messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Evaluate: ((-147 + 252/6) * 6) + 7^2 - log_7(117649) + (-208 mod 29) + ceil(-135/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-147 + 252/6) * 6) + 7^2 - log_7(117649) + (-208 mod 29) +... | {
"type": "arithmetic",
"seed": 1337,
"index": 5100
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in 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} & \... | {
"type": "optimization",
"seed": 1337,
"index": 5101
} |
[
{
"role": "user",
"content": "Find the solution to this equation and report it. Compute the indefinite integral: $\\int 3\\sin(4x) + -8\\cos(9x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 3\\sin(4x) + -8\\cos(9x)\\,dx $\n\nUse standard antiderivatives... | {
"type": "calculus",
"seed": 1337,
"index": 5102
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. -199x + 788 = 944"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -199x + 788 = 944 $\n\nMove all terms to the left (standard form):\n$ -199x + -156 = 0 $\n\nIsolate the $x$ term:\n$ -199x = 156 $\n\n... | {
"type": "linear_equation",
"seed": 1337,
"index": 5103
} |
[
{
"role": "user",
"content": "Evaluate the equation and solve it. -4x + 321 = 42"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -4x + 321 = 42 $\n\nMove all terms to the left (standard form):\n$ -4x + 279 = 0 $\n\nIsolate the $x$ term:\n$ -4x = -279 $\n\nDivide both sides by the c... | {
"type": "linear_equation",
"seed": 1337,
"index": 5104
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. Prove that $f(x,y)=\\frac12(2x^2+6y^2)+0x+5y+-2$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(2x^2+6y^2)+0x+5y+-2 $\n\n$ \\frac{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 5105
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n -330 + 214 = 106x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -330 + 214 = 106x $\n\nMove all terms to the left (standard form):\n$ -106x + -116 = 0 $\n\nIsolate the $x$ term:\n$ -106x = 116 $\... | {
"type": "linear_equation",
"seed": 1337,
"index": 5106
} |
[
{
"role": "user",
"content": "Please compute the answer to this equation. Solve for $x$ (in radians): tan(x) = \\frac{1}{2}."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = \\frac{1}{2} $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan... | {
"type": "trigonometry",
"seed": 1337,
"index": 5107
} |
[
{
"role": "user",
"content": "Find the correct answer to the equation below. Compute the indefinite integral: $\\int (2x + -6)\\sin(5x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int (2x + -6)\\sin(5x)\\,dx $\n\nUse standard antiderivatives (and integrati... | {
"type": "calculus",
"seed": 1337,
"index": 5108
} |
[
{
"role": "user",
"content": "Resolve the equation provided. 50x^2 - 176 + 51 = 60x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 50x^2 - 176 + 51 = 60x $\n\nMove all terms to the left (standard form):\n$ 50x^2 + -60x + -125 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5109
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. -876x - 974 = -816"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -876x - 974 = -816 $\n\nMove all terms to the left (standard form):\n$ -876x + -158 = 0 $\n\nIsolate the $x$ term:\n$ -876x = 158 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 5110
} |
[
{
"role": "user",
"content": "Solve the equation and calculate the result. Prove that $f(x)=7x^2+7x+-6$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=7x^2+7x+-6$.\n\nA twice-differentiable function on $\\mathbb{R}$ is convex if $f''(x)\\ge 0$ for all $x$.\n\n$ f'(x... | {
"type": "optimization",
"seed": 1337,
"index": 5111
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. 58 - 143 = -317x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 58 - 143 = -317x $\n\nMove all terms to the left (standard form):\n$ 317x + -85 = 0 $\n\nIsolate the $x$ term:\n$ 317x = 85 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 5112
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. 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$ \\alph... | {
"type": "trigonometry",
"seed": 1337,
"index": 5113
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. -871x + 385 = 337"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -871x + 385 = 337 $\n\nMove all terms to the left (standard form):\n$ -871x + 48 = 0 $\n\nIsolate the $x$ term:\n$ -871x = -48 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 5114
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Evaluate: ((-46 + 656/16) * 10) + 2^4 - log_9(531441) + (-423 mod 6) + floor(155/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-46 + 656/16) * 10) + 2^4 - log_9(531441) + (-423 mod 6) + floor(155... | {
"type": "arithmetic",
"seed": 1337,
"index": 5115
} |
[
{
"role": "user",
"content": "Determine the solution of the given equation. Find the modular inverse of 32 modulo 37, and justify your answer."
