1. One gram of salt is dissolved in water that is filled to a height of 5 cm in a beaker of diameter 10 cm. The accuracy of length measurement is 0.01 cm while that of mass measurement is 0.01 mg. When measuring the concentration C, the fractional error C/C is

(a) 0.8%

(b) 0.14%

(c) 0.5%

(d) 0.28%

Ans. (d)

Sol. m = 1 gm, h = 5 cm, d = 10 cm,

Concentration,

Volume,

Fractional error,

2. A system can have three energy levels: . The level E = 0 is doubly degenerate, while the others are non-degenerate. The average energy at inverse temperature is

(a)

(b)

(c) Zero

(d)

Ans. (d)

Sol. Average energy of the system

3. For a particular thermodynamics system the entropy S is related to the internal energy U and volume V by

S = c U^3/4V^1/4

where c is a constant. The Gibbs potential G = U – TS + pV for this system is

(a)

(b)

(c) zero

(d)

Ans. (c)

Sol. Entropy of the system,

S = cU^3/4 V^1/4

From combined form of first and second law of thermodynamics,

Gibbs potential,

4. An op-amp based voltage follower

(a) is useful for converting a low impedance source into a high impedance source

(b) is useful for converting a high impedance source into a low impedance source

(c) has infinitely high closed loop output impedance

(d) has infinitely high closed loop gain

Ans. (b)

Sol. Is useful for converting a high impedance source into a low impedance source. At the input op-amp impedance is high and at output impedance is low.

5. A particle of mass m in three dimensions is in the potential

Its ground state energy is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The energy eigenvalues of the given infinite sphereical square well potential is

(n = 1, 2, 3, .........)

The ground state energy will be

6. Which of the graphs below gives the correct qualitative behavior of the energy density of blackbody radiation of wavelength at two temperatures T₁ and T₂(T₁ < T₂)?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. At a consequence of Wien's displacement law, we can relate how the spectrum of black body radiation depends on the temperature of the body and corresponding mathematical equation is,

where, is the wavelength at which the energy density of black body radiation, is maximum at temperature T i.e. for two temperatures T₁ and T₂.

Since, and the total energy density is proportional to T⁴. Therefore,

7. Given that , the uncertainty in the ground state

of the hydrogen atom is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Given : Ground state wave function

Using the integration :

Momentum spread

8. An RC network produces a phase-shift of 30º. How many such RC network should be cascaded together and connected to a Common Emitter amplifier so that the final circuit behaves as an oscilator?

(a) 6

(b) 12

(c) 9

(d) 3

Ans. (a)

Sol. For an RC-phase shift oscillator 3 stages provide 180º. Hence, each stage provides a phase shift of 60º.

If each stage provides phase shift of 30º then 6 RC stages are used.

9. The free energy F of a system depends on a thermodynamics variable as

with a, b > 0. The value of , when the system is in thermodynamic equilibrium, is

(a) zero

(b) +(a/6b)^1/4

(c) +(a/3b)^1/4

(d) +(a/b)^1/4

Ans. (c)

Sol. Free energy of system,

, a, b > 0

In equilibrium,

Either,

Or,

At

But at

10. The inner shield of a triaxial conductor is driven by an (ideal) op-amp follower circuit as shown. The effective capacitance between the signal-carrying conductor and ground is

(a) unaffected

(b) doubled

(c) halved

(d) made zero

Ans. (d)

Sol. For a voltage follow ideal op-amp. The effective capacitance between signal carrying conductor and ground can be made either zero or uneffected (a) or (d).

11. Consider a system of two non-interacting identical fermions, each of mass m in an infinite square well potential of width a. (Take the potential inside the well to be zero and ignore spin). The composite wavefunction for the system with total energy

is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. A system of two non-interacting identical fermions have anti-symmetric wave function.

If we ignore the spin of the fermions, then space part of the wave function will be anti-symmetric.

