messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Please work out this equation. Solve for $x$ (in radians): cos(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arc... | {
"type": "trigonometry",
"seed": 1337,
"index": 6000
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Prove that $f(x,y)=\\frac12(1x^2+8y^2)+-5x+-5y+-5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+8y^2)+-5x+-5y+-5 $\n\n$ \\frac{\\partial^2... | {
"type": "optimization",
"seed": 1337,
"index": 6001
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. 617 - 201 = 421x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 617 - 201 = 421x $\n\nMove all terms to the left (standard form):\n$ -421x + 416 = 0 $\n\nIsolate the $x$ term:\n$ -421x = -416 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 6002
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. -867 + 991x = -840"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -867 + 991x = -840 $\n\nMove all terms to the left (standard form):\n$ 991x + -27 = 0 $\n\nIsolate the $x$ term:\n$ 991x = 27 $\n\nDivide both side... | {
"type": "linear_equation",
"seed": 1337,
"index": 6003
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 6004
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. 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": 6005
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. Prove that $f(x)=6x^2+-2x+1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=6x^2+-2x+1$.\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": 6006
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Differentiate with respect to $x$: $f(x) = \\ln(8x + 1)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(8x + 1) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{8}{8... | {
"type": "calculus",
"seed": 1337,
"index": 6007
} |
[
{
"role": "user",
"content": "Work through and solve this equation. 32x^2 - 121 + 13 = -23x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 32x^2 - 121 + 13 = -23x $\n\nMove all terms to the left (standard form):\n$ 32x^2 + 23x + -108 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6008
} |
[
{
"role": "user",
"content": "Find the answer 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$ \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 6009
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: 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 = \\arcsin(-\\frac{\... | {
"type": "trigonometry",
"seed": 1337,
"index": 6010
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Find the modular inverse of 17 modulo 18, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $17\\cdot a^{-1} \\equiv 1 \\pmod{18}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guar... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6011
} |
[
{
"role": "user",
"content": "Solve the following math equation. Evaluate: ((122 + -3/3) * 12) + 8^4 - log_8(32768) + (25 mod 30) + ceil(41/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((122 + -3/3) * 12) + 8^4 - log_8(32768) + (25 mod 30) + ceil(41/2) $\n\nCompute each ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6012
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Prove that $f(x)=3x^2+5x+-9$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=3x^2+5x+-9$.\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": 6013
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Prove that $f(x)=9x^2+9x+7$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=9x^2+9x+7$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(... | {
"type": "optimization",
"seed": 1337,
"index": 6014
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Prove that $f(x)=10x^2+7x+1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=10x^2+7x+1$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n... | {
"type": "optimization",
"seed": 1337,
"index": 6015
} |
[
{
"role": "user",
"content": "Can you compute this equation? Differentiate with respect to $x$: $f(x) = 7e^{-3x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 7e^{-3x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-21e^{-3x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 6016
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Compute the determinant: $A=\\begin{pmatrix}-2&0\\\\-1&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-2&0\\\\\\-1&3\\end{pmatrix} $\n\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6017
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. 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 $\n\nTh... | {
"type": "trigonometry",
"seed": 1337,
"index": 6018
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Evaluate: ((-35 + 6/3) * 6) + 11^2 - log_3(9) + (394 mod 21) + round(-168/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-35 + 6/3) * 6) + 11^2 - log_3(9) + (394 mod 21) + round(-168/2) $\n\nCompute each com... | {
"type": "arithmetic",
"seed": 1337,
"index": 6019
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? 39x^2 - 194x + 193 = -86"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 39x^2 - 194x + 193 = -86 $\n\nMove all terms to the left (standard form):\n$ 39x^2 + -194x + 279 = 0 $\n\nApply the quadratic f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6020
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Compute $(139+178) \\bmod 5$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (139+178) \\bmod 5 $\n\nUse the rule that addition is compatible with re... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6021
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Solve for $x$ (in radians): sin(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 0 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(0) = 0 $\n\nThe general ... | {
"type": "trigonometry",
"seed": 1337,
"index": 6022
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Evaluate: ((-195 + -80/5) * 11) + 5^5 - log_2(2) + (-14 mod 5) + round(-132/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-195 + -80/5) * 11) + 5^5 - log_2(2) + (-14 mod 5) + round(-132/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 6023
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Prove that $f(x,y)=\\frac12(4x^2+7y^2)+-2x+6y+5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(4x^2+7y^2)+-2x+6y+5 $\n\n$ \\frac{\\partial^2 f}{\\partia... | {
"type": "optimization",
"seed": 1337,
"index": 6024
