messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Solve the following math equation. Find the modular inverse of 19 modulo 20, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $19\\cdot a^{-1} \\equiv 1 \\pmod{20}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees inte... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7600
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Prove that $f(x,y)=\\frac12(3x^2+5y^2)+-2x+2y+8$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+5y^2)+-2x+2y+8 $\n\n$ \\frac{\\partial^2 f}{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 7601
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Find the modular inverse of 5 modulo 8, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{8}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7602
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Find the modular inverse of 37 modulo 40, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $37\\cdot a^{-1} \\equiv 1 \\pmod{40}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gu... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7603
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:\n$ 4x(x^... | {
"type": "optimization",
"seed": 1337,
"index": 7604
} |
[
{
"role": "user",
"content": "Compute the final answer to 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{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 7605
} |
[
{
"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": 7606
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. -42x^2 + 115 - 45x = -9"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -42x^2 + 115 - 45x = -9 $\n\nMove all terms to the left (standard form):\n$ -42x^2 + -45x + 124 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7607
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Evaluate: ((-40 + -506/22) * 8) + 6^4 - log_7(117649) + (-347 mod 14) + ceil(54/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-40 + -506/22) * 8) + 6^4 - log_7(117649) + (-347 mod 14) + ceil(5... | {
"type": "arithmetic",
"seed": 1337,
"index": 7608
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Compute the determinant: $A=\\begin{pmatrix}8&-1\\\\9&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}8&-1\\\\\\9&5\\end{pmatrix} $\n\nUse... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7609
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Differentiate with respect to $x$: $f(x) = 5e^{3x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 5e^{3x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{15e^{3x}} $"
... | {
"type": "calculus",
"seed": 1337,
"index": 7610
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. 24 - 106 - 154x^2 = 17x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 24 - 106 - 154x^2 = 17x $\n\nMove all terms to the left (standard form):\n$ -154x^2 + -17x + -82 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7611
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Find the modular inverse of 37 modulo 38, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $37\\cdot a^{-1} \\equiv 1 \\pmod{38}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7612
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Find the modular inverse of 9 modulo 14, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $9\\cdot a^{-1} \\equiv 1 \\pmod{14}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guara... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7613
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Evaluate: ((75 + 1250/25) * 3) + 5^4 - log_8(64) + (-103 mod 6) + floor(131/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((75 + 1250/25) * 3) + 5^4 - log_8(64) + (-103 mod 6) + floor(131/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 7614
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 7615
} |
[
{
"role": "user",
"content": "Can you compute this equation? -5 + 127x - 59 = 66x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -5 + 127x - 59 = 66x^2 $\n\nMove all terms to the left (standard form):\n$ -66x^2 + 127x + -64 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7616
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Evaluate: ((-139 + -35/7) * 11) + 7^5 - log_5(5) + (195 mod 10) + floor(-20/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-139 + -35/7) * 11) + 7^5 - log_5(5) + (195 mod 10) + floor(-20/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 7617
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. 559x - 921 = 53"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 559x - 921 = 53 $\n\nMove all terms to the left (standard form):\n$ 559x + -974 = 0 $\n\nIsolate the $x$ term:\n$ 559x = 974 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7618
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. Differentiate with respect to $x$: $f(x) = -8e^{3x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -8e^{3x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-24e^{3x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 7619
} |
[
{
"role": "user",
"content": "solve: Prove that $f(x)=9x^2+-7x+3$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=9x^2+-7x+3$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 18x + -7 $\n$ f''(x) = 18 $\n\nSi... | {
"type": "optimization",
"seed": 1337,
"index": 7620
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. Prove that $(\\mathbb{Z}_19, +)$ is a group under addition modulo 19."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_19=\\{0,1,\\dots,18\\}$ with operation $+$ modulo $19$.\n\n**Closure:** If $a,b\\in\\ma... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7621
