messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Prove that $f(x)=5x^2+6x+-4$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+6x+-4$.\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": 7900
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. 26 + 186x + 156x^2 = 163"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 26 + 186x + 156x^2 = 163 $\n\nMove all terms to the left (standard form):\n$ 156x^2 + 186x + -137 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7901
} |
[
{
"role": "user",
"content": "Can you compute this equation? Compute the determinant: $A=\\begin{pmatrix}2&-8\\\\4&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}2&-8\\\\\\4&6\\end{pmatrix} $\n\nUse $\\det(A)=ad-b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7902
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Solve for $x$ (in radians): sin(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 0 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(0) = 0 $\n\nThe general... | {
"type": "trigonometry",
"seed": 1337,
"index": 7903
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. 513 + 919x = -295"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 513 + 919x = -295 $\n\nMove all terms to the left (standard form):\n$ 919x + 808 = 0 $\n\nIsolate the $x$ term:\n$ 919x = -808 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 7904
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$... | {
"type": "trigonometry",
"seed": 1337,
"index": 7905
} |
[
{
"role": "user",
"content": "Can you compute this equation? Evaluate: ((-131 + 180/12) * 13) + 7^4 - log_8(8) + (488 mod 14) + ceil(-175/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-131 + 180/12) * 13) + 7^4 - log_8(8) + (488 mod 14) + ceil(-175/2) $\n\nCompute each ... | {
"type": "arithmetic",
"seed": 1337,
"index": 7906
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Differentiate with respect to $x$: $f(x) = -5\\sin(9x) + -7\\cos(7x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -5\\sin(9x) + -7\\cos(7x) $\n\nApply the standard differentiation r... | {
"type": "calculus",
"seed": 1337,
"index": 7907
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-9&1\\\\2&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -9 & 1 \\\\ 2 & 1 \\end{pmatrix} $\n\nCompute the determi... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7908
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Solve for $x$ (in radians): cos(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 7909
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Find the modular inverse of 16 modulo 19, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $16\\cdot a^{-1} \\equiv 1 \\pmod{19}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guaran... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7910
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 676 - 885 = -663x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 676 - 885 = -663x $\n\nMove all terms to the left (standard form):\n$ 663x + -209 = 0 $\n\nIsolate the $x$ term:\n$ 663x = 209 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7911
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. 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 $\n\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 7912
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. -629 - 14x = 590"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -629 - 14x = 590 $\n\nMove all terms to the left (standard form):\n$ -14x + -1219 = 0 $\n\nIsolate the $x$ term:\n$ -14x = 1219 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 7913
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Find the modular inverse of 5 modulo 26, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{26}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integers $x,y$ with:\n$ 5... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7914
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. 44 - 95x^2 + 68x = 129"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 44 - 95x^2 + 68x = 129 $\n\nMove all terms to the left (standard form):\n$ -95x^2 + 68x + -85 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7915
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Evaluate: ((-144 + -304/8) * 3) + 3^4 - log_8(4096) + (-270 mod 14) + floor(188/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-144 + -304/8) * 3) + 3^4 - log_8(4096) + (-270 mod 14) + floor(188/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 7916
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. -41 - 44x + 87x^2 = -125"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -41 - 44x + 87x^2 = -125 $\n\nMove all terms to the left (standard form):\n$ 87x^2 + -44x + 84 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7917
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Prove that $f(x,y)=\\frac12(5x^2+8y^2)+3x+-3y+-1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(5x^2+8y^2)+3x+-3y+-1 $\n\n$ \\frac{\\partial^2 f}{\\part... | {
"type": "optimization",
"seed": 1337,
"index": 7918
} |
[
{
"role": "user",
"content": "Work out the result of this 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 $a,b\\in\\mathbb{Z}_n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7919
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:\n$... | {
"type": "optimization",
"seed": 1337,
"index": 7920
