iit-jam 2015 solutions

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JAM JOINT ADMISSION TEST FOR MSc 2015

SECTION A: MCQ

Q1 Q10 carry one mark each.

Q1. A system consists of N number of particles, 1>>N . Each particle can have only one of the two energies 1E or ( )01 >+ E . If the system is in equilibrium at a temperature T , the average number of particles with energy 1E is

(a) 2N (b) / 1kT

Ne + (c) / 1kT

Ne + (d)

kTNe /

Ans: (c)

Solution: ( ) ( )1 12 1 E EE EkT kT kTN Ne e N Ne

+ = = =

Q2. A mass m , lying on a horizontal, frictionless surface, is connected to one end of a spring.

The other end of the spring is connected to a wall, as shown in the figure. At 0=t , the mass is given an impulse.

The time dependence of the displacement and the velocity of the mass (in terms of non-

zero constants A and B ) are given by

(a) ( ) ( ) tBtvtAtx cos,sin == (b) ( ) ( ) tBtvtAtx sin,sin == (c) ( ) ( ) tBtvtAtx sin,cos == (d) ( ) ( ) tBtvtAtx cos,cos ==

Ans: (a)

Solution: At time 0t = , the mass m is at rest. Thus, displacement will be zero at time 0t = . ( )sinx A t = Velocity is cos cosdxv A t B t

dt = = =

Thus, sinx A t= and ( ) cosV t B t=

m Impulse





Q3. A particle with energy E is incident on a potential given by

( )0

0, 0, 0

xV x

V xx (in terms of positive constants BA, and k ) is

(a) kxkx BeAe + (b) kxAe (c) ikxikx BeAe + (d) Zero Ans: (b)

Solution: For 0x > ; 22

0 0;2d V E E V

m d + =





Q5. Consider the coordinate transformation 2

,2

yxyyxx =+= . The relation between the

area elements ydxd and dxdy is given by jdxdyydxd = . The value of j is (a) 2 (b) 1 (c) 1 (d) 2

Ans: (c)

Solution: ,2 2

x y x yx y+ = =

dx dy J dxdy = 1 1

1 12 2 11 1 2 22 2

x xx y

Jy yx y

= = = =

Q6. The trace of a 22 matrix is 4 and its determinant is 8 . If one of the eigenvalues is ( )i+12 , the other eigenvalue is

(a) ( )i12 (b) ( )i+12 (c) ( )1 2i+ (d) ( )1 2i Ans: (a)

Solution: ( )1 2 1 22 2 , 2 1 4i i = + = + = and 1 2 8 = Q7. Temperature dependence of resistivity of a metal can be described by

(a) (b)

(c) (d)

Ans: (a)

R

T

R

TR

T

R

T





Solution: Electrical resistivity of metal varies as

5T (For DT > ) where D is the Debye temperature. Thus, correct answer is option (a)

Q8. A proton from outer space is moving towards earth with velocity c99.0 as measured in

earths frame. A spaceship, traveling parallel to the proton, measures protons velocity to

be c97.0 . The approximate velocity of the spaceship in the earths frame, is

(a) c2.0 (b) c3.0 (c) c4.0 (d) c5.0

Ans: (d)

Solution: Velocity of proton w.r.t. spaceship 0.97 c= 0.99 , , 0.97x xu c v v u c = = =

2

0.990.97 0.50.9711

xx

x

u v c vu c v cu v vcc

+ = = = +

Q9. A charge q is at the center of two concentric spheres. The outward electric flux through

the inner sphere is while that through the outer sphere is 2 . The amount of charge contained in the region between the two spheres is

(a) q2 (b) q (c) q (d) q2 Ans: (b)

Solution: 0

q = , 02q q q q

+ = = =

Q10. At room temperature, the speed of sound in air is 340 m/sec. An organ pipe with both

ends open has a length cmL 29= . An extra hole is created at the position 2/L . The lowest frequency of sound produced is

(a) Hz293 (b) Hz586 (c) Hz1172 (d) Hz2344

Ans: (c)

Ev

s s0.99p c=

0.99p c=





Solution: The fundamental frequency in organ pipe with both end open is 2vfL

=

with additional rate at 2L , the fundamental frequency becomes

2340 / sec 117222 29 10

2

v v v mf HzLL L m= = = = =

Q11 Q30 carry two marks each.

