messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Determine the solution for this math problem. Prove that $f(x,y)=\\frac12(8x^2+6y^2)+1x+-1y+9$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+6y^2)+1x+-1y+9 $\n\n$ \\frac{\\partial^2 f}{... | {
"type": "optimization",
"seed": 1337,
"index": 500
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Evaluate: ((23 + -360/9) * 12) + 2^3 - log_5(15625) + (480 mod 27) + ceil(55/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((23 + -360/9) * 12) + 2^3 - log_5(15625) + (480 mod 27) + ceil(55/2) $\n\nCompute eac... | {
"type": "arithmetic",
"seed": 1337,
"index": 501
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Prove that $(\\mathbb{Z}_21, +)$ is a group under addition modulo 21."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_21=\\{0,1,\\dots,20\\}$ with operation $+$ modulo $21$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, then... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 502
} |
[
{
"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": 503
} |
[
{
"role": "user",
"content": "Please evaluate and compute 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": 504
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Evaluate: ((-72 + -324/9) * 9) + 12^4 - log_10(10) + (186 mod 21) + round(43/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-72 + -324/9) * 9) + 12^4 - log_10(10) + (186 mod 21) + round(43... | {
"type": "arithmetic",
"seed": 1337,
"index": 505
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Evaluate: ((100 + -368/16) * 12) + 3^3 - log_6(1296) + (-476 mod 21) + floor(106/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((100 + -368/16) * 12) + 3^3 - log_6(1296) + (-476 mod 21) + floor(106/... | {
"type": "arithmetic",
"seed": 1337,
"index": 506
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 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": 507
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 219x + 197 = -156"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 219x + 197 = -156 $\n\nMove all terms to the left (standard form):\n$ 219x + 353 = 0 $\n\nIsolate the $x$ term:\n$ 219x = -353 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 508
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical 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 val... | {
"type": "trigonometry",
"seed": 1337,
"index": 509
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Prove that $(\\mathbb{Z}_25, +)$ is a group under addition modulo 25."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_25=\\{0,1,\\dots,24\\}$ with operation $+$ modulo $25$.\n\n**Closure:** If $a,b\\in\\mathb... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 510
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. 30x^2 + 34 + 27 = -96x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 30x^2 + 34 + 27 = -96x $\n\nMove all terms to the left (standard form):\n$ 30x^2 + 96x + 61 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 511
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. Find $A^{-1}$ for $A=\\begin{pmatrix}1&0\\\\6&8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 1 & 0 \\\\ 6 & 8 \\end{pmatrix} $\n\nCompute the de... | {
"type": "linear_algebra",
"seed": 1337,
"index": 512
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Evaluate: ((-98 + -336/7) * 3) + 9^4 - log_8(262144) + (10 mod 23) + floor(-65/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-98 + -336/7) * 3) + 9^4 - log_8(262144) + (10 mod 23) + floor(-65/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 513
} |
[
{
"role": "user",
"content": "solve Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}4&0&-4\\\\\\1&-4&-3\\\\\\0&4&-1\\end{pmatrix},\\quad b=\\begin{pmatrix}12\\\\\\13\\\\\\-13\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., by Gaussian elimination).\n$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 514
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Prove that $(\\mathbb{Z}_5, +)$ is a group under addition modulo 5."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_5=\\{0,1,\\dots,4\\}$ with operation $+$ modulo $5$.\n\n**Closure:** If $a,b\\in\\mathbb{Z... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 515
} |
[
{
"role": "user",
"content": "Solve this equation accurately. -49x^2 + 45 + 179 = 138x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -49x^2 + 45 + 179 = 138x $\n\nMove all terms to the left (standard form):\n$ -49x^2 + -138x + 224 = 0 $\n\nApply the quadratic formula:\n$ x = \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 516
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Prove that $f(x,y)=\\frac12(4x^2+3y^2)+-1x+-2y+-7$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(4x^2+3y^2)+-1x+-2y+-7 $\n\n$ \\frac{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 517
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Evaluate: ((-77 + -792/24) * 3) + 9^2 - log_7(16807) + (-463 mod 20) + ceil(-140/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-77 + -792/24) * 3) + 9^2 - log_7(16807) + (-463 mod 20) + ceil(-140/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 518
} |
[
{
"role": "user",
"content": "Evaluate the following 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(\\frac{1}{... | {
"type": "trigonometry",
"seed": 1337,
"index": 519
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Solve for $x$ (in radians): cos(x) = -\\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -\\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ ... | {
