messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Prove that $f(x)=6x^2+-2x+-6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=6x^2+-2x+-6$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x... | {
"type": "optimization",
"seed": 1337,
"index": 6600
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 859 + 241x = -238"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 859 + 241x = -238 $\n\nMove all terms to the left (standard form):\n$ 241x + 1097 = 0 $\n\nIsolate the $x$ term:\n$ 241x = -1097 $\n\nDivide both side... | {
"type": "linear_equation",
"seed": 1337,
"index": 6601
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 45x - 9 + 164x^2 = 186"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 45x - 9 + 164x^2 = 186 $\n\nMove all terms to the left (standard form):\n$ 164x^2 + 45x + -195 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6602
} |
[
{
"role": "user",
"content": "Resolve the equation provided. 288x + 152 = -229"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 288x + 152 = -229 $\n\nMove all terms to the left (standard form):\n$ 288x + 381 = 0 $\n\nIsolate the $x$ term:\n$ 288x = -381 $\n\nDivide both sides by th... | {
"type": "linear_equation",
"seed": 1337,
"index": 6603
} |
[
{
"role": "user",
"content": "Solve the following math equation. Solve for $x$ (in radians): cos(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \... | {
"type": "trigonometry",
"seed": 1337,
"index": 6604
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Find the modular inverse of 13 modulo 36, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $13\\cdot a^{-1} \\equiv 1 \\pmod{36}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guar... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6605
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. 106 - 673 = 237x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 106 - 673 = 237x $\n\nMove all terms to the left (standard form):\n$ -237x + -567 = 0 $\n\nIsolate the $x$ term:\n$ -237x = 567 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 6606
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 6607
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. 23x - 49 - 123x^2 = -131"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 23x - 49 - 123x^2 = -131 $\n\nMove all terms to the left (standard form):\n$ -123x^2 + 23x + 82 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6608
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. 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).... | {
"type": "trigonometry",
"seed": 1337,
"index": 6609
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? 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": 6610
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Evaluate: ((16 + -550/25) * 10) + 8^3 - log_10(10) + (35 mod 6) + round(89/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((16 + -550/25) * 10) + 8^3 - log_10(10) + (35 mod 6) + round(89/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 6611
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Compute $(-124+-23) \\bmod 7$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-124+-23) \\bmod 7 $\n\nUse the rule that addition is compatible with reduction... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6612
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. Differentiate with respect to $x$: $f(x) = -4\\sin(3x) + 1\\cos(5x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -4\\sin(3x) + 1\\cos(5x) $\n\nApply the standard differentiation rules:\... | {
"type": "calculus",
"seed": 1337,
"index": 6613
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. -333 - 750x = -863"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -333 - 750x = -863 $\n\nMove all terms to the left (standard form):\n$ -750x + 530 = 0 $\n\nIsolate the $x$ term:\n$ -750x = -530 $\n\nDiv... | {
"type": "linear_equation",
"seed": 1337,
"index": 6614
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Compute the indefinite integral: $\\int (-5x + 7)\\sin(2x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-5x + 7)\\sin(2x)\\,dx $\n\nUse standard antiderivatives (and integ... | {
"type": "calculus",
"seed": 1337,
"index": 6615
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Prove that $(\\mathbb{Z}_26, +)$ is a group under addition modulo 26."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_26=\\{0,1,\\dots,25\\}$ with operation $+$ modulo $26$.\n\n**Closure:** If $a,b\\in\\math... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6616
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}8&6\\\\\\8&4\\end{pmatrix},\\quad b=\\begin{pmatrix}26\\\\\\28\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}8&6\\\\\\8&4\\end... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6617
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Solve for $x$ (in radians): tan(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 6618
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. 20x + 32 - 24 = -63x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 20x + 32 - 24 = -63x^2 $\n\nMove all terms to the left (standard form):\n$ 63x^2 + 20x + 8 = 0 $\n\nApply the quadratic formula:\n$ x = \\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6619
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Differentiate with respect to $x$: $f(x) = -3x^3 + 3x^2 + 15x + -13$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -3x^3 + 3x^2 + 15x + -13 $\n\nApply the standard differentiation rules:\n... | {