Energy of the system

Given : n₁ = 1, n₂ = 2

The space part wavefunction of the system

12. A particle of mass m in the potential , is in an eigenstate of energy . The corresponding un-normalized eigenfunction is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

It is a 2-D anisotropic harmonic oscillator having and

Energy eigenvalue (where, n = 2n_x + n_y)

Given :

The corresponding wavefunction will be,

13. A particle of mass m and coordinate q has the Lagrangian

where is a constant. The Hamiltonian for the system is given by

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

We know that

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

14. If and C is the circle of unit radius in the plane defined by z = 1, with the centre on the z-axis, then the value of the integral is

(a)

(b)

(c)

(d) 0

Ans. (d)

Sol. i.e. field is conservative.

Since, line integral of a conservative field along a closed path is zero, then

15. Given, , for |t| < 1, the value of P₅(–1) is

(a) 0.26

(b) 1

(c) 0.5

(d) –1

Ans. (d)

Sol.

Putting x = –1 in the above equation, we get

Comparing the coefficient t⁵ on the both sides, we get P₅(–1) = –1

16. A charged particle is at a distance d from an infinite conducting plane maintained at zero potential. When released from rest, the particle reaches a speed u at a distance d/2 from the plane. At what distance from the plane will the particle the speed 2u?

(a) d/6

(b) d/3

(c) d/4

(d) d/5

Ans. (d)

Sol. Let x is the distance of charge particle from the plane. So, image charge will appear at x distance behind the plane.

17. Consider the matrix

The eigenvalues of M are

(a) –5, –2, 7

(b) –7, 0, 7

(c) –4i, 2i, 2i

(d) 2, 3, 6

Ans. (b)

Sol. Sum of the eigenvalues of the matrix M = trace of the matrix M = 0

Product of the eigenvalues of the matrix M = determinant of the matrix M = 0

Only –7, 0, 7 satisfies both the relation.

18. Consider the differential equation with the initial conditions x(0) = 0 and . The solution x(t) attains its maximum value when 't' is

(a) 1/2

(b) 1

(c) 2

(d)

Ans. (b)

Sol.

Assume the trial solution : x = ce^mt

General solution = x(t) = (c₁ + c₂t)e^–t

Checking for maxima and minima :

For maxima or minima, t = 1

At t = 1,

Therefore, x(t) attains its maximum value at t = 1

19. A light source is switched on and off at a constant frequency f. An observer moving with a velocity u with respect to the light source will observe the frequency of the switching to be

(a)

(b)

(c)

(d)

Ans. (d)

Sol. From time dilation formula

20. If C is the contour defined by , the value of the integral

is

(a)

(b)

(c) 0

(d)

Ans. (c)

Sol. has poles of order 2 at

Only z = 0 will be within the circle

The given function, can be expanded into Lourrent series about z = 0 as following:

Residue of f(z) at (z = 0) = coefficient of in the Laurent series expansion of f(z) about (z = 0).

Therefore, the value of the integral = 0

21. The time period of a simple pendulum under the influence of he acceleration due to gravity g is T. The bob is subjected to an additional acceleration of magnitude in the horizontal direction. Assuming small oscillations, the mean position and time period of oscillation, respectively, of the bob will be

(a) 0º to the vertical and

(b) 30º to the vertical and T/2

(c) 60º to the vertical and

(d) 0º to the vertical and

Ans. (c)

Sol.

22. Consider an electromagnetic wave at the interface between two homogeneouos dielectric media of the dielectric constants and . Assuming > and non charges on the surface, the electric field vector and the displacement vector D in the two media satisfy the following inequalities

(a)

(b)

(c)

(d)

Ans. (c)

Sol. According Boundary conditions

Equ (1), (2), (3), (4) put in equ (5),

So we can say that E₁ > E₂ and D₂ > D₁

23. If the electrostatic potential in spherical polar coordinates is

where and r₀ are constants, then the charge density at a distance r = r₀ will be

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

24. A current i_P flows through the primary coil of a transformer. The graph of i_P(t) as a function of time 't' is shown in figure below

