1. Consider the differential equation with the initial condition y(0) = 0. Then the Laplace transform Y(s) of the solution y(t) is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Given

Taking Laplace transform of both sides

We obtain

Since, y(0) = 0, we obtain

2. Consider the matrix equation

The condition for existence of a non-trivial solution, and the corresponding normalised solution (upto a sign) is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. We know that the matrix equation, AX = 0, where A is the given matrix and X is a column vector has a non-zero solution if and only if |A| = 0

we do not need to perform further calculation.

3. Consider the real function f(x) = 1/(x² + 4). The Taylor expansion of f(x) about x = 0 converges

(a) for all values of x

(b) for all values of x except x = +2

(c) in the region –2 < x < 2

(d) for x > 2 and x < –2

Ans. (c)

Sol.

Thus the Taylor's series of f(x) is times the binomial series of

Now, the binomial series converges if

Since |x| + 2 is always greater than 0,

4. Let A be a non-singular 3 × 3 matrix, the columns of which are denoted by the vectors and , respectively. Similarly, and denote the vectors that form the corresponding columns of (A^T)^–1. Which of the following is true?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. We can take any 3 × 3 non singular matrix in order to avoid long calculation.

We see that

5. The number of linearly independent power series solutions, around x = 0, of the second order linear differential equation , is

(a) 0 (this equation does not have a power series solution)

(b) 1

(c) 2

(d) 3

Ans. (c)

Sol.

Similar to the form Bessel D.E. (n = 0)

The given differential equation will have two linearly independent power series solution i.e.

6. A disc of mass m is free to rotate in a plane parallel to the xy plane with an angular velocity about a massless rigid rod suspended from the roof of a stationary car (as shown in the figure below). The rod is free to orient itself along any direction.

The car accelerates in the positive x-direction with an acceleration a > 0. Which of the following statements is true for the coordinates of the centre of mass of the disc in the reference frame of the car?

(a) only the x and the z coordinates change

(b) only the y and the z coordinates change

(c) only the x and the y coordinates change

(d) all the three coordinates change

Ans. (d)

Sol. Initial torque on the disc

Since, the disc has angular momentum, it will move in y-direction.

So, that change in angular momentum occurs in y-direction due to initial torque.

After a litte later,

Since. torque is in all directions, therefore disc will move in all direction. So all cordinates will change.

7. A cyclist, weighing a total of 80 kg with the bicycle, pedals at a speed of 10 m/s. She stops pedalling at an instant which is taken to be t = 0. Due to the velocity dependent frictional force, her velocity is found to very as , where t is measured in seconds. When the velocity drops to 8 m/s, she starts pedalling again to maintain a constant speed. The energy expended by her in 1 minute at this (new) speed, is

(a) 4 kJ

(b) 8 kJ

(c) 16 kJ

(d) 32 kJ

Ans. (b)

Sol. m = 80 kg, v_i = 10 m/s

When velocity is 8 m/s

Force at the moment, when velocity is 8 m/s

To maintain constant speed, the cyclist must apply the this much force

Therefore, power applied

Energy expended in one minute

8. A light signal travels from a point A to a point B, both within a glass slab that is moving with uniform velocity (in the same direction as the light) with speed 0.3c with respect to an external observer. If the refractive index of the slab is 1.5, then the observer will measure the speed of the signal as

(a) 0.67c

(b) 0.81c

(c) 0.97c

(d) c

Ans. (b)

Sol.

9. A monoatomic gas of volume V is in equilibrium in a uniform vertical cylinder, the lower end of which is closed by a rigid wall and the other by a frictionless piston. The piston is pressed lightly and released. Assume that the gas is a poor conductor of heat and the cylinder and piston are perfectly insulating. If the cross-sectional area of the cylinder is A, the angular frequency of small oscillations of the piston about the point of equilibrium, is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Piston at equilibrium position

mg = PA …(1)

For adiabatic process

If x be displacement of piston from equilibrium position, dV = Ax

Restoring force,

10. The normalized wavefunction of a particle in three dimensions is given by , where a > 0 is a constant. The ratio of the most probable distance from the origin to the mean distance from the origin, is. [You may use .]

(a) 1/3

(b) 1/2

(c) 3/2

(d) 2/3

Ans. (d)

Sol.

Method-1:

Radial probability density,

At most probable radial distance r_mp,

11. The state vector of a one-dimensional simple harmonic oscillator of angular frequency , at time to = 0, is given by , where and are the normalized ground state and the second excited state, respectively. The minimum time t after which the state vector is orthogonal to , is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

12. The normalized wavefunction in the momentum space of a particle in one dimension is , where and are real constants. The uncertainty in measuring its position is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. We can easily solve the question using dimension analysis.

