data_source stringclasses 1
value | answer stringlengths 1 204 | source_dataset stringclasses 4
values | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|---|
numina_math | 4.5e33 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A white dwarf star has an effective temperature, $T_{e}=50,000$ degrees Kelvin, but its radius, $R_{\\mathrm{WD}}$, is comparable to that of the Earth. Take $R_{\\mathrm{WD}}=10^... | math | {
"ground_truth": "4.5e33",
"style": "rule"
} | {
"index": 500,
"split": "test"
} |
numina_math | 41.8 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A prism is constructed from glass and has sides that form a right triangle with the other two angles equal to $45^{\\circ}$. The sides are $L, L$, and $H$, where $L$ is... | math | {
"ground_truth": "41.8",
"style": "rule"
} | {
"index": 501,
"split": "test"
} |
numina_math | 20.39 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A particular star has an absolute magnitude $M=-7$. If this star is observed in a galaxy that is at a distance of $3 \\mathrm{Mpc}$, what will its apparent magnitude be? Let's th... | math | {
"ground_truth": "20.39",
"style": "rule"
} | {
"index": 502,
"split": "test"
} |
numina_math | 0.006 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the gravitational acceleration due to the Sun at the location of the Earth's orbit (i.e., at a distance of $1 \\mathrm{AU}$ ). Give your answer in meters per second squared... | math | {
"ground_truth": "0.006",
"style": "rule"
} | {
"index": 503,
"split": "test"
} |
numina_math | np.arcsin(10/13) | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A collimated light beam propagating in water is incident on the surface (air/water interface) at an angle $\\theta_w$ with respect to the surface normal.\n\nSubproblem ... | math | {
"ground_truth": "np.arcsin(10/13)",
"style": "rule"
} | {
"index": 504,
"split": "test"
} |
numina_math | 0.49 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the theoretical limiting angular resolution (in arcsec) of a commercial 8-inch (diameter) optical telescope being used in the visible spectrum (at $\\lambda=5000 \\AA=500 \\... | math | {
"ground_truth": "0.49",
"style": "rule"
} | {
"index": 505,
"split": "test"
} |
numina_math | 100 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A star has a measured parallax of $0.01^{\\prime \\prime}$, that is, $0.01$ arcseconds. How far away is it, in parsecs? Let's think step by step and output the final answer withi... | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 506,
"split": "test"
} |
numina_math | 0.022 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An extrasolar planet has been observed which passes in front of (i.e., transits) its parent star. If the planet is dark (i.e., contributes essentially no light of its own) and ha... | math | {
"ground_truth": "0.022",
"style": "rule"
} | {
"index": 507,
"split": "test"
} |
numina_math | 9.6 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the Bohr energy levels scale as $Z^{2}$, where $Z$ is the atomic number of the atom (i.e., the charge on the nucleus), estimate the wavelength of a photon that results from a ... | math | {
"ground_truth": "9.6",
"style": "rule"
} | {
"index": 508,
"split": "test"
} |
numina_math | 3.83e35 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the Sun's absolute magnitude is $+5$, find the luminosity of a star of magnitude $0$ in ergs/s. A useful constant: the luminosity of the sun is $3.83 \\times 10^{33}$ ergs/s. ... | math | {
"ground_truth": "3.83e35",
"style": "rule"
} | {
"index": 509,
"split": "test"
} |
numina_math | 300 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A spectrum is taken of a single star (i.e., one not in a binary). Among the observed spectral lines is one from oxygen whose rest wavelength is $5007 \\AA$. The Doppler... | math | {
"ground_truth": "300",
"style": "rule"
} | {
"index": 510,
"split": "test"
} |
numina_math | \frac{2\pic^{2}R^{2}}{\lambda^{5}[e^{hc/(\lambdakT)}-1]d^{2}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The differential luminosity from a star, $\\Delta L$, with an approximate blackbody spectrum, is given by:\n\\[\n\\Delta L=\\frac{8 \\pi^{2} c^{2} R^{2}}{\\lambda^{5}\\left[e^{h ... | math | {
"ground_truth": "\\frac{2\\pic^{2}R^{2}}{\\lambda^{5}[e^{hc/(\\lambdakT)}-1]d^{2}}",
"style": "rule"