"type": "calculus",
"seed": 1337,
"index": 6620
} |
[
{
"role": "user",
"content": "Can you compute this equation? Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 6621
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. -526 - 194x = 817"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -526 - 194x = 817 $\n\nMove all terms to the left (standard form):\n$ -194x + -1343 = 0 $\n\nIsolate the $x$ term:\n$ -194x = 1343 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 6622
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Compute the indefinite integral: $\\int -1e^{2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -1e^{2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if need... | {
"type": "calculus",
"seed": 1337,
"index": 6623
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Find $A^{-1}$ for $A=\\begin{pmatrix}-6&-6\\\\6&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & -6 \\\\ 6 & 3 \\end{pmatrix} $\n\nCompute t... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6624
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Evaluate: ((-71 + -418/22) * 3) + 6^2 - log_3(81) + (-63 mod 13) + floor(-96/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-71 + -418/22) * 3) + 6^2 - log_3(81) + (-63 mod 13) + floor(-96/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 6625
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Differentiate with respect to $x$: $f(x) = 3x^3 + 7x^2 + -13x + -2$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 3x^3 + 7x^2 + -13x + -2 $\n\nApply the standard differentiation rules:\n$... | {
"type": "calculus",
"seed": 1337,
"index": 6626
} |
[
{
"role": "user",
"content": "Please solve this math problem. Solve for $x$ (in radians): cos(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\... | {
"type": "trigonometry",
"seed": 1337,
"index": 6627
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Evaluate: ((-95 + -108/4) * 20) + 12^4 - log_2(2) + (-176 mod 24) + ceil(73/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-95 + -108/4) * 20) + 12^4 - log_2(2) + (-176 mod 24) + ceil(73/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6628
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. 690x + 694 = 210"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 690x + 694 = 210 $\n\nMove all terms to the left (standard form):\n$ 690x + 484 = 0 $\n\nIsolate the $x$ term:\n$ 690x = -484 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 6629
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}3&-9\\\\\\-7&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}-30\\\\\\46\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}3&-9\\\\\\-7&-3\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6630
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, he... | {
"type": "optimization",
"seed": 1337,
"index": 6631
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Compute $(127+-143) \\bmod 2$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (127+-143) \\bmod 2 $\n\nUse the rule that addition is compatible with reduction ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6632
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. 956x + 719 = -428"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 956x + 719 = -428 $\n\nMove all terms to the left (standard form):\n$ 956x + 1147 = 0 $\n\nIsolate the $x$ term:\n$ 956x = -1147 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6633
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Prove that $f(x,y)=\\frac12(10x^2+6y^2)+-3x+-2y+6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(10x^2+6y^2)+-3x+-2y+6 $\n\n$ \\frac{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 6634
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Evaluate: ((-6 + 192/6) * 14) + 6^3 - log_2(8) + (-310 mod 2) + floor(12/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-6 + 192/6) * 14) + 6^3 - log_2(8) + (-310 mod 2) + floor(12/2) $\n\nCompute each... | {
"type": "arithmetic",
"seed": 1337,
"index": 6635
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: 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(\\frac{1}{2}) = 1.0472 $\n\... | {
"type": "trigonometry",
"seed": 1337,
"index": 6636
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. + 47 + 168x^2 = 34x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ + 47 + 168x^2 = 34x $\n\nMove all terms to the left (standard form):\n$ 168x^2 + -34x + 47 = 0 $\n\nApply the quadratic formula:\n$ x = \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6637
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. -102x + 854 = 100"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -102x + 854 = 100 $\n\nMove all terms to the left (standard form):\n$ -102x + 754 = 0 $\n\nIsolate the $x$ term:\n$ -102x = -754 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6638
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Solve for $x$ (in radians): cos(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = 0 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(0) = 1.5708 $\n\nThe gen... | {
"type": "trigonometry",
"seed": 1337,
"index": 6639
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:\n$ 4x(x... | {
"type": "optimization",
"seed": 1337,
"index": 6640
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Prove that $(\\mathbb{Z}_9, +)$ is a group under addition modulo 9."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_9=\\{0,1,\\dots,8\\}$ with operation $+$ modulo $9$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6641