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Compute the indefinite integral: $\\int (2x + 6)e^{6x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (2x + 6)e^{6x}\\,dx $\n\nUse standard antiderivatives (and integration by part... | {
"type": "calculus",
"seed": 1337,
"index": 7622
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Prove that $f(x)=10x^2+4x+0$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=10x^2+4x+0$.\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": 7623
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. -31 + 190 - 94x^2 = 162x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -31 + 190 - 94x^2 = 162x $\n\nMove all terms to the left (standard form):\n$ -94x^2 + -162x + 159 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7624
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. 12x - 869 = 110"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 12x - 869 = 110 $\n\nMove all terms to the left (standard form):\n$ 12x + -979 = 0 $\n\nIsolate the $x$ term:\n$ 12x = 979 $\n\nDivide both sides... | {
"type": "linear_equation",
"seed": 1337,
"index": 7625
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Find $A^{-1}$ for $A=\\begin{pmatrix}-4&5\\\\7&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -4 & 5 \\\\ 7 & 9 \\end{pmatrix} $\n\nCompute the... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7626
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Evaluate: ((-118 + -496/16) * 17) + 2^3 - log_6(36) + (-420 mod 6) + round(102/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-118 + -496/16) * 17) + 2^3 - log_6(36) + (-420 mod 6) + round(102/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 7627
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Evaluate: ((8 + -192/12) * 8) + 2^4 - log_4(16) + (318 mod 12) + round(-55/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((8 + -192/12) * 8) + 2^4 - log_4(16) + (318 mod 12) + round(-55/2) $\n\nCompute each ... | {
"type": "arithmetic",
"seed": 1337,
"index": 7628
} |
[
{
"role": "user",
"content": "Solve the following math equation. Prove that $(\\mathbb{Z}_29, +)$ is a group under addition modulo 29."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_29=\\{0,1,\\dots,28\\}$ with operation $+$ modulo $29$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, t... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7629
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute the determinant: $A=\\begin{pmatrix}-2&-2\\\\5&0\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-2&-2\\\\\\5&0\\end{pmatrix} $\n\nUse $\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7630
} |
[
{
"role": "user",
"content": "Solve the following math equation. Evaluate: ((171 + 648/18) * 15) + 9^4 - log_5(625) + (440 mod 18) + round(-157/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((171 + 648/18) * 15) + 9^4 - log_5(625) + (440 mod 18) + round(-157/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 7631
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Find the modular inverse of 14 modulo 27, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $14\\cdot a^{-1} \\equiv 1 \\pmod{27}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarant... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7632
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Evaluate: ((-29 + 735/15) * 15) + 12^3 - log_4(4096) + (-451 mod 21) + round(16/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-29 + 735/15) * 15) + 12^3 - log_4(4096) + (-451 mod 21) + round(16/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 7633
} |
[
{
"role": "user",
"content": "Resolve this equation and give 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$ \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 7634
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Prove that $f(x,y)=\\frac12(4x^2+10y^2)+5x+-3y+0$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(4x^2+10y^2)+5x+-3y+0 $\n\n$ \\frac{\\partial^2 f}... | {
"type": "optimization",
"seed": 1337,
"index": 7635
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Compute the indefinite integral: $\\int \\ln(6x + 0)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(6x + 0)\\,dx $\n\nUse standard antiderivatives (and integration by parts i... | {
"type": "calculus",
"seed": 1337,
"index": 7636
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Differentiate with respect to $x$: $f(x) = 8\\sin(3x) + 8\\cos(1x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 8\\sin(3x) + 8\\cos(1x) $\n\nApply the standard differentiation rules:\n$ f'... | {
"type": "calculus",
"seed": 1337,
"index": 7637
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 322x - 436 = 119"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 322x - 436 = 119 $\n\nMove all terms to the left (standard form):\n$ 322x + -555 = 0 $\n\nIsolate the $x$ term:\n$ 322x = 555 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 7638
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Compute the indefinite integral: $\\int \\ln(2x + -1)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(2x + -1)\\,dx $\n\nUse standard antiderivatives (and integration by parts if ne... | {
"type": "calculus",
"seed": 1337,
"index": 7639
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Compute the indefinite integral: $\\int \\ln(8x + 9)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(8x + 9)\\,dx $\n\nUse standard antiderivatives (and integration by parts if need... | {
"type": "calculus",
"seed": 1337,
"index": 7640
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Compute the indefinite integral: $\\int -9\\sin(5x) + 8\\cos(7x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -9\\sin(5x) + 8\\cos(7x)\\,dx $\n\nUse standard antiderivatives (and integratio... | {