Q11. A system comprises of three electrons. There are three single particle energy levels

accessible to each of these electrons. The number of possible configurations for this

system is

(a) 1 (b) 3 (c) 6 (d) 7

Ans: (c)

Solution: For electron spin is 12

. So in one single state two electrons can be adjusted the number

of ways are

Ground First Second

1 2 1 0

2 2 0 1

3 1 2 0

4 1 0 2

5 0 1 2

6 0 2 1

So, number of ways are 6 .

/2L

L





Q12. A rigid and thermally isolated tank is divided into two compartments of equal volumeV ,

separated by a thin membrane. One compartment contains one mole of an ideal gas A

and the other compartment contains one mole of a different ideal gas B . The two gases

are in thermal equilibrium at a temperature T . If the membrane ruptures, the two gases

mix. Assume that the gases are chemically inert. The change in the total entropy of the

gases on mixing is

(a) 0 (b) 2lnR (c) 2ln23 R (d) 2ln2R

Ans: (d)

Solution: For A , number of microstate after mixing is 2

For A , number of microstate before mixing is 1

ln 2 ln1 ln 2AS R R R = = Similarly, forB ln 2BS R =

2 ln 2A BS S S R = + = Q13. A Zener regulator has an input voltage in the range VV 2015 and a load current in the

range of 5 20mA mA . If the Zener voltage is V8.6 , the value of the series resistor should be

(a) 390 (b) 420 (c) 440 (d) 460 Ans: Some data is missing. (No answer is possible)

A B

SR0V

V2015 + V8.6





Q14. The variation of binding energy per nucleon with respect to the mass number of nuclei is

shown in the figure.

Consider the following reactions:

(i) 238 20692 82 10 22U Pb P n + + (ii) 238 206 492 82 28 6U Pb He e + + Which one of the following statements is true for the given decay modes of U23892 ?

(a) Both (i) and (ii) are allowed (b) Both (i) and (ii) are forbidden

(c) (i) is forbidden and (ii) is allowed (d) (i) is allowed and (ii) is forbidden

Ans: (c)

Q15. A rigid triangular molecule consists of three non-collinear atoms joined by rigid rods.

The constant pressure molar specific heat ( )pC of an ideal gas consisting of such molecules is

(a) R6 (b) R5 (c) R4 (d) R3

Ans: (c)

Solution: D.O.F 6= 62RTU = 3V

V

UC RT

= = 4P VC C R R = + = Q16. A satellite moves around the earth in a circular orbit of radius R centered at the earth. A

second satellite moves in an elliptic orbit of major axis R8 , with the earth at one of the

foci. If the former takes 1 day to complete a revolution, the latter would take

(a) 21.6 days (b) 8 days (c) 3 hours (d) 1.1 hour

Ans: (a)

0

12

4

5

6

7

8

9

20 40 60 80 100 120 140 160 180 200 220 240

3

Number of nucleons in nucleus, A

Ave

rage

bin

ding

ene

rgy

per

nucl

eon

(MeV

)





Solution: ( )2 3

3/ 212 1

2

8 228

T R T T daysT R = =

Q17. A positively charged particle, with a charge q , enters a region in which there is a uniform

electric field EG

and a uniform magnetic field BG

, both directed parallel to the positive

y -axis. At 0=t , the particle is at the origin and has a speed 0v directed along the positive x - axis. The orbit of the particle, projected on the zx- plane, is a circle. Let T

be the time taken to complete one revolution of this circle. The y -coordinate of the

particle at Tt = is given by

(a) 22

2qBmE (b) 2

22qBmE (c)

20

2

v mmEqBqB + (d)

qBmv02

Ans: (b)

Solution: 2 2

22

1 1 2 22 2y y

qE m mEy u t a t ym qB qB

= + = =

Q18. Vibrations of diatomic molecules can be represented as those of harmonic oscillators.

Two halogen molecules 2X and 2Y have fundamental vibrational frequencies 1216.7 10Xv Hz= and 1226.8 10Yv Hz= , respectively. The respective force constants

are mNKX /325= and mNKY /446= . The atomic masses of ClF , and Br are 5.35,0.19 and 79.9 atomic mass unit respectively. The halogen molecules 2X and 2Y are

(a) 22 FX = and 22 ClY = (b) 22 ClX = and 22 FY = (c) 22 BrX = and 22 FY = (d) 22 FX = and 22 BrY =

Ans: (b)

Solution: The oscillation frequency of diatomic molecule with reduce mass is

2 21 1

2 4k kf

f = = where k is force constant.

