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```fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
HCU-2015
Q1.

The unit vector parallel to the resultant of the vectors r1  2iˆ  4 ˆj  5kˆ and

r2  iˆ  2 ˆj  3kˆ is given by
1
1
1
(a) iˆ  ˆj  kˆ
3
6
2
(c)
Q2.
(b)
2ˆ 3 ˆ 6 ˆ
i  j k
7
7
7
(d) 3iˆ  6 ˆj  2kˆ
The solution to the differential equation
t3
A
3
t4
 At  B
(c) x 
12
where A and B are constants.
(a) x 
Q3.
Q4.
3ˆ 6 ˆ 2 ˆ
i  j k
7
7
7
d 2x
 t 2 is given by
2
dt
t3
 At  B
3
t3 t2
(d) x    A
3 2
(b) x 
If T  sin x sin y sin z,  2T is given by
(a) 0
(b)  2 sin x sin y sin z
(c) 3 sin x sin y sin z
(d)  3 sin x sin y sin z


a force F   A x 2i  yj is irrotational. The potential energy function associated with it is
given by
Q5.
 x2 y2 


(a) U  A
2 
 3
 x y2 

(b) U  A 
3 2 
 x2 y 
(c) U  A  
 3 2
 x y3 

(d) U  A 
3 2 
A bead moves outward with constant speed u along the spoke of a wheel. It starts from
the centre at    t . The angular position of the spoke is given by wt, where  is a
constant. The acceleration of the bead is given by
(a) ut 2 rˆ  uˆ
(b) urˆ  u tˆ
(c) u t 2 rˆ  utˆ
(d)  ut 2 rˆ  2uˆ
fiziks, H.No. 23, G.F, Jia Sarai,
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Phone: 011-26865455/+91-9871145498
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Email: [email protected]
Branch office
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28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
1
fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Q6.
A solid cylinder and a hollow cylinder are placed at the same height at the top of a long
incline and released at the same time.
(a) Both the cylinders reach the bottom together.
(b) Solid cylinder reaches the bottom first.
(c) Hollow cylinder reaches the bottom first
(d) Any one of them can reach the bottom first.
Q7.
A large vertical drum spins so fast that a ball inside it, is pinned to the wall. What
is the minimum steady angular velocity  which prevents the ball from falling to the
bottom of the drum? (The radius of the drum is R , the mass of the ball is M and 
is the co-efficient of friction.)
(a)  2 
Q8.
Mg
R
(b)  2 
g
R
(c)  2 
g
MR
(d)  2 
g
R
 
 
The equation of traveling wave is given as: y  10 cos
x
t  . What is the speed
20s 
 5cm
of the wave?
(a) 1cm / s
Q9.
(b) 0.5 cm / s
(c) 0.25cm / s
(d) 0.1cm / s
Two sound sources oscillate in phase with a frequency of 100 Hz . At a point 5m from
one source and 5.85 m from the other the amplitude of the sound from each source
separately is A . What is the phase difference of the two waves at that point (Assume that
the speed of sound is 343m / s .)
(a) 99o
Q10.
(b) 45o
(d) 89o
(c) 60o
It is given that surface tension of water is 0.072N / m and its density 103 kg / m3 . What is
the height to which water will rise in a tube of diameter 0.2 m m?
(a) 14.7 cm
Q11.
(b) 34.2 cm
(c) 29.4 cm
(d) 1.27cm
An air bubble of diameter 2 mm rises steadily through a solution of density 1750kg / m3
at the rate of 0.35cm / sec . If the density of air is negligible, what is the coefficient of
viscosity of the solution’?
(a) 1 poise
(b) 11 poise
(c) 20 poise
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(d) 4 poise
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Q12.
Q13.
Internal energy of an idea gas changes when
(a) temperature changes.
(b) volume changes.
(c) entropy changes
(d) pressure changes
A gas expands adiabatically and reversibly to a temperature TA . If the same expansion
happens adiabatically, but irreversibly, the final temperature is T . Then
Q14.
(a) T  TA
(b) T  TA
(c) T  TA
(d) T can take any value independent of TA
A cylinder fitted with a piston contains an ideal gas at 500 Kilo Pascals and occupies a
volume of 0.2m 3 . The gas expands isothermally to a pressure of’100 Kilo Pascals. The
work done by the gas is
(a) 160.9 KJ
Q15.
(b) 1609.0 KJ
(d) 5 104 KJ
(c) 16.09 KJ
The sun delivers about 1000W / cm 2 of electromagnetic flux to the earth’s surface.
Assume 7% conversion efficiency of the solar panel of dimensions 8m  20 m . If the
radiation is incident normally the solar power converted for use is
(a) 11.42  104 W
Q16.
(b) 11.35 104 W
(c) 1.56  104 W
(d) 1.12  104 W
The angle of refraction for a light beam incident on a heavy flint glass n  1.65 at
the po1arizing angle is given by
(a) 31.22o
Q17.
(b) 30.25o
(c) 28.20o
(d) 33.41o
A radio pulse from a doppler radar reflects off an aircraft in mid-flight. If the frequency
of the reflected radio pulse is less than the original one, then which of the following
statements is true?
(a) The aircraft is not moving.
(b)The aircraft is moving away from the radar
(c) The aircraft is moving towards the radar.
(d) Something is wrong with the radar
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Phone: 011-26865455/+91-9871145498
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Email: [email protected]
Branch office
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28-B/6, Jia Sarai, Near IIT
Hauz Khas, New Delhi-16
3
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Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Q18.
The total charge within a sphere of radius r in a charge cloud is given by:
r2
q 2
a
2 r
 ar
e  e a



