1. Which of the following equations is best represented by the graph below

A. xy where a is a constant
B. y mx where m and C are constants
C. y exp(x) where a is a constant
D. y axm where a and m are constants


2. The equation of the plane tangent to a sphere x2 y2 Z2 1 at the point is given by
A.
B. V3
C. x Y z 3V3
D. 2x 2y 2z 1
3. The set of all n x n complex matrices form a vector space under matrix addition. Which of the following is not a subspace?
A. The set of n x n complex matrices with all the diagonal elements equal to zero
B. The set of n x n complex matrices with the elements in the first column equal to zero
C. The set of n x n complex matrices with the elements in the first row equal to zero
D. The set of all n x n complex matrices with the determinant equal to zero

are given by 4. The eigenvalues of the matrix
A. 0,1,1
B. ±y'2
C. ±y'2
D. 0, 0,2
5. y sin x/x is a solution of the differential equation
A. Y cos x
B. xy' y sin x
C. xy cos x
D. xy sin x
6. If sin(z2/4) is expanded in powers of the series will converge for
A. all values of z
B. Izi<2
c.
Izi 1
1


D.


Izi <"2
7. The equation of the parabola shown in the figure is
y
A. y x2 a+b
B. y a
C. y
D. y a

8. The function z-2 has a singular point at z O. The nature of this singu­larity is
A. pole of order 2
B. essential singularity
C. pole of order 2 and an essential singularity
D. pole of order 2 and a branch point
9. Zl and Z2 are complex numbers. The modulus IZI z21 is
A. IZll IZ21
B. IZII IZ21
C. IZII-lz21
D. IZII ·IZ21
10. The value of the integral,
1
+00 -00

where is Dirac delta function, is given by
A.
a
B. aj(O)
1

C.
D. lalf(O)
11. The graph shown corresponds to the equation
A. y 2eX y
B. y e-X
C. y eX
D. y e-xl x

12. Suppose that a variable Q only takes on five possible values: the value "I" with probability 0.25, the value "2" with probability 0.25, the value "3" with probability 0.05, the value "4" with probability 0.30, and the value "5" with probability 0.20. What is the probability that Q is an odd number?
A. 0.45
B. 0.55
C. 0.0025
D. 0.50
13. The number of root(s) to the equation tanx x is
A. one
B. two
C. zero
D. infinite
14. In the Kepler problem the conserved quantites are
A. Energy only
B. Energy and angular momentum only
C. Energy and velocity
D. Energy, velocity and angular momentum
15. For a scattering experiment of a particle by a central potential what happens as the impact parameter decreases?
A. scattering angle decreases
B. scattering angle increases
C. scattering angle is unchanged
D. scattering angle may increase or decrease
16. Consider a system described by the Hamiltonian qi, t). If qi, and qi, are two conserved quantities defined on the phase space of this system, which of the following statements is correct?
A. and AB are conserved
B. is conserved but not AB
C. is not conserved but AB is
D. neither nor AB is conserved
where is the Poisson bracket of A and B.
17. A particle with spin 1 has orbital angular momentum 2. The allowed values of total angular momentum are
A. 3 only
B. ± 3only
C.
D. ±2, ±1, 0
2
p1
18. Consider a particle moving under a SHO whose Hamiltonian is Ho 2m "2mw2x2
(one-dimensional motion). If there is perturbation to the Hamiltonian by a term V AX, the change in the ground state energy, to first order in is
A. zero
B. A
C. -A
A