} | {
"index": 511,
"split": "test"
} |
numina_math | 8.7e8 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \\mathrm{kpc}$. The star has a temperature $T = 6 \\times 10^{5} K$ and produces a flux of $... | math | {
"ground_truth": "8.7e8",
"style": "rule"
} | {
"index": 512,
"split": "test"
} |
numina_math | 0.003 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A star is at a distance from the Earth of $300 \\mathrm{pc}$. Find its parallax angle, $\\pi$, in arcseconds to one significant figure. Let's think step by step and output the fi... | math | {
"ground_truth": "0.003",
"style": "rule"
} | {
"index": 513,
"split": "test"
} |
numina_math | 4e33 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The Sun's effective temperature, $T_{e}$, is 5800 Kelvin, and its radius is $7 \\times 10^{10} \\mathrm{~cm}\\left(7 \\times 10^{8}\\right.$ m). Compute the luminosity (power out... | math | {
"ground_truth": "4e33",
"style": "rule"
} | {
"index": 514,
"split": "test"
} |
numina_math | 4.49 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Use the Bohr model of the atom to compute the wavelength of the transition from the $n=100$ to $n=99$ levels, in centimeters. [Uscful relation: the wavelength of $L \\alpha$ ( $\... | math | {
"ground_truth": "4.49",
"style": "rule"
} | {
"index": 515,
"split": "test"
} |
numina_math | 0.01 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A radio interferometer, operating at a wavelength of $1 \\mathrm{~cm}$, consists of 100 small dishes, each $1 \\mathrm{~m}$ in diameter, distributed randomly within a $... | math | {
"ground_truth": "0.01",
"style": "rule"
} | {
"index": 516,
"split": "test"
} |
numina_math | 3.3e12 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Orbital Dynamics: A binary system consists of two stars in circular orbit about a common center of mass, with an orbital period, $P_{\\text {orb }}=10$ days. Star 1 is ... | math | {
"ground_truth": "3.3e12",
"style": "rule"
} | {
"index": 517,
"split": "test"
} |
numina_math | 10 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If a star cluster is made up of $10^{4}$ stars, each of whose absolute magnitude is $-5$, compute the combined apparent magnitude of the cluster if it is located at a distance of... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 518,
"split": "test"
} |
numina_math | 6630 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A galaxy moves directly away from us with a speed of $3000 \\mathrm{~km} \\mathrm{~s}^{-1}$. Find the wavelength of the $\\mathrm{H} \\alpha$ line observed at the Earth, in Angst... | math | {
"ground_truth": "6630",
"style": "rule"
} | {
"index": 519,
"split": "test"
} |
numina_math | 1.2 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The Spitzer Space Telescope has an effective diameter of $85 \\mathrm{cm}$, and a typical wavelength used for observation of $5 \\mu \\mathrm{m}$, or 5 microns. Based on this inf... | math | {
"ground_truth": "1.2",
"style": "rule"
} | {
"index": 520,
"split": "test"
} |
numina_math | 3e6 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "It has long been suspected that there is a massive black hole near the center of our Galaxy. Recently, a group of astronmers determined the parameters of a star that is orbiting ... | math | {
"ground_truth": "3e6",
"style": "rule"
} | {
"index": 521,
"split": "test"
} |
numina_math | 7e37 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \\mathrm{kpc}$. The star has a temperature $T = 6 \\times 10^{5} K$ and produces a flux of $... | math | {
"ground_truth": "7e37",
"style": "rule"
} | {
"index": 522,
"split": "test"
} |
numina_math | 50 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A large ground-based telescope has an effective focal length of 10 meters. Two astronomical objects are separated by 1 arc second in the sky. How far apart will the two correspon... | math | {
"ground_truth": "50",
"style": "rule"
} | {
"index": 523,
"split": "test"
} |
numina_math | \frac{aM^{1/3}}{GM^{2/3}+b} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The equation of state for cold (non-relativistic) matter may be approximated as:\n\\[\nP=a \\rho^{5 / 3}-b \\rho^{4 / 3}\n\\]\nwhere $P$ is the pressure, $\\rho$ the density, and... | math | {
"ground_truth": "\\frac{aM^{1/3}}{GM^{2/3}+b}",
"style": "rule"
} | {
"index": 524,