.


The electric field at the surface of the above sphere is given by
r
 ar

q
a 

e

e
(a)

4 0 a 2 

2 r
 ar
e  e a






2 r
 ar

q
a 

e

e
(d)

4 0 q 2 


The electric field in the x  y plane is given by E  iˆ8x  ˆj 4 y . Then the equation for the
q
(c)
0 a 2
Q19.
q
(b)
0 a 2
r
 a2 r

 e  e 2a 




lines of force is given by
(a) xy 2  constant
(b) x 2 y  constant
(c) x 2  y  constant
Q20.
(d) xy  constant

 
B
Consider Maxwell’s equation:   E  
. If  is the scalar potential and A is the
t

vector potential, then E can be written as



 
A
(a)      A
(b)   
t
(c) 
Q21.

  
 A
t

  
 A
(d)   
t

What is the current I 1 , flowing in the circuit shown in the following figure?
10 k  10 k 
(a) Infinite
15k 
20 k 
(b) 1 m A
30V
(c) 0.5 m A
10 k 
60 k 
10 k 
(d) zero
Q22.
A 50 Hz sinusoidal signal is applied to the input of a full-wave rectifier The frequency of
the output signal is
(a) 100Hz
(b) 50 Hz
(c) 70.7 Hz
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(d) 0 Hz
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Q23.
In a Bipolar Junction Transistor (BJT), the base-width is kept thinner than Emitter and
Collector width. This is to
Q24.
(a) reduce the base current
(b) increase the base current
(c) increase the base current
(d) decrease the collector current
If 1 and  2 are two independent solutions of the time-independent Schrödinger
equation, which of the following is also a solution of the same Schrodinger equation?
(a)  1  2
Q25.
(b)
1
2
(d)  1 2 
(c)  1   2
1/ 2
If H 1 and H 2 are Hamiltonians of two non-interacting systems with wave functions
 1 and  2 and energies E1 and E 2 respectively, the total wave function  and energy
E of the composite system are given by
Q26.
(a)    1  2 , E  E1  E2
(b)    1 2 , E  E1 E2
(c)    1  2 , E  E1 E2
(d)    1 2 , E  E1  E2
1 0

What value of  will make the matrix  0 
0 

2
(a)
Q27.
(b) 1
0 

   orthogonal?
 
(c)
1
(d)
2
1
2
The value of the integral of the function f x, y   x 2  y 2 integrated along a straight
line from 1, 0 to (0,1) as shown in the figure is
(a) 
(c)
2
3
1
2
Y
(b) 
2
3
(d) 
1
2
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0,1
1, 0
X
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Q28.
a b c 
If abc  P and A   c a b  , where A is an orthogonal matrix, then the value of
b c a 
a  b  c is
(a) 2
Q29.
(b) P
(d)  1
(c) 2 P
The general solution to the differential equation
d 2 y 2dy

 2 y  0 , is
dx
dx2
(a) e x A cos x  iB sin x
(b) e ix A cos x  iB sin x
(c) e ix A cos x  iB sin x
(d) e x A cos x  iB sin x
where A and B are arbitrary constants.
Q30.
In the complex z  plane, the equation z  2 z  1 represents
2
3
3
2
(d) a straight line through 0,0 and 1,1 .

