messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Please evaluate and compute this equation. 608x - 684 = -979"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 608x - 684 = -979 $\n\nMove all terms to the left (standard form):\n$ 608x + 295 = 0 $\n\nIsolate the $x$ term:\n$ 608x = -295 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 6900
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Prove that $f(x)=9x^2+3x+-1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=9x^2+3x+-1$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 18... | {
"type": "optimization",
"seed": 1337,
"index": 6901
} |
[
{
"role": "user",
"content": "Can you evaluate this math 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... | {
"type": "optimization",
"seed": 1337,
"index": 6902
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. 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).... | {
"type": "trigonometry",
"seed": 1337,
"index": 6903
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((114 + 408/17) * 12) + 3^4 - log_8(64) + (308 mod 27) + round(13/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((114 + 408/17) * 12) + 3^4 - log_8(64) + (308 mod 27) + round(13/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 6904
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value... | {
"type": "trigonometry",
"seed": 1337,
"index": 6905
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? -45 - 125 - 40x^2 = -161x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -45 - 125 - 40x^2 = -161x $\n\nMove all terms to the left (standard form):\n$ -40x^2 + 161x + -170 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6906
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Compute $(180+166) \\bmod 16$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (180+166) \\bmod 16 $\n\nUse the rule that addition is compatible with reduction... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6907
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? 21 + 81x^2 - 91 = -36x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 21 + 81x^2 - 91 = -36x $\n\nMove all terms to the left (standard form):\n$ 81x^2 + 36x + -70 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6908
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Evaluate: ((-54 + 81/3) * 3) + 3^5 - log_9(81) + (-255 mod 6) + round(-172/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-54 + 81/3) * 3) + 3^5 - log_9(81) + (-255 mod 6) + round(-172/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 6909
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. 38x^2 - 148 + 171x = 168"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 38x^2 - 148 + 171x = 168 $\n\nMove all terms to the left (standard form):\n$ 38x^2 + 171x + -316 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6910
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 6911
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Compute the indefinite integral: $\\int 1x^3 + 3x^2 + -2x + 14\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1x^3 + 3x^2 + -2x + 14\\,dx $\n\nUse standard antiderivatives (a... | {
"type": "calculus",
"seed": 1337,
"index": 6912
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Compute the indefinite integral: $\\int \\ln(4x + 5)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(4x + 5)\\,dx $\n\nUse standard antiderivatives (and integration by parts if n... | {
"type": "calculus",
"seed": 1337,
"index": 6913
} |
[
{
"role": "user",
"content": "Please determine the value of 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": 6914
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. 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)... | {
"type": "trigonometry",
"seed": 1337,
"index": 6915
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Compute the determinant: $A=\\begin{pmatrix}5&3\\\\6&0\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}5&3\\\\\\6&0\\end{pmatrix} $\n\nUse... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6916
} |
[
{
"role": "user",
"content": "Please calculate this equation. -91 - 719 = -977x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -91 - 719 = -977x $\n\nMove all terms to the left (standard form):\n$ 977x + -810 = 0 $\n\nIsolate the $x$ term:\n$ 977x = 810 $\n\nDivide both sides by t... | {
"type": "linear_equation",
"seed": 1337,
"index": 6917
} |
[
{
"role": "user",
"content": "Can you determine the solution 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 $a,b\\in\\m... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6918
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n Evaluate: ((51 + -418/19) * 3) + 5^4 - log_2(4) + (-345 mod 5) + floor(161/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((51 + -418/19) * 3) + 5^4 - log_2(4) + (-345 mod 5) + floor(161/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 6919
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Evaluate: ((187 + 700/25) * 3) + 9^3 - log_7(343) + (-48 mod 9) + floor(-49/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((187 + 700/25) * 3) + 9^3 - log_7(343) + (-48 mod 9) + floor(-49/2) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6920