"split": "test"
} |
numina_math | 7.5e7 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Take the total energy (potential plus thermal) of the Sun to be given by the simple expression:\n\\[\nE \\simeq-\\frac{G M^{2}}{R}\n\\]\nwhere $M$ and $R$ are the mass and radius... | math | {
"ground_truth": "7.5e7",
"style": "rule"
} | {
"index": 525,
"split": "test"
} |
numina_math | \frac{dM}{dt}=\frac{10^{5}L_{\odot}}{0.007c^{2}M_{\odot}^{6}}M^{6} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Once a star like the Sun starts to ascend the giant branch its luminosity, to a good approximation, is given by:\n\\[\nL=\\frac{10^{5} L_{\\odot}}{M_{\\odot}^{6}} M_{\\... | math | {
"ground_truth": "\\frac{dM}{dt}=\\frac{10^{5}L_{\\odot}}{0.007c^{2}M_{\\odot}^{6}}M^{6}",
"style": "rule"
} | {
"index": 526,
"split": "test"
} |
numina_math | [P_{0}^{2/5}-\frac{2}{5}gK^{-3/5}z]^{5/2} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A star of radius, $R$, and mass, $M$, has an atmosphere that obeys a polytropic equation of state:\n\\[\nP=K \\rho^{5 / 3} \\text {, }\n\\]\nwhere $P$ is the gas pressure, $\\rho... | math | {
"ground_truth": "[P_{0}^{2/5}-\\frac{2}{5}gK^{-3/5}z]^{5/2}",
"style": "rule"
} | {
"index": 527,
"split": "test"
} |
numina_math | 1.75 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An eclipsing binary consists of two stars of different radii and effective temperatures. Star 1 has radius $R_{1}$ and $T_{1}$, and Star 2 has $R_{2}=0.5 R_{1}$ and $T_{2}=2 T_{1... | math | {
"ground_truth": "1.75",
"style": "rule"
} | {
"index": 528,
"split": "test"
} |
numina_math | \sqrt{4\piG\rho_{0}r_{0}^{2}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: It has been suggested that our Galaxy has a spherically symmetric dark-matter halo with a density distribution, $\\rho_{\\text {dark }}(r)$, given by:\n\\[\n\\rho_{\\te... | math | {
"ground_truth": "\\sqrt{4\\piG\\rho_{0}r_{0}^{2}}",
"style": "rule"
} | {
"index": 529,
"split": "test"
} |
numina_math | 0.33 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The Very Large Array (VLA) telescope has an effective diameter of $36 \\mathrm{~km}$, and a typical wavelength used for observation at this facility might be $6 \\mathrm{~cm}$. ... | math | {
"ground_truth": "0.33",
"style": "rule"
} | {
"index": 530,
"split": "test"
} |
numina_math | 27.39 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: A particular star has an absolute magnitude $M=-7$. If this star is observed in a galaxy that is at a distance of $3 \\mathrm{Mpc}$, what will its apparent magnitud... | math | {
"ground_truth": "27.39",
"style": "rule"
} | {
"index": 531,
"split": "test"
} |
numina_math | 24.4 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the distance modulus to the Andromeda galaxy (M31). Take the distance to Andromeda to be $750 \\mathrm{kpc}$, and answer to three significant figures. Let's think step by st... | math | {
"ground_truth": "24.4",
"style": "rule"
} | {
"index": 532,
"split": "test"
} |
numina_math | 0.05 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The Hubble Space telescope has an effective diameter of $2.5 \\mathrm{~m}$, and a typical wavelength used for observation by the Hubble might be $0.6 \\mu \\mathrm{m}$, or 600 na... | math | {
"ground_truth": "0.05",
"style": "rule"
} | {
"index": 533,
"split": "test"
} |
numina_math | \arcsin{1.3\sin{\theta_w}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A collimated light beam propagating in water is incident on the surface (air/water interface) at an angle $\\theta_w$ with respect to the surface normal.\n\nIf the inde... | math | {
"ground_truth": "\\arcsin{1.3\\sin{\\theta_w}}",
"style": "rule"
} | {
"index": 534,
"split": "test"
} |
numina_math | 0.15 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What fraction of the rest mass energy is released (in the form of radiation) when a mass $\\Delta M$ is dropped from infinity onto the surface of a neutron star with $M=1 M_{\\od... | math | {
"ground_truth": "0.15",
"style": "rule"
} | {
"index": 535,
"split": "test"
} |
numina_math | 2e27 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The density of stars in a particular globular star cluster is $10^{6} \\mathrm{pc}^{-3}$. Take the stars to have the same radius as the Sun, and to have an average spee... | math | {
"ground_truth": "2e27",