z

y

x0v

,E BJG JG





For 2X molecule: 2x x x

x x

m m mm m

= =+

( ) ( )2 22 2 121 1 325 /

2 2 3.14 16.7 10x

xx

k N mmf Hz = =

27 2759.07 10 35.5 1.67 10 35.5 . . .xm kg kg a mu = = =

This is the atomic mass of chlorine ( )Cl . For 2Y molecule: 2

y y y

y y

m m mm m

= =+

( ) ( ) ( )2 2 22 121 1 446 /

2 2 3.14 26.8 10y

y

y

k N mmf Hz

= =

27 2731.73 10 19 1.67 10ym kg kg = = 19 . . .a mu=

This is the atomic mass of F . Thus, correct answer is option (b)

Q19. A hollow, conducting spherical shell of inner radius 1R and

outer radius 2R encloses a charge q inside, which is located at a

distance ( )1Rd < from the centre of the spheres. The potential at the centre of the shell is

(a) Zero (b) 0

14

qd

(c) 0 1

14

q qd R

(d)

0 2 2

14

q q qd R R

+

Ans: (d)

Solution: 0 1 2

14

q q qVd R R

= +

q d

1R

2R





Q20. Doppler effect can be used to measure the speed of blood through vessels. Sound of

frequency 1.0522 MHz is sent through the vessels along the direction of blood flow. The

reflected sound generates a beat signal of frequency 100 Hz. The speed of sound in blood

is 1545 m/sec. The speed of blood through the vessel, in m/sec, is

(a) 14.68 (b) 1.468 (c) 0.1468 (d) 0.01468

Ans: (d)

Solution: Consider ,b soundV V are velocities of blood cell and sound in blood. The sound of

frequency ( )0f is traveling towards blood cell where blood cell is moving away with velocity bV

Frequency of sound observed on blood cell is

0 sound bsound

V Vf fV

= (i)

Sound from blood cell of frequency f reflect back.

The frequency observed by observer is soundsound b

Vf f

V V = + (ii)

From equation (i) and (ii), we get 0sound b sound

sound sound b

V V Vf f

V V V = +

0 sound bsound b

V Vf fV V = = +

(iii)

Now, 0 0 0 sound bsound b

V Vf f f f fV V = = +

02 b

sound b

VfV V = +

0f

soundV bV

observerbV

f





0

2 bsound b

V fV V f

=+02sound b

b

V V fV f+ = 02 1

soundb

VV

ff

=

Given 601545 / sec, 1.0522 10 , 100soundV m f Hz f Hz= = =

6

1545 1545 0.073210432 1.0522 10 1

100

bV = = = 0.073 / secbV m =

Thus the best suitable answer is option (d).

Q21. Which of the following circuits represent the Boolean expression

PQQRPS ++= (a) (b)

(c) (d)

Ans: (b)

Q22. A conducting wire is in the shape of a regular hexagon, which is

inscribed inside an imaginary circle of radius R , as shown. A current

I flows through the wire The magnitude of the magnetic field at the

center of the circle is

(a) RI

23 0 (b)

RI

320 (c)

RI

03 (d)

RI

23 0

Ans: (c)

Solution: 0 3cos602

d R R= =

PQ S

PQ S

PQ

SR

PQ

SR

C

RI





( )0 2 1sin sin4IBd

= 0 00 0 0

1 2sin 30 2sin 304 3 2 342

I I IBd RR

= = =

0 0 01

3 36 62 3 3

I I IB BRR R

= = = =

Q23. An observer is located on a horizontal, circular turntable which rotates about a vertical

axis passing through its center, with a uniform angular speed of 2 rad/sec . A mass of

10 grams is sliding without friction on the turntable. At an instant when the mass is at a

distance of 8 cm from the axis it is observed to move towards the center with a speed of

6 cm/sec. The net force on the mass, as seen by the observer at that instant, is

(a) N0024.0 (b) N0032.0 (c) N004.0 (d) N006.0

Ans: (c)

Solution: Two forces will act on the particle

First coriolis force 52 ( ) 240 10cF m v N = = (in tangential direction) Another force is centrifugal force 2 5320 10rF m r N = = (in radial direction) Total force 2 2 0.04c c rF F F N= + =

Q24. Miller indicates of a plane in cubic structure that contains all the directions [ ] [ ]011,100 and [ ]111 are (a) ( )011 (b) ( )101 (c) ( )100 (d) ( )110

Ans: (a)

