1. A matrix is given by . The eigenvalues of the M are

(a) Real and positive

(b) Purely imaginary with modulus 1

(c) Complex with modulus 1

(d) Real and negative

Ans. (c)

Sol.

Eigenvalue equation for M:

2. The value of at which the first-order peak in X-ray ( = 1.53 Å) diffraction corresponding to (111) plane of a simple cubic structure with the lattice constant, a = 2.65 Å, is approximately

(a) 15º

(b) 30º

(c) 45º

(d) 60º

Ans. (b)

Sol.

3. The magnetic field associated with the electric field vector is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

4. Consider the following truth table:

The logic expression for F is:

(a) AB + BC + CA

(b)

(c)

(d)

Ans. (d)

Sol. According to truth table

5. The following are the plots of the temperature dependence of the magnetic susceptibility for three different samples.

The plots A, B and C correspond to

(a) Ferromagnet, paramagnet and diamagnet, respectively

(b) Paramagnet, diamagnet and ferromagnet, respectively

(c) Ferromagnet, diamagnet and paramagnet, respectively

(d) Diamagnet, paramagnet and ferromagnet, respectively

Ans. (b)

Sol. Magnetic susceptibility,

6. The equation of a surface of revolution is . The unit normal to the surface at the point is:

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Equation of surface,
[i.e. F(x, y, z) = constant]

Unit normal to surface,

Unit normal at

7. A gas of molecules each having mass 'm' is in thermal equilibrium at a temperature T. Let v_x, v_y, v_z to be the Cartesian components of velocity, , of the molecule. The mean value of (v_x – v_y + v_z)² is:

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

8. A circular platform is rotating with a uniform angular speed counterclock wise about an axis passing through its centre and perpendicular to its plane as shown in the figure. A person of mass 'm' walks radially inward with a uniform speed 'v' on the platform. The magnitude and the direction of the Coriolis force (with respect to the direction along with the person walks) is:

(a) 2mv towards his left

(b) 2mv towards his front

(c) 2mv towards his right

(d) 2mv towards his back

Ans. (c)

Sol. Coriolis force, . Magnitude of force, towards his right.

9. A quarter-wave plate is placed in between a polarizer and a photo-detector. When the optic axis of the quarter-wave plate is kept initially parallel to the pass axis of the polarizer and perpendicular to the direction of light propagation. The intensity of light passing through the quarter-wave plate is measured to be I₀ (see figure). If the quarter wave plate is now rotated by 45° about an axis parallel to the light propagation, what would be the intensity of the emergent light measured by the photo-detector?

(a)

(b)

(c)

(d) I₀

Ans. (d)

Sol. When an unpolarized light incident on a polariser, it becomes plane polarized. Let polarized light be given by

Case-1: When optical axis of polariser and QWP is parallel i.e. = 0º, plane polarized light emerge through QWP again as plane polarised i.e. intensity is I₀.

Case-2: When optical axis of polariser and QWP is making angle = 45º. The plane polarised light split into E-ray and O-ray, and phase difference between these ray is .

Light coming out of QWP is given by

The resultant light is circularly polarised as amplitude is equal and phase difference is .

The amplitude of resultant wave is

Thus, intensity of light, I = E² = I₀ remain same.

10. A particle of mass 'm', moving with a velocity , collides elastically with another particle of mass '2m' which is at rest initially. Here, v₀ is a constant. Which of the following statements is correct?

(a) The direction along which the centre of mass moves before collision is .

(b) The speed of the particle of mass 'm' before collision in the centre of mass frame is .

(c) After collision the speed of the particle with mass 2m in the centre of mass frame is

(d) The speed of the particle of mass 2m before collision in the centre of mass frame is .

Ans. (c)

Sol. Velocity of centre of mass,

Speed of particle of mass 'm' before collision in centre of mass frame

Speed of particle of mass '2m' before collision in centre of mass frame.