"type": "calculus",
"seed": 1337,
"index": 7641
} |
[
{
"role": "user",
"content": "Compute the solution for 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(-... | {
"type": "trigonometry",
"seed": 1337,
"index": 7642
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Evaluate: ((176 + -378/14) * 20) + 7^2 - log_6(36) + (-217 mod 23) + round(32/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((176 + -378/14) * 20) + 7^2 - log_6(36) + (-217 mod 23) + round(32/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 7643
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Evaluate: ((-95 + 32/2) * 1) + 8^5 - log_7(49) + (211 mod 9) + round(-108/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-95 + 32/2) * 1) + 8^5 - log_7(49) + (211 mod 9) + round(-108/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 7644
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. -529x + 693 = -646"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -529x + 693 = -646 $\n\nMove all terms to the left (standard form):\n$ -529x + 1339 = 0 $\n\nIsolate the $x$ term:\n$ -529x = -1339 $\n\nDi... | {
"type": "linear_equation",
"seed": 1337,
"index": 7645
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. -27 - 113x^2 - 150x = 13"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -27 - 113x^2 - 150x = 13 $\n\nMove all terms to the left (standard form):\n$ -113x^2 + -150x + -40 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7646
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Find the modular inverse of 11 modulo 13, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $11\\cdot a^{-1} \\equiv 1 \\pmod{13}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarante... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7647
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Compute the determinant: $A=\\begin{pmatrix}-1&0\\\\-1&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-1&0\\\\\\-1&9\\end{pmatrix} $\n\nUse $\\det(A)... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7648
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Prove that $f(x)=6x^2+-4x+-1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=6x^2+-4x+-1$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(... | {
"type": "optimization",
"seed": 1337,
"index": 7649
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Evaluate: ((44 + -432/24) * 8) + 6^4 - log_8(512) + (-238 mod 7) + round(-9/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((44 + -432/24) * 8) + 6^4 - log_8(512) + (-238 mod 7) + round(-9/2) $\n\nCompute each ... | {
"type": "arithmetic",
"seed": 1337,
"index": 7650
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Differentiate with respect to $x$: $f(x) = 1\\sin(8x) + 3\\cos(6x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 1\\sin(8x) + 3\\cos(6x) $\n\nApply the standard differentiation r... | {
"type": "calculus",
"seed": 1337,
"index": 7651
} |
[
{
"role": "user",
"content": "Please calculate this equation. Find the modular inverse of 5 modulo 6, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{6}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integers $x... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7652
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Prove that $f(x)=5x^2+9x+-5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+9x+-5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 10x... | {
"type": "optimization",
"seed": 1337,
"index": 7653
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Evaluate: ((-38 + 12/12) * 11) + 2^2 - log_5(625) + (386 mod 13) + round(155/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-38 + 12/12) * 11) + 2^2 - log_5(625) + (386 mod 13) + round(155/... | {
"type": "arithmetic",
"seed": 1337,
"index": 7654
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-6&2\\\\\\4&-6\\end{pmatrix},\\quad b=\\begin{pmatrix}20\\\\\\10\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-6&2\\\\\\4&-6... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7655
} |
[
{
"role": "user",
"content": "Solve the following math equation. Differentiate with respect to $x$: $f(x) = \\ln(4x + 1)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(4x + 1) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{4}{4x + 1}... | {
"type": "calculus",
"seed": 1337,
"index": 7656
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. -743 + 797 = 724x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -743 + 797 = 724x $\n\nMove all terms to the left (standard form):\n$ -724x + 54 = 0 $\n\nIsolate the $x$ term:\n$ -724x = -54 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 7657
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. 769x + 101 = -671"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 769x + 101 = -671 $\n\nMove all terms to the left (standard form):\n$ 769x + 772 = 0 $\n\nIsolate the $x$ term:\n$ 769x = -772 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7658
} |
[
{
"role": "user",
"content": "Work through and solve this equation. 972x - 97 = -395"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 972x - 97 = -395 $\n\nMove all terms to the left (standard form):\n$ 972x + 298 = 0 $\n\nIsolate the $x$ term:\n$ 972x = -298 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7659
} |
[
{
"role": "user",
"content": "solve: \n\n Differentiate with respect to $x$: $f(x) = (1x + -3)\\sin(1x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (1x + -3)\\sin(1x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{1\\sin(1x) + (1x + -3)\\cdot... | {
"type": "calculus",
