messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Compute the outcome of this equation. Compute the indefinite integral: $\\int -5\\sin(9x) + 3\\cos(2x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -5\\sin(9x) + 3\\cos(2x)\\,dx $\n\nUse standard antiderivatives (and integr... | {
"type": "calculus",
"seed": 1337,
"index": 400
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Solve for $x$ (in radians): sin(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 401
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-1&-3&2\\\\\\-3&-2&-2\\\\\\1&4&-4\\end{pmatrix},\\quad b=\\begin{pmatrix}-16\\\\\\-12\\\\\\22\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 402
} |
[
{
"role": "user",
"content": "Evaluate the following equation. 679x - 756 = -625"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 679x - 756 = -625 $\n\nMove all terms to the left (standard form):\n$ 679x + -131 = 0 $\n\nIsolate the $x$ term:\n$ 679x = 131 $\n\nDivide both sides by ... | {
"type": "linear_equation",
"seed": 1337,
"index": 403
} |
[
{
"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": 404
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Evaluate: ((39 + -440/10) * 18) + 8^5 - log_5(3125) + (107 mod 22) + round(175/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((39 + -440/10) * 18) + 8^5 - log_5(3125) + (107 mod 22) + round(175/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 405
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Differentiate with respect to $x$: $f(x) = 4e^{-2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 4e^{-2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-8e^{-2x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 406
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. 10 + 65x - 125 = 199x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 10 + 65x - 125 = 199x^2 $\n\nMove all terms to the left (standard form):\n$ -199x^2 + 65x + -115 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 407
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Compute the indefinite integral: $\\int -4e^{2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -4e^{2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n... | {
"type": "calculus",
"seed": 1337,
"index": 408
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. 22 - 117 - 6x^2 = 70x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 22 - 117 - 6x^2 = 70x $\n\nMove all terms to the left (standard form):\n$ -6x^2 + -70x + -95 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 409
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. 8 + 127x^2 - 5 = -120x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 8 + 127x^2 - 5 = -120x $\n\nMove all terms to the left (standard form):\n$ 127x^2 + 120x + 3 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 410
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Compute the indefinite integral: $\\int \\ln(4x + -1)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(4x + -1)\\,dx $\n\nUse standard antiderivatives (and integration ... | {
"type": "calculus",
"seed": 1337,
"index": 411
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Evaluate: ((42 + 615/15) * 5) + 8^2 - log_4(4) + (-332 mod 29) + floor(48/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((42 + 615/15) * 5) + 8^2 - log_4(4) + (-332 mod 29) + floor(48/2) $\n\nCompute e... | {
"type": "arithmetic",
"seed": 1337,
"index": 412
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? 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": 413
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Prove that $f(x,y)=\\frac12(1x^2+1y^2)+-2x+6y+-3$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+1y^2)+-2x+6y+-3 $\n\n$ \\frac{\\partial^2 f}{\\... | {
"type": "optimization",
"seed": 1337,
"index": 414
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}6&4\\\\-6&2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 6 & 4 \\\\ -6 & 2 \\end{pmatrix} $\n\nCompute the determi... | {
"type": "linear_algebra",
"seed": 1337,
"index": 415
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. 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 = \\arccos(-... | {
"type": "trigonometry",
"seed": 1337,
"index": 416
} |
[
{
"role": "user",
"content": "solve: \n\n Prove that $f(x,y)=\\frac12(3x^2+3y^2)+-2x+2y+-1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+3y^2)+-2x+2y+-1 $\n\n$ \\frac{\\partial^2 f}{\\partial x^2} = 3,\\quad \\frac... | {
"type": "optimization",
"seed": 1337,
"index": 417
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Evaluate: ((-80 + -544/16) * 16) + 3^2 - log_3(243) + (317 mod 16) + floor(-177/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-80 + -544/16) * 16) + 3^2 - log_3(243) + (317 mod 16) + floor(-177/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 418
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Differentiate with respect to $x$: $f(x) = 9\\sin(3x) + 4\\cos(8x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 9\\sin(3x) + 4\\cos(8x) $\n\nApply the standard differentiation rules:\n$ f'(x) =... | {
"type": "calculus",
"seed": 1337,
"index": 419
} |
[
{
"role": "user",
"content": "Find the solution to the following 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": 420
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. -212 + 689x = 721"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -212 + 689x = 721 $\n\nMove all terms to the left (standard form):\n$ 689x + -933 = 0 $\n\nIsolate the $x$ term:\n$ 689x = 933 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 421
