messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Determine the correct answer to this equation. -16 - 155x^2 + 16 = -141x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -16 - 155x^2 + 16 = -141x $\n\nMove all terms to the left (standard form):\n$ -155x^2 + 141x + 0 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7000
} |
[
{
"role": "user",
"content": "solve: \n\n 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 \\Rightarrow x\... | {
"type": "optimization",
"seed": 1337,
"index": 7001
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Evaluate: ((122 + 301/7) * 19) + 5^2 - log_9(9) + (-372 mod 27) + ceil(9/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((122 + 301/7) * 19) + 5^2 - log_9(9) + (-372 mod 27) + ceil(9/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 7002
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Find $A^{-1}$ for $A=\\begin{pmatrix}5&9\\\\-7&-1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 5 & 9 \\\\ -7 & -1 \\end{pmatrix} $\n\nCompute t... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7003
} |
[
{
"role": "user",
"content": "Please determine the outcome 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 $f'(x)=0$, he... | {
"type": "optimization",
"seed": 1337,
"index": 7004
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. 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": 7005
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Compute the indefinite integral: $\\int -1x^3 + -7x^2 + 7x + 5\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -1x^3 + -7x^2 + 7x + 5\\,dx $\n\nUse standard antiderivatives (and integration by ... | {
"type": "calculus",
"seed": 1337,
"index": 7006
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. 183x + 957 = 321"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 183x + 957 = 321 $\n\nMove all terms to the left (standard form):\n$ 183x + 636 = 0 $\n\nIsolate the $x$ term:\n$ 183x = -636 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 7007
} |
[
{
"role": "user",
"content": "Please solve this math problem. 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$ \\alpha = \\ar... | {
"type": "trigonometry",
"seed": 1337,
"index": 7008
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Prove that $f(x)=7x^2+7x+-6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+7x+-6$.\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": 7009
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}2&-4&-2\\\\\\-2&3&5\\\\\\-5&-1&1\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\6\\\\\\-2\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $A... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7010
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Evaluate: ((-120 + 376/8) * 2) + 5^5 - log_9(81) + (19 mod 8) + round(-50/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-120 + 376/8) * 2) + 5^5 - log_9(81) + (19 mod 8) + round(-50/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 7011
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Differentiate with respect to $x$: $f(x) = -3e^{2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -3e^{2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-6e^{2x}} $"... | {
"type": "calculus",
"seed": 1337,
"index": 7012
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Compute the indefinite integral: $\\int (-1x + 9)\\sin(6x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-1x + 9)\\sin(6x)\\,dx $\n\nUse standard antiderivatives (and integration by... | {
"type": "calculus",
"seed": 1337,
"index": 7013
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Prove that $(\\mathbb{Z}_19, +)$ is a group under addition modulo 19."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_19=\\{0,1,\\dots,18\\}$ with operation $+$ modulo $19$.\n\n**Closure:** If $a,b\\in\\mathbb{... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7014
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Find the modular inverse of 3 modulo 20, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $3\\cdot a^{-1} \\equiv 1 \\pmod{20}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7015
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. -752 - 851 = -451x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -752 - 851 = -451x $\n\nMove all terms to the left (standard form):\n$ 451x + -1603 = 0 $\n\nIsolate the $x$ term:\n$ 451x = 1603 ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7016
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Compute the indefinite integral: $\\int 3e^{-2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 3e^{-2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if need... | {
"type": "calculus",
"seed": 1337,
"index": 7017
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Differentiate with respect to $x$: $f(x) = \\ln(5x + -3)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(5x + -3) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{5}{5x + -... | {
"type": "calculus",
"seed": 1337,
"index": 7018
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n 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 = \\arc... | {
"type": "trigonometry",
"seed": 1337,
"index": 7019
} |
[
{
"role": "user",
"content": "Please work out this equation. Solve for $x$ (in radians): tan(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = 0 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(0) = 0 $\n\nThe general solutions ... | {
