messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Find the numerical 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": 2800
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Prove that $(\\mathbb{Z}_18, +)$ is a group under addition modulo 18."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_18=\\{0,1,\\dots,17\\}$ with operation $+$ modulo $18$.\n\n**Closure:** If $a,b\\in\\ma... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2801
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. 286 - 769x = -992"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 286 - 769x = -992 $\n\nMove all terms to the left (standard form):\n$ -769x + 1278 = 0 $\n\nIsolate the $x$ term:\n$ -769x = -1278 $\n\nD... | {
"type": "linear_equation",
"seed": 1337,
"index": 2802
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Evaluate: ((-67 + -391/23) * 1) + 9^5 - log_8(64) + (205 mod 18) + floor(52/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-67 + -391/23) * 1) + 9^5 - log_8(64) + (205 mod 18) + floor(52/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 2803
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{... | {
"type": "optimization",
"seed": 1337,
"index": 2804
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&5&0\\\\\\5&4&-2\\\\\\4&-2&0\\end{pmatrix},\\quad b=\\begin{pmatrix}-23\\\\\\0\\\\\\14\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2805
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Prove that $f(x,y)=\\frac12(9x^2+10y^2)+2x+6y+0$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(9x^2+10y^2)+2x+6y+0 $\n\n$ \\frac{\\partial^2 f}{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 2806
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. -852x + 870 = 480"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -852x + 870 = 480 $\n\nMove all terms to the left (standard form):\n$ -852x + 390 = 0 $\n\nIsolate the $x$ term:\n$ -852x = -390 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2807
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Find $A^{-1}$ for $A=\\begin{pmatrix}-3&6\\\\-6&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -3 & 6 \\\\ -6 & -5 \\end{pmatrix} $\n\nCompute ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2808
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Find the modular inverse of 19 modulo 31, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $19\\cdot a^{-1} \\equiv 1 \\pmod{31}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guaran... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2809
} |
[
{
"role": "user",
"content": "Solve this equation accurately. -714 - 706 = 309x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -714 - 706 = 309x $\n\nMove all terms to the left (standard form):\n$ -309x + -1420 = 0 $\n\nIsolate the $x$ term:\n$ -309x = 1420 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2810
} |
[
{
"role": "user",
"content": "Compute the solution for 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": 2811
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Find the modular inverse of 19 modulo 32, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $19\\cdot a^{-1} \\equiv 1 \\pmod{32}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guar... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2812
} |
[
{
"role": "user",
"content": "Calculate the answer for 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$ \\al... | {
"type": "trigonometry",
"seed": 1337,
"index": 2813
} |
[
{
"role": "user",
"content": "Find the numerical solution to 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": 2814
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n 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": 2815
} |
[
{
"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": 2816
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Prove that $f(x)=8x^2+10x+7$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+10x+7$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\... | {
"type": "optimization",
"seed": 1337,
"index": 2817
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Evaluate: ((-91 + 392/8) * 14) + 9^5 - log_3(81) + (103 mod 25) + round(-106/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-91 + 392/8) * 14) + 9^5 - log_3(81) + (103 mod 25) + round(-106/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 2818
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. 335 + 28x = -67"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 335 + 28x = -67 $\n\nMove all terms to the left (standard form):\n$ 28x + 402 = 0 $\n\nIsolate the $x$ term:\n$ 28x = -402 $\n\nDivide both sid... | {
"type": "linear_equation",
"seed": 1337,
"index": 2819
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. 15 - 197x + 134 = -151x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 15 - 197x + 134 = -151x^2 $\n\nMove all terms to the left (standard form):\n$ 151x^2 + -197x + 149 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2820
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Evaluate: ((-36 + 0/24) * 17) + 5^3 - log_7(2401) + (47 mod 24) + ceil(-22/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-36 + 0/24) * 17) + 5^3 - log_7(2401) + (47 mod 24) + ceil(-22/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 2821
} |
[
{
"role": "user",
"content": "Please calculate this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial^2 f}{... | {
"type": "optimization",
"seed": 1337,
"index": 2822
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Find the modular inverse of 5 modulo 6, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{6}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantee... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2823
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. -701 - 45 = 5x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -701 - 45 = 5x $\n\nMove all terms to the left (standard form):\n$ -5x + -746 = 0 $\n\nIsolate the $x$ term:\n$ -5x = 746 $\n\nDivide both sides by th... | {
"type": "linear_equation",
"seed": 1337,
"index": 2824
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Prove that $f(x)=5x^2+1x+-8$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+1x+-8$.\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": 2825
