ((110 + -833/17) * 20) + 9^5 - log_6(46656) + (102 mod 12) + round(33/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((110 + -833/17) * 20) + 9^5 - log_6(46656) + (102 mod 12) + round(33/... | {
"type": "arithmetic",
"seed": 1337,
"index": 6217
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Find the modular inverse of 9 modulo 22, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $9\\cdot a^{-1} \\equiv 1 \\pmod{22}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guara... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6218
} |
[
{
"role": "user",
"content": "Solve this equation step by step. 965 + 931 = 131x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 965 + 931 = 131x $\n\nMove all terms to the left (standard form):\n$ -131x + 1896 = 0 $\n\nIsolate the $x$ term:\n$ -131x = -1896 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6219
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((23 + -240/8) * 2) + 3^2 - log_6(6) + (461 mod 27) + floor(187/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((23 + -240/8) * 2) + 3^2 - log_6(6) + (461 mod 27) + floor(187/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6220
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Evaluate: ((-145 + 448/16) * 2) + 8^5 - log_2(8) + (-369 mod 7) + round(9/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-145 + 448/16) * 2) + 8^5 - log_2(8) + (-369 mod 7) + round(9/2) $\n\nCompute each comp... | {
"type": "arithmetic",
"seed": 1337,
"index": 6221
} |
[
{
"role": "user",
"content": "Please work out this equation. 40 + 101x + 22 = 98x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 40 + 101x + 22 = 98x^2 $\n\nMove all terms to the left (standard form):\n$ -98x^2 + 101x + 62 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6222
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Compute the indefinite integral: $\\int -9e^{5x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -9e^{5x}\\,dx $\n\nUse standard antiderivatives (and integration by parts ... | {
"type": "calculus",
"seed": 1337,
"index": 6223
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. 8 - 128x - 31x^2 = -65"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 8 - 128x - 31x^2 = -65 $\n\nMove all terms to the left (standard form):\n$ -31x^2 + -128x + 73 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6224
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given 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... | {
"type": "optimization",
"seed": 1337,
"index": 6225
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Find the modular inverse of 13 modulo 14, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $13\\cdot a^{-1} \\equiv 1 \\pmod{14}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantee... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6226
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. 644 - 241x = 328"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 644 - 241x = 328 $\n\nMove all terms to the left (standard form):\n$ -241x + 316 = 0 $\n\nIsolate the $x$ term:\n$ -241x = -316 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 6227
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Compute the indefinite integral: $\\int (-3x + 5)\\sin(6x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-3x + 5)\\sin(6x)\\,dx $\n\nUse standard antiderivatives (and integratio... | {
"type": "calculus",
"seed": 1337,
"index": 6228
} |
[
{
"role": "user",
"content": "Evaluate 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 $a,b\\in\\mathbb{Z}_n$, the... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6229
} |
[
{
"role": "user",
"content": "Please work out 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 $\n\nThe general s... | {
"type": "trigonometry",
"seed": 1337,
"index": 6230
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. -50x^2 - 82x + 17 = -90"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -50x^2 - 82x + 17 = -90 $\n\nMove all terms to the left (standard form):\n$ -50x^2 + -82x + 107 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6231
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Prove that $(\\mathbb{Z}_24, +)$ is a group under addition modulo 24."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_24=\\{0,1,\\dots,23\\}$ with operation $+$ modulo $24$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6232
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. 13x + 66 + 35x^2 = 84"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 13x + 66 + 35x^2 = 84 $\n\nMove all terms to the left (standard form):\n$ 35x^2 + 13x + -18 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6233
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n 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": 6234
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Evaluate: ((162 + 1008/21) * 17) + 2^5 - log_9(59049) + (-177 mod 28) + round(27/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((162 + 1008/21) * 17) + 2^5 - log_9(59049) + (-177 mod 28) + round(27/2) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6235
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Prove that $f(x)=8x^2+-10x+-5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+-10x+-5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $... | {
"type": "optimization",
"seed": 1337,
"index": 6236
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Evaluate: ((96 + -374/17) * 8) + 8^5 - log_2(2) + (477 mod 21) + round(-196/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((96 + -374/17) * 8) + 8^5 - log_2(2) + (477 mod 21) + round(-196/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 6237
} |
[
{
"role": "user",