"type": "trigonometry",
"seed": 1337,
"index": 7020
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. -975 - 877x = 580"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -975 - 877x = 580 $\n\nMove all terms to the left (standard form):\n$ -877x + -1555 = 0 $\n\nIsolate the $x$ term:\n$ -877x = 1555 ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7021
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Compute the indefinite integral: $\\int \\ln(5x + 7)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(5x + 7)\\,dx $\n\nUse standard antiderivatives (and integration by parts if ne... | {
"type": "calculus",
"seed": 1337,
"index": 7022
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Solve for $x$ (in radians): sin(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 1 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(1) = 1.5708 $\n\nThe... | {
"type": "trigonometry",
"seed": 1337,
"index": 7023
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n Evaluate: ((-79 + -702/18) * 6) + 8^2 - log_6(46656) + (96 mod 6) + ceil(199/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-79 + -702/18) * 6) + 8^2 - log_6(46656) + (96 mod 6) + ceil(199/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 7024
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Prove that $f(x,y)=\\frac12(1x^2+6y^2)+1x+-6y+-6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+6y^2)+1x+-6y+-6 $\n\n$ \\frac{\\partial^2 f}{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 7025
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Differentiate with respect to $x$: $f(x) = (4x + 9)e^{1x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (4x + 9)e^{1x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{4e^{1x} ... | {
"type": "calculus",
"seed": 1337,
"index": 7026
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. -36 + 49x + 32x^2 = 63"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -36 + 49x + 32x^2 = 63 $\n\nMove all terms to the left (standard form):\n$ 32x^2 + 49x + -99 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7027
} |
[
{
"role": "user",
"content": "Solve the following math 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": 7028
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. 30 - 128x^2 + 193 = 8x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 30 - 128x^2 + 193 = 8x $\n\nMove all terms to the left (standard form):\n$ -128x^2 + -8x + 223 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7029
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Differentiate with respect to $x$: $f(x) = (-6x + -8)e^{2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-6x + -8)e^{2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-6e^{2... | {
"type": "calculus",
"seed": 1337,
"index": 7030
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Solve for $x$ (in radians): sin(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ ... | {
"type": "trigonometry",
"seed": 1337,
"index": 7031
} |
[
{
"role": "user",
"content": "Can you compute this equation? Evaluate: ((-178 + 494/13) * 4) + 9^4 - log_9(531441) + (52 mod 23) + ceil(-184/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-178 + 494/13) * 4) + 9^4 - log_9(531441) + (52 mod 23) + ceil(-184/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 7032
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Differentiate with respect to $x$: $f(x) = -2e^{-4x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -2e^{-4x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{8e^{-4x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 7033
} |
[
{
"role": "user",
"content": "Evaluate the following 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": 7034
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Prove that $(\\mathbb{Z}_22, +)$ is a group under addition modulo 22."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_22=\\{0,1,\\dots,21\\}$ with operation $+$ modulo $22$.\n\n**Closure:** If $a,b\\in\\mathbb... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7035
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. 39 + 183x - 83 = 143x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 39 + 183x - 83 = 143x^2 $\n\nMove all terms to the left (standard form):\n$ -143x^2 + 183x + -44 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7036
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}1&5&-1\\\\\\-2&0&-3\\\\\\0&0&-1\\end{pmatrix},\\quad b=\\begin{pmatrix}5\\\\\\5\\\\\\-1\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7037
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Compute $(-17+61) \\bmod 20$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-17+61) \\bmod 20 $\n\nUse the rule that addition is compatible with reduction... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7038
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&-2&-2\\\\\\4&-5&1\\\\\\1&3&3\\end{pmatrix},\\quad b=\\begin{pmatrix}26\\\\\\-1\\\\\\-24\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7039
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Prove that $f(x,y)=\\frac12(1x^2+1y^2)+1x+1y+6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+1y^2)+1x+1y+6 $\n\n$ \\frac{\\partial^2 f}{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 7040
