messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Find the modular inverse of 6 modulo 31, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $6\\cdot a^{-1} \\equiv 1 \\pmod{31}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees in... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5800
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Evaluate: ((-131 + -99/3) * 2) + 6^2 - log_2(64) + (-482 mod 18) + ceil(-74/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-131 + -99/3) * 2) + 6^2 - log_2(64) + (-482 mod 18) + ceil(-74/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 5801
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Differentiate with respect to $x$: $f(x) = \\ln(8x + 0)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(8x + 0) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed... | {
"type": "calculus",
"seed": 1337,
"index": 5802
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Compute the indefinite integral: $\\int 1\\sin(5x) + 2\\cos(9x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1\\sin(5x) + 2\\cos(9x)\\,dx $\n\nUse standard antiderivatives (a... | {
"type": "calculus",
"seed": 1337,
"index": 5803
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Compute the indefinite integral: $\\int 5\\sin(8x) + 7\\cos(6x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 5\\sin(8x) + 7\\cos(6x)\\,dx $\n\nUse standard antiderivatives (an... | {
"type": "calculus",
"seed": 1337,
"index": 5804
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}7&7\\\\-4&-2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 7 & 7 \\\\ -4 & -2 \\end{pmatrix} $\n\nCompute ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5805
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. 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$... | {
"type": "trigonometry",
"seed": 1337,
"index": 5806
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. 20x^2 + 46 - 163x = 61"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 20x^2 + 46 - 163x = 61 $\n\nMove all terms to the left (standard form):\n$ 20x^2 + -163x + -15 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5807
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Differentiate with respect to $x$: $f(x) = (-6x + -10)\\sin(2x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-6x + -10)\\sin(2x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \... | {
"type": "calculus",
"seed": 1337,
"index": 5808
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n 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... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5809
} |
[
{
"role": "user",
"content": "Please calculate this equation. Differentiate with respect to $x$: $f(x) = (5x + -4)\\sin(5x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (5x + -4)\\sin(5x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{5\\sin(5... | {
"type": "calculus",
"seed": 1337,
"index": 5810
} |
[
{
"role": "user",
"content": "Please solve this math problem. Prove that $f(x,y)=\\frac12(2x^2+9y^2)+1x+4y+1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(2x^2+9y^2)+1x+4y+1 $\n\n$ \\frac{\\partial^2 f}{\\partial x^2} =... | {
"type": "optimization",
"seed": 1337,
"index": 5811
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Evaluate: ((-170 + 510/15) * 19) + 10^4 - log_8(64) + (183 mod 11) + floor(24/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-170 + 510/15) * 19) + 10^4 - log_8(64) + (183 mod 11) + floor(24/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 5812
} |
[
{
"role": "user",
"content": "Solve the following math equation. Differentiate with respect to $x$: $f(x) = -3\\sin(7x) + 5\\cos(3x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -3\\sin(7x) + 5\\cos(3x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \... | {
"type": "calculus",
"seed": 1337,
"index": 5813
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Compute the determinant: $A=\\begin{pmatrix}8&4\\\\-6&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}8&4\\\\\\-6&5\\end{pmatrix} $\n\nUse $\\de... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5814
} |
[
{
"role": "user",
"content": "Work out the result of this equation. 42 + 170x^2 - 147 = -79x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 42 + 170x^2 - 147 = -79x $\n\nMove all terms to the left (standard form):\n$ 170x^2 + 79x + -105 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5815
} |
[
{
"role": "user",
"content": "Please find the value of this math 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 = \\arc... | {
"type": "trigonometry",
"seed": 1337,
"index": 5816
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. -32 + 120 + 74x = 139x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -32 + 120 + 74x = 139x^2 $\n\nMove all terms to the left (standard form):\n$ -139x^2 + 74x + 88 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5817
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Differentiate with respect to $x$: $f(x) = 5\\sin(9x) + 9\\cos(9x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 5\\sin(9x) + 9\\cos(9x) $\n\nApply the standard differentiation rules:\n$ ... | {
"type": "calculus",
"seed": 1337,
"index": 5818
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. -837 - 790x = -327"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -837 - 790x = -327 $\n\nMove all terms to the left (standard form):\n$ -790x + -510 = 0 $\n\nIsolate the $x$ term:\n$ -790x = 510 $\n\... | {