"content": "Can you compute this equation? -25 - 11x + 54x^2 = 100"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -25 - 11x + 54x^2 = 100 $\n\nMove all terms to the left (standard form):\n$ 54x^2 + -11x + -125 = 0 $\n\nApply the quadratic formula:\n$ x = \\fra... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6238
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. -402 - 312 = 398x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -402 - 312 = 398x $\n\nMove all terms to the left (standard form):\n$ -398x + -714 = 0 $\n\nIsolate the $x$ term:\n$ -398x = 714 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 6239
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Compute $(79+119) \\bmod 14$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (79+119) \\bmod 14 $\n\nUse the rule that addition is compatible with reduction modul... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6240
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. -557 + 713 = -901x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -557 + 713 = -901x $\n\nMove all terms to the left (standard form):\n$ 901x + 156 = 0 $\n\nIsolate the $x$ term:\n$ 901x = -156 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 6241
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}5&2\\\\\\9&9\\end{pmatrix},\\quad b=\\begin{pmatrix}-46\\\\\\-126\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}5&2\\\\\\9&9\\end... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6242
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Compute the determinant: $A=\\begin{pmatrix}0&7\\\\2&2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}0&7\\\\\\2&2\\end{pmatrix} $\n\nUse $\\det(A)... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6243
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-5&-2&4\\\\\\-5&-4&4\\\\\\-3&-5&0\\end{pmatrix},\\quad b=\\begin{pmatrix}-32\\\\\\-40\\\\\\-32\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6244
} |
[
{
"role": "user",
"content": "Please calculate this equation. 16x - 90 + 84 = 89x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 16x - 90 + 84 = 89x^2 $\n\nMove all terms to the left (standard form):\n$ -89x^2 + 16x + -6 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6245
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Find the modular inverse of 17 modulo 40, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $17\\cdot a^{-1} \\equiv 1 \\pmod{40}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gua... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6246
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Compute $(119+62) \\bmod 15$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (119+62) \\bmod 15 $\n\nUse the rule that addition is compatible with reduction modulo $... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6247
} |
[
{
"role": "user",
"content": "Please find the solution 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 $f'(x)=0$, hence:... | {
"type": "optimization",
"seed": 1337,
"index": 6248
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Evaluate: ((-142 + 210/21) * 18) + 12^2 - log_5(5) + (-489 mod 29) + round(78/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-142 + 210/21) * 18) + 12^2 - log_5(5) + (-489 mod 29) + round(78/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6249
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}4&-4&-1\\\\\\-5&-4&-2\\\\\\3&2&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\19\\\\\\-11\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solv... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6250
} |
[
{
"role": "user",
"content": "Please solve this math problem. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}3&-7\\\\\\9&8\\end{pmatrix},\\quad b=\\begin{pmatrix}-1\\\\\\-61\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}3&-7\\\\\\9&8\\end{pmatrix}... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6251
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Evaluate: ((-15 + 170/17) * 20) + 4^5 - log_10(1000) + (-68 mod 20) + round(-134/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-15 + 170/17) * 20) + 4^5 - log_10(1000) + (-68 mod 20) + round(-134... | {
"type": "arithmetic",
"seed": 1337,
"index": 6252
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Prove that $f(x)=3x^2+-6x+-10$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=3x^2+-6x+-10$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\... | {
"type": "optimization",
"seed": 1337,
"index": 6253
} |
[
{
"role": "user",
"content": "Work out the result of this equation. 27 + 111x^2 - 166x = 77"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 27 + 111x^2 - 166x = 77 $\n\nMove all terms to the left (standard form):\n$ 111x^2 + -166x + -50 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6254
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Prove that $f(x)=4x^2+8x+0$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=4x^2+8x+0$.\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": 6255
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Evaluate: ((88 + -936/24) * 19) + 5^3 - log_6(216) + (337 mod 16) + round(-37/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((88 + -936/24) * 19) + 5^3 - log_6(216) + (337 mod 16) + round(-37/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6256
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((-199 + -672/21) * 20) + 8^4 - log_10(100000) + (-489 mod 11) + ceil(9/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-199 + -672/21) * 20) + 8^4 - log_10(100000) + (-489 mod 11) + ceil(9/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 6257
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Evaluate: ((-138 + 40/10) * 19) + 5^4 - log_10(10) + (400 mod 3) + round(-133/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-138 + 40/10) * 19) + 5^4 - log_10(10) + (400 mod 3) + round(-133/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 6258
