1. Isothermal compressibility of a substance is defined as . Its value for n mole of an ideal gas will be

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

2. A space crew has a life support system that can last only for 1000 hours. What minimum speed would be required for safe travel of the crew between two space station separated by a fixed distance of 1.08 × 10¹² km?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. = proper life time of the life support system = 1000 hrs = 3.6 × 10⁶ sec.

= lifetime of the life support system measured from lab frame

where v is the velocity of space crew w.r.t. lab frame

Maximum time required for travel between space stations measured in lab frame =

3. An oscillating voltage V(t) = V₀ cos is applied across a parallel plate capacitor having a plate separation d. The displacement current density through the capacitor is:

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

4. Two spherical nuclei have mass number 216 and 64 with their radii R₁ and R₂ respectively. The ratio is

(a) 1.0

(b) 1.5

(c) 2.0

(d) 2.5

Ans. (b)

Sol. For spherical nucleus, radius of the nucleus is given by

R = R₀A^1/3 (R₀ is constant and A is mass number)

5. In the Fourier series of the periodic function (shown in the figure)

f(x) = |sin x |

Which of the following coefficients are nonzero?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

6. A particle is moving in space with O as the origin. Some possible expressions for its position, velocity and acceleration in cylindrical coordinates are given below. Which one of these is correct?

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Let P be a point whose coordinates in cylindrical system and cartesian system (x, y, z)

Now, from figure we can write

 

7. Which one of the following is an incorrect bolean expression?

(a)

(b)

(c) P(P + Q) = Q

(d)

Ans. (c)

Sol.

Q is correct.

P is correct.

 

P is correct.

 

 

8. Monochromatic X rays of wavelength 1 Å are incident on a simple cubic crystal. The first order Brag reflection from (311) plane occurs at an angle of 30º from the plane. The lattice parameter of the crystal in Å is

(a) 1

(b)

(c)

(d)

Ans. (d)

Sol.

9. An electric field exists in space. What will be the total charge enclosed in a sphere of unit radius centered at the origin?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

[r = 1 on the surface of sphere]

10. A thin massless rod of length 2l has equal point masses 'm' attached at its ends (see figure). The rod is rotating about an axis passing through its centre and making angle with it. The magnitude of the rate of change of its angular momentum is:

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

11. A battery with a constant emf and internal resistance r_i provides power to an external circuit with a load resistance made up by combining resistance R_L and 2R_L in parallel. For what value of R_L will the power delivered to the load be maximum?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Power delivered is maximum when, internal resistance = External resistance

12. Among the following displacement versus time plots, which ones may represent an over damped oscillator?

(a) Only P and Q

(b) Only P and R

(c) Only P and S

(d) Only P, R and S

Ans. (d)

Sol. Over damped oscillator does not have oscillatory motion, Therefore any of the (P), (R) and (S) can represent its motion.

13. A wave packet in a certain medium is constructed by superposing waves of frequency around = 100 and the corresponding wave number k with k₀ = 10 as given in the table below

Find the ratio v_g / g_p of the group velocity v_g and the phase velocity v_P.

(a)

(b) 1

(c)

(d) 2

Ans. (d)

Sol.

14. A box containing 2 moles of diatomic ideal gas at temperature T₀ is connected to another identical box containing 2 moles of a monoatomic ideal gas at temperature 5T₀. There are no thermal losses and the heat capacity of the box is negligible. Find the final temperature of the mixture of the gases (ignore the vibrational degrees of freedom for the diatomic molecules)

(a) T₀

(b) 1.5 T₀

(c) 2.5 T₀

(d) 3 T₀

Ans. (c)

Sol. Total internal energy = constant

15. Moment of inertia of a solid cylinder of mass m, height 'h' and radius 'r' about an axis (shown in the figure by dashed line) passing through its centre of mass and perpendicular to its symmetry axis is:

(a)

(b)

(c)

(d)

Ans. (a)

Sol. M.I. of elementary disc about the axis of rotation

16. A parallel beam of light of diameter 1.8 cm contains two wavelengths 4999.75 Å and 5000.25 Å. The light is incident perpendicular on a large diffraction grating with 5000 lines per centimeter.

(a) Using Rayleigh criterition, find the least order at which the two wavelengths are resolved.

(b) What will be the angular separation (in radius) of the two wavelengths at order n = 2?

Sol. (a) Resolving power required to resolve two wavelengths =

So, resolving power of grating must be greater than 10000.

