messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Determine the solution for this math problem. Prove that $(\\mathbb{Z}_6, +)$ is a group under addition modulo 6."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_6=\\{0,1,\\dots,5\\}$ with operation $+$ modulo $6$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7700
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Prove that $f(x,y)=\\frac12(6x^2+8y^2)+5x+-4y+-1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(6x^2+8y^2)+5x+-4y+-1 $\n\n$ \\frac{\\partial^2... | {
"type": "optimization",
"seed": 1337,
"index": 7701
} |
[
{
"role": "user",
"content": "Determine the solution of the given 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 7702
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. -41 - 198x^2 + 17x = -198"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -41 - 198x^2 + 17x = -198 $\n\nMove all terms to the left (standard form):\n$ -198x^2 + 17x + 157 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7703
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. 47 + 194x + 107 = 24x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 47 + 194x + 107 = 24x^2 $\n\nMove all terms to the left (standard form):\n$ -24x^2 + 194x + 154 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7704
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Compute the indefinite integral: $\\int -4\\sin(3x) + -4\\cos(5x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -4\\sin(3x) + -4\\cos(5x)\\,dx $\n\nUse standard antiderivatives... | {
"type": "calculus",
"seed": 1337,
"index": 7705
} |
[
{
"role": "user",
"content": "Please work out this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}1&4\\\\\\-3&8\\end{pmatrix},\\quad b=\\begin{pmatrix}-9\\\\\\27\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}1&4\\\\\\-3&8\\end{pmatrix} ,... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7706
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Evaluate: ((-149 + 720/18) * 8) + 8^3 - log_5(5) + (-91 mod 7) + ceil(-133/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-149 + 720/18) * 8) + 8^3 - log_5(5) + (-91 mod 7) + ceil(-133/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 7707
} |
[
{
"role": "user",
"content": "solve: \n\n Compute the indefinite integral: $\\int 1e^{2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1e^{2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \\int 1e^{2x}\\,dx = \\boxed{\... | {
"type": "calculus",
"seed": 1337,
"index": 7708
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Evaluate: ((-121 + 273/7) * 5) + 6^5 - log_7(2401) + (92 mod 7) + ceil(-173/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-121 + 273/7) * 5) + 6^5 - log_7(2401) + (92 mod 7) + ceil(-173/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 7709
} |
[
{
"role": "user",
"content": "solve 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 = \\arccos(-\\frac{\\sqrt{3}}... | {
"type": "trigonometry",
"seed": 1337,
"index": 7710
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. 41 + 67x + 143 = 22x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 41 + 67x + 143 = 22x^2 $\n\nMove all terms to the left (standard form):\n$ -22x^2 + 67x + 184 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7711
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. 42 + 92x^2 + 175 = -92x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 42 + 92x^2 + 175 = -92x $\n\nMove all terms to the left (standard form):\n$ 92x^2 + 92x + 217 = 0 $\n\nApply the quadratic ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7712
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Evaluate: ((-27 + -77/7) * 14) + 10^3 - log_7(16807) + (93 mod 4) + floor(191/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-27 + -77/7) * 14) + 10^3 - log_7(16807) + (93 mod 4) + floor(191/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 7713
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Compute the indefinite integral: $\\int \\ln(2x + 4)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(2x + 4)\\,dx $\n\nUse standard antiderivatives (and integration by parts if... | {
"type": "calculus",
"seed": 1337,
"index": 7714
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}0&3\\\\\\1&-7\\end{pmatrix},\\quad b=\\begin{pmatrix}-9\\\\\\29\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}0&3\\\\\\1&-7\\end{p... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7715
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-1&1&1\\\\\\-2&1&4\\\\\\1&2&2\\end{pmatrix},\\quad b=\\begin{pmatrix}-2\\\\\\5\\\\\\2\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7716
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Evaluate: ((152 + -300/12) * 6) + 10^5 - log_10(1000000) + (59 mod 18) + floor(36/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((152 + -300/12) * 6) + 10^5 - log_10(1000000) + (59 mod 18... | {
"type": "arithmetic",
"seed": 1337,
"index": 7717
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Differentiate with respect to $x$: $f(x) = \\ln(1x + 3)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(1x + 3) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{1}{1x + 3}} ... | {
"type": "calculus",
"seed": 1337,
"index": 7718
} |
[
{
"role": "user",
"content": "solve: Prove that $f(x)=2x^2+-6x+-5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+-6x+-5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 4x + -6 $\n$ f''(x) = 4 $\n\nSi... | {
"type": "optimization",
"seed": 1337,
