1. One of the eigenvalues of the matrix e^A is e^a, where . The product of the other two eigenvalues of e^A is

(a) e^2a

(b) e^–a

(c) e^–2a

(d) 1

Ans. (d)

Sol. Eigenvalues of matrix A are a, a and –a. The product of two other eigenvalues of A are

e^ae^–a = 1

Alternatively

2. The polynomial f(x) = 1 + 5x + 3x² is written as a linear combination of the Legendre polynomials

as . The value of c₀ is

(a) 1/4

(b) 1/2

(c) 2

(d) 4

Ans. (c)

Sol. f(x) = 1 + 5x + 3x²

1 = P₀(x) x = P₁(x)

f(x) = P₀(x) + 5P₁(x) + 2P₂(x) + P₀(x)

= 2P₀(x) + 5P₁(x) + 2P₂(x)

= c₀P₀(x) + c₁P₁(x) + c₂P₂(x) c₀ = 2

3. The value of the integral , where C is a circle of radius , traversed counter-clockwise, with centre at z = 0, is

(a) 4

(b) 4i

(c) 2i

(d) 0

Ans. (b)

Sol.

4. A particle of mass m, moving along the x-direction, experiences a damping force , where is a constant and v is its instantaneous speed. If the speed at t = 0 is v₀, the speed at time t is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. From Newton's second law

integrating both sides gives

where c is a constant of integration

Since v = v₀ at t = 0, we obtain

5. The integral is evaluated from the point (–1, 0) to (1, 0) along the contour C, which is an arc of the parabola y = x² – 1, as shown in the figure.

The value of I is

(a) 0

(b) 2 sinh 1

(c) e²ⁱ sinh 1

(d) e + e^–1

Ans. (b)

Sol.

6. In terms of arbitrary constant A and B, the general solution to the differential equation

, is

(a)

(b)

(c) y = Ax + Bx³

(d)

Ans. (d)

Sol. The given equation is Euler-Cauchy differential equation. The characteristic equation of

Therefore the general solution is

7. In the attractive Kepler problem described by the central potential , where k is a positive constant), a particle of mass m with a non-zero angular momentum can never reach the centre due to the centrifugal barrier. If we modify the potential to

one finds that there is a critical value of the angular momentum v below which there is no centrifugal barrier. This value of is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

when introduce new potential

For critical value

For critical value

8. The time period of a particle of mass m, undergoing small oscillations around x = 0, in the potential , is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

9. Consider the decay A B + C of a relativistic spin –½ particle A. Which of the following statements is true in the rest frame of the particle A?

(a) The spin of both B and C may be 1/2

(b) The sum of the masses of B and C is greater than the mass of A

(c) The energy of B is uniquely determined by the masses of the particles

(d) The spin of both B and C may be integral

Ans. (c)

Sol. Correct option is (c)

10. Two current-carrying circular loops, each of radius R, are placed perpendicular to each other, as shown in the figure below.

The loop in the xy-plane carries a current I₀ while that in the xz-plane carries a current 2I₀. The resulting magnetic field at the origin is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Field due to loop in xy plane is

Field due to loop in xz plane is

11. An electric dipole of dipole moment is placed at the origin in the vicinity of two charges +q and –q at (L, b) and (L, –b), respectively, as shown in the figure below.

The electrostatic potential at the point is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Potential due to dipole

Potential due to +q charge

Potential due to –q charge

Resultant

12. A monochromatic and linearly polarized light is used in a Young's double slit experiment. A linear polarizer, whose pass axis is at an angle 45º to the polarization of the incident wave, is placed in front of one of the slits. If I_max and I_min, respectively, denote the maximum and minimum intensities of the interference pattern on the screen, the visibility, defined as the ratio , is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

13. An electromagnetic wave propagates in a nonmagnetic medium with relative permitivity = 4. The magnetic field for this wave is

,

where H₀ is a constant. The corresponding electric field is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

14. The ground state energy of an anisotropic harmonic oscillator described by the potential

(a) 5/2

(b) 7/2

(c) 3/2

(d) 1/2

Ans. (b)

Sol.

For ground state

n_x = 0, n_y = 0, n_z = 0

15. The product of uncertainties in the position and momentum of a simple harmonic oscillator of mass m and angular frequency in the ground state , is . The value of the product in the state , where l is a constant and is the momentum operator) is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Correct option is (c)

16. Let the wavefunction of the electron in a hydrogen atom be

where are the eigenstates of the Hamiltonian in the standard notation. The expectation value of the energy in this state is

(a) –10.8 eV

(b) –6.2 eV

(c) –9.5 eV

(d) –5.1 eV

Ans. (d)

Sol.

