PYQ 2012 - VedPrep

GATE PHYSICS 2012
Previous Year Question Paper with Solution.

1. Identify the CORRECT statement for the following vectors

(a) The vectors and are linearly independent

(b) The vectors and are linearly dependent

(d) The vectors and are normalized

Ans. (a)

Sol. Given vectors are

There, and are linearly independent.

Further and are not orthogonal

not normalized

2. Two uniform thin rods of equal length L, and masses and M₂ are joined together along the length. The moment of inertia of the combined rod of length 2L about an axis passing through the mid-point perpendicular to the length of the rod is,

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Moment of inertia of a thin rod about an axis passing through its one end and perpendicular to its length is , where m = mass, L = length of rod

Therefore, for given system

3. The space-time dependence of the electric field of a linearly polarized light in free space is given by where and k are the amplitude, the angular frequency and the wavevector, respectively. The time averaged energy density associated with the electric field is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

4. If the peak output voltage of a full wave rectifier is 10 V, its d.c. voltage is

(a) 10.0V

(b) 7.07V

(d) 3.18V

Ans. (c)

Sol. V_DC for full wave rectifier

5. A particle of mass m is confined in a two dimensional square well potential of dimension a. This potential V(x, y) is given by

The energy of the first excited state for this particle is given by,

(a)

(b)

(c)

(d)

Ans.

Sol. Energy eigenvalues of the particle confined in a two dimensional square well potential of width 2a is

Energy of the first excited state of the particle is

Note: The question in the original paper is wrong.

6. The isothermal compressibility, of an ideal gas at temperature T₀ and volume V₀, is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The isothermal compressibility is defined as .

It actually explains how does the volume of a substance change when pressure on it changes at constant temperature explains how easily a material can be compressed.

7. The ground state of sodium atom (¹¹Na) is a ²S_1/2 state. The difference in energy levels arising in the presence of a weak external magnetic field B, given in terms of Bohr magneton, µ_B is

(a) µ_BB

(b) 2µ_BB

(d) 6µ_BB

Ans. (b)

Sol.

For weak field, this state will split into two (2J + 1 = 2) energy levels.

The energy of splitted levels by

Therefore, the difference in energy levels = µ_BB + µ_BB = 2µ_BB.

8. For an ideal Fermi gas in three dimensions, the electron velocity V_F at he Fermi surface is related to electron concentration n as,

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The Fermi energy of a Femi gas can be given as

Now, this Fermi energy is related to velocity v_F as the Fermi surface as

9. Which one of the following sets corresponds to fundamental particles ?

(a) proton, electron and neutron

(b) proton, electron and photon

(d) quark, electron and meson

Ans.

Sol. Electron, photon and neutrino do not have substructures and hence are the fundamental particles.

10. In case of a Geiger-Muller (GM) counter, which one of the following statements is CORRECT?

(a) Multiplication factor of the detector is of the order of 10¹⁰.

(b) Type of the particles detected can be identified.

(d) Operating voltage of the detector is few tens of Volts.

Ans. (a)

Sol. Correct option is (a)

11. A plane electromagnetic wave traveling in free space is incident normally on a glass plate of refractive index 3/2. If there is no absorption by the glass, Its reflectivity is

(a) 4%

(b) 16 %

(d) 50%

Ans. (a)

Sol. Reflectivity for normal incidence is given as

12. A Ge semiconductor is doped with acceptor impurity concentration of 10¹⁵atoms /cm³. For the given hole mobility of 1800 cm²/V-s, the resistivity of this material is:

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

13. A classical gas of molecules each of mass m, is in thermal equilibrium at the absolute temperature, T. The velocity components of the molecules along the Cartesian axes are v_x, v_y and v_z. The mean value of (v_x + v_y)² is:

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

From equipartition theorem,

14. In a central force field the trajectory of a particle of mass m and angular momentum L in plane polar coordinates is given by

Where, is the eccentricity of the particle's motion. Which one of the following choices for gives rise to a parabolic trajectory ?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. From mathematical concepts of conic section it is known that for parabola = 1

15. Identify the CORRECT energy band diagram for silicon doped with Arsenic. Here CB, VB, E_D and E_F conduction band, valence band, impurity level and Fermi level, respectively.

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Arsenic is penta-valent.

Note: In N-type semi conductor as doping increases Fermi level moves towards the conduction band.

