messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Can you find the solution to this equation? Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&2&4\\\\\\-2&-4&-2\\\\\\4&-4&4\\end{pmatrix},\\quad b=\\begin{pmatrix}-10\\\\\\0\\\\\\12\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2400
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Evaluate: ((131 + 468/13) * 4) + 8^5 - log_6(216) + (-78 mod 10) + round(-30/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((131 + 468/13) * 4) + 8^5 - log_6(216) + (-78 mod 10) + round(-30/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 2401
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. -25 - 136x + 43 = 7x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -25 - 136x + 43 = 7x^2 $\n\nMove all terms to the left (standard form):\n$ -7x^2 + -136x + 18 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2402
} |
[
{
"role": "user",
"content": "Please solve this mathematical 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": 2403
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n -27 + 79x + 186 = 20x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -27 + 79x + 186 = 20x^2 $\n\nMove all terms to the left (standard form):\n$ -20x^2 + 79x + 159 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2404
} |
[
{
"role": "user",
"content": "Solve the following math equation. -306 + 394x = 306"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -306 + 394x = 306 $\n\nMove all terms to the left (standard form):\n$ 394x + -612 = 0 $\n\nIsolate the $x$ term:\n$ 394x = 612 $\n\nDivide both sides b... | {
"type": "linear_equation",
"seed": 1337,
"index": 2405
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Compute $(-35+-185) \\bmod 21$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-35+-185) \\bmod 21 $\n\nUse the rule that addition is compatible with reduct... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2406
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Evaluate: ((143 + 1078/22) * 14) + 7^3 - log_3(3) + (-165 mod 21) + round(-123/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((143 + 1078/22) * 14) + 7^3 - log_3(3) + (-165 mod 21) + round(-123/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 2407
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Compute $(28+77) \\bmod 17$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (28+77) \\bmod 17 $\n\nUse the rule that addition is compatible with reductio... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2408
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Evaluate: ((152 + -1050/25) * 14) + 2^2 - log_7(2401) + (-462 mod 16) + floor(168/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((152 + -1050/25) * 14) + 2^2 - log_7(2401) + (-462 mod 16) + floor(168... | {
"type": "arithmetic",
"seed": 1337,
"index": 2409
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Prove that $(\\mathbb{Z}_6, +)$ is a group under addition modulo 6."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_6=\\{0,1,\\dots,5\\}$ with operation $+$ modulo $6$.\n\n**Closure:** If $a,b\\in\\m... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2410
} |
[
{
"role": "user",
"content": "solve: \n\n 856 + 364x = 441"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 856 + 364x = 441 $\n\nMove all terms to the left (standard form):\n$ 364x + 415 = 0 $\n\nIsolate the $x$ term:\n$ 364x = -415 $\n\nDivide both sides by the coefficient of $x$:... | {
"type": "linear_equation",
"seed": 1337,
"index": 2411
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. 32x + 154x^2 - 166 = -148"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 32x + 154x^2 - 166 = -148 $\n\nMove all terms to the left (standard form):\n$ 154x^2 + 32x + -18 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2412
} |
[
{
"role": "user",
"content": "Solve the math equation and present 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": 2413
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Evaluate: ((10 + 387/9) * 20) + 4^5 - log_7(16807) + (-498 mod 27) + ceil(-38/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((10 + 387/9) * 20) + 4^5 - log_7(16807) + (-498 mod 27) + ceil(-38/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 2414
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}1&8\\\\\\3&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}25\\\\\\-12\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}1&8\\\\\\3&... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2415
} |
[
{
"role": "user",
"content": "Please work out this equation. -566 - 726 = 625x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -566 - 726 = 625x $\n\nMove all terms to the left (standard form):\n$ -625x + -1292 = 0 $\n\nIsolate the $x$ term:\n$ -625x = 1292 $\n\nDivide both sides b... | {
"type": "linear_equation",
"seed": 1337,
"index": 2416
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Evaluate: ((159 + -92/23) * 11) + 10^2 - log_7(16807) + (-259 mod 27) + ceil(-166/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((159 + -92/23) * 11) + 10^2 - log_7(16807) + (-259 mod 27) + ceil(-166... | {
"type": "arithmetic",
"seed": 1337,
"index": 2417
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Evaluate: ((-100 + -100/5) * 11) + 8^4 - log_5(3125) + (269 mod 25) + ceil(21/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-100 + -100/5) * 11) + 8^4 - log_5(3125) + (269 mod 25) + ceil(21/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 2418
