1. A point mass m, is constrained to move on the inner surface of a paraboloid of revolution (where a > 0 is a constant). When it spirals down the surface, under the influence of gravity (along –z direction), the angular speed about the z-axis is proportional to

(a) 1 (independent of z)

(b) z

(c) z^–1

(d) z^–2

Ans. (c)

Sol. Using Lagrangian in cylindrical coordinate

2. Two coupled oscillators in a potential can be decoupled into two independent harmonic oscillators (coordinates: x', y') by means of an appropriate transformation . The transformation matrix S is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The normal mode of given potential is in the basis of normal mode the potential can be diagonalise.

3. A heavy particle of rest mass M while moving along the positive z-direction, decays into two identical light particles with rest mass m (where M > 2m). The maximum value of the momentum that any one of the lighter particles can have in a direction perpendicular to the z-direction, is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Let P be the momentum of heavy mass M. And let P₁ be the momentum of the light particles of mass m in the direction perpendicular to z and P₂ be the momentum in z -direction.

According to conservation of momentum,

4. A frictionless horizontal circular table is spinning with a uniform angular velocity about the vertical axis through its centre. If a ball of radius a is placed on it at a distance r from the centre of the table, its linear velocity will be

(a)

(b)

(c)

(d) 0 (zero)

Ans. (d)

Sol. Since table is frictionless then there is not any tangential force, so ball will have zero speed.

5. An inductor L, a capacitor C and a resistor R are connected in series to an AC source, . If the net current is found to depend only on R, then

(a) C = 0

(b) L = 0

(c)

(d)

Ans. (c)

Sol. The net current is found to depend only on R,

6. Three point charges q are placed at the corners of an equilateral triangle. Another point charge –Q is placed at the centroid of the triangle. If the force on each of the charges q vanishes, then the ratio Q/q is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Force on charge q is zero so

7. Three infinitely long wires, each carrying equal current are placed in the xy-plane along x = 0, +d and –d. On the xy-plane, the magnetic field vanishes at

(a)

(b)

(c)

(d)

Ans. (d)

Sol. B₁ + B₂ + B₃

8. The following figure shows the intensity of the interference pattern in the Young's double-slit experiment with two slits of equal width is observed on a distant screen

If the separation between the slits is doubled and the width of each of the slits is halved, then the new interference pattern is best represented by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. (i)

As d is increased to 2d, so will be halved.

(ii) As slit width e is reduced to e/2 so width of central envelop will be increased.

9. Let , where is a constant, be the electric field of an electromagnetic wave travelling in vacuum. Which of the following vectors is a valid choice for ?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

10. Two time dependent non-zero vectors and , which are not initially parallel to each other, satisfy at all time t. If the area of the parallelogram formed by and be A(t) and the unit normal vector to it be , then

(a) A(t) increases linearly with t, but is a constant

(b) A(t) increases linearly with t, and rotates about

(c) A(t) is a constant, but rotates about

(d) A(t) and are constants

Ans. (d)

Sol.

11. A basket consists of an infinite number of red and black balls in the proportion p : (1 – p). Three balls are drawn at random without replacement. The probability of their being two red and one black is a maximum for

(a) p = 3/4

(b) p = 3/5

(c) p = 1/2

(d) p = 2/3

Ans. (d)

Sol.

12. The eigenvalues of the 3 × 3 matrix are

(a) a² + b² + c², 0, 0

(b) b² + c², a², 0

(c) a² + b², c², 0

(d) a² + c², b², 0

Ans. (a)

Sol. 2^nd and 3^rd raws are the multiplies of row 1

Row 2 = b/a row 1 and row 3 = c/a row 1

In such cases, the eigen values are trace and zeroes.

13. A function of a complex variable z is defined by the integral , where is a circular contour of radius 3, centred at origin, running counter-clockwise in the w-plane. The value of the function at z = (2 – i) is

(a) 0

(b) 1 – 4i

(c)

(d)

Ans. (c)

Sol.

