JEST 2018 Question Paper Physics

FOR JEST EXAM PREPARATION

JEST 2018
Question Paper ·
Physics
EXAM YEAR TYPE SUBJECT

JEST 2018 Question Paper Physics

Notes · Sample Papers · Previous Year Papers · Mock Tests

co m .
. e
em las

as fiziks ag

JEST 2018
PART A: ONE MARK QUESTIONS

m
Q1. When a collection of two-level systems is in equilibrium at temperature T0 , the ratio of

m . c o
o
T , the ratio is 8 :1..cThen m
the population in the lower and upper levels is 2 :1 . When the temperature is changed to

m (b) T  2T s e
(a) T  2T se
g l a
a m starting front rest, fails a vertical distance h before striking a vertical
lmass a
(c) T  3T (d) T  4T

g
0 0 0 0

Q2.
a
A ball of
spring, which it compresses by a length  . What is the spring constant of the spring?
(Hint: Measure all the vertical distances from the point where the ball first touches the
uncompressed spring, i.e., set this point as the origin of the vertical axis.)
2mg 2mg
(a) h    (b) h  
 2
3

o mh
c
.
2mg 2mg
h  
m
(c) (d)

e each of magnitude  , is subjected to a
 2 2

a s
l Each magnetic moment has a doubly degenerate
g
Q3. A collection of N interacting magnetic moments,

a
magnetic field H along the z direction.
level of energy zero and two non-degenerate levels of energies   H and  H
respectively. The collection is in thermal equilibrium at temperatureT . The total energy
E T , H  of the collection is

 H 
 HN sinh 
m


c. o
 k BT   HN
m
(a)  (b) 

c. o
 H    H 
1  cosh   2 1  cosh   
e m
em
 kbT  
la
 kbT  
s
l as ag
ag
 H   H 
 HN cosh   sinh  
(c)  k T  B
(d)   HN  k BT 
 H   H 
1  cosh   cosh  
 kbT   kbT 

m . c
c. o
s e m
m a
se l

g
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Q4. For which of the following conditions does the integral  1 Pm  x  Pn  x  dx vanish for
0

m  n , where Pm  x  and Pn  x  are the Legendre polynomials of order m and n

respectively?
(a) all m, m  n (b) m  n is an odd integer
(c) m  n is a nonzero even integer (d) n  m  1
Q5. If  q, p  is a canonically conjugate pair, which of the following is not a canonically

conjugate pair?
 pq 1   qp 1 
(a)  q 2 ,  (b)  p 2 ,  
 2   2 


(c) pq 1 ,  q 2 
 q 
(d)  f  p    where f   p  is the derivative of f  p  with respect to p .
 f   p  

Q6. A Germanium diode is operated at a temperature of 27 degree C . The diode terminal
voltage is 0.3 V when the forward current is 10 mA . What is the forward current (in mA)
if the terminal voltage is 0.4 V ?
(a) 477.3 (b) 577.3 (c) 47.73 (d) 57.73
Q7. If   x  is an infinitely differentiable function, then D̂  x  , where the operator

 d 
Dˆ  exp  ax  , is
 dx 

(a)   x  a  (b)   aea  x  (c)   ea x  (d) e a  x 

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Q8. Consider a particle of mass m moving under the effect of an attractive central potential
given as V  kr 3 where k 0 . For a given angular momentum

m
L, r0  3km / L2 corresponds to the radius of the possible circular orbit and the

mE  L . The particle is released from r  r with an inward
corresponding energyois . co
.c e m
2

e m  6mr 
a s
0 0

l
2

s g
0

g la 
a
reachar
velocity, energy E E and angular0momentum L . How long will be particle take to

0

(a) zero (b) 2mr02 L1 (c) 2 mr02 L1 (d) Infinite

Q9. What. is the difference between the maximum and the minimum eigenvalues of a system
   
of two electrons whose Hamiltonian is H  JS1.S 2 , where S1 and S 2 are the

m
co
corresponding spin angular momentum operators of the two electrons?