"style": "rule"
} | {
"index": 536,
"split": "test"
} |
numina_math | -1./3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For a gas supported by degenerate electron pressure, the pressure is given by:\n\\[\nP=K \\rho^{5 / 3}\n\\]\nwhere $K$ is a constant and $\\rho$ is the mass density. If a star is... | math | {
"ground_truth": "-1./3",
"style": "rule"
} | {
"index": 537,
"split": "test"
} |
numina_math | 0.033 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A galaxy moves directly away from us with speed $v$, and the wavelength of its $\\mathrm{H} \\alpha$ line is observed to be $6784 \\AA$. The rest wavelength of $\\mathrm{H} \\alp... | math | {
"ground_truth": "0.033",
"style": "rule"
} | {
"index": 538,
"split": "test"
} |
numina_math | 0.5613 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A candle has a power in the visual band of roughly $3$ Watts. When this candle is placed at a distance of $3 \\mathrm{~km}$ it has the same apparent brightness as a certain star.... | math | {
"ground_truth": "0.5613",
"style": "rule"
} | {
"index": 539,
"split": "test"
} |
numina_math | \frac{v_{0}}{r_{0}}\frac{1}{(1+r/r_{0})^{3/2}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A galaxy is found to have a rotation curve, $v(r)$, given by\n\\[\nv(r)=\\frac{\\left(\\frac{r}{r_{0}}\\right)}{\\left(1+\\frac{r}{r_{0}}\\right)^{3 / 2}} v_{0}\n\\]\nw... | math | {
"ground_truth": "\\frac{v_{0}}{r_{0}}\\frac{1}{(1+r/r_{0})^{3/2}}",
"style": "rule"
} | {
"index": 540,
"split": "test"
} |
numina_math | 2.75e11 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Orbital Dynamics: A binary system consists of two stars in circular orbit about a common center of mass, with an orbital period, $P_{\\text {orb }}=10$ days. Star 1 is ... | math | {
"ground_truth": "2.75e11",
"style": "rule"
} | {
"index": 541,
"split": "test"
} |
numina_math | 6e13 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The density of stars in a particular globular star cluster is $10^{6} \\mathrm{pc}^{-3}$. Take the stars to have the same radius as the Sun, and to have an average spee... | math | {
"ground_truth": "6e13",
"style": "rule"
} | {
"index": 542,
"split": "test"
} |
numina_math | 1e-5 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A radio interferometer, operating at a wavelength of $1 \\mathrm{~cm}$, consists of 100 small dishes, each $1 \\mathrm{~m}$ in diameter, distributed randomly within a $... | math | {
"ground_truth": "1e-5",
"style": "rule"
} | {
"index": 543,
"split": "test"
} |
numina_math | -10 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If a star cluster is made up of $10^{6}$ stars whose absolute magnitude is the same as that of the Sun (+5), compute the combined magnitude of the cluster if it is located at a d... | math | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": 544,
"split": "test"
} |
numina_math | 15625 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A certain red giant has a radius that is 500 times that of the Sun, and a temperature that is $1 / 2$ that of the Sun's temperature. Find its bolometric (total) luminosity in uni... | math | {
"ground_truth": "15625",
"style": "rule"
} | {
"index": 545,
"split": "test"
} |
numina_math | 1e-3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose air molecules have a collision cross section of $10^{-16} \\mathrm{~cm}^{2}$. If the (number) density of air molecules is $10^{19} \\mathrm{~cm}^{-3}$, what is the collis... | math | {
"ground_truth": "1e-3",
"style": "rule"
} | {
"index": 546,
"split": "test"
} |
numina_math | 3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two stars have the same surface temperature. Star 1 has a radius that is $2.5$ times larger than the radius of star 2. Star 1 is ten times farther away than star 2. What is the a... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 547,
"split": "test"
} |
numina_math | -3./2 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the slope of a $\\log N(>F)$ vs. $\\log F$ curve for a homogeneous distribution of objects, each of luminosity, $L$, where $F$ is the flux at the observer, and $N$ is the... | math | {
"ground_truth": "-3./2",
"style": "rule"
} | {
"index": 548,
"split": "test"
} |