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. 8 + 88x - 185x^2 = -125"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 8 + 88x - 185x^2 = -125 $\n\nMove all terms to the left (standard form):\n$ -185x^2 + 88x + 133 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 422
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. 35x - 963 = -102"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 35x - 963 = -102 $\n\nMove all terms to the left (standard form):\n$ 35x + -861 = 0 $\n\nIsolate the $x$ term:\n$ 35x = 861 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 423
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Differentiate with respect to $x$: $f(x) = 5x^3 + 2x^2 + 11x + -13$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 5x^3 + 2x^2 + 11x + -13 $\n\nApply the standard differentiation rules... | {
"type": "calculus",
"seed": 1337,
"index": 424
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. 10 - 108x^2 + 168 = -77x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 10 - 108x^2 + 168 = -77x $\n\nMove all terms to the left (standard form):\n$ -108x^2 + 77x + 178 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 425
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Evaluate: ((-142 + 126/7) * 2) + 9^5 - log_2(64) + (344 mod 17) + floor(-113/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-142 + 126/7) * 2) + 9^5 - log_2(64) + (344 mod 17) + floor(-113/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 426
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Evaluate: ((23 + 637/13) * 4) + 8^5 - log_4(4096) + (-139 mod 25) + floor(-68/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((23 + 637/13) * 4) + 8^5 - log_4(4096) + (-139 mod 25) + floor(-68/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 427
} |
[
{
"role": "user",
"content": "solve 35 - 101x^2 - 143x = 20"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 35 - 101x^2 - 143x = 20 $\n\nMove all terms to the left (standard form):\n$ -101x^2 + -143x + 15 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\pm \\sqrt{b^2 - ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 428
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. 844 - 234x = -14"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 844 - 234x = -14 $\n\nMove all terms to the left (standard form):\n$ -234x + 858 = 0 $\n\nIsolate the $x$ term:\n$ -234x = -858 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 429
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Find $A^{-1}$ for $A=\\begin{pmatrix}4&-6\\\\-5&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 4 & -6 \\\\ -5 & 6 \\end{pmatrix} $\n\nCompute the... | {
"type": "linear_algebra",
"seed": 1337,
"index": 430
} |
[
{
"role": "user",
"content": "solve Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}0&9\\\\\\-7&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}45\\\\\\11\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}0&9\\\\\\-7&-2\\end{pmatrix} ,\\quad b=\\begin{pmatr... | {
"type": "linear_algebra",
"seed": 1337,
"index": 431
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. 804 - 754 = -931x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 804 - 754 = -931x $\n\nMove all terms to the left (standard form):\n$ 931x + 50 = 0 $\n\nIsolate the $x$ term:\n$ 931x = -50 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 432
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Differentiate with respect to $x$: $f(x) = \\ln(7x + -7)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(7x + -7) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{7}{7x ... | {
"type": "calculus",
"seed": 1337,
"index": 433
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Prove that $(\\mathbb{Z}_28, +)$ is a group under addition modulo 28."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_28=\\{0,1,\\dots,27\\}$ with operation $+$ modulo $28$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 434
} |
[
{
"role": "user",
"content": "Evaluate the following equation. -487 - 663 = 231x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -487 - 663 = 231x $\n\nMove all terms to the left (standard form):\n$ -231x + -1150 = 0 $\n\nIsolate the $x$ term:\n$ -231x = 1150 $\n\nDivide both sides... | {
"type": "linear_equation",
"seed": 1337,
"index": 435
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Differentiate with respect to $x$: $f(x) = (-6x + -4)e^{6x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-6x + -4)e^{6x} $\n\nApply the standard differentiation rules:\n$ f'(x) = ... | {
"type": "calculus",
"seed": 1337,
"index": 436
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((-15 + 494/19) * 8) + 11^5 - log_10(1000) + (-258 mod 25) + floor(-185/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-15 + 494/19) * 8) + 11^5 - log_10(1000) + (-258 mod 25) + f... | {
"type": "arithmetic",
"seed": 1337,
"index": 437
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Compute the indefinite integral: $\\int \\ln(3x + -7)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(3x + -7)\\,dx $\n\nUse standard antiderivatives (and integration by parts if... | {
"type": "calculus",
"seed": 1337,
"index": 438
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Prove that $(\\mathbb{Z}_28, +)$ is a group under addition modulo 28."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_28=\\{0,1,\\dots,27\\}$ with operation $+$ modulo $28$.\n\n**Closure:** If $a,b\\in... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 439
