Physics - 2021 Past Year Paper | IIT JAM Past Year Papers and Model Test Paper (All Branches) PDF Download

 Page 1


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2021 
SECTION – A 
cos x
e
Multiple Choice Questions (MCQ) 
Q1 – Q10 Carry One Mark Each 
Q1. The function  is Taylor expanded about 0 x = . The coefficient of 
2
x is  
(a) 
1
2
-  (b) 
2
e
-  (c) 
2
e
  (d) Zero  
Ans. : (b)  
Solution: ( )
cos x
fx e = 
Taylor expansion about 
0
xx = is 
( ) ( ) ( ) ( )
( )
( )
2
0
0 0 0 0
...
2!
xx
fx fx x x f x f x
-
' '' = + - + +8 
? Taylor expansion about 0 x = 
( ) ( ) ( )
( )
2
0
00
2!
f
f x f xf x
''
' = + + + - -8 
Coefficient of 
2
x is 
( ) 0
2!
f ''
 
( )
cos x
fx e = ( )
cos
sin
x
f x xe ' ? =- 
( ) ( ) ( )
cos cos 2 cos
cos sin sin cos sin
xx x
f x xe x xe x x e
??
'' =- + - =- +
??
 
( ) ( )
1
0 10 f e e '' ? = -+ =- 
? coefficients of 
( )
2
0
2! 2
f
e
x
''
-
= =  
Q2. Let M be a 22 × matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues 
are  
(a) 2 and 4  (b) 3 and 3  (c) 2 and 6  (d) 2 - and 3 -  
Ans. : (a)  
Solution: M is a 22 × matrix  
We know trace = sum of eigenvalues  
and determinant = product of eigenvalues  
Let eigenvalues  are
1
? and 
2
? . 
12
6 ?? ?+ =  and  
12
8 ?? = 
1
2 ? = , 
2
4 ? =  
Page 2


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2021 
SECTION – A 
cos x
e
Multiple Choice Questions (MCQ) 
Q1 – Q10 Carry One Mark Each 
Q1. The function  is Taylor expanded about 0 x = . The coefficient of 
2
x is  
(a) 
1
2
-  (b) 
2
e
-  (c) 
2
e
  (d) Zero  
Ans. : (b)  
Solution: ( )
cos x
fx e = 
Taylor expansion about 
0
xx = is 
( ) ( ) ( ) ( )
( )
( )
2
0
0 0 0 0
...
2!
xx
fx fx x x f x f x
-
' '' = + - + +8 
? Taylor expansion about 0 x = 
( ) ( ) ( )
( )
2
0
00
2!
f
f x f xf x
''
' = + + + - -8 
Coefficient of 
2
x is 
( ) 0
2!
f ''
 
( )
cos x
fx e = ( )
cos
sin
x
f x xe ' ? =- 
( ) ( ) ( )
cos cos 2 cos
cos sin sin cos sin
xx x
f x xe x xe x x e
??
'' =- + - =- +
??
 
( ) ( )
1
0 10 f e e '' ? = -+ =- 
? coefficients of 
( )
2
0
2! 2
f
e
x
''
-
= =  
Q2. Let M be a 22 × matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues 
are  
(a) 2 and 4  (b) 3 and 3  (c) 2 and 6  (d) 2 - and 3 -  
Ans. : (a)  
Solution: M is a 22 × matrix  
We know trace = sum of eigenvalues  
and determinant = product of eigenvalues  
Let eigenvalues  are
1
? and 
2
? . 
12
6 ?? ?+ =  and  
12
8 ?? = 
1
2 ? = , 
2
4 ? =  
 
 
  
 
 
 
 
 
  
 
   
 
 
Q3. A planet is in a highly eccentric orbit about a star. The distance of its closest approach is 
300 times smaller than its farthest distance from the star. If the corresponding speeds are 
c
v and 
f
v , then 
c
f
v
v
 is 
(a) 
1
300
  (b) 
1
300
  (c) 300  (d) 300  
Ans. : (d)    
Solution: Using conservation of angular momentum  
300
300
f
cc
c c f f
f c c
r
vr
mv r mv r
vr r
= ?= = = 
Q4. An object of density ? is floating in a liquid with 75% of its volume submerged. The 
density of the liquid is   
(a) 
4
3
?  (b) 
3
2
?  (c) 
8
5
?  (d) 2 ?  
Ans. : (a)    
Solution:  Weight of object equal to buoyancy force  
 
