data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \\cos (3 t)$, which we will assume is a solution of the differential equation $m \\ddot{x}+b \\dot{x}+k x=0$. \n\nWhat is $b$ in terms of $m$? Write $b$ as a constant times a function of $m$.\n\nL... | math | {
"style": "rule",
"ground_truth": "m"
} | {
"index": 100,
"split": "test",
"question": "Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \\cos (3 t)$, which we will assume is a solution of the differential equation $m \\ddot{x}+b \\dot{x}+k x=0$. \n\nWhat is $b$ in terms of $m$? Write $b$ as a constant times a function of... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: The following subproblems refer to the differential equation. $\\ddot{x}+4 x=\\sin (3 t)$\n\nSubproblem 0: Find $A$ so that $A \\sin (3 t)$ is a solution of $\\ddot{x}+4 x=\\sin (3 t)$.\n\n\nSolution: We can find this by brute force. If $x=A \\sin (3 t)$, then $\\ddot... | math | {
"style": "rule",
"ground_truth": "-\\sin (3 t) / 5+ C_{1} \\sin (2 t)+C_{2} \\cos (2 t)"
} | {
"index": 101,
"split": "test",
"question": "Preamble: The following subproblems refer to the differential equation. $\\ddot{x}+4 x=\\sin (3 t)$\n\nSubproblem 0: Find $A$ so that $A \\sin (3 t)$ is a solution of $\\ddot{x}+4 x=\\sin (3 t)$.\n\n\nSolution: We can find this by brute force. If $x=A \\sin (3 t)$, th... |
math-ai/minervamath | [
{
"role": "user",
"content": "What is the smallest possible positive $k$ such that all functions $x(t)=A \\cos (\\omega t-\\phi)$---where $\\phi$ is an odd multiple of $k$---satisfy $x(0)=0$? \\\\\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 102,
"split": "test",
"question": "What is the smallest possible positive $k$ such that all functions $x(t)=A \\cos (\\omega t-\\phi)$---where $\\phi$ is an odd multiple of $k$---satisfy $x(0)=0$? \\\\"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: The following subproblems refer to the differential equation $\\ddot{x}+b \\dot{x}+x=0$.\\\\\n\nWhat is the characteristic polynomial $p(s)$ of $\\ddot{x}+b \\dot{x}+x=0$?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "s^{2}+b s+1"
} | {
"index": 103,
"split": "test",
"question": "Preamble: The following subproblems refer to the differential equation $\\ddot{x}+b \\dot{x}+x=0$.\\\\\n\nWhat is the characteristic polynomial $p(s)$ of $\\ddot{x}+b \\dot{x}+x=0$?"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \\cos (3 t)$, which we will assume is a solution of the differential equation $m \\ddot{x}+b \\dot{x}+k x=0$. \n\nSubproblem 0: What is $b$ in terms of $m$? Write $b$ as a constant times a functio... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{4} m"
} | {
"index": 104,
"split": "test",
"question": "Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \\cos (3 t)$, which we will assume is a solution of the differential equation $m \\ddot{x}+b \\dot{x}+k x=0$. \n\nSubproblem 0: What is $b$ in terms of $m$? Write $b$ as a constant times... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: In the following problems, take $a = \\ln 2$ and $b = \\pi / 3$. \n\nSubproblem 0: Given $a = \\ln 2$ and $b = \\pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers. \n\n\nSolution: Using Euler's formula, we find that the answer is $\\boxe... | math | {
"style": "rule",
"ground_truth": "-8-8 \\sqrt{3} i"
} | {
"index": 105,
"split": "test",
"question": "Preamble: In the following problems, take $a = \\ln 2$ and $b = \\pi / 3$. \n\nSubproblem 0: Given $a = \\ln 2$ and $b = \\pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers. \n\n\nSolution: Using Euler's formula, we find that the answer ... |
math-ai/minervamath | [
{
"role": "user",
"content": "Rewrite the function $\\operatorname{Re} \\frac{e^{i t}}{2+2 i}$ in the form $A \\cos (\\omega t-\\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4} \\cos (t-\\pi / 4)"
} | {
"index": 106,
"split": "test",
"question": "Rewrite the function $\\operatorname{Re} \\frac{e^{i t}}{2+2 i}$ in the form $A \\cos (\\omega t-\\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: The following subproblems refer to the differential equation $\\ddot{x}+b \\dot{x}+x=0$.\\\\\n\nSubproblem 0: What is the characteristic polynomial $p(s)$ of $\\ddot{x}+b \\dot{x}+x=0$?\n\n\nSolution: The characteristic polynomial is $p(s)=\\boxed{s^{2}+b s+1}$.\n\nFi... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 107,
"split": "test",
"question": "Preamble: The following subproblems refer to the differential equation $\\ddot{x}+b \\dot{x}+x=0$.\\\\\n\nSubproblem 0: What is the characteristic polynomial $p(s)$ of $\\ddot{x}+b \\dot{x}+x=0$?\n\n\nSolution: The characteristic polynomial is $p(s)=\\boxed{s^{2}+b s+... |
math-ai/minervamath | [
{
"role": "user",
"content": "Find the general (complex-valued) solution of the differential equation $\\dot{z}+2 z=e^{2 i t}$, using $C$ to stand for any complex-valued integration constants which may arise.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{e^{2 i t}}{(2+2 i)}+C e^{-2 t}"
} | {
"index": 108,
"split": "test",
"question": "Find the general (complex-valued) solution of the differential equation $\\dot{z}+2 z=e^{2 i t}$, using $C$ to stand for any complex-valued integration constants which may arise."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Consider the first-order system\n\\[\n\\tau \\dot{y}+y=u\n\\]\ndriven with a unit step from zero initial conditions. The input to this system is \\(u\\) and the output is \\(y\\). \n\nDerive and expression for the settling time \\(t_{s}\\), where the settling is to wi... | math | {
"style": "rule",
"ground_truth": "-\\tau \\ln \\Delta"
} | {
"index": 109,
"split": "test",
"question": "Preamble: Consider the first-order system\n\\[\n\\tau \\dot{y}+y=u\n\\]\ndriven with a unit step from zero initial conditions. The input to this system is \\(u\\) and the output is \\(y\\). \n\nDerive and expression for the settling time \\(t_{s}\\), where the settlin... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Consider the first-order system\n\\[\n\\tau \\dot{y}+y=u\n\\]\ndriven with a unit step from zero initial conditions. The input to this system is \\(u\\) and the output is \\(y\\). \n\nSubproblem 0: Derive and expression for the settling time \\(t_{s}\\), where the set... | math | {
"style": "rule",
"ground_truth": "2.2 \\tau"
} | {
"index": 110,
"split": "test",
"question": "Preamble: Consider the first-order system\n\\[\n\\tau \\dot{y}+y=u\n\\]\ndriven with a unit step from zero initial conditions. The input to this system is \\(u\\) and the output is \\(y\\). \n\nSubproblem 0: Derive and expression for the settling time \\(t_{s}\\), whe... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :\n\n$y(t)=e^{-a t}$\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{s+a}"
} | {
"index": 111,
"split": "test",
"question": "Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :\n\n$y(t)=e^{-a t}$"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For each Laplace Transform \\(Y(s)\\), find the function \\(y(t)\\) :\n\nSubproblem 0: \\[\nY(s)=\\boxed{\\frac{1}{(s+a)(s+b)}}\n\\]\n\n\nSolution: We can simplify with partial fractions:\n\\[\nY(s)=\\frac{1}{(s+a)(s+b)}=\\frac{C}{s+a}+\\frac{D}{s+b}\n\\]\nfind the co... | math | {
"style": "rule",
