messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Resolve the equation provided. 170x - 786 = 401"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 170x - 786 = 401 $\n\nMove all terms to the left (standard form):\n$ 170x + -1187 = 0 $\n\nIsolate the $x$ term:\n$ 170x = 1187 $\n\nDivide both sides by th... | {
"type": "linear_equation",
"seed": 1337,
"index": 3400
} |
[
{
"role": "user",
"content": "Solve this equation step by step. 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^... | {
"type": "optimization",
"seed": 1337,
"index": 3401
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Evaluate: ((-36 + -228/12) * 14) + 6^2 - log_7(343) + (53 mod 27) + round(73/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-36 + -228/12) * 14) + 6^2 - log_7(343) + (53 mod 27) + round(73/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 3402
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Find the modular inverse of 3 modulo 14, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $3\\cdot a^{-1} \\equiv 1 \\pmod{14}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarante... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3403
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial... | {
"type": "optimization",
"seed": 1337,
"index": 3404
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Compute the indefinite integral: $\\int \\ln(9x + 9)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(9x + 9)\\,dx $\n\nUse standard antiderivatives (and integration by parts if... | {
"type": "calculus",
"seed": 1337,
"index": 3405
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Compute the indefinite integral: $\\int 1e^{-5x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1e^{-5x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if neede... | {
"type": "calculus",
"seed": 1337,
"index": 3406
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}1&9\\\\\\7&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}-90\\\\\\-18\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}1&9\\\\\\7&... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3407
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Differentiate with respect to $x$: $f(x) = 1x^3 + 0x^2 + -10x + 5$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 1x^3 + 0x^2 + -10x + 5 $\n\nApply the standard differentiation rules:\n$ f'(x... | {
"type": "calculus",
"seed": 1337,
"index": 3408
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Compute the indefinite integral: $\\int 2x^3 + -5x^2 + 11x + 13\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + -5x^2 + 11x + 13\\,dx $\n\nUse standard antiderivatives (and in... | {
"type": "calculus",
"seed": 1337,
"index": 3409
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Evaluate: ((89 + 266/7) * 14) + 6^4 - log_3(729) + (-134 mod 26) + floor(-183/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((89 + 266/7) * 14) + 6^4 - log_3(729) + (-134 mod 26) + floor(-183/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 3410
} |
[
{
"role": "user",
"content": "Solve the following mathematical 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 $f'(x)=0$, hence:\... | {
"type": "optimization",
"seed": 1337,
"index": 3411
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. 17 + 137x^2 - 191 = -186x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 17 + 137x^2 - 191 = -186x $\n\nMove all terms to the left (standard form):\n$ 137x^2 + 186x + -174 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3412
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-9&4\\\\\\-5&7\\end{pmatrix},\\quad b=\\begin{pmatrix}85\\\\\\52\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-9&4\\\\\\-5&7\\en... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3413
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. -760x - 148 = -41"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -760x - 148 = -41 $\n\nMove all terms to the left (standard form):\n$ -760x + -107 = 0 $\n\nIsolate the $x$ term:\n$ -760x = 107 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 3414
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Evaluate: ((188 + -135/3) * 18) + 3^2 - log_6(7776) + (457 mod 20) + floor(-62/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((188 + -135/3) * 18) + 3^2 - log_6(7776) + (457 mod 20) + floor(-62/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 3415
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Evaluate: ((-65 + 28/2) * 16) + 5^2 - log_5(3125) + (-457 mod 2) + floor(-59/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-65 + 28/2) * 16) + 5^2 - log_5(3125) + (-457 mod 2) + floor(-59/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 3416
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-1&3&3\\\\\\-4&-1&0\\\\\\-5&-3&4\\end{pmatrix},\\quad b=\\begin{pmatrix}0\\\\\\16\\\\\\39\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solv... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3417
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Differentiate with respect to $x$: $f(x) = 1x^3 + 0x^2 + 18x + -6$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 1x^3 + 0x^2 + 18x + -6 $\n\nApply the standard differentiation rules:\n$ f'(x... | {
"type": "calculus",
"seed": 1337,
"index": 3418
