mathematical physics solutions · fiziks institute for net/jrf, gate, iit‐jam, m.sc. entrance,...

fiziks Institute 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 Phone: 011-26865455/+91-9871145498

Website: www.physicsbyfiziks.com | Email: [email protected]

1

MATHEMATICAL PHYSICS SOLUTIONS

GATE-2010

Q1. Consider an anti-symmetric tensor ijP with indices i and j running from 1 to 5. The

number of independent components of the tensor is

(a) 3 (b) 10 (c) 9 (d) 6

Ans: (b)

Solution: The number of independent components of the tensor

= 105252

1

2

1 2 NN

5N

Q2. The value of the integral

2

sinz

C

e zdz

z , where the contour C is the unit circle: 12 z ,

is

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

Ans: (d)

Solution:

2 1z 1 3z i.e. the pole 0z does not lie inside the contour.

2

sin2 0 0

z

C

e zdz i

z .

Q3. The eigenvalues of the matrix

100

023

032

are

(a) 5, 2, -2 (b) -5, -1, -1 (c) 5, 1, -1 (d) -5, 1, 1

Ans: (c)

Solution: The characteristic equation of the matrix A , 0 IA

0

100

023

032

IA 21 2 9 0 1, 2 3

1,1,5

Q4. If

3for 3

,3for 0

xx

xxf then the Laplace transform of f(x) is

(a) 2 3ss e (b) 2 3ss e (c) 2s (d) 2 3ss e




2

Ans: (d)

Solution:

0

dxxfexfL sx

33

3

0

3 dxexdxxfedxxfe sxsxxs

2 3

3 33 3

1 13 1 0

sx sx sxsx se e e

L f x x dx e dx s es s s s s

Q5. The solution of the differential equation for )cosh(2:2

2

tydt

ydty , subject to the

initial conditions 00 y and 00

tdt

dy, is

(a) ttt sinhcosh2

1 (b) ttt coshsinh

(c) tt cosh (d) tt sinh

Ans: (d)

Solution: For C.F 012 yD 1m tt eCeCFC 21..

tttt

eD

eD

ee

Dt

DIP

1

1

1

1

22

1

1cosh2

1

1..

2222 tt e

te

t 22

tttt et

et

eCeCy 2221

1 2As, 0 0 0............ 1y C C

tttttt eet

eet

eCeCdt

dy 2

1

22

1

221

0

Also, 0t

dy

dt

1 2 1 2

1 10 0 0 0........... 2

2 2C C C C

From equation (1) and (2),

1 20, 0C C .

Thus 2 2

t tt ty e e tty sinh




3

GATE-2011

Q6. Two matrices A and B are said to be similar if B = P-1AP for some invertible matrix P.

Which of the following statements is NOT TRUE?

(a) Det A = Det B (b) Trace of A = Trace of B

(c) A and B have the same eigenvectors (d) A and B have the same eigenvalues

Ans: (c)

Solution: If A and B be square matrices of the same type and if P be invertible matrix, then

matrices A and B = P-1AP have the same characteristic roots.

Then, IPPAPPIB 11 PIAP 1

where I is identity matrix.

PIAPIB 1 PIAP 1 PPIA 1 1 PPIA IA

Thus, the matrices A and B (= P-1AP) have the same characteristic equation and hence

same characteristic roots or eigen values. Since, the sum of the eigen values of a matrix

and product of eigen values of a matrix is equal to the determinant of matrix, hence third

alternative is incorrect.

Q7. If a force F is derivable from a potential function V(r), where r is the distance from the

origin of the coordinate system, it follows that

(a) 0 F (b) 0 F (c) 0V (d) 02 V

Ans: (a)

Solution: Since, F is derivative of potential V(r) and rVF

0 VF .

Q8. A 33 matrix has elements such that its trace is 11 and its determinant is 36. The

eigenvalues of the matrix are all known to be positive integers. The largest eigenvalues of

the matrix is

(a) 18 (b) 12 (c) 9 (d) 6

Ans: (d)

Solution: We know that for any matrix

1. The product of eigenvalues is equals to the determinant of that matrix.

2. .......321 Trace of matrix

11321 and 36321 . Hence, the largest eigen value of the matrix is 6.




4

Q9. The unit vector normal to the surface x2 + y2 – z = 1 at the point P(1, 1, 1) is

(a) 3

ˆˆˆ kji (b)

6

ˆˆˆ2 kji (c)

6

ˆˆ2ˆ kji (d)

3

ˆˆ2ˆ2 kji

Ans: (d)

Solution: The equation of the system is 01,, 22 zyxzyxf

1ˆˆˆ 22

zyxkz

jy

ix

f

ˆˆ ˆ2 2xi yj k

Hence, unit normal vector at (1, 1, 1) f

f

3

ˆˆ2ˆ2 kji .

Q10. Consider a cylinder of height h and radius a, closed at both ends, centered at the origin.

Let zkyjxir ˆˆˆ be the position vector and n be a unit vector normal to the surface.

The surface integral dsnrS ˆ over the closed surface of the cylinder is

(a) 2πa2 (a + h) (b) 3πa2h (c) 2 πa2h (d) zero

Ans: (b)

Solution: haddrdsnrVS V

233.ˆ.

Q11. The solutions to the differential equation 1

y

x

dx

dy

are a family of

(a) circles with different radii

(b) circles with different centres

(c) straight lines with different slopes

(d) straight lines with different intercepts on the y-axis

Ans: (a)

Solution: 1

y

x

dx

dy0 dyydyxdx 1

22

22Cy

yx 1

22 22 Cyyx

O

x

y

z




5

CCyx 1210 1

22

which is a family of circles with different radii.

Q12. Which of the following statements is TRUE for the function 2

sin

z

zzzf ?

(a) f z is analytic everywhere in the complex plane

(b) f z has a zero at z

(c) f z has a pole of order 2 at z

(d) f z has a simple pole at z

Ans: (c)

Solution: 2

sinz zf z

z

has a pole of order 2 at z

Q13. Consider a counterclockwise circular contour 1z about the origin. Let 2

sin

z

zzzf ,

then the integral dzzf over this contour is

(a) –iπ (b) zero (c) iπ (d) 2iπ

Ans: (b)

Solution: Since, pole z does not lie inside the contour, hence

0f z dz

GATE-2012

Q14. Identify the correct statement for the following vectors jia ˆ2ˆ3

and jib ˆ2ˆ

(a) The vectors a

and b

are linearly independent

(b) The vectors a

and b

are linearly dependent

(c) The vectors a

and b

are orthogonal

(d) The vectors a

and b

are normalized

Ans: (a)

Solution: If jibjia ˆ2ˆ,ˆ2ˆ3

are linearly dependent, then




6

,0 bma

for some values of m but here,

3 + m = 0 and 2 + 2m = 0 , do not have any solution. So, they are linearly independent.

0ba

(Not orthogonal); 0ba

(Not normalized)

Q15. The number of independent components of the symmetric tensor Aij with indices

, 1, 2,3i j is

(a) 1 (b) 3 (c) 6 (d) 9

Ans: (c)

Solution: For symmetric tensor,

333231

232221

131211

AAA

AAA

AAA

Aij

12 21 23 32 13 31, ,A A A A A A , hence there are six independent components.

Q16. The eigenvalues of the matrix

010101010

are

(a) 0, 1, 1 (b) 2,2,0

(c) 0,2

1,

2

1 (d) 0,2,2

Ans: (b)

Solution: 0 IA 0

10

11

01

2,2,0012

GATE-2013

Q17. If A

and B

are constant vectors, then rBA

is

(a) BA

(b) BA

(c) r

(d) zero

Ans: (d)

Solution: Let zyxAA ˆˆˆ0

, zyxBB ˆˆˆ0 and zzyyxxr ˆˆˆ .

000 ˆˆˆ BxyzBxzyByzxrB 0 rBA

.




7

Q18. For the function 213

16

zz

zzf , the residue at the pole 1z is (your answer

should be an integer) ____________.

Ans: 3

Solution: At 1z , pole is of order 2. So, residue is

22 1

22 1

1

1 161

2 1 3 1z

z zd

dz z z

=3 .

Q19. The degenerate eigenvalue of the matrix

411

141

114

is (your answer should be an

integer) ____________

Ans: 2,5,5

4 1 1

1 4 1 0

1 1 4

1 1 1

(2 ) 0 5 0

0 0 5

= 2(2 )(5 ) 0 2,5,5 .

Q20. The number of distinct ways of placing four indistinguishable balls into five

distinguishable boxes is ___________.

Ans: 120

Solution: 544 C =120 ways

GATE-2014

Q21. The unit vector perpendicular to the surface 3222 zyx at the point (1, 1, 1) is

(a)3

ˆˆˆ zyx (b)

3

ˆˆˆ zyx (c)

3

ˆˆˆ zyx (d)

3

ˆˆˆ zyx

Ans: (d)

Solution: Let, 2 2 2 3 0f x y z ˆ ˆ ˆ2 2 2f xx yy zz

ˆ ˆ ˆ ˆˆ ˆ2 2 2ˆ 1,1,1

12 3

f x y z x y zn at

f




8

Q22. The matrix

11

11

3

1

i

iA is

(a) orthogonal (b) symmetric (c) anti-symmetric (d) Unitary

Ans. : (d)

Solution: Unitary †A A I


Cz

dze

z

1

2

where C is the circle 4z , is

(a) i2 (b) i22 (c) i34 (d) i24

Ans. : (c)

Solution: Pole 1ze 2 1i mze e where 0,1,2,3.....m

For z i ,

22Res lim

iz i

z

z e

Similarly, for 2,Resz i

2 2 32 4I i i

Q24. The solution of the differential equation 02

2

ytd

yd, subject to the boundary conditions

10 y and 0y is

(a) tt sincos (b) tt sinhcosh

(c) tt sincos (d) cosh sinht t

Ans: (d)

Soluiton:

2 1 0D 1D 1 2t ty t c e c e

Applying boundary condition,

10 y 1 21 c c and 1 20 0y c e c e 1 20, 1c c

ty t e cosh sinhy t t t




9

GATE-2015

Q25. Consider a complex function

11

cos2

f zz z z

. Which one of the following

statements is correct?

(a) zf has simple poles at 0z and 2

1z

(b) zf has second order pole at 2

1z

(c) zf has infinite number of second order poles

(d) zf has all simple poles

Ans.: (a)

Solution:

11

cos2

f zz z z

For thn order pole, Res. lim finiten

z az a f z

At 0z ,

0limz

zf z finite

0z is a simple pole.

At 1

2z ,

2

1 1

2 2

1 12 2

lim lim1 cos

cos2

z z

z z

z zz z z

1

2

1lim

1.cos . sinz z z z

1

2

1lim

cos sinz z z z

1 2

2

finite

f z has second order pole at 1

2z

Q26. The value of dttt 3

0

2 63 is_______________ (upto one decimal place)

Ans.: 1.33

Solution: 3 3 3

2 2 2

0 0 0

1 43 6 3 2 2

3 3t t dt t t dt t t dt




10

Q27. If 2xexf and 2xexxg , then

(a) f and g are differentiable everywhere

(b) f is differentiable everywhere but g is not

(c) g is differentiable everywhere but f is not

(d) g is discontinuous at 0x

Ans. (b)

Solution: 2

( ) xf x e is differentiable but 2

( ) xg x x e is not differentiable.

2

2

; 0( )

; 0

x

x

xe xg x

xe x

Left hand Limit 2

0lim

x h

hg x h x h e

Right hand Limit 2

0lim

x h

hg x h x h e

0 0

lim limh h

g x h g x h

Q28. Consider yxivyxuzfw ,, to be an analytic function in a domain D . Which

one of the following options is NOT correct?

(a) yxu , satisfies Laplace equation in D

(b) yxv , satisfies Laplace equation in D

(c) 2

1

z

z

dzzf is dependent on the choice of the contour between 1z and 2z in D

(d) zf can be Taylor expended in D

Ans.: (c)

Solution: ( ) , ,w f z u x y iv x y to be an analytic function in a domain D , 2

1

z

z

f z dz is

independent of the choice of the contour between 1z and 2z in D .




11

Q29. The Heaviside function is defined as 1, for 0

1, for 0

tH t

t

and its Fourier transform

is given by /2i . The Fourier transform of 2/12/12

1 tHtH is

(a)

2

2sin

(b)

2

2cos

(c)

2sin

(d) 0

Ans.: (a)

Solution: 2i ftH f H t e dt

, for a function H t , 2iH f

For 0H t t , Fourier Transform is 02i fte H f

Shifting Theorem

For 1 1 1

2 2 2H t H t

2 2 2 21 2 1 2

2 2

i i i ii ie e e e i

i

The Fourier transform of sin

1 21/ 2 1/ 2

22

H t H t

.

Q30. A function zy satisfies the ordinary differential equation 2

2

10,

my y y

z z where

.....,3,2,1,0m Consider the four statements P, Q, R, S as given below.

P: mz and mz are linearly independent solutions for all values of m

Q: mz and mz are linearly independent solutions for all values of 0m

R: zln and 1 are linearly independent solutions for 0m

S: mz and zln are linearly independent solutions for all values of m

The correct option for the combination of valid statements is

(a) P, R and S only (b) P and R only (c) Q and R only (d) R and S only

Ans.: (c)

Solution: 2

2

10

my y y

z z 2 2 0z y zy m y , 0,1, 2,3,..., ,x d

m z e Ddx




12

If 0m ; 2 0z y zy , 1 0D D D y 2 0D D D y

2 0D y 1 2y c c x 1 2 lny c c z ( R is correct)

And if 0, 0m m , then 0m , then 2 2 0D m y D m

1 2mx mxy c e c e log log

1 2 1 2m z m z m mc e c e c z c z

or if 0, 0m m , then

1 2cosh log sinh logy c m z ic m x , 0m

GATE-2016

Q31. Consider the linear differential equation xydx

dy . If 2y at 0x , then the value of y at

2x is given by

(a) 2e (b) 22 e (c) 2e (d) 22e

Ans.: (d)

Solution: xydx

dy

1dy xdx

y

2

ln ln2

xy c

2 / 2xy ce

If 2y at 0x 2c 2 / 22 xy e .

The value of y at 2x is given by 22y e

Q32. Which of the following is an analytic function of z everywhere in the complex plane?

(a) 2z (b) 2*z (c) 2

z (d) z

Ans.: (a)

Solution: 22 2 2 2z x iy x y i xy 2 2 and 2u x y v xy

Cauchy Riemann equations 2 , 2u v v u

x yx y x y

satisfies.

Q33. The direction of f

for a scalar field 22

2

1

2

1,, zxyxzyxf at the point 2,1,1P is

(a)

5

ˆ2ˆ kj (b)

5

ˆ2ˆ kj (c)

5

ˆ2ˆ kj (d)

5

ˆ2ˆ kj

Ans.: (b)




13

Solution: ˆˆ ˆf x y i xj zk

1,1,2

ˆˆ 2ˆ

5

f j kn

f

Q34. A periodic function xf of period 2 is defined in the interval x

x

xxf

0,1

0,1

The appropriate Fourier series expansion for xf is

(a) ...5/5sin3/3sinsin4

xxxxf

(b) ..5/5sin3/3sinsin4

xxxxf

(c) ...5/5cos3/3coscos4

xxxxf

(d) ...5/5cos3/3coscos4

xxxxf

Ans.: (a)

Solution:

x

xxf

0,1

0,1

Let 01

cos sinn nn

f x a a nx b nx

0

1

2a f x dx

0 0

0 00

1 1 11 1 0

2 2 2a f x dx dx dx x x

This can also be seen without integration, since the area under the curve of xf between

to is zero.

1cosna f x nxdx

0

0

11 cos 1 cosna nxdx nxdx

0

0

1 sin sin0

nx nx

n n




14

1sinnb f x nxdx

0

0

11 sin 1 sinnb nxdx nxdx

0

0

1 cos cosn

nx nxb

n n

1 1 2 11 1 1 1 2n n n

n n n n n n

0;

4;n

n evenb

n oddn

Thus, Fourier series is 4 1 1sin sin 3 sin 5 ...

3 5f x x x x

GATE-2017

Q35. The contour integral 21

dz

z evaluated along a contour going from to along the

real axis and closed in the lower half-plane circle is equal to………….. (up to two

decimal places).

Ans. :

Solution: 2 2 2

1 1 1

1 1 1C Cdz dx dz

z x z

Poles, 21 0z z i , z i is inside C

1

Res limz i

z i z iz i z i

1 1

2i i i

2

1 12

1 2dx i

x i

(Since, here we use lower half plane i.e., we traversed in clockwise direction, hence we

have to take 2 i )

Q36. The coefficient of ikxe in the Fourier expansion of 2sinu x A x for 2k is

(a) 4

A (b)

4

A (c)

2

A (d)

2

A

Ans.: (b)




15

Solution: Since, sin2

i x i xe ex

i

2 2

2 2sin

4

i x i xe ex

Since, 2 k , hence 2 2

sin4

ikx ikxe ex

Hence, 2sin2k

Ac x dx

28

ikx ikx ikx ikx ikxAe e dx e dx e e dx

2 28

ikx ikxAe dx e dx dx

The first two integrals are zero and the third integral has the value 2 .

Thus,

28 4k

A Ac

Q37. The imaginary part of an analytic complex function is , 2 3v x y xy y . The real part of

the function is zero at the origin. The value of the real part of the function at 1 i

is ……………... (up to two decimal places)

Ans. : 3

Solution: The imaginary part of the given analytic function is , 2 3v x y xy y . From the

Cauchy – Riemann condition

2 3v u

xy x

Integrating partially gives

2, 3u x y x x g y

From the second Cauchy – Riemann condition

u v

y x

, we obtain 22 , ,

uy x y y g x

y

2dg y

ydy

2g y y c

Hence, 2 2, 3u x y x x y c

Since, the real part of the analytic function is zero at the origin.




16

Hence, 0 0 0 0 0c c

Thus, 2 2, 3u x y x x y

2 23 2 3f z x x y i xy y

Thus, the value of real part when

21 ,i.e. 1 and 1 is , 1 3 1 1 3z i x y u x y .

Q38. Let X be a column vector of dimension 1n with at least one non-zero entry. The

number of non-zero eigenvalues of the matrix TM XX is

(a) 0 (b) n (c) 1 (d) 1n

Ans. : (c)

Solution: Let

0

0

0

0

0

aX

, then 0 0 ... 0TX a

Here, X is an 1n column vector with the entry in the thi row equal to a. TX is a row

vector having entry in the thi column equal to a. Then, TXX is an 1n matrix having

the entry in the thi row and thi column equal to 2a .

Hence,

0 0 0...0...0 00 0 0...0...0 00 0 0...0...0 0..........................................0 0 0...0...0 0

TXX

Since this matrix is diagonal, its eigenvalues are 2 ,0,0.....0a . Hence, the number of non

zero eigenvalues of the matrix is 1TXX .

Q39. Consider the differential equation tan cosdy

y x xdx

. If 0 0,3

y y

is …………... (up to two decimal places)

th rowi

th rowi




17

Ans.: 0.52

Solution: The given differential equation is a linear differential equation of the form

cosdy

p x y xdx

Integrating factor p x dx

e

Thus integrating factor tan x dx

e

lnsec secxI F e x

Thus the general solution of the given differential equation is

sec sec cosy x x xdx c

secy x x c -(i)

It is given that 0 0y 0 sec0 0 0c c

Thus the solution satisfying the given condition is

secsec

xy x x y

x

Thus the value of 3

y

is

/ 3 / 3

0 52sec / 3 2 6

y




18

GATE-2018

Q40. The eigenvalues of a Hermitian matrix are all

(a) real (b) imaginary (c) of modulus one (d) real and positive

Ans. : (a)

Solution: Eigenvalue of Hermitian matrix must be real.

Q41. In spherical polar coordinates , ,r , the unit vector at 10, / 4, / 2 is

(a) k (b) 1 ˆˆ2

j k (c) 1 ˆˆ2

j k (d) 1 ˆˆ2

j k

Ans. : (d)

Solution: 0 0 ˆˆ ˆcos 45 sin 45j k

1 ˆˆ ˆ2

j k

Q42. The scale factors corresponding to the covariant metric tensor i jg in spherical polar

coordinates are

(a) 2 2 21, , sinr r (b) 2 21, ,sinr (c) 1,1,1 (d) 1, , sinr r

Ans. : (d)

Q43. Given 1ˆ ˆV i j

and 2

ˆˆ ˆ2 3 2V i j k

, which one of the following 3V

makes 1 2 3, ,V V V

a complete set for a three dimensional real linear vector space?

(a) 3ˆˆ ˆ 4V i j k

(b) 3

ˆˆ ˆ2 2V i j k

(c) 3ˆˆ ˆ2 6V i j k

(d) 3

ˆˆ ˆ2 4V i j k

Ans. : (d)

Solution: Let A be the matrix formed by taking 1 2,V V

and 3V

as column matrix i.e.,

1 2 3A V V V1 2 2

1 3 1 2

0 2 4

A

. Here 3ˆˆ ˆ2 4V i j k

Since, 0A , hence, 1 2,V V

and 3V

form a three dimensional real vector space.

Hence, option (d) is correct.

r

z

y

/4




19

Q44. Given

2

22 0

d f x df xf x

dx dx ,

and boundary conditions 0 1f and 1 0f , the value of 0.5f is __________ (up

to two decimal places) .

Ans. : 0.81

Solution:

2

22 0

d f x df xf x

dx dx

Auxiliary equation is,

2 2 1 0m m 21 0 1,1m m

Hence, the solution is

1 2xf x c c x e

using boundary condition,

01 10 1f c e c (i)

1 21 0f c c e (ii)

From (i) and (ii), 2 1c

Hence, 0.51 0.5 1 0.5 0.81xf x x e f e

Q45. The absolute value of the integral

3 2

2

5 3

4

z zdz

z

,

over the circle 1.5 1z in complex plane, is __________ (up to two decimal places).

Ans. : 81.64

Solution: 3 25 3

2 2

z zf z

z z

Pole, 2, 2z

2z is outside the center

2 1.5 1 So, will not be considered




20

Now,

3 2

2

5 3Re 2 lim 2

2 2z

z zs z

z z

3 252 32

4

40 12

4

13

2I i residue 2 13 26 3.14i 81.64I

GATE-2019

Q46. For the differential equation 2

2 21 0

d y yn n

dx x , where n is a constant, the product of

its two independent solutions is

(a) 1

x (b) x (c) nx (d)

1

1nx

Ans. : (b)

Q47. During a rotation, vectors along the axis of rotation remain unchanged. For the rotation

matrix

0 1 0

0 0 1

1 0 0

, the vector along the axis of rotation is

(a) 1 ˆˆ ˆ2 23

i j k (b) 1 ˆˆ ˆ3

i j k

(c) 1 ˆˆ ˆ3

i j k (d) 1 ˆˆ ˆ2 23

i j k

Ans. : (b)

Q48. The pole of the function cotf z z at 0z is

(a) a removable pole (b) an essential singularity

(c) a simple pole (d) a second order pole

Ans. : (c)

Solution: cotf z z at 0z

10

tanf z z

z is a simple pole 21 1

1 ....3

f z zz


2 2

cos kxdx

x a

, where 0k and 0a , is

(a) kaea

(b) 2 kaea

(c) 2

kaea

(d) 3

2kae

a




21

Ans. : (a)

Solution: 2 2

cos kxdx

x a

2 2

ikx ikze ef z

z a z ia z ia

Re.22

ik ia kae eI i

ia a

Q50. Let be a variable in the range . Now consider a function

1 for2 2

0 otherwise

if its Fourier-series is written as immm

C e , then the value of 2

3C

(rounded off to three decimal places) is__________




1

CLASSICAL MECHANICS SOLUTIONS

GATE- 2010

Q1. For the set of all Lorentz transformations with velocities along the x -axis consider the

two statements given below:

P: If L is a Lorentz transformation, then, 1L is also a Lorentz transformation.

Q: If 1L and 2L are Lorentz transformations, then 1 2L L is necessarily a Lorentz

transformation.

Choose the correct option

(a) P is true and Q is false (b) Both P and Q are true

(c) Both P and Q are false (d) P is false and Q is true

Ans: (b)

Q2. A particle is placed in a region with the potential 32

32

1xkxxV

, where , 0k .

Then,

(a) 0x and k

x are points of stable equilibrium

(b) 0x is a point of stable equilibrium and k

x is a point of unstable equilibrium

(c) 0x and k

x are points of unstable equilibrium

(d) There are no points of stable or unstable equilibrium

Ans: (b)

Solution: 32

1 32 x

kxV

02

xkxx

V k

xx ,0 .

xkx

V 22

2

vex

VxAt

2

2

,0 (Stable) and 2

2at ,

k Vx ve

x

(unstable)




2

Q3. A 0 meson at rest decays into two photons, which moves along the x -axis. They are

both detected simultaneously after a time, 10t s . In an inertial frame moving with a

velocity 0.6v c in the direction of one of the photons, the time interval between the two

detections is

(a) 15c (b) 0 s (c) 10 s (d) 20 s

Ans: (a)

Solution:

c

vc

v

tt

1

1

01

1 0.610

1 0.6

10 2 20sec ,

c

vc

v

tt

1

1

02

1 0.610

1 0.6

1

10 5sec2

1 2t t 15sec

Statement for Linked Answer Questions 4 and 5:

The Lagrangian for a simple pendulum is given by cos12

1 22 mglmlL

Q4. Hamilton’s equations are then given by

(a) 2

;sinml

pmglp

(b) 2

;sinml

pmglp

(c) m

pmp

; (d) ml

p

l

gp

;

Ans: (a)

Solution: cos12 2

2

mglml

PH sin ;

HP P mgl

2ml

P

P

H

.

Q5. The Poisson bracket between and is

(a) 1, (b) 2

1,

ml

(c) m

1, (d)

l

g ,

Ans: (b)

Solution:

2,,ml

P where2ml

P

P

PPml 2

10

11

2

ml 2

1

ml .




3

GATE- 2011

Q6. A particle is moving under the action of a generalized potential 2

1,

q

qqqV

. The

magnitude of the generalized force is

(a)

3

12

q

q (b)

3

12

q

q (c)

3

2

q (d)

3q

q

Ans: (c)

Solution: q

d V VF

dt q q

3

2qF

q

Q7. Two bodies of mass m and 2m are connected by a spring constant k . The frequency of

the normal mode is

(a) mk 2/3 (b) mk / (c) mk 3/2 (d) mk 2/

Ans: (a)

Solution:

k

m

kmk

2

3

32

where reduce mass mm

mm

2

23

2m .

Q8. Let ,p q and ,P Q be two pairs of canonical variables. The transformation

pqQ cos , pqP sin

is canonical for

(a) 1

2,2

(b) 2, 2 (c) 1, 1 (d) 1

, 22

Ans: (d)

Solution: 1

q

P

p

Q

p

P

q

Q

1sinsincoscos 11 pqpqpqpq

11sincos 122212 qppq 2,2

1 .

m m2k




4

Q9. Two particles each of rest mass m collide head-on and stick together. Before collision, the

speed of each mass was 0.6 times the speed of light in free space. The mass of the final

entity is

(a) 5 / 4m (b) 2m (c) 5 / 2m (d) 25 / 8m

Ans: (c)

Solution: From conservation of energy

21

2

2

2

2

2

2

11

cm

c

v

mc

c

v

mc

21

2

2

2

1

2cm

c

v

mc

Since cv 6.0 2/51 mm

GATE- 2012

Q10. In a central force field, the trajectory of a particle of mass m and angular momentum L in

plane polar coordinates is given by,

2

11 cos

m

r l

where, is the eccentricity of the particle’s motion. Which one of the following choice

for gives rise to a parabolic trajectory?

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

Ans: (b)

Solution: 2

11 cos

m

r l

For parabolic trajectory 1 .

Q11. A particle of unit mass moves along the x-axis under the influence of a potential,

22 xxxV . The particle is found to be in stable equilibrium at the point 2x . The

time period of oscillation of the particle is

(a) 2

(b) (c)

2

3 (d) 2

Ans: (b)

Solution: 22V x x x 2

2 2 2 0V

x x xx

2

2,3

x x




5

2

22 2 2 2 2

Vx x x

x

2

2

2

2 2 4x

V

x

2

2

2x

V

x

2

2T

T

Q12. A rod of proper length 0l oriented parallel to the x-axis moves with speed 2 / 3c along the

x -axis in the S -frame, where c is the speed of light in free space. The observer is also

moving along the x -axis with speed / 2c with respect to the S -frame. The length of the

rod as measured by the observer is

(a) 00.35l (b) 00.48l (c) 00.87 l (d) 00.97 l

Ans: (d)

Solution: 2

0 021 0.97xu

l l lc

Q13. A particle of mass m is attached to a fixed point O by a weightless

inextensible string of length a . It is rotating under the gravity as

shown in the figure. The Lagrangian of the particle is

cossin2

1, 2222 mgamaL

where and are the polar

angles. The Hamiltonian of the particles is

(a)

cos

sin2

12

22

2mga

pp

maH

(b)

cos

sin2

12

22

2mga

pp

maH

(c) cos2

1 222

mgappma

H (d) cos2

1 222

mgappma

H

Ans: (b)

Solution: LPPH cossin2

1 2222 mgamaPP

P

L

22

Pma P

ma

and 2 22 2

sinsin

PLP ma

ma

Put the value of and

z

m

g

O

a




6

22

2 22 2 2 2 2 2

1sin cos

sin 2 sin

P PP PH P P ma mga

ma ma ma ma

cossin2sin2 22

2

22

2

2

2

2

2

mgama

P

ma

P

ma

P

ma

PH

22

2 2

1cos

2 sin

PH P mga

ma


Q14. A particle of mass m slides under the gravity without friction along the parabolic path

2axy , as shown in the figure. Here a is a constant.