16. The first stokes line of a rotational Raman spectrum is observed at 12.96 cm^–1. Considering the rigid rotor approximation, the rotational constant is given by

(a) 6.48 cm^–1

(b) 3.24 cm^–1

(d) 1.62 cm^–1

Ans. (c)

Sol. The position of first strokes line of a rotational Ramam spectrum 6B =12.96 cm^–1 B = 2.16 cm^–1

Hence, the rotational constant B = 2.16 cm^–1

17. The total energy, E of an ideal non-relativistic Fermi gas in three dimensions is given by where N is the number of particles and V is the volume of the gas. Identify the CORRECT equation of state ( P being the pressure),

(a)

(b)

(d)

Ans. (b)

Sol.

18. Consider the wavefunction for a fermionic system consisting of two spin-half particles. The spatial part of the wavefunction is given by

Where and are single particle states. The spin part of the wavefunction with spin states should be

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For a system of identical Fermions, total wave function should be anti-symmetric in nature. Since, the given spatial part of the wave function is symmetric under particle exchange, therefore the spin part of the wave function is anti-symmetric under particle exchange.

19. The electric and the magnetic fields and , respectively corresponding to the scalar potential and vector potential are

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

20. Consider the following OP-AMP circuit.

Which one of the following correctly represents the output V_out corresponding to the input V_in ?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

When V_in is greater than V_a; output V₀ = + V_SAT

21. Deuteron has only one bound state with spin parity 1⁺, isospin 0 and electric quadrupole moment 0.286 efm². These data suggest that the nuclear forces are having

(a) only spin and isospin dependence

(b) no spin dependence and no tensor components

(d) spin dependence along with tensor components

Ans. (d)

Sol. The nuclear forces are same for neutron-neutron, neutron-proton and proton-proton inside nucleus. Therefore, it suggest that these forces are isopin dependent. The spin and quadruple moment is non-zero, therefore, they have spin dependence along with tensor components.

22. A particle of unit mass moves along the x-axis under the influence of a potential, V(x) = x(x – 2)². The particle is found to be in stable equilibrium at the point x = 2 The time period of oscillation of the particle is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Given : V(x) = x(x – 2)², m = 1

= [2(x – 2) + 2 (x – 2) + 2x]_{x = 2} = 4

23. Which one of the following CANNOT be explained by considering a harmonic approximation for the lattice vibrations in solids ?

(a) Debye's law

(b) Dulong petit's law

(d) Thermal expansion

Ans. (d)

Sol. Expression for potential of thermal expansion is given by U(x) = (cx² – gx³ – fx⁴). This is a non-harmonic approximation in the lattice vibration in solids. Here x is the displacement of their equilibrium position.

24. A particle is constrained to move in a truncated harmonic potential well (x > 0) as shown in the figure. Which one of the following statements is CORRECT ?

(a) The parity of the first excited state is even

(b) The parity of the ground state is even

(d) The first excited state energy is

Ans. (d)

Sol. Since the potential under which the particle is moving i.e. truncated harmonic potential, is not system metric about x = 0, therefore the wave function for the different states of the particle should not have a definite parity.

Energy eignvalues of a particle moving under a truncated harmonic potential are

Therefore, ground state energy and first excited state energy will be and respectively.

25. The number of independent components of the symmetric tensor A_ij with indices i, j = 1, 2, 3 is

(a) 1

(b) 3

(d) 9

Ans. (c)

Sol. For symmetric tensor, A_ij = A_ji

Therefore, if 3 of the off-diagonal elements are known then the other 3 off-diagonal elements are also known. Hence, number of independent = 3 diagonal + 3 off-diagonal = 6

26. Consider a system in the unperturbed state described by the Hamiltonian, . The system is subjected to a perturbation of the form , where . The energy eigenvalues of the perturbed system using the first order perturbation approximation are

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Energy eigenvalues corresponding to unperturbed Hamiltonian is E₀ = 1, 1.

Since the unperturbed Hamiltonian and perturbing Hamiltonian commutes with each other, therefore the eigenvalues of the perturbing Hamiltonian will give the first order correction to energy.

Eigenvalues equation of H' ?

Energy eigenvalues of the perturbd system using the first order perturbation approximation, will be 1 + 2 and 1 respectively.

27. Inverse susceptibility as a function of temperature, T for a material undergoing paramagnetic to ferromagnetic transition is given in the figure, where O is the origin. the values of the Curie constant, C and the Weiss molecular field constant, , in CGS units, are

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The magnetic susceptibility of a ferromagnetic material is given by

28. A plane polarized electromagnetic wave in free space at time t = 0 is given by . The magnetic field is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

29. The eigenvalues of the matrix are

(a) 0,1,1

(b)

(c)

(d)

Ans. (b)

Sol.

Only option (b) satisfies this.