Solution: The name of the plane containing all the directions

[ ] [ ] [ ]100 , 011 & 111 is ( )011 or ( )011 The best suitable answer is option (a)

[ ]011

y [ ]111

[ ]100x

z





Q25. Seven uniform disks, each of mass m and radius r , are inscribed

inside a regular hexagon as shown. The moment of inertia of this

system of seven disks, about an axis passing through the central disk

and perpendicular to the plane of the disks, is

(a) 227 mr (b) 27mr

(c) 22

13 rm (d) 22

55 rm

Ans: (d)

Solution: 2 2 2 2 2

2 54 556 42 2 2 2 2

mr mr mr mr mrmr + + = + =

Q26. A nucleus has a size of m1510 . Consider an electron bound within a nucleus. The

estimated energy of this electron is of the order of

(a)1 MeV (b) MeV210 (c) MeV410 (d) MeV610

Ans: (d)

Solution: 34

1915

6.6 10 6.6 10 / sec10

hp kgm

= = = 2 38

731

44 10 2.4 102 2 9.1 10e

pE Joulem

= = =

712 6

19

2.4 10 1.5 10 1.5 101.6 10

E eV eV MeV

= = =

Q27. Consider a vector field 3 F yi xz j zyk= + JG . Let C be the circle 422 =+ yx on the plane 2=z , oriented counter-clockwise. The value of the contour integral

CF dr JG Gv is

(a) 28 (b) 4 (c) 4 (d) 28 Ans: (a)

Solution: ( ). .C S

F dr F da= JG G JG JG Gv





3

/ / /x y z

F x y zy xz zy

=

JG JG

( ) ( ) ( ) ( )3 3 xz xzyz zyy yF x y zy z z x x y

= + + JG JG

( ) ( ) ( )2 3 3 0 0 1F x z xz y z z = + + JG JG ( ) 2 2 12 7z F x x z= = + +JG JG

da rdrd z= G ( ) ( ) . 2 12 7 . 7F da x x z rdrd z rdrd = + + = JG JG G ( ) 2 2

0 0

. 7 28S

F da rdr d = = JG JG G

Q28. Consider the equation xy

dxdy 2= with the boundary condition ( ) 11 =y . Out of the

following the range of x in which y is real and finite is

(a) 3 x (b) 03 x (c) 30 x (d) x3

Ans: Solution of the differential equation is satisfied by options (c) and (d).

Q29. The Fourier series for an arbitrary periodic function with period L2 , is given by

( )Lxnb

Lxna

axf

n nn n sincos

2 110 == ++= . For the particular periodic function

shown in the figure the value of 0a is

(a) 0 (b) 5.0 (c) 1 (d) 2

Ans: (c)

Solution: The wavefunction of the given function can be written as

( )xf1

2/1

2 1 0 1 2 x





( ) 0 11 0

x xf x

x x<





SECTION B: MSQ Q1 Q10 carry two marks each. Q1. For an electromagnetic wave traveling in free space, the electric field is given by

( )mVjkxtE 10cos100 8 +=G . Which of the following statements are true?

(a) The wavelength of the wave in meter is 6 (b) The corresponding magnetic field is directed along the positive z direction

(c) The Poynting vector is directed along the positive z direction

(d) The wave is linearly polarized

Ans: (a) and (d)

Solution: ( )8 100cos 10 /E t kx j V m= +G Option (a) is true

88 8

8

2 2 3 1010 10 610

c = = = =

Option (b) is wrong

( ) ( ) B k E x y z JG JG Option (c) is wrong

S k x JG Option (d) is true

Q2. In an ideal Op-Amp circuit shown below 1 23 , 1R k R k= = and 0.5siniV t= (in Volt). Which of the following statements are true?

(a) The current through =1R The current through 2R (b) The potential at P is

1

20 RRV

(c) The amplitude of 0V is V2

(d) The output voltage 0V is in phase with iV

Ans: (b), (c) and (d)

Option (a) is wrong

P

iV

1R

0V

2R

+





Current through 1R is 11

o iV VIR= and Current through 2R is 2

2

iVIR

=

Option (b) is true

The potential at P is 1

20 RRV (voltage divider rule)

Option (c) is true

20

1

31 1 0.5sin 2sin1i

RV V t tR

= + = + = 2mV V =

Option (d) is true

Q3. A particle of mass m is moving in yx plane. At any given time t , its position vector is given by ( ) cos sinr t A t i B t j = +G where BA, and are constants with BA . Which of the following statements are true?