The Lagrangian for this particle is given by

(a) 22

2

1mgaxxmL (b) 222241

2

1mgaxxxamL

(c) 22

2

1mgaxxmL (d) 222241

2

1mgaxxxamL

Ans: (b)

Solution: Equation of constrain is given by 2 2 21, . .,

2y ax K E T m x y

2 2 2 212 4

2y axx T m x a x x 22 41

2

1axxm

2V mgy mgax .

2 2 2 211 4

2L T V L m a x x mgax

m

x

y




7

Q15. The Lagrange’s equation of motion of the particle for above question is given by

(a) gaxx 2 (b) 2 2 2 21 4 2 4m a x x mgax ma xx

(c) 2222 4241 xxmamgaxxxam (d) gaxx 2

Ans: (b)

Solution: 2 2 2 21 4 4 2d dL dL

mx a x ma xx mgaxdt dx dx

GATE- 2013

Q16. In the most general case, which one of the following quantities is NOT a second order

tensor?

(a) Stress (b) Strain

(c) Moment of inertia (d) Pressure

Ans: (b)

Solution: Strain is not a tensor.

Q17. An electron is moving with a velocity of c85.0 in the same direction as that of a moving

photon. The relative velocity of the electron with respect to photon is

(a) c (b) c (c) c15.0 (d) c15.0

Ans: (b)

Q18. The Lagrangian of a system with one degree of freedom q is given by 22 qqL ,

where and are non-zero constants. If qp denotes the canonical momentum

conjugate to q then which one of the following statements is CORRECT?

(a) qpq 2 and it is a conserved quantity.

(b) qpq 2 and it is not a conserved quantity.

(c) qpq 2 and it is a conserved quantity.

(d) qpq 2 and it is not a conserved quantity.

Ans: (d)

Solution: As, q

Lp

q

but 0

L

q

. Thus, it is not a conserved quantity.




8

Q19. The relativistic form of Newton’s second law of motion is

(a) dt

dv

vc

mcF

22 (b)

dt

dv

c

vcmF

22

(c) dt

dv

vc

mcF

22

2

(d)

dt

dv

c

vcmF

2

22

Ans: (c)

Solution:

2

2

1c

v

mvP

3/2 22 2

2 2

1 1 1 2

21 1

dP dv v dvF m mv

dt dt c dtv vc c

2 2

2 2

322 2 2

22 2

11 1 212

11 1

v vdv dvc cF m mdt dtvv v

cc c

1/ 22 2 2

1/ 2 2 22 2 2 2

1 /

1 / 1 /

v cdv mc dvm

dt dtc vv c v c

Q20. Consider two small blocks, each of mass M, attached to two identical springs. One of the

springs is attached to the wall, as shown in the figure. The spring constant of each spring

is k . The masses slide along the surface and the friction is negligible. The frequency of

one of the normal modes of the system is,

(a) M

k

2

23

(b) M

k

2

33

(c) M

k

2

53

(d) M

k

2

63

Ans: (c)

M Mk k




9

Solution: 22

21 2

1

2

1xmxmT ,

212

21 2

1

2

1xxkkxV 12

21

22

21 2

2

1

2

1xxxxkkx 2 2

1 2 2 11

2 22

k x x x x

0 2;

0

m k kT V

m k k

2

2

20

k m k

k k m

2 2 22 0k m k m k

3 5

2

k

m

GATE- 2014

Q21. If the half-life of an elementary particle moving with speed 0.9c in the laboratory frame is

,105 8 s then the proper half-life is _______________ smcs /103.10 88

Ans: 2.18

Solution:

2

2

0

1c

v

tt

, 2

2

0 1c

vtt 85 10 0.19 82.18 10 s

Q22. Two masses m and m3 are attached to the two ends of a massless spring with force

constant K . If gm 100 and 0.3 /K N m , then the natural angular frequency of

oscillation is ________ Hz .

Ans: 0.318

Solution: 1

2

kf

, 1 2

1 2

. 3 . 3,

4 4

m m m m m

m m m

42

3

k

m 0.318f Hz

Q23. The Hamilton’s canonical equation of motion in terms of Poisson Brackets are

(a) HppHqq ,;, (b) pHpqHq ,;,

(c) pHppHq ,;, (d) HqpHpq ,;, Ans: (a)

Solution: . .df f q f p f

dt q t p t t

. .df f H f H f

dt q p p q t

,df f

f Hdt t

,dq

q Hdt

and ,dp

p Hdt




10

Q24. A bead of mass m can slide without friction along a massless rod kept at o45 with the

vertical as shown in the figure. The rod is rotating about the vertical axis with a constant

angular speed . At any instant r is the distance of the bead from the origin. The

momentum conjugate to r is

(a) rm

(b) rm 2

1

(c) rm2

1

(d) rm2

Ans: (a)

Solution: cos)sin(2

1 222222 mgrrrrmL

Equation of constrain is 4

and it is given

mgrrrmL2

1)

2

1(

2

1 222

Thus the momentum conjugate to r is r

Lpr

rp mr

Q25. A particle of mass m is in a potential given by

2

033

araV r

r r

where a and 0r are positive constants. When disturbed slightly from its stable

equilibrium position it undergoes a simple harmonic oscillation. The time period of

oscillation is

(a) a

mr

22

30 (b)

a

rm 302 (c)

a

rm 302

2 (d) 3

04mr

a

Ans: (a)

Solution:

2

033

araV r

r r ,

For equilibrium 2

02 4

30

3

arV a

r r r

, 0r r

0

2 220 0

2 3 5 3 5 30 0 0

4 42 2 2

r

ar arV a a a

r r r r r r

z

m

r

x

o45




11

0

2

2 302

2r

Vr mr

Tm a

Q26. A planet of mass m moves in a circular orbit of radius 0r in the gravitational potential

r

krV , where k is a positive constant. The orbit angular momentum of the planet is

(a) kmr02 (b) kmr02 (c) kmr0 (d) 0r km

Ans: (d)

Solution: r

k

mr

JVeffctive

2

2

2

2

3 20effectdV J k

dr mr r at 0rr

so kmrJ 0

Q27. Given that the linear transformation of a generalized coordinate q and the corresponding

momentum p , 4 ,Q q ap pqP 2 is canonical, the value of the constant a is ____

Ans: 0.25

Solution: . . 1Q P Q P

q p p q

1 2 4 1 1a 0.25a

Q28. The Hamiltonian of particle of mass m is given by22

22 q

m

pH

. Which one of the

following figure describes the motion of the particle in phase space?

(a) (b)

(c) (d)

p

q

p

q

p

q

p

q




12

Ans: (d)

GATE- 2015

Q29. A satellite is moving in a circular orbit around the Earth. If VT , and E are its average

kinetic, average potential and total energies, respectively, then which one of the

following options is correct?

(a) TETV ;2 (b) 0; ETV

(c) 2

;2

TE

TV (d)

2;

2

3 TE

TV

Ans.: (a)

Solution: From Virial theorem 1

2

nT V

where 1nV r

12

kV V n

r r

2V T

Q30. In an inertial frame S , two events A and B take place at 0,0 AA rct

and

yrct BB ˆ2,0

, respectively. The times at which these events take place in a frame

S moving with a velocity ˆ0.6c y with respect to S are given by

(a) 2

3;0 BA tctc (b) 0;0 BA tctc

(c) 2

3;0 BA tctc (d)

2

1;0 BA tctc

Ans.: (a)

Solution: Velocity of 'S with respect to S is 0.6v c

2'

2

21

A

A

vt y

ctv

c

For event A, 0, 0At y . So ' 0Act

2'

2

21

B

B

vt y

ctv

c




13

For event B, 0, 2Bt y . So ' 3

2Bct

Q31. The Lagrangian for a particle of mass m at a position r

moving with a velocity v

is given

by rVvrCvm

L

.2

2 , where rV is a potential and C is a constant. If cp

is the

canonical momentum, then its Hamiltonian is given by

(a) rVrCpm c 2

2

1 (b) rVrCp

m c 2

2

1

(c) rVm

pc 2

2

(d) rVrCpm c 222

2

1

Ans.: (b)

Solution: 2.

2

mL v Cr v V r

where v r

c cH r p L rp L

cc

p CrLp mr Cr r

r m

2

2c c c

c

p Cr p Cr p CrmH p cr V r

m m m

2

2c c

c

p Cr p CrmH p Cr V r

m m

2 2

2c cp Cr p Cr

H V rm m

21

2 cH p Cr V rm

Q32. The Hamiltonian for a system of two particles of masses 1m and 2m at 1r

and 2r

having

velocities 1v

and 2v

is given by

2 21 1 2 2 1 22

1 2

1 1ˆ

2 2

CH m v m v z r r

r r

, where C is

constant. Which one of the following statements is correct?

(a) The total energy and total momentum are conserved

(b) Only the total energy is conserved

(c) The total energy and the z - component of the total angular momentum are conserved

(d) The total energy and total angular momentum are conserved

Ans.: (c)




14

Solution: Lagrangian is not a function of time, so energy is conserved and component of 1 2r r

is only in z direction means potential is symmetric under , so zL is conserved.

Q33. A particle of mass 0.01 kg falls freely in the earth’s gravitational field with an initial

velocity 1100 ms . If the air exerts a frictional force of the form, kvf , then for

sNmk 105.0 , the velocity (in 1ms ) at time st 2.0 is _________ (upto two decimal

places). (use 210 msg and 72.2e )

Ans.: 4.94

Solution: dv

m mg kvdt

dv k

g vdt m

dv

dtk

g vm

0.2

10 0

u dvdt

kg v

m

0.2

010

lnu

m kg v t

k m

10ln ln 0.2

m k kg u g

k m m

0.05 .05

ln 10 ln 10 10 0.20.01 .01

mu

k

ln 10 5 ln 40 0.2m

uk

8 0.2ln

2

k

u m

8 0.2 8ln

2 2

ke

u m u

82 4.94 /u m s

e

Q34. Consider the motion of the Sun with respect to the rotation of the Earth about its axis. If

cF

and CoF

denote the centrifugal and the Coriolis forces, respectively, acting on the Sun,

then

(a) cF

is radially outward and cCo FF

(b) cF

is radially inward and cCo FF

2

(c) cF


2 (d) cF


2

Ans.: (b)

Q35. A particle with rest mass M is at rest and decays into two particles of equal rest masses

M10

3 which move along the z axis. Their velocities are given by




15

(a) zcvv ˆ8.021

(b) zcvv ˆ8.021

(c) zcvv ˆ6.021

(d) zcvzcv ˆ8.0;ˆ6.0 21

Ans.: (b)

Solution: 3 3

10 10M M M

From momentum conservation

1 2 1 2 1 20 P P P P P P

From energy conservation

1 2E E E

2 2

2

2 2

2 2

3 3

10 101 1

Mc McMc

v vc c

2

2

2

2

3

51

McMc

vc

2 2

2 2

9 161 0.8

25 25

v vv c

c c

GATE-2016

Q36. The kinetic energy of a particle of rest mass 0m is equal to its rest mass energy. Its

momentum in units of cm0 , where c is the speed of light in vacuum, is _______.

(Give your answer upto two decimal places)

Ans. : 1.73

Solution: 2 2 20 0 02m c E m c E m c

2

2002

2

32

21

m cm c v c

v

c

2 2 2 2 4 2 4 2 4 2 20 0 0 0 04 3 1.732E p c m c m c m c p c p m c m c

Q37. In an inertial frame of reference S , an observer finds two events occurring at the same

time at coordinates 01 x and dx 2 . A different inertial frame S moves with velocity

v with respect to S along the positive x -axis. An observer in S also notices these two




16

events and finds them to occur at times 1t and 2t and at positions 1x and 2x respectively.

If 1212 , xxxttt and

2

2

1

1

c

v

, which of the following statements is true?

(a) dxt ,0 (b) d

xt ,0

(c) dxc

vdt

,2

(d)

dx

c

vdt

,

2

Ans.: (c)

Solution:

2 12 12 2

2 1 2 2

2 21 1

vx vxt t

c ct tv v

c c

2

v xt t

c

It is given, 0,t x d

2 2

v x vdt

c c

2 2 1 12 1 2 2

2 21 1

x vt x vtx x

v v

c c

x x v t

x d .

Q38. The Lagrangian of a system is given by

cossin2

1 2222 mglmlL , where lm, and g are constants.

Which of the following is conserved?

(a) 2sin (b) sin (c)

sin

(d)

2sin

Ans.: (a)

Solution: As is cyclic coordinate, so 2 2sinL

p ml

, is a constant since lm, and g are

constants. Thus 2sin is conserved.




17

Q39. A particle of rest mass M is moving along the positive x -direction. It decays into two

photons 1 and 2 as shown in the figure. The energy of 1 is GeV1 and the energy of

2 is GeV82.0 . The value of M (in units of 2c

GeV) is ________. (Give your answer

upto two decimal places)

Ans.: 1.44

Solution: 2 2 2 41 2 1.82p c M c E E GeV

1 21 2

1 1 0.82 1cos cos

22

E E GeV GeVp

c c c c

1.11GeV

c

2 2 2 4 3.312p c m c 2 4 3.312 1.23 2.08m c

2.076 1.44m

GATE- 2017

Q40. If the Lagrangian 2

2 20

1 1

2 2

dqL m m q

dt

is modified to 0

dqL L q

dt

, which one

of the following is TRUE?

(a) Both the canonical momentum and equation of motion do not change

(b) Canonical momentum changes, equation of motion does not change

(c) Canonical momentum does not change, equation of motion changes

(d) Both the canonical momentum and equation of motion change

Ans. : (b)

Solution: For Lagrangian 2

2 20

1 1

2 2

dqL m m q

dt

canonical momentum is p mq and

equation of motion is given by 0d L L

dt q q

2 0mq m q

M 045060

2

1




18

For Lagrangian 2

2 20

1 1

2 2

dq dqL L q L m m q qq

dt dt

Canonical momentum is p mq q

Equation of motion is,

0d L L

dt q q

2 0mq m q

Q41. Two identical masses of 10 gm each are connected by a massless spring of spring

constant 1 /N m . The non-zero angular eigenfrequency of the system is…………rad/s.

(up to two decimal places)

Ans. : 14.14

Solution: ,k

where 10 1

2 2 1000 200

m

and 1 /k N m , 14.14

Q42. The phase space trajectory of an otherwise free particle bouncing between two hard walls

elastically in one dimension is a

(a) straight line (b) parabola (c) rectangle (d) circle

Ans. : (c)

Solution: 2

2

pE

m , 2p mE

Q43. The Poisson bracket , y xx xp yp is equal to

(a) x (b) y (c) 2 xp (d) yp

Ans. : (b)

Solution: , , , 0 ,y x y x xx xp yp x xp x yp y x p y

Q44. An object travels along the x -direction with velocity 2

c in a frame O . An observer in a

frame O sees the same object travelling with velocity 4

c. The relative velocity of O

with respect to O in units of c is…………….. (up to two decimal places). Ans. : 0.28




19

Solution: ' ,2 4xc c

u v , '

'

21

xx

x

u vu

u v

c

2

22 4 0.281 71 . .

2 4

c cc

cc c

c

Q45. A uniform solid cylinder is released on a horizontal surface with speed 5 /m s without

any rotation (slipping without rolling). The cylinder eventually starts rolling without

slipping. If the mass and radius of the cylinder are 10 gm and 1cm respectively, the final

linear velocity of the cylinder is…………… /m s . (up to two decimal places).

Ans. : 3.33

Solution: 21 3 2 103.33 / sec

2 2 3 3cm

cm cm cm cm cmv

mvr mv r I mv r mr v v v v mr

Q46. A person weighs pw at Earth’s north pole and ew at the equator. Treating the Earth as a

perfect sphere of radius 6400 km , the value

100p e

p

w w

w

is………….. (up to two

decimal places). (Take 210g ms ). Ans. : 0.33

Solution: pg g , 2eg g R

2

100 p e

p

w w R

w g

Now, 210 / secg m and 36400 10R m 2 2

24 3600T

Then 100 0.33p e

p

w w

w




20

GATE - 2018

Q47. In the context of small oscillations, which one of the following does NOT apply to the

normal coordinates?

(a) Each normal coordinate has an eigen-frequency associated with it

(b) The normal coordinates are orthogonal to one another

(c) The normal coordinates are all independent

(d) The potential energy of the system is a sum of squares of the normal coordinates with

constant coefficients

Ans. : (b)

Solution: Normal co-ordinate must be independent. It is not necessary that it should orthogonal.

Q48. A spaceship is travelling with a velocity of 0.7c away from a space station. The

spaceship ejects a probe with a velocity 0.59c opposite to its own velocity. A person in

the space station would see the probe moving at a speed Xc , where the value of X is

___________ (up to three decimal places).

Ans.: 0.187c

Solution: 0 7c , 0 59xu c ,

2

1

xx

x

uu

u

c

0 59 0 7

1 0 7 0 59x

c cu

0.11 0.11

0.1871 0.413 0.587

c cc

Q49. An interstellar object has speed v at the point of its shortest distance R from a star of

much larger mass M . Given 2 2 /v GM R , the trajectory of the object is

(a) circle (b) ellipse (c) parabola (d) hyperbola

Ans. : (c)

Solution: At shortest distance 2

22

J GMmE

mR R

Since, 2 2 2 2mvR J J m v R

Now, 2 2 2J m GMR 22GMm R (Given that 2 2GMv

R )

SpacestationSpaceship

0 7c 0.59xu c

Prob




21

2

2

2

2

GMm R GMmE

mR R 0

GMm GMm

R R

For Kepler’s potential, if energy is zero, then the shape is parabola.

Q50. A particle moves in one dimension under a potential V x x with some non-zero

total energy. Which one of the following best describes the particle trajectory in the phase

space?

(a) (b)

(c) (d)

Ans.: (a)

Solution: 2

2

pE x

m

For 0x , 2

2

pE x

m

2 2p m E x

For 0x , 2

2

pE x

m

2 2p m E x

Q51. If H is the Hamiltonian for a free particle with mass m , the commutator , ,x x H is

(a) 2 / m (b) 2 / m (c) 2 / 2m (d) 2 / 2m

Ans. : (b)

Solution: For free particle, potential is zero.

2

2xP

Hm

p

x

p

x

p

x

p

x

xp

x

V x

x




22

Now, 2 2

, ,2 2

xx

P ix H x P

m m

22

, , ,2 x

i ix x H x P i

m m m

Q52. For the transformation

1 2 12 cos , 2 sinQ q e p P q e p

(where is a constant) to be canonical, the value of is _________.

Ans. : 2

Solution: 1 2 12 cos , 2 .sinQ qe p P qe p

Since, , 1Q P

1Q P Q P

q p p q

1 1

1 2 1 1 2 .12 21 2

2 . cos 2 cos 2 sin . sin 12 2

q e p qe p qe p q e p

2 2 2 0. cos sin 1e p p e

2

Q53. A uniform circular disc of mass m and radius R is rotating with angular

speed about an axis passing through its centre and making an angle

030 with the axis of the disc. If the kinetic energy of the disc is

2 2m R , the value of is__________ (up to two decimal places).

Ans. : 0.21 Solution: The kinetic energy of the disc is,

1

2T L

Where L

is angular momentum and is angular velocity

2

0 21 1 3 1 3cos30

2 2 2 2 2 2

mRT L I

2 2 2 230.21

8T m R m R 2 2 2 20.21m R m R

Hence, 0.21




23

GATE-2019

Q54. Consider a transformation from one set of generalized coordinate and momentum ( ,q p )

to another set ( ,Q P ) denoted by,

;s rQ pq P q

where s and r are constants. The transformation is canonical if

(a) 0s and 1r (b) 2s and 1r

(c) 0s and 1r (d) 2s and 1r

Ans. : (b)

Solution: 1. . 1 0 1s rQ P Q Pq rq

q p p q

1 1r srq 2s and 1r

Q55. The Hamiltonian for a particle of mass m is 2

2

pH kqt

m where q and p are the

generalized coordinate and momentum, respectively, t is time and k is a constant. For

the initial condition, 0q and 0p at 0,t q t t . The value of is ________

Ans. : 3

Solution: H p

qp m

....(1)

2

2

H ktp kt p

q

….(2)

2 3

2 6

dq kt ktq

dt 3q t so 3




24

Q56. A ball bouncing of a rigid floor is described by the potential energy function

for 0

for 0

mgx xV x

x

Which of the following schematic diagrams best represents the phase space plot of the

ball?

(a) (b)

(c) (d)

Ans. : (b)

Solution: 2

2 22

pE mgx p m E mgx

m which is equation of parabola

Q57. Consider the Hamiltonian 2 4

2,

2

ap qH q p

q

, where and are parameters with

appropriate dimensions, and q and p are the generalized coordinate and momentum,

respectively. The corresponding Lagrangian ,L q q is

(a) 2

4 2

1

2

q

q q

(b) 2

4 2

1

2

q

q q

(c) 2

4 2

1 q

q q

(d) 2

4 2

1

2

q

q q

Ans. : (a)

Solution: 2 4

22

ap qL pq H pq

q

from Hamiltonian equation of motion

4

H qq p

p aq

2mE

2mE

E

mgx

2mE

2mE

E

mg x

E

mg

2mE

2mE

Emg

x

2mE

2mE

Emg

x




25

2

4 2

1

2

qL

q q

Q58. A projectile of mass 1kg is launched at an angle of 030 from the horizontal direction at

0t and takes time T before hitting the ground. If its initial speed is 110 ms , the value

of the action integral for the entire flight in the units of 2 1kgm s (round off to one

decimal place) is___________. [Take 210g ms ]

Ans. : 33.3

Solution: 2 sin

1secv

Tg

2 21

2L m x y mgy

1cos 5 3x v ms sin 5 10y v gt t

2 2 2 21 1 1 1sin 10. 10 5 5

2 2 2 2y ut gt v t gt t t t t

2 2 211 5 3 5 10 1 10 5 5

2L t t t

2100 100 50L t t

1

2

0 0100 100 50 33.3

TA Ldt t t dt

Q59. Two spaceships A and B , each of the same rest length L , are moving in the same

direction with speeds 4

5

c and

3

5

c, respectively, where c is the speed of light. As

measured by B , the time taken by A to completely overtake B [see figure below] in

units of /L c (to the nearest integer) is _____________

(i) (ii)

Ans. : 5

A

B 3 / 5c

4 / 5c A

B 3 / 5c

4 / 5c




26

Solution: ,

2

4 355 5 5

4 3 1 13 131 . .5 5 25

A B

cc c

u cc c

c

Kinematic equation is given by

5 25 5

1 513 169

Lc t L L t

c

Q60. Two events, one on the earth and the other one on the Sun, occur simultaneously in the

earth’s frame. The time difference between the two events as seen by an observer in a

spaceship moving with velocity 0.5c in the earth’s frame along the line joining the earth

to the Sun is t , where c is the speed of light. Given that light travels from the Sun to

the earth in 8.3 minutes in the earth’s frame, the value of t in minutes (rounded off to

two decimal places) is____________

(Take the earth’s frame to be inertial and neglect the relative motion between the earth

and the sun)

Ans. : 4.77

Solution: ' '2 1 0t t ' ' 8

2 1 8.3 3 10 60x x 0.5v c

' '' '2 1 ' '' '2 12 2 2 12 1

2 1 22 2 2 2

2 2 2 2

4.77 min

1 1 1 1

vx vxt t x xt t vc ct t t

cv v v v

c c c c



Website: www.physicsbyfiziks.com | Email: [email protected] 1

ELECTROMAGNETIC THEORY SOLUTIONS

GATE- 2010

Q1. An insulating sphere of radius a carries a charge density

arrar ;cos220 .

The leading order term for the electric field at a distance d, far away from the charge

distribution, is proportional to

(a) 1d (b) 2d (c) 3d (d) 4d

Ans: (c)

Solution:

V

dr

dr

rV cos11

2,

Ist term, 2

2 2 20

0 0 0

cos sin 0a

d a r r drd d

IInd term, 2

2 2 2 20

0 0 0

cos cos sin 0a

d a r r drd d

.

2 3

1 1V

r rE

Q2. Two magnetic dipoles of magnitude m each are placed in a plane as shown in figure.

The energy of interaction is given by

(a) Zero (b) 3

20

4 d

m

(c) 3

20

2

3

d

m

(d) 3

20

8

3

d

m

Ans: (d)

Solution: rmrmmmr

U ˆˆ34 21213

0

,

Since 02121 mmmm

003

0 45cos45cos34

mmd

U

3

20

8

3

d

mU

.

o45

o45

m2

d

m 1





Consider the propagation of electromagnetic waves in a linear, homogeneous and

isotropic material medium with electric permittivity ε and magnetic permeability μ.

Q3. For a plane wave of angular frequency ω and propagation vector k propagating in the

medium Maxwell’s equations reduce to

(a) EHkHEkHkEk ;;0;0

(b) EHkHEkHkEk ;;0;0

(c) EHkHEkHkEk ;;0;0

(d) EHkHEkHkEk ;;0;0 Ans: (d)

Q4. If and assume negative values in a certain frequency range, then the directions of

the propagation vector k and the Poynting vector S in that frequency range are related as

(a) k and S are parallel

(b) k and S are anti-parallel

(c) k and S are perpendicular to each other

(d) k and S makes an angle that depends on the magnitude of |ε| and |μ|

Ans: (a)

Q5. Consider a conducting loop of radius a and total loop resistance R placed in a region with

a magnetic field B thereby enclosing a flux 0. The loop is connected to an electronic

circuit as shown, the capacitor being initially uncharged

If the loop is pulled out of the region of the magnetic field at a constant speed u, the final

output voltage Vout is independent of

(a) 0 (b) u (c) R (d) C

Ans: (a)

outV

C




GATE-2011

Q6. If a force F is derivable from a potential function V(r), where r is the distance from the

origin of the coordinate system, it follows that

(a) 0 F (b) 0 F (c) 0V (d) 02 V

Ans: (a)

Q7. Two charges q and 2q are placed along the x -axis in front of a grounded, infinite

conducting plane, as shown in the figure. They

are located respectively at a distance of 0.5 m and

1.5 m from the plane. The force acting on the

charge q is

(a) 2

7

4

1 2

0

q

(b) 2

0

24

1q

(c) 2

04

1q

(d)

24

1 2

0

q

Ans: (a)

Solution: Using method of Images we can draw equivalent figure as shown below:

2

2 2 20 0 0

2 2 7 1 7

4 4 2 4 21 1 2

q q q q q q qF

Q8. A uniform surface current is flowing in the positive y-direction over an infinite sheet

lying in x-y plane. The direction of the magnetic field is

(a) along i for z > 0 and along i for z < 0

(b) along k for z > 0 and along k for z < 0

(c) along i for z > 0 and along i for z < 0

(d) along k for z > 0 and along k for z < 0

Ans: (a)

m5.0 q q2

m5.1

x

m5.1m5.1

xm5.0 q q2m5.0

qq2




Q9. A magnetic dipole of dipole moment m is placed in a non-uniform magnetic field B . If

the position vector of the dipole is r , the torque acting on the dipole about the origin is

(a) Bmr (b) Bmr

(c) Bm (d) BmrBm

Ans: (c)

Q10. A spherical conductor of radius a is placed in a uniform electric field kEE ˆ0 . The

potential at a point P(r, θ) for r > a, is given by

Φ(r, θ) = constant – coscos2

30

0r

aErE

where r is the distance of P from the centre O of the sphere and θ is the angle OP makes

with the z-axis

The charge density on the sphere at θ = 30o is

(a) 2/33 00 E (b) 2/3 00 E

(c) 2/3 00 E (d)) 2/00 E

Ans: (a)

Solution: .cos2

cos3

30

000

arar r

aEE

r

V

000

0000000 2

3330cos3cos3cos2cos EEEEE

Q11. Which of the following expressions for a vector potential A DOES NOT represent a

uniform magnetic field of magnitude B0 along the z-direction?

(a) 0,,0 0 xBA (b) 0,0,0 yBA

(c)

0,

2,

200 yBxB

A (d)

0,

2,

200 xByB

A

Ans: (c)

Solution: AB

.

r

O

P

k




Statement for Linked Questions 12 and 13:

A plane electromagnetic wave has the magnetic field given by

ktk

yxBtzyxB ˆ2

sin,,, 0

where k is the wave number and kji ˆ andˆ,ˆ are the Cartesian unit vectors in x, y and z

directions respectively.

