messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Please compute and solve 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$ \\al... | {
"type": "trigonometry",
"seed": 1337,
"index": 600
} |
[
{
"role": "user",
"content": "solve -542x - 983 = -246"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -542x - 983 = -246 $\n\nMove all terms to the left (standard form):\n$ -542x + -737 = 0 $\n\nIsolate the $x$ term:\n$ -542x = 737 $\n\nDivide both sides by the coefficient of $x$... | {
"type": "linear_equation",
"seed": 1337,
"index": 601
} |
[
{
"role": "user",
"content": "Calculate the solution of the equation below. 381 + 4x = 911"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 381 + 4x = 911 $\n\nMove all terms to the left (standard form):\n$ 4x + -530 = 0 $\n\nIsolate the $x$ term:\n$ 4x = 530 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 602
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. -13x + 175 - 42x^2 = -168"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -13x + 175 - 42x^2 = -168 $\n\nMove all terms to the left (standard form):\n$ -42x^2 + -13x + 343 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 603
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Find the modular inverse of 5 modulo 12, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $5\\cdot a^{-1} \\equiv 1 \\pmod{12}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantee... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 604
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Find $A^{-1}$ for $A=\\begin{pmatrix}9&-2\\\\4&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 9 & -2 \\\\ 4 & -7 \\end{pmatrix} $\n\nCompute the... | {
"type": "linear_algebra",
"seed": 1337,
"index": 605
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Differentiate with respect to $x$: $f(x) = 1e^{1x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 1e^{1x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{1e^{1x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 606
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Evaluate: ((180 + -6/6) * 20) + 7^4 - log_3(27) + (196 mod 8) + ceil(135/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((180 + -6/6) * 20) + 7^4 - log_3(27) + (196 mod 8) + ceil(135/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 607
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. 483x - 446 = 241"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 483x - 446 = 241 $\n\nMove all terms to the left (standard form):\n$ 483x + -687 = 0 $\n\nIsolate the $x$ term:\n$ 483x = 687 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 608
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Differentiate with respect to $x$: $f(x) = 3\\sin(6x) + -9\\cos(6x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 3\\sin(6x) + -9\\cos(6x) $\n\nApply the standard differentiation rules:\n$... | {
"type": "calculus",
"seed": 1337,
"index": 609
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Solve for $x$ (in radians): tan(x) = -\\frac{\\sqrt{3}}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = -\\frac{\\sqrt{3}}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n... | {
"type": "trigonometry",
"seed": 1337,
"index": 610
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. -754x - 117 = -924"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -754x - 117 = -924 $\n\nMove all terms to the left (standard form):\n$ -754x + 807 = 0 $\n\nIsolate the $x$ term:\n$ -754x = -807 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 611
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Evaluate: ((-71 + -594/18) * 16) + 2^4 - log_7(2401) + (-458 mod 12) + ceil(20/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-71 + -594/18) * 16) + 2^4 - log_7(2401) + (-458 mod 12) + ceil(20/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 612
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. 32 + 35 + 97x^2 = 185x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 32 + 35 + 97x^2 = 185x $\n\nMove all terms to the left (standard form):\n$ 97x^2 + -185x + 67 = 0 $\n\nApply the quadratic for... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 613
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Evaluate: ((115 + -1100/25) * 14) + 6^4 - log_5(625) + (89 mod 29) + floor(119/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((115 + -1100/25) * 14) + 6^4 - log_5(625) + (89 mod 29) + floor(... | {
"type": "arithmetic",
"seed": 1337,
"index": 614
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Compute the indefinite integral: $\\int -4x^3 + -8x^2 + -12x + -11\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -4x^3 + -8x^2 + -12x + -11\\,dx $\n\nUse standard antiderivatives (... | {
"type": "calculus",
"seed": 1337,
"index": 615
} |
[
{
"role": "user",
"content": "Determine the correct solution for the given equation. Evaluate: ((69 + 432/9) * 20) + 6^2 - log_9(59049) + (387 mod 3) + ceil(49/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((69 + 432/9) * 20) + 6^2 - log_9(59049) + (387 mod 3) + ceil(49/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 616
