1. In the scattering of some elementary particles, the scattering cross-section is found to depend on the total energy E and the fundamental constants h (Planck's constant) and c (the speed of light in vacuum). Using dimensional analysis, the dependence of on these quantities is given by

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Scattering cross-section s has a dimension of area i.e. [Length]²

Dimension of

2. If , then x is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Given:

3. The function of a complex variable z has

(a) a simple pole at 0 and poles of order 2 at for n = 1, 2, 3...

(b) a simple pole at 0 and poles of order 2 at and for n = 1, 2, 3...

(c) poles of order 2 at , n = 0, 1, 2, 3...

(d) poles of order 2 at +n, n = 0, 1, 2, 3...

Ans. (b)

Sol. Given:

Conditions of singularity:

For n = 0, singular point will be z = 0. The Laurent series expansion of f(z) about z = 0, contains the hightest negative power to be Therefore, z = 0 is a simple pole of f(z).

For n = 1, 2, 3........., singular points will be The corresponding Laurent series expansion contains the highest negative to be

Therefore, are poles of order 2.

4. The Fourier transform of f(x) is . If , where is the Dirac delta-function (and prime denotes derivative), what is ?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Given: Fourier transform of

Fourier transform of

Fourier transform of

5. The solution of the differential equation , with initial condition x = 0 at t = 0 is

(a)

(b)

(c)

(d) x = 1 – cos 2t, t > 0

Ans. (c)

Sol. Given:

Applying the condition i.e. at

So,

For the given options, x = sin 2t

Since, given is continuous is nature, then the solution of the differential equation and its derivative should be continuous in nature. Only option (c) satisfies this condition.

6. A particle moves in three-dimensional space in a central potential V(r) = kr⁴, where k is a constant. The angular frequency for a circular orbit depends on its radius R as

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

for Circular motion,

7. Two masses, m each, are placed at the points (x, y) = (a, a) and (–a, –a), and two masses, 2m each, are placed at the points (a, –a) and (–a, a). the principal moments of inertia of the system are

(a) 2ma², 4ma²

(b) 4ma², 8ma²

(c) 4ma², 4ma²

(d) 8ma², 8ma²

Ans. (b)

Sol. Axes 1 and 2 shown in the figure are the two principal axes.

Therefore, principal moment of inertial are 4ma², 8ma².

8. The Lagrangian of a system is given by where m and k are positive constants. The frequencies of its normal modes are

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

For frequencies of normal modes :

9. Consider a particle of mass m moving with a speed v. If T_R denotes the relativistic kinetic energy and T_N its non-relativistic approximation, then the value of (T_R – T_N)/T_R for v = 0.01c, is

(a) 1.25 × 10^–5

(b) 5.0 × 10^–5

(c) 7.5 × 10^–5

(d) 1.0 × 10^–4

Ans. (c)

Sol.

10. A hollow metallic sphere of radius a, which is kept at a potential V₀, has a charge Q at its centre. The potential at a point outside the sphere, at a distance r from the centre, is

(a) V₀

(b)

(c)

(d)

Ans. (d)

Sol. Laplace equation in the region r > a

Equation (2), (3) put in equation (1)

11. Consider a charge Q at the origin of 3-dimensional coordinate system. The flux of the electric field through the curved surface of a cone that has a height h and a circular base of radius R (as shown in the figure) is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Flux through a disc due to point charge is

Therefore, flux through base of cone

Therefore, using Gauss law, we get

12. Given a uniform magnetic field (where B₀ is a constant), a possible choice for the magnetic vector potential A is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. We know that,

13. A beam of unpolarized light in a medium with dielectric constant is reflected from a plane interface formed with another medium of dielectric constant . The two media have identical magnetic permeability. If the angle of incidence is 60º, then the reflected light

(a) is plane polarized perpendicular to the plane of incidence

(b) is plane polarized parallel to the plane of incidence

(c) is circularly polarized

(d) has the same polarization as the incident light

Ans. (a)

Sol. Brewster angle for the given case

Since, angle of incidence is equal to Brewster angle and the reflected wave will be plane polarized perpendicular to the plane of incidence.

