messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Solve the equation and calculate the result. 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": 3800
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Evaluate: ((-95 + 100/4) * 7) + 8^4 - log_7(117649) + (146 mod 19) + floor(127/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-95 + 100/4) * 7) + 8^4 - log_7(117649) + (146 mod 19) + floor(127/... | {
"type": "arithmetic",
"seed": 1337,
"index": 3801
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. 657 - 588x = 6"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 657 - 588x = 6 $\n\nMove all terms to the left (standard form):\n$ -588x + 651 = 0 $\n\nIsolate the $x$ term:\n$ -588x = -651 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 3802
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. 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$... | {
"type": "trigonometry",
"seed": 1337,
"index": 3803
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Evaluate: ((166 + 828/18) * 18) + 11^3 - log_9(59049) + (-464 mod 18) + ceil(-56/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((166 + 828/18) * 18) + 11^3 - log_9(59049) + (-464 mod 18) + ceil(-56/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 3804
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Evaluate: ((-99 + -407/11) * 10) + 12^4 - log_2(8) + (321 mod 16) + ceil(197/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-99 + -407/11) * 10) + 12^4 - log_2(8) + (321 mod 16) + ceil(197/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 3805
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. -615 + 296 = 6x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -615 + 296 = 6x $\n\nMove all terms to the left (standard form):\n$ -6x + -319 = 0 $\n\nIsolate the $x$ term:\n$ -6x = 319 $\n\nDivide both sides... | {
"type": "linear_equation",
"seed": 1337,
"index": 3806
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. Solve for $x$ (in radians): cos(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = 0 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(0) = 1.5708 $\n\nThe gene... | {
"type": "trigonometry",
"seed": 1337,
"index": 3807
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Compute the indefinite integral: $\\int 1x^3 + -3x^2 + 4x + 16\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 1x^3 + -3x^2 + 4x + 16\\,dx $\n\nUse standard antiderivatives (and integ... | {
"type": "calculus",
"seed": 1337,
"index": 3808
} |
[
{
"role": "user",
"content": "Evaluate and solve the given 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{\\partia... | {
"type": "optimization",
"seed": 1337,
"index": 3809
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}4&-5&-3\\\\\\3&2&3\\\\\\-3&2&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}-24\\\\\\-7\\\\\\9\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3810
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. 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": 3811
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Evaluate: ((-13 + -210/5) * 14) + 6^5 - log_10(1000000) + (103 mod 12) + round(-79/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-13 + -210/5) * 14) + 6^5 - log_10(1000000) + (103 mod 12) + rou... | {
"type": "arithmetic",
"seed": 1337,
"index": 3812
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. 999 - 448 = -11x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 999 - 448 = -11x $\n\nMove all terms to the left (standard form):\n$ 11x + 551 = 0 $\n\nIsolate the $x$ term:\n$ 11x = -551 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 3813
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Evaluate: ((151 + -200/25) * 18) + 11^3 - log_7(49) + (-110 mod 21) + round(64/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((151 + -200/25) * 18) + 11^3 - log_7(49) + (-110 mod 21) + round(64/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 3814
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-2&-1\\\\\\9&7\\end{pmatrix},\\quad b=\\begin{pmatrix}23\\\\\\-121\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-2&-1... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3815
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. Prove that $(\\mathbb{Z}_12, +)$ is a group under addition modulo 12."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_12=\\{0,1,\\dots,11\\}$ with operation $+$ modulo $12$.\n\n**Closure:** If $a,b\\in... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3816
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Compute $(-175+49) \\bmod 8$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-175+49) \\bmod 8 $\n\nUse the rule that addition is compatible with reduction mod... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3817
} |
[
{
"role": "user",
"content": "Calculate the result of the given equation. Evaluate: ((-170 + -725/25) * 14) + 12^2 - log_4(4096) + (-335 mod 10) + ceil(187/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-170 + -725/25) * 14) + 12^2 - log_4(4096) + (-335 mod 10) + ceil(18... | {
"type": "arithmetic",
"seed": 1337,
"index": 3818
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. 3 + 83 + 84x = -85x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 3 + 83 + 84x = -85x^2 $\n\nMove all terms to the left (standard form):\n$ 85x^2 + 84x + 86 = 0 $\n\nApply the quadratic formula:\n$ ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3819
