JEE Mains 31 August 2021 Question Paper Shift 1 | JEE Main & Advanced Previous Year Papers PDF Download

 Page 1


 
    1 
FINAL JEE –MAIN EXAMINATION – AUGUST, 2021 
(Held On Tuesday 31
st
 August, 2021)              TIME : 9 : 00 AM  to  12 : 00 NOON 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. A helicopter is flying horizontally with a speed 'v' 
at an altitude 'h' has to drop a food packet for a 
man on the ground. What is the distance of 
helicopter from the man when the food packet is 
dropped? 
 (1) 
?
2
2
2ghv 1
h
 (2) ?
22
2ghv h 
 (3) ?
2
2
2v h
h
g
 (4) ?
2
2
2gh
h
v
  
 Official Ans. by NTA (3) 
Sol. 
 
v 
helicopter 
h D 
Man 
R 
 
 
2h
Rv
g
?? 
 
22
D R h ?? 
 
2
2
2h
vh
g
??
? ? ?
??
??
??
 
 
2
2
2hv
Dh
g
?? 
 Option (3) is correct  
2. In the following logic circuit the sequence of the 
inputs A, B are (0, 0), (0,1), (1, 0) and (1, 1). The 
output Y for this sequence will be : 
 
A 
B 
P 
Q 
Y 
 
 (1) 1, 0, 1, 0  (2) 0, 1, 0, 1  
 (3) 1, 1, 1, 0  (4) 0, 0, 1, 1  
 Official Ans. by NTA (3) 
Sol. 
 
A 
B 
Y 
A·B 
A+B 
 
 ? ? ? ? Y A B A B ? ? ? ? 
 ?
? ? 0,0
Y1 ? 
 ?
? ? 0,1
Y1 ? 
 ?
? ? 1,0
Y1 ? 
 ?
? ? 1,1
Y0 ? 
 Option (3) is correct 
3. Two particles A and B having charges 20 ?C and  
–5 ?C respectively are held fixed with a separation 
of 5 cm. At what position a third charged particle 
should be placed so that it does not experience a 
net electric force? 
 
–5?C 
B 
5 cm 
A 
20?C 
 
 (1) At 5 cm from 20 ?C on the left side of system 
 (2) At 5 cm from – 5 ?C on the right side  
 (3) At 1.25 cm from – 5 ?C between two charges 
 (4) At midpoint between two charges  
 Official Ans. by NTA (2) 
Sol.
  
–5?C 20?C  
 Null point is possible only right side of –5µC 
  
N 
x 
20µC –5µC 
 
 
? ? ? ?
? ?
N 2 2
k 5 C k 20 C
E0
x
5x
? ? ?
? ? ? ?
?
 
 x = 5 cm 
 ? option (2) is correct 
Page 2


 
    1 
FINAL JEE –MAIN EXAMINATION – AUGUST, 2021 
(Held On Tuesday 31
st
 August, 2021)              TIME : 9 : 00 AM  to  12 : 00 NOON 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. A helicopter is flying horizontally with a speed 'v' 
at an altitude 'h' has to drop a food packet for a 
man on the ground. What is the distance of 
helicopter from the man when the food packet is 
dropped? 
 (1) 
?
2
2
2ghv 1
h
 (2) ?
22
2ghv h 
 (3) ?
2
2
2v h
h
g
 (4) ?
2
2
2gh
h
v
  
 Official Ans. by NTA (3) 
Sol. 
 
v 
helicopter 
h D 
Man 
R 
 
 
2h
Rv
g
?? 
 
22
D R h ?? 
 
2
2
2h
vh
g
??
? ? ?
??
??
??
 
 
2
2
2hv
Dh
g
?? 
 Option (3) is correct  
2. In the following logic circuit the sequence of the 
inputs A, B are (0, 0), (0,1), (1, 0) and (1, 1). The 
output Y for this sequence will be : 
 
A 
B 
P 
Q 
Y 
 
 (1) 1, 0, 1, 0  (2) 0, 1, 0, 1  
 (3) 1, 1, 1, 0  (4) 0, 0, 1, 1  
 Official Ans. by NTA (3) 
Sol. 
 
A 
B 
Y 
A·B 
A+B 
 
 ? ? ? ? Y A B A B ? ? ? ? 
 ?
? ? 0,0
Y1 ? 
 ?
? ? 0,1
Y1 ? 
 ?
? ? 1,0
Y1 ? 
 ?
? ? 1,1
Y0 ? 
 Option (3) is correct 
3. Two particles A and B having charges 20 ?C and  
–5 ?C respectively are held fixed with a separation 
of 5 cm. At what position a third charged particle 
should be placed so that it does not experience a 
net electric force? 
 
–5?C 
B 
5 cm 
A 
20?C 
 
 (1) At 5 cm from 20 ?C on the left side of system 
 (2) At 5 cm from – 5 ?C on the right side  
 (3) At 1.25 cm from – 5 ?C between two charges 
 (4) At midpoint between two charges  
 Official Ans. by NTA (2) 
Sol.
  
