messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Solve the equation and show the result. 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$ \\alph... | {
"type": "trigonometry",
"seed": 1337,
"index": 4000
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Compute $(37+180) \\bmod 21$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (37+180) \\bmod 21 $\n\nUse the rule that addition is compatible with re... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4001
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}3&-7\\\\\\6&7\\end{pmatrix},\\quad b=\\begin{pmatrix}28\\\\\\35\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}3&-7\\\\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4002
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Evaluate: ((186 + -40/4) * 7) + 3^3 - log_3(27) + (370 mod 23) + round(141/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((186 + -40/4) * 7) + 3^3 - log_3(27) + (370 mod 23) + round(141/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 4003
} |
[
{
"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": 4004
} |
[
{
"role": "user",
"content": "solve: \n\n Compute the indefinite integral: $\\int -1e^{-2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -1e^{-2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \\int -1e^{-2x}\\,dx = \\b... | {
"type": "calculus",
"seed": 1337,
"index": 4005
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Evaluate: ((126 + 504/12) * 19) + 4^5 - log_2(8) + (243 mod 19) + round(-95/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((126 + 504/12) * 19) + 4^5 - log_2(8) + (243 mod 19) + round(-95/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 4006
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Compute $(-170+-50) \\bmod 17$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-170+-50) \\bmod 17 $\n\nUse the rule that addition is compatible with reduction modulo... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4007
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. 677 + 451 = 232x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 677 + 451 = 232x $\n\nMove all terms to the left (standard form):\n$ -232x + 1128 = 0 $\n\nIsolate the $x$ term:\n$ -232x = -1128 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 4008
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-8&-7\\\\\\-1&8\\end{pmatrix},\\quad b=\\begin{pmatrix}-55\\\\\\2\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-8&-7\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4009
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. -431x + 341 = 333"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -431x + 341 = 333 $\n\nMove all terms to the left (standard form):\n$ -431x + 8 = 0 $\n\nIsolate the $x$ term:\n$ -431x = -8 $\n\nDivide both sides... | {
"type": "linear_equation",
"seed": 1337,
"index": 4010
} |
[
{
"role": "user",
"content": "Determine the correct answer to 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$... | {
"type": "trigonometry",
"seed": 1337,
"index": 4011
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. -28x + 170x^2 - 23 = -63"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -28x + 170x^2 - 23 = -63 $\n\nMove all terms to the left (standard form):\n$ 170x^2 + -28x + 40 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4012
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Compute the indefinite integral: $\\int (-3x + -5)e^{3x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-3x + -5)e^{3x}\\,dx $\n\nUse standard antiderivatives (and integratio... | {
"type": "calculus",
"seed": 1337,
"index": 4013
} |
[
{
"role": "user",
"content": "solve -426 - 986 = -351x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -426 - 986 = -351x $\n\nMove all terms to the left (standard form):\n$ 351x + -1412 = 0 $\n\nIsolate the $x$ term:\n$ 351x = 1412 $\n\nDivide both sides by the coefficient of $x$... | {
"type": "linear_equation",
"seed": 1337,
"index": 4014
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Evaluate: ((-42 + 156/4) * 11) + 12^5 - log_8(32768) + (-149 mod 27) + round(133/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-42 + 156/4) * 11) + 12^5 - log_8(32768) + (-149 mod 27) + round(1... | {
"type": "arithmetic",
"seed": 1337,
"index": 4015
} |
[
{
"role": "user",
"content": "solve: \n\n 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}{2}) = 0.523599 $\n\nT... | {
"type": "trigonometry",
"seed": 1337,
"index": 4016
} |
[
{
"role": "user",
"content": "Determine the correct solution for 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... | {
"type": "optimization",
"seed": 1337,
"index": 4017
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. 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": 4018
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Compute $(-116+20) \\bmod 17$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-116+20) \\bmod 17 $\n\nUse the rule that addition is compatible with reduction mod... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4019
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}6&4\\\\\\-2&4\\end{pmatrix},\\quad b=\\begin{pmatrix}42\\\\\\18\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}6&4\\\\\\-2&... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4020