(a) Orbit of the particle is an ellipse

(b) Speed of the particle is constant

(c) At any given time t the particle experiences a force towards origin

(d) The angular momentum of the particle is kABm Ans: (a), (c) and (d)

Solution: (a) ( ) cos sinr t A t i B t j = +K cos , sinx A t y B t = = cos , sinx yt tA B

= = 2 2

2 2 1x yA B

+ = (Ellipse)

(b) sin cosdr A t i B t jdt

= +G

Speed 2 2 2 2 2 2sin cosdr A t B tdt

= = +G

. Speed is function of time, so not constant.

(c) 2

2 22

cos sindr A t i B t jdt

= G

2 r= G . Force act towards origin.





(d) ( )L r p= G G

cos sin 0sin cos 0

i j km A t B t

A t B t

=

L m ABk =

Q4. A rod is hanging vertically from a pivot. A partic1e traveling in horizontal

direction, collides with the rod as shown in the figure. For the rod-particle

system, consider the linear momentum and the angular momentum about

the pivot .Which of the following statements are NOT true?

(a) Both linear momentum and angular momentum are conserved

(b) Linear momentum is conserved but angular momentum is not

(c) Linear momentum is not conserved but angular momentum is conserved

(d) Neither linear momentum nor annular momentum are conserved

Ans: (b), (c) and (d)

Q5. A particle is moving in a two dimensional potential well

( ), 0, 0 , 0 2

, elsewhereV x y x L y L=

= which of the following statements about the ground state energy 1E and ground state eigenfunction 0 are true? (a) 2

22

1 mLE == (b)

2 2

1 2

58

EmL= =

(c) 02 sin sin

2x y

L L L = (d)

Ly

Lx

L 2coscos20

= Ans: (b) and (c)

Solution: 222 2

2 22 4yx

n

nnEm L L

= + =

Ground state 1, 1x yn n= =2 2 2 2

2 2 2

1 1 52 4 8x

Em L L mL

= + = = =

Wave function 2 2 sin sin2 2

x yL L L L

=





Q6. Consider the circuit, consisting of an AC function generator ( ) vtVtV 2sin0= with VV 50 = an inductor mHL 0.8= , resistor = 5R and a capacitor FC 100= . Which of

the following statements are true if we vary the frequency?

(a) The current in the circuit would be maximum at 178Hz = (b) The capacitive reactance increases with frequency

(c) At resonance, the impedance of the circuit is equal to the resistance in the circuit

(d) At resonance, the current in the circuit is out of phase with the source voltage

Ans: (a) and (c)

Solution: Option (a) is true

( )( )3 61 1 178

2 2 3.14 8 10 100 10Hz

LC = = =

Option (b) is wrong

1CX C= CX as

Option (c) is true

Option (d) is wrong

Q7. Muons are elementary particles produced in the upper atmosphere. They have a life time

of s2.2 . Consider muons which are traveling vertically towards the earths surface at a speed of c998.0 . For an observer on earth, the height of the atmosphere above the

surface of the earth is km4.10 . Which of the following statements are true?

(a) The muons can never reach earths surface

(b) The apparent thickness of earths atmosphere in muons frame of reference is km96.0

(c) The lifetime of muons in earths frame of reference is s8.34 (d) Muons traveling at a speed greater than c998.0 reach the earths surface

Ans: (c) and (d)

R

L

C





Solution: 02

21

ttvc

= ( )

66

2

2.2 10 34.8 10 sec1 0.998

t

= =

Now distance will be 6 834.8 10 0.998 3 10 10.4192t v km= = = Apparent thickness 6 82.2 10 0.998 3 10 0.658X t v km = = =

Q8. As shown in the VP diagram AB and CD are two isotherms at temperatures 1T and 2T , respectively ( )21 TT > . AC and BD are two reversible adiabats. In this Carnot cycle,

which of the following statements are true?

(a) 1 21 2

Q QT T

=

(b) The entropy of the source decreases

(c) The entropy of the system increases

(d) Work done by the system 21 QQW = Ans: (a), (b) and (d)

Q9. The following figure shows a double slit Fraunhofer diffraction pattern produced by two

slits, each of width a separated by a distance bab





sin sin na na = = where a is the width of slit.

Reducing a increases the separation between diffraction minima i.e. increases the

separation between consecutive primary maxima.

The condition of interference maxima is

sin sin mb mb = = where b is the separation between slits.

The position of interference maxima gives the separation between secondary maxima.

