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Mathematical Methods

IIT-JAM 2005

Q1. Which of the following is INCORRECT for the matrix0 1

1 0M

(a) It is its own inverse (b) It is its own transpose

(c) It is non-orthogonal (d) It has eigen values ± 1

Ans. : (c)

Solution: The inverse of the given matrix is 1 0 11

1 0M

M

0 1 0 11

1 0 1 01M

Thus the given matrix is its own inverse.

The transpose of M is, 0 1

1 0TM M

The given matrix is orthogonal as each row vector is a unit vector and the two rows are

orthogonal.

The eigenvalues of orthogonal matrix are 1 or 1 . For the given matrix

1 2 10 1 and 2 1

Thus option (c) is correct option.

Q2. A periodic function can be expressed in a Fourier series of the form,

0

sincosn

nn nxbnxaxf . The functions 21 cosf x x and 2

2 sinf x x are

expanded in their respective Fourier series. If the coefficients for the first series are 1na

and 1nb , and the coefficients for the second series are 2

na and 2nb , respectively, then

which of the following is correct?

(a)

2

1

2

1 22

12

banda (b)

2

1

2

1 22

12

aandb

(c)

2

1

2

1 22

12

aanda (d)

2

1

2

1 22

12

bandb

Ans. : (c)




Solution: 1

1 1cos 2

2 2f x x and 2

1 1cos 2

2 2f x x

Hence, 12

1

2a and 2

2

1

2a

All the nb ’s of each of the series are zero. As there is no sine terms in any of the two

given functions.

Thus the correct option is (c).

IIT-JAM 2006

Q3. The symmertric part of 2a

P a bb

is

(a)

2112

2

2

bbabaa (b)

22

bbbaa

(c) 2

1 1

1

a a b a

b a b

(d) 2

2 1

1

a a b a

b a b

Ans. : (d)

Solution: The given matrix can be written as 2

22

2

a a abaP a b

b a bb

The transpose of P is,

2

2 2T a a b aP

a b b

Hence the symmetric part of P is,

2

2 2 2 21

2 2 2 2 2

T a a ab bP P

ab b b

2

2 1

1

a a b a

b a b

Hence the correct option is (d).




IIT-JAM 2007

Q4. 5 7

157 3

xx y

y

The matrix equation of above represents

(a) a circle of radius 15 (b) an ellipse of semi major axis 5

(c) an ellipse of semi major axis 5 (d) a hyperbola

Ans. : (b)

Solution: 5 7

157 3

xx y

y

5 715

7 3

x yx y

x y

2 25 7 7 3 15x xy xy y 2 25 3 15x y 2 2

13 5

x y

Thus the given equation represents an ellipse with semi-major axis 5 .

Q5. f x is a periodic function of x with a period of 2 . In the interval ,x f x

is given by

xx

xxf

0,sin

0,0

In the expansion of f x as a Fourier series of sine and cosine functions, the coefficient

of cos 2x is

(a) 32

(b) 1

(c) 0 (d) 32

Ans. : (d)

Solution: The coefficients of cos 2x is 2a .

Thus, 0

2

0

10 cos 2 sin cos 2a xdx x xdx

2

0

1sin 3 sin

2a x x dx

0

1 cos3cos

2 3

xx

1 cos3 1

cos 12 3 3

x

1 1 1 21 1

2 3 3 3




IIT-JAM 2008

Q6. The product PQ of any two real, symmetric matrices P and Q is

(a) symmetric for all P and Q (b) never symmetric

(c) symmetric, if PQ QP (d) anti-symmetric for all P and Q

Ans. : (c)

Solution: A matrix is symmetric, if its transpose is equal to the matrix itself.

Hence for the matrix PQ , T T TPQ Q P (since T T TAB B A )

Since, Q and P are symmetric matrices; ,T TQ Q P P

Hence, TPQ QP

It is easily seen that TPQ will be equal to PQ , only if QP PQ . Hence (c) is correct

option.

Q7. The work done by a force in moving a particle of mass m from any point ,x y to a

neighboring point ,x dx y dy is given by 22dW xydx x dy . The work done for a

complete cycle around a unit circle is

(a) 0 (b) 1 (c) 3 (d) 2

Ans. : (a)

Solution: Let us write the co-ordinates x and y as,

1 cos , 1 sinx y cos and sinx y .

Thus, sindx d and cosdy d

Thus, the given work dW can be written as,

22 cos sin sin cos cosdW d d

2 32sin cos cosd d

Thus the total work done along the complete circle is

2 2

2 3

0 0

2 sin cos cosW d d

It can be easily checked that the value of each of these integrals is 0 . Hence, the correct

option is (a).

1r ,x y

unit circle




IIT-JAM 2009

Q8. In the Fourier series of the periodic function (shown in the figure)

0

sincos

sin

nnn nxnx

xxf

Which of the following coefficients are non-zero?

(a) n for odd n (b) n for even n

(c) n for odd n (d) n for even n

Ans. : (b)

Solution: The given function is an even function (assuming the basic interval of definition to be

symmetric about the origin)

Hence, all the nB s are 0 .

0

2sin cosn x nx dx

0

2sin cosx nx dx

0

2sin 1 sin 1

2n x n x dx

0

cos 1 cos 11

1 1

n x n x

n n

For odd n ,

1 1 1 1 1

1 1 1 1n n n n n

0n , for odd n .

For even n ,

1 1 1 1 1

1 1 1 1n n n n n

2 1 1

1 1n n n

2

2 1 1

1

n n

n

2

4

1n

Thus for even n , n is nonzero. Hence the correct option is (b).

2




IIT-JAM 2010

Q9. A matrix is given by11

12

iM

i

. The eigenvalues of M are

(a) real and positive (b) purely imaginary with modulus 1

(c) complex with modulus 1 (d) real and negative

Ans. : (c)

Solution: We know that if is an eigenvalue of matrix A , then k is the eigenvalue of matrix

kA . Hence Let as evaluate the eigenvalue of matrix

1

1

iM

i

For the calculation of eigenvalues

1

01

i

i

2

1 0 1i i

1, 1i i

Thus the eigenvalues of the given matrix M are

1

1 1 11

2 2 2i i and 2

1 1 11

2 2 2i i

We see that 1 2 1 . Thus the correct option is (c).

Q10. The equation of a surface of revolution is 22

2

3

2

3yxz . The unit vector normal to

the surface at the point

1,0,

3

2A is

(a) ki ˆ10

2ˆ5

3 (b) ki ˆ

10

2ˆ5

3 (c) ki ˆ

5

2ˆ5

3 (d) ki ˆ

10

2ˆ10

3

Ans. : (b)

Solution: 2 2 2 2 2 2 2 23 3 3 33 3 2 0

2 2 2 2z x y z x y x y z

Let 2 2 23 3 2V x y z , taking gradient ˆ ˆ ˆ6 6 4V xx yy zz

.




The unit normal to the surface at the point 2

,0,13

A

is ˆV

nV

. Thus

2 2ˆ ˆ ˆˆ ˆ6 6 0 4 1 6 4

3 23 3ˆ ˆ ˆ52 40 10

36 163

x y z x zn x z

IIT-JAM 2011

Q11. The line integral B

A

ldF

, where yyx

yx

yx

xF ˆˆ

2222

, along the semi-circular path as

shown in the figure below is

(a) -2 (b) 0 (c) 2 (d) 4

Ans. : (b)

Solution: 2 2 1x y xdx ydy and ˆ ˆdl dxx dyy

2 2 2 2. 0

xdx ydyF dl

x y x y

0

B

A

F dl

xdx ydy

Q12. Given two n n matrices P and Q such that P is Hermitian and Q is skew (anti)-

Hermitian. Which one of the following combinations of P and Q is necessarily a

Hermitian matrix?

(a) QP ˆˆ (b) QPi ˆˆ (c) QiP ˆˆ (d) QP ˆˆ

Ans.: (c)

Solution: Any matrix is hermitian if its conjugate transpose is equal to the matrix itself.

For, ˆPQ , we have * * *ˆ ˆ ˆ ˆˆ ˆ ˆ ˆPQ Q P Q P QP

Thus, ˆPQ is not hermitian.

For matrix ˆîPQ , we have *ˆ ˆ ˆ ˆˆ ˆ ˆ îPQ i PQ i Q P iQP

x 1,0B 1,0A

y




Thus, ˆîPQ is not hermitian.

For matrix ˆP iQ , we have

* **ˆ ˆˆ ˆP iQ P iQ *ˆ ˆ ˆˆ ˆ ˆP i Q P i Q P iQ

Thus ˆP iQ is hermitian.

For ˆP Q , we have * **ˆ ˆ ˆˆ ˆ ˆP Q P Q P Q

Thus, ˆP Q is not hermitian.

Note: In this question “*” symbol has been used to denote the conjugate transpose of a

matrix.

IIT-JAM 2012

Q13. If F

is a constant vector and r

is the position vector then rF would be

(a) Fr

(b) F

(c) rF

(d) Fr

Ans.: (b)

Solution: Let 0 ˆ ˆ ˆF F x y z

and ˆ ˆ ˆr xx yy zz

0.F r F x y z

.

Thus 0 ˆ ˆ ˆF r F x y z F

IIT-JAM 2013

Q14. The inverse of the matrix

001

100

110

M is

(a) IM (b) IM 2 (c) 2MI (d) MI

where I is the identity matrix.

Ans.: (b)

Solution: Given

0 1 1

0 0 1

1 0 0

M




The characteristics equation is,

1 1

0 1 0

1 0

2 0 1 1 0 3 31 0 1 0

Thus the cayley-Hamilton theorem gives

3 0M M I

Multiply both sides by 1M gives

2 1 1 20M I M M M I . Thus option (b) is correct option.

Q15. The value of ii , where ,1i is

(a) 0 (b) 2

1 (c) 2 (d) 2

Ans.: (a)

Solution: i i i i

i ii i

2 2 2i i i i

i i

0i i i i

i i i i

Q16. The solution of the differential equation 0,,, dyyxyzdxyxxzyxdz is………

Ans.: 2 2 / 2x yCe

Given differential equation can be written as,

, , 0dz x y z x y xdx ydy dz

xdx ydyz

Integrating both sides gives

2 2

ln ln2 2

x yz c

2 2

ln2

x yz

c

2 2 / 2x yze

c

2 2 / 2x y

z ce

.




Q17. Given that ,11,11 ff and ,11" f the value of 21f is …………..

Ans.: 0.606

Solution: Let xf x ke

In order to satisfy each of the three given conditions 1

ke

.

Thus xe

f xe

Hence, 1/ 2 1

1/ 2 0.606e

fe e

.

IIT-JAM 2014

Q18. For vectors kjibkja ˆ5ˆ3ˆ2,ˆˆ

and kjc ˆˆ

, the vector product cba

is

(a) in the same direction as c

(b) in the direction opposite to c

(c) in the same direction as b

(d) in the direction opposite to b

Ans.: (a) ˆˆ ˆ

ˆ ˆˆ ˆ ˆ ˆ2 3 5 3 5 2 0 2 0 2 2 2

0 1 1

i j k

b c i j k i j k

ˆˆ ˆ

ˆ ˆˆ ˆ ˆ0 1 1 2 2 0 2 0 2 2 2 2

2 2 2

i j k

a b c i j k j k c

Q19. The value of

0

sinn

n nr for r = 0.5 and 3

is

(a) 3

1 (b)

3

2 (c)

2

3 (d) 3

Ans.: (a)

Solution: 2 3

0

sin 0 sin sin 2 sin 3n

n

r n r r r

Let, cos siniZ re r i

2 2 2 2 cos 2 sin 2iZ r e r i and so on.




Thus, we can see,

0 0

sin Img part ofn n

n n

r n z

0 1 1

in

in

z rez

z re

/ 3

/3/ 2

/ 2

12

i

i

e

e

0 0

0 0

1/ 2 cos 60 sin 60

11 cos 60 sin 60

2

i

i

1 3 1 31/ 22 2 4 4

3 31 1 31

4 42 2 2

i i

ii

1 3 3 31 3

3 3 3 3 3 3

i ii

i i i

0

3 3 3 3 3 4 3

12 12 12n

n

i i iz

Thus, 0

1sin

3n

n

r n

Q20. If the surface integral of the field kzjyixzyxA ˆ3ˆˆ2,,

over the closed

surface of an arbitrary unit sphere is to be zero, then the relationship between , and

is

(a) 06/ (b) 02/6/3/

(c) 03/2/ (d) 0/3/1/2

Ans.: (b)

Solution: It is given that . 0S

A da 0

V

A d

(From Divergence Theorem)

. 0 2 3 0 03 6 2V

A d

Q21. The line integral ldA

of a vector field 2

1 ˆ ˆ,A x y yi xjr

where 2 2 2r x y is

taken around a square (see figure) of side of unit length and centered at 00 , yx with

2

10 x and .

2

10 y If the value of the integral is L, then

00 , yx

x

y




(a) L depends on 00 , yx

(b) L is independent of 00 , yx and its value is -1

(c) L is independent of 00 , yx and its value is 0

(d) L is independent of 00 , yx and its value is 2

Ans.: (c)

Solution:

2 2 2 2

ˆ ˆ ˆ

0

x y z

Ax y z

y x

x y x y

2 2 2 2ˆ ˆ ˆ= 0 0 0 0

x yx y z

x x y y x y

2 2 2 2 2 2 2 2

2 2 22 2 2 2 2 2

2 2ˆ ˆ 0

x y x x x y y y y x x yA z z

x y x y x y

Thus, 0A dl

.




IIT-JAM 2015

Q22. Consider the coordinate transformation2

,2

yxy

yxx

. 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 21

1 1 2 2

2 2

x x

x yJ

y y

x y

Q23. The trace of a 22 matrix is 4 and its determinant is 8 . If one of the eigenvalues is

i12 , the other eigenvalue is

(a) i12 (b) i12 (c) 1 2i (d) 1 2i

Ans.: (a)

Solution: 1 2 1 22 2 , 2 1 4i i and 1 2 8

Q24. Consider a vector field 3 ˆˆ ˆF yi xz j zyk

. Let C be the circle 422 yx on the

plane 2z , oriented counter-clockwise. The value of the contour integral CF d r is

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

Ans.: (a)

Solution: . .C S

F dr F da

3

ˆ ˆ ˆ

/ / /

x y z

F x y z

y xz zy

3 3

ˆ ˆ ˆxz xzyz zyy y

F x y zy z z x x y

2 3ˆ ˆ ˆ3 0 0 1F x z xz y z z

x

y

2




ˆ ˆ2 2 12 7z F x x z

ˆda rdrd z

ˆ ˆ ˆ. 2 12 7 . 7F da x x z rdrd z rdrd

2 2

0 0

. 7 28S

F da rdr d

Q25. Consider the equation x

y

dx

dy 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.: (d)

Solution: x

y

dx

dy 2

2

1ln

dy dxx C

y x y

11 y1

ln1 11

C C 1

ln 1xy

1

1 lny

x

At 0,x y and ln x is not defined for negative values of x .

Thus, correct option is (d).

Q26. The Fourier series for an arbitrary periodic function with period L2 , is given by

L

xnb

L

xna

axf

n nn n

sincos

2 11

0

. 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

0 1

1 0

x xf x

x x

Coefficient 0a is defined as

xf

1

2/1

2 1 0 1 2x




0 1

0 1 01 1a x dx x dx

1 2 22 2

1 0

1 1 1 10 0 1

2 2 2 2 2 2

x x

0 1a

Q27. The phase of the complex number 1 i i in the polar representation is

(a) 4

(b)

2

(c)

4

3 (d)

4

5

Ans.: (c)

Solution: 1 1z i i z i for z x iy

1 3tan 1 tan ( 1)

4

y

x

IIT-JAM 2016

Q28. Which of the following points represent the complex number1

1 i

?

(a) (b)

(c) (d)

Ans.: (a)

Solution: 1 1 1 1 1 1

1 1 1 1 1 2 2

i ii

i i i

y

1

0.5

0.51

1

0.50.5 1

x

y

1

0.5

0.51

1

0.50.5 1

x

y

1

0.5

0.51

1

0.50.5 1

x

y

1

0.5

0.51

1

0.50.5 1

x




Q29. The eigenvalues of the matrix representing the following pair of linear equations

0x iy 0ix y

are

(a) 1 , 1i i (b) 1 , 1i i (c) 1, i (d) 1 , 1i i

Ans.: (d)

Solution: Characteristic equation is 0A I

2 2210 1 1 1 0 1

1

ii i

i

1 , 1i i

Q30. For the given set of equations

1x y , 1y z , 1x z ,

which one of the following statements is correct?

(a) Equations are inconsistent

(b) Equations are consistent and a single non-trivial solution exists

(c) Equations are consistent and many solutions exist

(d) Equations are consistent and only a trivial solution exists.

Ans.: (b)

Solution: The augmented matrix of the system can be written as

1 1 0 1

0 1 1 1

1 0 1 1

M

Row reduction gives

1 1 0 1 1 1 0 1

0 1 1 1 0 1 1 1

0 1 1 0 0 0 2 1

M

Thus, 1x y , 1y z and 2 1z

The last equation gives 1/ 2z . Using first two equations we find 1/ 2x y . Thus the

system has a single non trivial solution. The correct option is (b)




Q31. The tangent line to the curve 2 5 0x xy at 1,1 is represented by

(a) 3 2y x (b) 3 4y x

(c) 3 2x y (d) 3 4x y

Ans.: (b)

Solution: Given 2 5 0x xy 2 0dy

x y xdx

2x ydy

dx x

At 31,1 , 3

1

dy

dx

Hence the equation of tangent line is 1 3 1 3 4y x y x

Q32. Fourier series of a given function f x in the interval 0 to L is

01 1

2 2cos sin

2 n nn n

a nx nxf x a b

L L

.

If f x x in the region 20, ,b ………………

Ans.: 0 5

Solution: Here, 2 / 2l l

2

0

2 2sin

/ 2

xb x dx

0

2sin 4x xdx

0

2 cos 4 1sin 4

4 16

x xx

2 cos 1sin 4 0

4 16

2 10 5

4 2

.

Q33. Consider a function 3 3,f x y x y , where y represents a parabolic curve 2 1x . The

total derivative of f with respect to x , at 1x is………………….

Ans.: 27

Solution: 3 3,f x y x y . Also given that, 2 1y x

Hence, 33 2, 1f x y f x x x

22 2,

3 3 1 2df x y df x

x x xdx dx

Hence, the total derivative at 1x is 223 1 3 1 1 2 1 3 6 4 27




Q34. Consider a closed triangular contour traversed in counter-clockwise

direction, as shown in the figure.

The value of the integral, F dl evaluated along this contour, for a

vector field, ˆ ˆx yF ye xe

, is………….. ( ˆ ˆ,x ye e and ˆze are unit vectors

in Cartesian-coordinate system).

Ans.: 2

Solution: ˆ ˆx yF ye xe

ˆ2F z

and ˆda dxdyz . 2F da dxdy

1. 2 2 2 1 2

2F dl F da dxdy

Q35. A hemispherical shell is placed on the -x y 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……………… .

( da

is the elemental surface area, ˆ ˆ,x ye e and ˆze are unit vectors in Cartesian-coordinate

system)

Ans.: 2

Solution: 2 2ˆ ˆ /x yE ye xe x y

0E

except at origin.

. .S line

E da E dl

We have to take line integral around circle 2 2 2x y r in 0z plane. Let us use

cylindrical coordinate and use cos , sinx r y r sin , cosdx r d dy r d .

2 22

sin sin cos cos. /

r r d r r dE dl ydx xdy x y d

r

2

0. 2

line

E dl d

O 2,0P

y

x4

Q

4




IIT-JAM 2017

Q36. For the three matrices given below, which one of the choices is correct?

1

0 1

1 0

2

0

0

i

i

3

1 0

0 1

(a) 1 2 3i (b) 1 2 3i (c) 1 2 2 1 I (d) 3 2 1i

Ans. : (b)

Solution: These are pauli spin matrix which will satisfied 1 2 3i and 1 2 2 1 0

Q37. The integral of the vector 40ˆ, , cosA z

(standard notation for cylindrical

coordinates is used) over the volume of a cylinder of height L and radius 0R is:

(a) 0ˆ ˆ20 R L i j (b) 0 (c) 0

ˆ40 R L j (d) 0ˆ40 R Li

Ans. : (d)

Solution: By seeing the options lets calculate

0 2

0 0 0

40 ˆ ˆcos cos sinR L

V

Ad i j d d dz

ˆ ˆˆ cos sini j

2

0 0

0

ˆ ˆ ˆ40 cos cos sin 40V

Ad R L i j d R L i

Q38. Which one of the following graphs represents the derivative dff x

dx of the function

2

1

1f x

x

most closely (graphs are schematic and not drawn to scale)?

(a) (b)

(c) (d)

f x

x x

f x

f x

x

f x

x




Ans. : (a)

Solution: 2

1

1f x

x

and 2

2

1

df xf x

dx x

anti-symmetric function but f x is positive

and f x is positive

Q39. For the Fourier series of the following function of period 2

0 0

1 0

xf x

x

the ratio (to the nearest integer) of the Fourier coefficients of the first and the third

harmonic is:

(a) 1 (b) 2 (c) 3 (d) 6

Ans. : (c)

Solution: 0

0

1 1 11

2 2 2a dx

00

1 11 cos sin 0na nx dx nx

n

0

0

1 1 1 21 sin cos 1 1nb nx dx nx

n n n

Hence, 1

3

2 33

2

b

b

Q40. The volume integral of the function 2, , cosf r r over the region

0 2, 0 / 3 and 0 2r is………….

(Specify your answer upto two digits after the decimal point)

Ans. : 15.07

Solution: / 32 /3 2 5

2 2

00 0 0

2 1 cos 2cos sin 2

5 2 2V

I fd r r drd d

/ 3

0

32 1 24cos 2 / 3 cos 0 2 15.07

5 4 5I

f x

x




Q41. Consider two particles moving along the x - axis. In terms of their coordinates 1x and 2x ,

their velocities are given as 12 1

dxx x

dt and 2

1 2

dxx x

dt , respectively. When they start

moving from their initial locations of 1 0 1x and 2 0 1x , the time dependence of

both 1x and 2x contains a term of the form a te , where a is a constant. The value of a

(an integer) is………………

Ans. : 2

Solution: From the given relations we can write

1 2dx dx

dt dt

Integrating both sides with respect to t gives, 1 2 1x x c , where c being a constant of

integration

At 10, 1t x and 2 1x

Hence, 0c

Thus, 1 2x x (i)

Using equation (i) the first equation can be written as

1 11

1

2 2dx dx

x dtdt x

1 1ln 2 lnx t k 211 1

1

ln 2 txt x t k e

k

Using 1 0 1x , we obtain 1 1k , thus 21

tx e

Using equation (ii) the second equation can be written as

2 22

2

2 2dx dx

x dtdt x

Integrating gives

222 2

2

ln 2 ln ln txx t k e

k

Thus, 22 2

tx k e




Using 2 0 1x , we obtain 2 1k

Thus, 22

tx t e

Hence, the value of a is 2 .

Q42. Consider the differential equation 2 0y y y . If 0 0y and 0 1y , then the

value of 2y is……………….

(Specify your answer to two digits after the decimal point)

Ans. : 0.27

Solution: The characteristic equation is 2 2 1 0m m 21 0m

Thus 1m is a repeated root

Thus the general solution is

1 2xy c c x e

since 1 10 0 0 0y c c

Thus we can write 2xy c xe 2

x xy c e xe

since 0 1y

2 21 1 0 1c c

xy xe

222

2 22 2 0.27

2.72y e

e




IIT-JAM 2018

Q43. Let 3 3, 2f x y x y . The curve along which 2 0f is

(a) 2x y (b) 2x y

(c) 6x y (d) 2

yx

Ans.: (b)

Solution: 2 2

2 3 3 3 32 2

2 2 0f x y x yx y

2 6 12f x y

2 0 6 12 0f x y 2x y

Q44. A curve is given by 2 3 ˆˆ ˆr t ti t j t k

. The unit vector of the tangent to the curve at

1t is

(a) ˆˆ ˆ

3

i j k (b)

ˆˆ ˆ

6

i j k (c)

ˆˆ ˆ2 2

3

i j k (d)

ˆˆ ˆ2 3

14

i j k

Ans.: (d)

Solution: Let n be a unit vector tangent to the curve at .t

2 2

ˆˆ ˆ/ 2 3ˆ

/ 1 4 5

dr dt i tj tkn

dr dt t t

at

ˆˆ 2 3ˆ1,

14

i j kt n

Q45. The function , 0

, 0

x xf x

x x

is expanded as a Fourier series of the form

0 1 1cos sinn nn n

a a nx b nx

. Which of the following is true?

(a) 0 0, 0na b (b) 0 0, 0na b

(c) 0 0, 0na b (d) 0 0, 0na b

Ans.: (b)

Solution:- , 0

, 0

x xf x

x x

0

0 0

1

2 2a xdx xdx

0 0a




0

0

1sin sinnb x nxdx x nxdx

0

2 20

1 cos sin cos sinx nx nx x nx nx

n n n n

1 2 cos n

n

2; even

2; odd

n

nnb

nn

Thus, 0nb

Q46. Which one of the following curves correctly represents (schematically) the solution for

the equation 2 3: 0 0?df

f fdx

(a) (b)

(c) (d)

Ans.: (b)

Solution:- 2 3; 0 0df

f fdx

1

ln 3 23 2 2

dfdx f x A

f

Since, 10 0 ln 3

2f A

1 3

ln2 3 2

xf

231

2xf e

Now, we can see, at 0, 0,x f at 3

,2

x f

Thus option (b) is correct one.

0

1

2

f x

x

f x

3

2

0 x

f x

3

2

0 x

f x

1

2

0 x




Q47. Consider the transformation to a new set of coordinates , from rectangular Cartesian

coordinates ,x y , where 2 3x y and 3 2x y . In the , coordinate system,

the area element dxdy is

(a) 1

13d d (b)

2

13d d (c) 5d d (d)

3

5d d

Ans.: (a)

Solution:-

2 3,13

3 2,

J x y

J x y

x y

, 1

, 13

J x yJ

J

Since, area element in system is, 1

13dA J d d d d

Q48. Let matrix 4

6 9

xM

. If det 0M , then

(a) M is symmetric (b) M is invertible

(c) one eigenvalue is 13 (d) Its eigenvectors are orthogonal

Ans.: (a), (c), (d)

Solution:- Since, 4

6 9

xM

,

If 0 36 6 0 6M x x

Hence, 4

6 9

xM

(a) Here, M M , so it is symmetric matrix

(b) Determinant 0M , so noninvertible matrix

(c) For eigenvalue-

0M I 4 6

06 9




(4 )(9 ) 36 0 0, 13

(d) Eigen vectors for distinct eigen values for a symmetric matrix are orthogonal.

Q49. Let 6 23 2 8f x x x . Which of the following statements is (are) true?

(a) The sum of all its roots is zero

(b) The product of its roots is 8

3

(c) The sum of all its roots is 2

3

(d) Complex roots are conjugates of each other.

Ans.: (a), (b), (d)

Solution:- 6 23 2 8f x x x

Now, 6 23 2 8 0x x 6 22 8

03 3

x x

6 5 4 3 2 0Ax Bx Cx Dx Ex Fx G

6 5 4 3 22 80. 0. 0. 0. 0

3 3x x x x x x

Here, sum of roots 0B

A

And product of roots 8

3

G

A

Since all coefficient are real, then complex roots are conjugate to each other.

Hence, options (a), (b) and (d) are correct.

Q50. The coefficient of 3x in the Taylor expansion of sin sin x around 0x is ______.


Ans.: 0.33

Solution:- Let sin sinf x x

cos sin .cosf x x x

sin sin .cos .cos sin .cos sinf x x x x x x

2cos .sin sin sin .cos sinx x x x




32cos sin sin sin cos cos sin cos cos sin sin sin sin cosf x x x x x x x x x x x

3 1sin 2 .sin sin cos .cos sin cos .cos sin sin 2 .sin sin

2x x x x x x x x

at 0,x

0 1 1 2f

Hence,

2 3

1 0 1 0 00 1 .....