.
em
J J 3J
(a) (b) (c) (d) J

a s
4 2 4

g l
a
Q10. Two dielectric spheres of radius R are separated by a distance a such that a  R . One
of the spheres (sphere 1 ) has a charge q and the other is neutral. If the linear dimensions
of the systems are scaled up by a factor two, by what factor should be change on the
sphere 1 be changed so that the force between the two spheres remain unchanged?
(a) 2 (b) 4 2 (c) 4 (d) 2 2
Q11. An electric charge distribution produces an electric field

m
c. o
 

m  
 r r
E  1 e

c. o where  and  are constants. The net charge within a sphere of radiussem centered at
r3

em the origin is l a
1

las a g
ag (a) 4  1  e  0
1
(b) 4  1  e  0
1

1 1
(c) 4 0 (d) 4 0
e e

m . c
c. o
s e m
m a
se l

g
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Q12. The Laplace transform of
 sin  at   at cos  at   is
 2a 
3

2as s2  a2
(a) (b)
s  a  s  a 
2
2 2 2 2 2

1 1
(c) (d)
s  a s  a 
2 2
2 2

Q13. Two of the eigenvalues of the matrix
 a 3 0
 
A   3 2 0
 0 0 1
 
are 1 and 1 . What is the third eigenvalue?
(a) 2 (b) 5 (c) 2 (d) 5
Q14. In a thermodynamic process the volume of one mole of an ideal is varied as where
V  aT 1 a is a constant. The adiabatic exponent of the gas is  . What is the amount of
heat received by the gas if the temperature of the gas increases by T in the process?
RT RT 2
(a) RT (b) (c) (d) RT
1  2  1

  exp   x    sin  x   dx , where  ... is Dirac distribution, is

Q15.


e 1 e 1 e
(a) 1 (b) (c) (d)
e 1 e 1 e 1
Q16. The integral
 cos x
 x  1 dx is
 2


(a) (b)  e2 (c)  (d) zero
e

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Q17. A one dimensional harmonic oscillator (mass m and frequency  ) is in a state  such

that the only possible outcomes of an energy measurement are E0 , E1 or E2 , where En is

m
co
the energy of the n -th excited state. If H is the Hamiltonian of the oscillator,
3
.c om H   11  , then the probability that the energy e m .
2 2

s
 H 
m
2
and

s e l a
2 4

l a a g
1 g
measurement yields E0 is

(a) a
1 1
(b) (c) (d) 0
2 4 8
Q18. The charge density as a function of the radial distance r is given by
R2  r 2
  r   0 for r  R and zero otherwise. The electric flux over the surface of an
R2

m
ellipsoid with axes 3R, 4 R and 5R centered at the origin is
4 8
c.15o8  R
em
(a) 0 R 3 (b) 0 R 3 (c) 3
(d) zero
3 0 9 0
s
0

a
0

g l
a
Q19. A quantum particle of mass m is moving on a horizontal circular path of radius a . The
particle is prepared in a quantum state described by the wavefunction

4
 cos 2  ,
3
 being the azimuthal angle. If a measurement of the z -component of orbital angular
momentum of die particle is carried out, the possible outcomes and the corresponding

m
c. o
probabilities are

m
c. o e m
1 1 1
(a) Lz  0,  , 2 with 0 P  0   , P      and P  2  

em (b) L  0 with P  0  1
5 5 5
la s
las ag
g
z

a 1
(c) L  0,   with P  0   and P     
z
3
1
3
2 1
(d) Lz  0, 2 with P  0   and P  2  
3 6

m . c
c. o
s e m
m a
se l

g
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Q20. Consider two canonically conjugate operators X̂ and Yˆ such that  Xˆ , Yˆ   iI , where I
is identity operator. If Xˆ  11Qˆ1  12Qˆ 2 , Yˆ   21Qˆ1   22Qˆ 2 , where  ij are complex
numbers and Qˆ1 , Qˆ 2   zI , the value of 11 22  12 21 is
i
(a) iz (b) (c) i (d) z
z
Q21. Suppose the spin degree of freedom of two particles (nonzero rest mass and nonzero spin)
is described completely by a Hilbert space of dimension twenty one. Which of the
following could be the spin of one of the particles?
3 1
(a) 2 (b) (c) 1 (d)
2 2
Q22. For a classical system of non-interacting particles in the presence of a spherically
symmetric potential V  r    r 3 , what is the mean energy per particle?  is a constant.

