messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Compute the indefinite integral: $\\int (-3x + 1)\\sin(4x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-3x + 1)\\sin(4x)\\,dx $\n\nUse standard antiderivatives (and integratio... | {
"type": "calculus",
"seed": 1337,
"index": 5700
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Compute the indefinite integral: $\\int (-1x + 9)e^{3x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-1x + 9)e^{3x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed)... | {
"type": "calculus",
"seed": 1337,
"index": 5701
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. -50x + 116 + 181x^2 = 155"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -50x + 116 + 181x^2 = 155 $\n\nMove all terms to the left (standard form):\n$ 181x^2 + -50x + -39 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5702
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. -151 - 552 = 469x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -151 - 552 = 469x $\n\nMove all terms to the left (standard form):\n$ -469x + -703 = 0 $\n\nIsolate the $x$ term:\n$ -469x = 703 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 5703
} |
[
{
"role": "user",
"content": "Solve the following math equation. Prove that $f(x)=3x^2+10x+6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=3x^2+10x+6$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 6x + 1... | {
"type": "optimization",
"seed": 1337,
"index": 5704
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. 851 + 376x = -277"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 851 + 376x = -277 $\n\nMove all terms to the left (standard form):\n$ 376x + 1128 = 0 $\n\nIsolate the $x$ term:\n$ 376x = -1128 $\n\nDivide both sides... | {
"type": "linear_equation",
"seed": 1337,
"index": 5705
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. -643 + 628x = -804"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -643 + 628x = -804 $\n\nMove all terms to the left (standard form):\n$ 628x + 161 = 0 $\n\nIsolate the $x$ term:\n$ 628x = -161 $\n\... | {
"type": "linear_equation",
"seed": 1337,
"index": 5706
} |
[
{
"role": "user",
"content": "solve 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^2-1)=0 \\Rightarrow x\\in\\... | {
"type": "optimization",
"seed": 1337,
"index": 5707
} |
[
{
"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": 5708
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Differentiate with respect to $x$: $f(x) = 7\\sin(8x) + 3\\cos(6x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 7\\sin(8x) + 3\\cos(6x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxe... | {
"type": "calculus",
"seed": 1337,
"index": 5709
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Solve for $x$ (in radians): sin(x) = -\\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = -\\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\ar... | {
"type": "trigonometry",
"seed": 1337,
"index": 5710
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Find the modular inverse of 11 modulo 32, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $11\\cdot a^{-1} \\equiv 1 \\pmod{32}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees integ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5711
} |
[
{
"role": "user",
"content": "Determine the solution for 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$ ... | {
"type": "trigonometry",
"seed": 1337,
"index": 5712
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Compute the determinant: $A=\\begin{pmatrix}6&-6\\\\-7&2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}6&-6\\\\\\-7&2\\end{pmatrix} $\n\nUse $\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5713
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Compute $(43+-176) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (43+-176) \\bmod 8 $\n\nUse the rule that addition is compatible with reduction m... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5714
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n + 167x + 187x^2 = 27"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ + 167x + 187x^2 = 27 $\n\nMove all terms to the left (standard form):\n$ 187x^2 + 167x + -27 = 0 $\n\nApply the quadratic f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5715
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. -29x + 3x^2 + 131 = -6"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -29x + 3x^2 + 131 = -6 $\n\nMove all terms to the left (standard form):\n$ 3x^2 + -29x + 137 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5716
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}4&9\\\\\\-3&2\\end{pmatrix},\\quad b=\\begin{pmatrix}57\\\\\\36\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}4&9\\\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5717
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 5718
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. Prove that $(\\mathbb{Z}_21, +)$ is a group under addition modulo 21."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_21=\\{0,1,\\dots,20\\}$ with operation $+$ modulo $21$.\n\n**Closure:** If $a,b\\in\\mathb... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5719
} |
[
{
"role": "user",
"content": "Solve the equation shown 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^2-... | {
"type": "optimization",
"seed": 1337,
"index": 5720
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Compute the determinant: $A=\\begin{pmatrix}3&-9\\\\-4&9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}3&-9\\\\\\-4&9\\end{pmatrix} $\n\nUse $\\det(A)... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5721
