GATE question papers: Physics 2008 (PH)

Q. 1 - Q. 20 carry one mark each.

1. For arbitrary matrices E, F, G and H, if EF - FE = 0 then Trace(EFGH) is equal to

(A) Trace (HGFE) (B) Trace (E) Trace (F) Trace (G) Trace (H)

2. An unitary matrix is given, where a, b, c, d, a and b are real. The inverse of the matrix is

(A) (B) (C) (D)

3. The curl of a vector field is . Identify the appropriate vector field from the choices given below.

(A) = (B) =

4. A rigid body is rotating about its centre of mass, fixed at the origin, with an angular velocity and angular acceleration. If the torque acting on it is and its angular momentum is , the rate of change of its kinetic energy is

(A) × (B) ×

5. A cylinder of mass M and radius R is rolling down without slipping on an inclined plane of angle of inclinationq. The number of generalized coordinates required to describe the motion of this system is

(A) 1 (B) 2 (C) 4 (D) 6

6. A parallel plate capacitor is being discharged. What is the direction of the energy flow in terms of the Poynting vector in the space between the plates?

(A) Along the wire in the positive z axis (B) Radially inward (-)

7. Unpolarized light falls from air to a planar air-glass interface (refractive index of glass is 1.5) and the reflected light is observed to be plane polarized. The polarization vector and the angle of incidence q, are

(A) perpendicular to the plane of incidence and q_i = 42°

(B) parallel to the plane of incidence and q_i = 56°

(D) parallel to the plane of incidence and q_i = 42°

8. A finite wave train, of an unspecified nature, propagates along the positive x axis with a constant speed v and without any change of shape. The differential equation among the four listed below, whose solution it must be, is

(A) (B)

9. Let denote the ground state of the hydrogen atom. Choose the correct statement from those given below:

(A) [L_x, L_y] (B) J² = 0 (C) ¹ 0 (D) [S_x, S_y]

10. Thermodynamic variables of a system can be volume V, pressure P, temperature T, number of particles N, internal energy E and chemical potentialm, etc. For a system to be specified by Microcanonical (MC), Canonical (CE) and Grand Canonical (GC) ensembles, the parameters required for the respective ensembles are:

(A) MC: (N,V,T); CE: (E,V,N); GC: (V,T,m) (B) MC: (E,V,N); CE: (N,V,T); GC: (V,T,m)

11. The pressure versus temperature diagram of a given system at certain low temperature range is found to be parallel to the temperature axis in the liquid-to-solid transition region. The change in the specific volume remains constant in this region. The conclusion one can get from the above is

(A) the entropy of solid is zero in this temperature region.

(B) the entropy increases when the system goes from liquid to solid phase in this temperature region.

(D) the change in entropy is zero in the liquid-to-solid transition region.

12. The radial wave function of the electrons in the state of n = 1 and l = 0 in a hydrogen atom is
R₁₀ = , a₀ is the Bohr radius. The most probable value of r for an electron is

(A) a₀ (B) 2a₀ (C) 4a₀ (D) 8a₀

13. The last two terms of the electronic configuration of manganese (Mn) atom is 3d⁵4s². The term factor of Mn⁴⁺ ion is

(A) ⁴D_1/2 (B) ⁴F_3/2 (C) ³F_9/2 (D) ³D_7/2

14. The coherence length of laser light is

(A) directly proportional to the length of the active lasing medium.

(B) directly proportional to the width of the spectral line.

(D) inversely proportional to the length of the active lasing medium.

15. Metallic monovalent sodium crystallizes in body centered cubic structure. If the length of the unit cell is 4 ´ 10^-8 cm, the concentration of conduction electrons in metallic sodium is

(A) 6.022 ´ 10²³ cm^-3 (B) 3.125 ´ 10²² cm^-3

16. The plot of inverse magnetic susceptibility 1/c versus temperature T of an antiferromagnetic sample corresponds to

17. According to the quark model, the K⁺ meson is composed of the following quarks:

(A) u u d (B) u (C) u (D) s

18. An O¹⁶ nucleus is spherical and has a charge radius R and a volume V º pR³. According to the empirical observations of the charge radii, the volume of the ₅₄Xe¹²⁸ nucleus, assumed to be spherical, is

(A) 8V (B) 2V (C) 6.75V (D) 1.89V

19. A common emitter transistor amplifier circuit is operated under a fixed bias. In this circuit, the operating point

(A) remains fixed with an increase in temperature.

(B) moves towards cut-off region with an increase in temperature.