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Find the modular inverse of 35 modulo 37, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $35\\cdot a^{-1} \\equiv 1 \\pmod{37}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guaran... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3820
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Find the modular inverse of 17 modulo 18, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $17\\cdot a^{-1} \\equiv 1 \\pmod{18}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3821
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Evaluate: ((-2 + -242/11) * 17) + 5^5 - log_7(49) + (449 mod 21) + round(58/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-2 + -242/11) * 17) + 5^5 - log_7(49) + (449 mod 21) + round(58/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 3822
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Compute the determinant: $A=\\begin{pmatrix}0&9\\\\3&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}0&9\\\\\\3&-7\\end{pmatrix} $\n\nUse $\\det(A)=ad-b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3823
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. Solve for $x$ (in radians): sin(x) = \\frac{\\sqrt{2}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = \\frac{\\sqrt{2}}{2} $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \... | {
"type": "trigonometry",
"seed": 1337,
"index": 3824
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Evaluate: ((-69 + 154/11) * 7) + 3^5 - log_8(4096) + (137 mod 3) + ceil(-74/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-69 + 154/11) * 7) + 3^5 - log_8(4096) + (137 mod 3) + ceil(-74/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 3825
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}2&-1\\\\\\4&-9\\end{pmatrix},\\quad b=\\begin{pmatrix}-1\\\\\\61\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}2&-1\\\\\\4&-9\\end{pma... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3826
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. -898 + 208 = -151x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -898 + 208 = -151x $\n\nMove all terms to the left (standard form):\n$ 151x + -690 = 0 $\n\nIsolate the $x$ term:\n$ 151x = 690 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 3827
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}0&-1&-2\\\\\\-1&5&4\\\\\\-1&4&4\\end{pmatrix},\\quad b=\\begin{pmatrix}-4\\\\\\5\\\\\\7\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g.... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3828
} |
[
{
"role": "user",
"content": "Find the correct solution for this 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": 3829
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. 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": 3830
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Evaluate: ((13 + -296/8) * 13) + 2^4 - log_3(81) + (-128 mod 22) + round(177/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((13 + -296/8) * 13) + 2^4 - log_3(81) + (-128 mod 22) + round(177/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 3831
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Evaluate: ((-194 + 98/2) * 15) + 11^2 - log_5(3125) + (-393 mod 2) + round(-190/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-194 + 98/2) * 15) + 11^2 - log_5(3125) + (-393 mod 2) + round(-190/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 3832
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. 540x - 391 = 726"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 540x - 391 = 726 $\n\nMove all terms to the left (standard form):\n$ 540x + -1117 = 0 $\n\nIsolate the $x$ term:\n$ 540x = 1117 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 3833
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Compute $(193+-158) \\bmod 21$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (193+-158) \\bmod 21 $\n\nUse the rule that addition is compatible with ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3834
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Prove that $(\\mathbb{Z}_3, +)$ is a group under addition modulo 3."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_3=\\{0,1,\\dots,2\\}$ with operation $+$ modulo $3$.\n\n**Closure:** If $a,b\\in\\mathb... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3835
} |
[
{
"role": "user",
"content": "Compute the final answer to this 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$ \... | {
"type": "trigonometry",
"seed": 1337,
"index": 3836
} |
[
{
"role": "user",
"content": "Please work out this equation. 42 - 40 + 9x^2 = -82x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 42 - 40 + 9x^2 = -82x $\n\nMove all terms to the left (standard form):\n$ 9x^2 + 82x + 2 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\pm... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3837
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Find $A^{-1}$ for $A=\\begin{pmatrix}0&8\\\\-4&2\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 0 & 8 \\\\ -4 & 2 \\end{pmatrix} $\n\nCompute the dete... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3838
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Compute $(-132+144) \\bmod 20$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-132+144) \\bmod 20 $\n\nUse the rule that addition is compatible with reduction modulo $... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3839