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Find $A^{-1}$ for $A=\\begin{pmatrix}5&-3\\\\-1&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 5 & -3 \\\\ -1 & 5 \\end{pmatrix} $\n\nCompute the determinant... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6259
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. -13x^2 + 160x + 40 = -22"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -13x^2 + 160x + 40 = -22 $\n\nMove all terms to the left (standard form):\n$ -13x^2 + 160x + 62 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6260
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Compute the determinant: $A=\\begin{pmatrix}8&8\\\\-6&-3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}8&8\\\\\\-6&-3\\end{pmatrix} $\n\nUse $\\de... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6261
} |
[
{
"role": "user",
"content": "Please calculate and resolve 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{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 6262
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Prove that $(\\mathbb{Z}_13, +)$ is a group under addition modulo 13."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_13=\\{0,1,\\dots,12\\}$ with operation $+$ modulo $13$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6263
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Evaluate: ((13 + 528/16) * 16) + 6^4 - log_7(49) + (-326 mod 30) + floor(152/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((13 + 528/16) * 16) + 6^4 - log_7(49) + (-326 mod 30) + floor(15... | {
"type": "arithmetic",
"seed": 1337,
"index": 6264
} |
[
{
"role": "user",
"content": "Evaluate the following equation. -602 - 901x = 648"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -602 - 901x = 648 $\n\nMove all terms to the left (standard form):\n$ -901x + -1250 = 0 $\n\nIsolate the $x$ term:\n$ -901x = 1250 $\n\nDivide both sides... | {
"type": "linear_equation",
"seed": 1337,
"index": 6265
} |
[
{
"role": "user",
"content": "solve: Evaluate: ((-140 + -144/12) * 13) + 4^2 - log_10(1000) + (390 mod 16) + round(-185/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-140 + -144/12) * 13) + 4^2 - log_10(1000) + (390 mod 16) + round(-185/2) $\n\nCompute each component:\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6266
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. 855 + 524x = -4"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 855 + 524x = -4 $\n\nMove all terms to the left (standard form):\n$ 524x + 859 = 0 $\n\nIsolate the $x$ term:\n$ 524x = -859 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6267
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Differentiate with respect to $x$: $f(x) = \\ln(5x + 6)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(5x + 6) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{5}{5x +... | {
"type": "calculus",
"seed": 1337,
"index": 6268
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. 42x - 122 + 102x^2 = 39"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 42x - 122 + 102x^2 = 39 $\n\nMove all terms to the left (standard form):\n$ 102x^2 + 42x + -161 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6269
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Evaluate: ((-136 + 667/23) * 15) + 9^2 - log_10(1000) + (-185 mod 11) + round(189/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-136 + 667/23) * 15) + 9^2 - log_10(1000) + (-185 mod 11) + round(189/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 6270
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. -30x - 191 - 100x^2 = 65"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -30x - 191 - 100x^2 = 65 $\n\nMove all terms to the left (standard form):\n$ -100x^2 + -30x + -256 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6271
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. -543x + 912 = -988"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -543x + 912 = -988 $\n\nMove all terms to the left (standard form):\n$ -543x + 1900 = 0 $\n\nIsolate the $x$ term:\n$ -543x = -1900 $\n\nDiv... | {
"type": "linear_equation",
"seed": 1337,
"index": 6272
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 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 $\n\nTh... | {
"type": "trigonometry",
"seed": 1337,
"index": 6273
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Evaluate: ((167 + 58/2) * 18) + 6^4 - log_2(8) + (-349 mod 21) + floor(-138/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((167 + 58/2) * 18) + 6^4 - log_2(8) + (-349 mod 21) + floor(-138/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6274
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Prove that $f(x)=3x^2+-5x+3$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=3x^2+-5x+3$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 6x +... | {
"type": "optimization",
"seed": 1337,
"index": 6275
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Find $A^{-1}$ for $A=\\begin{pmatrix}-1&8\\\\6&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -1 & 8 \\\\ 6 & -5 \\end{pmatrix} $\n\nCompute the determinant:... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6276
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&-4&4\\\\\\-4&0&2\\\\\\4&0&1\\end{pmatrix},\\quad b=\\begin{pmatrix}-24\\\\\\-14\\\\\\11\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6277
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute $(194+134) \\bmod 23$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (194+134) \\bmod 23 $\n\nUse the rule that addition is compatible with reduction mod... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6278
