Iit jam 2016 physics solutions BY Trajectoryeducation

1
TRAJECTORYEDUCATION
IIT JAM-2016 PHYSICS-SOLUTIONS
1. For an infinitely long wire with uniform line charge density, , along the z-axis, the electric field at a
point (a, b, 0) away from the origin is
( ˆ ˆ, ye e and ˆze are unit vectors in Cartesian-Coordinate system.)
(a) 2 2
0
ˆ ˆ( )
2
x ye e
a b


 
(b)  2 2
0
ˆ ˆ( )
2
x yae be
a b


 
(c)  2 2
0
ˆ
2
xe
a b

 
(d)  2 2
0
ˆ
2
ze
a b

 
Ans. (b)
y
( , ,0)a b
re
x
E= ˆ
2
re
r


r= 2 2
a b
ˆre = 2 2
ˆ ˆx yae ber
r a b




E

= ˆ ˆ( )
2
x yae be




2. A 1W point source at origin emits light uniformly in all the directions. If the units for both the axes are
measured in centimetre, then the Poynting vector at the point (1, 1, 0) in W/cm2
is
(a)
1
ˆ ˆ( )
8 2
x ye e

(b)
1
ˆ ˆ( )
16
x ye e

(c)
1
ˆ ˆ( )
16 2
x ye e

(d)
1
ˆ ˆ( )
4 2
x ye e

Ans. (a)
S

= 2
ˆ
4
r
P
e
r
r= 2
ˆre =
ˆ ˆ
2
x ye e

2 | JAM-2016 PHYSICS TRAJECTORY EDUCATION
TRAJECTORYEDUCATION
S

=
ˆ ˆ( )1
4 2 2
x ye e

=
1
ˆ ˆ( )
8 2
x ye e

3. A charged particle in a uniform magnetic field 0
ˆzB B e

starts moving from the origin with velocity
ˆ ˆ(3 2 )m /s.x zv e e 

The trajectory of the particle and the time t at which it reaches 2 meters above the
xy-plane are
( ˆ ˆ, ,x ye e and ˆze are unit vectors in Cartesian-Coordinate system.)
(a) Helical path; t = 1s (b) Helical path; t = 2/3s
(c) Circular path; t = 1s (d) Circular path; t = 2/3s
Ans. (a)
Its velocity component in z-direction is 2 m/s. Its path will be helical and will reach 2m above xy plane
is 1 sec.
4. Consider the particle mass m following a trajectory given 0 1cosx x t  and 0 2siny y t  where
0 0 1, ,x y  and 2 are constant of appropriate dimensions. The force on the particle is
(a) central only if 1 2   (b) central only if x0 = y0 and 1 2  
(c) always central (d) central only if x0 = y0 and 1 2  
Ans. (a)
r
 = 1 2
ˆ ˆcos sinx ti y tj   
a
 =
2
2 2
1 1 2 22
ˆ ˆcos 4 sin
d r
x ti y tj
dt
      

If 1 2   then a

= 2
1 r

F

= 2
1ma m r  
 
which will be central
5. Which one of the following points represent the complex number =
1
?
1 i
(a)
–1 –0.5 0.5 1
0.5
1
–0.5
–1
y
x
(b)
–1 –0.5 0.5 1
0.5
1
–0.5
–1
y
x
(c)
–1 –0.5 0.5 1
0.5
1
–0.5
–1
y
x
(d)
–1 –0.5 0.5 1
0.5
1
–0.5
–1
y
x

TRAJECTORY EDUCATION JAM-2016 PHYSICS | 3
TRAJECTORYEDUCATION
Ans. (a)
z=
1 (1 ) 1 1 1
1 (1 )(1 ) 2 2 2
i i
i
i i i
 
   
  
–1 –0.5 0.5 1
0.5
1
–0.5
–1
y
x
6. The eigenvalues of the matrix representing the following pair of linear equations x + iy = 0, ix + y = 0
are
(a) 1 + i, 1 + i (b) 1 – i, 1 – i
(c) 1, i (d) 1 + i, 1 – i
Ans. (d)
x + iy=0
ix + y=0
1
1
i x
i y
   
   
   