"type": "trigonometry",
"seed": 1337,
"index": 520
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-1&1&0\\\\\\3&3&2\\\\\\-5&-1&4\\end{pmatrix},\\quad b=\\begin{pmatrix}2\\\\\\22\\\\\\-6\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=... | {
"type": "linear_algebra",
"seed": 1337,
"index": 521
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Compute the indefinite integral: $\\int -6e^{-5x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -6e^{-5x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed... | {
"type": "calculus",
"seed": 1337,
"index": 522
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. - - 33x^2 + 118x = 173"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ - - 33x^2 + 118x = 173 $\n\nMove all terms to the left (standard form):\n$ -33x^2 + 118x + -173 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 523
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \... | {
"type": "trigonometry",
"seed": 1337,
"index": 524
} |
[
{
"role": "user",
"content": "solve: \n\n 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 $a,b\\in\\mathbb{Z}_n$, then $(a+b)\\bmod n\\in\... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 525
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Prove that $f(x)=1x^2+-6x+-1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+-6x+-1$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\... | {
"type": "optimization",
"seed": 1337,
"index": 526
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-5&3&3\\\\\\-4&-3&-2\\\\\\0&1&-4\\end{pmatrix},\\quad b=\\begin{pmatrix}32\\\\\\5\\\\\\-1\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 527
} |
[
{
"role": "user",
"content": "solve: \n\n Compute $(-200+-180) \\bmod 23$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-200+-180) \\bmod 23 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ (a+b)\\bmod n... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 528
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Prove that $f(x)=4x^2+9x+5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=4x^2+9x+5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\... | {
"type": "optimization",
"seed": 1337,
"index": 529
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-3&-1&2\\\\\\1&-1&-1\\\\\\-3&4&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}-6\\\\\\7\\\\\\-6\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "S... | {
"type": "linear_algebra",
"seed": 1337,
"index": 530
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. 50 + 199 - 32x^2 = -197x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 50 + 199 - 32x^2 = -197x $\n\nMove all terms to the left (standard form):\n$ -32x^2 + 197x + 249 = 0 $\n\nApply the quadra... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 531
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. -39x^2 - 145 - 45 = -11x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -39x^2 - 145 - 45 = -11x $\n\nMove all terms to the left (standard form):\n$ -39x^2 + 11x + -190 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 532
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Evaluate: ((78 + -481/13) * 5) + 8^5 - log_7(343) + (-252 mod 4) + floor(107/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((78 + -481/13) * 5) + 8^5 - log_7(343) + (-252 mod 4) + floor(107/2) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 533
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. -261x = 0"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -261x = 0 $\n\nMove all terms to the left (standard form):\n$ -261x + 0 = 0 $\n\nIsolate the $x$ term:\n$ -261x = 0 $\n\nDivide both sides by the coef... | {
"type": "linear_equation",
"seed": 1337,
"index": 534
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Prove that $f(x)=8x^2+9x+1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+9x+1$.\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": 535
} |
[
{
"role": "user",
"content": "solve Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-2&1\\\\\\7&-6\\end{pmatrix},\\quad b=\\begin{pmatrix}-5\\\\\\0\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-2&1\\\\\\7&-6\\end{pmatrix} ,\\quad b=\\begin{pmatri... | {
"type": "linear_algebra",
"seed": 1337,
"index": 536
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-2&-3&-3\\\\\\-1&-2&5\\\\\\3&-2&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}6\\\\\\-23\\\\\\17\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (... | {
"type": "linear_algebra",
"seed": 1337,
"index": 537
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Find the modular inverse of 12 modulo 37, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $12\\cdot a^{-1} \\equiv 1 \\pmod{37}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guara... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 538
} |
[
{
"role": "user",
"content": "Please calculate this equation. Compute the determinant: $A=\\begin{pmatrix}9&-1\\\\-4&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}9&-1\\\\\\-4&1\\end{pmatrix} $\n\nUse $\\det(A)=a... | {
"type": "linear_algebra",
"seed": 1337,
"index": 539
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. -571x + 190 = 503"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -571x + 190 = 503 $\n\nMove all terms to the left (standard form):\n$ -571x + -313 = 0 $\n\nIsolate the $x$ term:\n$ -571x = 313 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 540