"index": 7719
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Compute the indefinite integral: $\\int -9\\sin(6x) + -7\\cos(9x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -9\\sin(6x) + -7\\cos(9x)\\,dx $\n\nUse standard antiderivatives ... | {
"type": "calculus",
"seed": 1337,
"index": 7720
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. 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$... | {
"type": "trigonometry",
"seed": 1337,
"index": 7721
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Evaluate: ((140 + 378/18) * 14) + 2^4 - log_7(49) + (-104 mod 27) + floor(-154/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((140 + 378/18) * 14) + 2^4 - log_7(49) + (-104 mod 27) + floor(-1... | {
"type": "arithmetic",
"seed": 1337,
"index": 7722
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. -7x - 41 - 48x^2 = -18"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -7x - 41 - 48x^2 = -18 $\n\nMove all terms to the left (standard form):\n$ -48x^2 + -7x + -23 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7723
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Compute $(-61+-187) \\bmod 17$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-61+-187) \\bmod 17 $\n\nUse the rule that addition is compatible with r... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7724
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Compute the indefinite integral: $\\int (6x + -2)\\sin(6x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (6x + -2)\\sin(6x)\\,dx $\n\nUse standard antiderivatives (and integratio... | {
"type": "calculus",
"seed": 1337,
"index": 7725
} |
[
{
"role": "user",
"content": "Can you compute this equation? Compute the indefinite integral: $\\int 1\\sin(8x) + 4\\cos(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1\\sin(8x) + 4\\cos(4x)\\,dx $\n\nUse standard antiderivatives (and integration by ... | {
"type": "calculus",
"seed": 1337,
"index": 7726
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Evaluate: ((-158 + -484/11) * 17) + 3^5 - log_4(4096) + (-113 mod 20) + ceil(10/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-158 + -484/11) * 17) + 3^5 - log_4(4096) + (-113 mod 20) + ceil(10/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 7727
} |
[
{
"role": "user",
"content": "solve: Compute $(198+-2) \\bmod 19$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (198+-2) \\bmod 19 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ (a+b)\\bmod n = ((a\\bm... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7728
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. 467 - 861 = 371x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 467 - 861 = 371x $\n\nMove all terms to the left (standard form):\n$ -371x + -394 = 0 $\n\nIsolate the $x$ term:\n$ -371x = 394 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 7729
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. -308 - 985 = 759x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -308 - 985 = 759x $\n\nMove all terms to the left (standard form):\n$ -759x + -1293 = 0 $\n\nIsolate the $x$ term:\n$ -759x = 1293 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7730
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}2&4&-5\\\\\\-4&3&1\\\\\\5&-4&-4\\end{pmatrix},\\quad b=\\begin{pmatrix}-7\\\\\\-17\\\\\\19\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7731
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Compute the indefinite integral: $\\int (2x + 8)e^{2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (2x + 8)e^{2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts ... | {
"type": "calculus",
"seed": 1337,
"index": 7732
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Differentiate with respect to $x$: $f(x) = (6x + 4)\\sin(5x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (6x + 4)\\sin(5x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed... | {
"type": "calculus",
"seed": 1337,
"index": 7733
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. 916 + 578x = 907"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 916 + 578x = 907 $\n\nMove all terms to the left (standard form):\n$ 578x + 9 = 0 $\n\nIsolate the $x$ term:\n$ 578x = -9 $\n\nDivide both sides by... | {
"type": "linear_equation",
"seed": 1337,
"index": 7734
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. 15 - 31x + 77x^2 = 149"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 15 - 31x + 77x^2 = 149 $\n\nMove all terms to the left (standard form):\n$ 77x^2 + -31x + -134 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7735
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Prove that $(\\mathbb{Z}_13, +)$ is a group under addition modulo 13."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_13=\\{0,1,\\dots,12\\}$ with operation $+$ modulo $13$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7736
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}8&6\\\\-5&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 8 & 6 \\\\ -5 & -5 \\end{pmatrix} $\n\nCompute the de... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7737
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 10 + 58 - 11x^2 = 74x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 10 + 58 - 11x^2 = 74x $\n\nMove all terms to the left (standard form):\n$ -11x^2 + -74x + 68 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7738