},
{
"role": "assistant",
"content": "We want $a^{-1}$ such that $32\\cdot a^{-1} \\equiv 1 \\pmod{37}$.\n\nSince $\\gcd(a,m)=1$, Bézout's identity guar... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5116
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. -50 + 195 - 46x^2 = 180x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -50 + 195 - 46x^2 = 180x $\n\nMove all terms to the left (standard form):\n$ -46x^2 + -180x + 145 = 0 $\n\nApply the quadra... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5117
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Evaluate: ((-196 + 52/13) * 1) + 9^2 - log_4(64) + (99 mod 4) + ceil(181/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-196 + 52/13) * 1) + 9^2 - log_4(64) + (99 mod 4) + ceil(181/2) $\n\nCompute... | {
"type": "arithmetic",
"seed": 1337,
"index": 5118
} |
[
{
"role": "user",
"content": "Solve the equation below and give the answer. -295 - 976 = -274x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -295 - 976 = -274x $\n\nMove all terms to the left (standard form):\n$ 274x + -1271 = 0 $\n\nIsolate the $x$ term:\n$ 274x = 1271 $\n\nDivi... | {
"type": "linear_equation",
"seed": 1337,
"index": 5119
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Compute the determinant: $A=\\begin{pmatrix}4&6\\\\-3&-1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}4&6\\\\\\-3&-1\\end{pmatrix} $\n\nUse $\\d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5120
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. 28x^2 + 116x + 9 = 157"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 28x^2 + 116x + 9 = 157 $\n\nMove all terms to the left (standard form):\n$ 28x^2 + 116x + -148 = 0 $\n\nApply the quadratic formula:\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5121
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Prove that $f(x,y)=\\frac12(3x^2+6y^2)+6x+6y+-5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(3x^2+6y^2)+6x+6y+-5 $\n\n$ \\frac{\\partial^2 f}{\... | {
"type": "optimization",
"seed": 1337,
"index": 5122
} |
[
{
"role": "user",
"content": "Find the value that satisfies this equation. -762x - 675 = 73"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -762x - 675 = 73 $\n\nMove all terms to the left (standard form):\n$ -762x + -748 = 0 $\n\nIsolate the $x$ term:\n$ -762x = 748 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 5123
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. -664 - 881 = -913x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -664 - 881 = -913x $\n\nMove all terms to the left (standard form):\n$ 913x + -1545 = 0 $\n\nIsolate the $x$ term:\n$ 913x = 1545 $\n\nDi... | {
"type": "linear_equation",
"seed": 1337,
"index": 5124
} |
[
{
"role": "user",
"content": "Solve this equation accurately. Differentiate with respect to $x$: $f(x) = 9e^{-2x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 9e^{-2x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{-18e^{-2x}} $"
}
] | {
"type": "calculus",
"seed": 1337,
"index": 5125
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Compute the indefinite integral: $\\int 3e^{-2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 3e^{-2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\... | {
"type": "calculus",
"seed": 1337,
"index": 5126
} |
[
{
"role": "user",
"content": "Find the final answer to the given equation. Find $A^{-1}$ for $A=\\begin{pmatrix}-6&3\\\\4&0\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} -6 & 3 \\\\ 4 & 0 \\end{pmatrix} $\n\nCompute the ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5127
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Evaluate: ((157 + -76/19) * 17) + 7^5 - log_5(5) + (-232 mod 16) + floor(-199/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((157 + -76/19) * 17) + 7^5 - log_5(5) + (-232 mod 16) + floor(-199/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 5128
} |
[
{
"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": 5129
} |
[
{
"role": "user",
"content": "Calculate the answer for this equation. Differentiate with respect to $x$: $f(x) = 2x^3 + 9x^2 + -2x + 20$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = 2x^3 + 9x^2 + -2x + 20 $\n\nApply the standard differentiation rules:\n$ f'(x) = ... | {
"type": "calculus",
"seed": 1337,
"index": 5130
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. Compute the determinant: $A=\\begin{pmatrix}-4&3\\\\-9&-4\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-4&3\\\\\\-9&-4\\end{pmatrix} $\n\nU... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5131
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Differentiate with respect to $x$: $f(x) = \\ln(4x + 2)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(4x + 2) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxe... | {