"ground_truth": "\\omega_{n}^{2} e^{-\\zeta \\omega_{n} t} \\cos \\left(\\omega_{n} \\sqrt{1-\\zeta^{2}} t\\right)-\\frac{\\zeta \\omega_{n}^{2}}{\\sqrt{1-\\zeta^{2}}} e^{-\\zeta \\omega_{n} t} \\sin \\left(\\omega_{n} \\sqrt{1-\\zeta^{2}} t\\right)"
} | {
"index": 112,
"split": "test",
"question": "Preamble: For each Laplace Transform \\(Y(s)\\), find the function \\(y(t)\\) :\n\nSubproblem 0: \\[\nY(s)=\\boxed{\\frac{1}{(s+a)(s+b)}}\n\\]\n\n\nSolution: We can simplify with partial fractions:\n\\[\nY(s)=\\frac{1}{(s+a)(s+b)}=\\frac{C}{s+a}+\\frac{D}{s+b}\n\\]\nf... |
math-ai/minervamath | [
{
"role": "user",
"content": "A signal \\(x(t)\\) is given by\n\\[\nx(t)=\\left(e^{-t}-e^{-1}\\right)\\left(u_{s}(t)-u_{s}(t-1)\\right)\n\\]\nCalculate its Laplace transform \\(X(s)\\). Make sure to clearly show the steps in your calculation.\n\nLet's think step by step and output the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{s+1}-\\frac{e^{-1}}{s}-\\frac{e^{-1} e^{-s}}{s+1}+\\frac{e^{-1} e^{-s}}{s}"
} | {
"index": 113,
"split": "test",
"question": "A signal \\(x(t)\\) is given by\n\\[\nx(t)=\\left(e^{-t}-e^{-1}\\right)\\left(u_{s}(t)-u_{s}(t-1)\\right)\n\\]\nCalculate its Laplace transform \\(X(s)\\). Make sure to clearly show the steps in your calculation."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: You are given an equation of motion of the form:\n\\[\n\\dot{y}+5 y=10 u\n\\]\n\nSubproblem 0: What is the time constant for this system?\n\n\nSolution: We find the homogenous solution, solving:\n\\[\n\\dot{y}+5 y=0\n\\]\nby trying a solution of the form $y=A \\cdot e... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 114,
"split": "test",
"question": "Preamble: You are given an equation of motion of the form:\n\\[\n\\dot{y}+5 y=10 u\n\\]\n\nSubproblem 0: What is the time constant for this system?\n\n\nSolution: We find the homogenous solution, solving:\n\\[\n\\dot{y}+5 y=0\n\\]\nby trying a solution of the form $y=... |
math-ai/minervamath | [
{
"role": "user",
"content": "A signal \\(w(t)\\) is defined as\n\\[\nw(t)=u_{s}(t)-u_{s}(t-T)\n\\]\nwhere \\(T\\) is a fixed time in seconds and \\(u_{s}(t)\\) is the unit step. Compute the Laplace transform \\(W(s)\\) of \\(w(t)\\). Show your work.\n\nLet's think step by step and output the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{s}-\\frac{1}{s} e^{-s T}"
} | {
"index": 115,
"split": "test",
"question": "A signal \\(w(t)\\) is defined as\n\\[\nw(t)=u_{s}(t)-u_{s}(t-T)\n\\]\nwhere \\(T\\) is a fixed time in seconds and \\(u_{s}(t)\\) is the unit step. Compute the Laplace transform \\(W(s)\\) of \\(w(t)\\). Show your work."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Assume that we apply a unit step in force separately to a mass \\(m\\), a dashpot \\(c\\), and a spring \\(k\\). The mass moves in inertial space. The spring and dashpot have one end connected to inertial space (reference velocity \\(=0\\) ), and the force is applied ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{m} t"
} | {
"index": 116,
"split": "test",
"question": "Preamble: Assume that we apply a unit step in force separately to a mass \\(m\\), a dashpot \\(c\\), and a spring \\(k\\). The mass moves in inertial space. The spring and dashpot have one end connected to inertial space (reference velocity \\(=0\\) ), and the force i... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :\n\nSubproblem 0: $y(t)=e^{-a t}$\n\n\nSolution: This function is one of the most widely used in dynamic systems, so we memorize its transform!\n\\[\nY(s)=\\boxed{\\frac{1}{s+a}}\n\\]\n\nFinal answer... | math | {
"style": "rule",
"ground_truth": "\\frac{s+\\sigma}{(s+\\sigma)^{2}+\\omega_{d}^{2}}"
} | {
"index": 117,
"split": "test",
"question": "Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :\n\nSubproblem 0: $y(t)=e^{-a t}$\n\n\nSolution: This function is one of the most widely used in dynamic systems, so we memorize its transform!\n\\[\nY(s)=\\boxed{\\frac{1}{s+a}}\n\\]\n\nFi... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :\n\nSubproblem 0: $y(t)=e^{-a t}$\n\n\nSolution: This function is one of the most widely used in dynamic systems, so we memorize its transform!\n\\[\nY(s)=\\boxed{\\frac{1}{s+a}}\n\\]\n\nFinal answer... | math | {
"style": "rule",
"ground_truth": "\\frac{\\omega_{d}}{(s+\\sigma)^{2}+\\omega_{d}^{2}}"
} | {
"index": 118,
"split": "test",
"question": "Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :\n\nSubproblem 0: $y(t)=e^{-a t}$\n\n\nSolution: This function is one of the most widely used in dynamic systems, so we memorize its transform!\n\\[\nY(s)=\\boxed{\\frac{1}{s+a}}\n\\]\n\nFi... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Consider the mass \\(m\\) sliding horizontally under the influence of the applied force \\(f\\) and a friction force which can be approximated by a linear friction element with coefficient \\(b\\). \n\nFormulate the state-determined equation of motion for the velocity... | math | {
"style": "rule",
"ground_truth": "m \\frac{d v}{d t}+b v=f"
} | {
"index": 119,
"split": "test",
"question": "Preamble: Consider the mass \\(m\\) sliding horizontally under the influence of the applied force \\(f\\) and a friction force which can be approximated by a linear friction element with coefficient \\(b\\). \n\nFormulate the state-determined equation of motion for th... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Consider the rotor with moment of inertia \\(I\\) rotating under the influence of an applied torque \\(T\\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \\(B\\).\n\nSubproblem 0: Formul... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 120,
"split": "test",
"question": "Preamble: Consider the rotor with moment of inertia \\(I\\) rotating under the influence of an applied torque \\(T\\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \\(B\\).\n\nSubproblem... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Consider the mass \\(m\\) sliding horizontally under the influence of the applied force \\(f\\) and a friction force which can be approximated by a linear friction element with coefficient \\(b\\). \n\nSubproblem 0: Formulate the state-determined equation of motion fo... | math | {
"style": "rule",
"ground_truth": "0.10"
} | {
"index": 121,
"split": "test",
"question": "Preamble: Consider the mass \\(m\\) sliding horizontally under the influence of the applied force \\(f\\) and a friction force which can be approximated by a linear friction element with coefficient \\(b\\). \n\nSubproblem 0: Formulate the state-determined equation of... |
math-ai/minervamath | [
{
"role": "user",
"content": "Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = -\\frac{(4 s-10)}{s(s+2)(s+5)}$.\nUse $u(t)$ to denote the unit step function.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1 - 3e^{-2t} + 2e^{-5t}) u(t)"
} | {
"index": 122,
"split": "test",
"question": "Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = -\\frac{(4 s-10)}{s(s+2)(s+5)}$.\nUse $u(t)$ to denote the unit step function."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "A signal has a Laplace transform\n\\[\nX(s)=b+\\frac{a}{s(s+a)}\n\\]\nwhere \\(a, b>0\\), and with a region of convergence of \\(|s|>0\\). Find \\(x(t), t>0\\).\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "b \\delta(t)+1-e^{-a t}"
