source stringclasses 1
value | id stringclasses 173
values | question stringlengths 10 4.67k | options listlengths 4 4 | answer stringclasses 4
values | reasoning null |
|---|---|---|---|---|---|
MMLU | MMLU-57 | Which one of the following statements is true: | [
"Large and positive g shifts are observed when the LUMO – HOMO gap is small and λ is positive.",
"Large deviations in g occurs when the energy difference between the SOMO and lowest lying excited state is large.",
"A large deviation of g from ge is expected when λ is large and the energy difference between the ... | C | null |
MMLU | MMLU-58 | What is the magnetic moment of 205Tl? | [
"[ɣ (205TI = 1.569 x 108 T-1 s-1, I = ½]",
"0.717 x 10^-26 J T-1",
"0.827 x 10^-26 J T-1",
"1.433 x 10^-26 J T-1"
] | C | null |
MMLU | MMLU-59 | A molecule has a rotational correlation time of 1 ns. At what magnetic field would protons in this molecule have the fastest spin-lattice relaxation? [Use eqns 5.3 and 5.4.] | [
"3.74 T",
"5.19 T",
"6.08 T",
"9.49 T"
] | A | null |
MMLU | MMLU-60 | Of the following ionic substances, which has the greatest lattice enthalpy? | [
"MgO",
"MgS",
"NaF",
"NaCl"
] | A | null |
MMLU | MMLU-61 | Which of the following compounds has a 1H resonance approximately 1.55 kHz away from TMS on a spectrometer with a 12.0 T magnet? | [
"CH3F",
"CH3Cl",
"CH3Br",
"CH3I"
] | B | null |
MMLU | MMLU-62 | Calculate the equilibrium polarization of 13C nuclei in a 20.0 T magnetic field at 300 K. | [
"10.8 x 10^-5",
"4.11 x 10^-5",
"3.43 x 10^-5",
"1.71 x 10^-5"
] | D | null |
MMLU | MMLU-63 | The magnetic moment (μI) of an unknown nuclide is 2.884 x 10^-27 J T-1. Given the nuclear spin is known to be 1, identify the unknown nuclide. | [
"14N",
"2H",
"19F",
"6Li"
] | B | null |
MMLU | MMLU-64 | Which of the following is (are) characteristic of mass spectrometry?
I. Analyte molecules are converted to gaseous ions.
II. The ions are separated according to their mass-to-charge ratio.
III. In addition to compound identification, mass spectra can be utilized to determine precise isotopic masses and isotopic ratios. | [
"II only",
"I and II only",
"I and III only",
"I, II, and III"
] | D | null |
MMLU | MMLU-65 | Which of the following types of spectroscopy is a light-scattering technique? | [
"Nuclear magnetic resonance",
"Infrared",
"Raman",
"Ultraviolet-visible"
] | C | null |
MMLU | MMLU-66 | When the Heisenberg uncertainty principle is applied to a quantum mechanical particle in the lowest energy level of a one-dimensional box, which of the following is true? | [
"Momentum is known exactly, but no information about position can be known.",
"Position is known exactly, but no information about momentum can be known.",
"No information about either position or momentum can be known.",
"Neither position nor momentum can be known exactly."