"type": "calculus",
"seed": 1337,
"index": 5132
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. -705 - 947 = 780x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -705 - 947 = 780x $\n\nMove all terms to the left (standard form):\n$ -780x + -1652 = 0 $\n\nIsolate the $x$ term:\n$ -780x = 1652 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 5133
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. 28x^2 + 106 + 21x = 151"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 28x^2 + 106 + 21x = 151 $\n\nMove all terms to the left (standard form):\n$ 28x^2 + 21x + -45 = 0 $\n\nApply the quadratic formul... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5134
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Compute the indefinite integral: $\\int 4x^3 + 3x^2 + -12x + 18\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 4x^3 + 3x^2 + -12x + 18\\,dx $\n\nUse standard antiderivatives (and ... | {
"type": "calculus",
"seed": 1337,
"index": 5135
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? -249x - 303 = -82"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -249x - 303 = -82 $\n\nMove all terms to the left (standard form):\n$ -249x + -221 = 0 $\n\nIsolate the $x$ term:\n$ -249x = 221 $\n\nDiv... | {
"type": "linear_equation",
"seed": 1337,
"index": 5136
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Evaluate: ((200 + -528/24) * 9) + 2^4 - log_6(46656) + (379 mod 7) + ceil(-63/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((200 + -528/24) * 9) + 2^4 - log_6(46656) + (379 mod 7) + ceil(-63/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 5137
} |
[
{
"role": "user",
"content": "Please calculate and resolve this equation. Evaluate: ((152 + 282/6) * 13) + 3^2 - log_9(531441) + (431 mod 5) + round(14/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((152 + 282/6) * 13) + 3^2 - log_9(531441) + (431 mod 5) + round(14/2) $\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 5138
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}0&4\\\\\\6&9\\end{pmatrix},\\quad b=\\begin{pmatrix}0\\\\\\-24\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}0&4\\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5139
} |
[
{
"role": "user",
"content": "Please solve this math problem. Evaluate: ((105 + 408/12) * 14) + 4^2 - log_3(729) + (411 mod 7) + floor(-128/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((105 + 408/12) * 14) + 4^2 - log_3(729) + (411 mod 7) + floor(-128/2) $\n\nCompute ea... | {
"type": "arithmetic",
"seed": 1337,
"index": 5140
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-2&1&-3\\\\\\-3&-4&3\\\\\\4&-3&0\\end{pmatrix},\\quad b=\\begin{pmatrix}-11\\\\\\19\\\\\\-1\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5141
} |
[
{
"role": "user",
"content": "Please find and report the solution to this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-3&4&1\\\\\\-2&-3&-1\\\\\\-4&-1&-1\\end{pmatrix},\\quad b=\\begin{pmatrix}5\\\\\\-4\\\\\\-2\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5142
} |
[
{
"role": "user",
"content": "Please work out this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}0&-8\\\\7&3\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 0 & -8 \\\\ 7 & 3 \\end{pmatrix} $\n\nCompute the determinant: $... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5143
} |
[
{
"role": "user",
"content": "Please find the value of this math equation. 20 - 116x + 186 = -191x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 20 - 116x + 186 = -191x^2 $\n\nMove all terms to the left (standard form):\n$ 191x^2 + -116x + 206 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5144
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}5&3\\\\\\3&8\\end{pmatrix},\\quad b=\\begin{pmatrix}46\\\\\\71\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}5&3\\\\\\3&8\\end{pmatrix} ... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5145
} |
[
{
"role": "user",
"content": "Solve the following math equation. Compute the indefinite integral: $\\int 4e^{-2x}\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 4e^{-2x}\\,dx $\n\nUse standard antiderivatives (and integration by parts if needed):\n$ \\int... | {
"type": "calculus",
"seed": 1337,
"index": 5146
} |
[
{
"role": "user",
"content": "Compute the solution for this equation. Differentiate with respect to $x$: $f(x) = \\ln(8x + -4)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = \\ln(8x + -4) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{\\frac{8}{... | {
"type": "calculus",
"seed": 1337,
"index": 5147
} |
[
{
"role": "user",