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n -31 + 32x - 15x^2 = 126"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -31 + 32x - 15x^2 = 126 $\n\nMove all terms to the left (standard form):\n$ -15x^2 + 32x + -157 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2419
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. x + 107 - 71x^2 = -196"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ x + 107 - 71x^2 = -196 $\n\nMove all terms to the left (standard form):\n$ -71x^2 + 1x + 303 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2420
} |
[
{
"role": "user",
"content": "Please solve this math problem. Solve for $x$ (in radians): cos(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\... | {
"type": "trigonometry",
"seed": 1337,
"index": 2421
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Find the modular inverse of 5 modulo 39, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{39}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees in... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2422
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}0&-3&1\\\\\\-1&-2&-5\\\\\\2&-3&2\\end{pmatrix},\\quad b=\\begin{pmatrix}12\\\\\\10\\\\\\8\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2423
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Compute the indefinite integral: $\\int -9e^{3x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -9e^{3x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if neede... | {
"type": "calculus",
"seed": 1337,
"index": 2424
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Find the modular inverse of 7 modulo 9, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $7\\cdot a^{-1} \\equiv 1 \\pmod{9}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarante... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2425
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Evaluate: ((185 + 300/12) * 5) + 12^2 - log_8(262144) + (-82 mod 2) + floor(8/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((185 + 300/12) * 5) + 12^2 - log_8(262144) + (-82 mod 2) + floor(8/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 2426
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}1&3&-2\\\\\\-1&-4&2\\\\\\5&2&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}9\\\\\\-10\\\\\\11\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2427
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. 41 - 123x - 129x^2 = 140"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 41 - 123x - 129x^2 = 140 $\n\nMove all terms to the left (standard form):\n$ -129x^2 + -123x + -99 = 0 $\n\nApply the quadratic ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2428
} |
[
{
"role": "user",
"content": "solve: \n\n -868 - 331x = 84"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -868 - 331x = 84 $\n\nMove all terms to the left (standard form):\n$ -331x + -952 = 0 $\n\nIsolate the $x$ term:\n$ -331x = 952 $\n\nDivide both sides by the coefficient of $x... | {
"type": "linear_equation",
"seed": 1337,
"index": 2429
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Evaluate: ((-120 + 552/24) * 19) + 7^3 - log_8(512) + (469 mod 23) + floor(-128/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-120 + 552/24) * 19) + 7^3 - log_8(512) + (469 mod 23) + floor(-128/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 2430
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. 41 - 101x^2 - 145x = 73"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 41 - 101x^2 - 145x = 73 $\n\nMove all terms to the left (standard form):\n$ -101x^2 + -145x + -32 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2431
} |
[
{
"role": "user",
"content": "Please work out this equation. Evaluate: ((-60 + -216/12) * 3) + 6^2 - log_3(81) + (400 mod 8) + round(-16/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-60 + -216/12) * 3) + 6^2 - log_3(81) + (400 mod 8) + round(-16/2) $\n\nCompute each co... | {
"type": "arithmetic",
"seed": 1337,
"index": 2432
} |
[
{
"role": "user",
"content": "solve 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 = \\arcsin(-\\frac{\\sqrt{2}}... | {
"type": "trigonometry",
"seed": 1337,
"index": 2433
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. -938 - 343 = -761x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -938 - 343 = -761x $\n\nMove all terms to the left (standard form):\n$ 761x + -1281 = 0 $\n\nIsolate the $x$ term:\n$ 761x = 1281 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 2434
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. 326 - 505 = -844x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 326 - 505 = -844x $\n\nMove all terms to the left (standard form):\n$ 844x + -179 = 0 $\n\nIsolate the $x$ term:\n$ 844x = 179 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 2435
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n -507 + 744x = 326"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -507 + 744x = 326 $\n\nMove all terms to the left (standard form):\n$ 744x + -833 = 0 $\n\nIsolate the $x$ term:\n$ 744x = 833 $\n\... | {
"type": "linear_equation",
"seed": 1337,
"index": 2436
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Compute $(167+-3) \\bmod 5$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (167+-3) \\bmod 5 $\n\nUse the rule that addition is compatible with reduction ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2437
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Find $A^{-1}$ for $A=\\begin{pmatrix}-8&-6\\\\9&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -8 & -6 \\\\ 9 & 3 \\end{pmatrix} $\n\nC... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2438