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Evaluate: ((-130 + 288/8) * 17) + 5^2 - log_2(16) + (257 mod 18) + round(115/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-130 + 288/8) * 17) + 5^2 - log_2(16) + (257 mod 18) + round(1... | {
"type": "arithmetic",
"seed": 1337,
"index": 4021
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Evaluate: ((175 + 484/22) * 3) + 2^3 - log_7(117649) + (175 mod 6) + floor(49/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((175 + 484/22) * 3) + 2^3 - log_7(117649) + (175 mod 6) + floor(49/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 4022
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}3&-5&5\\\\\\2&-1&4\\\\\\0&1&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}-1\\\\\\-3\\\\\\-1\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4023
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? 43x - 122 + 6 = 27x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 43x - 122 + 6 = 27x^2 $\n\nMove all terms to the left (standard form):\n$ -27x^2 + 43x + -116 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4024
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Compute the indefinite integral: $\\int 8e^{-3x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 8e^{-3x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \\int 8e^{-3x}\\,dx = \\... | {
"type": "calculus",
"seed": 1337,
"index": 4025
} |
[
{
"role": "user",
"content": "Work through and solve 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$ \\alpha ... | {
"type": "trigonometry",
"seed": 1337,
"index": 4026
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. 6x + 45 + 5 = -158x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 6x + 45 + 5 = -158x^2 $\n\nMove all terms to the left (standard form):\n$ 158x^2 + 6x + 50 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4027
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-1&-2\\\\\\-9&5\\end{pmatrix},\\quad b=\\begin{pmatrix}2\\\\\\-74\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-1&-2\\\\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4028
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. 8 + 3 - 144x = -19x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 8 + 3 - 144x = -19x^2 $\n\nMove all terms to the left (standard form):\n$ 19x^2 + -144x + 11 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4029
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. -25x^2 + 73 + 54 = 18x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -25x^2 + 73 + 54 = 18x $\n\nMove all terms to the left (standard form):\n$ -25x^2 + -18x + 127 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4030
} |
[
{
"role": "user",
"content": "Provide the solution to the following 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": 4031
} |
[
{
"role": "user",
"content": "Please solve this math problem. -127x + 499 = 865"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -127x + 499 = 865 $\n\nMove all terms to the left (standard form):\n$ -127x + -366 = 0 $\n\nIsolate the $x$ term:\n$ -127x = 366 $\n\nDivide both sides by... | {
"type": "linear_equation",
"seed": 1337,
"index": 4032
} |
[
{
"role": "user",
"content": "Determine the correct answer to 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": 4033
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. -787 - 898x = 203"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -787 - 898x = 203 $\n\nMove all terms to the left (standard form):\n$ -898x + -990 = 0 $\n\nIsolate the $x$ term:\n$ -898x = 990 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 4034
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Evaluate: ((-14 + 782/17) * 13) + 8^5 - log_4(16) + (-255 mod 14) + floor(-102/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-14 + 782/17) * 13) + 8^5 - log_4(16) + (-255 mod 14) + floor(-102/... | {
"type": "arithmetic",
"seed": 1337,
"index": 4035
} |
[
{
"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": 4036
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Compute the determinant: $A=\\begin{pmatrix}9&2\\\\7&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}9&2\\\\\\7&5\\end{pmatrix} $\n\nUse $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4037
} |
[
{
"role": "user",
"content": "Evaluate 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 $f'(x)=0$, hence:\n$ 4x(x^2... | {
"type": "optimization",
"seed": 1337,
"index": 4038
} |
[
{
"role": "user",
"content": "solve: Prove that $f(x,y)=\\frac12(7x^2+10y^2)+0x+6y+5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(7x^2+10y^2)+0x+6y+5 $\n\n$ \\frac{\\partial^2 f}{\\partial x^2} = 7,\\quad \\frac{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 4039
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. 32x - 82x^2 - 95 = -150"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 32x - 82x^2 - 95 = -150 $\n\nMove all terms to the left (standard form):\n$ -82x^2 + 32x + 55 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4040
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. -149 + 507 = -697x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -149 + 507 = -697x $\n\nMove all terms to the left (standard form):\n$ 697x + 358 = 0 $\n\nIsolate the $x$ term:\n$ 697x = -358 $\n... | {