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. Differentiate with respect to $x$: $f(x) = (1x + -10)\\sin(4x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (1x + -10)\\sin(4x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \... | {
"type": "calculus",
"seed": 1337,
"index": 3419
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Prove that $(\\mathbb{Z}_16, +)$ is a group under addition modulo 16."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_16=\\{0,1,\\dots,15\\}$ with operation $+$ modulo $16$.\n\n**Closure:** If $a,b\\in\\mathb... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3420
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Evaluate: ((-46 + 30/6) * 15) + 7^2 - log_9(531441) + (-327 mod 5) + round(25/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-46 + 30/6) * 15) + 7^2 - log_9(531441) + (-327 mod 5) + round(25/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 3421
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Prove that $f(x,y)=\\frac12(1x^2+9y^2)+-5x+6y+-4$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+9y^2)+-5x+6y+-4 $\n\n$ \\frac{\\partial^2 f}{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 3422
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 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": 3423
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Find the modular inverse of 26 modulo 29, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $26\\cdot a^{-1} \\equiv 1 \\pmod{29}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gu... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3424
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Evaluate: ((-30 + -484/11) * 13) + 7^5 - log_2(4) + (-185 mod 30) + ceil(21/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-30 + -484/11) * 13) + 7^5 - log_2(4) + (-185 mod 30) + ceil(21/2) $\n\nCompute each component:\n$ -48... | {
"type": "arithmetic",
"seed": 1337,
"index": 3425
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Compute the determinant: $A=\\begin{pmatrix}6&-7\\\\-8&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}6&-7\\\\\\-8&-4\\end{pmatrix}... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3426
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. -830 + 642x = 764"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -830 + 642x = 764 $\n\nMove all terms to the left (standard form):\n$ 642x + -1594 = 0 $\n\nIsolate the $x$ term:\n$ 642x = 1594 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 3427
} |
[
{
"role": "user",
"content": "Solve the equation and return 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": 3428
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial^... | {
"type": "optimization",
"seed": 1337,
"index": 3429
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Compute $(-2+156) \\bmod 13$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-2+156) \\bmod 13 $\n\nUse the rule that addition is compatible with reductio... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3430
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Compute the determinant: $A=\\begin{pmatrix}9&8\\\\-7&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}9&8\\\\\\-7&-7\\end{pmatrix} $\n\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3431
} |
[
{
"role": "user",
"content": "Solve the following math equation. Find the modular inverse of 13 modulo 23, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $13\\cdot a^{-1} \\equiv 1 \\pmod{23}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees inte... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3432
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Compute $(108+77) \\bmod 13$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (108+77) \\bmod 13 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ (a+b)\\bmod n =... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3433
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. 2 + 147 - 44x^2 = 75x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 2 + 147 - 44x^2 = 75x $\n\nMove all terms to the left (standard form):\n$ -44x^2 + -75x + 149 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3434
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Evaluate: ((-6 + 798/19) * 19) + 5^5 - log_7(49) + (-79 mod 30) + ceil(191/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-6 + 798/19) * 19) + 5^5 - log_7(49) + (-79 mod 30) + ceil(191/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 3435
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. -426 - 769x = -487"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -426 - 769x = -487 $\n\nMove all terms to the left (standard form):\n$ -769x + 61 = 0 $\n\nIsolate the $x$ term:\n$ -769x = -61 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 3436
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Prove that $(\\mathbb{Z}_2, +)$ is a group under addition modulo 2."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_2=\\{0,1,\\dots,1\\}$ with operation $+$ modulo $2$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, the... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3437
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. 41x^2 + 4 - 34x = 94"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 41x^2 + 4 - 34x = 94 $\n\nMove all terms to the left (standard form):\n$ 41x^2 + -34x + -90 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3438