Q31.
1
,
2
The power series
  1
n
n 2 x n converges for
n 1
(a)  1  x  1
Q32.
(b)  1  x  1
(c)  1  x  1
(d)  1  x 
On changing variable from x to t where t  log x , the differential equation
x2
d2y
dy
  et
 y  0 , becomes
2
dt
dt
(a) e 2t
(c)
d2y
dy
 e t
 y  0
2
dt
dt
d2y
dy

     y  0
2
dt
dt
(b)
d2y
dy

 y  0
2
dt
dt
(d)
d2y
dy
   1  y  0
2
dt
dt
fiziks, H.No. 23, G.F, Jia Sarai,
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Q33.
A uniform drum of radius 0.5 m , mass 40 kgs rolls (without slipping) down an incline of
angle 5o If the drum starts from rest from a height of 29.4 metres, the speed of its centre
of mass will be
(a) 39.2 m / s
Q34.
(b) 19.6 m / s
(c) 32.6 m / s
(d) 25.5 m / s
A mass M is moving along a closed rectangular path of
length l
along the
Y
y  direction and a along the
x  direction (as shown in the figure) under the influence

 x2 
of the force F  b 0 1  2  ˆj . The work done is given
 a 
O
(a) zero
(b) b 0 l
(c) b 0 a
(d) b0 al
B
l
by
Q35.
a
A
C
X

A force field F  3x  y  z iˆ  x  y  z 2 ˆj  3x  2 y  4z kˆN is acting on a particle of


mass m . The particle moves in a circular path of radius 5m with a constant speed
of 5m / s in the x  y plane. The cent of the orbit is the origin. The change in the kinetic
energy as a particle completes one rotation is given by
(a) 25 J
Q36.
(b) 25 J
(c) 50 J
(d) 50 J
A particle moves along a trajectory whose displacement is given by
r t   cost iˆ  sint  ˆj  tkˆ with   1
The path of the particle is a
(a) parabola, with direction of motion to the right of the origin.
(b) parabola, with direction of motion to the left of the origin
(c) hyperbola with direction of motion to the right ‘of the origin.
(d) hyperbola, with direction of motion to the left of the origin.
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Q37.
The centre of mass of a solid hemisphere (as shown in
Z
figure) of unit radius is given by
Q38.
3

(a)  0, 0,

8

3


(b)  0,
, 0
8


3


(c) 1,
, 1
8



(d)  0, 0,

1
2

3
X
Solid Hemisphere
A particle of mass m starts at rest from the top of a smooth fixed hemisphere of radius a .
The angle at which the particle leaves the hemisphere is equal to
1
(a) cos1  
 3
Q39.
(b)

2
The Lagrangian of Atwood’s machine (shown in the figure, where m2  m1 ) is given by
(a)
1
1
m1 x12  m2 x x2  m1 gx1  m2 gx 2
2
2
1
1
(b) m1 x12  m2 x 1x  m1 gx1  m2 g 2
2
2
(c)
(d)
Q40.
2
(d) sin 1  
3
2
(c) cos1  
3
R
X1
1
m1  m2 x12  m1  m2 gx1
2
X2
m1
1
m1  m2 x12  m1  m2 gx1
2
m2
Consider a rope of mass per unit length  length a , suspended just above a table as
shown in figure. If the rope is released from rest at the top, the force on the table when a
length x of the rope has dropped to the table is
X
(a) xg
(b) 2 xg
a
(c) 3xg
(d) 4 xg
T able top
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Q41.
An object of unit mass orbits in a central potential given by U r  . Its orbit is
r  a exp b  with b  0 . The relationship between the angular momentum and the total
energy E is given by


L2 2
b  1  U r 
2
L
(c) E  b  1  U r 
r
(a) E 
Q42.


L2 2
b  1  U r 
2r 2
a 2 L2
(d) E  2 b  1  U r 
r
(b) E 
A wave is propagating in a string of tension T and mass per unit 1ength  . The traveling
wave can be described as yx, t   A sin kx   t  . What is the kinetic energy in one
wavelength   of the traveling wave
(a)
(c)
Q43.
2TA 2 2
(b)