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Prove that $(\\mathbb{Z}_12, +)$ is a group under addition modulo 12."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_12=\\{0,1,\\dots,11\\}$ with operation $+$ modulo $12$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7041
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Differentiate with respect to $x$: $f(x) = -9e^{1x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -9e^{1x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-9e^{1x}} $"
... | {
"type": "calculus",
"seed": 1337,
"index": 7042
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. 20x^2 + 1x - 5 = 59"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 20x^2 + 1x - 5 = 59 $\n\nMove all terms to the left (standard form):\n$ 20x^2 + 1x + -64 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7043
} |
[
{
"role": "user",
"content": "solve: Prove that $f(x)=7x^2+0x+2$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+0x+2$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 14x + 0 $\n$ f''(x) = 14 $\n\nSince... | {
"type": "optimization",
"seed": 1337,
"index": 7044
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute the indefinite integral: $\\int 1e^{2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1e^{2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \... | {
"type": "calculus",
"seed": 1337,
"index": 7045
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Prove that $f(x,y)=\\frac12(3x^2+1y^2)+-6x+3y+-1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+1y^2)+-6x+3y+-1 $\n\n$ \\frac{\\partial^2 f}{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 7046
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Evaluate: ((-19 + -1250/25) * 4) + 4^2 - log_10(100) + (-498 mod 8) + ceil(-131/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-19 + -1250/25) * 4) + 4^2 - log_10(100) + (-498 mod 8) + ceil(-131/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 7047
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Solve for $x$ (in radians): sin(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -1 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(-1) = -1.5708 $\... | {
"type": "trigonometry",
"seed": 1337,
"index": 7048
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Differentiate with respect to $x$: $f(x) = 5\\sin(6x) + -3\\cos(6x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 5\\sin(6x) + -3\\cos(6x) $\n\nApply the standard differentiation rules:\n... | {
"type": "calculus",
"seed": 1337,
"index": 7049
} |
[
{
"role": "user",
"content": "Can you compute this equation? -7x^2 + 51 + 142 = 108x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -7x^2 + 51 + 142 = 108x $\n\nMove all terms to the left (standard form):\n$ -7x^2 + -108x + 193 = 0 $\n\nApply the quadratic formula:\n$ x = \\fra... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7050
} |
[
{
"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$... | {
"type": "optimization",
"seed": 1337,
"index": 7051
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. 13x + 28 - 35x^2 = -113"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 13x + 28 - 35x^2 = -113 $\n\nMove all terms to the left (standard form):\n$ -35x^2 + 13x + 141 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7052
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. -41x - 100x^2 + 137 = -23"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -41x - 100x^2 + 137 = -23 $\n\nMove all terms to the left (standard form):\n$ -100x^2 + -41x + 160 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7053
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Prove that $(\\mathbb{Z}_9, +)$ is a group under addition modulo 9."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_9=\\{0,1,\\dots,8\\}$ with operation $+$ modulo $9$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, th... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7054
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Evaluate: ((65 + -228/6) * 19) + 2^4 - log_8(512) + (-277 mod 14) + ceil(-30/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((65 + -228/6) * 19) + 2^4 - log_8(512) + (-277 mod 14) + ceil(-30/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 7055
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Evaluate: ((-96 + -176/16) * 14) + 12^2 - log_6(1296) + (-187 mod 14) + round(19/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-96 + -176/16) * 14) + 12^2 - log_6(1296) + (-187 mod 14) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 7056
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Differentiate with respect to $x$: $f(x) = -4e^{5x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -4e^{5x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-20e^{5x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 7057
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? -21 - 143x - 69x^2 = -90"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -21 - 143x - 69x^2 = -90 $\n\nMove all terms to the left (standard form):\n$ -69x^2 + -143x + 69 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7058
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. 21 + 134x^2 - 108x = -195"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 21 + 134x^2 - 108x = -195 $\n\nMove all terms to the left (standard form):\n$ 134x^2 + -108x + 216 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7059