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Evaluate: ((-84 + -816/17) * 17) + 7^5 - log_9(6561) + (-361 mod 10) + floor(190/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-84 + -816/17) * 17) + 7^5 - log_9(6561) + (-361 mod 10) + floor(190/... | {
"type": "arithmetic",
"seed": 1337,
"index": 2439
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 8x + 172 + 127 = -192x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 8x + 172 + 127 = -192x^2 $\n\nMove all terms to the left (standard form):\n$ 192x^2 + 8x + 299 = 0 $\n\nApply the quadratic formula:\n$ x = ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2440
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. 41 - 3x + 173x^2 = 32"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 41 - 3x + 173x^2 = 32 $\n\nMove all terms to the left (standard form):\n$ 173x^2 + -3x + 9 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2441
} |
[
{
"role": "user",
"content": "Please solve this math problem. -13x^2 - 16 - 48 = 68x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -13x^2 - 16 - 48 = 68x $\n\nMove all terms to the left (standard form):\n$ -13x^2 + -68x + -64 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2442
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 11 - 99 + 166x = 190x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 11 - 99 + 166x = 190x^2 $\n\nMove all terms to the left (standard form):\n$ -190x^2 + 166x + -88 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2443
} |
[
{
"role": "user",
"content": "Can you compute this equation? Evaluate: ((0 + 40/10) * 16) + 4^2 - log_9(9) + (-400 mod 4) + floor(76/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((0 + 40/10) * 16) + 4^2 - log_9(9) + (-400 mod 4) + floor(76/2) $\n\nCompute each component:... | {
"type": "arithmetic",
"seed": 1337,
"index": 2444
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Compute the indefinite integral: $\\int 7\\sin(6x) + -7\\cos(6x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 7\\sin(6x) + -7\\cos(6x)\\,dx $\n\nUse standard antiderivatives (a... | {
"type": "calculus",
"seed": 1337,
"index": 2445
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}7&8\\\\\\1&-1\\end{pmatrix},\\quad b=\\begin{pmatrix}-9\\\\\\3\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}7&8\\\\\\1&-1\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2446
} |
[
{
"role": "user",
"content": "Resolve the equation provided. 18x + 24 + 180 = -129x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 18x + 24 + 180 = -129x^2 $\n\nMove all terms to the left (standard form):\n$ 129x^2 + 18x + 204 = 0 $\n\nApply the quadratic formula:\n$ x = \\fr... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2447
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Compute the determinant: $A=\\begin{pmatrix}2&4\\\\9&7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}2&4\\\\\\9&7\\end{pmatrix} $\n\nUse $\\d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2448
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. -12x^2 + 55x - 162 = 36"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -12x^2 + 55x - 162 = 36 $\n\nMove all terms to the left (standard form):\n$ -12x^2 + 55x + -198 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2449
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute the determinant: $A=\\begin{pmatrix}3&-8\\\\5&2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}3&-8\\\\\\5&2\\end{pmatrix} $\n\nUse $\\d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2450
} |
[
{
"role": "user",
"content": "Can you compute this equation? Differentiate with respect to $x$: $f(x) = 3\\sin(2x) + 8\\cos(4x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 3\\sin(2x) + 8\\cos(4x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed... | {
"type": "calculus",
"seed": 1337,
"index": 2451
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Compute the indefinite integral: $\\int \\ln(2x + -4)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(2x + -4)\\,dx $\n\nUse standard antiderivatives (and integration by... | {
"type": "calculus",
"seed": 1337,
"index": 2452
} |
[
{
"role": "user",
"content": "Determine the correct answer to 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 $f'(x)=0$, he... | {
"type": "optimization",
"seed": 1337,
"index": 2453
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Prove that $f(x,y)=\\frac12(3x^2+7y^2)+4x+4y+5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+7y^2)+4x+4y+5 $\n\n$ \\frac{\\partial^2 f}{\\partia... | {
"type": "optimization",
"seed": 1337,
"index": 2454
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Evaluate: ((6 + 0/25) * 20) + 12^2 - log_7(7) + (484 mod 11) + ceil(-97/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((6 + 0/25) * 20) + 12^2 - log_7(7) + (484 mod 11) + ceil(-97/2) $\n\nCompute each component:\n$ 0/25 = 0 $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 2455
} |
[
{
"role": "user",
"content": "solve: \n\n Compute the indefinite integral: $\\int (-1x + 4)\\sin(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-1x + 4)\\sin(4x)\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \\int (-... | {
"type": "calculus",
"seed": 1337,
"index": 2456
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Evaluate: ((130 + 13/13) * 10) + 8^2 - log_2(64) + (285 mod 17) + round(-41/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((130 + 13/13) * 10) + 8^2 - log_2(64) + (285 mod 17) + round(-41/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 2457
} |
[
{
"role": "user",
"content": "Work out the result of 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$ \\alph... | {