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Differentiate with respect to $x$: $f(x) = -8\\sin(6x) + 3\\cos(7x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -8\\sin(6x) + 3\\cos(7x) $\n\nApply the standard differentiation rules:\n$... | {
"type": "calculus",
"seed": 1337,
"index": 6279
} |
[
{
"role": "user",
"content": "Solve the following math equation. Differentiate with respect to $x$: $f(x) = -2\\sin(3x) + 2\\cos(6x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -2\\sin(3x) + 2\\cos(6x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \... | {
"type": "calculus",
"seed": 1337,
"index": 6280
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Evaluate: ((41 + 441/21) * 14) + 6^5 - log_9(531441) + (-349 mod 19) + ceil(-50/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((41 + 441/21) * 14) + 6^5 - log_9(531441) + (-349 mod 19) + ceil(-50/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 6281
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. 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 = \\ar... | {
"type": "trigonometry",
"seed": 1337,
"index": 6282
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((91 + 384/8) * 15) + 10^4 - log_3(3) + (24 mod 2) + floor(132/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((91 + 384/8) * 15) + 10^4 - log_3(3) + (24 mod 2) + floor(132/2) $\n\nCompute eac... | {
"type": "arithmetic",
"seed": 1337,
"index": 6283
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Evaluate: ((-101 + 180/20) * 3) + 11^4 - log_2(16) + (-314 mod 9) + ceil(178/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-101 + 180/20) * 3) + 11^4 - log_2(16) + (-314 mod 9) + ceil(178/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 6284
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Find $A^{-1}$ for $A=\\begin{pmatrix}-4&2\\\\5&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -4 & 2 \\\\ 5 & -4 \\end{pmatrix} $\n\nCompute... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6285
} |
[
{
"role": "user",
"content": "solve: Compute $(-34+87) \\bmod 20$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-34+87) \\bmod 20 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ (a+b)\\bmod n = ((a\\bm... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6286
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Evaluate: ((105 + -216/18) * 4) + 7^4 - log_3(3) + (145 mod 29) + ceil(-145/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((105 + -216/18) * 4) + 7^4 - log_3(3) + (145 mod 29) + ceil(-145/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 6287
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? -116x + 681 = 800"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -116x + 681 = 800 $\n\nMove all terms to the left (standard form):\n$ -116x + -119 = 0 $\n\nIsolate the $x$ term:\n$ -116x = 119 $\n\nDivide both sid... | {
"type": "linear_equation",
"seed": 1337,
"index": 6288
} |
[
{
"role": "user",
"content": "Please calculate this equation. Compute the indefinite integral: $\\int -8\\sin(6x) + -7\\cos(6x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -8\\sin(6x) + -7\\cos(6x)\\,dx $\n\nUse standard antiderivatives (and integratio... | {
"type": "calculus",
"seed": 1337,
"index": 6289
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Differentiate with respect to $x$: $f(x) = \\ln(7x + 3)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(7x + 3) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\fr... | {
"type": "calculus",
"seed": 1337,
"index": 6290
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. 47x - 97 + 91x^2 = -45"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 47x - 97 + 91x^2 = -45 $\n\nMove all terms to the left (standard form):\n$ 91x^2 + 47x + -52 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6291
} |
[
{
"role": "user",
"content": "Can you 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{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 6292
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Evaluate: ((-78 + -204/6) * 12) + 9^3 - log_3(9) + (-57 mod 3) + floor(52/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-78 + -204/6) * 12) + 9^3 - log_3(9) + (-57 mod 3) + floor(52/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6293
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}0&5\\\\\\-7&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}20\\\\\\-57\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}0&5\\\\\\-7... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6294
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Evaluate: ((-145 + 550/22) * 18) + 2^2 - log_3(729) + (-306 mod 3) + floor(-148/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-145 + 550/22) * 18) + 2^2 - log_3(729) + (-306 mod 3) + floor(-148/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 6295
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-3&2&2\\\\\\-5&-3&-2\\\\\\1&-3&5\\end{pmatrix},\\quad b=\\begin{pmatrix}-14\\\\\\-14\\\\\\31\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solv... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6296
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Evaluate: ((38 + 165/11) * 7) + 4^4 - log_9(729) + (-349 mod 28) + round(-48/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((38 + 165/11) * 7) + 4^4 - log_9(729) + (-349 mod 28) + round(-48/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6297
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. 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{\\part... | {
"type": "optimization",
"seed": 1337,
"index": 6298
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Differentiate with respect to $x$: $f(x) = -3e^{-2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -3e^{-2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{6e^{-2x}} $"
... | {
"type": "calculus",
"seed": 1337,
"index": 6299
} |
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