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Evaluate: ((-119 + -136/8) * 5) + 5^4 - log_3(3) + (214 mod 22) + round(-198/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-119 + -136/8) * 5) + 5^4 - log_3(3) + (214 mod 22) + round(-198/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6025
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Find the modular inverse of 17 modulo 32, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $17\\cdot a^{-1} \\equiv 1 \\pmod{32}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees intege... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6026
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Find the modular inverse of 5 modulo 8, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{8}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gua... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6027
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Evaluate: ((-13 + 814/22) * 1) + 8^2 - log_8(512) + (-373 mod 29) + round(163/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-13 + 814/22) * 1) + 8^2 - log_8(512) + (-373 mod 29) + round(163/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 6028
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}0&2\\\\\\7&-4\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\9\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}0&2\\\\\\7&-4\\e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6029
} |
[
{
"role": "user",
"content": "Can you compute this equation? Evaluate: ((111 + -893/19) * 5) + 12^3 - log_7(7) + (277 mod 9) + round(48/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((111 + -893/19) * 5) + 12^3 - log_7(7) + (277 mod 9) + round(48/2) $\n\nCompute each comp... | {
"type": "arithmetic",
"seed": 1337,
"index": 6030
} |
[
{
"role": "user",
"content": "Determine the solution to the math 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 val... | {
"type": "trigonometry",
"seed": 1337,
"index": 6031
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. 710 - 256x = 333"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 710 - 256x = 333 $\n\nMove all terms to the left (standard form):\n$ -256x + 377 = 0 $\n\nIsolate the $x$ term:\n$ -256x = -377 $\n\nDi... | {
"type": "linear_equation",
"seed": 1337,
"index": 6032
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-2&9\\\\\\3&-1\\end{pmatrix},\\quad b=\\begin{pmatrix}-9\\\\\\-24\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-2&9\\\\\\3&-1\\e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6033
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. 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": 6034
} |
[
{
"role": "user",
"content": "Solve the equation shown below. 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": 6035
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Compute the indefinite integral: $\\int \\ln(2x + -2)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(2x + -2)\\,dx $\n\nUse standard antiderivatives (and integration by parts if n... | {
"type": "calculus",
"seed": 1337,
"index": 6036
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Evaluate: ((-4 + -30/6) * 4) + 12^3 - log_7(16807) + (374 mod 17) + floor(197/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-4 + -30/6) * 4) + 12^3 - log_7(16807) + (374 mod 17) + floor(197/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 6037
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Compute the determinant: $A=\\begin{pmatrix}7&6\\\\-5&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}7&6\\\\\\-5&9\\end{pmatrix} $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6038
} |
[
{
"role": "user",
"content": "Solve the equation shown below. 8 + 106x^2 + 29 = -200x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 8 + 106x^2 + 29 = -200x $\n\nMove all terms to the left (standard form):\n$ 106x^2 + 200x + 37 = 0 $\n\nApply the quadratic formula:\n$ x = \\fra... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6039
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((-193 + 130/10) * 16) + 11^4 - log_4(4) + (-419 mod 13) + ceil(-146/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-193 + 130/10) * 16) + 11^4 - log_4(4) + (-419 mod 13) + ceil(-... | {
"type": "arithmetic",
"seed": 1337,
"index": 6040
} |
[
{
"role": "user",
"content": "Can you compute this equation? 49x + 892 = -915"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 49x + 892 = -915 $\n\nMove all terms to the left (standard form):\n$ 49x + 1807 = 0 $\n\nIsolate the $x$ term:\n$ 49x = -1807 $\n\nDivide both sides by the ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6041
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 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": 6042
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Solve for $x$ (in radians): tan(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\ar... | {
"type": "trigonometry",
"seed": 1337,
"index": 6043
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. 38x^2 - 112 + 56x = 37"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 38x^2 - 112 + 56x = 37 $\n\nMove all terms to the left (standard form):\n$ 38x^2 + 56x + -149 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6044
} |
[
{
"role": "user",
"content": "Solve this equation accurately. 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 6045
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Compute $(180+152) \\bmod 10$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (180+152) \\bmod 10 $\n\nUse the rule that addition is compatible with reduction ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6046
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Evaluate: ((-11 + -8/4) * 18) + 10^4 - log_2(8) + (147 mod 14) + floor(-62/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-11 + -8/4) * 18) + 10^4 - log_2(8) + (147 mod 14) + floor(-62/2) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6047
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 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 $\n\nThe ge... | {
"type": "trigonometry",
"seed": 1337,
"index": 6048
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Evaluate: ((-197 + -42/6) * 18) + 2^5 - log_5(5) + (491 mod 27) + round(-187/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-197 + -42/6) * 18) + 2^5 - log_5(5) + (491 mod 27) + round(-187/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 6049