14. The temperatures of two perfect black bodies A and B are 400 K and 200 K, respectively. If the surface area of A is twice that of B, the ratio of total power emitted by A to that by B is

(a) 4

(b) 2

(c) 32

(d) 16

Ans. (c)

Sol. Stefan's law, e =

Energy density energy per unit time per unit area

Power (P) and Area (A), then

15. Two ideal gases in a box are initially separated by a partition. Let N₁, V₁ and N₂, V₂ be the numbers of particles and volumes occupied by the two systems. When the partition is removed, the pressure of the mixture at an equilibrium temperature T, is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For an ideal gas, PV = Nk_BT

16. An idealised atom has a non-degenerate ground state at zero energy and a g-fold degenerate excited state of energy E. In a non-interacting system of N such atoms, the population of the excited state may exceed that of the ground state above a temperature . The minimum value of g for which this is possible is

(a) 8

(b) 4

(c) 2

(d) 1

Ans. (b)

Sol. Let N_g and N_e e the number of particles in ground state and excited state respectively.

According to Boltzmann distribution law N_g > N_e.

So, less energy means more number of particles.

Equation (2) divided by (1), then

17. The Hamiltonian of a system of N non-interacting particles, each of mass m, in one dimension is where > 0 is a constant and p_i and x_i are the momentum and position respectively of the i-th particle. The average internal energy of the system is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

18. A 10 V battery is connected in series to a resistor R and a capacitor C, as shown the figure.

The initial charge on the capacitor is zero. The switch is turned on and the capacitor is allowed to charge to its full capacity. The total work done by the battery in this process is

(a) 10^–3 J

(b) 2 × 10^–3 J

(c) 5 × 10^–4 J

(d) 47 × 10^–2 J

Ans. (a)

Sol. The total work done by the battery in this process is

W = qV = (CV) V = CV² 10 × 10^–6 × (10)² = 10^–3 Joules

19. In the 3-bit register shown below, Q₁ and Q₃ are the least and the most significant bits of the output, respectively.

If Q₁, Q₂ and Q₃ are set to zero initially, then the output after the arrival of the second falling clock (CLK) edge is

(a) 001

(b) 100

(c) 011

(d) 110

Ans. (c)

Sol.

20. The Boolean equation is to be implemented using only two-input NAND gates. The minimum number of gates required is

(a) 3

(b) 4

(c) 5

(d) 6

Ans. (b)

Sol.

Implementing Ex-OR Gate

So minimum 4 number of gates are required.

21. The temperature variation of the resistivity of four materials are shown in the following graphs.

The material that would make the most sensitive temperature sensor, when used at temperatures between T₁ and T₂, is

(a) A

(b) B

(c) C

(d) D

Ans. (c)

Sol. For the temperature sensor, the variation in the resistivity of material should be as large as possible without any local maximum or minimum. Option (A) and (D) shows minimum while in (B) gradient is very low in comparison to (C). Thus option (C) is the correct answer.

22. Let denote the energy eigenstates of a particle in a one-dimensional simple harmonic potential . If the particle is initially prepared in the state state is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

23. For the one dimensional potential wells A, B and C, as shown in the figure, let E_A, E_B and E_C denote the ground state energies of a particle, respectively.

The correct ordering of the energies is

(a) E_C > E_B > E_A

(b) E_A > E_B > E_C

(c) E_B > E_C > E_A

(d) E_B > E_A > E_C

Ans. (a)

Sol. Correct option is (a)

24. An angular momentum eigenstate is rotated by an infinitesimally small angle about the positive y-axis in the counter clockwise direction. The rotated state, to order (upto a normalisation constant), is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

25. The wavelength of the first Balmer line of hydrogen is 656 nm. The wavelength of the corresponding line for a hydrogenic atom with Z = 6 and nuclear mass of 19.92 × 10^–27 kg is

(a) 18.2 nm

(b) 109.3 nm

(c) 143.5 nm

(d) 393.6 nm

Ans. (a)

Sol.