1 2 3

x x f x x x f x x xf x f x f x

Hence, coefficient of 3x is _____

1 2 12 0.33

3 3 2 1 3

IIT-JAM 2019

Q51. The function 2

8

9

xf x

x

is continuous everywhere except at

(a) 0x (b) 9x (c) 9x i (d) 3x i

Ans. : (d)

Solution: We know that a rational function is discontinuous at a point where the denominator is

0 . Therefore,

2 9 0 3x x i

Q52. If , ,x y z is a scalar function which satisfies the Laplace equation, then the gradient of

is

(a) Solenoidall and irrigational (b) Solenoidall but not irrotational

(c) Irrotational but not solenoid (d) Neither Solenoidall nor irrotational

Ans. : (a)

Solution: 0 0 0, 0, 0E E E




Q53. A unit vector perpendicular to the plane containing ˆˆ ˆ 2A i j k

and ˆˆ ˆ2B i j k

is

(a) 1 ˆˆ ˆ3 426

i j k (b) 1 ˆˆ ˆ3 319

i j k

(c) 1 ˆˆ ˆ5 335

i j k (d) 1 ˆˆ ˆ5 335

i j k

Ans. : (d)

Solution: ˆ 0A n

and ˆ 0B n

Verify option d : 1ˆ 1 5 6 0

35A n

1ˆ 2 5 3 0

35B n

Q54. The eigenvalues of

3 0

3 0

0 0 6

i

i

are

(a) 2,4and 6 (b) 2 ,4 and 6i i (c) 2 ,4and 8i (d) 0,4and 8

Ans. : (a)

Solution: For calculation of eigenvalues

3 0

3 0 0

0 0 6

i

i

3 3 6 6 0i i

3 3 6 6 0

or 26 3 1 0

or 26 6 8 0

or 6 2 4 0 . Therefore, 6 or 2 or 4 .




Q55. The gradient of scalar field , ,S x y z has the following characteristic(s)

(a) Line integral of a gradient is path-independent

(b) Closed line integral of a gradient is zero

(c) Gradient of S is a measure of the maximum rate of change in the field S

(d) Gradient of S is a scalar quantity

Ans.: (a), (b), (c)

Q56. The flux of the function 2 2ˆ ˆ ˆ3 4F y x xy z y yz z

passing through the surface

ABCD along n is_________

(Round off to 2 decimal places)

Ans. : 1.17

Solution: 1y plane

ˆS

F da F dxdzy 23xy z dxdz

1 1

2

0 0

3x z dxdz 13

1

00

33

z

zxz dx

1

0

13

3z dx

12

0

32 3

x x

3 1 9 2 71.17

2 3 6 6

Q57. The value of 23

2

0

i

z dz

, along the line 3y x , where z x iy is______


Ans. : 111.1

Solution: 23

2

0

3i

z dz y x

z x iy

xA

By

n1

C

D

0, 0, 0

1

z

1




3z y iy

z x iy 3y iy 3 i y

3dz dy idy 3 i dy

21

2

0

3 3 3i i i y dy

212

0

10001000 1 111.11

9y dy




Mechanics and General Properties of Matter

IIT-JAM 2005

Q1. A solid sphere of mass m and radius a is rolling with a linear speed v on a flat surface

without slipping. The magnitude of the angular momentum of the sphere on the surface is

(a) mav5

2 (b) mav

5

7 (c) mav (d) mav

2

3

Ans.: (b)

Solution: 22 2 7

5 5 5

v mvaL I mva ma mva mva mva

a

IIT-JAM 2007

Q2. In terms of the basic units of mass M , length L , time T and charge Q , the dimensions

of magnetic permeability of vacuum 0 are

(a) 2MLQ (b) 2 1 2ML T Q (c) 1LTQ (d) 1 1LT Q

Ans.: (a)

Solution: 0,I

F I dl B Bd

2

0F I 2

2 20 0

QMLT MLQ

T

Q3. A projectile is fired from the origin O at an angle of 045 from the horizontal. At the

highest point P of its trajectory the radial and transverse components of its acceleration

in terms of the gravitational acceleration g are

(a) 5

,5

2 ga

gar (b)

5,

5

2 ga

gar

(c) 5

2,

5

ga

gar (d)

5

2,

5

ga

gar

Ans.: (d)

Solution: Maximum 2 2

maxsin

2

vh

g

and range

2 sin 2vR

g

where

4

max 1tan

22

hR

xO

Py




From the figure cos 90 sin5

rg

a g g

2

cos5

ga g

Q4. A satellite moves around a planet in a circular orbit at a distance R from its centre. The

time period of revolution of the satellite is T . If the same satellite is taken to an orbit of

radius 4R around the same planet, the time period would be

(a) 8T (b) 4T (c) 4

T (d)

8

T

Ans.: (a)

Solution: 2 3T R

3

222

48 8

T RT T

T R

IIT-JAM 2008

Q5. EFGH is a thin square plate of uniform density and

side 4a . Four point masses, each of mass m , are placed

on the plate as shown in the figure. In the moment of

inertia matrix I of the composite system,

(a) only xyI is zero

(b) only xzI and yzI are zero

(c) all the product of inertia terms are zero

(d) none of the product of inertia terms is zero

Ans.: (c)

Solution: 0, 0, 0xy i i i xz i i i yz i i ii i i

I m x y I m x z I m y z

F

GH

m m

)0,,( aa )0,,( aa

)0,,( aa )0,,( aa

mm

y

x




Q6. A circular disc (in the horizontal xy -plane) is spinning about a vertical axis through its

center O with a constant angular velocity

. As viewed from the reference frame of the

disc, a particle is observed to execute uniform circular motion, in the anticlockwise sense,

centered at P . When the particle is at the point Q , which of the following figures

correctly represents the directions of the Coriolis force corF

and the centrifugal force cfgF

?

(a) (b)

(c) (d)

Ans.: (c)

Solution: 1 1 1 1 ˆ ˆˆ ˆcentifugalF r z z rr r

ˆ ˆˆ2 2corrF m v m z v r

cfgr FF ,

O

PQ

rF

O

PQ cfgF

rF

O

PQ

cfgFrF

O

PQ

cfgF




IIT-JAM 2009

Q7. A space crew has a life support system that can last only for 1000 hours. What minimum

speed would be required for safe travel of the crew between two space stations separated

by a fixed distance of 121.08 10 km ?

(a) 3

c (b)

2

c (c)

2

c (d)

5

c

Ans.: (b)

Solution: 12

2

2

1000 36001.08 10 1000

1

v

v

c

512

2 2

2 2

10 361080 10

21 1

v v cc v

v v

c c

Q8. A particle is moving in space with O as the origin. Some possible expression for its

position, velocity and acceleration in cylindrical coordinates , , z are given below.

Which one of these is correct?

(a) Position vector zzr ˆˆˆ and velocity z

dt

dz

dt

d

dt

dv ˆˆˆ

(b) Velocity zdt

dz

dt

d

dt

dv ˆˆˆ

and acceleration zdt

zd

dt

d

dt

d

dt

da ˆˆˆ

2

2

2

2

(c) Position vector zzr ˆˆ and velocity z

dt

dz

dt

d

dt

dv ˆˆˆ

(d) Position vector zzr ˆˆˆ and velocity z

dt

dz

dt

d

dt

dv ˆˆˆ

Ans.: (d)




Q9. A thin massless rod of length 2l has equal point masses m attached at its ends (see

figure). The rod is rotating about an axis passing through its centre and making angle

with it. The magnitude of the rate of change of its angular momentum dt

Ld

is

(a) 2 22 sin cosml (b) 2 22 sinml

(c) 2 2 22 sinml (d) 2 2 22 cosml

Ans.: (a)

Solution: 22 sin 2 sinJ m r r m r r m r l m l

2 22 sin 90 sin 2 cos sindL

Torque J m l m ldt

Q10. Moment of inertia of a solid cylinder of mass M , height l and radius R about an axis

(shown in the figure by dashed line) passing through its centre of mass and perpendicular

to its symmetry axis is

(a) 22

12

1

4

1MlMR

(b) 22

8

1

2

1MlMR

(c) 22

12

1

2

1MlMR

(d) 22

4

1

2

1MlMR

Ans.: (a)

Solution: If R be the radius, l , the length and M , the mass of the solid cylinder, supposed to be

uniform and of a homogeneous composition, we have its mass per unit length = M

l.

Now, imagining the cylinder to be made up of a number of discs each of radius R ,

placed adjacent to each other, and considering one such disc of thickness dx and at a

distance x from the centre O of the cylinder, (figure), we have

m

l

R

l




Mass of the disc = M

dxl

and radius = R

And M.I. of the disc about its diameter 2

4

M RAB dx

l and its M.I. about the parallel

axis YOY , passing through the centre O of the cylinder and perpendicular to its axis of

cylindrical symmetry (or its length), in accordance with the principle of parallel axes,

2

2

4

M R Mdx dx x

l l

Hence, M.I. of the whole cylinder about this axis, i.e. I twice the integral of the above

expression between the limits 0x and 2

lx ,

i.e., 2 2

/ 2 / 22 2

0 0

22

4 4l lM R M M R

I dx x dx dx x dxl l l

/ 22 3

0

2

4 3

lM R x x

l

or 2 3 2 3 2 22 2

4 2 8 3 8 24 4 12

M R l l M R l l R lI M

l l

X R

Y

O

B

dxX

A/2l

x

lY




IIT-JAM 2010

Q11. A circular platform is rotating with a uniform angular speed counterclockwise about

an axis passing through its centre and perpendicular to its plane as shown in the figure. A

person of mass m walks radially inward with a uniform speed v on the platform. The

magnitude and the direction of the Coriolis force (with respect to the direction along

which the person walks) is

(a) 2m v towards his left (b) 2m v towards his front

(c) 2m v towards his right (d) 2m v towards his back

Ans.: (c)

Solution: ˆˆˆ2 2 2F m v m v z r m v

2m v towards his right

Q12. A particle of mass m , moving with a velocity jivv ˆˆ0

, collides elastically with

another particle of mass 2m which is at rest initially. Here, 0v is a constant. Which of the

following statements is correct?

(a) The direction along which the centre of mass moves before collision is

2

ˆˆ ji

(b) The speed of the particle of mass m before collision in the center of mass frame

is 02v .

(c) After collision the speed of the particle with mass 2m in the centre of mass frame

is 03

2v .

(d) The speed of the particle of mass 2m before collision in the center of mass frame

is 02v .

Ans. : (c)




Solution: Velocity of center of mass is 0 0ˆ ˆ ˆ ˆ2 0

3 3

mv i j m v i j

m m

so option a is wrong

Velocity of mass m with respect to center of mass before collision

0 01 0

ˆ ˆ ˆ ˆ2ˆ ˆ

3 3cm

v i j v i ju v v i j

so speed is 0

4 4

9 9v 0

8

9v

Velocity of mass 2m with respect to center of mass before collision

0 02

ˆ ˆ ˆ ˆ0

3 3cm

v i j v i ju v

so speed is 0

0 021 1 2

9 9 9 3

vv v which is

also speed after the collision with respect to center of mass

IIT-JAM 2011

Q13. A rain drop falling vertically under gravity gathers moisture from the atmosphere at a rate

given by 2ktdt

dm , where m is the instantaneous mass, t is time and k is a constant. The

equation of motion of the rain drop is mgdt

dmv

dt

dvm

If the drop starts falling at 0t , with zero initial velocity and initial mass 0m

( 30 2 , 12 /m gm k gm s and 21000 /g cm s ), the velocity v of the drop after one

second is

(a) 250 /cm s (b) 500 /cm s (c) 750 /cm s (d) 1000 /cm s

Ans.: (b)

Solution: 3

2 23

dm ktkt m

dt

dv dmm v mg

dt dt 2dv

m vkt mgdt

2 2

3

3

6

dv kt dv ktv g v g

dt m dt kt

which is linear differential equation

So 2

33

3. exp ( 6)

3 6

ktI F dt kt

kt




3 36 6v kt g kt dt c 0, 0 0t v c

4

3

24

4 6

g kt tv

kt

Put 2gm, k = 12 gm/s3 and g = 1000 cm/s2),and 1t

1000 12 24500 / sec

4 12 6v cm

Q14. A particle of mass m is moving in a potential 2

220 22

1

mx

axmxV where 0 and a

are positive constants. The angular frequency of small oscillations for the simple

harmonic motion of the particle about a stable minimum of the potential V x is

(a) 02 (b) 2 0 (c) 4 0 (d) 024

Ans.: (b)

Solution: 2

220 22

1

mx

axmxV

2 40 3 2 2

0

0dV a a

m x xdx mx m

0

2 222 2 200 0 02 4

334

x x

md V am m m

mdx mx

0

2

2 20

04

2x x

d V

dx m

m m




IIT-JAM 2012

Q15. Three masses m , 2m and 3m are moving in xy - plane with speeds 3 ,2u u and u ,

respectively, as shown in the figure. The three masses collide at the same time at P and

stick together. The velocity of the resulting mass would be

(a) yxu

ˆ3ˆ12

(b) yxu

ˆ3ˆ12

(c) yxu

ˆ3ˆ12

(d) yxu

ˆ3ˆ12

Ans.: (d)

Solution: 3 2 .2 cos 60 3 cos 60ˆ ˆ

6 12cmm u m u mu u

u i x

0 2 .2 sin 60 3 sin 60 3ˆ ˆ6 12cm

m m u mu uu i y

m

The combined mass will move with same velocity as initial center of mass is moving

yxu

v ˆ3ˆ12

Q16. The figure shows a thin square sheet of metal of uniform density along with possible

choices for a set of principal axes (indicated by dashed lines) of the moment of inertia,

lying in the plane of the sheet. The correct choice(s) for the principal axes would be

(a) ,p q and r (b) p and r (c) p and q (d) p only

Ans.: (c)

Solution: zI about center of mass due to p and q are same

So, p and q are correct choice.

m2

xm

u3

u26060

u

m3

y

P

q rp




IIT-JAM 2013

Q17. A particle is released at 1x in a force field 0,ˆ2

2 2

2

xe

x

xxF x

. Which one of

the following statements is FALSE?

(a) xF

is conservative

(b) The angular momentum of the particle about the origin is constant

(c) The particle moves towards 2x

(d) The particle moves towards the origin

Ans.: (c)

Solution: 0,ˆ2

2 2

2

xe

x

xxF x

0F

are zero so force is conservative

0r F

so angular momentum is conserved.

For equilibrium point 2

2

20 0 2, 2

2

xF F x

x

0x so equilibrium

point is 2

To check stable and unstable equilibrium point 3

4 20 0

2

dF x

dx x

At 2x , 3

4 2 40 2

2 2 2

dF xve

dx x

so it is unstable point so particle

moves away from point 2x so it is obvious that it will moves towards origin

Q18. If the dimensions of mass, length, time and charge are TLM ,, and C respectively, the

dimensions of the magnetic induction field B

is

(a) 112 CTML (b) 11 CMT (c) CTL 12 (d) CTL 11

Ans.: (b)

Solution: ,F I dl B

2 1 1CMLT LB B MT C

T




Q19. The path of a particle of mass m , moving under the influence of a central force, in plane

polar coordinates is given by kerr 0 , where 0r and k are positive constants of

appropriate dimensions. The angular momentum of the particle is L and its total energy

is zero. The potential energy functions rV , in terms of Lm, and k is……………..

Ans.:

Solution: 2 2 2

2

1L u d uu f

m ud

kerr 0 ,0

1 ku er

, 2 2

20 0

1 kkd u k e

k k er rd

2 2 2

2

1L u d uu f

m ud

2 2 2 2 2 2 2 3

22 3

0 0 0 00 0

.k k k k

k kL u k e k L e k e k L ee e k k

m r r m r rr mr

=1

fu

2

31/ 1L

f u u k km

2

3

1 1L k kf r

m r

2

3

1 1L k kdV

dr m r

2

3

1 1L k kdV dr

m r

2

2

1 1

2

L k kV

r

Total energy, 22

22 2

11 1

2 2 2

L k kLE mr

mr r

Also, 0kr r e 0

kr kr e 0 2k L kL

kr emr mr

22 2 2

2 2 2 2

11 10

2 2 2

L k kk L LE m

m r mr r

22 2 2

2 1

2 2 2

L k k mk L Lr

m m m

2 2 2 2 1

2

k L L mL k k

m




2

2 2 2

1 2

2 1 1

L k k mV

L k mL k k

2

1

1 1

mk kV

mk k k

IIT-JAM 2014

Q20. A spherical ball of ice has radius 0R and is rotating with an angular speed about an

axis passing through its centre. At time ,0t it starts acquiring mass because the

moisture (at rest) around it starts to freeze on it uniformly. As a result its radius increases

as ,0 tRtR where is a constant. The curve which best describes its angular

speed with time is

(a) (b)

(c) (d)

Ans.: (b). constantI t t

constantM t R t

2

0

constantt

M t R t

2

1t

t

Q21. The moment of inertia of a disc about one of its diameters is .MI The mass per unit area

of the disc is proportional to the distance from its centre. If the radius of the disc is R and

its mass is M , the value of MI is

(a) 2

2

1MR (b) 2

5

2MR (c) 2

10

3MR (d) 2

5

3MR

Ans.: (c)

t

t

t

t




Solution: Mass density is cr 3

0

3, .

2

r MM cr rdrd c

R

52 2

0 0

2.

5

R R

zc R

I r dm r cr rdrd put value of c 23

5zI MR

From perpendicular axis theorem z x yI I I and x yI I so 23

10x y MI I I MR

Q22. Two points N and S are located in the northern and southern hemisphere, respectively,

on the same longitude. Projectiles P and Q are fired from N and S , respectively,

towards each other. Which of the following options is correct for the projectiles as they

approach the equator?

(a) Both P and Q will move towards the east

(b) Both P and Q will move towards the west

(c) P will move towards the east and Q towards the west

(d) P will move towards the west and Q towards the east

Ans.: (b)

Solution: Coriolis force

2F m v

z

For both particles ˆ ˆr zV V r V z

So, ˆˆˆ ˆ ˆ2 2r z rF m z v r z v z m v

So both will move towards

Earth revolves from west to east. So the particle will move towards west.




Q23. Two particles A and B of mass m and one particle C of mass M are kept on the x axis

in the order ABC . Particle A is given a velocity .iv Consequently there are two

collisions, both of which are completely inelastic. If the net energy loss because of these

collisions is 8

7 of the initial energy, the value of M is (ignore frictional losses)

(a) m8 (b) m6 (c) m4 (d) m2

Ans.: (b)

Q24. What is the maximum height above the dashed line attained by the water stream coming

out at B from a thin tube of the water tank assembly shown in the figure? Assume

10 , 2h m L m and .30

(a) 10 m (b) 2m (c) 1.2m (d) 3.2m

Ans.: (d)

Solution: Velocity at the end of thin tube 21

2mgh mv

Distance at the end of thin tube 010 sin 30L

9h m

219 18

2mg mv v g

2 2sin 18 10 1/ 4

2 252 2 10

vh m

g

Height from the dashed line 02.25 2sin 30 3.25m m

A

hL B

0y




Q25. At an instant shown, three point masses , 2m m and 3 ,m rest on a horizontal surface and

are at the vertices of an equilateral triangle

of unit side length. Assuming that G is the

gravitational constant, the magnitude and

direction of the torque on the mass 3m ,

about the point O , at that instant is

(a) Zero (b) ,32

3 2mG going into the paper

(c) ,33 2mG coming out of the paper (d) ,34

3 2mG going into the paper

Ans.: (d) Force on 3m mass

2 0 2 0 2 0 2 0ˆ ˆ6 sin 30 3 sin 30 6 cos30 3 cos30F Gm Gm x Gm Gm y

2 23 9 3ˆ ˆ

2 2F Gm x Gm y

Radius vector for 3m mass

3

ˆ ˆ02

r x y

23 3ˆ ˆ 0

2 2r f Gm y x 23 3

ˆ4

Gm z .

m3

m2mO




IIT-JAM 2015

Q26. 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 0t , 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 cosdx

v A t B tdt

Thus, sinx A t and cosV t B t

Q27. 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)

Solution: 2 3

3/ 212 1

2

8 228

T RT T days

T R

m Impulse




Q28. 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 / seccm . 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

Q29. 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) 2

2

7mr (b) 27mr

(c) 2

2

13rm (d) 2

2

55rm

Ans.: (d)

Solution: 2 2 2 2 2

2 54 556 4

2 2 2 2 2

mr mr mr mr mrmr




Q30. A particle of mass m is moving in yx plane. At any given time t , its position vector is

given by ˆcos sinr t A t i B t j 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 ˆ

Ans.: (a), (c) and (d)

Solution: (a) ˆ ˆcos sinr t A t i B t j

cos , sinx A t y B t

cos , sinx y

t tA B

2 2

2 21

x y

A B (Ellipse)

(b) ˆ ˆsin cosdr

A t i B t jdt

Speed 2 2 2 2 2 2sin cosdr

A t B tdt

. Speed is function of time, so not constant.

(c) 2

2 22

ˆ ˆcos sindr

A t i B t jdt

2 r

. Force act towards origin.

(d) L r p

ˆˆ

cos sin 0

sin cos 0

i j k

m A t B t

A t B t

L m ABk

Q31. 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)




Q32. 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 x

vl l

c 2

0 cos 1 0.8l 0 0

10.6 0.3

2xl l l and 00

3sin

2y

ll l

New length 2

2 00 0 0

3 30.3 0.09 0.916

2 4

ll l l l

% change in length 0

0

1 0.91100 0.09 100 9%

l

l

Q33. 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 I

R R

3.2663

ga

Q34. 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………….

cm50




Ans.: 17.5

Solution: According to Bernoulli’s equation

2 21 1

2 2b b b t t tP gh V P gh V

2 21

2b 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 / sec

1000 / 10tt

kg kgV

A kg m m

5 / sectV m

According to equation of continuity

tt t b b b t

b

AAV A V V V

A

2

2

15 / sec 0.2 / sec

25b

cmV m m

cm

22 211000 10 50 10 1000 5 0.2

2b tP P P

5000 500 25 0.04P 25000 12480 17480 /N m 17.5 aP kP

Q35. A block of mass 2kg is at rest on a horizontal table The coefficient of friction between

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

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

Ans.: 3.225

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

Applied force is more than friction

3 2 1ma F N 21 10.5 /

2a m s

m

2 2v u as 2 2 0.5 10 10 3.225 /v as m s

0, 10u s m

50t bh h cm

, ,t t tP V A

, ,b b bP V A




Q36. 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

M

R. Hence the mass of the

semicircular element

2 / 2

Mdm rdrd

R

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

The y coordinate of centre of mass

1

sinY r dmM

2

0 0

1 2sin

R Mr rdrd

M R

2

20 0

2sin

R

r dr dR

3

2

2 42 1.3

3 3

R R

R

IIT-JAM 2016

Q37. 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 8r m from the origin,

the magnitude of the instantaneous velocity of the particle in /m s is

(a) 8 15 (b) 20 (c) 2 37 (d) 10

Ans.: (b)

Solution: 12 /rv m s 2 8 16 / secv r m

2 2rv v v 144 256 400 20 / secm

0 m3

r dr r

R




Q38. A cylindrical rod of length L has a mass density distribution given by 0 1x

xL

,

where x is measured from one end of the rod and 0 is a constant of appropriate

dimensions. The centre of mass of the rod is

(a) 5

9L (b)

4

9L (c)

1

9L (d)

1

2L

Ans.: (a)

Solution:

2 3 2 32

000 0 0

2 2

00 0 0 0

1 11

2 32 3 2 31

1 122 2

LL L L

cm L L L L

x x x L Lx dx Lxdm x dxL L Lxx x Ldm dx dx Lx LL L L

556

3 92

cm

Lx L

Q39. An incompressible, non-viscous fluid is injected into a conical pipe at

its orifice as schematically shown in the figure. The pressure at the

orifice of area 0A is 0P . Neglecting the effect of gravity and assuming

streamline flow, which one of the following plots correctly predicts the

pressure along axis of the cone?

(a) (b)

(c) (d)

Ans. : (a)

/4x

0 0 0;X P v

P

0P

0X X

P

0P

0X X

P

0P

0X X

P

0P

0X X




Solution: The area cross-section at B is

22 tan

4xA r x

2xA x

Velocity at B is 0 0 x xA v A v 0 0 0 0 0 02 2x x

x

A v A v A vv v

A x x

According to Bernauli equation,

2 2

2 2 2 2 2 0 00 0 0 0 0 0 2 4

1 1 1 1

2 2 2 2x x x x

A vP v P v P P v v P v

x

22 0

0 0 4 4

11

2x

AP P v

x

Graph (a) correctly represent the variation of xP w.r.t. x

Q40. 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 varies both in magnitude and direction

(d) about 0,0,1 is constant in direction but not in magnitude

Ans.: (a) and (b)

Solution: Angular momentum will be constant in x y plane

O 0x x ,x xP v 0 0,P v

0A

A/4

xA

B




IIT-JAM 2017

Q41. Consider a uniform thin circular disk of radius R and mass M . A concentric square of

side / 2R is cut out from the disk (see figure). What is the moment of inertia of the

resultant disk about an axis passing through the centre of the disk and perpendicular to it?

(a) 2 1

14 48

MRI

(b) 2 1

12 48

MRI

(c) 2 1

14 24

MRI

(d) 2 1

12 24

MRI

Ans. : (b)

Solution: ' 2 22

2 12disc square

M a aMRI I I

'2 2 2 4

M R R MM

R and

2

Ra

2 11

2 48

MRI

Q42. The linear mass density of a rod of length L varies from one end to the other as

2

0 21

x

L

, where x is the distance from one end with tensions 1T and 2T in them (see

figure), and 0 is a constant. The rod is suspended from a ceiling by two massless strings.

Then, which of the following statement(s) is (are) correct?

(a) The mass of the rod is 02

3

L

(b) The centre of gravity of the rod is located at 9

16

Lx

(c) The tension 1T in the left string is 07

12

Lg

(d) The tension 2T in the right string is 03

2

Lg

Ans. : (b), (c)

R

/ 2R

1T2T

0 L x




Solution: The mass of rod is 2

00 2

0

41

3

L Lxm dx

L

so (a) is wrong

The centre of gravity of the rod is located at

2

0 20

2

0 20

19

161

L

cm L

xx dx

L Lx

xdx

L

Force equation 01 2

4

3

LgT T

and torque equation

1 2 1 2 2 19 9 9 7 9

16 16 16 16 7

L L L LT T L T T T T

put value of 2 19

7T T in equation 0

1 24

3

LgT T

0 01 1

4 716

7 3 12

Lg LgT T

0 02 1

7 99 9

7 7 12 12

Lg LgT T

Q43. An object of mass m with non-zero angular momentum is moving under the influence

of gravitational force of a much larger mass (ignore drag). Which of the following

statement(s) is (are) correct?

(a) If the total energy of the system is negative, then the orbit is always circular

(b) The motion of m always occurs in a two-dimensional plane

(c) If the total energy of the system is 0 , then the orbit is a parabola

(d) If the area of the particle’s bound orbit is S , then its time period is 2 /mS

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

Solution: The eccentricity of curve 2

2

21

Ele

mk where 1 2k Gm m and E is energy .

If total energy is negative then orbit can be either elliptical or circular so (a) is wrong

In two body central force problem motion is confine in plane so (b) is correct

If the total energy of the system is 0 , then the orbit is a parabola one can calculate

1e so (c) is correct




From second law of thermodynamics 2

2 2

dA l S l mST

dt m T m l so (d) is correct

Q44. A particle of mass m fixed in space is observed from a frame rotating about its z - axis

with angular speed . The particle is in the frame’s xy plane at a distance R from its

origin. If the Coriolis and centrifugal forces on the particle are CORF

and CFGF

,

respectively, then (all the symbols have their standard meaning and refer to the rotating

frame),

(a) 0COR CFGF F

(b) 2 ˆCOR CFGF F m Rr

(c) 2 ˆ2CORF m Rr

(d) 2 ˆCFGF m Rr

Ans. : (b), (c)

Solution: 2 ˆcorr centipetalF F mr Rr

22 , 2corF m v v r m R

Q45. A particle of unit mass is moving in a one-dimensional potential 2 4V x x x . The

minimum mechanical energy (in the same units as V x ) above which the motion of the

particle cannot be bounded for any given initial condition is……………


Ans. : 0.25

Solution: 2 4V x x x

The potential is mimima at 0x and maxima at 1

2x fig

The value 0V x 0x

The value 0.25V x 1

2x

minimum mechanical energy above which the motion of the particle cannot be bounded

is .25fE




Q46. Sand falls on a conveyor belt at the rate of 1.5 /kg s . If the belt is moving with a constant

speed of 7 /m s , the power needed to keep the conveyor belt running is…………….

(Specify your answer in Watts to two digits after the decimal point)

Ans. : 73.4

Solution:

2

2

11 22

. .2 2

d mvdW dm dv

P v mvdt dt dt dt

if v is constant 0

dv

dt

21

1 1. . 1.5 49 36.7

2 2

dmP v watt

dt

Work done due to friction .W mg d , 21

2d at ,

v vv at t

a g

21

2

vd g

g

221 1

.2 2

vW mg g mv

g

22

1 1. . 1.5 49 36.7

2 2

dmP v watt

dt

1 2 36.7 36.7 73.4 .P P P watt watt watt

Q47. In planar polar co-ordinates, an object’s position at time t is given

as , , 8 radtr e m t . The magnitude of its acceleration in 2/m s at 0t (to the

nearest integer) is……………….

Ans. : 9

Solution: 2 8t tra r r e e at 0t , 21 8 1 7 / secm

2 0 2 8 2 8t ta r r e e 2 27 4 8 49 32 81 9 / seca m




Q48. At 0t , a particle of mass m having velocity 0v starts moving through a liquid kept in a

horizontal tube and experiences a drag force d

dxF k

dt

. It covers a distance L before

coming to rest. If the times taken to cover the distances / 2L and / 4L are 2t and 4t

respectively, then the ratio 2 4/t t (ignoring gravity) is……….