Given : Momentum space between wave function,

Dimension of p = Dimension of

13. Let x denote the position operator and p the canonically conjugate momentum operator of a particle. The commutator , where and are constants, is zero if

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

14. Two point charges +3Q and –Q are placed at (0, 0, d) and (0, 0, 2d), respectively, above an infinite grounded conducting sheet kept in the xy plane. At a point (0, 0, z), where z >> d, the electrostatic potential of this charge configuration would approximately be

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The electrostatic potential at (0, 0, z)

15. A rectangular piece of dielectric material is inserted partially into the (air) gap between the plates of a parallel plate capacitor. The dielectric piece will

(a) remain stationary where it is placed

(b) be pushed out from the gap between the plates

(c) be drawn inside the gap between the plates and its velocity does not change sign

(d) execute an oscillatory motion in the region between the plates

Ans. (d)

Sol. The capacitor has width a and length l.

The combined capacitance of the capacitor is

Therefore, the electrostatic potential energy of the system is

Therefore, the force on the dielectric slab is

Therefore, the force on the slab will be inwards and when it reaches at the other end the force on it will be opposite direction. So, slab will execute oscillations.

16. An electromagnetic wave is travelling in free space (of permittivity ) with electric field .

The average power (per unit area) crossing planes parallel to 4x + 3y = 0 will be

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Now, the pointing vector,

Therefore, the average power per unit area crossing planes parallel to 4x + 3y = 0 is

17. A plane electromagnetic wave from within a dielectric medium (with and ) is incident on its boundary with air, at z = 0. The magnetic field in the medium is , where and k are positive constants.

The angles of reflection and refraction are, respectively,

(a) 45º and 60º

(b) 30º and 90º

(c) 30º and 60º

(d) 60º and 90º

Ans. (b)

Sol.

18. The dispersion relation of a gas of spin fermions in two dimensions is , where E is the energy, is the wave vector and v is a constant with the dimension of velocity. If the Fermi energy at zero temperature is , the number of particles per unit area is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

The total number of particle is

At zero temperature (T = 0K), we have

Therefore, the number of particles per unit area is

19. The relation between the internal energy U, entropy S, temperature T, pressure p, volume V, chemical potential µ and number of particles N of a thermodynamic system is dU = TdS – pdV + µdN. That U is an exact differential implies that

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Given dU = TdS – pdV + µdN

Since, dU is an exact different, we have

20. The number of microstates of a gas of N particles in a volume V and of internal energy U, is given by

,

(where a and b are positive constants). Its pressure P, volume V and temperature T, are related by

(a)

(b)

(c) PV = Nk_BT

(d) P(V – Nb) = Nk_BT

Ans. (d)

Sol.

The Boltzmann's entropy is

For combined I and II low of thermodynamics, we have

TdS = dU + PdV – µdN

21. Consider a system of identical atoms in equilibrium with blackbody radiation in a cavity at temperature T. The equilibrium probabilities for each atom being in the ground state and an excited state are P₀ and P₁, respectively. Let n be the average number of photons in a mode in the cavity that causes transition between the two states. Let and denote, respectively, the squares of the matrix elements corresponding to the atomic transitions and . which of the following equations hold in equilibrium?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Here the transition will 3 different types:

   1. Stimulated absorbtion transition: It depends on the number of atom in state , photon number inside the cavity and the square of the matrix elements. Therfore, the rate of transition of atom from is .

   2. Simulated emission transition: It depends on the number of atom in state , photon number inside the vacity and the square of the matrix elements. Therfore, the rate of transition of atom from is .

   3. Spontaneous emission transition: It does not depend on the number of photo inside the cavity. It depends on the number of atom in state and the square of the matrix elements. Therefore, the rate of transition of atom from is .

At equilibrium the rate of transition of atom to the is equal to .

Therefore,

22. In the circuit below the voltages V_BB and V_CC are kept fixed, the voltage measured at B is a constant, but that measured at A fluctuates between a few µV to a few mV.

From these measurements it may be inferred that the

(a) base is open internally

(b) emitter is open internally

(c) collector resistor is open

(d) base resistor is open

Ans. (d)

Sol. The correct option is (d).

23. The full scale voltage of an n-bit Digital-to-Analog Converter is V. The resolution that can be achieved in it is

(a) V/(2ⁿ – 1)

(b) V/(2ⁿ + 1)

(c) V/2²ⁿ

(d) V/n

Ans. (a)

Sol.

n = number of bits.