} | {
"index": 123,
"split": "test",
"question": "A signal has a Laplace transform\n\\[\nX(s)=b+\\frac{a}{s(s+a)}\n\\]\nwhere \\(a, b>0\\), and with a region of convergence of \\(|s|>0\\). Find \\(x(t), t>0\\)."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For each Laplace Transform \\(Y(s)\\), find the function \\(y(t)\\) :\n\n\\[\nY(s)=\\boxed{\\frac{1}{(s+a)(s+b)}}\n\\]\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{b-a}\\left(e^{-a t}-e^{-b t}\\right)"
} | {
"index": 124,
"split": "test",
"question": "Preamble: For each Laplace Transform \\(Y(s)\\), find the function \\(y(t)\\) :\n\n\\[\nY(s)=\\boxed{\\frac{1}{(s+a)(s+b)}}\n\\]"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Consider the rotor with moment of inertia \\(I\\) rotating under the influence of an applied torque \\(T\\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \\(B\\).\n\nFormulate the state-... | math | {
"style": "rule",
"ground_truth": "I \\frac{d \\omega}{d t}+2 B \\omega=T"
} | {
"index": 125,
"split": "test",
"question": "Preamble: Consider the rotor with moment of inertia \\(I\\) rotating under the influence of an applied torque \\(T\\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \\(B\\).\n\nFormulate ... |
math-ai/minervamath | [
{
"role": "user",
"content": "Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = \\frac{4}{s^2(s^2+4)}$.\nUse $u(t)$ to denote the unit step function.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(t + \\frac{1}{2} \\sin{2t}) u(t)"
} | {
"index": 126,
"split": "test",
"question": "Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = \\frac{4}{s^2(s^2+4)}$.\nUse $u(t)$ to denote the unit step function."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$. We measure the voltage $v_{o}$ across the capacitor. $v_{i}$ and $v_{o}$ share a ground reference.\n\nCalculate the tr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{L C s^{2}+R C s+1}"
} | {
"index": 127,
"split": "test",
"question": "Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$. We measure the voltage $v_{o}$ across the capacitor. $v_{i}$ and $v_{o}$ share a ground reference.\n\nCalcul... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: You are given an equation of motion of the form:\n\\[\n\\dot{y}+5 y=10 u\n\\]\n\nWhat is the time constant for this system?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.2"
} | {
"index": 128,
"split": "test",
"question": "Preamble: You are given an equation of motion of the form:\n\\[\n\\dot{y}+5 y=10 u\n\\]\n\nWhat is the time constant for this system?"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$. We measure the voltage $v_{o}$ across the capacitor. $v_{i}$ and $v_{o}$ share a ground reference.\n\nSubproblem 0: Ca... | math | {
"style": "rule",
"ground_truth": "1e-8"
} | {
"index": 129,
"split": "test",
"question": "Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$. We measure the voltage $v_{o}$ across the capacitor. $v_{i}$ and $v_{o}$ share a ground reference.\n\nSubpro... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Here we consider a system described by the differential equation\n\\[\n\\ddot{y}+10 \\dot{y}+10000 y=0 .\n\\]\n\nWhat is the value of the natural frequency \\(\\omega_{n}\\) in radians per second?\n\nLet's think step by step and output the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 130,
"split": "test",
"question": "Preamble: Here we consider a system described by the differential equation\n\\[\n\\ddot{y}+10 \\dot{y}+10000 y=0 .\n\\]\n\nWhat is the value of the natural frequency \\(\\omega_{n}\\) in radians per second?"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Consider a circuit in which a voltage source of voltage in $v_{i}(t)$ is connected in series with an inductor $L$ and capacitor $C$. We consider the voltage across the capacitor $v_{o}(t)$ to be the output of the system.\nBoth $v_{i}(t)$ and $v_{o}(t)$ share ground r... | math | {
"style": "rule",
"ground_truth": "\\frac{d^{2} v_{o}}{d t^{2}}+\\frac{v_{o}}{L C}=\\frac{v_{i}}{L C}"
} | {
"index": 131,
"split": "test",
"question": "Preamble: Consider a circuit in which a voltage source of voltage in $v_{i}(t)$ is connected in series with an inductor $L$ and capacitor $C$. We consider the voltage across the capacitor $v_{o}(t)$ to be the output of the system.\nBoth $v_{i}(t)$ and $v_{o}(t)$ shar... |
math-ai/minervamath | [
{
"role": "user",
"content": "Write (but don't solve) the equation of motion for a pendulum consisting of a mass $m$ attached to a rigid massless rod, suspended from the ceiling and free to rotate in a single vertical plane. Let the rod (of length $l$) make an angle of $\\theta$ with the vertical. Gravity... | math | {
"style": "rule",
"ground_truth": "m l \\ddot{\\theta}(t)-m g \\sin \\theta(t)=f(t) \\cos \\theta(t)"
} | {
"index": 132,
"split": "test",
"question": "Write (but don't solve) the equation of motion for a pendulum consisting of a mass $m$ attached to a rigid massless rod, suspended from the ceiling and free to rotate in a single vertical plane. Let the rod (of length $l$) make an angle of $\\theta$ with the vertical... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Here we consider a system described by the differential equation\n\\[\n\\ddot{y}+10 \\dot{y}+10000 y=0 .\n\\]\n\nSubproblem 0: What is the value of the natural frequency \\(\\omega_{n}\\) in radians per second?\n\n\nSolution: $\\omega_{n}=\\sqrt{\\frac{k}{m}}$\nSo\n$\... | math | {
"style": "rule",
"ground_truth": "99.9"
} | {
"index": 133,
"split": "test",
"question": "Preamble: Here we consider a system described by the differential equation\n\\[\n\\ddot{y}+10 \\dot{y}+10000 y=0 .\n\\]\n\nSubproblem 0: What is the value of the natural frequency \\(\\omega_{n}\\) in radians per second?\n\n\nSolution: $\\omega_{n}=\\sqrt{\\frac{k}{m}... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Here we consider a system described by the differential equation\n\\[\n\\ddot{y}+10 \\dot{y}+10000 y=0 .\n\\]\n\nSubproblem 0: What is the value of the natural frequency \\(\\omega_{n}\\) in radians per second?\n\n\nSolution: $\\omega_{n}=\\sqrt{\\frac{k}{m}}$\nSo\n$\... | math | {
"style": "rule",
"ground_truth": "0.05"
} | {
"index": 134,
"split": "test",
"question": "Preamble: Here we consider a system described by the differential equation\n\\[\n\\ddot{y}+10 \\dot{y}+10000 y=0 .\n\\]\n\nSubproblem 0: What is the value of the natural frequency \\(\\omega_{n}\\) in radians per second?\n\n\nSolution: $\\omega_{n}=\\sqrt{\\frac{k}{m}... |
math-ai/minervamath | [
{
"role": "user",
"content": "What is the speed of light in meters/second to 1 significant figure? Use the format $a \\times 10^{b}$ where a and b are numbers.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3e8"
} | {
"index": 135,
"split": "test",
"question": "What is the speed of light in meters/second to 1 significant figure? Use the format $a \\times 10^{b}$ where a and b are numbers."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nSubproblem 0: Age of our universe when most He nuclei were formed in minutes: \n\n\nSolution: \\boxed{1} minute.\n\nFinal answer: The final answer is 1. I hope it is corre... | math | {
"style": "rule",
"ground_truth": "1e11"
} | {
"index": 136,
"split": "test",