Ans. : 2.41

Solution: Using Newton’s second law we obtain

2

2

dx d xk m

dt dt

2

20

d x dxm k

dt dt

2

20

d x k dx

dt m dt

The characteristic equation is

2 0 0k k

m m

, 0 and

k

m

General solution:

1 2

kt

mx t c c e

(i)

Using the condition 0x , we obtain

1 2 0c c (ii)

2

kt

mkv t c e

m

(iii)

Using the condition 00v v

0 2 2 0

k mv c c v

m k and 1 0

mc v

k

0 1k

tm

mvx t e

k

(iv)

00

0

k k kt t t

m m mmv k v

v t e v e ek m v

(v)




Using equations (iv) and (v), we obtain 0

0

1mv v

x tk v

,

0 01mv mv

L Lk k

From the question

2 20

0

1 12 2

k kt t

m mmvL kL

e ek mv

(vi)

4 40

0

1 14 4

k kt t

m mmvL kL

e ek mv

(vii)

Taking Logarithm on both sides of equations (vi) and (vii)

20

ln 12

k kLt

m mv

and 4

0

ln 14

k kLt

m mv

Thus,

0

02

4 0

0

ln 12 ln 1/ 2

2 41ln 3/ 4

ln 14

mvkmv kt

t mvkmv k

.

IIT-JAM 2018

Q49. There are three planets in circular orbits around a star at distances , 4a a and 9a ,

respectively. At time 0t t , the star and the three planets are in a straight line. The period

of revolution of the closest planet is T . How long after 0t will they again be in the same

straight line?

(a) 8T (b) 27T (c) 216T (d) 512T

Ans. : (c)

Solution: 3/ 21T ka T , 3/ 2

2 4 8T k a T , 3/ 2

3 9 27T k a T

Common time that all three star will meet again is 0 1 2 3 216t T T T T , which is LCM

of all time period.




Q50. A disc of radius 1R having uniform surface density has a concentric hole of radius

2 1R R . If its mass is M , the principal moments of inertia are

(a) 2 2 2 2 2 2

1 2 1 2 1 2, ,

2 4 4

M R R M R R M R R

(b) 2 2 2 2 2 2

1 2 1 2 1 2, ,

2 4 4

M R R M R R M R R

(c) 2 2 2 2 2 2

1 2 1 2 1 2, ,

2 4 8

M R R M R R M R R

(d) 2 2 2 2 2 2

1 2 1 2 1 2, ,

2 4 8

M R R M R R M R R

Ans. : (b)

Solution:

2 2

1 1

2 22 12 2

2 22 1

2 .2

R R

zz ZZ

R R

M R RMI dm r r r dr I

R R

xx yy zzI I I

By symmetry xx yyI I . Therefore, 2 2

2 1

4xx yy

M R RI I

Q51. A raindrop falls under gravity and captures water molecules from atmosphere. Its mass

charges at the rate m t , where is a positive constant and m t is the instantaneous

mass. Assume that acceleration due to gravity is constant and water molecules are at rest

with respect to earth before capture. Which of the following statements is correct?

(a) The speed of the raindrop increases linearly with time

(b) The speed of the raindrop increases exponentially with time

(c) The speed of the raindrop approaches a constant value when 1t

(d) The speed of the raindrop approaches a constant value when 1t

Ans.: (c)

Solution: Applying impulse momentum m equation.

mg dt m dm v dv mv mdv dm

mg vdt dt




or, dmdv

g vdt mdt

0,dv dm

g vdt mdt

0 0

v t

v

dvdt

g v

at 00,t V V 0ln lng v g v t

0

tg ve

g v

0

tg gV V e

As, 1, 0tt e , so g

V

(which is constant)

Q52. A particle P of mass m is constrained to move on the surface of cylinder under a force

kr

as shown in figure ( k is the positive constant). Which of the following statements is

correct? (Neglect friction.)

(a)Total energy of the particle is not conserved.

(b) The motion along z direction is simple harmonic.

(c) Angular momentum of the particle about O increases with time.

(d) Linear momentum of the particle is conserved.

Ans. : (b)

Solution: ˆ ˆF kr k rr zz

rF kr , 0F

zF kz mz kz the motion along z is simple harmonic motion.

Q53. Two projectiles of identical mass are projected from the ground with same

initial angle with respect to earth surface and same initial velocity u

in the same plane. They collide at the highest point of their trajectories and

stick to each other. Which of the following statements is (are) correct?

(a) The momentum of the combined object immediately after collision is zero.

(b) Kinetic energy is conserved in the collision

(c) The combined object moves vertically downward.

(d) The combined object moves in a parabolic path.

Ans. : (a), (c)

z

x

r

P

O y

u u




Solution: At the highest point there is only Horizontal velocity. In horizontal direction there is

not any External force. So momentum in the horizontal direction is conserved.

(c) After collision whole system will full under gravitation.

Q54. A particle of mass m is moving along the positive x direction under a potential

22

1

2 2V x kx

x

( k and are positive constants). If the particle is slightly displaced

from its equilibrium position, it oscillates with an angular frequency ______.

(Specify your answer in units of k

m as an integer.)

Ans. : 2

Solution: 2

22 2

kxV x

x

3

0V

kxx x

1/ 44

0x xx x

2

2 4

34

Vk k

x x

0

2

24

2x x

Vx k

m m

2

Q55. A planet has average density same as that of the earth but it has only 1

8 of the mass of the

Earth. If the acceleration due to gravity at the surface is pg and eg for the planet and

Earth, respectively, then p

e

g

g ____________.

(Specific your answer upto one digit after the decimal point.)

Ans. : 0.5

Solution: 2

GMg

R and

M

V

1/3

3M M MV R R

2/ 3

/

Mg

M 1/ 3g M




1/3 1/3

10.5

8 2p p e

e e e

g M M

g M M

10.5

2

Q56. A body of mass 1 kg is moving under a central force in an elliptic orbit with semi major

axis 1000 m and semi minor axis 100 m . The orbital angular momentum of the body is

2 1100 kg m s . The time period of motion of the body is _________ hours.

(Specific your answer upto two digits after the decimal point)

Ans. : 1.74

Solution: 2

2

dA L mT ab

dt m L

2 1

3.14 1000 100100

6 28 1000

6280

3600 1 744hr

Q57. The moon moves around the earth in a circular orbit with a period of 27 days. The radius

of the earth R is 66.4 10 m and the acceleration due to gravity on the earth surface is

29.8 ms . If D is the distance of the moon from the center of the earth, the value of D

R

will be _________. (Specific your answer upto two digits after the decimal point)

Ans. : 59.6

Solution: For circular orbit 22

GMmm D

D

2 23

22 22

GM GMT RD

R

2

3 22 24

GM TD R

R

2 2

24

gT R R

3 2 2

2 2 6

9.8 (27 24 60 60)

4 4 6.4 10

D gT

R R

113 3

2 6

5.34 1059.5

4 6.4 10

D

R




Q58. A syringe is used to exert 1.5 atmospheric pressure to release water horizontally. The

speed of water immediately after ejection is________ (take 1 atmospheric pressure 510

Pascal, density of water 3 310 kg m ) (Specify your answer in 1ms as an integer)

Ans: 10

Solution: Apply Bernoulli’s equation,

2 21 1 1 2 2 2

1 1

2 2P v gh ev gh

1 2 1, 0h h v

1 21.5 atm, 1 atmP P

5

1 22

2 2 1 5 0.5 10

1000

P Pv

2 10 /v m s

Q59. A particle of mass m is moving in a circular orbit given by cosx R t ; siny R t ,

as observed in an inertial frame 1S . Another inertial frame 2S moves with uniform

velocity ˆv Ri with respect to 1 1 2. andS S S are related by Galilean transformation,

such that the origins coincide at 0t . The magnitude of the angular momentum of the

particle at 2

t

, as observed in 2S about its origin is expressed as 2mR x . Then x

is _______.


Ans. : 5.28

Solution: From 2S Frame

' ˆ ˆr x vt i yi ˆ ˆcos sinR t vt i R ti

ˆ ˆv x v i yj ˆ ˆsin cosR t v i R tj

ˆ ˆcos sinL r P m R t vt i R ti ˆ ˆsin cosR t v i R tj

2 2 2 2 ˆcos cos sin sinm R t vtR t R t vR t k

12

V

Syringe




2 cos sinm R tR t vR t

at 2

t

, L

2 21 2 5.28m R m R 25.28L m R

IIT-JAM 2019

Q60. The mass per unit length of a rod (length 2m ) varies as =3x kg/m . The moment of

inertia (in kg 2m ) of the rod about a perpendicular-axis passing through the tip of the rod

(at 0x )

(a) 10 (b) 12 (c) 14 (d) 16

Ans. : (b)

Solution:

2422 2

0 0 0

33 12

4

l xI x dx x xdx

Q61. The amount of work done to increases the speed of an electron nom / 3c to 2 / 3c is

( 83 10 m/sc and rest mass of electron is 0..511 MeV )

(a) 56.50keV (b) 143.58 keV (c) 168.20keV (d) 511.00 keV

Ans. : (b)

Solution: Change in kinetic energy is equal to work done

2 2 2 2

2 20 0 0 00 02 2 2 2

2 1 2 12 2 2 2

1 1 1 1

m c m c m c m cW m c m c

v v v v

c c c c

put 1 2/ 3, 2 / 3v c v c 20 0.511m c

143.58 keVW

Q62. If the motion of a particle is described by 5cos 8 , 5sin 8x t y t and 5z t , then

the trajectory of the particle is

(a) Circular (b) Elliptical (c) Helical (d) Spiral

Ans. : (c)

Solution: 5cos 8 , 5sin 8x t y t and 5z t , 2 2 25 , 5x y z t motion is Helical




Q63. A ball of mass m is falling freely under gravity through a viscous medium in which the

drag force is proportional to the instantaneous velocity v of the ball. Neglecting the

buoyancy force of the medium, which one of the following figures best describes the

variation of v as a function of time t ?

(a) (b) (c) (d)

Ans. : (d)

Solution: F V

ma KV

mdV

KVdt

2V t

Q64. Consider an object moving with a velocity v

in a frame which rotates with a constant

angular velocity. The Coriolis force experienced by the object is

(a) Along v

(b) Along

(c) Perpendicular to both v

and

(d) always directed towards the axis of rotation

Ans. : (c) Solution: 2cF m v

Q65. Consider a classical particle subjected to an attractive inverse-square force field. The total

energy of the particle is E and the eccentricity is . The particle will follow a parabolic

orbit if

(a) 0and 1E (b) 0and 1E

(c) 0and 1E (d) 0and 1E

Ans. : (c)

Solution: 2

2

21

EJ

mk for parabolic orbit 0and 1E

v

0 t

v

0 t

v

0 t

v

0 t




Q66. If the wavelength of 2K X -ray line of an element is o

1.544 A . Then the atomic number

Z of the element is_______

(Rydberg constant 7 -11.097 10 mR and velocity of light 83 10 m/sc )

Ans. : 29

Solution: According to Mosely’s formula, the frequency of K X - ray line is related to atomic

number by the formula

215 33.29 10 1

4f K z Hz

or 215 33.29 10 1

4

cz

or 8

2510

3 10 33.29 10 1

41.544 10z

Therefore, 1 28.06z

or 29.06z

Since atomic number must be an integer

29z

Q67. If the diameter of the Earth is increased by 4% without changing the mass, then the

length of the day is _______ hours.

(Take the length of the day before the increment as 24 hours. Assume the Earth to be a

sphere with uniform density). (Round off to 2 decimal places)

Ans. : 25.95

Solution: 221 1 2 2

1 2

2 2.04I I MR M R R

T T

2 22 1 1.04 24 1.04 25.95T T




Oscillations, Waves and Optics

IIT-JAM 2005

Q1. Consider a beam of light of wavelength incident on a system of a polarizer and an

analyzer. The analyzer is oriented at 045 to the polarizer. When an optical component is

introduced between them, the output intensity becomes zero. (Light is incident normally

on all components). The optical component is

(a) a full-wave plate (b) a half-wave plate

(c) a quarter-wave plate (d) an ordinary glass plate

Ans.: (b)

Solution: Half wave plate introduce phase difference of , if incidence wave is plane polarized

than after passing through HWP the wave is also plane polarized. If electric field of the

incidence wave makes angle 045 with optic axis of HWP than plane polarized at output

will be at 045 , as a result it will incidence on polarizer at 090 . According to malus law

intensity at output will be

0)2/(coscos 20

20 III

Q2. A combination of two thin convex lenses of equal focal lengths, is kept separated along

the optic axes by a distance of 20cm between them. The combination behaves as a lens

system of infinite focal length. If an object is kept at 10cm from the first lens, its image

will be formed on the other side at a distance x from the second lens. The value of x is

(a) 10cm (b) 20cm (c) 6.67cm (d) infinite

Ans.: (a)

Solution: As we see in the figure that the image is formed 10cm apart from the second lens.

f

10f cm 10f cm

f

10 cm 20 cm 10 cm




IIT-JAM 2006

Q3. At a given point in space the total light wave is composed of three phasors 1P a ,

iea

P22 and ie

aP

23 . The intensity of light at this point is

(a)

2cos4 22

a (b)

2cos4 42

a

(c) 22 cosa (d) 2cos4 22a

Ans.: (b)

Solution: 1 2 3 2 cos sin cos sin2 2 2

i ia a aP P P P a e e i i

21 cos 2 cos2

a a

2 2 44 cos2

I P a

Q4. A spring-mass system has undamped natural angular frequency 10 100 rad s . The

solution x t at critical damping is given by ttxtx 000 exp1 , where 0x is a

constant. The system experiences the maximum damping force at time

(a) 0.01s (b) 0.1s (c) 0.01 s (d) 0.1 s

Ans.: (a)

Solution: Damping force, dt

dxbFd

For maximum damping force, 0002

2

2

2

dt

xd

dt

xdb

dt

dFd

0 0 00 0 0 0 0 0 0 0 0 01 1t t tdx

x t e x e x x t edt

0 0 0

22

0 0 0 0 0 0 0 0 0 0 0 021 1 0t t td x

x x t e x e x t edt

00

11 0 0.01sect t t




Q5. kztijiAtzyxE expˆ4ˆ3,,,

represents an electromagnetic wave. Possible

directions of the fast axis of a quarter wave plate which convert this wave into a

circularly polarized wave are

(a) jiandji ˆ7ˆ2

1ˆˆ72

1 (b) jiandji ˆ3ˆ4

2

1ˆ4ˆ32

1

(c) jiandji ˆ3ˆ42

1ˆ4ˆ32

1 (d) jiandji ˆ7ˆ

2

1ˆˆ72

1

Ans.: (a)

Solution: The fast axis of the quarter wave plate must make angle of 045 with the direction of

vibration of electric field so that amplitude of ordinary ray and extra-ordinary ray is equal

to produce circularly polarized light.

kztiAEkztijiAtzyxE expexpˆ4ˆ3,,, 0

Where jiE ˆ4ˆ30

Let us calculate the angle between 0E

and jiBandjiA ˆ7ˆ2

1ˆˆ72

1

00

0

1 121 4 25

12 2cos 452525 50 / 2 2

E A

E A

and

00

0

1 13 28 25

12 2cos 452525 50 / 2 2

E B

E B




IIT-JAM 2007

Q6. When two simple harmonic oscillations represented by

0 cosx A t and 0 cosy B t are superposed

at right angles, the resultant is an ellipse with its major axis

along the y -axis as shown in the figure. The conditions

which correspond to this are

(a) 00 2;2

BA (b) 00;

4BA

(c) 002;2

BA (d) 00;

4BA

Ans.: (c)

Q7. Three polarizers P , Q and R are placed parallel to each other with their planes

perpendicular to the z -axis. Q is placed between P and R . Initially the polarizing

directions of P and Q are parallel, but that of R is perpendicular to them. In this

arrangement when unpolaized light of intensity 0I is incident on P , the intensity coming

out of R is zero. The polarizer Q is now rotated about the z -axis. As a function of angle

of rotation, the intensity of light coming out of R is best represented by

(a) (b)

(c) (d)

Ans.: (b)

4oI

8oI

2

2

3 2

40I

80I

2

2

3 2

x

y

2

1O

1 2

40I

80I

2

2

3 2

40I

80I

2

2

3 2




Solution: 01 2

II

202 cos

2

II

2 203 cos cos 90

2

II

30 0I

2 20 0 03

1 145 cos 45cos 45

2 2 2 2 8

I I II

0390 0I

IIT-JAM 2008

Q8. The instantaneous position x t of a small block performing one-dimensional damped

oscillations cosrtx t Ae t a . Here is the angular frequency, the damping

coefficient, A the initial amplitude and the initial phase. If ,00

0v

dt

dxandx

tt

the values of A and (with 0,1,2,....n ) are

(a)

2

12,

2

nvA (b)

n

vA ,

(c)

2

12,

nvA (d)

2

12,

2

nvA

Ans.: (c)

Solution:

0

2 1cos 0 cos 0

2t

nx A

cos sint tdxAe t Ae t

dt

cos sintAe t t

0

cos sint

dx vv A v A v A

dt

2

P Q R

0I 1 0 /2I I

1 0

202 cos

2

II 3I

3 90




IIT-JAM 2009

Q9. Among the following displacement versus time plots, which ones may represent an over-

damped oscillator?

(P) (Q)

(R) (S)

(a) only (P) and (Q) (b) only (P) and (R)

(c) only (P) and (S) (d) only (P), (R) and (S)

Ans.: (a) IIT-JAM 2010

Q10. A quarter-wave plate is placed in between a polarizer and a photo-director. When the

optic axis of the quarter-wave plate is kept initially parallel to the pass axis of the

polarizer and perpendicular to the direction

of light propagation. The intensity of light

passing through the quarter-wave plate is

measured to be 0I (see figure). If the quarter

wave plate is now rotated by 045 about an

axis parallel to the light propagation, what

would be the intensity of the emergent light

measured by the photo-director?

(a) 2oI

(b) 20I

(c) 22

0I (d) 0I

Ans.: (d)

Solution: After passing through QWP plane polarized light of intensity 0I will convert into

circularly polarized with intensity 0I .

Polaroid plate veQuarter wa detector-Photo

platevequarter wa ofrotation

ofDirection

tx t tx

t

tx

t

tx

t




IIT-JAM 2011

Q11. Six simple harmonic oscillations each of same frequency and equal amplitude are

superposed. The phase difference between any two consecutive oscillations i.e.,

1n n is constant, where n is the phase of the thn oscillation. If the resultant

amplitude of the superposition is zero, what is the phase difference ?

(a) 6

(b)

3

(c)

2

(d) 2π

Ans.: (d)

Solution: Resultant amplitude of the superposition of n SHM is

2/sin

2/sin

na

R , where, n

n

sin / 2

0 sin / 2 0 / 2sin / 2

a nR n n

22

Q12. Intensity of three different light beams after passing through an analyzer is found to vary

as shown in the following graphs. Identify the option giving the correct states of

polarization of the incident beams from graphs.

1

00 1 2 3 4 5 6

5.0

1Graph

Analyzer Orientation (Radians)

2Graph

1

00 1 2 3 4 5 6

5.0





(a) Graph1: Linear Polarization Graph2: Circular Polarization, Graph3: Elliptic

Polarization

(b) Graph 1: Circular Polarization, Graph 2: Linear Polarization, Graph 3: Elliptic

Polarization

(c) Graph 1: Unpolarized Graph 2: Circular Polarization, Graph 3: Linear Polarization

(d) Graph 1: Unpolarized Graph 2: Elliptic Polarization, Graph 3: Circular Polarization

Ans.: (b)

Solution: Graph I UPL or CPL

Graph II LPL

Graph III EPL or UPL + LPL or CPL + LPL

Hence answer is (b)

IIT-JAM 2012

Q13. A lightly damped harmonic oscillator loses energy at the rate of 1% per minute. The

decrease in amplitude of the oscillator per minute will be closest to

(a) 0.5% (b) 1% (c) 1.5% (d) 2%

Ans.: (d)

Solution: Decay of energy is governed by equation, teEE 20

Decay of amplitude is governed by equation, tA ae

3Graph

1

00 1 2 3 4 5 6

5.0





Q14. Group I contains x- and y- components of the electric field and Group II contains the type

of polarization of light.

Group I Group II

P. kztEE

kztE

E

y

x

sin

cos20

0

1. Linearly Polarized

Q. kztEE

kztEEy

x

cossin

0

0 2. Circularly Polarized

R. kztEE

kztEEy

x

sinsin

2

1 3. Unpolarized

S.

4sin

sin

0

0

kztEE

kztEE

y

x

4. Elliptically Polarized

The correct set of matches is

(a) 1;4;2;4 SRQP (b) 4;1;3;1 SRQP

(c) 4;1;2;4 SRQP (d) 2;3;1;3 SRQP

Ans.: (c)

Solution: P. kztEEandkztE

E yx sincos2

00

The phase difference between Ex and Ey is /2 with different amplitude. Therefore the

resultant will be elliptically polarized.

Q. kztEEandkztEE yx cossin 00

The phase difference between Ex and Ey is /2 with same amplitude. Therefore the

resultant will be circularly polarized.

R. kztEEandkztEE yx sinsin 21

The phase difference between Ex and Ey is 0 with different amplitude. Therefore the

resultant will be linarly polarized.

S.

4sinsin 00

kztEEandkztEE yx

The phase difference between Ex and Ey is /4 with same amplitude. Therefore the

resultant will be elliptically polarized.




IIT-JAM 2013

Q15. A traveling pulse is given by

2

22222exp,

c

tbxaabxtAtxf where baA ,, and c

are positive constants of appropriate dimensions. The speed of the pulse is

(a) a

b (b)

a

b2 (c)

a

cb (d)

a

b

2

Ans.: (a)

Solution:

2

2

2

2222

exp2

exp,c

btaxA

c

tbxaabxtAtxf

Phase factor is constant

22

2

2

cconstbtaxconstc

btax

Taking differentiation, we get

abdtdxbdtadxbdtadxbtax //00

Velocity of the pulse is b/a

IIT-JAM 2014

Q16. A collimated beam of light of diameter 1 mm is propagating along the x-axis. The beam

is to be expanded to a collimated beam of diameter 10 mm using a combination of two

convex lenses. A lens of focal length of 50 mm and another lens with focal length F are

to be kept at a distance d between them. The values of F and d respectively, are

(a) 450 mm and 10 mm (b) 400 mm and 500 mm

(c) 550 mm and 600 mm (d) 500 mm and 550 mm

Ans.: (d)

Solution: *AO P and A O P (Refer to the figure)

*AO OP

A O OP

2

0.5 50

5

mm mm

mm f 2 500f mm

50 500 550d OO mm




Q17. The electric fields of two light sources with nearby frequencies 1 and ,2 and wave

vectors 1k and ,2k are expressed as tzkieiEE 11ˆ101

and tzkieiEE 22ˆ202

,

respectively. The interference pattern on the screen is photographed at ;0tt denote

02121 tzkk by . For this pattern

(a) a bright fringe will be obtained for 1cos

(b) a bright fringe intensity is given by 220

210 EE

(c) a dark fringe will be obtained for 1cos

(d) a drak fringe intensity is given by 22010 EE

Ans.: (d)

Solution: A bright fringe will be obtained for cos 1

A bright fringe intensity is given by 2

10 20E E

A dark fringe will be obtained for cos 1

Q18. White light is incident on a grating 1G with groove density 600 lines/mm and width

mm.50 A small portion of the diffracted light is incident on another grating 2G with

groove density 1800 lines/mm and width 15 mm. The resolving power of the combined

system is

(a) 3103 (b) 31057 (c) 71081 (d) 510108

Ans.: (c)

Solution: 1 1 1 1 11 600 50R n N N N

2 2 2 2 21 1800 15R n N N N

71 2 81 10R R R

A1 50f mm

O

2f

10 mm1 mm *O

50

2f

A

P




Q19. A stationary source (see figure) emits sound waves of

frequency f towards a wall. If an observer moving

with speed u in a direction perpendicular to the wall,

measures a frequency ff8

9 at the instant shown,

then u is related to the speed of sound sV as

(a) sV (b) 2/sV (c) 4/sV (d) 8/sV

Ans.: (c)

Solution: Velocity component along sound wave

0cos 602

uu

2s

s

uV

f fV

9 28

s

s

uV

f fV

9 8 4s sV V u

44

ss

VV u u

IIT-JAM 2015

Q20. 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)

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

vf

L

/2L

L

WallSource

Observer

30

30

u




with additional rate at 2

L, the fundamental frequency becomes

2

340 / sec1172

22 29 102

v v v mf Hz

LL L m

Q21. 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 mNK X /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 2

1 1

2 4

k kf

f

where k is force constant.

For 2X molecule: 2

x x x

x x

m m m

m m

2 22 2 12

1 1 325 /

2 2 3.14 16.7 10

xx

x

k N mm

f Hz

27 2759.07 10 35.5 1.67 10 35.5 . . .xm kg kg a m u

This is the atomic mass of chlorine Cl .

For 2Y molecule: 2

y y y

y y

m m m

m m

2 2 22 12

1 1 446 /

2 2 3.14 26.8 10

yy

y

k N mm

f Hz

27 2731.73 10 19 1.67 10ym kg kg 19 . . .a m u

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




Q22. 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 / secm . The speed of blood through the vessel, in / secm , 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

0sound b

sound

V Vf f

V

(i)

Sound from blood cell of frequency f reflect back.

The frequency observed by observer is sound

sound 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

0sound b

sound b

V Vf f

V V

(iii)

Now, 0 0 0sound b

sound b

V Vf f f f f

V V

0

2 b

sound b

Vf

V V

0

2 b

sound b

V f

V V f

02sound b

b

V V f

V f

021

soundb

VV

f

f

0f

soundVbV

observerbV

f




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

6

1545 15450.073

210432 1.0522 101

100

bV

0.073 / secbV m

Thus the best suitable answer is option (d).

Q23. The following figure shows a double slit Fraunhofer diffraction pattern produced by two

slits, each of width a separated by a distance bab , .

Which of the following statements are correct?

(a) Reducing a increases the separation between consecutive primary maxima

(b) Reducing a increases the separation between consecutive secondary maxima

(c) Reducing b increases the separation between consecutive primary maxima

(d) Reducing b increases the separation between consecutive secondary maxima

Ans.: (a) and (d)

Solution: The minima condition for double slit Fraunhofer diffraction is

sin sinn

a 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 sinm

b 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).

maximaPrimary

maximaSecondary




Q24. 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 Snell’s law sin

sin

in

r

For ordinary ray 050 , 1.6584oi n

1sin sinsin sino o

o o

i ir r

n n

0

1 1 1sin 50 0.766sin sin sin 0.462

1.6584oo

rn

0

0 27.51r

For extra-ordinary ray 050 , 1.4864ei n

1sin sinsin sine e

e e

i ir r

n n

0

1 1 1sin 50 0.766sin sin sin 0.515

1.4864ee

rn

031.02er

Thus, the angular separation between the o - ray and e - ray is 03.51e or r

050

Calciteaxis Optic

Air

050i

or

er

- rayo

- raye




Q25. 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 point P .

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

(in 2/mW m ) is ___________.

Ans.: 9.23

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

The total intensity at is

* 2 2 22 2 cos 2 cos cos 1I EE A B A AB a a where

2 2 2sin

d d d y

D

3 3

7

3.22 10 15 102 505.7

6 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

mm22.3

mm04.2C

B

A

P

mm15

m1

ad

d

A

B

C

D O

y

p




IIT-JAM 2016

Q26. Consider a particle of mass m following a trajectory given by 0 1cosx x t and

0 2siny y t , where 0 0 1, ,x y and 2 are constants of appropriate dimensions. The force on

the particle is

(a) central only if 1 2 (b) central only if 0 0x y and 1 2

(c) always central (d) central only if 0 0x y and 1 2

Ans.: (c)

Solution: 20 1 1cosx x t 2

0 2 2cosy y t

2 20 2 2 0 1 1

ˆ ˆ ˆ ˆcos cosr xi yi r y ti x tj

If 1 2 then 2 2 20 2 2 0 1 1

ˆ ˆcos cosr y ti x tj r

Q27. Two sinusoidal signals of frequency x and y having same

amplitude are applied to x - and y - channels of a cathode ray

oscilloscope (CRO), respectively. The following stationary figure

will be observed when

(a) y x (b) phase difference is 0

(c) 2y x (d) phase difference is / 2

Ans.: (b)

Solution: number of cuts on -axis 4

number of cuts on -axis 2x

y

y

x

2x y

and this lissajous figure appears when phase difference is 0 . Thus correct option is (b)

Q28. Light traveling 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.: (c)

Solution: According to Fermat’s principle, the ray will correspond to that path for which the time

taken is an extremum in comparison to nearby paths i.e. it is either a minimum or a

maximum or stationary. Thus correction option is (c).