numina_math | 2.7778e-6 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Comparison of Radio and Optical Telescopes.\n\nThe Very Large Array (VLA) is used to make an interferometric map of the Orion Nebula at a wavelength of $10 \\mathrm{~cm... | math | {
"ground_truth": "2.7778e-6",
"style": "rule"
} | {
"index": 549,
"split": "test"
} |
numina_math | -7 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A globular cluster has $10^{6}$ stars each of apparent magnitude $+8$. What is the combined apparent magnitude of the entire cluster? Let's think step by step and output the fina... | math | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": 550,
"split": "test"
} |
numina_math | 48 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \\mathrm{kpc}$. The star has a temperature $T = 6 \\times 10^{5} K$ and produces a flux of $... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 551,
"split": "test"
} |
numina_math | 16 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A Boolean function $F(A, B)$ is said to be universal if any arbitrary boolean function can be constructed by using nested $F(A, B)$ functions. A universal function is useful, sin... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 552,
"split": "test"
} |
numina_math | 0.192 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Unfortunately, a mutant gene can turn box people into triangles late in life. A laboratory test has been developed which can spot the gene early so that the dreaded triangle tran... | math | {
"ground_truth": "0.192",
"style": "rule"
} | {
"index": 553,
"split": "test"
} |
numina_math | 0.5978 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Buzz, the hot new dining spot on campus, emphasizes simplicity. It only has two items on the menu, burgers and zucchini. Customers make a choice as they enter (they are not allow... | math | {
"ground_truth": "0.5978",
"style": "rule"
} | {
"index": 554,
"split": "test"
} |
numina_math | 11000 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Given the following data from an Experimental Forest, answer the following questions. Show your work and units.\n$\\begin{array}{ll}\\text { Total vegetative biomass } ... | math | {
"ground_truth": "11000",
"style": "rule"
} | {
"index": 555,
"split": "test"
} |
numina_math | 0.53 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A population of 100 ferrets is introduced to a large island in the beginning of 1990 . Ferrets have an intrinsic growth rate, $r_{\\max }$ of $1.3 \\mathrm{yr}^{-1}$.\n... | math | {
"ground_truth": "0.53",
"style": "rule"
} | {
"index": 556,
"split": "test"
} |
numina_math | 15000 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Given the following data from an Experimental Forest, answer the following questions. Show your work and units.\n$\\begin{array}{ll}\\text { Total vegetative biomass } ... | math | {
"ground_truth": "15000",
"style": "rule"
} | {
"index": 557,
"split": "test"
} |
numina_math | 1538 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The Peak District Moorlands in the United Kingdom store 20 million tonnes of carbon, almost half of the carbon stored in the soils of the entire United Kingdom (the Moo... | math | {
"ground_truth": "1538",
"style": "rule"
} | {
"index": 558,
"split": "test"
} |
numina_math | 4.4e7 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A population of 100 ferrets is introduced to a large island in the beginning of 1990 . Ferrets have an intrinsic growth rate, $r_{\\max }$ of $1.3 \\mathrm{yr}^{-1}$.\n... | math | {
"ground_truth": "4.4e7",
"style": "rule"
} | {
"index": 559,
"split": "test"
} |
numina_math | I(0) | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The following subproblems refer to a circuit with the following parameters. Denote by $I(t)$ the current (where the positive direction is, say, clockwise) in the circui... | math | {
"ground_truth": "I(0)",
"style": "rule"
} | {
"index": 560,
"split": "test"
} |
numina_math | y^{\prime}+ry-rx(t)=0 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the following \"mixing problem.\" A tank holds $V$ liters of salt water. Suppose that a saline solution with concentration of $c \\mathrm{gm} /$ liter is added at the ra... | math | {
"ground_truth": "y^{\\prime}+ry-rx(t)=0",
"style": "rule"
} | {
"index": 561,
"split": "test"
} |