Q12. The electric field tzyxE ,,, corresponding to the above wave is given by

(a) 2

ˆˆ

2sin0

jit

kyxcB

(b)

2

ˆˆ

2sin0

jit

kyxcB

(c) itk

yxcB ˆ2

sin0

(d) jt

kyxcB ˆ

2sin0

Ans: (a)

Solution: 0

ˆ ˆˆsin

2 2

k i j x y kc cE k B B t k

k k

0

ˆ ˆsin

2 2

i jkE cB x y t

Q13. The average Poynting vector is given by

(a)

2

ˆˆ

2 0

20 jicB

(b)

2

ˆˆ

2 0

20 jicB

(c)

2

ˆˆ

2 0

20 jicB

(d)

2

ˆˆ

2 0

20 jicB

Ans: (d)

Solution: 2 2 20 0 0

0 0 0

ˆ ˆ ˆ ˆˆ2 2 22 2

cB cB cBi j i jS k




GATE-2012

Q14. The space-time dependence of the electric field of a linearly polarized light in free space

is given by 0ˆ cosxE t kz where E0, ω and k are the amplitude, the angular frequency

and the wavevector, respectively. The time average energy density associated with the

electric field is

(a) 2004

1E (b) 2

002

1E (c) 2

00 E (d) 2002 E

Ans: (a)

Solution: 200

220

20 4

1cos

2

1

2

1EukzwtEEu EE

Q15. A plane electromagnetic wave traveling in free space is incident normally on a glass plate

of refractive index 3/2. If there is no absorption by the glass, its reflectivity is

(a) 4% (b) 16% (c) 20% (d) 50%

Ans: (a)

Solution: %404.25

4

4

1

2/31

2/3122

21

21 ornn

nnR

Q16. The electric and the magnetic field tzE ,

and tzB , , respectively corresponding to the

scalar potential 0, tz and vector potential tzitzA ˆ,

are

(a) tj-Bandˆ

ziE (b) tjBandˆ

ziE

(c) tj-Bandˆ

ziE (d) tj-Bandˆ

ziE Ans: (d)

Solution: .ˆ,ˆ tjABzit

A

t

AE

Q17. A plane polarized electromagnetic wave in free space at time t=0 is given

by ˆ, 10 exp 6 8E x z j i x z

. The magnetic field tzxB ,,

is given by

(a) ctzxiikc

tzxB 1086expˆ8ˆ61

,,

(b) ctzxiikc

tzxB 1086expˆ8ˆ61

,,

(c) ctzxiikc

tzxB 86expˆ8ˆ61

,,

(d) ctzxiikc

tzxB 86expˆ8ˆ61

,,




Ans: (a)

Solution: trkijki

cE

k

k

cEk

cB

.expˆ10

10

ˆ8ˆ611ˆ1

.10,1086expˆ8ˆ61

cctzxiikc

B

Q18. Two infinitely extended homogeneous isotopic dielectric media (medium-1and medium-2

with dielectric constant 5and20

2

0

1

, respectively)

meet at the z = 0 plane as shown in the figure. A uniform

electric field exists everywhere. For z ≥ 0, the electric field

is given by kjiE ˆ5ˆ3ˆ21

. The interface separating the

two media is charge free. The electric displacement vector

in the medium-2 is given by

(a) kjiD ˆ10ˆ15ˆ1002 (b) kjiD ˆ10ˆ15ˆ1002

(c) kjiD ˆ10ˆ6ˆ402 (d) kjiD ˆ10ˆ6ˆ402

Ans: (b)

Solution: jiEEE ˆ3ˆ2221

and 0f kkEEDD ˆ2ˆ5

521

2

1221

kjiE ˆ2ˆ3ˆ22

.ˆ10ˆ15ˆ100222 kjiED

GATE-2013

Q19. At a surface current, which one of the magnetostatic boundary condition is NOT

CORRECT?

(a) Normal component of the magnetic field is continuous.

(b) Normal component of the magnetic vector potential is continuous.

(c) Tangential component of the magnetic vector potential is continuous.

(d) Tangential component of the magnetic vector potential is not continuous.

Ans: (d)

medium - 1

medium - 2 z = 0




Q20. Interference fringes are seen at an observation plane 0z , by the superposition of two

plane waves trkiA

11 exp and trkiA

22 exp , where 1A and 2A are real

amplitudes. The condition for interference maximum is

(a) 1221 mrkk

(b) mrkk 221

(c) 1221 mrkk

(d) mrkk 221

Ans: (b)

Q21. For a scalar function satisfying the Laplace equation, has

(a) zero curl and non-zero divergence

(b) non-zero curl and zero divergence

(c) zero curl and zero divergence

(d) non-zero curl and non-zero divergence

Ans: (c)

Solution: 02 0. and 0 .

Q22. A circularly polarized monochromatic plane wave is incident on a dielectric interface at

Brewaster angle. Which one of the following statements is correct?

(a) The reflected light is plane polarized in the plane of incidence and the transmitted

light is circularly polarized.

(b) The reflected light is plane polarized perpendicular to the plane of incidence and the

transmitted light is plane polarized in the plane of incidence.

(c) The reflected light is plane polarized perpendicular to the plane of incidence and the

transmitted light is elliptically polarized.

(d) There will be no reflected light and the transmitted light is circularly polarized.

Ans: (c)

Q23. A charge distribution has the charge density given by 00 xxxxQ . For

this charge distribution the electric field at 0,0,2 0x

(a) 2009

ˆ2

x

xQ

(b)

3004

ˆ

x

xQ

(c)

2004

ˆ

x

xQ

(d)

20016

ˆ

x

xQ

Ans:




Solution: Potential ' ' '

22 3

0

1....

4

a a a

a a a

x x xV r dx x dx x dx

x x x

First term, total charge

00

0

0

0

00

QQxdxxQxdxxQxdxQx

x

x

x

T

Second term, dipole moment

00000 20

0

0

0

QxxQQxxdxxxQxdxxxQxdxxpx

x

x

x

xx

Qx

x

Qxx

x

Qxx

x

VE

x

QxV ˆ

8ˆ

24

4ˆ

4

4ˆ

4

22

03

00

0

30

0

20

0

0

Q24. A monochromatic plane wave at oblique incidence undergoes reflection at a dielectric

interface. If ri kk ˆ,ˆ and n are the unit vectors in the directions of incident wave, reflected

wave and the normal to the surface respectively, which one of the following expressions

is correct?

(a) 0ˆˆˆ nkk ri (b) 0ˆˆˆ nkk ri (c) 0ˆˆˆ ri knk (d) 0ˆˆˆ ri knk

Ans: (c)

Q25. In a constant magnetic field of 0.6 Tesla along the z direction, find the value of the path

integral dlA

in the units of (Tesla 2m ) on a square loop of side length 2/1 meters.

The normal to the loop makes an angle of 060 to the z-axis, as shown in the figure. The

answer should be up to two decimal places. ___________

Ans: 0.15

zo60




Solution: 2

2

0 .15.02

1

2

16.060cos.. mTBAadBadAdlA

SS

GATE-2014

Q26. Which one of the following quantities is invariant under Lorentz transformation?

(a) Charge density (b) Charge (c) Current (d) Electric field

Ans: (b)

Q27. An unpolarized light wave is incident from air on a glass surface at the Brewster angle.

The angle between the reflected and the refracted wave is

(a) o0 (b) o45 (c) o90 (d) o120

Ans: (c)

Q28. The electric field of a uniform plane wave propagating in a dielectric non-conducting

medium is given by 7ˆ10cos 6 10 0.4 /E x t z V m

. The phase velocity of the

wave is _________ sm /108

Ans: 1.5

Solution: 7

86 101.5 10 / sec

0.4v m

k

Q29. If the vector potential zzyyxxA ˆ3ˆ2ˆ

, satisfies the Coulomb gauge, the value of the

constant is _______

Ans: 1

Solution: Coulomb gauge condition . 0A

2 3 0 1

Q30. A ray of light inside Region 1 in the xy -plane is incident

at the semicircular boundary that carries no free charges.

The electric field at the point 0 ,4

P r

in plane polar

coordinates is 1 ˆ ˆ7 3rE e e

where re and e are the unit

vectors. The emerging ray in Region 2 has the electric

field 2E

parallel to x -axis. If 1 and 2 are the dielectric

constants of Region-1 and Region-2 respectively, then 1

2

is ________

y

O1 2 x

4/,0 rP

1Region 2Region




Ans: 2.32

Solution: 1 ˆ ˆ7 3rE e e

10ˆ ˆ ˆ7 3 . 7cos 45 3sin 45

2x rE e e x

4ˆ ˆ ˆ7 3 . 7sin 45 3sin 45

2y rE e e y

Thus 1E

makes an angle 1 1 04tan tan 21.8

10y

x

E

E

2 2 2

1 1 1

tan tan 452.32

tan tan 23.2

. where 01 45 and 0

2 45

Q31. The value of the magnetic field required to maintain non-relativistic protons of energy

MeV1 in a circular orbit of radius 100 mm is _______Tesla

(Given: 27 191.67 10 , 1.6 10pm kg e C )

Ans: 1.44

Solution:

2 219 2 13 272 2 213 2

227 219

1.6 10 0.1 1.6 10 2 1.67 101.6 10

2 2 1.67 10 1.6 10 0.1p

Bq B RE B

m

13 27 402

4038

10 2 1.67 10 3.34 102.08

1.6 101.6 10 0.01B

2.08 1.44B Tesla Tesla

Q32. In an interference pattern formed by two coherent sources, the maximum and minimum

intensities are 09I and 0I respectively. The intensities of the individual wave are

(a) 00 and3 II (b) 00 and4 II (c) 00 4and5 II (d) 00 and9 II

Ans: (b)

Solution: 2

max 1 2I I I and 2

min 1 2I I I

2

0 1 29I I I and 2

0 1 2I I I 1 0 2 04I I and I I

Q33. The intensity of a laser in free space is 2/150 mmW . The corresponding amplitude of the

electric field of the laser is _________m

V 2212

0 ./10854.8 mNC

4/,0 rP

1 2

1

O

y

x1Region 2Region

2 2E

1E




Ans: 10.6

Solution: 3

20 0 0 8 12

0

1 2 2 150 1010.6 /

2 3 10 8.854 10

II c E E V m

c

GATE-2015

Q34. A point charge is placed between two semi-infinite conducting plates which are inclined

at an angle of o30 with respect to each other. The number of image charges

is___________.

Ans.: 11

Solution: 360 360

1 1 1130

n

Q35. Given that the magnetic flux through the closed loop PQRSP is . If 1R

P

ldA

along

PQR , the value of R

P

ldA

along PSR is

(a) 1 (b) 1 (c) 1 (d) 1

Ans.: (b)

Solution: . .R P

s P RB d a A dl A dl A dl 1 1

R R

P PA dl A dl

Q36. The space between two plates of a capacitor carrying charges Q and Q is filled with

two different dielectric materials, as shown in the figure. Across the interface of the two

dielectric materials, which one of the following statements is correct?

(a) E

and D

are continuous

(b) E

is continuous and D

is discontinuous

(c) D

is continuous and E

is discontinuous

(d) E

and D

are discontinuous

Ans.: (d)

P

QR

S

Q Q




Q37. Four forces are given below in Cartesian and spherical polar coordinates

(i) rR

rKF ˆexp

2

2

1

(ii) zyyxKF ˆˆ 33

2

(iii) yyxxKF ˆˆ 333

(iv)

rKF

4

where K is a constant Identify the correct option

(a) (iii) and (iv) are conservative but (i) and (ii)are not

(b) (i) and (ii) are conservative but (iii) and (iv) are not

(c) (ii) and (iii) are conservative but (i) and (iv) are not

(d) (i) and (iii) are conservative but (ii) and (iv) are not

Ans.: (d)

Solution:

1 2

2

2

ˆ sin

10

sin

k exp 0 0

r r r

Fr r

r

R

2 2 2 22

3 3

ˆ ˆ3 0 3 0 3 3

0

x y z

F x ky z kx ky x kx zx y z

kx ky

3

3 3

0

0

x y z

Fx y z

kx ky

4 2 2

sin

1 1cos

sin sin

0 0 sin

r r r

F r kr r r

kr

r




Q38. A monochromatic plane wave (wavelength nm600 ) tkziE exp0 is incident

normally on a diffraction grating giving rise to a plane wave trkiE

11 exp in the

first order of diffraction. Here

zxkkandEE ˆ

2

3ˆ

2

11101

. The period (in m ) of

the diffraction grating is ______________ (upto one decimal place) Ans.: 1.2

Solution: sin 1sin

d n d n

and 1 1

1 3ˆ ˆ

2 2k k x z

01

1

1 3ˆˆ ˆ

2 2 1sin 30

21 3 1 34 4 4 4

z x zk k

k k

6001200 1.2

sin 30d nm nm m

Q39. A long solenoid is embedded in a conducting medium and is insulated from the medium.

If the current through the solenoid is increased at a constant rate, the induced current in

the medium as a function of the radial distance r from the axis of the solenoid is

proportional to

(a) 2r inside the solenoid and r

1 outside (b) r inside the solenoid and

2

1

r outside

(c) 2r inside the solenoid and 2

1

r outside (d) r inside the solenoid and

r

1 outside

Ans.: (d)

Solution: ;B

E dl dat

,For r R

2

0 0

0

22 2

2

r

r

dI dI rE r n r dr n

dt dt

0

1

2

dIE n r

dt

,For r R

2

0 0

0

22 2

2

R

r

dI dI RE r n r dr n

dt dt

20

1

2

dIE n R

r dt

grating

1k

1 ˆk k z z




Q40. A plane wave tkziEyix expˆˆ 0 after passing through an optical element emerges

as tkziEyix expˆˆ 0 , where k and are the wavevector and the angular

frequency, respectively. The optical element is a

(a) quarter wave plate (b) half wave plate

(c) polarizer (d) Faraday rotator

Ans.: (b)

Solution: Incident wave: 0 0 0cos sinix i y E e E x E y

Left circular polarization with phase angle 1ie

Emergent wave: 0 0 0ˆcos sinix i y E e E x E y

Right circular polarization with phase angle 01

ie

Thus there is phase change of and hence path difference is 2

.

Q41. A charge q is distributed uniformly over a sphere, with a positive charge q at its center

in (i). Also in (ii), a charge q is distributed uniformly over an ellipsoid with a positive

charge q at its center. With respect to the origin of the coordinate system, which one of

the following statements is correct?

(a) The dipole moment is zero in both (i) and (ii)

(b) The dipole moment is non-zero in (i) but zero in (ii)

(c) The dipole moment is zero in (i) but non-zero in (ii)

(d) The dipole moment is non-zero in both (i) and (ii)

Ans.: (a)

Solution: 0i ip q r in both cases.

X

Z

Y)i(

X

Z

Y)ii(




GATE-2016

Q42. Which of the following magnetic vector potentials gives rise to a uniform magnetic field

kB ˆ0 ?

(a) kzB ˆ0 (b) jxB ˆ

0 (c) jxiyB ˆˆ2

0 (d) jxiyB ˆˆ2

0

Ans.: (c)

Solution: (a) 0A

(b) 0ˆA B k

(c) 0ˆA B k

(d) 0A

Q43. The magnitude of the magnetic dipole moment associated with a square shaped loop

carrying a steady current I is m . If this loop is changed to a circular shape with the same

current I passing through it, the magnetic dipole moment becomespm

. The value of p

is ______.

Ans.: 4

Solution: Magnetic dipole moment associated with a square shaped loop (let side is a) carrying a

steady current I is 2m Ia .

Magnetic dipole moment associated with a circular shaped loop (let radius is r) carrying a

steady current I is 2m I r .

Here 4 2a r2a

r

2 2

2 2 4 4a Ia mm I r I

Q44. In a Young’s double slit experiment using light, the apparatus has two slits of unequal

widths. When only slit-1 is open, the maximum observed intensity on the screen is 04I .

When only slit- 2 is open, the maximum observed intensity is 0I . When both the slits are

open, an interference pattern appears on the screen. The ratio of the intensity of the

principal maximum to that of the nearest minimum is ________.

Ans.: 9




Solution:

2 2 2

1 2 0 0 0 0max 02 2 2

min 01 2 0 0 0 0

4 2 99

4 2

I I I I I II I

I II I I I I I

Q45. An infinite, conducting slab kept in a horizontal plane carries a uniform charge density .

Another infinite slab of thickness t, made of a linear dielectric material of dielectric

constant k , is kept above the conducting slab. The bound charge density on the upper

surface of the dielectric slab is

(a) k2

(b)

k

(c)

k

k

2

2 (d)

k

k 1

Ans.: (d)

Solution:

Electric field due to infinite, conducting slab inside the dielectric is 0

ˆ ˆE z zk

Polarisation 0 0

0

1ˆ ˆ1e

kP E k z z

k k

1

1ˆ.

kP z

k

Q46. The electric field component of a plane electromagnetic wave travelling in vacuum is

given by itkzEtzE ˆcos, 0

. The Poynting vector for the wave is

(a) jtkzEc ˆcos2

220

0

(b) ktkzEc ˆcos2

220

0

(c) jtkzEc ˆcos2200 (d) ktkzEc ˆcos22

00

Ans.: (d)

Solution: itkzEtzE ˆcos, 0

01 ˆˆ , cosE

B z E z t kz t jc c

The Poynting vector for the wave is

2

2 2 200 0

0 0

1 ˆ ˆcos cosE

S E B kz t k c E kz t kc

Q47. The yx plane is the boundary between free space and a magnetic material with relative

permeability r . The magnetic field in the free space is ˆˆx zB i B k . The magnetic field in

the magnetic material is

(a) kBiB zxˆˆ (b) kBiB zrx

ˆˆ (c) kBiB zxr

ˆˆ1

(d) kBiB zxr

ˆˆ

k 1

1z




Ans.: (d)

Solution: 1 2ˆ

zB B k B and 1 2H H 1 2

0 0 r

B B

2 1ˆ

r r xB B B i

The magnetic field in the magnetic material is kBiB zxrˆˆ

GATE- 2017

Q48. Identical charges q are placed at five vertices of a regular hexagon of side a . The

magnitude of the electric field and the electrostatic potential at the centre of the hexagon

are respectively

(a) 0,0 (b) 2

0 0

,4 4

q q

a a

(c) 2

0 0

5,

4 4

q q

a a (d)

20 0

5 5,

4 4

q q

a a

Ans. : (c)

Solution: The resultant field at P is2

04

qE

a

The electrostatic potential at P is 0

5

4

qV

a

Q49. A parallel plate capacitor with square plates of side 1m separated by 1 micro meter is

filled with a medium of dielectric constant of 10 . If the charges on the two plates are 1C

and 1C , the voltage across the capacitor is………….. kV . (up to two decimal places).

( 120 8.854 10 /F m )

Ans. : 11.29

Solution: 6

012

0

1 1 1011.29

8.854 10 10 1r

r

A qdq CV V V kV

d A

Q50. Light is incident from a medium of refractive index 1.5n onto vacuum. The smallest

angle of incidence for which the light is not transmitted into vacuum is…………...

degrees. (up to two decimal places)

Ans. : 41.8

Solution: 12

1

1 1sin sin

1.5 1.5C C

n

n

41.8C

aP

q

q q

q q




Q51. A monochromatic plane wave in free space with electric field amplitude of 1 /V m is

normally incident on a fully reflecting mirror. The pressure exerted on the mirror

is……………… 1210 Pa . (up to two decimal places) ( 120 8.854 10 /F m )

Ans. : 8.85

Solution: 22 2 12 120 0 0 0

2 2 18.854 10 1 8.85 10

2

IP c E E Pa

c c

Q52. Three charges 2 , 1 , 1C C C are placed at the vertices of an equilateral triangle of side

1m as shown in the figure. The component of the electric dipole moment about the

marked origin along the y direction is………C m .

Ans. : 1.73

Solution: ˆ ˆ ˆ ˆ1 1 1 2 2 1.5 1 0.25p x x x y

Along the y direction 2 1 0.25 1.73

Q53. An infinite solenoid carries a time varying current 2I t At , with 0A . The axis of

the solenoid is along the z direction. r and are the usual radial and polar directions in

cylindrical polar coordinates. ˆˆ ˆr zB B r B B z

is the magnetic field at a point outside

the solenoid. Which one of the following statements is true?

(a) 0, 0, 0r zB B B (b) 0, 0, 0r zB B B

(c) 0, 0, 0r zB B B (d) 0, 0, 0r zB B B

Ans. : (d)

Q54. A uniform volume charge density is placed inside a conductor (with resistivity 210 m ).

The charge density becomes

1

2.718 of its original value after time…….Fermi seconds

(up to two decimal places) ( 120 8.854 10 /F m )

Ans. : 88.54

01.5m1C

2C

x1C

1m

y




Solution: 0/0 tt e 0

1/ ln ln 1

0 2.718

tt

12 2 150 8.854 10 10 88.54 10 sec 88.54t fs

Q55. Consider a metal with free electron density of 22 36 10 cm . The lowest frequency of

electromagnetic radiation to which this metal is transparent, is 161.38 10 Hz . If this

metal had a free electron density of 23 31.8 10 cm instead, the lowest frequency

electromagnetic radiation to which it would be transparent is…………… 1610 Hz (up to

two decimal places).

Ans. : 2.39

Solution: Cut-off frequency is f n .

Thus 23

16 162 2 22 1 2 22

1 1 1

1.8 101.38 10 2.39 10

6 10

f n nf f f Hz

f n n

GATE- 2018

Q56. Among electric field ( E

), magnetic field ( B

), angular momentum ( L

) and vector

potential ( A

), which is/are odd under parity (space inversion) operation?

(a) E

only (b) E

and A

only

(c) E

and B

only (d) B

and L

only

Ans. : (b)

Solution: Under parity operation r r

VE

r

; :E P E

B I r

; :B P B

L r p

; :L P L

AE

t

; :A P A




Q57. An infinitely long straight wire is carrying a steady current I . The ratio of magnetic

energy density at distance 1r to that at 2 12r r from the wire is ___________.

Ans. : 4

Solution: 22

11 22 2 2

0 2 1 1

214

2B

BB

ru rBu

r u r r

Q58. A light beam of intensity 0I is falling normally on a surface. The surface absorbs 20%

of the intensity and the rest is reflected. The radiation pressure on the surface is given by

0 /X I c , where X is __________ (up to one decimal place). Here c is the speed of light.

Ans. : 1.8

Solution: Radiation pressure 0 0 00.8 1.8I I I

c c c

Q59. The number of independent components of a general electromagnetic field tensor

is__________

Ans. : 6

Solution: In Cartesian co-ordinate, three Independent coordinate for electric field, , ,x y zE E E

and three Independent co-ordinate for magnetic field , ,x y zB B B .

Q60. Consider an infinitely long solenoid with N turns per unit length, radius R and carrying

a current cosI t t , where is a constant and is the angular frequency. The

magnitude of electric field at the surface of the solenoid is

(a) 0

1sin

2NR t (b) 0

1cos

2NR t

(c) 0 sinNR t (d) 0 cosNR t

Ans. : (a)

Solution: 0 ˆ, inside

0 , outside

NI t zB

Since, line

BE dl da

t

202 sinE R N t R




0

1sin

2E NR t

Q61. A constant and uniform magnetic field 0ˆB B k

pervades all space. Which one of the

following is the correct choice for the vector potential in Coulomb gauge?

(a) 0ˆB x y i (b) 0

ˆB x y j (c) 0ˆB xj (d) 0

1 ˆ ˆ2

B xi yj

Ans. : (c)

Solution: Check option (c),

0ˆ0,A B A B k

Q62. A long straight wire, having radius a and resistance per unit length r , carries a current

I . The magnitude and direction of the Poynting vector on the surface of the wire is

(a) 2 / 2I r a , perpendicular to axis of the wire and pointing inwards

(b) 2 / 2I r a , perpendicular to axis of the wire and pointing outwards

(c) 2 /I r a , perpendicular to axis of the wire and pointing inwards

(d) 2 /I r a , perpendicular to axis of the wire and pointing outwards

Ans. : (a)

Solution: 0

0 0

1 1

2 2

IV IR IS E B

l a l a

,R

V IR rl

2

2

I rS

a

Q63. A quarter wave plate introduces a path difference of / 4 between the two components

of polarization parallel and perpendicular to the optic axis. An electromagnetic wave with

0ˆ ˆ i kz tE x y E e

is incident normally on a quarter wave plate which has its optic axis

making an angle 0135 with the x - axis as shown.

The emergent electromagnetic wave would be

(a) elliptically polarized

(b) circularly polarized

(c) linearly polarized with polarization as that of incident wave

(d) linearly polarized but with polarization at 090 to that of the incident wave

Ans. : (c)

Optic axis

y

0135x




Q64. An electromagnetic plane wave is propagating with an intensity 5 21.0 10I Wm in a

medium with 03 and 0 . The amplitude of the electric field inside the medium

is _________ 3 110 Vm (up to one decimal place).

( 12 2 1 2 7 2 8 10 08.85 10 , 4 10 , 3 10C N m NA c ms )

Ans. : 6.6

Solution: 2 21 2 22

12

I II v E E I

v

7

2 5 5 4012

0

4 102 10 2 10 4363.4 10

3 3 8.8 10E

2 366 10 6.6 10 /E V m

GATE-2019

Q65. The electric field of an electromagnetic wave is given by ˆ3sinE kz t x

ˆ4cos kz t y . The wave is

(a) linearly polarized at an angle 1 4tan

3

from the x - axis

(b) linearly polarized at an angle 1 3tan

4

from the x - axis

(c) elliptically polarized in clockwise direction when seen travelling towards the observer

(d) elliptically polarized in counter-clockwise direction when seen travelling towards the

observer

Ans. : (d)

Solution: At 0, 3sin , 4cosx yz E t E t

At 0, 0, 4x yt E E

At , 3, 02 x yt E E

y

x




Q66. An infinitely long thin cylindrical shell has its axis coinciding with the z -axis. It carries a

surface charge density 0 cos , where is the polar and 0 is a constant. The

magnitude of the electric field inside the cylinder is

(a) 0 (b) 0

02

(c) 0

03

(d) 0

04

Ans. : (b)

Solution: 0 0

0 0 0

cos cos

2 2 2

RdddE

R R

Along axis of cylinder 2

20 0

0 00

cos cos2 2x xdE dE E d

Q67. A circular loop made of a thin wire has radius 2cm and resistance 2 . It is placed

perpendicular to a uniform magnetic field of magnitude 0 0.01B

Tesla. At time 0t

the field starts decaying as 0/0

t tB B e

, where 0 1t s . The total charge that passes

through a cross section of the wire during the decay is Q . The value of Q in C

(rounded off to two decimal places) is____________

Ans. : 6.28

Solution: 1

,d AdB d

Idt dt R dt R

0/2 20 0 0 1t t td d

r B e r B e tdt dt

22

00

0 0 01

tt r Br e

Q I t dt B e dtR R

223.14 2 10 0.01 6.28 C

Q68. The electric field of an electromagnetic wave in vacuum is given by

90 ˆcos 3 4 1.5 10E E y z t x

The wave is reflected from the 0z surface. If the pressure exerted on the surface is

20E , the value of (rounded off to one decimal place) is___________

Ans. : 0.8




Solution: ˆ ˆ3 4K y z

3

tan4

yR

z

K

K

20 0

2 1 42 cos

2 5R

IP cE

c c

20 00.8P E

Q69. A solid cylinder of radius R has total charge Q distributed uniformly over its volume. It

is rotating about its axis with angular speed . The magnitude of the total magnetic

moment of the cylinder is

(a) 2QR (b) 21

2QR (c) 21

4QR (d) 21

8QR

Ans. : (c)

Solution: Magnetic moment due to disc 4

4

R

Due to cylinder 4

4

Rd dz

dz

4 4

204 4

LR Q Q Rdz

R L

Q70. An infinitely long wire parallel to the x -axis is kept at z d and carries a current I in

the positive x direction above a superconductor filling the region 0z (see figure). The

magnetic field B

inside the superconductor is zero so that the field just outside the

superconductor is parallel to its surface. The magnetic field due to this configuration at a

point , , 0x y z is

(a)

0

22

ˆˆ

2

z d j ykI

y z d

(b)

0

2 22 2

ˆ ˆˆ ˆ

2

z d j yk z d j ykI

y z d y z d

(c)

0

2 22 2

ˆ ˆˆ ˆ

2

z d j yk z d j ykI

y z d y z d

dx

l

z

superconductor




(d)

02 22 2

ˆ ˆˆ ˆ

2

yj z d k yj z d kI

y z d y z d

Ans. : (b)

Solution: Verify that 0B

, when 0d

Q71. The vector potential inside a long solenoid with n turns per unit length and carrying

current I , written in cylindrical coordinates is 0 ˆ, ,2

nIA s z s

. If the term

0 ˆ ˆcos sin2

nIs s

, where 0, 0 is added to , ,A S z

, the magnetic

field remains the same if

(a) (b) (c) 2 (d) 2

1ˆ ˆ ˆUseful formulae: ;

1 1ˆˆ ˆs sz z

t t tt S z

S S z

svv v vv vv s z

s z z s s s

Ans. : (d)

Solution: 0

ˆˆ ˆ

1ˆ

0r

r r z

B A nIzr r z

A rA

0

ˆˆ ˆ

1 cosˆcos 1

2

0r

r r z

B A nI zr r z

A rA

Equate 0

coscos 1

2B B nI

cos cos2

2




Q72. For a given load resistance 4.7LR ohm, the power transfer efficiencies load

total

P

P

of

a dc voltage source and a dc current source with internal resistances 1R and 2R ,

respectively, are equal. The product 1 2R R in units of 2ohm (rounded off to one decimal

place) is___________

Ans. : 22.09

Solution: For dc voltage source

2

1total

L

VP

R R

and

2

1LR L

L

VP R

R R

1

LR Ldc vol

total L

P R

P R R

For dc current source

2 2

2

Ltotal

L

R RP I

R R

and 2

2 2

2LR L L L

L

R IP I R R

R R

2

2

LRdc curr

total L

P R

P R R

Since dc vol dc curr

2

1 2

L

L L

R R

R R R R

2 2 1L L LR R R R R R 2

1 2 LR R R

2 21 2 4.7 22.09R R

Q73. Consider a system of three charges as shown in the figure below:

d d

d

2

q

2

q

y

,r z

q

LR2RI

V

1R

LR




For 10 ; 60r m degrees; 610q Coulomb, and 310d m , the electric dipole

potential in volts (rounded off to three decimal places) at a point ,r is _________

[Use: 2

92

0

19 10

4

Nm

C

]

Ans. : 0.045

Solution: Monopole moment 02 2

q qq

ˆ ˆ ˆ2 2

q qp dy dy q dz

ˆp qdz

2 20 0

1 1 cos,

4 4

p r qdV r

r r

6 3 09

2

10 10 cos 60, 9 10

10V r

9

9 109 10 0.045

2 100

Q74. The electric field of an electromagnetic wave is given by ˆ3sinE kz t x

ˆ4cos kz t y . The wave is

(a) linearly polarized at an angle 1 4tan

3

from the x - axis

(b) linearly polarized at an angle 1 3tan

4

from the x - axis

(c) elliptically polarized in clockwise direction when seen travelling towards the observer

(d) elliptically polarized in counter-clockwise direction when seen travelling towards the

observer

Ans. : (d)

Solution: At 0, 3sin , 4cosx yz E t E t

At 0, 0, 4x yt E E

y

x




At , 3, 02 x yt E E

Q75. In a set of N successive polarizers, the thm polarizer makes an angle 2

m

N

with the

vertical. A vertically polarized light beam of intensity 0I is incident on two such sets with

1N N and 2N N , where 2 1N N . Let the intensity of light beams coming out be

1I N and 2I N , respectively. Which of the following statements is correct about the

two outgoing beams?

