IIT JAM-2016 PHYSICS-SOLUTIONS
N O I T A C U D E Y R O T C E J A R T
the z -axis, -axis, the electric field at a , along the z
1. For an infinitely long wire w ire with uniform line charge density, density,
point (a (a, b, b, 0) away from the origin is
( eˆ, eˆ y and eˆ z are unit vectors in Cartesian-Coordinate system.) system.)
(a)
20 a 2 b 2
(eˆ x
(c)
20 a 2
b2
eˆy )
(b)
20 a 2 b 2
(aeˆx
eˆ x
(d)
20 a 2
b2
beˆy )
eˆ z
Ans. (b)
y
e r
(a, b,0)
x
E = E =
eˆr 2 r
r = r =
a2
eˆr =
r
r
b2
aeˆ x
a2
b2
E =
beˆy
2 ( a 2 b 2 )
(aeˆx beˆy )
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/cm 2 is
(a) (c)
1
8 2
(eˆx
1
16 2
eˆy )
(eˆx
(b)
eˆy )
(d)
1
16
(eˆx
1
4 2
eˆy )
(eˆx
eˆy )
Ans. (a)
P S = 4r 2 eˆr r = r =
eˆr =
2
eˆ x
eˆy 2
1
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TRAJECTORY EDUCATION
(eˆ x eˆy )
1
N O I T A C U D E Y R O T C E J A R T S =
=
4 2 1
8 2
2
(eˆx
eˆy )
3. A charged charged particle in a uniform magnetic field B B0eˆ z starts moving from the origin with velocity
v
time t at at which it reaches 2 meters above the (3eˆ x 2eˆz ) m / s . The trajectory of the particle and the time t
xy-plane xy-plane are
( eˆ x , eˆy and eˆ z are unit vectors in Cartesian-Coordinate system.) system.)
(a) (a) Heli Helica call path path;; t = = 1 s (b) (b) Heli Helica call path path;; t = = 2/3 s (c) (c) Circ Circul ular ar path path;; t = = 1 s (d) (d) Circ Circula ularr path path;; t = t = 2/3 s Ans. (a) Its velocity component in z in z -direction -direction is 2 m 2 m// s. s. Its path will be helical and will reach 2 m above xy above xy plane plane is 1 sec. 4. Consider the particle mass m following m following a trajectory given x x0 cos 1t and y y0 sin 2t where
x0 , y0 , 1 and
2 are constant of appropriate dimensions. The force on the particle is
(a) (a) cent centra rall onl only y if if 1
2
(c) always central
(b) (b) cent centra rall only only if x x0 = y0 and
1 2
(d) central only if x 0 = y = y0 and
1 2
Ans. (a)
r = x cos 1tiˆ y sin 2 tjˆ
a=
If 1
2 then
d 2 r 2
dt
12 x cos 1tiˆ 22 y sin 2tjˆ
12 r
a=
2 will be central F = ma m1 r which
5. Which one of the f ollowing points represent the complex number =
(a)
y
(b)
1
1 i
?
y
1
1
0.5
0.5
x
–1 –0.5 –0.5
x
0.5 1
–1 –0.5 –0.5
–1
(c)
0.5 1
–1
y
(d)
1
y
1
0.5
0.5
x
x
–1 –0.5 –0.5 –1
0.5 1
–1 –0.5 –0.5 –1
0.5 1
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 3
Ans. (a)
N O I T A C U D E Y R O T C E J A R T z =
1
1 i
(1 i )
(1 i )(1 i )
1 i 2
1
1
2
2
i
y 1
0.5
x
–1 –0.5 –0.5
0.5 1
–1
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 i x 0 i 1 y = 0
let
1 & 2 are eigenvalues
1 2 = 2
1 2 = 2 1 = 1 i, 2 1 i
7. The solution of the Boolean equation Y
A B AB is
(a) 1
(b) AB
(d) A B
(c) AB Ans. (d)
Y = A B AB = AB A B = A( B 1) B
= A B 8. A spherical closed container with smooth in ner wall contains a monoatomic ideal gas. If the collisions between the wall and the atoms are elastic, then the Maxwell speed distribution function the atoms is best represented by : (a)
(b)
dnv
dnv
dv
dv
0
v
0
v
dnv for dv
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TRAJECTORY EDUCATION
N O I T A C U D E Y R O T C E J A R T (c)
(d)
dnv
dnv
dv
dv
0
0
v
v
Ans. (c)
x and y having same amplitude are applied to x- and y-
9. Two sinusoidal signals of frequencies
channels of a cathode ray oscilloscope (CRO) respectively. The following stationary figure will be observed when y 1
–1
1
x
–1
(a)
y x
(b) Phase difference is 0
(c)
y 2x
(d) phase difference is
/ 2.