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Evaluate: ((-66 + 714/17) * 18) + 4^4 - log_5(125) + (390 mod 20) + round(64/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-66 + 714/17) * 18) + 4^4 - log_5(125) + (390 mod 20) + round(64/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 6921
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}8&-3\\\\\\1&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}29\\\\\\-1\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}8&-3\\\\\\1&-5\\end... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6922
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Evaluate: ((-137 + 396/18) * 15) + 4^3 - log_4(64) + (-422 mod 21) + floor(182/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-137 + 396/18) * 15) + 4^3 - log_4(64) + (-422 mod 21) + floor(182/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 6923
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. 21 + 132x^2 = -82"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 21 + 132x^2 = -82 $\n\nMove all terms to the left (standard form):\n$ 132x^2 + 0x + 103 = 0 $\n\nApply the quadratic formula:\n$ x = \\fra... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6924
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Prove that $(\\mathbb{Z}_25, +)$ is a group under addition modulo 25."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_25=\\{0,1,\\dots,24\\}$ with operation $+$ modulo $25$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6925
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. -23 + 84 - 173x = -34x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -23 + 84 - 173x = -34x^2 $\n\nMove all terms to the left (standard form):\n$ 34x^2 + -173x + 61 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6926
} |
[
{
"role": "user",
"content": "Please find and report the solution to 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 ... | {
"type": "trigonometry",
"seed": 1337,
"index": 6927
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. 17x^2 + 138 - 3x = 200"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 17x^2 + 138 - 3x = 200 $\n\nMove all terms to the left (standard form):\n$ 17x^2 + -3x + -62 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6928
} |
[
{
"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": 6929
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. 35 + 134x^2 - 166 = -108x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 35 + 134x^2 - 166 = -108x $\n\nMove all terms to the left (standard form):\n$ 134x^2 + 108x + -131 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6930
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Compute the indefinite integral: $\\int (-3x + -10)e^{2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-3x + -10)e^{2x}\\,dx $\n\nUse standard antiderivatives (and integrati... | {
"type": "calculus",
"seed": 1337,
"index": 6931
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Evaluate: ((-195 + -40/2) * 8) + 3^4 - log_10(10000) + (-314 mod 15) + ceil(-123/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-195 + -40/2) * 8) + 3^4 - log_10(10000) + (-314 mod 15) + ceil(-12... | {
"type": "arithmetic",
"seed": 1337,
"index": 6932
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 733x - 103 = 341"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 733x - 103 = 341 $\n\nMove all terms to the left (standard form):\n$ 733x + -444 = 0 $\n\nIsolate the $x$ term:\n$ 733x = 444 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 6933
} |
[
{
"role": "user",
"content": "Compute and 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)=... | {
"type": "optimization",
"seed": 1337,
"index": 6934
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Prove that $f(x,y)=\\frac12(1x^2+7y^2)+-2x+-1y+3$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+7y^2)+-2x+-1y+3 $\n\n$ \\frac{\\partial^2 f}{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 6935
} |
[
{
"role": "user",
"content": "Compute the correct solution 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 6936
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. -39x^2 - 179 - 81 = -118x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -39x^2 - 179 - 81 = -118x $\n\nMove all terms to the left (standard form):\n$ -39x^2 + 118x + -260 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6937
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Evaluate: ((-41 + -625/25) * 4) + 5^2 - log_10(1000000) + (-16 mod 16) + round(97/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-41 + -625/25) * 4) + 5^2 - log_10(1000000) + (-16 mod 16) + round(97/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6938
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Prove that $(\\mathbb{Z}_17, +)$ is a group under addition modulo 17."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_17=\\{0,1,\\dots,16\\}$ with operation $+$ modulo $17$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6939
} |
[
{
"role": "user",
"content": "Resolve the equation provided. 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) = 0 $\n\nThe general solutions ... | {
"type": "trigonometry",
"seed": 1337,
"index": 6940
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Evaluate: ((35 + 300/15) * 10) + 6^5 - log_8(8) + (-84 mod 12) + ceil(-117/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((35 + 300/15) * 10) + 6^5 - log_8(8) + (-84 mod 12) + ceil(-117/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 6941