3 5 3 3
(a) k BT (b) k BT (c)  k BT (d)  k BT
2 2 2 2
Q23. A particle of mass 1 kg is undergoing small oscillation about the equilibrium point in the
1 1
potential V  x   12
 6 for x  0 meters. The time period (in seconds) of the
2x x
oscillation is -
 
(a) (b) (c) 1.0 (d) 
2 3
Q24. A block of mass M is moving on a frictionless inclined surface of a
wedge of mass m under the influence of gravity. The wedge is lying on
a rigid frictionless horizontal surface. The configuration can be 
  r1 
described using the radius vectors r1 and r2 shown in the figure. How r2

many constraints are present and what are the types?
(a) One constraint; holonomic and scleronomous
(b) Two constraints; Both are holonomic; one is scleronomous and rheonomous
(c) Two constraints; Both are scleronomous; one is holonomic and other is non-
holonomic.
(d) Two constraints; Both are holonomic and scleronomous

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Q25. An electromagnetic wave of wavelength  is incident normally on a dielectric slab of
thickness t . If K is the dielectric constant of the slab. the change in phase of the

m
emergent wave compared with the case of propagation in the absence of the dielectric
slab is
om . co
.c (b) 2 e m
s
2 t 2 t
(a) K 1
e m (c) (d)  K  1
l a
s g
 

l a a
ag

m
. co
s em
g l a
a

m
m c. o
c. o s e m
s em la
gla ag
a

m . c
c. o
s e m
m a
se l

g
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PART B (THREE MARKS QUESTIONS)
Q1. An electronic circuit with 10000 components performs its intended function success fully
with a probability 0.99 if there are no faulty components in the circuit. The probability
that there are faulty components is 0.05 . if there are faulty components, the circuit
perform successfully with a probability 0.3 . The probability that the circuit performs
x
successfully is . What is x ?
10000
Q2. If an abelian group is constructed with two distinct elements a and b such that
a 2  b 2  I , where I is the group identity. What is the order order of the smallest abelian
group containing a, b and I ?
Q3. In the circuit shown below, the capacitor is initially unchanged. Immediately after the key
K is closed, the reading in the ammeter is 27 mA .
2R
R
C A
K
4R 3R

What will the reading (in mA ) be a long time later?
Q4. The normalized eigenfunctions and eigenvalues of the Hamiltonian of a Particle confined
to move between 0  x  a in one dimension are
2 n x n 2 2  2
 n  x   sin and En 
a a 2ma 2
respectively. Here 1, 2,3... . Suppose the state of the particle is

x    x 
  x   A sin   1  cos  
 a   a 
where A is the normalization constant. If the energy of the particle is measured, the
 22 x
probability to get the result as 2
is . What is the value of x ?
2ma 100

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Q5. Consider the transistor circuit shown in the figure. Assume VBEQ  0.7 V , VBB  6V
iC

m
om . co
RC

.c e m
s
C

m
V

s e  CC
g l a
a a

l
RB

ag
B
VCE
iB 

V E
 BB iE

and the leakage current is negligible. What is the required value of RB in kilo-ohms if

the base current is to be 4  A ?

m
Q6. A harmonic oscillator has the following Hamiltonian
pˆ 2 1
. co
em
H0   m 2 xˆ 2

s
2m 2

g l a
It is perturbed with a potential V   xˆ 4 Some of the matrix elements of x̂ 2 in terms of its
a
expectation value in the ground state are given as follows:

0 xˆ 2 0  C

0 xˆ 2 2  2C

1 xˆ 2 1  3C

m
m
1 xˆ 2 3  6C
c. o
c. o where n is the normalized eigenstate of H corresponding s e m to thee
s em g l a 0

gla  1 
a energy correction of
a
eigenvalue E    n   . Suppose E and E denote the
n 0 1
 2

O    to thee ground state and the first excited state, respectively. What is the fraction