34
43
V g Vdg d ?? = ?= 
Q5. An experiment with a Michelson interferometer is performed in vacuum using a laser of 
wavelength 610nm . One of the beams of the interferometer passes through a small glass 
cavity 1.3cm long. After the cavity is completely filled with a medium of refractive 
index ,472 n dark fringes are counted to move past a reference line. Given that the speed 
of light is 
8
3 10 / ms × , the value of n is  
(a) 1.01  (b) 1.04  (c) 1.06  (d) 1.10  
Ans. : (a)  
Solution: Change in optical path xm ? ?= ( ) 2 1 n tm ? ? -= 
9
2
472 610 10
1
2 2 1 3 10
n
n
t
?
-
-
××
? -= =
× · ×
 
1 0 011 n ? -= · 1 011 n ?= · 
Page 3


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2021 
SECTION – A 
cos x
e
Multiple Choice Questions (MCQ) 
Q1 – Q10 Carry One Mark Each 
Q1. The function  is Taylor expanded about 0 x = . The coefficient of 
2
x is  
(a) 
1
2
-  (b) 
2
e
-  (c) 
2
e
  (d) Zero  
Ans. : (b)  
Solution: ( )
cos x
fx e = 
Taylor expansion about 
0
xx = is 
( ) ( ) ( ) ( )
( )
( )
2
0
0 0 0 0
...
2!
xx
fx fx x x f x f x
-
' '' = + - + +8 
? Taylor expansion about 0 x = 
( ) ( ) ( )
( )
2
0
00
2!
f
f x f xf x
''
' = + + + - -8 
Coefficient of 
2
x is 
( ) 0
2!
f ''
 
( )
cos x
fx e = ( )
cos
sin
x
f x xe ' ? =- 
( ) ( ) ( )
cos cos 2 cos
cos sin sin cos sin
xx x
f x xe x xe x x e
??
'' =- + - =- +
??
 
( ) ( )
1
0 10 f e e '' ? = -+ =- 
? coefficients of 
( )
2
0
2! 2
f
e
x
''
-
= =  
Q2. Let M be a 22 × matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues 
are  
(a) 2 and 4  (b) 3 and 3  (c) 2 and 6  (d) 2 - and 3 -  
Ans. : (a)  
Solution: M is a 22 × matrix  
We know trace = sum of eigenvalues  
and determinant = product of eigenvalues  
Let eigenvalues  are
1
? and 
2
? . 
12
6 ?? ?+ =  and  
12
8 ?? = 
1
2 ? = , 
2
4 ? =  
 
 
  
 
 
 
 
 
  
 
   
 
 
Q3. A planet is in a highly eccentric orbit about a star. The distance of its closest approach is 
300 times smaller than its farthest distance from the star. If the corresponding speeds are 
c
v and 
f
v , then 
c
f
v
v
 is 
(a) 
1
300
  (b) 
1
300
  (c) 300  (d) 300  
Ans. : (d)    
Solution: Using conservation of angular momentum  
300
300
f
cc
c c f f
f c c
r
vr
mv r mv r
vr r
= ?= = = 
Q4. An object of density ? is floating in a liquid with 75% of its volume submerged. The 
density of the liquid is   
(a) 
4
3
?  (b) 
3
2
?  (c) 
8
5
?  (d) 2 ?  
Ans. : (a)    
Solution:  Weight of object equal to buoyancy force  
 
34
43
V g Vdg d ?? = ?= 
Q5. An experiment with a Michelson interferometer is performed in vacuum using a laser of 
wavelength 610nm . One of the beams of the interferometer passes through a small glass 
cavity 1.3cm long. After the cavity is completely filled with a medium of refractive 
index ,472 n dark fringes are counted to move past a reference line. Given that the speed 
of light is 
8
3 10 / ms × , the value of n is  
(a) 1.01  (b) 1.04  (c) 1.06  (d) 1.10  
Ans. : (a)  
Solution: Change in optical path xm ? ?= ( ) 2 1 n tm ? ? -= 
9
2
472 610 10
1
2 2 1 3 10
n
n
t
?
-
-
××
? -= =
× · ×
 
1 0 011 n ? -= · 1 011 n ?= · 
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
Q6. For a semiconductor material, the conventional flat band energy diagram is shown in the 
figure. The variables , YX respectively, are 
(a) Energy, Momentum  
(b) Energy, Distance 
(c) Distance, Energy  
(d) Momentum, Energy  
Ans. : (b)  
Solution: Along the y-axis energy varies while along x-axis distance is variable.  
Q7. For the given circuit, 
D
V is the threshold voltage of the diode. The graph that best 
deposits variation of 
0
V with 
i
V is 
 