=
0
0
 
 
 
let 1 2 are eigenvalues
1 2   =2
1 2  =2
1 = 21 , 1i i   
7. The solution of the Boolean equation Y A B AB   is
(a) 1 (b) AB
(c) AB (d) A B
Ans. (d)
Y= A B AB 
= A B A B  
= ( 1)A B B 
= A B
8. A spherical closed container with smooth inner wall contains a monoatomic ideal gas. If the collisions
between the wall and the atoms are elastic, then the Maxwell speed distribution function vdn
dv
 
 
 
for
the atoms is best represented by :
(a)
vdn
dv
v0
(b)
vdn
dv
v0

TRAJECTORYEDUCATION
(c)
vdn
dv
v0
(d)
vdn
dv
v0
Ans. (c)
9. Two sinusoidal signals of frequencies x and y having same amplitude are applied to x- and y-
channels of a cathode ray oscilloscope (CRO) respectively. The following stationary figure will be
observed when
y
1
–1
x
–1 1
(a) y x   (b) Phase difference is 0
(c) 2y x   (d) phase difference is / 2.
Ans. (b)
y
x


=
No. of timesfig. cuts axis
No. of timesfig. cuts axis
x
y
n x
n y

=
2 1
4 2

x = 2 y
Phase difference = 0 or .
10. The phase difference ( ) between input and output voltage for the following circuits (i) and (ii)
vi
C
C vi Cvo vo
R
~ ~
(i) (ii)
will be
(a) 0 and 0 (b) / 2 and 0 / 2    respectively
(c) / 2 and / 2 (d) 0 and 0 / 2    respectively
Ans. (d)

TRAJECTORYEDUCATION
(i) Vi = i t
V e 

Z=
1 1 2
j C j C j C
 
  
V° =
1
2
j tV
i e
j C



phase diff.=0
(ii) Z= 2
2 2
1 1 i
R R e
j C C
 
  
 
1 1
tan
CR
 
  
 
i=
( )
2
2 2
1
j t
iV V e
Z
R
C
 




V° =
2
2
2 2
1
1
j t
V e
i
j C
R
C
 
  
 

 



0
2

  
11. For the given set of equations : x + y = 1, x + z = 1, x + z = 1, which one of the following statements is
correct?
(a) Equations are inconsistent
(b) Equations are consistent and a single non-trivial solution exist.
(c) Equations are consistent and many solution exist.
(d) Equations are consistent and only a trivial solution exist
Ans. (b)
x + y=1
y + z=1
x + z=1
1 1 0
0 1 1
1 0 1
x
y
z
   
   
   
      
=
1
1
1
 
 
 
  
1 1 0 1 1 0 1 1 0
0 1 1 ~ 0 1 1 ~ 0 1 1
1 0 1 0 1 1 0 0 2
     
     
     
          
rank = 3. So, it will have unique solution.
12. The tangent line to the curve x2
+ xy + 5 = 0 at (1, 1) is represented by
(a) y = 3x – 2 (b) y = –3x + 4
(c) x = 3y – 2 (d) x = –3y + 4
Ans. (b)
x2
+ xy + 5=0
Taking derivative
2
dy
x x y
dx
  =0

TRAJECTORYEDUCATION
At (1, 1) 3
dy
dx
 
y=–3x + C
It passes through (1, 1)
y=–3x + 4
13. In the following RC circuit, the capacitor was charged in two different ways.
(i) The capacitor was first charged to 5V by moving the toggle switch to position P and then it was
charged to 10V by moving the toggle switch to position Q
(ii) The capacitor was directly charged to 10 V, by keeping the toggle switch at position Q
Assuming the capacitor to be ideal, which one of the following statement is correct?
R
P
Q
10V
5V
C
(a) The energy dissipation in cases (i) and (ii) will be equal and non-zero
(b) The energy dissipation for case (i) will be more than that for case (ii)
(c) The energy dissipation for case (i) will be less than that for case (ii)
(d) The energy will not be dissipated in either case.
Ans. (c)
Similar problem is Samvedna Book.
14. If a constant voltage +V is applied to the input of the following OPAMP circuit for a time t, then the
output voltage V° will approach
C
V0
R
+V
–
+
(a) +V exponentially (b) –V exponentially
(c) +V linearly (d) –V linearly
Ans. (d)
It is integrator circuit
15. In the following RC network, for an input signal frequency
1
,
2
f
RC


the voltage gain
o
i
v
v and the
phase angle  between vo and vi respectively are

TRAJECTORYEDUCATION
vi
C
vo
R
~
C
R
(a)
1
2
and 0 (d)
1
3
and 0
(c)
1
2
and
2

(d)
1
3
and
2

Ans. (b)
Let Vi = i t
V e 

z=
1
1
1
R
j C
R
j C R
j C


 
 