"seed": 1337,
"index": 7660
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. -48x + 169 - 193 = 175x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -48x + 169 - 193 = 175x^2 $\n\nMove all terms to the left (standard form):\n$ -175x^2 + -48x + -24 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7661
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Evaluate: ((-158 + -228/19) * 10) + 6^3 - log_7(2401) + (-427 mod 17) + round(145/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-158 + -228/19) * 10) + 6^3 - log_7(2401) + (-427 mod 17) + round(145... | {
"type": "arithmetic",
"seed": 1337,
"index": 7662
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. -770 - 444 = 239x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -770 - 444 = 239x $\n\nMove all terms to the left (standard form):\n$ -239x + -1214 = 0 $\n\nIsolate the $x$ term:\n$ -239x = 1214 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 7663
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Evaluate: ((-33 + -28/4) * 3) + 4^2 - log_9(9) + (-408 mod 19) + floor(139/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-33 + -28/4) * 3) + 4^2 - log_9(9) + (-408 mod 19) + floor(139/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 7664
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Compute the indefinite integral: $\\int (-3x + -5)e^{1x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-3x + -5)e^{1x}\\,dx $\n\nUse standard antiderivatives (and integration by parts i... | {
"type": "calculus",
"seed": 1337,
"index": 7665
} |
[
{
"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": 7666
} |
[
{
"role": "user",
"content": "Work through and solve this equation. -45 - 40 - 57x = 24x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -45 - 40 - 57x = 24x^2 $\n\nMove all terms to the left (standard form):\n$ -24x^2 + -57x + -85 = 0 $\n\nApply the quadratic formula:\n$ x = ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7667
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Compute $(-100+-67) \\bmod 13$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-100+-67) \\bmod 13 $\n\nUse the rule that addition is compatible with reductio... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7668
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Evaluate: ((73 + -94/2) * 5) + 4^5 - log_2(64) + (-269 mod 10) + ceil(44/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((73 + -94/2) * 5) + 4^5 - log_2(64) + (-269 mod 10) + ceil(44/2) $\n\nCompute e... | {
"type": "arithmetic",
"seed": 1337,
"index": 7669
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Find the modular inverse of 21 modulo 29, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $21\\cdot a^{-1} \\equiv 1 \\pmod{29}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees inte... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7670
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:\n$ 4x(x^... | {
"type": "optimization",
"seed": 1337,
"index": 7671
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Prove that $f(x)=8x^2+0x+1$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=8x^2+0x+1$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 16x + 0 $... | {
"type": "optimization",
"seed": 1337,
"index": 7672
} |
[
{
"role": "user",
"content": "Please work out this equation. Compute the indefinite integral: $\\int 3\\sin(2x) + -2\\cos(9x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 3\\sin(2x) + -2\\cos(9x)\\,dx $\n\nUse standard antiderivatives (and integration b... | {
"type": "calculus",
"seed": 1337,
"index": 7673
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Compute the indefinite integral: $\\int \\ln(3x + -1)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int \\ln(3x + -1)\\,dx $\n\nUse standard antiderivatives (and integration by... | {
"type": "calculus",
"seed": 1337,
"index": 7674
} |
[
{
"role": "user",
"content": "Can you find 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$ \\alpha = \\arccos... | {
"type": "trigonometry",
"seed": 1337,
"index": 7675
} |
[
{
"role": "user",
"content": "solve: \n\n Compute $(132+72) \\bmod 13$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (132+72) \\bmod 13 $\n\nUse the rule that addition is compatible with reduction modulo $n$:\n$ (a+b)\\bmod n = ((a... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7676
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Compute the indefinite integral: $\\int 8\\sin(9x) + 3\\cos(2x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 8\\sin(9x) + 3\\cos(2x)\\,dx $\n\nUse standard antiderivatives (and int... | {
"type": "calculus",
"seed": 1337,
"index": 7677
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Compute the determinant: $A=\\begin{pmatrix}-6&-6\\\\-9&-3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-6&-6\\\\\\-9&-3\\end{pm... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7678
} |
[
{
"role": "user",
"content": "solve: Differentiate with respect to $x$: $f(x) = (2x + 9)e^{6x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (2x + 9)e^{6x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{2e^{6x} + (2x + 9)\\cdot 6e^{6x}} $"
}... | {
"type": "calculus",
"seed": 1337,
"index": 7679
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Prove that $(\\mathbb{Z}_17, +)$ is a group under addition modulo 17."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_17=\\{0,1,\\dots,16\\}$ with operation $+$ modulo $17$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, th... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7680