"question": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nSubproblem 0: Age of our universe when most He nuclei were formed in minutes: \n\n\nSolution: \\boxed{1} minute.\n\nFinal answer: The final answer is 1. I hope i... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: In a parallel universe, the Boston baseball team made the playoffs.\n\nManny Relativirez hits the ball and starts running towards first base at speed $\\beta$. How fast is he running, given that he sees third base $45^{\\circ}$ to his left (as opposed to straight to h... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}c"
} | {
"index": 137,
"split": "test",
"question": "Preamble: In a parallel universe, the Boston baseball team made the playoffs.\n\nManny Relativirez hits the ball and starts running towards first base at speed $\\beta$. How fast is he running, given that he sees third base $45^{\\circ}$ to his left (as opposed to str... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: In the Sun, one of the processes in the He fusion chain is $p+p+e^{-} \\rightarrow d+\\nu$, where $d$ is a deuteron. Make the approximations that the deuteron rest mass is $2 m_{p}$, and that $m_{e} \\approx 0$ and $m_{\\nu} \\approx 0$, since both the electron and th... | math | {
"style": "rule",
"ground_truth": "m_{p} c^{2}\\left(\\gamma^{2}-1\\right) \\sin ^{2} \\theta"
} | {
"index": 138,
"split": "test",
"question": "Preamble: In the Sun, one of the processes in the He fusion chain is $p+p+e^{-} \\rightarrow d+\\nu$, where $d$ is a deuteron. Make the approximations that the deuteron rest mass is $2 m_{p}$, and that $m_{e} \\approx 0$ and $m_{\\nu} \\approx 0$, since both the elect... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: In a parallel universe, the Boston baseball team made the playoffs.\n\nSubproblem 0: Manny Relativirez hits the ball and starts running towards first base at speed $\\beta$. How fast is he running, given that he sees third base $45^{\\circ}$ to his left (as opposed to... | math | {
"style": "rule",
"ground_truth": "\\lambda_{\\text {red}} / \\sqrt{2}"
} | {
"index": 139,
"split": "test",
"question": "Preamble: In a parallel universe, the Boston baseball team made the playoffs.\n\nSubproblem 0: Manny Relativirez hits the ball and starts running towards first base at speed $\\beta$. How fast is he running, given that he sees third base $45^{\\circ}$ to his left (as ... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nSubproblem 0: Age of our universe when most He nuclei were formed in minutes: \n\n\nSolution: \\boxed{1} minute.\n\nFinal answer: The final answer is 1. I hope it is corre... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 140,
"split": "test",
"question": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nSubproblem 0: Age of our universe when most He nuclei were formed in minutes: \n\n\nSolution: \\boxed{1} minute.\n\nFinal answer: The final answer is 1. I hope i... |
math-ai/minervamath | [
{
"role": "user",
"content": "How many down quarks does a tritium ($H^3$) nucleus contain?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 141,
"split": "test",
"question": "How many down quarks does a tritium ($H^3$) nucleus contain?"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "How many up quarks does a tritium ($H^3$) nucleus contain?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 142,
"split": "test",
"question": "How many up quarks does a tritium ($H^3$) nucleus contain?"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nAge of our universe when most He nuclei were formed in minutes:\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 143,
"split": "test",
"question": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nAge of our universe when most He nuclei were formed in minutes:"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nSubproblem 0: Age of our universe when most He nuclei were formed in minutes: \n\n\nSolution: \\boxed{1} minute.\n\nFinal answer: The final answer is 1. I hope it is corre... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 144,
"split": "test",
"question": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nSubproblem 0: Age of our universe when most He nuclei were formed in minutes: \n\n\nSolution: \\boxed{1} minute.\n\nFinal answer: The final answer is 1. I hope i... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nSubproblem 0: Age of our universe when most He nuclei were formed in minutes: \n\n\nSolution: \\boxed{1} minute.\n\nFinal answer: The final answer is 1. I hope it is corre... | math | {
"style": "rule",
"ground_truth": "400000"
} | {
"index": 145,
"split": "test",
"question": "Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. \n\nSubproblem 0: Age of our universe when most He nuclei were formed in minutes: \n\n\nSolution: \\boxed{1} minute.\n\nFinal answer: The final answer is 1. I hope i... |
math-ai/minervamath | [
{
"role": "user",
"content": "Potassium metal can be used as the active surface in a photodiode because electrons are relatively easily removed from a potassium surface. The energy needed is $2.15 \\times 10^{5} J$ per mole of electrons removed ( 1 mole $=6.02 \\times 10^{23}$ electrons). What is the longes... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"index": 146,
"split": "test",
"question": "Potassium metal can be used as the active surface in a photodiode because electrons are relatively easily removed from a potassium surface. The energy needed is $2.15 \\times 10^{5} J$ per mole of electrons removed ( 1 mole $=6.02 \\times 10^{23}$ electrons). What is ... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For red light of wavelength $(\\lambda) 6.7102 \\times 10^{-5} cm$, emitted by excited lithium atoms, calculate:\n\nSubproblem 0: the frequency $(v)$ in Hz, to 4 decimal places. \n\n\nSolution: $c=\\lambda v$ and $v=c / \\lambda$ where $v$ is the frequency of radiatio... | math | {
"style": "rule",
"ground_truth": "671.02"
} | {
"index": 147,
"split": "test",
"question": "Preamble: For red light of wavelength $(\\lambda) 6.7102 \\times 10^{-5} cm$, emitted by excited lithium atoms, calculate:\n\nSubproblem 0: the frequency $(v)$ in Hz, to 4 decimal places. \n\n\nSolution: $c=\\lambda v$ and $v=c / \\lambda$ where $v$ is the frequency o... |
math-ai/minervamath | [
{
"role": "user",
"content": "What is the net charge of arginine in a solution of $\\mathrm{pH} \\mathrm{} 1.0$ ? Please format your answer as +n or -n.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "+2"
} | {
"index": 148,
"split": "test",
"question": "What is the net charge of arginine in a solution of $\\mathrm{pH} \\mathrm{} 1.0$ ? Please format your answer as +n or -n."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For red light of wavelength $(\\lambda) 6.7102 \\times 10^{-5} cm$, emitted by excited lithium atoms, calculate:\n\nSubproblem 0: the frequency $(v)$ in Hz, to 4 decimal places. \n\n\nSolution: $c=\\lambda v$ and $v=c / \\lambda$ where $v$ is the frequency of radiatio... | math | {
"style": "rule",
"ground_truth": "1.4903e4"
} | {
"index": 149,
"split": "test",