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Prove that $f(x,y)=\\frac12(1x^2+10y^2)+3x+1y+7$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+10y^2)+3x+1y+7 $\n\n$ \\frac{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 617
} |
[
{
"role": "user",
"content": "Please work out this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-7&-3\\\\\\0&-2\\end{pmatrix},\\quad b=\\begin{pmatrix}49\\\\\\14\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-7&-3\\\\\\0&-2\\end{pmatri... | {
"type": "linear_algebra",
"seed": 1337,
"index": 618
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Prove that $f(x,y)=\\frac12(1x^2+8y^2)+-1x+-5y+2$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(1x^2+8y^2)+-1x+-5y+2 $\n\n$ \\frac{\\partial^2 f}... | {
"type": "optimization",
"seed": 1337,
"index": 619
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. Evaluate: ((-177 + 48/3) * 1) + 10^4 - log_3(243) + (130 mod 5) + floor(-92/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-177 + 48/3) * 1) + 10^4 - log_3(243) + (130 mod 5) + floor(-92/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 620
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-5&-3&3\\\\\\-4&-3&3\\\\\\5&-1&5\\end{pmatrix},\\quad b=\\begin{pmatrix}-2\\\\\\-1\\\\\\-2\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=... | {
"type": "linear_algebra",
"seed": 1337,
"index": 621
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Prove that $f(x)=5x^2+2x+-8$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=5x^2+2x+-8$.\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": 622
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 46x^2 - 91x + 127 = -155"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 46x^2 - 91x + 127 = -155 $\n\nMove all terms to the left (standard form):\n$ 46x^2 + -91x + 282 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 623
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. -50 + 49x - 101x^2 = -132"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -50 + 49x - 101x^2 = -132 $\n\nMove all terms to the left (standard form):\n$ -101x^2 + 49x + 82 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 624
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. -565x - 388 = -384"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -565x - 388 = -384 $\n\nMove all terms to the left (standard form):\n$ -565x + -4 = 0 $\n\nIsolate the $x$ term:\n$ -565x = 4 $\n\nDivid... | {
"type": "linear_equation",
"seed": 1337,
"index": 625
} |
[
{
"role": "user",
"content": "Solve the equation and return the answer. Evaluate: ((16 + 54/18) * 15) + 8^5 - log_5(625) + (438 mod 11) + ceil(-156/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((16 + 54/18) * 15) + 8^5 - log_5(625) + (438 mod 11) + ceil(-156/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 626
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. 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$ \\alp... | {
"type": "trigonometry",
"seed": 1337,
"index": 627
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Evaluate: ((-22 + -315/21) * 13) + 3^3 - log_10(10) + (-221 mod 13) + floor(-75/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-22 + -315/21) * 13) + 3^3 - log_10(10) + (-221 mod 13) + floor(-75/2... | {
"type": "arithmetic",
"seed": 1337,
"index": 628
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}0&-4&-4\\\\\\-3&0&-5\\\\\\5&1&1\\end{pmatrix},\\quad b=\\begin{pmatrix}8\\\\\\15\\\\\\-2\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (e.g., by Gaussian eliminati... | {
"type": "linear_algebra",
"seed": 1337,
"index": 629
} |
[
{
"role": "user",
"content": "Compute the final answer to this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}6&-3\\\\\\-6&2\\end{pmatrix},\\quad b=\\begin{pmatrix}-51\\\\\\44\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}6&-3\\\\\\-6&2\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 630
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Differentiate with respect to $x$: $f(x) = 2e^{-4x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 2e^{-4x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-8e^{-4x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 631
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Evaluate: ((151 + 84/3) * 6) + 11^4 - log_7(16807) + (-343 mod 18) + floor(-53/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((151 + 84/3) * 6) + 11^4 - log_7(16807) + (-343 mod 18) + floor(-53/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 632
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. 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": 633
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. -20 - 90 + 141x^2 = -36x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -20 - 90 + 141x^2 = -36x $\n\nMove all terms to the left (standard form):\n$ 141x^2 + 36x + -110 = 0 $\n\nApply the quadratic form... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 634
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. Prove that $f(x)=4x^2+3x+-3$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=4x^2+3x+-3$.\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": 635
