messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Prove that $(\\mathbb{Z}_7, +)$ is a group under addition modulo 7."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_7=\\{0,1,\\dots,6\\}$ with operation $+$ modulo $7$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, th... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6300
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Differentiate with respect to $x$: $f(x) = 9e^{-2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 9e^{-2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-18e^{-2x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 6301
} |
[
{
"role": "user",
"content": "Can you compute this equation? Compute the indefinite integral: $\\int \\ln(6x + -2)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(6x + -2)\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$... | {
"type": "calculus",
"seed": 1337,
"index": 6302
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. Compute the indefinite integral: $\\int (-2x + -5)\\sin(1x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-2x + -5)\\sin(1x)\\,dx $\n\nUse standard antiderivatives (and integrati... | {
"type": "calculus",
"seed": 1337,
"index": 6303
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Find the modular inverse of 7 modulo 22, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $7\\cdot a^{-1} \\equiv 1 \\pmod{22}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identit... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6304
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Evaluate: ((-33 + 88/11) * 17) + 8^3 - log_2(32) + (12 mod 15) + round(-77/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-33 + 88/11) * 17) + 8^3 - log_2(32) + (12 mod 15) + round(-77/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 6305
} |
[
{
"role": "user",
"content": "Please calculate this equation. Differentiate with respect to $x$: $f(x) = \\ln(9x + 0)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(9x + 0) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{9}{9x + 0}} $... | {
"type": "calculus",
"seed": 1337,
"index": 6306
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Evaluate: ((-21 + -54/6) * 17) + 9^5 - log_8(32768) + (417 mod 24) + round(141/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-21 + -54/6) * 17) + 9^5 - log_8(32768) + (417 mod 24) + rou... | {
"type": "arithmetic",
"seed": 1337,
"index": 6307
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. 3 + 125 + 180x^2 = 198x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 3 + 125 + 180x^2 = 198x $\n\nMove all terms to the left (standard form):\n$ 180x^2 + -198x + 128 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6308
} |
[
{
"role": "user",
"content": "Please calculate this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial^2 f}{... | {
"type": "optimization",
"seed": 1337,
"index": 6309
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Compute the indefinite integral: $\\int 5x^3 + -5x^2 + 16x + 16\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 5x^3 + -5x^2 + 16x + 16\\,dx $\n\nUse standard antiderivatives (an... | {
"type": "calculus",
"seed": 1337,
"index": 6310
} |
[
{
"role": "user",
"content": "solve Evaluate: ((89 + 400/16) * 1) + 7^4 - log_9(81) + (-155 mod 15) + ceil(119/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((89 + 400/16) * 1) + 7^4 - log_9(81) + (-155 mod 15) + ceil(119/2) $\n\nCompute each component:\n$ 400/16 = 25 $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6311
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given 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 va... | {
"type": "trigonometry",
"seed": 1337,
"index": 6312
} |
[
{
"role": "user",
"content": "Solve the following math equation. Compute the determinant: $A=\\begin{pmatrix}1&-3\\\\-5&-8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}1&-3\\\\\\-5&-8\\end{pmatrix} $\n\nUse $\\det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6313
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. -780x + 596 = -367"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -780x + 596 = -367 $\n\nMove all terms to the left (standard form):\n$ -780x + 963 = 0 $\n\nIsolate the $x$ term:\n$ -780x = -963 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 6314
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Find the modular inverse of 5 modulo 34, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{34}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees int... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6315
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. 45x^2 - 20 - 81 = 60x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 45x^2 - 20 - 81 = 60x $\n\nMove all terms to the left (standard form):\n$ 45x^2 + -60x + -101 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6316
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Evaluate: ((5 + 60/3) * 1) + 6^5 - log_6(216) + (-383 mod 24) + ceil(162/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((5 + 60/3) * 1) + 6^5 - log_6(216) + (-383 mod 24) + ceil(162/2) $\n\nCompute each compon... | {
"type": "arithmetic",
"seed": 1337,
"index": 6317
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Compute $(-146+33) \\bmod 10$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-146+33) \\bmod 10 $\n\nUse the rule that addition is compatible with reduction modulo $n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6318
