For the function to be continuous and differentiable at any point, a and b should have the values
A. 0
B. a b 1/2
C. a b 3/2
D. a 3/2
2. The sum of the squares of the eigenvalues of a 3x 3 matrix A is 20. The corre­sponding sum for the matrix 2A is
A. 40
B. 60
C. 120
D. 80
3. The differential equation and its general.solution for the roots of the characteristics equations 4 are given by
A. ylll -8y" 16y' y C1 (C2 C3X) cosh 4x
B. ylll_ 8y" 16y' y Cl (C2 c3x )e4x
C. ylll -8y" 16y' y C1 (C2 sin4x
D. ylll -8y" 16y' Cr (C2 c3x )e4x
4. If 0 is real, the function sin(02) is
A. a periodic function of 0 with period 21r
B. a periodic function of 0 with period
C. a periodic function of 0 with period 1r2
D. not a periodic function of 0
3

5. One mole of a monatomic ideal gas initially at temperature To expands from volume to 2Vat constant pressure. The work of expansion and the heat absorbed
o
by the gas are:
A. W RTo, Q RTo
B. W RToln2, Q RTo
C. W 3RTo, Q
D. W
where R is the gas constant.

6. A body of constant heat capacity Cp and a temperature Ti is brought into contact with a reservoir at temperature Tf. If the equilibrium between the body and the reservoir is established at constant pressure and Ti the total entropy change of the body plus the reservoir is
A. CpIn(Tf
B. Cp(T.
C. Cp[ln(Tf (TilTf
D. Cp[(Td1j) -In(Tf/Ti
7. If Z is the canonical partition function of a system, the entropy of the system is
A. kBlnZ
B. kBT(8InZ/8Vh
C. kBTlnZ
D. kB[lnZ +T(8InZjaT)p]
8. If the interatomic potential energy in a solid is given by where a
Tn Tm
and bare positive constants and n the equilibrium interatomic spacing, To, is
A. To
n ml
B. To ­



C. To
D. To
[m C)rm nl
4


U-Sr

9. For non-magnetic metals, specific heat at low temperatures « eD, the Debye temperature) varies with temperature as
A. aT
B. a+bT
C.
D. a+bT2

10. A particle of mass m whose total energy is twice its rest energy collides with an identical particle at rest. If they stick together, the mass 11:1 and the velocity of the resulting composite particle are given by
A. M v c/V3
B. M V3m, v cJ2/3
C. M 2m, v c/5f3
D. M V2m, v cJ773

11. In a Compton scattering experiment a photon of energy Eo is scattered by an electron (of rest mass at rest. The energies of the photons scattered at 00 900 and 1800 with respect to the incident photon are in the proportion
mc2 mc2
A.
. 2Eo+ mc2 • Eo mc2
mc2 mc2

B. 1: Eo+mc2 2Eo+mc2
2

mc2 mc
C. 1: 2E+ mc2 Eo-mc2
o
mc2 V2rn.c2

D. 1:Eo mc2 Eo(V2 V2mc2

12. Consider 4 infinite sheets of surface charge density as shown in the figure.
5
4
3
2
z

The field in region 2 is given by
A. 0
B. (J /E02
C. (2Eo)2

D. 5


13. Consider the following charge distributions. All the polygons are regular with edge length d.
q/3
q

2q

d d dd




q
d d


qf2
qf3



The dipole moment is independent of the choice of origin for configurations
A. and
B. and
C. (iii) and
D. and
14. A hollow metallic waveguide can support
A. TE, TM and TEM waves
B. Either TE or TM waves
C. Both TE and TM waves
D. only TEM waves
15. To use a transistor as an amplifier
A. emitter-base junction should be forward-biased and the collector-base junction reverse-biased
B. both junctions should be forward-biased
C. both junctions should be reverse-biased
D. it does not matter how the transistor is biased, it always works as an amplifier
16. Which of the following logic gates can be used as a controlled inverter?
A .. AND gate
B. OR gate
C. INVERTER gate
D. XOR gate
6


17. The neutrinos emitted in a decay process are
A. right handed
B. left handed
C. both left and right handed
D. spin-zero and massless particles

18. In the fission of 235U92, the mass ratio of the two fission fragments is 1.5. What is
the ratio of the velocities of these fragments?

A. 3:2
B. 2:3
C. 9:4
D. 4:9

19. Alpha decay is an example of
A. nuclear fission
B. Barrier penetration
C. Coulomb repulsion
D. nuclear ionization

20. Let In denote the normalised eigenstates of the Hamiltonian H describing a linear
harmonic oscillator corresponding to the eigenvalues The normalised state
1
vg[211 -12
is

A. an eigenstate of H corresponding to the eigenvalue
B. an eigenstate of H corresponding to the eigenvalue 111iw.
2
C. an eigenstate of H corresponding to the eigenvalue 0
D. not an eigenstate of H.