11. A trapped air bubble of volume V₀ is released from a depth 'h' measured from the water surface in a large water tank. The volume of the bubble grows to 2V₀ as it reaches just below the surface. The temperature of the water and the pressure above the surface of water (10⁵ N/m²) remain constant throguhout the process. If the density of water is 1000 kg/m³ and the acceleration due to gravity is 10 m/s², then the depth 'h' is:

(a) 1 m

(b) 10 m

(c) 50 m

(d) 100 m

Ans. (b)

Sol. According to Boyle's law

12. A particle of mass 'm' is confined in a two-dimensional infinite square well potential of side 'a'. The energy of the particle in a given state is . The state is:

(a) 4-fold degenerate

(b) 3-fold degenerate

(c) 2-fold degenerate

(d) Non-degenerate

Ans. (b)

Sol. Energy eigenvalue of a particle of mass m confined in a 2-D infinite potential web

The possible value of n_x, n_y is (1, 7) (7, 1), (5, 5) i.e. the state is 3-fold degenerate

13. For a wave in a medium the angular frequency and the wave vector are related by, , where and c are constants. The product of group and phase velocities, i.e. v_gv_p is:

(a) 0.25c²

(b) 0.4c²

(c) 0.5c²

(d) c²

Ans. (d)

Sol.

14. Three identical non-interacting particles, each of spin ½ and mass 'm', are moving in a one-dimensional infinite potential well given by

The energy of the lowest energy state of the system is:

(a)

(b)

(c)

(d)

Ans. (c)

Sol. E = 2 × E₁ + 1 × E₂

15. Assume that z = 0 plane is the interface between two linear and homogeneous dielectrics (see figure). The relative permitivities are = 5 for z > 0 and = 4 for z < 0. The electric field in the region z > 0 is kV/m. If there are no free charges on the interface, the electric field in the region z < 0 is given by

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

16. A particle of mass 2/3 kg is subjected to a potential energy function V(x) = (3x² – 2x³)J, where
x > 0 and expressed in meters.

(a) Sketch V(x) with respect to x in the range 0 to 2m. Mark the positions of all the maxima and minima. What is the maximum value of the potential energy?

(b) Supposing the particle is released at x = 4/3 m, find its velocity when it reaches x = 3/2 m.

Sol. V(x) = (3x² – 2x³) J = x² (3 – 2x)J

The maxima and minima

Putting x = 1, we get, V_max = 1 J.

17. A vector field is given by:

Here and r₀ are two constants.

(a) Find the curl of this field both the regions.

(b) Find the line integral along the closed semicircle path of radius 2r₀ as shown in the figure below.

Sol. A vector field is defined as

18. A solid cylinder C of mass 10 kg rolls without slipping in an inclined plane which offers friction. The angle of the inclined plane with the horizontal is 30º as shown in the figure below. A massless inextensible string is wrapped around the cylinder and it passes over a frictionless pulley, P. The other end of the string holds a block B of mass 2 kg as shown in the figure. Take g = 10 m/s².

(a) Draw the free body diagram of the cylinder and the block. If the block B moves down by 0.01 m, how much distance does C move along the incline?

(b) If the block moves with an acceleration of 0.05 g upward, find the magnitude of the frictional force.

Sol. Let us the first find out direction of friction.

T – 10g sin 30º = 10 a ... (i)

2g – T = 2a ... (ii)

Solving (i) and (ii),

So, the point of constant ships down the incline, so, friction will act up the incline.

Free Body Diagram:

Let V_cm be the velocity of cylinder. Velocity of upper most point on the cylinder

v = v_cm +

= 2v_cm (Because v_cm = )

Velocity of the block B will be same as the velocity at uppermost point of the cylinder.

So, V_cm of cylinder C is half the velocity of block B. So, it B moves down by 0.1 m, C will move by distance

19. (a) A photon of initial momentum p₀ collides with an electron of rest mass 'm₀' moving with relativistic momentum P and energy E. The change in wavelength of the photon after scattering by an angle is given by, , where c is the speed of the light and is the wavelength of the incident photon before scattering. What will be the value of when the electron is moving in a direction opposite to that of the incident photon with momentum P and energy E? Show that the value of becomes independent of the wavelength of the incident photon when the electron is at rest before collision.

(b) In a Compton experiment experiment, the ultraviolet light of the wavelength 2000Å is scattered from an electron at rest. What should be the minimum resolving power of an optical instrument to measure the Compton shift, if the observation is made at 90º with respect to the direction of the incident light?