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. -921x + 597 = 189"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -921x + 597 = 189 $\n\nMove all terms to the left (standard form):\n$ -921x + 408 = 0 $\n\nIsolate the $x$ term:\n$ -921x = -408 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7739
} |
[
{
"role": "user",
"content": "Can you compute this equation? Compute the determinant: $A=\\begin{pmatrix}-9&-6\\\\-1&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-9&-6\\\\\\-1&5\\end{pmatrix} $\n\nUse $\\det(A)=... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7740
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Evaluate: ((-108 + -744/24) * 10) + 10^5 - log_10(1000000) + (-259 mod 24) + round(146/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-108 + -744/24) * 10) + 10^5 - log_10(1000000) + (-259 mod 24)... | {
"type": "arithmetic",
"seed": 1337,
"index": 7741
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Find the modular inverse of 2 modulo 39, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $2\\cdot a^{-1} \\equiv 1 \\pmod{39}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7742
} |
[
{
"role": "user",
"content": "Please solve this math problem. Differentiate with respect to $x$: $f(x) = -7\\sin(7x) + -1\\cos(2x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -7\\sin(7x) + -1\\cos(2x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\... | {
"type": "calculus",
"seed": 1337,
"index": 7743
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}3&5\\\\\\5&-9\\end{pmatrix},\\quad b=\\begin{pmatrix}-5\\\\\\-43\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}3&5\\\\\\5&-9\\end{p... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7744
} |
[
{
"role": "user",
"content": "Can you compute this equation? -872 + 123 = 644x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -872 + 123 = 644x $\n\nMove all terms to the left (standard form):\n$ -644x + -749 = 0 $\n\nIsolate the $x$ term:\n$ -644x = 749 $\n\nDivide both sides by ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7745
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Find the modular inverse of 29 modulo 33, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $29\\cdot a^{-1} \\equiv 1 \\pmod{33}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7746
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Differentiate with respect to $x$: $f(x) = -4x^3 + 3x^2 + -13x + 11$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -4x^3 + 3x^2 + -13x + 11 $\n\nApply the standard differentiation rules... | {
"type": "calculus",
"seed": 1337,
"index": 7747
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}1&-8\\\\\\-1&-7\\end{pmatrix},\\quad b=\\begin{pmatrix}47\\\\\\43\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}1&-8\\\\\\-1... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7748
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. -35 - 48x - 178x^2 = -152"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -35 - 48x - 178x^2 = -152 $\n\nMove all terms to the left (standard form):\n$ -178x^2 + -48x + 117 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7749
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. -602 + 400x = 155"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -602 + 400x = 155 $\n\nMove all terms to the left (standard form):\n$ 400x + -757 = 0 $\n\nIsolate the $x$ term:\n$ 400x = 757 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 7750
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Compute the determinant: $A=\\begin{pmatrix}-3&8\\\\8&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-3&8\\\\\\8&6\\end{pmatrix} $\n... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7751
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Compute the indefinite integral: $\\int -2x^3 + -6x^2 + 5x + -18\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -2x^3 + -6x^2 + 5x + -18\\,dx $\n\nUse standard antiderivatives (and integration ... | {
"type": "calculus",
"seed": 1337,
"index": 7752
} |
[
{
"role": "user",
"content": "Please solve this math problem. Find the modular inverse of 22 modulo 35, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $22\\cdot a^{-1} \\equiv 1 \\pmod{35}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integer... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7753
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}8&3\\\\\\-2&5\\end{pmatrix},\\quad b=\\begin{pmatrix}41\\\\\\-39\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}8&3\\\\\\-2... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7754
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. -2x^2 + 27 + 50x = 170"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -2x^2 + 27 + 50x = 170 $\n\nMove all terms to the left (standard form):\n$ -2x^2 + 50x + -143 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7755
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. 33x - 41 - 48 = 93x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 33x - 41 - 48 = 93x^2 $\n\nMove all terms to the left (standard form):\n$ -93x^2 + 33x + -89 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7756
} |
[
{
"role": "user",
"content": "Find the correct solution for 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}} $"
}
... | {
"type": "calculus",
"seed": 1337,
"index": 7757
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Evaluate: ((42 + 690/23) * 8) + 8^3 - log_2(4) + (221 mod 10) + floor(80/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((42 + 690/23) * 8) + 8^3 - log_2(4) + (221 mod 10) + floor(80/2) $\n\nCompute each c... | {