–5?C 20?C  
 Null point is possible only right side of –5µC 
  
N 
x 
20µC –5µC 
 
 
? ? ? ?
? ?
N 2 2
k 5 C k 20 C
E0
x
5x
? ? ?
? ? ? ?
?
 
 x = 5 cm 
 ? option (2) is correct 
 
2 
 
 
4. A reversible engine has an efficiency of 
1
4
. If the 
temperature of the sink is reduced by 58°C, its 
efficiency becomes double. Calculate the 
temperature of the sink : 
 (1) 174°C  (2) 280°C 
 (3) 180.4°C  (4) 382°C 
 Official Ans. by NTA (1) 
 Official Ans. by ALLEN (Bonus) 
Sol. T
2
 = sink temperature 
 
2
1
T
1
T
? ? ? 
 
2
1
T 1
1
4T
?? 
 
2
1
T 3
T4
? … (i) 
 
2
1
T 58 1
1
2T
?
?? 
 
2
11
T 58 1
T T 2
?? 
 
1
3 58 1
4 T 2
?? 
 
1
1
1 58
T 232
4T
? ? ? 
 
2
3
T 232
4
?? 
 T
2
 = 174 K 
5. An object is placed at the focus of concave lens 
having focal length f. What is the magnification 
and distance of the image from the optical centre 
of the lens? 
 (1) 1, ?  (2) Very high, ?  
 (3) 
1
,
22
f
  (4) 
1
,
44
f
  
 Official Ans. by NTA (3) 
Sol. 
 
f 
 
 U = – f 
 
1 1 1 1 2
V U f V f
? ? ? ? ?
?
 
 
f
V
2
?
? 
 
V1
m
U2
?? 
 distance = 
f
2
 
 Option (3) 
6. A sample of a radioactive nucleus A disintegrates 
to another radioactive nucleus B, which in turn 
disintegrates to some other stable nucleus C. Plot 
of a graph showing the variation of number of 
atoms of nucleus B vesus time is : 
 (Assume that at t = 0, there are no B atoms in the 
sample) 
 (1) 
 
time 
No. of atoms 
 (2) 
 
time 
No. of atoms 
  
 (3) 
 
time 
No. of atoms 
 (4) 
 
time 
No. of atoms 
 
 Official Ans. by NTA (2) 
Sol. 
 
A B 
C (stable) 
 
 Initially no. of atoms of B = 0 after t = 0, no. of 
atoms of B will starts increasing & reaches 
maximum value when rate of decay of B = rate of 
formation of B. 
 After that maximum value, no. of atoms will starts 
decreasing as growth & decay both are exponential 
functions, so best possible graph is (2) 
 Option (2)  
Page 3


 
    1 
FINAL JEE –MAIN EXAMINATION – AUGUST, 2021 
(Held On Tuesday 31
st
 August, 2021)              TIME : 9 : 00 AM  to  12 : 00 NOON 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. A helicopter is flying horizontally with a speed 'v' 
at an altitude 'h' has to drop a food packet for a 
man on the ground. What is the distance of 
helicopter from the man when the food packet is 
dropped? 
 (1) 
?
2
2
2ghv 1
h
 (2) ?
22
2ghv h 
 (3) ?
2
2
2v h
h
g
 (4) ?
2
2
2gh
h
v
  
 Official Ans. by NTA (3) 
Sol. 
 
v 
helicopter 
h D 
Man 
R 
 
 
2h
Rv
g
?? 
 
22
D R h ?? 
 
2
2
2h
vh
g
??
? ? ?
??
??
??
 
 
2
2
2hv
Dh
g
?? 
 Option (3) is correct  
2. In the following logic circuit the sequence of the 
inputs A, B are (0, 0), (0,1), (1, 0) and (1, 1). The 
output Y for this sequence will be : 
 
A 
B 
P 
Q 
Y 
 
 (1) 1, 0, 1, 0  (2) 0, 1, 0, 1  
 (3) 1, 1, 1, 0  (4) 0, 0, 1, 1  
 Official Ans. by NTA (3) 
Sol. 
 
A 
B 
Y 
A·B 
A+B 
 
 ? ? ? ? Y A B A B ? ? ? ? 
 ?
? ? 0,0
Y1 ? 
 ?
? ? 0,1
Y1 ? 
 ?
? ? 1,0
Y1 ? 
 ?
? ? 1,1
Y0 ? 
 Option (3) is correct 
3. Two particles A and B having charges 20 ?C and  
–5 ?C respectively are held fixed with a separation 
of 5 cm. At what position a third charged particle 
should be placed so that it does not experience a 
net electric force? 
 
–5?C 
B 
5 cm 
A 
20?C 
 
 (1) At 5 cm from 20 ?C on the left side of system 
 (2) At 5 cm from – 5 ?C on the right side  
 (3) At 1.25 cm from – 5 ?C between two charges 
 (4) At midpoint between two charges  
 Official Ans. by NTA (2) 
Sol.
  