First Balmer line

26. The state of an electron in a hydrogen atom is

where denotes common eigenstates of and operators in the standard notation. In a measurement of for the electron in this state, the result is recorded to be 0. Subsequently a measurement of energy is performed. the probability that the result is E₂ (the energy of the n = 2 state) is

(a) 1

(b) 1/2

(c) 2/3

(d) 1/3

Ans. (c)

Sol. We will use postulates 4 first then use postulate 2 and 3.

If L_z is measured and measurement is 0 then state is proportionate to

27. A particle with incoming wave vector , after being scattered by the potential , goes out with wave vector . The differential scattering cross-section, calculated in the first Born approximation, depends on , as

(a) 1/q²

(b) 1/q⁴

(c) 1/q

(d) 1/q^3/2

Ans. (a)

Sol. Using Born Approximation for high energy

28. A quantum particle in a one-dimensional infinite potential well, with boundaries at 0 and a, is perturbed by adding to the initial Hamiltonian. The correction to the energies of the ground and the first excited states (to first order in ) are respectively

(a) 0 and 0

(b) 2/a and 0

(c) 0 and 2/a

(d) 2/a and 2/a

Ans. (b)

Sol.

29. Spin fermions of mass m and 4m are in a harmonic potential . Which configuration of 4 such particles has the lowest value of the ground state energy?

(a) 4 particles of mass m

(b) 4 particles of mass 4m

(c) 1 particle of mass m and 3 particles of mass 4m

(d) 2 particles of mass m and 2 particles of mass 4m

Ans. (d)

Sol.

30. Falling drops of rain break up and coalesce with each other and finally achieve an approximately spherical shape in the steady state. The radius of such a drop scales with the surface tension as

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

31. The velocity v(x) of a particle moving in one dimension is given by , where v₀ and x₀ are positive constants of appropriate dimensions. If the particle is initially at x/x₀ = , where , then, in the long time, it

(a) executes an oscillatory motion around x = 0

(b) tends towards x = 0

(c) tends towards x = x₀

(d) executes an oscillatory motion around x = x₀

Ans. (c)

Sol.

So motion is not oscillatory.

x = Ae^kt if we assume k small and t is large we can assume x is some fixed quantity so

32. A pendulum executes small oscillations between angles +₀ and –₀. If d is the time spent between and + d, then is best represented by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Correct option is (b)

33. Consider a particle with total energy E moving in one dimension in a potential V(x) as shown in the figure below.

Which of the following figures best represents the orbit of the particle in the phase space?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Use concept T = E – V where T is kinetic energy E is total energy and v is potential energy.

34. The energy density I of a black body radiation at temperature T is given by the Planck's distribution function , where v is the frequency. The function I(v, T) for two different temperatures T₁ and T₂ are shown below.

If the two curves coincide when is plotted against then the values of a and b are, respectively,

(a) 2 and 1

(b) –2 and 2

(c) 3 and –1

(d) –3 and 1

Ans. (d)

Sol.

For a = – 3, b = 1

Both graphs are now same

35. For an ideal gas consisting of N distinguishable particles in a volume V, the probability of finding exactly 2 particles in a volume << V, in the limit N, V , is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

36. The Hamiltonian of a system of 3 spins is H = J(S₁S₂ + S₂S₃), where S_i = ±1 for i = 1,2,3. Its canonical partition function, at temperature T, is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Number of states 2³ = 8

H = J (S₁S₂ + S₂S₃)

37. A certain two-dimensional solid crystallises to a square monoatomic lattice with lattice constant a. Each atom can contribute an integer number of free conduction electrons. The minimum number of electrons each atom must contribute such that the free electron Fermi circle at zero temperature encloses the first Brillouin zone completely, is

(a) 3

(b) 4

(c) 4

(d) 2

Ans. (c)

Sol. Brillouin zone of the square lattice of lattice constant 'a' is also a square as shown below

The radius of the Fermi circle in two-dimension is

The fermi circle will enclose the 1^st B.Z completely

Since N should be integer, therefore, minimum number of electron (N) is 4.