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}1&1&-4\\\\\\-5&-2&-4\\\\\\3&2&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}4\\\\\\4\\\\\\5\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6050
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. -23 - 125x - 150x^2 = 38"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -23 - 125x - 150x^2 = 38 $\n\nMove all terms to the left (standard form):\n$ -150x^2 + -125x + -61 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6051
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 6052
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}6&7\\\\-5&-6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 6 & 7 \\\\ -5 & -6 \\end{pmatrix} $\n\nCompute the deter... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6053
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Prove that $f(x)=7x^2+0x+-6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+0x+-6$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\... | {
"type": "optimization",
"seed": 1337,
"index": 6054
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}3&-3&-2\\\\\\-5&-4&1\\\\\\-5&1&2\\end{pmatrix},\\quad b=\\begin{pmatrix}-9\\\\\\24\\\\\\19\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6055
} |
[
{
"role": "user",
"content": "Solve this equation accurately. 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": 6056
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. 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 = \\arcco... | {
"type": "trigonometry",
"seed": 1337,
"index": 6057
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Find the modular inverse of 11 modulo 28, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $11\\cdot a^{-1} \\equiv 1 \\pmod{28}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantee... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6058
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Compute the indefinite integral: $\\int (-3x + -4)\\sin(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-3x + -4)\\sin(4x)\\,dx $\n\nUse standard antiderivatives (and integrat... | {
"type": "calculus",
"seed": 1337,
"index": 6059
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. 19 + 152 - 173x = -155x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 19 + 152 - 173x = -155x^2 $\n\nMove all terms to the left (standard form):\n$ 155x^2 + -173x + 171 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6060
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. 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": 6061
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Find the modular inverse of 17 modulo 20, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $17\\cdot a^{-1} \\equiv 1 \\pmod{20}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6062
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. 31 + 36x^2 - 198x = 12"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 31 + 36x^2 - 198x = 12 $\n\nMove all terms to the left (standard form):\n$ 36x^2 + -198x + 19 = 0 $\n\nApply the quadratic f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6063
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Evaluate: ((98 + -220/10) * 15) + 8^5 - log_7(7) + (126 mod 22) + ceil(-159/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((98 + -220/10) * 15) + 8^5 - log_7(7) + (126 mod 22) + ceil(-159/2) ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6064
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Find $A^{-1}$ for $A=\\begin{pmatrix}5&3\\\\0&7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 5 & 3 \\\\ 0 & 7 \\end{pmatrix} $\n\nComp... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6065
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Compute the indefinite integral: $\\int 2e^{4x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2e^{4x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if ne... | {
"type": "calculus",
"seed": 1337,
"index": 6066
} |
[
{
"role": "user",
"content": "Please calculate this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}1&-9\\\\4&-8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 1 & -9 \\\\ 4 & -8 \\end{pmatrix} $\n\nCompute the determinant... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6067
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Compute the indefinite integral: $\\int 2x^3 + 6x^2 + -10x + -9\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + 6x^2 + -10x + -9\\,dx $\n\nUse standard antiderivatives (and integra... | {
"type": "calculus",
"seed": 1337,
"index": 6068
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Evaluate: ((-159 + -370/10) * 11) + 4^4 - log_6(1296) + (-216 mod 30) + round(-67/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-159 + -370/10) * 11) + 4^4 - log_6(1296) + (-216 mod 30) + ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6069
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Evaluate: ((-15 + 282/6) * 4) + 7^2 - log_10(10) + (-84 mod 16) + round(189/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-15 + 282/6) * 4) + 7^2 - log_10(10) + (-84 mod 16) + round(189/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 6070
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, henc... | {
"type": "optimization",
"seed": 1337,
"index": 6071
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Compute the indefinite integral: $\\int 6\\sin(4x) + 5\\cos(9x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 6\\sin(4x) + 5\\cos(9x)\\,dx $\n\nUse standard antiderivatives (and inte... | {
"type": "calculus",
"seed": 1337,
"index": 6072
} |
[
{
"role": "user",
"content": "Please find the value of this math 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$... | {
"type": "trigonometry",
"seed": 1337,
"index": 6073
} |
[
{
"role": "user",
"content": "Please solve this math problem. Evaluate: ((4 + -40/8) * 12) + 3^2 - log_10(1000) + (-97 mod 9) + floor(65/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((4 + -40/8) * 12) + 3^2 - log_10(1000) + (-97 mod 9) + floor(65/2) $\n\nCompute each com... | {
"type": "arithmetic",
"seed": 1337,
"index": 6074
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: -371x + 276 = -284"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -371x + 276 = -284 $\n\nMove all terms to the left (standard form):\n$ -371x + 560 = 0 $\n\nIsolate the $x$ term:\n$ -371x = -560 $\n\nDivide both sides by the coefficie... | {