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Solve for $x$ (in radians): cos(x) = -\\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -\\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 5722
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. 445 + 102x = -310"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 445 + 102x = -310 $\n\nMove all terms to the left (standard form):\n$ 102x + 755 = 0 $\n\nIsolate the $x$ term:\n$ 102x = -755 $\n\nDivide both sid... | {
"type": "linear_equation",
"seed": 1337,
"index": 5723
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 483x - 196 = -567"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 483x - 196 = -567 $\n\nMove all terms to the left (standard form):\n$ 483x + 371 = 0 $\n\nIsolate the $x$ term:\n$ 483x = -371 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 5724
} |
[
{
"role": "user",
"content": "Can you solve the equation below? -17x^2 + 53 - 91x = 121"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -17x^2 + 53 - 91x = 121 $\n\nMove all terms to the left (standard form):\n$ -17x^2 + -91x + -68 = 0 $\n\nApply the quadratic formula:\n$ x = \\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5725
} |
[
{
"role": "user",
"content": "Find the numerical solution to 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$ ... | {
"type": "trigonometry",
"seed": 1337,
"index": 5726
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Solve for $x$ (in radians): sin(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(\\f... | {
"type": "trigonometry",
"seed": 1337,
"index": 5727
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. 116 + 990 = 586x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 116 + 990 = 586x $\n\nMove all terms to the left (standard form):\n$ -586x + 1106 = 0 $\n\nIsolate the $x$ term:\n$ -586x = -1106 $\n\nDivide both ... | {
"type": "linear_equation",
"seed": 1337,
"index": 5728
} |
[
{
"role": "user",
"content": "Please work out this equation. -199 - 603 = 807x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -199 - 603 = 807x $\n\nMove all terms to the left (standard form):\n$ -807x + -802 = 0 $\n\nIsolate the $x$ term:\n$ -807x = 802 $\n\nDivide both sides by ... | {
"type": "linear_equation",
"seed": 1337,
"index": 5729
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Evaluate: ((165 + 150/10) * 16) + 6^3 - log_9(729) + (-104 mod 29) + floor(-126/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((165 + 150/10) * 16) + 6^3 - log_9(729) + (-104 mod 29) + fl... | {
"type": "arithmetic",
"seed": 1337,
"index": 5730
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. Find $A^{-1}$ for $A=\\begin{pmatrix}0&-7\\\\3&-5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 0 & -7 \\\\ 3 & -5 \\end{pmatrix} $\n\nCompute the de... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5731
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Solve for $x$ (in radians): cos(x) = -1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = -1 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(-1) = 3.14159 $\n\nThe ... | {
"type": "trigonometry",
"seed": 1337,
"index": 5732
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Solve for $x$ (in radians): sin(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 1 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(1) = 1.5708... | {
"type": "trigonometry",
"seed": 1337,
"index": 5733
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. 17x + 191 - 37x^2 = -52"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 17x + 191 - 37x^2 = -52 $\n\nMove all terms to the left (standard form):\n$ -37x^2 + 17x + 243 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5734
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Prove that $(\\mathbb{Z}_15, +)$ is a group under addition modulo 15."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_15=\\{0,1,\\dots,14\\}$ with operation $+$ modulo $15$.\n\n**Closure:** If $a,b\\... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5735
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. 5x^2 + 102x - 167 = 122"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 5x^2 + 102x - 167 = 122 $\n\nMove all terms to the left (standard form):\n$ 5x^2 + 102x + -289 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5736
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. -13x^2 - 117x - 9 = -66"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -13x^2 - 117x - 9 = -66 $\n\nMove all terms to the left (standard form):\n$ -13x^2 + -117x + 57 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5737
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. 34 - 74x - 93x^2 = 74"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 34 - 74x - 93x^2 = 74 $\n\nMove all terms to the left (standard form):\n$ -93x^2 + -74x + -40 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5738
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}0&-2\\\\\\9&1\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\67\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}0&-2\\\\\\9&1\\e... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5739
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Evaluate: ((54 + 408/17) * 4) + 6^3 - log_6(6) + (-256 mod 23) + floor(-180/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((54 + 408/17) * 4) + 6^3 - log_6(6) + (-256 mod 23) + floor(-180... | {
"type": "arithmetic",
"seed": 1337,
"index": 5740