"type": "trigonometry",
"seed": 1337,
"index": 2458
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Find $A^{-1}$ for $A=\\begin{pmatrix}-2&-7\\\\1&8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -2 & -7 \\\\ 1 & 8 \\end{pmatrix} $\n\nCompute the de... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2459
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Compute the indefinite integral: $\\int -1\\sin(4x) + -8\\cos(1x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -1\\sin(4x) + -8\\cos(1x)\\,dx $\n\nUse standard antiderivatives (a... | {
"type": "calculus",
"seed": 1337,
"index": 2460
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Prove that $f(x,y)=\\frac12(8x^2+1y^2)+4x+5y+1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+1y^2)+4x+5y+1 $\n\n$ \\frac{\\partial^2 f}{\\part... | {
"type": "optimization",
"seed": 1337,
"index": 2461
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Prove that $(\\mathbb{Z}_10, +)$ is a group under addition modulo 10."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_10=\\{0,1,\\dots,9\\}$ with operation $+$ modulo $10$.\n\n**Closure:** If $a,b\\in\\mat... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2462
} |
[
{
"role": "user",
"content": "Calculate the answer for 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{\\parti... | {
"type": "optimization",
"seed": 1337,
"index": 2463
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Evaluate: ((-75 + -385/11) * 1) + 6^2 - log_8(512) + (37 mod 25) + round(-79/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-75 + -385/11) * 1) + 6^2 - log_8(512) + (37 mod 25) + round(-79/2) $\n\nCompute each component:\n$ -... | {
"type": "arithmetic",
"seed": 1337,
"index": 2464
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. 614x - 908 = -490"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 614x - 908 = -490 $\n\nMove all terms to the left (standard form):\n$ 614x + -418 = 0 $\n\nIsolate the $x$ term:\n$ 614x = 418 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 2465
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Find the modular inverse of 2 modulo 11, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $2\\cdot a^{-1} \\equiv 1 \\pmod{11}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2466
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & ... | {
"type": "optimization",
"seed": 1337,
"index": 2467
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. -626x - 47 = 926"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -626x - 47 = 926 $\n\nMove all terms to the left (standard form):\n$ -626x + -973 = 0 $\n\nIsolate the $x$ term:\n$ -626x = 973 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 2468
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Find the modular inverse of 2 modulo 19, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $2\\cdot a^{-1} \\equiv 1 \\pmod{19}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guaran... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2469
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Compute the indefinite integral: $\\int \\ln(8x + -6)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(8x + -6)\\,dx $\n\nUse standard antiderivatives (and integration by parts if n... | {
"type": "calculus",
"seed": 1337,
"index": 2470
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Compute the indefinite integral: $\\int 2e^{1x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2e^{1x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):... | {
"type": "calculus",
"seed": 1337,
"index": 2471
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Differentiate with respect to $x$: $f(x) = \\ln(9x + 1)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(9x + 1) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\fr... | {
"type": "calculus",
"seed": 1337,
"index": 2472
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Evaluate: ((182 + -10/2) * 10) + 6^3 - log_2(2) + (-65 mod 16) + ceil(114/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((182 + -10/2) * 10) + 6^3 - log_2(2) + (-65 mod 16) + ceil(114/2) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 2473
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Find the modular inverse of 7 modulo 36, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $7\\cdot a^{-1} \\equiv 1 \\pmod{36}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2474
} |
[
{
"role": "user",
"content": "Can you compute this equation? -264x + 348 = 123"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -264x + 348 = 123 $\n\nMove all terms to the left (standard form):\n$ -264x + 225 = 0 $\n\nIsolate the $x$ term:\n$ -264x = -225 $\n\nDivide both sides by ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2475
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. -967x - 587 = -679"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -967x - 587 = -679 $\n\nMove all terms to the left (standard form):\n$ -967x + 92 = 0 $\n\nIsolate the $x$ term:\n$ -967x = -92 $\n\nDivide both side... | {
"type": "linear_equation",
"seed": 1337,
"index": 2476
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Prove that $f(x)=6x^2+7x+-10$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=6x^2+7x+-10$.\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": 2477
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Evaluate: ((103 + -299/13) * 3) + 10^5 - log_4(1024) + (216 mod 21) + floor(-44/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((103 + -299/13) * 3) + 10^5 - log_4(1024) + (216 mod 21) + floor(-4... | {