Resolving power of grating R = Nn > 10000; = 1.11; n_min = 2

17. A block of mass 'M' is free to slide on a frictionless horizontal floor. The block has a cylinderical cavity of radius R in the middle of it. The centre of mass (CM) of the block lies on the dashed line passing through the centre of the cavity (see figure). Initially the CM of the block is at horizontal distance X₁ from the origin. Now a point particle of mass 'm' is released from A into the cavity. There is negligible friction between the particle and the cavity surface. When the particle reaches point B, the CM of the block is at a distance X₂ from the origin. Find (X₂–X₁).

Sol. Let x´ be the initial position at centre of mass of combine system

and x´´ be the final position of centre of mass of combine system

Since we do not apply any external force

18. The electric field of an electromagnetic wave propagating through vacuum is given by

(a) What is the wave vector ? Hence, find the value of .

(b) At the time t = 0 there is a point charge 'q' with velocity at the origin. What is the instantaneous Lorentz force acting on the particle?

Sol.

Now, B = ?

19. Consider two infinitely long wires parallel to the z-axis carrying the same current I. One wire passes through the point L with coordinates (–1, 1) and the other through M with coordinates (–1, –1) in the XY plane as shown in the figure. The direction of the current in both the wires is in the positive z-direction.

(a) Find the value of along the semicircular closed path of the radius 2 units shown in the figure.

(b) Third long wire carrying current I and also perpendicular to the XY plane is placed at a point N with coordinates (x, 0) so that the magnetic field at the origin is doubled. Find x and the direction of the current in the third wire.

Sol. (a) According to Ampere's law,

Net current enclosed by the loop, i = 2I. So,

(b) the magnetic field due to wires L & M

The horizontal component of will cancel out vertical component

So, resultant magnetic field due to two wires L & M.

The magnetic field due to third wire should also be . So that net magnetic field at origin should get doubled. So, magnetic field due to the 3rd wire will be

So, x = 1

20. A particle of mass 'm' is confined in a one dimensional box of unit length. At time t = 0 the wavefunction of the particle is where A is the normalization constant.

(a) Write the wavefunction at a later time t.

(b) Find the expectation values of momentum and energy at t = 0.

Sol. For a particle of mass 'm' confined in a 1-D box of length 'l', the wave function of the particle in n^th state, is

Given wave function / state of the particle

Applying normalization condition,

 

21. For the transistor circuit shown in the figure = 100 and V_BE = 0.7 V. Determine the base current I_B, the collector-emitter voltage V_CE, the emitter voltage V_E, the base voltage V_B and the saturation current I_Csat.

Sol. Applying Kirchoff's law to base circuit.

Applying Kirchoff's law to collector circuit.

20 – I_CR_C – V_CE – I_ER_E = 0

So, collector emitter voltage. V_CE = 9.46 volt

Emitter voltage V_E = I_ER_E = 3.54 × 1 = 3.54 v

Base voltage V_B = V_E + V_BE = 3.54 + 0.7 = 4.24 V

22. The equation of state of one mole of a van der Waals gas and its internal energy U(T, V) is given by U(T, V) = U₀ + C_VT – , where U₀ and C_V can be taken as constants.

(a) Prove that in a reversible adiabatic process the temperature and volume satisfy the equation = constant.

(b) Calculate the change in entropy of the gas when it undergoes a reversible isothermal expansion from volume V₀ to 2V₀.

Sol. (a) Equation of state of one mode of vander waals gas is

For reversible adiabatic process dQ = 0

For isothermal expansion dT = 0

23. (a) Find the normalized eigenvector of the matrix , corresponding to its psotive eigenvalue.

(b) The Normalized eigenvectors of the matrix are and with the eigenvalues and respectively and > . If the eigenvector obtained in part (a) is expressed as = P + Q . Find the constants P and Q.

Sol.

Similarly as before, normalised eigenvectors corresponds to are

 

 

24. A particle of mass 'm' is shown is thrown vertically up from the ground with initial speed v₀. As it moves it experiences a drag force , where v is the speed of the particle and k is a constant UP to what height does the particle go and what its speed when it reaches the ground again?

Sol.

Let the maximum height be H.

At h = H, v = 0

25. Consider the radiactive transformation A B C with decay constants, and for elements A and B; C is a stable element. Assume that at t = 0, N_A = N₀, N_B = 0 and N_C = 0, where N_A, N_B and N_C are the number of atoms of A, B and C, respectively.

(a) Show that at any later time t the number of atoms N_A of element B will be

(b) Sketch the qualitatively the variation of N_A, N_B and N_C with time on three separate plots.

Sol.

= Rate of product – Rate of decay

Adding (1), (2) and (3)