Ans. (b)
y = x =
n x
n y
2 4
No. of timesfig. cuts x axis
No. of timesfig. cuts y axis
1
2
x = 2 y
Phase difference = 0 or .
10. The phase difference () between input and output voltage for the following circuits (i) and (ii) R
C
vi ~
C
vo
vi ~
(i)
will be (a) 0 and 0
(c) Ans. (d)
/ 2 and / 2
C
vo
(ii)
/ 2 and 0 / 2 respectively (d) 0 and 0 / 2 respectively
(b)
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 5
V i = V eit
(i)
N O I T A C U D E Y R O T C E J A R T Z =
V ° =
1
j C
1
j C
2
j C
V i e jt j C 2 1
phase diff. = 0
Z = R
(ii)
i=
V ° =
0
V i
Z
1
jC
1
j C
R2
1
C 2
2
e i
tan 1 1 CR
V e j ( t )
R 2
1
C 2 2
j t
i
V e
R 2
2
1
C 2 2
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 0 1
1 0 1
1 0
1
0
1 1 1 ~ 0 1 1 0 1
1 0 x 1
0
0
1 1 ~ 0 1 0
1 y 1 z
1 = 1 1
0
1
1
1
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 = 3 x – 2 (b) y = –3 x + 4 (c) x = 3 y – 2 (d) x = –3 y + 4 Ans. (b) x2 + xy + 5=0 Taking derivative 2 x x
dy
dx
y =0
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dy
TRAJECTORY EDUCATION
N O I T A C U D E Y R O T C E J A R T At (1, 1)
dx
3
y = –3 x + C
It passes through (1, 1)
y = –3 x + 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? C
R
5V
P
10V
Q
(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
R
+V
(a) +V exponentially (c) +V linearly Ans. (d) It is integrator circuit
– +
V 0
(b) –V exponentially (d) –V linearly
15. In the following RC network, for an input signal frequency f
phase angle
between vo and vi respectively are
vo , the voltage gain vi and the 2 RC
1
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 7 R
C
N O I T A C U D E Y R O T C E J A R T C
vi ~
(a) (c)
1
2
1
2
and 0 and
vo
R
(d)
(d)
2
1
and 0
3
1
and
3
2
Ans. (b)
Let
V i = Veit
1
z = R
jC
j
= R
f =
R
C
1
2 RC
R
1
jC 1
jC
R
1 j CR
2 f
Z = R jR = R jR
1
RC
C
1 j
(1 j )R 2
3 3 j 3 R(1 j ) 3 R 2 2 2 2
i=
z
2V
3R
e
i t
4
R 1 jR
V ° =
R 1 j
i= =
V
V i
=
2V
j /4
3 2e
1
3
, 0
R
R
= R Vi
1
2V
3R
e j ( t /4)
e j ( t /4)
V 3
e jt
2e j 4
3R
2
e i/4
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TRAJECTORY EDUCATION
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)
N O I T A C U D E Y R O T C E J A R T
17. Consider a free electron (e) and a photon ( ph) both having 10 eV of energy. If wavelength and momentum respectively, then (mass of electron = 9.1 × 10 –31 kg; speed of light = 3 × 10 8 m/s)
(a)
e ph and P e = P ph
(b)
e ph and P e > P ph
(c)
e ph and P e < P ph
(d)
e ph and P e < P ph
Ans. (b)
e =
ph =
h
p
h
2mE
hc
E
e h E 1 E 1 10 1.6 10 –19 . = –91 ph 2mE hc c 2m c 2 9.110
So,
1
3 10
8
1
10–6
1
e ph
pe =
p ph =
p ph pe
=
2mE
E c
E
c 2mE
1
E
c
2m
1
p ph < pc 18. For an ideal gas, which one of the following T-S diagram is valid? (a)
(b)
T
T
Isochor
Isobar
Isochor
Isobar
S
S
(c)
(d)
T
T Isochor
Isobar
S
Isochor
Isobar
S
and p represent
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 9
Ans. (a) For isobaric process
N O I T A C U D E Y R O T C E J A R T dS =
dQ
C p dT
T T S = C plogT + C 1 S C 1 C p
T = e
For Isochoric process
dS =
dQ
CV dT
T T S = C V logT + C 2 S C 2 C
T = 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 train is V ° when train approaches the station
V V
f 1 = 484 f V
where V = 330 m/ s
V V V
f 2 = 442 f
when train leaves the station
f
So,
484 f
&
(1) + (2)
442
= =
V
V
V
V
V
V
1 1 =2 484 442
f
f ° = 462 Hz
1
V
1
V
V
V
...(1) ...(2)
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TRAJECTORY EDUCATION
Putting f ° in (2)
N O I T A C U D E Y R O T C E J A R T V =14.93 m/ s 14.93
3600 53.75km / s 1000 21. The minimum length of a plane mirror to the see the entire fu ll length image of an object is half of the object’s height. Suppose is the distance between eye and top of the head of t he 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 (h ) / 2 above the floor
(c) when the bottom edge of mirror is kept (h ) / 2 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 Ans. (b)