Reducing b increases the separation between consecutive secondary maxima.

The correct answer is option (a) and (d).

Q10. A unit cube made of a dielectric material has a polarization jiP 43 +=G units. The edges of the cube are parallel to the Cartesian axes. Which of the following statements are true?

(a) The cube carries a volume bound charge of magnitude 5 units

(b) There is a charge of magnitude 3 units on both the surfaces parallel to the zy plane (c) There is a charge of magnitude 4 units on both the surfaces parallel to the zx plane (d) There is a net non-zero induced charge on the cube

Ans: (b) and (c)

Solution: 3 4P i j= +JG Option (a) is wrong

. 0b P = =JG JG

Option (b) is true

At 0x = , ( ) ( ) . 3 4 . 3b P n i j i = = + = JG At 1x = , ( ) ( ) . 3 4 . 3b P n i j i = = + =JG Option (c) is true

At 0y = , ( ) ( ) . 3 4 . 4b P n i j j = = + = JG At 1y = , ( ) ( ) . 3 4 . 4b P n i j j = = + =JG Option (d) is wrong





SECTION C: NAT

Q1 Q10 carry one mark each.

Q1. The power radiated by sun is W26108.3 and its radius is km5107 . The magnitude of

the Poynting vector (in 2cmW ) at the surface of the sun is _________________

Ans: 6174

Solution: ( )26

2 210

3.8 10 / 6174 /4 7 10

PI W cm W cmA

= = =

Q2. A particle is in a state which is a superposition of the ground state 0 and the first excited state 1 of a one-dimensional quantum harmonic oscillator. The state is given by

10 52

51 += . The expectation value of the energy of the particle in this state (in

units of ,= being the frequency of the oscillator) is ________ Ans: 1.3

Solution: 12n

E n = + = and 1

2 5P =

= , 3 42 5

P = =

1 3 4 13 1.32 5 2 5 10

E = + = == = = = Q3. In an experiment on charging of an initially uncharged capacitor, an RC circuit is made

with the resistance = kR 10 and the capacitor FC 1000= along with a voltage source of V6 . The magnitude of the displacement current through the capacitor (in A ), 5 seconds after the charging has started, is _______________

Ans: 364

Solution: 3 6/ 5/10 10 1000 10 5/10

3 4 44

6 6 6 6 36410 10 10 1.65 1010

t RCVI e e e AR e

= = = = = =





Q4. In the given circuit 10CCV V= and 100= for npn transistor. The collector voltage CV (in volts) is __________.

Ans: 5.7

Solution: 535 0.7 4.3 10 4.3

100 10B C BI A I I mA= = = =

10 4.3 5.7C CC C CV V I R V = = = Q5. Unpolarized light is incident on a calcite plate at an angle of incidence o50 as shown in

the figure. Take 6584.10 =n and 1.4864en = for calcite. The angular separation ( in degrees) between the two emerging rays within the plate is

Ans: 3.51

Solution: Inside the crystal incident light split into two components, ordinary ray and extra-

ordinary ray

According to Snells law sinsin

i nr=

For ordinary ray 050 , 1.6584oi n= = 1sin sinsin sino o

o o

i ir rn n

= =

K100

+

K1

CCV

CV

V5

050

Calciteaxis Optic

Air

050i =

or

er

- rayo- raye





[ ]01 1 1sin 50 0.766sin sin sin 0.4621.6584o o

rn

= = = 0

0 27.51r =

For extra-ordinary ray 050 , 1.4864ei n= = 1sin sinsin sine e

e e

i ir rn n

= =

[ ]01 1 1sin 50 0.766sin sin sin 0.5151.4864e e

rn

= = = 031.02er =

Thus, the angular separation between the o - ray and e - ray is 03.51e or r = = Q6. In the hydrogen atom spectrum. the ratio of the longest wavelength in the Lyman series

(final state 1=n ) to that in the Balmer series (final State 2=n ) is ____________ Ans: 0.185