30. Match the typical spectroscopic regions specified in List-I with the corresponding type of transitions in List-II and find the correct answer using the codes given below the list :

List-I List-II

P. Infrared region 1. Electronic transitions involving valence

electrons

Q. Ultraviolet visible region 2. Nuclear transitions

R. X-ray region 3. Vibrational transitions of molecules

S. - rays region 4. Transitions involving inner shell

electrons

Codes :

P Q R S

(a) 1 3 2 4

(b) 2 4 1 3

(d) 4 1 2 3

Ans. (c)

Sol. P. Infrared region 3. Vibration transitions of molecules

Q. Ultraviolet visible region 1. Electronic transitions involving valence electrons.

R. X-ray region 4. Transitions involving inner shell electrons

S. -rays region 2. Nuclear transitions

31. In the following circuit, for the output voltage to V₀ = (–V₁ + V₂/2) the ratio R₁/R₂ is

(a) 1/2

(b) 1

(d) 3

Ans. (d)

Sol.

By applying voltage division rule

Applying KCL at node - (a)

2V_a – V₁ =V₀; Put V_a value

Compare this equation with given V₀ value

32. The terms arising from 2s¹3d¹ electronic configuration in J–J scheme are

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The given electronic configuration = 2s¹3d¹

Thus, the total angular momentum (J₁) for 2s¹ electron

The total angular momentum (J₂) for 3d¹ electron

Thus, total angular momentum J in j-j, coupling is given by

33. In the following circuit, the voltage drop across the ideal diode in forward bias condition is 0.7 V.

The current passing through the diode is

(a) 0.5 mA

(b) 1.0 mA

(d) 2.0 mA

Ans. (b)

Sol.

Applying KVL rule,

Loop-(1): 24 = 12 i₁ + 6(i₁ – i₂) …(i)

Loop-(2): –0.7 = 3.3 i₂ + 6(i₂ – i₁) …(i)

Solving (i) and (ii), we get i₂ = 1 mA

34. Choose the CORRECT statement form the following

(a) Neutron interacts through electromagnetic interaction

(b) Electron does not interact through weak interaction

(d) Quark interacts through strong interaction but not through weak interaction

Ans. (a)

Sol. Charged particles interact through electromagnetic. In weak all particles interact and neutrino interact through weak.

35. A rod of proper length oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Velocity of rod with respect to observer

length of rod in observer's frame is

36. A simple cubic crystal with lattice parameter a_c undergoes transition into a tetragonal structure with lattice parameter , below a certain temperature. The ratio of the interplanar spacings of (101) planes for the cubic and the tetragonal structure is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Relation between lattice parameters : a, b and c; and interplaner spacing of (hkl) plane is given by

For simple cubic crystal, a = b = c = a_c, d₍₁₀₁₎ =?

37. Consider the following circuit in which the current gain of the transistor is 100.

Which one of the following correctly represents the load line (collector current I_c with respect to collectr-emitter voltage V_CE) and Q-point of this circuit?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

For drawing DC load line High

Applying KVL at emitter collector loop.

15 = 900 I_C + V_CE + 100 I_C; Load line equation

Put V_CE = 0 I_C = 15 mA

Put I_C = 0 V_CE = 15 V

Now KVL at Base-Emitter loop

15 = 100 × I_B + 0.7 +0.1 × IE

I_E = I_B; I_E = ( + 1)I_B; I_E = 101 × I_B

I_B = 0.13 mA

I_C = × I_B = 100 × 0.13

I_C =13 mA

Now KVL at o/a (output)

15 = 0.9 × I_C + V_CE + 0.1 × I_E

Put value of I_C and I_E

V_CE = 2V

38. Consider a system whose three energy levels are given by 0, The energy level is two-fold degenerate and the other two are non-degenerate. The partition function of the system with is given by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The partition function is

39. Two infinitely extended homogeneous isotropic dielectric media (medium-1 and medium –2 with dielectric constants and , respectively) meet at the z = 0 plane as shown in the figure. A uniform electric field exists everywhere. For the electric field is given by The interface separating the two media is charge free.

The electric displacement vector in the medium-2 is given by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The correct option is (b)

40. The ground state wavefunction for the hydrogen atom is given by , where is the Bohr radius.

The plot of the radial probability density, P(r) for the hydrogen atom in the ground state is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Radial probability density for the ground state of Hydrogen atom is

For small 'r', the term 'r²' will dominant and for large 'r', the term e^–2r/a0, will dominate.