"type": "linear_equation",
"seed": 1337,
"index": 6075
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Solve for $x$ (in radians): tan(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \... | {
"type": "trigonometry",
"seed": 1337,
"index": 6076
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n -7 + 142 - 118x^2 = 43x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -7 + 142 - 118x^2 = 43x $\n\nMove all terms to the left (standard form):\n$ -118x^2 + -43x + 135 = 0 $\n\nApply the quadra... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6077
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Evaluate: ((-75 + -414/9) * 20) + 11^2 - log_7(49) + (206 mod 24) + floor(-90/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-75 + -414/9) * 20) + 11^2 - log_7(49) + (206 mod 24) + floor(-90/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 6078
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. -51x - 485 = 723"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -51x - 485 = 723 $\n\nMove all terms to the left (standard form):\n$ -51x + -1208 = 0 $\n\nIsolate the $x$ term:\n$ -51x = 1208 $\n\n... | {
"type": "linear_equation",
"seed": 1337,
"index": 6079
} |
[
{
"role": "user",
"content": "Compute the solution for 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{\\parti... | {
"type": "optimization",
"seed": 1337,
"index": 6080
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Solve for $x$ (in radians): tan(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -1 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(-1) = -0.785398 $\n\... | {
"type": "trigonometry",
"seed": 1337,
"index": 6081
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. 27x - 166x^2 - 31 = 156"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 27x - 166x^2 - 31 = 156 $\n\nMove all terms to the left (standard form):\n$ -166x^2 + 27x + -187 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6082
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Find $A^{-1}$ for $A=\\begin{pmatrix}-3&-5\\\\7&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -3 & -5 \\\\ 7 & 6 \\end{pmatrix} $\n\nCompute th... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6083
} |
[
{
"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": 6084
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Find $A^{-1}$ for $A=\\begin{pmatrix}2&5\\\\9&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 2 & 5 \\\\ 9 & -5 \\end{pmatrix} $\n\nCompute the... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6085
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. 5x + 131x^2 - 155 = 118"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 5x + 131x^2 - 155 = 118 $\n\nMove all terms to the left (standard form):\n$ 131x^2 + 5x + -273 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6086
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. 480x - 566 = 701"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 480x - 566 = 701 $\n\nMove all terms to the left (standard form):\n$ 480x + -1267 = 0 $\n\nIsolate the $x$ term:\n$ 480x = 1267 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 6087
} |
[
{
"role": "user",
"content": "solve 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 = \\arctan(-\\frac{1}{2}) = -0.463648 $\n\nTh... | {
"type": "trigonometry",
"seed": 1337,
"index": 6088
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. 37 - 166 + 40x^2 = -191x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 37 - 166 + 40x^2 = -191x $\n\nMove all terms to the left (standard form):\n$ 40x^2 + 191x + -129 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6089
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Find $A^{-1}$ for $A=\\begin{pmatrix}-2&6\\\\4&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -2 & 6 \\\\ 4 & -4 \\end{pmatrix} $\n\nCompute the determi... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6090
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. -44 - 105x^2 + 131x = 112"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -44 - 105x^2 + 131x = 112 $\n\nMove all terms to the left (standard form):\n$ -105x^2 + 131x + -156 = 0 $\n\nApply the quadratic ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6091
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Differentiate with respect to $x$: $f(x) = (-6x + -3)e^{5x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-6x + -3)e^{5x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-6... | {
"type": "calculus",
"seed": 1337,
"index": 6092
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Find the modular inverse of 2 modulo 37, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $2\\cdot a^{-1} \\equiv 1 \\pmod{37}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantee... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6093
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. 32 - 107 - 15x^2 = -36x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 32 - 107 - 15x^2 = -36x $\n\nMove all terms to the left (standard form):\n$ -15x^2 + 36x + -75 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6094
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Differentiate with respect to $x$: $f(x) = 1e^{-4x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 1e^{-4x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-4e^{-4x}} $"
... | {
"type": "calculus",
"seed": 1337,
"index": 6095
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. -358 + 371x = 620"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -358 + 371x = 620 $\n\nMove all terms to the left (standard form):\n$ 371x + -978 = 0 $\n\nIsolate the $x$ term:\n$ 371x = 978 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 6096
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Evaluate: ((-132 + 192/12) * 1) + 2^4 - log_2(32) + (334 mod 29) + ceil(-136/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-132 + 192/12) * 1) + 2^4 - log_2(32) + (334 mod 29) + ceil(-136/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6097
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. 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": 6098
} |
[
{
"role": "user",
"content": "Please determine the value 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 $f'(x)=0$, henc... | {
"type": "optimization",
"seed": 1337,
"index": 6099
} |
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