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Differentiate with respect to $x$: $f(x) = 2x^3 + 4x^2 + 9x + 14$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 2x^3 + 4x^2 + 9x + 14 $\n\nApply the standard differentiation rules:\n$... | {
"type": "calculus",
"seed": 1337,
"index": 6642
} |
[
{
"role": "user",
"content": "solve: \n\n Evaluate: ((-60 + 300/6) * 19) + 3^3 - log_4(4096) + (111 mod 4) + floor(189/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-60 + 300/6) * 19) + 3^3 - log_4(4096) + (111 mod 4) + floor(189/2) $\n\nCompute each component:\n$ 300/6... | {
"type": "arithmetic",
"seed": 1337,
"index": 6643
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Evaluate: ((165 + 80/2) * 1) + 3^5 - log_6(36) + (444 mod 4) + ceil(49/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((165 + 80/2) * 1) + 3^5 - log_6(36) + (444 mod 4) + ceil(49/2) $\n\nCompute each ... | {
"type": "arithmetic",
"seed": 1337,
"index": 6644
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Evaluate: ((-90 + 372/12) * 10) + 9^2 - log_4(64) + (-423 mod 18) + floor(13/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-90 + 372/12) * 10) + 9^2 - log_4(64) + (-423 mod 18) + floor(13/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6645
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Evaluate: ((-179 + -64/4) * 19) + 2^2 - log_6(1296) + (381 mod 6) + floor(4/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-179 + -64/4) * 19) + 2^2 - log_6(1296) + (381 mod 6) + floor(4/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 6646
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Evaluate: ((29 + 533/13) * 2) + 3^5 - log_6(46656) + (-82 mod 27) + floor(130/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((29 + 533/13) * 2) + 3^5 - log_6(46656) + (-82 mod 27) + floor(130/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 6647
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-4&-3&1\\\\\\-1&-4&2\\\\\\1&5&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}7\\\\\\13\\\\\\-17\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Sol... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6648
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n Evaluate: ((-91 + -190/19) * 7) + 9^4 - log_9(6561) + (-287 mod 14) + round(-144/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-91 + -190/19) * 7) + 9^4 - log_9(6561) + (-287 mod 14) + round(-1... | {
"type": "arithmetic",
"seed": 1337,
"index": 6649
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. -23x + 39x^2 + 80 = 21"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -23x + 39x^2 + 80 = 21 $\n\nMove all terms to the left (standard form):\n$ 39x^2 + -23x + 59 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6650
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n 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": 6651
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Evaluate: ((184 + 300/12) * 13) + 10^5 - log_9(59049) + (-106 mod 22) + ceil(-173/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((184 + 300/12) * 13) + 10^5 - log_9(59049) + (-106 mod 22) + ceil(-173... | {
"type": "arithmetic",
"seed": 1337,
"index": 6652
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Solve for $x$ (in radians): tan(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -1 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(-1) = -0.785398 $\n\nTh... | {
"type": "trigonometry",
"seed": 1337,
"index": 6653
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Compute the indefinite integral: $\\int (2x + 5)e^{2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (2x + 5)e^{2x}\\,dx $\n\nUse standard antiderivatives (and integration by par... | {
"type": "calculus",
"seed": 1337,
"index": 6654
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. -220x - 229 = 508"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -220x - 229 = 508 $\n\nMove all terms to the left (standard form):\n$ -220x + -737 = 0 $\n\nIsolate the $x$ term:\n$ -220x = 737 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6655
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. -3x^2 - 134 - 135 = -138x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -3x^2 - 134 - 135 = -138x $\n\nMove all terms to the left (standard form):\n$ -3x^2 + 138x + -269 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6656
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Evaluate: ((-107 + 90/9) * 15) + 10^4 - log_9(729) + (272 mod 14) + round(166/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-107 + 90/9) * 15) + 10^4 - log_9(729) + (272 mod 14) + round(166/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 6657
} |
[
{
"role": "user",
"content": "Solve this equation step by step. 11x^2 - 153x - 90 = -189"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 11x^2 - 153x - 90 = -189 $\n\nMove all terms to the left (standard form):\n$ 11x^2 + -153x + 99 = 0 $\n\nApply the quadratic formula:\n$ x = \... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6658
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. 133x - 994 = 291"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 133x - 994 = 291 $\n\nMove all terms to the left (standard form):\n$ 133x + -1285 = 0 $\n\nIsolate the $x$ term:\n$ 133x = 1285 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 6659