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Evaluate: ((89 + 234/18) * 8) + 2^2 - log_5(625) + (74 mod 14) + round(78/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((89 + 234/18) * 8) + 2^2 - log_5(625) + (74 mod 14) + round(78/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 5741
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-8&-4\\\\9&7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -8 & -4 \\\\ 9 & 7 \\end{pmatrix} $\n\nCompute ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5742
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Evaluate: ((-27 + -680/17) * 5) + 3^2 - log_6(1296) + (-308 mod 26) + floor(67/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-27 + -680/17) * 5) + 3^2 - log_6(1296) + (-308 mod 26) + floor(67/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 5743
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Prove that $f(x,y)=\\frac12(3x^2+1y^2)+-1x+6y+-9$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+1y^2)+-1x+6y+-9 $\n\n$ \\frac{\\partial^2 f}{\\... | {
"type": "optimization",
"seed": 1337,
"index": 5744
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Prove that $f(x)=7x^2+-7x+5$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+-7x+5$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x... | {
"type": "optimization",
"seed": 1337,
"index": 5745
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Evaluate: ((-73 + 176/4) * 3) + 4^5 - log_10(1000000) + (-206 mod 27) + floor(-159/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-73 + 176/4) * 3) + 4^5 - log_10(1000000) + (-206 mod 27) + floor(-... | {
"type": "arithmetic",
"seed": 1337,
"index": 5746
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. -45 + 41x + 143 = 191x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -45 + 41x + 143 = 191x^2 $\n\nMove all terms to the left (standard form):\n$ -191x^2 + 41x + 98 = 0 $\n\nApply the quadratic formula:\n$ x = ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5747
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Compute $(-171+56) \\bmod 15$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-171+56) \\bmod 15 $\n\nUse the rule that addition is compatible with reduction... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5748
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. -19x^2 - 119 + 143x = 168"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -19x^2 - 119 + 143x = 168 $\n\nMove all terms to the left (standard form):\n$ -19x^2 + 143x + -287 = 0 $\n\nApply the qu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5749
} |
[
{
"role": "user",
"content": "solve: Solve for $x$ (in radians): sin(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 1 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(1) = 1.5708 $\n\nThe general solutions for $\\sin(x)=k$ a... | {
"type": "trigonometry",
"seed": 1337,
"index": 5750
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. -207 - 4 = 533x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -207 - 4 = 533x $\n\nMove all terms to the left (standard form):\n$ -533x + -211 = 0 $\n\nIsolate the $x$ term:\n$ -533x = 211 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 5751
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. Evaluate: ((87 + 148/4) * 2) + 12^4 - log_10(1000000) + (-479 mod 27) + ceil(-94/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((87 + 148/4) * 2) + 12^4 - log_10(1000000) + (-479 mod 27) + ceil(-94/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 5752
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Compute the indefinite integral: $\\int 2e^{-1x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 2e^{-1x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):... | {
"type": "calculus",
"seed": 1337,
"index": 5753
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Prove that $f(x,y)=\\frac12(4x^2+7y^2)+-5x+4y+0$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(4x^2+7y^2)+-5x+4y+0 $\n\n$ \\frac{\\partial^2 f}{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 5754
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. Evaluate: ((-61 + -640/20) * 9) + 6^2 - log_7(7) + (-238 mod 7) + ceil(73/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-61 + -640/20) * 9) + 6^2 - log_7(7) + (-238 mod 7) + ceil(73/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 5755
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Evaluate: ((-53 + -63/3) * 1) + 12^4 - log_3(81) + (419 mod 20) + floor(-57/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-53 + -63/3) * 1) + 12^4 - log_3(81) + (419 mod 20) + floor(-57/... | {
"type": "arithmetic",
"seed": 1337,
"index": 5756
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\pa... | {
"type": "optimization",
"seed": 1337,
"index": 5757
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. 341x - 822 = 749"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 341x - 822 = 749 $\n\nMove all terms to the left (standard form):\n$ 341x + -1571 = 0 $\n\nIsolate the $x$ term:\n$ 341x = 1571 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 5758
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. -89 - 359x = -487"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -89 - 359x = -487 $\n\nMove all terms to the left (standard form):\n$ -359x + 398 = 0 $\n\nIsolate the $x$ term:\n$ -359x = -398 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 5759
} |
[
{
"role": "user",
"content": "Solve this mathematical equation for me. Evaluate: ((151 + 24/3) * 8) + 3^3 - log_9(59049) + (-419 mod 27) + ceil(-174/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((151 + 24/3) * 8) + 3^3 - log_9(59049) + (-419 mod 27) + ceil(-174/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 5760