=
1
j R
R
C j CR
 
  
f=
1 1 1
2
2
f C
RC RC R
       

Z=
1
R
R jR
j
 

=
(1 )
2
j R
R jR

 
= 4 /43 3 3 3 3
(1 ) 2
2 2 2 2 2
j iR R
R j R j e e    
     
 
i= 42
3
i t
iV V
e
z R
 
  
  

V° =
1
R
j R
 
   
i= ( /4)2
1 3
j tR V
e
j R
  
  
=
( /4)
/4
2
33 2
j t j t
j
V V
e e
e
   


i
V
V

=
1
, 0
3
 

TRAJECTORYEDUCATION
16. Light travelling between two points takes a path for which
(a) time of flight is always minimum (b) distance is always minimum
(c) time of flight is extremum (d) distance is extremum
Ans. (a)
17. Consider a free electron (e) and a photon (ph) both having 10 eV of energy. If  and p represent
wavelength and momentum respectively, then
(mass of electron = 9.1 × 10–31
kg; speed of light = 3 × 108
m/s)
(a) e ph   and Pe = Pph (b) e ph   and Pe Pph
(c) e ph   and Pe Pph (d) e ph   and Pe Pph
Ans. (b)
e =
2
h h
p mE

ph =
hc
E
e
ph


=
–19
–91
1 1 10 1.6 10
.
2 2 9.1 102
h E E
hc c m cmE
 
 
 
8 –6
1 1
1
3 10 10
  

So, e ph  
pe = 2mE
pph =
E
c
ph
e
p
p
=
1
1
22
E E
c mc mE
 
pph pc
18. For an ideal gas, which one of the following T-S diagram is valid?
(a)
T
Isochor
Isobar
S
(b)
T
Isobar
Isochor
S
(c)
T Isochor
Isobar
S
(d)
T
Isochor
Isobar
S

TRAJECTORYEDUCATION
Ans. (a)
For isobaric process
dS=
pC dTdQ
T T

S=CplogT + C1
T=
1
p
S C
C
e

For Isochoric process
dS=
VC dTdQ
T T

S=CVlogT + C2
T=
2S C
C
e

19. If U, F, H, and G represent internal energy, Helmholtz free energy, enthalpy, and Gibbs free energy
respectively, then which one of the following is a correct thermodynamic relation?
(a) dU = PdV – TdS (b) dH = VdP + TdS
(c) dF = –PdV + SdT (d) dG = VdP + SdT
Ans. (b)
dU=TdS – PdV
H=U + PV
dH=dU + PdV + VdP
=TdS – pdV + PdV + VdP
=TdS + VdP
20. A train passes through a station with a constant speed. A stationary observer at the station at platform
measure the tone of the train whistle as 484 Hz when it approaches the station and 442 Hz when it
leaves the station. If the sound velocity in air is 330 m/s, then the tone of the whistle and the speed of
the train are
(a) 462 Hz, 54 km/h (b) 463 Hz, 52 km/h
(c) 463 Hz, 56 km/h (d) 464 Hz, 52 km/h
Ans. (a)
Let speed of trains is V°
when train approaches the station
f1 = 484
V
f
V V


 
   
where V = 330 m/s
when train leaves the station f2 = 442
V
f
V V


 
   
So,
484
f
= 1
V V V
V V
 
  ...(1)