21. A, Band 6 are three Hermitian operators. Which of the following statements cannot be correct?
A.
B.
C. i6
D. A2+ B2 62
7 22. The allowed energies for a particle in a one-dimensional square well potential be­tween and a are




The wave function shown in the figure

corresponds to
...
.t"l.. n

B. n=2
C. n=4
D. n=7
23. With the positive z-axis chosen to be upwards, the Lagrangian of a particle of mass m falling under gravity is
A. -mgz
B. -mgz
C. +mgz
D. +m.gz
24. An aircraft executes a horizontal loop of radius 1 km with a steady speed of 900 km/h. The ratio of horizontal acceleration of the aircraft to the acceleration due to gravity (g=9.8 rn/sec2 is about
A. 0
B. 5
C. 6
D. 7
25. Which of the following has non-vanishing Poisson bracket with L the Z-cornponent of the angular momentum
A. a(x2 +y2 z2)
B. +bz2
C. ax2 +by2 +az2
D. a(x2
8



SECTION


26. The value of the integral
2
i:dx x
5(2x

where denotes the Dirac delta function, is
A. 0
B. 1/4
C. 1
D. 1/8
27. If II and denote the values of the integral of evaluated along the semicircle
2

0 1 and O2 as shown, then II -12 is

A. 0
B. 21ri x


C. 21ri x
C)..

D. 21r
28. For a matrix A satisfying A3 which of the following cannot be a possible set of eigenvalues?
A.
B. e21l"i/3)
C. e21l"i/3, e41l"i/3)
D. e41l"i/3)
29. It is impossible to find a real 4 x 4 matrix with the eigenvalues given by
A.
B.
C.
D.
30. The value of the integral

e-x2 H3(X)P2


where and are the Hermite and the Legendre polynomials, is
A. 23. 3!J1f
B. 2/5
C. )23. 3!J1f .
D. 0
z

31. For the function fez) 2)2 the order of the pole at z and the residue
a.t
the pole are respectively given by

A.


B.
2,1

C.


D.
1,1


32. The Bernoulli equation given by m =f. m can be reduced to a linear equation for z by substitution
A. z yl-m
B. z ym-l
C. Z yl-2m
D. z y2m-l
33. The rank of the matrix

11 0 1



1

00 is
A. 1
B. 2
C. 3
D. 4
10



34. Let In denote the eigenstates of a linear harmonic oscillator corresponding to the eigenvalues If +11 denotes the state of a harmonic oscillator at t 0 and that at time the 12 first becomes
zero at
A. t 7f/8w
B. t 7f/4w
C. t 7f/2w
D. t 7f/w
35. Which of the following pairs of wave function (with A describe the same physical state of a free particle
Aeikx Be-ikx _Aeikx Be-ikx

A.·
Aeikx Be-ikx Aeikx

B. 1P2(X) -Be-ikx
Aeikx Be-ikx _Aeikx Be-ikx

C.
D. Aeikx Be-ikx Beikx Ae-ikx
36. For a particle in the state the expectation value of 5z z is


A. a+cL


B.
C.
D. 0
37. Two spin-! particles with spins 51 and 52 are in a singlet state. The value of the
1
potential V1 n2S1 . for such a state will be

A.
B.
C. 0
D.


38. Ifil and prespectively denote the Hamiltonian and the momentum operator for a free particle, then the functions sin kx and cos kx are eigenfunctions
A. of both il and p
B. only of H but not of p
C. only of p but not of H
D. of neither il nor p
39. If the Gibbs free energy of a system is given by where a and R are constants, the specific heat at constant pressure, C is
A.
B. QR
2

C.
D.
40. The entropy of a magnetic system changes by an amount 6.S -CH6.H/T2 (where C is a constant) if its temperature is held constant at T and the magnetic field is changed from H to H The magnetization 111 of such a system depends
on temperature T and field H as
A. M=CH2/T
B. M=CT/H
C. 111
D. M=CH/T
Hint: the change in the Gibbs free energy is given by dG -SdT -MdH
41. A system of two energy levels Eo and E1 (E1 Eo) is populated by N particles at
temperature T in accordance with the Boltzmann distribution law. The average energy per particle is
A. Eo +E1
B. E1 ±
C. [Eo
D. [Eo +E1
12


u-c;q...
42. A monatomic gas consists of atoms with two internal energy levels: a ground state of degeneracy 91 and an excited state of degeneracy 92 at an energy E above the ground state (chosen as the zero of energy scale). The specific heat of the gas is
3