"type": "linear_equation",
"seed": 1337,
"index": 5819
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. 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 $\n\nT... | {
"type": "trigonometry",
"seed": 1337,
"index": 5820
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. -7x - 25x^2 - 117 = 48"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -7x - 25x^2 - 117 = 48 $\n\nMove all terms to the left (standard form):\n$ -25x^2 + -7x + -165 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5821
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Compute $(13+54) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (13+54) \\bmod 8 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5822
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Evaluate: ((158 + 296/8) * 16) + 11^2 - log_8(64) + (-462 mod 17) + ceil(124/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((158 + 296/8) * 16) + 11^2 - log_8(64) + (-462 mod 17) + ceil(124... | {
"type": "arithmetic",
"seed": 1337,
"index": 5823
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}0&0&1\\\\\\-4&2&4\\\\\\-5&-1&0\\end{pmatrix},\\quad b=\\begin{pmatrix}1\\\\\\8\\\\\\19\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5824
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}8&5\\\\\\2&-6\\end{pmatrix},\\quad b=\\begin{pmatrix}-75\\\\\\32\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}8&5\\\\\\2&-6\\end{pma... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5825
} |
[
{
"role": "user",
"content": "solve: \n\n Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-2&0&2\\\\\\-4&5&-4\\\\\\-2&-2&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}12\\\\\\-12\\\\\\10\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., by Gaussian eliminati... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5826
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Find $A^{-1}$ for $A=\\begin{pmatrix}-2&9\\\\-4&7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -2 & 9 \\\\ -4 & 7 \\end{pmatrix} $\n\nCompute t... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5827
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. -10 - 37 - 4x^2 = 155x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -10 - 37 - 4x^2 = 155x $\n\nMove all terms to the left (standard form):\n$ -4x^2 + -155x + -47 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5828
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Compute the determinant: $A=\\begin{pmatrix}-9&1\\\\-3&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-9&1\\\\\\-3&6\\end{pmatrix} $\n\nUse ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5829
} |
[
{
"role": "user",
"content": "solve: Evaluate: ((22 + 128/8) * 15) + 10^4 - log_8(262144) + (-248 mod 16) + round(94/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((22 + 128/8) * 15) + 10^4 - log_8(262144) + (-248 mod 16) + round(94/2) $\n\nCompute each component:\n$ 128... | {
"type": "arithmetic",
"seed": 1337,
"index": 5830
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. 48 + 137x^2 + 170 = -91x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 48 + 137x^2 + 170 = -91x $\n\nMove all terms to the left (standard form):\n$ 137x^2 + 91x + 218 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5831
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Prove that $f(x)=8x^2+-3x+-5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+-3x+-5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 1... | {
"type": "optimization",
"seed": 1337,
"index": 5832
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Compute $(28+113) \\bmod 12$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (28+113) \\bmod 12 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5833
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. 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": 5834
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Solve for $x$ (in radians): sin(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 1 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(1) = 1.5708 $\n\nThe ... | {
"type": "trigonometry",
"seed": 1337,
"index": 5835
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? 554 - 879 = -906x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 554 - 879 = -906x $\n\nMove all terms to the left (standard form):\n$ 906x + -325 = 0 $\n\nIsolate the $x$ term:\n$ 906x = 325 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 5836
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Differentiate with respect to $x$: $f(x) = (5x + 0)e^{4x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (5x + 0)e^{4x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\b... | {
"type": "calculus",
"seed": 1337,
"index": 5837
} |
[
{
"role": "user",
"content": "Compute and determine the solution to 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\\ma... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5838
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-4&-1\\\\0&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -4 & -1 \\\\ 0 & 1 \\end{pmatrix} $\n\nCompute ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5839
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. 8 + 138 - 137x = -177x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 8 + 138 - 137x = -177x^2 $\n\nMove all terms to the left (standard form):\n$ 177x^2 + -137x + 146 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5840