(D) moves towards the saturation region with an increase in temperature.

20. Under normal operating conditions, the gate terminal of an n-channel junction field effect transistor (JEET) and an n-channel metal oxide semiconductor field effect transistor (MOSFET) are

(A) both biased with positive potentials.

(B) both biased with negative potentials.

(D) biased with negative and positive potentials, respectively.

Q. 21 to Q. 75 carry two marks each.

21. The eigenvalues of the matrix are

(A) when q = 45° (B) when q = 30°

22. If the Fourier transform F [d(x-a)] = exp (-i2pv a), then F^-1 (cos 2p a v) will correspond to

(A) d(x - a) - d(x + a) (B) a constant

23. If I º Ln(z), where C is the unit circle taken anticlockwise and Ln (z) is the principal branch of the Logarithm function, which one of the following is correct?

(A) I = 0 by residue theorem.

(B) I is not defined since Ln (z) has a branch cut

(D) Ln(z²) = 2I

24. The value of is

(A) o (B) 2pi (C) -2pi (D) (1 + 2i)p

25. Consider the Bessel equation (v = 0), . Which one of the following statements is correct?

(A) Equation has regular singular points at z = 0 and z = ¥.

(B) Equation has 2 linearly independent solutions that are entire.

at z = 0.

(D) Limit z ® ¥, taken along x axis, exists for both the linearly independent solutins.

26. Under a certain rotation of coordinate axes, a rank-1 tensor v_a (a=1,2,3) transforms according t the orthogonal transformation defined by the relations ; ; . Under the same rotation a rank-2 tensor T_a,b would transform such that

(A) (B)

27. The Lagrangian of a system is given by L = .It describes the motion of

(A) a harmonic oscillator. (B) a damped harmonic oscillator.

28. The moment of inertia tensor of a rigid body is given by I = . The magnitude of the moment of inertia about an axis is

(A) 6 (B) 5 (C) 2 (D) 8/3

29. A hoop of radius R is pivoted at a point on the circumference. The period of small oscillations in the plane of hoop is

(A) 2p (B) 2p (C) 2p (D) 2p

30. A mass m is constrained to move on a horizontal frictionless surface. It is set in circular motion with radius r₀ and angular speed w₀ by an applied force communicated through an inextensible thread the passes through a hole on the surface as shown in the figure. This force is then suddenly doubled. The magnitude of the radial velocity of the mass

(A) increases till the mass falls into the hole.

(B) decreases till the mass falls into the hole.

(D) becomes zero at a radius r₁ where 0 < r₁ <r₀.

31. For a simple harmonic oscillator the Lagrangian is given by L = If A(p, q) = and H(p, q) is the Hamiltonian of the system, the poisson bracket {A(p, q) H(p, q)} is given by

(A) iA(p, q) (B) (C) (D)

32. A plane electromagnetic wave is given by E₀ (exp {i (k z - wt)}. At a given location, the number of times vanishes in one second is

(A) An integer near when d = np and zero when d ¹ np , n is integer

(B) An integer near and is independent of d

(D) An integer near and is independent of d

33. A dielectric sphere is placed in a uniform electric field directed along the positive y-axis. Which one of the following represents the correct equipotential surfaces?

34. A rod of length L with uniform charge density l per unit length is in the xy-plane and rotating about z-axis passing through one of its edge with an angular velocity as shown in the figure below. refer to the unit vectors at Q, is the vector potential at a distance d from the origin O along z-axis for d >> L and is the current density due to the motion of the rod. Which one of the following statements is correct?

(A) along ; along ; || µ (B) along ; along ; || µ

35. A circular disc of radius a on the xy plane has a surface charge density .The electric dipole moment of this charge distribution is

(A) (B) (C) s (D)

36. At time t = 0, a charge distribution exists within an ideal homogeneous conductor of permittivity e and conductivity s. At a later time is given by

(A) = exp

(B) =

(D) =

37. A nonrelativistic charged particle moves along the positive x-axis with a constant positive acceleration . The particle is the origin at t = 0. Radiation is observed at t = 0 at a distant point (0, d, 0) on the y-axis. Which one of the following statements is correct?

(A) The radiation is unpolarized.

(B) The radiation is plane polarized with polarization parallel to the x-axs.

(D) The radiation is elliptically polarized.

38. For a physical system, two observables O₁ and O₂ are known to be compatible. Choose the correct implication from amongst those given below:

(A) Every eigenstate of O₁ must necessarily be an eigenstate of O₂.