] | D | null |
MMLU | MMLU-67 | The dissociation energy for a hydrogen-bromine bond is defined as the change in enthalpy, DH, for which of the following reactions? | [
"2 HBr(g) → H2(g) + Br2(l)",
"HBr(g) → H+(g) + Br−(g)",
"H(g) + Br(g) → HBr(g)",
"HBr(g) → H(g) + Br(g)"
] | D | null |
MMLU | MMLU-68 | 23Na has an NMR frequency of 198.52 MHz in a 750 MHz NMR spectrometer. What is its magnetogyric ratio? | [
"101.1 x 10^7 T-1 s-1",
"26.75 x 10^7 T-1 s-1",
"7.081 x 10^7 T-1 s-1",
"0.9377 x 10^7 T-1 s-1"
] | C | null |
MMLU | MMLU-69 | Exact solutions of the Schrödinger equation CANNOT be obtained for a | [
"simple harmonic oscillator",
"particle in a one-dimensional box",
"rigid rotor",
"helium atom"
] | D | null |
MMLU | MMLU-70 | What is the limiting high-temperature molar heat capacity at constant volume (C_V) of a gas-phase diatomic molecule? | [
"1.5R",
"2R",
"2.5R",
"3.5R"
] | D | null |
MMLU | MMLU-71 | The +1 oxidation state is more stable than the +3 oxidation state for which group 13 element? | [
"B",
"Al",
"In",
"Tl"
] | D | null |
MMLU | MMLU-72 | The fact that the infrared absorption frequency of deuterium chloride (DCl) is shifted from that of hydrogen chloride (HCl) is due to the differences in their | [
"electron distribution",
"dipole moment",
"force constant",
"reduced mass"
] | D | null |
MMLU | MMLU-73 | The number of allowed energy levels for the 55Mn nuclide are: | [
"3",
"5",
"8",
"4"
] | A | null |
MMLU | MMLU-74 | A substance that is NOT generally considered to be a toxic pollutant in water is | [
"carbonic acid",
"a halogenated hydrocarbon",
"lead",
"mercury"
] | A | null |
MMLU | MMLU-75 | All proteins absorb electromagnetic radiation of wavelength around 190 nm, which corresponds to a π → π* excitation in the protein molecule. In which region of the spectrum is this wavelength found? | [
"X-ray",
"Ultraviolet",
"Visible",
"Infrared"
] | B | null |
MMLU | MMLU-76 | The rotational correlation time of a spherical object of radius r immersed in a liquid with viscosity η is given by the Stokes equation: τc = 4πηr3/3kBT. A small molecule with relative molecular mass Mr = 100 in water at 298 K has τc = 50 ps. Assuming that Mr is proportional to the molecular volume, estimate τc for a s... | [
"420 ns",
"42 ns",
"4.2 ns",
"420 ps"
] | C | null |
MMLU | MMLU-77 | Chlorofluorocarbons (CFCs) such as F3CCCl3 are implicated in the decomposition of stratospheric ozone. Which of the following methods would be best suited to measurement of trace amounts (sub-ppb) of CFCs in an air sample? | [
"Gas chromatographic separation of the air sample on a capillary column followed by electron capture detection",
"Gas chromatographic separation of the air sample on a packed column followed by thermal conductivity detection",
"Gas chromatographic separation of the air sample on a capillary column followed by f... | A | null |
MMLU | MMLU-78 | The solid-state structures of the principal allotropes of elemental boron are made up of which of the following structural units? | [
"B12 icosahedra",
"B8 cubes",
"B6 octahedra",
"B4 tetrahedra"
] | A | null |
MMLU | MMLU-79 | A buffer is made from equal concentrations of a weak acid and its conjugate base. Doubling the volume of the buffer solution by adding water has what effect on its pH? | [
"It has little effect.",
"It significantly increases the pH.",
"It significantly decreases the pH.",
"It changes the pH asymptotically to the pKa of the acid."
] | A | null |
MMLU | MMLU-80 | Which of the following is classified as a conjugate acid-base pair? | [
"HCl / NaOH",
"H3O+ / H2O",
"O2 / H2O",
"H+ / Cl−"
] | B | null |
MMLU | MMLU-81 | Which of the following metal ions cannot be used as a paramagnetic quencher? | [
"Ti3+",
"Cr3+",
"Fe3+",
"Zn2+"
] | B | null |
MMLU | MMLU-82 | An electron and a proton are placed in a 1.0 T magnetic field at 300 K. What is the ratio of the equilibrium polarizations: |pe / pH| ? | [
"[|ɣe| = 1.761 x 10^11 T^-1 s^-1]",
"820",
"658",
"329"
] | B | null |
MMLU | MMLU-83 | The amplitude of a free induction decay drops to 25% of its initial intensity after 1.0 s. Assuming exponential relaxation and Ω = 0, determine the value of T2. | [
"0.721 s",
"0.750 s",
"1.386 s",
"1.661 s"
] | D | null |
MMLU | MMLU-84 | The molecular geometry of thionyl chloride, SOCl2, is best described as | [
"trigonal planar",
"T-shaped",
"tetrahedral",
"trigonal pyramidal"
] | D | null |
MMLU | MMLU-85 | Which of the following procedures tend(s) to minimize the influence of random errors on measured results?