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Evaluate: ((-180 + 186/6) * 8) + 11^5 - log_2(4) + (117 mod 16) + round(-79/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-180 + 186/6) * 8) + 11^5 - log_2(4) + (117 mod 16) + round(-79/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 2826
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Compute the determinant: $A=\\begin{pmatrix}2&2\\\\0&0\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}2&2\\\\\\0&0\\end{pmatrix} $\n\nUse... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2827
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Find the modular inverse of 15 modulo 22, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $15\\cdot a^{-1} \\equiv 1 \\pmod{22}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gu... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2828
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Evaluate: ((-51 + -66/6) * 6) + 4^5 - log_2(32) + (339 mod 13) + ceil(-22/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-51 + -66/6) * 6) + 4^5 - log_2(32) + (339 mod 13) + ceil(-22/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 2829
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Prove that $f(x)=2x^2+-6x+-8$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+-6x+-8$.\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": 2830
} |
[
{
"role": "user",
"content": "Find the numerical solution to 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 = \\arct... | {
"type": "trigonometry",
"seed": 1337,
"index": 2831
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-1&8\\\\\\-5&7\\end{pmatrix},\\quad b=\\begin{pmatrix}50\\\\\\52\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-1&8\\\\\\-5&7\\en... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2832
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. -44x - 104 - 119 = 98x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -44x - 104 - 119 = 98x^2 $\n\nMove all terms to the left (standard form):\n$ -98x^2 + -44x + -223 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2833
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Evaluate: ((161 + -297/11) * 2) + 3^2 - log_5(25) + (401 mod 8) + floor(-46/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((161 + -297/11) * 2) + 3^2 - log_5(25) + (401 mod 8) + floor(-46/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 2834
} |
[
{
"role": "user",
"content": "Please calculate this equation. Compute the determinant: $A=\\begin{pmatrix}-1&-3\\\\-6&0\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-1&-3\\\\\\-6&0\\end{pmatrix} $\n\nUse $\\det(A)... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2835
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n Compute the determinant: $A=\\begin{pmatrix}3&5\\\\2&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}3&5\\\\\\2&6\\end{pmatrix} $\n\nUse ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2836
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:\n$ 4... | {
"type": "optimization",
"seed": 1337,
"index": 2837
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Find $A^{-1}$ for $A=\\begin{pmatrix}-5&-6\\\\7&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -5 & -6 \\\\ 7 & 3 \\end{pmatrix} $\n\nCompute the... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2838
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:\n... | {
"type": "optimization",
"seed": 1337,
"index": 2839
} |
[
{
"role": "user",
"content": "Solve this equation step by step. -333 + 550x = -782"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -333 + 550x = -782 $\n\nMove all terms to the left (standard form):\n$ 550x + 449 = 0 $\n\nIsolate the $x$ term:\n$ 550x = -449 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2840
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. 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": 2841
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Evaluate: ((39 + 637/13) * 2) + 2^5 - log_8(4096) + (98 mod 19) + ceil(121/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((39 + 637/13) * 2) + 2^5 - log_8(4096) + (98 mod 19) + ceil(121/2) $\n\nCompute each ... | {
"type": "arithmetic",
"seed": 1337,
"index": 2842
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Find $A^{-1}$ for $A=\\begin{pmatrix}9&-9\\\\2&-3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 9 & -9 \\\\ 2 & -3 \\end{pmatrix} $\n\nCompute the determ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2843
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Compute the determinant: $A=\\begin{pmatrix}-3&4\\\\8&-3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-3&4\\\\\\8&-3\\end{pmatrix} $\n\nUse $\\d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2844
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Compute $(-91+-80) \\bmod 7$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-91+-80) \\bmod 7 $\n\nUse the rule that addition is compatible with redu... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2845
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, henc... | {
"type": "optimization",
"seed": 1337,
"index": 2846
} |
[
{
"role": "user",
"content": "Can you solve the equation below? 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(\\frac{1}... | {
"type": "trigonometry",
"seed": 1337,
"index": 2847
} |
[
{
"role": "user",
"content": "solve 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 = \\arcsin(-\\frac{\\sqrt{3}}... | {
"type": "trigonometry",
"seed": 1337,
"index": 2848
} |
[
{
"role": "user",
"content": "Please work out 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$ 4x(x^2-1... | {
"type": "optimization",
"seed": 1337,
"index": 2849
} |
[
{
"role": "user",
"content": "Resolve the equation provided. 77 - 845 = -914x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 77 - 845 = -914x $\n\nMove all terms to the left (standard form):\n$ 914x + -768 = 0 $\n\nIsolate the $x$ term:\n$ 914x = 768 $\n\nDivide both sides by the ... | {
"type": "linear_equation",
"seed": 1337,
"index": 2850
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Find the modular inverse of 5 modulo 8, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{8}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2851