"content": "Solve the following math equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-2&-5\\\\\\-7&4\\end{pmatrix},\\quad b=\\begin{pmatrix}-51\\\\\\-28\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-2&-5\\\\\\-7&4\\end{... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5148
} |
[
{
"role": "user",
"content": "Can you evaluate this math equation? Evaluate: ((46 + -567/21) * 11) + 12^4 - log_4(64) + (-80 mod 30) + floor(-37/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((46 + -567/21) * 11) + 12^4 - log_4(64) + (-80 mod 30) + floor(-37/2) $\n\nCompu... | {
"type": "arithmetic",
"seed": 1337,
"index": 5149
} |
[
{
"role": "user",
"content": "Compute the correct solution for this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}-7&-8\\\\\\5&-6\\end{pmatrix},\\quad b=\\begin{pmatrix}-15\\\\\\-1\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}-7&-8\\\\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5150
} |
[
{
"role": "user",
"content": "Solve the mathematical equation provided below. Evaluate: ((34 + -18/9) * 16) + 5^5 - log_8(32768) + (-90 mod 30) + floor(-59/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((34 + -18/9) * 16) + 5^5 - log_8(32768) + (-90 mod 30) + floor(-59/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 5151
} |
[
{
"role": "user",
"content": "Solve the following math equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}0&8\\\\\\-5&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}8\\\\\\-3\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}0&8\\\\\\-5&-3\\end{pmatr... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5152
} |
[
{
"role": "user",
"content": "Solve the following mathematical problem. 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\\in\\mathbb{Z... | {
"type": "abstract_algebra",
"seed": 1337,
"index": 5153
} |
[
{
"role": "user",
"content": "Solve the equation and state the result. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}-2&3&-5\\\\\\1&-5&1\\\\\\4&-4&1\\end{pmatrix},\\quad b=\\begin{pmatrix}-22\\\\\\15\\\\\\22\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$ (... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5154
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. Compute the determinant: $A=\\begin{pmatrix}-6&6\\\\6&-9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Compute the determinant of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix}-6&6\\\\\\6&-9\\end{pmatrix} $\n\nUse $\\d... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5155
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? Evaluate: ((52 + -224/7) * 1) + 11^5 - log_7(7) + (391 mod 7) + ceil(187/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((52 + -224/7) * 1) + 11^5 - log_7(7) + (391 mod 7) + ceil(187/2) $\n\nCompute ... | {
"type": "arithmetic",
"seed": 1337,
"index": 5156
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Solve for $x$ (in radians): sin(x) = 0."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ sin(x) = 0 $\n\nLet $\\alpha = \\arcsin(k)$ (principal value).\n$ \\alpha = \\arcsin(0) = 0 $\n\nThe general s... | {
"type": "trigonometry",
"seed": 1337,
"index": 5157
} |
[
{
"role": "user",
"content": "Solve the given mathematical expression. 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$ \\alp... | {
"type": "trigonometry",
"seed": 1337,
"index": 5158
} |
[
{
"role": "user",
"content": "Please evaluate and compute this equation. -209 + 970x = -669"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -209 + 970x = -669 $\n\nMove all terms to the left (standard form):\n$ 970x + 460 = 0 $\n\nIsolate the $x$ term:\n$ 970x = -460 $\n\nDivide bo... | {
"type": "linear_equation",
"seed": 1337,
"index": 5159
} |
[
{
"role": "user",
"content": "Solve the equation shown below. -686 - 143x = 461"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -686 - 143x = 461 $\n\nMove all terms to the left (standard form):\n$ -143x + -1147 = 0 $\n\nIsolate the $x$ term:\n$ -143x = 1147 $\n\nDivide both sides ... | {
"type": "linear_equation",
"seed": 1337,
"index": 5160
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Differentiate with respect to $x$: $f(x) = (6x + 4)e^{3x}$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (6x + 4)e^{3x} $\n\nApply the standard differentiation rules:\n$ f'(x) = \\boxed{6e^... | {
"type": "calculus",
"seed": 1337,
"index": 5161
} |
[
{
"role": "user",
"content": "Solve the equation shown below. Compute the indefinite integral: $\\int 9\\sin(7x) + 3\\cos(3x)\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int 9\\sin(7x) + 3\\cos(3x)\\,dx $\n\nUse standard antiderivatives (and integration by... | {
"type": "calculus",
"seed": 1337,
"index": 5162
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. Differentiate with respect to $x$: $f(x) = (3x + 8)\\sin(4x)$."