"type": "arithmetic",
"seed": 1337,
"index": 2478
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 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 ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2479
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. -3x^2 - 114 - 134x = -82"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -3x^2 - 114 - 134x = -82 $\n\nMove all terms to the left (standard form):\n$ -3x^2 + -134x + -32 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2480
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Compute $(80+135) \\bmod 4$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (80+135) \\bmod 4 $\n\nUse the rule that addition is compatible with reduction m... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2481
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}2&3&1\\\\\\-5&1&3\\\\\\5&-2&5\\end{pmatrix},\\quad b=\\begin{pmatrix}-9\\\\\\17\\\\\\-25\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2482
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. -33x - 32 - 106 = -10x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -33x - 32 - 106 = -10x^2 $\n\nMove all terms to the left (standard form):\n$ 10x^2 + -33x + -138 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2483
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. -658 - 88 = -245x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -658 - 88 = -245x $\n\nMove all terms to the left (standard form):\n$ 245x + -746 = 0 $\n\nIsolate the $x$ term:\n$ 245x = 746 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 2484
} |
[
{
"role": "user",
"content": "Solve this equation step by step. 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$ 4x(x^... | {
"type": "optimization",
"seed": 1337,
"index": 2485
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Compute $(-48+104) \\bmod 12$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-48+104) \\bmod 12 $\n\nUse the rule that addition is compatible with reduct... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2486
} |
[
{
"role": "user",
"content": "Solve the equation shown below. 27x^2 - 1 - 174 = -113x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 27x^2 - 1 - 174 = -113x $\n\nMove all terms to the left (standard form):\n$ 27x^2 + 113x + -175 = 0 $\n\nApply the quadratic formula:\n$ x = \\fr... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2487
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Solve for $x$ (in radians): tan(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(\\fr... | {
"type": "trigonometry",
"seed": 1337,
"index": 2488
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Find the modular inverse of 23 modulo 36, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $23\\cdot a^{-1} \\equiv 1 \\pmod{36}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gua... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2489
} |
[
{
"role": "user",
"content": "Can you compute this equation? Evaluate: ((-15 + 750/25) * 17) + 10^5 - log_5(625) + (-82 mod 12) + round(-140/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-15 + 750/25) * 17) + 10^5 - log_5(625) + (-82 mod 12) + round(-140/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 2490
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}4&-4\\\\4&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 4 & -4 \\\\ 4 & -5 \\end{pmatrix} $\n\nCompute... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2491
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Differentiate with respect to $x$: $f(x) = (-2x + 0)e^{2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-2x + 0)e^{2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-2e^{... | {
"type": "calculus",
"seed": 1337,
"index": 2492
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. Evaluate: ((55 + -483/21) * 16) + 9^3 - log_7(117649) + (216 mod 2) + ceil(49/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((55 + -483/21) * 16) + 9^3 - log_7(117649) + (216 mod 2) + ceil(49/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 2493
} |
[
{
"role": "user",
"content": "solve: 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 solutions for $\\tan(x)=k$ are:\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 2494
} |
[
{
"role": "user",
"content": "Please solve 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 $\n\nThe general ... | {
"type": "trigonometry",
"seed": 1337,
"index": 2495
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. 293x - 421 = 914"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 293x - 421 = 914 $\n\nMove all terms to the left (standard form):\n$ 293x + -1335 = 0 $\n\nIsolate the $x$ term:\n$ 293x = 1335 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 2496
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Compute the indefinite integral: $\\int 1\\sin(8x) + 8\\cos(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1\\sin(8x) + 8\\cos(4x)\\,dx $\n\nUse standard antiderivatives (and i... | {
"type": "calculus",
"seed": 1337,
"index": 2497
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Find the modular inverse of 17 modulo 18, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $17\\cdot a^{-1} \\equiv 1 \\pmod{18}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guara... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2498
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. 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 value).\n$ ... | {
"type": "trigonometry",
"seed": 1337,
"index": 2499
} |
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