(d) 10
eˆ ˆr r V = re
= 12eˆr 16eˆ
| V | = 122 162
20m /s
23. A cylindrical rod of length L has a mass density distribution given by
measured from one end of the rod and
x ( x) 0 1 , where x is L
0 is a constant of appropriate dimensions. The centre of mass
of the rod is (a)
5 L 9
(b)
4 L 9
(c)
1 L 9
(d)
1 L 2
Ans. (a)
xcm =
xdm xdx dm dx
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 11 L
x
x
x.1 L dx
N O I T A C U D E Y R O T C E J A R T =
0 L
.1 L dx 0
x
=
2
2
x
x
3
L
3 L 0
x3
L
5 L. 9
2 L 0 24. A particle travels in a medium along a horizontal linear path. The initial velocity of the particle is v 0 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)
(b)
v(t )
v(t )
t
t
(c)
(d)
v(t )
v(t )
t
t
Ans. (d)
m
dv
dt
dv v
= – kv =
k
m
dt
v = Ve kt / m
25. A lightly damped harmonic oscillator with natural frequency
0 is driven by a periodic force of
. The amplitude of oscillation is maximum when (b) 0 is slightly lower than 0 (d) the force is in phase with the displacement is slightly higher than 0
frequency (a) (c)
Ans. (a) Amplitude maximum takes place at
2 2r 2
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)
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TRAJECTORY EDUCATION
N O I T A C U D E Y R O T C E J A R T
(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
0.4
10m /s
v = 10cos t
velocity,
=
x vdt =
k
m
10
640
0.4
40
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 t hat 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 surro unded 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
1 2
4
(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)
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 13
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 P 0. Neglecting the effect of gravity and assuming streamline flow, which one of the following plots correctly predicts the pressure along axis of the cone?
N O I T A C U D E Y R O T C E J A R T /4
(a) p
^
x
(b)
p0
x0
(c)
p0
x0
x
p
(d)
p0
x0
p
x
p
p0
x0
x
x
Ans. (a) Applying equation of continuity
V
x
2 AV = ( x )V
V =
AV
x 2
Applying Bernoulli equation
P
1
2
1
V 2 = P V 2 2
P = P
1
A2 1 2 4 x
V 2 31. Consider a spherical dielectric material of radius ‘ a’ centered at origin. If the polarization vector 2
P P0 eˆ x where P 0 is a constant of appropriate dimensions, then
( eˆ x , eˆy and eˆ z are unit vectors in Cartesian-coordinate system)
14 | JAM-2016 PHYSICS
TRAJECTORY EDUCATION
(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),
N O I T A C U D E Y R O T C E J A R T P 2n – P 1n =
b
P 2n = 0
z
y
x
b = 0
So,
b = P
dP dx
0
b = 0
32. For an electric dipole with moment P P0 eˆ z placed at the origin, ( p0 is a constant of appropriate
dimensions and eˆ x , eˆy and eˆ z are unit vectors in Cartesian coordinate system) 1
, where r is the distance from origin. r 2 (b) a spherical surface centered at origin is an equipotential surface (c) electric flux through a spherical surface enclosing the origin is zero (a) potential fall as
(d) radial component of E is zero of the xy-plane Ans. (a,c,d) V =
p cos
4 r 2
z
y
x 33. Three infinitely long conductors carrying current I 1, I 2 and I 3 lies perpendicular to the plane of the paper as shown below :
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 15 C 3
N O I T A C U D E Y R O T C E J A R T C 2
I 3
I 2
C 1
I 1
If the value of the integral B.dl for the loops, C 1, C 2 and C 3 are 2 0 , 40 and
0 in the units of
C
N / A respectively, then (a) I 1 = 3 A into the paper (c) I 3 = 0 Ans. (a,b)
(b) I 2 = 5 A out of the paper (d) I 3 = 1 A out of the paper
B dl = 2µ° C 1
I 1 + I 2 = 2
...(1)
B dl = 4µ°
I 2 + I 3 = 4
...(2)
B dl = µ°
I 1 + I 2 + I 3 = 1
...(3)
From (1), (2), & (3) I 3 = –1 i.e. 1A into the paper I 1 = –3 i.e. 3A into the paper I 2 = 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 f ollowing statement(s) is/are true?