14. A Hermitian operator has two normalised eigenstates and with eigenvalues 1 and 2, respectively. The two states and are such that and . Which of the following are possible values of and ?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

15. The ground state energy of a particle of mass m in the potential , where L and V₀ are constants (with ) is approximately

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Since, the contribution from the terms above in can be neglected.

It is a 1-D linear harmonic oscillator

With angular frequency,

Therefore, ground state energy will be

16. Let denote the eigenstates of a hydrogen atom in the usual notation. The state is an eigenstate of

(a) L², but not of the Hamiltonian or L_z

(b) the Hamiltonian, but not of L² or L_z

(c) the Hamiltonian, L² and L_z

(d) L² and L_z, but not of the Hamiltonian

Ans. (b)

Sol.

Given state is an eigenstate of Hamiltonian, but not of and .

17. The Hamiltonian for a spin-½ particle at rest is given by , where and are Pauli spin matrices and E₀ and are constants. The eigenvalues of this Hamiltonian are

(a)

(b)

(c) E₀ (doubly degenerate)

(d)

Ans. (a)

Sol. Given:

Eigenvalue equation :

18. The heat capacity of (the interior of) a refrigerator is 4.2 kJ/K. The minimum work that must be done to lower the internal temperature from 18ºC to 17ºC when the outside temperature is 27ºC will be

(a) 2.20 kJ

(b) 0.80 kJ

(c) 0.30 kJ

(d) 0.14 kJ

Ans. (d)

Sol. Heat removed from inside of refreigerator

Q₂ = 4.2(18 – 17) = 4.2 kJ

Heat given to outside atmosphere

[For minimum work, we can take ]

The minimum work = Q₁ – Q₂ = 4.3298 – 4.2 = 0.1298 kJ = 0.13 or 0.14 kJ

19. For a system of independent non-interacting one-dimensional oscillators, the value of the free energy per oscillator, in the limit T 0, is

(a)

(b)

(c)

(d) 0

Ans. (a)

Sol. For a system of N independent non-interacting linear oscillators.

Partition function,

Average internal energy,

Helmholtz free energy of the system, F = U – TS as

Helmholtz free energy per oscillator as

20. The partition function of a system of N Ising is , where and are functions of temperature, but are independent of N. If , the free energy per spin in the limit is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The partition function of a system of N Ising spins,

Helmholtz free energy,

Helmholtz free energy per oscillator

21. The Hamiltonian of a system of N non-interacting spin-½ particles is , where are the components of i^th spin along an external magnetic field B. At a temperature T such that , the specific heat per particle is

(a)

(b)

(c) k_B (ln 2)²

(d)

Ans. (d)

Sol. The Hamiltonian of i^th spin

H_i = –µ₀BS_i, S_i = +1

The possible internal energy of the i^th spin

22. If the reverse bias voltage of a silicon varactor is increased by a factor of 2, the corresponding transition capacitance

(a) increases by a factor of

(b) increases by a factor of 2

(c) decreases by a factor of

(d) decreases by a factor of 2

Ans. (c)

Sol. Transition capacitance,

If

CT decreases by a factor .

23. In the schematic figure given below, the initial values of 4 bit shift registers A and B are 1011 and 0010 respectively. The values of SO_A and SO_B after the pulse T₂ are respectively.

(a) 1110 and 1001

(b) 1101 and 1001

(c) 1101 and 1100

(d) 1110 and 1100

Ans. (d)

Sol.

After the pulse T₂ shift becomes low. Hence, values of SO_A and SO_B after the pulse T₂ will be 1110 and 1000 respectively.

24. If the parameters y and x are related by y = log(x), then the circuit that can be used to produce an output voltage V₀ varying linearly with x is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. y = log x

To produce output voltage varying linearly with x. The expression of output will be exponential.

Only option (c) satisfies this condition.

25. Two data sets A and B consist of 60 and 10 readings of a voltage measured using voltmeters of resolution of 1mV and 0.5mV respectively. The uncertainty in the mean voltage obtained from the data sets A and B are U_A and U_B, respectively. If the uncertainty of the mean of the combined data sets is U_AB, then which of the following statements is correct?