I. Signal averaging
II. Use of internal standards
III. Averaging the results from multiple samples | [
"I only",
"II only",
"III only",
"I and III only"
] | D | null |
MMLU | MMLU-86 | A PO3- radical produces a pair of lines separated by 280 MHz with giso = 2.005. Calculate the expected resonant field positions of both lines in the X-band EPR spectrum (ν = 9.5 GHz). | [
"Bres for mI = -½ = 325.0 mT; Bres for mI = +½ = 335.0 mT",
"Bres for mI = -½ = 123.5 mT; Bres for mI = +½ = 124.5 mT.",
"Bres for mI = -½ = 333.5 mT; Bres for mI = +½ = 343.5 mT.",
"Bres for mI = -½ = 0.218 mT; Bres for mI = +½ = 0.418 mT."
] | D | null |
MMLU | MMLU-87 | Calculate the relative occupancies of the α and β spin energy levels for a radical species with g = 2.05, at L- and W-band frequencies (take TS = 300 K). | [
"Nα/Nβ = 0.9800 at L-band; Nα/Nβ = 0.9509 at W-band",
"Nα/Nβ = 0.9950 at L-band; Nα/Nβ = 0.9609 at W-band",
"Nα/Nβ = 0.9910 at L-band; Nα/Nβ = 0.9709 at W-band",
"Nα/Nβ = 0.9850 at L-band; Nα/Nβ = 0.9809 at W-band"
] | D | null |
MMLU | MMLU-88 | Which of the following is a primary standard for use in standardizing bases? | [
"Ammonium hydroxide",
"Sulfuric acid",
"Acetic acid",
"Potassium hydrogen phthalate"
] | D | null |
MMLU | MMLU-89 | Which of the following is lower for argon than for neon? | [
"Melting point",
"Boiling point",
"Polarizability",
"First ionization energy"
] | D | null |
MMLU | MMLU-90 | The 99Ru chemical shift of [Ru(H2O)6]2+ is 16,050 ppm relative to [Ru(CN)6]2- at 0 ppm. The electronic excitation energies of [Ru(H2O)6]2+, [Ru(CN)6]2- and [Ru(NH3)6]2+ are 18,900, 40,000 and 25,600 cm-1 respectively. Assuming that the chemical shift differences are dominated by paramagnetic currents, determine the che... | [
"7,530 ppm",
"8,090 ppm",
"11,070 ppm",
"14,840 ppm"
] | B | null |
MMLU | MMLU-91 | During a delay, spins with an offset frequency Ω = 250 rad s-1 precess through an angle of 60°. How long is the delay? | [
"4.19 ms",
"26.3 ms",
"240 ms",
"1510 ms"
] | A | null |
MMLU | MMLU-92 | Which of the following is always true of a spontaneous process? | [
"The process is exothermic.",
"The process does not involve any work.",
"The entropy of the system increases.",
"The total entropy of the system plus surroundings increases."
] | D | null |
MMLU | MMLU-93 | Nitronyl nitroxides are stable radicals in which the unpaired electron is coupled to two equivalent nitrogen nuclei. How many lines will appear in the EPR spectrum of a solution of a rigid nitronyl nitroxide diradical with J >> a? | [
"3 lines",
"9 lines",
"5 lines",
"7 lines"
] | D | null |
MMLU | MMLU-94 | Redox enzyme catalysis involves the cyclic oxidation and reduction of metal ions that have at least two stable positive oxidation states. Which of the following groups of metals could be found at the active site of redox enzymes? | [
"Cu, Fe, Co",
"Zn, Ca, Ga",
"Sr, Ga, Mg",
"Na, Ba, Al"
] | A | null |
MMLU | MMLU-95 | In the vibrational-rotational spectrum of a diatomic molecule, the R-branch of the spectrum is the result of which of the following transitions? | [
"ΔJ = 0; Δu = 0",
"ΔJ = 1; Δu = 0",
"ΔJ = 2; Δu = 0",
"ΔJ = 1; Δu = 1"
] | D | null |
MMLU | MMLU-96 | Of the following atoms, which has the lowest electron affinity? | [
"F",
"Si",
"O",
"Ca"
] | D | null |
MMLU | MMLU-97 | Of the following ions, which has the smallest radius? | [
"K+",
"Ca2+",
"Sc3+",
"Rb+"
] | C | null |
MMLU | MMLU-98 | What is the NMR frequency of 31P in a 20.0 T magnetic field? | [
"54.91 MHz",
"239.2 MHz",
"345.0 MHz",
"2167 MHz"
] | C | null |
MMLU | MMLU-99 | Predict the number of lines in the EPR spectrum of a solution of dimethylnitroxide (CH3)2NO• assuming the lines do not overlap. | [
"21",
"3",
"7",
"24"
] | B | null |
MMLU | MMLU-100 | Of the following compounds, which is LEAST likely to behave as a Lewis acid? | [
"BeCl2",
"MgCl2",
"ZnCl2",
"SCl2"
] | D | null |
MMLU | MMLU-1 | Let k be the number of real solutions of the equation e^x + x - 2 = 0 in the interval [0, 1], and let n be the number of real solutions that are not in [0, 1]. Which of the following is true? | [