E1
?
E0

m . c
c. o
s e m
m a
se l

g
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Q7. A person on Earth observes two rockets A and B directly approaching each other with
speeds 0.8 c and 0.6 c respectively. At a time when the distance between the rockets is

observed to be 4.2 108 m , the clocks of the rockets and the Earth are synchronized to
t  0 s . The time of collision (in seconds) of the two rockets as measured in rocket A ' s
x
frame is . What is x ?
10
Q8. Two parallel rails of a railroad track are insulated from each other and from the ground.
The distance between the rails is 1 meter. A voltmeter is electrically connected between
the rails. Assume the vertical component of the earth’s magnetic field to the 0.2 gauss.
What is the voltage developed between the rails when a train travels at a speed of
180 km / h along the track? Give the answer in milli-volts.
Q9. Consider a simple pendulum in three dimensional space. It consists of a string length
l  20 cm and bob mass m  15 kg attached to it as shown in the figure below The
acceleration due to gravity is downwards as shown in the figure with a magnitude
g  10 ms 2 . x

y
l  20cm
z

m  15kg

g  10ms 2
The pendulum is pulled in the x  z plane to a position where the string makes an angle

 with the z -axis. It is then released an angular velocity  radians per second
3
about the z-axis. What should be the value of  in radians per second so that the angle
lie siring makes with the z -axis does not change with time?
Q10. Two conductors are embedded in a material of conductivity 104 ohm  m and dielectric

constant  80 0 The resistance between the two conductors is 106 ohm. What is the

capacitance(in pF ) between the two conductors? Ignore the decimal part of the answer.

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PART C (THREE MARKS QUESTIONS)
Q1. An ideal fluid is subjected to a thermodynamic process described by   CV  and

m
co
P  n  where  is energy density and P is pressure. For what values of n and  the

.c om
process is adiabatic if the volume is changed slowly?
e m .
m
e
(a)     1, n  1 (b)   1   , n  
l a s
l a s a g
g
(c)   1, n    1 (d)    , n  1  

Q2. If y  a
x  satisfies

dy
 y 1   log y  
2

dx  

 
and y  0   1 for x  0 then y   is
2

o m 

.c mass and heat capacity divides a
(a) 0 (b) 1 (c) (d) infinity

m
2

se
vertical, insulated cylinder of height l2aH and cross sectional area A into two halves.
Q3. A frictionless heat conducting piston of negligible

g
Each half contains one mole ofa an ideal gas at temperature T and pressure P 0 0

corresponding to STP. The heat capacity ratio   C p / Cv is given. A load of weight W

is tied to the piston and suddenly released. After the system comes to equilibrium, the
piston is at rest and the temperatures of the gases in the two compartments are equal.

m
What is the final displacement y of the piston from its initial position, assuming

m c. o
c. o (a) 2H
yW  T0Cv ?

s em
em la 
H 2H

s
(b) H 

ag
(c) (d)

la
 

ag

m . c
c. o
s e m
m a
se l

g
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Q4. An apparatus is made from two concentric conducting cylinders of radii a and b
respectively, where a  b . The inner cylinder is grounded and the outer cylinder is at a
positive potential V . The space between the cylinders has a uniform magnetic field
H directed along the axis of the cylinders. Electrons leave the inner cylinder with zero
speed and travel towards the outer cylinder. What is the threshold value of V below
which the electrons cannot reach the outer cylinder?

   
2
eH 2 b 2  a 2 eH 2 b 2  a 2
(a) (b)
8 mc 2 8 mc 2b 2

  b  a 
2
eH 2 b 2  a 2 eH 2b 2 2

(c) (d)
8 mc 2 a 2 8 mc 2
Q5. A theoretical model for a real (non-ideal) gas gives the following expressions for the
internal energy U  and the pressure  P  ,

2 5/ 3 2 1/ 3 2
U T , V   aV 2 / 3  bV 2 / 3T 2 and P T ,V  
aV  bV T
3 3
where a and b are constants. Let V0 and T0 be the initial volume and initial temperature
respectively. If the gas expands adiabatically, the volume of the gas is proportional to
(a) T (b) T 3/ 2 (c) T 3/ 2 (d) T 2
Q6. Consider two coupled harmonic oscillators of mass m in each. The Hamiltonian
describing the oscillators is
pˆ 2 pˆ 2 1

Hˆ  1  2  m 2 xˆ12  xˆ22   xˆ1  xˆ2 
2m 2m 2
2

The eigenvalues of Ĥ are given by (with n1 and n2 being non-negative integers)