442
f
= 1
V V V
V V
 
  ...(2)
(1) + (2) 
1 1
484 442
f
 
  
=2
f° =462 Hz

TRAJECTORYEDUCATION
Putting f° in (2)
V=14.93 m/s
=
14.93
3600 53.75 /
1000
km s 
21. The minimum length of a plane mirror to the see the entire full length image of an object is half of the
object’s height. Suppose  is the distance between eye and top of the head of the person of height h.
The person will be able to see his entire full length image with a mirror of height h/2 fixed on the wall
(a) when the bottom edge of mirror is kept h/2 above the floor
(b) when the bottom edge of mirror is kept ( ) / 2h   above the floor
(c) when the bottom edge of mirror is kept ( ) / 2h   above the floor
(d) when the centre of the mirror is at the same height as centre of the person
Ans. (c)
h-

h-
2
22. A particle is moving in a plane with a constant radial velocity of 12 m/s and constant angular velocity
of 2 rad/s. When the particle is at a distance r = 8 m from the origin, the magnitude of the intantaneous
velocity of the particle in m/s is
(a) 8 15 (b) 20
(c) 2 37 (d) 10
Ans. (b)
V

= ˆ ˆrre r e 
= ˆ ˆ12 16re e
| |V

= 2 2
12 16 20 /m s 
23. A cylindrical rod of length L has a mass density distribution given by 0( ) 1 ,
x
x
L
 
    
 
where x is
measured from one end of the rod and 0 is a constant of appropriate dimensions. The centre of mass
of the rod is
(a)
5
9
L (b)
4
9
L
(c)
1
9
L (d)
1
2
L
Ans. (a)
xcm =
xdm x dx
dm dx



 
 

TRAJECTORYEDUCATION
= 0
0
. 1
. 1
L
L
x
x dx
L
x
dx
L
 
  
 
 
  
 


=
2 3
0
3
0
52 3 .
9
2
L
L
x x
L L
x
x
L



24. A particle travels in a medium along a horizontal linear path. The initial velocity of the particle is v0
and the viscous force acting on it is proportional to its instantaneous velocity. In the absence of any
other forces, which one of the following figures correctly represents the velocity of the particle as a
function of time?
(a)
v t( )
t
(b)
v t( )
t
(c)
v t( )
t
(d)
v t( )
t
Ans. (d)
dv
m
df
=–kv
dv
v
=
k
df
m

v= /
. k m
V e
25. A lightly damped harmonic oscillator with natural frequency 0 is driven by a periodic force of
frequency . The amplitude of oscillation is maximum when
(a)  is slightly lower than 0 (b) 0  
(c)  is slightly higher than 0 (d) the force is in phase with the displacement
Ans. (a)
Amplitude maximum takes place at
2 2
2r   
26. A block of mass 0.38 kg is kept at rest on a frictionless surface and attached to a wall with a spring of
negligible mass. A bullet weighing 0.02 kg moving with a speed of 200 m/s hits the block at time
t = 0 and gets stuck to it. The displacement of the block (in metre) with respect to the equilibrium
(Spring constant = 640 N/m)

TRAJECTORYEDUCATION
(a) 2sin5t (b) cos10t
(c) 0.4cos25t (d) 0.25sin40t
Ans. (d)
Let instantaneous velocity imparted to the block is v
Applying conservation of linear momentum
0.02×200=(0.385 + 0.01)v
v=
4
10 /
0.4
m s
velocity, v= 10cos t
 =
640
40
0.4
k
m
 
x vdt  =
10
sin t

=0.25 sin40t
27. One mole of an ideal gas with average molecular speed v0 is kept in a container of fixed volume. If the
temperature of the gas is increased such that the average speed get doubled, then
(a) the mean free path of the gas molecule will increase
(b) the mean free path of the gas molecule will not change
(c) the mean free path of the gas molecule will decrease
(d) the collision frequency of the gas molecule with wall of the container remains unchanged
Ans. (b)
28. An arbitrarily shaped conductor encloses a charge q and is surrounded by a conducting hollow sphere
as shown in the figure. Four different regions of space, 1, 2, 3, and 4, are indicated in the figure.
Which one of the following statement is correct?
q
4
2
1
(a) The electric field lines in region 2 are not affected by the position of the charge q
(b) The surface charge density on the inner wall of the hollow sphere is uniform
(c) The surface charge density on the outer surface of the sphere is always uniform irrespective of
the position of charge q in region 1.
(d) The electric field in region 2 has a radial symmetry
Ans. (c)
29. Consider a small bar magnet undergoing simple harmonic motion (SHM) along the x-axis. A coil
whose plane is perpendicular to the x-axis is placed such that the magnet passes in and out of it during
its motion. Which one of the following statements is correct? Neglect damping effects.
(a) Induced e.m.f. is minimum when the centre of the bar magnet crosses the coil
(b) The frequency of the induced current in the coil is half of the frequency of the SHM
(c) Induced e.m.f. in the coil will not change with the velocity of the magnet
(d) The sign of the e.m.f. depends on the ple (N or S) face of the magnet which enters into the coil.
Ans. (a)