"question": "Preamble: For red light of wavelength $(\\lambda) 6.7102 \\times 10^{-5} cm$, emitted by excited lithium atoms, calculate:\n\nSubproblem 0: the frequency $(v)$ in Hz, to 4 decimal places. \n\n\nSolution: $c=\\lambda v$ and $v=c / \\lambda$ where $v$ is the frequency o... |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine the atomic weight of ${He}^{++}$ in amu to 5 decimal places from the values of its constituents.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.03188"
} | {
"index": 150,
"split": "test",
"question": "Determine the atomic weight of ${He}^{++}$ in amu to 5 decimal places from the values of its constituents."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Determine the following values from a standard radio dial. \n\nSubproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer. \n\n\nSolution: \\[\n\\mathrm{c}=v \\lambda, \\therefore \\lambda_{\\min }=\\frac{\\mathrm... | math | {
"style": "rule",
"ground_truth": "566"
} | {
"index": 151,
"split": "test",
"question": "Preamble: Determine the following values from a standard radio dial. \n\nSubproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer. \n\n\nSolution: \\[\n\\mathrm{c}=v \\lambda, \\therefore \\lambda_{\\min }=\\fra... |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine the wavelength of radiation emitted by hydrogen atoms in angstroms upon electron transitions from $n=6$ to $n=2$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4100"
} | {
"index": 152,
"split": "test",
"question": "Determine the wavelength of radiation emitted by hydrogen atoms in angstroms upon electron transitions from $n=6$ to $n=2$."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Determine the following values from a standard radio dial. \n\nSubproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer. \n\n\nSolution: \\[\n\\mathrm{c}=v \\lambda, \\therefore \\lambda_{\\min }=\\frac{\\mathrm... | math | {
"style": "rule",
"ground_truth": "2.78"
} | {
"index": 153,
"split": "test",
"question": "Preamble: Determine the following values from a standard radio dial. \n\nSubproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer. \n\n\nSolution: \\[\n\\mathrm{c}=v \\lambda, \\therefore \\lambda_{\\min }=\\fra... |
math-ai/minervamath | [
{
"role": "user",
"content": "Calculate the \"Bohr radius\" in angstroms to 3 decimal places for ${He}^{+}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.264"
} | {
"index": 154,
"split": "test",
"question": "Calculate the \"Bohr radius\" in angstroms to 3 decimal places for ${He}^{+}$."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For red light of wavelength $(\\lambda) 6.7102 \\times 10^{-5} cm$, emitted by excited lithium atoms, calculate:\n\nthe frequency $(v)$ in Hz, to 4 decimal places.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.4677"
} | {
"index": 155,
"split": "test",
"question": "Preamble: For red light of wavelength $(\\lambda) 6.7102 \\times 10^{-5} cm$, emitted by excited lithium atoms, calculate:\n\nthe frequency $(v)$ in Hz, to 4 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Electromagnetic radiation of frequency $3.091 \\times 10^{14} \\mathrm{~Hz}$ illuminates a crystal of germanium (Ge). Calculate the wavelength of photoemission in meters generated by this interaction. Germanium is an elemental semiconductor with a band gap, $E_{g}$, of $0.7 \\m... | math | {
"style": "rule",
"ground_truth": "1.77e-6"
} | {
"index": 156,
"split": "test",
"question": "Electromagnetic radiation of frequency $3.091 \\times 10^{14} \\mathrm{~Hz}$ illuminates a crystal of germanium (Ge). Calculate the wavelength of photoemission in meters generated by this interaction. Germanium is an elemental semiconductor with a band gap, $E_{g}$, o... |
math-ai/minervamath | [
{
"role": "user",
"content": "What is the energy gap (in eV, to 1 decimal place) between the electronic states $n=3$ and $n=8$ in a hydrogen atom?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.3"
} | {
"index": 157,
"split": "test",
"question": "What is the energy gap (in eV, to 1 decimal place) between the electronic states $n=3$ and $n=8$ in a hydrogen atom?"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine for hydrogen the velocity in m/s of an electron in an ${n}=4$ state. Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5.47e5"
} | {
"index": 158,
"split": "test",
"question": "Determine for hydrogen the velocity in m/s of an electron in an ${n}=4$ state. Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: A pure crystalline material (no impurities or dopants are present) appears red in transmitted light.\n\nSubproblem 0: Is this material a conductor, semiconductor or insulator? Give the reasons for your answer.\n\n\nSolution: If the material is pure (no impurity states... | math | {
"style": "rule",
"ground_truth": "1.9"
} | {
"index": 159,
"split": "test",
"question": "Preamble: A pure crystalline material (no impurities or dopants are present) appears red in transmitted light.\n\nSubproblem 0: Is this material a conductor, semiconductor or insulator? Give the reasons for your answer.\n\n\nSolution: If the material is pure (no impur... |
math-ai/minervamath | [
{
"role": "user",
"content": "Calculate the minimum potential $(V)$ in volts (to 1 decimal place) which must be applied to a free electron so that it has enough energy to excite, upon impact, the electron in a hydrogen atom from its ground state to a state of $n=5$.\n\nLet's think step by step and output th... | math | {
"style": "rule",
"ground_truth": "13.1"
} | {
"index": 160,
"split": "test",
"question": "Calculate the minimum potential $(V)$ in volts (to 1 decimal place) which must be applied to a free electron so that it has enough energy to excite, upon impact, the electron in a hydrogen atom from its ground state to a state of $n=5$."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For light with a wavelength $(\\lambda)$ of $408 \\mathrm{~nm}$ determine:\n\nSubproblem 0: the frequency in $s^{-1}$. Please format your answer as $n \\times 10^x$, where $n$ is to 3 decimal places. \n\n\nSolution: To solve this problem we must know the following rel... | math | {
"style": "rule",
"ground_truth": "4080"
} | {
"index": 161,
"split": "test",
"question": "Preamble: For light with a wavelength $(\\lambda)$ of $408 \\mathrm{~nm}$ determine:\n\nSubproblem 0: the frequency in $s^{-1}$. Please format your answer as $n \\times 10^x$, where $n$ is to 3 decimal places. \n\n\nSolution: To solve this problem we must know the fol... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Reference the information below to solve the following problems. \n$\\begin{array}{llll}\\text { Element } & \\text { Ionization Potential } & \\text { Element } & \\text { Ionization Potential } \\\\ {Na} & 5.14 & {Ca} & 6.11 \\\\ {Mg} & 7.64 & {Sc} & 6.54 \\\\ {Al}... | math | {
"style": "rule",
"ground_truth": "1.22"
} | {
"index": 162,
"split": "test",
"question": "Preamble: Reference the information below to solve the following problems. \n$\\begin{array}{llll}\\text { Element } & \\text { Ionization Potential } & \\text { Element } & \\text { Ionization Potential } \\\\ {Na} & 5.14 & {Ca} & 6.11 \\\\ {Mg} & 7.64 & {Sc} & 6.54... |
math-ai/minervamath | [
{
"role": "user",
"content": "Light of wavelength $\\lambda=4.28 \\times 10^{-7} {~m}$ interacts with a \"motionless\" hydrogen atom. During this interaction it transfers all its energy to the orbiting electron of the hydrogen. What is the velocity in m/s of this electron after interaction? Please format yo... | math | {