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Evaluate: ((-154 + -392/14) * 16) + 9^2 - log_5(15625) + (295 mod 3) + floor(45/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-154 + -392/14) * 16) + 9^2 - log_5(15625) + (295 mod 3) + floor(4... | {
"type": "arithmetic",
"seed": 1337,
"index": 3439
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Solve for $x$ (in radians): sin(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(\\fra... | {
"type": "trigonometry",
"seed": 1337,
"index": 3440
} |
[
{
"role": "user",
"content": "Work out the result of this equation. -35x^2 - 160x + 101 = 179"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -35x^2 - 160x + 101 = 179 $\n\nMove all terms to the left (standard form):\n$ -35x^2 + -160x + -78 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3441
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. -21 + 85x^2 + 187x = -165"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -21 + 85x^2 + 187x = -165 $\n\nMove all terms to the left (standard form):\n$ 85x^2 + 187x + 144 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3442
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-5&6\\\\\\2&1\\end{pmatrix},\\quad b=\\begin{pmatrix}68\\\\\\0\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-5&6\\\\\\2&1\\e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3443
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Compute $(-120+-90) \\bmod 24$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-120+-90) \\bmod 24 $\n\nUse the rule that addition is compatible with reduction ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3444
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. 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 $a,b\\in\\mathbb{Z}_n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3445
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Compute the indefinite integral: $\\int \\ln(9x + 9)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(9x + 9)\\,dx $\n\nUse standard antiderivatives (and integration by p... | {
"type": "calculus",
"seed": 1337,
"index": 3446
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Evaluate: ((-47 + 210/7) * 19) + 8^3 - log_2(8) + (142 mod 19) + ceil(-159/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-47 + 210/7) * 19) + 8^3 - log_2(8) + (142 mod 19) + ceil(-159/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 3447
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. 245 - 670 = -805x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 245 - 670 = -805x $\n\nMove all terms to the left (standard form):\n$ 805x + -425 = 0 $\n\nIsolate the $x$ term:\n$ 805x = 425 $\n\n... | {
"type": "linear_equation",
"seed": 1337,
"index": 3448
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. -271x + 874 = -199"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -271x + 874 = -199 $\n\nMove all terms to the left (standard form):\n$ -271x + 1073 = 0 $\n\nIsolate the $x$ term:\n$ -271x = -1073 $\n\nDi... | {
"type": "linear_equation",
"seed": 1337,
"index": 3449
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. -324x - 412 = 136"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -324x - 412 = 136 $\n\nMove all terms to the left (standard form):\n$ -324x + -548 = 0 $\n\nIsolate the $x$ term:\n$ -324x = 548 $\n\n... | {
"type": "linear_equation",
"seed": 1337,
"index": 3450
} |
[
{
"role": "user",
"content": "solve: Differentiate with respect to $x$: $f(x) = 2e^{-5x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 2e^{-5x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-10e^{-5x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 3451
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Prove that $f(x)=1x^2+8x+-10$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=1x^2+8x+-10$.\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": 3452
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Solve for $x$ (in radians): tan(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$... | {
"type": "trigonometry",
"seed": 1337,
"index": 3453
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=... | {
"type": "optimization",
"seed": 1337,
"index": 3454
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Evaluate: ((200 + 312/12) * 10) + 2^2 - log_6(46656) + (114 mod 28) + ceil(168/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((200 + 312/12) * 10) + 2^2 - log_6(46656) + (114 mod 28) + ceil(168/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 3455
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. -17 + 70x^2 + 16x = -141"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -17 + 70x^2 + 16x = -141 $\n\nMove all terms to the left (standard form):\n$ 70x^2 + 16x + 124 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3456
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. 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 valu... | {
"type": "trigonometry",
"seed": 1337,
"index": 3457
} |
[
{
"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": 3458
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-8&4\\\\\\2&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}4\\\\\\-8\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-8&4\\\\\\2&-2\\e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3459