"k = 0 and n = 1",
"k = 1 and n = 0",
"k = n = 1",
"k > 1"
] | B | null |
MMLU | MMLU-2 | Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G ? | [
"0",
"1",
"2",
"3"
] | D | null |
MMLU | MMLU-3 | Suppose P is the set of polynomials with coefficients in Z_5 and degree less than or equal to 7. If the operator D sends p(x) in P to its derivative p′(x), what are the dimensions of the null space n and range r of D? | [
"n = 1 and r = 6",
"n = 1 and r = 7",
"n = 2 and r = 5",
"n = 2 and r = 6"
] | D | null |
MMLU | MMLU-4 | The shortest distance from the curve xy = 8 to the origin is | [
"4",
"8",
"16",
"2sqrt(2)"
] | A | null |
MMLU | MMLU-5 | There are 25 suitcases, 5 of which are damaged. Three suitcases are selected at random. What is the probability that exactly 2 are damaged? | [
"2/69",
"1/30",
"2/23",
"12/125"
] | C | null |
MMLU | MMLU-6 | Let (Z_{10} , +, x) be the ring of integers modulo 10, and let S be the subset of Z_{10} represented by {0, 2, 4, 6, 8}. Which of the following statements is FALSE? | [
"(S, +, x) is closed under addition modulo 10.",
"(S, +, x) is closed under multiplication modulo 10.",
"(S, +, x) has an identity under addition modulo 10.",
"(S, +, x) has no identity under multiplication modulo 10."
] | D | null |
MMLU | MMLU-7 | A ladder 9 meters in length is leaning against a vertical wall on level ground. As the bottom end of the ladder is moved away from the wall at a constant rate of 2 meters per second, the top end slides downward along the wall. How fast, in meters per second, will the top end of the ladder be sliding downward at the mom... | [
"12*sqrt(2)",
"6*sqrt(2)",
"4*sqrt(2)",
"1/(2*sqrt(2))"
] | C | null |
MMLU | MMLU-8 | Consider a segment of length 10. Points A and B are chosen randomly such that A and B divide the segment into three smaller segments. What is the probability that the three smaller segments could form the sides of a triangle? | [
"0",
"10%",
"25%",
"50%"
] | C | null |
MMLU | MMLU-9 | Let V be a finite-dimensional real vector space and let P be a linear transformation of V such that P^2 = P. Which of the following must be true?
I. P is invertible.
II. P is diagonalizable.
III. P is either the identity transformation or the zero transformation. | [
"None",
"I only",
"II only",
"III only"
] | C | null |
MMLU | MMLU-10 | Which of the following is true? | [
"Every compact space is complete",
"Every complete space is compact",
"Neither (a) nor (b).",
" Both (a) and (b)."
] | A | null |
MMLU | MMLU-11 | A discrete graph is complete if there is an edge connecting any pair of vertices. How many edges does a complete graph with 10 vertices have? | [
"10",
"20",
"25",
"45"
] | D | null |
MMLU | MMLU-12 | Suppose X is a discrete random variable on the set of positive integers such that for each positive integer n, the probability that X = n is 1/2^n. If Y is a random variable with the same probability distribution and X and Y are independent, what is the probability that the value of at least one of the variables X and ... | [
"1/64",
"15/64",
"1/4",
"3/8"
] | B | null |
MMLU | MMLU-13 | Let V and W be 4-dimensional subspaces of a 7-dimensional vector space X. Which of the following CANNOT be the dimension of the subspace V intersect W? | [
"0",
"1",
"2",
"3"
] | A | null |
MMLU | MMLU-14 | A large class is to be divided into teams and each student must be a member of exactly one team. However, each student dislikes three of their classmates. Dislike between students need not be mutual. If the teams do not need to be equally sized, how many must be created so that no student is the teammate of someone the... | [
"4",
"7",
"10",
"13"
] | B | null |
MMLU | MMLU-15 | Sofia and Tess will each randomly choose one of the 10 integers from 1 to 10. What is the probability that neither integer chosen will be the square of the other? | [
"0.64",
"0.72",
"0.81",
"0.95"
] | D | null |
MMLU | MMLU-16 | Let K be a nonempty subset of R^n, where n > 1. Which of the following statements must be true?