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Evaluate: ((-44 + 600/20) * 1) + 9^5 - log_7(16807) + (-465 mod 24) + round(168/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-44 + 600/20) * 1) + 9^5 - log_7(16807) + (-465 mod 24) + round(168/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 2852
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Compute the determinant: $A=\\begin{pmatrix}-4&-3\\\\-6&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-4&-3\\\\\\-6&9\\end{pmatrix} $\n\nUse $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2853
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. 307 - 480 = 950x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 307 - 480 = 950x $\n\nMove all terms to the left (standard form):\n$ -950x + -173 = 0 $\n\nIsolate the $x$ term:\n$ -950x = 173 $\n\nDiv... | {
"type": "linear_equation",
"seed": 1337,
"index": 2854
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Evaluate: ((8 + 1050/21) * 20) + 2^5 - log_7(49) + (115 mod 30) + ceil(-9/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((8 + 1050/21) * 20) + 2^5 - log_7(49) + (115 mod 30) + ceil(-9/2) $\n\nCompute each... | {
"type": "arithmetic",
"seed": 1337,
"index": 2855
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. -32x^2 - 182 - 145x = -41"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -32x^2 - 182 - 145x = -41 $\n\nMove all terms to the left (standard form):\n$ -32x^2 + -145x + -141 = 0 $\n\nApply the quad... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2856
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Evaluate: ((55 + -506/22) * 11) + 10^3 - log_7(343) + (385 mod 13) + floor(-49/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((55 + -506/22) * 11) + 10^3 - log_7(343) + (385 mod 13) + floor(-49/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 2857
} |
[
{
"role": "user",
"content": "Please calculate 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(\\frac{1}{2... | {
"type": "trigonometry",
"seed": 1337,
"index": 2858
} |
[
{
"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$... | {
"type": "optimization",
"seed": 1337,
"index": 2859
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Evaluate: ((95 + 279/9) * 2) + 3^5 - log_2(2) + (132 mod 10) + round(-88/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((95 + 279/9) * 2) + 3^5 - log_2(2) + (132 mod 10) + round(-88/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 2860
} |
[
{
"role": "user",
"content": "Please calculate this equation. 46 + 195x - 30x^2 = 93"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 46 + 195x - 30x^2 = 93 $\n\nMove all terms to the left (standard form):\n$ -30x^2 + 195x + -47 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2861
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Prove that $f(x)=5x^2+-6x+5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+-6x+5$.\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": 2862
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}3&3&-4\\\\\\2&-4&-1\\\\\\-1&2&-4\\end{pmatrix},\\quad b=\\begin{pmatrix}-12\\\\\\21\\\\\\-24\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2863
} |
[
{
"role": "user",
"content": "Evaluate the following equation. -463 + 443x = 47"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -463 + 443x = 47 $\n\nMove all terms to the left (standard form):\n$ 443x + -510 = 0 $\n\nIsolate the $x$ term:\n$ 443x = 510 $\n\nDivide both sides by th... | {
"type": "linear_equation",
"seed": 1337,
"index": 2864
} |
[
{
"role": "user",
"content": "Please compute the answer to 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 $\n\nTh... | {
"type": "trigonometry",
"seed": 1337,
"index": 2865
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. 50 - 11x^2 - 110 = 170x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 50 - 11x^2 - 110 = 170x $\n\nMove all terms to the left (standard form):\n$ -11x^2 + -170x + -60 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2866
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Differentiate with respect to $x$: $f(x) = (-1x + -3)\\sin(1x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-1x + -3)\\sin(1x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \... | {
"type": "calculus",
"seed": 1337,
"index": 2867
} |
[
{
"role": "user",
"content": "Please compute 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": 2868
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Prove that $(\\mathbb{Z}_14, +)$ is a group under addition modulo 14."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_14=\\{0,1,\\dots,13\\}$ with operation $+$ modulo $14$.\n\n**Closure:** If $a,b\\in\\math... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2869
} |
[
{
"role": "user",
"content": "Please calculate this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial^2 f}{... | {
"type": "optimization",
"seed": 1337,
"index": 2870
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? 44x + 73x^2 - 42 = -64"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 44x + 73x^2 - 42 = -64 $\n\nMove all terms to the left (standard form):\n$ 73x^2 + 44x + 22 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2871
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Compute $(113+184) \\bmod 23$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (113+184) \\bmod 23 $\n\nUse the rule that addition is compatible with reduct... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2872
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-5&-2&3\\\\\\-3&2&1\\\\\\2&2&4\\end{pmatrix},\\quad b=\\begin{pmatrix}-15\\\\\\-5\\\\\\2\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., by Ga... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2873
} |
[
{
"role": "user",
"content": "Can you compute this equation? Evaluate: ((-3 + -396/12) * 14) + 9^5 - log_2(2) + (-200 mod 22) + round(105/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-3 + -396/12) * 14) + 9^5 - log_2(2) + (-200 mod 22) + round(105/2) $\n\nCompute each ... | {
"type": "arithmetic",
"seed": 1337,
"index": 2874