} |
[
{
"role": "user",
"content": "Please compute the 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": 636
} |
[
{
"role": "user",
"content": "Solve this equation and return the solution. Find $A^{-1}$ for $A=\\begin{pmatrix}9&4\\\\7&7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 9 & 4 \\\\ 7 & 7 \\end{pmatrix} $\n\nCompute the de... | {
"type": "linear_algebra",
"seed": 1337,
"index": 637
} |
[
{
"role": "user",
"content": "Please work out this equation. -943 + 98 = -289x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -943 + 98 = -289x $\n\nMove all terms to the left (standard form):\n$ 289x + -845 = 0 $\n\nIsolate the $x$ term:\n$ 289x = 845 $\n\nDivide both sides by th... | {
"type": "linear_equation",
"seed": 1337,
"index": 638
} |
[
{
"role": "user",
"content": "Solve the given equation and provide the solution. Find the modular inverse of 4 modulo 5, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $4\\cdot a^{-1} \\equiv 1 \\pmod{5}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity gua... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 639
} |
[
{
"role": "user",
"content": "solve: \n\n Evaluate: ((55 + -630/14) * 4) + 10^3 - log_5(15625) + (-68 mod 24) + floor(-195/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((55 + -630/14) * 4) + 10^3 - log_5(15625) + (-68 mod 24) + floor(-195/2) $\n\nCompute each component:\... | {
"type": "arithmetic",
"seed": 1337,
"index": 640
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. Prove that $f(x,y)=\\frac12(8x^2+9y^2)+0x+-3y+1$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(8x^2+9y^2)+0x+-3y+1 $\n\n$ \\frac{\\partial... | {
"type": "optimization",
"seed": 1337,
"index": 641
} |
[
{
"role": "user",
"content": "Find the solution to the following equation. Evaluate: ((189 + 322/14) * 12) + 6^2 - log_6(46656) + (-127 mod 10) + floor(-50/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((189 + 322/14) * 12) + 6^2 - log_6(46656) + (-127 mod 10) + floor(-50... | {
"type": "arithmetic",
"seed": 1337,
"index": 642
} |
[
{
"role": "user",
"content": "Please evaluate the 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$ \\alpha = \\arcsin(-\... | {
"type": "trigonometry",
"seed": 1337,
"index": 643
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Compute the indefinite integral: $\\int (-4x + -9)\\sin(5x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-4x + -9)\\sin(5x)\\,dx $\n\nUse standard antiderivatives (and integration by parts ... | {
"type": "calculus",
"seed": 1337,
"index": 644
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. 390 - 118 = -702x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 390 - 118 = -702x $\n\nMove all terms to the left (standard form):\n$ 702x + 272 = 0 $\n\nIsolate the $x$ term:\n$ 702x = -272 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 645
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Differentiate with respect to $x$: $f(x) = 3\\sin(9x) + -9\\cos(7x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 3\\sin(9x) + -9\\cos(7x) $\n\nApply the standard differentiation rules:\n$ f'(... | {
"type": "calculus",
"seed": 1337,
"index": 646
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Differentiate with respect to $x$: $f(x) = 9\\sin(4x) + 6\\cos(7x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 9\\sin(4x) + 6\\cos(7x) $\n\nApply the standard differentiation rules:\n$ f'(x) =... | {
"type": "calculus",
"seed": 1337,
"index": 647
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-3&-3\\\\-2&-7\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -3 & -3 \\\\ -2 & -7 \\end{pmatrix} $\n\nCompute th... | {
"type": "linear_algebra",
"seed": 1337,
"index": 648
} |
[
{
"role": "user",
"content": "Find the numerical solution to this equation. Evaluate: ((-102 + 529/23) * 5) + 3^4 - log_2(64) + (-50 mod 8) + ceil(-152/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-102 + 529/23) * 5) + 3^4 - log_2(64) + (-50 mod 8) + ceil(-152/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 649
} |
[
{
"role": "user",
"content": "Please solve this mathematical 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 gen... | {
"type": "trigonometry",
"seed": 1337,
"index": 650
} |
[
{
"role": "user",
"content": "Please calculate 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(-\\frac{1... | {
"type": "trigonometry",
"seed": 1337,
"index": 651
} |
[
{
"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": 652
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Prove that $f(x)=2x^2+-3x+6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=2x^2+-3x+6$.\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": 653
} |
[
{
"role": "user",
"content": "Compute the outcome of this equation. -6 + 55 + 76x^2 = -180x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -6 + 55 + 76x^2 = -180x $\n\nMove all terms to the left (standard form):\n$ 76x^2 + 180x + 49 = 0 $\n\nApply the quadratic formula:\n$ x = ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 654