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Find $A^{-1}$ for $A=\\begin{pmatrix}-3&1\\\\7&2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -3 & 1 \\\\ 7 & 2 \\end{pmatrix} $\n\nCompute the det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6319
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Prove that $f(x,y)=\\frac12(8x^2+10y^2)+4x+-2y+-6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+10y^2)+4x+-2y+-6 $\n\n$ \\frac{\\partial... | {
"type": "optimization",
"seed": 1337,
"index": 6320
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Differentiate with respect to $x$: $f(x) = (-1x + 0)\\sin(2x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-1x + 0)\\sin(2x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed... | {
"type": "calculus",
"seed": 1337,
"index": 6321
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Prove that $f(x)=5x^2+4x+-5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+4x+-5$.\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": 6322
} |
[
{
"role": "user",
"content": "Please find the value of this math 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 = \\arc... | {
"type": "trigonometry",
"seed": 1337,
"index": 6323
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Compute $(-89+-2) \\bmod 16$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-89+-2) \\bmod 16 $\n\nUse the rule that addition is compatible with reduction mod... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6324
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}5&2&5\\\\\\-1&-3&-1\\\\\\1&-3&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}24\\\\\\3\\\\\\3\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6325
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Prove that $f(x,y)=\\frac12(7x^2+8y^2)+0x+2y+2$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(7x^2+8y^2)+0x+2y+2 $\n\n$ \\frac{\\partial^2 f}{\\... | {
"type": "optimization",
"seed": 1337,
"index": 6326
} |
[
{
"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": 6327
} |
[
{
"role": "user",
"content": "solve: \n\n Find $A^{-1}$ for $A=\\begin{pmatrix}3&-4\\\\1&7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 3 & -4 \\\\ 1 & 7 \\end{pmatrix} $\n\nCompute the determinant: $\\det(A)=ad-bc$.\n$... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6328
} |
[
{
"role": "user",
"content": "Work out the result of this equation. 20 - 32x^2 - 15x = -181"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 20 - 32x^2 - 15x = -181 $\n\nMove all terms to the left (standard form):\n$ -32x^2 + -15x + 201 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6329
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. -713 + 773x = 708"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -713 + 773x = 708 $\n\nMove all terms to the left (standard form):\n$ 773x + -1421 = 0 $\n\nIsolate the $x$ term:\n$ 773x = 1421 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6330
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. 564x - 848 = -559"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 564x - 848 = -559 $\n\nMove all terms to the left (standard form):\n$ 564x + -289 = 0 $\n\nIsolate the $x$ term:\n$ 564x = 289 $\n\n... | {
"type": "linear_equation",
"seed": 1337,
"index": 6331
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. 17 - 161x^2 - 187 = 67x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 17 - 161x^2 - 187 = 67x $\n\nMove all terms to the left (standard form):\n$ -161x^2 + -67x + -170 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6332
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? -45 + 24 + 77x^2 = 177x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -45 + 24 + 77x^2 = 177x $\n\nMove all terms to the left (standard form):\n$ 77x^2 + -177x + -21 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6333
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Compute the indefinite integral: $\\int \\ln(7x + 7)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(7x + 7)\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed... | {
"type": "calculus",
"seed": 1337,
"index": 6334
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Solve for $x$ (in radians): sin(x) = -\\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(-\\frac{1... | {
"type": "trigonometry",
"seed": 1337,
"index": 6335
} |
[
{
"role": "user",
"content": "Please work out this equation. Compute the indefinite integral: $\\int 9\\sin(1x) + 4\\cos(2x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 9\\sin(1x) + 4\\cos(2x)\\,dx $\n\nUse standard antiderivatives (and integration by ... | {
"type": "calculus",
"seed": 1337,
"index": 6336
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Compute the indefinite integral: $\\int -6\\sin(9x) + 6\\cos(9x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -6\\sin(9x) + 6\\cos(9x)\\,dx $\n\nUse standard antiderivatives (and integra... | {
"type": "calculus",
"seed": 1337,
"index": 6337
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. 125 + 86 = 866x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 125 + 86 = 866x $\n\nMove all terms to the left (standard form):\n$ -866x + 211 = 0 $\n\nIsolate the $x$ term:\n$ -866x = -211 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 6338
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:... | {