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-2&6\\\\3&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -2 & 6 \\\\ 3 & 5 \\end{pmatrix} $\n\nCompute the determi... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7060
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Differentiate with respect to $x$: $f(x) = (-6x + 6)\\sin(3x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-6x + 6)\\sin(3x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{... | {
"type": "calculus",
"seed": 1337,
"index": 7061
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. 47x + 189x^2 - 34 = -107"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 47x + 189x^2 - 34 = -107 $\n\nMove all terms to the left (standard form):\n$ 189x^2 + 47x + 73 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7062
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}0&-5&-4\\\\\\1&-5&1\\\\\\0&5&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}12\\\\\\20\\\\\\-26\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7063
} |
[
{
"role": "user",
"content": "Solve the equation and provide 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 $f'(x)=0$, hence:... | {
"type": "optimization",
"seed": 1337,
"index": 7064
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Differentiate with respect to $x$: $f(x) = (-2x + 9)e^{4x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-2x + 9)e^{4x} $\n\nApply the standard differentiation rules:\n$ f'(x) = ... | {
"type": "calculus",
"seed": 1337,
"index": 7065
} |
[
{
"role": "user",
"content": "solve: Evaluate: ((-84 + 34/17) * 3) + 6^5 - log_5(15625) + (-117 mod 4) + floor(-193/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-84 + 34/17) * 3) + 6^5 - log_5(15625) + (-117 mod 4) + floor(-193/2) $\n\nCompute each component:\n$ 34/17... | {
"type": "arithmetic",
"seed": 1337,
"index": 7066
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Compute the determinant: $A=\\begin{pmatrix}3&-3\\\\6&8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}3&-3\\\\\\6&8\\end{pmatrix} $\n\n... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7067
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Evaluate: ((-129 + -517/11) * 5) + 3^2 - log_8(32768) + (-275 mod 3) + round(-136/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-129 + -517/11) * 5) + 3^2 - log_8(32768) + (-275 mod 3) + round... | {
"type": "arithmetic",
"seed": 1337,
"index": 7068
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. -11x - 54 - 184x^2 = -126"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -11x - 54 - 184x^2 = -126 $\n\nMove all terms to the left (standard form):\n$ -184x^2 + -11x + 72 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7069
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Compute the determinant: $A=\\begin{pmatrix}-1&-8\\\\-8&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-1&-8\\\\\\-8&-5\\end{pmatrix} $\n\nUse $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7070
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. 836x + 716 = -955"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 836x + 716 = -955 $\n\nMove all terms to the left (standard form):\n$ 836x + 1671 = 0 $\n\nIsolate the $x$ term:\n$ 836x = -1671 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 7071
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. 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": 7072
} |
[
{
"role": "user",
"content": "solve: \n\n 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^{5x}\\,dx = \\boxe... | {
"type": "calculus",
"seed": 1337,
"index": 7073
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Prove that $f(x)=8x^2+5x+1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+5x+1$.\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": 7074
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n 263 + 342x = -286"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 263 + 342x = -286 $\n\nMove all terms to the left (standard form):\n$ 342x + 549 = 0 $\n\nIsolate the $x$ term:\n$ 342x = -549 $\n\... | {
"type": "linear_equation",
"seed": 1337,
"index": 7075
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. -694 - 961 = -827x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -694 - 961 = -827x $\n\nMove all terms to the left (standard form):\n$ 827x + -1655 = 0 $\n\nIsolate the $x$ term:\n$ 827x = 1655 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 7076
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((-165 + 475/25) * 19) + 9^3 - log_5(25) + (-448 mod 8) + round(-155/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-165 + 475/25) * 19) + 9^3 - log_5(25) + (-448 mod 8) + round(-155/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 7077
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. 25 - 55 - 74x^2 = 187x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 25 - 55 - 74x^2 = 187x $\n\nMove all terms to the left (standard form):\n$ -74x^2 + -187x + -30 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7078
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Prove that $f(x,y)=\\frac12(6x^2+9y^2)+5x+-4y+6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(6x^2+9y^2)+5x+-4y+6 $\n\n$ \\frac{\\partial^2 f... | {