–5?C 20?C  
 Null point is possible only right side of –5µC 
  
N 
x 
20µC –5µC 
 
 
? ? ? ?
? ?
N 2 2
k 5 C k 20 C
E0
x
5x
? ? ?
? ? ? ?
?
 
 x = 5 cm 
 ? option (2) is correct 
 
2 
 
 
4. A reversible engine has an efficiency of 
1
4
. If the 
temperature of the sink is reduced by 58°C, its 
efficiency becomes double. Calculate the 
temperature of the sink : 
 (1) 174°C  (2) 280°C 
 (3) 180.4°C  (4) 382°C 
 Official Ans. by NTA (1) 
 Official Ans. by ALLEN (Bonus) 
Sol. T
2
 = sink temperature 
 
2
1
T
1
T
? ? ? 
 
2
1
T 1
1
4T
?? 
 
2
1
T 3
T4
? … (i) 
 
2
1
T 58 1
1
2T
?
?? 
 
2
11
T 58 1
T T 2
?? 
 
1
3 58 1
4 T 2
?? 
 
1
1
1 58
T 232
4T
? ? ? 
 
2
3
T 232
4
?? 
 T
2
 = 174 K 
5. An object is placed at the focus of concave lens 
having focal length f. What is the magnification 
and distance of the image from the optical centre 
of the lens? 
 (1) 1, ?  (2) Very high, ?  
 (3) 
1
,
22
f
  (4) 
1
,
44
f
  
 Official Ans. by NTA (3) 
Sol. 
 
f 
 
 U = – f 
 
1 1 1 1 2
V U f V f
? ? ? ? ?
?
 
 
f
V
2
?
? 
 
V1
m
U2
?? 
 distance = 
f
2
 
 Option (3) 
6. A sample of a radioactive nucleus A disintegrates 
to another radioactive nucleus B, which in turn 
disintegrates to some other stable nucleus C. Plot 
of a graph showing the variation of number of 
atoms of nucleus B vesus time is : 
 (Assume that at t = 0, there are no B atoms in the 
sample) 
 (1) 
 
time 
No. of atoms 
 (2) 
 
time 
No. of atoms 
  
 (3) 
 
time 
No. of atoms 
 (4) 
 
time 
No. of atoms 
 
 Official Ans. by NTA (2) 
Sol. 
 
A B 
C (stable) 
 
 Initially no. of atoms of B = 0 after t = 0, no. of 
atoms of B will starts increasing & reaches 
maximum value when rate of decay of B = rate of 
formation of B. 
 After that maximum value, no. of atoms will starts 
decreasing as growth & decay both are exponential 
functions, so best possible graph is (2) 
 Option (2)  
 
 
3 
 
 
7. A coil having N turns is wound tightly in the form 
of a spiral with inner and outer radii 'a' and 'b' 
respectively. Find the magnetic field at centre, 
when a current I passes through coil : 
 (1) 
? ??
??
?
??
0
e
IN b
log
2(b a) a
 (2) 
? ? ??
??
?
??
0
I ab
8 a b
  
 (3) 
? ??
?
??
?
??
0
I 11
4(a b) a b
 (4) 
? ? ??
??
?
??
0
I ab
8 a b
 
 Official Ans. by NTA (1) 
Sol. 
 
i 
a 
b 
 
 No. of turns in dx width = 
N
dx
ba ?
 
 
b
0
a
i N
dB dx
b a 2x
?
??
?
??
?
??
??
 
 
? ?
0
Ni b
Bn
2 b a a
?
??
?
??
?
??
 
 Option (1) 
8. A body of mass M moving at speed V
0
 collides 
elastically with a mass 'm' at rest. After the 
collision, the two masses move at angles ?
1
 and ?
2
 
with respect to the initial direction of motion of the 
body of mass M. The largest possible value of the 
ratio M/m, for which the angles ?
1
 and ?
2
 will be 
equal, is : 
 (1) 4 (2) 1 (3) 3 (4) 2  
 Official Ans. by NTA (3) 
Sol. 
 
M 
m 
V=0 
M 
?
1 
? 
m 
?
2 
V
2
 
V
1
 
V
0
 
 
 given ?
1
 = ?
2 
= ? 
 from momentum conservation 
 in x-direction 
0 1 2
MV MV cos mV cos ? ? ? ? 
 in y-direction 
12
0 MV sin mV sin ? ? ? ? 
 Solving above equations 
 
1
2 0 1
MV
V , V 2V cos
m
? ? ? 
 From energy conservation 
 
2 2 2
0 1 2
1 1 1
MV MV MV
2 2 2
?? 
 Substituting value of V
2
 & V
0
, we will get  
 
2
M
1 4cos 4
m
? ? ? ? 
 
M
3
m
? 
 Option (3) 
9. The masses and radii of the earth and moon are 
(M
1
, R
1
) and (M
2
, R
2
) respectively. Their centres 
are at a distance 'r' apart. Find the minimum escape 
velocity for a particle of mass 'm' to be projected 
from the middle of these two masses: 
 (1) 
?
?
12
4G(M M ) 1
V
2r
 
 (2) 
?
?
12
4G(M M )
V
r
 
 (3) 
?
?
12
2G(M M ) 1
V
2r
 
 (4) 
?
?
12
2G (M M )
V
r
 
 Official Ans. by NTA (2) 
Sol. 
 