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Evaluate: ((65 + -216/8) * 5) + 4^3 - log_2(2) + (-181 mod 22) + round(130/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((65 + -216/8) * 5) + 4^3 - log_2(2) + (-181 mod 22) + round(130/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 541
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. -10x^2 - 98 + 82 = -90x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -10x^2 - 98 + 82 = -90x $\n\nMove all terms to the left (standard form):\n$ -10x^2 + 90x + -16 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 542
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Compute the indefinite integral: $\\int 2\\sin(5x) + -5\\cos(5x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2\\sin(5x) + -5\\cos(5x)\\,dx $\n\nUse standard antiderivatives (and i... | {
"type": "calculus",
"seed": 1337,
"index": 543
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Differentiate with respect to $x$: $f(x) = 2\\sin(4x) + 1\\cos(8x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 2\\sin(4x) + 1\\cos(8x) $\n\nApply the standard differentiation ru... | {
"type": "calculus",
"seed": 1337,
"index": 544
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. -740 - 460x = 269"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -740 - 460x = 269 $\n\nMove all terms to the left (standard form):\n$ -460x + -1009 = 0 $\n\nIsolate the $x$ term:\n$ -460x = 1009 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 545
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Differentiate with respect to $x$: $f(x) = -2e^{-4x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -2e^{-4x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{8e^{... | {
"type": "calculus",
"seed": 1337,
"index": 546
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Evaluate: ((-199 + 672/21) * 10) + 9^5 - log_7(343) + (-40 mod 19) + round(75/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-199 + 672/21) * 10) + 9^5 - log_7(343) + (-40 mod 19) + round(75/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 547
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Evaluate: ((-184 + -144/3) * 18) + 11^4 - log_8(512) + (-412 mod 21) + round(-179/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-184 + -144/3) * 18) + 11^4 - log_8(512) + (-412 mod 21) + round(-... | {
"type": "arithmetic",
"seed": 1337,
"index": 548
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Solve for $x$ (in radians): cos(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 549
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Find the modular inverse of 2 modulo 21, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $2\\cdot a^{-1} \\equiv 1 \\pmod{21}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 550
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. 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": 551
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. -23 + 131x + 92x^2 = 178"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -23 + 131x + 92x^2 = 178 $\n\nMove all terms to the left (standard form):\n$ 92x^2 + 131x + -201 = 0 $\n\nApply the quadratic f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 552
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. -449 - 167 = 774x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -449 - 167 = 774x $\n\nMove all terms to the left (standard form):\n$ -774x + -616 = 0 $\n\nIsolate the $x$ term:\n$ -774x = 616 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 553
} |
[
{
"role": "user",
"content": "solve: \n\n Evaluate: ((-12 + 40/4) * 10) + 5^2 - log_7(343) + (165 mod 3) + floor(-53/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-12 + 40/4) * 10) + 5^2 - log_7(343) + (165 mod 3) + floor(-53/2) $\n\nCompute each component:\n$ 40/4 = 10... | {
"type": "arithmetic",
"seed": 1337,
"index": 554
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Prove that $f(x)=2x^2+9x+0$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+9x+0$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 4x + 9 $\... | {
"type": "optimization",
"seed": 1337,
"index": 555
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\ar... | {
"type": "trigonometry",
"seed": 1337,
"index": 556
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. -26x - 35 - 157 = 179x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -26x - 35 - 157 = 179x^2 $\n\nMove all terms to the left (standard form):\n$ -179x^2 + -26x + -192 = 0 $\n\nApply the quadratic ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 557
} |
[
{
"role": "user",
"content": "Evaluate the following equation. 690 + 875 = 769x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 690 + 875 = 769x $\n\nMove all terms to the left (standard form):\n$ -769x + 1565 = 0 $\n\nIsolate the $x$ term:\n$ -769x = -1565 $\n\nDivide both sides b... | {
"type": "linear_equation",
"seed": 1337,
"index": 558
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Prove that $f(x,y)=\\frac12(4x^2+5y^2)+0x+-2y+-10$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(4x^2+5y^2)+0x+-2y+-10 $\n\n$ \\frac{\\partial^2 f}{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 559
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. -14 + 167x - 130 = -167x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -14 + 167x - 130 = -167x^2 $\n\nMove all terms to the left (standard form):\n$ 167x^2 + 167x + -144 = 0 $\n\nApply the quadrat... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 560
} |
[
{
"role": "user",