"style": "rule",
"ground_truth": "2.19e6"
} | {
"index": 163,
"split": "test",
"question": "Light of wavelength $\\lambda=4.28 \\times 10^{-7} {~m}$ interacts with a \"motionless\" hydrogen atom. During this interaction it transfers all its energy to the orbiting electron of the hydrogen. What is the velocity in m/s of this electron after interaction? Please... |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine the minimum potential in V (to 2 decimal places) that must be applied to an $\\alpha$-particle so that on interaction with a hydrogen atom, a ground state electron will be excited to $n$ $=6$.\n\nLet's think step by step and output the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "6.62"
} | {
"index": 164,
"split": "test",
"question": "Determine the minimum potential in V (to 2 decimal places) that must be applied to an $\\alpha$-particle so that on interaction with a hydrogen atom, a ground state electron will be excited to $n$ $=6$."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Reference the information below to solve the following problems. \n$\\begin{array}{llll}\\text { Element } & \\text { Ionization Potential } & \\text { Element } & \\text { Ionization Potential } \\\\ {Na} & 5.14 & {Ca} & 6.11 \\\\ {Mg} & 7.64 & {Sc} & 6.54 \\\\ {Al}... | math | {
"style": "rule",
"ground_truth": "0.822"
} | {
"index": 165,
"split": "test",
"question": "Preamble: Reference the information below to solve the following problems. \n$\\begin{array}{llll}\\text { Element } & \\text { Ionization Potential } & \\text { Element } & \\text { Ionization Potential } \\\\ {Na} & 5.14 & {Ca} & 6.11 \\\\ {Mg} & 7.64 & {Sc} & 6.54... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For \"yellow radiation\" (frequency, $v,=5.09 \\times 10^{14} s^{-1}$ ) emitted by activated sodium, determine:\n\nSubproblem 0: the wavelength $(\\lambda)$ in m. Please format your answer as $n \\times 10^x$, where n is to 2 decimal places.\n\n\nSolution: The equatio... | math | {
"style": "rule",
"ground_truth": "1.70e4"
} | {
"index": 166,
"split": "test",
"question": "Preamble: For \"yellow radiation\" (frequency, $v,=5.09 \\times 10^{14} s^{-1}$ ) emitted by activated sodium, determine:\n\nSubproblem 0: the wavelength $(\\lambda)$ in m. Please format your answer as $n \\times 10^x$, where n is to 2 decimal places.\n\n\nSolution: T... |
math-ai/minervamath | [
{
"role": "user",
"content": "Subproblem 0: In the balanced equation for the reaction between $\\mathrm{CO}$ and $\\mathrm{O}_{2}$ to form $\\mathrm{CO}_{2}$, what is the coefficient of $\\mathrm{CO}$?\n\n\nSolution: \\boxed{1}.\n\nFinal answer: The final answer is 1. I hope it is correct.\n\nSubproblem 1: ... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 167,
"split": "test",
"question": "Subproblem 0: In the balanced equation for the reaction between $\\mathrm{CO}$ and $\\mathrm{O}_{2}$ to form $\\mathrm{CO}_{2}$, what is the coefficient of $\\mathrm{CO}$?\n\n\nSolution: \\boxed{1}.\n\nFinal answer: The final answer is 1. I hope it is correct.\n\nSubp... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.\n\n$\\mathrm{NH}_{4} \\mathrm{OH}$\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35.06"
} | {
"index": 168,
"split": "test",
"question": "Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.\n\n$\\mathrm{NH}_{4} \\mathrm{OH}$"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Subproblem 0: In the balanced equation for the reaction between $\\mathrm{CO}$ and $\\mathrm{O}_{2}$ to form $\\mathrm{CO}_{2}$, what is the coefficient of $\\mathrm{CO}$?\n\n\nSolution: \\boxed{1}.\n\nFinal answer: The final answer is 1. I hope it is correct.\n\nSubproblem 1: ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 169,
"split": "test",
"question": "Subproblem 0: In the balanced equation for the reaction between $\\mathrm{CO}$ and $\\mathrm{O}_{2}$ to form $\\mathrm{CO}_{2}$, what is the coefficient of $\\mathrm{CO}$?\n\n\nSolution: \\boxed{1}.\n\nFinal answer: The final answer is 1. I hope it is correct.\n\nSubp... |
math-ai/minervamath | [
{
"role": "user",
"content": "Magnesium (Mg) has the following isotopic distribution:\n\\[\n\\begin{array}{ll}\n24_{\\mathrm{Mg}} & 23.985 \\mathrm{amu} \\text { at } 0.7870 \\text { fractional abundance } \\\\\n25_{\\mathrm{Mg}} & 24.986 \\mathrm{amu} \\text { at } 0.1013 \\text { fractional abundance } \\... | math | {
"style": "rule",
"ground_truth": "24.310"
} | {
"index": 170,
"split": "test",
"question": "Magnesium (Mg) has the following isotopic distribution:\n\\[\n\\begin{array}{ll}\n24_{\\mathrm{Mg}} & 23.985 \\mathrm{amu} \\text { at } 0.7870 \\text { fractional abundance } \\\\\n25_{\\mathrm{Mg}} & 24.986 \\mathrm{amu} \\text { at } 0.1013 \\text { fractional abun... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Electrons are accelerated by a potential of 10 Volts.\n\nDetermine their velocity in m/s. Please format your answer as $n \\times 10^x$, where $n$ is to 2 decimal places.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.87e6"
} | {
"index": 171,
"split": "test",
"question": "Preamble: Electrons are accelerated by a potential of 10 Volts.\n\nDetermine their velocity in m/s. Please format your answer as $n \\times 10^x$, where $n$ is to 2 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine the frequency (in $s^{-1}$ of radiation capable of generating, in atomic hydrogen, free electrons which have a velocity of $1.3 \\times 10^{6} {~ms}^{-1}$. Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places.\n\nLet's think step by step and ... | math | {
"style": "rule",
"ground_truth": "4.45e15"
} | {
"index": 172,
"split": "test",
"question": "Determine the frequency (in $s^{-1}$ of radiation capable of generating, in atomic hydrogen, free electrons which have a velocity of $1.3 \\times 10^{6} {~ms}^{-1}$. Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "In the balanced equation for the reaction between $\\mathrm{CO}$ and $\\mathrm{O}_{2}$ to form $\\mathrm{CO}_{2}$, what is the coefficient of $\\mathrm{CO}$?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 173,
"split": "test",
"question": "In the balanced equation for the reaction between $\\mathrm{CO}$ and $\\mathrm{O}_{2}$ to form $\\mathrm{CO}_{2}$, what is the coefficient of $\\mathrm{CO}$?"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Electrons are accelerated by a potential of 10 Volts.\n\nSubproblem 0: Determine their velocity in m/s. Please format your answer as $n \\times 10^x$, where $n$ is to 2 decimal places. \n\n\nSolution: The definition of an ${eV}$ is the energy gained by an electron whe... | math | {
"style": "rule",
"ground_truth": "3.89e-10"
} | {
"index": 174,
"split": "test",
"question": "Preamble: Electrons are accelerated by a potential of 10 Volts.\n\nSubproblem 0: Determine their velocity in m/s. Please format your answer as $n \\times 10^x$, where $n$ is to 2 decimal places. \n\n\nSolution: The definition of an ${eV}$ is the energy gained by an el... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: In all likelihood, the Soviet Union and the United States together in the past exploded about ten hydrogen devices underground per year.\n\nIf each explosion converted about $10 \\mathrm{~g}$ of matter into an equivalent amount of energy (a conservative estimate), how... | math | {