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Evaluate: ((98 + 200/10) * 13) + 3^3 - log_2(8) + (-35 mod 9) + ceil(166/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((98 + 200/10) * 13) + 3^3 - log_2(8) + (-35 mod 9) + ceil(166/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 6660
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Evaluate: ((20 + -276/6) * 18) + 9^4 - log_8(32768) + (-183 mod 27) + round(-34/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((20 + -276/6) * 18) + 9^4 - log_8(32768) + (-183 mod 27) + round(-34/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 6661
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Compute the indefinite integral: $\\int (-6x + 10)\\sin(5x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-6x + 10)\\sin(5x)\\,dx $\n\nUse standard antiderivatives (and integrat... | {
"type": "calculus",
"seed": 1337,
"index": 6662
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. 38x - 176 + 173 = 95x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 38x - 176 + 173 = 95x^2 $\n\nMove all terms to the left (standard form):\n$ -95x^2 + 38x + -3 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6663
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 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": 6664
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Prove that $f(x)=4x^2+1x+5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=4x^2+1x+5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 8x ... | {
"type": "optimization",
"seed": 1337,
"index": 6665
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Differentiate with respect to $x$: $f(x) = 6x^3 + -7x^2 + -1x + -18$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 6x^3 + -7x^2 + -1x + -18 $\n\nApply the standard differentiation rules:\n$ f... | {
"type": "calculus",
"seed": 1337,
"index": 6666
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. -10x - 104x^2 + 115 = 12"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -10x - 104x^2 + 115 = 12 $\n\nMove all terms to the left (standard form):\n$ -104x^2 + -10x + 103 = 0 $\n\nApply the quadratic f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6667
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 6668
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Evaluate: ((-107 + 96/4) * 12) + 10^5 - log_9(531441) + (-99 mod 20) + ceil(27/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-107 + 96/4) * 12) + 10^5 - log_9(531441) + (-99 mod 20) + ceil(27/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6669
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Evaluate: ((-179 + 143/11) * 13) + 2^4 - log_3(3) + (-459 mod 16) + ceil(-81/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-179 + 143/11) * 13) + 2^4 - log_3(3) + (-459 mod 16) + ceil(-81/2) $\n\nCompute each component:\n$ 1... | {
"type": "arithmetic",
"seed": 1337,
"index": 6670
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Compute the indefinite integral: $\\int 2x^3 + -9x^2 + 14x + 18\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + -9x^2 + 14x + 18\\,dx $\n\nUse standard antiderivatives (and integr... | {
"type": "calculus",
"seed": 1337,
"index": 6671
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-9&6\\\\\\-8&6\\end{pmatrix},\\quad b=\\begin{pmatrix}-99\\\\\\-90\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-9&6\\\\\\-8&6\\end{... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6672
} |
[
{
"role": "user",
"content": "Solve this equation step by step. -27 - 193 + 96x^2 = 190x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -27 - 193 + 96x^2 = 190x $\n\nMove all terms to the left (standard form):\n$ 96x^2 + -190x + -220 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6673
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 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{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 6674
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. -3x^2 + 187 + 114x = 41"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -3x^2 + 187 + 114x = 41 $\n\nMove all terms to the left (standard form):\n$ -3x^2 + 114x + 146 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6675
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. 35x^2 + 100x + 23 = -185"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 35x^2 + 100x + 23 = -185 $\n\nMove all terms to the left (standard form):\n$ 35x^2 + 100x + 208 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6676
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Evaluate: ((45 + -52/2) * 6) + 11^5 - log_8(512) + (-104 mod 28) + ceil(-192/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((45 + -52/2) * 6) + 11^5 - log_8(512) + (-104 mod 28) + ceil(-192/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6677
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-5&-2\\\\-3&4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -5 & -2 \\\\ -3 & 4 \\end{pmatrix} $\n\nCompute the det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6678
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. -229x + 456 = -852"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -229x + 456 = -852 $\n\nMove all terms to the left (standard form):\n$ -229x + 1308 = 0 $\n\nIsolate the $x$ term:\n$ -229x = -1308 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 6679
} |
[
{
"role": "user",