},
{
"role": "assistant",
"content": "Differentiate the function:\n$ f(x) = (3x + 8)\\sin(4x) $\n\nApply the standard differentiation rules:\n$ f'(x) = \\... | {
"type": "calculus",
"seed": 1337,
"index": 5163
} |
[
{
"role": "user",
"content": "Resolve this equation and give the result. Find $A^{-1}$ for $A=\\begin{pmatrix}9&-6\\\\-4&-1\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 9 & -6 \\\\ -4 & -1 \\end{pmatrix} $\n\nCompute th... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5164
} |
[
{
"role": "user",
"content": "Calculate and solve this equation. -89 + 434x = -231"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -89 + 434x = -231 $\n\nMove all terms to the left (standard form):\n$ 434x + 142 = 0 $\n\nIsolate the $x$ term:\n$ 434x = -142 $\n\nDivide both sides b... | {
"type": "linear_equation",
"seed": 1337,
"index": 5165
} |
[
{
"role": "user",
"content": "Can you find the solution to this equation? -800x - 523 = 109"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ -800x - 523 = 109 $\n\nMove all terms to the left (standard form):\n$ -800x + -632 = 0 $\n\nIsolate the $x$ term:\n$ -800x = 632 $\n\nDivide b... | {
"type": "linear_equation",
"seed": 1337,
"index": 5166
} |
[
{
"role": "user",
"content": "Find the correct solution for this equation. Evaluate: ((157 + 247/13) * 2) + 2^3 - log_2(8) + (160 mod 15) + floor(128/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((157 + 247/13) * 2) + 2^3 - log_2(8) + (160 mod 15) + floor(128/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 5167
} |
[
{
"role": "user",
"content": "Determine the solution to the math equation provided. Evaluate: ((-18 + 275/11) * 15) + 5^3 - log_9(9) + (-78 mod 8) + round(-134/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-18 + 275/11) * 15) + 5^3 - log_9(9) + (-78 mod 8) + round(-134/... | {
"type": "arithmetic",
"seed": 1337,
"index": 5168
} |
[
{
"role": "user",
"content": "Find the result of the given mathematical equation. -24 - 69x^2 - 132 = 198x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -24 - 69x^2 - 132 = 198x $\n\nMove all terms to the left (standard form):\n$ -69x^2 + -198x + -156 = 0 $\n\nApply the quadra... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5169
} |
[
{
"role": "user",
"content": "Determine the result of this mathematical equation. Evaluate: ((-116 + -931/19) * 19) + 12^2 - log_9(531441) + (-107 mod 27) + floor(100/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-116 + -931/19) * 19) + 12^2 - log_9(531441) + (-107 mod ... | {
"type": "arithmetic",
"seed": 1337,
"index": 5170
} |
[
{
"role": "user",
"content": "Determine the value produced by this equation. Prove that $f(x,y)=\\frac12(7x^2+4y^2)+2x+-6y+5$ is convex on $\\mathbb{R}^2$."
},
{
"role": "assistant",
"content": "Compute the Hessian matrix $H$.\n\n$ f(x,y)=\\frac12(7x^2+4y^2)+2x+-6y+5 $\n\n$ \\frac{\\partial^2 f}... | {
"type": "optimization",
"seed": 1337,
"index": 5171
} |
[
{
"role": "user",
"content": "Can you compute this equation? 14 - 76x^2 - 7 = 98x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 14 - 76x^2 - 7 = 98x $\n\nMove all terms to the left (standard form):\n$ -76x^2 + -98x + 7 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac{-b \\p... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5172
} |
[
{
"role": "user",
"content": "Evaluate and solve the given equation. -26x^2 - 66 - 12x = 33"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -26x^2 - 66 - 12x = 33 $\n\nMove all terms to the left (standard form):\n$ -26x^2 + -12x + -99 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5173
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. 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{\\p... | {
"type": "optimization",
"seed": 1337,
"index": 5174
} |
[
{
"role": "user",
"content": "Please work through this equation and solve it. Evaluate: ((42 + -240/12) * 14) + 2^3 - log_5(25) + (-312 mod 5) + round(189/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((42 + -240/12) * 14) + 2^3 - log_5(25) + (-312 mod 5) + round(189/2) $... | {
"type": "arithmetic",
"seed": 1337,
"index": 5175
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. Solve the linear system $Ax=b$:\n$A=\\begin{pmatrix}0&-6\\\\\\7&-3\\end{pmatrix},\\quad b=\\begin{pmatrix}-30\\\\\\48\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b$.\n$ A=\\begin{pmatrix}0&-6\... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5176
} |
[
{
"role": "user",
"content": "Please calculate this equation. -5 + 64x^2 + 178 = -22x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -5 + 64x^2 + 178 = -22x $\n\nMove all terms to the left (standard form):\n$ 64x^2 + 22x + 173 = 0 $\n\nApply the quadratic formula:\n$ x = \\frac... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5177
} |
[
{
"role": "user",
"content": "Find the solution of the math equation below. Evaluate: ((-140 + 350/7) * 14) + 2^2 - log_3(243) + (298 mod 18) + floor(-181/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-140 + 350/7) * 14) + 2^2 - log_3(243) + (298 mod 18) + floor(-181/2)... | {
"type": "arithmetic",
"seed": 1337,
"index": 5178
} |
[
{
"role": "user",
"content": "Please resolve the math equation below. Find all stationary points of $f(x)=x^4-2x^2$ and classify them."