detector
P
Q
(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 i deal gas to reach the final state f from an initial state i. Which among the following statement(s) is/are correct?
i
P
3
2
f
1
V
(a) The work done by the gas is maximum along path 3 (b) Minimum change in the internal energy occurs along path –2
16 | JAM-2016 PHYSICS
TRAJECTORY EDUCATION
(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 E 1, starting from r >> a, moves towards the origin, then
N O I T A C U D E Y R O T C E J A R T U(r )
m1
E 1
r
0
c
U min
(a) (b) (c) (d)
a
b
there is only one turning point for the particle velocity of the particle starts to increase x = a and reaches its maximum at r = c. velocity of the particle at r = a and r = b are equal the particle gets bounded if it elastically collides with a stationary particle of mass m 2 at r = c,
imparting a momentum greater than 2m2 E 1 Ans. (a,b,c,d) If m1 collides with particle of mass m2 at r = c Let E 1 is is energy of particle m1 after collision conserving energy. 0 E 1 = E 1
p 2
2m2
The particle of m1 will get bounded if E 1 < 0 E 1
p 2
2m2
<0
p > 2m2 E 1 37. A particle moves in a circular path in the xy-plane centered at the origin. If the speed of the parti cle 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) z
y
x
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 17
38. A pn junction was formed with a heavily doped (10 18 cm –3) p-region and lightly doped (10 14 cm –3) nregion. 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 beam of 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 r apidly 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)
N O I T A C U D E Y R O T C E J A R T 2 2 m ( EV ) L
T e 41. Fourier series of a given function f ( x) in the interval 0 to L is
f ( x) =
If f ( x) = x in the region (0, ), b2 Solution :
2nx 2nx n1 an cos n 1 bn sin L L 2
a0
_________ . bn = b2 =
2
0 2
f sin nx dx
0
x sin2 x dx
2 x
= cos 2 x sin 2 x 2 0 4 2 = 1 2 1
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
= 3 x 2 3 y 2 2x
= 3 x2 + 6 x( x2 + 1)2
at x = 1,
df dx
= 3 + 24 = 27
Ans. 27
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TRAJECTORY EDUCATION
43. A rectangular area ( A1) is formed by two vectors x and y as shown in figure (i). A new set of vectors,
N O I T A C U D E Y R O T C E J A R T
representing the area ( A2) as shown in figure (ii), are given as : u1 x; u2
kx y, where k is a
dimensionless constant.
u2
y
x
u1
(i)
(ii)
The Jacobian of the frame (u1 , u2 ) with respect to ( x , y ) is ___________. Solution :
u1 u1 , u2 x = x, y u2 x
u1 y 1 0 1 u2 k 1 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 = N e t
dN =
Ne t dt
Probability of decay in dt time
dN
t N = e dt
Probability of decay in 15 days 15
e
t
dt = e
t 15 0
e
ln 2 5
15
t
0
0
= 1 – e –3 log 2 = 1
In percentage, probability of decay is
700 8
1 8
7 8
87.5%
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 1.8 V , the value of the stopping potential for sodium will be ________ V . Solution : Let stopping potential of Na is V S
hc
=2.3 eV + eV S = 4.5 eV + 1.8 eV
eV S = 4 eV V S = 4 V
Ans. 4
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 19
46. The addition of two binary numbers 1000.01 and 0001.11 in binary representing is _________. Solution :
N O I T A C U D E Y R O T C E J A R T 1000.01
0001.11
1010.00
Ans. 1010
47. The number of second nearest neighbour ions to a Na+ ion in NaCl crystal is __________. Solution : Ans. 12 12 48. The output voltage V 0 of the OPAMP circuit given below is _______ V . 2 R
R
– +
R R R
1V 2V 3V
V 0
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 = nRT log
V 2
V 1
= 0.5 8.31 300 log
4
Ans. –358.6J = –358.6 J 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 = 0.5m
1
f
1 = 2 1 R
1
R1 1
R1 = R, R2 = – R 2
µ – 1=
2 1 R
2 0.75
2 Ans. 1.75 µ=1.75 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, F __________. ( eˆ x , eˆy and eˆ z are unit vectors in Cartesian-coordinate system.)