(B) Every non-degenerateof of O₁ must necessarily be an eigenstate of O₂.

(C) When an observation of O₁ is carried out on an arbitrary state of the physical system, a subsequent observation of O₂ leads to an unambiguous result.

(D) Observation of O₁ and O₂ carried out on an arbitrary state of the physical system, lead to the identical results irrespective of the order in which the observations are made.

39. An exact measurement of the position of a simple harmonic oscillator (SHO) is made with the result x = x₀. [The SHO has energy levels E_n (n = 0, 1, 2.....) and associated normalized wave -function] . Subsequently, an exact measurement of energy E is made. Using the general notation Pr(E = E^') denoting the probability that a result E^' is obtained for this measurement, the following statements are written. Written one of the following statements is correct?

(A) Pr(E = E₀) = 0

(B) Pr(E = E₀) = 1 for some value of n.

(D) Pr(E > E₀) > 0 for any E^''.

40. Consider the combined system of proton an electron in the hydrogen atom in its (electronic) ground state. Let I denote the quantum number associated with the total angular momentum and let

41. A particle is placed in a one dimensional box of size L along the x-axis (0<x<L). Which of the following is true?

(A) In the ground state, the probability of finding the particle in the interval (L/4, 3L/4) is half.

(B) In the first excited state, the probability of finding the particle in the interval (L/4, 3L/4) is half .

This also holds for states with n = 4, 6, 8.......

(D) In the ground state, the particle has a definite momentum.

42. An inelastic ball of mass m has been thrown vertically upwards from the ground at z = 0. The initial kinetic energy of the ball is E. The phase trajectory of the ball after successive bouncing on the ground is

43. A system contain N non-interacting localized particles of spin 1/2 and magnetic moment m each is kept in constant external magnetic field B and in thermal equilibrium at temperature T. The magnetization of the system is

(A) Nm coth (B) Nm tanh (C) Nm sinh (D) Nm cosh

44. Two identical particles have to be distributed among three energy levels. Let r_B, r_F and r_C represent the ratios of probability of finding two particles to that of finding one particle in a given energy state. The subscripts B, F and C correspond to whether the particles are bosons, fermions and classical particles, respectively. Then r_B : r_F : r_C is equal to

(A) : 0 : 1 (B) 1 :: 1 (C) 1 :: (D) 1 : 0 :

45. A photon gas is at thermal equilibrium at temperature T. The mean number of photons in an energy state e = is

(A) (B)

46. Consider a system of N atoms of an ideal gas of type A at temperature T and volume V. It is kept in diffusive contact with another system of N atoms of another ideal gas of type B at the same temperature T and volume V. Once the combined system reaches equilibrium,

(A) the total entropy of the final system is the same as the sum of the entropy of the individual system always.

(B) the entropy mixing is 2Nk_B In 2.

(D) the entropy of mixing is non-zero when the atoms A and B are of the same type.

47. Consider a system of two non-interacting classical particles which can occupy any of the three energy levels with energy values E = 0, e and 2e having degeneracies g(E) = 1, 2 and 4 respectively.The mean energy of the system is

(A) (B)

48. Three consecutive absorption lines at 64.275 cm^-1 and 89.985 cm^-1 have been observed in a microwave spectrum for a linear rigid diatomic molecule. The moments fo inertia l_A and l_B are (l_A is with respect to the bond axis passing through the center of mass and l_B is with respect to an axis passing through the centre of mass and perpendicular to bond axis)

(A) both equal to cm² (B) zero and cm²

49. A pure rotational Raman spectrum of a linear diatomic molecule is recorded using electromagnetic radiation of frequency V_e. The frequency of two consecutive stokes lines are

(A) v_e - 10B, v_e - 14B (B) v_e - 2B, v_e - 4B

50. Which one of the following statement is INCORRECT in vibrational spectroscopy with angarmonicity?

(A) The selection rule for vibrational spectroscopy is Dv = ± 1, ± 2,....

(B) Anharmonicity leads to multiple absorption lines.

(D) The frequencies of hot band lines are smaller than the fundamental absorption.

51. The molecular spectra of two linear molecules O-C-O and O-C-S are recorded in the microwave region. Which one of the following statement is correct?

(A) Both the molecules would show absorption lines.

(B) Both the molecules not would show absorption lines.

(D) O-C-S would show absorption lines, but not O-C-O.