M
1
R
1
 
M
2
R
2
 
r/2 
r 
 
 
2 12
GM m GM m 1
mV 0
2 r / 2 r / 2
? ? ? 
 ? ?
2
12
1 2Gm
mV M M
2r
?? 
 
? ?
12
4G M M
V
r
?
? 
 Option (2) 
Page 4


 
    1 
FINAL JEE –MAIN EXAMINATION – AUGUST, 2021 
(Held On Tuesday 31
st
 August, 2021)              TIME : 9 : 00 AM  to  12 : 00 NOON 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. A helicopter is flying horizontally with a speed 'v' 
at an altitude 'h' has to drop a food packet for a 
man on the ground. What is the distance of 
helicopter from the man when the food packet is 
dropped? 
 (1) 
?
2
2
2ghv 1
h
 (2) ?
22
2ghv h 
 (3) ?
2
2
2v h
h
g
 (4) ?
2
2
2gh
h
v
  
 Official Ans. by NTA (3) 
Sol. 
 
v 
helicopter 
h D 
Man 
R 
 
 
2h
Rv
g
?? 
 
22
D R h ?? 
 
2
2
2h
vh
g
??
? ? ?
??
??
??
 
 
2
2
2hv
Dh
g
?? 
 Option (3) is correct  
2. In the following logic circuit the sequence of the 
inputs A, B are (0, 0), (0,1), (1, 0) and (1, 1). The 
output Y for this sequence will be : 
 
A 
B 
P 
Q 
Y 
 
 (1) 1, 0, 1, 0  (2) 0, 1, 0, 1  
 (3) 1, 1, 1, 0  (4) 0, 0, 1, 1  
 Official Ans. by NTA (3) 
Sol. 
 
A 
B 
Y 
A·B 
A+B 
 
 ? ? ? ? Y A B A B ? ? ? ? 
 ?
? ? 0,0
Y1 ? 
 ?
? ? 0,1
Y1 ? 
 ?
? ? 1,0
Y1 ? 
 ?
? ? 1,1
Y0 ? 
 Option (3) is correct 
3. Two particles A and B having charges 20 ?C and  
–5 ?C respectively are held fixed with a separation 
of 5 cm. At what position a third charged particle 
should be placed so that it does not experience a 
net electric force? 
 
–5?C 
B 
5 cm 
A 
20?C 
 
 (1) At 5 cm from 20 ?C on the left side of system 
 (2) At 5 cm from – 5 ?C on the right side  
 (3) At 1.25 cm from – 5 ?C between two charges 
 (4) At midpoint between two charges  
 Official Ans. by NTA (2) 
Sol.
  
–5?C 20?C  
 Null point is possible only right side of –5µC 
  
N 
x 
20µC –5µC 
 
 
? ? ? ?
? ?
N 2 2
k 5 C k 20 C
E0
x
5x
? ? ?
? ? ? ?
?
 
 x = 5 cm 
 ? option (2) is correct 
 
2 
 
 
4. A reversible engine has an efficiency of 
1
4
. If the 
temperature of the sink is reduced by 58°C, its 
efficiency becomes double. Calculate the 
temperature of the sink : 
 (1) 174°C  (2) 280°C 
 (3) 180.4°C  (4) 382°C 
 Official Ans. by NTA (1) 
 Official Ans. by ALLEN (Bonus) 
Sol. T
2
 = sink temperature 
 
2
1
T
1
T
? ? ? 
 
2
1
T 1
1
4T
?? 
 
2
1
T 3
T4
? … (i) 
 
2
1
T 58 1
1
2T
?
?? 
 
2
11
T 58 1
T T 2
?? 
 
1
3 58 1
4 T 2
?? 
 
1
1
1 58
T 232
4T
? ? ? 
 
2
3
T 232
4
?? 
 T
2
 = 174 K 
5. An object is placed at the focus of concave lens 
having focal length f. What is the magnification 
and distance of the image from the optical centre 
of the lens? 
 (1) 1, ?  (2) Very high, ?  
 (3) 
1
,
22
f
  (4) 
1
,
44
f
  
 Official Ans. by NTA (3) 
Sol. 
 
f 
 
 U = – f 
 
1 1 1 1 2
V U f V f
? ? ? ? ?
?
 
 
f
V
2
?
? 
 