1

11

1

x

y




Q29. A train passes through a station with a constant speed. A stationary observer at the station

platform measures 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 , 54 /Hz km h (b) 463 , 52 /Hz km h

(c) 463 , 56 /Hz km h (d) 464 , 52 /Hz km h

Ans.: (a) Solution: Let of = original frequency of the whistle

af = observed frequency when train approaches platform

rf = observed frequency when train recedes platform

tv = velocity of train

v = velocity of sound in air

a ot

vf f

v v

and r ot

vf f

v v

Now, a t a rt

r t a r

f v v f fv v

f v v f f

484 442330 15 / sec 54 /

484 442tv m km hr

and 330 15

484 462330

a to

f v vf Hz

v

Q30. The minimum length of a plane mirror to see the entire full-length image of an object is

half of the object’s height. Suppose is the distance between eye and top of the head of

a person of height h . The person will be able to see his entire full-length image with a

mirror of height / 2h fixed on the wall

(a) when the bottom edge of mirror is kept / 2h 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)




Solution: Let BH h is the height of person

HE , where H represents top of head and E represents eye.

In the mirror, distance between eye and top of head will be 2

.

Since total height of mirror is 2

h,

therefore, for diagram 2 2

hB E

and BE h

2BH h

and 2 2 2 2 2

h hBB BH B H h

Q31. A particle travels in a medium along a horizontal linear path. The initial velocity of the

particle is 0v 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)

(c) (d)

Ans.: (d)

Solution: Viscous force instantaneous velocity

mdv t dv t bF bv t bv t dt

dt v t m

Integrating on both sides

ln

dv t b bdt v t t c

v t m m

h

B

E

H

/2

H

E / 2h

B

v t

t

v t

t

v t

t

v t

t




where 00,t v t v 0lnc v

0

0 0

ln ln lnb

tm

v t v tb bv t t v t e

m v m v

Thus graph (d) correctly represent the variation of v t w.r.t. time.

Q32. A lightly damped harmonic oscillator with natural frequency 0 is driven by a periodic

force of frequency . The amplitude of oscillation is maximum when

(a) is slightly lower than 0

(b) 0

(c) is slightly higher than 0

(d) The force is in phase with the displacement

Ans.: (a)

Solution: Amplitude in driven oscillator is

0

22 2 2 20

/

4

F mA

b

To find the condition for maximum amplitude, differentiate above equation w.r.t. and

put 0dA

d

i.e.

2 2 200

1/ 222 2 2 20

2 2 810

24

bFdA

d mb

2 2 2 2 20 02 0 2b b

Thus is slightly lower than 0 . Correct option is (a).




Q33. A block of mass 0.38kg is kept at rest on a frictionless surface and attached to a wall

with a spring of negligible mass. A bullet weighing 0.02kg moving with a speed of

200 /m s hits the block at time 0t and gets stuck to it. The displacement of the block

(in metre) with respect to the equilibrium position is given by

(Spring constant 640 /N m )

(a) 2sin 5t (b) cos10t (c) 0.4cos 25t (d) 0.25sin 40t

Ans.: (d)

Solution: 640

16000.38 0.02

k k

m m m

40 rad/sec

Let v be the velocity acquired by the block m where bullet m strikes it and comes to

rest in it.

By conservation of momentum

0.02200

0.38 0.02

mm m v m v v v

m m

0.02

200 10 / sec0.4

m

The block is set in oscillation about it mean position with maximum amplitude A

sin cosdx

x A t A tdt

In the mean position, the velocity is maximum

10 1010 0.25

40A A

0.25sin 40x t




Q34. In the optical arrangement as shown below, the axes

of two polarizing sheets P and Q are oriented such

that no light is detected. Now when a third polarizing

sheet R is placed in between P and Q , then light is

detected. Which of the following statement s is (are) true?

(a) Polarization axes of P and Q are perpendicular to each other.

(b) Polarization axis of R is not parallel to P

(c) Polarization axis of R is not parallel to Q

(d) Polarization axes of P and Q are parallel to each other.

Ans.: (a), (b) and (c)

Solution: According to Malu’s law

20 cos2

II where is angle between pass axis of P and Q

where 0I , 090

i.e. P and Q are perpendicular to each other. Thus option (a) is correct.

If third polarizer R is introduced between P and Q making angle 1 w.r.t. pass axis of

P and 0190 w.r.t. .

2 201 1cos cos 90

2

II

If 1 0 , then 0I thus R can’t be parallel to P . Now, If 01 90 , then again 0I .

Thus R can’t be parallel to also.

Thus options (a), (b) and (c) are correct.

Q35. 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 0.5m above it. If the radius of curvature of the lens is

0.75m then, the refractive index of the material is…………….

Ans.: 1.75

Solution: For bi-convex lens

detector

QP

unpolarized light

QP

unpolarized light

I0I

1R

2R0.5m f




1 2

1 1 11

R R f

1 2

1 1 1 11 0.75 1.75

1 10.51 10.75 0.75

f

R R

IIT-JAM 2017

Q36. The dispersion relation for electromagnetic waves travelling in a plasma is given as

2 2 2 2pc k , where c and p are constants. In this plasma, the group velocity is:

(a) proportional to but not equal to the phase velocity

(b) inversely proportional to the phase velocity

(c) equal to the phase velocity

(d) a constant

Ans. : (b)

Solution: 2 2 2 2pc k

22 2 2

2 2 2p g

p

d c kc k v

dk c k

and 2 2 2

p

p

c kv

k k

2

gp

cv

v

Q37. Which of the following is due to inhomogeneous refractive index of earth’s atmosphere?

(a) Red colour of the evening Sun

(b) Blue colour of the sky

(c) Oval shape of the evening Sun

(d) Large apparent size of the evening Sun

Ans. : (c)

Solution: At sunrise and sunset, the Sun is near the horizon. Due to inhomogenous refractive

index of atmosphere, the rays from the upper and lower part of the periphery of the Sun

bend unequally on traveling through earth’s atmosphere. That is why the Sun appears

oval at the time of sunrise and sunset.




Q38. A pendulum is made of a massless string of length L and a small bob

of negligible size and mass m . It is released making an angle

0 1rad from the vertical. When passing through the vertical, the

string slips a bit from the pivot so that its length increases by a small

amount L in negligible time. If it swings up to angle 1 on the other side before

starting to swing back, then to a good approximation which of the following expressions

is correct?

(a) 1 0 (b) 1 0 12L

(c) 1 0 1L

(d) 1 0

31

2L

Ans. : (b)

Solution: 1 cos 1 cosi fmgl mg l

2 22sin 2sin2 2

fil l

1/ 2 1/ 2

1 1i f f il l

0 01 , 12 2f i i fl l

Q39. Consider two coherent point sources ( 1S and 2S ) separated by a small distance along a

vertical line and two screens 1P and 2P as shown in Figure. Which one of the choices

represents the shapes of the interference fringes at the central regions on the screens?

(a) Circular on 1P and straight line on 2P

(b) Circular on 1P and circular on 2P

(c) Straight lines on 1P and straight lines on 2P

(d) Straight lines on 1P and circular on 2P

Ans. : (a)

0 1L L

mm

1P

2P1S2S




Solution: The locus of constant path difference on plate 2P is straight line. Therefore on plate 2P

interference fringes are straight line in nature. Whereas on plate 1P the locus of constant

path difference is circular, therefore fringes are circular

Q40. Unpolarized light is incident on a combination of polarizer, a 2

plate and a

2

and a

4

plate kept one after the other. What will be the output polarization for the following

configurations?

Configuration 1: Axes of the polarizer, the 2

plate and the

4

plate are all parallel to

each other

Configuration 2: The 2

plate is rotated by 045 with respect to configuration 1.

Configuration 3: The 4

plate is rotated by 045 with respect to configuration 1.

(a) Linear for configuration 1 linear for configuration 2, circular for configuration 3.

(b) Linear for configuration 1 circular for configuration 2, circular for configuration 3.

(c) Circular for configuration 1 circular for configuration 2, circular for configuration 3.

(d) Circular for configuration 1 linear for configuration 2, circular for configuration 3.

Ans. : (b)

Solution: The output of polarization is 2 2

22 2

2cos sin

x y xy

a b ab

where 0 0cos , sin and a E b E is phase

Difference between o and e -rays at the exit of plate

Configuration 1:

P

LP

/ 2 / 4

LP

00 00

(Circularly Polarized)LP

X

y

0E




Configuration 2:

Configuration 3:

Thus correct option is (b)

Q41. Unpolarized light of intensity 0I passes through a polarizer 1P . The light coming out of

the polarizer falls on a quarter-wave plate with its optical axis at 045 with respect to the

polarization axis of 1P and then passes through another polarizer 2P with its polarization

axis perpendicular to that of 1P . The intensity of the light coming out of 2P is I . The

ratio 0 /I I is………….


Ans. : 4.00

Solution The intensity at the emerged beam at the exit of second polaroid 2P is

20 sin 24

II

045 045

P

LP

/ 2 / 4

CP (Circularly Polarized)LP

/ 2

00 045

P

LP

/ 2 / 4

CP (Circularly Polarized)

/ 2




when 045

2 00 0sin 904 4

I II

0 4I

I

Q42. An anti-reflection film coating of thickness 0.1 m is to be deposited on a glass plate for

normal incidence of light of wavelength 0.5 m . What should be the refractive index of

the film?


Ans. : 1.25

Solution The condition for constructive interference is1

22

t m

, where 0,1,2m

for 0m

6

6

/ 2 0.5 10

2 4 4 0.1 10

m

t t m

5

1.254

Q43. Intensity versus distance curve for a double slit diffraction experiment is shown in the

figure below. If the width of each of the slits is 0.7 m , what is the separation between

the two slits in micrometers? (Specify your answer to two digits after the decimal point)

Ans. : 3.50

Inte

nsit

y

Distance

1P2P

0I

/ 4




Solution: The condition for the absent order is d n

b m

where, for 1,m 5th interference

maxima is absent (from figure)

1, 5m n

5 5 5 0.7 3.5d

d b m mb

IIT-JAM 2018

Q44. Two vehicles A and B are approaching an observer O at rest with equal speed as shown

in the figure. Both vehicles have identical sirens blowing at a frequency sf . The observer

hears these sirens at frequency Af and Bf , respectively from the two vehicles. Which

one of the following is correct?

(a) A B sf f f (b) A B sf f f

(c) A B sf f f (d) A B sf f f

Ans.: (b)

Solution: Doppler shift

sA s

s A

vf f

v v

, sB s

s B

vf f

v v

A B sV V V

A B sf f f

b

e d

A

BO

A

B

O




Q45. Which of the following arrangements of optical components can be used to distinguish

between an unpolarised light and a circularly polarised light?

(a) (b)

(c) (d)

Ans.: (c)

Solution: (i) In configuration (A), output will be linearly polarized for both

(ii) In configuration (B), output will be linearly polarized for both

(iii) In configuration (C), output will be linearly polarized of constant intensity if input is

unpolarised whereas it is linearly polarized with intensity varying from zero to maximum

if input is circularly polarized.

(iv) In configuration (D) output will be linearly polarized for both.

Q46. The plane of polarisation of a plane polarized light rotates by 060 after passing through a

wave plate. The pass-axis of the wave plate is at an angle with respect to the plane of

polarization of the incident light. The wave plate and are

(a) 0,604

(b) 0,30

2

(c) 0,120

2

(d) 0,30

4

Ans.: (b)

Solution: When plane polarized light is incident on the / 4 plate, it converts it into circularly

polarized light, whereas / 2 plate rotates is by angle 2 , where is angle between

fast axis and polarization direction.

Given, 0 02 60 30 .

Light

/ 2plate analyser

Light

/ 4plate analyser

Light

/ 4plate analyserpolariser

Light

/ 2plate analyserpolariser




Q47. Consider two waves 1 cosy a t kz and 2 cosy a t k k z . The

group velocity of the superposed wave will be ( and k k )

(a) k k

(b) 2

2k k

(c) k

(d) k k

Ans. : (c)

Solution: 1 cosy a t kz , 2 cosy a t k k z

1 2

2 22 cos cos

2 2 2

t kz k ky y y a t z

/ 2

/ 2gvk k

Q48. Consider a convex lens of focal length f . A point object moves towards the lens along

its axis between 2 f and f . If the speed of the object is 0V , then its image would move

with speed IV . Which of the following is correct?

(a) I oV V ; the image moves away from the lens.

(b) I oV V ; the image moves away from the lens.

(c) I oV V ; the image moves away from the lens.

(d) I oV V ; the image moves away from the lens.

Ans. : (c)

Solution: 2

0

1 1 1andI

fV V

f u v u f

For, 0, andiu f v V u decreases than v increases.

0IV V and image moves away from the lens.

Q49. Two beams of light in the visible range 400 700nm nm interfere with each other at a

point. The optical path difference between them is 5000 nm . Which of the following

wavelengths will interfere constructively at the given point?

(a) 416.67nm (b) 555.55 nm (c) 625 nm (d) 666.66 nm

Ans.: (a),(b) and (c)




Solution: 2.p d

For constructive interference 2n where n is integer

22 .n p d

. 5000p d nm

n n

for, 8, 625n nm

9, 555 55n nm

10, 500n nm

11, 454 5n nm

12, 416 67n nm

Thus, correct options are (a),(b) and (c)

Q50. Consider a convex lens of focal length f . The lens is cut along a diameter in two parts.

The two lens parts and an object are kept as shown in the figure. The images are formed

at following distances from the object:

(a) 2 f (b) 3 f (c) 4 f (d)

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

Solution: For first lens 1 1 1

v u f

For second lens 1 1 1

v u f

(i) if , ,u v f v (ii) if 2 , 2 , 2u f v f v f

(iii) if 2 , 2u f v f No image (iv) if 2 , 2 , 2u f v f v f

(v) if , ,u f v v (vi) ,u f v ve , No image

Thus, V cannot be 2 f . The correct options are (b),(c) and (d)

0 f2 f

x




Q51. In a grating with grating constant d a b , where a is the slit width and b is the

separation between the slits, the diffraction pattern has the fourth order missing. The

value of b

a is______. (Specific your answer as an integer.)

Ans. : 3

Solution: 1d a b b

n m m ma a a

For, 1, 4m n 4 1 3b b

a a

Q52. Consider a slit of width 18 m which is being illuminated simultaneously with light of

orange color (wavelength 600nm ) and of blue color (wavelength 450nm ) . The

diffraction pattern is observed on a screen kept at a distance in front of the slit. The

smallest angle at which only the orange color is observed is 1 and the smallest angle at

which only the blue color is observed is 2 . The angular difference 2 1 (in degrees)

is___________


Ans.: 0.48

Solution: sind

9

1 1 011 6

600 10sin sin 1.91

18 10d

9

1 1 022 6

450 10sin sin 1.43

18 10d

02 1 0.48 .




IIT-JAM 2019

Q53. A thin lens of refractive index 3

2 is kept inside a liquid of refractive index

4

3. If the focal

length of the lens in air is 10 cm , then the focal length inside the liquid is

(a) 10 cm (b) 30cm (c) 40cm (d) 50cm

Ans. : (c)

Solution: 1 2

1 3 1 11

2af R R

1 2

1 3 / 2 1 11

4 / 3lf R R

31

24

91

8

l

a

f

f

4 4 10 40l af f cm

Q54. Light of wavelength (in free space) propagates through a dispersive medium with

refractive index 1.5 0.6n . The group velocity of a wave travelling inside this

medium in units of 810 m/s is

(a) 1.5 (b) 2.0 (c) 3.0 (d) 4.0

Ans. : (b)

Solution: 2

gd d d

v kdk d dk

2

2

2

2

d dk

d d

2 2

2 p

cd c ckn

d n v

2 1dc

d n

22 2

0 1d

n ndc

n

2

2 2

1d

n nd

cn




2 2

0.6 1.5 1.2

1.5 2.6

nc c

n

710

1.5

cm

822 10

3c

Q55. The maximum number of intensity minima that can be observed I the Fraunhofer

diffraction pattern of a single slit (width 10 m ) illuminated by a laser bean

(wavelength 0.630 m ) will be

(a) 4 (b) 7 (c) 12 (d) 15

Ans. : (d)

Solution: sine n

max10

15.87 150.63

e mn

m

Q56. For an under damped harmonic oscillator with velocity v t

(a) Rate of energy dissipation varies linearly with v t

(b) Rate of energy dissipation varies as square of v t

(c) The reduction in the oscillator frequency, compared to the undamped case, is independent of v t

(d) For weak damping, the amplitude decays exponentially to zero

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

Solution: Displacement sinrtx Ae t

Velocity cosrtdxv A e t

dt

Energy 2 2 2 2 2 21 1 1,

2 2 2rtE mv m x A e

Power dissipation, 2 2 212

2rtdE

P m A e rdt

2P v

Power dissipation is proportional to 2v , thus option (a) is wrong and option (b) is correct.




Also, displacement sinrtx Ae t decays exponentially to zero, thus option (d) is

also correct.

The damped oscillation frequency is

2 20 r

It is independent of v t . Thus option (c) is also correct.

Q57. An object of 2cm height is placed at a distance of 30cm in front of a concave mirror

with radius of curvature 40cm . The height of the image is ________cm.

Ans. : 4

Solution: 30u cm

20f cm

1 1 1

v u f

1 1 1 1 1

20 30v f u

1 1

60v 60v cm

I v

mO u

602 4

30I cm cm

Q58. The 7th bright fringe in the Young’s double slit experiment using a light of wavelength

550nm shifts to the central maxima after covering the two slits with two sheets of

different refractive indices 1n and 2n but having same thickness 6 m . The value of

1 2n n is _______.


Ans. : 0.64

Solution: 1 21 1 7n t n t

9

1 2 6

7 7 550 100.64

6 10n n

t

R O F20cm

30cm

O

2cm




Q59. Light of wavelength 680 nm is incident normally on a diffraction grating having

4000 lines/cm . The diffraction angle (in degrees) corresponding to the third-order

maximum is ______


Ans. : 055

Solution: sine d n

2

910sin 3 680 10

4000

1 0sin 0.82 55




Electricity and Magnetism

IIT-JAM 2005

Q1. A small loop of wire of area 20.01A m , 40N turns and resistance 20R is

initially kept in a uniform magnetic field B in such a way that the field is normal to the

loop. When it is pulled out of the magnetic field, a total charge of 52 10Q C flows

through the coil. The magnitude of magnetic field B is

(a) 31 10 T (b) 34 10 T

(c) zero (d) unobtainable, as the data is insufficient

Ans.: (a)

Solution: Magnetic flux through the loop NBA

Induced e.m.f dt

d and induced currentdt

dQ

dt

d

Ri

1dQd

R 1

.

Thus , 510201.04020

1 B TB 3101 .

Q2. Two point charges 1q and 2q are fixed with a finite distance d between them. It is

desired to put a third charge 3q in between these two charges on the line joining them so

that the charge 3q is in equilibrium. This is

(a) possible only if 3q is positive

(b) possible only if 3q is negative

(c) possible irrespective of the sign of 3q

(d) not possible at all

Ans. : (c)

Solution: If 3q is positive,

If 3q is negative,

In both case there is possibility that charge 3q may be in equilibrium.

1q 3q

2F 1Fd

3q2q

1q 3q

1F 2Fd

3q2q




IIT-JAM 2006

Q3. Two electric dipoles 1P and 2P are placed at 0,0,0 and 1,0,0 respectively with both

of them pointing in the z direction. Without changing the orientations of the dipoles 2P

is moved to 0, 2,0 . The ratio of the electrostatic potential energy of the dipoles after

moving to that before moving is

(a) 1

16 (b)

2

1 (c)

4

1 (d)

8

1

Ans. : (d)

Solution: Electrostatic potential energy3

1

rU

8

13

2

31

1

2 r

r

U

U

Q4. A small magnetic dipole is kept at the origin in the -x y plane. One wire 1L is located at

az in the -x z plane with a current I flowing in the positive x direction. Another

wire 2L is at az in -y z plane with the same current I as in 1L , flowing in the

positive y -direction. The angle made by the magnetic dipole with respect to the

positive x -axis is

(a) 0225 (b) 0120 (c) 045 (d) 0270

Ans.: (a)

Solution: Magnetic field at 0z due to wire at az is yBB ˆ .

Magnetic field at 0z due to wire at az is xBB ˆ .

Resultant magnetic field at 0z makes an angle of 045 with x and 0225 with x .




IIT-JAM 2007

Q5. A uniform and constant magnetic field B coming out of the

plane of the paper exists in a rectangular region as shown in

the figure. A conducting rod PQ is rotated about O with a

uniform angular speed in the plane of the paper. The emf

PQE induced between P and Q is best represented by the

graph

(a) (b)

(c) (d)

Ans.: (a)

Solution: When point P is inside due to motional emf , potential PQ is positive. When point Q

is inside potential QP is positive or potential PQ is negative.

IIT-JAM 2008

Q6. If the electrostatic potential at a point ,x y is given by 2 4V x y volts, the

electrostatic energy density at that point 3/in J m is

(a) 05 (b) 010 (c) 020 (d) 20 42

2

1yx

Ans.: (b)

B

P

O

Q

tO

PQE

tO

PQE

tO

PQE

tO

PQE




Solution: yxVE ˆ4ˆ2 20 /E V m

Electrostatic energy density 300

2

0 /10202

1

2

1mJE

IIT-JAM 2009

Q7. An oscillating voltage 0 cosV t V t is applied across a parallel

plate capacitor having a plate separation d . The displacement current

density through the capacitor is

(a) d

tV cos00 (b) d

tV cos000

(c) d

tV cos000 (d) d

tV sin00

Ans.: (d)

Solution: Displacement current density

d

tV

t

tV

dt

EJ d

sin0000

Q8. An electric field ˆcossinˆ rrE

exists in space. What will be the total charge

enclosed in a sphere of unit radius centered at the origin?

(a) 04 (b) 04 (c) 04 (d) 04

Ans.: (a)

Solution: 02

00 4ˆsinˆcossinˆ rddrradEQenc

tVtV cos0

d




IIT-JAM 2010

Q9. The magnetic field associated with the electric field vector jtkzEE ˆsin0

is given

by

(a) itkzc

EB ˆsin0

(b) itkzc

EB ˆsin0

(c) jtkzc

EB ˆsin0

(d) ktkzc

EB ˆsin0

Ans.: (a)

Solution:

0ˆˆ sinkz E kz t jk E

B

0 0ˆ ˆsin sinkE E

kz t i kz t ic

Q10. Assume that 0z plane is the interface between two linear and homogeneous dielectrics

(see figure). The relative permittivities are 5r for

0z and 4r for 0z . The electric field in the

region 0z is mVkkjiE ˆ4ˆ5ˆ31 . If there are

no free charges on the interface, the electric field in

the region 0z is given by

(a) mVkkjiE

ˆˆ

4

5ˆ4

32 (b) mVkkjiE ˆˆ5ˆ32

(c) mVkkjiE ˆ5ˆ5ˆ32 (d) mVkkjiE ˆ5ˆ5ˆ32

Ans.: (d)

Solution: 1 2 2ˆ ˆ3 5E E E i j

and 0f kkEEDD ˆ5ˆ44

51

2

1221

mVkkjiE ˆ5ˆ5ˆ32

z

0z

4r

5r




Q11. A closed Gaussian surface consisting of a hemisphere and a circular disc of radius R , is

placed in a uniform electric field E

, as shown in the figure. The circular disc makes an

angle 030 with the vertical. The flux of the electric field vector coming out of the

curved surface of the hemisphere is

(a) 21

2R E

(b) 23

2R E

(c) 2R E

(d) 22 R E

Ans.: (b)

Solution: xEzExEzEE ˆ2

1ˆ

2

3ˆ30sinˆ30cos

rddRxEzEadE

S

E ˆsinˆ2

1ˆ

2

3 2

2/

0

2

0

2 sincossin2

1cos

2

3

ddEERE

2/

0

2

0

2 sincos2

3

ddERE

2/

0

2

0

22 cossin2

1

ddER

ERERE22

2

30

2

12

2

3

OR

0 2cos30E

S

E da E R 23

2R E

E

z

x

030

E




IIT-JAM 2011

Q12. Equipotential surface corresponding to a particular charge distribution are given by

22 24 2 ix y z V , where the values of iV are constants. The electric field E

at the

origin is

(a) 0E

(b) ˆ2E x

(c) ˆ4E y

(d) ˆ4E y

Ans.: (d)

Solution: ˆ ˆ ˆˆ8 2 2 2 0,0,0 4E V xx y y zz E y

IIT-JAM 2012

Q13. A parallel plate air-gap capacitor is made up of two plates of area 210cm each kept at a

distance of 0.88mm . A sine wave of amplitude 10V and

frequency 50 Hz is applied across the capacitor as shown in the

figure. The amplitude of the displacement current density (in

2/mA m ) between the plates will be closest to

(a) 0.03 (b) 0.30 (c) 3.00 (d) 30.00

Ans.: (a)

Solution: Displacement current density,

d

tV

t

tV

dt

EJ d

sin0000

Amplitude of the displacement current density (in mA/m2) , 0 0 0 00

2d

V fVJ

d d

200 0 9 5

1 50 104 0.03 /

2 9 10 2 88 10d

fVJ mA m

d

~




Q14. A segment of a circular wire of radius R, extending from 0 to 2/ , carries a constant

linear charge density . The electric field at origin O is

(a) yxR

ˆˆ4 0

(b)

yx

Rˆ

2

1ˆ

2

1

4 0

(c)

yx

Rˆ

2

1ˆ

2

1

4 0

(d) 0

Ans.: (a)

Solution: ˆ ˆx yE E x E y

where cosx

line

E dE , siny

line

E dE .

and 2

0

1

4

dldE

R

.

/2

2 20 0 0

1cos cos

4 4x

line

dl RdE

R R

/2

00 0

sin4 4xE

R R

Similarly/2

2 20 0 0

1sin sin

4 4y

line

dl RdE

R R

/2

00 0

cos4 4yE

R R

Thus 0

ˆ ˆ ˆ ˆ4x yE E x E y x y

R

O

R

y

x

y

x

d E

O

R

dl




IIT-JAM 2014

Q15. A particle of mass m carrying charge q is moving in a circle in a magnetic field B .

According to Bohr’s model, the energy of the particle in the thn level is

(a)

m

hqB

n 2

1 (b)

m

hqBn

(c)

m

hqBn

2 (d)

m

hqBn

4

Ans.: (d)

Solution: 2 2 2

2n

n

q B rE

m 2n

n n n n nn

mv m n nmv r n and r r r

qB qB mr qB

2 2 2 2 2

2 2 4n

n

q B r q B n qBhE n

m m qB m

Q16. A conducting slab of copper PQRS is kept on the -x y plane in a uniform magnetic field

along x - axis as indicted in the

figure. A steady current I flows

through the cross section of the slab

along the y - axis. The direction of

the electric field inside the slab,

arising due to the applied magnetic

field is along the

(a) negative Y direction (b) positive Y direction

(c) negative Z direction (d) positive Z direction

Ans.: (c) Q17. In a parallel plate capacitor the distance between the plates is 10cm . Two dielectric slabs

of thickness 5cm each and dielectric constants 21 K and 42 K respectively, are

inserted between the plates. A potential of 100V is applied across the capacitor as shown

in the figure. The value of the net bound surface charge density at the interface of the two

dielectrics is

(a) 03

2000 (b) 03

1000 (c) 0250 (d) 03

2000

P

S

Q

B

X

Z

Y

R

I

cm10 V1004K 2

2K1




Ans.: (a)

Solution: 1 21 2 0 0 0

3

2 4 4V E d E d d d d d d

2100 , 5 10V volts d cm 4

0 002

4 4 4 10100

3 3 5 10 15V

d

1 0 1 0 1 1 1 00

12 2eP E K E

2 0 2 0 2 2 2 00

31 3

4 4eP E K E

4

1 2 0 0

3 1 4 10 2000

2 4 4 4 15 3

Q18. A rigid uniform horizontal wire PQ of mass M , pivoted at P , carries a constant current

I .

It rotates with a constant angular speed in a

uniform vertical magnetic field B . If the current

were switched off, the angular acceleration of

the wire, in terms of B, M and I would be

(a) 0 (b) M

BI

3

2 (c)

M

BI

2

3 (d)

M

BI

Ans.: (c)

Solution: Torque mr F I

d r dF l IBdl

F I dl B dF IBdl

2

0 2

L IBLIB ldl

Moment of inertia about point P , 2

3m

MLI

mI 2 2

2 3

IBL ML 3

2

BI

M

2

11

24K 2

1K 2

P Q




Q19. A steady current in a straight conducting wire produces a surface charge on it. Let outE

and inE be the magnitudes of the electric fields just outside and just inside the wire,

respectively. Which of the following statements is true for these fields?

(a) outE is always greater than inE

(b) outE is always smaller than inE

(c) outE could be greater or smaller than inE

(d) outE is equal to inE

Ans.: (a)

Solution: In this case 0, 0in outE E . So out inE E

Q20. A small charged spherical shell of radius 0.01m is at a potential of 30V . The

electrostatic energy of the shell is

(a) J10 10 (b) J105 10 (c) J105 9 (d) J10 9

Ans.: (b)

Solution: 04

qV

R and

2

08

qW

R .