52. When the refractive index m of the active medium change by Dm in a laser resonator of length L, the change in the spectral spacing between the longitudinal modes of the laser is (c is the speed of light in free space)

(A) (B) (C) (D) zero

53. The primitive translation vectors of the body centered cubic lattice are and The primitive translation vectors of the reciprocal lattice are

(A)

(B)

(C)

(D)

54. The structure factor of a single cell of identical atoms of form factor f is given by S_hkt = is the coordinate of an atom, and hkt are the Miller indices, which one of the following statement is correct for the diffraction peaks of body centered cubic (BCC) and face centered cubic (FCC) lattices?

(A) BCC : (200); (110); (222)

FCC : (111); (311); (400)

(B) BCC : (210); (110); (222)

FCC : (111); (311); (400)

FCC : (111); (211); (400)

(D) BCC : (200); (110); (222)

FCC : (111); (211); (400)

55. The lattice specific heat C of a crystalline solid can be obtained using the Dulong petit model. Einstein model and Debye model. At low temperature , which one of the following statements is true (a and A are constants)

(A) Dulong petit C :µ exp(-a/T) ; Einstein : C = . constant ; Debye : C µ

(B) Dulong petit : C = . constant ; Einstein : C µ ; Debye :µ exp(-a/T)

(D) Dulong petit : C µ ; Einstein : C µ ; Debye : C = . constant

56. A linear diatomic lattice constant a with masses M and m (M > m) are coupled by a force constant C. The dispersion relation is given by

Which one of the following statements is INCORRECT?

(A) The atoms vibrating in transverse mode correspond to the optical branch.

(B) The maximum frequency of the acoustic branch depends on the mass of the lighter atom m.

(D) No normal modes exist in the acoustic branch for any frequency greater than the maximum frequency at k = p /a.

57. The kinetic energy of a free electron at a corner of the first Brillouin zone of a two demensional square lattice is larger than that of an electron at the mid-point of a side of the zone by a factor b. The value of b is

(A) b = (B) b = 2 (C) b = 4 (D) b = 8

58. An intrinsic semiconductor with mass of a hole m_h and mass of an electron m_e is at a finite temperature T. If the top of the valence band energy is E_v and the bottom of the conduction band energy is E_c , the Fermi energy of the semiconductor is

(A) E_F = (B) E_F =

59. Choose the correct statement from the following:

(A) The reaction K⁺K^- ® can proceed irrespective of the kinetic energies of K⁺ and K^-.

(B) The reaction K⁺K^- ® is forbidden by the baryon number conservation.

(D) The decay K^o ® p⁺p^- proceeds via weak interactions.

60. The following gives a list of pairs containing (i) a nucleus (ii) one of its properties. Find the pair which is INAPPROPRIATE.

(A) (i) ₁₀Ne²⁰ nucleus; (ii) stable nucleus

(B) (i) A spheroidal nucleus; (ii) an electric quadrupole moment

(D) (i) U²³⁸ nucleus; (ii) Binding energy = 1785 Me V (approximately)

61. The four possible configurations of neutrons in the ground state of ₄Be⁹ nucleus, according to the shell model, and the associated nuclear spin are listed below. Choose the correct one:

(A) (Is_1/2)²(Ip_3/2)³ : j = 3/2 (B) (Is_1/2)²(Ip_1/2)²(Ip_1/2)¹ : j = 3/2

62. The mass difference between the pair of mirror nuclei ₆C¹¹ and ₅B¹¹ is given to be DMeV/c². According to the semi-empirical mass formula, the mass difference between the pair of mirror nuclei ₉F¹⁷ and ₈O¹⁷ will approximately be (rest mass of proton m_p = 938.27 MeV/c² and rest mass of neutron m_n = 938.57 MeV/c² )

(A) 1.39 DMeV/c² (B) (1.39D + 0.5) MeV/c²

63. A heavy nucleus is found to contain more neutrons than protons. This fact is related to which one of the following statements.

(A) The nuclear force between neutrons is stronger than that between protons.

(B) The nuclear force between protons is of a shorter range than those between neutrons, so that a smaller number of protons are held together by the nuclear force

(D) It costs more energy to add a proton t a (heavy) nucleus than a neutron because of the Coulomb repulsion between protons.

64. A neutral pi meson (p⁰) has a rest-mass of approximately 140 Me V/c² and a lifetime of sec. A p⁰ produced in the laboratory is found to decay after 1.25 sec into two photons. Which of the following sets represents a possible set of energies of the two photons as seen in the laboratory?