V1
m
U2
?? 
 distance = 
f
2
 
 Option (3) 
6. A sample of a radioactive nucleus A disintegrates 
to another radioactive nucleus B, which in turn 
disintegrates to some other stable nucleus C. Plot 
of a graph showing the variation of number of 
atoms of nucleus B vesus time is : 
 (Assume that at t = 0, there are no B atoms in the 
sample) 
 (1) 
 
time 
No. of atoms 
 (2) 
 
time 
No. of atoms 
  
 (3) 
 
time 
No. of atoms 
 (4) 
 
time 
No. of atoms 
 
 Official Ans. by NTA (2) 
Sol. 
 
A B 
C (stable) 
 
 Initially no. of atoms of B = 0 after t = 0, no. of 
atoms of B will starts increasing & reaches 
maximum value when rate of decay of B = rate of 
formation of B. 
 After that maximum value, no. of atoms will starts 
decreasing as growth & decay both are exponential 
functions, so best possible graph is (2) 
 Option (2)  
 
 
3 
 
 
7. A coil having N turns is wound tightly in the form 
of a spiral with inner and outer radii 'a' and 'b' 
respectively. Find the magnetic field at centre, 
when a current I passes through coil : 
 (1) 
? ??
??
?
??
0
e
IN b
log
2(b a) a
 (2) 
? ? ??
??
?
??
0
I ab
8 a b
  
 (3) 
? ??
?
??
?
??
0
I 11
4(a b) a b
 (4) 
? ? ??
??
?
??
0
I ab
8 a b
 
 Official Ans. by NTA (1) 
Sol. 
 
i 
a 
b 
 
 No. of turns in dx width = 
N
dx
ba ?
 
 
b
0
a
i N
dB dx
b a 2x
?
??
?
??
?
??
??
 
 
? ?
0
Ni b
Bn
2 b a a
?
??
?
??
?
??
 
 Option (1) 
8. A body of mass M moving at speed V
0
 collides 
elastically with a mass 'm' at rest. After the 
collision, the two masses move at angles ?
1
 and ?
2
 
with respect to the initial direction of motion of the 
body of mass M. The largest possible value of the 
ratio M/m, for which the angles ?
1
 and ?
2
 will be 
equal, is : 
 (1) 4 (2) 1 (3) 3 (4) 2  
 Official Ans. by NTA (3) 
Sol. 
 
M 
m 
V=0 
M 
?
1 
? 
m 
?
2 
V
2
 
V
1
 
V
0
 
 
 given ?
1
 = ?
2 
= ? 
 from momentum conservation 
 in x-direction 
0 1 2
MV MV cos mV cos ? ? ? ? 
 in y-direction 
12
0 MV sin mV sin ? ? ? ? 
 Solving above equations 
 
1
2 0 1
MV
V , V 2V cos
m
? ? ? 
 From energy conservation 
 
2 2 2
0 1 2
1 1 1
MV MV MV
2 2 2
?? 
 Substituting value of V
2
 & V
0
, we will get  
 
2
M
1 4cos 4
m
? ? ? ? 
 
M
3
m
? 
 Option (3) 
9. The masses and radii of the earth and moon are 
(M
1
, R
1
) and (M
2
, R
2
) respectively. Their centres 
are at a distance 'r' apart. Find the minimum escape 
velocity for a particle of mass 'm' to be projected 
from the middle of these two masses: 
 (1) 
?
?
12
4G(M M ) 1
V
2r
 
 (2) 
?
?
12
4G(M M )
V
r
 
 (3) 
?
?
12
2G(M M ) 1
V
2r
 
 (4) 
?
?
12
2G (M M )
V
r
 
 Official Ans. by NTA (2) 
Sol. 
 
M
1
R
1
 
M
2
R
2
 
r/2 
r 
 
 
2 12
GM m GM m 1
mV 0
2 r / 2 r / 2
? ? ? 
 ? ?
2
12
1 2Gm
mV M M
2r
?? 
 
? ?
12
4G M M
V
r
?
? 
 Option (2) 
 
4 
 
 
10. A small square loop of side 'a' and one turn is 
placed inside a larger square loop of side b and one 
turn (b >> a). The two loops are coplanar with their 
centres coinciding. If a current I is passed in the 
square loop of side 'b', then the coefficient of 
mutual inductance between the two loops is : 
 (1) 
?
?
2
0
a
82
4b
 (2) 
?
?
0
82
4a
 
 (3) 
?
?
2
0
b
82
4a
 (4) 
?
?
0
82
4b
 
 Official Ans. by NTA (1) 
Sol.
  
I 
a 
b 
 
 
0
I
B 2sin 45 4
4 b / 2
???
? ? ?
??
?
??
 
 
2 0
I
2 2 a
b
?
? ? ?
?
 
 
2 2
00
2 2 a a
M 8 2
I b 4 b
?? ?
? ? ? ?
??
 