"content": "Please work out this equation. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arc... | {
"type": "trigonometry",
"seed": 1337,
"index": 6680
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Evaluate: ((161 + 65/5) * 5) + 11^2 - log_7(16807) + (-418 mod 2) + floor(-112/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((161 + 65/5) * 5) + 11^2 - log_7(16807) + (-418 mod 2) + floor(-112/... | {
"type": "arithmetic",
"seed": 1337,
"index": 6681
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}9&-2\\\\\\2&4\\end{pmatrix},\\quad b=\\begin{pmatrix}-72\\\\\\24\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}9&-2\\\\\\2&4\\end{pmatri... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6682
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. -170x + 50 = -268"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -170x + 50 = -268 $\n\nMove all terms to the left (standard form):\n$ -170x + 318 = 0 $\n\nIsolate the $x$ term:\n$ -170x = -318 $\n\nD... | {
"type": "linear_equation",
"seed": 1337,
"index": 6683
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Evaluate: ((81 + 231/11) * 15) + 8^4 - log_5(5) + (65 mod 13) + round(159/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((81 + 231/11) * 15) + 8^4 - log_5(5) + (65 mod 13) + round(159/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 6684
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. 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 $a,b\\in\\mathbb{Z}_n$,... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6685
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Evaluate: ((151 + 143/11) * 3) + 5^4 - log_5(25) + (-165 mod 27) + floor(40/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((151 + 143/11) * 3) + 5^4 - log_5(25) + (-165 mod 27) + floor(40/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 6686
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Find the modular inverse of 31 modulo 40, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $31\\cdot a^{-1} \\equiv 1 \\pmod{40}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gu... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6687
} |
[
{
"role": "user",
"content": "solve: \n\n -15x + 74x^2 + 76 = -54"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -15x + 74x^2 + 76 = -54 $\n\nMove all terms to the left (standard form):\n$ 74x^2 + -15x + 130 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\pm \\sqrt{b^2... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6688
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. Prove that $f(x,y)=\\frac12(3x^2+9y^2)+-1x+4y+-10$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+9y^2)+-1x+4y+-10 $\n\n$ \\frac{\\partia... | {
"type": "optimization",
"seed": 1337,
"index": 6689
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Prove that $f(x,y)=\\frac12(6x^2+5y^2)+-3x+-6y+2$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(6x^2+5y^2)+-3x+-6y+2 $\n\n$ \\frac{\\partial^2 f}{\\partial ... | {
"type": "optimization",
"seed": 1337,
"index": 6690
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Evaluate: ((-163 + -108/4) * 1) + 3^4 - log_5(125) + (-499 mod 8) + round(69/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-163 + -108/4) * 1) + 3^4 - log_5(125) + (-499 mod 8) + round(69/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 6691
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. -733 + 191 = 73x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -733 + 191 = 73x $\n\nMove all terms to the left (standard form):\n$ -73x + -542 = 0 $\n\nIsolate the $x$ term:\n$ -73x = 542 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 6692
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Solve for $x$ (in radians): cos(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = 0 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(0) = 1.5708 $\n\nThe ge... | {
"type": "trigonometry",
"seed": 1337,
"index": 6693
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. 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 $a,b\\in\\mathbb{Z}_n$, ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6694
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Differentiate with respect to $x$: $f(x) = \\ln(2x + 0)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(2x + 0) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{2}{2x + 0}} ... | {
"type": "calculus",
"seed": 1337,
"index": 6695
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Solve for $x$ (in radians): tan(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alph... | {
"type": "trigonometry",
"seed": 1337,
"index": 6696
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Find $A^{-1}$ for $A=\\begin{pmatrix}-1&2\\\\-8&-2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -1 & 2 \\\\ -8 & -2 \\end{pmatrix} $\n\nCompute ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 6697
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Compute $(-154+-23) \\bmod 3$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-154+-23) \\bmod 3 $\n\nUse the rule that addition is compatible with reduction ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 6698
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. -12 - 92x^2 + 48x = -3"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -12 - 92x^2 + 48x = -3 $\n\nMove all terms to the left (standard form):\n$ -92x^2 + 48x + -9 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 6699
} |
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