(a) 
 
(b) 
 
(c) 
 
(d) 
 
 
0
i
V
0
V
0
i
V
0
V
0
i
V
0
V
D
V 0
i
V
0
V
i
V
1k ?
1k ?
0
V
+
-
X
valence band
conduction band
Y
Page 4


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2021 
SECTION – A 
cos x
e
Multiple Choice Questions (MCQ) 
Q1 – Q10 Carry One Mark Each 
Q1. The function  is Taylor expanded about 0 x = . The coefficient of 
2
x is  
(a) 
1
2
-  (b) 
2
e
-  (c) 
2
e
  (d) Zero  
Ans. : (b)  
Solution: ( )
cos x
fx e = 
Taylor expansion about 
0
xx = is 
( ) ( ) ( ) ( )
( )
( )
2
0
0 0 0 0
...
2!
xx
fx fx x x f x f x
-
' '' = + - + +8 
? Taylor expansion about 0 x = 
( ) ( ) ( )
( )
2
0
00
2!
f
f x f xf x
''
' = + + + - -8 
Coefficient of 
2
x is 
( ) 0
2!
f ''
 
( )
cos x
fx e = ( )
cos
sin
x
f x xe ' ? =- 
( ) ( ) ( )
cos cos 2 cos
cos sin sin cos sin
xx x
f x xe x xe x x e
??
'' =- + - =- +
??
 
( ) ( )
1
0 10 f e e '' ? = -+ =- 
? coefficients of 
( )
2
0
2! 2
f
e
x
''
-
= =  
Q2. Let M be a 22 × matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues 
are  
(a) 2 and 4  (b) 3 and 3  (c) 2 and 6  (d) 2 - and 3 -  
Ans. : (a)  
Solution: M is a 22 × matrix  
We know trace = sum of eigenvalues  
and determinant = product of eigenvalues  
Let eigenvalues  are
1
? and 
2
? . 
12
6 ?? ?+ =  and  
12
8 ?? = 
1
2 ? = , 
2
4 ? =  
 
 
  
 
 
 
 
 
  
 
   
 
 
Q3. A planet is in a highly eccentric orbit about a star. The distance of its closest approach is 
300 times smaller than its farthest distance from the star. If the corresponding speeds are 
c
v and 
f
v , then 
c
f
v
v
 is 
(a) 
1
300
  (b) 
1
300
  (c) 300  (d) 300  
Ans. : (d)    
Solution: Using conservation of angular momentum  
300
300
f
cc
c c f f
f c c
r
vr
mv r mv r
vr r
= ?= = = 
Q4. An object of density ? is floating in a liquid with 75% of its volume submerged. The 
density of the liquid is   
(a) 
4
3
?  (b) 
3
2
?  (c) 
8
5
?  (d) 2 ?  
Ans. : (a)    
Solution:  Weight of object equal to buoyancy force  
 
34
43
V g Vdg d ?? = ?= 
Q5. An experiment with a Michelson interferometer is performed in vacuum using a laser of 
wavelength 610nm . One of the beams of the interferometer passes through a small glass 
cavity 1.3cm long. After the cavity is completely filled with a medium of refractive 
index ,472 n dark fringes are counted to move past a reference line. Given that the speed 
of light is 
8
3 10 / ms × , the value of n is  
(a) 1.01  (b) 1.04  (c) 1.06  (d) 1.10  
Ans. : (a)  
Solution: Change in optical path xm ? ?= ( ) 2 1 n tm ? ? -= 
9
2
472 610 10
1
2 2 1 3 10
n
n
t
?
-
-
××
? -= =
× · ×
 
1 0 011 n ? -= · 1 011 n ?= · 
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
Q6. For a semiconductor material, the conventional flat band energy diagram is shown in the 
figure. The variables , YX respectively, are 
(a) Energy, Momentum  
(b) Energy, Distance 
(c) Distance, Energy  
(d) Momentum, Energy  
Ans. : (b)  
Solution: Along the y-axis energy varies while along x-axis distance is variable.  
Q7. For the given circuit, 
D
V is the threshold voltage of the diode. The graph that best 
deposits variation of 
0
V with 
i
V is 
 
(a) 
 
(b) 
 
(c) 
 
(d) 
 