(a) U_AB < U_A and U_AB > U_B

(b) U_AB < U_A and U_AB < U_B

(c) U_AB > U_A and U_AB < U_B

(d) U_AB > U_A and U_AB > U_B

Ans.

Sol.

26. The Hermite polynomial H_n(x) satisfies the differential equation . The corresponding generating function satisfies the equation

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Given: Generating function

27. A function f(x) satisfies the differential equation , where is positive. The Fourier transform of f, and the solution of the equation are, respectively,

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Given:

For the equation becomes

Region-1:

Region-2:

The above form of solution is taken because for the existence of fourier transform, as f(x) should be continuous at x = a i.e.

Then

We get,

General form:

So, the solution of the differential equation will be

Fourier transform of f(x) is

28. For a dynamical system governed by the equation , with |x| < 1,

(a) x = –1 and x = 1 are both unstable fixed points

(b) x = –1 and x = 1 are both stable fixed points

(c) x = –1 is an unstable fixed point and x = 1 is a stable fixed point

(d) x = –1 is stable fixed point and x = 1 is an unstable fixed point

Ans. (c)

Sol.

For fixed points,

Let . Therefore,

At unstable fixed point

At stable fixed point

Obviously, at x = –1. So, x = –1 is unstable fixed point and at x = ±1.

So, x = ±1 is stable fixed point.

29. The value of the integral , evaluated using Simpson's rule with h = 2, is

(a) 0.565

(b) 0.620

(c) 0.698

(d) 0.736

Ans. (a)

Sol. Given: a = 0, b = 8. h = 2, number of sub-intervals

For

According to simpon's rule,

= 0.565 (approx.)

30. A canonical transformation (p, q) (P, Q) is performed on the Hamiltonian via the generating function . If Q(0) = 0, which of the following graphs shows schematically the dependence of Q(t) on t?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

From equation (1), we get

New Hamiltonian,

Hamiltonian's equation,

31. A distant source, emitting radiation of frequency , moves with a velocity 4c/5 in a certain direction with respect to a receiver (as shown in the figure).

The upper cut-off frequency of the receiver is 3/2. Let be the angle as shown. For the receiver to detect the radiation, should at least be

(a)

(b)

(c)

(d)

Ans. (b)

Sol. According to relativistic doppler effect.

32. The Lagrangian of a particle moving in a plane is given in Cartesian coordinates as . In polar coordinates the expression for the canonical momentum p_r (conjugate to the radial coordinate r) is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. In plane polar coordinates :

Eq. (1), (2) put in Eq. (3)

33. A small magnetic needle is kept at (0, 0) with its moment along the x-axis. Another small magnetic needle is at the point (1, 1) and is free to rotate in the xy-plane. In equilibrium the angle between their magnetic moments is such that

(a) tan = 1/3

(b) tan = 0

(c) tan = 3

(d) tan = 1

Ans. (c)

Sol. Second dipole will not experience torque due to first dipole if its dipole moment is parallel to magnetic field of first dipole.

Angle between and

34. A dipole of moment , oscillating at frequency , radiates spherical waves. The vector potential at large distance is . To order (1/r) the magnetic field at a point is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

35. The frequency dependent dielectric constant of a material is given by where A is a positive constant, the resonant frequency and the damping coefficient. For an electromagnetic wave of angular frequency , which of the following is true? (Assume that )

(a) There is negligible absorption of the wave

(b) The wave propagation is highly dispersive

(c) There is strong absorption of the electromagnetic wave

(d) The group velocity and the phase velocity will have opposite sign

Ans. (a)

Sol.

Since, is almost real quantity. So, the obsorption will be very less.

36. A hydrogen atom is subjected to the perturbation where a₀ is the Bohr radius. The change in the ground state energy to first order in is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Given: (a₀ is Bohr radius)

First order energy correction to ground state energy

[We know ground state state eigenfunction is ]

Using the standard integration,

37. A positron is suddenly absorbed by the nucleus of a tritium atom to turn the latter into a He⁺ ion. If the electron in the tritium atom was initially in the ground state, the probability that the resulting He⁺ ion will be in its ground state is

(a) 1

(b)

(c)

(d)

Ans. (d)

Sol. Wave function for ground state of H-like atom is

Initial state ground state of atom (Z = 1)

Final state = ground state of He⁺ ion (Z = 2)

Required probability

Now,

Using the standard integration,

So, probability

38. The product f the uncrtainties for a particle in the state (where denotes an eigenstate of L² and L_z) will be a minimum for

(a) a = + ib

(b) a = 0 and b = 1

(c)

(d) a = +b

Ans. (d)

Sol.

