1A
Q.1. A cylinder with moment of inertia Io rotates with angular velocity ωo. A second
cylinder with moment of inertia I1 , not rotating initially, drops onto the first cylinder and
the two reach the same final angular velocity ωf. The value of ωf is
A:
ωf = ωo
B:
ωf = ωo ( Io/ I1)
C:
ωf = ωo Io / (Io + I1)
D:
ωf = ωo I1 / Io
E:
ωf = ωo (Io + I1)/ Io
2A
Q.2.
Two equal masses m1= m2= m are connected by a spring having a spring constant k. If
the equilibrium separation is lo and the spring rests on a frictionless table then the angular
frequency is
A;
√
B:
√
C:
√
D:
2√𝑚
E
√𝑙
𝑘
𝑚
2𝑘
𝑚
3𝑘
𝑚
𝑘
𝑔
𝑜
3A
Q.3. An object orbits a star in an elliptical orbit. The distance at the apehelion is 2a and the
distance at the perihelion is a. The ratio of the object’s speed at perihelion to apehelion is
A: 2
B: 3
C: 1
D: √2
E: √3
4A
Q.4. A spherical mass m is dropped vertically off a building, starting from rest and y=0. The
mass experiences a resistive force FR= -bv. Let γ=b/m. The position y(t) of the mass is
𝑔𝑡
A: y=
𝛾
𝑔𝑡 2
B: y=
2
𝑔
C: y= 𝛾 (1 − 𝑒𝑥𝑝−γt )
D: 𝑦 =
𝑔𝑡
γ
𝑔
− 𝛾2 (1 − 𝑒𝑥𝑝−γt )
𝑔
E: y=− 𝛾2 (1−𝑒𝑥𝑝−γt)
5A
Q.5. A particle of mass m is restricted to move on the surface of a sphere of radius R near the
earth’s surface. (Potential energy U=mgz). The Lagrangian in spherical coordinates is
A:
B:
C:
D:
E:
1
2
1
2
1
2
1
2
1
2
𝑚(𝑅 2 + 𝑅 2 𝑠𝑖𝑛2 𝜃) − 𝑚𝑔𝑅𝑐𝑜𝑠𝜃
𝑚(𝑅 2 𝜃̇ 2 + 𝑅 2 𝜑̇ 2 𝑠𝑖𝑛2 𝜃) + 𝑚𝑔𝑅𝑐𝑜𝑠𝜃
𝑚(𝑅 2 + 𝑅 2 𝑠𝑖𝑛2 𝜃) + 𝑚𝑔𝑅𝑐𝑜𝑠𝜃
𝑚 (𝑅 2 𝜃̇ 2 + 𝑅 2 𝜑̇ 2
) − 𝑚𝑔𝑅𝑐𝑜𝑠𝜃
𝑚(𝑅 2 𝜃̇ 2 + 𝑅 2 𝜑̇ 2 𝑠𝑖𝑛2 𝜃) − 𝑚𝑔𝑅𝑐𝑜𝑠𝜃
6A
Q.6. Consider a one-dimensional particle which moves along the x-axis and whose Hamiltonian is
d2
H   2  16x 2 , (where ε is a real constant having the dimensions of energy) then which one
dx


of the following is correct ( e x dx 

2

)

A) The expectation value of position observable x is nonzero
B) The total probability of finding the particle anywhere along the negative x-axis is
less than
1
.
2
C)  ( x)  Ae 2 x is an eigenfunction of H with eigenvalue 4ε.
2
D) If 𝜑(𝑥) is another eigenfunction of H given by 𝜑(𝑥) = 2𝑥𝜓(𝑥)then 𝜑(𝑥) and 𝜓(𝑥) are not
orthogonal.
E) The total probability of finding the particle anywhere along the positive x-axis is
more than
1
.
2
7A
Q.7. Consider a beam of Helium atoms excited to 3P2 state and is passed through the SternGerlach experimental setup. Which one of the following is correct?
A: The beam would split into three components corresponding to Ms = ±1 and Ms =0 states
B: The beam would split into three components corresponding to Ml =±1 and Ml=0 states
C: A continuous distribution would be observed.
D: The beam would split into five components corresponding to Mj=±2, Mj=±1 and Mj = 0
states
E: The Stern Gerlach setup can be used only to observe the splitting for the S =
1
state
2
8A
Q.8. The lowest energy bound state of a one electron bound system can be described by which
one of the following wave function
A) 𝐴 𝑒𝑥𝑝−𝑘𝑟
B)
𝑘
𝑟
C) sinkr
D) Both A and B
E) Both A and C
9A
Q.9. The energy levels of a particle of mass m moving in a one dimensional potential
V ( x )  , x  0
1
V ( x)  m 2 x 2 , x  0
2
are given by
1