"content": "Solve the following math equation. Differentiate with respect to $x$: $f(x) = \\ln(2x + -5)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(2x + -5) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{2}{2x + ... | {
"type": "calculus",
"seed": 1337,
"index": 561
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}3&3\\\\\\-3&-6\\end{pmatrix},\\quad b=\\begin{pmatrix}-18\\\\\\12\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}3&3\\\\\\-3&-6\\end{pmatri... | {
"type": "linear_algebra",
"seed": 1337,
"index": 562
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Find $A^{-1}$ for $A=\\begin{pmatrix}-6&9\\\\8&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & 9 \\\\ 8 & -4 \\end{pmatrix} $\n\nCompute the d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 563
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Prove that $f(x)=2x^2+-4x+-6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+-4x+-6$.\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": 564
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}5&-5\\\\\\-5&-9\\end{pmatrix},\\quad b=\\begin{pmatrix}70\\\\\\28\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}5&-... | {
"type": "linear_algebra",
"seed": 1337,
"index": 565
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. 6 - 160x + 80 = 130x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 6 - 160x + 80 = 130x^2 $\n\nMove all terms to the left (standard form):\n$ -130x^2 + -160x + 86 = 0 $\n\nApply the quadratic ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 566
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Find $A^{-1}$ for $A=\\begin{pmatrix}9&5\\\\0&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 9 & 5 \\\\ 0 & -5 \\end{pmatrix} $\n\nCompute th... | {
"type": "linear_algebra",
"seed": 1337,
"index": 567
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. Evaluate: ((-149 + 260/13) * 9) + 5^3 - log_9(729) + (381 mod 16) + round(4/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-149 + 260/13) * 9) + 5^3 - log_9(729) + (381 mod 16) + round(4/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 568
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Compute the indefinite integral: $\\int \\ln(6x + -5)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(6x + -5)\\,dx $\n\nUse standard antiderivatives (and integration by parts... | {
"type": "calculus",
"seed": 1337,
"index": 569
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. 13x - 189 + 111 = -72x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 13x - 189 + 111 = -72x^2 $\n\nMove all terms to the left (standard form):\n$ 72x^2 + 13x + -78 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 570
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Compute $(17+5) \\bmod 6$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (17+5) \\bmod 6 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 571
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Evaluate: ((-81 + -336/24) * 2) + 10^4 - log_8(64) + (-165 mod 2) + floor(174/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-81 + -336/24) * 2) + 10^4 - log_8(64) + (-165 mod 2) + floor(174/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 572
} |
[
{
"role": "user",
"content": "Can you solve the equation below? -16x^2 - 119 - 145 = -6x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -16x^2 - 119 - 145 = -6x $\n\nMove all terms to the left (standard form):\n$ -16x^2 + 6x + -264 = 0 $\n\nApply the quadratic formula:\n$ x = \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 573
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Evaluate: ((-158 + 752/16) * 20) + 3^2 - log_10(10000) + (83 mod 9) + ceil(147/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-158 + 752/16) * 20) + 3^2 - log_10(10000) + (83 mod 9) + ceil(147/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 574
} |
[
{
"role": "user",
"content": "Please calculate this equation. Compute the determinant: $A=\\begin{pmatrix}4&-4\\\\1&-2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}4&-4\\\\\\1&-2\\end{pmatrix} $\n\nUse $\\det(A)=a... | {
"type": "linear_algebra",
"seed": 1337,
"index": 575
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Compute the indefinite integral: $\\int 2x^3 + 5x^2 + -4x + -9\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + 5x^2 + -4x + -9\\,dx $\n\nUse standard antiderivatives (and integ... | {
"type": "calculus",
"seed": 1337,
"index": 576
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. -19 + 4x + 32x^2 = -55"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -19 + 4x + 32x^2 = -55 $\n\nMove all terms to the left (standard form):\n$ 32x^2 + 4x + 36 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 577
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. 4x^2 - 147x - 90 = 199"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 4x^2 - 147x - 90 = 199 $\n\nMove all terms to the left (standard form):\n$ 4x^2 + -147x + -289 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 578
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. 249 - 78x = 758"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 249 - 78x = 758 $\n\nMove all terms to the left (standard form):\n$ -78x + -509 = 0 $\n\nIsolate the $x$ term:\n$ -78x = 509 $\n\nDivide both sid... | {
"type": "linear_equation",
"seed": 1337,
"index": 579
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Find $A^{-1}$ for $A=\\begin{pmatrix}4&5\\\\-2&2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 4 & 5 \\\\ -2 & 2 \\end{pmatrix} $\n\nCompute the det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 580