},
{
"role": "assistant",
"content": "Compute the derivative:\n$ f(x)=x^4-2x^2 $\n$ f'(x)=4x^3-4x=4x(x^2-1). $\n\nStationary points satisfy $f'(x)=0$, hence:\n$... | {
"type": "optimization",
"seed": 1337,
"index": 5179
} |
[
{
"role": "user",
"content": "Can you determine the solution of this equation? Evaluate: ((-65 + 230/5) * 16) + 10^3 - log_6(6) + (-37 mod 14) + floor(-47/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-65 + 230/5) * 16) + 10^3 - log_6(6) + (-37 mod 14) + floor(-47/2) $\... | {
"type": "arithmetic",
"seed": 1337,
"index": 5180
} |
[
{
"role": "user",
"content": "Please solve the equation presented here. Evaluate: ((-181 + -60/4) * 1) + 9^4 - log_2(16) + (331 mod 3) + floor(-36/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-181 + -60/4) * 1) + 9^4 - log_2(16) + (331 mod 3) + floor(-36/2) $\n\nComput... | {
"type": "arithmetic",
"seed": 1337,
"index": 5181
} |
[
{
"role": "user",
"content": "Solve the mathematical problem presented. 40 + 99 + 69x^2 = 51x"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 40 + 99 + 69x^2 = 51x $\n\nMove all terms to the left (standard form):\n$ 69x^2 + -51x + 139 = 0 $\n\nApply the quadratic formula:\n$ x =... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5182
} |
[
{
"role": "user",
"content": "Determine the value of the unknown in this equation. -14 - 40x + 6x^2 = 103"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ -14 - 40x + 6x^2 = 103 $\n\nMove all terms to the left (standard form):\n$ 6x^2 + -40x + -117 = 0 $\n\nApply the quadratic fo... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5183
} |
[
{
"role": "user",
"content": "Calculate the solution to this math equation. Compute the indefinite integral: $\\int -2x^3 + 5x^2 + -5x + -14\\,dx$."