numina_math | cx^{-2}-\frac{\cos(2x)}{2x^{2}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the general solution of $x^{2} y^{\\prime}+2 x y=\\sin (2 x)$, solving for $y$. Note that a general solution to a differential equation has the form $x=x_{p}+c x_{h}$ where ... | math | {
"ground_truth": "cx^{-2}-\\frac{\\cos(2x)}{2x^{2}}",
"style": "rule"
} | {
"index": 562,
"split": "test"
} |
numina_math | \frac{dx}{dt}=kx-a | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An African government is trying to come up with good policy regarding the hunting of oryx. They are using the following model: the oryx population has a natural growth rate of $k... | math | {
"ground_truth": "\\frac{dx}{dt}=kx-a",
"style": "rule"
} | {
"index": 563,
"split": "test"
} |
numina_math | 4 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the complex number $z$ is given by $z = 1+\\sqrt{3} i$, what is the magnitude of $z^2$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "u... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 564,
"split": "test"
} |
numina_math | 2 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the polar representation $(r, \\theta)$ of the complex number $z=1+\\sqrt{3} i$, what is $r$? Let's think step by step and output the final answer within \\boxed{}.",
"rol... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 565,
"split": "test"
} |
numina_math | 1+\sqrt{3}i | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: In the following problems, take $a = \\ln 2$ and $b = \\pi / 3$. \n\nGiven $a = \\ln 2$ and $b = \\pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are r... | math | {
"ground_truth": "1+\\sqrt{3}i",
"style": "rule"
} | {
"index": 566,
"split": "test"
} |
numina_math | 0 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: Find the general solution of the differential equation $y^{\\prime}=x-2 y$ analytically using integrating factors, solving for $y$. Note that a function $u(t)$ such... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 567,
"split": "test"
} |
numina_math | -0.875 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The following subproblems relate to applying Euler's Method (a first-order numerical procedure for solving ordinary differential equations with a given initial value) o... | math | {
"ground_truth": "-0.875",
"style": "rule"
} | {
"index": 568,
"split": "test"
} |
numina_math | \sqrt{2}\cos(2t-\pi/4) | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Rewrite the function $f(t) = \\cos (2 t)+\\sin (2 t)$ in the form $A \\cos (\\omega t-\\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$. Let's thin... | math | {
"ground_truth": "\\sqrt{2}\\cos(2t-\\pi/4)",
"style": "rule"
} | {
"index": 569,
"split": "test"
} |
numina_math | \frac{1}{2a}(\exp{a*t}-\exp{-a*t}) | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the ordinary differential equation $\\ddot{x}-a^{2} x=0$, where $a$ is a nonzero real-valued constant, find a solution $x(t)$ to this equation such that $x(0) = 0$ and $\\d... | math | {
"ground_truth": "\\frac{1}{2a}(\\exp{a*t}-\\exp{-a*t})",
"style": "rule"
} | {
"index": 570,
"split": "test"
} |
numina_math | x_{0}\cos(\omegat)+\dot{x}_{0}\sin(\omegat)/\omega | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find a solution to the differential equation $\\ddot{x}+\\omega^{2} x=0$ satisfying the initial conditions $x(0)=x_{0}$ and $\\dot{x}(0)=\\dot{x}_{0}$. Let's think step by step a... | math | {
"ground_truth": "x_{0}\\cos(\\omegat)+\\dot{x}_{0}\\sin(\\omegat)/\\omega",
"style": "rule"
} | {
"index": 571,
"split": "test"
} |
numina_math | \ln2+i\pi/3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the complex number $a+b i$ with the smallest possible positive $b$ such that $e^{a+b i}=1+\\sqrt{3} i$. Let's think step by step and output the final answer within \\boxed{}... | math | {
"ground_truth": "\\ln2+i\\pi/3",
"style": "rule"
} | {
"index": 572,
"split": "test"
} |
numina_math | e^{t}/3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: Find the general solution of the differential equation $\\dot{x}+2 x=e^{t}$, using $c$ for the arbitrary constant of integration which will occur.\n\n\nSolution: We... | math | {
"ground_truth": "e^{t}/3",
"style": "rule"
} | {
"index": 573,
"split": "test"
} |
numina_math | 2 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: For $\\omega \\geq 0$, find $A$ such that $A \\cos (\\omega t)$ is a solution of $\\ddot{x}+4 x=\\cos (\\omega t)$.\n\n\nSolution: If $x=A \\cos (\\omega t)$, then ... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 574,