D. ­
4
19. 'Zero point energy' is a result of
A. Energy conservation
B. Relativistic effects
C. Uncertainty principle
D. Pauli's exclusion principle
20. The Second law of thermodynamics says (qualitatively) that
A. Energy is always conserved
B. Energy is always greater than entropy
C. Entropy increases as disorder increases
D. Hot water added to cold water produces warm water
21. The plot below represents
A. Bose-Einstein distribution at non-zero temperature
B. Bose-Einstein distribution at zero temperature
C. Fermi-Dirac distribution at non-zero temperature
D. Fermi-Dirac distribution at zero temperature
22. In a canonical ensemble, if Pr represents the probability that a system has energy Er then the entropy S is given by (here k is the Botlzmann's constant)
A. k I:r In Pr
B. -k I:r Pr2 ln Pr
C. kI:rPrlnPr
D. -kI:r Pr In Pr
23. The energy of a uniformly charged spherical shell of total charge q and radius R is proportional to
A. q/R
B. q2/R3/2
C. q/R2
D. q2/R
24. The magnetic field at P due to an infinite straight wire carrying steady current I is
A. Mol/27rZo pointing out of the page
B. Mol/27rZo pointing into the page
C. Mol/27rZo along Zo
D. Mol/27rZo parallel to the wire
25. Consider two oppositely charged overlapping spheres with same radius and with charge densities and -po Which one of the following statements is true?
A. Field is nowhere uniform
B. Field is everywhere uniform
C. Field in the region of overlap (shaded) is uniform
D. Field in the region of nonoverlap (unshaded) is uniform

SECTION

26. The curve parametrically given by x 1+2 y t E represents
A. an ellipse
B. a hyperbola
C. the right hand branch of a hyperbola
D. the left hand branch of hyperbola
27. If C is a unit circle in the complex z-plane with origin as the centre and traversed in an anti-clockwise direction, then the value of the integral dz is
r
lc z 4
A. 21f
B. 0
C. 1f/4
D. 1f
28. Urn 1 contains 4 red and 6 green balls while urn 2 contains 6 red and 3 green balls. A ball is selected at random from urn 1 and transferred to urn 2. Then a ball is selected at random from urn 2. The probability that the ball from urn 2 is green is
A. 0
B. 9/25
C. 1/25
D. 6/25
29. Which of the following equations has damped oscillatory solution?
d2y dy
A. dt2 dt Y 0
d2y dy
B. dt2 dt 2y 0
d2y dy

C. dt2 -10 dt 25y 0
d2y dy
D. dt2 5 dt 25y 0
2
.1
3z
30. The value of the mtegral 2 dz,
c z -4z 9
where c is the circle J3z -21 1 oriented in anticlockwise direction is
A. zero
B. 27fi
C. 47fi
D. 67fi
31. The initial estimate of the root of the equation found by using Newton­Raphson method, is XQ 5. The angle, the tangent to the function makes at x is 57 0 with respect to the x-axis. The next estimate of the root, Xl most nearly is
A. -3.2470
B. -0.2470
C. 3.2470
D. 6.2470
32. Which of the following is not a basis for the complex vector space C3 consisting of all complex 3-tuples
A.
B.
C.
D.
33. Which of the following matrices is non-invertible?
A.
-1

B. 1


-1

C.
-1

D.
-1

34. Vectors ri and is can be combined to find scalar and vector products, ri .is and ri x is respectively. Which of the following statements is true?
A. (ri x is?
B. f2)2
C. (ri x is)2 10
D. r2?
35. The fourier transform of a gaussian function, e-x2 in k-space is
A. a gaussian in 11k, like k2
B. a powerlaw in like l/k2
C. a sinusoidal function like sin k or cos k
D. a gaussian in k like e-k2
36. The probability of coins falling 'heads or tails' is The probability of a cubic dice falling with side or or 3,· . " or 6 facing upwards, is i. The probability when tossing a coin and rolling a dice of having the result coin-heads and dice 4 is given by
1 1 3
A.
11 2

B. 2+
11 1

C.
26 12
111

D. 2x 8
37. Which of the following series does not diverge for x 1
A. 1-x x2 x3 ...
x23

x
B.
x2 x4 x6
C.
D. 1+ x3 x6 x9
38. x G(rj is the curl of a 3D vector field with r z). For G(rj the curl F(rj is:
A.
B.
C.
D.
39. The limit of the sequence
is
A. 1
B. 2
C. 2v'2
D. 00
40. If Mlj aMij . then VfM'T is proportional to
a12
A.
B. a
C. an/ 2
D. an
41. The principal value of (where In denotes the natural logarithm) is:
A. In2
7r

B. In2+ "2
C. InV5 17ri
D. In 37ri
4

42. The matrix M which diagonalizes the matrix H is given by
A. 1i)
B.
1/V2
c.


D.