TA2 2
2
(d)
TA2 2

TA2 2
4
Consider the figure as shown. A and B have equal masses m and all the springs have
the same spring constant k . What are the normal frequencies for the system, if the
oscillations are assumed to be small?
k
A
X1
Q44.
k
k
B
X2
(a)
k 2k
,
m m
(b)
k
k
,
2m m
(c)
k 3k
,
m m
(d)
k
k
,
1
m m
A violin string is held under tension T . What will be fractional change in the frequency
of its fundamental mode of vibration if the tension is increased by the amount T ?
(a)
T
T2
(b)
T
T
(c)
T
T
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(d)
T
2T
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Q45.
One can estimate the resonant frequencies of the human ear canal of length 2.2 cm , that
are in the range of human hearing. How many resonant frequencies lie in this range?
Speed of sound is given as 343m / s .
(a) 3
Q46.
(b) 4
(c) 5
(d) 6
One end of a 100cm long wire (without load) is fixed and a mass of 2kg is attached to
the other end. The mass is kept moving with uniform speed in a horizontal circle of
radius 60cm . What is the strain of the wire it the radius of the wire is 0.032cm and
Young’s modulus is 2 1011 / dynes/cm2 ?
(a) 0.032 102
Q47.
(b) 32  102
(c) 0.32  102
(d) 3.2  102
What will be density of lead under a pressure of 20,000N / cm2 ? (Density of lead
 11.5gm / cm3 and bulk modulus of lead  0.80  1010 N / m 2 )
Q48.
(a) 11.69 gm/ cc
(b) 14.19gm/ cc
(c) 13.89gm/ cc
(d) 15.98 gm/ cc
The excess of pressure inside a spherical soap bubble of radius 1cm is balanced by that
due to a column of oil of specific gravity 0.9 gm / cm3 and height 1.36 m m. What is the
surface tension T ?
Q49.
(a) T  3.06  103 N / m
(b) T  323.0  102 N / m
(c) T  3.06  102 N / m
(d) T  3.06  104 N / m
A big drop is formed by coalescing 1000 small droplets of water. By how many times
will the surface energy decrease?
(a) 100
Q50.
(b) 10
(c) 5
(d) 30
A plate of area 100cm2 arid thickness 2 mm is placed on the upper surface of some
castor oil. If the coefficient of viscosity is 15.5 poise, what. is the horizontal force
necessary to move the plate with a velocity 3cm / sec ?
(a) 10.2 N
(b) 0.13 N
(c) 2.3 N
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(d) 0.23 N
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Q51.
The velocity of water in a river is 10km / hr near the surface. If the river is 5m what is the
shearing stress between the horizontal layers of water? (The coefficient of viscosity of
water 102 poise.)
(b) 105 N / m 2
(a) 103 N / m 2
Q52.
(d) 101 N / m 2
(c) 10N / m 2
The figures shown represent two engines in T  S phase plane. Let  a arid b the
efficiencies of the engines (i) and (ii), respectively. Then
T
T
T2
T2
T1
T1
S1
(a) b   a
Q53.
(b)  a  b
i
S2
S
(c)  a  b
S
S2
S1
ii 
(d)  a  b
The relation between pressure and volume of an ideal gas in a reversible process is given
by P  aV  b, where a 
31
255
Pascal/met er 3 , b 
Pascals . The volume at which the
56
7
temperature attains maximum is
(a) 32.9m 3
Q54.
(b) 329.7 m 3
(d) 20.7 m3
(c) 0.329m 3
Clausius-Clapeyron equation for liquid-gas transition is given by:
d
1
. If

d T v g  vl 
v g  vl and vg  RT , the expression for saturated vapour pressure is given by
(a)   constant e
1
RT
(b)   constant e
(d)   constant
(c)   0
Q55.
-1
RT
For a nonmagnetic insulator, the specific heat is proportional to
(a) T
(b) T
2
(c) T
3
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(d) e

k BT
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Q56.
Consider the superposition of two sinusoidal waves given by: y1  4 sin 3x  2t cm and
y2  4 sin3x  2t cm . The maximum displacement of the resultant motion at x  2.3 cm
is
(a) 8.65cm
Q57.
(b) 1.61cm
(c) 4.63cm
(d) 7.60cm
A total of 25 fringes move across the field of view when one of the mirrors of’ the
Michelson interferrometer is moved by a distance d . If the interferromneter is illuminated
by a laser of wavelength 632.8 nm , the distance by which the mirror is moved is
(b) 38.6m
(a) 79.1 m
Q58.
(d) 158.2 m
(c) 57.3 
A parallel light beam of diameter d is incident on a convex lens of focal length f . The
emerging beam from this lens passes again through another lens of focal length 2 f , kept
at a distance 3 f from first lens. The diameter of the output beam is
Q59.
(a) d
(b) 2 d
(c) not related with d
(d)
1
d
2
Consider a uniformly charged disc of radius a and surface charge density  . Consider a
point P on the axis of the disc at a distance z from the disc. The potential at P is given
by
(a)
(c)
 1
0 z
(b)
2
a