(a) En1 ,n2    n1  n2  1

 1 1  1
(b) En1 ,n2    n1      n2  
 2 3  2

 1  1
(c) En1 ,n2    n1    3   n2  
 2  2
1
(d) En1 ,n2    n1  n2  1
3

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Q7. A ball comes in from the left with speed 1 (in arbitrary units) and causes a series of
collisions. The other four balls shown in the figure are initially at rest. The initial motion

m
is shown below (the number in the circle indicate the object’s relative mass). This initial

om
velocities of the balls shown in the figure are represented as 1, 0, 0, 0, 0 .
. co
.c1 e m
e m l a s
s g
2

a
1 1 1

g a
l sign means that the velocity is directed to the left. All collisions are elastic.
a
A negative
Which of the following indicates the velocities of the balls after all the collisions are
completed?
 1 1 1  1 2
(a)   ,  , 0, 0,  (b)   , 0, 0, 0, 
 2 2 2  3 3

 1 3  1 1
m
(c)   , 0, 0, 0,  (d)   , 0, 0, 0, 

co
 2 4  2 2
.
qas
em
Q8. Consider the Lagrangian

l
g2
2

a
L  1  1  q 2 

of a particle executing oscillations whose amplitude is A . If p denotes the momentum of

the particle, then 4 p 2 is

 
(a) A2  q 2 4  A2  q 2   
(b) A2  q 2 4  A2  q 2 
 
(c) A2  q 2 4  A2  q 2   
(d) A2  q 2 4  A2  q 2 
o m
m c
.between the
A block of mass M rests on a plane inclined at an angle  with respect to the horizontal.

c. o A horizontal force F  Mg is applied to the block If  is the static friction
Q9.

e m
em l s
a is
las g
(b) 1    cot   1 a
block and the plane, the range of  so that the block remains stationary

ag (a)    tan     mu
1  1  1  1 
(c)  tan   (d)  cot  
1  mu 1  mu 1  mu 1  mu

m . c
c. o
s e m
m a
se l

g
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13

fiziks

Q10. The coordinate q and the momentum p of a particle satisfy
dq dp
 p,  3q  4 p
dt dt
If A  t  is the area of any region of points moving in the  q, p  -space, then the ratio

A t 
is
A 0

(a) 1 (b) exp  3t  (c) exp  4t  (d) exp  3t / 4 

Q11. The elastic wave on a stretched rectangular membrane of size L  2 L in the x  y plane is
described by the function
x y
A sin sin cos t   
L L
where A and  are constants. The speed of the elastic waves is v . The angular
frequency  is

5v 2 v 5 v 17 v
(a) (b) (c) (d)
L L 2L 2L
Q12. A large cylinder of radius R filled with particles of mass m . The cylinder spins about its
axis at an angular speed  radians per second, providing an acceleration g for the
particles at the rim. If tile temperature T is constant inside the cylinder, what is the ratio
of air pressure P0 at the axis to the pressure Pc , at the rim?

 mgR   mgR  mgR 2kbT
(a) exp   (b) exp    (c) (d)
 2kbT   2kbT  2kbT mgR

Q13. In an experiment, certain quantity of an ideal gas at temperature T0 pressure P0 and

volume V0 is heated by a current flowing through a Wire for a duration of t seconds. The

volume is kept constant and the pressure changes to P1 . If the experiment is performed at

constant pressure starting with the same initial conditions, the volume changes from V0

to V1 . The ratio of the specific heats at constant pressure and constant volume is

P1  P0 V0 P1  P0 V1 PV
1 1 PV
0 0
(a) (b) (c) (d)
V1  V0 P0 V1  V0 P1 PV
0 0 PV
1 1

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14

co m .
. e
em las

as fiziks ag

Q14. Consider a wavepacket defined by

  x    dkf  k  exp i  kx  


m
om . co
K K
Further, f  k   0 for k  and f  k   a for k  . Then, the form of normalized

.c e m
2 2
  x  is
e m l a s
a
l Kxs a g
8gK
(a) a
Kx
sin
2 2
sin (b)
x 2 K x
Kx
sin
8 K Kx 2 2
(c) cos (d)
x 2 K x
Q15. If F  x, y   x 2  y 2  xy , its Legendre transformed function G  u , v  , upto a

o m
c
(c) .u  v
multiplicative constant, is

e m (d)  u  v 
2
(a) u  v  uv (b) u  v  uv
s
2 2 2 2 2 2

g la
a

m
m c. o
c. o s e m
s em la
gla ag
a

m . c
c. o
s e m
m a
se l

g
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15