"type": "arithmetic",
"seed": 1337,
"index": 7758
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Differentiate with respect to $x$: $f(x) = 7\\sin(3x) + -9\\cos(3x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 7\\sin(3x) + -9\\cos(3x) $\n\nApply the standard differentiation rules:\n$ ... | {
"type": "calculus",
"seed": 1337,
"index": 7759
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. 3x + 69x^2 - 151 = -46"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 3x + 69x^2 - 151 = -46 $\n\nMove all terms to the left (standard form):\n$ 69x^2 + 3x + -105 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7760
} |
[
{
"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": 7761
} |
[
{
"role": "user",
"content": "Please find the solution of 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 = \\arcco... | {
"type": "trigonometry",
"seed": 1337,
"index": 7762
} |
[
{
"role": "user",
"content": "Compute the correct solution for this 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": 7763
} |
[
{
"role": "user",
"content": "Compute and determine the solution to this equation. Find the modular inverse of 3 modulo 25, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $3\\cdot a^{-1} \\equiv 1 \\pmod{25}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7764
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. 15 + 90 - 35x = -111x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 15 + 90 - 35x = -111x^2 $\n\nMove all terms to the left (standard form):\n$ 111x^2 + -35x + 105 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7765
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Differentiate with respect to $x$: $f(x) = 6x^3 + -9x^2 + -13x + -17$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 6x^3 + -9x^2 + -13x + -17 $\n\nApply the standard differentiation rules:\n$ f'... | {
"type": "calculus",
"seed": 1337,
"index": 7766
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. -44x - 643 = 465"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -44x - 643 = 465 $\n\nMove all terms to the left (standard form):\n$ -44x + -1108 = 0 $\n\nIsolate the $x$ term:\n$ -44x = 1108 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 7767
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. -22 - 627x = -400"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -22 - 627x = -400 $\n\nMove all terms to the left (standard form):\n$ -627x + 378 = 0 $\n\nIsolate the $x$ term:\n$ -627x = -378 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 7768
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 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": 7769
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. -778 - 768 = -481x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -778 - 768 = -481x $\n\nMove all terms to the left (standard form):\n$ 481x + -1546 = 0 $\n\nIsolate the $x$ term:\n$ 481x = 1546 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 7770
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. 823 - 904x = 191"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 823 - 904x = 191 $\n\nMove all terms to the left (standard form):\n$ -904x + 632 = 0 $\n\nIsolate the $x$ term:\n$ -904x = -632 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 7771
} |
[
{
"role": "user",
"content": "Can you determine the solution of 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) = 0 $\n\nThe ... | {
"type": "trigonometry",
"seed": 1337,
"index": 7772
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. -27x + 38x^2 + 138 = 99"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -27x + 38x^2 + 138 = 99 $\n\nMove all terms to the left (standard form):\n$ 38x^2 + -27x + 39 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7773
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Find the modular inverse of 2 modulo 39, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $2\\cdot a^{-1} \\equiv 1 \\pmod{39}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarante... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7774
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Compute $(-82+81) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-82+81) \\bmod 8 $\n\nUse the rule that addition is compatible with reduct... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7775
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Evaluate: ((2 + -589/19) * 19) + 11^4 - log_7(2401) + (-108 mod 21) + ceil(-2/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((2 + -589/19) * 19) + 11^4 - log_7(2401) + (-108 mod 21) + ceil(-2/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 7776
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}1&0&-1\\\\\\3&3&-2\\\\\\-1&-1&1\\end{pmatrix},\\quad b=\\begin{pmatrix}1\\\\\\-3\\\\\\1\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7777
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. 10 - 39x - 30x^2 = 132"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 10 - 39x - 30x^2 = 132 $\n\nMove all terms to the left (standard form):\n$ -30x^2 + -39x + -122 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7778
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Compute the indefinite integral: $\\int 2x^3 + 8x^2 + 0x + -6\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2x^3 + 8x^2 + 0x + -6\\,dx $\n\nUse standard antiderivatives (and integra... | {
"type": "calculus",
"seed": 1337,
"index": 7779