I. If K is compact, then every continuous real-valued function defined on K is bounded.
II. If every continuous real-valued function defined on K is bounded, then K is compact.
III. If K is compact, then K is connected. | [
"I only",
"II only",
"III only",
"I and II only"
] | D | null |
MMLU | MMLU-17 | Suppose V is a real vector space of finite dimension n. Call the set of matrices from V into itself M(V). Let T be in M(V). Consider the two subspaces U = {X ∈ M(V) : TX = XT} and W = {TX − XT : X ∈ M(V)}. Which of the following must be TRUE?
I. If V has a basis containing only eigenvectors of T then U = M(V).
II. dim ... | [
"I only",
"II only",
"III only",
"I and II only"
] | B | null |
MMLU | MMLU-18 | Let F be a constant unit force that is parallel to the vector (-1, 0, 1) in xyz-space. What is the work done by F on a particle that moves along the path given by (t, t^2, t^3) between time t=0 and time t=1? | [
"-1/4",
"-1/(4sqrt(2))",
"0",
"sqrt(2)"
] | C | null |
MMLU | MMLU-19 | Let X be a topological space, and let A, B, and C be nonempty subsets of X such that A c B c C; that is, A is a proper subset of B, and B is a proper subset of C. Which of the following statements is (are) true?
I. If A and C are connected, then B is connected.
II. If A and C are compact, then B is compact.
III. If A a... | [
"I and II only",
"III only",
"I and III only",
"I, II, and III"
] | D | null |
MMLU | MMLU-20 | Let x_1 = 1 and x_(n+1) = sqrt(3+2x_n) for all positive integers n. If it is assumed that {x_n} converges, then lim x_n = | [
"3",
"e",
"sqrt(5)",
"0"
] | A | null |
MMLU | MMLU-21 | If v is an eigenvector of an invertible matrix A, then which of the following is I are necessarily
true?
I. v is also an eigenvector of 2A.
II. v is also an eigenvector of A^2.
III. v is also an eigenvector of A^(-1). | [
"I only",
"II only",
"III only",
"I, II, and III"
] | D | null |
MMLU | MMLU-22 | For how many positive integers k does the ordinary decimal representation of the integer k! end in exactly 99 zeros? | [
"None",
"One",
"Four",
"Five"
] | D | null |
MMLU | MMLU-23 | Let R be a ring with a multiplicative identity. If U is an additive subgroup of R such that ur in U for all u in U and for all r in R, then U is said to be a right ideal of R. If R has exactly two right ideals, which of the following must be true?
I. R is commutative.
II. R is a division ring (that is, all elements exc... | [
"I only",
"II only",
"III only",
"I and II only"
] | B | null |
MMLU | MMLU-24 | A six-sided die (whose faces are numbered 1 through 6, as usual) is known to be counterfeit: The probability of rolling any even number is twice the probability of rolling any odd number. What is the probability that if this die is thrown twice, the first roll will be a 5 and the second roll will be a 6? | [
"2/81",
"1/18",
"2/27",
"1/9"
] | A | null |
MMLU | MMLU-25 | Statement 1 | Suppose {s_n} and {t_n} are sequences of complex numbers such that lim_n→∞ s_n = s and lim_n→∞ tn = t. lim_n→∞ s_nt_n = st. Statement 2 | There exists a bounded sequence of real numbers with no convergent subsequence. | [
"True, True",
"True, False",
"False, True",
"False, False"
] | B | null |
MMLU | MMLU-26 | Define a binary operation on Z, the set of integers, by the equation m • n = m + n + mn. Which of
the following statements is (are) true about the binary structure (Z, • )?