(a) 2 1I N I N ; the polarization in each case is vertical

(b) 2 1I N I N ; the polarization in each case is vertical

(c) 2 1I N I N ; the polarization in each case is horizontal

(d) 2 1I N I N ; the polarization in each case is horizontal

Ans. : (c)

Solution: 12

1 01

/ 2cos

Nn

I N IN

,

22

2 02

/ 2cos

Nn

I N IN

2 1I N I N

For last polarization, pass axis will be horizontal.

Ex: 1 5N

10

0 05 cos 18 0.605I I I

2 10N

20

0 010 cos 9 0.780I I I

10 5I I




QUANTUM MECHANICS SOLUTIONS

GATE- 2010

Q1. Which of the following is an allowed wavefunction for a particle in a bound state? N is

a constant and , 0 .

(a) 3r

eN

r

(b) reN 1

(c) 222 zyxxeNe (d)

Rr

Rr if0 ifconstant zero-non

Ans: (c)

Q2. A particle of mass m is confined in the potential

2 21

, for 02

, for 0

m x xV x

x

Let the wavefunction of the particle be given by

105

2

5

1 x ,

where 0 and 1 are the eigenfunctions of the ground state and the first excited state

respectively. The expectation value of the energy is

(a) 10

31 (b)

10

25 (c)

10

13 (d)

10

11

Ans: (a)

Solution: For half parabolic potential

2

30 E ,

2

71 E

1 3 4 7 31

5 2 5 2 10E

.

Q3. For a spin-s particle, in the eigen basis of 2

S , xS the expectation value 2xsm S sm is

(a)

2

1 22 mss (b) 22 21 mss

(c) 22 1 mss (d) 22m

Ans: (a)

Solution: 2xsm S sm 21

4sm S S sm 2 21

4sm S S S S S S sm

1

4sm S S S S sm 2

2

12

mss

2 22 zS S S S S S




Q4. A particle of mass m is confined in an infinite potential well:

0, if 0 ,

, otherwise.

x LV x

It is subjected to a perturbing potential

L

xVxV op

2sin

within the well. Let 1E and 2E be corrections to the ground

state energy in the first and second order in 0V , respectively.

Which of the following are true?

(a) 1 20; 0E E (b) 1 20; 0E E

(c) 1 20;E E depends on the sign of 0V (d) 1 20; 0E E

Ans: (a)

Solution: 02

sin2

0

011

L

dxL

xV

LE

;

1 1

212

1

m m

Pm

EE

VE

mEE 1 so veE 2

1 .

GATE- 2011

Q5. The quantum mechanical operator for the momentum of a particle moving in one

dimension is given by

(a) dx

di (b)

dx

di (c)

ti (d)

2

22

2 dx

d

m

Ans: (b)

Q6. An electron with energy E is incident from left on a potential

barrier, given by

0

0, for 0

, for 0

xV x

V x

as shown in the figure. For E < V0, the space part of the

wavefunction for x > 0 is of the form

(a) axe (b) axe (c) iaxe (d) iaxe

Ans: (b)

Solution: 0VE , so there is decaying wave function.

xV

xVp

L0

E

x0

xV

0V




Q7. If Lx, Ly and Lz are respectively the x, y and z components of angular momentum operator

L. The commutator [Lx Ly, Lz] is equal to

(a) 22yx LLi (b) zLi2 (c) 22

yx LLi (d) 0

Ans: (c)

Solution: zyx LLL , = yzxzyx LLLLLL , = 22yx LLi

Q8. The normalized ground state wavefunciton of a hydrogen atom is given by

area

r /2/3

2

4

1

, where a is the Bohr radius and r is the distance of the electron

from the nucleus, located at the origin. The expectation value 2

1

r is

(a) 2

8

a

(b)

2

4

a

(c)

2

4

a (d)

2

2

a

Ans: (d)

Solution: 2

1

r

2 22

3 20 0 0

4 1sin

4

r

ar e dr d da r

2

2

a

Q9. The normalized eigenstates of a particle in a one-dimensional potential well

0 if 0

otherwise

x aV x

are given by

a

xn

axn

sin2

, where n = 1, 2, 3,….

The particle is subjected to a perturbation

0 cos , for 0' 2

0 , otherwise

x aV x

V x a

The shift in the ground state energy due to the perturbation, in the first order perturbation

theory,

(a) 3

2 oV (b)

3oV

(c) 3oV

(d) 3

2 oV

Ans: (a)

Solution: dxxVEa

1

2/

0

*1

11

2/

0

02 cossin

2 a

dxa

xV

a

x

a

2/

0

3

0

3

sin2

a

a

a

x

Va

3

2 0V

Common data questions Q10 and Q11




In a one-dimensional harmonic oscillator, 0 1, and 2 are respectively the ground, first

and the second excited states. These three states are normalized and are orthogonal to one

another 1 and 2 are two states defined by

1 0 1 22 3 , 2 0 1 2 2 0 1 2,

where is a constant

Q10. The value of which 2 is orthogonal to 1 is

(a) 2 (b) 1 (c) – 1 (d) – 2

Ans: (c)

Solution: For orthogonal condition scalar product 2 1, 0 , so 1 2 3 0 1

Q11. For the value of α determined in Q10, the expectation value of energy of the oscillator in

the state 2 is

(a) (b) 3 / 2 (c) 3 (d) 9 / 2

Ans: (b)

Solution: 2 0 1 2 put 1 ,22

22

H

H 3

2

5

2

3

2

2

3

GATE- 2012

Q12. A particle of mass m is confined in a two dimensional square well potential of

dimension a. This potential V(x, y) is given by

V(x, y) = 0 for –a < x < a and –a < y < a

= ∞ elsewhere

The energy of the first excited state for this particle is given by,

(a) 2

22

ma

(b)

2

222

ma

(c)

2

22

8

5

ma

(d)

2

224

ma

Ans: (c)

Solution:

2 22 2

22 2

x yE n nm a

2

2222

8mann yx

2

22

8

5

ma

2,1 yx nn .




Q13. Consider the wavefunction srr 21 ,

for a fermionic system consisting of two spin-

half particles. The spatial part of the wavefunction is given by

21122211212

1, rrrrrr

where 21and are single particle states. The spin part χs of the wavefunction with spin

states 1/2-and2/1 should be

(a) 2

1 (b)

2

1 (c) αα (d) ββ

Ans: (b)

Solution: Since 21,rr is symmetric the total wavefunction must be antisymmetric for fermions

so spin part must be antisymmetric.

Q14. A particle is constrained to move in a truncated harmonic potential well (x > 0) as shown

in the figure. Which one of the following statements is CORRECT?

(a) The parity of the first excited state is even

(b) The parity of the ground state is even

(c) the ground state energy is 2

1

(d) The first excited state energy is 2

7

Ans: (d)

Solution: There is only odd parity. Ground state is 2

3and first excited

2

7

Q15. Consider a system in the unperturbed state described by the Hamiltonian,

10

010H .

The system is subjected to a perturbation of the form 'H

, where 1 . The

energy eigenvalues of the perturbed system using the first order perturbation

approximation are

(a) 1 and (1 + 2δ) (b) (1 + δ) and (1 - δ)

(c) (1+ 2δ) and (1 - 2δ) (d) (1+ δ) and (1 - 2δ)

Ans: (a)

xV

x




Solution: HH 0 , H0 is degenerate so after using degenerate perturbation through diagonalized

H one will get 2 0

0 0H

, 1 0 2 0

0 1 0 0H

.

So 21E and 01 .

Q16. The ground state wavefunction for the hydrogen atom is given by

0/

2/3

0100

1

4

1 area

, where 0a is the Bohr radius. The plot of the radial probability

density, P(r) for the hydrogen atom in the ground state is

(a) (b)

(c) (d)

Ans: (d)

Solution: The ground state is given by 0/

2/3

0100

1

4

1 area

Radial probability function

22rrP = 02 /2

30

1 1

4r ar e

a

Common Data for Questions 17–18

The wavefunction of particle moving in free space is given by, 2ikx ikxe e

Q17. The energy of the particle is

(a) m

k

2

5 22 (b)

m

k

4

3 22 (c)

m

k

2

22 (d)

m

k 22

Ans: (c)

P(r)

0r/a

P(r)

0r/a

P(r)

0r/a

P(r)

0r/a

P(r)

0r/a




Solution: EH , 2

22

2 xmH

2

22

ikx ikxik ik e ik ik em

2 22( 2 )

2ikx ikxk

H e em

2 2

2

k

m

Q18. The probability current density for the real part of the wavefunction is

(a) 1 (b) m

k (c)

m

k

2

(d) 0

Ans: (d)

Solution: The real part of the wave function kxkxreal cos2cos

Current density for real part of wave function = 0

GATE- 2013

Q19. Which one of the following commutation relations is NOT CORRECT? Here, symbols

have their usual meanings.

(a) 0,2 zLL (b) zyx LiLL ,

(c) LLLz , (d) LLLz ,

Ans: (d)

Q20. A proton is confined to a cubic box, whose sides have length m1210 . What is the

minimum kinetic energy of the proton? The mass of proton is kg271067.1 and

Planck’s constant is Js341063.6 .

(a) J17101.1 (b) J17103.3 (c) J17109.9 (d) J17106.6

Ans: (c)

Solution: 2 2

172

39.9 10

2ma

Q21. A spin-half particle is in a linear superposition 6.08.0 of its spin-up and spin-

down states. If and are the eigenstates of z , then what is the expectation value

up to one decimal place, of the operator xz 510 ? Here, symbols have their usual

meanings. _______________

Ans: 7.6




Solution: 6.8.1 0 0.8

0.8 0.60 1 0.6

Operator 10 5z xA 1 0 0 1

10 50 1 1 0

10 5

5 10A

AA 10 5 0.8

0.8 0.65 10 0.6

= 8.8 1.2 7.6

Q22. Consider the wave function rrAe rki /0 , where A is the normalization constant.

For 02rr , the magnitude of probability current density up to two decimal places, in

units of mkA /2 is _____________

Ans: 0.25

Solution: 2

2 2 0rk kJ A

m r m

2

2 2 20

0

0.252 4

r k k kJ A J A A

r m m m

Common data questions 23 and 24

To the given unperturbed Hamiltonian

200

052

025

we add a small perturbation given by

111

111

111

where is small quantity.

Q23. The ground state eigenvector of the unperturbed Hamiltonian is

(a) 0,21,2/1 (b) 0,2/1,2/1

(c) 1,0,0 (d) 0,0,1

Ans: (c)

111

111

111

,

200

052

025

0 PHH




Eigen value of 0H is 7,3,2 321 EEE and the Eigen vector corresponds

to 1

0

0

1

, 2

11

12

0

, 3

11

12

0

.

Q24. A pair of eigenvalues of the perturbed Hamiltonian, using first order perturbation theory,

is

(a) 27,23 (b) 2,23 (c) 27,3 (d) 22,3

Ans: (c)

Solution: 1 1 1 1PE H 121 E

2 2 2PE H

0

1

1

2

1.

111

111

111

.0112

1 1

0 0 1 1 0

0

3 3 3

1 1 1 11 1

1 1 0 . 1 1 1 . 12 2

1 1 1 0PE H

=

0

1

1

.0222

1.

242

13 E

121 E , 032 E , 273 E .

GATE- 2014

Q25. The recoil momentum of an atom is Ap when it emits an infrared photon of wavelength

nm1500 , and it is Bp when it emits a photon of visible wavelength nm500 . The ratio

B

A

p

pis

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

Ans: (c)

Solution: h

p , A

B

B

A

p

p

, 1500

500

A

B

=1 : 3

Q26. The ground state and first excited state wave function of a one dimensional infinite

potential well are 1 and 2 respectively. When two spin-up electrons are placed in this




potential which one of the following with 1x and 2x denoting the position of the two

electrons correctly represents the space part of the ground state wave function of the

system?

(a) 222112112

1xxxx (b) 12212211

2

1xxxx

(c) 222112112

1xxxx (d 12212211

2

1xxxx )

Ans: (d)

Solution: From the given information only possible spin configuration is symmetric in nature so

space part will anti symmetric

122122112

1xxxx

Q27. If L

is the orbital angular momentum and S is the spin angular momentum, then SL

.

does not commute with

(a) zS (b) 2L (c) 2S (d) 2SL

Ans: (d)

Q28. An electron in the ground state of the hydrogen atom has the wave function

0

30

1 a

r

ea

r

, where 0a is constant. The expectation value of the operator

22ˆ rzQ , where cosrz is (Hint:

1 10

1 !ar nn n

nne r dr

a a

)

(a) 2

20a

(b) 20a (c)

2

3 20a

(d) 202a

Ans: (d)

Solution: 22ˆ rzQ 2 2 20 0 03 2a a a

Q29. A particle of mass m is subjected to a potential

yxyxmyxV ,,

2

1, 222




The state with energy 4 is g fold degenerate. The value of g is ______

Ans: 4

Solution: This is two isotropic dimensional harmonic oscillator the energy eigen value for nth

state is )1( nEn with degeneracy )1( ngn so degeneracy for 4 is 4 .

Q30. A hydrogen atom is in the state

200 310 321

8 3 4

21 7 21 ,

where mln ,, in mnl denote the principal, orbital and magnetic quantum numbers,

respectively. If L

is the angular momentum operator, then the average value of 2L

is_______ 2

Ans: 2

Solution: If 2L will measure on state the measurement is 20 , 22 and 26 with probability

21

8,

3,

7 21

4so,

21

46

7

32 222 L = 22

Q31. 21and are two orthogonal states of a spin 2

1system. It is given that

,1

0

3

2

0

1

3

11

where

0

1and

1

0represent the spin-up and spin-down states,

respectively. When the system is in the state 2 its probability to be in the spin-up state

is_______

Ans: 3

2

Solution: If is ,1

0

3

2

0

1

3

11

then ,

1

0

3

1

0

1

3

22

so probability that 2 is in up state is 3

2

Q32. A particle is confined to a one dimensional potential box, with the potential

0, 0

, otherwise

x aV x




If particle is subjected to a perturbation within the box. xW . Where is small

constant, the first order correction to the ground state energy is

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

Ans: (c)

Solution: First order energy correction is xW . The average value of position in ground

state is 2

ax so answer is 2/a

Q33. A one dimensional harmonic oscillator is in the superposition of number state n given

by 32

32

2

1 .

The average energy of the oscillator in the given state is______ .

Ans: 3.25

Solution: Average energy will

1 5 3 74 2 4 2 3.25

1 34 4

Q34. If L and L are the angular momentum ladder operators then the expectation value of

LLLL in the state 1,1 ml of an atom is _____ 2

Ans: 2

Solution: 2222 ))1.((2)(2 mllLLLLLL z = 22




GATE- 2015

Q35. An operator for a spin 1

2particle is given by A B

, where

ˆ ˆ ,2

BB x y

denotes Pauli matrices and is a constant. The eigenvalues of A are

(a) 2

B (b) B (c) B,0 (d) B,0

Ans.: (b)

Solution: ˆ ˆ ˆ,2

BA B x y

ˆ ˆx x y y z z x x y yA B B B A B B

0 1 0ˆ1 0 02 2

iB BA

i

0 1ˆ1 02

iBA

i

10 0

12

iBA I

i

B

Q36. The Pauli matrices for three spin2

1 particles are 21 ,

and 3

, respectively. The

dimension of the Hilbert space required to define an operator 321ˆ

O is_______

Ans.: 8

Solution: 32 has dimension of 4 and 1. 32 has dimension of 2 4 8

Q37. Let L

and p

be the angular and linear momentum operators, respectively, for a a particle.

The commutator yx pL , gives

(a) zi p (b) 0 (c) xi p (d) zi p

Ans.: (d)

Solution: , ,x y z y yL p yp zp p , ,z y y yyp p zp p , y zy p p

, 0y yp p and , 0yz p ,x y zL p i p , yy p i




Q38. Consider a system of eight non-interacting, identical quantum particles of spin2

3 in a

one dimensional box of length L . The minimum excitation energy of the system, in units

of 2

22

2mL

is ________

Ans.: 5

Solution: spin 3

2 degeneracy 3

2 1 2 1 42

S

2 2 2 2 2 2

2 2 2

4 204 4

2 2 2groundEmL mL mL

2 2 2 2 2 2 2 2

2 2 2 24 3 4 1 9 25

2 2 2 2

stIexcitedE

mL mL mL mL

Now minimum excitation energy stI

excited groundE E E 2 2 2 2

2 225 20

2 2mL mL

2 2

25

2mL

Q39. A particle is confined in a box of length L as shown in the figure. If the

potential 0V is treated as a perturbation, including the first order

correction, the ground state energy is

(a) 02

22

2V

mLE

(b)

220

2

22 V

mLE

(c) 42

02

22 V

mLE

(d)

220

2

22 V

mLE

Ans.: (d)

Solution: 2

1 2 20 0

02

2sin 0 sin

LL

L

x xE V dx dx

L L L

2 21 0 00

0 0

2 2 21 cos sin

2 2

L L

V Vx x LE dx x

L L L L

1 00 2

VE

2 20

22 2

VE

mL

0V2/L




Q40. Let the Hamiltonian for two spin-½ particles of equal masses m , momenta 1p

and 2p

and positions 1r

and 2r

be 212

22

122

221 2

1

2

1

2

1 krrmp

mp

mH , where 1

and 2 denote the corresponding Pauli matrices, eV1.0 and eVk 2.0 . If the

ground state has net spin zero, then the energy (in eV ) is ___________

Ans.: 0.3

Solution: 2 2 2 2 21 21 2 1 2

1 1 1.

2 2 2H p p m r r k

m m

21

2 22 2 2 2

1 2 1 21 2 1 22 . 2 .

1 2 1 22 . 0 3 3 6 . 3I I I

Now energy 32 3

2E k 3 0.1 0.2 3 0.3 eV

Q41. Suppose a linear harmonic oscillator of frequency and mass m is in the state

1

20

2

1

ie at 0t where 0 and 1 are the ground and the first

excited states, respectively. The value of x in the units of m

at 0t is _____

Ans. : 0

Solution: 20 1

1

2

i

e

†

2x a a

m

†

2x a a

m

20

1

2

i

a e

and † 21 2

12

2

i

a e

2 20 0 1 1

1 1 1 1

2 2 2 2 2

i i

x e em

0 02

xm




GATE-2016

Q42. Which of the following operators is Hermitian?

(a)dx

d (b)

2

2

dx

d (c)

2

2

dx

di (d)

3

3

dx

d

Ans. : (b)

Q43. The scattering of particles by a potential can be analyzed by Born approximation. In

particular, if the scattered wave is replaced by an appropriate plane wave, the

corresponding Born approximation is known as the first Born approximation. Such an

approximation is valid for

(a) large incident energies and weak scattering potentials.

(b) large incident energies and strong scattering potentials.

(c) small incident energies and weak scattering potentials.

(d) small incident energies and strong scattering potentials.

Ans.: (a)

Q44. Consider an elastic scattering of particles in 0l states. If the corresponding phase shift

0 is 090 and the magnitude of the incident wave vector is equal to 12 fm then the

total scattering cross section in units of 2fm is _______.

Ans.: 2

Solution: 020

42 1 sin

ll

k

for 0l , it is given 00 90 and 12 fmk

4sin 90 2

2

Q45. A hydrogen atom is in its ground state. In the presence of a uniform electric field

0 ˆE E z

, the leading order change in its energy is proportional to 0

nE . The value of

the exponent n is _______.

Ans.: 2

Solution: First order energy correction is zero 1,0,0 0 1,0,0cos 0E r

So one need to find correction of second

2

, , 0 1,0,0 200 0

1 1

cosn l m

n m

E rE

E E

So value of 2n




Q46. If 1s

and 2s

are the spin operators of the two electrons of a He atom, the value of

21.ss

for the ground state is

(a) 2

2

3 (b) 2

4

3 (c) 0 (d) 2

4

1

Ans.: (b)

Solution: 1 2s s s

, 1 11 1

, , 0,12 2

s s s ,

2

111.

222

211

2

21

ssssssss

For

2 2 2

21 2

3 32 34 41,

2 4s s s

2 2 2

21 2

3 30 34 40,

2 4s s s

Q47. A two-dimensional square rigid box of side L contains six non-interacting electrons at

KT 0 . The mass of the electron is m . The ground state energy of the system of

electrons, in units of 2

22

2mL

is _________.

Ans.: 24

Solution: 2 2 2 2 2 2 2 2 2 2

2 2 2

1 1 2 1 242 4

2 2 2mL mL mL

Q48. yx , and z are the Pauli matrices. The expression xyyx 2 is equal to

(a) zi3 (b) zi (c) zi (d) zi3

Ans.: (c)

Solution: zyxxyyxyxxyyx i 2

Q49. If x and p are the x components of the position and the momentum operators of a

particle respectively, the commutator 22 , px is

(a) pxxpi (b) pxxpi 2 (c) pxxpi (d) pxxpi 2

Ans.: (d)

Solution: xpipxippxpxppx 22,, 2222 pxxpi 2




Q50. Let ml, be the simultaneous eigenstates of 2L and zL . Here L

is the angular

momentum operator with Cartesian components lLLL zyx ,,, is the angular momentum

quantum number and m is the azimuthal quantum number. The value of

1,0 ( ) 1, 1x yL iL is

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

Ans.: (c)

Solution: 1,0 ( 1, 1 1,0 1, 1 2 1,0 1,0 2x yL iL L

Q51. For the parity operator P , which of the following statements is NOT true?

(a) †P P (b) PP 2 (c) IP 2 (d) † 1P P

Ans.: (b)

Q52. The state of a system is given by 321 32 , where 1 2 3, and form

an orthonormal set. The probability of finding the system in the state 2 is ________.

(Give your answer upto two decimal places)

Ans. : 0.28

Solution: Probability that in state 2

2 2 2 2

2

1 2 3

4 4 2

0.281 4 9 14 7

Q53. A particle of mass m and energy E , moving in the positive x

direction, is incident on a step potential at 0x , as indicated in the

figure. The height of the potential is 0V , where EV 0 . At 0xx ,

where 00 x , the probability of finding the electron is e

1 times the

probability of finding it at 0x . If

202

EVm

, the value of 0x is

(a) 2

(b) 1

(c) 21

(d) 41

Ans.: (c)

Solution: 0 02 210

1 1

2x xe e e x

e

E0V

0x x0x




GATE- 2017

Q54. The Compton wavelength of a proton is…………….. fm. (up to two decimal places).

Ans. : 83 10

Solution: 27 34 19 8 11.67 10 , 6.626 10 , 1.602 10 , 3 10pm kg h Js e C c ms

Q55. A one dimensional simple harmonic oscillator with Hamiltonian 2

20

1

2 2

pH kx

m is

subjected to a small perturbation, 3 41H x x x . The first order correction to the

ground state energy is dependent on

(a) only (b) and (c) and (d) only

Ans. : (d)

Solution: 3 41H x x x , 1 3 4

gE x x x , 3 40, 0, 0x x x

Q56. For the Hamiltonian 0 .H a I b

where 0 ,a R b

is a real vector, I is the 2 2

identity matrix, and are the Pauli matrices, the ground state energy is

(a) b (b) 02a b (c) 0a b (d) 0a

Ans. : (c)

Solution: 0

0 00

1 0 0 1 0 1 0.

0 1 1 0 0 0 1

z x yx y z

x y z

a b b ibia I b a b b b

i b ib a b

0

00

. z x y

x y z

a b b ibH a I b

b ib a b

For eigen value 0

0

0z x y

x y z

a b b ib

b ib a b

2 20 0 0z z x ya b a b b b

1 0 1 0,a b a b

Q57. The degeneracy of the third energy level of a 3-dimensional isotropic quantum harmonic

oscillator is

(a) 6 (b) 12 (c) 8 (d) 10

Ans. : (a)




Solution: First energy level is 0n

Second energy level is 1n

Third energy level is 2n

Degeneracy of third level 1 2 3 4

62 2

n n

Q58. A free electron of energy 1eV is incident upon a one-dimensional finite potential step of

height 0.75eV . The probability of its reflection from the barrier is…………. (up to two

decimal places).

Ans. : 0.11

Solution:

2 2 20

0

1 0.25 1 0.50.11

1 0.51 0.25

E E VR

E E V

Q59. Consider a one-dimensional potential well of width 3nm . Using the uncertainty principle

2x p

, an estimate of the minimum depth of the well such that it has at least one

bound state for an electron is ( 31 349.31 10 , 6.626 10 ,em kg h Js 191.602 10e C )

(a) 1 eV (b) 1meV (c) 1eV (d) 1MeV

Ans. : (b)

Solution: 2

2

pE

m ,

2 2p p

x a

So, 2

28E

ma 2342

192 2 31 18

6.6 10.001 10 1

32 32 10 9.31 10 9 10

hJ meV

ma

Q60. The integral 22

0

xx e dx

is equal to……….. (up to two decimal places).

Ans. : 0.44

Solution: The given integral is 22

0

xx e dx

Let 2 then 22

dtx t xdx dt dx

t




Thus, the given integral can be written as

1/ 2

0 0

1

22t tdt

t e e t dtt

31

2

0

1

2te t dt

1 3 1 1 1

2 2 2 2 2 4

Hence the value of the integral up to two decimal places is 0.44 .

Q61. Which one of the following operators is Hermitian?

(a) 2 2

2x xp x x p

i

(b) 2 2

2x xp x x p

i

(c) xi p ae (d) xi p ae

Ans. : (a)

Solution:

† †2 22 2 2 2

†,2 2 2

x xx x x xp x x pp x x p p x x p

A i A i i

GATE-2018

Q62. The ground state energy of a particle of mass m in an infinite potential well is 0E . It

changes to 30 1 10E , when there is a small potential pump of height

2 2

0 250V

mL

and width /100a L , as shown in the figure. The value of is ________ (up to two

decimal places).

Ans. : 0.81

Solution: 1 2, ,2 2 2 2 100

L a L a La

2

1

2

21 0

2sin

xE V dx

L L

22

11

0 02 21 cos sin

2

V Vx L xdx x

L L L L

a

0V

L

V x

x




0

2 2sin sin

2 2 2

L a L aV La

L L L

0 sin sin100 2

V L L a a

L L L

0

1 10.0314 0.0314

100 2V

3 30 0

2010 10 10 10

25V E

3 3

0 010 0.81 10E E

Hence, 0.81

Q63. A two-state quantum system has energy eigenvalues corresponding to the normalized

states . At time 0t , the system is in quantum state 1

2 . The

probability that the system will be in the same state at / 6t h is _________ (up to

two decimal places).

Ans. : 0.25

Solution: 10

2

And 1

2

i t i t

t e e

At ,6

t

2 2

6 61

2

i h i h

h ht e e

3 3

1

2

i i

e e

Now, probability in same state

20t

P

2/3 /31

4i ie e

21

2cos4 3

21 1

24 2

0.25




GATE-2019

Q64. An electric field 0 ˆE E z

is applied to a Hydrogen atom in 2n excited state. Ignoring

spin the 2n state is fourfold degenerate, which in the ,l m basis are given by

0,0 , 1,1 , 1,0 and 1, 1 . If H is the interaction Hamiltonian corresponding to the

applied electric field, which of the following matrix elements is nonzero?

(a) 0,0 0,0H (b) 0,0 1,1H

(c) 0,0 1,0H (d) 0,0 1, 1H

Ans. : (c)

Q65. For a spin 1

2 particle, let and denote its spin up and spin down states

respectively. If 1

2a and 1

2b are composite

states of two such particles, which of the following statements is true for their total spin S ?

(a) 1S for a and b is not an eigenstate of the operator 2S

(b) a is not an eigenstate of the operator 2S and 0S for b

(c) 0S for a , and 1S for b

(d) 1S for a , and 0S for b

Ans. : (d)

Solution: 1S is triplet a , and 0S for singlet for b

Q66. The Hamiltonian for a quantum harmonic oscillator of mass m in three dimensions is

2

2 21

2 2

pH m r

m

where is the angular frequency. The expectation value of 2r in the first excited state of

the oscillator in units of m

(rounded off to one decimal place) is___________

Ans. : 2.5

Solution: 2 2 2 2r x y z




2 1 2 1 2 12 x y zn n n

m

For first excited state 1, 0, 0x y zn n n

Hence it is triply degenerate one can take

0, 1, 0x y zn n n or 0, 0, 1x y zn n n

putting any one combination, expectation value of 2 52.5

2r

m m

Q67. Let 2

1

0

, 2

0

1

represent two possible states of a two-level quantum system.