17. Three identical spin-½ particles of mass m are confined to a one-dimensional box of length L, but are otherwise free. Assuming that they are non-interacting, the energies of the lowest two energy eigenstates, in units of , are

(a) 3 and 6

(b) 6 and 9

(c) 6 and 11

(d) 3 and 9

Ans. (b)

Sol.

For ground state configuration 2 particle has engine E₀ and 1 particle has engine 4E₀

Total energy is 2 × E₀ + 1 × 4E₀ = 6E₀

For first excited state configuration, 1 particles has engine E₀ and 2 particle has engine 4E₀

Total energy 1 × E₀ + 2 × 4E₀ = 9E₀

Lowest two energy levels are 6E₀, 9E₀ respectively, where

18. The heat capacity C_V at constant volume of a metal, as a function of temperature, is , where and are constants. the temperature dependence of the entropy at constant volume is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

19. The rotational energy levels of a molecule are where l = 0, 1, 2,.... and I₀ is its moment of inertia. The contribution of the rotational motion to the Helmholtz free energy per molecule, at low temperature in a dilute gas of these molecules, is approximately

(a)

(b)

(c) –k_BT

(d)

Ans. (d)

Sol.

For low temperature, higher temperature can be neglected

20. The vibrational motion of a diatomic molecule may be considered to be that of a simple harmonic oscillator with angular frequency . If a gas of these molecules is at a temperature T, what is the probability that a randomly picked molecule will be found in its lowest vibrational state?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

21. Consider an ideal Fermi gas in a grand canonical ensemble at a constant chemical potential. The variance of the occupation number of the single particle energy level with mean occupation number is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Note: This may also be divided using simple Bernoulli distribution.

22. Consider the following circuit, consisting of an RS flip-flop and two AND gates.

Which of the following connections will allow the entire circuit to act as a JK flip-flop?

(a) connect Q to pin 1 and to pin 2

(b) connect Q to pin 2 and to pin 1

(c) connect Q to K input and to J input

(d) connect Q to J input and to K input

Ans. (b)

Sol. Correct option is (b)

23. The truth table below gives the value Y(A, B, C), where A, B and C are binary variables.

The output Y can be represented by

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

24. A sinusoidal signal is an input to the following circuit:

Which of the following graphs best describes the output waveform?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. In CE transistor output has phase charge of

25. A sinusoidal voltage having a peak value of V_P is an input to the following circuit, in which the DC voltage is V_b.

Assuming an ideal diode, which of the following best describes the output waveform?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Correct option is (c)

26. The Green's function G(x, x') for the equation , with the boundary values y(0) = 0 and y(1) = 0, is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

p(x') = 1

x₁ = 1, y₂ = x

y₁ = x, y₂ = 1 – x

A = –1

27. A 4 × 4 complex matrix A satisfies the relation , where I is the 4 × 4 identify matrix. The number of independent real parameters of A is

(a) 32

(b) 10

(c) 12

(d) 16

Ans. (d)

Sol. Given that

Let A = 2B then

Therefore,

This shows that B is a unitary matrix. The number of independent real parameters needed to specify an n×n unitary matrix is n². Thus, the number of independent parameter needed to specify matrix B is 4² = 16.

Now, the number of independent parameters needed to specify matrix A is same as that of matrix B.

Thus the number of independent parameters needed to specify A is 16

28. The contour C of the following integral , in the complex z-plane is shown in the figure below.

This integral is equivalent to an integral along the contours

(a)

(b)

(c)

(d)

Ans. (c)

Sol. z = 1, 3 are branch points ∞ is not a branch point 1 branch cut 3

29. The value of the integral , evaluated using the trapezoidal rule with a step size of 0.2, is

(a) 0.30

(b) 0.39

(c) 0.34

(d) 0.27

Ans. (c)

Sol.