Thus, 2 2 2

0 9 1009

0

4 4 900 100.5 10 5 10 Joules

8 2 9 10 2

VR V RW

R

Q21. A ring of radius R carries a linear charge density . It is rotating with angular speed .

The magnetic field at its center is

(a) 2

3 0 (b)

20

(c) 0 (d) 0

Ans.: (b)

Solution: 0

2

IB

R

, where I v R . Thus 0

2B

.




IIT-JAM 2015

Q22. The electric field of a light wave is given by

4sinˆsinˆ

0

kztjkztiEE

.

The polarization state of the wave is

(a) Left handed circular (b) Right handed circular

(c) Left handed elliptical (d) Right handed elliptical

Ans.: (c)

Solution: 0 0sin , sin4x yE E t kz E E t kz

.

Thus resultant is elliptically polarized wave.

At 0 00, sin , sin4x yz E E t E E t

When 00, 0,2

x y

Et E E and when 0, , 0

4 2x y

Et E E

Q23. 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

, 0

2q q

q q

Q24. A positively charged particle, with a charge q , enters a region in which there is a uniform

electric field E

and a uniform magnetic field B

, both directed parallel to the positive

y -axis. At 0t , 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) 2

2

2qB

mE (b)

2

22

qB

mE (c)

20

2

v mmE

qBqB

(d)

qB

mv02




Ans.: (b)

Solution: 2 2

22

1 1 2 2

2 2y y

qE m mEy u t a t y

m qB qB

Q25. 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

1

4

q

d

(c) 0 1

1

4

q q

d R

(d) 0 1 2

1

4

q q q

d R R

Ans.: (d)

Solution: Charge induced on inner surface is q and charge induced on outer surface is q .

Thus, 0 1 2

1

4

q q qV

d R R

.

Q26. 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) R

I

2

3 0 (b) R

I

320 (c)

R

I

03

(d) R

I

2

3 0

Ans.: (c)

Solution: 0 3cos30

2d R R

02 1sin sin

4

IB

d

0 00 0 01 2sin 30 2sin 30

4 3 2 34

2

I I IB

d RR

z

y

x0v

,E B

q d

1R

2R

C

RI

060

I

R d

C




The magnitude of the magnetic field at center of the circle is

0 0 01

3 36 6

2 3 3

I I IB B

RR R

Q27. For an electromagnetic wave traveling in free space, the electric field is given

by m

VjkxtE ˆ10cos100 8

. 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 ˆ100cos 10 /E t kx j V m

88 8

8

2 2 3 1010 10 6

10

c

. Option (a) is true

ˆ ˆ ˆ ˆB k E x y z

. Option (b) is wrong

ˆ ˆS k x

. Option (c) is wrong. Option (d) is true.

Q28. 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)

R

L

C




Solution: 3 6

1 1178

2 2 3.14 8 10 100 10Hz

LC

. Option (a) is true.

1CX

C CX as . Option (b) is wrong

Option (c) is true

Option (d) is wrong

Q29. A unit cube made of a dielectric material has a polarization jiP ˆ4ˆ3

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: ˆ ˆ3 4P i j

. 0b P

. Option (a) is wrong

At 0x , ˆ ˆ ˆˆ. 3 4 . 3b P n i j i

, At 1x , ˆ ˆ ˆˆ. 3 4 . 3b P n i j i

Option (b) is true

At 0y , ˆ ˆ ˆˆ. 3 4 . 4b P n i j j

, At 1y , ˆ ˆ ˆˆ. 3 4 . 4b P n i j j

Option (c) is true.

Option (d) is wrong

Q30. The power radiated by sun is W26108.3 and its radius is km5107 . The magnitude of

the Poynting vector (in2cm

W) at the surface of the sun is………………

Ans.: 6174

Solution:

262 2

210

3.8 10/ 6174 /

4 7 10

PI W cm W cm

A




Q31. 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 6364

10 10 10 1.65 1010t RCV

I e e e AR e

Q32. 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 m

t t

Q33. 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: 1

1 2

010 0

1 2 0

22 2 1.690.93

1.69 2.25

rTT I

I r r

EnE E

n n E




IIT-JAM 2016

Q34. For an infinitely long wire with uniform line-charge density, along the z - axis, the

electric field at a point , ,0a b away from the origin is

( ˆ ˆ,x ye e and ˆze are unit vectors in Cartesian – coordinate system)

(a) 2 2

0

ˆ ˆ2

x ye ea b

(b) 2 2

0

ˆ ˆ2

x yae bea b

(c) 2 2

0

ˆ2

xea b

(d) 2 2

0

ˆ2

zea b

Ans.: (b)

Solution: 2 2 2

0 0 0

ˆ ˆ ˆ2 2 2

x yE r r ae ber r a b

2 2r a b

Q35. A 1 W point source at origin emits light uniformly in all the directions. If the units for

both the axes are measured in centimeter, then the Poynting vector at the point 1,1,0 in

2

W

cm is

(a) 1ˆ ˆ

8 2x ye e

(b) 1

ˆ ˆ16 x ye e

(c) 1ˆ ˆ

16 2x ye e

(d) 1

ˆ ˆ4 2

x ye e

Ans.: (a)

Solution: 2 3

1 1ˆ ˆ ˆ ˆ ˆ

4 4 4 2 2 8 2x y x y

P P r PI S r r e e e e

A r r r

2 21 1 2r




Q36. A charged particle in a uniform magnetic field 0 ˆzB B e

starts moving from the origin

with velocity ˆ ˆ3 2 /x zv e e m s

. The trajectory of the particle and the time t at which it

reaches 2 meters above the xy - plane are

( ˆ ˆ,x ye e and ˆze are unit vectors in Cartesian-coordinate system)

(a) Helical path; 1t s (b) Helical path; 2 / 3t s

(c) Circular path; 1t s (d) Circular path; 2 / 3t s

Ans.: (a)

Solution: 3 /v m s and 2 /v m s , thus 2

1 secm

tv

Q37. The phase difference between input and output voltage for the following circuits (i)

and (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)

Solution: (i) Co i

C C

Xv v

X X

1

2o

i

v

v , phase difference is 0 .

(ii) Co i

C

Xv v

R X

2

1 1 1

1 / 1 1

i CRo

i C

ve

v R X i CR CR

Phase difference is 0 / 2 .

iv

C

C ov

(i)

iv

R

C ov

(ii)




Q38. 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 to10V , by keeping the toggle switch at positionQ .

Assuming the capacitor to be ideal, which one of the following statements is correct?

(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)

Solution: The energy dissipation in cases (i) is 2 21 15 10 5 25

2 2C C C

The energy dissipation in cases (ii) is 2110 50

2C C

Q39. In the following RC network, for an input signal frequency1

2f

RC , the voltage gain

o

i

v

v and the phase angle between ov and iv respectively are

(a) 1

2 and 0 (b)

1

3 and 0 (c)

1

2 and

2

(d)

1

3 and

2

Ans.: (b)

Solution: 1

2f

RC , then

1

2CX jRj fC

2 1

1 2C

PC

j j RRX jR jRZ

R X R jR j

and 1S CZ R X R jR R j

P 5V

R

10V

Q

C

ivR

R

ov

C

C




Po i

P S

Zv v

Z Z

11 1 11 2 1 2 11 1 11 1

2

o

Si

P

j j RvZ R j R jv jR R R jZ j j R j j R

1 1 1 1

2 1 3 3 3 1 3o

i

j j R j j R jv

v jR R R j jR R j

, and phase angle 0

Q40. 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 statements is correct?

(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)

Solution: From the given statement only option (c) is correct.

Q41. 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 center 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 pole ( N or S ) face of the magnet which enters

into the coil

Ans.: (a)

Solution: From the given statement only option (a) is correct.

q

1

2

3 4




Q42. Consider a spherical dielectric material of radius ‘ a ’ centered at origin. If the

polarization vector, 0 ˆxP P e

, where 0P �is a constant of appropriate dimensions, then

( ˆ ˆ,x ye e , and ˆze are unit vectors in Cartesian- coordinate system)

(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 changes over the surface.

Ans.: (a), (b), (d)

Solution: . 0b P

0 0ˆ ˆ ˆ. . sin cos 0b xP n P e r P

at 0,0, a 0 .

Q43. For an electric dipole with momentum 0 ˆzP p e

placed at the origin, ( 0p is a constant of

appropriate dimensions and ˆ ˆ,x ye e and ˆze are unit vectors in Cartesian coordinate system)

(a) potential falls 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 on the xy - plane.

Ans.: (a), (c), (d)

Solution: 2 2

ˆ. cos,

4 4dipo o

r p pV r

r r

.

30

ˆˆ, 2cos sin4

dipp

E r rr

.




Q44. Three infinitely-long conductors carrying currents 1 2,I I and 3I

lie perpendicular to the plane of the paper as shown in the figure.

If the value of the integral .C

B dl

for the loops 1 2,C C and

3C are 0 02 , 4 and 0 in the units of N

A respectively, then

(a) 1 3I A into the paper (b) 2 5I A out of the paper

(c) 3 0I . (d) 3 1I A out of the paper

Ans.: (a), (b)

Solution: 0. enc

C

B dl I

1 2 2I I , 2 3 4I I , 1 2 3 1I I I

1 3I A , 2 5I A and 3 1I A .

Q45. The shape of a dielectric lamina is defined by the two curves 0y and 21y x . If the

charge density of the lamina 215 /y C m , then the total charge on the lamina

is……………..C .

Ans.: 8

Solution: Total charge on the lamina is

21 1

1 22

01 1

1515 1

2

x

S

Q da ydxdy x dx

11 5 3

4 2

1 1

15 151 2 2

2 2 5 3

x xQ x x dx x

15 1 2 1 2 15 2 41 1 2

2 5 3 5 3 2 5 3Q

15 168

2 15Q C

3C

2C3I

2I

1C1I

x

y

11 0




IIT-JAM 2017

Q46. A current 10I A flows in an infinitely long wire along the axis of

hemisphere (see figure). The value of B ds

over the

hemispherical surface as shown in the figure is:

(a) 010 (b) 05 (c) 0 (d) 07.5

Ans. : (a)

Solution: 00 0. 2 2 10

2

IB ds B dl B r r I

r

Q47. Consider two, single turn, co-planar, concentric coils of radii 1R and 2R

with 1 2R R . The mutual inductance between the two coils is

proportional to

(a) 1

2

R

R (b) 2

1

R

R (c)

22

1

R

R (d)

21

2

R

R

Ans. : (c)

Solution:

20 122 2

2 1 2 1 22 21 1 21

1 1 1 1

2

IR

B R R RM I M

I I I R

Q48. Consider a thin long insulator coated conducting wire carrying current I . It is now wound

once around an insulating thin disc of radius R to bring

the wire back on the same side, as shown in the figure.

The magnetic field at the centre of the disc is equal to:

(a) 0

2

I

R

(b) 0 2

34

I

R

(c) 0 2

14

I

R

(d) 0 1

12

I

R

Ans. : (d)

Solution: From R.H.R. magnetic field is pointing inwards, 0 0 0 12 1

4 2 2

I I IB

R R R

10I A

2R

1R

RI

I




Q49. The electric field of an electromagnetic wave is given by

3 7ˆ ˆ2 3 10 sin 10 2 3E k j x y z t

.

The value of is ( c is the speed of light):

(a) 14 c (b) 12 c (c) 10 c (d) 7 c

Ans. : (a)

Solution: 3 7ˆ ˆ2 3 10 sin 10 2 3E k j x y z t

7 7 7ˆ ˆ ˆ10 2 3 10 14, 10k x y z k

,7

7

1014

10 14c c

k

Q50. A rectangular loop of dimension L and width w moves with a constant velocity v away

from an infinitely long straight wire carrying a current I in the plane of the loop as

shown in the figure below. Let R be the resistance of the loop. What is the current in the

loop at the instant the near –side is at a distance r from the wire?

(a)

0

2 2

IL wv

R r r w

(b)

0

2 2

IL wv

R r w

(c)

0

2

IL wv

R r r w

(d)

0

2 2

IL wv

R r r w

Ans. : (c)

Solution: 0 0. ln2 2

r

B

S r

I IL r wB da Ldr

r r

0 01 1 1

2 2B IL ILwvd dr

IR dt R r w r dt Rr r w

v

wR

LI

r




Q51. For a point dipole of dipole moment ˆp pz

located at the origin, which of the following

is (are) correct?

(a) The electric field at 0, ,0b is zero

(b) The work done in moving a charge q from 0, ,0b to 0,0,b is 2

04

qp

b

(c) The electrostatic potential at ,0,0b is zero

(d) If a charge q is kept at 0,0,b it will exert a force of magnitude 3

04

qp

b on the

dipole.

Ans. : (b) and (c)

Solution: 2

0

cos

4

pV

r

and 30

ˆˆ2cos sin4

pE r

r

(a) At 0, ,0b ; 2

0E

(b) The work done in moving a charge q from 0, ,0b to 0,0,b

2 20 0

0,0, 0, ,0 04 4

p qpW q V b V b q

b b

(c) The electrostatic potential at ,0,0b is ,0,0 0V b

(d) At 0,0,b ; 0 3

0

2ˆ

4

pE r

b

If a charge q is kept at 0,0,b it will exert a force of magnitude3

0

2

4

qp

b .

Q52. A dielectric sphere of radius R has constant polarization 0 ˆP P z

, so that the field inside

the sphere is 0

0

ˆ3in

PE z

. Then, which of the following is (are) correct?

(a) The bound surface charge density is 0 cosP

(b) The electric field at a distance r on the z - axis varies as 2

1

r for r R

(c) The electric potential at a distance 2R on the z - axis is 0

012

P R

(d) The electric field outside is equivalent to that of a dipole at the origin

Ans. : (a), (c) and (d)




Solution: 0 0ˆ ˆˆ. . cosb P n P z r P

3 30 0

22 20 0 0 0

ˆ ˆ.ˆ ˆ1 . 1 4 . 1 4

4 4 3 4 3 122dip

P z z P Rp r R P r RV

r r R

Q53. Consider a circular parallel plate capacitor of radius R with separation d between the

plates d R . The plates are placed symmetrically about the origin. If a sinusoidal

voltage 0 sinV V t is applied between the plates, which of the following statement(s) is

(are) true?

(a) The maximum value of the Poynting vector at r R is 2

0 024

V R

d

(b) The average energy per cycle flowing out of the capacitor is zero

(c) The magnetic field inside the capacitor is constant

(d) The magnetic field lines inside the capacitor are circular with the curl being

independent of r .

Ans. : (a), (b) and (d)

Solution: 0 sinV tVE

d d

and 20 0 0 0 0

0

cos

2 2 2dI R V tE

B RR R t d

2 20 0 0 0 0 0 0

2 20

cos sin sin cos1sin 2

2 2 4

R V t V t RV t t RVS EB t

d d d d

20 0

max 24

RVS

d

; 0S , 0

2, inside

2dI r

BR

Q54. In a coaxial cable, the radius of the inner conductor is 2mm and that of the outer one is

5mm . The inner conductor is at a potential of10V , while the

outer conductor is grounded. The value of the potential at a

distance of 3.5mm from the axis is…………

(Specify your answer in volts to two digits after the decimal

point)

Ans. : 3.8

10V




Solution: 2 0V

In Cylindrical coordinate system, 1

0 lnV

r V A r Br r r

Thus 10 ln 2A B and 0 ln 5A B

10

10 ln 2 ln 5 10.86ln 5 / 2

A A A and

10ln 517.39

ln 5 / 2B

3.5 ln 3.5 3.8V r A B V

Q55. The wave number of an electromagnetic wave incident on a metal surface is

120 750 i m inside the metal, where 1i . The skin depth of the wave in the

metal is………(Specify your answer in mm to two digits after the decimal point).

Ans. : 1.33

Solution: 120 750k k i i m

Skin depth, 1 1 1000

1.33750 750

d m mm mm

Q56. A sphere of radius R has a uniform charge density . A sphere of

smaller radius / 2R is cut out from the original sphere, as shown in the

figure below. The center of the cut out sphere lies at / 2z R . After the

smaller sphere has been cut out, the magnitude of the electric field at

/ 2z R is 0/R n . The value of the integer n is……………

Ans. : 8

Solution: Electric field inside a uniformly charge solid sphere of radius R is 0

ˆ3

rE r

Electric field outside a uniformly charge solid sphere of radius R is 3

20

ˆ3

RE r

r

Electric field at 2

Rz is

3

20 0 0

/ 2/ 2

3 3 8

RR RE

R

8n

R

/2R




IIT-JAM 2018

Q57. A current I is flowing through the sides of an equilateral triangle of side a . The

magnitude of the magnetic field at the centroid of the triangle is

(a) 09

2

I

a

(b) 0I

a

(c) 03

2

I

a

(d) 03 I

a

Ans.: (a)

Solution: 2 2 3/ 4

2RS a a a and

3

3 6

RSOS a

For segment PQ

00 032sin 60

234

6

PQ QR RP

I IB B B

aa

093

2PQ

IB B

a

Q58. Three infinite plane sheets carrying uniform charge densities , 2 ,3 are parallel to

the x z plane at ,3 ,4y a a a , respectively. The electric field at the point 0, 2 ,0a is

(a) 0

4j

(b) 0

3j

(c) 0

2j

(d) 0

j

Ans.: (b)

Solution: The electric field at the point 0, 2 ,0P a is

0 0 0

2 3 ˆ2 2 2

E j

0

3j

R

P QS

a IO

x

z

y

a

3a

2

4a

3P




Q59. A rectangular loop of dimensions l and w moves with a constant

speed of v through a region containing a uniform magnetic field B

directed into the paper and extending a distance of 4w . Which of

the following figures correctly represents the variation of emf

with the position x of the front end of the loop?

(a) (b)

(c) (d)

Ans.: (c)

Solution:

Case-I: at 10,x Blw and at, 2,x dx Bl w dx

Bldx d

Blvdt

Case-II: Blv and direction will be opposite.

When loop is inside there is no flux change so, 0 .

v

lw

0 4w

B

x

Bwv

Bwv0 0 w

4w x

Bwv

Bwv0

0 w4w

x

Blv

Blv

0 0 w4w x

Blv

Blv

00 w

4wx

dxl

v B

Case-I

vB

Case-II

l




Q60. A long solenoid is carrying a time dependent current such that the

magnetic field inside has the form 20

ˆB t B t k

, where k is along

the axis of the solenoid. The displacement current at the point P on

a circle of radius r in a plane perpendicular to the axis

(a) is inversely proportional to r and radially outward

(b) is inversely proportional to r and tangential

(c) increases linearly with time and is tangential.

(d) is inversely proportional to 2r and tangential

Ans.: (b)

Solution: d B

E dl dldt

202 2E r B t R

20B tR

Er

0d

EJ

t

2

0 0 1d d

B RJ J

r r

Q61. Given a spherically symmetric charge density 2 ,

0,

kr r Rr

r R

( k being a constant),

the electric field for r R is (take the total charge as Q )

(a) 3

50

ˆ4

Qrr

R (b)

2

40

3ˆ

4

Qrr

R (c)

3

50

5ˆ

8

Qrr

R (d)

50

ˆ4

Qr

R

Ans.: (a)

Solution: 0

. enc

S

QE d a

2 2 2

0 0

14 4

r

E r kr r dr

5

2

0

14 4

5

rE r k

E

3

05

krE

5

2 2

0

4 45

R RQ kr r dr k

5

5

4

Qk

R

3 3

5 50 0

5

4 5 4

Q r QrE

R R

r p

k




Q62. An infinitely long solenoid, with its axis along k , carries a current I . In addition there is

a uniform line charge density along thee axis. If S

is the energy flux, in cylindrical

coordinates ˆˆˆ , , k , then

(a) S

is along

(b) S

is along k

(c) S

has non zero components along and k

(d) S

is along ˆˆ k

Ans. : (d)

Solution: Ê E

ˆB Bk

S E B

ˆˆS k

Q63. Let the electric field in some region R be given by 2 2ˆ ˆy xE e i e j

. From this we may

conclude that

(a) R has a non-uniform charge distribution

(b) R has no charge distribution

(c) R has a time dependent magnetic field.

(d) The energy flux in R is zero everywhere.

Ans.: (b), (c)

Solution: 0E

and 0E

,

Thus R has no charge distribution and R has a time dependent magnetic field.

B

Ê




Q64. In presence of a magnetic field ˆBj and an electric field Ê k , a particle moves

undeflected . Which of the following statements is (are) correct?

(a) The particle has positive charge, velocity Êi

B

(b) The particle has positive charge, velocity Êi

B

(c) The particle has negative charge, velocity Êi

B

(d) The particle has negative charge, velocity Êi

B

Ans.: (b), (d)

Solution: 0F q E v B

E

vB

For ve charge: â k

Ê

v xB

For ve charge: â k

Ê

v xB

Q65. Consider an electromagnetic plane wave 0

2ˆ ˆ cos 3E E i bj ct x y

, where

is the wavelength, c is the speed of light and b is a constant. The value of b is

_________. (Specific your answer upto two digits after the decimal point)

Ans. : 0.577

Solution: 0ˆ ˆ ˆˆ ˆcosE E n t k r n i bj

2ˆ ˆ ˆ3k i j

2ˆ 0 1 3 0k n b

10.577

3b




IIT-JAM 2019

Q66. A small spherical ball having charge q and mass m , is tied to a thin

massless non-conducting string of length l . The other end of the

string is fixed to an infinitely extended thin non-conducting sheet

with uniform surface charge density . Under equilibrium the

string makes an angle o45 with the sheet as shown in the figure.

Then is given by ( g is the acceleration due to gravity and 0 is

the permittivity of free space)

(a) 0mg

q

(b) 02

mg

q

(c) 02mg

q

(d) 0

2

mg

q

Ans. : (c)

Solution: 0

tan tan2

F qE q

mg mg mg

02tan

mg

q

00 02 2tan 45

mg mg

q q

Q67. Consider the normal incidence of a plane electromagnetic wave with electric field given

by 0 1 êxpE E k z t x

over an interface at 0z separating two media [wave

velocities 1v and 2 2 1v v v and wave vectors 1k and 2k , respectively] as shown in

figure. The magnetic field vector of the reflected wave is ( is the angular frequency)

(a) 01

1

êxpE

i k z t yv

(b) 01

1

êxpE

i k z t yv

(c) 01

1

êxpE

i k z t yv

(d) 0

11

êxpE

i k z t yv

045

l

q

m

z

2v

2k Medium2Medium11v

1k

y

O

x




Ans. : (c)

Q68. During the charging of a capacitor C in a series RC circuit, the typical variations in the

magnitude of the charge q t deposited on one of the capacitor plates, and the current

i t in the circuit, respectively are best represented by

(a) Figure I and figure II (b) Figure I and Figure IV

(c) Figure III and figure II (d) Figure III and figure IV

Ans. : (a)

Q69. Which one of the following is an impossible magnetic field B

?

(a) 2 2 3ˆ ˆ3 2B x z x xz z

(b) 3

2ˆ ˆ ˆ2 23

zB xyx yz y yz z

(c) 2

3 3ˆ ˆ ˆ42

zB xz y x yx y x z z

(d) 2ˆ ˆ6 3B xzx yz y

Ans. : (d)

Solution: Check that 0B

(a) 2 26 6 0B xz xz

(b) 2 22 2 0B y z y z

(c) 3 3 0B z x x z

(d) 26 3 0B z z

q

0 t

Fig.Ii

0 t

Fig.IVq

0 t

Fig.IIIi

0 t

Fig.II

z

Medium2Medium1

y

O

x

E

B

E

B




Q70. Which of the following statement(s) is/are true?

(a) Newton’s laws of motion and Maxwell’s equations are both invariant under Lorentz

transformations

(b) Newton’s laws of motion and Maxwell’s equations are both invariant under Galilean

transformations

(c) Newton’s laws of motion are invariant under Galilean transformations and Maxwell’s

equations are invariant under Lorentz transformations

(d) Newton’s laws of motion are invariant under Lorenz transformations and Maxwell’s

equations are invariant under Galilean transformations

Ans. : (c)

Q71. Out of the following statements, choose the correct option(s) about a perfect conductor.

(a) The conductor has an equipotential surface

(b) Net charge, if any, resides only on the surface of conductor

(c) Electric field cannot exist inside the conductor

(d) Just outside the conductor, the electric field is always perpendicular to its surface

Ans.: (a), (b), (c), (d)

Q72. The electrostatic energy (in units of 0

1

4J

) of a uniformly charged spherical shell of

total charge 5C and radius 4 m is______. (Round off to 3 decimal places)

Ans.: 3.125

Solution: 2 2

0 0

1

8 4 2

q qW

R R

0

1 25

4 2 4W Joules

0

13.125

4Joules

Q73. An infinitely long very thin straight wire carries uniform line charge

density 28 10 /C m . The magnitude of electric displacement vector at a point located

20 mm away from the axis of the wire is ___________ 2/C m .

Ans. : 2




Solution: 2 20

0

8 10 / ,2 2

c m E D Er r

2

2 23

8 10 4/ 2 /

2 20 10 2D c m C m

Q74. A surface current ˆ100 A/mK x

flows on the surface 0z , which separates two media

with magnetic permeabilities 1 and 2 as shown in the figure. If the magnetic field in

the region 1 is 1 ˆ ˆ ˆ4 6 2B x y z mT

, then the magnitude of the normal component of 2B

will be ________ mT

Ans. : 2

Solution: 2 1 ˆ2B B zmT (Since 1 ˆ2B zmT )

ˆ100 /K x A m

n2B

62 10 10 /H m

0z

x0z

1 ˆ ˆ ˆ4 6 2B x y z mT

61 5 10 /H m

0z




Kinetic Theory, Thermodynamics

IIT-JAM 2005

Q1. The molar specific heat of a gas as given from the kinetic theory is .2

5R If it is not

specified whether it is PC or VC , one could conclude that the molecules of the gas

(a) are definitely monatomic (b) are definitely rigid diatomic

(c) are definitely non-rigid diatomic (d) can be monatomic or rigid diatomic

Ans. : (d)

Solution: If molecule is mono atomic, then 5

2p

RC

And if molecule is rigid diatomic then 5

2V

RC

Q2. The value of entropy at absolute zero of temperature would be

(a) zero for all the materials

(b) finite for all the materials

(c) zero for some materials and non-zero for others

(d) unpredictable for any material

Ans. : (a)

Solution: If system will achieve absolute zero then it is perfectly ordered system then entropy

will be zero.

Q3. Which of the following statements is INCORRECT?

(a) Indistinguishable particles obey Maxwell-Boltzmann statistics

(b) All particles of an ideal Bose gas occupy a single energy state at 0T

(c) The integral spin particles obey Bose-Einstein statistics

(d) Protons obey Fermi-Dirac statistics

Ans. : (a)

Solution: Distinguishable particles obey Maxwell-Boltzmann statistics.




IIT-JAM 2006

Q4. A solid melts into a liquid via first order phase transition. The relationship between the

pressure P and the temperature T of the phase transition is 02P T P , where 0P is a

constant. The entropy change associated with the phase transition is 1 11.0 J mole K . The

Clausius-Clapeyron equation for the latent heat is vdT

dPTL

. Here solidliquid vvv

is the change in molar volume at the phase transition. The correct statement relating the

values of the volumes is

(a) solidliquid vv (b) 1 solidliquid vv

(c) 2

1 solidliquid vv (d) 2 solidquid vv

Ans. : (c)

Solution: Since 02P T P 2dP

dT

It is given vdT

dPTL

2L T v 2

dLv

dT

Since -1 -11.0 Jmole KdS , 1dQ mdL

dST dT

1

1 22

v v

IIT-JAM 2007

Q5. Experimental measurements of heat capacity per mole of Aluminum at low temperatures

show that the data can be fitted to the formula, 3VC aT bT , where

2 1 5 4 10.00135 , 2.48 10a JK mole b JK mole and T is the temperature in Kelvin.

The entropy of a mole of Aluminum at such temperatures is given by the formula

(a) ,3

3 cTb

aT where 0c is a constant (b) ,42

3 cTbaT

where 0c is a constant

(c) 3

3T

baT (d) 3

42T

baT

Ans. : (a)

Solution: vc dTds

T

3aT bTs dT

T

3

3

baT T c




IIT-JAM 2008

Q6. The chemical potential of an ideal Bose gas at any temperature is

(a) necessarily negative (b) either zero or negative

(c) necessarily positive (d) either zero or positive

Ans. : (b)

Solution: The chemical potential of an ideal Bose gas at any temperature is either zero or

negative.(zero in case of photon ).

Q7. A thermodynamic system is maintained at constant temperature and pressure. In

thermodynamic equilibrium, its

(a) Gibbs free energy is minimum (b) enthalpy is maximum

(c) Helmholtz free energy is minimum (d) internal energy is zero

Ans. : (a)

Solution: A thermodynamic system is maintained at constant temperature and pressure can be

defined by Gibbs energy 0dG SdT VdP i.e. Gibbs free energy is minimum.