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. -44 - 187x - 92 = -157x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -44 - 187x - 92 = -157x^2 $\n\nMove all terms to the left (standard form):\n$ 157x^2 + -187x + -136 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7780
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Solve for $x$ (in radians): tan(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = 1 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(1) = 0.785398 $\n\nT... | {
"type": "trigonometry",
"seed": 1337,
"index": 7781
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Evaluate: ((-160 + -25/5) * 7) + 3^5 - log_6(1296) + (438 mod 25) + ceil(53/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-160 + -25/5) * 7) + 3^5 - log_6(1296) + (438 mod 25) + ceil(53/2) $\n\nCompute each... | {
"type": "arithmetic",
"seed": 1337,
"index": 7782
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. 609x - 861 = 414"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 609x - 861 = 414 $\n\nMove all terms to the left (standard form):\n$ 609x + -1275 = 0 $\n\nIsolate the $x$ term:\n$ 609x = 1275 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 7783
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Evaluate: ((-20 + 140/10) * 7) + 7^5 - log_7(7) + (-189 mod 24) + floor(-94/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-20 + 140/10) * 7) + 7^5 - log_7(7) + (-189 mod 24) + floor(-94/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 7784
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. -10x^2 + 62x + 79 = -66"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -10x^2 + 62x + 79 = -66 $\n\nMove all terms to the left (standard form):\n$ -10x^2 + 62x + 145 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7785
} |
[
{
"role": "user",
"content": "Provide the solution to the following 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).... | {
"type": "trigonometry",
"seed": 1337,
"index": 7786
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Prove that $f(x)=3x^2+-7x+6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=3x^2+-7x+6$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 6... | {
"type": "optimization",
"seed": 1337,
"index": 7787
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}4&5&2\\\\\\5&1&-3\\\\\\5&-1&-4\\end{pmatrix},\\quad b=\\begin{pmatrix}4\\\\\\5\\\\\\5\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7788
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\n\n 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$... | {
"type": "trigonometry",
"seed": 1337,
"index": 7789
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. -26x^2 + 146 - 107x = 47"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -26x^2 + 146 - 107x = 47 $\n\nMove all terms to the left (standard form):\n$ -26x^2 + -107x + 99 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7790
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. -41 - 62x + 106 = 136x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -41 - 62x + 106 = 136x^2 $\n\nMove all terms to the left (standard form):\n$ -136x^2 + -62x + 65 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7791
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Compute the indefinite integral: $\\int -3\\sin(4x) + 6\\cos(2x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -3\\sin(4x) + 6\\cos(2x)\\,dx $\n\nUse standard antiderivatives (and inte... | {
"type": "calculus",
"seed": 1337,
"index": 7792
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Compute the indefinite integral: $\\int (-4x + -9)\\sin(5x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-4x + -9)\\sin(5x)\\,dx $\n\nUse standard antiderivatives (and integr... | {
"type": "calculus",
"seed": 1337,
"index": 7793
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. 36 + 128x - 141x^2 = 21"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 36 + 128x - 141x^2 = 21 $\n\nMove all terms to the left (standard form):\n$ -141x^2 + 128x + 15 = 0 $\n\nApply the quadrati... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7794
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Differentiate with respect to $x$: $f(x) = (5x + -6)e^{6x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (5x + -6)e^{6x} $\n\nApply the standard differentiation rules:\n$ f'(x) =... | {
"type": "calculus",
"seed": 1337,
"index": 7795
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. -268 + 738 = -308x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -268 + 738 = -308x $\n\nMove all terms to the left (standard form):\n$ 308x + 470 = 0 $\n\nIsolate the $x$ term:\n$ 308x = -470 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7796
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. -862x + 659 = -889"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -862x + 659 = -889 $\n\nMove all terms to the left (standard form):\n$ -862x + 1548 = 0 $\n\nIsolate the $x$ term:\n$ -862x = -1548 $\n\nDiv... | {
"type": "linear_equation",
"seed": 1337,
"index": 7797
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-8&-9\\\\\\-6&4\\end{pmatrix},\\quad b=\\begin{pmatrix}46\\\\\\-30\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-8&-9\\\\\\-... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7798
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Compute the determinant: $A=\\begin{pmatrix}-5&-6\\\\3&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-5&-6\\\\\\3&-7\\end{pmatrix} $\n\nUs... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7799
} |
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