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Solve for $x$ (in radians): cos(x) = \\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = \\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 3840
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((-42 + -136/8) * 13) + 2^5 - log_8(32768) + (-313 mod 5) + round(30/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-42 + -136/8) * 13) + 2^5 - log_8(32768) + (-313 mod 5) + round(30/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 3841
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. 17x^2 - 157 + 87 = 15x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 17x^2 - 157 + 87 = 15x $\n\nMove all terms to the left (standard form):\n$ 17x^2 + -15x + -70 = 0 $\n\nApply the quadratic formula:... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3842
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. 393 + 777 = 946x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 393 + 777 = 946x $\n\nMove all terms to the left (standard form):\n$ -946x + 1170 = 0 $\n\nIsolate the $x$ term:\n$ -946x = -1170 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 3843
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Evaluate: ((170 + -148/4) * 9) + 10^4 - log_2(64) + (-283 mod 30) + round(141/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((170 + -148/4) * 9) + 10^4 - log_2(64) + (-283 mod 30) + round(141/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 3844
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. -324 - 501 = -891x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -324 - 501 = -891x $\n\nMove all terms to the left (standard form):\n$ 891x + -825 = 0 $\n\nIsolate the $x$ term:\n$ 891x = 825 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 3845
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Prove that $f(x,y)=\\frac12(8x^2+4y^2)+3x+5y+6$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+4y^2)+3x+5y+6 $\n\n$ \\frac{\\partial^2 f}{\\parti... | {
"type": "optimization",
"seed": 1337,
"index": 3846
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. 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... | {
"type": "optimization",
"seed": 1337,
"index": 3847
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. 49x + 15 - 154x^2 = -118"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 49x + 15 - 154x^2 = -118 $\n\nMove all terms to the left (standard form):\n$ -154x^2 + 49x + 133 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3848
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. 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^... | {
"type": "calculus",
"seed": 1337,
"index": 3849
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Differentiate with respect to $x$: $f(x) = \\ln(2x + -8)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(2x + -8) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{2}{2x + -8... | {
"type": "calculus",
"seed": 1337,
"index": 3850
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Compute the determinant: $A=\\begin{pmatrix}-1&3\\\\2&8\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-1&3\\\\\\2&8\\end{pmatrix} $\n\nUse $\\d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3851
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Solve for $x$ (in radians): cos(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ cos(x) = 1 $\n\nLet $\\alpha = \\arccos(k)$ (principal value).\n$ \\alpha = \\arccos(1) = 0 $\n\nThe g... | {
"type": "trigonometry",
"seed": 1337,
"index": 3852
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Evaluate: ((67 + 816/17) * 1) + 4^2 - log_3(3) + (400 mod 28) + floor(-4/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((67 + 816/17) * 1) + 4^2 - log_3(3) + (400 mod 28) + floor(-4/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 3853
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-3&-6\\\\\\3&-5\\end{pmatrix},\\quad b=\\begin{pmatrix}-33\\\\\\11\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-3&-6... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3854
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. Evaluate: ((84 + -30/6) * 10) + 2^5 - log_5(5) + (-160 mod 29) + round(-107/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((84 + -30/6) * 10) + 2^5 - log_5(5) + (-160 mod 29) + round(-107/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 3855
} |
[
{
"role": "user",
"content": "Solve the following mathematical 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$ \\al... | {
"type": "trigonometry",
"seed": 1337,
"index": 3856
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Evaluate: ((-141 + 400/8) * 8) + 2^4 - log_6(1296) + (-85 mod 27) + ceil(-28/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-141 + 400/8) * 8) + 2^4 - log_6(1296) + (-85 mod 27) + ceil(-28/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 3857
} |
[
{
"role": "user",
"content": "Please calculate this equation. Prove that $(\\mathbb{Z}_22, +)$ is a group under addition modulo 22."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_22=\\{0,1,\\dots,21\\}$ with operation $+$ modulo $22$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, then... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3858
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}0&4\\\\\\-6&7\\end{pmatrix},\\quad b=\\begin{pmatrix}28\\\\\\103\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}0&4\\\\\\-6&7... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3859
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. 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": 3860