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Compute the indefinite integral: $\\int -5\\sin(6x) + 3\\cos(6x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -5\\sin(6x) + 3\\cos(6x)\\,dx $\n\nUse standard antiderivatives (and... | {
"type": "calculus",
"seed": 1337,
"index": 6942
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Prove that $(\\mathbb{Z}_28, +)$ is a group under addition modulo 28."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_28=\\{0,1,\\dots,27\\}$ with operation $+$ modulo $28$.\n\n**Closure:** If $a,b\\in\\math... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6943
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. 37x^2 - 124 + 117x = -32"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 37x^2 - 124 + 117x = -32 $\n\nMove all terms to the left (standard form):\n$ 37x^2 + 117x + -92 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6944
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Evaluate: ((-125 + 123/3) * 10) + 4^2 - log_10(100000) + (252 mod 5) + floor(175/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-125 + 123/3) * 10) + 4^2 - log_10(100000) + (252 mod 5) + floor(175/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6945
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Differentiate with respect to $x$: $f(x) = -5\\sin(9x) + -3\\cos(1x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -5\\sin(9x) + -3\\cos(1x) $\n\nApply the standard differentiation rules:\n$ f'(x... | {
"type": "calculus",
"seed": 1337,
"index": 6946
} |
[
{
"role": "user",
"content": "Please evaluate and solve 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\nThe gen... | {
"type": "trigonometry",
"seed": 1337,
"index": 6947
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Prove that $f(x)=5x^2+-6x+-2$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+-6x+-2$.\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": 6948
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Evaluate: ((-89 + 312/12) * 8) + 6^3 - log_4(16) + (-340 mod 28) + floor(-130/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-89 + 312/12) * 8) + 6^3 - log_4(16) + (-340 mod 28) + floor(-130/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 6949
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}6&-6\\\\\\2&-6\\end{pmatrix},\\quad b=\\begin{pmatrix}-6\\\\\\6\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}6&-6\\\\\\2&-6\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6950
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Evaluate: ((19 + -182/13) * 16) + 4^4 - log_5(15625) + (-251 mod 2) + round(-44/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((19 + -182/13) * 16) + 4^4 - log_5(15625) + (-251 mod 2) + round(-44/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6951
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Find the modular inverse of 33 modulo 40, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $33\\cdot a^{-1} \\equiv 1 \\pmod{40}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guaran... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6952
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, he... | {
"type": "optimization",
"seed": 1337,
"index": 6953
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Find the modular inverse of 19 modulo 23, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $19\\cdot a^{-1} \\equiv 1 \\pmod{23}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarant... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6954
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? 170x - 540 = -953"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 170x - 540 = -953 $\n\nMove all terms to the left (standard form):\n$ 170x + 413 = 0 $\n\nIsolate the $x$ term:\n$ 170x = -413 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 6955
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. 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) = 3.1... | {
"type": "trigonometry",
"seed": 1337,
"index": 6956
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Find $A^{-1}$ for $A=\\begin{pmatrix}-1&-7\\\\8&-2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -1 & -7 \\\\ 8 & -2 \\end{pmatrix} $\n\nCompute the d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6957
} |
[
{
"role": "user",
"content": "Please calculate this equation. Compute $(-63+42) \\bmod 7$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-63+42) \\bmod 7 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ (... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6958
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. 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": 6959
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. 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": 6960
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Differentiate with respect to $x$: $f(x) = -1x^3 + 8x^2 + 9x + -14$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -1x^3 + 8x^2 + 9x + -14 $\n\nApply the standard differentiation rules:\n... | {
"type": "calculus",
"seed": 1337,
"index": 6961