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Evaluate: ((-154 + -588/12) * 2) + 6^3 - log_9(729) + (-472 mod 12) + floor(-80/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-154 + -588/12) * 2) + 6^3 - log_9(729) + (-472 mod 12) + floor(-80... | {
"type": "arithmetic",
"seed": 1337,
"index": 7921
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. 25 - 115x - 50 = 16x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 25 - 115x - 50 = 16x^2 $\n\nMove all terms to the left (standard form):\n$ -16x^2 + -115x + -25 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7922
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-4&8\\\\\\-3&5\\end{pmatrix},\\quad b=\\begin{pmatrix}-40\\\\\\-26\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-4&8\\\\\\-3&5\\en... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7923
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$... | {
"type": "trigonometry",
"seed": 1337,
"index": 7924
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Compute the indefinite integral: $\\int 1\\sin(3x) + 6\\cos(7x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1\\sin(3x) + 6\\cos(7x)\\,dx $\n\nUse standard antiderivatives (and integration ... | {
"type": "calculus",
"seed": 1337,
"index": 7925
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-3&-3&5\\\\\\3&4&4\\\\\\0&0&4\\end{pmatrix},\\quad b=\\begin{pmatrix}23\\\\\\13\\\\\\16\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7926
} |
[
{
"role": "user",
"content": "Can you solve the equation below? 381 - 824 = -586x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 381 - 824 = -586x $\n\nMove all terms to the left (standard form):\n$ 586x + -443 = 0 $\n\nIsolate the $x$ term:\n$ 586x = 443 $\n\nDivide both sides by... | {
"type": "linear_equation",
"seed": 1337,
"index": 7927
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. -33x - 21 + 56x^2 = 115"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -33x - 21 + 56x^2 = 115 $\n\nMove all terms to the left (standard form):\n$ 56x^2 + -33x + -136 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7928
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Evaluate: ((-84 + -441/21) * 15) + 9^2 - log_7(2401) + (31 mod 7) + round(-52/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-84 + -441/21) * 15) + 9^2 - log_7(2401) + (31 mod 7) + round(-52/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 7929
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Prove that $f(x)=7x^2+-7x+5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+-7x+5$.\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": 7930
} |
[
{
"role": "user",
"content": "Please work out this equation. 14x + 75 - 191 = -115x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 14x + 75 - 191 = -115x^2 $\n\nMove all terms to the left (standard form):\n$ 115x^2 + 14x + -116 = 0 $\n\nApply the quadratic formula:\n$ x = \\f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7931
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. -31 - 193 + 91x = -112x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -31 - 193 + 91x = -112x^2 $\n\nMove all terms to the left (standard form):\n$ 112x^2 + 91x + -224 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7932
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Solve for $x$ (in radians): tan(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value... | {
"type": "trigonometry",
"seed": 1337,
"index": 7933
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Prove that $f(x)=2x^2+-4x+-2$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+-4x+-2$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$... | {
"type": "optimization",
"seed": 1337,
"index": 7934
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Differentiate with respect to $x$: $f(x) = \\ln(4x + 9)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(4x + 9) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac... | {
"type": "calculus",
"seed": 1337,
"index": 7935
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. 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} & \\fra... | {
"type": "optimization",
"seed": 1337,
"index": 7936
} |
[
{
"role": "user",
"content": "solve: \n\n Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}4&3&0\\\\\\0&3&0\\\\\\-1&2&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}25\\\\\\9\\\\\\8\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., by Gaussian elimination).\n$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7937
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Evaluate: ((174 + -481/13) * 18) + 9^4 - log_6(1296) + (234 mod 11) + floor(38/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((174 + -481/13) * 18) + 9^4 - log_6(1296) + (234 mod 11) + floor(38/... | {
"type": "arithmetic",
"seed": 1337,
"index": 7938
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. -866x + 586 = -195"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -866x + 586 = -195 $\n\nMove all terms to the left (standard form):\n$ -866x + 781 = 0 $\n\nIsolate the $x$ term:\n$ -866x = -781 $\n\nDi... | {
"type": "linear_equation",
"seed": 1337,
"index": 7939
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. -20 + 59x^2 + 25 = -32x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -20 + 59x^2 + 25 = -32x $\n\nMove all terms to the left (standard form):\n$ 59x^2 + 32x + 5 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7940