The state obtained by the incoherent superposition of 1 and 2 is given by a density

matrix that is defined as 1 1 1 2 2 2c c . If 1 0.4c and 2 0.6c , the matrix

element 22 (rounded off to one decimal place) is __________

Ans. : 0.6

Solution: 2,2 2 2 1 2 1 1 2 2 2 2 2 2c c

2 0.6c

Q68. The wave function x of a particle is as shown below

Here K is a constant, and a d . The position uncertainty x of the particle is

(a)2 23

12

a d (b)

2 23

12

a d (c)

2

6

d (d)

2

24

d

Ans. : (b)

x

x

dd

K

/ 2a/ 2a




Solution:

,2 2 2 2

0,2 2 2 2

,2 2 2 2

0, 02 2

a d a dk x

a d a dx

xa d a d

k x

a d

1

2 2 2 2

2 2

2 2 2 2

1

a d a d

a d a d

k dx k dx

2 2 12 2 2 2 2 2 2 2

a d a d a d a dk k

2 11

2 2 2 2 2

d d d dk k

d

Hence wavefunction is symmetric about 0x , so 0x

2 2 2 2

2 2 2 2 2

2 2 2 2

a d a d

a d a d

x k x dx k x dx

2 2 2 2

2 2 2 2

23 3

3

a d a d

a d a d

kx x

2

3 3 3 3

3 8

ka d a d a d a d

2

3 3 2 2 3 3 2 2 3 3 2 23 3 3 3 3 324

ka d a d ad a d a d ad a d a d ad

3 3 3a d ad a d

2 22 2 2

2 3 24 3 3

4 1224 24 2 12

d d ak a dx a a d

d




2 2

22 3

12

a dx x x

Q69. Consider the motion of a particle along the x - axis in a potential V x F x . Its ground

state energy 0E is estimated using the uncertainty principle. Then 0E is proportional to

(a) 1/ 3F (b) 1/ 2F (c) 2/ 5F (d) 2/ 3F

Ans. : (d)

Solution: 2

2

pE F x

m

2

2

pE Fx

m for 0x

2

02

pE Fx

m from uncertainty theory

.x p px

2 2

22 2

pE F x E F x

m m x

For minimum energy,

1/ 32 2

30

dEF x

d x mFm x

1/ 32 / 32 2

2 / 322

mFF E F

m mF

Q70. The Hamiltonian operator for a two-level quantum system is 1

2

0

0

EH

E

. If the state

of the system at 0t is given by 11

012

then 2

0 t at a later time t

is

(a) 1 2 /11

2E E te (b) 1 2 /1

12

E E te

(c) 1 2

11 cos /

2E E t (d) 1 2

11 cos /

2E E t

Ans. : (c)

Solution: 11

012

1

2

exp1

2 exp

iE t

tiE t




2

21 2

1 2

1 10 exp exp 1 cos /

4 2

iE t iE tt E E t

Q71. Consider a potential barrier V x of the form:

where 0V is a constant. For particles of energy 0E V incident on this barrier from the

left which of the following schematic diagrams best represents the probability density

2x as a function of x ?

(a) (b)

(c) (d)

Ans. : (a)

Q72. The Hamiltonian of a system is 1

1H

with 1 . The fourth order contribution

to the ground state energy of H is 4 . The value of (rounded off to three decimal

places) is_________.

Ans. : 0.125

Solution: 1

1H

the eigen value of the hamiltonion is 2 21 , 1g fE E

0x x a x b x

V x V x

0V

2x

0x x a x b x

2x

0x x a x b x

2x

0x x a x b x

2x

0x x a x b x




The ground state is 21gE

Taylor expansion of 2 4 2 4

21 1 ..... 1 .....2 8 2 8

1

0.1258

Q73. Electrons with spin in the z - direction z are passed through a Stern-Gerlach (SG) set

up with the magnetic field at 060 from z . The fraction of electrons that will emerge

with their spin parallel to the magnetic field in the SG set up (rounded off to two decimal

places) is___________

0 1 0 1 0

, ,1 0 0 0 1x y z

i

i

Ans. : 0.25

Solution: 0

0

1/ 2cos 60

3 / 2sin 60

state related to up state is 11 1

,02 2

The fraction of electrons that will emerge with their spin parallel to the magnetic field

2 10.25

4




THERMODYNAMICS AND STATISTICAL PHYSICS SOLUTIONS

GATE 2010

Q1. A system of N non-interacting classical point particles is constrained to move on the two-

dimensional surface of a sphere. The internal energy of the system is

(a) TNkB2

3 (b) TNkB2

1 (c) TNkB (d) TNkB2

5

Ans: (c)

Solution: There are 2 N degree of freedom.

The internal energy of the system is TNkTNkTNk

BBB

22

Q2. Which of the following atoms cannot exhibit Bose-Einstein condensation, even in

principle?

(a) 1H1 (b) 4H2 (c) 23Na11 (d) 30K19

Ans: (d)

Solution: For Bose-Einstein condensation:

Number of electron + number of proton + number of neutron = Even

For 30 K19

Number of proton = 19, Number of electron = 19, Number of neutron = 11.

19 + 19 + 11 = 49 this is odd. So it will not exhibit Bose-Einstein condensation.

Q3. For a two-dimensional free electron gas, the electronic density n, and the Fermi energy EF,

are related by

(a) 3/ 2

2 3

2

3FmE

n

(b) 2FmE

n

(c) 22

FmEn (d)

3/ 22 FmE

n

Ans: (c)

Solution: 0

( ) ( )FE

n g E f E dE , 2

2( )

mg E dE dE

h

At T=0 , 1,

0,F

F

if E Ef E

if E E

2

2

h

mEn F

2 22FmE




Q4. Which among the following sets of Maxwell relations is correct? (U-internal energy,

H-enthalpy, A-Helmholtz free energy and G-Gibbs free energy)

(a) VS S

UP

V

UT

and (b) PS S

HT

P

HV

and

(c) ST P

GV

V

GP

and (d) VT P

AS

S

AP

and

Ans: (b)

Solution: VdPTdSdH VP

HT

S

H

SP

,

Q5. Partition function for a gas of photons is given as, 32

3 3ln

45BV k T

ZC

. The specific heat

of the photon gas varies with temperature as

(a) (b)

(c) (d)

Ans: (a)

Solution: 2B

ln zU K T ,

T

V

v

UC

T

3

VC T .

Q6. From Q. no. 5, the pressure of the photon gas is

(a)

33

32

15 C

TkB

(b)

33

42

8 C

TkB

(c)

33

42

45 C

TkB

(d)

33

2/32

45 C

TkB

Ans: (c)

Solution: Since, F

PV

ln

T

zP KT

V

42

03 345

k T

C

VC

T

VC

T

VC

T

VC

T




GATE 2011

Q7. A Carnot cycle operates on a working substance between two reservoir at temperatures T1

and T2 with T1 > T2. During each cycle, an amount of heat Q1 is extracted from the

reservoir at T1 and an amount Q2 is delivered in the reservoir at T2. Which of the

following statements is INCORRECT?

(a) Work done in one cycle is Q1 – Q2

(b) 2

2

1

1

T

Q

T

Q

(c) Entropy of the hotter reservoir decreases

(d) Entropy of the universe (consisting of the working substance and the two reservoirs)

increases

Ans: (c)

Solution: Entropy of hotter reservoirs decreases.

Q8. In a first order phase transition, at the transition temperature, specific heat of the system

(a) diverges and its entropy remains the same

(b) diverges and its entropy has finite discontinuity

(c) remains unchanged and its entropy has finite discontinuity

(d) has finite discontinuity and its entropy diverges

Ans: (b)

Q9. A system of N non-interacting and distinguishable particle of spin 1 is in thermodynamic

equilibrium. The entropy of the system is

(a) 2kB ln N (b) 3kB ln N (c) NkB ln 2 (d) NkB ln 3

Ans: (d)

Solution: Bi

S k ln , =3 is number of microstate. zS 1; S 1, 0, 1

The entropy of the system is NkB ln 3.




Q10. A system has two energy levels with energies ε and 2ε. The lower level is 4-fold

degenerate while the upper level is doubly degenerate. If there are N non-interacting

classical particles in the system, which is in thermodynamic equilibrium at a temperature

T, the fraction of particles in the upper level is

(a) TkBe /1

1

(b) TkBe /21

1

(c) TkTk BB ee /2/ 42

1

(d) TkTk BB ee /2/ 42

1

Ans: (b)

Solution: Partition function kTkT eeZ // 24 2 /

/ 2 /

22

4 2

kT

kT kT

eP

e e

kTe /21

1

GATE 2012

Q11. The isothermal compressibility, of an ideal gas at temperatures 0T and 0V is given by

(a) 00

1

TP

V

V

(b) 00

1

TP

V

V

(c) 0

0TV

PV

(d) 0

0TV

PV

Ans: (c)

Solution: Isothermal compressibilityT

PV

V

Q12. For an ideal Fermi gas in three dimensions, the electron velocity VF at the Fermi surface

is related to electron concentration n as,

(a) 3/2nVF (b) nVF (c) 2/1nVF (d) 3/1nVF

Ans: (d)

Solution: 2

2

1FF mVE

3/2nEF 3/22 nVF 3/1nVF .

Q13. A classical gas of molecules, each of mass m, is in thermal equilibrium at the absolute

temperature T. The velocity components of the molecules along the Cartesian axes are

yx vv , and zv . The mean value of 2yx vv is

(a) m

TkB (b) m

TkB

2

3 (c)

m

TkB

2

1 (d)

m

TkB2

Ans: (d)




Solution: 2yx VV 2 2 2x y x yv v v v 2 2 2x y x yv v v v B2k T

m

0x yv v and 22yx VV

2 Bk T

m .

Q14. The total energy, E of an ideal non-relativistic Fermi gas in three dimensions is given by

3/2

3/5

V

NE , where N is the number of particles and V is the volume of the gas. Identify the

CORRECT equation of state (P being the pressure),

(a) EPV3

1 (b) EPV

3

2 (c) EPV (d) EPV

3

5

Ans: (b)

Solution:

5

3

N

E 2 NP

V 3 V

53

23

2 N 2PV E

3 3V .

Q15. Consider a system whose three energy levels are given by 0, ε and 2ε. The energy level ε

is two-fold degenerate and the other two are non-degenerate. The partition function of the

system withTkB

1 is given by

(a) e21 (b) 22 ee (c) 2)1( e (d) 21 ee

Ans: (c)

Solution: 2,,0 321 EEE ; 1,2,1 321 ggg where 21, gg and 3g are degeneracy.

The partition function 321321

EEE egegegZ 221 ee 21 e

GATE 2013

Q16. If Planck’s constant were zero, then the total energy contained in a box filled with

radiation of all frequencies at temperature T would be ( k is the Boltzmann constant and

T is nonzero)

(a) zero (b) Infinite (c) kT2

3 (d) kT

Ans: (d)

Solution: If Planck’s constant were zero, then the system behaved as a classical system and the

energy is kT .




Q17. Across a first order phase transition, the free energy is

(a) proportional to the temperature

(b) a discontinuous function of the temperature

(c) a continuous function of the temperature but its first derivative is discontinuous

(d) such that the first derivative with respect to temperature is continuous

Ans: (c)

Q18. Two gases separated by an impermeable but movable partition are allowed to freely

exchange energy. At equilibrium, the two sides will have the same

(a) pressure and temperature (b) volume and temperature

(c) pressure and volume (d) volume and energy

Ans: (a)

Q19. The entropy function of a system is given by EEaEES 0 where a and 0E are

positive constants. The temperature of the system is

(a) negative for some energies (b) increases monotonically with energy

(c) decreases monotonically with energy (d) Zero

Ans: (a)

Solution: From first and second law of thermodynamics

TdS dU PdV 1dS dU PdV

T

1

V

S

E T

E U

EEaEES 0 0 0 2V

Sa E E aE a E E

E

0

1

2T

a E E

.

Q20. Consider a linear collection of N independent spin ½ particles, each at a fixed location.

The entropy of this system is ( k is the Boltzmann constant)

(a) zero (b) Nk (c) Nk2

1 (d) 2lnNk

Ans: (d)

Solution: There are two microstates possible for spin 1

2 particle, so entropy is given by 2lnNk .




Q21. Consider a gas of atoms obeying Maxwell-Boltzmann statistics. The average value of

pae over all the moments p

of each of the particles (where a

is a constant vector and a

is the magnitude, m is the mass of each atom, T is temperature and k is Boltzmann’s

constant) is,

(a) one (b) zero (c) mkTa

e2

2

1

(d) mkTa

e2

2

3

Ans: (c)

Solution: . ., ,p a p ax y z x y ze f p p p e dp dp dp

where , ,x y zf p p p is Maxwell probability

distribution at temperature T. 22 2

. 2 2 2yx z

y yx x z z

pp pp ap a p ap a mkT mkT mkT

x x y y z ze A e e dp A e e dp A e e dp

22 2 2 2 2( )( ) ( ) ( ). 2 2 22

y yx y z x x z zp mkTaa a a mkT p mkTa p mkTa

p a mkT mkT mkTx x y y z ze e A e dp A e dp A e dp

2 2 2( ). 2 .1.1.1

x y za a a mkTp ae e

= mkTa

e2

2

1

Common Data for Questions 22 and 23: There are four energy levels ,E ,2E E3 and

E4 (where 0E ). The canonical partition function of two particles is, if these particles

are

Q22. Two identical fermions

(a) EEEE eeee 8642 (b) EEEEE eeeee 76543

(c) 2432 EEEE eeee (d) EEEE eeee 8642

Ans: (b)

Solution: The possible value of Energy for two Fermions

EEEEEEEEEE 7,6,5,4,3 54321

The partition function is EEEEE eeeeeZ 76543 2 , then the answer may

be option (b).

Q23. Two distinguishable particles

(a) EEEE eeee 8642 (b) EEEEE eeeee 76543

(c) 2432 EEEE eeee (d) EEEE eeee 8642




Ans: (c)

Solution: When two particles are distinguishable then minimum value of Energy is E2 and

maximum value is E8 .

So from checking all four options 2432 EEEE eeeeZ

GATE 2014

Q24. For a gas under isothermal condition its pressure P varies with volume V as 3/5VP .

The bulk modules B is proportional to

(a) 2/1V (b) 3/2V (c) 5/3V (d) 3/5V

Ans: (d)

Solution: 3/5 KVP , dV

dPVB 3/5VB

Q25. At a given temperature T , the average energy per particle of a non-interacting gas of

two-dimensional classical harmonic oscillators is _________ TkB

( Bk is the Boltzmann constant)

Ans: 2

Q26. Which one of the following is a fermion?

(a) particle (b) 24 Be nucleus (c) Hydrogen atom (d) deuteron

Ans (d)

Solution: If total number of particles i.e., electron, proton and neutron is odd, then it is a

fermions: 3P N E

Q27. For a free electron gas in two dimensions the variations of the density of states. EN as a

function of energy E , is best represented by

(a) (b)

(c) (d)

EN

E

EN

E

EN

E

EN

E




Ans. : (c)

Solution: 0EEN

Q28. For a system of two bosons each of which can occupy any of the two energy levels 0 and

. The mean energy of the system at temperature T with Tk

1 is given by

(a)

2

2

21

2

ee

ee (b)

22

1

ee

e (c)

2

2

2

2

ee

ee (d)

2

2

2

2

ee

ee

Ans. : None of the options are matched.

Solution: If both particle will in ground state the energy will 0 , which is non-degenerate. If one

particle is in ground state and other is in first excited state then energy is and non

degenerate. If both particles will in first excited state, then energy will 2 , which is

non-degenerate.

Then partition function is 1 exp exp 2Z

Average value of energy exp 2 exp 2

1 exp exp 2

No one answer is correct, but answer may be (a).

Q29. Consider a system of 3 fermions which can occupy any of the 4 available energy states

with equal probability. The entropy of the system is

(a) 2lnBk (b) 2ln2 Bk (c) 4ln2 Bk (d) 4ln3 Bk

Ans: (b)

Solution: Number of ways that 3 fermions will adjust in 4 available energy is 434 C so

entropy is 4lnBk = 2ln2 Bk

GATE 2015

Q30. In Boss-Einstein condensation, the particles

(a) have strong interparticle attraction

(b) condense in real space

(c) have overlapping wavefunctions

(d) have large and positive chemical potential

Ans.: (c)

Solution: In Bose- Einstein condensates, the particles have overlapping wave function.




Q31. For a black body radiation in a cavity, photons are created and annihilated freely as a

result of emission and absorption by the walls of the cavity. This is because

(a) the chemical potential of the photons is zero

(b) photons obey Pauli exclusion principle

(c) photons are spin-1 particles

(d) the entropy of the photons is very large

Ans.: (a)

Solution: The chemical potential of photon is zero

Q32. Consider a system of N non-interacting spin2

1 particles, each having a magnetic

moment , is in a magnetic field zBB ˆ

. If E is the total energy of the system, then

number of accessible microstates is given by

(a)

!2

1!

2

1

!

B

EN

B

EN

N

(b)

!

!

B

EN

B

EN

(c) !2

1!

2

1

B

EN

B

EN

(d)

!

!

B

EN

N

Ans.: (a)

Solution: Number of microstate is 1

NnC , where 1n is number of particle in

1

2 state and

2 1n N n is number of state in 1

2 state.

where 1 2

1 1,

2 2

E En N n N

B B

So, number of microstate 1 12 2

N

E EN N

B B




Q33. The average energy U of a one dimensional quantum oscillator of frequency and in

contact with a heat bath at temperature T is given by

(a)

2

1coth

2

1U (b)

2

1sinh

2

1U

(c)

2

1tanh

2

1U (d)

2

1cosh

2

1U

Ans.: (a)

Solution: 1

2

0

inE

i

Z e e

where 1

2E n

1

2sinh2

Z

1ln ln

2sinh2

U Z

coth2 2

Q34. The entropy of a gas containing N particles enclosed in a volume V is given by

2/5

2/3

lnN

aVENkS B , where E is the total energy, a is a constant and Bk is the

Boltzmann constant. The chemical potential of the system at a temperature T is given

by

(a)

2

5ln

2/5

2/3

N

aVETkB (b)

2

3ln

2/5

2/3

N

aVETkB

(c) 3/ 2

3/ 2

5ln

2B

aVEk T

N

(d)

2

3ln

2/3

2/3

N

aVETkB

Ans.: (a)

Solution: 3/ 2

5/ 2lnB

P

G aVES Nk

T N

3/ 2

5/ 2lnB

aVES Nk

N

3/ 2

5/ 2ln lnB

aVEG Nk T A

N

3/ 2 5/ 23/ 2

5/ 2 3/ 2 7 / 2

5 / 2ln .B B

G aVE Nk T Nk T aVE

N N aVE N

3/ 2

5

2

5ln

2B

aVEk T

N




GATE-2016

Q35. The total power emitted by a spherical black body of radius R at a temperature T is 1P .

Let 2P be the total power emitted by another spherical black body of radius 2

R kept at

temperature T2 . The ratio, 2

1

P

P is _______. (Give your answer upto two decimal places)

Ans.: 0.25

Solution:

2 4 2 44 1 1 1

2 4 242 2 2

4 10.25

16 42

2

P R T R TP AT

P R T RT

Q36. The entropy S of a system of N spins, which may align either in the upward or in the

downward direction, is given by ln 1 (1 )BS k N p p p In p . Here Bk is the

Boltzmann constant. The probability of alignment in the upward direction is p. The value

of p, at which the entropy is maximum, is _______. (Give your answer upto one decimal

place)

Ans.: 0.5

Solution: ln 1 (1 )BS k N p p p In p

For maximum entropy, 0dS

dp 1 1

ln ln 1 1 1 01

p p p pp p

ln 1 ln 1 1 0 ln 0 1 0.51

pp p p p p

p

Q37. For a system at constant temperature and volume, which of the following statements is

correct at equilibrium?

(a) The Helmholtz free energy attains a local minimum.

(b) The Helmholtz free energy attains a local maximum.

(c) The Gibbs free energy attains a local minimum.

(d) The Gibbs free energy attains a local maximum.

Ans.: (a)

Solution: dF SdT PdV




Q38. N atoms of an ideal gas are enclosed in a container of volume V . The volume of the

container is changed to V4 , while keeping the total energy constant. The change in the

entropy of the gas, in units of 2lnBNk , is _______, where Bk is the Boltzmann constant.

Ans.: 2

Solution: 1 21

ln1, ln4B BS Nk S Nk 2 1 ln 4 2 ln 2B BS S S Nk Nk

Q39. Consider a system having three energy levels with energies 2,0 and 3 ,with

respective degeneracies of 2,2 and 3 . Four bosons of spin zero have to be

accommodated in these levels such that the total energy of the system is 10 . The

number of ways in which it can be done is ______.

Ans.: 18

Solution: The system have energy 10 , if out of four boson two boson are in energy level

2 and two boson are in energy level 3 and

1

1i i

i i i

n gW

n g

, 1 12, 2n g and 2 22, 3n g

2 2 1 2 3 13 6 18

2 2 1 2 3 1W

Q40. A two-level system has energies zero and E . The level with zero energy is non-

degenerate, while the level with energy E is triply degenerate. The mean energy of a

classical particle in this system at a temperature T is

(a) Tk

E

Tk

E

B

B

e

Ee

31

(b) Tk

E

Tk

E

B

B

e

Ee

1

(c)Tk

E

Tk

E

B

B

e

Ee

1

3 (d)

Tk

E

Tk

E

B

B

e

Ee

31

3

Ans.: (d)

Solution:

0

0

0 3 3

3 1 3

i

B

i

B

EEEkT

k Ti i kT kTi

E E E

kT kT k TkTi

i

g E ee E e Ee

E

e eg e e




GATE 2017

Q41. Consider a triatomic molecule of the shape shown in the figure in three

dimensions. The heat capacity of this molecule at high temperature

(temperature much higher than the vibrational and rotational energy scales

of the molecule but lower than its bond dissociation energies) is:

(a) 3

2 Bk (b) 3 Bk (c) 9

2 Bk (d) 6 Bk

Ans. : (d)

Solution: If given molecules are at lower temperature i.e. atoms are attached to rigid rod then

degree of freedom is 6 , so internal energy is 6

2Bk T

, but at high temperature, vibration

mode will active, so there are three extra vibration mode will active, so total energy

3 3 6B B BU k T k T k T

6V BV

UC k

T

Q42. A reversible Carnot engine is operated between temperatures 1T and 2 2 1T T T with a

photon gas as the working substance. The efficiency of the engine is

(a) 1

2

31

4

T

T (b) 1

2

1T

T (c)

3/ 4

1

2

1T

T

(d) 4/3

1

2

1T

T

Ans. : (b)

Solution: Efficiency of Carnot engine does not depends on nature of working substance rather

depends on temperature of source and sink

1

2

1T

T

Q43. Water freezes at 00 C at atmospheric pressure 51.01 10 Pa . The densities of water and

ice at this temperature and pressure are 31000 /kg m and 3934 /kg m respectively. The

latent heat of fusion is 53.34 10 /J kg . The pressure required for increasing the melting

temperature of ice by 010 C is…………… GPa . (up to two decimal places)

Ans. : 20.01 10




Solution:

2 2

1 1

22 1

2 1 2 1 2 1 1

lnp T

P TV

TdP L L dT LdP P P

dT T v v v v T v v T

5 22

2 12 1 1

ln 1 10 0.01 10TL

P P Pa GPav v T

Q44. Consider N non- interacting, distinguishable particles in a two-level system at

temperature T . The energies of the levels are 0 and , where 0 . In the high

temperature limit Bk T , what is the population of particles in the level with energy

?

(a) 2

N (b) N (c)

4

N (d)

3

4

N

Ans. : (a)

Solution: exp

1 exp

kTP

kT

, population of particle in the level with energy is

exp

1 exp

kTNP N

kT

, for Bk T ,

exp 1

1 1 21 exp

NkTNP N N

kT

Q45. The energy density and pressure of a photon gas are given by 4u aT and 3

uP . Where

T is the temperature and a is the radiation constant. The entropy per unit volume is given

by 3aT . The value of is…………… (up to two decimal places)

Ans. : 1.33

Solution: T T

S UTdS dU PdV T P

V V

4 431 4

1.333 3T T

S U P aT aTaT

V T V T T T




Q46. Consider two particles and two non-degenerate quantum levels 1 and 2. Level 1 always

contains a particle. Hence, what is the probability that level 2 also contains a particle for

each of the two cases:

(i) when the two particles are distinguishable and (ii) when the two particles are bosons?

(a) (i) 1

2 and (ii)

1

3 (b) (i)

1

2 and (ii)

1

2

(c) (i) 2

3 and (ii)

1

2 (d) (i) 1 and (ii) 0

Ans. : (c)

Solution: (I): For distinguishable particle: , 22

3P

(II): For indistinguishable particle (Bosons): , 12

2P

GATE-2018

Q47. A microcanonical ensemble consists of 12 atoms with each taking either energy 0 state,

or energy state. Both states are non-degenerate. If the total energy of this ensemble is

4 , its entropy will be _________ Bk (up to one decimal place), where Bk is the

Boltzmann constant.

Ans. : 6.204

Solution: The number of ways having total energy 4 , out of 12 atom is

124

12 12 11 10 9495

4 8 4 3 2C

Hence, entropy, ln ln 495B BS k w k 6.204Bk 6.204 Bk

Q48. An air-conditioner maintains the room temperature at 027 C while the outside temperature

is 047 C . The heat conducted through the walls of the room from outside to inside due to

temperature difference is 7000 W . The minimum work done by the compressor of the

air-conditioner per unit time is__________ W .

B A

A B AB

A

A AA




Ans. : 466.67

Solution: 2 1Q W Q

Coefficient of performance of refrigerator AC 2Q

W

Also, coefficient of performance of refrigerator, 2

1 2

T

T T

300 7000

47 27 W

7000 20/

300W J s

1400466.67

3W

Q49. Two solid spheres A and B have same emissivity. The radius of A is four times the

radius of B and temperature of A is twice the temperature of B . The ratio of the rate of

heat radiated from A to that from B is __________.

Ans. : 256

Solution:

2 4

2 4

Rate of heat radiation from solid sphere 4

Rate of heat radiation from solid sphere 4A A

B B

A R T

B R T

4A BR R and 2A BT T

2 42 4

2 4 2 4

4 2416 16 256

4B BA A

B B B B

R TR T

R T R T

Q50. The partition function of an ensemble at a temperature T is

2coshN

B

Zk T

where Bk is the Boltzmann constant. The heat capacity of this ensemble at B

Tk

is

BX Nk , where the value of X is __________ (up to two decimal places).

Ans. : 0.42

Solution: The partition function, 2cosh

N

B

zk T

The average energy, 2 ln

B

zE k T

T

1 47 273 323T k

1Q

W

2 7000 /Q J S

2 27 273 300T k 2Q Heat coming in room




222sinh

2cosh

BB B

B

Nk Tk T k T

k T

tanhB

Nk T

22

sec .B B

d EC N h

dT k T k T

At Tk

,

2

2

2 2sec 1

/

NC h

k k

2sec 1 0.42 BNk h Nk

GATE-2019

Q51. Consider a one-dimensional gas of N non-interacting particles of mass m with the

Hamiltonian for a single particle given by

2

2 212

2 2

pH m x x

m

The high temperature specific heat in units of BR Nk ( Bk is the Boltzmann constant) is

(a) 1 (b) 1.5 (c) 2 (d) 2.5

Ans. : (c)

Solution: 2

2 2 21 12

2 2 2

pH m x m x

m 02 2

NkT NkTU

H NkT

VH

C NkTT

Q52. A large number N of ideal bosons, each of mass m , are trapped in a three-dimensional

potential 2 2

2

m rV r

. The bosonic system is kept at temperature T which is much

lower than the Bose-Einstein condensation temperature CT . The chemical potential

satisfies

(a) 3

2 (b)

32

2

(c) 3 2 (d) 3




Ans. : (a)

Q53. At temperature T Kelvin K , the value of the Fermi function at an energy 0.5eV

above the Fermi energy is 0.01 . Then T , to the nearest integer, is __________

( 58.62 10 /Bk eV K )

Ans.: 1262

Solution:

/

/

1 11

1F B

F B

E E k T

E E k TF E e

F Ee

/ 1 1lnF BE E k T F

B

E EF Fe

F k T F

1ln

F

B

E ET

Fk

F

5

0.5 0.50.99 8.62 ln 99

8.62 10 ln0.01

T

50.5 10

1262.38.62 4.595

K

Q54. In a thermally insulated container, 0.01 kg of ice at 273 K is mixed with 0.1 kg of

water at 300 K . Neglecting the specific heat of the container, the change in the entropy

of the system in /J K on attaining thermal equilibrium (rounded off to two decimal

places) is____________

Ans. : 1.03

Solution: 290.29eqT K (Heat gain Heat lost)

273ice icem L m C T 300m C T

290.29T K

ice waters s s

lnice iiceice

ice ice

m L Ts m C

T T 14.85 /J K

290.29ln 13.82 /

300waterS m C J K

1.03 /S J K




Q55. Consider two system A and B each having two distinguishable particles. In both the

systems, each particle can exist in states with energies 0,1,2 and 3 units with equal

probability. The total energy of the combined system is 5 units. Assuming that the

system A has energy 3 units and the system B has energy 2 units, the entropy of the

system is lnBk . The value of is__________

Ans. : 12

Solution:

4 3 12

lnS ln12Bk

12 .

AB

0123

BA

B A

2BE 3AE

A

AA

AB

BB

B

4A 3B




ELECTRONICS SOLUTIONS

GATE-2010

Q1. The voltage resolution of a 12-bit digital to analog converter (DAC), whose output varies

from V10 to V10 is, approximately

(a) mV1 (b) mV5 (c) mV20 (d) mV100

Ans: (b)

Solution: Voltage resolution= mVV

8.412

2012

Q2. The figure shows a constant current source charging a capacitor that is initially uncharged.

If the switch is closed at t = 0, which of the following plots depicts correctly the output

voltage of the circuit as a function of time?

(a) (b)

(c) (d)

Ans: (d)

Solution: dt

CdVI 0

0 tC

IV 0

0

outV

t

outV

t

outV

t

outV

t

outV




Q3. In one of the following circuits, negative feedback does not operate for a negative input.

Which one is it? The opamps are running from ± 15 V supplies.

(a) (b)

(c) (d)

Ans: (c)

Q4. For any set of inputs, A and B, the following circuits give the same output, Q, except one.

Which one is it?

(a) (b)

(c) (d)

Ans. : (d)

Q

Q

QQ




GATE-2011

Q5. Which of the following statements is CORRECT for a common emitter amplifier circuit?

