Iit jam 2016 physics solutions BY Trajectoryeducation

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IIT JAM-2016 PHYSICS-SOLUTIONS1. 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 20

ˆ ˆ( )2

x ye ea b

(b) 2 2

0

ˆ ˆ( )2 x yae be

a b

(c) 2 20

ˆ2

xea b

(d) 2 20

ˆ2

zea b

Ans. (b)

y

( , ,0)a b�re

x

E= ˆ2 re

r

r= 2 2a b

ˆre = 2 2

ˆ ˆx yae berr 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 aremeasured 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 rP er

r= 2

ˆre =ˆ ˆ

2x ye e

2 | JAM-2016 PHYSICS TRAJECTORY EDUCATION

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S =

ˆ ˆ( )14 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 thexy-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 planeis 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 2ˆ ˆcos sinx ti y tj

a =2

2 21 1 2 22

ˆ ˆcos 4 sind r x ti y tjdt

If 1 2 then a = 21 r

F

= 21ma 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

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Ans. (a)

z=1 (1 ) 1 1 1

1 (1 )(1 ) 2 2 2i 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 = 0are(a) 1 + i, 1 + i (b) 1 – i, 1 – i(c) 1, i (d) 1 + i, 1 – i

Ans. (d)x + iy= 0ix + y= 0

11i x

i y

=00

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 B8. 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 vdndv

for

the atoms is best represented by :

(a)

vdndv

v0

(b)

vdndv

v0


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(c)

vdndv

v0

(d)

vdndv

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 beobserved when

y1

–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 axisNo. of timesfig. cuts axis

x

y

n xn y

=2 14 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)


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(i) Vi =i tV e

Z=1 1 2

j C j C j C

V° =1

2j tVi e

j C

phase diff.= 0

(ii) Z= 22 2

1 1 iR R ej C C

1 1tanCR

i =( )

22 21

j tiV V e

ZR

C

V° =2

22 2

11

j tV ei

j CR

C

02

11. For the given set of equations : x + y = 1, x + z = 1, x + z = 1, which one of the following statements iscorrect?(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= 1y + z= 1x + z= 1

1 1 00 1 11 0 1

xyz

=111

1 1 0 1 1 0 1 1 00 1 1 ~ 0 1 1 ~ 0 1 11 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 dyx x ydx

= 0


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At (1, 1) 3dydx

y= –3x + CIt passes through (1, 1)

y= –3x + 413. 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 wascharged 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 QAssuming 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 theoutput 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 frequency1 ,

2f

RC

the voltage gaino

i

vv and the

phase angle between vo and vi respectively are


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vi

Cvo

R

~

C

R

(a)12

and 0 (d)13

and 0

(c)12

and2

(d)13

and2

Ans. (b)

Let Vi =i tV e

z=

11

1

Rj CR

j C Rj C

=1

j RRC j CR

f =1 1 12

2f C

RC RC R

Z=1

RR jRj

=(1 )

2j RR jR

= 4 /43 3 3 3 3(1 ) 22 2 2 2 2

j iR RR j R j e e

i = 423

i tiV V e

z R

V° =1

Rj R

i = ( /4)21 3

j tR V ej R

= ( /4)/4

233 2

j t j tj

V Ve ee

i

VV =

1 , 03


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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 hp mE

ph =hcE

e

ph

=–19

–91

1 1 10 1.6 10.2 2 9.1 102

h E Ehc c m cmE

8 –6

1 1 13 10 10

So, e ph

pe = 2mE

pph =Ec

ph

e

pp =

1 122

E Ec mc mE

pph < pc18. For an ideal gas, which one of the following T-S diagram is valid?

(a)

TIsochor

Isobar

S

(b)

TIsobar

Isochor

S

(c)

T Isochor

Isobar

S

(d)

T Isochor

Isobar

S


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Ans. (a)For isobaric process

dS= pC dTdQT T

S= CplogT + C1

T=1

p

S CCe

For Isochoric process

dS= VC dTdQT T

S= CVlogT + C2

T=2S C

Ce

19. If U, F, H, and G represent internal energy, Helmholtz free energy, enthalpy, and Gibbs free energyrespectively, 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 + PVdH=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 platformmeasure the tone of the train whistle as 484 Hz when it approaches the station and 442 Hz when itleaves the station. If the sound velocity in air is 330 m/s, then the tone of the whistle and the speed ofthe 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 VfV V

where V = 330 m/s

when train leaves the station f2 = 442 VfV V

So,484

f = 1V V VV V

...(1)