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Compute the indefinite integral: $\\int -3e^{4x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -3e^{4x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed)... | {
"type": "calculus",
"seed": 1337,
"index": 5841
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Evaluate: ((-106 + -350/7) * 6) + 8^4 - log_8(512) + (-455 mod 30) + floor(-142/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-106 + -350/7) * 6) + 8^4 - log_8(512) + (-455 mod 30) + floor(-142/... | {
"type": "arithmetic",
"seed": 1337,
"index": 5842
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. -308x - 649 = 37"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -308x - 649 = 37 $\n\nMove all terms to the left (standard form):\n$ -308x + -686 = 0 $\n\nIsolate the $x$ term:\n$ -308x = 686 $\n\nDivide both sid... | {
"type": "linear_equation",
"seed": 1337,
"index": 5843
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Prove that $(\\mathbb{Z}_20, +)$ is a group under addition modulo 20."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_20=\\{0,1,\\dots,19\\}$ with operation $+$ modulo $20$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, th... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5844
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-4&-2\\\\-1&-8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -4 & -2 \\\\ -1 & -8 \\end{pmatrix} $\n\nCompute the... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5845
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? -49x - 125 + 160x^2 = 9"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -49x - 125 + 160x^2 = 9 $\n\nMove all terms to the left (standard form):\n$ 160x^2 + -49x + -134 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5846
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. -27x^2 + 171 - 166 = 45x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -27x^2 + 171 - 166 = 45x $\n\nMove all terms to the left (standard form):\n$ -27x^2 + -45x + 5 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5847
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Compute the indefinite integral: $\\int -6e^{3x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -6e^{3x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed)... | {
"type": "calculus",
"seed": 1337,
"index": 5848
} |
[
{
"role": "user",
"content": "Solve the following math equation. Evaluate: ((89 + 240/12) * 7) + 2^5 - log_5(125) + (205 mod 15) + floor(-120/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((89 + 240/12) * 7) + 2^5 - log_5(125) + (205 mod 15) + floor(-120/2) $\n\nCompute e... | {
"type": "arithmetic",
"seed": 1337,
"index": 5849
} |
[
{
"role": "user",
"content": "solve: Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-4&4\\\\\\-1&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\-14\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-4&4\\\\\\-1&-3\\end{pmatrix} ,\\quad b=\\begin{p... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5850
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. 970 - 479 = 395x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 970 - 479 = 395x $\n\nMove all terms to the left (standard form):\n$ -395x + 491 = 0 $\n\nIsolate the $x$ term:\n$ -395x = -491 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 5851
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. 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 = \\arcs... | {
"type": "trigonometry",
"seed": 1337,
"index": 5852
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Compute the indefinite integral: $\\int 2x^3 + 5x^2 + -18x + -19\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + 5x^2 + -18x + -19\\,dx $\n\nUse standard antiderivatives (an... | {
"type": "calculus",
"seed": 1337,
"index": 5853
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&3&1\\\\\\-3&2&-5\\\\\\3&1&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}-1\\\\\\16\\\\\\11\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5854
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Evaluate: ((70 + 598/23) * 15) + 8^3 - log_6(46656) + (191 mod 19) + floor(-145/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((70 + 598/23) * 15) + 8^3 - log_6(46656) + (191 mod 19) + floor(-145/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 5855
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Compute the determinant: $A=\\begin{pmatrix}5&5\\\\8&4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}5&5\\\\\\8&4\\end{pmatrix} $\n\nUse $\\det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5856
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Solve for $x$ (in radians): cos(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(\\... | {
"type": "trigonometry",
"seed": 1337,
"index": 5857
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Compute the indefinite integral: $\\int -4e^{1x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -4e^{1x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):... | {
"type": "calculus",
"seed": 1337,
"index": 5858
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. - + 84x^2 + 117 = -140x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ - + 84x^2 + 117 = -140x $\n\nMove all terms to the left (standard form):\n$ 84x^2 + 140x + 117 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5859
} |
[
{
"role": "user",
"content": "Find the value that satisfies this 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": 5860