"type": "linear_equation",
"seed": 1337,
"index": 4041
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Find the modular inverse of 14 modulo 17, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $14\\cdot a^{-1} \\equiv 1 \\pmod{17}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity g... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4042
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Evaluate: ((64 + 560/14) * 16) + 4^3 - log_9(59049) + (-305 mod 29) + round(-159/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((64 + 560/14) * 16) + 4^3 - log_9(59049) + (-305 mod 29) + round(-15... | {
"type": "arithmetic",
"seed": 1337,
"index": 4043
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Solve for $x$ (in radians): sin(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 4044
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, he... | {
"type": "optimization",
"seed": 1337,
"index": 4045
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Compute the indefinite integral: $\\int \\ln(8x + -4)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(8x + -4)\\,dx $\n\nUse standard antiderivatives (and integration by parts ... | {
"type": "calculus",
"seed": 1337,
"index": 4046
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((122 + -136/8) * 5) + 10^5 - log_9(729) + (44 mod 9) + floor(122/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((122 + -136/8) * 5) + 10^5 - log_9(729) + (44 mod 9) + floor(122/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 4047
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Evaluate: ((194 + -900/20) * 14) + 5^2 - log_6(216) + (380 mod 29) + round(47/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((194 + -900/20) * 14) + 5^2 - log_6(216) + (380 mod 29) + round(47/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 4048
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. 13 + 98x^2 + 43x = -11"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 13 + 98x^2 + 43x = -11 $\n\nMove all terms to the left (standard form):\n$ 98x^2 + 43x + 24 = 0 $\n\nApply the quadratic formula:\n$ x = \\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4049
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Compute the indefinite integral: $\\int 5\\sin(2x) + -1\\cos(7x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 5\\sin(2x) + -1\\cos(7x)\\,dx $\n\nUse standard antiderivatives (and integration ... | {
"type": "calculus",
"seed": 1337,
"index": 4050
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. -25x^2 + 42 - 191x = -179"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -25x^2 + 42 - 191x = -179 $\n\nMove all terms to the left (standard form):\n$ -25x^2 + -191x + 221 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4051
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Find $A^{-1}$ for $A=\\begin{pmatrix}-6&-8\\\\4&0\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & -8 \\\\ 4 & 0 \\end{pmatrix} $\n\nCompute the d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4052
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Differentiate with respect to $x$: $f(x) = 3x^3 + -4x^2 + 20x + -17$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 3x^3 + -4x^2 + 20x + -17 $\n\nApply the standard differentiation rules:\n$ f'(x) = \\... | {
"type": "calculus",
"seed": 1337,
"index": 4053
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Evaluate: ((142 + -594/18) * 3) + 5^4 - log_10(10000) + (-80 mod 27) + ceil(197/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((142 + -594/18) * 3) + 5^4 - log_10(10000) + (-80 mod 27) + ceil(197... | {
"type": "arithmetic",
"seed": 1337,
"index": 4054
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Find the modular inverse of 13 modulo 20, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $13\\cdot a^{-1} \\equiv 1 \\pmod{20}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantee... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4055
} |
[
{
"role": "user",
"content": "Can you compute this equation? Compute the indefinite integral: $\\int 3e^{5x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 3e^{5x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \\int 3e^{5... | {
"type": "calculus",
"seed": 1337,
"index": 4056
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given 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 va... | {
"type": "trigonometry",
"seed": 1337,
"index": 4057
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:... | {
"type": "optimization",
"seed": 1337,
"index": 4058
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Find $A^{-1}$ for $A=\\begin{pmatrix}-6&-5\\\\-1&-6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & -5 \\\\ -1 & -6 \\end{pmatrix} $\n\nCo... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4059
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Differentiate with respect to $x$: $f(x) = \\ln(4x + 9)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(4x + 9) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\f... | {
"type": "calculus",
"seed": 1337,
"index": 4060
} |
[
{
"role": "user",
"content": "Resolve the equation provided. -489 - 943 = -512x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -489 - 943 = -512x $\n\nMove all terms to the left (standard form):\n$ 512x + -1432 = 0 $\n\nIsolate the $x$ term:\n$ 512x = 1432 $\n\nDivide both sides b... | {