(a) The output is taken from the emitter

(b) There is 180o phase shift between input and output voltages

(c) There is no phase shift between input and output voltages

(d) Both p-n junctions are forward biased

Ans: (b)

Q6. For an intrinsic semiconductor, me* and mh

* are respectively the effective masses of

electrons and holes near the corresponding band edges. At a finite temperature the

position of the Fermi level

(a) depends on me* but not on mh

* (b) depends on mh* but not on me

*

(c) depends on both me* and mh

* (d) depends neither on me* nor on mh

*

Ans: (c)

Q7. In the following circuit, the voltage across and the current through the 2 kΩ resistance are

(a) 20 V, 10 mA (b) 20 V, 5 mA (c) 10 V, 10 mA (d) 10 V, 5 mA

Ans: (d)

Q8. In the following circuit, Tr1 and Tr2 are identical transistors having VBE = 0.7 V. The

current passing through the transistor Tr2 is

(a) 57 mA (b) 50 mA (c) 48 mA (d) 43 mA

500 k1

k2

V10V20

V30

100Tr2

Tr1

V 5




Ans: (d)

Solution: Current through mAI 43100

7.05,100

mAIIII CBC 432 .

Q9. Consider the following circuit

Which of the following correctly represents the output Vout corresponding to the input Vin?

(a) (b)

(c) (d)

Ans: (a)

Solution: .21041

1,210

41

1VVVV ltut

k1 k4

V10

V10inV outV

time

V5

V2

V2

V5

inV

outVtime

10V-

10V

time

V5

V2

V2

V5

inV

outVtime

10V-

10V

time

V5

V2

V2

V5

inV

outVtime

10V-

10V

time

V5

V2

V2

V5

inV

outVtime

10V-

10V




Q10. The following Boolean expression

Y A B C D A B C D A B C D A B C D A B C D A B C D can

be simplified to

(a) DACBA (b) DACBA

(c) DACBA (d) DACBA

Ans: (c)

GATE-2012

Q11. If the peak output voltage of a full wave rectifier is 10 V, its d.c. voltage is

(a) 10.0 V (b) 7.07 V (c) 6.36 V (d) 3.18 V

Ans: (c)

Solution: VV

V mdc 36.6

11

70

22

1014

7/22

1022

Q12. A Ge semiconductor is doped with acceptor impurity concentration of 1015 atoms/cm3.

For the given hole mobility of 1800 cm2/V-s, the resistivity of the material is

(a) 0.288 Ω cm (b) 0.694 Ω cm (c) 3.472 Ω cm (d) 6.944 Ω cm

Ans: (c)

Solution: cmueN hA

47.31800106.110

1111915

1 1

11

1 1

CBA

DC DC CD DCDA

BA

BA

AB

BA




Q13. Identify the CORRECT energy band diagram for silcon doped with Arsenic. Here CB,

VB, ED and EF are conduction band, valence band, impurity level and Fermi level,

respectively.

(a) (b)

(c) (d)

Ans: (b)

Solution: N-type material ( Si doped with SA ).

Q14. In the following circuit, for the output voltage to be 2/210 VVV the ratio R1/R2 is

(a) 1/2

(b) 1

(c) 2

(d) 3

Ans: (d)

Solution: When 1012 ,0 VvV

when 221

2021 1,0 V

RR

R

R

RvV

Since 32

12

2 2

1

21

2210

R

R

RR

RVVV

B C

DE

FE

B V

B C

DEFE

B V

B C

DEFE

B V

B C

DE

FE

B V

CCV

oV

CCV-

R

R1V

2V1R

2R




Q15. Consider the following OP-AMP circuit.

Which one of the following correctly represents the output Vout

corresponding to the input Vin?

(a) (b)

(c) (d)

Ans: (a)

Solution: Voltage at inverting input .1541

12 VV

When CCin VvVv 0,1 and when CCin VvVv 0,1

5V

V1V0

inV

10VoutV

10V

t

t

5V

V0

inV

10VoutV

10V

t

t

5V

V1V0

inV

10VoutV

10V

t

t

5V

V0

inV

10VoutV

10V

t

t

V 5

inV

10V

outV

10V-1k

4k




Q16. Consider the following circuit in which the current gain βdc of the transistor is 100.

Which one of the following correctly represents the load line (collector current IC with

respect to collector-emitter voltage VCE) and Q-point of this circuit?

(a) (b)

(c) (d)

Ans: (a)

Solution: .100

3.14

10010100

7.0153

mARR

VVI

EB

BECCB

mAmAII BC 133.14 , .2101310090015 3 VRRIVV ECCCCCE

.151000

15

R

V

C

CC, mA

RI

ESatC

900

V 15

100

k 100

CEV

mA) 13 V,(2point-QmA 15

I C →

V15

mA) 10 V,(2point-QmA 13I C

→

CEV V15

mA) 7.5 V,5.7(point-Q

mA 15

I C →

CEV V15

mA) 6.5 V,5.7(point-Q

mA 13

I C →

CEV V15




Q17. In the following circuit, the voltage drop across the ideal diode in forward bias condition

is 0.7V. The current passing through the diode is

(a) 0.5 mA

(b) 1.0 mA

(c) 1.5 mA

(d) 2.0 mA

Ans: (b)

Solution: Let current through k12 is I and through diode is DI

Then 63.37.0 DD III (1)

and 061224 DIII (2)

From (1) and (2) .1mAI D

GATE-2013

Q18. What should be the clock frequency of a DAbit /6 converter so that its maximum

conserved time is s32 ?

(a) 1 MHz (b) MHz2 (c) MHz5.0 (d) MHz4

Ans: (c)

Q19. A phosphorous doped silicon semiconductor (doping density: 1017/cm3) is heated from

100C to 200C. Which one of the following statements is CORRECT?

(a) Position of Fermi level moves towards conduction band

(b) Position of dopant level moves towards conduction band

(c) Position of Fermi level moves towards middle of energy gap

(d) Position of dopant level moves towards middle of energy gap

Ans: (c)

12k

Volt 24

k 6 k 3.3





Consider the following circuit

Q20. For this circuit the frequency above which the gain will decrease by 20 dB per decade is

(a) kHz9.15 (b) kHz2.1

(c) kHz6.5 (d) kHz5.22

Ans: (a)

Solution: kHzRC

f H 162

1

Q21. At kHz2.1 the closed loop gain is

(a) 1 (b) 1.5 (c) 3 (d) 0.5

Ans: (b) 5.1

1

1

2

10

H

F

in

ff

RR

v

v

GATE-2014

Q22. The input given to an ideal OP-AMP integrator circuit is

The correct output of the integrator circuit is

(a) (b)

(c) (d)

Ans: (a)

inVk10

1000pF

k1

k2

outV

0V

V

0t t

0V

V

0t t

0V

V

0t t

0V

V

0t t

0V

V

0t t




Q23. The minimum number of flip-flops required to construct a mod-75 counter is

__________

Ans: 7

Q24. The donor concentration in a sample of n -type silicon is increased by a factor of 100.

The shift in the position of the Fermi level at 300K, assuming the sample to non

degenerate is ________ meV . KatmeVTkB 30025

Ans: 115.15

Solution: ln cC F

d

NE E kT

N

and ln ln ln 100

100c c

C Fd d

N NE E kT kT kT

N N

Thus shift is ln 100 25ln 100 115.15E kT meV meV

Q25. The current gain of the transistor in the following circuit is 100dc . The value of

collector current CI is_________ mA

Ans: 1.6

Solution:

12 00.016 1.6

150 100 3 3CC BE

B C BB C E

V VI mA I I mA

R R R

Q26. In order to measure a maximum of V1 with a resolution of mV1 using a n bit

D

Aconverter working under the principle of ladder network the minimum value of n

is________

Ans: 10

Solution: 3 11 10 10

2 1nn

k150

V12

k3 F20

0V

F20iV

k3




Q27. A low pass filter is formed by a resistance R and a capacitanceC . At the cut-off angular

frequency 1

C RC the voltage gain and the phase of the output voltage relative to the

input voltage respectively are

(a) o45and71.0 (b) o45and71.0 (c) o90and5.0 (d) o90and5.0

Ans: (b)

Solution: 0 1 1

11

C

in C

C

v XRv R X j CR

X

At 1

C RC

0

0

450

45

1 1 1

1 22j

jin

ve

v j e

GATE-2015

Q28. The band gap of an intrinsic semiconductor is eVEg 72.0 and * *6h nm m . At ,300 K the

Fermi level with respect to the edge of the valence band (in eV ) is at _______(upto three

decimal places) 1231038.1 JKkB

Ans.: 0.395

Solution: *

*

3ln

2 4c h

in

E E mE kT

m

/ 2/ gi v E kTE E kTi V c vn N e N N e / 2/ gi v E kTE E kT c

v

Ne e

N / 2/ gi v E kTE E kT v

c

Ne e

N

ln2

gi v v

c

EE E N

kT N kT

3* 4

*ln

2gh

e

Em

m kT

3

ln 64 2

gi v

EE E kT

3 0.720.026 1.7917

4 2i vE E 0.3949 0.395eV eV

Q29. Which one of the following DOES NOT represent an exclusive OR operation for inputs

A and B ?

(a) ABBA (b) ABBA (c) BABA (d) ABBA

Ans.: (d)




Solution: (a) ( ) ( )( )A B AB A B A B AB AB

(b) AB AB

(c) AB AB

(d) A B AB AB

Q30. Consider the circuit shown in the figure, where 1RC . For an input signal iV shown

below, choose the correct 0V from the options:

(a) (b)

(c) (d)

Ans.: (b)

Solution: 00idv vC

dt R

0 0

in inin

dv dvv RC v v t

dt dt

inv t 0 1v V and inv t 0 1v V

C

R

0ViV

R

iV

1

1 2 3 t

0V

1

1

1 2 3 t

0V

1

1

1 2 3 t

1.0

1.0

0V

1 2 3 t

0V

1

1 2 3 t




Q31. In the simple current source shown in the figure, 1Q and 2Q are identical transistors with

current gain 100 and VVBE 7.0

The current mAI in 0 is __________ (upto two decimal places)

Ans.: 5.86

Solution: 0CC C C BEV I R V , 30 0.7 29.3

5.865 5CI mA

Q32. In the given circuit, if the open loop gain 510A the feedback configurations and the

closed loop gain fA are

(a) series-shunt, 9fA (b) series-series, 10fA

(c) series-shunt, 10fA (d) shunt-shunt, 10fA

Ans.: (c)

Solution: 1

1 1 9 10.FF

RA

R

Q33. In the given circuit, the voltage across the source resistor is1 V . The drain voltage (inV )

is ___________

Ans.: 15

Solution: 1 1

25 5000500 500S D S D D DD D DV I R I A V V I R 15DV V

5k

VVice 30

0I

2Q1Q

k9k1

0ViV

LR

V25

k5

500M2




GATE-2016

Q34. The number density of electrons in the conduction band of a semiconductor at a given

temperature is 319102 m . Upon lightly doping this semiconductor with donor

impurities, the number density of conduction electrons at the same temperature

becomes 320104 m . The ratio of majority to minority charge carrier concentration

is________.

Ans : 400

Solution: Intrinsic carrier concentration is 19 32 10in m

Majority carrier concentration is 20 34 10n m

Minority carrier concentration is 2192

18 320

2 1010

4 10in

p mn

The ratio of majority to minority charge carrier concentration is20

18

4 10400

10

n

p

Q35. For the digital circuit given below, the output X is

(a) CBA . (b) CBA . (c) CBA . (d) CBA .

Ans.: (b)

Q36. For the transistor shown in the figure, assume VVBE 7.0 and 100dc . If outin VVV ,5

(in Volts) is _________. (Give your answer upto one decimal place)

Ans.: 5.7

A

BC

X

inV

200k

1k

outV

3k

10V




Solution: 5 0.7 4.3

200 100 300in BE

BB E

V VI mA

R R

, 1.433C BI I mA

out CC C CV V I R 10 1.433 3 5.7outV V

GATE-2017

Q37. The best resolution that a 7 bit A/D convertor with 5V full scale can achieve

is…………… mV . (up to two decimal places)

Ans. : 39.37

Solution: Resolution7

539.37

2 1mV

Q38. In the figure given below, the input to the primary of the transformer is a voltage varying

sinusoidally with time. The resistor R is connected to the centre tap of the secondary.

Which one of the following plots represents the voltage across the resistor R as a

function of time?

(a) (b)

(c) (d)

Ans. : (a)

Solution:

Full wave rectifier with RC filter.

R

C

0

V

t

0

V

t

0

V

t

0

V

t




Q39. The minimum number of NAND gates required to construct an OR gate is:

(a) 2 (b) 4 (c) 5 (d) 3

Ans. : (d)

Q40. For the transistor amplifier circuit shown below with 1 2 310 , 10 , 1R k R k R k ,

and 99 . Neglecting the emitter diode resistance, the input impedance of the amplifier

looking into the base for small ac signal is…………. k . (up to two decimal places)

Ans. : 4.75

Solution: i bZ Z R where 3 99bZ R k and 1 2 5R R R k

4.75i bZ Z R k

Q41. Consider an ideal operational amplifier as shown in the figure below with

1 25 , 1 , 100LR k R k R k . For an applied input voltage 10V mV , the current

passing through 2R is…………….. A . (up to two decimal places)

Ans. : 10.0

Solution: 22

1010

1

VI A

R

inV

1R

2R3R

C

CCV

B

EoutV

V 1R

2RLR




GATE – 2018

Q42. The logic expression ABC ABC ABC ABC can be simplified to

(a) XORA C (b) AND A C (c) 0 (d) 1

Ans. : (a)

Solution: Y ABC ABC ABC ABC AC B B AC B B

XORY AC AC A C

Q43. In a 2-to-1 multiplexer as shown below, the output 0X A if 0C and 1X A if 1C .

Which one of the following is the correct implementation of this multiplexer?

(a) (b)

(c) (d)

Ans. : (a)

Solution: Check option (a),

0 1X A C AC

If 00C X A

If 11C X A

Q44. For an operational amplifier (ideal) circuit shown below,

X

1A

C0A

X

1A

C0A

X

1A

C0A

X

1A

C0A

4k

LR5k

10V

10V2k

0V1V

2V




If 1 1V V and 2 2V V , the value of 0V is __________V (up to one decimal place).

Ans. : 3.6

Solution: 0 01 02

4 41 2

2 5V V V V V

0 2 1.6 3.6V V

Q45. A p - doped semiconductor slab carries a current 100I mA in a magnetic field

0.2B T as shown. One measures 0.25yV mV and 2xV mV . The mobility of holes

in the semiconductor is___________ 2 1 1m V s (up to two decimal places)

Ans. : 1.55

Q46. An n - channel FET having Gate-Source switch-off voltage OFF 2GSV V is used to

invert a 0 5V square-wave signal as shown. The maximum allowed value of R would

be _________ k (up to two decimal places).

Ans. : 0.70

z4w mm

I

B x

xV

yV

10l mm

1t mmy

5V

0V

inV R

1k

12V

100

5V

5k

5V

0V

outV




GATE-2019

Q47. For the following circuit, what is the magnitude of outV if in 1.5V V ?

(a) 0.015V (b) 0.15V (c) 15V (d) 150V

Ans. : (c)

Solution: 0

1001.5 150 15out

RV V V V

R

Q48. Consider the following Boolean expression:

A B A B C A B C

It can be represented by a single three-input logic gate. Identify the gate

(a) AND (b) OR (c) XOR (d) NAND

Ans. : (d)

Solution: Y A B A B C A B C

Y A B A B C AB AC

A B A BC AB AC

A ABC AB BC AB AC

A ABC BC AB AB AC

A BC B AC A B AC

A AC B A AC B A C B

Y ABC

Q49. A 3 - bit analog-to-digital converter is designed to digitize analog signals ranging from

0V to 10V . For this converter, the binary output corresponding to an input of 6 V is

(a) 011 (b) 101 (c) 100 (d) 010

inVR

100 R

15V

15V

outV




Ans. : (c)

Solution: 0 000 0V

101 001 1.42

7V

202 010 2.8

7V

303 011 4.28

7V

404 100 5.71

7V

505 101 7.14

7V

606 110 8.57

7V

707 111 10

7V

Q50. For the following circuit, the correct logic values for the entries 2X and 2Y in the truth

table are

(a) 1 and 0 (b) 0 and 0 (c) 0 and 1 (d) 1 and 1

Ans. : (a)

B

G

A

C

P

Y

X

G A B P C X Y

1 0 1 0 0 0 1

0 0 0 1 0 2X 2Y

1 0 0 0 1 0 1




ATOMIC AND MOLECULAR PHYSICS SOLUTIONS

GATE-2010

Q1. To detect trace amounts of gaseous species in a mixture of gases, the preferred probing

tool is

(a) Ionization spectroscopy with X-rays (b) NMR spectroscopy

(c) ESR spectroscopy (d) Laser spectroscopy

Ans: (a)

Q2. A collection of N atoms is exposed to a strong resonant electromagnetic radiation with Ng

atoms in the ground state and Ne atoms in the excited state, such that Ng+Ne=N. This

collection of two-level atoms will have the following population distribution:

(a) g eN N (b) g eN N

(c) / 2g eN N N (d) / 2g eN N N

Ans: (c)

Solution: In two level lair population inversion is possible to achieve at any power level. The

maximum possible situation can be 2

NNN eg

Q3. Two states of an atom have definite parities. An electric dipole transition between these

states is

(a) Allowed if both the sates have even parity

(b) Allowed if both the states have odd parity

(c) Allowed if the two states have opposite parities

(d) Not allowed unless a static electric field is applied

Ans: (c)

Q4. The spectrum of radiation emitted by a black body at a temperature 1000 K peaks in the

(a) Visible range of frequencies (b) Infrared range of frequencies

(c) Ultraviolet range of frequencies (d) Microwave range of frequencies

Ans: (a)




Q5. The three principal moments of inertia of a methanol (CH3OH) molecule have the

property Ix = Iy = I and Iz ≠ I. The rotation energy eigenvalues are

(a)

III

mll

I z

11

21

2

21

22 (b) 1

2

2

llI

(c)

III

m

z

11

2

21

2 (d)

II

mll

I z

11

21

2

21

22

Ans: (a)

Solution: OHCH 3 is example of symmetric rotator, where zyx III , ( III yx and II z )

The classical expression for energy is 222

2

1

2

1z

zyx J

IJJ

IE .

This can be expressed in term of 2222zyx JJJJ by adding and subtracting 2

zJ

22

2

1

2

1

2

1z

z

JII

JI

E

.

Quantum mechanically

II

mJJ

IE

z

J 11

21

2

222

Q6. Match the typical spectra of stable molecules with the corresponding wave-number range

1. Electronic spectra (i) 106 cm-1 and above

2. Rotational spectra (ii) 105 – 106 cm-1

3. Molecule dissociation (iii) 108 – 102 cm-1

(a) 1 – ii, 2 – i, 3 – iii (b) 1 – ii, 2 – iii, 3 – i

(b) 1 – iii, 2 – ii, 3 – i (d) 1 – i, 2 – ii, 3 – iii

Ans: (b)

Q7. Consider the operations rrP : (parity) and T: t → - t (time reversal). For the electric

and magnetic fields E and B , which of the following set of transformations is correct?

(a) ;,: BBEEP (b) ;,: BBEEP

BBEET ,: BBEET ,:

(c) ;,: BBEEP (d) ;,: BBEEP

BBEET ,: BBEET ,:

Ans: (b)




Common Data Questions 8 and 9:

In the presence of a weak magnetic field, atomic hydrogen undergoes the transition:

2 21/ 2 1/ 2P S , by emission of radiation

Q8. The number of distinct spectral lines that are observed in the resultant Zeeman spectrum

is

(a) 2 (b) 3 (c) 4 (d) 6

Ans: (c)

Solution: 2 21/ 2 1/ 2p S is sodium D1 lines and it has total 4 zeeman components.

Q9. The spectral line corresponding to the transition

2 21 1/ 2

2

1 1

2 2j jP m S m

is observed along the direction of the applied magnetic field. The emitted electromagnetic

field is

(a) Circularly polarized (b) Linearly polarized

(c) Unpolarized (d) Not emitted along the magnetic field direction

Ans: (a)

Solution: For 2 21/ 2 1/ 2

1 1

2 2j jP m S m

Here 1 jm gives component.

In longitudinal observation is circularly polarized.

GATE-2011

Q10. The population inversion in a two layer material CANNOT be achieved by optical

pumping because

(a) the rate of upward transitions is equal to the rate of downward transitions

(b) the upward transitions are forbidden but downward transitions are allowed

(c) the upward transitions are allowed but downward transitions are forbidden

(d) the spontaneous decay rate of the higher level is very low

Ans: (a)




Q11. A heavy symmetrical top is rotating about its own axis of symmetry (the z-axis). If I1, I2

and I3 are the principal moments of inertia along x, y and z axes respectively, then

(a) I2 = I3; I1 ≠ I2 (b) I1 = I3; I1 ≠ I2 (c) I1 = I2; I1 ≠ I3 (d) I1 ≠ I2 ≠ I3

Ans: (c)

Q12. A neutron passing through a detector is detected because of

(a) the ionization it produces (b) the scintillation light it produces

(c) the electron-hole pairs it produces

(d) the secondary particles produced in a nuclear reaction in the detector medium

Ans: (b)

Q13. An atom with one outer electron having orbital angular momentum l is placed in a weak

magnetic field. The number of energy levels into which the higher total angular

momentum state splits, is

(a) 2l + 2 (b) 2l + 1 (c) 2l (d) 2l – 1

Ans: (b)

Q14. For a multi-electron atom l, L and S specify the one-electron orbital angular momentum,

total orbital angular momentum and total spin angular momentum, respectively. The

selection rules for electric dipole transition between the two electronic energy levels,

specified by l, L and S are

(a) ∆L = 0, ±1; ∆S = 0; ∆l = 0, ±1 (b) ∆L = 0, ±1; ∆S = 0; ∆l = ±1

(c) ∆L = 0, ±1; ∆S = ±1; ∆l = 0, ±1 (d) ∆L = 0, ±1; ∆S = ±1; ∆l = ±1

Ans: (b)

Q15. The lifetime of an atomic state is 1 nanosecond. The natural line width of the spectral line

in the emission spectrum of this state is of the order of

(a) 10-10 eV (b) 10-9 eV (c) 10-6 eV (d) 10-4 eV

Ans: (c)

Solution: eVeVSJ

t

hhE 6

19

25

9

34

1014.4106.1

10625.6

10

10625.6

Q16. The degeneracy of an excited state of nitrogen atom having electronic configuration

1s22s22p23d1 is

(a) 6 (b) 10 (c) 15 (d) 150

Ans: (b)




Solution: Degeneracy = 2 (2l + 1)

Q17. The far infrared rotational absorption spectrum of a diatomic molecule shows equilibrium

lines with spacing 20 cm-1. The position of the first Stokes line in the rotational Raman

spectrum of this molecule is

(a) 20 cm-1 (b) 40 cm-1 (c) 60 cm-1 (d) 120 cm-1

Ans: (c)

Solution: Given 2B = 20 cm-1 B = 10 cm-1

The position of the first stokes line in the rotational Raman spectrum = 6B

1601066 cmB .

GATE-2012

Q18. The ground state of sodium atom ( Na11 ) is a 2/12S state. The difference in energy levels

arising in the presence of a weak external magnetic field B, given in terms of Bohr

magnet on, B , is

(a) BB (b) BB2 (c) BB4 (d) BB6

Ans: (b)

Solution: The energy separation in the Zeeman level is BgME BJ

For 2/12S state; 2g and

2

1JM . Therefore BE B 1 and BE B 2 .

Thus BE B2

Q19. The first Stokes line of a rotational Raman spectrum is observed at 112.96 cm .

Considering the rigid rotor approximation, the rotational constant is given by

(a) 6.48 cm-1 (b) 3.24 cm-1 (c) 2.16 cm-1 (d) 1.62 cm-1

Ans: (c)

Solution: The first Stoke line of the Rotational Raman spectrum lies at = 6B

Thus 196.126 cmB 12.16B cm .




Q20. Match the typical spectroscopic regions specified in Group I with the corresponding type

of transitions in Group II.

Group I Group II

(P) Infra-red region (i) electronic transitions involving valence electrons

(Q) Ultraviolet-visible region (ii) nuclear transitions

(R) X-ray region (iii) vibrational transitions of molecules

(S) γ-ray region (iv) transitions involving inner shell electrons

(a) (P, i); (Q, iii); (R, ii); (S, iv) (b) (P, ii); (Q, iv); (R, i); (S, iii)

(c) (P, iii); (Q, i); (R, iv); (S, ii) (d) (P, iv); (Q, i); (R, ii); (S, iii)

Ans: (c)

Q21. The term Jjj 21 , arising from 1132 ds electronic in j-j coupling scheme are

(a) 2,31,2 2

5,

2

1and

2

3,

2

1

(b)

1,20,1 2

3,

2

1and

2

1,

2

1

(c) 2,30,1 2

5,

2

1and

2

1,

2

1

(d)

2,31,2 2

5,

2

1and

2

1,

2

3

Ans: (a)

Q22. The equilibrium vibration frequency for an oscillator is observed at 2990 cm-1. The ratio

of the frequencies corresponding to the first and the fundamental spectral lines is 1.96.

Considering the oscillator to be anharmonic, the anharmonicity constant is

(a) 0.005 (b) 0.02 (c) 0.05 (d) 0.1

Ans: (b)

Solution: 1299021 cmxee and

2 1 31.96

1 2e e

e e

x

x

1 3

0.981 2

e

e

x

x

02.0 ex .

GATE-2013

Q23. The number of spectral lines allowed in the spectrum for the PD 22 33 transition in

sodium is _____________.

Ans: 28

Solution: The numbers of Zeeman components for 2D5/2 → 2P3/2 transition = 12

The numbers of Zeeman components for 2D3/2 → 2P3/2 transition = 10

The numbers of Zeeman components for 2D3/2 → 2P1/2 transition = 6




Q24. In a normal Zeeman Effect experiment, spectral splitting of the line at the wavelength

643.8 nm corresponding to the transition 11

21 55 PD of cadmium atoms is to be

observed. The spectrometer has a resolution of 0.01 nm. Minimum magnetic field needed

to observe this is smcCekgme /103,6.1,101.9 81931

(a) T26.0 (b) T52.0 (c) T6.2 (d) T2.5

Ans: (b)

Solution: Separation of Zeeman Components

m

eB

4

m

eB

cc

4

22

31 8 9

22 19 9

4 4 3.14 9.1 10 3 10 0.01 100.514

1.6 10 643.8 10

mcB T

e

Q25. The spacing between vibrational energy levels in CO molecule is found to

be eV21044.8 . Given that the reduced mass of CO is ,1014.1 26 kg Planck’s constant

is Js3410626.6 and JeV 19106.11 . The force constant of the bond in CO

molecule is

(a) 1.87 N/m (b) 18.7 N/m (c) 187 N/m (d) 1870 N/m

Ans: (c)

Solution: The energy of the quantum harmonic oscillator is

,........2,1,0,2

1

nnhE

The frequency of oscillation is

k

2

1 .

Where k = Spring constant and = reduced mass

The energy levels are equally spaced with energy separation of

khhE

2

mNEh

k /7.1861014.1106.11044.810626.6

14.322 262

19234

2




GATE-2014

Q26. The number of normal Zeeman splitting components of 1 1P D transition is

(a) 3 (b) 4 (c) 8 (d) 9

Ans: (a)

Solution: This is singlet transition.

Q27. The moment of inertia of a rigid diatomic molecule A is 6 times that of another rigid

diatomic molecule B . If the rotational energies of the two molecules are equal, then the

corresponding values of the rotational quantum numbers AJ and BJ are

(a) 1,2 BA JJ (b) 1,3 BA JJ

(c) 0,5 BA JJ (d) 1,6 BA JJ

Ans: (b)

Solution:

1

1 6A A B B

B B A B

J J I I

J J I I

6, 1A BJ J

Q28. The value of the magnetic field required to maintain non-relativistic protons of energy

MeV1 in a circular orbit of radius 100mm is_______Tesla

(Given: 27 191.67 10 , 1.6 10pm kg e C )

Ans: 1.44

Solution: 2

21 2, 1.44

2

mv mEqvB E mv B

r qr

Q29. Neutrons moving with speed sm /103 are used for the determination of crystal structure.

If the Bragg angle for the first order diffraction is o30 the interplannar spacing of the

crystal is ______ 0

A . (Given: 27 341.675 10 6.626 10 .nm kg h J s )

Ans: 4

Solution: 2 sinh

dmv

34

027 3

6.62 102 sin 30

1.67 10 10d

04Ad

Q30. The emission wavelength for the transition 32 FD is 3122 Ǻ. The ratio of population

of the final to the initial states at a temperature K5000 is

34 8 236.626 10 - , 3 10 / , 1.380 10 /Bh J s c m s k J K

(a) 51003.2 (b) 51002.4 (c) 51002.7 (d) 51083.9




Ans: (c)

Solution:

9.227641144 52 1 5

7.02 102 1 7

B

hcf k TF

I i

JNe e

N J

GATE-2015

Q31. In a rigid rotator of mass M , if the energy of the first excited state is (1 meV ), then the

fourth excited state energy (in meV ) is ____________.

Ans.: 10

Solution: 1E J J where 0,1,2,3..J

44 1

1

4 4 110 10

1 1 1

EE E meV

E

, where 0,1,2,3..J

Q32. The binding energy per molecule of NaCl (lattice parameter is nm563.0 ) is eV956.7 .

The repulsive term of the potential is of the form 9r

K, where K is a constant. The value

of the Modelung constant is ___________ (upto three decimal places)

(Electron charge 212120

19 10854.8;106.1 mNCCe )

Ans.: 2.80

Solution: The total energy of one ion due to the presence of all others in NaCl crystal is

(considering univalent ions)

2

04 n

Ae KU r

r r , where A is Modelung Constant.