 Option (1) 
11. Choose the correct waveform that can represent the 
voltage across R of the following circuit, assuming 
the diode is ideal one: 
  
R 
D 
3V 
+ 
– 
V
i
=10 sin?t ~ 
 
 (1) 
 
V 
+3 
–3 
t 
 
 (2) 
 
V 
+3 
–3 
t 
  
 (3) 
 
V 
+3 
–3 
t 
   
 (4) 
 
V 
+3 
–3 
t 
  
 Official Ans. by NTA (3) 
 Official Ans. by ALLEN (1) 
Sol. When V
i
 > 3 volt, V
R
 > 0 
 Because diode will be in forward biased state 
 When 
i
V 3volt ? ; V
R
 = 0 
 Because diode will be in reverse biased state. 
12. A uniform heavy rod of weight 10 kg ms
–2
, cross-
sectional area 100 cm
2
 and length 20 cm is hanging 
from a fixed support. Young modulus of the 
material of the rod is 2 × 10
11
 Nm
–2
. Neglecting the 
lateral contraction, find the elongation of rod due 
to its own weight. 
 (1) 2 × 10
–9
 m (2) 5 × 10
–8
 m  
 (3) 4 × 10
–8
 m (4) 5 × 10
–10
 m  
 Official Ans. by NTA (4) 
Sol. 
  
 We know, 
 
WL
2AY
?? 
 
4 11
10 1
100 10 2 10
25
?
?
? ? ? ? ? ?
?
 
 
9 10
1
10 5 10 m
2
??
? ? ? ? ? 
 Option (4) 
Page 5


 
    1 
FINAL JEE –MAIN EXAMINATION – AUGUST, 2021 
(Held On Tuesday 31
st
 August, 2021)              TIME : 9 : 00 AM  to  12 : 00 NOON 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. A helicopter is flying horizontally with a speed 'v' 
at an altitude 'h' has to drop a food packet for a 
man on the ground. What is the distance of 
helicopter from the man when the food packet is 
dropped? 
 (1) 
?
2
2
2ghv 1
h
 (2) ?
22
2ghv h 
 (3) ?
2
2
2v h
h
g
 (4) ?
2
2
2gh
h
v
  
 Official Ans. by NTA (3) 
Sol. 
 
v 
helicopter 
h D 
Man 
R 
 
 
2h
Rv
g
?? 
 
22
D R h ?? 
 
2
2
2h
vh
g
??
? ? ?
??
??
??
 
 
2
2
2hv
Dh
g
?? 
 Option (3) is correct  
2. In the following logic circuit the sequence of the 
inputs A, B are (0, 0), (0,1), (1, 0) and (1, 1). The 
output Y for this sequence will be : 
 
A 
B 
P 
Q 
Y 
 
 (1) 1, 0, 1, 0  (2) 0, 1, 0, 1  
 (3) 1, 1, 1, 0  (4) 0, 0, 1, 1  
 Official Ans. by NTA (3) 
Sol. 
 
A 
B 
Y 
A·B 
A+B 
 
 ? ? ? ? Y A B A B ? ? ? ? 
 ?
? ? 0,0
Y1 ? 
 ?
? ? 0,1
Y1 ? 
 ?
? ? 1,0
Y1 ? 
 ?
? ? 1,1
Y0 ? 
 Option (3) is correct 
3. Two particles A and B having charges 20 ?C and  
–5 ?C respectively are held fixed with a separation 
of 5 cm. At what position a third charged particle 
should be placed so that it does not experience a 
net electric force? 
 
–5?C 
B 
5 cm 
A 
20?C 
 
 (1) At 5 cm from 20 ?C on the left side of system 
 (2) At 5 cm from – 5 ?C on the right side  
 (3) At 1.25 cm from – 5 ?C between two charges 
 (4) At midpoint between two charges  
 Official Ans. by NTA (2) 
Sol.
  
–5?C 20?C  
 Null point is possible only right side of –5µC 
  
N 
x 
20µC –5µC 
 
 
? ? ? ?
? ?
N 2 2
k 5 C k 20 C
E0
x
5x
? ? ?
? ? ? ?
?
 
 x = 5 cm 
 ? option (2) is correct 
 
2 
 
 
4. A reversible engine has an efficiency of 
1
4
. If the 
temperature of the sink is reduced by 58°C, its 
efficiency becomes double. Calculate the 
temperature of the sink : 
 (1) 174°C  (2) 280°C 
 (3) 180.4°C  (4) 382°C 
 Official Ans. by NTA (1) 
 Official Ans. by ALLEN (Bonus) 
Sol. T
2
 = sink temperature 
 
2
1
T
1
T
? ? ? 
 
2
1
T 1
1
4T
?? 
 
2
1
T 3
T4
? … (i) 
 
2
1
T 58 1
1
2T
?
?? 
 
2
11
T 58 1
T T 2
?? 
 
1
3 58 1
4 T 2
?? 
 
1
1
1 58
T 232
4T
? ? ? 
 