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Compute $(137+-37) \\bmod 9$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (137+-37) \\bmod 9 $\n\nUse the rule that addition is compatible with reduction modulo... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 440
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n Evaluate: ((140 + -81/3) * 19) + 8^2 - log_7(7) + (-32 mod 23) + round(-27/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((140 + -81/3) * 19) + 8^2 - log_7(7) + (-32 mod 23) + round(-27/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 441
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Prove that $f(x,y)=\\frac12(5x^2+6y^2)+-6x+2y+-7$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(5x^2+6y^2)+-6x+2y+-7 $\n\n$ \\frac{\\partial^2 f}... | {
"type": "optimization",
"seed": 1337,
"index": 442
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Prove that $f(x)=6x^2+-5x+-6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=6x^2+-5x+-6$.\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": 443
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. 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 $\n\nTh... | {
"type": "trigonometry",
"seed": 1337,
"index": 444
} |
[
{
"role": "user",
"content": "Work through and solve this equation. -3x - 35 + 158 = 116x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -3x - 35 + 158 = 116x^2 $\n\nMove all terms to the left (standard form):\n$ -116x^2 + -3x + 123 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 445
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. 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$ \\alph... | {
"type": "trigonometry",
"seed": 1337,
"index": 446
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. -4x + 249 = 686"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -4x + 249 = 686 $\n\nMove all terms to the left (standard form):\n$ -4x + -437 = 0 $\n\nIsolate the $x$ term:\n$ -4x = 437 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 447
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. -44 + 198x^2 - 109 = 123x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -44 + 198x^2 - 109 = 123x $\n\nMove all terms to the left (standard form):\n$ 198x^2 + -123x + -153 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 448
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Differentiate with respect to $x$: $f(x) = (1x + -3)e^{1x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (1x + -3)e^{1x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{1... | {
"type": "calculus",
"seed": 1337,
"index": 449
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Compute $(-120+75) \\bmod 3$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-120+75) \\bmod 3 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 450
} |
[
{
"role": "user",
"content": "Please find the value of 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{\\... | {
"type": "optimization",
"seed": 1337,
"index": 451
} |
[
{
"role": "user",
"content": "Evaluate the following equation. -47 - 110 - 81x^2 = 81x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -47 - 110 - 81x^2 = 81x $\n\nMove all terms to the left (standard form):\n$ -81x^2 + -81x + -157 = 0 $\n\nApply the quadratic formula:\n$ x = \\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 452
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Prove that $f(x)=8x^2+7x+8$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+7x+8$.\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": 453
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Compute the indefinite integral: $\\int 3\\sin(4x) + -6\\cos(3x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 3\\sin(4x) + -6\\cos(3x)\\,dx $\n\nUse standard antiderivatives (and ... | {
"type": "calculus",
"seed": 1337,
"index": 454
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Prove that $f(x,y)=\\frac12(3x^2+3y^2)+-2x+-2y+3$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+3y^2)+-2x+-2y+3 $\n\n$ \\frac{\\partial^2 f}{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 455
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Solve for $x$ (in radians): sin(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 1 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(1) = 1.5708 $\n\nThe... | {
"type": "trigonometry",
"seed": 1337,
"index": 456
} |
[
{
"role": "user",
"content": "Solve the following math equation. 48 + 170x + 98 = 178x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 48 + 170x + 98 = 178x^2 $\n\nMove all terms to the left (standard form):\n$ -178x^2 + 170x + 146 = 0 $\n\nApply the quadratic formula:\n$ x = ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 457
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. -365 + 260 = -694x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -365 + 260 = -694x $\n\nMove all terms to the left (standard form):\n$ 694x + -105 = 0 $\n\nIsolate the $x$ term:\n$ 694x = 105 $\... | {
"type": "linear_equation",
"seed": 1337,
"index": 458
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Solve for $x$ (in radians): tan(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = 0 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(0) = 0 $\n\nThe general sol... | {
"type": "trigonometry",
"seed": 1337,
"index": 459