"type": "linear_equation",
"seed": 1337,
"index": 4061
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. -28x + 120x^2 - 92 = -197"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -28x + 120x^2 - 92 = -197 $\n\nMove all terms to the left (standard form):\n$ 120x^2 + -28x + 105 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4062
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}5&-1\\\\\\9&9\\end{pmatrix},\\quad b=\\begin{pmatrix}25\\\\\\45\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}5&-1\\\\\\9&9\\end{pmatrix} ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4063
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: 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)=0 \\Rightarro... | {
"type": "optimization",
"seed": 1337,
"index": 4064
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Compute the indefinite integral: $\\int 3x^3 + 7x^2 + 16x + -10\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 3x^3 + 7x^2 + 16x + -10\\,dx $\n\nUse standard antiderivatives (and ... | {
"type": "calculus",
"seed": 1337,
"index": 4065
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Prove that $f(x)=4x^2+-9x+0$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=4x^2+-9x+0$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 8x +... | {
"type": "optimization",
"seed": 1337,
"index": 4066
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Evaluate: ((-108 + 228/6) * 1) + 12^3 - log_3(81) + (-73 mod 23) + round(87/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-108 + 228/6) * 1) + 12^3 - log_3(81) + (-73 mod 23) + round(87/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 4067
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 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 $\n\nThe genera... | {
"type": "trigonometry",
"seed": 1337,
"index": 4068
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. 6 + 231 = -368x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 6 + 231 = -368x $\n\nMove all terms to the left (standard form):\n$ 368x + 237 = 0 $\n\nIsolate the $x$ term:\n$ 368x = -237 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 4069
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? 156x - 960 = -774"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 156x - 960 = -774 $\n\nMove all terms to the left (standard form):\n$ 156x + -186 = 0 $\n\nIsolate the $x$ term:\n$ 156x = 186 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 4070
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Evaluate: ((146 + -260/10) * 8) + 6^4 - log_4(256) + (-289 mod 3) + floor(-172/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((146 + -260/10) * 8) + 6^4 - log_4(256) + (-289 mod 3) + floor(-172/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 4071
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Evaluate: ((3 + -950/19) * 3) + 4^4 - log_9(6561) + (272 mod 17) + ceil(-187/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((3 + -950/19) * 3) + 4^4 - log_9(6561) + (272 mod 17) + ceil(-187/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 4072
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. -2x + 50 - 111 = -156x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -2x + 50 - 111 = -156x^2 $\n\nMove all terms to the left (standard form):\n$ 156x^2 + -2x + -61 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4073
} |
[
{
"role": "user",
"content": "Please calculate this equation. Find the modular inverse of 2 modulo 11, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $2\\cdot a^{-1} \\equiv 1 \\pmod{11}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integers ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4074
} |
[
{
"role": "user",
"content": "solve Prove that $(\\mathbb{Z}_27, +)$ is a group under addition modulo 27."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_27=\\{0,1,\\dots,26\\}$ with operation $+$ modulo $27$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, then $(a+b)\\bmod n\\in\\{0,1... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4075
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Differentiate with respect to $x$: $f(x) = 5x^3 + -6x^2 + -3x + 12$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 5x^3 + -6x^2 + -3x + 12 $\n\nApply the standard differentiation rules:\n$ f'... | {
"type": "calculus",
"seed": 1337,
"index": 4076
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. 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 ... | {
"type": "calculus",
"seed": 1337,
"index": 4077
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. 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": 4078
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. -839x + 494 = -215"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -839x + 494 = -215 $\n\nMove all terms to the left (standard form):\n$ -839x + 709 = 0 $\n\nIsolate the $x$ term:\n$ -839x = -709 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 4079
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. -775x - 977 = 395"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -775x - 977 = 395 $\n\nMove all terms to the left (standard form):\n$ -775x + -1372 = 0 $\n\nIsolate the $x$ term:\n$ -775x = 1372 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 4080