IIT-JAM 2009

Q8. A box containing 2 moles of a diatomic ideal gas at temperature 0T is connected to

another identical box containing 2 moles of a monatomic ideal gas at temperature 05T .

There are no thermal losses and the heat capacity of the boxes is negligible. Find the final

temperature of the mixture of gases (ignore the vibrational degrees of freedom for the

diatomic molecules).

(a) 0T (b) 01.5T (c) 02.5T (d) 03T

Ans. : (c)

Solution: Internal energy of the system remains conserve i.e. monoatomic diatomic mixtureU U U

11 1monotomic VU n C T , 22 2diatomic VU n C T

1

5

2v

RC ,

2

3

2v

RC , 1 2 2n n , 1 0T T , 2 05T T




Let the common temperature of mixture is T and specific heat is 1 21 2

1 2

V VV

n C n CC

n n

and

number of moles of mixture is 1 2n n n , then

1 21 1 2 2 02.5V V Vn C T n C T nC T T T

Q9. Isothermal compressibility Tk of a substance is defined as 1

TT

Vk

V P

. Its value for

n moles of an ideal gas will be

(a) P

1 (b)

P

n (c)

P

1 (d)

P

n

Ans. : (c)

Solution: PV nRT and 1

TT

Vk

V P

1

P .

IIT-JAM 2010

Q10. A gas of molecules each having mass m is in thermal equilibrium at a temperature T .

Let , ,x y zv v v be the Cartesian components of velocity, v

, of a molecules. The mean value

of 2zyx vvv is

(a) m

TkB221 (b) m

TkB221

(c) m

TkB22 (d) m

TkB22

Ans. : (a)

Solution: 2 2 2 2 2 2 2 2 2x y z x y y x y z x y zv v v v v v v v v v v v

2 2 2 2 2 2 2 2 2x y z x y y x y z x y zv v v v v v v v v v v v

2 2 2 Bx y z

k Tv v v

m and 0x y zv v v

2 2 21 Bx y z

k Tv v v

m




Q11. A trapped air bubble of volume 0V is released from a depth h measured from the water

surface in a large water tank. The volume of the bubble grows to 02V as it reaches just

below the surface. The temperature of the water and the pressure above the surface of

water ( 5 210 /N m ) remain constant throughout the process. If the density of water is

31000 /kg m and the acceleration due to gravity is 210 /m s , then the depth h is

(a) 1m (b) 10m (c) 50m (d) 100 m

Ans. : (b)

Solution: At depth h pressure 1 0P P gh and volume 1 0V V

At surface pressure 2 0P P and volume 2 02V V

Then, 1 1 2 2PV PV

5

0 0 00 0 0 0 3

0

1 102 10

10 10

PV PP gh V P V h m

gV g

5 20 10 /P N m , 210 /g m s , 31000 / 10kg m h m

IIT-JAM 2011

Q12. Consider free expansion of one mole of an ideal gas in an adiabatic container from

volume 1V to 2V . The entropy change of the gas, calculated by considering a reversible

process between the original state 1,V T to the final state 2 ,V T , where T is the

temperature of the system is denoted by 1S . The corresponding change in the entropy of

the surrounding is 2S . Which of the following combinations is correct?

(a) 1 1 2 2 1 2ln / , ln /S R V V S R V V

(b) 1 1 2 2 1 2ln / , ln /S R V V S R V V

(c) 1 2 1 2ln / , 0S R V V S

(d) 1 2 1 2ln / , 0S R V V S

Ans. : (c)




Solution: Free expansion is irreversible process when gas expand 1V to 2V which can be

explained by choosing any path between two state (because entropy is state function). So

one can choose reversible isothermal process.

So, 21

1

lnV

S RV

. Hence it is free expansion so entropy of surrounding is 2 0S .

Q13. A gas of molecular mass m is at temperature T . If the gas obeys Maxwell-Boltzmann

velocity distribution, the average speed of molecules is given by

(a) Bk T

m (b)

2 Bk T

m (c)

2 Bk T

m (d)

8 Bk T

m

Ans. : (d)

Solution: The velocity distribution function

3/ 2 2

24 , 02 2

m mvf v v dv v

kT kT

Now,

23/ 2

22

0

, 42

mv

kTmv v e v dv

kT

23 / 2 3 / 2 23 2

0

1 24 4 2

2 2 2

mv

kTm m kTv e dv

kT kT m

2

1 2 2 84 1 2

2 2 2 2

m m kT m kT kT

kT kT m kT m m




IIT-JAM 2012

Q14. For a liquid to vapour phase transition at trT , which of the following plots between

specific Gibbs free energy g and temperature T is correct?

(a) (b)

(c) (d)

Ans. : (a)

Solution: dG VdP SdT

Q15. A tiny dust particle of mass 111.4 10 kg is floating in air at 300 K . Ignoring gravity, its

rms speed (in /m s ) due to random collisions with air molecules will be closest to

(a) 0.3 (b) 3 (c) 30 (d) 300

Ans. : (c)

Solution: 23

611

3 3 1.38 10 30030 10 /

1.4 10rms

kTv m s

m

Q16. When the temperature of a blackbody is doubled, the maximum value of its spectral

energy density, with respect to that at initial temperature, would become

(a) 16

1 times (b) 8 times (c) 16 times (d) 32 times

Liquid Vapour

TtrT

g

Liquid Vapour

TtrT

g

Liquid Vapour

TtrT

g

Liquid Vapour

TtrT

g




Ans. : (c)

Solution: 4 4

4 1 1 22 1 14 4

2 2 1

16U T T

U T U U UU T T

IIT-JAM 2013

Q17. A blackbody at temperature T emits radiation at a peak wavelength . If the temperature

of the blackbody becomes T4 , the new peak wavelength is

(a) 256

1 (b)

64

1 (c)

16

1 (d)

4

1

Ans. : (d)

Solution: From wein Law, 1max 11max 1 2max 2 2max

2 4 4

T TT T

T T

Q18. Let FDBEMB NNN ,, denote the number of ways in which two particles can be distributed

in two energy states according to Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac

statistics respectively. Then FDBEMB NNN :: is

(a) 1:3:4 (b) 3:2:4 (c) 3:3:4 (d) 2:3:4

Ans. : (a)

Solution: 2, 2,N g 2n

For Maxwell, Boltzmann, 222 4

2nN

W gn

For Boson, 2, 2, 2N g n ; 1 2 2 1

31 2 2 1

n gW

n g

For Fermion, 2, 2,N g 2n ; 2

12 1

gW

g g n

Q19. Two thermally isolated identical systems have heat capacities which vary as

3TCv (where 0 ). Initially one system is at 300 K and the other at 400 K . The

systems are then brought into thermal contact and the combined system is allowed to

reach thermal equilibrium. The final temperature of the combined system is………….

Ans. : 357 360K T K




Solution: There is not a unique value of temperature rather range of temperature

The maximum temperature when work done is zero so 1 2 0dQ dQ

max max4 44

3 3 max

300 400

300 4000 2 0

4 4 4

T T Tm T dT m T dT

4 44

max max

300 400360

2T T K

The minimum temperature of system, when process is reversible so change in entropy of

system is zero

1 0 0S S

min min3 3

300 4000

T Tm T m TdT dT

T T

min

2 23

min

300 400357

2T T K

So 357 360K T K

IIT-JAM 2014

Q20. In 1 - dimension, an ensemble of N classical particles has energy of the form

.2

1

22

2

kxm

PE x The average internal energy of the system at temperature T is

(a) TNkB2

3 (b) TNkB2

1 (c) TNkB3 (d) TNkB

Ans. : (d)

Solution: Since, 2

21

2 2nP

E kxm

Now, 2 21 1

2 2 2 2x

kT kTE P k x kT

m

And for N - classical particle, E NkT

Q21. A solid metallic cube of heat capacity S is at temperature 300 K . It is brought in contact

with a reservoir at 600 K . If the heat transfer takes place only between the reservoir and

the cube, the entropy change of the universe after reaching the thermal equilibrium is

(a) 0.69 S (b) 0.54 S (c) 0.27 S (d) 0.19S

Ans. : (d)




Solution: Heat taken by cube 600 300 300S S J

entropy change of res300

0.5600

QS S

T

entropy change of cube 600

5ln 0.69300

S

0.5 0.69 0.19univS S S S

Q22. A real gas has specific volume V at temperature T . Its coefficient of volume expansion

and isothermal compressibility are and ,Tk respectively. Its molar specific heat at

constant pressure pC and molar specific heat at constant volume vC are related as

(a) RCC vp (b) T

vp k

TvCC

(c) T

vp k

TvCC

2 (d) vp CC

Ans. : (c)

Q23. At atmospheric pressure ,Pa105 aluminium melts at 550 K . As it melts, its density

decreases from 33 kg/m103 to .kg/m109.2 33 Latent heat of fusion of aluminium is

324 10 J/kg . The melting point of aluminium at a pressure of Pa107 is closest to

(a) 551.3K (b) 552.6 K (c) 558.7 K (d) 547.4 K

Ans. : (a)

Solution:

240001 1

2900 3000

fu

fi

Ldp

dT T V V T

As 1

V

62088 10dp

dT T

62088 10

dTdp

T

or 62088 10 lnp T C

At 510 Pa , 550T K

5 610 2088 10 ln 550 C




913.18 10C

At 710 Pa , 7 6 910 2088 10 ln 13.18 10T

T antilog 6.31

On solving, 550.1T K

IIT-JAM 2015

Q24. A system consists of N number of particles, 1N . 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) 2

N (b)

/ 1kT

N

e (c)

/ 1kT

N

e (d) kTNe /

Ans. : (d)

Solution: 1 12 1 E EE E

kT kT kTN Ne Ne N Ne

Q25. 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) 2ln2

3R (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, for B ln 2BS R 2 ln 2A BS S S R

A B




Q26. 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 6

2

RTU 3V

V

UC R

T

4P VC C R R

Q27. 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 2

1 2

Q Q

T 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)

Q28. 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 ,2A

B B A B

PT K V V P , 8.314 /R J K

Process B C is isobaric

V

2T

1TB

1QP

C2Q D

A

2P

V V2

1P

C B

A

Volume

Pre

ssur

e




2A

C B

PP P , C AV V , 300CT K ,

51,

2n

C A is Isochoric

For process A B , 1 ln 4610BA

A

VQ nRT J

V

2 300 8001 1P

n R TQ nC T R

= 10392 J

3 800 3001

RQ

500 6235.5

1

RJ

Total heat exchange is 1 2 3 452Q Q Q

Q29. 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: 1

1 80

273

MLS

T

,

302

2 273

dTS MC

T 2

3021 1ln

273S

1 2S S S 80 302

1.1ln 0.29 0.1 0.39273 273

S




IIT-JAM 2016

Q30. A spherical closed container with smooth inner wall contains a monoatomic ideal gas. If

the collisions between the wall and the atoms are elastic, then the Maxwell speed-

distribution function vdn

dv

for the atoms is best represented by:

(a) (b)

(c) (d)

Ans. : (c)

Solution: 2

2 exp2

dn mvv

dv kT

Q31. For an ideal gas, which one of the following -T S diagram is valid?

(a) (b)

(c) (d)

Ans. : (a)

Solution: ,V PV P

S SC T C T

T T

, P VC C from the slope of -T S diagram one can plot

isochoric and isobaric plot.

vdn

dv

0 v

vdn

dv

0 v

vdn

dv

0 v

vdn

dv

0 v

TIsochor

Isobar

S

T

Isochor

Isobar

S

T Isochor

Isobar

S

T Isochor

Isobar

S




Q32. 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) Solution: H U PV dH dU PdV VdP TdS PdV PdV VdP dH VdP TdS

Q33. One mole of an ideal gas with average molecular speed 0v is kept in a container of fixed

volume. If the temperature of the gas is increased such that the average speed gets

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)

Solution: For fixed volume if temperature is increased then pressure is also increased by same

amount so mean free path will not change.

22

kT

d P

Q34. 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 statements s is (are) correct?

(a) The work done by the gas is maximum along path -3 .

(b) Minimum change in the internal energy occurs along path - 2 .

(c) Maximum heat transfer is for path - 1

(d) Heat transfer is path independent

Ans. : (a)

Pi

32

1 f

V




Solution: Work done is area under curve so it is maximum in path 3 Hence change in internal

energy is same in all path so heat exchange is also maximum in path 3

Q35. A cylinder contains 16 g of 2O . The work done when the gas is compressed to 75% of

the original volume at constant temperature of 027 C is………………. J .

[Universal gas constant 8.31 / moleR J K ]

160.5, 300

32n T K

Ans. : 358

Solution: .75 75 1

ln 8.31 300 ln 0.75 358100 2

V

V

dVW PdV nRT nRT J

V

Q36. An aluminum plate of mass 0.1 kg at 095 C is immersed in 0.5 litre of water at 020 C

kept inside an insulating container and is then removed. If the temperature of the water is

found to be 023 C , then the temperature of the aluminum plate is………… 0C

(The specific heat of water and aluminum are 4200 /J kg K and 900 /J kg K

respectively, the density of water is 31000 /kg m ).

Ans. : 94.36

Solution: f i f ia a a a w w w wM S T T M S T T 0.1 4200 368 0.5 900 296 293afT

4502100 368 450 3 368 0.64

700af afT T

368 .64 367.36afT 367.36 273 94.36

Q37. If there is a 10% decrease in the atmospheric pressure at a hill compared to the pressure

at sea level, then the change in the boiling point of water is………………. 0C

(Take latent heat of vaporisation of water as 2270 /kJ kg and the change in the specific

volume at the boiling point to be 31.2 /m kg )

Ans. : 2

Solution: 5

2 02 13

2 1

1 0.1 1.01 10 373 1.20.02 10 2

2270 10

V VdP LdT dP T C

dT T V V L




IIT-JAM 2017

Q38. Consider a system of N particles obeying classical statistics, each of which can have an

energy 0 or E . The system is in thermal contact with a reservoir maintained at a

temperature T . Let k denote the Boltzmann constant. Which one of the following

statements regarding the total energy U and the heat capacity C of the system is correct?

(a) /1 E kT

NEU

e

and

/

2/1

E kT

E kT

NE eC k

kT e

(b) /1 E kT

NE EU

kT e

and

/

2/1

E kT

E kT

NE eC k

kT e

(c) /1 E kT

NEU

e

and

2 /

2 2/1

E kT

E kT

NE eC k

kT e

(d) /1 E kT

NEU

e

and

2 /

2 2/1

E kT

E kT

NE eC k

kT e

Ans. : (c), (d) Solution: Since, 1 20,E E E , For one particle,

1 21 2

1 2

exp exp

exp exp

E EE E

kT kTU

E EkT kT

exp

1 exp

EE

kTEkT

1 exp

EEkT

So for N Particle 1 exp

NEU

EkT

V

dUC

dT

2 /

2 2/1

E kT

E kT

NE ek

kT e

Q39. Consider two identical, finite, isolated systems of constant heat capacity C at

temperature 1T and 2 1 2T T T . An engine works between them until their temperatures

become equal. Taking into account that the work performed by the engine will be

maximum maxW if the process is reversible (equivalently, the entropy change of the

entire system is zero), the value of maxW is:

(a) 1 2C T T (b) 1 2

2

C T T (c) 1 2 1 2C T T TT (d) 2

1 2C T T




Ans. : (d)

Solution: For maximum, 2

1 21 2

ln 0ff

TdS C T T T

T T

max 1 2 1 2 2f f fdW C T T C T T C T T T

2

1 2 2 1 1 22C T T T T C T T

Q40. A white dwarf star has V and contains N electrons so that the density of electrons is

Nn

V . Taking the temperature of the star to be 0 K , the average energy per electron in

the star is 2

2/320

33

10n

m

, where m is the mass of the electron. The electronic

pressure in the star is:

(a) 0n (b) 02n (c) 0

1

3n (d) 0

2

3n

Ans. : (d)

Solution: Since, 2 2

2/3 2 /32 20 2/3

3 3 13 3

10 10n N

m m V

and 0U N

2

2/32 5/300

3 2 23

10 3 3

ddUP N N N V n

dV dV m

Q41. In the radiation emitted by a black body, the ratio of the spectral densities at frequencies

2 and will vary with as:

(a) 1/ 1Bh k Te

(b) 1/ 1Bh k Te

(c) / 1Bh k Te (d) / 1Bh k Te

Ans. : (b) Solution: The ratio of spectral density at the frequencies 2 and

12

exp 1 exp 1 exp 1 11 2

exp 1 exp 1 exp 1 exp 1exp 1

h h hkT kT kT

h h h hh kT kT kT kTkT

1/ 1Bh k Te




Q42. An isolated box is divided into two equal compartments by a partition (see figure). One

compartment contains a van der Waals gas while the other compartment is empty. The

partition between the two compartments is now removed. After the gas has filled the

entire box and equilibrium has been achieved, which of the following statement(s) is (are)

correct?

(a) Internal energy of the gas has not changed

(b) Internal energy of the gas has decreased

(c) Temperature of the gas has increased

(d) Temperature of the gas has decreased

Ans. : (a) and (d)

Solution: It is the example of free expansion, so Internal energy of the gas has not changed

Va

dU C T dVV

For van der Waal gas, 2V

adU C dT dV

V

For keeping internal energy constant, if dV increases then dT must decrease

Q43. Consider a Carnot engine operating between temperature of 600 K and 400 K . The

engine performs 1000 J of work per cycle. The heat (in Joules) extracted per cycle from

the high temperature reservoir is…………..


Ans. : 3000

Solution: 21

1 1 1 1

400 1000 1000 21 1 3000

600 6

T WQ J

T Q Q Q




IIT-JAM 2018

Q44. Two boxes A and B contain an equal number of molecules of the same gas. If the

volumes are AV and BV and A and B denote respective mean free paths, then

(a) A B (b) A B

A BV V

(c)

1/ 2 1/ 2A B

A BV V

(d) A A B BV V

Ans. : (b)

Solution: 2 2 2

1

2 2 2

kT VV

d P d n d N

, where

Nn

V

,A A B BKV KV

A B

A BV V

Q45. Which one of the figures correctly represents the T S diagram of a Carnot engine?

(a) (b)

(c) (d)

Ans. : (b)

Solution:

A B Isothermal expansion

B C Adiabatic expansion

C D Isothermal compression

D A Adiabatic compression

T

S

T

S

T

S

T

S

P

V

AB

CD

T

S

A B

CD




Q46. The equation of state for one mole of a non-ideal gas is given by 1B

PV AV

, where

the coefficient A and B are temperature dependent. If the volume changes from 1V to 2V

in an isothermal process, the work done by the gas is

(a) 1 2

1 1AB

V V

(b) 2

1

lnV

ABV

(c) 2

1 1 2

1 1ln

VA AB

V V V

(d) 2 1

1

lnV V

A BV

Ans. : (c)

Solution: 1B

PV AV

2

A ABP

V V

2 2

1 1

2

V V

V V

A ABW PdV dV dV

V V 2

1 1 2

1 1ln

VA AB

V V V

Q47. An ideal gas consists of three dimensional polyatomic molecules. The temperature is

such that only one vibrational mode is excited. If R denotes the gas constant, then the

specific heat at constant volume of one mole of the gas at this temperature is

(a) 3R (b) 7

2R (c) 4R (d)

9

2R

Ans.: (c)

Solution: For a polyatomic gas

4PC f R

3VC f R

As, 1, 4Vf C R

Q48. Consider an ensemble of thermodynamic systems each of which is characterized by the

same number of particles, pressure and temperature. The thermodynamic function

describing the ensemble is

(a) Enthalpy (b) Helmholtz free energy

(c) Gibbs free energy (d) Entropy




Ans. : (c)

Solution: The variable

G H TS

dG dH TdS SdT

TdS VdP TdS SdT

dG VdP SdT

Q49. Which of the following relations is (are) true for thermodynamic variables?

(a) VV

PTdS C dT T dV

T

(b) P

P

VTdS C dT T dP

T

(c) dF SdT PdV (d) dG SdT VdP

Ans. : (b), (d)

Solution: ,S S T V V T

S SdS dT dV

T V

VV

PTdS C dT T dV

T

(a) is correct.

,S S T P p T

S SdS dT dP

T P

P T

S STdS T dT T dP

T P

pP

VC dT T dP

T

, (b) is correct.

dF SdT PdV so (c) is incorrect

dG SdT PdV so (d) is correct

Q50. Consider a monoatomic ideal gas operating in a closed cycle as

shown in the P V diagram given below. The ratio 1

2

P

P is _______.

(Specific your answer upto two digits after the decimal point)

Ans. : 0.16

Solution: For monoatomic gas 5

3r

1P

2P

P

13V1VV

adiabatic




For adiabatic problems 1 1 2 2r rPV PV

5/3

1 1

2 1 3

P V

P V

5

33 0 16

IIT-JAM 2019

Q51. The Fermi-Dirac distribution function n is

( Bk is the Boltzmann constant, T is the temperature and F is the Fermi energy)

(a) 1

1F

Bk T

n

e

(b) 1

1F

Bk T

n

e

(c) 1

1F

Bk T

n

e

(d) 1

1F

Bk T

n

e

Ans. : (c)

Q52. In a heat engine based on the Carnot cycle, heat is added to the working substance at

constant

(a) Entropy (b) Pressure

(c) Temperature (d) Volume

Ans. : (c)

Q53. Isothermal compressibility is given by

(a) 1

T

V

V P

(b)1

T

P

P V

(c)1

T

V

V P

(d)

1

T

P

P V

Ans. : (c)

Q54. A red star having radius Rr at a temperature RT and a white star having radius wr at a

temperature wT , radiate the same total power. If these stars radiate as perfect black bodies,

then

(a) andR w R wr r T T (b) andR w R wr r T T

(c) andR w R wr r T T (d) andR w R wr r T T

13V

P

2P

1S

1VV




Ans. : (c)

Solution: 4 1E AT 2 4 2 44 4W W R Rr T r T as W Rr r

2

RW R W R

W

rT T T T

r

Q55. During free expansion of an ideal gas under adiabatic condition, the internal energy of the

gas.

(a) Decreases (b) Initially decreases and then increases

(c) Increases (d) Remains constant

Ans. : (d)

Solution: As W U Q

0Q W U

Work is done at the expense of internal energy.

Q56. In the given phase diagram for a pure substance regions I, II, III, IV,

respectively represent

(a) Vapour, Gas, Solid, Liquid (b) Gas, Vapour, Liquid, solid

(c) Gas, Liquid, Vapour, solid (d) Vapour, Gas, Liquid, Solid

Ans. : (b)

Solution: IV – Solid

III – Liquid

II – Vapour

I – Gas (superheated dry vapour)

Q57. A thermodynamic system is described by the , ,P V T coordinates. Choose the valid

expression(s) for the system.

(a) T P V

P V P

V T T

(b)

T P V

P V P

V T T

(c) P V T

V T V

T P P

(d)

P V T

V T V

T P P

Ans. : (a), (c)

P

IV

T

CriticalPoint

IIIII I




Q58. Two gases having molecular diameters 1D and 2D and mean free paths 1 and 2 ,

respectively, are trapped separately in identical containers. If 2 12D D , then

1

2

_______.

(Assume there is no change in other thermodynamic parameters)

Ans. : 4

Solution: 2

1 22

2 1

14

x dx

d x d

Q59. A di-atomic gas undergoes adiabatic expansion against the piston of a cylinder. As a

result, the temperature of the gas drops from 1150 K to 400 K . The number of moles of

the gas required to obtain 2300 J of work from the expansion is ______. (The gas

constant -1 18.314 molR J K .)


Ans. : 0.1475

Solution: 7

5

2 1

1

nR T TW

V

400 1150

2300 8.3141 1.4

n

0.1475n

2

1

2 /d N V

2

1

d




Modern Physics

IIT-JAM 2005

Q1. If Hpe MandMM , are the rest masses of electron, proton and hydrogen atom in the

ground state (with energy 13.6 eV ), respectively, which of the following is exactly true?

( c is the speed of light in free space)

(a) epH MMM

(b) 2

6.13

c

eVMMM epH

(c) 2

6.13

c

eVMMM epH

(d) H p eM M M K , where 2

13.6 eVK

c or zero

Ans. : (c)

Solution: 22

. .. . p e H H p e

B EB E M M M c M M M

c where eVEB 6.13.. .

IIT-JAM 2006

Q2. Electrons of energy E coming from x impinge upon a potential barrier of width

2a and height 0V centered at the origin with EV 0 , as shown in the figure below. Let

EVmk

02

. In the region a x a , the electrons is a linear combination of

(a) kxe and kxe (b) ikxe and kxe (c) ikxe and ikxe (d) ikxe and kxe

Ans. : (a)

Solution: Since, EV 0 in region a x a . Thus, Schrodinger equation is given by

a a x

V

0V




2 2

22 oV Em x

2

2 2

2 ( )0om V E

x

22

20k

x

where

EVmk

02

.

Thus, the solution of the wave equation is kxe and kxe , which is exponential in nature.

Q3. The relation between angular frequency and wave number k for given type of waves

is 2 3k k . The wave number 0k for which the phase velocity equals the group

velocity is,

(a)

3 (b)

3

1 (c)

(d)

2

1

Ans. : (c)

Solution: Group velocity, g

dV

dk

and phase velocity is pV

k

2 3 = k + k ………(A)

Differentiating both sides we get 22 . 3d

kdk

Now dividing both sides by k we will get

2 . 3d

kk dk k

2 . 3p gV V kk

For 0k k and p gV V

20

0

2 3pV kk

12

0

0

3

2 2p

kV

k

From equation (A) pVk

12

00

kk

Thus,

1 12 2

00

0 0

3

2 2

kk

k k

0

0

02 2

k

k

0k




Q4. A particle of rest mass 0m is moving uniformly in a straight line with relativistic velocity

c , where c is the velocity of light in vacuum and 0 1 . The phase velocity of the de Broglie wave associated with the particle is,

(a) c (b) c

(c) c (d) 2

c

Ans. : (b)

Solution: 2 2 2 2 40E p c m c

2 22 2 . gdE

E pc E v pcdp

2 2

p pg

E c c cv v

p v c

Q5. A neutron of mass, 2710nm kg is moving inside a nucleus to be a cubical box of size

1410 m with impenetrable walls. Take 3410 Js and 131 10MeV J . An estimate of

the energy in MeV of the neutron is,

(a) 80 MeV (b) MeV8

1 (c) 8MeV (d) MeV

80

1

Ans:

Solution:

2342 2 68

2 2 27 2827 14

3 10 103 3 10 10

2 2 10 102 10 10n

Em a

13 13 1315 10 15 10 10 15J MeV MeV

IIT-JAM 2007

Q6. The following histogram represents the binding energy per particle ( . . /B E A ) in MeV as

a function of the mass number A of a

nucleus. A nucleus with mass number

180A fissions into two nuclei of

equal masses. In this process

(a) 180 MeV of energy is released

(b) 180 MeV of energy is absorbed

(c) 360 MeV of energy is released

(d) 360 MeV of energy is absorbed

2

4

6

8

40 80 120 160 200

AB.E.

A




Ans. : (c)

Solution: 909018022

orAA

A

Product B.E MeV1080690690

B.E. of nucleus A MeV7204180

Since, B.E of the product nucleus is greater than the nucleus A, hence in this process

energy is released and that is MeVMeV 3607201080 .

Q7. The black body spectrum of an object 1O is such that its radiant intensity (i.e., intensity

per unit wavelength interval) is maximum at a wavelength of 200nm . Another object 2O

has the maximum radiant intensity at 600nm . The ratio of power emitted per unit area by

1O to that of 2O is

(a) 81

1 (b)

9

1 (c) 9 (d) 81

Ans. : (d)

Solution: From Wein’s law T k , where k is a constant. Thus, 1 2

2 1

T

T

1

2

3T

T

Power ( P ) is proportional to 4T4

1 14

2 2

81P T

P T

Q8. A particle is confined in a one dimensional box with impenetrable walls at x a . Its

energy eigenvalue is 2eV and the corresponding eigenfunction is as shown below.

The lowest possible energy of the particle is

(a) 4eV (b) 2eV (c) 1eV (d) 0.5eV

Ans. : (d)

Solution: The given state is representation of first exited state whose energy is 2eV .

If nE is energy of thn state and 0E is energy of ground state then, 20nE n E .

So, 2 04 2E E eV 0 0.5E eV

a 0 a




IIT-JAM 2008

Q9. A photon of wavelength is incident on a free electron at rest and is scattered in the

backward direction. The functional shift in its wavelength in terms of the Compton

wavelength c of the electron is,

(a) 2

C (b) 3

2 C (c) 2

3 C (d) C2

Ans. : (d)

Solution: (1 cos )c

When photon scattered in backward direction then . So, 2 c

Functional shift is

=C2

Q10. In an inertial frame S , a stationary rod makes an angle with the x -axis. Another

inertial frame S moves with a velocity v with respect to S along the common -x x

axis. As observed from S the angle made by the rod with the x - axis is . Which of

the following statement is correct?