"type": "optimization",
"seed": 1337,
"index": 6339
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. 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": 6340
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Compute $(72+54) \\bmod 7$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (72+54) \\bmod 7 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6341
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. 6 - 158 - 189x = -176x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 6 - 158 - 189x = -176x^2 $\n\nMove all terms to the left (standard form):\n$ 176x^2 + -189x + -152 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6342
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Compute $(114+-92) \\bmod 22$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (114+-92) \\bmod 22 $\n\nUse the rule that addition is compatible with reduction... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6343
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 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": 6344
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Prove that $(\\mathbb{Z}_14, +)$ is a group under addition modulo 14."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_14=\\{0,1,\\dots,13\\}$ with operation $+$ modulo $14$.\n\n**Closure:** If $a,b\\in... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6345
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Evaluate: ((185 + 690/15) * 17) + 10^4 - log_6(1296) + (-232 mod 3) + floor(-65/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((185 + 690/15) * 17) + 10^4 - log_6(1296) + (-232 mod 3) + floor(-65/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 6346
} |
[
{
"role": "user",
"content": "solve: \n\n Find $A^{-1}$ for $A=\\begin{pmatrix}0&3\\\\-2&-6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 0 & 3 \\\\ -2 & -6 \\end{pmatrix} $\n\nCompute the determinant: $\\det(A)=ad-bc$.\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6347
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Evaluate: ((-126 + 100/10) * 2) + 9^2 - log_10(1000000) + (-497 mod 6) + floor(-70/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-126 + 100/10) * 2) + 9^2 - log_10(1000000) + (-497 mod 6) + flo... | {
"type": "arithmetic",
"seed": 1337,
"index": 6348
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Evaluate: ((11 + 144/24) * 5) + 5^5 - log_10(100) + (115 mod 9) + ceil(-2/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((11 + 144/24) * 5) + 5^5 - log_10(100) + (115 mod 9) + ceil(-2/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6349
} |
[
{
"role": "user",
"content": "Can you solve the equation below? -2 + 116x + 108 = -104x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -2 + 116x + 108 = -104x^2 $\n\nMove all terms to the left (standard form):\n$ 104x^2 + 116x + 106 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6350
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Evaluate: ((76 + 91/7) * 6) + 9^5 - log_2(16) + (92 mod 21) + round(4/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((76 + 91/7) * 6) + 9^5 - log_2(16) + (92 mod 21) + round(4/2) $\n\nCompute each component:\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6351
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Compute $(125+193) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (125+193) \\bmod 8 $\n\nUse the rule that addition is compatible with re... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6352
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Evaluate: ((44 + 416/16) * 14) + 7^2 - log_6(36) + (413 mod 18) + round(85/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((44 + 416/16) * 14) + 7^2 - log_6(36) + (413 mod 18) + round(85/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 6353
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Evaluate: ((145 + -900/25) * 20) + 10^5 - log_3(3) + (-109 mod 16) + floor(77/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((145 + -900/25) * 20) + 10^5 - log_3(3) + (-109 mod 16) + floor(77/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6354
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Differentiate with respect to $x$: $f(x) = \\ln(6x + -5)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(6x + -5) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{6}{6... | {
"type": "calculus",
"seed": 1337,
"index": 6355
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical 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} & \\... | {
"type": "optimization",
"seed": 1337,
"index": 6356
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. 492 + 314 = 782x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 492 + 314 = 782x $\n\nMove all terms to the left (standard form):\n$ -782x + 806 = 0 $\n\nIsolate the $x$ term:\n$ -782x = -806 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 6357
} |
[
{
"role": "user",
"content": "Please calculate this equation. Evaluate: ((114 + 1056/22) * 19) + 6^4 - log_3(3) + (-408 mod 7) + floor(-158/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((114 + 1056/22) * 19) + 6^4 - log_3(3) + (-408 mod 7) + floor(-158/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 6358
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute $(-179+26) \\bmod 15$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-179+26) \\bmod 15 $\n\nUse the rule that addition is compatible with reduction mod... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6359