A. '2kB
3 91e-€/kBT

B. -kB T
21 92e-€ B
3 99 E2ef/kBT
· 12

C• -kB
2 kB T2(92 91e€/kBT)2

9192E'2 e-f/kBT

D. kB T(92 91ef/kBT)2
43. Consider an ideal gas consisting of N particles obeying classical statistics. Suppose that the energy of one particle E is proportional to the magnitude of its momentum i.e., E Cpo The partition function of the gas is proportional to
A. 3N
B.
C. N


(CkBT)N

D.
44. If in a diffraction experiment, X-ray photon of wavevector k gets scattered to the final state with wave vector where 0 is a reciprocal lattice vector, the Bragg diffraction condition is given by
A.
B. k·0+G2
C. k·0=(0·0)/2
D. 2(k .0)

45. If al a2 and a3 are the primitive translation vectors of a cubic lattice of size (lattice parameter) a and 2 are the
orthogonal unit vectors parallel to the cube edges, the primitive translation vectors bI b2 ,b3 of the reciprocal lattice are given by
A. bI 21f b2 21f b3 27r
a a· a

B
• /. .Tn 3-a "T
a a

C "b-21f(A 21f( 27r( A.
•
aa a

D b-21f( 21f( 27r(
• 3=7
a a

46. If the atoms vibrating in a lattice at finite temperature are considered to be a
collection of N identical harmonic oscillators, as Einstein did, the heat capacity at constant volume for such a solid is given by

where x fiw/kBT
47. Consider a system of N weakly interacting particles, each of spin and magnetic moment exposed to an external magnetic field H. If the system is in thermal contact with a heat reservoir at temperature the mean energy of the system as a function of T and H is given by
A. N kBT)
B. N kBT)
C. N kBT)
D.
14



48. Which of the following relations is consistent with the London equation curl]
c ­
-411"Ai B for a superconductor, where AL is the London penetration depth.
A. VR RIAL
B. V 2R R/AL
C. V2 ii/Ai
D. vii ii/Ai

49. The output Va of the circuit is

A. Va 0
B. Va (R21R1)V1
C. Va R2 R1)V1

D. Va R1)V1
50. In the following figure, the transistor has (3dc of 80 and VCE(sat) of 0.1 V. RB is adjusted to get transistor into saturation. The value of IC(sat) and the corresponding value of RB are
+25V

2N4401


A. 114 rnA and 17 kO
B. 1.43 rnA and 17 kO
C. 114 rnA and 1.7 kD

D. 1.43 rnA and 17.48 kO

51. In emitter follower configuration of a transistor
A. the emitter current follows the collector current and both are in same phase
B. the collector current follows the emitter current and both are in same phase
C. the base current follows the emitter current and is in opposite phase
D. the emitter current follows the base current and both are in same phase

52. Binary equivalent of (8.2ho is
.. 1-1 1 1

PL. VV1 . VV
B. (1 000. 001 1)2
C. (1 000. 011 0)2
D. (0111.011 0)2

53. In a pure germanium the intrinsic concentration of charge at room temperature
2
is 2.5 x 1013 cm3. The electron and hole mobilities are given by f-tn 3800 cm and f-tp 1800 cm2/V-s respectively. The resistivity of the intrinsic germanium is
A. 22.3 n-cm
B. 66.9 n-cm
C. 44.6 n-cm
D. 11.2 n-cm

54. The change in isospin in the process AD p
A. 6.1 0,1
B. 6.1=
C. 6.1
D. 6.£ 1,2

55. A spin-O particle A decaying in its rest frame into particles Band through strong interaction. Which of the following statement is not true?
A. 0+
B. 0+
C.
D. 0­
where JP= or 0-representing the standard convention for labeling the spin and intrinsic parity of a particle.


56. Which of the following reactions represents a strong interaction
A. p KO
B. pO-t nO nO
C. nO
D. p P-t nO

57. In the nuclear shell model, is a good closed-shell nucleus and has spin parity JP 0+. What are the predicted JP values for and rO?
A.
B.
2
C.
2


D.