} |
[
{
"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": 7941
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Find the modular inverse of 12 modulo 19, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $12\\cdot a^{-1} \\equiv 1 \\pmod{19}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity g... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7942
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Prove that $f(x)=8x^2+-6x+7$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+-6x+7$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 16... | {
"type": "optimization",
"seed": 1337,
"index": 7943
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. -196 + 523x = -872"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -196 + 523x = -872 $\n\nMove all terms to the left (standard form):\n$ 523x + 676 = 0 $\n\nIsolate the $x$ term:\n$ 523x = -676 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 7944
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. Prove that $(\\mathbb{Z}_16, +)$ is a group under addition modulo 16."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_16=\\{0,1,\\dots,15\\}$ with operation $+$ modulo $16$.\n\n**Closure:** If $a,b\\in\\mathb... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7945
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Evaluate: ((167 + -1080/24) * 5) + 9^2 - log_2(2) + (-341 mod 10) + round(-191/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((167 + -1080/24) * 5) + 9^2 - log_2(2) + (-341 mod 10) + roun... | {
"type": "arithmetic",
"seed": 1337,
"index": 7946
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Evaluate: ((100 + -420/12) * 18) + 3^3 - log_3(81) + (279 mod 26) + round(33/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((100 + -420/12) * 18) + 3^3 - log_3(81) + (279 mod 26) + round(33/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 7947
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. 28x + 23 - 151 = 55x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 28x + 23 - 151 = 55x^2 $\n\nMove all terms to the left (standard form):\n$ -55x^2 + 28x + -128 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7948
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. 25 + 87x^2 - 33x = 57"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 25 + 87x^2 - 33x = 57 $\n\nMove all terms to the left (standard form):\n$ 87x^2 + -33x + -32 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7949
} |
[
{
"role": "user",
"content": "Please determine the solution 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 $f'(x)=0$, h... | {
"type": "optimization",
"seed": 1337,
"index": 7950
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute $(-169+-40) \\bmod 11$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-169+-40) \\bmod 11 $\n\nUse the rule that addition is compatible with reduction m... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7951
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Compute the indefinite integral: $\\int \\ln(5x + -8)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(5x + -8)\\,dx $\n\nUse standard antiderivatives (and integration by parts... | {
"type": "calculus",
"seed": 1337,
"index": 7952
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Evaluate: ((169 + 352/11) * 1) + 11^3 - log_5(3125) + (466 mod 28) + floor(-150/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((169 + 352/11) * 1) + 11^3 - log_5(3125) + (466 mod 28) + floor(-150/... | {
"type": "arithmetic",
"seed": 1337,
"index": 7953
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. 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).... | {
"type": "trigonometry",
"seed": 1337,
"index": 7954
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Evaluate: ((-57 + 228/19) * 13) + 2^2 - log_9(531441) + (-236 mod 22) + round(15/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-57 + 228/19) * 13) + 2^2 - log_9(531441) + (-236 mod 22) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 7955
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Evaluate: ((-100 + 154/7) * 1) + 8^4 - log_7(49) + (396 mod 11) + ceil(27/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-100 + 154/7) * 1) + 8^4 - log_7(49) + (396 mod 11) + ceil(27/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 7956
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-5&2\\\\\\-6&8\\end{pmatrix},\\quad b=\\begin{pmatrix}2\\\\\\-20\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-5&2\\\\\\-6&8... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7957
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. 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 parts if ne... | {
"type": "calculus",
"seed": 1337,
"index": 7958
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n 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).\... | {
"type": "trigonometry",
"seed": 1337,
"index": 7959
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Prove that $(\\mathbb{Z}_3, +)$ is a group under addition modulo 3."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_3=\\{0,1,\\dots,2\\}$ with operation $+$ modulo $3$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, the... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7960
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. 11x + 87 - 39x^2 = -180"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 11x + 87 - 39x^2 = -180 $\n\nMove all terms to the left (standard form):\n$ -39x^2 + 11x + 267 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7961