39. The ground energy of a particle in the potential V(x) = g|x|, estimated using the trial wave function

(where g and c are constants) is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Given:

Applying the normalization condition i.e.

40. An ensemble of non-interacting spin-½ particles is in contact with a heat bath at temperature T and is subjected to an external magnetic field. Each particle can be in one of the two quantum states of energies . If the mean energy particle is , then the free energy per particle is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The energy of each particle

The mean energy of a particle

The Helmholtz free energy of a particle,

41. Which of the following graphs shows the qualitative dependence of the free energy f(h,T) of a ferromagnet in an external magnetic field h, and at a fixed temperature T < T_c, where T_c is the critical temperature?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Change of free energy,

At fixed temperature,

[For ferromagnet ]

42. Consider a random walker on a square lattice. At each step the walker moves to as nearest neighbour site with equal probability for each of the four sites. The walker starts at the origin and takes 3 steps. The probability that during this walk no site is visited more than once is

(a) 12/27

(b) 27/64

(c) 3/8

(d) 9/16

Ans. (d)

Sol. In each step, the random walker can move to 4 different postions. After 3 steps, there will be 4³ = 64 ways in which the random wlaker can move in different ways.

Step-1: Random wilker can move in 4 directions i.e. North, south, east or west.

Step-2: For a specific direction in step-1, the random walker can move in 3 different direction such that no site more than once.

Step-3: For a specific direction in second step, the random walker can move in 3 different directions number of favourable ways = 4 × 3 × 3 = 36

Required probability

43. Consider an n-MOSFET with the following parameters: current drive strength K = 60 µA/V², breakdown voltage BV_DS = 10V, ratio of effective gate width to the channel length and threshold voltage V_th = 0.5V. In the circuit given below, this n-MOSFET is operating in the

(a) ohmic region

(b) cut-off region

(c) saturation region

(d) breakdown region

Ans. (c)

Sol. For saturation region V_Ds> V_gs – V_T

Hence, operating in saturation.

44. The state diagram corresponding to the following circuit is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

For xy = 00, If Q_n = 0 then Q_{n + 1} = 1

And If Q_n = 1 then Q_{n + 1} = 0

For, xy = 11, Q_n = 0 Q_{n + 1} = 0

45. A sinusoidal signal of peak to peak amplitude 1V and unknown time period is input to the following circuit for 5 seconds duration. If the counter measures a value (3E8)_H in hexadecimal then the time period of the input signal is

(a) 2.5 ms

(b) 4 ms

(c) 10 ms

(d) 5 ms

Ans. (d)

Sol. (3EB)_H = (1000)_D

Time duration = 5 sec and in 5 sec counter reads = 1000

Now, time period

46. The first order diffraction peak of a crystalline solid occurs at a scattering angle of 30º when the diffraction pattern is recorded using an x-ray beam of wavelength 0.15nm. If the error in measurements of the wavelength and the angle are 0.01 nm and 1º respectively, then the error in calculating the inter-planar spacing will approximately be

(a) 1.1 × 10^–2 nm

(b) 1.3 × 10^–4 nm

(c) 2.5 × 10^–2 nm

(d) 2.0 × 10^–3 nm

Ans. (a)

Sol.

Bragg's law,

Error in the estimation of d is given by

47. The dispersion relation of electrons in a 3-dimensional lattice in the tight binding approximation is given by, where a is the lattice constant and are constants with dimension of energy. The effective mass tensor at the corner of the first Brillouin zone is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

48. A thin metal film of dimension 2 mm × 2 mm contains 4 × 10¹² electrons. The magnitude of the Fermi wavevector of the system, in the free electron approximation, is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. 2D fermi energy is given by

where, N = Number of free electron

49. For an electron moving through a one-dimensional periodic lattice of periodicity a, which of the following corresponds to an energy eigenfunction consistent with Bloch's theorem?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. According to Block's theorem, will be periodic function if it satisfies:

Only option (b) satisfies this condition.