2
 z2  z

0
2
 z2  a2  z2

(d)
0
Q60.
a
2

a


2
 z2  a

0
A charge q is distributed uniformly over the surface of a thin circular insulating
of radius a . The potential at the rim of the disc is given by
(a)
(c)
q
 a 0
2
q
a 0
2
(b)
q
4 0 a
(d)
q
4 a 2 0
2
2
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Q61.
Current I is passing through a conical shaped copper
I
wire of resistivity  as shown in the figure. If L is
a
b
the length of the wire, b is the front radius and a is
the back radius of the wire, what is the resistance of
L
the wire?
(a)
Q62.
L
 ab
(b)
L
(c)
 a  b 
L
(d)
ab

 2 

L
ab

 ab 

A very long wire carrying charge q per unit length is held parallel to an infinite
conducting plane at a distance  from it. What is the force of attraction per unit length?
(a)
q
4 0 
(b)
 q2
4 0 
(c)
q2
2 0 
(d)
 q2
2 0 
r
Q63.
The potential in a medium is given by  r  
q e
. The charge density for r  0 is
4 0 r
given by
r
q gl
e
(a) 
0
Q64.
r
2 r
r
q
q
q
(c) 
e
e
e  (d) 
2
2
4 0
4 0 r 
4 r 

What is the magnetic field B produced at point P (center of the arcs) due to the current
(b) 

I in the close loop as shown in figure?
   0 I
(a) B 
4
1
1 ˆ
  k
 R1 R2 
   0 I
(b) B 
4
1
1 ˆ
  k
 R1 R2 
   0 I  1
1 ˆ
(c) B 
  k
4  R2 R1 
   0 I  1
1 ˆ
(d) B 
  k
4  R2 R1 
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I
I
R2

R1
P
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Q65.
In the circuit given below, the diode can be considered as ideal. The peak secondary
voltage and the D.C. load voltage respectively are then given by
10.1
(a) 31.1V and 9.9 V
(b) 22V and 9.9 V
10 
(c) 22V and 7.0 V
(d) 31.1V and 7.0 V
Q66.
What is the effective resistance between points A and B ?
(a) Rab  10
20 
(b) Rab  11
10 
(c) Rab  15
10 
(d) Rab  12
Q67.
20 
20 
20 
10 
10 
What are the values of R4 and L4 , if the a.c. bridge shown in figure is balanced?
600
(a) R4  720 and L4  12 mH
100
R1
120m H
R2
L1
(b) R4  3000 and L4  24 mH
R4
R3
L4
(c) R4  120 and L4  600mH
500
(d) R4  3000 and L4  600mH
Q68.
What is the operating point VCE , I C  for the transistor circuit shown in figure? (Assume
I C  I E and VBE  0.7V )
RC 700
(a) 5V , 2m A
VCC
15V
(b) 7.5V , 1.5 mA
V BB
5V
(c) 7.5V , 1mA
RE 4.3 k 
(d) 10V , 1 m A
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Q69.
A particle of mass 1.40105 kg
has a relativistic momentum of magnitude
3.151013 kg m / s . How fast is the particle traveling?
(a) 2.68108 m/ s
Q70.
(b) 1.8108 m / s
(c) 3.31016 m / s
(d) 3108 m / s
A proton of rest mass 1.681027 kg is traveling at 2.5108 m / s . What is kinetic energy
of the proton?
(a) 1.181010 J
Q71.
(b) 0.181010 J
(c) 2.181027 J
(d) 1.321027 J
A spaceship passes you at a speed of 0.85c . You measure its length to be 48.2 m . What
is its length at rest?
(a) 80 m
Q72.
(b) 100m
(d) 81m
Which of the following functions is the eigen-function of the operator,  i
(a) expikx
Q73.
(c) 91.5 m
(c) sin kx
(b) coskx
d
?
dx
(d) coskx  sinkx
In a one dimensional problem, the normalized wave function for the ground state is given
by   Ne x , where N is the normalization constant. What is the value of N ?
2


(a)
Q74.
2
(c)

(d)


Commutator of two Hermitian operators has to be
(a) Hermitian
Q75.

2
(b)
(b) anti-Hermitian
(c) unitary
(d) orthogonal
A sample of radioactive isotopes contains two different nuclides, labeled A and B .
Initially, the sample composition is 1 : 1 . The half-life of A is 3 hours and that of B is 6
hours. What is the expected ratio
(a)
1
6
(b)
1
4
A
after 18 hours?
B
(c)
1
8
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(d)
1
2
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