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Find the modular inverse of 7 modulo 18, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $7\\cdot a^{-1} \\equiv 1 \\pmod{18}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees i... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 655
} |
[
{
"role": "user",
"content": "Solve the equation and provide the answer. 44x^2 + 118 - 62 = 154x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 44x^2 + 118 - 62 = 154x $\n\nMove all terms to the left (standard form):\n$ 44x^2 + -154x + 56 = 0 $\n\nApply the quadratic formula:\n... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 656
} |
[
{
"role": "user",
"content": "Solve this equation accurately. -24 - 105x^2 + 102x = -78"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -24 - 105x^2 + 102x = -78 $\n\nMove all terms to the left (standard form):\n$ -105x^2 + 102x + 54 = 0 $\n\nApply the quadratic formula:\n$ x = ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 657
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Evaluate: ((157 + 660/15) * 9) + 4^2 - log_2(8) + (-426 mod 18) + floor(172/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((157 + 660/15) * 9) + 4^2 - log_2(8) + (-426 mod 18) + floor(172/2) $\n\... | {
"type": "arithmetic",
"seed": 1337,
"index": 658
} |
[
{
"role": "user",
"content": "Compute the solution for this 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{4}{4x... | {
"type": "calculus",
"seed": 1337,
"index": 659
} |
[
{
"role": "user",
"content": "Compute the solution to this equation. -767 - 890x = 952"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -767 - 890x = 952 $\n\nMove all terms to the left (standard form):\n$ -890x + -1719 = 0 $\n\nIsolate the $x$ term:\n$ -890x = 1719 $\n\nDivide both... | {
"type": "linear_equation",
"seed": 1337,
"index": 660
} |
[
{
"role": "user",
"content": "Please evaluate and solve this equation. Evaluate: ((-175 + 575/23) * 6) + 6^5 - log_5(25) + (-444 mod 27) + floor(32/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-175 + 575/23) * 6) + 6^5 - log_5(25) + (-444 mod 27) + floor(32/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 661
} |
[
{
"role": "user",
"content": "Please find the solution of this equation. -394 + 158 = -806x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -394 + 158 = -806x $\n\nMove all terms to the left (standard form):\n$ 806x + -236 = 0 $\n\nIsolate the $x$ term:\n$ 806x = 236 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 662
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. -32x + 132 + 81x^2 = -66"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -32x + 132 + 81x^2 = -66 $\n\nMove all terms to the left (standard form):\n$ 81x^2 + -32x + 198 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 663
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-7&1\\\\\\-3&4\\end{pmatrix},\\quad b=\\begin{pmatrix}-58\\\\\\-7\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-7&1\\\\\\-3&... | {
"type": "linear_algebra",
"seed": 1337,
"index": 664
} |
[
{
"role": "user",
"content": "Can you solve the equation below? Compute the determinant: $A=\\begin{pmatrix}9&-4\\\\-6&-1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}9&-4\\\\\\-6&-1\\end{pmatrix} $\n\nUse $\\det(... | {
"type": "linear_algebra",
"seed": 1337,
"index": 665
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Compute $(-163+96) \\bmod 17$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (-163+96) \\bmod 17 $\n\nUse the rule that addition is compatible with reduction modulo ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 666
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. 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 general ... | {
"type": "trigonometry",
"seed": 1337,
"index": 667
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Compute the indefinite integral: $\\int (-4x + 8)e^{5x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (-4x + 8)e^{5x}\\,dx $\n\nUse standard antiderivatives (and integrati... | {
"type": "calculus",
"seed": 1337,
"index": 668
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Find the modular inverse of 6 modulo 7, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $6\\cdot a^{-1} \\equiv 1 \\pmod{7}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity g... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 669
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Find $A^{-1}$ for $A=\\begin{pmatrix}-5&2\\\\6&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -5 & 2 \\\\ 6 & -4 \\end{pmatrix} $\n\nCompute ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 670
} |
[
{
"role": "user",
"content": "Work through and solve this equation. Differentiate with respect to $x$: $f(x) = -5x^3 + 9x^2 + 2x + 10$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -5x^3 + 9x^2 + 2x + 10 $\n\nApply the standard differentiation rules:\n$ f'(x) = \\... | {
"type": "calculus",
"seed": 1337,
"index": 671
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. -46x^2 - 20 + 7x = 198"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -46x^2 - 20 + 7x = 198 $\n\nMove all terms to the left (standard form):\n$ -46x^2 + 7x + -218 = 0 $\n\nApply the quadratic formula:\n$ x = \\f... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 672