41. Total binding energies of O¹⁵, O¹⁶ and O¹⁷ are 111.96 MeV, 127.62 MeV and 131.76 MeV, respectively. The energy gap between ¹p_1/2 and ¹d_1/2 neutron shells for the nuclei whose mass number is close to 16, is:

(a) 4.1 MeV

(b) 11.5 MeV

(d) 19.8 MeV

Ans. (b)

Sol. For (1p_1/2), O¹⁵ ₀n¹ O¹⁶

Therefore, binding energy of n in (1p_1/2)

= B.E. (O¹⁶) – B.E. (O¹⁵) (as n has zero B.E.)

= 127.62 – 111.96 = 15.66 MeV

For (1d_5/2), O¹⁶ + ₀n¹ O¹⁷

Therefore, binding energy of n in (¹d_5/2)

= B.E.(O¹⁷) – B.E.(O¹⁶) = 131.76 – 127.62 = 4.14 MeV

Therefore, energy difference = 15.66 – 4.14 = 11.52 MeV

42. A particle of mass m is attached to a fixed point 'O' by a weightless inextensible string of length a. It is rotating under the gravity as shown in the figure.

The Lagrangian of the particle is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Hamiltonian is defined as

43. Given where is a constant vector and is the position vector. The value of , where C is a circle of unit radius centered at origin is,

(a) 0

(b)

(c)

(d) 1

Ans. (c)

Sol.

44. The value of the integral , using the contour C of circle with unit radius |z| = 1 is:

(a) 0

(b)

(c)

(d)

Ans. (d)

Sol.

f(z) has an essential singular point at z = 0.

45. A paramagnetic system consisting of N spin-half particles, is placed in an external magnetic field. It is found that N/2 spins are aligned parallel and the remaining N/2 spins are aligned antiparallel to the magnetic field. The statistical entropy of the system is,

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The number of microstates when N particles are distributed into N₁ and N₂ particles are:

46. The equilibrium vibration frequency for an oscillator is observed at 2990 cm^–1. The ratio of the frequencies corresponding to the first and the fundamental spectral lines is 1.96. Considering the oscillator to be anharmonic, the anharmonicity constant is

(a) 0.005

(b) 0.02

(d) 0.1

Ans. (b)

Sol. The vibrational terms (energies in wave number unit m^–1 or cm^–1) of diatomic are given by

is the wave-number spacing of energy levels that would occur if the potential were a parabola.

x_e is called an harmonicity constant that arise if potential curve is not a perfect parabola.

Fundamental band corresponds to the transition, v (= 0) v(1), therefore wave number

First spectral lines (or first overtone band) corresponds to

47. At a certain temperature T, the average speed of nitrogen molecules in air is found to be 400 m/s. The most probable and the root mean square speeds of the molecules are, respectively,

(a) 355 m/s, 434 m/s

(b) 820 m/s, 917 m/s

(d) 422 m/s, 600 m/s

Ans. (a)

Sol.

Now from equation (i) and (iii),

Common data for Q.48 and Q.49

The wavefunction of a particle moving in free space is given by,

48. The energy of the particle is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

49. The probability current density for the real part of the wavefunction is

(a) 1

(b)

(c)

(d) 0

Ans. (d)

Sol. Real part of the wave function is Re() = cox kx + 2 cos kx.

Probability current density for the real part of the wave function is

Common data for Q.50 and Q. 51

The dispersion relation for a one dimensional monatomic crystal with lattice spacing a, which interacts via nearest neighbour harmonic potential is given by

where A is a constant of a appropriate unit.

50. The group velocity at the boundary of the first Brillouin zone is

(a) 0

(b) 1

(c)

(d)

Ans. (a)

Sol.

At the boundary of the first Brillouin zone, i.e. at , equation (i) is gives

At the boundary of first Brillouin zone i.e. for , equation (ii) gives

Standing waves produced at Billouin zone boundary.

51. The force constant between the nearest neighbour of the lattice is (M is the mass of the atom)

(a)

(b)

(d) 2MA²

Ans. (a)

Sol. The dispersion relation ( versus K) for one-dimensional monatomic lattice is given by

Comparing equation (iv) and equation (i), we get (where, C is force constant)

Statement for Linked Answer Q.52 and Q.53

In a hydrogen atom, consider that the electronic charge is uniformly distributed in a spherical volume of radius a(= 0.5 × 10^–10 m) around the proton. The atom is placed in a uniform electric field E = 30 × 10⁵ V/m. Assume that the spherical distribution of the negative charge remains undistorted under the electric field.

52. In the equilibrium condition, the separation between the positive and the negative charge centers is

(a) 8.66 × 10^–16 m

(b) 2.60 × 10^–15 m

(d) 8.66 × 10^–15 m

Ans. (c)

Sol. If d be separation (iv) and equation (i), we get (where, C is force constant)