TRAJECTORYEDUCATION
30. An incompressible non-viscous fluid is injected into a conical pipe at it orifice as schematically shown
in the figure. The pressure at the orifice of area A0 is P0. Neglecting the effect of gravity and assuming
streamline, flow, which one of the following plots correctly predicts the pressure along axis of the
cone?
/4 x
^
(a)
x0
x
p0
p (b)
x0
x
p0
p
(c)
x0
x
p0
p (d)
x0
x
p0
p
Ans. (a)
Applying equation of continuity
x
V
AV  = 2
( )x V
V= 2
AV
x
 

Applying Bernoulli equation
21
2
P V   =
21
2
P V 
P=
2
2
2 4
1
1
2
A
P V
x

 
 
   
 
31. Consider a spherical dielectric material of radius ‘a’ centered at origin. If the polarization vector
0
ˆxP P e

where P0 is a constant of appropriate dimensions, then
( ˆ ˆ,x ye e and ˆze are unit vectors in Cartesian-coordinate system)

TRAJECTORYEDUCATION
(a) the bound volume charge density is zero
(b) the bound surface charge density is zero at (0, 0, a)
(c) the electric field is zero inside the dielectric
(d) the sign of the surface charge density charge over the surface
Ans. (a,b,d)
At (0,0,a),
P2n–P1n = b
P2n =0
x
y
z
So, b =0
b = 0
dP
P
dx

  

b =0
32. For an electric dipole with moment 0
ˆzP P e

placed at the origin, (p0 is a constant of appropriate
dimensions and ˆ ˆ,x ye e and ˆze are unit vectors in Cartesian coordinate system)
(a) potential fall as 2
1
,
r
where r is the distance from origin.
(b) a spherical surface centered at origin is an equipotential surface
(c) electric flux through a spherical surface enclosing the origin is zero
(d) radial component of E

is zero of the xy-plane
Ans. (a,c,d)
V= 2
cos
4
p
r


x
y
z
33. Three infinitely long conductors carrying current I1, I2 and I3 lies perpendicular to the plane of the
paper as shown below :

TRAJECTORYEDUCATION
C3
C2
C1
I1
I2
I3
If the value of the integral .
C
B dl

 for the loops, C1, C2 and C3 are 0 02 ,4  and 0 in the units of
N/A respectively, then
(a) I1 = 3A into the paper (b) I2 = 5A out of the paper
(c) I3 = 0 (d) I3 = 1A out of the paper
Ans. (a,b)
1C
B dl

 =2µ°
 I1 + I2 =2 ...(1)
B dl

 =4µ°
 I2 + I3 =4 ...(2)
B dl

 =µ°
 I1 + I2 + I3 =1 ...(3)
From (1), (2), (3)
I3 = –1 i.e. 1A into the paper
I1 = –3 i.e. 3A into the paper
I2 = 5A i.e. 5A out of the paper
34. In the optical arrangement as shown below, the axes of two polarizing sheets P and Q are oriented
such that no light is detected. Now when a third polarizing sheet (R) is placed in between P and Q,
then light is detected. Which of the following statement(s) is/are true?
P Q
detector
(a) Polarization axes of P and Q are perpendicular to each other
(b) Polarization axes of R is not parallel to P.
(c) Polarization axes of R is not parallel to Q
(d) Polarization axes of P and Q are parallel to each other
Ans. (a, b, c)
35. The P-V diagram below shows three possible paths for an ideal gas to reach the final state f from an
initial state i. Which among the following statement(s) is/are correct?
V
P
i
1
2
3
f
(a) The work done by the gas is maximum along path 3
(b) Minimum change in the internal energy occurs along path –2