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. Evaluate: ((36 + 235/5) * 10) + 6^2 - log_5(25) + (-420 mod 10) + ceil(-133/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((36 + 235/5) * 10) + 6^2 - log_5(25) + (-420 mod 10) + ceil(-133/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 7962
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Differentiate with respect to $x$: $f(x) = -6\\sin(3x) + 6\\cos(9x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -6\\sin(3x) + 6\\cos(9x) $\n\nApply the standard differentiation rules:... | {
"type": "calculus",
"seed": 1337,
"index": 7963
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? -27 - 76x + 79 = 169x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -27 - 76x + 79 = 169x^2 $\n\nMove all terms to the left (standard form):\n$ -169x^2 + -76x + 52 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7964
} |
[
{
"role": "user",
"content": "Please calculate this equation. 461 - 809 = 119x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 461 - 809 = 119x $\n\nMove all terms to the left (standard form):\n$ -119x + -348 = 0 $\n\nIsolate the $x$ term:\n$ -119x = 348 $\n\nDivide both sides by t... | {
"type": "linear_equation",
"seed": 1337,
"index": 7965
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Evaluate: ((158 + 64/2) * 5) + 7^3 - log_3(9) + (155 mod 20) + floor(138/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((158 + 64/2) * 5) + 7^3 - log_3(9) + (155 mod 20) + floor(138/2) $\n\nCompute each... | {
"type": "arithmetic",
"seed": 1337,
"index": 7966
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. 9 + 131 + 167x^2 = 47x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 9 + 131 + 167x^2 = 47x $\n\nMove all terms to the left (standard form):\n$ 167x^2 + -47x + 140 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7967
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}6&2\\\\\\-8&0\\end{pmatrix},\\quad b=\\begin{pmatrix}34\\\\\\-48\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}6&2\\\\\\-8&0\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7968
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Evaluate: ((126 + -160/16) * 8) + 7^5 - log_7(343) + (173 mod 22) + round(-43/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((126 + -160/16) * 8) + 7^5 - log_7(343) + (173 mod 22) + round(-4... | {
"type": "arithmetic",
"seed": 1337,
"index": 7969
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Solve for $x$ (in radians): cos(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = 0 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(0) = 1.5708 $\n\nThe ge... | {
"type": "trigonometry",
"seed": 1337,
"index": 7970
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Differentiate with respect to $x$: $f(x) = 4\\sin(5x) + -3\\cos(2x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 4\\sin(5x) + -3\\cos(2x) $\n\nApply the standard differentiation rules:\n... | {
"type": "calculus",
"seed": 1337,
"index": 7971
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Find the modular inverse of 29 modulo 34, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $29\\cdot a^{-1} \\equiv 1 \\pmod{34}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guar... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7972
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Compute the indefinite integral: $\\int -6\\sin(7x) + 6\\cos(5x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -6\\sin(7x) + 6\\cos(5x)\\,dx $\n\nUse standard antiderivatives (an... | {
"type": "calculus",
"seed": 1337,
"index": 7973
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Evaluate: ((101 + 500/10) * 3) + 9^5 - log_3(243) + (-244 mod 19) + floor(-78/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((101 + 500/10) * 3) + 9^5 - log_3(243) + (-244 mod 19) + floor(-78/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 7974
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Find $A^{-1}$ for $A=\\begin{pmatrix}-9&1\\\\-9&-1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -9 & 1 \\\\ -9 & -1 \\end{pmatrix} $\n\nCo... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7975
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Solve for $x$ (in radians): sin(x) = -\\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(-\... | {
"type": "trigonometry",
"seed": 1337,
"index": 7976
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. -35 + 73x - 97 = 121x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -35 + 73x - 97 = 121x^2 $\n\nMove all terms to the left (standard form):\n$ -121x^2 + 73x + -132 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7977
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Differentiate with respect to $x$: $f(x) = -2\\sin(4x) + -7\\cos(9x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -2\\sin(4x) + -7\\cos(9x) $\n\nApply the standard differentiation rules:\n$ f... | {
"type": "calculus",
"seed": 1337,
"index": 7978
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. 411x + 146 = -983"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 411x + 146 = -983 $\n\nMove all terms to the left (standard form):\n$ 411x + 1129 = 0 $\n\nIsolate the $x$ term:\n$ 411x = -1129 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7979