1/V2
i/V2
43. The Laplace transform of the function sin2t is
2

A. 8(82+
1

B.
82+4
c.
2sin(8)cos(8)
21


D.



82+4 8
44. A box contains 10 screws needed in a certain order for assembling a product. Out of these, 6 are alike right handed screws, and the remaining 4 are alike left-landed screws. Now, if the 6 right-handed screws are needed first and the 4 left-handed screws are needed later, the probability that random drawing yields the screws in the desired order is:
A.


B. ­
10!


c.
10! 10!
10!

D.

45. The function cos z cot z has poles at z mf where
A. n 0 only
B. n is an odd integer only
C. n is an even integer only
D. n is any integer
46. The set forms a group under addition, modulo 2. The inverses of elements 0 and 1 are respectively
A. 0,1
B. 1,0
C. 0, 0
D. 1, 1
47. The function f with real, is
A. a periodic function of with period 27f/3
B. a periodic function of with period
C. a periodic function of with period 127f2
D. not a periodic function of
48. The error function is defined as
lx
2
erf(x) y'ir 0 e-t2 dt.
Which of the following statements is correct?
A. erf(x) is an even function
B. erf(x) is an odd function
C. erf(O) 1
D. erf(oo) 1
49. Consider three operators Z on a three dimensional complex vector space de­fined by
a Ib+ el, Ie al)
A. all the three operators Z are linear operators
B. only X is a linear operator
C. only Y is a linear operator
D. only Z is a linear operator
50. The function given by
ax 1f/2 sinx 1f/2
is continuous if
A. a sin b
2b

B.
1f
b 1fa
C.2
D. b sin a
51. Roots of an equation -cos can be found as an intersection of two graphs. If one of the graphs is fl x2, then the other graph is
A. cos x
B. -2cosx
C. cos2 x
D. -cos2 X
52. The transformation is given to be canonical where Q ay'qet cos p f3y'2q e-t sinp. Then a and f3 should satisfy
A. af3 1
B. a f3 0
C. af3
D. af3
53. The Hamiltonian corresponding to the Lagrangian
m df df df.... m.... 2

2 dtdt dt 2
is
p.p .... (....
A· 2m w x rj,
P .... (.... m(....
B· wxrj+-wxrj,
2m 2
.... (.... m(.....
C· 2m w x rj w x rj
P .... (.... ....)
D· 2m r x
54. For a particle of mass moving under the influence of a central potential, the Poisson bracket {L1 L2,L3 is (here L1, L2, and L3 are the components of the angular momentum)
A. zero
B. Ld}
C.
D.
.... kf
55. A particle of mass m moves in a central force field defined by F If E is
r
the total energy supplied to the particle, then its speed is given by
A. [kr2+ 2E/m]2
B. [kr2+ 2Emj1/2
2E] 1/2

C.
mr2 m
(km 2E)2
D. r2 m
56. The mutual potential energy V of two particles depends on their spatial separation according to
a b
V b 0.
rr
For what separation, are the particles in static equilibrium?
A. a/b
B. a/2b
C. a2/b
D. 2a/b
57. If the Hamiltonian for a system in two dimensions is H The Poisson
2m r
bracket is
A. zero
B. Px/m
C. -k/r2
D. -kx/r3
58. A particle of mass m has uncertainty a in its position. Its kinetic energy must be equal to or greater than
A. fi2/(2ma)
B. fi2/(8ma2)
C. fi2/(2ma2)
D. fi2a2/2m
59. Given that Sz and the spin state of a particle being subse­quent measurement of spin along z-direction would yield the spin value as
B.
C. zero
D. half of the measurements yield and the other half yield
60. Consider a state in which the coordinate x has zero uncertainty. Then a possible form of the wave function is
e-a1xl
A.
B. b(x
C. 'ljJ e-x 2/a 2
eikx
D.
61. On the Hilbert space of square integrable functions 1:dxl'ljJ(xW 00, the action of the parity operator P is defined as follows