Sol. The change in wavelength of photon after scattering.

If electron is at rest, then P = 0

(b) For UV light of wavelength 2000Å scattered at 90º.

20. A conducting spherical shell of radius R₁ carries a total charge Q. A spherical layer of a linear, homogeneous and isotropic dielectric of dielectic constant K and outer radius R₂ (>R₁) covers the shell as shown in the figure.

(a) Find the electric field and the polarization vector inside the dielectric. From this , calculate the surface bound charge density, on the outer surface of the dielectric layer and the volume bound charge density, inside the dielectric.

(b) Calculate the electrosdtatic energy stored in the region R₁
< r < R₂.

Sol.

21. Two spaceships A and B of proper length 50 m each are moving with relativistic speeds 0.8c and 0.6c, respectively, in the same direction with respect to a stationary observer S. Here c is the speed of light. At t = 0, the spaceship A is just behind the spaceship B as shown below.

(a) Find the time taken by the spaceship A to completely overtake the spaceship B (that is the back of A is in the line with the front of B) as seen by the observer S.

(b) Find the time taken by the spaceship A to completely overtake the spaceship B as seen by observer in the spaceship B.

Sol. Proper length of spaceship A w.r.t. rest frame of A =

Proper length of spaceship B w.r.t. rest frame of B =

1. Time taken by the spaceship A to completely overtake the spaceship B, measured in S frame.
   

   
   

   
   

1. Time taken by spaceship A to completely overtake spaceship B, measured in frame B
   

22. A free particle of mass 'm' with energy V₀/2 is incident from left on a step potential of height V₀ as shown in the figure below.

Writing down the time independent. Schrodinger equation in both the regions, obtain the corresponding general solutions. Apply the boundary conditions to find the wave functions in both the regions. Determine the reflection coefficient R. What is the transmission coefficient T?

Sol. Time independent Schrodinger equation

Solution of (5) and (6)

Applying boundary condition,

Adding (9) and (10)

Substracting (10) and (9)

Transition coefficient, T = 1 – R = 0

So, R + T = 1

23. For the transistor shown in the figure, the dc current gain b_dc = 50 and V_BE = 0.7 V. The switch S is initially open.

(a) Calculate the voltage at point A. If the switch S is now closed, what would be the voltage at point A?

(b) Draw the dc load line and find the Q-point of the circuit with the switch S remaining closed.

Sol.

S is open:

Applying Kirchoff's law

If S is closed: Let us draw Thevenin equivalent of the circuit

V_A = V_Th – I_B × R_th = 2.5 – 0.0986 × 5V = 2V

Applying Kirchoff's voltage law to output circuit.

Q. point of the circuit = (V_CE, I_C)

= (4.93 V, 1.25 mA)

Equation of DC load line

24. Consider a capacitor placed in free space, consisting of two concentric circular parallel plates of radii r. The separation z between the plates oscillates with a constant frequency , i.e. z(t) = z₀ + z₁ cos t. Here z₀ and z₁(< z₀) are constants. The separation z(t)(<<r) is varied in such a way that the voltage V₀ across the capacitor remains constant.

(a) Calculate the displacement current density and the displacement current between the plates through a concentric circle of radius r/2.

(b) Calculate the magnetifc field vector between the plates at a distance r/2 from the axis of the capacitor.

Sol.

Displacement current through a circle of radius .

Let us take a loop of radius 'x' with centre on the axis.

Current inclosed,

Applying Maxwell's equation

25. One mole of an ideal monatomic gas in an initial state with pressure, P_i and volume V_i, is to be taken to a final state d with P_f = B²P_i and V_f = V_i/B through the path a b c d as shown in the figure below for a particular value of B(>1). Here a b and c d are adiabatic while b c is an isotherm with temperature T₀. States b and c correspond to (p₁, V₁) and (p₂, V₂), respectively.

Find the ratio and the total work done by the gas in terms of P_i, V_i, T₀ and B.

Sol.

Dividing (2) and (1)

Work done by gas

w = w_ab + w_bc + w_cd