24. The spring constant k, of a spring of mass m_s, is determined experimentally by loading the spring with mass M and recording the time period T, for a single oscillation. If the experiment is carried out for different masses, then the graph that correctly represents the result is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. If mass of the spring is m_s and a block of mass M is attached with it. The equivalent mass of the system is equal to

Therefore, the time period of oscillation is given by

25. A Zener diode with an operating voltage of 10V at 25ºC has a positive temperature coefficient of 0.07% per ºC of the operating voltage. The operating voltage of this Zener diode at 125ºC is

(a) 12.0 V

(b) 11.7 V

(c) 10.7 V

(d) 9.3 V

Ans. (c)

Sol. Given, V_z = 10 volt

Therefore, the final zener voltage,

F_{z final} = V_z + = (10 + 0.7) volt = 10.7 volt

26. Consider an element of the group SU(2), where is any one of the parameters of the group. Under an infinitesimal change , it changes as . To order , the matrix should always be

(a) positive definite

(b) real symmetric

(c) hermitian

(d) anti-hermitian

Ans. (d)

Sol.

Here, m is one of the Pauli spin matrices, since Pauli matrices are hermitian,

taken complex conjugate, so matrix should anti-hermitian.

27. The differential equation , with the initial condition y(0) = 0, is solved using Euler's method. If y_E(x) is the exact solution and y_N(x) the numerical solution obtained using n steps of equal length, then the relative error is proportional to

(a) 1/n²

(b) 1/n³

(c) 1/n⁴

(d) 1/n

Ans. (d)

Sol.

Euler's method, y_i = y_{i – 1} + hf (x_{i – 1}, y_{i – 1})

Since, 0, 5, 14, 30, .... different from square terms

Ar, x₀ = 0 x₁ = x₀ + h = h x₂ = x₀ + 2h = 2h x₃ = x₀ + 3h = 3h

x_{n – 1} = x₀ + (n – 1)h = (n – 1)h. Now, x_n = nh

28. The interval [0, 1] is divided into n parts of equal length to calculate the integral using the trapezoidal rule. The minimum value of n for which the result is exact, is

(a) 2

(b) 3

(c) 4

(d)

Ans. (a)

Sol.

29. The generating function G(t, x) for the Legendre polynomials P_n(t) is

If the function f(x) is defined by the integral equation , it can be expressed as

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Differentiating both sides,

30. A particle moves in one dimension in a potential where k and are constants. Which of the following curves best describes the trajectories of this system in phase space?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Since, threre is minimum (potential well). So, motion can be bounded.

Therefore, phase spacae trajectory will be closed loop for energies less than the height of potential well. However for higher energies will not be closed loops.

Further, E = K.E. + P.E.

E = T + V(x)

T = E – V(x)

31. Let (x, p) be the generalized coordinate and momentum of a Hamiltonian system. If new variables (X, P) are defined by and , where and are constants, then the conditions for it to be a canonical transformation, are

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

The transformation will be canonical if the Poisson bracket of X and P is equal to 1.

Therefore, [X, P]_{x, P} = 1

Both satisfied simultaneously. But none of the given option satisfied the above two conditions simultaneously. However, only option (c) satifies = 1. So, it may be taken as correct option.

32. Consider a set of particles which interact by a pair potential V = ar⁶, where r is the inter-particle separation and a > 0 is a constant. If a system of such particles has reached virial equilibrium, the ratio of the kinetic to the total energy of the system is

(a) 1/2

(b) 1/3

(c) 3/4

(d) 2/3

Ans. (c)

Sol. Using virial theorem,

33. In an inertial frame S, the magnetic vector potential in a region of space is given by (where a is a constant) and the scalar potential is zero. The electric and magnetic fields seen by an inertial observer moving with a velocity with respect to S, are, respectively [In the following .]

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

We know that

According to Lorentz transformation,

E'_x = E_x = 0 B'_x = B_x = 0

34. In the rest frame S₁ of a point particle with electric charge q₁, another point particle with elecric charge q₂ moves with a speed v parallel to the x-axis at a perpendicular distance l. The magnitude of the electromagnetic force felt by q₁ due to q₂ when the distance between them is minimum, is

[In the following .]