&442

f = 1V V VV V

...(2)

(1) + (2) 1 1

484 442f

= 2

f° = 462 Hz


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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 theobject’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 velocityof 2 rad/s. When the particle is at a distance r = 8 m from the origin, the magnitude of the intantaneousvelocity of the particle in m/s is(a) 8 15 (b) 20

(c) 2 37 (d) 10Ans. (b)

V = ˆ ˆrre r e

= ˆ ˆ12 16re e

| |V

= 2 212 16 20 /m s

23. A cylindrical rod of length L has a mass density distribution given by 0( ) 1 ,xxL

where x is

measured from one end of the rod and 0 is a constant of appropriate dimensions. The centre of massof the rod is

(a)59

L (b)49

L

(c)19

L (d)12

L

Ans. (a)

xcm =xdm x dx

dm dx


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0

. 1

. 1

L

L

xx dxLx dxL

=

2 3

03

0

52 3 .9

2

L

L

x xL L

xxL

24. A particle travels in a medium along a horizontal linear path. The initial velocity of the particle is v0and the viscous force acting on it is proportional to its instantaneous velocity. In the absence of anyother forces, which one of the following figures correctly represents the velocity of the particle as afunction of time?

(a)

v t( )

t

(b)

v t( )

t

(c)

v t( )

t

(d)

v t( )

tAns. (d)

dvmdf = –kv

dvv =

k dfm

v= /. k mV e

25. A lightly damped harmonic oscillator with natural frequency 0 is driven by a periodic force offrequency . 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 displacementAns. (a)

Amplitude maximum takes place at2 22r

26. A block of mass 0.38 kg is kept at rest on a frictionless surface and attached to a wall with a spring ofnegligible mass. A bullet weighing 0.02 kg moving with a speed of 200 m/s hits the block at timet = 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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(c) 0.4cos25t (d) 0.25sin40tAns. (d)

Let instantaneous velocity imparted to the block is vApplying conservation of linear momentum

0.02×200= (0.385 + 0.01)v

v=4 10 /

0.4m s

velocity, v= 10cos t

=640 400.4

km

x vdt =10 sin t

= 0.25 sin40t27. 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?

q4

21

(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 duringits 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)


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30. An incompressible non-viscous fluid is injected into a conical pipe at it orifice as schematically shownin the figure. The pressure at the orifice of area A0 is P0. Neglecting the effect of gravity and assumingstreamline, flow, which one of the following plots correctly predicts the pressure along axis of thecone?

/4 x^

(a)

x0x

p0

p (b)

x0x

p0

p

(c)

x0x

p0

p (d)

x0x

p0

p

Ans. (a)Applying equation of continuity

xV

AV = 2( )x V

V= 2

AVx

Applying Bernoulli equation

212

P V = 212

P V

P=2

22 4

1 12

AP Vx

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)


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(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 = bP2n = 0

x

y

z

So, b = 0

b = 0dPPdx

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-planeAns. (a,c,d)

V= 2

cos4p

r

x

y

z

33. Three infinitely long conductors carrying current I1, I2 and I3 lies perpendicular to the plane of thepaper as shown below :


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C3

C2

C1

I1

I2I3

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 paperI1 = –3 i.e. 3A into the paperI2 = 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 orientedsuch 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

Pi

1

23

f

(a) The work done by the gas is maximum along path 3(b) Minimum change in the internal energy occurs along path –2


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(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 EAns. (a,b,c,d)

If m1 collides with particle of mass m2 at r = cLet 1E is is energy of particle m1 after collisionconserving energy.

10 E s=2

122

pEm

The particle of m1 will get bounded if1E < 0

2

122

pEm

<0

p> 2 12m E37. 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


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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 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 thefollowing 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 LT e

41. Fourier series of a given function f(x) in the interval 0 to L is

f(x)= 01 1

2 2cos sin2 n n n na nx nxa b

L L

If f(x) = x in the region 2(0, ), _________.b Solution :

bn =0

2 sinf nx dx

b2 =0

2 sin 2x x dx

=0

2 1cos2 sin 22 4x 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 derivativeof f with respect to x, at x = 1 is __________.