The potential energy will be minimum at the equilibrium spacing 0r .

Thus 0

2 120

2 10 0 0 0

04 4

n

nr r

Ae rdU Ae KnK

dr r r n

Thus, Binding energy of molecule or lattice energy is

0

2 120

00 0 0 04 4

n

nr r

Ae rAeU U

r nr

2

0 0

1

4

Ae n

r n

Given repulsive term of the potential is 99

Kn

r

Also binding energy per molecule is 0 7.95U eV




The Modelung constant is 0 00 2

4

1

r nA U

e n

and the lattice parameter is

0.563a nm . Thus the interatomic separation is 0 0.282

ar nm .

12 2 1 2 9

19219

4 3.14 8.85 10 0.282 10 97.95 1.67 10

81.67 10

C N mA J

J

27.95 1.67 4 3.14 8.85 0.282 910

1.67 8A

2.80A

Q33. Match the phrases in Group I and Group II and identify the correct option.

Group I Group II

(P) Electron spin resonance (ESR) (i) radio frequency

(Q) Nuclear magnetic resonance (NMR) (ii) visible range frequency

(R) Transition between vibrational states of a molecule (iii) microwave frequency

(S) Electronic transition (iv) far-infrared range

(a) (P-i), (Q-ii), (R-iii), (S-iv) (b) (P-ii), (Q-i), (R-iv), (S-iii)

(c) (P-iii), (Q-iv), (R-i), (S-ii) (d) (P-iii), (Q-i), (R-iv), (S-ii)

Ans.: (d)

Solution: (P) Electron spin resonance (ESR) is achieved by Microwave frequency (iii)

(Q): Nuclear magnetic resonance (NMR) is achieved by Radio frequency (i)

(R): Transition between vibrational states of a molecule is achieved by radiation of far

infrared range (iv)

(S): Electronic transition is achieved by visible radiation (ii)

Q34. The excitation wavelength of laser in a Raman effect experiment is nm546 . If the

Stokes’ line is observed at nm552 , then the wavenumber of the anti-Stokes’ line (in

1cm ) is ___________

Ans.: 18514

Solution: Raman displacement is

0 0AS S or 0 0

1 1 1 1

AS S

where SAS ,, 0 are wavelength of anti-stoke, exciting & stoke line.

From above relation we can write




0

0

0

0

000 2

211211111

S

SAS

S

S

ASSASSAS

9 9

9

9 9

546 10 552 10 546 55210

5582 552 10 546 10AS

m mm

m m

9 7540.129 10 540.129 10AS m cm

Anti-stoke wavenumber is 17

1 118514

540.129 10ASAS

cmcm

Q35. The number of permitted transitions from 2/12

2/32 SP in the presence of a weak

magnetic field is ________________

Ans. : 6

Solution: Zeeman splitting of 2/32 P and 2/1

2 S is shown below

The selection rule for Zeeman transactions are

0, 1 0 0 if 0JM J

There are total six transition in accordance with above selection rules.

GATE-2016

Q36. The molecule 217 O

(a) Raman active but not NMR (nuclear magnetic resonance) active.

(b) Infrared active and Raman active but not NMR active.

(c) Raman active and NMR active.

(d) Only NMR active.

Ans.: (c)

Solution: (i) Molecule 172O can not absorb infrared as there is no change in dipole moment

during vibration. Thus 172O is infrared inactive.

23/2P

21/2S

1/2

1/2

3/21/2

1/23/2

JM




(ii) Molecule 172O shows change in polaraziability during rotation. Thus it is Raman

active molecule.

(iii) The nucleus of 17 O has spin 5

2, therefore it is NMR active.

Q37. There are four electrons in the d3 shell of an isolated atom. The total magnetic moment

of the atom in units of Bohr magneton is ________.

Ans.: 0

Solution: The configuration leads to 2S and 2L

Since it is the case of less than half filled sub shell, thus according to Hund’s rules, lower

J will be in ground state.

0J L S 2J

eg J

m

.

Thus, 0

Q38. Which of the following transitions is NOT allowed in the case of an atom, according to

the electric dipole radiation selection rule?

(a) ss 12 (b) sp 12 (c) sp 22 (d) pd 23

Ans.: (a)

Solution: In electron dipole transition, 1l . Thus in transition 2 1 , 0s s l . It violate the

selection rule and hence not allowed.

Q39. The number of spectroscopic terms resulting from the SL. coupling of a p3 electron and

a d3 electron is _______.

Ans.: 12

Solution: For 1 13 3p d : 1 21 1

, 0,12 2

s s S

1 21, 2 1,2,3l l L

110, 1 1 TermS L J P

120, 2 2 TermS L J D

130, 3 3 TermS L J F

3 3 30 1 21, 1 0,1, 2 Terms , ,S L J P P P

43d

2LM 1 0 1 2




3 3 31 2 31, 2 1, 2,3 Terms , ,S L J D D D

3 3 32 3 41, 3 2,3, 4 Terms , ,S L J F F F

Thus total number of spectroscopic terms are 12.

GATE-2017

Q40. The wavefunction of which orbital is spherically symmetric:

(a) xp (b) yp (c) s (d) xyd

Ans. : (c)

Solution: For s orbital 0l

Q41. The total energy of an inert-gas crystal is given by 12 6

0.5 1E R

R R (in eV ), where R

is the inter-atomic spacing in Angstroms. The equilibrium separation between the atoms

is Angstroms. (up to two decimal places)

Ans. : 1

Solution: Given that 12 6

0.5 1E R

R R

For equilibrium separation

0dE

dR

13 7

12 0.5 60

dE

dR R R

6 6

1 66 0 1R

R R

Q42. Which one of the following gases of diatomic molecules is Raman, infrared, and NMR

active?

(a) 1 1-H H (b) 12 16-C O (c) 1 35-H Cl (d) 16 16-O O

Ans. : (c)

Solution: (a) 1 1H H Infrared inactive

(b) 12 16C O NMR Inactive

(c) 1 35H Cl Raman, infrared & NMR active

(d) 16 16O O Infrared , Raman inactive




Q43. Using Hund’s rule the total angular momentum quantum number J for the electronic

ground state of the nitrogen atom is

(a) 1

2 (b)

3

2 (c) 0 (d) 1

Ans. : (b)

Solution: 2 2 3: 7 :1 2 2N s s p

For 3 :p

spectral term 2 1 43/ 2

sJL s

Q44. Positronium is an atom made of an electron and a positron. Given the Bohr radius for the

ground state of the Hydrogen atom to be 0.53 Angstroms, the Bohr radius for the ground

state of positronium is…………Angstroms. (up to two decimal places).

Ans. : 1.06

Solution: 0e

n

mr a

When 2

2 2e e e e

e e e

m m m m

m m m

002 2 0.53 1.06nr a A

LM 1 0 1




GATE-2018

Q45. Which one of the following represents the 3p radial wave function of hydrogen atom?

( 0a is the Bohr radius)

(a) (b)

(c) (d)

Ans. : (b)

Solution: 3p radial wave function is 0331

0

16

r

arR r e

a

Q46. Given the following table,

Group I Group II

P: Stern-Gerlach experiment 1: Wave nature of particles

Q: Zeeman effect 2: Quantization of energy of electrons in the atoms

R: Frank-Hertz experiment 3: Existence of electron spin

S: Davisson-Germer experiment 4: Space quantization of angular momentum

Which one of the following correctly matches the experiments from Group I to their

inferences in Group II?

(a) P-2, Q-3, R-4, S-1 (b) P-1, Q-3, R-2, S-4

(c) P-3, Q-4, R-2, S-1 (d) P-2, Q-1, R-4, S-3

Ans. : (c)

00/r a

R r

00/r a

R r

00/r a

R r

0 0/r a

R r




Q47. The expression for the second overtone frequency in the vibrational absorption spectra of

a diatomic molecule in terms of the harmonic frequency e and anharmonicity constant

ex is

(a) 2 1e ex (b) 2 1 3e ex (c) 3 1 2e ex (d) 3 1 4e ex

Ans. : (d)

Solution:2

1 1

2 2V e e ev x v

Second overtone 0 3v v

2 2

3 0

7 7 1

2 2 2 2e

v v e e e e ev x x

3 12e e ex 3 1 4e ew x

Q48. Match the physical effects and order of magnitude of their energy scales given below,

where 2

04

e

c

is fine structure constant; em and pm are electron and proton mass,

respectively. Group I Group II

P: Lamb shift 1: 2 2~ eO m c

Q: Fine structure 2: 4 2~ eO m c

R: Bohr energy 3: 4 2 2~ /e pO m c m

S: Hyperfine structure 4: 5 2~ eO m c

(a) P-3, Q-1, R-2, S-4 (b) P-2, Q-3, R-1, S-4

(c) P-4, Q-2, R-1, S-3 (d) P-2, Q-4, R-1, S-3

Ans. : (c)

Solution:- Bohr energy 2 2eE m c

Fine structure 4 2eE m c

Lamb straight 5 2eE m c

Hyperfine structure 4 2

e

p

m cE

m




Q49. The intrinsic/permanent electric dipole moment in the ground state of hydrogen atom is

( 0a is the Bohr radius)

(a) 03ea (b) zero (c) 0ea (d) 03ea

Ans. : (b)

Solution: For dipole moment energy is coseEr

11 cos cos 0E eEr eE r [ cos 0 ]

Q50. Which one of the following is an allowed electric dipole transition?

(a) 1 30 1S S (b) 2 2

3/ 2 5/ 2P D (c) 2 25/ 2 1/ 2D P (d) 3 5

0 0P D

Ans. : (b)

Solution: For electric dipole transition

0, 1 0 0L , 0. 1J , 0S

Only option (b) satisfies above selection rules

Q51. The term symbol for the electronic ground state of oxygen atom is

(a) 10S (b) 1

2D (c) 30P (d) 3

2P

Ans. : (d)

Solution: 2 2 4: 1 , 2 , 2O s s p

Here, 1S , 2L

According to Hund’s rule, for ground state energy

2J L S 2 1 32

SJL P

Q52. 4 MeV - rays emitted by the de-excitation of 19F are attributed, assuming spherical

symmetry, to the transition of protons from 3/ 21d state to 5/ 21d state. If the contribution

of spin-orbit term to the total energy is written as C l s

, the magnitude of C is ______

MeV (up to one decimal place).

Ans. : 6.1

Solution: 1 2

1 3 5ˆ ˆ1, , ,2 2 2

l s j j

j l s 2 2 2

2 2 2 22

j l sj l s l s l s

1 0 1LM




21 1 1

2

j j l s sl s

5/ 2 3/ 2

E l s l s 25 7 3 5

2 2 2 2 2

220 20

8 8C

204

8E C MeV

32, 1.6

20C MeV C MeV .

Q53. An atom in its singlet state is subjected to a magnetic field. The Zeeman splitting of its

650 nm spectral line is 0.03 nm . The magnitude of the field is ___________ Tesla (up

to two decimal places).

( 19 31 8 11.60 10 , 9.11 10 , 3.0 10ee C m kg c ms )

Ans. : 1.52

Solution: 2

4

eB

c m

2

4c mB

e

8 31

92 199

3 10 4 9.1 100.03 10 1.52

1.6 10650 10T

GATE-2019

Q54. The spin-orbit interaction term of an electron moving in a central field is written as

f r l s

, where r is the radial distance of the electron from the origin. If an electron

moves inside a uniformly charged sphere, then

(a) f r constant (b) 1f r r (c) 2f r r (d) 3f r r

Ans. : (a)

Solution: The electric potential of a uniformly charged sphere at r R is

2

23

2

kQ rV

R R

where Q is the electric charge on the sphere of radius R and k is a constant.

The interaction energy is W f r l s

, where for central potential V , 1 Vf r

r r

3 3

1 kQr kQf r

r R R

constant. Thus option (a) is correct.




Q55. The ground state electronic configuration of the rare-earth ion ( 3Nd ) is 3 2 64 5 5Pd f s p .

Assuming LS coupling, the Lande g - factor of this ion is 8

11. The effective magnetic

moment in units of Bohr magneton B (rounded off to two decimal places) is

____________

Ans.: 3.62

Solution: For 34 f 6, 3 / 2, 9 / 2L S J

8 9 91 1

11 2 2J B Bg J J

8 9 11

3.6211 2 2 B B

3 2 1 0 1 2 3LM




SOLID STATE PHYSICS SOLUTIONS

GATE-2010

Q1. The valence electrons do not directly determine the following property of a metal

(a) Electrical conductivity (b) Thermal conductivity

(c) Shear modulus (d) Metallic luster

Ans: (c)

Q2. Consider X-ray diffraction from a crystal with a face-centered-cubic (fcc) lattice. The

lattice plane for which there is NO diffraction peak is

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

Ans: (a)

Q3. The Hall coefficient, RH, of sodium depends on

(a) The effective charge carrier mass and carrier density

(b) The charge carrier density and relaxation time

(c) The charge carrier density only

(d) The effective charge carrier mass

Ans: (c)

Q4. The Bloch theorem states that within a crystal, the wavefunction, ψ( r ), of an electron has

the form

(a) rkierur . where u( r ) is an arbitrary function and k is an arbitrary vector

(b) rGierur where u( r ) is an arbitrary function and G is a reciprocal lattice vector

(c) rGierur where ,ruru is a lattice vector and G is a reciprocal

lattice vector

(d) rkierur . where ,ruru is a lattice vector and k is an arbitrary

vector

Ans: (d)




Q5. In an experiment involving a ferromagnetic medium, the following observations were

made. Which one of the plots does NOT correctly represent the property of the medium?

(TC is the Curie temperature)

(a) (b)

(c) (d)

Ans: (c)

Q6. The thermal conductivity of a given material reduces when it undergoes a transition from

its normal state to the superconducting state. The reason is:

(a) The Cooper pairs cannot transfer energy to the lattice

(b) Upon the formation of Cooper pairs, the lattice becomes less efficient in heat transfer

(c) The electrons in the normal state lose their ability to transfer heat because of their

coupling to the Cooper pairs

(d) The heat capacity increases on transition to the superconducting state leading to a

reduction in thermal conductivity

Ans: (d)

Q7. For a two-dimensional free electron gas, the electronic density n, and the Fermi energy

EF, are related by

(a)

32

23

3

2

FmE

n (b) 2FmE

n (c) 22

FmEn (d)

31

31

2 FmEn

Ans: (b)

Solution: For two dimensional gas, the number of possible k-states between k and k+dk is

CT/1 T/1

CT T CT T




dkkL

dkkL

dkkg

22

222

22

it is multiplied by 2 for electron gas

Since 2

2 2

mE

k dEm

dkkdEm

dkk22

22

22

dEmL

dEEg2

22

22

The total number of electrons at KT 00 is

FF EE

dEEgEFdEEgN00

2 2

2 2 20

2 22 2

2 4

FE

F

m L m LdE E

FELm

N 22

n

mL

N

mEF

2

2

2

2FmE

n

Q8. Far away from any of the resonance frequencies of a medium, the real part of the

dielectric permittivity is

(a) Always independent of frequency (b) Monotonically decreasing with frequency

(c) Monotonically increasing with frequency (d) A non-monotonic function of frequency

Ans: (a)

Solution:

2

1

frequency

selectronic

ionic

dipolar




GATE-2011

Q9. The temperature (T) dependence of magnetic susceptibility (χ) of a ferromagnetic

substance with a Curie temperature (Tc) is given by

(a) cc

TTTT

C

for , (b) c

c

TTTT

C

for ,

(c) cc

TTTT

C

for , (d)

cTT

C

, for all temperatures

where C is constant .

Ans: (b)

Q10. The order of magnitude of the energy gap of a typical superconductor is

(a) 1 MeV (b) 1 KeV (c) 1 eV (d) 1 meV

Ans: (d)

Q11. For a three-dimensional crystal having N primitive unit cells with a basis of p atoms, the

number of optical branches is

(a) 3 (b) 3p (c) 3p – 3 (d) 3N – 3p

Ans: (c)

Q12. For an intrinsic semiconductor, me* and mh

* are respectively the effective masses of

electrons and holes near the corresponding band edges. At a finite temperature the

position of the Fermi level

(a) depends on me* but not on mh

* (b) depends on mh* but not on me

*

(c) depends on both me* and mh

* (d) depends neither on me* nor on mh

*

Ans: (c)

Solution: The Fermi level for intrinsic semicondutor is

*

*

ln4

3

2 e

hB

vcF m

mTk

EEE

Q13. A metal with body centered cubic (bcc) structure show the first (i.e. smallest angle)

diffraction peak at a Bragg angle of θ = 30o. The wavelength of X-ray used is 2.1 Ǻ. The

volume of the PRIMITIVE unit cell of the metal is

(a) 26.2 (Ǻ)3 (b) 13.1(Ǻ)3 (c) 9.3 (Ǻ)3 (d) 4.6 (Ǻ)3

Ans: (b)




Solution: According to Bragg’s law sin2d where 222 lkh

ad

For BCC structure the first diffraction peak appear for (110) plane.

2

ad 000 1.230sin230sin

2

2Aa

a

00 1.221.22

12 AaAa 097.2 Aa .

The volume primitive unit cell of BCC is volume 30303

1.132

2.26

2AA

a

Common Data for Questions 14 and 15:

The tight binding energy dispersion (E-k) relation for electrons in a one-dimensional

array of atoms having lattice constant a and total length L is

E = E0 – β – 2γ cos (ka),

where E0, β and γ are constants and k is the wave vector.

Q14. The density of states of electrons (including spin degeneracy) in the band is given by

(a) kaa

L

sin (b) kaa

L

sin2 (c) kaa

L

cos2 (d) kaa

L

cos

Ans: (a)

Solution: 1 2 1

2 2 22 / 2 2 sin

L LD E

dE dk a ka

2

2 sin

L

a ka

Q15. The effective mass of electrons in the band is given by

(a) kaa cos2

2

(b) kaa cos2 2

2

(c) kaa sin2

2

(d) kaa sin2 2

2

Ans: (b)

Solution: Effective mass kaa

dk

Edm

cos2 2

2

2

2

2*

kaa cos2 2

2




GATE-2012

Q16. For an ideal Fermi gas in three dimensions, the electron velocity VF at the Fermi surface

is related to electron concentration n as,

(a) 3/2nVF (b) nVF (c) 2/1nVF (d) 3/1nVF

Ans: (d)

Solution: 3/123 nm

VF

Q17. The total energy, E of an ideal non-relativistic Fermi gas in three dimensions is given by

3/2

3/5

V

NE where N is the number of particles and V is the volume of the gas. Identify the

CORRECT equation of state (P being the pressure),

(a) EPV3

1 (b) EPV

3

2 (c) EPV (d) EPV

3

5

Ans: (b)

Q18. Which one of the following CANNOT be explained by considering a harmonic

approximation for the lattice vibrations in solids?

(a) Deby’s T3 law (b) Dulong Petit’s law

(c) Optical branches in lattices (d) Thermal expansion

Ans: (d)

Solution: Thermal expansion in solid can only be explained if solid behave as a anharmonic

oscillator.

Q19. A simple cubic crystal with lattice parameter ca undergoes transition into a tetragonal

structure with lattice parameters ctt aba 2 and ct ac 2 , below a certain temperature.

The ratio of the interplanar spacing of (1 0 1) planes for the cubic and the tetragonal

structure is

(a) 6

1 (b)

6

1 (c)

8

3 (d)

8

3

Ans: (c)

Solution: For Cubic Lattice2222

cc

a

lkh

ad

For Tetragonal lattice3

2

2

2

2

22

ct

a

c

l

a

kh

ad

. Therefore, the ratio is 8

3

t

c

d

d




Q20. Inverse susceptibility (1/χ) as a function of temperature, T for a

material undergoing paramagnetic to ferromagnetic transition is

given in the figure, where O is the origin. The values of the

Curie constant, C, and the Weiss molecular field constant, λ, in

CGS units, are

(a) 25 103,105 C (b) 52 105,103 C

(c) 42 102,103 C (d) 24 103,102 C

Ans: (c)

Solution: C

TT C

1

and CTC . Here KTC 600 and 41021

Thus 2103 C and 4102 . Common Data for Questions 21–22

The dispersion relation for a one dimensional monoatomic crystal with lattice spacing a,

which interacts nearest neighbour harmonic potential is given by

2

sinKa

A

where A is a constant of appropriate unit.

Q21. The group velocity at the boundary of the first Brillouin zone is

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

2Aa (d)

22

1 2Aa

Ans: (a)

Solution: At the first Brillouin zone the frequency is maximum and the group velocity which is

the derivative of the angular frequency is zero.

Q22. The force constant between the nearest neighbour of the lattice is (M is the mass of the

atom)

(a) 4

2MA (b)

2

2MA (c) MA2 (d) 2MA2

Ans: (a)

Solution: 4

sin2

ka

M

24

4

MAA

M

O T

K 6001

unit) CGS(102 4




GATE-2013

Q23. A phosphorous doped silicon semiconductor (doping density: 1017/cm3) is heated from

100C to 200C. Which one of the following statements is CORRECT?

(a) Position of Fermi level moves towards conduction band

(b) Position of dopant level moves towards conduction band

(c) Position of Fermi level moves towards middle of energy gap

(d) Position of dopant level moves towards middle of energy gap Ans: (c)

Solution: Phosphorous doped silicon semiconductors behave as a n-type semiconductor. In

n-type semiconductor Fermi level lies near conduction band and moves toward middle of

the band gap upon heating. At a very high temperature the Fermi level is near the middle

of the band gap and semiconductor behaves as intrinsic semiconductor.

Q24. Considering the BCS theory of superconductors, which one of the following statements is

NOT CORRECT? ( h is the Plank’s constant and e is the electronic charge)

(a) Presence of energy gap at temperature below the critical temperature

(b) Different critical temperature for isotopes

(c) Quantization of magnetic flux in superconduction ring in the unit of

e

h

(d) Presence of Meissner effect Ans: (c)

Solution: Quantization of magnetic flux in superconduction ring in the unit of

e

h

2

Q25. Group I contains elementary excitations in solids. Group II gives the associated field with

these excitations. MATCH the excitations with their associated field and select your

answer as per codes given below.

Group I Group II

(P) phonon (i) photon + lattice vibration

(Q) plasmon (ii) electron +elastic deformation

(R) polaron (iii) collective electron oscillations

(S) polariton (iv) elastic wave

Codes (a) iiSiRiiiQivP ,,, (b) iSiiRiiiQivP ,,,

(c) ivSiiRiiiQiP ,,, (d) iSiiRivQiiiP ,,,




Ans: (b)

Solution: Phonon: Quantum of energy of the elastic wave in solid, produced due to the vibration

of atoms in solid.

Plasmon: Quantum of energy of the wave produced due to the oscillation of plasma,

which contains charged particles (positive ions and negative electrons or ions).

Polaron: A charge placed in a polarizable medium will be screened. The induced

polarization will follow the charge carrier when it is moving through the medium. The

carrier together with the induced polarization is considered as one entity, which is called

a polaron.

Polariton: A polariton is a quasiparticle resulting from the mixing of a photon with

phonon.

Q26. A lattice has the following primitive vector Åin : ,ˆˆ2 kja ,ˆˆ2 ikb

jic ˆˆ2

.

The reciprocal lattice corresponding to the above lattice is

(a) BCC lattice with cube edge of 1-Å2

(b) BCC lattice with cube edge of -1Å2

(c) FCC lattice with cube edge of 1-Å2

(d) FCC lattice with cube edge of -1Å2

Ans: (a)

Solution: The reciprocal lattice vectors are

-1ˆ ˆ ˆ2 -i j k Å2

b ca

a b c

-1ˆ ˆ ˆ2 i j k Å2

c ab

a b c

-1ˆ ˆ ˆ2 i j k Å2

a bc

a b c




Q27. The total energy of an ionic solid is given by an expression 9

0

2

4 r

B

r

eE

where

is Madelung constant, r is the distance between the nearest neighbours in the crystal and

B is a constant. If 0r is the equilibrium separation between the nearest neighbours then

the value of B is

(a) 0

80

2

36 re

(b) 0

80

2

4 re

(c) 0

100

2

9

2

re

(d) 0

100

2

36 re

Ans: (a)

Solution: At 0rr , 100

200

2 9

40

0r

B

r

e

dr

dE

rr

0

80

2

36 re

B

GATE-2014

Q28. The Miller indices of a plane passing through the three points having coordinates (0, 0, 1)

4

1,

2

1,

2

10,0,1 are

(a) (212) (b) (111) (c) (121) (d) (211)

Ans: (a)

Solution: The equation of plane is determined from following determinant:

1 0 0

1 0 1 0

1 1 1

2 2 4

x y z

1 1 1 11 1 0

2 4 2 2x y z

1

02 4 2 2

x y z 2 2 2 0x y z , 2 0hx ky lz . Miller indices are 2 1 2

Q29. The plot of specifies heat versus temperature across the superconducting transition

temperature cT is most appropriately represented by

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

Ans: (a)

Solution: 2kTVC e

pC

CT T

pC

CT T

pC

CT T

pC

CT T




Q30. The energy k for band electrons as a function of the wave vector k in the first Brillouin

zone

ak

a

of a one dimensional monoatomic lattice is shown as ( a is lattice

constant)

The variation of the group velocity gv is most appropriately represented by

(a) (b)

Ans: (b)

Solution: 0 cosE E ka

1sing

dE aV ka

dk

Q31. For Nickel the number density is 323 /108 cmatoms and electronic configuration is

2862622 4333221 sdpspss . The value of the saturation magnetization of Nickel in its

ferromagnetic state is _____________ mA /109 .

(Given the value of Bohr magneton 2211021.9 AmB )

Ans: 4.42

Solution: Component of magnetic dipoles in a solid material are in the direction of external field.

a/ a/O

k

k

a/a/O

gv

k a/a/O

gv

k

a/a/O

gv

ka/

a/O

gv

k




SM (Magnetic dipole moment) B N 21 20.6 9.21 10 Am N .

(For : 0.6iN magnetic dipole moment , :2.22, :1.2Fe For Cu )

29 38 10 /A

N

NN m

A

, 219.21 10 /B A m

21 29 90.6 9.21 10 8 10 4.42 10 /SM A m , nA atomic weight

GATE-2015

Q31. The energy dependence of the density of states for a two dimensional non-relativistic

electron gas is given by, nCEEg , where C is constant. The value of n

is____________

Ans.: 0

Solution: We know that

1/ 2g E E for 3 D , 0g E E for 2 D , 1/ 2g E E for 1 D

0n for 2 D

Q32. The lattice parameters cba ,, of an orthorhombic crystal are related by cba 32 . In

units of a the interplanar separation between the 110 planes is ____________. (Upto

three decimal places)

Ans.: 0.447

Solution: 1102 2 2

222 2 2

1 10.447

1 1 50

2

hkl

ad d

h k laa b c a

2 3a b c

Q33. The dispersion relation for phonons in a one dimensional monoatomic Bravais lattice

with lattice spacing a and consisting of ions of masses M is given by

kaM

ck cos1

2 , where is the frequency of oscillation, k is the wavevector

and C is the spring constant. For the long wavelength modes a , the ratio of the

phase velocity to the group velocity is_________

Ans.: 1

Solution: 21 cos

Ck ka

M




For long wavelength modes a

2

cos 12

kaka 2

21 1

2

kaC Ck a k

M M

Phase velocity P

Cv a

k M

and Group velocity g

d Cv a

dk M

1P

g

v

v

Q34. In a Hall effect experiment, the hall voltage for an intrinsic semiconductor is negative.

This is because (symbols carry usual meaning)

(a) pn (b) pn (c) n h (d) * *h nm m

Ans.: (c)

Solution: The Hall voltage is JBRV HH

where :J current density, :B magnetic field and :HR Hall constant

2 2 2 2 2

2 2 2 2 2

1 p n n pH

n p n p

p n p n BR

e n p p n B

For intrinsic semiconductor inpn np

np

iH en

R

1

In Intrinsic semiconductor pn , therefore Hall voltage is negative.

Q35. Given that the Fermi energy of gold is eV54.5 , the number density of electrons is

__________ 28 310 m (upto one decimal place)

(Mass of electron jeVsjhkg 193431 106.11;10626.6;1011.9 )

Ans.: 5.9

Solution: Relation between electron density )(n and Fermi energy FE is

3/222

32

nm

EF

3/ 2

3/ 22 3

21

3 F

mn E

3/ 231

3/ 2192 334

2 9.1 1015.54 1.6 10

3 3.14 1.0546 10 -

kgn J

J s

45 283

102

1 2.45 10 8.35 10

29.61 1.17 10n m

3291059.0 m 28 35.9 10n m




Q36. Which one of the following represents the electron occupancy for a superconductor in its

normal and superconducting states?

(a) (b)

(c) (d)

Ans. : (d)

Solution: In normal slide, some states below Fermi levels are empty and equal number of states

above Fermi levels are filled. If material converts into a superconductor, electrons above

the Fermi Level makes cooper pair and they fall back below level Fermi level as same

energy released during cooper pair formation. Therefore, correct option is (d).

GATE-2016

Q37. Consider a metal which obeys the Sommerfield model exactly. If FE is the Fermi energy

of the metal at KT 0 and HR is its Hall coefficient, which of the following statements

is correct?

(a) 2

3

FH ER (b) 3

2

FH ER

(c) 2

3

FH ER (d) HR is independent of FE .

Ans.: (c)

Solution: 1

HRne

, where 3/ 23/ 22 2 / 32 3 / 22 2

23

2 3F

F H F

EmE n n R E

m

Ef

E

stateNormal

Ef

E

statectingSupercondu

Ef

E


Ef

E

stateNormal

Ef

E

stateNormal

Ef

E


Ef

E


Ef

E

stateNormal




Q38. A one-dimensional linear chain of atoms contains two types of atoms of masses 1m and

2m (where 12 mm ), arranged alternately. The distance between successive atoms is the

same. Assume that the harmonic approximation is valid. At the first Brillouin zone

boundary, which of the following statements is correct?