"split": "test"
} |
numina_math | \frac{\cos(2t)}{15}+C_{1}e^{t}+C_{2}e^{-t}+C_{3}\cos(t)+C_{4}\sin(t) | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: Find a purely sinusoidal solution of $\\frac{d^{4} x}{d t^{4}}-x=\\cos (2 t)$.\n\n\nSolution: We choose an exponential input function whose real part is $\\cos (2 t... | math | {
"ground_truth": "\\frac{\\cos(2t)}{15}+C_{1}e^{t}+C_{2}e^{-t}+C_{3}\\cos(t)+C_{4}\\sin(t)",
"style": "rule"
} | {
"index": 575,
"split": "test"
} |
numina_math | \frac{1}{4-\omega^{2}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For $\\omega \\geq 0$, find $A$ such that $A \\cos (\\omega t)$ is a solution of $\\ddot{x}+4 x=\\cos (\\omega t)$. Let's think step by step and output the final answer within \\... | math | {
"ground_truth": "\\frac{1}{4-\\omega^{2}}",
"style": "rule"
} | {
"index": 576,
"split": "test"
} |
numina_math | \frac{\cos(2t)+\sin(2t)}{4} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find a solution to $\\dot{x}+2 x=\\cos (2 t)$ in the form $k_0\\left[f(k_1t) + g(k_2t)\\right]$, where $f, g$ are trigonometric functions. Do not include homogeneous solutions t... | math | {
"ground_truth": "\\frac{\\cos(2t)+\\sin(2t)}{4}",
"style": "rule"
} | {
"index": 577,
"split": "test"
} |
numina_math | -0.2 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The following subproblems refer to the differential equation. $\\ddot{x}+4 x=\\sin (3 t)$\n\nFind $A$ so that $A \\sin (3 t)$ is a solution of $\\ddot{x}+4 x=\\sin (3 t... | math | {
"ground_truth": "-0.2",
"style": "rule"
} | {
"index": 578,
"split": "test"
} |
numina_math | x/2-1/4+ce^{-2x} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the general solution of the differential equation $y^{\\prime}=x-2 y$ analytically using integrating factors, solving for $y$. Note that a function $u(t)$ such that $u \\dot... | math | {
"ground_truth": "x/2-1/4+ce^{-2x}",
"style": "rule"
} | {
"index": 579,
"split": "test"
} |
numina_math | \frac{e^{-2t}}{15}+C_{1}e^{t}+C_{2}e^{-t}+C_{3}\cos(t)+C_{4}\sin(t) | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: Find a purely exponential solution of $\\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$.\n\n\nSolution: The characteristic polynomial of the homogeneous equation is given by $p... | math | {
"ground_truth": "\\frac{e^{-2t}}{15}+C_{1}e^{t}+C_{2}e^{-t}+C_{3}\\cos(t)+C_{4}\\sin(t)",
"style": "rule"
} | {
"index": 580,
"split": "test"
} |
numina_math | s^{2}+\omega^{2} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Consider the differential equation $\\ddot{x}+\\omega^{2} x=0$. \\\\\n\nA differential equation $m \\ddot{x}+b \\dot{x}+k x=0$ (where $m, b$, and $k$ are real constants... | math | {
"ground_truth": "s^{2}+\\omega^{2}",
"style": "rule"
} | {
"index": 581,
"split": "test"
} |
numina_math | 2\cos(\pit+\pi/3) | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Rewrite the function $\\cos (\\pi t)-\\sqrt{3} \\sin (\\pi t)$ in the form $A \\cos (\\omega t-\\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$. L... | math | {
"ground_truth": "2\\cos(\\pit+\\pi/3)",
"style": "rule"
} | {
"index": 582,
"split": "test"
} |
numina_math | 2\pi/\omega | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The following subproblems refer to the damped sinusoid $x(t)=A e^{-a t} \\cos (\\omega t)$.\n\nWhat is the spacing between successive maxima of $x(t)$? Assume that $\\o... | math | {
"ground_truth": "2\\pi/\\omega",
"style": "rule"
} | {
"index": 583,
"split": "test"
} |
numina_math | \frac{4A}{3+3i}e^{it} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The following subproblems refer to a spring/mass/dashpot system driven through the spring modeled by the equation $m \\ddot{x}+b \\dot{x}+k x=k y$. Here $x$ measures th... | math | {
"ground_truth": "\\frac{4A}{3+3i}e^{it}",
"style": "rule"
} | {
"index": 584,
"split": "test"
} |
numina_math | I(0)e^{-\frac{t}{RC}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The following subproblems refer to a circuit with the following parameters. Denote by $I(t)$ the current (where the positive direction is, say, clockwise) in the circui... | math | {
"ground_truth": "I(0)e^{-\\frac{t}{RC}}",
"style": "rule"
} | {
"index": 585,
"split": "test"
} |