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}7&4\\\\\\6&0\\end{pmatrix},\\quad b=\\begin{pmatrix}-27\\\\\\-30\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}7&4\\\\\\6&0\\end{... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7980
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-6&6\\\\1&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & 6 \\\\ 1 & -7 \\end{pmatrix} $\n\nCompute the det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7981
} |
[
{
"role": "user",
"content": "solve: Evaluate: ((21 + -105/5) * 7) + 6^4 - log_2(4) + (11 mod 14) + floor(-37/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((21 + -105/5) * 7) + 6^4 - log_2(4) + (11 mod 14) + floor(-37/2) $\n\nCompute each component:\n$ -105/5 = -21 $\n$... | {
"type": "arithmetic",
"seed": 1337,
"index": 7982
} |
[
{
"role": "user",
"content": "Can you solve the equation below? 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": 7983
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Find $A^{-1}$ for $A=\\begin{pmatrix}-6&-4\\\\0&2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & -4 \\\\ 0 & 2 \\end{pmatrix} $\n\nCompute the de... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7984
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Solve for $x$ (in radians): tan(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -1 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(-1) = -0.785398 $\n\... | {
"type": "trigonometry",
"seed": 1337,
"index": 7985
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Find the modular inverse of 31 modulo 38, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $31\\cdot a^{-1} \\equiv 1 \\pmod{38}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarant... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7986
} |
[
{
"role": "user",
"content": "Please determine the value of this 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 = \\arc... | {
"type": "trigonometry",
"seed": 1337,
"index": 7987
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-6&4\\\\\\7&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}12\\\\\\-38\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-6&4\\\\\\7&-2\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7988
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. 23x^2 - 120x - 165 = 48"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 23x^2 - 120x - 165 = 48 $\n\nMove all terms to the left (standard form):\n$ 23x^2 + -120x + -213 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7989
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Solve for $x$ (in radians): cos(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -1 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(-1) = 3.14159 $\n\nThe... | {
"type": "trigonometry",
"seed": 1337,
"index": 7990
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 524x + 13 = -694"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 524x + 13 = -694 $\n\nMove all terms to the left (standard form):\n$ 524x + 707 = 0 $\n\nIsolate the $x$ term:\n$ 524x = -707 $\n\nDivide both sides by... | {
"type": "linear_equation",
"seed": 1337,
"index": 7991
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Evaluate: ((-135 + 9/9) * 13) + 12^5 - log_7(2401) + (90 mod 19) + floor(122/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-135 + 9/9) * 13) + 12^5 - log_7(2401) + (90 mod 19) + floor(122/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 7992
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. -21x - 37 + 81x^2 = 62"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -21x - 37 + 81x^2 = 62 $\n\nMove all terms to the left (standard form):\n$ 81x^2 + -21x + -99 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7993
} |
[
{
"role": "user",
"content": "Can you find the solution to this 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": 7994
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. 282x + 504 = -102"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 282x + 504 = -102 $\n\nMove all terms to the left (standard form):\n$ 282x + 606 = 0 $\n\nIsolate the $x$ term:\n$ 282x = -606 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 7995
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. -447x + 344 = 519"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -447x + 344 = 519 $\n\nMove all terms to the left (standard form):\n$ -447x + -175 = 0 $\n\nIsolate the $x$ term:\n$ -447x = 175 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 7996
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Solve for $x$ (in radians): sin(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ ... | {
"type": "trigonometry",
"seed": 1337,
"index": 7997
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Evaluate: ((38 + 38/19) * 10) + 3^4 - log_8(4096) + (-51 mod 8) + ceil(-133/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((38 + 38/19) * 10) + 3^4 - log_8(4096) + (-51 mod 8) + ceil(-133/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 7998
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. 118 - 386 = -36x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 118 - 386 = -36x $\n\nMove all terms to the left (standard form):\n$ 36x + -268 = 0 $\n\nIsolate the $x$ term:\n$ 36x = 268 $\n\nDivide both sides b... | {
"type": "linear_equation",
"seed": 1337,
"index": 7999
} |
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