(a)

(b)

(c) if v is negative and if v is positive

(d) if v is negative and if v is positive

Ans. : (b)

Solution: 2

0 2cos 1x

vl l

c , 0 sinyl l

2

2

tantan

1

y

x

l

l v

c




Q11. The activity of a radioactive sample is decreased to 75% of the initial value after 30 days.

The half-life (in days) of the sample is approximately

[You may use ln 3 1.1, ln 4 1.4 ]

(a) 38 (b) 45 (c) 59 (d) 69

Ans. : (d)

Solution: 100

11.14.1

30

1

3

4ln

30

1

4/3ln

30

1ln

1

0

00

R

R

R

R

t

1/ 2

0.693 0.69369.3

1/100T

day.

IIT-JAM 2009

Q12. A wave packet in a certain medium is constructed by superposing waves of frequency

around 0 100 and the corresponding wave-number k with 0 10k as given in the

table below,

k

81.00 9.0

90.25 9.5

100.00 10.0

110.25 10.5

121.00 11.0

Find the ratio /g pv v of the group velocity gv and the phase velocity pv .

(a) 2

1 (b) 1 (c)

2

3 (d) 2

Ans. : (d)

Solution: For 0 100 and 0 10k k the phase velocity is 0

0pv

k

=10

The group velocity is 2 1

2 1gv

k k k

=110.25 90.25

2010.5 9.5

202

10g

p

v

v




Q13. Two spherical nuclei have mass numbers 216 and 64 with their radii 1R and 2R ,

respectively. The ratio 2

1

R

R is

(a) 1.0 (b) 1.5 (c) 2.0 (d) 2.5

Ans. : (d)

Solution: 5.14

6

64

2163/13/1

2

1

2

1

A

A

R

R

IIT-JAM 2010

Q14. A particle of mass m is confined in a two-dimensional infinite square well potential of

side a . The eigen-energy of the particle in a given state is 2

2225

maE

. The state is

(a) 4 -fold degenerate (b) 3 -fold degenerate

(c) 2 -fold degenerate (d) Non-degenerate

Ans. : (d)

Solution: The eigen-energy of the particle in a given state is given by

2 22 2

2( )

2 x yE n nma

where 1,2,3...xn 1, 2,3...yn

2 2

2

25E

ma

can be obtained by 5xn and 5yn which is non degenerate .

Q15. For a wave in a medium the angular frequency and the wave vector k

are related by,

2220

2 kc , where 0 and c are constants. The product of group and phase

velocities, i.e., pg vv . is

(a) 20.25c (b) 20.4c (c) 20.5c (d) 2c

Ans. : (d)

Solution: 2220

2 kc

22 2d

c kdk

2dc

k dk

2.p gv v c




Q16. Three identical non-interacting particles, each of spin 1

2 and mass m , are moving in a

one-dimensional infinite potential well given by,

0 for 0

for 0 and

x aV x

x x a

The energy of the lowest energy state of the system is

(a) 2

22

ma

(b)

2

222

ma

(c)

2

223

ma

(d)

2

22

2

5

ma

Ans. : (c)

Solution: Spin 1

2s means particles are fermions and it will obey Pauli Exclusion Principle.

Degeneracy, 2 1 2g s g means in every state maximum 2 identical particle can

be adjusted. If we have three fermions, then in ground state two fermions will be adjusted

and one fermion in next higher level will be adjusted. Thus, the energy of the lowest

energy state of the system is 2 2 2 2 2 2

2 2 2

4 62

2 2 2ma ma ma

=

2 2

2

3

ma

IIT-JAM 2011

Q17. The wave function of a quantum mechanical particle is given by

xxx 21 5

4

5

3

where 1 x and 2 x are eigenfunctions with corresponding energy eigenvalues

1eV and 2eV , respectively. The energy of the particle in the state is

(a) eV25

41 (b) eV5

11 (c) eV25

36 (d) eV5

7

Ans. : (a)

Solution: H

E

= 9 16

1 225 25

eV eV = eV25

41




Q18. Light described by the equation 15 190 / sin 6.28 10E V m s t

15 1sin 12.56 10 s t is incident on a metal surface. The work function of the metal is

2.0eV . Maximum kinetic energy of the photoelectrons will be,

(a) 2.14eV (b) 4.28eV (c) 6.28eV (d) 12.56eV

Ans. : (c)

Solution: maxK W

For given wave maximum kinetic energy is for highest so 15 112.56 10 sec

34 15 1 19

19

6.6 10 s 12.56 10 82.8 108.24

2 6.28 1.6 10

J s JeV eV

max 8.24 2 6.24K W eV eV eV

IIT-JAM 2012

Q19. Light takes 4 hours to cover the distance from Sun to Neptune. If you travel in a

spaceship at a speed 0.99c (where c is the speed of light in vacuum), the time (in

minutes) required to cover the same distance measured with a clock on the spaceship will

be approximately

(a) 34 (b) 56 (c) 85 (d) 144

Ans. : (a)

Solution: 22

0 2 2

0.991 4 60 60 1 4 60 60 .14

cvl ct c c m

c c

4 60 60 .14min 33.9 34min

.99 60

ct

c

Q20. Co6027 is a radioactive nucleus of half-life s8102ln2 . The activity of 10 g of Co60

27 in

disintegrations per second is,

(a) 10105

1 (b) 10105 (c) 1410

5

1 (d) 14105




Ans. : (d)

Solution: NR , where 23 23106 10 10

60N

198888

2/1

10510386.1

693.0

10386.1

693.0

103010.0303.22

693.0

102ln2

693.0693.0

sT

Thus, 9 23 14105 10 6 10 5 10

60R .

IIT-JAM 2013

Q21. Electric field component of an electromagnetic radiation varies with time as,

tttaE 00 cossincos , where a is a constant and the values of and 0 are

115101 s and 115105 s respectively. This radiation falls on a metal of work function

eV2 . The maximum kinetic energy (in eV ) of photoelectrons is

(a) 64.0 (b) 30.1 (c) 70.1 (d) 95.1

Ans. : (b)

Solution: maxK W

For given wave, maximum kinetic energy is for highest , so 15 10 5 10 sec

34 15 1 19

0 19

6.6 10 s 5 10 33 103.28

2 6.28 1.6 10

J s JeV

max 3.28 2 1.28K W eV eV eV

Q22. A free particle of mass m is confined to a region of length L . The de Broglie wave

associated with the particle is sinusoidal in nature as given in the figure. The energy of

the particle is

Ans. :

Solution: If wavelength of standing wave is and length of wall is L then from the figure

3

2L

2

3

L .

03/L 3/2L L




If p is momentum and is wavelength, then from de-Broglie hypothesis h

p

, thus

3

2

hp

L .

When particle is confined into a box then total energy is only kinetic energy which is

given by 2

2

pE

m put the value of

3

2

hp

L one will get

2

2

9

8

hE

mL .

IIT-JAM 2014

Q23. In a photoelectric effect experiment, ultraviolet light of wavelength 320 nm falls on the

photocathode with work function of 2.1 eV. The stopping potential should be close to

(a) 1.8 V (b) 1.6 V (c) 2.2 V (d) 2.4V

Ans. : (a)

Solution: Since, .K E eV W hc WV

e e

Now, 9320 10 , 2.1m W eV

34 8

19 9

6.6 10 3 102.1

1.6 10 320 10V

3.867 2.1 3.9 2.1V 1.8 V .

Thus, option (a) is correct.

Q24. Four particles of mass m each are inside a two dimensional square box of side L . If each

state obtained from the solution of the Schrodinger equation is occupied by only one

particle, the minimum energy of the system in units of 2

2

mL

h is

(a) 2 (b) 2

5 (c)

2

11 (d)

4

25

Ans. : (b)

Solution: For 2 - Dimensional box, possible configurations are 1,1 , 2,1 , 2, 2

Now, ground state energy 2 2

2 222 x yn n

mL

; let 2 2

0 22E

mL




0 0 02 1 2 5 1 8E E E 2 2

0 220 20

2E

mL

2

2

5

2E

mL

Thus, option (b) is correct

Q25. Two frames, O and ,O are in relative motion as shown.

O is moving with speed / 2c , where c is the speed of light.

In frame O , two separate events occur at 11 , tx and

., 22 tx In frame ,O these events occur simultaneously.

The value of 1212 / ttxx is

(a) 4/c (b) 2/c (c) c2 (d) c

Ans.: (c) 2 22 2

21

x vtx

vc

1 1 2 11 2 12 2

2 21 1

x vt x xx x x

v vc c

1 2 12 12 2

2 2 12 2

2 2

,

1 1

Vn vxt t

c ct t tv vc c

2 2

2 2 221

v cx t t

c v

2 2

2 1 2 1 21

v cx x t t

c v

2 2

2 1 2

2 1 2

2

1

1

v ct t

c vx xvc

2 2 2

2 12 2 1

2 1

22y

h hc c cx x t t c

v t t v c

xOO

c/2




IIT-JAM 2015

Q26. A particle with energy E is incident on a potential given by

0

0, 0

, 0

xV x

V x

.

The wave function of the particle for 0VE in the region 0x (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;2

dV E E V

m d

kx kxBe Ae , where

0

2

2m V Ek

0 as x 0A kxAe

Q27. 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. : (d)

Solution: For electron spin is1

2. 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

7 1 1 1

So, number of ways are 7.




Q28. 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 4

92 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)

Solution: In reaction (i) all conservation laws are valid. In reaction (ii) charge is not conserved.

Q29. 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

6.6 106.6 10 / sec

10

hp kgm

2 387

31

44 102.4 10

2 2 9.1 10e

pE Joule

m

712 6

19

2.4 101.5 10 1.5 10

1.6 10E eV eV MeV

0

1

2

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

)




Q30. A proton from outer space is moving towards earth with velocity c99.0 as measured in

earth’s frame. A spaceship, traveling parallel to the proton, measures proton’s velocity to

be c97.0 . The approximate velocity of the spaceship in the earth’s 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 0.5

0.9711

xx

x

u v c vu c v c

u v vcc

Q31. A particle is moving in a two dimensional potential well

0, 0 , 0 2

,, elsewhere

x L y LV x y

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

5

8E

mL

(c) 0

2sin sin

2

x y

L L L

(d) L

y

L

x

L 2coscos

20

Ans. : (b) and (c)

Solution: 222 2

2 22 4yx

n

nnE

m L L

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

2 2 2

1 1 5

2 4 8xEm L L mL

Wave function 2 2 sin sin

2 2

x y

L L L L

Ev

s s0.99p c

0.99p c




Q32. 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 earth’s 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 earth’s surface

(b) The apparent thickness of earth’s atmosphere in muon’s frame of reference is km96.0

(c) The lifetime of muons in earth’s frame of reference is s8.34

(d) Muons traveling at a speed greater than c998.0 reach the earth’s surface

Ans.: (c) and (d)

Solution: 0

2

21

tt

vc

66

2

2.2 1034.8 10 sec

1 0.998t

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

Q33. 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

105

2

5

1 . 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: 1

2nE n

and 1

2 5P

,

3 4

2 5P

1 3 4 131.3

2 5 2 5 10E

Q34. In the hydrogen atom spectrum. the ratio of the longest wavelength in the Lyman series

(final state 1n ) to that in the Balmer series (final State 2n ) is…………..

Ans. : 0.185




Solution: According to Bohr Theory 2 2

1 1 1

L f i

Rn n

The longest wavelength in the Lyman series is

2

1 1 1 3 4

1 2 4 3LL

R RR

The longest wavelength in the Balmer series is

2 2

1 1 1 1 1 9 4

2 3 4 9 36B

R R R

1 5 36

36 5BB

RR

4 5 50.185

3 36 27L

B

R

R

Q35. 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 0.2412

2 2nm

Kinetic Energy of scattered electron

34 89

1 1 1. . 6.6 10 3 10

0.24 0.2412 10

hc hcK E Joules

26 26

209 9

19.8 10 19.8 10. . 4.17 4.15 0.02 396 10

10 10K E Joules

20

19

396 10. . 24.75

1.6 10K E eV eV

3n

2n

1n L

H




IIT-JAM 2016

Q36. Consider a free electron e and a photon ph both having 10eV of energy. If and

P represents wavelength and momentum respectively, then

(mass of electron 319.1 10 kg ; speed of light 83 10 /m s )

(a) e ph and e phP P (b) e ph and e phP P

(c) e ph and e phP P (d) e ph and e phP P

Ans. : (c)

Solution: For photon phE

pc

, phh hc

p E

For electron2 2 4

eE m c

pc

,

2 2 4eh hc

p E m c

Q37. 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 0.1 kV

passes through the slit, then, which of the following statement s is (are) correct?

(a) The uncertainty in the position of the electrons in x -direction before passing the slit is

zero

(b) The momentum of electrons in x - direction is d

immediately after passing the slit

(c) The uncertainty in the position of electrons 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) and (d)

Solution: The electrons beam before slit is

collimated in y - direction as shown in

figure. Thus, before slit

yP P and 0xP

yP

xP

x

yd




also x as 0xP

Thus options (a) and (c) are not correct.

Now, after the slit xP d as a result xPx d

i.e., xPd

Thus, option (b) is correct.

Whereas presence of slit does not affect the uncertainty in momentum in y - direction.

Thus option (d) is also correct.

Q38. 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 coefficients is always one

Ans. : (b) and (d)

Solution: 1R T

2E V E

RE V E

Q39. A particular radioisotope has a half-life of 5 days. In 15 days the probability of decay in

percentage will be…………..

Ans. : 87.5

1/ 2/ 15/5

00 0

1 1

2 2 8

t TN

N N N

In 15 days the probability of decay 0

0

7100 100 87.5%

8

N N

N




Q40. In photoelectric experiment both sodium (work function 2.3eV ) 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 be1.8V , the value of the stopping

potential for sodium will be………….. V .

Ans. : 4

Solution: For tungsten 1.8 4.5 6.3s t s teV h W h eV W

For sodium 6.3 2.3 4s seV h W eV

Q41. The de Broglie wavelength of a relativistic electron having 1 MeV of energy

is………….. 1210 m . (Take the rest mass energy of the electron to be 0.5 MeV . Plank

constant 346.63 10 Js , speed of light 83 10 /m s , Electronic charge 191.6 10 C )

Ans. : 1.43

Solution: 22 22 02 2 2 2

0 2

1 .25 .75E m c MeVE p c m c p

c cc

As, 34 8 13

13 1213

6.6 10 3 10 19.8 1014.34 10 1.43 10

1.38.75 1.6 10

hm m

p

Q42. X -ray of 20 keV energy is scattered inelastically from a carbon target. The kinetic

energy transferred to the recoiling electron by photons scattered at 090 with respect to the

incident beam is……………. keV .

(Planck constant = 346.6 10 Js, Speed of light = 83 10 / ,m s electron mass =

319.1 10 .kg Electronic charge = 191.6 10 C )

Ans. : 0.77

Solution: 1 cosh

mc '

2

h

mc

'

2

1

hc hc mc

' 2 ' 2

1 1 1 1 1 1 1 1

20 .5E E keV MeVE mc E mc

' 19.23E keV

Recoil velocity of electron ' 0.77E E keV




IIT-JAM 2017

Q43. Consider an inertial frame S moving at speed 2

c away from another inertial frame S

along the common -x x axis, where c is the speed of light. As observed from S , a

particle is moving with speed 2

c in the y direction, as shown in the figure. The speed of

the particle as seen from S is:

(a) 2

c (b)

2

c (c)

7

4

c (d)

3

5

c

Ans. : (c)

Solution: ˆ2

cv i ' ' '0, , 0

2x y zc

u u u

'

'

2

21

xx

x

u v cu

u v

c

,

2'

2

'

2

11

1 32 4 4

1

y

xx

vu

c ccuu v

c

,

2'

2

'

2

10

1

z

zx

vu

cuu v

c

2 23 7

4 16 4

c c cu

Q44. Consider Rydberg (hydrogen-like) atoms in a highly excited state with n around 300 .

The wavelength of radiation coming out of these atoms for transitions to the adjacent

states lies in the range:

(a) Gamma rays pm (b) UV nm

(c) Infrared m (d) RF m

Ans. : (d)

Solution: 2 2

1 1 1

f i

Rn n

y

S x

y

S x

/ 2c

/ 2c




where 7 11.097 10R m

299fn and 300in 2 2

2 2

1 i f

i T

n nR

n n

2 22 2

2 22 2

300 2991 1

300 299

i f

i f

n n

R n n R

2 2

7

300 2991

1.097 10 599

7

7

1.34 101.22

1.097 10m

1.22m

This wavelength corresponds to RF Thus correct option is (d)

Q45. A photon of frequency strikes an electron of mass m initially at rest. After scattering

at an angle , the photon loses half of its energy. If the electron recoils at an angle ,

which of the following is (are) true?

(a) 2

cos 1mc

h

(b) 2

sin 1mc

h

(c) The ratio of the magnitudes of momenta of the recoiled electron and scattered photon

is sin

sin

(d) Change in photon wavelength is 1 2cosh

mc

Ans. : (a), (c)

Solution: 1 cos 1 cosh c c h

mc mc

22

1 cos 1 cos cos 1c c h c h mc

mc mc h

From the conservation of momentum in y direction

sinsin sin

sin

h pp

hcc

300n

0E




Q46. For an atomic nucleus with atomic number Z and mass number A , which of the

following is (are) correct?

(a) Nuclear matter and nuclear charge are distributed identically in the nuclear volume

(b) Nuclei with 83Z and 209A emit - radiation

(c) The surface contribution to the binding energy is proportional to 2/3A

(d) - decay occurs when the proton to neutron ratio is large, but not when it is small

Ans. : (b) and (c)

Solution: From given statement only (b) and (c) are correct.

Q47. Consider a one-dimensional harmonic oscillator of angular frequency . If 5 identical

particles occupy the energy levels of this oscillator at zero temperature, which of the

following statement(s) about their ground state energy 0E is (are) correct?

(a) If the particles are electrons, 0

13

2E

(b) If the particles are protons, 0

25

2E

(c) If the particles are spin-less fermions, 0

25

2E

(d) If the particles are bosons, 0

5

2E

Ans. : (a), (c) and (d)

Solution: If particles are electrons and protons then ground state energy

03 5 13

2 2 12 2 2 2

E

If the particles are spin-less fermions, then energy is

0

3 5 7 9 25

2 2 2 2 2 2E

If the particles are bosons 0

1 55

2 2E




Q48. A particle of mass m is placed in a three-dimensional cubic box of side a . What is the

degeneracy of its energy level with energy 2 2

214

2ma

?

(Express your answer as an integer)

Ans. : 6

Solution: 2 2 2 14x y zn n n

1, 2, 3

1, 3, 2

2, 3, 1

2, 3, 1

3, 1, 2

3, 2, 1

x y z

x y z

x y z

x y z

x y z

x y z

n n n

n n n

n n n

n n n

n n n

n n n

So degeneracy is 6

Q49. For a proton to capture an electron to form a neutron and a neutrino (assumed massless),

the electron must have some minimum energy. For such an electron the de-Broglie

wavelength in pictometers is……………


Ans. : 1.02

Solution: From conservation of energy

22 2 27 8 131.675 1.673 10 3 10 1.8 10e e e n pE m c K m m c Joules

2 22 2 4e eE pc m c pc 220.6 10 . / seceE

p kg mc

2epc m c

34

1222

6.6 101.1 10 1.1

0.6 10

hm pm

p




IIT-JAM 2018

Q50. Let gT and eT be the kinetic energies of the electron in the ground and the third excited

states of a hydrogen atom, respectively. According to the Bohr model, the ratio g

e

T

T is

(a) 3 (b) 4 (c) 9 (d) 16

Ans. : (d)

Solution: From Bohr model the kinetic energy and Total energy E and kinetic energy T

2

ET where 0

1g

EE , 0

16e

EE

1616 :1

1g g

e e

T E

T F

Q51. The mean momentum p

of a nucleon in a nucleus of mass number A and atomic

number Z depends on ,A Z as

(a) 1

3p A

(b) 1

3p Z

(c)1

3p A

(d) 2

3p AZ

Ans. : (c)

Solution: The radius of a nucleus can be combined as 2

(greater than the wavelength of

electron)

The moment h

p

1/30R R A which implies 1/3

0

hp A

R .

As, 1/ 3p A

Q52. A particle of mass m is in a one dimensional potential 0, 0

, otherwise

x LV x

.

At some instant its wave function is given by 1 2

1 2

33x x i x , where

1 x and 2 x are the ground and the first excited states, respectively. Identify the

correct statement.

(a) 2 2

2

3;

2 2

Lx E

m L

(b)

2 2

2

2;

3 2

Lx E

m L

(c) 2 2

2

8;

2 2

Lx E

m L

(d)

2 2

2

2 4;

3 2 3

Lx E

m L




Ans.: (a)

Solution: 0

0 0

0

91 24

3 3 3 31 2 33 3 3

EE E

E E

Where, 2 2

0 22E

mL

2 2 2 2

2 2

3 3

2 2E

mL mL

1 1 2 1 1 2 2 1

1 2 2 2

3 3 3 33 3

i iX X X X X

1 2

3 2 3 2

L L

2

L

Q53. A system of 8 non-interacting electrons is confined by a three dimensional potential

2 21

2V r m r . The ground state energy of the system in units of is ______

(Specify your answer as an integer.)

Ans. : 18

Solution: 0n is non degenerate so there will 2 electron in the ground state.

1n is triple degenerate so there is 6 electron in the first excited state

3 5

2 62 2

E

3 15 18

Q54. Rod 1R has a rest length 1m and rod 2R has a rest length of 2m . 1R and 2R are moving

with respect to the laboratory frame with velocities ˆvi and ˆvi , respectively. If 2R has

a length of 1m in the rest frame of 1R , v

c is given by__________


Ans. : 0.48

Solution:

vS S

1R

v

v

2R




V v , xu v

21

xx

x

u Vu

u V

c

2

2

2

1

v

vc

2

0 21 xu

l lc

2

21 2 1 xu

c

2

2 2 2 2

22

2

41 /1 4 / 3

14 4

1

vv c v c

vcc

2 2

2 2

3 34

4 4

v v

c c

2 2

13 3 12

4 4 52

v v

c c

12

52

v

c 0.479

v

c 0.48 .

Q55. Two events 1E and 2E take place in an inertial frame S with respective time space

coordinates (in SI units): 1 1 10, 0E t r

and 82 2 2 20, 10 , 0, 0zE t x y z . Another

inertial frame S is moving with respect to S with a velocity ˆ0.8v ci

. The time

difference 2 1t t as observed in S is ___________ s .

(Specify your answer in seconds upto two digits after the decimal point)

Ans. : 0.44

Solution: 2 1 0t t and 82 1 10x x

2 1 2 1

2 1 2 222 21 / 1 /

t t x x vt t

cv c v c

2 1 2

2 21 /

vx x

cv c

2 1 2

2 21 /

vx x

cv c

882

8

0.810 8 10

6 3 101 0.64

cc

80.44sec

18 .




IIT-JAM 2019

Q56. A classical particle has total energy E . The plot of

potential energy U as a function of distance r

from the centre of force located at 0r is shown in

the figure. Which of the regions are forbidden for

the particle?

(a) I and II (b) II and IV

(c) I an IV (d) I and III

Ans. : (d)

Solution: In the region I and III potential energy is more than total energy .

Q57. In the thermal neutron induced fission of 235U , the distribution of relative number of the

observed fission fragments (Yield) versus mass number A is given by

(a) (b) (c) (d)

Ans. : (a)

Q58. For a quantum particle confined inside a cubic box of side L , the ground state energy is

given by 0E . The energy of the first excited state is

(a) 02E (b) 02E (c) 03E (d) 06E

Ans. : (d)

Solution:

2 2 2 2 2

2 2 2, , 022x y z

x y zn n n x y z

n n nE n n n E

ma

2 2

2,1,1 1,2,1 1,1,2 02

4 1 16

2E E E E

ma

80 120 160

Yie

ld

A80 120 160

Yie

ld

A80 120 160

Yie

ld

A80 120 160

Yie

ld

A

0 r

IV

III

II

IE

U




Q59. A -ray photon emitted from a 137 Cs source collides with an electron at rest. If the

Compton shift of the photon is 133.25 10 m , then the scattering angle is closets to

(Planck’s constant 346.626 10 Jsh , electron mass 319.109 10 kgm and velocity

of light in free space 83 10 m/sc )

(a) o45 (b) o60 (c) o30 (d) o90

Ans. : (c)

Solution: .1 cos cos 1 e

e

m ch

m c h

13 31 8

34

3.25 10 9.109 10 3 10 31 0.866

26.6 10

030

Q60. The relation between the nuclear radius R and the mass number A , given

by 1/31.2 fmR A , implies that

(a) The central density of nuclei is independent of A

(b) The volume energy per nucleon is a constant

(c) The attractive part of the nuclear force has a long range

(d) The nuclear force is charge dependent

Ans. : (a), (b), (d)

Q61. An atomic nucleus X with half-life XT decays to a nucleusY , which has half-life YT .

The condition (s) for secular equilibrium is (are)

(a) X YT T (b) X YT T (c) X YT T (d) x YT T

Ans. : (d)

Q62. In a typical human body, the amount of radioactive 40 K is 53.24 10 percent of its mass.

The activity due to 40 K in a human body of mass 70kg is ______ kBq.


(Half-life of 40 163.942 10 SK , Avogadro’s number 23 16.022 10 molAN

Ans. : 6.0




Solution: dN

Ndt

3 5

6

70 100.693 3.24 106.022 10

40 1003.942 10 s

136.0 10 disintegrations / s

136.0 10 Bq

106.0 10 kBq

Q63. A proton is confined within a nucleus of size 1310 cm . The uncertainty in its velocity is

_________ 810 m/s .


(Planck’s constant 346.626 10h J and proton mass 271.672 10 kgPm )

Ans. : 0.31

Solution: 4

hp x

4

hv

m x

34

27 15

6.6 10

4 3.14 1.672 10 10

80.31 10 /m s

Q64. Given the wave function of a particle 2sin 0x x x L

L L

and 0 elsewhere

the probability of finding the particle between 0x and 2

Lx is _______.


Ans. : 0.5

Solution: 2sin 0x x x L

L L

, / 2 2

0

10

2 2

LLp x dx




Solid State Physics, Devices and Electronics

IIT-JAM 2005

Q1. A circuit and the signal applied at its input terminals ( iV ) are shown in figure below.

Which one of the options correctly describes the output waveform ( 0V ). (Assume all the

devices used are ideal).

(a) (b)

(c) (d)

Ans. : (c)

Solution: Its clamper circuit in which peak to peak remains fixed and voltage level will shift in

the direction of diode current.

Q2. The susceptibility of a diamagnetic material is

(a) positive and proportional to temperature

(b) negative and inversely proportional to temperature

(c) negative and independent of temperature

(d) positive and inversely proportional to temperature

Ans. : (b)

D oVC

iVt

2V

2V

0

t

2V

2V

0 t

2V

0

t

4V

0

t

4V-

0




Q3. Which of the following statements is correct for NaCl crystal structure?

(a) It is simple cubic lattice with one atom basis

(b) It is a face-centered cubic lattice with one atom basis

(c) It is a simple cubic lattice with two atom basis

(d) It is a face-centered cubic lattice with two atom basis

Ans. : (d)

IIT-JAM 2006

Q4. In a crystalline solid, the energy band structure ( -E k relation) for an electron of mass m

is given by

m

kkE

2

322

. The effective mass of the electron in the crystal is

(a) m (b) m3

2 (c)

2

m (d) 2m

Ans. : (c)

Solution: The expression of effective mass of electron in solid is 2

2

2*

dkEd

m

2 2 2 2

2

24 3 4

2 2

dE d Ek

dk m dk m m *

2

mm

Q5. The truth table for the given circuit is

(a) (b)

(c) (d)

J

Q

K

J K Q 0 0 0 1 1 0 1 1

1 0 0 1

J K Q 0 0 0 1 1 0 1 1

0 1 0 1

J K Q 0 0 0 1 1 0 1 1

0 1 1 0

J K Q 0 0 0 1 1 0 1 1

1 0 1 0




Ans. : (c)

Solution: . .K KQ J K J

Q6. In an intrinsic semiconductor, the free carrier

concentration n (in 3cm ) varies with temperature

T (in Kelvin) as shown in the figure below. The

band gap of the semiconductor is (use Boltzmann

constant 5 18.625 10Bk eVK )

(a) 1.44eV

(b) 0.72eV

(c) 1.38eV

(d) 0.69eV

Ans. : (d)

Solution: Since exp2

gi c v

En n N N

kT

1

2 2 1

1 1exp

2gEn

n k T T

51

2 2 1

1 12 ln 2 8.625 10 36.5 32.5 0.003 0.0002g

nE k

n T T

5 32 8.625 10 4 1 10 0.69gE eV

IIT-JAM 2007

Q7. Fermi energy of a certain metal 1M is 5eV . A second metal 2M has an electron which is

6% higher than that of 1M . Assuming that the free electron theory is valid for both the

metals, the Fermi energy of 2M is closet to

(a) 5.6eV (b) 5.2eV (c) 4.8eV (d) 4.4eV

Ans. : (b)

Solution: The expression of Fermi energy is 3/222

32

nm

EF

Let us consider that 11,EFn is the concentration of electron and Fermi energy in metal M1

and 22 ,EFn is in metal M2.