Solution: According to Bohr Theory 2 21 1 1

L f i

Rn n

=

The longest wavelength in the Lyman series is

2

1 1 1 3 41 2 4 3LL

R RR

= = =

The longest wavelength in the Balmer series is

2 21 1 1 1 1 9 4

2 3 4 9 36BR R R

= = = 1 5 36

36 5BBR

R

= =

4 5 5 0.1853 36 27

L

B

RR

= = =

Q7. A rod is moving with a speed of c8.0 in a direction at o60 to its own length. The

percentage contraction in the length of the rod is __________

Ans: 9

Solution: 2

0 21x xvl lc

= ( )20 cos 1 0.8l = 0 01 0.6 0.32xl l l = = and 003

sin2y

ll l = =

3n =2n =

1n =L

H





New length ( )2

2 00 0 0

3 30.3 0.09 0.9162 4ll l l l

= + = + =

% change in length ( ) 0

0

1 0.91100 0.09 100 9%

ll

= =

Q8. X rays of wavelength 0.24 nm are Compton scattered and the scattered beam is observed at an angle of o60 relative to the incident beam. The Compton wavelength of

the electron is nm00243.0 . The kinetic energy of scattered elections in eV

is___________

Ans: 25

Solution: 0.24 , 0.00243Cnm = = and 060 = ( ) ( )1 cos 1 cosC C = = +

1 10.24 0.00243 1 0.24 0.00243 0.24 0.00121 0.24122 2

nm = + = + = + = Kinetic Energy of scattered electron

34 8 91 1 1. . 6.6 10 3 10

0.24 0.2412 10hc hcK E Joules

= =

( )26 26 209 919.8 10 19.8 10. . 4.17 4.15 0.02 396 1010 10K E Joules

= = =

20

19

396 10. . 24.751.6 10

K E eV eV

= =

Q9. A diode at room temperature ( )eVkT 025.0= with a current of A1 has a forward bias voltage VVF 4.0= . For VVF 5.0= , the value of the diode current (in A ) is _________

Ans: 54.5

Solution: ( ) ( )( )( )( )

( )( )

2

1

/ 0.5/ 0.025 20/ 2

0 20.4 / 0.025 16/1

1 1 11 54.5 54.5

1 11

T

T

T

V VV V

V V

e e eII I e I AI e ee

= = = = = = Q10. GaAs has a diamond structure. The number of Ga-As bonds per atom which have to be

broken to fracture the crystal in the ( )001 plane is _______ Ans: 4





Solution: Diamond structure has tetrahedral bond. To fracture the diamond structure along

( )0 0 1 plane, four bonds need to be broken. Q.11 Q.20 Carry two marks each.

Q11. In the thermodynamic cycle shown in the figure, one mole of a monatomic ideal gas is

taken through a cycle. AB is a reversible isothermal expansion at a temperature of K800

in which the volume of the gas is doubled. BC is an isobaric

contraction to the original volume in which the temperature is

reduced to K300 . CA is a constant volume process in which

the pressure and temperature return to their initial values. The

net amount of heat (in Joules) absorbed by the gas in one

complete cycle is _____________

Ans: 452

Solution: Process A B is isothermal expansion 800 , ,A A AT K V P= and 800 , 2 , 2

AB B A B

PT K V V P= = = ProcessB C is isobaric

2A

C BP

P P= = , C AV V= , 300CT K= C A is Isochoric

1 ln 4602BAA

VQ nRT JV = =

( ) ( )2 300 8001 1Pn R TQ nC T R

= = = = 10344J

( ) ( )3 800 3001RQ = ( ) 500 61941

R J= =

Total heat exchange is 1 2 3 452Q Q Q+ + =

2P

V V2

1P

C B

A

Volume

Pres

sure





Q12. In a region of space, a time dependent magnetic field ( ) ttB 4.0= Tesla points vertically upwards. Consider a horizontal, circular loop of radius 2cm in this region. The

magnitude of the electric field (in mmV / ) induced in the loop is ___________.

Ans: 4

Solution: 2

2 2 102 0.4 4 /2 2

B r BE r r E mV mt t

= = = =

JG JG

Q13. A plane electromagnetic wave of frequency Hz14105 and amplitude 310 /V m traveling in a homogeneous dielectric medium of dielectric constant 1.69, is incident normally at

the interface with a second dielectric medium of dielectric constant 2.25. The ratio of the

amplitude of the transmitted wave to that of the incident wave is __________.

Ans: 0.93

Solution: 1010 01 2 0 1 2

22 2 1.69 0.931.69 2.25

rTT I

I r r

EnE En n E

= = = = + + +

Q14. For the arrangement given in the following figure, the coherent light sources ,A B and C

have individual intensities of 2 22 / , 2 /mW m mW m and 25 /mW m respectively at pointP .

The wavelength of each of the sources is 600 nm . The resultant intensity at point P

(in 2/mW m ) is ___________.

Ans: 29.23 /mw m

Solution: The electric field on the screen is the sum of the fields

produced by the slits individually.