"style": "rule",
"ground_truth": "9e11"
} | {
"index": 175,
"split": "test",
"question": "Preamble: In all likelihood, the Soviet Union and the United States together in the past exploded about ten hydrogen devices underground per year.\n\nIf each explosion converted about $10 \\mathrm{~g}$ of matter into an equivalent amount of energy (a conservative esti... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.\n\nSubproblem 0: $\\mathrm{NH}_{4} \\mathrm{OH}$\n\n\nSolution: $\\mathrm{NH}_{4} \\mathrm{OH}$ :\n$5 \\times 1.01=5.05(\\mathrm{H})$\n$1 \\times 14.01=14.01(\\mathr... | math | {
"style": "rule",
"ground_truth": "46.08"
} | {
"index": 176,
"split": "test",
"question": "Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.\n\nSubproblem 0: $\\mathrm{NH}_{4} \\mathrm{OH}$\n\n\nSolution: $\\mathrm{NH}_{4} \\mathrm{OH}$ :\n$5 \\times 1.01=5.05(\\mathrm{H})$\n$1 \\times 14.01=14.... |
math-ai/minervamath | [
{
"role": "user",
"content": "Subproblem 0: In the balanced equation for the reaction between $\\mathrm{CO}$ and $\\mathrm{O}_{2}$ to form $\\mathrm{CO}_{2}$, what is the coefficient of $\\mathrm{CO}$?\n\n\nSolution: \\boxed{1}.\n\nFinal answer: The final answer is 1. I hope it is correct.\n\nSubproblem 1: ... | math | {
"style": "rule",
"ground_truth": "56.0"
} | {
"index": 177,
"split": "test",
"question": "Subproblem 0: In the balanced equation for the reaction between $\\mathrm{CO}$ and $\\mathrm{O}_{2}$ to form $\\mathrm{CO}_{2}$, what is the coefficient of $\\mathrm{CO}$?\n\n\nSolution: \\boxed{1}.\n\nFinal answer: The final answer is 1. I hope it is correct.\n\nSubp... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For \"yellow radiation\" (frequency, $v,=5.09 \\times 10^{14} s^{-1}$ ) emitted by activated sodium, determine:\n\nthe wavelength $(\\lambda)$ in m. Please format your answer as $n \\times 10^x$, where n is to 2 decimal places.\n\nLet's think step by step and output t... | math | {
"style": "rule",
"ground_truth": "5.89e-7"
} | {
"index": 178,
"split": "test",
"question": "Preamble: For \"yellow radiation\" (frequency, $v,=5.09 \\times 10^{14} s^{-1}$ ) emitted by activated sodium, determine:\n\nthe wavelength $(\\lambda)$ in m. Please format your answer as $n \\times 10^x$, where n is to 2 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "For a proton which has been subjected to an accelerating potential (V) of 15 Volts, determine its deBroglie wavelength in m. Please format your answer as $n \\times 10^x$, where $n$ is to 1 decimal place.\n\nLet's think step by step and output the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "7.4e-12"
} | {
"index": 179,
"split": "test",
"question": "For a proton which has been subjected to an accelerating potential (V) of 15 Volts, determine its deBroglie wavelength in m. Please format your answer as $n \\times 10^x$, where $n$ is to 1 decimal place."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For light with a wavelength $(\\lambda)$ of $408 \\mathrm{~nm}$ determine:\n\nthe frequency in $s^{-1}$. Please format your answer as $n \\times 10^x$, where $n$ is to 3 decimal places.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.353e14"
} | {
"index": 180,
"split": "test",
"question": "Preamble: For light with a wavelength $(\\lambda)$ of $408 \\mathrm{~nm}$ determine:\n\nthe frequency in $s^{-1}$. Please format your answer as $n \\times 10^x$, where $n$ is to 3 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine in units of eV (to 2 decimal places) the energy of a photon ( $h v)$ with the wavelength of $800$ nm.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.55"
} | {
"index": 181,
"split": "test",
"question": "Determine in units of eV (to 2 decimal places) the energy of a photon ( $h v)$ with the wavelength of $800$ nm."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine for barium (Ba) the linear density of atoms along the $<110>$ directions, in atoms/m.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.39e9"
} | {
"index": 182,
"split": "test",
"question": "Determine for barium (Ba) the linear density of atoms along the $<110>$ directions, in atoms/m."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "A photon with a wavelength $(\\lambda)$ of $3.091 \\times 10^{-7} {~m}$ strikes an atom of hydrogen. Determine the velocity in m/s of an electron ejected from the excited state, $n=3$. Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places.\n\nLet's thin... | math | {
"style": "rule",
"ground_truth": "9.35e5"
} | {
"index": 183,
"split": "test",
"question": "A photon with a wavelength $(\\lambda)$ of $3.091 \\times 10^{-7} {~m}$ strikes an atom of hydrogen. Determine the velocity in m/s of an electron ejected from the excited state, $n=3$. Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For the element copper (Cu) determine:\n\nthe distance of second nearest neighbors (in meters). Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.61e-10"
} | {
"index": 184,
"split": "test",
"question": "Preamble: For the element copper (Cu) determine:\n\nthe distance of second nearest neighbors (in meters). Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "A line of the Lyman series of the spectrum of hydrogen has a wavelength of $9.50 \\times 10^{-8} {~m}$. What was the \"upper\" quantum state $\\left({n}_{{i}}\\right)$ involved in the associated electron transition?\n\nLet's think step by step and output the final answer within... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 185,
"split": "test",
"question": "A line of the Lyman series of the spectrum of hydrogen has a wavelength of $9.50 \\times 10^{-8} {~m}$. What was the \"upper\" quantum state $\\left({n}_{{i}}\\right)$ involved in the associated electron transition?"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine the diffusivity $\\mathrm{D}$ of lithium ( $\\mathrm{Li}$ ) in silicon (Si) at $1200^{\\circ} \\mathrm{C}$, knowing that $D_{1100^{\\circ} \\mathrm{C}}=10^{-5} \\mathrm{~cm}^{2} / \\mathrm{s}$ and $\\mathrm{D}_{695^{\\circ} \\mathrm{C}}=10^{-6} \\mathrm{~cm}^{2} / \\m... | math | {
"style": "rule",
"ground_truth": "1.45e-5"
} | {
"index": 186,
"split": "test",
"question": "Determine the diffusivity $\\mathrm{D}$ of lithium ( $\\mathrm{Li}$ ) in silicon (Si) at $1200^{\\circ} \\mathrm{C}$, knowing that $D_{1100^{\\circ} \\mathrm{C}}=10^{-5} \\mathrm{~cm}^{2} / \\mathrm{s}$ and $\\mathrm{D}_{695^{\\circ} \\mathrm{C}}=10^{-6} \\mathrm{~cm}... |
math-ai/minervamath | [
{
"role": "user",
"content": "By planar diffusion of antimony (Sb) into p-type germanium (Ge), a p-n junction is obtained at a depth of $3 \\times 10^{-3} \\mathrm{~cm}$ below the surface. What is the donor concentration in the bulk germanium if diffusion is carried out for three hours at $790^{\\circ} \\ma... | math | {
"style": "rule",
"ground_truth": "2.88e16"
} | {
"index": 187,
"split": "test",