TRAJECTORYEDUCATION
(c) Maximum heat transfer is for path-1
(d) Heat transfer is path independent
Ans. (a)
36. Potential energy U as a function of r is shown below. If a particle of mass m1 and energy E1, starting
from r a, moves towards the origin, then
0
U r( )
Umin
b a
c
r
m1
E1
(a) there is only one turning point for the particle
(b) velocity of the particle starts to increase x = a and reaches its maximum at r = c.
(c) velocity of the particle at r = a and r = b are equal
(d) the particle gets bounded if it elastically collides with a stationary particle of mass m2 at r = c,
imparting a momentum greater than 2 12m E
Ans. (a,b,c,d)
If m1 collides with particle of mass m2 at r = c
Let 1E is is energy of particle m1 after collision
conserving energy.
10 E s=
2
1
22
p
E
m
 
The particle of m1 will get bounded if
1E 0
2
1
22
p
E
m
 0
p 2 12m E
37. A particle moves in a circular path in the xy-plane centered at the origin. If the speed of the particle is
constant, then its angular momentum
(a) about the origin is constant both in magnitude and direction
(b) about (0, 0, 1) is constant in magnitude but not in direction
(c) about (0, 0, 1) is varies both in magnitude and direction
(d) about (0, 0, 1) is constant in direction but not in magnitude
Ans. (a,b)
x
y
z

TRAJECTORYEDUCATION
38. A pn junction was formed with a heavily doped (1018
cm–3
) p-region and lightly doped (1014
cm–3
) n-
region. Which of the following statement(s) is(are) correct?
(a) The width of the depletion layer will be more in the n-side of the junction
(b) The width of the depletion layer will be more in the p-side of the junction
(c) The width of the depletion layer will be same on both side of the junction
(d) If the pn junction is reverse biased, then the width of the depletion region increases.
Ans. (b)
39. A slit has width ‘d’ along the x-direction. If a beamof electrons, accelerated in y-direction to a particular
velocity by applying a potential difference of 100 0.1kV passes through the slit, then, which of the
following statement(s) is/are correct?
(a) The uncertainty in the position of electrons in x-direction before passing the slit is zero
(b) The momentum of electrons in x-direction is ~h/d immediately after passing the slit
(c) The uncertainty in the position of electron in y-direction before passing the slit is zero
(d) The presence of the slit does not affect the uncertainty in momentum of electrons in y-direction
Ans. (b,c,d)
40. A free particle of energy E collides with a one-dimensional square potential barrier of height V and
width W. Which one of the following statement (s) is/are correct?
(a) For E V, the transmission coefficient for the particle across the barrier will always be unity.
(b) For E V, the transmission coefficient changes more rapidly with W than with V.
(c) For E V, if V is doubled, the transmission coefficient will also be doubled.
(d) Sum of the reflection and the transmission coefficient is always one
Ans. (b,d)
2 2 ( )m E V L
T e  

41. Fourier series of a given function f(x) in the interval 0 to L is
f(x)=
0
1 1
2 2
cos sin
2
n n n n
a nx nx
a b
L L
 
 
    
    
   
 
If f(x) = x in the region 2(0, ), _________.b 
Solution :
bn =
0
2
sinf nxdx

 
b2 =
0
2
sin 2x xdx

 
=
0
2 1
cos2 sin 2
2 4
x
x x

 
    
=
2
1
2
 
   
  
Ans. –1
42. Consider a function f(x, y) = x3
+ y3
, where y represents a parabolic curve x2
+ 1. The total derivative
of f with respect to x, at x = 1 is __________.
Solution :
df
dx
=
f f dy
x y dx
 
 
 
= 2 2
3 3 2x y x 
=3x2
+ 6x(x2
+ 1)2
at x = 1,
df
dx
=3 + 24 = 27 Ans. 27

TRAJECTORYEDUCATION
43. Arectangular area (A1) is formed by two vectors x

and y

as shown in figure (i). Anew set of vectors,
representing the area (A2) as shown in figure (ii), are given as : 1 2; ,u x u kx y  
    
where k is a
dimensionless constant.
x
y 2u
1u
( )i ( )ii
The Jacobian of the frame 1 2( , )u u
 
with respect to ( , )x y
 
is ___________.
Solution :
 
 
1 2,
,
u u
x y


=
1 1
2 2
1 0
1
1
u u
x y
u u k
x y
 
 
 
 
 