} |
[
{
"role": "user",
"content": "Determine the solution of the equation provided. 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).\... | {
"type": "trigonometry",
"seed": 1337,
"index": 673
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. Evaluate: ((129 + -120/8) * 12) + 9^5 - log_5(25) + (-410 mod 7) + floor(68/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((129 + -120/8) * 12) + 9^5 - log_5(25) + (-410 mod 7) + floor(68/2) $\n\nComp... | {
"type": "arithmetic",
"seed": 1337,
"index": 674
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. 47 + 61x^2 + 65 = -114x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 47 + 61x^2 + 65 = -114x $\n\nMove all terms to the left (standard form):\n$ 61x^2 + 114x + 112 = 0 $\n\nApply the quadratic... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 675
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Evaluate: ((175 + -322/23) * 2) + 5^2 - log_8(4096) + (-44 mod 23) + round(49/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((175 + -322/23) * 2) + 5^2 - log_8(4096) + (-44 mod 23) + round(49/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 676
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. 41 + 200 - 75x^2 = 150x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 41 + 200 - 75x^2 = 150x $\n\nMove all terms to the left (standard form):\n$ -75x^2 + -150x + 241 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 677
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Differentiate with respect to $x$: $f(x) = (2x + 2)e^{5x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (2x + 2)e^{5x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{2... | {
"type": "calculus",
"seed": 1337,
"index": 678
} |
[
{
"role": "user",
"content": "Compute the result for the following equation. Evaluate: ((73 + -64/8) * 18) + 11^4 - log_5(5) + (-399 mod 22) + floor(-159/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((73 + -64/8) * 18) + 11^4 - log_5(5) + (-399 mod 22) + floor(-159/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 679
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. -180 - 974x = -999"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -180 - 974x = -999 $\n\nMove all terms to the left (standard form):\n$ -974x + 819 = 0 $\n\nIsolate the $x$ term:\n$ -974x = -819 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 680
} |
[
{
"role": "user",
"content": "Please determine the solution to this equation. 855x - 128 = 913"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 855x - 128 = 913 $\n\nMove all terms to the left (standard form):\n$ 855x + -1041 = 0 $\n\nIsolate the $x$ term:\n$ 855x = 1041 $\n\nDivide... | {
"type": "linear_equation",
"seed": 1337,
"index": 681
} |
[
{
"role": "user",
"content": "Can you find the solution to 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 $\n\nThe ... | {
"type": "trigonometry",
"seed": 1337,
"index": 682
} |
[
{
"role": "user",
"content": "Determine the result of the equation below. Find the modular inverse of 4 modulo 9, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $4\\cdot a^{-1} \\equiv 1 \\pmod{9}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guarantees... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 683
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Evaluate: ((163 + 516/12) * 4) + 10^2 - log_3(9) + (97 mod 8) + round(-159/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((163 + 516/12) * 4) + 10^2 - log_3(9) + (97 mod 8) + round(-159/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 684
} |
[
{
"role": "user",
"content": "Find the answer to the following equation. -868x + 142 = -391"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -868x + 142 = -391 $\n\nMove all terms to the left (standard form):\n$ -868x + 533 = 0 $\n\nIsolate the $x$ term:\n$ -868x = -533 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 685
} |
[
{
"role": "user",
"content": "Find the solution to this math problem. Compute the determinant: $A=\\begin{pmatrix}5&9\\\\4&1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}5&9\\\\\\4&1\\end{pmatrix} $\n\nUse $\\det(... | {
"type": "linear_algebra",
"seed": 1337,
"index": 686
} |
[
{
"role": "user",
"content": "SOLVE THIS NOW: Differentiate with respect to $x$: $f(x) = -1x^3 + 9x^2 + -19x + 3$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = -1x^3 + 9x^2 + -19x + 3 $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-3x^2 + 18x + ... | {
"type": "calculus",
"seed": 1337,
"index": 687
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. 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\\mathbb{Z}_... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 688
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Compute the determinant: $A=\\begin{pmatrix}9&2\\\\7&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}9&2\\\\\\7&-4\\end{pmatrix} $\n\nUse $\\det(A)=ad-... | {
"type": "linear_algebra",
"seed": 1337,
"index": 689
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Compute $(22+178) \\bmod 3$ and justify using modular arithmetic rules."