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The electric field at O due to point charge q₂ is

Therefore, the force left by the charge q₁ is

35. A circular current carrying loop of radius a carries a steady current. A constant electric charge is kept at the centre of the loop. The electric and magnetic fields, and respectively, at a distance d vertically above the centre of the loop satisfy

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Electric and magnetic field along the axis

36. A phase shift of 340º is observed when a beam of particles of energy 0.1 MeV is scattered by a target. When the beam energy is changed, the observed phase shift is 60º. Assuming that only s-wave scattering is relevant and that the cross-section does not change with energy, the beam energy is

(a) 0.4 MeV

(b) 0.3 MeV

(c) 0.2 MeV

(d) 0.15 MeV

Ans. (b)

Sol. According to partial wave analysis, total cross-section is given as

Assuming that, only s-wave scattering is relevant, we obtain

As does not change with energy 'E', therefore,

37. The Hamiltonian of a two-level quantum system is . A possible initial state in which the probability of the system being in that quantum state does not change with time, is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The possible initial state in which probability of the systembeing in that quantum statesdoes not change with time, wll be eigenstate of the Hamiltonian H i.e.

38. Consider a one-dimensional infinite square well

If a perturbation

is applied, then the correction to the energy of the first excited state, to first order in is nearest to

(a) V₀

(b) 0.16 V₀

(c) 0.2 V₀

(d) 0.33 V₀

Ans. (d)

Sol. Unperturbed system: 1-D infinite potential well

First order correction to energy

Nearest answer is 0.33 V_o.

39. The energy eigenvalues E_n of a quantum system in the potential V = cx⁶ (where c > 0 is a constant), for large values of the quantum number n, varies as

(a) n^4/3

(b) n^3/2

(c) n^5/4

(d) n^6/5

Ans. (b)

Sol. According to WKB approximation,

40. Consider a quantum system of non-interacting bosons in contact with a particle bath. The probability of finding no particle in a given single particle quantum state is 10^–6. The average number of particles in that state is of the order of

(a) 10³

(b) 10⁶

(c) 10⁹

(d) 10¹²

Ans. (b)

Sol. f_oc () = the probability of occupancy = the average number of particles in a state

f_un () = the probability of find in number of particles in a state = 10^–6

The order of average number of particles in a state = (10^–6 –1) ~ 10⁶

41. A closed system having three non-degenerate energy levels with energies , is at temperature T. For , the probability of finding the system in the state with energy E = 0, is

(a) 1/(1 + 2 cosh 2)

(b) 1/(2 cosh 2)

(c)

(d) 1/cosh 2

Ans. (a)

Sol. The probability of finding the system in the state with energy E = 0 is

42. Two non-degenerate energy levels with energies 0 and are occupied by N non-interacting particles at a temperature T. Using classical statistics, the average internal energy of the system is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The average internal energy of the system is

43. In the circuit below, D₁ and D₂ are two silicon diodes with the same characteristics.

If the forward voltage drop of a silicon diode is 0.7 V, then the value of the current is

(a) 18.6 mA

(b) 9.3 mA

(c) 13.95 mA

(d) 14.65 mA

Ans. (c)

Sol. Given D₁ and D₂

Two identical diotes

V_y(each) = 0.7 V

Equidistant figure

To find I₁ + I_D1 ?

I₁ + ID₂ + ID₂

ID₁ +ID₂ = 465 mA

I₁ + ID₁ = 9.3 + 4.65 mA = 13.95 mA

44. The circuit below comprises of D-flip flops. The output is taken from Q₃, Q₂, Q₁ and Q₀, as shown in the figure.

The binary number given by the string Q₃Q₂Q₁Q₀ changes for every clock pulse that is applied to the CLK input. If the output is initialized at 0000, then the corresponding sequence of decimal numbers that repeats itself, is

(a) 3, 2, 1, 0

(b) 1, 3, 7, 14, 12, 8

(c) 1, 3, 7, 15, 12, 14, 0

(d) 1, 3, 7, 15, 14, 12, 8, 0

Ans. (d)

Sol. Note Its a 4-bit Johnson counter

45. Two physical quantities T and M are related by the equation , where a and b are constant parameters. The variation of T as a function of M was recorded in an experiment to determine the value of a graphically. Let m be the slope of the straight line when T² is plotted vs M, and δm be the uncertainty in determining it. The uncertainty in determining a is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We have,

46. The sensitivity of a hot cathode pressure gauge is 10 mbar^–1. If the ratio between the numbers of the impinging charged particles to emitted electrons is 1:10, then the pressure is

(a) 10 mbar

(b) 10^–1 mbar

(c) 10^–2 mbar

(d) 10² mbar

Ans. (c)

Sol. The pressure (P) of a hot cathode pressure gauge

where, i⁺ current due to impinging particles

i^– current due to emitted electrons

c = sensitivity of gauge

47. The Zeeman shift of the energy of a state with quantum numbers L, S, J and m_J is

where B is the applied magnetic field, g_S is the g-factor for the spin and µ_B/h = 1.4 MHz-G^–1, where h is the Planck constant. The approximate frequency shift of the S = 0, L = 1 and m_J = 1 state, at a magnetic field of 1 G, is