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. 14 + 38x^2 + 28x = -194"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 14 + 38x^2 + 28x = -194 $\n\nMove all terms to the left (standard form):\n$ 38x^2 + 28x + 208 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2875
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. -16 - 64 - 64x^2 = 104x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -16 - 64 - 64x^2 = 104x $\n\nMove all terms to the left (standard form):\n$ -64x^2 + -104x + -80 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2876
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Prove that $f(x)=8x^2+-3x+2$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+-3x+2$.\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": 2877
} |
[
{
"role": "user",
"content": "Compute the outcome of this 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).\n$ \\alph... | {
"type": "trigonometry",
"seed": 1337,
"index": 2878
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. 462 + 634 = 67x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 462 + 634 = 67x $\n\nMove all terms to the left (standard form):\n$ -67x + 1096 = 0 $\n\nIsolate the $x$ term:\n$ -67x = -1096 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 2879
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Prove that $f(x,y)=\\frac12(3x^2+9y^2)+-5x+6y+-10$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+9y^2)+-5x+6y+-10 $\n\n$ \\frac{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 2880
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. -712 + 313 = -493x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -712 + 313 = -493x $\n\nMove all terms to the left (standard form):\n$ 493x + -399 = 0 $\n\nIsolate the $x$ term:\n$ 493x = 399 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 2881
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Find the modular inverse of 5 modulo 6, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{6}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2882
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. 24 - 136 + 39x = 27x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 24 - 136 + 39x = 27x^2 $\n\nMove all terms to the left (standard form):\n$ -27x^2 + 39x + -112 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2883
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. 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 $f'(x)=0$, hen... | {
"type": "optimization",
"seed": 1337,
"index": 2884
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. 24x^2 - 153 + 131 = 14x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 24x^2 - 153 + 131 = 14x $\n\nMove all terms to the left (standard form):\n$ 24x^2 + -14x + -22 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 2885
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Differentiate with respect to $x$: $f(x) = -5e^{1x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -5e^{1x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-5e^{1x}} $"
... | {
"type": "calculus",
"seed": 1337,
"index": 2886
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 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\nT... | {
"type": "trigonometry",
"seed": 1337,
"index": 2887
} |
[
{
"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": 2888
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Compute the determinant: $A=\\begin{pmatrix}8&1\\\\2&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}8&1\\\\\\2&-7\\end{pmatrix} $\n\nUse $\\de... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2889
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Compute the determinant: $A=\\begin{pmatrix}7&-3\\\\-7&-3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}7&-3\\\\\\-7&-3\\end{pmatrix} $\n\nU... | {
"type": "linear_algebra",
"seed": 1337,
"index": 2890
} |
[
{
"role": "user",
"content": "Can you solve the equation below? 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).\n$ \\alpha = ... | {
"type": "trigonometry",
"seed": 1337,
"index": 2891
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Differentiate with respect to $x$: $f(x) = 9e^{5x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 9e^{5x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{45e^{5x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 2892
} |
[
{
"role": "user",
"content": "Calculate and solve 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$ \\alpha = \... | {
"type": "trigonometry",
"seed": 1337,
"index": 2893
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Prove that $f(x)=3x^2+2x+0$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=3x^2+2x+0$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 6x + 2 $\... | {
"type": "optimization",
"seed": 1337,
"index": 2894
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. 559x + 641 = 878"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 559x + 641 = 878 $\n\nMove all terms to the left (standard form):\n$ 559x + -237 = 0 $\n\nIsolate the $x$ term:\n$ 559x = 237 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 2895
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Find the modular inverse of 5 modulo 21, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{21}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guaran... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2896
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Prove that $(\\mathbb{Z}_7, +)$ is a group under addition modulo 7."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_7=\\{0,1,\\dots,6\\}$ with operation $+$ modulo $7$.\n\n**Closure:** If $a,b\\in\\mathbb{... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 2897
} |
[
{
"role": "user",
"content": "Please work out this equation. Prove that $f(x,y)=\\frac12(8x^2+8y^2)+0x+-6y+-8$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+8y^2)+0x+-6y+-8 $\n\n$ \\frac{\\partial^2 f}{\\partial x^2... | {
"type": "optimization",
"seed": 1337,
"index": 2898
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Prove that $f(x,y)=\\frac12(10x^2+2y^2)+-2x+-4y+-9$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(10x^2+2y^2)+-2x+-4y+-9 $\n\n$ \\frac{\\partial^2... | {
"type": "optimization",
"seed": 1337,
"index": 2899
} |
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