yeˆ x xeˆy , is
20 | JAM-2016 PHYSICS
TRAJECTORY EDUCATION y
Q
N O I T A C U D E Y R O T C E J A R T
4
4
O
x
P (2. 0)
Solution :
Mdx Ndy =
N M
x y dx dy
ydx xdy = 2dxdy = 2
1
2
2 2sin
4
Ans. 2 2 = 2 2 52. A hemispherical shell is placed on the xy-plane centered at the origin. For a vector field
E ( yeˆ x xeˆy ) / (x 2 y 2 ), the value of the integral ( E ) da over the hemispherical surface is S
___________ . Solution :
z
y
C
x
On the bounding curve of hemisphere
E =
=
yiˆ xjˆ a sin iˆ a cos jˆ x 2 y 2 a2 1
a
eˆ
Using Stokes’s theorem
1
E nˆ dS = E dr eˆ dr a
S
=
1 a
1
a
dS
2
a d 2
Ans. 2
0
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
Solution :
15 yC / m 2 , then the total charge on the lamina is ___________C.
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 21 y
N O I T A C U D E Y R O T C E J A R T 2
y = 1 – x
x
q=
dS
1 1 x 2
= 15
ydy dx
1 1
= 15
1
0
y 2 2
1 x 2
dx
0
15
1
1 x 2 2
2
dx
1
1 2 1 8 5 3
= 15
54. In the circuit given below, the collector to emitter voltage V CE is __________ V.
(Neglect V BE , take
100 )
VCC = +10V
5k
5k
V CE
5k
10k
Solution :
VCC = +10V
5k
5k
10k
Ans. 8
22 | JAM-2016 PHYSICS
TRAJECTORY EDUCATION
R2V CC
5V
N O I T A C U D E Y R O T C E J A R T V th =
Rth =
R1 R2 R1 R2
5k
R1 R2
V th – I B Rth – V BE – I E R E = 0 I B =
Vth V BE
Rth 1 RE
I C = I B
5 0.7
5 101 10
0.0042 mA
0.42 mA
V CE = V CC – I C ( RC + R E ) =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 –34Js, Speed of light 3 × 108m/s, Electronic charge = 1.6 × 10 –19C) Solution : E 2 = m°2c2 + p2c2 1 = 0.25 + p2c2 p = =
0.75 MeV/c
0.866 1.6 10 13
3 108 = 4.619 × 10 –22
6.63 1034
h
= p 4.619 1022
Ans. 1.4354 = 1.4354 × 10 –12 m 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 :
2d hkl sin = n 2d hkl sin 20º = 0.15 × 1
d hkl =
a
h 2 k 2 l 2
0.15
2sin20º
(n = 1)
0.25 nm
=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 × 10 8 m/s, electron mass = 9.1 × 10 –31 kg, Electronic charge = 1.6 × 10 –19 C ) Solution : KE transferred, KE =
hc
hc
TRAJECTORY EDUCATION
JAM-2016 PHYSICS | 23
h
= mc 1 cos
N O I T A C U D E Y R O T C E J A R T = 2.246 pm 1240
= 20000 eV nm = 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 Ans. 5.63 = 5.63 KeV 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
Ans. 25 T = 95ºC – 70ºC = 25ºC 59. The maximum and minimum speeds of a comet that or bits 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 × 10 8 km with a speed of 30 km/s) Solution : mV ar a = mV pr p m × 10 × r a = m × 80 × r p
r a
r p
=8
a 1 e =8 a 1 e
1 e 1 e
=8
e=
7
9
(where a is semimajor axis)
24 | JAM-2016 PHYSICS
TRAJECTORY EDUCATION
apehilion of elliptical orbit is given by
N O I T A C U D E Y R O T C E J A R T L2 = ra GMm 2 1 e
...(1)
In case of circular orbit
mV 2
GMm
= r 2 r 2 GM = V r L
m
= V ar a
...(2) ...(3)
From (1), (2), (3)
Va2 r a2 = r a V 2 r 1 e
r a
r
=
V 2 2 a
V
1 e
30 30 10
7 1 9
Ans. 2 =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
=
dT = =
L
T V2 V 1 T V2 V 1 L
dP
373 1.2
2270 1000 =1.97ºC
0.1105
Ans. 1.97