Which of the following graph represents the current i_S in the secondary coil?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Current in secondary coil i_s is proportional to

Therefore,

Therefore, if

Then,

25. A time-dependent current (where K is a constant) is switched on at t = 0 in an infinite current-carrying wire. The magnetic vector potential at a perpendicular distance 'a' from the wire is given (for time t > a/c) by

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

26. The pressure of a non-relativistic free Fermi gas in three-dimensions depends, at T = 0, on the density of fermions n as

(a) n^5/3

(b) n^1/3

(c) n^2/3

(d) n^4/3

Ans. (a)

Sol. (g_s is called weight factor)

Pressure,

27. A double slit interference experiment uses a laser emitting light of two adjacent frequencies v₁ and v₂ (v₁ < v₂). The minimum path difference between the interfering beams for which the interference pattern disappears is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Relaxation time (Coherence time)

Coherence length

The fring width disappear if path difference is greater than half of the coherence length.

Therefore, minimum path difference for interference pattern to disappear is

28. The recently-discovered Higgs boson at the LHC experiment has a decay mode into a photon and a Z boson. If the rest masses of the Higgs and Z boson are 125 GeV/c² and 90 GeV/c² respectively, and the decaying Higgs particle is at rest, the energy of the photon will approximately be

(a)

(b) 35 GeV

(c) 30 GeV

(d) 15 GeV

Ans. (c)

Sol. The momentum of photon will be same (in magnitude) as Z-boson. If the energy of photon is E, then momentum

Now, energy of Z-boson where, m_z = rest mass of Z-boson.

Energy conservation :

Rest energy of Higgs boson = energy of Z-boson + energy of photon.

29. A permanently deformed even-even nucleus with J^P = 2⁺ has rotational energy 93 keV. The energy of the next excited state is

(a) 372 keV

(b) 310 keV

(c) 273 keV

(d) 186 keV

Ans. (b)

Sol. The energy , where, I is the moment of inertia.

So, for J = 2,

The next excited state is J^P = 4⁺

So, energy

30. How much does the total angular momentum quantum number J change in the transition of Cr(3d⁶) atom as it ionizes to Cr²⁺(3d⁴)?

(a) increases by 2

(b) decreases by 2

(c) decreases by 4

(d) does not change

Ans. (c)

Sol.

Since, 3d⁶ is greater than half filled, then ground state will correspond to maximum value of j, i.e. j = 4

Ground state term = ⁵D₄

Since, 3d⁴ is less than half filled, the ground state will correspond to minimum value of j, i.e. j = 0

Ground state term = ⁵D₀

Therefore, total angular momentum quantum number decreases by 4.

31. For the logic circuit shown in the figure below

a simplified equivalent circuit is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Output at point P = 0

Output at point K = ABC

Output at point x = 0 + ABC = ABC

32. A spectral line due to a transition from an electronic state p to an s state splits into three Zeeman lines in the presence of a strong magnetic field. At intermediate field strengths the number of spectral lines is

(a) 10

(b) 3

(c) 6

(d) 9

Ans. (a)

Sol.

There are 12 different combinations of m_j between final and initial level. But and is forbidden according to selection rule : .

Number of spectral lines = 12 – 2 = 10

33. A particle in the infinite square well

is prepared in a state with the wavefunction

The expectation value of the energy of the particle is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Since, is normalized, then

Therefore,

Expectation value of energy

34. The average local magnetic field acting on an Ising spin is H_int = αM, where M is the magnetization and a is positive constant. At a temperature T sufficiently close to (and above) the critical temperature T_c, the magnetic susceptibility at zero external field is proportional to (k_B is the Boltzmann constant)

(a) k_BT –

(b) (k_BT + )^–1

(c) (k_BT – )^–1

(d) tan h (k_BT + )

Ans. (c)

Sol. Total effective magnetic field

He = H₀ + H_in

Therefore, magnetization,

Taking the value of constant c = 1

35. In one dimension, a random walker takes a step with equal probability to the left or right. what is the probability that the walker returns to the starting point after 4 steps?