1 2 3E E E E= + + i iaA Ae Be = + +

where 2 sind =

mm22.3

mm04.2C

B

A

P

mm15

m1

ad

dA

B

C

D O

y

p





The total intensity at is ( ) ( )* 2 2 22 2 cos 2 cos cos 1I EE A B A AB a a = = + + + +

where 2 2 2sind d d yD

= = 3 3

7

3.22 10 15 102 505.76 10 1

= =

0145.8 = given, 2 2 22 / , 5 / , 3.22 , 2.04 , 0.6335A mw m B mw m d mm ad mm a mm= = = = =

( ) ( )3 3 3 32 2 10 5 10 2 2 10 cos 2 2 5 10 cos cos 1I a a = + + + + 3 29.23 10 /w m=

29.23 /I mw m= Q15. One gram of ice at Co0 is melted and heated to water at Co39 . Assume that the specific

heat remains constant over the entire process. The latent heat of fusion of ice is

80 Calories/gm. The entropy change in the process (in Calories per degree) is _________.

Ans: 0.39

Solution: 11 80273

MLST

= = , 3022 273dTS MCT

= 2 3021.1ln 273S = 1 2S S S = + 80 3021.1ln 0.29 0.1 0.39273 273S = + = + =

Q16. A uniform disk of mass m and radius R rolls, without slipping, down a fixed plane

inclined at an angle o30 to the horizontal. The linear acceleration of the disk (in 2sec/m )

is _____________.

Ans: 3.266

Solution: Equation of Motion sinmg f ma = Torque fR I= =

2

sin , ,2

I a mRmg ma IR R = = =

3.2663ga = =





Q17. A nozzle is in the shape of a truncated cone, as shown in the figure.

The area at the wide end is 225cm and the narrow end has an area

of 21 cm . Water enters the wider end at a rate of sec/500 gm . The

height of the nozzle is cm50 and it is kept vertical with the wider

end at the bottom. The magnitude of the pressure difference in kPa

( 23 /101 mNkPa = ) between the two ends of the nozzle is __________ Ans: 17.5

Solution: According to Bernoullis equation

2 21 12 2b b b t t t

P gh V P gh V + + = + +

( ) ( )2 212b t t b t bP P g h h V V = + Now given 500 / sect tAV gm =

3 3

3 4 2

500 10 / sec 500 10 / sec1000 / 10t t

kg kgVA kg m m

= =

5 / sectV m = According to equation of continuity

tt t b b b t

b

AAV AV V VA

= =2

2

1 5 / sec 0.2 / sec25bcmV m mcm

= =

( )22 211000 10 50 10 1000 5 0.22b t

P P P = = +

( )5000 500 25 0.04P = + 25000 12480 17480 /N m= + = 17.5 aP kP = Q18. A block of mass 2 kg is at rest on a horizontal table The coefficient of friction between

the block and the table is 0.1. A horizontal force 3N is applied to the block The speed of

the block (in m/s) after it has moved a distance 10m is ________________.

Ans: 3.225

Solution: 0.1 2 10 2rf N N= = = 2m kg=

cm50

50t bh h cm =

, ,t t tP V A

, ,b b bP V A





Applied force is more than friction

3 2 1ma F N= = = 21 1 0.5 /2

a m sm

= = =

2 2v u as= + 2 2 0.5 10 10 3.225 /v as m s = = = = 0, 10u s m= =

Q19. A homogeneous semi-circular plate of radius mR 3= is shown in the figure. The distance of the center of mass of the p1ate (in meter) from the point O is _______.

Ans: 1.3

Solution: In problem 3R m= The area of the shaded part is rdr . The area of the plate is 2 / 2R . As the plate is uniform, the mass per unit area is 2 / 2

MR . Hence the mass of the

semicircular element

( )2 22/ 2M MrdrrdrR R

= The y - coordinate of the centre of mass of this wire is 2 /r . The y - coordinate of the centre of the plate is, therefore,

3

2 20

1 2 2 1 4 4 4 1.33 3

R r Mr M R RY drM R M R

= = = = = The x - coordinate of the centre of mass is zero by symmetry.

Q20. Consider a m20 diameter np junction fabricated in silicon. The donor density is 1610 per 3cm . The charge developed on the n side is C13106.1 . Then the width

(in m ) of the depletion region on the n side of the np junction is _________. Ans:

0 m3

r dr+ rR

iit-jam 2015 solutions

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