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. -263 - 565 = 204x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -263 - 565 = 204x $\n\nMove all terms to the left (standard form):\n$ -204x + -828 = 0 $\n\nIsolate the $x$ term:\n$ -204x = 828 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 581
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Find the modular inverse of 7 modulo 10, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $7\\cdot a^{-1} \\equiv 1 \\pmod{10}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integers ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 582
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. 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 identit... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 583
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. 334x - 363 = -728"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 334x - 363 = -728 $\n\nMove all terms to the left (standard form):\n$ 334x + 365 = 0 $\n\nIsolate the $x$ term:\n$ 334x = -365 $\n\nDi... | {
"type": "linear_equation",
"seed": 1337,
"index": 584
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. 28x + 946 = 716"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 28x + 946 = 716 $\n\nMove all terms to the left (standard form):\n$ 28x + 230 = 0 $\n\nIsolate the $x$ term:\n$ 28x = -230 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 585
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. 907x + 685 = 12"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 907x + 685 = 12 $\n\nMove all terms to the left (standard form):\n$ 907x + 673 = 0 $\n\nIsolate the $x$ term:\n$ 907x = -673 $\n\nDivide both side... | {
"type": "linear_equation",
"seed": 1337,
"index": 586
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. -477x - 882 = -315"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -477x - 882 = -315 $\n\nMove all terms to the left (standard form):\n$ -477x + -567 = 0 $\n\nIsolate the $x$ term:\n$ -477x = 567 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 587
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. 221 + 842 = 248x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 221 + 842 = 248x $\n\nMove all terms to the left (standard form):\n$ -248x + 1063 = 0 $\n\nIsolate the $x$ term:\n$ -248x = -1063 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 588
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Evaluate: ((-104 + -76/2) * 11) + 5^4 - log_10(10) + (-93 mod 14) + ceil(81/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-104 + -76/2) * 11) + 5^4 - log_10(10) + (-93 mod 14) + ceil(81/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 589
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Differentiate with respect to $x$: $f(x) = 5x^3 + -9x^2 + 5x + 3$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 5x^3 + -9x^2 + 5x + 3 $\n\nApply the standard differentiation rules:\n$ f... | {
"type": "calculus",
"seed": 1337,
"index": 590
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Evaluate: ((-77 + 38/19) * 14) + 7^2 - log_5(625) + (-27 mod 22) + round(54/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-77 + 38/19) * 14) + 7^2 - log_5(625) + (-27 mod 22) + round(54/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 591
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute $(168+53) \\bmod 19$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (168+53) \\bmod 19 $\n\nUse the rule that addition is compatible with reduction modul... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 592
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Compute the indefinite integral: $\\int 8\\sin(9x) + 5\\cos(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 8\\sin(9x) + 5\\cos(4x)\\,dx $\n\nUse standard antiderivatives (an... | {
"type": "calculus",
"seed": 1337,
"index": 593
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. 704x + 795 = -912"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 704x + 795 = -912 $\n\nMove all terms to the left (standard form):\n$ 704x + 1707 = 0 $\n\nIsolate the $x$ term:\n$ 704x = -1707 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 594
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Differentiate with respect to $x$: $f(x) = \\ln(3x + -7)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(3x + -7) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{3}{3... | {
"type": "calculus",
"seed": 1337,
"index": 595
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Compute the determinant: $A=\\begin{pmatrix}7&-8\\\\-8&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}7&-8\\\\\\-8&-4\\end{pmatrix} $\n\n... | {
"type": "linear_algebra",
"seed": 1337,
"index": 596
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Find $A^{-1}$ for $A=\\begin{pmatrix}6&8\\\\1&0\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 6 & 8 \\\\ 1 & 0 \\end{pmatrix} $\n\nCompute the d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 597
} |
[
{
"role": "user",
"content": "Determine the solution of 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} & \\frac{\... | {
"type": "optimization",
"seed": 1337,
"index": 598
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Compute the determinant: $A=\\begin{pmatrix}-9&7\\\\5&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-9&7\\\\\\5&9\\end{pmatrix} $\n\nUse ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 599
} |
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