(a) 10 MHz

(b) 1.4 MHz

(c) 5 MHz

(d) 1.8 MHz

Ans. (b)

Sol. In Anomalous zeeman effect

48. The separations between the adjacent levels of a normal multiplet are found to be 22 cm^–1 and 33 cm^–1. Assume that the multiplet is described well by the L-S coupling scheme and the Lande's interval rule, namely E(J) – E(J –1) = AJ, where A is a constant. The term notations for this multiplet is

(a) ³P_{0, 1, 2}

(b) ³F_{2, 3, 4}

(c) ³G_{3, 4, 5}

(d) ³D_{1, 2, 3}

Ans. (d)

Sol. By Lande Interval Rule E(J) – E(J – 1) = AJ i.e. proportional to higher J.

Seperation between adjacement laves are 22 cm^–1 and 33 cm^–1, i.e. their ratio

The ratio is given by option (d) i.e. ³D_{1, 2, 3} as seperation between ³D¹, ³D₂ and ³D₃ are in the ration 2 : 3(i.e. proportional to higher J).

49. If the fine structure splitting between the 2²P_3/2 and 2²P_1/2 levels in the hydrogen atom is 0.4 cm^–1, the corresponding splitting in Li²⁺ will approximately by

(a) 1.2 cm^–1

(b) 10.8 cm^–1

(c) 32.4 cm^–1

(d) 36.8 cm^–1

Ans. (c)

Sol. Fine structure splitting is proportional to Z₄ i.e. . Nowfor H, Z = 1 and for Li⁺⁺, Z = 3.

50. A crystal of MnO has NaCl structure. It has a paramagnetic to anti-ferromagnetic transition at 120K. Below 120 K, the spins within a single [111] plane are parallel but the spins in adjacent [111] planes are antiparallel. If neutron scattering is used to determine the lattice constants, respectively, d and , below and above the transition temperature of MnO then

(a) d = d'/2

(b)

(c) d = 2d'

(d)

Ans. (c)

Sol. MnO has NaCl type structure which is paramagnetic at room temperature

T_c = 1200 K

T < T_c Antiferromagnetic (ordered)

T > T_c Paramagentic (disorderd)

[111] plane.

Lattice constant,

The distortion below, T_c = 120 K is apparently an exchange striction which compress the crystal along , increasing the nearest nieghbour and decreasing the next nearest neigherbour distances. In antiferromagnetic state the lattice parameter will be double paramagnetic state.

51. A metallic nanowire of length l is approximated as a one-dimensional lattice of N atoms with lattice spacing a. If the dispersion of electrons in the lattice is given as E(k) = E₀ – 2t cos ka, where E₀ and t are constants, then the density of states inside the nanowire depends on E as

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Given : E(k) = E₀ – 2t cos ka …(i)

The density of state is one-dimensional is

Also, the equation (i) gives, dE = 2 at sin kadk

Since, the number of atoms would be large in the nanowire, we have

52. Consider a two-dimensional material of length and width w subjected to a constant magnetic field B applied perpendicular to it. The number of charge carriers per unit area may be expressed as , where k is a positive real number and q is the carrier charge. Then the Hall resistivity is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Number of charge carrier per unit area,

Hall resistivity,

53. The spin-parity assignments for the ground and first excited states of the isotope , in the single particle shell model, are

(a) (1/2)^– and (3/2)^–

(b) (5/2)⁺ and (7/2)⁺

(c) (3/2)⁺ and (5/2)⁺

(d) (3/2)^– and (5/2)^–

Ans. (d)

Sol.

N = 29: (¹s_1/2 )² (¹p_3/2 )⁴ (¹p_1/2 )² (¹d_5/2 )⁶ (²s_1/2 )² (¹d_3/2 )⁴ (¹f_7/2 )⁸ (²p_3/2 )¹for ground state

54. The first excited state of the rotational spectrum of the nucleus has an energy 45 keV above the ground state. The energy of the second excited state (in keV) is

(a) 150

(b) 120

(c) 90

(d) 60

Ans. (a)

Sol. Rotational energy,

For first excited stage J = 2

Therefore, for second excited state J = 4, energy is

55. Which of the following processes is not allowed by the strong interaction but is allowed by the weak interaction?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Since strangeness is not conserved, so it is neither no electromagntic

In weak interactions, involving strange particles = 1 or = + 1 or –1. However, this does not necessarily hold in second order weak reactions, where there are mixes of K⁰ and mesons. Since this reaction is second order weak recation involving mixes of K⁰ and , thus = 2 is allowed and it is a weak interaction.