"question": "By planar diffusion of antimony (Sb) into p-type germanium (Ge), a p-n junction is obtained at a depth of $3 \\times 10^{-3} \\mathrm{~cm}$ below the surface. What is the donor concentration in the bulk germanium if diffusion is carried out for three hours at $790^{\\... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: One mole of electromagnetic radiation (light, consisting of energy packages called photons) has an energy of $171 \\mathrm{~kJ} /$ mole photons.\n\nDetermine the wavelength of this light in nm.\n\nLet's think step by step and output the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "700"
} | {
"index": 188,
"split": "test",
"question": "Preamble: One mole of electromagnetic radiation (light, consisting of energy packages called photons) has an energy of $171 \\mathrm{~kJ} /$ mole photons.\n\nDetermine the wavelength of this light in nm."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Two lasers generate radiation of (1) $9.5 \\mu {m}$ and (2) $0.1 \\mu {m}$ respectively.\n\nDetermine the photon energy (in eV, to two decimal places) of the laser generating radiation of $9.5 \\mu {m}$.\n\nLet's think step by step and output the final answer within \... | math | {
"style": "rule",
"ground_truth": "0.13"
} | {
"index": 189,
"split": "test",
"question": "Preamble: Two lasers generate radiation of (1) $9.5 \\mu {m}$ and (2) $0.1 \\mu {m}$ respectively.\n\nDetermine the photon energy (in eV, to two decimal places) of the laser generating radiation of $9.5 \\mu {m}$."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "At $100^{\\circ} \\mathrm{C}$ copper $(\\mathrm{Cu})$ has a lattice constant of $3.655 \\AA$. What is its density in $g/cm^3$ at this temperature? Please round your answer to 2 decimal places.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8.64"
} | {
"index": 190,
"split": "test",
"question": "At $100^{\\circ} \\mathrm{C}$ copper $(\\mathrm{Cu})$ has a lattice constant of $3.655 \\AA$. What is its density in $g/cm^3$ at this temperature? Please round your answer to 2 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine the atomic (metallic) radius of Mo in meters. Do not give the value listed in the periodic table; calculate it from the fact that Mo's atomic weight is $=95.94 \\mathrm{~g} /$ mole and $\\rho=10.2 \\mathrm{~g} / \\mathrm{cm}^{3}$. Please format your answer as $n \\tim... | math | {
"style": "rule",
"ground_truth": "1.39e-10"
} | {
"index": 191,
"split": "test",
"question": "Determine the atomic (metallic) radius of Mo in meters. Do not give the value listed in the periodic table; calculate it from the fact that Mo's atomic weight is $=95.94 \\mathrm{~g} /$ mole and $\\rho=10.2 \\mathrm{~g} / \\mathrm{cm}^{3}$. Please format your answer a... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Determine the following values from a standard radio dial. \n\nWhat is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "188"
} | {
"index": 192,
"split": "test",
"question": "Preamble: Determine the following values from a standard radio dial. \n\nWhat is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Consider a (111) plane in an FCC structure. How many different [110]-type directions lie in this (111) plane?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 193,
"split": "test",
"question": "Consider a (111) plane in an FCC structure. How many different [110]-type directions lie in this (111) plane?"
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Determine the velocity of an electron (in $\\mathrm{m} / \\mathrm{s}$ ) that has been subjected to an accelerating potential $V$ of 150 Volt. Please format your answer as $n \\times 10^x$, where $n$ is to 2 decimal places. \n(The energy imparted to an electron by an acceleratin... | math | {
"style": "rule",
"ground_truth": "7.26e6"
} | {
"index": 194,
"split": "test",
"question": "Determine the velocity of an electron (in $\\mathrm{m} / \\mathrm{s}$ ) that has been subjected to an accelerating potential $V$ of 150 Volt. Please format your answer as $n \\times 10^x$, where $n$ is to 2 decimal places. \n(The energy imparted to an electron by an a... |
math-ai/minervamath | [
{
"role": "user",
"content": "In a diffractometer experiment a specimen of thorium (Th) is irradiated with tungsten (W) $L_{\\alpha}$ radiation. Calculate the angle, $\\theta$, of the $4^{\\text {th }}$ reflection. Round your answer (in degrees) to 2 decimal places.\n\nLet's think step by step and output th... | math | {
"style": "rule",
"ground_truth": "28.71"
} | {
"index": 195,
"split": "test",
"question": "In a diffractometer experiment a specimen of thorium (Th) is irradiated with tungsten (W) $L_{\\alpha}$ radiation. Calculate the angle, $\\theta$, of the $4^{\\text {th }}$ reflection. Round your answer (in degrees) to 2 decimal places."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "A metal is found to have BCC structure, a lattice constant of $3.31 \\AA$, and a density of $16.6 \\mathrm{~g} / \\mathrm{cm}^{3}$. Determine the atomic weight of this element in g/mole, and round your answer to 1 decimal place.\n\nLet's think step by step and output the final ... | math | {
"style": "rule",
"ground_truth": "181.3"
} | {
"index": 196,
"split": "test",
"question": "A metal is found to have BCC structure, a lattice constant of $3.31 \\AA$, and a density of $16.6 \\mathrm{~g} / \\mathrm{cm}^{3}$. Determine the atomic weight of this element in g/mole, and round your answer to 1 decimal place."
} |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: Iron $\\left(\\rho=7.86 \\mathrm{~g} / \\mathrm{cm}^{3}\\right.$ ) crystallizes in a BCC unit cell at room temperature.\n\nCalculate the radius in cm of an iron atom in this crystal. Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places.\n\nLe... | math | {
"style": "rule",
"ground_truth": "1.24e-8"
} | {
"index": 197,
"split": "test",
"question": "Preamble: Iron $\\left(\\rho=7.86 \\mathrm{~g} / \\mathrm{cm}^{3}\\right.$ ) crystallizes in a BCC unit cell at room temperature.\n\nCalculate the radius in cm of an iron atom in this crystal. Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal pla... |
math-ai/minervamath | [
{
"role": "user",
"content": "Preamble: For the element copper (Cu) determine:\n\nSubproblem 0: the distance of second nearest neighbors (in meters). Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places.\n\n\nSolution: The answer can be found by looking at a unit cell of $\\mathrm{... | math | {
"style": "rule",
"ground_truth": "2.55e-10"
} | {
"index": 198,
"split": "test",
"question": "Preamble: For the element copper (Cu) determine:\n\nSubproblem 0: the distance of second nearest neighbors (in meters). Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places.\n\n\nSolution: The answer can be found by looking at a unit cell of ... |
math-ai/minervamath | [
{
"role": "user",
"content": "Subproblem 0: What is the working temperature for silica glass in Celsius?\n\n\nSolution: \\boxed{1950}.\n\nFinal answer: The final answer is 1950. I hope it is correct.\n\nSubproblem 1: What is the softening temperature for silica glass in Celsius?\n\n\nSolution: \\boxed{1700}... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"index": 199,
"split": "test",
"question": "Subproblem 0: What is the working temperature for silica glass in Celsius?\n\n\nSolution: \\boxed{1950}.\n\nFinal answer: The final answer is 1950. I hope it is correct.\n\nSubproblem 1: What is the softening temperature for silica glass in Celsius?\n\n\nSolution: \\b... |
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