numina_math | \frac{e^{2it}}{(2+2i)} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: 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 wh... | math | {
"ground_truth": "\\frac{e^{2it}}{(2+2i)}",
"style": "rule"
} | {
"index": 586,
"split": "test"
} |
numina_math | \frac{2-\omega^{2}-\omegai/4}{\omega^{4}-\frac{63}{16}\omega^{2}+4} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The following subproblems consider a second order mass/spring/dashpot system driven by a force $F_{\\text {ext }}$ acting directly on the mass: $m \\ddot{x}+b \\dot{x}+... | math | {
"ground_truth": "\\frac{2-\\omega^{2}-\\omegai/4}{\\omega^{4}-\\frac{63}{16}\\omega^{2}+4}",
"style": "rule"
} | {
"index": 587,
"split": "test"
} |
numina_math | x^{\prime}+\frac{r}{V}x-rc=0 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The following subproblems refer to the following \"mixing problem\": A tank holds $V$ liters of salt water. Suppose that a saline solution with concentration of $c \\ma... | math | {
"ground_truth": "x^{\\prime}+\\frac{r}{V}x-rc=0",
"style": "rule"
} | {
"index": 588,
"split": "test"
} |
numina_math | -t^2-t-3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the polynomial solution of $\\ddot{x}-x=t^{2}+t+1$, solving for $x(t)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-t^2-t-3",
"style": "rule"
} | {
"index": 589,
"split": "test"
} |
numina_math | -8 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $x + yi$, wher... | math | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": 590,
"split": "test"
} |
numina_math | \frac{\cos(2t)}{15} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find a purely sinusoidal solution of $\\frac{d^{4} x}{d t^{4}}-x=\\cos (2 t)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\cos(2t)}{15}",
"style": "rule"
} | {
"index": 591,
"split": "test"
} |
numina_math | -2+2\sqrt{3}i | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $x + yi$, wher... | math | {
"ground_truth": "-2+2\\sqrt{3}i",
"style": "rule"
} | {
"index": 592,
"split": "test"
} |
numina_math | \frac{t}{4}\sin(2t) | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find a solution of $\\ddot{x}+4 x=\\cos (2 t)$, solving for $x(t)$, by using the ERF on a complex replacement. The ERF (Exponential Response Formula) states that a solution to $p... | math | {
"ground_truth": "\\frac{t}{4}\\sin(2t)",
"style": "rule"
} | {
"index": 593,
"split": "test"
} |
numina_math | \frac{1}{2}(\exp{a*t}+\exp{-a*t}) | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the ordinary differential equation $\\ddot{x}-a^{2} x=0$, where $a$ is a nonzero real-valued constant, find a solution $x(t)$ to this equation such that $x(0) = 1$ and $\\d... | math | {
"ground_truth": "\\frac{1}{2}(\\exp{a*t}+\\exp{-a*t})",
"style": "rule"
} | {
"index": 594,
"split": "test"
} |
numina_math | \frac{e^{t}}{3}+ce^{-2t} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the general solution of the differential equation $\\dot{x}+2 x=e^{t}$, using $c$ for the arbitrary constant of integration which will occur. Let's think step by step and ou... | math | {
"ground_truth": "\\frac{e^{t}}{3}+ce^{-2t}",
"style": "rule"
} | {
"index": 595,
"split": "test"
} |
numina_math | (\frac{t^{2}}{2}-t+C)e^{-t} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find a solution of $\\ddot{x}+3 \\dot{x}+2 x=t e^{-t}$ in the form $x(t)=u(t) e^{-t}$ for some function $u(t)$. Use $C$ for an arbitrary constant, should it arise. Let's think s... | math | {
"ground_truth": "(\\frac{t^{2}}{2}-t+C)e^{-t}",
"style": "rule"
} | {
"index": 596,
"split": "test"
} |
numina_math | -2 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the complex number $z$ is given by $z = 1+\\sqrt{3} i$, what is the real part of $z^2$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "u... | math | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": 597,
"split": "test"
} |
numina_math | \frac{e^{-2t}}{15} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find a purely exponential solution of $\\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{e^{-2t}}{15}",
"style": "rule"
} | {
"index": 598,
"split": "test"
} |
numina_math | m | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 \\d... | math | {
"ground_truth": "m",
"style": "rule"
} | {
"index": 599,
"split": "test"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.