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. 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 $f'(x)=0$,... | {
"type": "optimization",
"seed": 1337,
"index": 460
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Evaluate: ((-1 + 80/20) * 2) + 5^5 - log_5(25) + (192 mod 8) + ceil(74/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-1 + 80/20) * 2) + 5^5 - log_5(25) + (192 mod 8) + ceil(74/2) $\n\nCompute each... | {
"type": "arithmetic",
"seed": 1337,
"index": 461
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-3&-1&5\\\\\\1&1&5\\\\\\-2&-5&0\\end{pmatrix},\\quad b=\\begin{pmatrix}-6\\\\\\2\\\\\\-4\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 462
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Find $A^{-1}$ for $A=\\begin{pmatrix}-9&-9\\\\-9&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -9 & -9 \\\\ -9 & 1 \\end{pmatrix} $\n\nCompute the determi... | {
"type": "linear_algebra",
"seed": 1337,
"index": 463
} |
[
{
"role": "user",
"content": "solve: \n\n Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}6&5\\\\\\6&-9\\end{pmatrix},\\quad b=\\begin{pmatrix}-23\\\\\\-93\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}6&5\\\\\\6&-9\\end{pmatrix} ,\\quad b=\\begin{... | {
"type": "linear_algebra",
"seed": 1337,
"index": 464
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. -275x + 268 = -124"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -275x + 268 = -124 $\n\nMove all terms to the left (standard form):\n$ -275x + 392 = 0 $\n\nIsolate the $x$ term:\n$ -275x = -392 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 465
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Evaluate: ((-98 + 88/8) * 13) + 4^5 - log_9(6561) + (257 mod 23) + floor(45/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-98 + 88/8) * 13) + 4^5 - log_9(6561) + (257 mod 23) + floor(45/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 466
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Evaluate: ((-13 + -182/14) * 2) + 6^3 - log_4(4) + (-181 mod 18) + floor(171/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-13 + -182/14) * 2) + 6^3 - log_4(4) + (-181 mod 18) + floor(171/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 467
} |
[
{
"role": "user",
"content": "Please work out this equation. Compute the indefinite integral: $\\int 9\\sin(4x) + 1\\cos(8x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 9\\sin(4x) + 1\\cos(8x)\\,dx $\n\nUse standard antiderivatives (and integration by ... | {
"type": "calculus",
"seed": 1337,
"index": 468
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}0&-4&2\\\\\\-1&2&-1\\\\\\1&-5&0\\end{pmatrix},\\quad b=\\begin{pmatrix}0\\\\\\0\\\\\\10\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g... | {
"type": "linear_algebra",
"seed": 1337,
"index": 469
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Find the modular inverse of 8 modulo 11, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $8\\cdot a^{-1} \\equiv 1 \\pmod{11}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guar... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 470
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x... | {
"type": "optimization",
"seed": 1337,
"index": 471
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. -39 - 138 + 14x = 138x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -39 - 138 + 14x = 138x^2 $\n\nMove all terms to the left (standard form):\n$ -138x^2 + 14x + -177 = 0 $\n\nApply the quadr... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 472
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 473
} |
[
{
"role": "user",
"content": "Work through and solve this equation. -56 - 287 = 337x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -56 - 287 = 337x $\n\nMove all terms to the left (standard form):\n$ -337x + -343 = 0 $\n\nIsolate the $x$ term:\n$ -337x = 343 $\n\nDivide both side... | {
"type": "linear_equation",
"seed": 1337,
"index": 474
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. -491x - 541 = 560"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -491x - 541 = 560 $\n\nMove all terms to the left (standard form):\n$ -491x + -1101 = 0 $\n\nIsolate the $x$ term:\n$ -491x = 1101 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 475
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Solve for $x$ (in radians): sin(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(\\frac{1}{2}) = 0.523599 $\... | {
"type": "trigonometry",
"seed": 1337,
"index": 476
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Differentiate with respect to $x$: $f(x) = 6\\sin(8x) + -2\\cos(2x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 6\\sin(8x) + -2\\cos(2x) $\n\nApply the standard differentiation rules:\n... | {
"type": "calculus",
"seed": 1337,
"index": 477
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. -2x^2 + 79 + 168x = 118"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -2x^2 + 79 + 168x = 118 $\n\nMove all terms to the left (standard form):\n$ -2x^2 + 168x + -39 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 478
} |
[
{
"role": "user",
"content": "Please calculate 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 = \\arctan(\\frac{1}{2... | {
"type": "trigonometry",
"seed": 1337,
"index": 479
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Evaluate: ((-14 + 264/11) * 5) + 8^3 - log_8(262144) + (-369 mod 16) + ceil(-148/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-14 + 264/11) * 5) + 8^3 - log_8(262144) + (-369 mod 16) + ceil(-14... | {