Ans. 1
44. A particle radioisotope has a half-life of 5 days. In 15 days the probability of decay in percentage will
be ___________.
Solution :
N= t
N e

dN= t
N e dt

Probability of decay in dt time
dN
N
= t
e dt

Probability of decay in 15 days
15
0
t
e dt
 =
2 15
ln
15
5
0 0
t
t
e e


  
=1 – e–3 log 2
=
1 7
1
8 8
 
In percentage, probability of decay is
700
87.5%
8
 Ans. 87.5
45. In a photoelectric experiment both sodium (work function = 2.3 eV) and tungsten (work function =
4.5eV) metals were illuminated by an ultraviolet light of same wavelength. If the stopping potential
for tungsten is measured to be 1.8V, the value of the stopping potential for sodium will be ________V.
Solution :
Let stopping potential of Na is VS
hc

=2.3 eV + eVS = 4.5 eV + 1.8 eV
eVS =4 eV
VS =4 V Ans. 4

TRAJECTORYEDUCATION
46. The addition of two binary numbers 1000.01 and 0001.11 in binary representing is _________.
Solution :
1000.01
0001.11
1010.00
Ans. 1010
47. The number of second nearest neighbor ions to a Na+
ion in NaCl crystal is __________.
Solution :
12 Ans. 12
48. The output voltage V0 of the OPAMP circuit given below is _______ V.
V0
R
–
+
2R
R
R
R
1V
2V
3V
Solution :
49. A cylinder contains 16 g of O2. The work done when the gas is compressed to 75% of the original
volume at constant temperature of 27°C is ___________J.
Solution :
W= 2
1
log
V
nRT
V
= 0.5 8.31 300 log
4

  
=–358.6 J Ans. –358.6J
50. When sunlight is focused on a paper using a bi-convex lens, it starts to burn in the shortest time if the
lens is kept 0.5 m above it. If the radius of curvature of the lens is 0.75 m then, the refractive index of
the material is _________.
Solution :
f=  
1 1
1 1
0.5 1m
R R
 
    
 
R1 =R, R2 = –R
f=
 2 1
0.5
R
 

µ – 1=
0.5 0.75
2

µ=1.1875 Ans. 1.1875
51. Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure
below.
The value of the integral, .F dl

 evaluated along this contour, for a vector field, ˆ ˆ ,x yF ye xe 

is
__________.
( ˆ ˆ,x ye e and ˆze are unit vectors in Cartesian-coordinate system.)

TRAJECTORYEDUCATION
y
x
O P(2. 0)
Q
4

4

Solution :
Mdx Ndy =
N M
dx dy
x y
  
   

ydx xdy = 2dxdy
=
1
2 2 2sin
2 4

   
= 2 2 Ans. 2 2
52. A hemispherical shell is placed on the xy-plane centered at the origin. For a vector field
2 2
ˆ ˆ( ) / ,x yE ye xe x y   

the value of the integral ( )
S
E da 
  
 over the hemispherical surface is
___________ .
Solution :
x
y
z
C
On the bounding curve of hemisphere
E

= 2 2 2
ˆ ˆ ˆ ˆsin cosyi xj a i a j
x y a
     


=
1
ˆe
a

Using Stokes’s theorem
ˆ
S
E n dS 

=
1 1
ˆE dr e dr dS
a a
     
  
 
=
2
0
1
2ad
a

   Ans. 2
53. The shape of a dielectric lamina is defined by the two curves y = 0 and y = 1 – x2
. If the charge density
of the lamina 2
15 / ,yC m  then the total charge on the lamina is ___________C.
Solution :

TRAJECTORYEDUCATION
y
x
y x= 1 –
2
q= dS
=
2
1 1
1 0
15
x
ydy dx


 
=  
2
11 12
22
1 10
15
15 1
2 2
x
y
dx x dx

 
  
=
1 2
15 1 8
5 3
 
    
Ans. 8
54. In the circuit given below, the collector to emitter voltage VCE is __________V.
(Neglect VBE, take 100  )
5k
5k
5k
10k
VCE
V VCC = +10
Solution :
5k
5k
10k
V VCC = +10

TRAJECTORYEDUCATION
Vth =
2
1 2
5CCR V
V
R R


Rth =
1 2
1 2
5
R R
k
R R
 

Vth – IBRth – VBE – IERE =0
IB =
 
5 0.7
0.0042 mA
1 5 101 10
th BE
th E
V V
R B R
 
 
   