},
{
"role": "assistant",
"content": "Compute the indefinite integral:\n$ \\int -2x^3 + 5x^2 + -5x + -14\\,dx $\n\nUse standard antiderivatives (an... | {
"type": "calculus",
"seed": 1337,
"index": 5184
} |
[
{
"role": "user",
"content": "Please solve this math problem. 48x - 158x^2 + 116 = 163"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 48x - 158x^2 + 116 = 163 $\n\nMove all terms to the left (standard form):\n$ -158x^2 + 48x + -47 = 0 $\n\nApply the quadratic formula:\n$ x = \\... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5185
} |
[
{
"role": "user",
"content": "Determine the correct answer to this equation. 34x - 80 + 183 = 180x^2"
},
{
"role": "assistant",
"content": "To solve the equation:\n$ 34x - 80 + 183 = 180x^2 $\n\nMove all terms to the left (standard form):\n$ -180x^2 + 34x + 103 = 0 $\n\nApply the quadratic formu... | {
"type": "quadratic_equation",
"seed": 1337,
"index": 5186
} |
[
{
"role": "user",
"content": "Please find and report the solution to 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 val... | {
"type": "trigonometry",
"seed": 1337,
"index": 5187
} |
[
{
"role": "user",
"content": "Please determine the value of this equation. 398 - 348x = -538"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 398 - 348x = -538 $\n\nMove all terms to the left (standard form):\n$ -348x + 936 = 0 $\n\nIsolate the $x$ term:\n$ -348x = -936 $\n\nDivide ... | {
"type": "linear_equation",
"seed": 1337,
"index": 5188
} |
[
{
"role": "user",
"content": "Please find and report the solution 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 $f'(x)... | {
"type": "optimization",
"seed": 1337,
"index": 5189
} |
[
{
"role": "user",
"content": "Solve the math equation and present the result. Evaluate: ((-188 + 252/7) * 3) + 2^5 - log_3(81) + (464 mod 14) + floor(37/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-188 + 252/7) * 3) + 2^5 - log_3(81) + (464 mod 14) + floor(37/2) $\n\n... | {
"type": "arithmetic",
"seed": 1337,
"index": 5190
} |
[
{
"role": "user",
"content": "Solve this equation step by step. Evaluate: ((-76 + 75/3) * 2) + 3^2 - log_9(729) + (82 mod 5) + round(155/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((-76 + 75/3) * 2) + 3^2 - log_9(729) + (82 mod 5) + round(155/2) $\n\nCompute each compo... | {
"type": "arithmetic",
"seed": 1337,
"index": 5191
} |
[
{
"role": "user",
"content": "Find the solution to the equation shown. Evaluate: ((123 + 330/15) * 1) + 5^5 - log_4(1024) + (370 mod 24) + ceil(-74/2)"
},
{
"role": "assistant",
"content": "Evaluate the expression:\n$ ((123 + 330/15) * 1) + 5^5 - log_4(1024) + (370 mod 24) + ceil(-74/2) $\n\nCom... | {
"type": "arithmetic",
"seed": 1337,
"index": 5192
} |
[
{
"role": "user",
"content": "Determine the answer to the equation shown. 206 + 455 = 750x"
},
{
"role": "assistant",
"content": "Solve the equation:\n$ 206 + 455 = 750x $\n\nMove all terms to the left (standard form):\n$ -750x + 661 = 0 $\n\nIsolate the $x$ term:\n$ -750x = -661 $\n\nDivide bot... | {
"type": "linear_equation",
"seed": 1337,
"index": 5193
} |
[
{
"role": "user",
"content": "Solve the equation and provide the final value. 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": 5194
} |
[
{
"role": "user",
"content": "solve the follow equation, show ALL of your work: \n\n 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 va... | {
"type": "trigonometry",
"seed": 1337,
"index": 5195
} |
[
{
"role": "user",
"content": "Please determine the outcome of this equation. Solve the $3\\times 3$ linear system $Ax=b$.\n$A=\\begin{pmatrix}1&2&-4\\\\\\5&-3&-4\\\\\\1&4&0\\end{pmatrix},\\quad b=\\begin{pmatrix}-9\\\\\\0\\\\\\-3\\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "Solve $Ax=b... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5196
} |
[
{
"role": "user",
"content": "Please compute the solution to this problem. Prove that $f(x)=10x^2+-5x+8$ is convex on $\\mathbb{R}$."
},
{
"role": "assistant",
"content": "Let $f(x)=10x^2+-5x+8$.\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": 5197
} |
[
{
"role": "user",
"content": "Please compute and solve this equation. Find $A^{-1}$ for $A=\\begin{pmatrix}4&2\\\\-6&-9\\end{pmatrix}$."
},
{
"role": "assistant",
"content": "Find the inverse of the $2\\times 2$ matrix:\n$ A=\\begin{pmatrix} 4 & 2 \\\\ -6 & -9 \\end{pmatrix} $\n\nCompute the det... | {
"type": "linear_algebra",
"seed": 1337,
"index": 5198
} |
[
{
"role": "user",
"content": "Evaluate the following equation. Solve for $x$ (in radians): tan(x) = 1."
},
{
"role": "assistant",
"content": "Solve the trigonometric equation:\n$ tan(x) = 1 $\n\nLet $\\alpha = \\arctan(k)$ (principal value).\n$ \\alpha = \\arctan(1) = 0.785398 $\n\nThe general s... | {
"type": "trigonometry",
"seed": 1337,
"index": 5199
} |
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