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Prove that $f(x,y)=\\frac12(7x^2+8y^2)+-5x+-3y+2$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(7x^2+8y^2)+-5x+-3y+2 $\n\n$ \\frac{\\partial^2 f}{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 4081
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. 862 + 176x = -462"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 862 + 176x = -462 $\n\nMove all terms to the left (standard form):\n$ 176x + 1324 = 0 $\n\nIsolate the $x$ term:\n$ 176x = -1324 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 4082
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Solve for $x$ (in radians): tan(x) = -\\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -\\frac{1}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arc... | {
"type": "trigonometry",
"seed": 1337,
"index": 4083
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Evaluate: ((119 + 660/20) * 18) + 2^3 - log_9(81) + (28 mod 2) + floor(175/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((119 + 660/20) * 18) + 2^3 - log_9(81) + (28 mod 2) + floor(175/2) $\n\nCompute e... | {
"type": "arithmetic",
"seed": 1337,
"index": 4084
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Prove that $f(x,y)=\\frac12(3x^2+4y^2)+6x+3y+-7$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+4y^2)+6x+3y+-7 $\n\n$ \\frac{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 4085
} |
[
{
"role": "user",
"content": "Find the answer to the following 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 $\n\nT... | {
"type": "trigonometry",
"seed": 1337,
"index": 4086
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. -18 + 37 + 192x = -191x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -18 + 37 + 192x = -191x^2 $\n\nMove all terms to the left (standard form):\n$ 191x^2 + 192x + 19 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4087
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Evaluate: ((-132 + 117/3) * 10) + 11^4 - log_3(243) + (103 mod 15) + ceil(82/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-132 + 117/3) * 10) + 11^4 - log_3(243) + (103 mod 15) + ceil(82/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 4088
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. 45x + 155x^2 + 89 = 41"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 45x + 155x^2 + 89 = 41 $\n\nMove all terms to the left (standard form):\n$ 155x^2 + 45x + 48 = 0 $\n\nApply the quadratic formula:\n$ x = \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 4089
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Prove that $f(x,y)=\\frac12(5x^2+8y^2)+-2x+3y+-9$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(5x^2+8y^2)+-2x+3y+-9 $\n\n$ \\frac{\\partial^2 f... | {
"type": "optimization",
"seed": 1337,
"index": 4090
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Find $A^{-1}$ for $A=\\begin{pmatrix}7&9\\\\9&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 7 & 9 \\\\ 9 & -7 \\end{pmatrix} $\n\nCompute the... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4091
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}1&-5\\\\3&-1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 1 & -5 \\\\ 3 & -1 \\end{pmatrix} $\n\nCompute... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4092
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Prove that $f(x)=5x^2+1x+-3$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+1x+-3$.\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": 4093
} |
[
{
"role": "user",
"content": "Solve the following math equation. Prove that $f(x)=5x^2+-8x+-5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+-8x+-5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 10x ... | {
"type": "optimization",
"seed": 1337,
"index": 4094
} |
[
{
"role": "user",
"content": "Please find the solution of this 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 = \\arcsi... | {
"type": "trigonometry",
"seed": 1337,
"index": 4095
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Prove that $(\\mathbb{Z}_11, +)$ is a group under addition modulo 11."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_11=\\{0,1,\\dots,10\\}$ with operation $+$ modulo $11$.\n\n**Closure:** If $a,b\\in\\m... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 4096
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Compute the indefinite integral: $\\int -6x^3 + -1x^2 + -10x + 16\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -6x^3 + -1x^2 + -10x + 16\\,dx $\n\nUse standard antiderivatives ... | {
"type": "calculus",
"seed": 1337,
"index": 4097
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-5&2\\\\3&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -5 & 2 \\\\ 3 & 5 \\end{pmatrix} $\n\nCompute the d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 4098
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Evaluate: ((76 + 110/5) * 4) + 11^4 - log_3(3) + (-481 mod 22) + ceil(127/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((76 + 110/5) * 4) + 11^4 - log_3(3) + (-481 mod 22) + ceil(127/2) $\n\nCompute e... | {
"type": "arithmetic",
"seed": 1337,
"index": 4099
} |
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