"type": "optimization",
"seed": 1337,
"index": 7079
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Differentiate with respect to $x$: $f(x) = -4\\sin(7x) + 0\\cos(4x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -4\\sin(7x) + 0\\cos(4x) $\n\nApply the standard differentiation rules:\n$ ... | {
"type": "calculus",
"seed": 1337,
"index": 7080
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Compute the indefinite integral: $\\int 7e^{-3x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 7e^{-3x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed... | {
"type": "calculus",
"seed": 1337,
"index": 7081
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}6&-5\\\\\\6&1\\end{pmatrix},\\quad b=\\begin{pmatrix}23\\\\\\53\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}6&-5\\\\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7082
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Prove that $f(x)=8x^2+-1x+-4$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+-1x+-4$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n... | {
"type": "optimization",
"seed": 1337,
"index": 7083
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Prove that $(\\mathbb{Z}_10, +)$ is a group under addition modulo 10."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_10=\\{0,1,\\dots,9\\}$ with operation $+$ modulo $10$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7084
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Evaluate: ((30 + -40/10) * 14) + 12^5 - log_8(32768) + (-142 mod 26) + ceil(52/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((30 + -40/10) * 14) + 12^5 - log_8(32768) + (-142 mod 26) + ceil(52/2) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 7085
} |
[
{
"role": "user",
"content": "Please work out 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$ \\alpha = \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 7086
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Differentiate with respect to $x$: $f(x) = \\ln(7x + -7)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(7x + -7) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\fra... | {
"type": "calculus",
"seed": 1337,
"index": 7087
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Differentiate with respect to $x$: $f(x) = -3x^3 + -6x^2 + -12x + 0$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -3x^3 + -6x^2 + -12x + 0 $\n\nApply the standard differentiation r... | {
"type": "calculus",
"seed": 1337,
"index": 7088
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Find $A^{-1}$ for $A=\\begin{pmatrix}-7&-2\\\\0&-2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -7 & -2 \\\\ 0 & -2 \\end{pmatrix} $\n\nCompute t... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7089
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Evaluate: ((-61 + -70/7) * 16) + 5^3 - log_5(3125) + (-221 mod 8) + round(-76/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-61 + -70/7) * 16) + 5^3 - log_5(3125) + (-221 mod 8) + round(-76/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 7090
} |
[
{
"role": "user",
"content": "Please evaluate and compute 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{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 7091
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Differentiate with respect to $x$: $f(x) = (5x + -5)e^{4x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (5x + -5)e^{4x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{5e^{4x} + (5x +... | {
"type": "calculus",
"seed": 1337,
"index": 7092
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Differentiate with respect to $x$: $f(x) = \\ln(5x + -3)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(5x + -3) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{... | {
"type": "calculus",
"seed": 1337,
"index": 7093
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Compute the indefinite integral: $\\int 2\\sin(1x) + -1\\cos(3x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2\\sin(1x) + -1\\cos(3x)\\,dx $\n\nUse standard antiderivatives (and integ... | {
"type": "calculus",
"seed": 1337,
"index": 7094
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}7&3\\\\\\1&-6\\end{pmatrix},\\quad b=\\begin{pmatrix}-6\\\\\\-33\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}7&3\\\\\\1&-... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7095
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Evaluate: ((-73 + -112/4) * 18) + 2^2 - log_8(262144) + (-460 mod 26) + round(-170/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-73 + -112/4) * 18) + 2^2 - log_8(262144) + (-460 mod 26) + round(-... | {
"type": "arithmetic",
"seed": 1337,
"index": 7096
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. 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": 7097
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. 50x^2 - 161 + 44x = -196"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 50x^2 - 161 + 44x = -196 $\n\nMove all terms to the left (standard form):\n$ 50x^2 + 44x + 35 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7098
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Prove that $f(x)=8x^2+7x+6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+7x+6$.\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": 7099
} |
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