(A) 70 MeV and 70 MeV (B) 35 MeV and 100 MeV

65. An a.c. voltage of 220 V_rms is applied to the primary of a 10:1 step-down transformer. The secondary of the transformer is centre tapped and connected to a full wave rectifier with a load resistance. The d.c. voltage appearing across the load is

(A) (B) (C) (D)

66. Let I₁ and I₂ represent mesh currents in the loop abcda and befcb respectively. The correct expression describing Kirchoff's voltage loop law in one of the following loops is,

(A) 30I₁ - 15I₂ = 10 (B) -15I₁+ 20I₂ = -20

67. The simplest logic gate circuit corresponding to the Boolean expression, Y = P + Q is

68. An analog voltage V is converted into 2-bit binary number. The minimum number of comparators required and their reference voltages are

(A) 3, (B) 3, (C) 4, (D) 4,

69. The following circuit (where R_L >> R) performs the operation of

(A) OR gate for a negative logic system (B) NAND gate for a negative logic system

70. In the T type master-slave JK flip flop is shown along with the clock and input waveforms. The Q_n output of flop flop was zero initially. Identify the correct output waveform.

Common Data for Questions

Common Data for Questions 71, 72 and 73:A beam of identical particles of mass m and energy E is

incident from left on a potential barrier of width L (between 0 < x < L) and height V₀ as shown in the figure

(E < V₀).

For x > L, there is tunneling with a transmission coefficient T > 0. Let A₀.A_R and A_T denote the amplitudes for

the incident, reflected and the transmitted waves, respectively.

71. Throughout 0 < x < L, the wave-function

(A) can be chosen to be real.

(B) is exponentially decaying

(D) is zero.

72. Let the probability current associated with the incident wave be S₀. Let R be the reflection coefficient. Then

(A) the probability current vanishes in the classically forbidden region.

(B) the probability current is TS₀ for x > L.

(D) for x > L, the probability current is complex.

73. The ratio of the reflected to the incident amplitude A_R|A₀ is

(A) I-A_T|A₀ (B) in magnitude

Common Data for Questions 74 and 75: Consider two concentric conducting spherical shells with inner

and outer radii a, b and c, d as shown in the figure. Both the shells are given Q amount of positive chares.

74. The electric field in different regions are

(A) 0 for r < a ; for a < r < b

0 for b < r < c ; for r >d

(B) for r < a ; 0 for a < r < b

for b < r < c ; for r >d

0 for b < r < c ; for r >d

(D) 0 for r < a ; 0 for a < r < b

0 for b < r < c ; for r >d

75. In order to have equal surface charge densities on the outer surfaces of both the shells, the following conditions should be satisfied

(A) d = 4b and c = 2a (B) d = 2b and c = a

Linked Answer Questions: Q.76 to Q.85 carry two marks each.

Statement for Linked Answer Questions 76 and 77:

Consider the b-decay of a free neutron at rest in the laboratory.

76. Which of the following configurations of the decay products correspond to the largest energy of the anti-neutrino ? ( rest mass of electron m_e = 0.51 MeV/c², rest mass of proton m_p = 938.27 MeV/c² and rest mass of neutron m_n = 939.57 MeV/c²)

(A) In the laboratory, proton is produced at rest.

(B) In the laboratory, momenta of proton, electron and the anti-neutrino all have the same magnitude.

(D) In the laboratory, electron is produced at rest.

77. Using the result of the above problem, answer the following, Which of the following represents approximately the maximum allowed energy of the anti-neutrino ?

(A) 1.3 MeV (B) 0.8 MeV (C) 0.5 MeV (D) 2.0 MeV

Statement for Linked Answer Questions 78 and 79:

Consider a two dimensional electron gas of N electrons of mass m each in system of size L ´ L.

78. The density of states between energy e and e + de is

(A) de (B) de (C) de (D) ede

79. The ground state energy E₀ of the system in terms of the Fermi energy E_F and the number of electrons N is given by

(A) NE_F (B) NE_F (C) NE_F (D) NE_F

Statement for Linked Answer Questions 80 and 81:

The rate of a clock in a spaceship "Suryashakti" is observed from earth to be 3/5 of the rate of the clocks on earth

80. The speed of the spaceship "Suryashakti" relative to earth is

(A) (B) (C) (D)

81. The rate of a clock in a spaceship ''Aakashganga'' is observed from earth to be 5/13 of the rate of the clocks on earth. If both Aakashganga and Suryashakti are moving in the same direction relative to someone on earth, then the speed of Aakashganga relative to Suryashakti is

(A) (B) (C) (D)