Which of the following statements concerning P is not correct?
A. P is a hermitian operator
B. P has eigenvalues and
C. P is a unitary operator
D. P is a projection operator
62. If In with n .. denote the eigenstates of the Hamiltonian H of a one dimensional quantum mechanical harmonic oscillator corresponding to the eigen­values )lU.u, then the expectation value of H in the state
1'ljJ 111 +212 -i13
is
A. 15lU.u
15lU.u

B.
6
C. 7lU.u
D.
6
1
63. The density operator corresponding to the state .J2[10 +ill is
1
A. 01+ 11]
1
B. "2[10 01-11 11]
C. 11+ill 01+ 11]
D. 11+ill 11]
64. The ratio of the population of two levels 1 and with the energy E1 and E2(E1 E2 of a classical system in contact with a heat bath at temperature T is given by (where E E1 -E2
A. E/kT
B.
C. log(E/kT)
D.
65. Consider N independent particles, each of which can be in two states, with energies aand E. The partition function of this system is
A. Z
B. Z N(l
C. Z N


D. Z

66. The entropy S of a system at fixed volume V depends on absolute temperature T as S -aT3 -bT2 80 where 80 are constants. The specific heat Cv is
A. -(aT2 bT)
B. 3aT2 2bT
C. -6aT -2b
D. 3aT3 2bT2
67. A quantum mechanical system has 3 energy levels, with energies Eo, 2Eo,3Eo and respective degeneracies 1,2 and 3. The partition function for this system is

kBT
A. e-{3Eo e-2{3Eo e-3{3Eo
B. e-{3Eo 2e-2{3Eo 3e-3{3Eo
e-{3Eo 2e-{3Eo 3e-3{3Eo
C. 6e-{3Eo
e-{3Eo 2C2{3Eo 3e-3{3Eo
D. 6e-6{3Eo
68. The probability distribution of points in phase space representing a microcanonical ensemble as a function of energy, is expressible as one of the following
A. uniform distribution
B. Gaussian
C. Lorentzian
D. Dirac's o-function
69. Given that p is the number density of the particles and A is the de-Broglie wave­length, the quantum statistics reduces to classical case· under the following condi­tions:
A. pA3 « 1
B. PA3
C. p)..3» 1
D. p=o
70. Which one of the following vector potentials corresponds to a constant magnetic field in the positive z direction
A. Ay
B. Ay
C. Ay,Az)
D. Ay,Az)
71. A point charge q is placed outside, at a distance d from the centre, of a grounded conducting sphere of radius r. The magnitude and the position of the image charge are
a
A. q d d
2

d

72. A thin, nonconducting ring of radius as shown below, has a charge Q uniformly spread out on it. The electric potential at a point which is located on the axis of symmetry at a distance x from the centre of the ring, is given by
Q
A.
47fEoX
Qx
B. 47fEo(R2 x2)
Q

C.
2
47fEoVR2 x
Q

D. 47fE(R2 x2)3/2
o
73. A rectangular waveguide has a cross-sectional dimension of 3 cm xl cm. It is filled with a material of dielectric constant 4. What is its cutoff frequency for the dominant mode?
A. 5 GHz.
B. 1.25 GHz.
C. 15 GHz.
D. 2.5 GHz.
74. A double slit interference arrangement is illuminated with green light of wavelength 546 nm. The two slits are 0.12 mm apart and the screen on which the interference pattern appears is 55 cms away. The angular position of the first minimum, in radians, is approximately
A. 0.023
B. 0.0034
C. 0.0023
D. 0.00023
75. To evaluate the dipole moments in the given configurations, one needs to have the knowledge of the origin
A. for both I and II

6+q
B. only for I and not for II S SITS
C. only for II and not for I S q s
D. knowledge of origin is irrelevant in both the cases