IC = 0.42 mABI 
VCE =VCC – IC(RC + RE)
=10 – 15 × 0.42 = 3.7 V Ans. 3.7
55. The de Broglie wavelength of a relativistic electron having 1 MeV of energy is ___________ × 10–12
m. (Take the rest mass energy of the electron to be 0.5MeV. Planck constant = 6.63 × 10–34
Js, Speed of
light 3 × 108
m/s, Electronic charge = 1.6 × 10–19
C)
Solution :
E2
=m°
2
c2
+ p2
c2
1=0.25 + p2
c2
p= 0.75 MeV/c
=
13
8
0.866 1.6 10
3 10

 

=4.619 × 10–22
 =
34
22
6.63 10
4.619 10
h
p





=1.4354 × 10–12
m Ans. 1.4354
56. X-ray diffraction of a cubic crystal gives an intensity maximum for Bragg angle of 20° corresponding
to the (110) plane. The lattice parameter of the crystal is _____________nm.
(Consider wavelength of X-ray = 0.15 nm)
Solution :
2 sinhkld  = n
2dhkl sin 20º=0.15 × 1 (n = 1)
dhkl =
0.15
0.25 nm
2sin 20º

2 2 2
a
h k l 
=0.22
a= 0.22 2 0.31 nm  Ans. 0.31
57. X-rays of 20 keV energy is scattered inelastically from a carbon target. The kinetic energy transferred
to the recoiling electron by photons scattered at 90° with respect to the incident beam is _________
keV.
(Planck constant = 6.6 × 10–34
Js, Speed of light = 3 × 108
m/s, electron mass = 9.1 × 10–31
kg,
Electronic charge = 1.6 × 10–19
C)
Solution :
KE transferred,
KE=
hc hc

   

TRAJECTORYEDUCATION
 =  1 cos
h
mc
 
=2.246 pm
 =
1240
eV nm
20000

=0.062 nm
 = 0.062 0.02426 0.0863 nm     
hc

=
1240
14368.48 eV
0.0863

=14.37 KeV
KE tranfered=
hc hc

 
=20 – 14.37
=5.63 KeV Ans. 5.63
58. An aluminum plate of mass 0.1 kg at 95°C is immersed in 0.5 litre of water at 20°C kept inside an
insulating container and is then removed. If the temperature of the water is found to be 23°C, then the
temperature of the aluminium plate is __________°C.
(The specific heat of water and aluminum are 4200 J/kg-K and 900 J/kg-K respectively, the density of
water is 1000 kg/m3
)
Solution :
Heat gained by water
Q1 = m S T 
=0.5 × 10–3
× 1000 × 4200 × 3
=6300 J
Heat lost by aluminium
=Q1 = 6300 J
6300= mS T
= 0.1 900 T 
T =70ºC
Temperature of aluminium plate
T=95ºC – 70ºC = 25ºC Ans. 25
59. The maximum and minimum speeds of a comet that orbits the Sun are 80 and 10 km/s respectively.
The ratio of the aphelion distance of the comet to the radius of the Earth’s orbit is ___________.
(Assume that Earth moves in a circular orbit of radius 1.5 × 108
km with a speed of 30 km/s)
Solution :
mVara =mVprp
m × 10 × ra =m × 80 × rp
a
p
r
r
=8

 
 
1
1
a e
a e


=8 (where a is semimajor axis)

1
1
e
e


=8
e=
7
9

TRAJECTORYEDUCATION
apehilion of elliptical orbit is given by
 
2
2
1
L
GMm e
=ra ...(1)
In case of circular orbit
2
GMm
r
=
2
mV
r
GM=V2
r ...(2)
L
m
=Vara ...(3)
From (1), (2), (3)
 
2 2
2
1
a aV r
V r e
=ra
ra
r
=  
2
2
30 30 7
1 1
10 9a
V
e
V
  
   
 
=2 Ans. 2
60. If there is a 10% decrease in the atmospheric pressured at a hill compared to the pressure at sea level,
then the change in the boiling point of water is _____________°C.
(Take latent heat of vaporisation of water is 2270 kJ/kg and the change in the specific volume at the
boiling point to be 1.2 m3
/kg.)
Solution :
dP
dT
=
 2 1
L
T V V
dT=
 2 1T V V
dP
L

=
5373 1.2
0.1 10
2270 1000

 

=1.97ºC Ans. 1.97


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