30. The motion of a particle in one-dimension is described by the Lagrangian

in suitable units. The value of the action along the classical path from x = 0 at t = 0 to x = x₀ at t = t₀, is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

From Lagrangian equation of motion

The solution is x = A sin t + B cos t

t = 0 x = 0 B = 0

x = A sin t

31. The Hamiltonian of a classical one-dimensional harmonic oscillator is , in suitable units. The total time derivative of the dynamical variable is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

32. A relativistic particle of mass m and charge e is moving in a uniform electric field of strength . Starting from rest at t = 0, how much time will it take to reach the speed c/2?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

33. In an inertial frame, uniform electric and magnetic fields and are perpendicular to each other and satisfy (in suitable units). In another inertial frame, which moves at a constant velocity with respect to the first frame, the magnetic field is . In the second frame, an electric field consistent with the previous observations is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

34. Electromagnetic wave of angular frequency is propagating in a medium in which, over a band of frequencies, the refractive index is , where is a constant. The ratio of the group velocity to the phase velocity at is

(a) 3

(b) 1/4

(c) 2/3

(d) 2

Ans. (a)

Sol.

35. A rotating spherical shell of uniform surface charge and mass density has total mass M and charge Q. If its angular momentum is L and magnetic moment is µ, then the ratio µ/L is

(a) Q/3M

(b) 2Q/3M

(c) Q/2M

(d) 3Q/4M

Ans. (c)

Sol.

36. Consider the operator A_x = L_yp_z – L_zp_y, where L_i and p_i denote, respectively, the components of the angular momentum and momentum operators. The commutator [A_x, x], where x is the x-component of the position operator, is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. A_x = L_yp_z – L_zp_y, L_y = zp_x – xp_z, L_z = xp_y – yp_x

[A_x, x] = [L_yp_z, x] – [L_zp_y, x] = [L_y, x]p_z – [L_z, x]p_y

= [zp_x, x]p_z + [yp_x, x]p_y = z[p_z, x]p_z + y[p_x, x]p_y

37. A one-dimensional system is described by the Hamiltonian , where > 0. The ground state energy varies as a function of as

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Using Bohr-Sommerfield theory,

38. If the position of the electron in the ground state of a hydrogen atom is measured, the probability that it will be found at a distance r > a₀ (a₀ being Bohr radius) is nearest to

(a) 0.91

(b) 0.66

(c) 0.32

(d) 0.13

Ans. (b)

Sol.

39. A system of spin-½ particles is prepared to be in the eigenstate of with eigenvalue +1. The system is rotated by an angle of 60º about the x-axis. After the rotation, the fraction of the particles that will be measured to be in the eigenstate of with eigenvalue +1 is

(a) 1/3

(b) 2/3

(c) 1/4

(d) 3/4

Ans. (d)

Sol. Rotation with angle about x axis

If is measure on , the measurement is +1 with probability and –1 with probability

40. The Hamiltonian of a one-dimensional using model of N spins (N large) is

where the spin and J is a positive constant. At inverse temperature , the correlation function between the nearest neighbour spins is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

For such an using model for N >> 1

41. At low temperatures, in the Debye approximation, the contribution of the phonons to the heat capacity of a two-dimensional solid is proportional to

(a) T²

(b) T³

(c) T^1/2

(d) T^3/2

Ans. (a)

Sol. The dispersion relation of phonons is

The phonon specific heat in d-dimension is

For 2 dimensional solid d = 2

42. A particle hops on a one-dimensional lattice with lattice spacing a. The probability of the particle to hop to the neighbouring site to its right is p, while the corresponding probability to hop to the left is q = 1 – p. The root-mean-squared deviation in displacement after N steps, is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The standard deviation of Binomial distribution

Step size = 2a (L & R)

Mean square displacement

43. The energy levels accessible to a molecule have energies E₁ = 0, E₂ = and E₃ = 2 (where is a constant). A gas of these molecules is in thermal equilibrium at temperature T. The specific heat at constant volume in the high temperature limit varies with temperature as

(a) 1/T³

(b) 1/T^3/2

(c) 1/T

(d) 1/T²

Ans. (d)

Sol.

44. The input V_i to the following circuit is a square wave as shown in the following figure.

Which of the waveforms best describes the output?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Differentiator circuit.

Correct option is (c)

45. The amplitude of a carrier signal of frequency f₀ is sinusoidally modulated at a frequency f' << f₀. Which of the following graphs best describes its power spectrum?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. 2sin A cos B = sin (A + B) + sin (A – B)

46. The standard deviation of the following set of data:

{10.0, 10.0, 9.9, 9.9, 9.8, 9.9, 9.9, 9.9, 9.8, 9.9}

(a) 0.10

(b) 0.07

(c) 0.01

(d) 0.04

Ans. (b)

Sol.

and standard deviation is

47. The diatomic molecule HF has an absorption line in the rotational band at 40 cm^–1 for the isotope ¹⁸F. The corresponding line for the isotope ¹⁹F will be shifted by approximately

(a) 0.05 cm^–1

(b) 0.11 cm^–1

(c) 0.33 cm^–1

(d) 0.01 cm^–1

Ans. (b)

Sol. For ¹HF¹⁸ : 2B₁ = 40 cm^–1 B₁ = 20 cm^–1 and reduce mass is

For ¹HF¹⁹ : The reduce mass is and rotational constant is B₂.