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. -16 - 814x = 792"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -16 - 814x = 792 $\n\nMove all terms to the left (standard form):\n$ -814x + -808 = 0 $\n\nIsolate the $x$ term:\n$ -814x = 808 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 3460
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. -8 + 192x^2 + 72x = 170"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -8 + 192x^2 + 72x = 170 $\n\nMove all terms to the left (standard form):\n$ 192x^2 + 72x + -178 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3461
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. 5 - 199x - 19x^2 = 47"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 5 - 199x - 19x^2 = 47 $\n\nMove all terms to the left (standard form):\n$ -19x^2 + -199x + -42 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3462
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Find $A^{-1}$ for $A=\\begin{pmatrix}7&-6\\\\4&4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 7 & -6 \\\\ 4 & 4 \\end{pmatrix} $\n\nCompute the determinant: $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3463
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: 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 general solutions for $\... | {
"type": "trigonometry",
"seed": 1337,
"index": 3464
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. -19 + 168x^2 + 27 = -136x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -19 + 168x^2 + 27 = -136x $\n\nMove all terms to the left (standard form):\n$ 168x^2 + 136x + 8 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3465
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. 839 + 358 = 186x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 839 + 358 = 186x $\n\nMove all terms to the left (standard form):\n$ -186x + 1197 = 0 $\n\nIsolate the $x$ term:\n$ -186x = -1197 $\n\... | {
"type": "linear_equation",
"seed": 1337,
"index": 3466
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Find $A^{-1}$ for $A=\\begin{pmatrix}-9&7\\\\3&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -9 & 7 \\\\ 3 & 9 \\end{pmatrix} $\n\nCompute the d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3467
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Find the modular inverse of 17 modulo 40, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $17\\cdot a^{-1} \\equiv 1 \\pmod{40}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity g... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3468
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. - - 118x + 16x^2 = 160"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ - - 118x + 16x^2 = 160 $\n\nMove all terms to the left (standard form):\n$ 16x^2 + -118x + -160 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3469
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n -976x + 181 = -828"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -976x + 181 = -828 $\n\nMove all terms to the left (standard form):\n$ -976x + 1009 = 0 $\n\nIsolate the $x$ term:\n$ -976x = -1009 $\n\nD... | {
"type": "linear_equation",
"seed": 1337,
"index": 3470
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Prove that $f(x,y)=\\frac12(1x^2+9y^2)+-1x+6y+2$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+9y^2)+-1x+6y+2 $\n\n$ \\frac{\\partia... | {
"type": "optimization",
"seed": 1337,
"index": 3471
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. 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 $... | {
"type": "trigonometry",
"seed": 1337,
"index": 3472
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Find the modular inverse of 20 modulo 33, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $20\\cdot a^{-1} \\equiv 1 \\pmod{33}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guara... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3473
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. 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": 3474
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Prove that $f(x)=5x^2+-7x+6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+-7x+6$.\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": 3475
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical 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": 3476
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Prove that $f(x)=4x^2+-3x+5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=4x^2+-3x+5$.\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": 3477
} |
[
{
"role": "user",
"content": "solve: \n\n Prove that $f(x)=4x^2+7x+2$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=4x^2+7x+2$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 8x + 7 $\n$ f''(x) = 8 $\n\nSin... | {
"type": "optimization",
"seed": 1337,
"index": 3478
} |
[
{
"role": "user",
"content": "solve: Compute $(-77+119) \\bmod 21$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-77+119) \\bmod 21 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ (a+b)\\bmod n = ((a\\... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3479
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Find $A^{-1}$ for $A=\\begin{pmatrix}7&-2\\\\-6&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 7 & -2 \\\\ -6 & -5 \\end{pmatrix} $\n\nCompute t... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3480
} |
[
{
"role": "user",