I. This structure is not a group since the operation is not associative.
II. This structure is not a group since there is no identity element.
III.... | [
"I and II only",
"I and III only",
"II and III only",
"III only"
] | D | null |
MMLU | MMLU-27 | It takes Kate k days to write a GRE math practice test. It takes John j days to write a GRE math practice test. If Kate and John work on a practice test in alternating 2-day shifts, it takes them 10 days when Kate starts and 10.5 days when John starts. How long would it take the two to complete a practice test if Kate ... | [
"9/2 days",
"5 days",
"41/8 days",
"36/7 days"
] | D | null |
MMLU | MMLU-28 | What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle? | [
"1/2",
"1",
"sqrt(2)",
"pi"
] | A | null |
MMLU | MMLU-29 | Let f(x,y) = x^2 - 2xy + y^3 for all real x and y. Which of the following is true? | [
"f has all of its relative extrema on the line x = y.",
"f has all of its relative extrema on the parabola x = y^2.",
"f has a relative minimum at (0, 0).",
"f has an absolute minimum at (2/3, 2/3)."
] | A | null |
MMLU | MMLU-30 | What is the largest order of an element in the group of permutations of 5 objects? | [
"5",
"6",
"12",
"15"
] | B | null |
MMLU | MMLU-31 | S(n) is a statement about positive integers n such that whenever S(k) is true, S(k + 1) must also be true. Furthermore, there exists some positive integer n0 such that S(n0) is not true. Of the following, which is the strongest conclusion that can be drawn? | [
"S(n0 + 1) is not true.",
"S(n0 - 1) is not true.",
"S(n) is not true for any n <= n0",
"S(n) is not true for any n >= n0"
] | C | null |
MMLU | MMLU-32 | In xyz-space, what are the coordinates of the point on the plane 2x + y + 3z = 3 that is closest to the origin? | [
"(0, 0, 1)",
"(3/7, 3/14, 9/14)",
"(7/15, 8/15, 1/15)",
"(5/6, 1/3, 1/3)"
] | B | null |
MMLU | MMLU-33 | For which integers n such that 3 <= n <= 11 is there only one group of order n (up to isomorphism)? | [
"For no such integer n",
"For 3, 5, 7, and 11 only",
"For 3, 5, 7, 9, and 11 only",
"For 4, 6, 8, and 10 only"
] | B | null |
MMLU | MMLU-34 | A tree is a connected graph with no cycles. How many nonisomorphic trees with 5 vertices exist? | [
"1",
"2",
"3",
"4"
] | C | null |
MMLU | MMLU-35 | Suppose A and B are n x n invertible matrices, where n > 1, and I is the n x n identity matrix. If A and B are similar matrices, which of the following statements must be true? I. A - 2I and B - 2I are similar matrices. II. A and B have the same trace. III. A^-1 and B^-1 are similar matrices. | [
"I only",
"II only",
"III only",
"I, II, and III"
] | D | null |
MMLU | MMLU-36 | Let C be the circle in the yz-plane whose equation is (y - 3)^2 + z^2 = 1. If C is revolved around the z-axis, the surface generated is a torus. What is the equation of this torus? | [
"x^2 + y^2 + z^2 + 8 = 6y",
"(x^2 + y^2 + z^2)^2 = 8 + 36(x^2 + z^2)",
"(x^2 + y^2 + z^2 + 8)^2 = 36(x^2 + z^2)",
"(x^2 + y^2 + z^2 + 8)^2 = 36(x^2 + y^2)"
] | D | null |
MMLU | MMLU-37 | Which of the following is an equation of the line tangent to the graph of y = x + e^x at x = 0 ? | [
"y = 2x + 1",
"y = 2x",
"y = x + 2",
"y = x + 1"
] | A | null |
MMLU | MMLU-38 | Let V be the real vector space of all real 2 x 3 matrices, and let W be the real vector space of all real 4 x 1 column vectors. If T is a linear transformation from V onto W, what is the dimension of the subspace kernel of T? | [
"2",
"3",
"4",
"5"
] | A | null |
MMLU | MMLU-39 | Statement 1 | Suppose ∑|a_i| diverges and ∑ a_i = 2. There is a rearrangement a_i_k of the terms such that ∑ a_i_k = 4. Statement 2 | There exists metric spaces X and Y with X closed and bounded and a continuous mapping f : X → Y such that f(X) is NOT “closed and bounded”. | [
"True, True",
"True, False",
"False, True",
"False, False"
] | A | null |
MMLU | MMLU-40 | The maximum number of acute angles in a convex 10-gon in the Euclidean plane is | [
"1",
"2",
"3",
"4"
] | C | null |
MMLU | MMLU-41 | If f(z) is an analytic function that maps the entire finite complex plane into the real axis, then the imaginary axis must be mapped onto | [
"the entire real axis",
"a point",
"a ray",
"an open finite interval"
] | B | null |
MMLU | MMLU-42 | A group G in which (ab)^2 = a^2b^2 for all a, b in G is necessarily | [
"finite",
"cyclic",
"of order two",
"abelian"
] | D | null |
MMLU | MMLU-43 | If f is a continuously differentiable real-valued function defined on the open interval (-1, 4) such that f(3) = 5 and f'(x) ≥ -1 for all x, what is the greatest possible value of f(0)? | [
"3",
"4",
"5",
"8"
] | D | null |
MMLU | MMLU-44 | Suppose A and B are n × n matrices with real entries. Which of the follow are TRUE?