Solution :dfdx =

f f dyx y dx

= 2 23 3 2x y x = 3x2 + 6x(x2 + 1)2

at x = 1,dfdx = 3 + 24 = 27 Ans. 27


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43. A rectangular area (A1) is formed by two vectors x and y as shown in figure (i). A new 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 ux y

=

1 1

2 2

1 01

1

u ux yu u kx y

Ans. 1

44. A particle radioisotope has a half-life of 5 days. In 15 days the probability of decay in percentage willbe ___________.

Solution :N= tN e

dN= tN e dt

Probability of decay in dt timedNN

= te dt

Probability of decay in 15 days15

0

te dt =2 15

ln15

50 0

tte e

= 1 – e–3 log 2

=1 718 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 potentialfor 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 eVVS = 4 V Ans. 4


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46. The addition of two binary numbers 1000.01 and 0001.11 in binary representing is _________.Solution :

1000.010001.111010.00

Ans. 1010

47. The number of second nearest neighbor ions to a Na+ ion in NaCl crystal is __________.Solution :

12 Ans. 1248. The output voltage V0 of the OPAMP circuit given below is _______ V.

V0

R–+

2R

RRR

1V2V3V

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

logVnRTV

= 0.5 8.31 300 log4

= –358.6 J Ans. –358.6J50. 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 ofthe material is _________.

Solution :

f = 1 1

1 10.5 1mR R

R1 = R, R2 = –R

f = 2 10.5

R

µ – 1=0.5 0.75

2

µ= 1.1875 Ans. 1.187551. 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.)


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y

xO P(2. 0)

Q

4

4

Solution :

Mdx Ndy =N M dx dyx y

ydx xdy = 2dx dy

=12 2 2sin2 4

= 2 2 Ans. 2 252. A hemispherical shell is placed on the xy-plane centered at the origin. For a vector field

2 2ˆ 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 jx y a

=1 ea

Using Stokes’s theorem

ˆS

E n dS

=1 1ˆE dr e dr dSa a

=2

0

1 2a da

Ans. 2

53. The shape of a dielectric lamina is defined by the two curves y = 0 and y = 1 – x2. If the charge densityof the lamina 215 / ,yC m then the total charge on the lamina is ___________C.

Solution :


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y

x

y x = 1 – 2

q= dS

=21 1

1 0

15x

ydy dx

= 211 12 22

1 10

1515 12 2

xy dx x dx

= 1 215 1 85 3

Ans. 8

54. In the circuit given below, the collector to emitter voltage VCE is __________V.(Neglect VBE, take 100 )

5k

5k5k

10k

VCE

V VCC = +10

Solution :

5k

5k

10k

V VCC = +10


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Vth = 2

1 2

5CCR V VR R

Rth = 1 2

1 2

5R R kR R

Vth – IBRth – VBE – IERE = 0

IB = 5 0.7 0.0042 mA

1 5 101 10th BE

th E

V VR B R

IC = 0.42 mABI VCE = VCC – IC(RC + RE)

= 10 – 15 × 0.42 = 3.7 V Ans. 3.755. 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 oflight 3 × 108m/s, Electronic charge = 1.6 × 10–19C)

Solution :E2 = m°

2c2 + p2c2

1= 0.25 + p2c2

p= 0.75 MeV/c

=13

8

0.866 1.6 103 10

= 4.619 × 10–22

=34

22

6.63 104.619 10

hp

= 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° correspondingto 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

ah k l

= 0.22

a= 0.22 2 0.31 nm Ans. 0.3157. 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


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= 1 coshmc

= 2.246 pm

=1240 eV nm20000

= 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 aninsulating container and is then removed. If the temperature of the water is found to be 23°C, then thetemperature 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 ofwater 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 = 70ºCTemperature of aluminium plate

T= 95ºC – 70ºC = 25ºC Ans. 2559. 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

rr = 8

11

a ea e

= 8 (where a is semimajor axis)

11

ee

= 8

e=79


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apehilion of elliptical orbit is given by

2

2 1L

GMm e = ra ...(1)

In case of circular orbit

2

GMmr =

2mVr

GM= V2r ...(2)Lm = Vara ...(3)

From (1), (2), (3)

2 2

2 1a aV r

V r e = ra

rar =

2

2

30 30 71 110 9a

V eV

= 2 Ans. 260. 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 theboiling point to be 1.2 m3/kg.)

Solution :dPdT = 2 1

LT V V

dT= 2 1T V V

dPL

= 5373 1.2 0.1 102270 1000

= 1.97ºC Ans. 1.97

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