Since,

Thus, 2B₂ = 39.889 cm^–1

Shift in spectral line = 2B₁ – 2B₂ = 40 – 39.889 = 0.11 cm^–1

48. The excited state (n = 4, l = 2) of an electron in an atom may decay to one or more of the lower energy levels shown in the diagram below.

Of the total emitted light, a fraction 1/4 comes from the decay to the state (n = 3, l = 1), will be

(a) 3/4

(b) 1/2

(c) 1/4

(d) 0

Ans. (a)

Sol. According to the selection rule for electric dipole, transition is , the transition from (n = 4, l = 2) to (n = 3, l = 0) and (n = 3, l = 2) is forbidden. If I is the intensity of the total emitted light and , therefore, .

49. The volume of an optimal cavity is 1 cm³. The number of modes it can support within a bandwidth of 0.1 nm, centred at = 500 nm, is of the order of

(a) 10³

(b) 10⁵

(c) 10¹⁰

(d) 10⁷

Ans. (c)

Sol. Number of Laser modes

50. Barium Titanate (BaTiO₃) crystal has a cubic perovskite structure, where the Ba²⁺ ions are at the vertices of a unit cube, the O^2– ions are at the centres of the faces while the Ti²⁺ is at the centre. The number of optical phonon modes of the crystal is

(a) 12

(b) 15

(c) 5

(d) 18

Ans. (a)

Sol. Effective number of atoms per unit cell is

Total degree of freedom 5 × 3 = 15

The number of Acoustical phonon modes = 3

The number of optical phonon modes 15 – 3 = 12

51. The dispersion relation of optical phonons in a cubic crystal is given by – ak², where and a are positive constants. The contribution to the density of states due to these phonons with frequencies just below is proportional to

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

52. A silicon crystal is doped with phosphorus atoms. (The binding energy of a H atom is 13.6 eV, the dielectric constant of silicon is 12 and the effective mass of electron in the crystal is 0.4 m_e). The gap between the donor energy level and the bottom of the conduction band is nearest to

(a) 0.01 eV

(b) 0.08 eV

(c) 0.02 eV

(d) 0.04 eV

Ans. (d)

Sol.

53. Assume that poin-nucleon scattering at low energies, in which isospin is conserved, is described by the effective interaction potential , where F(r) is a function of the radial separation r and and denote, respectively, the isospin vectors of a pion and the nucleon. The ratio of the scattering cross-sections corresponding to total isospins I = 3/2 and 1/2, is

(a) 3/2

(b) 1/4

(c) 5/4

(d) 1/2

Ans. (d)

Sol. The isospin of poin is = 1

The isospin of nucleon is

There are three different -mesons

and two nucleons, a proton and a neutron

we can write the states corresponding

The states corresponding to are

The best possible answer is option (d)

None of the options is matched.

54. A nucleus decays by the emission of a gamma ray from an excited state of spin-parity 2⁺ to the ground state with spin-parity 0⁺. What is the type of the corresponding radiation?

(a) magnetic dipole

(b) electric quadrupole

(c) electric dipole

(d) magnetic quadrupole

Ans. (b)

Sol. I_i = 2 I₊ = 0

L = 2 and parity change

The transition is of electric quadupole (E₂) nature.

55. The low lying energy levels due to the vibrational excitations of an even-even nucleus are shown in the figure below:

The spin-parity j^p of the level E₁ is

(a) 1⁺

(b) 1^–

(c) 2^–

(d) 2⁺

Ans. (d)

Sol. Quadrupole oscillations are the lowest order nuclear vibration mode. The quanta of vibrational energy are called phonons. A quadrupole phonon carries 2 units of angular momentum. Therefore, the parity is P = (–1)² = +ve

Also, the even-even ground state is O⁺. The 1 phonon excited state is 2⁺. The 2 phonons excited states are 0⁺, 2⁺, 4⁺. Thus correct option is (a)

0.56 ___2⁺:1-phonon

0 ___0⁺: Ground state

MeV