(a) 3/8

(b) 5/16

(c) ¼

(d) 1/16

Ans. (a)

Sol. In one dimension the probability of taking r steps right out of N steps.

After 4 steps, the Walker returns to the starting point i.e. r = 2

So,

36. Consider an electron in a b.c.c. lattice with lattice constant a. A single particle wavefunction that satisfies the Block theorem will have the form , with being

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The reciprocal of BCC will be FCC i.e. block theorem satisfy the wave equation.

Correct option is (b)

Note :E_n(K + G) = E_n(k). This indicates that E_nk is periodic with an period equal to the reciprocal lattice vector .

37. The dispersion relation for electrons in an f.c.c. crystal is given, in the tight binding approximation by

where 'a' is the lattice constant and is a constant with the dimension of energy. The x-component of the velocity of the electrons at is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The x-component of velocity

38. The following data is obtained in an expriment that measures the viscosity as a function of molecular weight M for a set of polymers.

M (Da) η(kPa – s)

990 0.28 + 0.03

5032 30 + 2

10191 250 + 10

19825 2000 + 200

The relation that best describes the dependence of η on M is

(a) ~ M^4/9

(b) ~ M^3/2

(c) ~ M²

(c) ~ M³

Ans. (d)

Sol. Taking the ratio of consecutive datas for and M.

39. The integral is to be evaluated up to 3 decimal places using Simpson's 3-point rule. If the interval [0, 1] is divided into 4 equal parts, the correct result is

(a) 0.683

(b) 0.667

(c) 0.657

(d) 0.638

Ans. (c)

Sol. Number of intervals = 4, width of interval = h = (b – a)/n = (1 – 0)/4 = 0.25

According to Simpon's rule.

40. In a classical model, a scalar (spin-0) meson consists of a quark and an antiquark bound by a potential

where a = 200 VeV fm^–1 and b = 100 MeV fm. If the masses of the quark and antiquark are negligible, the mass of the meson can be estimated as approximately

(a) 141 MeV/c²

(b) 283 MeV/c²

(c) 353 MeV/c²

(d) 425 MeV/c²

Ans. (b)

Sol. Bound state corresponds to minimum potential energy. For that

From

corresponds to minimum potential energy.

So, the potential energy of bound state

This is the total energy of meson because the mesons are spinless.

The mass

41. Let , where x₁ and x₂ are independent and identically distributed Gaussian random variables of mean µ and standard deviation . Then is

(a) 1

(b)

(c)

(d)

Ans. (b)

Sol.

The answers given for is a scalar number i.e. is independent of µ.

For simplicity, we can tanke µ = 0

y = (x₁ + x₂)/2

42. The graph of a real periodic function f(x) for the range is shown below

Which of the following graphs represents the real part of its Fourier transform?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The given graph in the question is of f(x) = cos

Therfore, the fourier transform of consist of two delta functions of same height at and k = –.

43. The matrices

satisfy the commutation relations

(a) [A, B] = B + C, [B, C] = 0, [C, A] = B + C

(b) [A, B] = C, [B, C] = A, [C, A] = B

(c) [A, B] = B, [B, C] = 0, [C, A] = A

(d) [A, B] = C, [B, C] = 0, [C, A] = B

Ans. (d)

Sol. The given matrix A, B, C satisfy the relations given in option (d)

44. The function satisfies the equation

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

As sin(x + iy) = sin x cos iy + cos x sin iy = sin x cosh y + i cos x cosh y

45. The coordinates and momenta x_i, p_i(i = 1, 2, 3) of a particle satisfy the canonical Poisson bracket relations {x_i, p_j} = . If C₁ = x₂p₃ + x₃p₂ and C₂ = x₁p₂ – x₂p₁ are constants of motion, and if C₃ = {C₁, C₂} = x₁p₃ + x₃p₁, then

(a) {C₂, C₃} = C₁ and {C₃, C₁} = C₂

(b) {C₂, C₃} = C₁ and {C₃, C₁} = C₂

(c) {C₂, C₃} = –C₁ and {C₃, C₁} = C₂

(d) {C₂, C₃} = C₁ and {C₃, C₁} = –C₂

Ans. (d)

Sol.