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-6&-7\\\\\\-5&-6\\end{pmatrix},\\quad b=\\begin{pmatrix}14\\\\\\13\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-6&-7\\\\\\-5&-6\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7681
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? 334 + 907 = -479x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 334 + 907 = -479x $\n\nMove all terms to the left (standard form):\n$ 479x + 1241 = 0 $\n\nIsolate the $x$ term:\n$ 479x = -1241 $\n\nDivide both sid... | {
"type": "linear_equation",
"seed": 1337,
"index": 7682
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Evaluate: ((-149 + -100/2) * 19) + 5^5 - log_7(16807) + (130 mod 27) + round(-69/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-149 + -100/2) * 19) + 5^5 - log_7(16807) + (130 mod 27) + round(-69/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 7683
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Find the modular inverse of 6 modulo 29, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $6\\cdot a^{-1} \\equiv 1 \\pmod{29}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guaran... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 7684
} |
[
{
"role": "user",
"content": "Work out the result of this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}0&-3\\\\1&2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 0 & -3 \\\\ 1 & 2 \\end{pmatrix} $\n\nCompute the determi... | {
"type": "linear_algebra",
"seed": 1337,
"index": 7685
} |
[
{
"role": "user",
"content": "Please work out this equation. 215x + 183 = -865"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 215x + 183 = -865 $\n\nMove all terms to the left (standard form):\n$ 215x + 1048 = 0 $\n\nIsolate the $x$ term:\n$ 215x = -1048 $\n\nDivide both sides by ... | {
"type": "linear_equation",
"seed": 1337,
"index": 7686
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Solve for $x$ (in radians): cos(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -1 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(-1) = 3.14159 $\n\nThe general ... | {
"type": "trigonometry",
"seed": 1337,
"index": 7687
} |
[
{
"role": "user",
"content": "solve: \n\n Evaluate: ((-38 + -270/9) * 17) + 2^4 - log_2(64) + (485 mod 27) + floor(4/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-38 + -270/9) * 17) + 2^4 - log_2(64) + (485 mod 27) + floor(4/2) $\n\nCompute each component:\n$ -270/9 = ... | {
"type": "arithmetic",
"seed": 1337,
"index": 7688
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. 3 - 102 + 44x = 53x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 3 - 102 + 44x = 53x^2 $\n\nMove all terms to the left (standard form):\n$ -53x^2 + 44x + -99 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7689
} |
[
{
"role": "user",
"content": "Compute the final answer to 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:... | {
"type": "optimization",
"seed": 1337,
"index": 7690
} |
[
{
"role": "user",
"content": "solve: Differentiate with respect to $x$: $f(x) = 8\\sin(5x) + 8\\cos(3x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 8\\sin(5x) + 8\\cos(3x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{40\\cos(5x) - 24\\sin(... | {
"type": "calculus",
"seed": 1337,
"index": 7691
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. -489 + 333 = 882x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -489 + 333 = 882x $\n\nMove all terms to the left (standard form):\n$ -882x + -156 = 0 $\n\nIsolate the $x$ term:\n$ -882x = 156 $\n\n... | {
"type": "linear_equation",
"seed": 1337,
"index": 7692
} |
[
{
"role": "user",
"content": "Solve the following math equation. Evaluate: ((73 + 775/25) * 6) + 2^3 - log_2(8) + (356 mod 15) + floor(48/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((73 + 775/25) * 6) + 2^3 - log_2(8) + (356 mod 15) + floor(48/2) $\n\nCompute each comp... | {
"type": "arithmetic",
"seed": 1337,
"index": 7693
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Differentiate with respect to $x$: $f(x) = 6x^3 + -1x^2 + -19x + -1$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 6x^3 + -1x^2 + -19x + -1 $\n\nApply the standard differentiation rul... | {
"type": "calculus",
"seed": 1337,
"index": 7694
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Prove that $f(x)=7x^2+2x+7$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+2x+7$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n... | {
"type": "optimization",
"seed": 1337,
"index": 7695
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? - - 136x - 105x^2 = 141"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ - - 136x - 105x^2 = 141 $\n\nMove all terms to the left (standard form):\n$ -105x^2 + -136x + -141 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7696
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. 49x + 169x^2 + 4 = 67"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 49x + 169x^2 + 4 = 67 $\n\nMove all terms to the left (standard form):\n$ 169x^2 + 49x + -63 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 7697
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. -675x - 562 = 228"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -675x - 562 = 228 $\n\nMove all terms to the left (standard form):\n$ -675x + -790 = 0 $\n\nIsolate the $x$ term:\n$ -675x = 790 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 7698
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. -610 - 747x = -586"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -610 - 747x = -586 $\n\nMove all terms to the left (standard form):\n$ -747x + -24 = 0 $\n\nIsolate the $x$ term:\n$ -747x = 24 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 7699
} |
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