31. A particle moves along the x-axis with an acceleration given by a(t) = -kv², where v is the instantaneous velocity and k is a positive constant. If the particle has an initial velocity v₀ at x = 0, what is the velocity as a function of position x?
(A) v(x) = v₀ - kx
(B) v(x) = v₀e⁻ᵏˣ
(C) v(x) =
(D) v(x) = v₀/(1 + kx)
(E) v(x) = v₀ - kx²

53. A circular coil 0.05 m in radius, with 50 turns of wire, lies on a horizontal plane. It carries a counterclockwise current of 8.0 A. The coil is in a uniform 1.50 T magnetic field directed toward the +Y direction. Find the torque on the coil. (Given: Unit vectors î, ĵ and k̂ point in the directions of the positive X-, Y- and Z- axes and have magnitude of 1) (A) -0.03 π î (B) -1.50 π î (C) 0.03 π k̂ (D) 1.5 π k̂ (E) 0.03 π ĵ

54. A beam of monochromatic light with a wavelength of λ = 600 nm is incident perpendicularly onto a diffraction grating with 500 lines/mm. What is the maximum diffraction order (m) that can be observed on the screen? (Note: Consider only the orders on one side of the central maximum, excluding m = 0) (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

55. Alice directs a beam of 0.12 nm x rays at certain planes of a silicon crystal. As she increases the angle of incidence of the beam from zero, the first strong interference maximum occurs when the beam makes an angle of θ = tan⁻¹() with the planes. How far apart are the planes? (A) 0.045 nm (B) 0.075 nm (C) 0.1 nm (D) 0.15 nm (E) 0.2 nm

56. A rectangular conducting loop with resistance R lies in the same plane as a long straight wire carrying a steady current i. The nearest edge of the loop is initially at a distance a from the wire, and the loop has the width w and length L as shown in the figure. The loop is then pulled directly away from the wire with constant speed v. As the motion just begins, the induced current in the loop is ______. (Given: permeability constant μ₀) (A)(B) (C) (D) (E)

57. A thin layer of oil with an index of refraction n = 1.5 coats a glass plate with an index of refraction of 1.8. White light from air (n = 1.0) is incident normal to the film. In the reflected light: (1) Fully constructive interference occurs at a wavelength of λ₁ = 600 nm. (2) Fully destructive interference occurs at a wavelength of λ₂ = 400 nm. Assume that both wavelengths correspond to the same interference order, what is the thickness of the oil film? (A) 100 nm (B) 133 nm (C) 167 nm (D) 200 nm (E) 266 nm

58. What is the de Broglie wavelength of an electron accelerated from rest through a potential difference of 50.0 V? (A) 0.100 nm (B) 0.139 nm (C) 0.174 nm (D) 0.834 nm (E) none of the above

59. The radioactive tracer has a half-life of T₁/₂ = 2.0×10³ years. A sample initially contains N₀ = 8.0×10¹⁵ nuclei. What is the activity of the remaining nuclei after 4.0×10³ years? (Given: 1 year = 3.15×10⁷ s, 1 Ci = 3.7×10¹⁰ decays/s, ln2 = 0.693) (A) 3.0×10⁻⁷ Ci (B) 5.9×10⁻⁷ Ci (C) 1.4×10⁻⁵ Ci (D) 5.9×10⁻⁵ Ci (E) 2.2×10⁻³ Ci

60. In the Bohr model of the hydrogen atom, the reference level of zero energy is taken to be that of a free electron at infinite separation from the nucleus. Let K denote the kinetic energy of the electron in a bound orbit and let E be its total mechanical energy. Which of the following correctly describes the relationship between K and E? (A) K = -E (B) K = E (C) K = E/2 (D) K = -E/2 (E) K = -2E

61. Temperature dependence of resistivity for three materials (A, B and C) is depicted below. Which of the following statements is (are) correct?I. A represents an insulator. II. B is a metal. III. C represents a superconductor. (A) I and II (B) II and III (C) I and III (D) II (E) III

62. Which of the following terms best describes the pair of compounds shown below? (A) same compound (B) constitution isomers (C) enantiomers (D) diastereomers (E) conformers

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