50. The LS configuration of the ground state of ¹²Mg, ¹³Al, ¹⁷Cl and ¹⁸Ar are, respectively.

(a) ³S₁, ²P_1/2, ²P_1/2 and ¹S₀

(b) ³S₁, ²P_3/2, ²P_3/2 and ³S₁

(c) ¹S₁, ²P_1/2, ²P_3/2 and ¹S₀

(d) ¹S₀, ²P_3/2, ²P_1/2 and ³S₁

Ans. (c)

Sol. Electronic configuration of ¹²Mg : 1s²2s²2p⁶3s²

Electronic configuration of ¹⁸Ar : 1s²2s²2p⁶3s²3p⁶

Any electronic configuration having closed subshells, has ground state term ¹S₀.

¹³Al : 1s²2s²2p⁶3s²3p¹

Ground state: ²P_1/2

¹⁷Cl : 1s²2s²2p⁶3s²3p⁵

Ground state : ²P_3/2

51. For a two levels system, the population of atoms in the upper and lower levels are 3 × 10¹⁸ and 0.7 × 10¹⁸, respectively. If the coefficient of stimulated emission is 3.0 × 10⁵ m³/W-s³ and the energy density is 9.0 J/m³-Hz, the rate of stimulated emission will be

(a) 6.3 × 10¹⁶ s^–1

(b) 4.1 × 10¹⁶ s^–1

(c) 2.7 × 10¹⁶ s^–1

(d) 1.8 × 10¹⁶ s^–1

Ans. (*)

Sol.

Given: Coefficient of stimulated emission, B₂₁ = 3 × 10⁵ m³/W-s³

Energy density u(v) = 9J/m³ – Hz

Rate of stimulated emission

None of the options is correct.

52. The first ionization potential of K is 4.34 eV, the electron affinity of Cl is 3.82 eV and the equilibrium separation of KCl is 0.3nm. The energy required to dissociate a KCl molecule into a K and a Cl atom is

(a) 8.62 eV

(b) 8.16 eV

(c) 4.28 eV

(d) 4.14 eV

Ans. (c)

Sol. Energy required to dissociate KCl atom into K⁺ and Cl^–

To convert K⁺ into K atom, an electron is to be added and to convert Cl^– into Cl atom, an electron is should be released.

Required dissociation energy = (4.8 – 4.34 + 3.82)eV = 4.28 eV

Correct option is (c)

53. Consider the following processes involving free particles

(i)

(ii)

(iii)

(iv)

Which of the following statements is true?

(a) Process (i) obeys all conservation laws

(b) Process (ii) conserves baryon number, but violates energy-momentum conservation

(c) Process (iii) is not allowed by strong interactions, but is allowed by weak interactions

(d) Process (iv) conserves baryon number, but violates lepton number conservation

Ans. (b)

Sol. (i)

Letpon number is not conserved.

(ii)

So, Baryon number is conserved, but energy and momentum cannot be conserved simultaneously. Since both and n are much heavier as compared to , if we consider and n at rest, initial momentum is zero but must have kinetic energy and this will violate momentum conservation,

(iii)

hence, this process is not allowed by any interaction.

(iv)

This reaction is allowed.

We see that statement (ii) is true.

54. The electric quadrupole moment of an odd proton nucleus is , where j is the total angular momentum. Given that R₀ = 1.2 fm, what is the value, in barn, of the quadrupole moment of the ²⁷Al nucleus in the shell model?

(a) 0.043

(b) 0.023

(c) 0.915

(d) 0

Ans. (a)

Sol. For ²⁷Al, Z = 13, N = 14

The nucleonic configuration for protons is

So,

Hence, the quadrupole moment,

55. Of the nuclei of mass number A = 125, the binding energy calculated from the liquid drop model (given that the coefficients for the Coulomb and the asymmetry energy are a_c = 0.7 MeV and a_sym = 22.5 MeV respectively) is a maximum for

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Binding energy,

For most stable nucleus,

For A = 125,