 
0
i
V
0
V
0
i
V
0
V
0
i
V
0
V
D
V 0
i
V
0
V
i
V
1k ?
1k ?
0
V
+
-
X
valence band
conduction band
Y
 
 
  
 
 
 
 
 
  
 
   
 
 
Ans. : (a)  
Solution: 
Q8.  Arrange the following telescopes, where D is the telescope diameter and ? is the 
wavelength, in order of decreasing resolving power:  
I. 100 , 21 D m cm ? = =                                II. 2 , 500 D m nm ? = = 
III. 1 , 100 D m nm ? = =                               IV. 2 , 10 D m mm ? = = 
(a) III, II, IV, I     (b) II, III, I, IV  
(c) IV, III, II, I     (d) III, II, I, IV  
Ans. : (d)  
Solution: 1.22 d
D
?
? = 
3
2.56 10
I
d ?
-
= × ;                         
7
3.05 10
II
d?
-
= × 
7
1.22 10
III
d ?
-
= × ;                       
3
6.1 10
IV
d ?
-
= × 
Resolving power 
1
d ?
= 
 
III II I IV
RP RP RP RP > >> 
Q9. Metallic lithium has bcc crystal structure. Each unit cell is a cube of side a . The number 
of atoms per unit volume is  
(a) 
3
1
a
   (b) 
3
2
2a
  (c) 
3
2
a
   (d) 
3
4
a
  
• •
• •
+
-
.. oc
.. sc i
V
1k ?
1k ?
0
V
+
-
Negative half Cycle
Postive half Cycle
.. oc
.. sc
i
V
1k ?
1k ?
0
V
+
-
+
-
• •
• • 0 =
0
1
1
ii
k
V VV
k
=-=
Page 5


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2021 
SECTION – A 
cos x
e
Multiple Choice Questions (MCQ) 
Q1 – Q10 Carry One Mark Each 
Q1. The function  is Taylor expanded about 0 x = . The coefficient of 
2
x is  
(a) 
1
2
-  (b) 
2
e
-  (c) 
2
e
  (d) Zero  
Ans. : (b)  
Solution: ( )
cos x
fx e = 
Taylor expansion about 
0
xx = is 
( ) ( ) ( ) ( )
( )
( )
2
0
0 0 0 0
...
2!
xx
fx fx x x f x f x
-
' '' = + - + +8 
? Taylor expansion about 0 x = 
( ) ( ) ( )
( )
2
0
00
2!
f
f x f xf x
''
' = + + + - -8 
Coefficient of 
2
x is 
( ) 0
2!
f ''
 
( )
cos x
fx e = ( )
cos
sin
x
f x xe ' ? =- 
( ) ( ) ( )
cos cos 2 cos
cos sin sin cos sin
xx x
f x xe x xe x x e
??
'' =- + - =- +
??
 
( ) ( )
1
0 10 f e e '' ? = -+ =- 
? coefficients of 
( )
2
0
2! 2
f
e
x
''
-
= =  
Q2. Let M be a 22 × matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues 
are  
(a) 2 and 4  (b) 3 and 3  (c) 2 and 6  (d) 2 - and 3 -  
Ans. : (a)  
Solution: M is a 22 × matrix  
We know trace = sum of eigenvalues  
and determinant = product of eigenvalues  
Let eigenvalues  are
1
? and 
2
? . 
12
6 ?? ?+ =  and  
12
8 ?? = 
1
2 ? = , 
2
4 ? =  
 
 
  
 
 
 
 
 
  
 
   
 
 
Q3. A planet is in a highly eccentric orbit about a star. The distance of its closest approach is 
300 times smaller than its farthest distance from the star. If the corresponding speeds are 
c
v and 
f
v , then 
c
f
v
v
 is 
(a) 
1
300
  (b) 
1
300
  (c) 300  (d) 300  
Ans. : (d)    
Solution: Using conservation of angular momentum  
300
300
f
cc
c c f f
f c c
r
vr
mv r mv r
vr r
= ?= = = 
Q4. An object of density ? is floating in a liquid with 75% of its volume submerged. The 
density of the liquid is   
(a) 
4
3
?  (b) 
3
2
?  (c) 
8
5
?  (d) 2 ?  
Ans. : (a)    
Solution:  Weight of object equal to buoyancy force  
 