N = 4

38. A tight binding model of electrons in one dimension has the dispersion relation = –2t(1 – cos ka), where t > 0, a is the lattice constant and . Which of the following figures best represents the density of states over the range ?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

39. A lattice is defined by the unit vectors and , where a > 0 is a constant. The spacing between the (100) planes of the lattice is

(a)

(b) a/2

(c) a

(d)

Ans. (a)

Sol. Interplaner spacing for Hexagonal lattice is

Here |a| = |a₁| = a, |b| = |a₂| = a and |c| = |a₃| = a

For (100) plane

40. A spacecraft of mass m = 1000 kg has a fully reflecting sail that is oriented perpendicular to the direction of the sun. The sun radiates 10²⁶ W and has a mass M = 10³⁰ kg. Ignoring the effect of the planets, for the gravitational pull of the sun to balance the radiation pressure on the sail, the area of the sail will be

(a) 10² m²

(b) 10⁴ m²

(c) 10⁸ m²

(d) 10⁶ m²

Ans. (d)

Sol. m 10³ kg

P = 10²⁶ W

Radiation pressure for fully reflecting surface

41. The electric field due to a uniformly charged infinite line along the z-axis, as observed in the rest frame S of the line charge, is . In a frame M moving with a constant speed v with respect to S along the z-direction, the electric field is (in the following = v/c and )

(a) E'_x = E_x and E'_y = E_y

(b)

(c)

(d)

Ans. (d)

Sol. M is moving in z-direction

42. A parallel plate capacitor with rectangular plates of length , breadth b and plate separation d, is held vertically on the surface of a dielectric liquid of dielectric constant and density as shown in the figure. The length and breadth are large enough for edge effects to be neglected.

The plates of the capacitor are kept at a constant voltage difference V. Ignoring effects of surface tension, the height h upto which the liquid level rises inside the capacitor, is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Upward force on dielectric

If liquid rises to height h, then

43. Using the following values of x and f(x)

the integral , evaluated by the Trapezoidal rule, is 5/16. The value of a is

(a) 3/4

(b) 3/2

(c) 7/4

(d) 19/24

Ans. (a)

Sol.

44. The Green's function for the differential equation , satisfying the initial conditions , is

The solution of the differential equation when the source (the Heaviside step function) is

(a) sin t

(b) 1 – sin t

(c) 1 – cos t

(d) cos²t – 1

Ans. (c)

Sol. Heaviside step function is

(D² + 1) = 0

D = + i and x(t) = C₁ cos t + C₂ sin t

x₁(0) = 0 then C₁ = 0 then x₁(t) = sin t

C'₂ = 0 then x₂(t) = C'₁ cos t

= sin (t) (– sin t) – cos t cos t

= – sin²t – cos²t = – 1

45. The solution of the differential equation , with the boundary conditions y(0) = 0 and y'(0) = –1, is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

46. If we take the nuclear spin I into account, the total angular momentum is , where and are the orbital and spin angular momenta of the electron. The Hamiltonian of the hydrogen atom is corrected by the additional interaction , where > 0 is a constant. The total angular momentum quantum number F of the p-orbital state with the lowest energy is

(a) 0

(b) 1

(c) 1/2

(d) 3/2

Ans. (b)

Sol.

For hydrogen atom, p-orbital electron

Out of these four possibilities lowest state is corresponding to following quantum number

47. The absorption lines arising from pure rotational effects of HCl are observed at 83.03 cm^–1, 103.73 cm^–1, 124.30 cm^–1, 145.03 cm^–1 and 165.51 cm^–1. The moment of inertia of the HCl molecule is (take )

(a) 1.1 × 10^–48 kg-m²

(b) 2.8 × 10^–47 kg-m²

(c) 2.8 × 10^–48 kg-m²

(d) 1.1 × 10^–42 kg-m²

Ans. (b)

Sol.