I. The trace of A^2 is nonnegative.
II. If A^2 = A, then the trace of A is nonnegative.
III. The trace of AB is the product of the traces of A and B. | [
"II only",
"III only",
"I and II only",
"II and III only"
] | A | null |
MMLU | MMLU-45 | What is the units digit in the standard decimal expansion of the number 7^25? | [
"1",
"3",
"5",
"7"
] | D | null |
MMLU | MMLU-46 | Statement 1 | f : X → Y is continuous and X is compact. f must be uniformly continuous. Statement 2 | If f(x), g(x) : R → R are everywhere differentiable f(g(x)) must be everywhere differentiable. | [
"True, True",
"True, False",
"False, True",
"False, False"
] | A | null |
MMLU | MMLU-47 | The region bounded by the curves y = x and y = x^2 in the first quadrant of the xy-plane is rotated about the y-axis. The volume of the resulting solid of revolution is | [
"π/12",
"π/6",
"π/3",
"2π/3"
] | B | null |
MMLU | MMLU-48 | In the xy-plane, the curve with parametric equations x = cos t and y = sin t, 0 <= t <= pi, has length | [
"3",
"pi",
"3pi",
"3/2"
] | B | null |
MMLU | MMLU-49 | Which of the following are valid intervals of convergence for a power series?
(I) [−1, 3)
(II) (−∞, 0]
(III) [2, 2] ∪ [3, 3]
(IV) (−∞,∞) | [
"Only (II)",
"Only (IV )",
"(I) and (IV )",
"(II) and (III)"
] | C | null |
MMLU | MMLU-50 | If f: R -> R is a bounded function that is Lebesgue integrable, then which of the following must be true? | [
"f is Riemann integrable.",
"There does not exist a countably infinite E of R such that f is nondifferentiable at every x \\in E.",
"There does not exist an uncountable subset E of R such that f is nondifferentiable at every x \\in E.",
"None of the above."
] | D | null |
MMLU | MMLU-51 | In the xy-plane, the graph of x^(log y) = y^log x is | [
"the open first quadrant",
"a closed curve",
"a ray in the open first quadrant",
"a single point"
] | A | null |
MMLU | MMLU-52 | For which p is the norm ||x||_p expressable as an inner product? | [
"1",
"2",
"p >= 1",
"infinity"
] | B | null |
MMLU | MMLU-53 | Let S be a compact topological space, let T be a topological space, and let f be a function from S onto T. Of the following conditions on f, which is the weakest condition sufficient to ensure the compactness of T? | [
"f is a homeomorphism.",
"f is continuous and one-to-one",
"f is continuous.",
"f is one-to-one"
] | C | null |
MMLU | MMLU-54 | What is the area of an equilateral triangle whose inscribed circle has radius 2? | [
"12",
"16",
"12*sqrt(3)",
"16*sqrt(3)"
] | C | null |
MMLU | MMLU-55 | For which value of n are there exactly two abelian groups of order n up to isomorphism? | [
"4",
"7",
"8",
"12"
] | A | null |
MMLU | MMLU-56 | Which of the following is not always true for x, y, z ∈ R^k? | [
"|x| ≥ 0",
"|x · y| ≤ |x||y|",
"|x| = 0 if and only if x = 0",
"|x + y| = |x| + |y|"
] | D | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.