34
43
V g Vdg d ?? = ?= 
Q5. An experiment with a Michelson interferometer is performed in vacuum using a laser of 
wavelength 610nm . One of the beams of the interferometer passes through a small glass 
cavity 1.3cm long. After the cavity is completely filled with a medium of refractive 
index ,472 n dark fringes are counted to move past a reference line. Given that the speed 
of light is 
8
3 10 / ms × , the value of n is  
(a) 1.01  (b) 1.04  (c) 1.06  (d) 1.10  
Ans. : (a)  
Solution: Change in optical path xm ? ?= ( ) 2 1 n tm ? ? -= 
9
2
472 610 10
1
2 2 1 3 10
n
n
t
?
-
-
××
? -= =
× · ×
 
1 0 011 n ? -= · 1 011 n ?= · 
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
Q6. For a semiconductor material, the conventional flat band energy diagram is shown in the 
figure. The variables , YX respectively, are 
(a) Energy, Momentum  
(b) Energy, Distance 
(c) Distance, Energy  
(d) Momentum, Energy  
Ans. : (b)  
Solution: Along the y-axis energy varies while along x-axis distance is variable.  
Q7. For the given circuit, 
D
V is the threshold voltage of the diode. The graph that best 
deposits variation of 
0
V with 
i
V is 
 
(a) 
 
(b) 
 
(c) 
 
(d) 
 
 
0
i
V
0
V
0
i
V
0
V
0
i
V
0
V
D
V 0
i
V
0
V
i
V
1k ?
1k ?
0
V
+
-
X
valence band
conduction band
Y
 
 
  
 
 
 
 
 
  
 
   
 
 
Ans. : (a)  
Solution: 
Q8.  Arrange the following telescopes, where D is the telescope diameter and ? is the 
wavelength, in order of decreasing resolving power:  
I. 100 , 21 D m cm ? = =                                II. 2 , 500 D m nm ? = = 
III. 1 , 100 D m nm ? = =                               IV. 2 , 10 D m mm ? = = 
(a) III, II, IV, I     (b) II, III, I, IV  
(c) IV, III, II, I     (d) III, II, I, IV  
Ans. : (d)  
Solution: 1.22 d
D
?
? = 
3
2.56 10
I
d ?
-
= × ;                         
7
3.05 10
II
d?
-
= × 
7
1.22 10
III
d ?
-
= × ;                       
3
6.1 10
IV
d ?
-
= × 
Resolving power 
1
d ?
= 
 
III II I IV
RP RP RP RP > >> 
Q9. Metallic lithium has bcc crystal structure. Each unit cell is a cube of side a . The number 
of atoms per unit volume is  
(a) 
3
1
a
   (b) 
3
2
2a
  (c) 
3
2
a
   (d) 
3
4
a
  
• •
• •
+
-
.. oc
.. sc i
V
1k ?
1k ?
0
V
+
-
Negative half Cycle
Postive half Cycle
.. oc
.. sc
i
V
1k ?
1k ?
0
V
+
-
+
-
• •
• • 0 =
0
1
1
ii
k
V VV
k
=-=
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
Ans. : (c)  
Solution: The effective number of atoms in BCC is  
1
81 2
8
eff
n = × += 
 The volume of the bcc unit cell of side a is 
3
a . 
 Thus, the number of atoms per unit volume is 
3
2
a
= .  
Therefore, the correct option is (c). 
Q10. The moment of inertia of a solid sphere (radius R and mass M ) about the axis which is 
at a distance of 
2
R
 from the centre is  
(a) 
2
3
20
MR  (b) 
2
1
2
MR  (c) 
2
13
20
MR  (d) 
2
9
10
MR 
Ans. : (c)  
Solution:  
Using parallel axis theorem  
 
2
0. C M
I I Md = + ; 
2
R
d = ,
2
.
2
5
C M
I MR = 
 
2
2
2 2 22
0
2 2 8 5 13
5 2 5 4 20 20
R MR
I MR M MR MR MR
+ ?? ? ?
= + = += =
?? ? ?
?? ? ?
 
Q11 – Q30. carry two marks each 
Q11. Let ( ) , xy denote the coordinates in a rectangular Cartesian coordinate system C . Let 
( ) , xy '' denote the coordinates in another coordinate system C ' defined by 
     23 x xy ' = + 
     34 y xy '=-+ 
The area element in C ' , is 
(a) 
1
17
dx dy ''  (b) 12dx dy ''  (c) dx dy ''  (d) x dx dy '' '  
Ans. : (a)  
Solution: ( ) ( ) ,, C xy C x y ' '' ? 
dxdy Jdx dy '' = 
where 
( )
( )
,
,
xy
J xy
xx
J
xy
Jx y
yy
??
'' ??
= =
?? ''
'' ??