48. The energies of the 3lowest states of an atom are E₀ = –14 eV, E₁ = –9 eV and E₂ = –7 eV. The Einstein coefficients are A₁₀ = 3 × 10⁸ s^–1, A₂₀ = 1.2 × 10⁸ s^–1 and A₂₁ = 8 × 10⁷ s^–1. If a large number of atoms are in the energy level E₂, the mean radiative lifetive of this excited state is

(a) 8.3 × 10^–9 s

(b) 1 × 10^–8 s

(c) 0.5 × 10^–8 s

(d) 1.2 × 10^–8 s

Ans. (c)

Sol. Rate of spontaneous decay from E₂ state

= (A₂₀ + A₂₁) N₁ = A₂N₁

A₂ = A₂₀ + A₂₁ = (1.2 × 10⁸ + 0.8 × 10⁸) s^–1 = 2.0 × 10⁸ s^–1

49. Two voltmeters A and B with internal resistances 2 and 0.1 are used to measure the voltage drops V_A and V_B, respectively, across the resistor R in the circuit shown below.

The ratio V_A/V_B is

(a) 0.58

(b) 1.73

(c) 1

(d) 2

Ans. (b)

Sol. Let us draw Thevenin's equivalent across point ab:

Voltmeter is connected across point ac in parallel.

Case A: Voltmeter internal resistance is 2M, so equivalent resistance across ac is

Case B: Voltmeter internal resistance 0.1 k, so equivalent resistance across ac is

50. The I-V characteristics of the diode D in the circuit below is given by

where I_s is the reverse saturation current, v is the voltage across the diode and T is the absolute temperature.

If the input voltage is V_in, then the output voltage V_out is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

51. A rod pivoted at one end is rotating clockwise 25 times a second in a plane. A video camera which records at a rate of 30 frames per second is used to film the motion. To someone watching the video, the apparent motion of the rod will seem to be

(a) 10 rotations per second in the clockwise direction

(b) 10 rotations per second in the anticlockwise direction

(c) 5 rotations per second in the clockwise direction

(d) 5 rotations per second in the anticlockwise direction

Ans. (d)

Sol. Correct option is (d)

52. In the circuit shown below, the gain of the op-amp in the middle of its bandwidth is 10⁵. A sinusoidal voltage with angular frequency = 100 rad/s is applied to the input of the op-amp.

The phase difference between the input and the output voltage is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

53. Charged pions decay to muons µ^– and anti-muon neutrinos . Take the rest masses of a muon and a pion to be 105 MeV and 140 MeV, respectively. The probability that the measurement of the muon spin along the direction of its momentum is positive, is closest to

(a) 0.5

(b) 0.75

(c) 1

(d) 0

Ans. (c)

Sol. Correct option is (c)

54. The binding energy B of a nucleus is approximated by the formula B = a₁A – a₂A^2/3 – a₃Z²A^–1/3 – a₄(A – 2Z)²A^–1 where Z is the atomic number and A is the mass number of the nucleus. If , the atomic number Z for naturally stable isobars (constant value of A) is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. B = a₁A – a₂A^2/3 – a₃Z² A^–1/3 –a₄ (A – 2Z)² A^–1

55. The magnetic moments of a proton and a neutron are 2.792 µ_N and –1.913 µ_N, where µ_N is the nucleon magnetic moment. The values of the magnetic moments of the mirror nuclei and , respectively, in the Shell model, are closest to

(a) 23.652 µ_N and –18.873 µ_N

(b) 26.283 µ_N and –16.983 µ_N

(c) –2.628 µ_N and 1.887 µ_N

(d) 2.628 µ_N and –1.887 µ_N

Ans. (d)

Sol.