2
3
T 232
4
?? 
 T
2
 = 174 K 
5. An object is placed at the focus of concave lens 
having focal length f. What is the magnification 
and distance of the image from the optical centre 
of the lens? 
 (1) 1, ?  (2) Very high, ?  
 (3) 
1
,
22
f
  (4) 
1
,
44
f
  
 Official Ans. by NTA (3) 
Sol. 
 
f 
 
 U = – f 
 
1 1 1 1 2
V U f V f
? ? ? ? ?
?
 
 
f
V
2
?
? 
 
V1
m
U2
?? 
 distance = 
f
2
 
 Option (3) 
6. A sample of a radioactive nucleus A disintegrates 
to another radioactive nucleus B, which in turn 
disintegrates to some other stable nucleus C. Plot 
of a graph showing the variation of number of 
atoms of nucleus B vesus time is : 
 (Assume that at t = 0, there are no B atoms in the 
sample) 
 (1) 
 
time 
No. of atoms 
 (2) 
 
time 
No. of atoms 
  
 (3) 
 
time 
No. of atoms 
 (4) 
 
time 
No. of atoms 
 
 Official Ans. by NTA (2) 
Sol. 
 
A B 
C (stable) 
 
 Initially no. of atoms of B = 0 after t = 0, no. of 
atoms of B will starts increasing & reaches 
maximum value when rate of decay of B = rate of 
formation of B. 
 After that maximum value, no. of atoms will starts 
decreasing as growth & decay both are exponential 
functions, so best possible graph is (2) 
 Option (2)  
 
 
3 
 
 
7. A coil having N turns is wound tightly in the form 
of a spiral with inner and outer radii 'a' and 'b' 
respectively. Find the magnetic field at centre, 
when a current I passes through coil : 
 (1) 
? ??
??
?
??
0
e
IN b
log
2(b a) a
 (2) 
? ? ??
??
?
??
0
I ab
8 a b
  
 (3) 
? ??
?
??
?
??
0
I 11
4(a b) a b
 (4) 
? ? ??
??
?
??
0
I ab
8 a b
 
 Official Ans. by NTA (1) 
Sol. 
 
i 
a 
b 
 
 No. of turns in dx width = 
N
dx
ba ?
 
 
b
0
a
i N
dB dx
b a 2x
?
??
?
??
?
??
??
 
 
? ?
0
Ni b
Bn
2 b a a
?
??
?
??
?
??
 
 Option (1) 
8. A body of mass M moving at speed V
0
 collides 
elastically with a mass 'm' at rest. After the 
collision, the two masses move at angles ?
1
 and ?
2
 
with respect to the initial direction of motion of the 
body of mass M. The largest possible value of the 
ratio M/m, for which the angles ?
1
 and ?
2
 will be 
equal, is : 
 (1) 4 (2) 1 (3) 3 (4) 2  
 Official Ans. by NTA (3) 
Sol. 
 
M 
m 
V=0 
M 
?
1 
? 
m 
?
2 
V
2
 
V
1
 
V
0
 
 
 given ?
1
 = ?
2 
= ? 
 from momentum conservation 
 in x-direction 
0 1 2
MV MV cos mV cos ? ? ? ? 
 in y-direction 
12
0 MV sin mV sin ? ? ? ? 
 Solving above equations 
 
1
2 0 1
MV
V , V 2V cos
m
? ? ? 
 From energy conservation 
 
2 2 2
0 1 2
1 1 1
MV MV MV
2 2 2
?? 
 Substituting value of V
2
 & V
0
, we will get  
 
2
M
1 4cos 4
m
? ? ? ? 
 
M
3
m
? 
 Option (3) 
9. The masses and radii of the earth and moon are 
(M
1
, R
1
) and (M
2
, R
2
) respectively. Their centres 
are at a distance 'r' apart. Find the minimum escape 
velocity for a particle of mass 'm' to be projected 
from the middle of these two masses: 
 (1) 
?
?
12
4G(M M ) 1
V
2r
 
 (2) 
?
?
12
4G(M M )
V
r
 
 (3) 
?
?
12
2G(M M ) 1
V
2r
 
 (4) 
?
?
12
2G (M M )
V
r
 
 Official Ans. by NTA (2) 
Sol. 
 
M
1
R
1
 
M
2
R
2
 
r/2 
r 
 
 
2 12
GM m GM m 1
mV 0
2 r / 2 r / 2
? ? ? 
 ? ?
2
12
1 2Gm
mV M M
2r
?? 
 
? ?
12
4G M M
V
r
?
? 
 Option (2) 
 
4 
 
 
10. A small square loop of side 'a' and one turn is 
placed inside a larger square loop of side b and one 
turn (b >> a). The two loops are coplanar with their 
centres coinciding. If a current I is passed in the 
square loop of side 'b', then the coefficient of 
mutual inductance between the two loops is : 
 (1) 
?
?
2
0
a
82
4b
 (2) 
?
?
0
82
4a
 
 (3) 
?
?
2
0
b
82
4a
 (4) 
?
?
0
82
4b
 
 Official Ans. by NTA (1) 
Sol.
  