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Evaluate: ((103 + 112/4) * 16) + 2^5 - log_6(36) + (-41 mod 22) + round(-82/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((103 + 112/4) * 16) + 2^5 - log_6(36) + (-41 mod 22) + round(-82/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6360
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Find the modular inverse of 27 modulo 32, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $27\\cdot a^{-1} \\equiv 1 \\pmod{32}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gua... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6361
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Compute the determinant: $A=\\begin{pmatrix}0&5\\\\-8&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}0&5\\\\\\-8&6\\end{pmatrix} $\n\nUse... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6362
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? -10 - 39 + 10x^2 = 51x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -10 - 39 + 10x^2 = 51x $\n\nMove all terms to the left (standard form):\n$ 10x^2 + -51x + -49 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6363
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Compute $(-113+-6) \\bmod 10$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-113+-6) \\bmod 10 $\n\nUse the rule that addition is compatible with reduction modulo $n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6364
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Evaluate: ((-18 + 301/7) * 3) + 5^4 - log_8(8) + (-191 mod 6) + floor(-114/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-18 + 301/7) * 3) + 5^4 - log_8(8) + (-191 mod 6) + floor(-114/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 6365
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((5 + -235/5) * 8) + 3^3 - log_8(64) + (-492 mod 9) + floor(143/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((5 + -235/5) * 8) + 3^3 - log_8(64) + (-492 mod 9) + floor(143/2) $\n\nCompute e... | {
"type": "arithmetic",
"seed": 1337,
"index": 6366
} |
[
{
"role": "user",
"content": "Compute and determine the solution to 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\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 6367
} |
[
{
"role": "user",
"content": "Find the solution of 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 $f'(x)=0$, hen... | {
"type": "optimization",
"seed": 1337,
"index": 6368
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Prove that $f(x,y)=\\frac12(8x^2+8y^2)+3x+-3y+-1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+8y^2)+3x+-3y+-1 $\n\n$ \\frac{\\partial^... | {
"type": "optimization",
"seed": 1337,
"index": 6369
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? -50x^2 + 72 - 183x = -9"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -50x^2 + 72 - 183x = -9 $\n\nMove all terms to the left (standard form):\n$ -50x^2 + -183x + 81 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6370
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Compute the determinant: $A=\\begin{pmatrix}-9&7\\\\3&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-9&7\\\\\\3&1\\end{pmatrix} $\n\nUse $\\det(... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6371
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Solve for $x$ (in radians): sin(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(\\frac{1}... | {
"type": "trigonometry",
"seed": 1337,
"index": 6372
} |
[
{
"role": "user",
"content": "Find the final answer to the given 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 gene... | {
"type": "trigonometry",
"seed": 1337,
"index": 6373
} |
[
{
"role": "user",
"content": "solve: \n\n Find $A^{-1}$ for $A=\\begin{pmatrix}2&2\\\\-8&-3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 2 & 2 \\\\ -8 & -3 \\end{pmatrix} $\n\nCompute the determinant: $\\det(A)=ad-bc$.\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6374
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n 5x^2 - 170 + 111 = -86x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 5x^2 - 170 + 111 = -86x $\n\nMove all terms to the left (standard form):\n$ 5x^2 + 86x + -59 = 0 $\n\nApply the quadratic ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6375
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Compute $(86+-199) \\bmod 11$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (86+-199) \\bmod 11 $\n\nUse the rule that addition is compatible with reduct... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6376
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-2&4\\\\\\-7&7\\end{pmatrix},\\quad b=\\begin{pmatrix}24\\\\\\21\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-2&4\\\\\\-... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6377
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Evaluate: ((-186 + -693/21) * 10) + 5^3 - log_3(243) + (-405 mod 2) + floor(-153/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-186 + -693/21) * 10) + 5^3 - log_3(243) + (-405 mod 2) + floor(-1... | {
"type": "arithmetic",
"seed": 1337,
"index": 6378
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Prove that $f(x,y)=\\frac12(1x^2+1y^2)+0x+6y+1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+1y^2)+0x+6y+1 $\n\n$ \\frac{\\partial^2 f}{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 6379