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. 574x + 812 = 717"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 574x + 812 = 717 $\n\nMove all terms to the left (standard form):\n$ 574x + 95 = 0 $\n\nIsolate the $x$ term:\n$ 574x = -95 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 5761
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. 387x + 437 = 401"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 387x + 437 = 401 $\n\nMove all terms to the left (standard form):\n$ 387x + 36 = 0 $\n\nIsolate the $x$ term:\n$ 387x = -36 $\n\nDivide both sides... | {
"type": "linear_equation",
"seed": 1337,
"index": 5762
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}0&-2&-5\\\\\\-5&-4&3\\\\\\4&3&5\\end{pmatrix},\\quad b=\\begin{pmatrix}-8\\\\\\-11\\\\\\8\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $A... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5763
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Evaluate: ((-74 + -288/24) * 13) + 12^2 - log_5(5) + (-464 mod 11) + floor(-189/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-74 + -288/24) * 13) + 12^2 - log_5(5) + (-464 mod 11) + floor(-189/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 5764
} |
[
{
"role": "user",
"content": "Please solve this math problem. Evaluate: ((66 + 66/11) * 19) + 3^2 - log_3(81) + (-36 mod 2) + ceil(87/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((66 + 66/11) * 19) + 3^2 - log_3(81) + (-36 mod 2) + ceil(87/2) $\n\nCompute each component... | {
"type": "arithmetic",
"seed": 1337,
"index": 5765
} |
[
{
"role": "user",
"content": "Please determine the solution 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... | {
"type": "optimization",
"seed": 1337,
"index": 5766
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Solve for $x$ (in radians): tan(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = 0 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(0) = 0 $\n\nThe... | {
"type": "trigonometry",
"seed": 1337,
"index": 5767
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. -13 - 116x^2 - 176x = -15"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -13 - 116x^2 - 176x = -15 $\n\nMove all terms to the left (standard form):\n$ -116x^2 + -176x + 2 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5768
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. -5 - 103x + 152 = -21x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -5 - 103x + 152 = -21x^2 $\n\nMove all terms to the left (standard form):\n$ 21x^2 + -103x + 147 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5769
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. 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$ \\alp... | {
"type": "trigonometry",
"seed": 1337,
"index": 5770
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Evaluate: ((76 + 406/14) * 13) + 5^4 - log_8(32768) + (-36 mod 26) + round(65/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((76 + 406/14) * 13) + 5^4 - log_8(32768) + (-36 mod 26) + round(65/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 5771
} |
[
{
"role": "user",
"content": "Can you compute this equation? Evaluate: ((10 + -60/4) * 18) + 7^5 - log_8(262144) + (-277 mod 2) + ceil(-60/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((10 + -60/4) * 18) + 7^5 - log_8(262144) + (-277 mod 2) + ceil(-60/2) $\n\nCompute eac... | {
"type": "arithmetic",
"seed": 1337,
"index": 5772
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Evaluate: ((-180 + 465/15) * 15) + 3^4 - log_7(49) + (246 mod 14) + round(24/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-180 + 465/15) * 15) + 3^4 - log_7(49) + (246 mod 14) + round(24/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 5773
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation.\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} & \\frac{... | {
"type": "optimization",
"seed": 1337,
"index": 5774
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Find the modular inverse of 31 modulo 36, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $31\\cdot a^{-1} \\equiv 1 \\pmod{36}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gua... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5775
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. -22 + 607 = -698x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -22 + 607 = -698x $\n\nMove all terms to the left (standard form):\n$ 698x + 585 = 0 $\n\nIsolate the $x$ term:\n$ 698x = -585 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 5776
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. -455 + 409x = 11"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -455 + 409x = 11 $\n\nMove all terms to the left (standard form):\n$ 409x + -466 = 0 $\n\nIsolate the $x$ term:\n$ 409x = 466 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 5777
} |
[
{
"role": "user",
"content": "Please calculate and resolve 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$ ... | {
"type": "trigonometry",
"seed": 1337,
"index": 5778
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. -6x - 10x^2 + 109 = -135"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -6x - 10x^2 + 109 = -135 $\n\nMove all terms to the left (standard form):\n$ -10x^2 + -6x + 244 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5779
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: -13 + 9x - 152x^2 = 63"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -13 + 9x - 152x^2 = 63 $\n\nMove all terms to the left (standard form):\n$ -152x^2 + 9x + -76 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\pm \\sqrt{b... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5780