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. 244x - 647 = 362"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 244x - 647 = 362 $\n\nMove all terms to the left (standard form):\n$ 244x + -1009 = 0 $\n\nIsolate the $x$ term:\n$ 244x = 1009 $\n\nDivide both s... | {
"type": "linear_equation",
"seed": 1337,
"index": 3861
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. Find the modular inverse of 19 modulo 27, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $19\\cdot a^{-1} \\equiv 1 \\pmod{27}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantee... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3862
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. 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": 3863
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Find $A^{-1}$ for $A=\\begin{pmatrix}1&-3\\\\3&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 1 & -3 \\\\ 3 & 1 \\end{pmatrix} $\n\nCompute the... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3864
} |
[
{
"role": "user",
"content": "Please calculate this equation. Evaluate: ((-93 + 288/18) * 4) + 11^4 - log_4(64) + (-462 mod 8) + round(44/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-93 + 288/18) * 4) + 11^4 - log_4(64) + (-462 mod 8) + round(44/2) $\n\nCompute each c... | {
"type": "arithmetic",
"seed": 1337,
"index": 3865
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Evaluate: ((-197 + 96/16) * 15) + 3^3 - log_4(1024) + (-17 mod 12) + ceil(93/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-197 + 96/16) * 15) + 3^3 - log_4(1024) + (-17 mod 12) + ceil(93/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 3866
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Prove that $f(x,y)=\\frac12(6x^2+2y^2)+2x+1y+3$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(6x^2+2y^2)+2x+1y+3 $\n\n$ \\frac{\\partial^2 f}{... | {
"type": "optimization",
"seed": 1337,
"index": 3867
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Evaluate: ((65 + 517/11) * 13) + 7^3 - log_2(64) + (-89 mod 8) + round(-142/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((65 + 517/11) * 13) + 7^3 - log_2(64) + (-89 mod 8) + round(-142/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 3868
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Show that $f(x,y)=x^2-y^2$ is not convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Consider $f(x,y)=x^2-y^2$.\n\nCompute the Hessian:\n$ H=\\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\parti... | {
"type": "optimization",
"seed": 1337,
"index": 3869
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. Prove that $(\\mathbb{Z}_11, +)$ is a group under addition modulo 11."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_11=\\{0,1,\\dots,10\\}$ with operation $+$ modulo $11$.\n\n**Closure:** If $a,b\\in\\ma... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3870
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. -54 - 953 = -767x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -54 - 953 = -767x $\n\nMove all terms to the left (standard form):\n$ 767x + -1007 = 0 $\n\nIsolate the $x$ term:\n$ 767x = 1007 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 3871
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. -7x^2 - 168x - 95 = -185"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -7x^2 - 168x - 95 = -185 $\n\nMove all terms to the left (standard form):\n$ -7x^2 + -168x + 90 = 0 $\n\nApply the quadratic formula... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3872
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-5&9\\\\\\-1&8\\end{pmatrix},\\quad b=\\begin{pmatrix}106\\\\\\77\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-5&9\\\\\\-1&8... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3873
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Prove that $f(x,y)=\\frac12(5x^2+8y^2)+-6x+2y+3$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(5x^2+8y^2)+-6x+2y+3 $\n\n$ \\frac{\\partial^2 f}{\\par... | {
"type": "optimization",
"seed": 1337,
"index": 3874
} |
[
{
"role": "user",
"content": "Can you compute this equation? Find $A^{-1}$ for $A=\\begin{pmatrix}8&9\\\\2&5\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 8 & 9 \\\\ 2 & 5 \\end{pmatrix} $\n\nCompute the determinant: $\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3875
} |
[
{
"role": "user",
"content": "Provide the solution to the following equation. -3 - 181x = -494"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -3 - 181x = -494 $\n\nMove all terms to the left (standard form):\n$ -181x + 491 = 0 $\n\nIsolate the $x$ term:\n$ -181x = -491 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 3876
} |
[
{
"role": "user",
"content": "Compute the solution of this mathematical equation. -36x^2 + 22x - 46 = 114"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -36x^2 + 22x - 46 = 114 $\n\nMove all terms to the left (standard form):\n$ -36x^2 + 22x + -160 = 0 $\n\nApply the quadratic ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3877
} |
[
{
"role": "user",
"content": "Solve the equation and show the result. 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$... | {
"type": "optimization",
"seed": 1337,
"index": 3878
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Find $A^{-1}$ for $A=\\begin{pmatrix}-6&-3\\\\-5&6\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & -3 \\\\ -5 & 6 \\end{pmatrix} $\n\nCompute... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3879