"content": "solve: \n\n Evaluate: ((-88 + 189/21) * 4) + 3^5 - log_2(2) + (-111 mod 12) + round(48/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-88 + 189/21) * 4) + 3^5 - log_2(2) + (-111 mod 12) + round(48/2) $\n\nCompute each component:\n$ 189/21 = ... | {
"type": "arithmetic",
"seed": 1337,
"index": 3481
} |
[
{
"role": "user",
"content": "Please calculate this equation. 876x + 77 = -762"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 876x + 77 = -762 $\n\nMove all terms to the left (standard form):\n$ 876x + 839 = 0 $\n\nIsolate the $x$ term:\n$ 876x = -839 $\n\nDivide both sides by the... | {
"type": "linear_equation",
"seed": 1337,
"index": 3482
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. 12x^2 - 115x + 100 = -165"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 12x^2 - 115x + 100 = -165 $\n\nMove all terms to the left (standard form):\n$ 12x^2 + -115x + 265 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3483
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Differentiate with respect to $x$: $f(x) = -8\\sin(2x) + 5\\cos(6x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -8\\sin(2x) + 5\\cos(6x) $\n\nApply the standard differentiation rules:... | {
"type": "calculus",
"seed": 1337,
"index": 3484
} |
[
{
"role": "user",
"content": "Please calculate this equation. Solve for $x$ (in radians): cos(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -1 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(-1) = 3.14159 $\n\nThe general ... | {
"type": "trigonometry",
"seed": 1337,
"index": 3485
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Evaluate: ((-39 + 140/7) * 12) + 7^2 - log_9(729) + (-98 mod 9) + round(-98/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-39 + 140/7) * 12) + 7^2 - log_9(729) + (-98 mod 9) + round(-98/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 3486
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. 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": 3487
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. -40 + 6x + 145x^2 = -36"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -40 + 6x + 145x^2 = -36 $\n\nMove all terms to the left (standard form):\n$ 145x^2 + 6x + -4 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3488
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Find $A^{-1}$ for $A=\\begin{pmatrix}2&-3\\\\7&-9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 2 & -3 \\\\ 7 & -9 \\end{pmatrix} $\n\nCompute... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3489
} |
[
{
"role": "user",
"content": "solve 44x - 122x^2 - 163 = -112"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 44x - 122x^2 - 163 = -112 $\n\nMove all terms to the left (standard form):\n$ -122x^2 + 44x + -51 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\pm \\sqrt{b^2... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3490
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Evaluate: ((180 + 132/4) * 17) + 6^3 - log_3(81) + (67 mod 16) + round(174/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((180 + 132/4) * 17) + 6^3 - log_3(81) + (67 mod 16) + round(174/2) $\n\nCompute e... | {
"type": "arithmetic",
"seed": 1337,
"index": 3491
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Prove that $f(x,y)=\\frac12(10x^2+9y^2)+-4x+-1y+7$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(10x^2+9y^2)+-4x+-1y+7 $\n\n$ \\frac{\\partial... | {
"type": "optimization",
"seed": 1337,
"index": 3492
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Compute the indefinite integral: $\\int -6x^3 + 4x^2 + 12x + -12\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -6x^3 + 4x^2 + 12x + -12\\,dx $\n\nUse standard antiderivatives (and... | {
"type": "calculus",
"seed": 1337,
"index": 3493
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Compute $(183+110) \\bmod 22$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (183+110) \\bmod 22 $\n\nUse the rule that addition is compatible with reduct... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3494
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. 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": 3495
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. 47 + 137x^2 - 53x = 0"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 47 + 137x^2 - 53x = 0 $\n\nMove all terms to the left (standard form):\n$ 137x^2 + -53x + 47 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3496
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Prove that $f(x,y)=\\frac12(8x^2+1y^2)+5x+-5y+6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+1y^2)+5x+-5y+6 $\n\n$ \\frac{\\partial^2 f}{\\parti... | {
"type": "optimization",
"seed": 1337,
"index": 3497
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. -19 - 43x + 96 = 181x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -19 - 43x + 96 = 181x^2 $\n\nMove all terms to the left (standard form):\n$ -181x^2 + -43x + 77 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3498
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Prove that $(\\mathbb{Z}_29, +)$ is a group under addition modulo 29."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_29=\\{0,1,\\dots,28\\}$ with operation $+$ modulo $29$.\n\n**Closure:** If $a,b\\in\\mathbb... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3499
} |
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