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. 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 $\n\nT... | {
"type": "trigonometry",
"seed": 1337,
"index": 5861
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. 141x + 431 = 129"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 141x + 431 = 129 $\n\nMove all terms to the left (standard form):\n$ 141x + 302 = 0 $\n\nIsolate the $x$ term:\n$ 141x = -302 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 5862
} |
[
{
"role": "user",
"content": "Please solve this mathematical 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{\\part... | {
"type": "optimization",
"seed": 1337,
"index": 5863
} |
[
{
"role": "user",
"content": "Please work out this equation. 229 - 15x = 978"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 229 - 15x = 978 $\n\nMove all terms to the left (standard form):\n$ -15x + -749 = 0 $\n\nIsolate the $x$ term:\n$ -15x = 749 $\n\nDivide both sides by the co... | {
"type": "linear_equation",
"seed": 1337,
"index": 5864
} |
[
{
"role": "user",
"content": "Work through and solve this equation. -36x^2 + 171 + 76 = 48x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -36x^2 + 171 + 76 = 48x $\n\nMove all terms to the left (standard form):\n$ -36x^2 + -48x + 247 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5865
} |
[
{
"role": "user",
"content": "Find the solution of 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{\... | {
"type": "optimization",
"seed": 1337,
"index": 5866
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Differentiate with respect to $x$: $f(x) = -5x^3 + -1x^2 + 11x + -16$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -5x^3 + -1x^2 + 11x + -16 $\n\nApply the standard differentiation rules:\n$... | {
"type": "calculus",
"seed": 1337,
"index": 5867
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Prove that $f(x)=1x^2+6x+8$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+6x+8$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n... | {
"type": "optimization",
"seed": 1337,
"index": 5868
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Prove that $f(x,y)=\\frac12(8x^2+8y^2)+-6x+6y+-7$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+8y^2)+-6x+6y+-7 $\n\n$ \\frac{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 5869
} |
[
{
"role": "user",
"content": "solve Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-2&-2\\\\\\-9&9\\end{pmatrix},\\quad b=\\begin{pmatrix}8\\\\\\126\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-2&-2\\\\\\-9&9\\end{pmatrix} ,\\quad b=\\begin{pma... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5870
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. -428 - 60x = -287"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -428 - 60x = -287 $\n\nMove all terms to the left (standard form):\n$ -60x + -141 = 0 $\n\nIsolate the $x$ term:\n$ -60x = 141 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 5871
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. -471 - 597 = 978x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -471 - 597 = 978x $\n\nMove all terms to the left (standard form):\n$ -978x + -1068 = 0 $\n\nIsolate the $x$ term:\n$ -978x = 1068 $\n\nDiv... | {
"type": "linear_equation",
"seed": 1337,
"index": 5872
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-7&7\\\\\\-1&0\\end{pmatrix},\\quad b=\\begin{pmatrix}-35\\\\\\0\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-7&7\\\\\\-1&0\\end{pmatrix... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5873
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. 19x^2 + 195x + 51 = -65"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 19x^2 + 195x + 51 = -65 $\n\nMove all terms to the left (standard form):\n$ 19x^2 + 195x + 116 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5874
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. 14x^2 - 71 - 52 = 108x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 14x^2 - 71 - 52 = 108x $\n\nMove all terms to the left (standard form):\n$ 14x^2 + -108x + -123 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5875
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Evaluate: ((-198 + -800/16) * 2) + 2^5 - log_7(49) + (85 mod 3) + ceil(199/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-198 + -800/16) * 2) + 2^5 - log_7(49) + (85 mod 3) + ceil(199/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 5876
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. -26x + 4 - 117 = 174x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -26x + 4 - 117 = 174x^2 $\n\nMove all terms to the left (standard form):\n$ -174x^2 + -26x + -113 = 0 $\n\nApply the quadrat... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5877
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Find the modular inverse of 18 modulo 29, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $18\\cdot a^{-1} \\equiv 1 \\pmod{29}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5878
} |
[
{
"role": "user",
"content": "Please solve this math problem. -x^2 - 85 - 191x = 75"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -x^2 - 85 - 191x = 75 $\n\nMove all terms to the left (standard form):\n$ -1x^2 + -191x + -160 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5879