5.32

5.33

5.34

5.35

5.36

2 3 /1000

nln




Given, 1112 06.106.0 nnnn

2

1

2 2/322/3 2/32

2/32 12/3 2/32 2/32 1 1

1

31.062

1.06 1.0396

32

F

F

nE n nm

E n nnm

2 11.0396 1.0396 5 5.2F FE E eV eV

Q8.

Fig. (i) Fig. (ii)

Figure (i) and (ii) represent respectively,

(a) NOR, NOR (b) NOR, NAND

(c) NAND, NAND (d) OR, NAND

Ans. : (c)

IIT-JAM 2008

Q9. The ratio of the second-neighbour distance to the nearest-neighbour distance in an fcc

lattice is

(a) 22 (b) 2 (c) 3 (d) 2

Ans. : (d)

Solution: In FCC lattice the first nearest is at distance 2 / 2 / 2a a

Whereas the second nearest at distance of a

The ratio of the second-neighbour distance to the nearest-neighbour distance is

22/

a

a

Q10. Consider a doped semiconductor having the electron and the hole mobilities n and p ,

respectively. Its intrinsic carrier density is in . The hole concentration p for which the

conductivity is minimum at a given temperature is

(a) p

nin

(b) n

pin

(c) n

pin

(d) p

nin




Ans. : (a)

Solution: 2

Conductivity in p n p

ne n p e p

p

For minimum conductivity, 0 /i n pd

p ndp

Q11. The logic expression for the output Y of the following circuit is

(a) SQRQP (b) SQRQP

(c) SQRQP (d) SQRQP

Ans. : (a)

Solution:

P

Q

R

S

Y

P QP

Q

R

S

Y

QR

P Q QR

P Q QR S

P Q QR S




IIT-JAM 2009

Q12. Monochromatic X -rays of wavelength 0

1A are incident on a simple cubic crystal. The

first order Bragg reflection from 311 plane occurs at angle of 030 from the plane. The

lattice parameter of the crystal in 0

A is

(a) 1 (b) 3 (c) 2

11 (d) 11

Ans. : (d)

Solution: Expression of Bragg´s law is nd sin2 ,

where, d is the inter-planar spacing defined for cubic lattice of lattice constant a in term

of Miller indices h k l as 222 lkh

ad

The lattice parameter a is

11112/12

1113

30sin2

1

sin2222

0222

lkha

Q13. Which one of the following is an INCORRECT Boolean expression?

(a) QPQQP (b) PQPQP

(c) QQPP (d) QRQPRQPRQPRQP

Ans. : (c)

Solution: (a) 1.PQ PQ P P Q Q Q

(b) P Q P Q P QQ P

(c) 1P P Q P PQ P Q P

(d) PQR PQR PQR PQR PQ R R PQ R R PQ PQ P P Q Q

Q14. A battery with a constant emf and internal resistance ir provides power to an external

circuit with a load resistance made up by combining resistance LR and 2 LR in parallel.

For what value of LR will the power delivered to the load be maximum?

(a) 4i

L

rR (b)

2i

L

rR (c) iL rR

3

2 (d) iL rR

2

3




Ans. : (d)

Solution: Since LR and 2 LR are in parallel so load2 2

2 3L L

LL L

R RR R

R R

.

Power through load

2

2

i

P I R Rr R

For maximum power through load

2 2 2

4

20 0 2 0i i

i i

i

r R R r RdPr R R R r

dR r R

Thus 2 3

3 2i L i L iR r R r R r

IIT-JAM 2010

Q15. The following are the plots of the temperature dependence of the magnetic susceptibility

for three different samples.

The plots ,a b and c correspond to

(a) ferromagnet, paramagnet and diamagnet, respectively

(b) paramagnet, diamagnet and ferromagnet, respectively

(c) ferromagnet, diamagnet and paramagnet, respectively

(d) diamagnet, paramagnet and ferromagnet, respectively

Ans. : (b)

Q16. The value of at which the first-order peak in X -ray (0

1.53A ) diffraction

corresponding to 111 plane of a simple cubic structure with the lattice constant,

0

2.65A , is approximately

(a) 015 (b) 030 (c) 045 (d) 060

)(T

T0

a c)(T

T0

b)(T

T0




Ans. : (b)

Solution: Expression of Bragg´s law is, nd sin2


of Miller indices (h k l) as 222 lkh

ad

2

13

65.22

53.1

22sin 222

lkh

ad

1 01sin 30

2

Q17. Consider the following truth table

The logic expression for F is

(a) AB BC CA

(b) CBCABA

(c) BABAC

(d) BABAC

Ans. : (d)

Solution: F AB C A B

A B C F

0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

1 0 0 0 1 1 1 0

C

BA

BA

AB

BA

00

01

11

10

1 0C

1

11

1

AB

AC

BC




IIT-JAM 2011

Q18. An X - ray diffraction XRD experiment is carried out on a crystalline solid having

FCC structure at room temperature. The solid undergoes a phase transformation on

cooling to 020 C and shows orthorhombic structure with small decrease in its unit cell

lengths as compared to the FCC unit cell lengths. As a result the 311 line of the XRD

pattern corresponding to the FCC system

(a) will split into a doublet

(b) will split into a triplet

(c) will remain unchanged

(d) will split into two separate doublets

Ans. : (b)

Solution: In FCC the inter-planar spacing is defined as 222 lkh

ad

whereas in Orthorhombic the inter-planar spacing is

2

2

2

2

2

2

1

c

l

b

k

a

hd

For 311 , numbers XRD peaks in FCC is derived from Bragg’s law as

112

113222

sin 222222

aalkh

ad

Thus only one peak appears in FCC.

In Orthorhombic,

2

2

2

2

2

2

22

sin

c

l

b

k

a

hd

For 311 , there will be three peaks corresponding to

1 1 11 2 32 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2

sin , sin sin3 1 1 1 3 1 1 1 3

2 2 2

and

a b c a b c a b c




Q19. Which of the following circuit does not satisfy the Boolean expression FBABA

(a) (b)

(c) (d)

Ans. : (d)

Solution: (a) F AB AB

(b) . .F A B AB A B A B AB AB

(c) .F A B A B AB AB

(d) F AB AB

IIT-JAM 2012

Q20. An X -ray beam of wavelength 0

1.54 A is diffracted from the 110 planes of a solid with

a cubic lattice of lattice constant 0

3.08A . The first-order Bragg diffraction occurs at

(a)

4

1sin 1 (b)

22

1sin 1 (c)

2

1sin 1 (d)

2

1sin 1

Ans. : (b)

Solution: Expression of Bragg´s law is nd sin2


of Miller indices ( h k l ) as 222 lkh

ad

22

12

08.32

54.1

22sin 222

lkh

ad

1 1sin

2 2

F

F

F

F




Q21. The Boolean expression QPP , where P and Q are the inputs to a circuit, represents

the following logic gate

(a) AND (b) NAND (c) NOT (d) OR

Ans. : (d)

Solution:

P Q PQ P PQ

0

0

1

1

0

1

0

1

0

1

0

0

0

1

1

1

IIT-JAM 2013

Q22. The fraction of volume unoccupied in the unit cell of the body centered cubic lattice is

(a) 8

38 (b)

8

3 (c)

6

26 (d)

23

Ans. : (a)

Solution: The effective number of atoms in bcc structure is

2102

18

8

1

2

1

8

1 ifceff nnnn

The radius of atom and lattice constant are related as

4

3ar

The volume occupide by the atoms in unit cell is

3

34 3 32

3 4 8

aa

Thus amount of volume unoccupied in the unit cell is

3 3 33 8 3volume of unit cell filled volume

8 8a a a

a

a

a

G

FE

D C




Thus, the fraction of volume unoccupied in the unit cell of the body centered cubic lattice

is

8

38

Q23. For an ideal op-amp circuit given below, the dc gain

and the cut off frequency respectively are

(a) 1 and 1kHz

(b) 1 and 100 Hz

(c) 11 and 1kHz

(d) 11 and 100 Hz

Ans. : (b)

Solution: DC Gain10

1 111

and 1

2Hf RC

11

12 1

2

kHz

Q24. A variable power supply 5 20V V is connected to a Zener diode specified by a

breakdown voltage of 10V (see figure). The ratio of the maximum power to the minimum

power dissipated across the load resistor is

Ans. : 9.02

Solution: When VV 5 open circuit voltage VVV Zi 1033.351500

1000

VVV iL 33.3L

iL R

VP

2

min, .

When VV 20 open circuit voltage VVV Zi 1033.13201500

1000

VVV zL 10L

ZL R

VP

2

max, 22

,max2

,min

109.02

3.33L Z

L i

P V

P V

k10k1

k1

F2

1

500

VV 205 k1




IIT-JAM 2014

Q25. Octal equivalent of decimal number 10478 is

(a) 8736 (b) 8673 (c) 8637 (d) 8367

Ans. : (a)

Solution: 2 1 08 10736 7 8 3 8 6 8 448 24 6 478

2 1 08 10673 6 8 7 8 3 8 384 56 3 443

2 1 08 10637 6 8 3 8 7 8 384 24 7 415

2 1 08 10367 3 8 6 8 7 8 192 48 7 247

Q26. In an ideal operational amplifier depicted below, the potential at node A is

(a) 1V (b) 0V (c) 5V (d) 25V

Ans. : (b)

Solution: Node A is virtually grounded.

Q27. To operate a npn transistor in active region, its emitter-base and collector- base junction

respectively, should be

(a) forward biased and reversed biased (b) forward biased and forward biased

(c) reversed biased and forward biased (d) reversed biased and reversed biased

Ans. : (a)

Solution: In active region: emitter-base junction is F.B.

: collector-base junction is R.B.

k 5

V 5

IV

A

_k 25




Q28. Diamond lattice can be considered as a combination of two fcc lattice displaced along the

body diagonal by one quarter of its length. There are eight atoms per unit cell. The

packing fraction of the diamond structure is

(a) 0.48 (b) 0.74 (c) 0.34 (d) 0.68

Ans. : (c)

Solution:

343effn r

P FV

Where, 3 3 88, and 2

4 3eff

a rn V a r a

3

3

48 33 0.34

168

3

rP F

r

.

Q29. Thermal neutrons (energy 300 , 0.025Bk eV ) are sometimes used for structural

determination of materials. The typical lattice spacing of a material for which these can

be used is

(a) 0.01nm (b) 0.05nm (c) 0.1 nm (d) 0.15nm

Ans. : (c)

Q30. A sine wave of 5V amplitude is applied at the input of the

circuit shown in the figure. Which of the following

waveforms represents the output most closely?

(a) (b)

(c) (d)

Ans. : (d)

Solution: If 3inV V , diode is OFF and 0 inV V .

If 3inV V , diode is ON and 0 3V V .

inV outVk1

V3 5V

-3V

3V

-5V

5V

-5V

3V

-3V




Q31. 1011 binary input have been applied at 0123 XXXX input in the shown logic circuit made

of XOR gates. The binary output 0123 YYYY of the circuit will be

(a) 1101 (b) 1010 (c) 1111 (d) 0101

Ans. : (a)

Solution:

IIT-JAM 2015

Q32. Temperature dependence of resistivity of a metal can be described by

(a) (b)

(c) (d)

3X

2X

1X

0X

3Y

2Y

1Y

0Y

3Y

2Y

1Y

0Y

2X

1X

0X

3X1

0

1

1

1

1

0

1

R

T

R

T

R

T

R

T




Ans. : (a)

Solution: Electrical resistivity of metal varies as

5T (For DT )

T (For DT )

where D is the Debye temperature. Thus, correct answer is option (a)

Q33. 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)

Q34. Which of the following circuits represent the Boolean expression

PQQRPS

(a) (b)

(c) (d)

SR0V

V2015

V8.6

PQ S

P

QS

PQ

SR

PQ

SR




Ans. : (b)

Solution: S P QR QP P QR QP P QR QP PQ

(a) (b)

(c) (d)

Q35. 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)

Q36. 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 R

RV

(c) The amplitude of 0V is V2

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

Ans. : (a), (c) and (d)

P

QS PQ

PQ S PQ P Q

PQ

SR

P Q

R

P Q R PQ

R

PQ

SR

PQR

011

y 111

100

x

z

P

iV

1R

0V

2R




Solution: Voltage at P is 0 2

1 2P

V RV

R R

.

Current through 1R is

0 2

0 1 01 21

1 1 1 1 2 1 2

oo P

V RV

V V V R VR RI

R R R R R R R

and Current through 2R is

02

2 1 2

P VVI

R R R

.

Thus 1 2I I . Option (a) is true

The potential at P is 0 2

1 2P

V RV

R R

. Option (b) is not true.

20

1

31 1 0.5sin 2sin

1i

RV V t t

R

2mV V . Option (c) is true

Option (d) is true

Q37. In the given circuit 10CCV V and 100 for npn transistor. The collector voltage

CV (in volts) is………….

Ans. : 5.7

Solution: 53

5 0.74.3 10 4.3

100 10B C BI A I I mA

10 4.3 5.7C CC C CV V I R V

Q38. 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

K100

K1

CCV

CV

V5




Solution:

2

1

/ 0.5/ 0.025 20

/ 20 20.4 / 0.025 16/

1

1 1 11 54.5 54.5

1 11

T

T

T

V V

V V

V V

e e eII I e I A

I e ee

Q39. 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.

IIT-JAM 2016

Q40. The solution of the Boolean equation Y A B AB is

(a) 1 (b) AB (c) A B (d) A B

Ans. : (b) and (d) both are correct

Solution: . 1Y A B AB A B A B A B B A B or AB

Q41. If a constant voltage V is applied to the input of the following OPAMP circuit for a

time t , then the output voltage 0V will approach

(a) V exponentially (b) V exponentially

(c) V linearly (d) V linearly

Ans. : (d)

Solution: R CI I 000 d VVC

R dt

0

0

dV V VV t c

dt RC RC

VR

C

oV




Q42. A pn junction was formed with a heavily doped 18 310 cm p -region and lightly doped

14 310 cm 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 the both side of the junction

(d) If the pn junction is reverse biased, then the width of the depletion region increase

Ans. : (a), (d)

Solution: Since p - region is heavily doped and n - region is lightly doped, on n - side width of

depletion layer will be more and on p - side width of the depletion layer will be less.

Q43. The addition of two binary numbers 1000.01 and 0001.11 in binary representation

is………….

Ans. : 1010

Solution: 1000.01

0001.11

1001.00

Q44. The number of second-nearest neighbor ions to a +Na ion in NaCl crystal is__________.

Ans. : 12

Solution: The 2nd nearest neighbour is at distance 2

2 2

a a

The number of 2nd nearest neighbour 3 8

122

Q45. The output voltage 0V of the OPAMP circuit given below is……………..V

1V

R

2R

oV

2V

3V

R

R

R

Cl

Na




Ans. : 6

Solution: 0 1 1

21 3

RV V V

R

Where, 1

/ 2 / 2 / 21 2 3

/ 2 / 2 / 2

R R RV

R R R R R R

1

1 1 11 2 3 2

3 3 3V V 0 6V V

Q46. In the circuit given below, the collector to emitter voltage CEV is……………V .

(Neglect BEV , take 100 )

Ans. : 2.5

Solution: 5

10 55 5BV V

5E B BEV V V V 5

0.510

EE

E

VI mA

R

10 0.5 5 10 2.5CE CC C C EV V I R R V

Q47. X -ray diffraction of a cubic crystal gives an intensity maximum for Bragg angle 020

corresponding to the 110 plane. The lattice parameter of the crystal is………….. nm .

(Consider wavelength of 0.15X ray nm )

Ans. : 0.31 Solution: According to Bragg’s law 2 sind n

For 1n , 9 9

0

0.5 10 0.15 102 sin 0.219

2sin 2 sin 20 2 0.342d d d nm

Now, 2 2 2 2

a ad

h k l

2 0.31 0.31a d nm a nm

CEV

5k

10CCV V

10 k

5k

5k

1V

R

2R

2V

3V

R

R

R

0V




IIT-JAM 2017

Q48. Consider the following circuit with two identical Si diodes. The input ac voltage

waveform has the peak voltage 2 ,PV V as shown

The voltage waveform across PQ will be represented by:

(a) (b)

(c) (d)

Ans. : (c)

Solution:

During positive half cycle 1D is ON, when input is more than 0.7V and 1D is OFF when

input is less than 0.7V.

During negative half cycle 1D is ON, when input is more negative than 0.7 IR and

1D is OFF when input is less negative than 0.7 IR .

Q49. If the Boolean function Z PQ PQR PQRS PQRST PQRSTU , then Z is

(a) PQ R S T U (b) PQ

(c) P Q (d) P Q R S T U

PQV

t

PQV

t

PQV

t

PQV

t

inV

2V

t

R

RQ

P

1D 2D

inV

2V

t

R

RQ

P

1D2D




Ans. : (c)

Solution: Z PQ PQR PQRS PQRST PQRSTU

1PQ PQR PQRS PQRST U PQ PQR PQRS PQRST

1PQ PQR PQRS T PQ PQR PQRS

1PQ PQR S PQ PQR 1PQ R PQ Z PQ P Q

Q50. Shown in the figure is a combination of logic gates. The output values at P and Q are

correctly represented by which of the following?

(a) 0 0 (b) 11 (c) 0 1 (d) 1 0

Ans. : (c)

Solution:

Q51. A plane in a cubic lattice makes intercepts of ,2

aa and

2

3

a with the three

crystallographic axes, respectively. The Miller indices for this plane are:

(a) 2 43 (b) 34 2 (c) 634 (d) 123

Ans. : (a)

Solution: Intercepts: a , 2

a,

2

3

a

Divide by a : 1, 1

2,

2

3

Reciprocal: 1, 2 , 3

2

LCM: 2 , 4 , 3

Miller Indices = 2 4 3

1

0Q

P

0

1

Q

P 0

10

1




Q52. Which one of the following schematic curves best represents the variation of conductivity

of a metal with temperature T ?

(a) (b)

(c) (d)

Ans. : (b)

Solution: Resistivity of metal is 0 ph , where 0 is temperature independent

And, 5ph T when DT and ph T when DT

The conductivity is defined as 1

Thus correct option is (b)

Q53. KCl has the NaCl type structure which is fcc with two-atom basis, one at 0,0,0 and

the other at 1/ 2,1/ 2,1/ 2 . Assume that the atomic form factors of K and Cl are

identical. In an X - ray diffraction experiment on KCl , which of the following h k l

peaks will be observed? (a) 100 (b) 110 (c) 111 (d) 200

Ans. : (d)

Solution In KCl the reflection from 111 layer containing K ions is exactly out of phase with

the reflection from the Cl close packed layers. The net effect is that the two reflections

cancel and 111 is absent. This mean the first reflection is from 200

0 T

0 T

0 T

0 T

S

T

0 T




Q54. An n p n transistor is connected in a circuit as shown

in the figure. If 1 , 50, 0.7C BEI mA V V and the

current through 2R is 10 BI where, BI is the base current.

Then the ratio 1 2/R R is:

(a) 0.375

(b) 0.25

(c) 0.5

(d) 0.275

Ans. : (b)

Solution: 1 2 2 10i BI I I I I , 1 , 50, 0.7C BEI mA V V

2 2 0.7 0.5 1.2B BE E EV I R V I R V

2

2

1.2 1.2 1.26

10 10 1/ 50B

V V VR k

I I

21

1 2 1

6 61.2 24

6CC

B

V RV R k

R R R

2

1

60.25

24

R

R

Q55. An intrinsic semiconductor of band gap 1.25eV has an electron concentration 10 310 cm

at 300 K . Assume that its band gap is independent of temperature and that the electron

concentration depends only exponentially on the temperature. If the electron

concentration at 200 K is 310 1 10; integerNY cm Y N , then the value of N

is…………

Ans. : 4

Solution: / 2gE kT

i c vn N N e1

2

/ 2 10 31

/ 22 2

10 1 1exp

2 300 200

g

g

E kTgi

E kTi i

En e cm

n n ke

1R

2R 0.5k0.5EV V

3CEV VCV

2.5k

6V6V

1I

2I

iI

B

E

C

1R

2R 0.5k0.5EV V

3CEV VCV

2.5k

6V




1910 10 10 6

2 23

1 1 1.25 1.6 10 110 exp 10 exp 10 5.7 10

2 300 200 2 1.38 10 100 6g

i

En

k

4 3

2 5.7 10in cm 4N

Q56. An OPAMP is connected in a circuit with a Zener diode as shown in the figure. The value

of resistance R in k for obtaining a regulated output 0 9V V is…………..


Ans. : 1.09

Solution: 0

11 4.7 9 1.09V V R k

R

IIT-JAM 2018

Q57. Which one of the following graphs shows the correct

variation of 0V with iV ? Here, dV is the voltage drop across

the diode and the OP-Amp is assumed to be ideal.

(a) (b)

(c) (d)

Ans. : (a)

Solution: During positive half cycle it behaves as voltage follower i.e. 0 iv v , during negative

half cycle 0 0v .

R

1k12inV V 4.7ZV V

oV

1k

iV dV0V

LR

0 iV

0V

0 iV

0V

dV

iV

0V

dV0

0V

iV0




Q58. The Boolean expression AB A B A B can be simplified to

(a) A B (b) AB (c) A B (d) AB

Ans.: (c)

Solution: Y AB AB A B AB AB AB

A B AB AB AB AB AB A B

Q59. In a pn junction, dopant concentration on the p -side is higher than that on the n -side.

Which of the following statements is (are) correct, when the junction is unbiased?

(a) The width of the depletion layer is larger on the n -side.

(b) At thermal equilibrium the Fermi energy is higher on the p -side.

(c) In the depletion region, number of negative charges per unit area on the p -side is

equal to number of positive charges per unit area on the n -side

(d) The value of the built-in potential barrier depends on the dopant concentration.

Ans. : (a), (c) and (d)

Q60. Which of the combinations of crystal structure and their coordination number is (are)

correct?

(a) body centered cubic 8 (b) face centred cubic 6

(c) diamond 4 (d) hexagonal closed packed 12

Ans. : (a), (c), (d)

Solution: Co-ordination number in different crystal structure are

(i) Body central cubic 8

(ii) Face central cubic 12

(iii) Diamond 4

(iv) Hexagonal closed packed 12




Q61. For the given circuit, value of the base current bI of the npn transistor will be ____mA.

( is the current gain and assume Op-Amp as ideal.)

(Specific your answer in mA upto two digits after the decimal point.)

Ans. : 0.1

Solution: 2 5 Ev V V

55

1EI mA

55 0.1

50B BI mA I mA mA

Q62. The lattice constant of unit cell of NaCl crystal is 0.563 nm . X -rays of wavelength

0.141 nm are diffracted by this crystal. The angle at which the first order maximum

occurs is _______degrees.

Ans. : 12.5

Solution: 2 sind

1 1 3sin sin

2 2d a

1 1 03 0.141sin sin 0.217 12.53

2 0.563

bI1k

10V

50

1k

5V

1k

bI

10V

50

1k

5V1v

2v

E

EI




Q63. For the following circuit, the collector voltage with respect to ground will be _________

V . (Emitter diode voltage is 0.7V and DC of the transistor is large)

(Specify your answer in volts upto one digit after the decimal point).

Ans. : 3.1

Solution: 0.7BEV V and largedc so 0BI

From input section;

0 0.7 1 3 0EI

2.3EI mA

From output section;

10 3 2.3 0CV

10 6.9 3.1CV V

Q64. In the following circuit, the time constant RC is much greater than the period of the

input signal. Assume diode as ideal and resistance R to be large. The dc output voltage

across resistance R will be ____________ V .

(Specify your answer in volts upto one digit after the decimal point)

Ans. : 68

Solution: It’s a voltage doubler circuit

2 2 2 2 2 24R m rmsV V V 68RV V

10V

1k

3k

3V

3k

24 rmsV

C

CR

0BI

10V

1k

3k

3V

3k

B

E

C




Q65. For a metal, the electron density is 28 36.4 10 m . The Fermi energy is __________ eV .

( 34 31 196.626 10 , 9.11 10 , 1 1.6 10eh J s m kg eV J )

(Specify your answer in electron volts eV upto one digit after the decimal point)

Ans. : 5.84

Solution: 2342

2/3 2/32 2 2831

1.05 103 3 6.4 10

2 2 9.11 10FE nm

39 20 196.1 10 1.53 10 9.34 10 J

19

19

9.34 105.84

1.6 10eV eV

.

IIT-JAM 2019

Q66. Which one of the following crystallographic planes represent 101 Miller indices of a

cubic unit cell?

(a) (b) (c) (d)

Ans. : (b)

Solution: Plane intercepts

: : : : : : : :1 0 1

a b c a b cx y z a c

h k l

, ,x a y z c

Plane is parallel to y - axis and intersecting x and z - axis at a and c . Thus, option (b)

is correct.

xm

z

x

y

z z

x

y

x

y

z




Q67. For using a transistor as an amplifier, choose the correct option regarding the resistances

of base-emitter BER and base-collector BCR junctions

(a) Both BER and BCR are very low (b) Very low BER and very high BCR

(c) Very high BER and very low BCR (d) Both BER and BCR are very high

Ans. : (b)

Q68. The output of following logic circuit can be simplified to

(a) X YZ (b) Y XZ (c) XYZ (d) X Y Z

Ans. : (b)

Solution: Output XY X Y Z Y Y Z XY XY XZ Y YZ

1XY XZ Y Y X XZ Y XZ

Q69. For a forward biased p-n junction diode, which one of the following energy-band

diagrams is correct ( F is the Fermi energy)

(a) (b)

(c) (d)

Ans. : (a)

Y

Z

X

Electron Energy

Conduction Band-typen-typep

F p

Valence Band

F n ElectronEnergy


Valence Band

F

Electron Energy


Valence Band

F n F p

ElectronEnergy


Valence Band F n

F p




Q70. The location of Cs and Cl ions inside the unit cell of cacl crystal is shown in the

figure. The Bravais lattice of CaCl is

(a) Simple cubic (b) Body centred orthorhomble

(c) Face centred cubic (d) Base centred orthorhombic

Ans. : (a)

Solution: Cesium-Chloride is made of two interpenetrating simple cubic lattices are displaced

diagonally by half of the diagonal length. Thus, Bravais lattice of CsCl is simple cubic.

The correct option is (a).

Q71. In the X -ray diffraction pattern recorded for a simple cubic solid (lattice) parameter

o1a A ) using X -rays of wavelength

o1A , the first order diffraction peak(s) would

appear for the

(a) 100 planes (b) 112 planes (c) 210 planes (d) 220 planes

Ans. : (a)

Solution: In simple cubic cell, planes are present. The first order diffraction peak would appear

for the first plane 100 .

Q72. Sodium Na exhibits body-centred cubic (BCC) crystal structure with atomic radius

0.186 nm . The lattice parameter of Na unit cell is______ nm .

Ans. : 0.43

Solution: For BCC, 3 4a r

4 4 0.186

0.433 3

ra a nm

a

a

ay

z

x

Cl

Cs




Q73. For the input voltage 1 200 sin 400V mV t , the amplitude of the output voltage 0V of

the given OPAMP circuit is __________V . (Round off to 2 decimal places)

Ans. : 11.03

Solution: 01

351 4 5 200 sin 400

10 iv v mV t

02

354 5 200 sin 400

10v mV t

0

35 354 5 200 sin 400

10 10v mV t

3.5 3.5 4.5 200 11.03mV mV Volts

Q74. The value of emitter current in the given circuit is _______ A .


Ans. : 444.9

Solution: 1

CC BEB

B E

V VI

R R

0V

fR

35k

10k10k

10k

1R2R

3R

iV

fRfR

35k35k

2 M 4kC

100

E

EC2k

B

10CCV V

0.3V




6 3

10 0 3

2 10 101 2 10BI

6

9.7

2.202 10A

1E BI I 9.7

1012.202

A 444.9 A

Q75. The Zener current zI for the given circuit is ______ mA .

Ans. : 1

Solution: Open circuit voltage 20 2

40 40 26.720 10 3i

kV V Volts

k k

i ZV V , Zener “ON”

20

120

ZL

L

VI mA

R and

40 202

10RI mA

1Z R LI I I mA

Q76. The decimal equivalent of the binary number 110.101 is _______.

Ans. : 6.625

Solution: 2 1 0 1 2 3110 101 1 1 2 0 2 1 2 0 2 1 2

1 14 2 0 0

2 8

6 0.5 0 0.125 6.625

10R k ZI

LV

RV

Vin

= 4

0 V

VZ =

20

V

RL =

20

k

Documents

Mathematical MethodsInstitute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016