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. 14x^2 + 33x - 129 = -188"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 14x^2 + 33x - 129 = -188 $\n\nMove all terms to the left (standard form):\n$ 14x^2 + 33x + 59 = 0 $\n\nApply the quadratic formula:\n$ x ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5781
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. x - 177 + 171 = -47x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ x - 177 + 171 = -47x^2 $\n\nMove all terms to the left (standard form):\n$ 47x^2 + 1x + -6 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5782
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. -928 - 77x = -851"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -928 - 77x = -851 $\n\nMove all terms to the left (standard form):\n$ -77x + -77 = 0 $\n\nIsolate the $x$ term:\n$ -77x = 77 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 5783
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Compute $(60+-91) \\bmod 7$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (60+-91) \\bmod 7 $\n\nUse the rule that addition is compatible with reduction modul... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5784
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Evaluate: ((-85 + -286/13) * 13) + 12^2 - log_5(125) + (320 mod 19) + ceil(18/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-85 + -286/13) * 13) + 12^2 - log_5(125) + (320 mod 19) + ceil(18/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 5785
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. 15 - 72x + 48x^2 = 167"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 15 - 72x + 48x^2 = 167 $\n\nMove all terms to the left (standard form):\n$ 48x^2 + -72x + -152 = 0 $\n\nApply the quadratic formula:\n$ x... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5786
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. -8 - 26x + 71x^2 = 187"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -8 - 26x + 71x^2 = 187 $\n\nMove all terms to the left (standard form):\n$ 71x^2 + -26x + -195 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5787
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. 102 - 349x = 275"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 102 - 349x = 275 $\n\nMove all terms to the left (standard form):\n$ -349x + -173 = 0 $\n\nIsolate the $x$ term:\n$ -349x = 173 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 5788
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. 963 - 838 = -285x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 963 - 838 = -285x $\n\nMove all terms to the left (standard form):\n$ 285x + 125 = 0 $\n\nIsolate the $x$ term:\n$ 285x = -125 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 5789
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-3&-4&5\\\\\\-3&1&5\\\\\\2&-2&-1\\end{pmatrix},\\quad b=\\begin{pmatrix}7\\\\\\2\\\\\\0\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., by G... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5790
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Evaluate: ((-199 + -143/11) * 4) + 8^3 - log_9(59049) + (-180 mod 4) + round(10/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-199 + -143/11) * 4) + 8^3 - log_9(59049) + (-180 mod 4) + rou... | {
"type": "arithmetic",
"seed": 1337,
"index": 5791
} |
[
{
"role": "user",
"content": "Solve the following math equation. Prove that $(\\mathbb{Z}_25, +)$ is a group under addition modulo 25."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_25=\\{0,1,\\dots,24\\}$ with operation $+$ modulo $25$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, t... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5792
} |
[
{
"role": "user",
"content": "Determine the correct 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{... | {
"type": "optimization",
"seed": 1337,
"index": 5793
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Differentiate with respect to $x$: $f(x) = (-1x + -9)e^{2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (-1x + -9)e^{2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxe... | {
"type": "calculus",
"seed": 1337,
"index": 5794
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Prove that $(\\mathbb{Z}_27, +)$ is a group under addition modulo 27."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_27=\\{0,1,\\dots,26\\}$ with operation $+$ modulo $27$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, th... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5795
} |
[
{
"role": "user",
"content": "Please solve this math problem. 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... | {
"type": "trigonometry",
"seed": 1337,
"index": 5796
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given 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 ... | {
"type": "trigonometry",
"seed": 1337,
"index": 5797
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}8&6\\\\\\2&8\\end{pmatrix},\\quad b=\\begin{pmatrix}-28\\\\\\32\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}8&6\\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5798
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Evaluate: ((-34 + -77/11) * 20) + 2^2 - log_6(216) + (-31 mod 12) + floor(192/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-34 + -77/11) * 20) + 2^2 - log_6(216) + (-31 mod 12) + floor(192/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 5799
} |
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