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. -43x^2 + 116 - 57 = 178x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -43x^2 + 116 - 57 = 178x $\n\nMove all terms to the left (standard form):\n$ -43x^2 + -178x + 59 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6962
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Evaluate: ((27 + -448/14) * 8) + 8^3 - log_5(125) + (-310 mod 4) + round(167/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((27 + -448/14) * 8) + 8^3 - log_5(125) + (-310 mod 4) + round(167/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 6963
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. 36 + 155 - 145x^2 = 93x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 36 + 155 - 145x^2 = 93x $\n\nMove all terms to the left (standard form):\n$ -145x^2 + -93x + 191 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6964
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((-19 + 688/16) * 11) + 12^2 - log_2(64) + (358 mod 29) + ceil(-125/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-19 + 688/16) * 11) + 12^2 - log_2(64) + (358 mod 29) + ceil(-12... | {
"type": "arithmetic",
"seed": 1337,
"index": 6965
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Compute the indefinite integral: $\\int -7e^{1x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -7e^{1x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed... | {
"type": "calculus",
"seed": 1337,
"index": 6966
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Evaluate: ((-78 + 592/16) * 4) + 11^4 - log_3(27) + (-423 mod 29) + round(14/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-78 + 592/16) * 4) + 11^4 - log_3(27) + (-423 mod 29) + round(14/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 6967
} |
[
{
"role": "user",
"content": "solve: Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-6&-6\\\\\\2&-6\\end{pmatrix},\\quad b=\\begin{pmatrix}-6\\\\\\50\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-6&-6\\\\\\2&-6\\end{pmatrix} ,\\quad b=\\begin{pm... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6968
} |
[
{
"role": "user",
"content": "Compute the outcome of 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... | {
"type": "optimization",
"seed": 1337,
"index": 6969
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Prove that $f(x,y)=\\frac12(8x^2+3y^2)+-2x+3y+-10$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+3y^2)+-2x+3y+-10 $\n\n$ \\frac{\\partial^... | {
"type": "optimization",
"seed": 1337,
"index": 6970
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Prove that $f(x,y)=\\frac12(8x^2+6y^2)+3x+3y+4$ 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+3y+4 $\n\n$ \\frac{\\partial^2 f}{\\partia... | {
"type": "optimization",
"seed": 1337,
"index": 6971
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Prove that $f(x,y)=\\frac12(3x^2+10y^2)+1x+-1y+-1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+10y^2)+1x+-1y+-1 $\n\n$ \\frac{\\partial^2 f}... | {
"type": "optimization",
"seed": 1337,
"index": 6972
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Compute the indefinite integral: $\\int 6e^{-2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 6e^{-2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed... | {
"type": "calculus",
"seed": 1337,
"index": 6973
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. 502x + 885 = 106"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 502x + 885 = 106 $\n\nMove all terms to the left (standard form):\n$ 502x + 779 = 0 $\n\nIsolate the $x$ term:\n$ 502x = -779 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6974
} |
[
{
"role": "user",
"content": "Resolve the equation provided. -29 - 195 - 10x^2 = 174x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -29 - 195 - 10x^2 = 174x $\n\nMove all terms to the left (standard form):\n$ -10x^2 + -174x + -224 = 0 $\n\nApply the quadratic formula:\n$ x = \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6975
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Find $A^{-1}$ for $A=\\begin{pmatrix}-7&-1\\\\-3&-2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -7 & -1 \\\\ -3 & -2 \\end{pmatrix} $\n\nCompute th... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6976
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n Evaluate: ((-32 + 235/5) * 10) + 9^5 - log_2(8) + (-338 mod 13) + floor(-15/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-32 + 235/5) * 10) + 9^5 - log_2(8) + (-338 mod 13) + floor(-15/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6977
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Evaluate: ((-62 + 198/18) * 3) + 2^5 - log_2(8) + (-93 mod 26) + round(179/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-62 + 198/18) * 3) + 2^5 - log_2(8) + (-93 mod 26) + round(179/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 6978
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n 702 - 265 = 620x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 702 - 265 = 620x $\n\nMove all terms to the left (standard form):\n$ -620x + 437 = 0 $\n\nIsolate the $x$ term:\n$ -620x = -437 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6979
} |
[
{
"role": "user",
"content": "Please solve this math problem. Evaluate: ((155 + 473/11) * 5) + 2^5 - log_8(262144) + (-86 mod 8) + round(71/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((155 + 473/11) * 5) + 2^5 - log_8(262144) + (-86 mod 8) + round(71/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 6980