"type": "arithmetic",
"seed": 1337,
"index": 480
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Compute the determinant: $A=\\begin{pmatrix}-7&1\\\\-9&-8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-7&1\\\\\\-9&-8\\end{pmat... | {
"type": "linear_algebra",
"seed": 1337,
"index": 481
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. 16 + 46x + 133x^2 = 98"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 16 + 46x + 133x^2 = 98 $\n\nMove all terms to the left (standard form):\n$ 133x^2 + 46x + -82 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 482
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Evaluate: ((-73 + 192/24) * 10) + 6^3 - log_6(7776) + (169 mod 29) + ceil(-4/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-73 + 192/24) * 10) + 6^3 - log_6(7776) + (169 mod 29) + ceil(-4/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 483
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. 10 - 27x + 131x^2 = -103"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 10 - 27x + 131x^2 = -103 $\n\nMove all terms to the left (standard form):\n$ 131x^2 + -27x + 113 = 0 $\n\nApply the quadrat... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 484
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Prove that $f(x)=10x^2+7x+7$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=10x^2+7x+7$.\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": 485
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. -21 - 10 + 154x = 98x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -21 - 10 + 154x = 98x^2 $\n\nMove all terms to the left (standard form):\n$ -98x^2 + 154x + -31 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 486
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Prove that $f(x)=5x^2+-2x+10$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+-2x+10$.\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": 487
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Compute the indefinite integral: $\\int -4\\sin(5x) + 1\\cos(8x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -4\\sin(5x) + 1\\cos(8x)\\,dx $\n\nUse standard antiderivati... | {
"type": "calculus",
"seed": 1337,
"index": 488
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. -35 - 64 - 99x^2 = -109x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -35 - 64 - 99x^2 = -109x $\n\nMove all terms to the left (standard form):\n$ -99x^2 + 109x + -99 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 489
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Evaluate: ((88 + 575/25) * 6) + 11^5 - log_4(16) + (15 mod 2) + ceil(9/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((88 + 575/25) * 6) + 11^5 - log_4(16) + (15 mod 2) + ceil(9/2) $\n\nCompute each comp... | {
"type": "arithmetic",
"seed": 1337,
"index": 490
} |
[
{
"role": "user",
"content": "Please work out this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}3&2&2\\\\\\2&-1&4\\\\\\-2&-1&3\\end{pmatrix},\\quad b=\\begin{pmatrix}-12\\\\\\-3\\\\\\12\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., by Ga... | {
"type": "linear_algebra",
"seed": 1337,
"index": 491
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Differentiate with respect to $x$: $f(x) = -5x^3 + 1x^2 + -19x + -9$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -5x^3 + 1x^2 + -19x + -9 $\n\nApply the standard differentiation rules:\n$ f'(... | {
"type": "calculus",
"seed": 1337,
"index": 492
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Find $A^{-1}$ for $A=\\begin{pmatrix}-9&-1\\\\-1&8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -9 & -1 \\\\ -1 & 8 \\end{pmatrix} $\n\nCompute t... | {
"type": "linear_algebra",
"seed": 1337,
"index": 493
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Prove that $f(x,y)=\\frac12(1x^2+2y^2)+-6x+-6y+6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+2y^2)+-6x+-6y+6 $\n\n$ \\frac{\\partial^2 f}{\\... | {
"type": "optimization",
"seed": 1337,
"index": 494
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Prove that $(\\mathbb{Z}_9, +)$ is a group under addition modulo 9."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_9=\\{0,1,\\dots,8\\}$ with operation $+$ modulo $9$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 495
} |
[
{
"role": "user",
"content": "Please evaluate and compute 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$ \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 496
} |
[
{
"role": "user",
"content": "Evaluate the following equation. -5 + 191x - 30 = 71x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -5 + 191x - 30 = 71x^2 $\n\nMove all terms to the left (standard form):\n$ -71x^2 + 191x + -35 = 0 $\n\nApply the quadratic formula:\n$ x = \\fra... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 497
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Solve for $x$ (in radians): cos(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -1 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(-1) = 3.14159 $\n\nT... | {
"type": "trigonometry",
"seed": 1337,
"index": 498
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? -761 + 521 = 675x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -761 + 521 = 675x $\n\nMove all terms to the left (standard form):\n$ -675x + -240 = 0 $\n\nIsolate the $x$ term:\n$ -675x = 240 $\n\nDiv... | {
"type": "linear_equation",
"seed": 1337,
"index": 499
} |
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