= p₂.x₃ – (–x₂)p₃ + (–p₁).0 – x₁.0 + 0.x₁ – 0.p₁

= p₂x₃ + x₂p₃ = C₁

= p₃.0 – x₃.0 + 0.x₃ – 0.p₃ + p₁x₂ – x₁.p₂

= p₁x₂ – p₂x₁

= –C₂

46. A canonical transformation relates the old coordinates (q, p) to the new ones (Q, P) by the relations Q = q² and P = p/2q. The corresponding time-independent generating function is

(a)

(b) q² P

(c) q²/P

(d) qP²

Ans. (b)

Sol. We know that

From given relation p = 2Pq

From given relation Q = q²

using equations (i) and (ii) we get F = q²P

47. The time evolution of a one-dimensional dynamical system is described by

If this has one stable and two unstable fixed points, then the parameter 'b' satisfies

(a) 0 < b < 1

(b) b > 1

(c) b < –1

(d) b =2

Ans. (b)

Sol. For fixed points,

(x + 1)(x² – b²) = 0

Let

f = – (x+ 1) (x² – b²)

At x = –1

At x = +b

At x = –b

For stable fixed point

For unstable fixed point

Obviously for b > 1, we get from equations (ii) and (iii)

48. A charge (–e) is placed in vacuum at the point (d, 0, 0), where d > 0. The region x < 0 is filled uniformly with a metal. The electric field at the point is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

49. An electron is in the ground state of a hydrogen atom. The probability that it is within the Bohr radius is approximately equal to

(a) 0.60

(b) 0.90

(c) 0.16

(d) 0.32

Ans. (d)

Sol. Probability that the electron will be within the Bohr radius

50. A beam of light of frequency is reflected from a dielectric-metal interface at normal incidence. The refractive index of the dielectric medium is n and that of the metal is n₂ = n(1 + i). If the beam is polarised parallel to the interface, then the phase change experienced by the light upon reflection is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

51. The scattering amplitude f() for the potential , where and µ are positive constants, is given, in the Born approximation by

(in the following and )

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Given : (spherically symmetric potential).

Using First Born approximation,

Scattering amplitude

Using, Laplace transform of r sinbr i.e.

Therefore,

52. The ground state eigenfunction for the potential , where is the delta function, is given by , where A and > 0 are constants. If a perturbation H' = bx² is applied, the first order correction to the energy of the ground state will be

(a)

(b)

(c)

(d)

Ans. (d)

Sol. First order correction to ground state energy

53. A thin infinitely long solenoid placed along the z-axis contains a magnetic flux . Which of the following vector potentials corresponds to the magnetic field at an arbitrary point (x, y, z)?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Magnetic vector potential outside the solenoid is

54. The van der Waals equation of state for a gas is given by

where, P, V and T represent the pressure, volume and temperature respectively, and a and b are constant parameters. At the critical point, where all the roots of the above cubic equation are degenerate, the volume is given by

(a)

(b)

(c)

(d) 3b

Ans. (d)

Sol. The van der waals equation of state for a gas is

at critical point,

V_c = 3b

55. An electromagnetically-shielded room is designed so that a frequency = 10⁷ rad/s the intensity of the external radiation that penerates the room is 1% of the incident radiation. If is the conductivity of the shielding material, its minimum thickness should be (given that ln 10 = 2.3)

(a) 4.60 mm

(b) 2.30 mm

(c) 0.23 mm

(d) 0.46 mm

Ans. (b)

Sol. Electromagnetically shielded wave equation defined as

where, = skin depth, x = thickness

Intensity, (taking proportionality constant to be 1)

I = E²

According to question,