} |
[
{
"role": "user",
"content": "Solve the following math equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-8&4\\\\-3&-9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -8 & 4 \\\\ -3 & -9 \\end{pmatrix} $\n\nCompute the determ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3880
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. Differentiate with respect to $x$: $f(x) = \\ln(2x + -5)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(2x + -5) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\fr... | {
"type": "calculus",
"seed": 1337,
"index": 3881
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Differentiate with respect to $x$: $f(x) = 2x^3 + -5x^2 + -19x + -10$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 2x^3 + -5x^2 + -19x + -10 $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{6x^2 + -10... | {
"type": "calculus",
"seed": 1337,
"index": 3882
} |
[
{
"role": "user",
"content": "Please calculate this equation. Prove that $f(x)=3x^2+5x+10$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=3x^2+5x+10$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x) = 6x + 5 $\... | {
"type": "optimization",
"seed": 1337,
"index": 3883
} |
[
{
"role": "user",
"content": "Compute the value of the following equation. Differentiate with respect to $x$: $f(x) = \\ln(4x + 3)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(4x + 3) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{... | {
"type": "calculus",
"seed": 1337,
"index": 3884
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Find $A^{-1}$ for $A=\\begin{pmatrix}0&-9\\\\-9&0\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 0 & -9 \\\\ -9 & 0 \\end{pmatrix} $\n\nCompute the d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 3885
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. 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$ \\alph... | {
"type": "trigonometry",
"seed": 1337,
"index": 3886
} |
[
{
"role": "user",
"content": "Compute the value of the following 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 $f'(x)=0$, henc... | {
"type": "optimization",
"seed": 1337,
"index": 3887
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. -186x - 960 = 115"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -186x - 960 = 115 $\n\nMove all terms to the left (standard form):\n$ -186x + -1075 = 0 $\n\nIsolate the $x$ term:\n$ -186x = 1075 ... | {
"type": "linear_equation",
"seed": 1337,
"index": 3888
} |
[
{
"role": "user",
"content": "Please work out this equation. 482 + 299 = -33x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 482 + 299 = -33x $\n\nMove all terms to the left (standard form):\n$ 33x + 781 = 0 $\n\nIsolate the $x$ term:\n$ 33x = -781 $\n\nDivide both sides by the co... | {
"type": "linear_equation",
"seed": 1337,
"index": 3889
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Compute the indefinite integral: $\\int -3x^3 + -7x^2 + -15x + -5\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -3x^3 + -7x^2 + -15x + -5\\,dx $\n\nUse standard antiderivatives (and ... | {
"type": "calculus",
"seed": 1337,
"index": 3890
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. -33x - 280 = -956"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -33x - 280 = -956 $\n\nMove all terms to the left (standard form):\n$ -33x + 676 = 0 $\n\nIsolate the $x$ term:\n$ -33x = -676 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 3891
} |
[
{
"role": "user",
"content": "Please work out this equation. Evaluate: ((-169 + -425/17) * 10) + 3^2 - log_10(10000) + (463 mod 25) + ceil(61/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-169 + -425/17) * 10) + 3^2 - log_10(10000) + (463 mod 25) + ceil(61/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 3892
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Evaluate: ((68 + 408/24) * 16) + 4^4 - log_7(49) + (-144 mod 2) + floor(-61/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((68 + 408/24) * 16) + 4^4 - log_7(49) + (-144 mod 2) + floor(-61/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 3893
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. 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$ \\a... | {
"type": "trigonometry",
"seed": 1337,
"index": 3894
} |
[
{
"role": "user",
"content": "Please evaluate the equation provided. Compute $(-66+-61) \\bmod 23$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-66+-61) \\bmod 23 $\n\nUse the rule that addition is compatible with reduction modul... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 3895
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this 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 ... | {
"type": "trigonometry",
"seed": 1337,
"index": 3896
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Prove that $f(x,y)=\\frac12(4x^2+2y^2)+-3x+1y+3$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(4x^2+2y^2)+-3x+1y+3 $\n\n$ \\frac{\\partial^2 f}{\\... | {
"type": "optimization",
"seed": 1337,
"index": 3897
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. 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": 3898
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. -24 + 44x + 152x^2 = -105"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -24 + 44x + 152x^2 = -105 $\n\nMove all terms to the left (standard form):\n$ 152x^2 + 44x + 81 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 3899
} |
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