(a) The atoms of mass 2m are at rest in the optical mode, while they vibrate in the

acoustical mode.

(b The atoms of mass 1m are at rest in the optical mode, while they vibrate in the

acoustical mode.

(c) Both types of atoms vibrate with equal amplitudes in the optical as well as in the

acoustical modes.

(d) Both types of atoms vibrate, but with unequal, non-zero amplitudes in the optical as

well as in the acoustical modes.

Ans.: (a)

Solution: In optical mode, at Brillouin zone boundary atom of heavier mass 2m is at rest,

whereas in Acoustic mode, atoms of lighter mass 1m is at rest.

Q39. A solid material is found to have a temperature independent magnetic susceptibility,

C . Which of the following statements is correct?

(a) If C is positive, the material is a diamagnet.

(b) If C is positive, the material is a ferromagnet.

(c) If C is negative, the material could be a type I superconductor.

(d) If C is positive, the material could be a type I superconductor.

Ans.: (b)

optical mode 1 2

1 12

m m

Acoustic

mode

2a

2a

2 12 / m

1 22 / m

k




Solution: Susceptibility is defined as 0

1

, where and 0 are permeability of medium

and vacuum respectively.

(i) For Diamagnet; 0 , thus 0 i.e. is negative

(ii) For Ferromagnet, 0 , thus 0 i.e. is positive

(iii) For superconductor, 0 , thus 1

Thus best answer is (b)

Q40. Atoms, which can be assumed to be hard spheres of radius R , are arranged in an fcc

lattice with lattice constant a , such that each atom touches its nearest neighbours. Take

the center of one of the atoms as the origin. Another atom of radius r (assumed to be

hard sphere) is to be accommodated at a position

0,2

,0a

without distorting the lattice.

The maximum value of R

ris ________. (Give your answer upto two decimal places)

Ans.: 0.41

Solution: The new atom location is 0, , 02

a

i.e. it is on the middle of y - axis.

If new atom of radius r fit without distorting the original lattice, then from figure (ii) we

get

2

aR r (i)

whereas for FCC 4

2 4 2 22

a R a R R (ii)

a

R

2a

r

2

a

figure ( )ii

newatom at 0, ,02

a

z

y

x Position of new

atom 0, ,02

a figure ( )i




Thus from (i) and (ii) 2 22 1

2R R r R r 2 1 1.414 1 0.414

r

R

Q41. The energy vs. wave vector kE relationship near the bottom of a band for a solid can

be approximated as 42 kaBkaAE , where the lattice constant 0

1.2 Aa . The

values of A and B are J19103.6 and J20102.3 , respectively. At the bottom of

the conduction band, the ratio of the effective mass of the electron to the mass of free

electron is _______. (Give your answer upto two decimal places)

(Take sJ 341005.1 , mass of free electron kg31101.9 ) Ans.: 0.22

Solution: 2 4E A ka B ka

2 4 32 4E

Aa k Ba kk

and 2

2 4 22

2 12E

Aa Ba kk

At the bottom of the band 0k

Thus effective mass 2 2

*2 2 2/ 2

mE k Aa

234

219 10

1.05 10

2 6.3 10 2.1 10

J s

J m

6829

39

1.1025 100.01984 10

55.57 10

3219.84 10 kg

* 321

31

19.84 102.18 10 0.218

9.1 10

m kg

m kg

0.22

Q42. The Fermi energies of two metals X and Y are eV5 and eV7 and their Debye

temperatures are K170 and K340 , respectively. The molar specific heats of these

metals at constant volume at low temperatures can be written as

3TATC XXXV and 3TATC YYYV where and A are constants. Assuming

that the thermal effective mass of the electrons in the two metals are same, which of the

following is correct?

(a) 8,5

7

Y

X

Y

X

A

A

(b) 8

1,

5

7

Y

X

Y

X

A

A

(c) 8

1,

7

5

Y

X

Y

X

A

A

(d) 8,7

5

Y

X

Y

X

A

A




Ans.: (a)

Solution: Heat capacity is defined as 3VC T AT

where 23 1

2 BF

NkE

and 4

3 3

12 1 1234

5 B BD D

A Nk Nk

Thus,

2

2

3 12 7 73 1 5 52

X Y

X

Y

BF FX

Y FB

F

NkE E eV

E eVNkE

and 33 3

3

3

1234

3402 8

1 170234

X Y

X

Y

BD DX

Y DB

D

NkA

A Nk

Thus, 7

5X

Y

and 8X

Y

A

A

GATE-2017

Q43. The atomic mass and mass density of Sodium are 23 and 30.968 g cm , respectively. The

number density of valence electrons is……………… 22 310 cm . (Up to two decimal

places) (Avogadro number, 236.022 10AN )

Ans. : 2.54

Solution: 3 3

eff eff A

A

n M n Nn

N a a M

where 30.968gcm , 236.022 10AN , 23M g

230.968 6.022 10

23n

22 32.54 10 cm

Q44. Consider a one dimensional lattice with a weak periodic potential 0

2cos

xU x U

a

.

The gap at the edge of the Brillouin zone ka

is:

(a) 0U (b) 0

2

U (c) 02U (d) 0

4

U

Ans. : (c)




Solution: 0

2cosU U x

a

Energy at the edge of Brillouin Zone is 0

2cos .t

aU U

a

Energy at the 0k is 0bU U Band gap 02t bU U U U

Q45. Consider a 2 - dimensional electron gas with a density of 19 210 m . The Fermi energy of

the system is………………… eV (up to two decimal places).

( 31 349.31 10 , 6.626 10 ,em kg h Js 191.602 10e C )

Ans. : 2.34

Solution: 2

22FE n

m

234

1931

1.055 102 3.142 10

2 9.31 10

J s

180.3756 10 0.2345 10 2.34J eV eV

Q46. The real space primitive lattice vectors are 1 â ax

and 2 ˆ ˆ32

aa x y

. The reciprocal

space unit vectors 1b

and 2b

for this lattice are, respectively

(a) ˆ2

ˆ3

yx

a

and

4ˆ

3y

a

(b)

ˆ2ˆ

3

yx

a

and

4ˆ

3y

a

(c) 2

ˆ3

xa

and

ˆ4ˆ

3

xy

a

(d)

2ˆ

3x

a

and

ˆ4ˆ

3

xy

a

Ans. (a)

Solution: 1 â ax

, 2 ˆ ˆ32

aa x y

. Assume, 3 â z

Now, 1 2 3 ˆ ˆ ˆ ˆ. 32

aa a a ax x y z

2 ˆˆ ˆ3

2

a xy x

2 230 3

2 2

a a

2 31

21 2 3

ˆ ˆ322 2

32

ax ya a

ba a a

a

ˆ2ˆ

3

yx

a

Similarly,

3 12

1 2 3

4ˆ2

3

a ab y

a a a a

. Thus correct option is (a)




GATE – 2018

Q47. For the given unit cells of a two dimensional square lattice, which option lists all the

primitive cells?

(a) (1) and (2) (b) (1), (2) and (3)

(c) (1), (2), (3) and (4) (d) (1), (2), (3), (4) and (5)

Ans. : (c)

Solution: For primitive cell, 1effN

In cell (1), (2), (3) and (4) 1effN , these are primitive cell

Whereas in cell (5), 2effN , this is non-primitive cell.

Q48. At low temperatures (T ), the specific heat of common metals is described by (with

and as constants)

(a) 3T T (b) 3T (c) exp /T (d) 5T T

Ans. : (a)

Solution: 3e pnC C C T T

Q49. The high temperature magnetic susceptibility of solids having ions with magnetic

moments can be described by 1

T

with T as absolute temperature and as

constant. The three behaviours i.e., paramagnetic, ferromagnetic and anti-ferromagnetic

are described, respectively, by

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

(c) 0, 0, 0 (d) 0, 0, 0

Ans. : (c)

1 5

4

3

2




Solution: Paramagnetism: C

T

Ferromagnetism: C

C

T T

Anti-ferromagnetism: C

C

T T

Q50. The energy dispersion for electrons in one dimensional lattice with lattice parameter a is

given by 0

1cos

2E k E W ka , where W and 0E are constants. The effective mass of

the electron near the bottom of the band is

(a) 2

2

2

Wa

(b)

2

2Wa

(c)

2

22Wa

(d)

2

24Wa

Ans. : (a)

Solution: 0

1cos

2E k E W ka

sin2

dE aWka

dk

2 2

2cos

2

d E a Wka

dk

2 2*

2 2

2 cos2

md E a W

kadk

2

2

2

Wa

[At bottom of the band, 0k ]

Q51. Amongst electrical resistivity , thermal conductivity , specific heat C , Young’s

modulus Y and magnetic susceptibility , which quantities show a sharp change at

the superconducting transition temperature?

(a) , , ,C Y (b) , ,C (c) , , ,C (d) , ,Y

Ans. : (b)




GATE-2019

Q52. The relative magnetic permeability of a type-I super conductor is

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

4

Ans.: (a)

Solution: 0 0 0 1B H M H H H H

0 1 x 0

1r

For type-I superconductor: 1

1 1 0r

Q53. In order to estimate the specific heat of phonons, the appropriate method to apply would

be

(a) Einstein model for acoustic phonons and Debye model for optical phonons

(b) Einstein model for optical phonons and Debye model for acoustic phonons

(c) Einstein model for both optical and acoustic phonons

(d) Debye model for both optical and acoustic phonons

Ans.: (b)

Solution: At low temperature, the optical branch phonons have energies higher than Bk T and

therefore, optical branch waves are not excited. And Debye model is not suitable for

optical branch instead it is suitable for acoustical branch. Whereas Einstein model is

useful for high temperature and therefore can be applied to optical branch.

Q54. Consider a three-dimensional crystal of N inert gas atoms. The total energy is given by

12 6

2U R N p qR R

, where 12.13, 14.45p q and R is the nearest

neighbour distance between two atoms. The two constants, and R , have the

dimensions of energy and length, respectively. The equilibrium separation between two

nearest neighbour atoms in units of (rounded off to two decimal places)

is____________

Ans.: 1.09




Solution: 12 6

2U R N p qR R

11 5

2 20 2 12 6 0

dUN p q

dR R R R R

12 6

13 712 6 0p q

R R

12 6

13 712 6p q

R R

6 612

6

pR

q

1/ 6

2 pR

q

given 12.13p , 14.45q

1/ 6

1/ 62 12.131.679 1.09

14.45R

Thus 1.09R

Q55. The energy-wavevector E k dispersion relation for a particle in two dimensions is

E Ck , where C is a constant. If its density of states D E is proportional to pE then

the value of p is____________

Ans.: 1

Solution: For sE k k . The density of states in d - dimension is 1

d

sD E E

Given, E Ck 1, 2s d

Thus 2

11D E E

1E

Q56. A conventional type-I superconductor has a critical temperature of 4.7 K at zero

magnetic field and a critical magnetic field of 0.3 Tesla at 0 K . The critical field in Tesla

at 2K (rounded off to three decimal places) is__________

Ans.: 0.246




Solution: 2 2

2

0

21 0.3 1 0.3 1 0.426

4.7cc

TH T H

T

0.3 1 0.181 0.3 0.819 0.246 Atm

Q57. A particle of mass m moves in a lattice along the x - axis in a periodic potential

V x V x d with periodicity d . The corresponding Brillouin zone extends from

0k to 0k with these two k - points being equivalent. If a weak force F in the x -

direction is applied to the particle, it starts a periodic motion with the time period T .

Using the equation of motion crystaldpF

dt for a particle moving in a band, where crystalp is

the crystal momentum of the particle, the period T is found to be ( h is Planck constant)

(a) 2md

F (b)

22

md

F (c)

2h

Fd (d)

h

Fd

Ans. : (d)

Solution: 00

dd

E E Fdx F x Fd

Using Heisenberg uncertainty E t h , h h

T tE Fd

. Thus correct option is (d)

Q58. In a certain two-dimensional lattice, the energy dispersion of the electrons is

1 32 cos 2cos cos

2 2x x yk t k a k a k a

where ,x yk k k

denotes the wave vector, a is the lattice constant and t is a constant

in units of eV . In this lattice the effective mass tensor ijm of electrons calculated at the

center of the Brillouin zone has the form 2

2

0

0ijmta

. The value of (rounded off

to two decimal places) is ____________

Ans.: 0.333

Solution: Effective mass tensor matrix 4




1 1 10

11 10

xx xy xx

ij

yyyx yy

m m mm

mm m

When 2

2 2/xxx

mE k

and 2

2 2/yyy

mE k

Now 1 3

2 sin sin cos2 2x x y

x

Et a k a a k a k a

k

2 2

22

1 32 cos cos cos

2 2 2x x yx

E at a k a k a k a

dk

At the Brillouin zone centre i.e. at 0x yk k

2

2 22

12 1 3

2x

Eta ta

k

Similarly, 1 3

2 3 cos sin2 2x y

y

Et a k a k a

k

2 2

2

3 1 32 cos cos

2 2 2x yy

E at k a k a

k

At the Brillouin zone centre i.e. at 0x yk k

2

22

3y

Eta

Thus 2 2

2 2 2/ 3xxx

mE k ta

and 2 2

2 2 2/ 3yyy

mE k ta

2

22

22

2

10 0

3 31

0033

ijtam

ta

ta

Thus 1

0.3333

.




NUCLEAR AND PARTICLE PHYSICS

GATE-2010

Q1. The basic process underlying the neutron β-decay is

(a) eeud (b) eud

(c) eeus (d) eedu

Ans: (a)

Q2. In the nuclear shell model the spin parity of 157 N is given by

(a) 2

1

(b) 2

1

(c) 2

3

(d) 2

3

Ans: (a)

Solution: 7Z ; 12/14

2/32

2/1 pps and 8N

2

1parity-spin,11parity

2

1,1 1Jl

Q3. Match the reactions on the left with the associated interactions on the right.

(1) π+ → μ+ + (i) Strong

(2) π0 → γ + γ (ii) Electromagnetic

(3) π0 + n → π- + p (iii) Weak

(a) (1, iii), (2, ii), (3, i) (b) (1, i), (2, ii), (3, iii)

(c) (1, ii), (2, i), (3, iii) (d) (1, iii), (2, i), (3, ii)

Ans: (a)

Q4. The ground state wavefunction of deuteron is in a superposition of s and d states. Which

of the following is NOT true as a consequence?

(a) It has a non-zero quadruple moment

(b) The neutron-proton potential is non-central

(c) The orbital wavefunction is not spherically symmetric

(d) The Hamiltonian does not conserve the total angular momentum

Ans: (d)




Q5. The first three energy levels of 90228Th are shown below

The expected spin-parity and energy of the next level are given by

(a) (6+; 400 keV) (b) (6+; 300 keV) (c) (2+; 400 keV) (d) (4+; 300 keV)

Ans: (a)

Solution:

keVE

E

E

JJ

JJ

E

E393

144

166

1

16

4

6

11

22

1

2

GATE-2011

Q6. The semi-empirical mass formula for the binding energy of nucleus contains a surface

correction term. This term depends on the mass number A of the nucleus as

(a) A-1/3 (b) A1/3 (c) A2/3 (d) A

Ans: (c)

Q7. According to the single particles nuclear shell model, the spin-parity of the ground state

of O178 is

(a)

2

1 (b)

2

3 (c)

2

3 (d)

2

5

Ans: (d)

Solution: 8Z and 12/52

2/14

2/32

2/1;9 dppsN

2

5parity-spin,11parity

2

5,2 2Jl

Q8. In the β-decay of neutron n→ p + e- + e , the anti-neutrino e , escapes detection. Its

existence is inferred from the measurement of

(a) energy distribution of electrons (b) angular distribution of electrons

(c) helicity distribution of electrons (d) forward-backward asymmetry of electrons

Ans: (a)

Q9. The isospin and the strangeness of baryon are

(a) 1, -3 (b) 0, -3 (c) 1, 3 (d) 0, 3

Ans: (b)

420

keV187keV5.57

keV0




GATE-2012

Q10. Deuteron has only one bound state with spin parity 1+, isospin 0 and electric quadrupole

moment 0.286 efm2. These data suggest that the nuclear forces are having

(a) only spin and isospin dependence

(b) no spin dependence and no tensor components

(c) spin dependence but no tensor components

(d) spin dependence along with tensor components

Ans: (d)

Q11. The quark content of pandK ,, is indicated:

uudpduusKuus ;;; .

In the process, Kp , considering strong interactions only, which of the

following statements is true?

(a) The process, is allowed because ∆S = 0

(b) The process is allowed because ∆I3 =0

(c) The process is not allowed because ∆S ≠ 0 and ∆I3 ≠ 0

(d) The process is not allowed because the baryon number is violated

Ans: (c)

Solution: kp

1100: S (not conserved)

12

1

2

11:3 I (not conserved)

For strong interaction S and I3 must conserve. Therefore this process is not allowed under

strong interaction

Q12. Which one of the following sets corresponds to fundamental particles?

(a) proton, electron and neutron

(b) proton, electron and photon

(c) electron, photon and neutrino

(d) quark, electron and meson

Ans: (a)




Q13. In case of a Geiger-Muller (GM) counter, which one of the following statement is

CORRECT?

(a) Multiplication factor of the detector is of the order of 1010

(b) Type of the particles detected can be identified

(c) Energy of the particles detected can be distinguished

(d) Operating voltage of the detector is few tens of Volts

Ans: (c)

Q14. Choose the CORRECT statement from the following

(a) Neutron interacts through electromagnetic interaction

(b) Electron does not interact through weak interaction

(c) Neutrino interacts through weak and electromagnetic interaction

(d) Quark interacts through strong interaction but not through weak interaction

Ans: (d)

GATE-2013

Q15. The decay process evepn violates

(a) Baryon number (b) lepton number (c) isospin (d) strangeness

Ans: (c)

Q16. The isospin I and baryon number B of the up quark is

(a) 1,1 BI (b) 3/1,1 BI

(c) 1,2/1 BI (d) 3/1,2/1 BI

Ans: (d)

Q17. In the decay process, the transition 32 , is

(a) allowed both by Fermi and Gamow-Teller selection rule

(b) allowed by Fermi and but not by Gamow-Teller selection rule

(c) not allowed by Fermi but allowed by Gamow-Teller selection rule

(d) not allowed both by Fermi and Gamow-Teller selection rule

Ans: (c)

Solution: According to Fermi Selection Rule:




ChangeNoParityI ,0

According to Gammow-Teller Selection Rule:

ChangeNoParityI ,1,0

In the decay process, the transition 32 ,

ChangeNoParityI ,1 .

GATE-2014

Q18. Which one of the following is a fermions’?

(a) -particle (b) 74 Be nucleus

(c) Hydrogen atom (d) deuteron

Ans: (b)

Solution: If a nucleus contains odd number of nucleons, it is fermions. If a nucleus contains even

number of nucleons, it is a boson.

Q19. Which one of the following three-quark states qqq denoted by X CANNOT be a

possible baryon? The corresponding electric charge is indicated in the superscript.

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

Ans: (d)

Solution: X qqq

2 2 2 62 two unit positive charge

3 3 3 3X uuu

2 2 1 4 11 single unit positive charge

3 3 3 3 3X uud

1 1 11 single unit negative charge

3 3 3X ddd

Not possible with X qqq . So the correct option is (d)

Q20. Consider the process . The minimum kinetic energy of the muons

in the centre of mass frame required to produce the pion pairs at rest is

______ MeV .

Ans: 81.7




Solution: Use conservation of energy and momentum in relativistic form.

2 2105 / 140 /m MeV c and m MeV c

22 2 2

2

m m c m m cE

m

2 2280 210

163.32 105

MeV MeVE MeV

For pair it will be 163.3

81.72

MeV MeV

Q21. A nucleus X undergoes a first forbidden -decay to nucleus Y . If the angular

momentum I and parity P , denoted by PI as 2

7

for X , which of the following is a

possible PI value forY ?

(a) 2

1

(b)2

1

(c)2

3

(d)2

3

Ans: (c)

Solution: For first forbidden -decay; 0,1 or 2I and Parity does change.

GATE-2015

Q22. The decay e is forbidden, because it violates

(a) momentum and lepton number conservations

(b) baryon and lepton number conservations

(c) angular momentum conservation

(d) lepton number conservation

Ans.: (d)

Solution: e . In this decay lepton number is not conserved.

Q23. A beam of X - ray of intensity 0I is incident normally on a metal sheet of thickness

2 mm . The intensity of the transmitted beam is 00.025 I . The linear absorption

coefficient of the metal sheet 1min is _______________ (upto one decimal place)

Ans.: 1844.4

Solution: 0 00 3 3

0

1 1 1ln ln ln 40

2 10 0.025 2 10

x I II I e

x I I

3103

2.303log 40 1.151 10 2 0.3010 1

2 10

11844.4 m




Q24. The mean kinetic energy of a nucleon in a nucleus of atomic weight A varies as nA ,

where n is____________(upto two decimal places)

Ans.: -0.67

Solution:

2 2 2 2220 0

32 2

0 0

14 4 2 2 4 42 2 2

4 / 34 4

R R

R R

d dr dr dr Rm dr r dr m mT

Rr dr r dr

2

32 22 1

330

1 1 1T A

RAR A

20.667 0.67

3n

Q25. The atomic masses of HSmEu 11

15262

15263 ,, and neutron are ,919756.151,921749.151

007825.1 and 008665.1 in atomic mass units (amu), respectively. Using the above

information, the Q - value of the reaction pSmnEu 15262

15263 is ___________ 310

amu (upto three decimal places)

Ans.: 2.833

Solution: 3152.930414 152.927581 2.833 10 . . .Q a m u

Q26. In the nuclear shell model, the potential is modeled as 0,2

1 22 SLrmrV

.

The correct spin-parity and isospin assignments for the ground state of 136 C is

(a) 2

1;

2

1

(b) 2

1;

2

1

(c) 2

1;

2

3

(d) 2

1;

2

3

Ans.: (a)

Solution: 136C , 7, 6N Z , for 7N ;

2 4 1

1 3 1

2 2 2

11 1 1

2S P P j and l

Thus spin- parity is1

2

.




GATE-2016

Q27. In the 3SU quark model, the triplet of mesons ,, 0 has

(a) Isospin 0 , Strangeness 0 (b) Isospin 1 , Strangeness 0

(c) Isospin 2

1 , Strangeness 1 (d) Isospin

2

1 , Strangeness 1

Ans.: (b)

Solution: 0, , are not strange particle thus strangness 0

Since meson group contain 3 particles, thus 1I

Q28. Consider the reaction XCreMn 5424

5425 . The particle X is

(a) (b) e (c) n (d) 0

Ans.: (b)

Q29. Which of the following statements is NOT correct?

(a) A deuteron can be disintegrated by irradiating it with gamma rays of energy MeV4 .

(b) A deuteron has no excited states.

(c) A deuteron has no electric quadrupole moment.

(d) The 01S state of deuteron cannot be formed.

Ans.: (c)

Q30. According to the nuclear shell model, the respective ground state spin-parity values of

O158 and O17

8 nuclei are

(a) 2

1,

2

1

(b) 2

5,

2

1

(c)2

5,

2

3

(d) 2

1,

2

3

Ans.: (b)

Solution: O158 ; 8Z and 7N ; 7 :N 2 4 1

1/ 2 3/ 2 1/ 2s p p

1

2j and 1l . Thus spin and parity

1

2

O178 ; 8Z and 9N ; 9 :N 2 4 2 1

1/ 2 3/ 2 1/ 2 5/ 2s p p d

5


5

2




GATE-2017

Q31. Which one of the following conservation laws is violated in the decay

(a) Angular momentum (b) Total Lepton number

(c) Electric charge (d) Tau number

Ans. : (d)

Solution:

1q 1 1 1 conserved

1L 1 1 1 conserved

1L 0 0 0 Not conserved

spin = 12

1 1 1

2 2 2 conserved

Tau number is not conserved

Q32. Electromagnetic interactions are:

(a) C conserving

(b) C non-conserving but CP conserving

(c) CP non-conserving but CPT conserving

(d) CPT non-conserving

Ans. : (a)

Solution: In electromagnetic interaction C is conserved

CPT: Conserved in all interaction

CP: Conserved in EM and Strong interactions

2

13.6nE eV

n

For 1n , 1 13.6E eV Ground state

For n , 0E Highest state

Thus, correct option is (a)

Q33. In the nuclear reaction 13 136 7eC N X , the particle X is

(a) an electron (b) an anti-electron

(c) a muon (d) a pion

Ans. : (a)




Solution: 13 136 7eC N X 13 13

6 7 eC N X

0eL 0 1 1

To conserve the Lepton number ,eL X should be e

Q34. PJ for the ground state of the 136C nucleus is

(a) 1 (b) 3

2

(c) 3

2

(d) 1

2

Ans. : (d)

Solution: 136 : 6, 7C Z N , 7 :N 2 4 1

1/ 2 3/ 2 1/ 2s p p

1


1

2

Q35. The decays at rest to and v . Assuming the neutrino to be massless, the

momentum of the neutrino is…………….. /MeV c . (up to two decimal places)

( 2 2139 / , 105 /m MeV c m MeV c )

Ans. : 29.84

Solution: 2 2 2

2

m m cE p c

m

So

2 219321 11025 29.84

2 2 139

m m cp MeV

m c c




GATE-2018 Q36. The elementary particle 0 is placed in the baryon decuplet, shown below, at

(a) P (b) Q (c) R (d) S

Ans. : (c)

Q37. In the decay, ee X , what is X ?

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

Ans. : (d) Solution:- e uu e

: 1 0 0 1uL

: 0 1 1 0eL

Q38. For nucleus 164Er , a 2J state is at 90 keV . Assuming 164Er to be a rigid rotor, the

energy of its 4 state is ___________ keV (up to one decimal place)

Ans. : 300

Solution: 1JE hcBJ J _________ 4

22 2 1E hc B and 4

4 4 1E hc B _________ 2

Then, 4

2

20

6

E

E

4

2090 300

6E keV keV

P R

S

3rd component of isospin

Str

ange

ness

Q

0

0

0

Q

P R




Q39. Inside a large nucleus, a nucleon with mass 2939 MeVc has Fermi momentum 11.40 fm

at absolute zero temperature. Its velocity is Xc , where the value of X is__________ (up

to two decimal places).

( 197 -fmc MeV )

Ans. : 29.0

Solution: Here, fermi – momentum or fermi radius, 11.40Fk fm and 197c Mev – fm

Now, Fermi velocity –

FF

kPV

m m

2

Fc k c

mc

197 1 40

939

c

275 8

939

c 0.29c

Q40. An particle is emitted by a 23090 Th nucleus. Assuming the potential to be purely

Coulombic beyond the point of separation, the height of the Coulomb barrier is________

MeV (up to two decimal places).

(2

00

1.44 -fm, 1.30fm4

eMeV r

)

Ans. : 995.25

Solution: The height of coulomb barrier for particle from

422688

23090 2HeXTh ( - particle)

R

zeVC

2

0

2

4

1

Here, fmMeVe

fmR 44.14

,3.10

2

0

And 3/10 ARR

Here, we consider pure Coulombic interection

67.758.109.64226 3/13/13/13/13/1 AAA XTh

67.73.13/10 ThARR

Hence, fm

MeVeVC 67.73.1

44.1180

67.73.1

902

4 0

2

MeVVC 995.25




GATE-2019

Q41. Considering baryon number and lepton number conservation laws, which of the

following process is/are allowed?

(i) 0ep e v

(ii) ee v v

(a) both (i) and (ii) (b) only (i) (c) only (ii) (d) neither (i) nor (ii)

Ans. : (c)

Solution: (i) 0eP e

: 1 0 0 0B : Not conserved

Therefore, this is not an allowed process

(ii) ee

: 1 0 1 0q : conserved

: 1/ 2 1/ 2 1/ 2 1/ 2spin : conserved

: 1 1 0 0eL : conserved

: 0 0 1 1L : conserved

Since neutrino is involve, therefore parity is violated. This is allowed through weak

interaction

Q42. A massive particle X in free space decays spontaneously into two photons. Which of the

following statements is true for X ?

(a) X is charged

(b) Spin of X must be greater than or equal to 2

(c) X is a boson

(d) X must be a baryon

Ans. : (c)

Solution: X r r

: 0 0 0q

: 0,1,2 1 1spin

Thus spin of X can be either 0,1 or 2 . (integer)

Therefore, option (b) is wrong while option (c) is correct.




Q43. The nuclear spin and parity of 4020 Ca in its ground state is

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

Ans.: (a)

Solution: 4020Ca is an even-even nuclei, therefore 0,I P ve

Spin-parity 0

Q44. Low energy collision ( s - wave scattering) of pion ( ) with deuteron ( d ) results in the

production of two proton ( d p p ). The relative orbital angular momentum (in

units of ) of the resulting two-proton system for this reaction is

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

Ans.: (b)

Solution: d p p

Parity: 1 1 ( 1)lp p

1 1l

p p

Since 1p 1 1l

Thus, 1l .

Q45. A radioactive element X has a half-life of 30 hours. It decays via alpha, beta and

gamma emissions with the branching ratio for beta decay being 0.75 . The partial half-life

for beta decay in unit of hours is ____________

Ans.: 40

Solution: Branching ratio is the fraction of particles (here ) which decays by an individual

decay mode with respect to the total number of particles which decays

1/ 2

1/ 2

x x

dNTdt

BRdt Tdt

1/ 21/ 2

3040

0.75x

TT

BR hours

mathematical physics solutions · fiziks institute for net/jrf, gate, iit‐jam, m.sc. entrance,...

Documents