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partia... | {
"type": "optimization",
"seed": 1337,
"index": 6380
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Evaluate: ((-156 + 234/13) * 20) + 8^4 - log_7(2401) + (24 mod 10) + ceil(-158/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-156 + 234/13) * 20) + 8^4 - log_7(2401) + (24 mod 10) + ceil(-158/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 6381
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-5&-4&4\\\\\\3&2&4\\\\\\-4&1&1\\end{pmatrix},\\quad b=\\begin{pmatrix}5\\\\\\-27\\\\\\-4\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6382
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((-21 + -150/10) * 4) + 5^4 - log_3(3) + (238 mod 14) + floor(-151/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-21 + -150/10) * 4) + 5^4 - log_3(3) + (238 mod 14) + floor(-151/... | {
"type": "arithmetic",
"seed": 1337,
"index": 6383
} |
[
{
"role": "user",
"content": "Compute the solution of 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 $\n\nT... | {
"type": "trigonometry",
"seed": 1337,
"index": 6384
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Evaluate: ((80 + -100/2) * 16) + 9^2 - log_7(2401) + (170 mod 9) + round(39/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((80 + -100/2) * 16) + 9^2 - log_7(2401) + (170 mod 9) + round(39/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6385
} |
[
{
"role": "user",
"content": "Please solve this math problem. Evaluate: ((103 + 243/9) * 2) + 7^3 - log_9(9) + (178 mod 13) + round(42/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((103 + 243/9) * 2) + 7^3 - log_9(9) + (178 mod 13) + round(42/2) $\n\nCompute each compone... | {
"type": "arithmetic",
"seed": 1337,
"index": 6386
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: 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 $a,b\\in\\mathbb{Z}_n$, then $(a+b)\\bmod n\... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6387
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. 49x^2 + 71 - 131x = 200"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 49x^2 + 71 - 131x = 200 $\n\nMove all terms to the left (standard form):\n$ 49x^2 + -131x + -129 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6388
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. Evaluate: ((105 + 276/23) * 10) + 8^5 - log_2(32) + (293 mod 22) + ceil(19/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((105 + 276/23) * 10) + 8^5 - log_2(32) + (293 mod 22) + ceil(19/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 6389
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. 801x - 804 = 141"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 801x - 804 = 141 $\n\nMove all terms to the left (standard form):\n$ 801x + -945 = 0 $\n\nIsolate the $x$ term:\n$ 801x = 945 $\n\nDivide both side... | {
"type": "linear_equation",
"seed": 1337,
"index": 6390
} |
[
{
"role": "user",
"content": "Calculate the solution to this math 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 = \\ar... | {
"type": "trigonometry",
"seed": 1337,
"index": 6391
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 6x - 5x^2 + 77 = 44"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 6x - 5x^2 + 77 = 44 $\n\nMove all terms to the left (standard form):\n$ -5x^2 + 6x + 33 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6392
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Compute $(182+37) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (182+37) \\bmod 8 $\n\nUse the rule that addition is compatible with reduction mo... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6393
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-3&-9\\\\8&-8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -3 & -9 \\\\ 8 & -8 \\end{pmatrix} $\n\nCompute th... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6394
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Prove that $f(x)=1x^2+-10x+3$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+-10x+3$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 2x... | {
"type": "optimization",
"seed": 1337,
"index": 6395
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. -6x - 52x^2 - 35 = -111"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -6x - 52x^2 - 35 = -111 $\n\nMove all terms to the left (standard form):\n$ -52x^2 + -6x + 76 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6396
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. -36x + 517 = -316"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -36x + 517 = -316 $\n\nMove all terms to the left (standard form):\n$ -36x + 833 = 0 $\n\nIsolate the $x$ term:\n$ -36x = -833 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 6397
} |
[
{
"role": "user",
"content": "Please work out this equation. -39x^2 + 32x + 56 = 41"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -39x^2 + 32x + 56 = 41 $\n\nMove all terms to the left (standard form):\n$ -39x^2 + 32x + 15 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6398
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. -20 - 189 + 82x^2 = -44x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -20 - 189 + 82x^2 = -44x $\n\nMove all terms to the left (standard form):\n$ 82x^2 + 44x + -209 = 0 $\n\nApply the quadrat... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6399
} |
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