I 
a 
b 
 
 
0
I
B 2sin 45 4
4 b / 2
???
? ? ?
??
?
??
 
 
2 0
I
2 2 a
b
?
? ? ?
?
 
 
2 2
00
2 2 a a
M 8 2
I b 4 b
?? ?
? ? ? ?
??
 
 Option (1) 
11. Choose the correct waveform that can represent the 
voltage across R of the following circuit, assuming 
the diode is ideal one: 
  
R 
D 
3V 
+ 
– 
V
i
=10 sin?t ~ 
 
 (1) 
 
V 
+3 
–3 
t 
 
 (2) 
 
V 
+3 
–3 
t 
  
 (3) 
 
V 
+3 
–3 
t 
   
 (4) 
 
V 
+3 
–3 
t 
  
 Official Ans. by NTA (3) 
 Official Ans. by ALLEN (1) 
Sol. When V
i
 > 3 volt, V
R
 > 0 
 Because diode will be in forward biased state 
 When 
i
V 3volt ? ; V
R
 = 0 
 Because diode will be in reverse biased state. 
12. A uniform heavy rod of weight 10 kg ms
–2
, cross-
sectional area 100 cm
2
 and length 20 cm is hanging 
from a fixed support. Young modulus of the 
material of the rod is 2 × 10
11
 Nm
–2
. Neglecting the 
lateral contraction, find the elongation of rod due 
to its own weight. 
 (1) 2 × 10
–9
 m (2) 5 × 10
–8
 m  
 (3) 4 × 10
–8
 m (4) 5 × 10
–10
 m  
 Official Ans. by NTA (4) 
Sol. 
  
 We know, 
 
WL
2AY
?? 
 
4 11
10 1
100 10 2 10
25
?
?
? ? ? ? ? ?
?
 
 
9 10
1
10 5 10 m
2
??
? ? ? ? ? 
 Option (4) 
 
 
5 
 
 
13. Two plane mirrors M
1
 and M
2
 are at right angle to 
each other shown. A point source 'P' is placed at 'a' 
and '2a' meter away from M
1
 and M
2
 respectively. 
The shortest distance between the images thus 
formed is : (Take 5 = 2.3) 
 
a P 
2a 
M
2
 
M
1
 
 
 (1) 3a  (2) 4.6 a  
 (3) 2.3 a  (4) 2 10 a  
 Official Ans. by NTA (2) 
Sol. 
 
a 
2a 
M
2
 
I
2
 
2a 
2a 
a 
I
1
 
2a 
I
3 
a 
a 
 
 Shortest distance is 2a between I
1
 & I
3
 
 But answer given is for I
1
 & I
2
 
 
? ? ? ?
22
4a 2a ? 
 a 20 
 4.47 a 
 Option (2) 
14. Match List-I with List-II.  
  List-I List-II 
 (a) Torque  (i) MLT
–1
   
 (b) Impulse  (ii) MT
–2
  
 (c) Tension  (iii) ML
2
T
–2
   
 (d) Surface Tension (iv) MLT
–2
  
 Choose the most appropriate answer from the 
option given below : 
 (1) (a)–(iii), (b)–(i), (c)–(iv), (d)–(ii)  
 (2) (a)–(ii), (b)–(i), (c)–(iv), (d)–(iii) 
 (3) (a)–(i), (b)–(iii), (c)–(iv), (d)–(ii)  
 (4) (a)–(iii), (b)–(iv), (c)–(i), (d)–(ii) 
 Official Ans. by NTA (1) 
Sol. torque 
22
ML T
?
?? (III) 
 Impulse 
1
I MLT
?
? (I) 
 Tension force 
2
MLT
?
? (IV) 
 Surface tension 
2
MT
?
? (II) 
 Option (1) 
15. For an ideal gas the instantaneous change in 
pressure 'p' with volume 'v' is given by the 
equation 
dp
dv
= –ap. If p = p
0 
at v = 0 is the given 
boundary condition, then the maximum 
temperature one mole of gas can attain is :  
 (Here R is the gas constant) 
 
 (1) 
0
p
aeR
  (2) 
0
ap
eR
  
 (3) infinity  (4) 0ºC  
 Official Ans. by NTA (1) 
Sol. 
0
p v
p0
dp
a dv
P
??
??
 
 
0
p
n av
p
??
??
??
??
 
 
av
0
p p e
?
? 
 For temperature maximum p-v product should be 
maximum 
 
av
0
p ve pv
T
nR R
?
?? 
 ? ? ? ?
av av 0
p dT
0 e ve a
dv R
??
? ? ? ? 
 
? ?
av
0
pe
1 av 0
R
?
?? 
 
1
v,
a
?? 
 
00
p 1 p
T
Rae Rae
?? 
 at v = ? 
 T = 0 
 Option (1)