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Compute $(153+105) \\bmod 21$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (153+105) \\bmod 21 $\n\nUse the rule that addition is compatible with reduction mo... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6981
} |
[
{
"role": "user",
"content": "Can you compute this equation? -22 + 147x + 4 = -94x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -22 + 147x + 4 = -94x^2 $\n\nMove all terms to the left (standard form):\n$ 94x^2 + 147x + -18 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6982
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Evaluate: ((-200 + -28/7) * 12) + 5^4 - log_8(262144) + (29 mod 8) + floor(-186/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-200 + -28/7) * 12) + 5^4 - log_8(262144) + (29 mod 8) + floor(-186/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6983
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-2&-2\\\\\\-8&4\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\-68\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-2&-2\\\\\\-... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6984
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n -759 + 371x = 572"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -759 + 371x = 572 $\n\nMove all terms to the left (standard form):\n$ 371x + -1331 = 0 $\n\nIsolate the $x$ term:\n$ 371x = 1331 $\... | {
"type": "linear_equation",
"seed": 1337,
"index": 6985
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Find the modular inverse of 11 modulo 36, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $11\\cdot a^{-1} \\equiv 1 \\pmod{36}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guar... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6986
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Compute the indefinite integral: $\\int 2x^3 + 2x^2 + 6x + -11\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + 2x^2 + 6x + -11\\,dx $\n\nUse standard antiderivative... | {
"type": "calculus",
"seed": 1337,
"index": 6987
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. - - 191x^2 + 167 = -128x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ - - 191x^2 + 167 = -128x $\n\nMove all terms to the left (standard form):\n$ -191x^2 + 128x + 167 = 0 $\n\nApply the quadratic f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6988
} |
[
{
"role": "user",
"content": "solve: 987 + 648x = -182"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 987 + 648x = -182 $\n\nMove all terms to the left (standard form):\n$ 648x + 1169 = 0 $\n\nIsolate the $x$ term:\n$ 648x = -1169 $\n\nDivide both sides by the coefficient of $x$:... | {
"type": "linear_equation",
"seed": 1337,
"index": 6989
} |
[
{
"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": 6990
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-5&-2&2\\\\\\1&-1&-2\\\\\\3&5&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}-30\\\\\\8\\\\\\37\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solv... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6991
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. 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": 6992
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-2&1\\\\\\4&-4\\end{pmatrix},\\quad b=\\begin{pmatrix}-4\\\\\\20\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-2&1\\\\\\4&-4\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6993
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute the indefinite integral: $\\int -1\\sin(1x) + 3\\cos(5x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -1\\sin(1x) + 3\\cos(5x)\\,dx $\n\nUse standard antiderivatives (and int... | {
"type": "calculus",
"seed": 1337,
"index": 6994
} |
[
{
"role": "user",
"content": "Please resolve the math 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$... | {
"type": "optimization",
"seed": 1337,
"index": 6995
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Evaluate: ((-56 + -330/11) * 3) + 12^3 - log_8(4096) + (-392 mod 17) + floor(-122/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-56 + -330/11) * 3) + 12^3 - log_8(4096) + (-392 mod 17) + floor(-122/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 6996
} |
[
{
"role": "user",
"content": "Please solve this math problem. Evaluate: ((103 + -196/7) * 13) + 2^4 - log_8(8) + (289 mod 16) + ceil(25/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((103 + -196/7) * 13) + 2^4 - log_8(8) + (289 mod 16) + ceil(25/2) $\n\nCompute each compo... | {
"type": "arithmetic",
"seed": 1337,
"index": 6997
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Differentiate with respect to $x$: $f(x) = (-5x + -6)\\sin(3x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-5x + -6)\\sin(3x) $\n\nApply the standard differentiation rules:\n$ f'(x... | {
"type": "calculus",
"seed": 1337,
"index": 6998
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. 16 + 111 + 191x^2 = 170x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 16 + 111 + 191x^2 = 170x $\n\nMove all terms to the left (standard form):\n$ 191x^2 + -170x + 127 = 0 $\n\nApply the quadrat... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6999
} |
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