58. A pion at rest decays to a muon and a neutrino. The magnitude of momentum of the outgoing muon is
m2 _m2
A. 11" J.l • c
2m7[
m2 +m2
B. J.l • c
2m1l"
m2 +m2
C. 11" J.l. c


D. ·c
2m7[

5.9. An experiment is performed to search for the evidence of the particle H through the strong reaction p+p H The electric charge Baryon number

and strangeness quantum number of the particle H will be

A.
Q B 0,5

B.
Q B 5


C. Q B=2 5=2
D.


60. Let a plane monochromatic wave be totally internally reflected at the interface z 0 between two dielectrics. Let x be the plane of incidence with x along the interface. The planes of constant amplitudes and constant phases in medium 2 are given by
A. Z CI, X C2
B. x CI, X C2
C. X CI, Z C2
D. x CI, X Z C2
(here CI and C2 are constants)
61. The transformation that reduces the wave equation
cPu(x, 82u(x, 8x2 c2 8t2 b

to t he £orm 8{8r IS gIven y
A. r
B. it r it where i
C.
D. t r t where
62. In the domain of anomalous dispersion, the group velocity (vg refractive index and speed of light in vacuum obey
A. vg cln
B. vg c/n
C. vg c/n
D. c/n vg c
63. The resolving power of a grating of a given width for a given order of diffraction can be increased by
A. making thinner grooves without changing the groove separation
B. making thinner grooves and proportionately decreasing the groove separation
C. making wider grooves without changing the groove separation
D. making wider grooves and proportionately increasing the groove separation
18



64. In a Newton's ring experiment, a plano-convex lens with radius of curvature of the convex surface 4.18m is placed on an optically fiat surface with its convex surface down. Interference fringes are observed using a travelling microscope and sodium light 589.3 nm). The radius of the tenth dark ring is
A. 4.96 mm
B. 6.92 mm
C. 1.63 mm
D. 2.45 mm

65. Consider a uniformly charged sphere of radius R with total charge q. Let the sphere be coated with a medium with charge density p air (where a r R). The electric field Eoutside the charged sphere will be independent of r when
A. q 21raR2/to, f
B. q 211'0:R2 f
C. q 11'aR2 E f
D. q f

66. Consider two infinite current carrying wires rigidly held parallel to the z-axis. The current through the wires is The distance between the wires is d. The magnetic field at the points marked P in the three cases shown in the figure below are
x x
x

I t
i •
I


2J-Lo/ O. 4J-Lo/
A.
A• 1 11 ,III y
0 2J-LoI 4J-Lo/
A A
.B. 1 ;11 111 311'd y
C. y
.2 I 4/
D.

19





67. Two infinite conducting planes intersect at the origin (see figure). All distances are in meters.
• (1.
•
q



A charge q is placed at The potential at the point will be
q 4V5

A. 117
41fEo 3y 10
q 2V5-6

B. 117
41fEo 3y 10
q 4V5 6

c. JIG
41fEo 3 10
q -2V5-6

D. 117
41fEo 3y 10

68. The mast of a sailboat makes an angle ewith respect to its deck. An observer standing on the dock sees the boat go by with a speed v. The angle made by the mast as seen by the observer is
A. tan-1 (Jl-v2jc2 tane)
1


B.


c.
tan-1 (tan(eJl-v2jc2))

D.
tan-1


20



69. The coherence matrix for light propagating along z-direction can be written down
as

1xy
EyE; EyE;
where Ex and Ey are the projections of the electric field along the x and y axes and
the overbars denote averaging over time for long enough observation periods. If the
determinant of the matrix equals 1z/4 (where 1 IExlz IEyIZ) then the light is
A. completely polarized
B. completely unpolarized
C. 50 partially polarized
D. 25 partially polarized

70. For a particle in a one dimensional potential

the time period of small oscillations about the equilibrium is
A. 21r
2Von z
21r8._.-
B.
maZ

C.
ma 0
D.
21rJlIo:Z

71. The Lagrangian of a point particle is

Its equation of motion is
A. mx ,Biz wZx 4ax3 0
B. mx -wZx 4ax3
C. 2,Bx)x (3iZ wZx 4ax3
D. ,B±Z wZx 4ax3 0


72.
A particle of mass m moves in a spherically symmetric potential and has angular momentum L. Which of the following is a constant of motion?

73.
A particle of mass m and angular momentum L2 lOml/aR2 moves in a potential V Va o. The radius of the stable circular orbit is



A. R
B. R/2
C. 2R
D. 3R
..
74. The Poisson bracket of Lx and p2, where rxp denotes the angular momentum,
is
A. 2Lx
B. 2px
C. LxPx
D. 0

75. Consider a diatomic molecule undergoing a vibrational electronic transition from V" to Which one of the following is correct?
t


A. The most intense line will be transition
B. The most intense line will be transition
C. The most intense line will be transition
D. The most intense line will be transition

23