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. -15 + 191x^2 + 103x = 43"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -15 + 191x^2 + 103x = 43 $\n\nMove all terms to the left (standard form):\n$ 191x^2 + 103x + -58 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5880
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Differentiate with respect to $x$: $f(x) = \\ln(8x + -9)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(8x + -9) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\fra... | {
"type": "calculus",
"seed": 1337,
"index": 5881
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Evaluate: ((-192 + 405/15) * 14) + 2^4 - log_6(46656) + (314 mod 3) + round(-187/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-192 + 405/15) * 14) + 2^4 - log_6(46656) + (314 mod 3) + round(-187/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 5882
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Evaluate: ((62 + 190/5) * 11) + 6^2 - log_7(343) + (-331 mod 29) + ceil(-50/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((62 + 190/5) * 11) + 6^2 - log_7(343) + (-331 mod 29) + ceil(-50/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 5883
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? -38x + 91 + 182x^2 = -82"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -38x + 91 + 182x^2 = -82 $\n\nMove all terms to the left (standard form):\n$ 182x^2 + -38x + 173 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5884
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Compute the determinant: $A=\\begin{pmatrix}-6&1\\\\-7&-1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-6&1\\\\\\-7&-1\\end{pmatrix} $\n\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5885
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Differentiate with respect to $x$: $f(x) = -6e^{-2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -6e^{-2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{12e^{-2x}} $"
... | {
"type": "calculus",
"seed": 1337,
"index": 5886
} |
[
{
"role": "user",
"content": "Calculate the solution of the 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{\... | {
"type": "optimization",
"seed": 1337,
"index": 5887
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Prove that $f(x,y)=\\frac12(6x^2+1y^2)+5x+4y+-3$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(6x^2+1y^2)+5x+4y+-3 $\n\n$ \\frac{\\partial^2 f}{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 5888
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Differentiate with respect to $x$: $f(x) = (1x + 10)e^{5x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (1x + 10)e^{5x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{... | {
"type": "calculus",
"seed": 1337,
"index": 5889
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Prove that $f(x,y)=\\frac12(8x^2+6y^2)+-3x+5y+-6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+6y^2)+-3x+5y+-6 $\n\n$ \\frac{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 5890
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}6&-3\\\\-2&-9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 6 & -3 \\\\ -2 & -9 \\end{pmatrix} $\n\nCompute ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5891
} |
[
{
"role": "user",
"content": "Work out the result of this equation. -42 - 92x^2 + 82x = 58"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -42 - 92x^2 + 82x = 58 $\n\nMove all terms to the left (standard form):\n$ -92x^2 + 82x + -100 = 0 $\n\nApply the quadratic formula:\n$ x = ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5892
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given 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 ... | {
"type": "trigonometry",
"seed": 1337,
"index": 5893
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Evaluate: ((-34 + 288/12) * 3) + 10^4 - log_6(1296) + (221 mod 13) + floor(-76/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-34 + 288/12) * 3) + 10^4 - log_6(1296) + (221 mod 13) + floor(-76/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 5894
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Find the modular inverse of 5 modulo 9, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{9}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees in... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5895
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Compute $(-157+-75) \\bmod 14$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-157+-75) \\bmod 14 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ (a+b)\\bmod... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5896
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. 40 + 5x^2 + 157x = 184"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 40 + 5x^2 + 157x = 184 $\n\nMove all terms to the left (standard form):\n$ 5x^2 + 157x + -144 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5897
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. 450 - 394x = -815"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 450 - 394x = -815 $\n\nMove all terms to the left (standard form):\n$ -394x + 1265 = 0 $\n\nIsolate the $x$ term:\n$ -394x = -1265 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 5898
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. -913 - 572x = -596"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -913 - 572x = -596 $\n\nMove all terms to the left (standard form):\n$ -572x + -317 = 0 $\n\nIsolate the $x$ term:\n$ -572x = 317 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 5899
} |
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