A) E n   n   , n  2, 4, 6....
2

n 
B) E n    1 , n  0,1,2....
2 
1

C) E n   n   , n  1, 3, 5, 7....
2

3

D) E n   n   , n  0,1,2....
2

E) None of the above
10A
Q.10.. Consider a system whose wave function is given by
1
2
 ( x, y, z )  Y00 
1
1
1
Y11  Y1, 1 
Y22
2
3
6
Which of the following is a false statement
A)  ( x, y, z ) is normalized
B)  ( x, y, z ) is not an eigenstate of L2
C)  ( x, y, z ) is not an eigenstate of LZ
D) If a measurement of the z-component is carried out the probabilities of finding 0,  ,  and 2
are
1 1 1
1
, , and respectively.
4 3 4
6
E) Parity of ( x, y, z ) is odd.
11A
Q. 11. A uniformly charged ring with total charge q and radius c is concentric and coplanar to
another one with charge –q and radius b where b<c. The potential on the axis of the ring at
a distance r (r>>c) from the center of the ring is
q  b 2  c 2 
A) V=


4 0 r 3  2 
B) V=
 b2  c2 


4 0 r 3  2 
q
q  b 2 
C) V=


4 0 r  2r 2 

b2 
1



4 0 r  2r 2 
E) Zero
D) V=
q
12A
Q.12. When the current in an R-L circuit is decreasing, approximately what fraction of the
original energy stored in the inductor has been dissipated after 2.3 time constants?
A) 1.0%
B) 10%
C) 50%
D) 30%
E) 99.0%
13A
Q.13. A conducting sphere of radius a, is surrounded by an isolated thick spherical conducting
shell of inner and outer radii b and c respectively. The outer shell is isolated and considered
to be initially uncharged. A charge +Q is to be placed on the inner sphere. The capacitance
of the conducting sphere is
 abc 
A) C  4 0 

bc 
 ac 
B ) C  4 0 

bc
abc


C ) C  4 0 

 ba  cb  ac 
 ac 
D ) C  4 0 ln 

bc
 a 2  c2  b2 
E ) C  4 0 
2 
 (a  b  c) 
14A
Q.14. A single turn loop is situated in air, with a uniform magnetic field normal to its plane. The
area of the loop is 5 m2. What is the emf appearing at the terminals of the loop if the rate of
change of flux density is 2 weber/m2/sec?
A) ε=-5 volts
B) ε=-10 volts
C) ε=-2.5 volts
D) ε=-25 volts
E) ε=-50 volts
15A
Q.15. Given a magnetic field in free space where there is neither charge nor current density.
ˆ sin(t  kx)  ˆjaky cos(t  kx)
B  ia
Where a, k and ω are constants. The time-dependent part of electric field, derived by
using Maxwell’s equation is
aky
A) E  kˆ
cos(t  kx)
0 0
akx
B ) E  kˆ
sin(t  kx)
0 0
ak 2 y
C ) E  kˆ
cos(t  kx)
0 0
aky
D) E  kˆ
[sin(t  kx)  cos(t  kx)]
0 0
ak 2 x
E ) E  kˆ
sin(t  kx)
0 0
16A
Q.16. A gas of free electrons is confined to 2 dimensions in a total area A. The density of states
D(E) for the system would be proportional to
A:
A
B:
AE
C:
A.1/E
D:
1/(A E)
E:
A E1/2
17A
Q.17. A system consists of n spin one (S=1) independent particles. In the absence of any applied
magnetic field the entropy of the system is
A: nkB
B: 3nkB
C: 3kB ln n
D: 0
E: nkB ln 3
18A
Q.18. A quantum harmonic oscillator has the vibration frequency ωo. Ignoring zero point
energy, at a temperature such that kT= 2ħωo , the partition function has the value
A:
B:
C:
1
1−𝑒𝑥𝑝−2
1
1−𝑒𝑥𝑝+2
1
1−𝑒𝑥𝑝−0.5
D:
exp +0.5
E:
1-exp-0.5
19A
Q.19. A gas consists of N diatomic molecules each of which can translate as well as vibrate in
3-d space. The specific heat of the gas in equilibrium at a temperature T would be
A:
3𝑁𝑘𝐵
2
B: 3𝑁𝑘𝐵 𝑇
C:
D:
E:
9𝑁𝑘𝐵 𝑇
2
9𝑁𝑘𝐵
2
9𝑁𝑘𝐵
20A
5: An ideal gas at temperature T is confined to one side of a chamber separated from a second
side by a movable piston. The gas is heated, expands and the piston moves such that the
temperature of the gas remains the same. At the end, the following statement about the
condition of the gas is true
A:
Its internal energy and entropy are both unchanged.
B:
Its internal energy is unchanged but entropy has increased.
C:
Its internal energy has decreased but the entropy is unchanged.
D:
If heat has been added the temperature cannot remain constant.
E:
Both internal energy and entropy have increased.
21A
Set A: Q. 1. C; Q.2. B; Q.3. A ; Q.4. D ; Q. 5. E;
Q.6 C ; Q.7 D ; Q.8 A; Q.9 C ; Q.10. E
Q.11.(B) ; Q.12.(E) ; Q.13(C) ; Q.14.( B) ; Q.15.(C).;
Q.16. A; Q.17. E; Q.18. C; Q.19. D; Q.20. B