},
{
"role": "assistant",
"content": "Compute the modular sum:\n$ (22+178) \\bmod 3 $\n\nUse the rule that addition is compatible with reduction ... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 690
} |
[
{
"role": "user",
"content": "Resolve the equation provided. Prove that $(\\mathbb{Z}_4, +)$ is a group under addition modulo 4."
},
{
"role": "assistant",
"content": "Consider $\\mathbb{Z}_4=\\{0,1,\\dots,3\\}$ with operation $+$ modulo $4$.\n\n**Closure:** If $a,b\\in\\mathbb{Z}_n$, then $(a+b... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 691
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Evaluate: ((67 + 120/8) * 13) + 7^2 - log_4(64) + (109 mod 30) + floor(-123/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((67 + 120/8) * 13) + 7^2 - log_4(64) + (109 mod 30) + floor(-123/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 692
} |
[
{
"role": "user",
"content": "Please solve this mathematical equation. -14 + 159x^2 - 84 = 72x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -14 + 159x^2 - 84 = 72x $\n\nMove all terms to the left (standard form):\n$ 159x^2 + -72x + -98 = 0 $\n\nApply the quadratic formula:\n$... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 693
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. 5 - 179 - 175x^2 = 152x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 5 - 179 - 175x^2 = 152x $\n\nMove all terms to the left (standard form):\n$ -175x^2 + -152x + -174 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 694
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. 944x + 240 = -353"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 944x + 240 = -353 $\n\nMove all terms to the left (standard form):\n$ 944x + 593 = 0 $\n\nIsolate the $x$ term:\n$ 944x = -593 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 695
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. 218x - 86 = 656"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 218x - 86 = 656 $\n\nMove all terms to the left (standard form):\n$ 218x + -742 = 0 $\n\nIsolate the $x$ term:\n$ 218x = 742 $\n\nDivide both si... | {
"type": "linear_equation",
"seed": 1337,
"index": 696
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Evaluate: ((98 + 420/20) * 3) + 4^5 - log_8(32768) + (-359 mod 2) + ceil(193/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((98 + 420/20) * 3) + 4^5 - log_8(32768) + (-359 mod 2) + ceil(193/2) $\n\nC... | {
"type": "arithmetic",
"seed": 1337,
"index": 697
} |
[
{
"role": "user",
"content": "Determine the solution for this math problem. Evaluate: ((80 + 527/17) * 15) + 8^5 - log_3(27) + (358 mod 20) + ceil(-85/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((80 + 527/17) * 15) + 8^5 - log_3(27) + (358 mod 20) + ceil(-85/2) $\n\nCo... | {
"type": "arithmetic",
"seed": 1337,
"index": 698
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. 10x^2 + 190 + 122 = 183x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 10x^2 + 190 + 122 = 183x $\n\nMove all terms to the left (standard form):\n$ 10x^2 + -183x + 312 = 0 $\n\nApply the quadratic ... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 699
} |
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