JEE Mains 31 August 2021 Question Paper Shift 2 | JEE Main & Advanced Previous Year Papers PDF Download

 Page 1


 
    1 
(Held On Tuesday 31
st
 August, 2021)                     TIME : 3 : 00 PM  to  6 : 00 PM 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. Four identical hollow cylindrical columns of mild 
steel support a big structure of mass 50 × 10
3
 kg, 
The inner and outer radii of each column are 50 cm 
and 100 cm respectively. Assuming uniform local 
distribution, calculate the compression strain of 
each column. [Use Y = 2.0 × 10
11
 Pa, g = 9.8 m/s
2
] 
 (1) 3.60 × 10
–8
 (2) 2.60 × 10
–7 
 (3) 1.87 × 10
–3
 (4) 7.07 × 10
–4 
 Official Ans. by NTA (2) 
Sol. Force on each column =  
mg
4
 
 Strain = 
mg
4AY
 
            
3
11
50 10 9.8
4 (1 0.25) 2 10
??
?
? ? ? ? ?
 
          = 2.6 × 10
–7
 
2. A current of 1.5 A is flowing through a triangle, of 
side 9 cm each. The magnetic field at the centroid 
of the triangle is : 
 (Assume that the current is flowing in the 
clockwise direction.) 
 (1) 3 × 10
–7
 T, outside the plane of triangle  
 (2) 23 × 10
–7
 T, outside the plane of triangle  
 (3) 23 × 10
–5
 T, inside the plane of triangle  
 (4) 3 × 10
–5
 T, inside the plane of triangle  
 Official Ans. by NTA (4) 
Sol. 
 
A 
? 
i 
i 
60° 
r 
 
 
0
i
B 3 (sin60 sin60 )
4r
???
? ? ? ?
??
?
??
 
 
/2
tan 60
r
?? 
 Where r = 
2
9 10
M
23
?
?
 
 ? B = 3 × 10
–5
 T 
 Current is flowing in clockwise direction so, B is 
inside plane of triangle by right hand rule. 
3. A system consists of two identical spheres each of 
mass 1.5 kg and radius 50 cm at the end of light 
rod. The distance between the centres of the two 
spheres is 5 m. What will be the moment of inertia 
of the system about an axis perpendicular to the 
rod passing through its midpoint ? 
 (1) 18.75 kgm
2
  
 (2) 1.905 × 10
5
 kgm
2 
 (3) 19.05 kgm
2 
 (4) 1.875 × 10
5
 kgm
2 
 Official Ans. by NTA (3) 
Sol. 
 
r 
C
1 
r 
C
2 
A 
5m 
d d 
 
 M = 1.5 kg,  r = 0.5 m,  d = 
5
2
m 
 
22
2
I 2 Mr Md
5
??
??
??
??
 
 = 19.05 kgm
2
 
Page 2


 
    1 
(Held On Tuesday 31
st
 August, 2021)                     TIME : 3 : 00 PM  to  6 : 00 PM 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. Four identical hollow cylindrical columns of mild 
steel support a big structure of mass 50 × 10
3
 kg, 
The inner and outer radii of each column are 50 cm 
and 100 cm respectively. Assuming uniform local 
distribution, calculate the compression strain of 
each column. [Use Y = 2.0 × 10
11
 Pa, g = 9.8 m/s
2
] 
 (1) 3.60 × 10
–8
 (2) 2.60 × 10
–7 
 (3) 1.87 × 10
–3
 (4) 7.07 × 10
–4 
 Official Ans. by NTA (2) 
Sol. Force on each column =  
mg
4
 
 Strain = 
mg
4AY
 
            
3
11
50 10 9.8
4 (1 0.25) 2 10
??
?
? ? ? ? ?
 
          = 2.6 × 10
–7
 
2. A current of 1.5 A is flowing through a triangle, of 
side 9 cm each. The magnetic field at the centroid 
of the triangle is : 
 (Assume that the current is flowing in the 
clockwise direction.) 
 (1) 3 × 10
–7
 T, outside the plane of triangle  
 (2) 23 × 10
–7
 T, outside the plane of triangle  
 (3) 23 × 10
–5
 T, inside the plane of triangle  
 (4) 3 × 10
–5
 T, inside the plane of triangle  
 Official Ans. by NTA (4) 
Sol. 
 
A 
? 
i 
i 
60° 
r 
 
 
0
i
B 3 (sin60 sin60 )
4r
???
? ? ? ?
??
?
??
 
 
/2
tan 60
r
?? 
 Where r = 
2
9 10
M
23
?
?
 
 ? B = 3 × 10
–5
 T 
 Current is flowing in clockwise direction so, B is 
inside plane of triangle by right hand rule. 
3. A system consists of two identical spheres each of 
mass 1.5 kg and radius 50 cm at the end of light 
rod. The distance between the centres of the two 
spheres is 5 m. What will be the moment of inertia 
of the system about an axis perpendicular to the 
rod passing through its midpoint ? 
 (1) 18.75 kgm
2
  
 (2) 1.905 × 10
5
 kgm
2 
 (3) 19.05 kgm
2 
 (4) 1.875 × 10
5
 kgm
2 
 Official Ans. by NTA (3) 
Sol. 
 
r 
C
1 
r 
C
2 
A 
5m 
d d 
 
 M = 1.5 kg,  r = 0.5 m,  d = 
5
2
m 
 
22
2
I 2 Mr Md
5
??
??
??
??
 
 = 19.05 kgm
2
 
 
2 
 
 
4. Statement I : 
 Two forces 
? ?
? PQ and 
? ?
? PQ where ? PQ , 
when act at an angle ?
1
 to each other, the 
magnitude of their resultant is ?
22
3(P Q ), when 
they act at an angle ?
2
, the magnitude of their 
resultant becomes ?
22
2(P Q ) . This is possible 
only when ?
1
 < ?
2
. 
 Statement II : 
 In the situation given above. 
 ?
1
 = 60° and ?
2
 = 90° 
 In the light of the above statements, choose the 
most appropriate answer from the options given 
below :- 
 (1) Statement-I is false but Statement-II is true 
 (2) Both Statement-I and Statement-II are true 
 (3) Statement-I is true but Statement-II is false 
 (4) Both Statement-I and Statement-II are false. 
 Official Ans. by NTA (2) 
Sol. A P Q ?? 
 B P – Q ? PQ ? 
 
22
| A| | B| P Q ? ? ? 
 
22
| A B| 2(P Q )(1 cos ) ? ? ? ? ? 
 For 
22
| A B| 3(P Q ) ? ? ? 
 ?
1
 = 60° 
 For 
22
| A B| 2(P Q ) ? ? ? 
 ?
2
 = 90° 
5. A free electron of 2.6 eV energy collides with a  
H
+
 ion. This results in the formation of a hydrogen 
atom in the first excited state and a photon is 
released. Find the frequency of the emitted photon. 
(h = 6.6 × 10
–34
 Js) 
 (1) 1.45 × 10
16
 MHz 
 (2) 0.19 × 10
15
 MHz 
 (3) 1.45 × 10
9
 MHz 
 (4) 9.0 × 10
27
 MHz 
 Official Ans. by NTA (3) 
Sol. For every large distance P.E. = 0 
 & total energy = 2.6 + 0 = 2.6 eV 
 Finally in first excited state of H atom total energy  
= –3.4 eV 
 Loss in total energy  = 2.6 – (–3.4) 
                = 6eV  
 It is emitted as photon 
 
1240
206 nm
6
? ? ? 
 
8
9
3 10
f
206 10
?
?
?
?
 = 1.45 × 10
15
 Hz 
      = 1.45 × 10
9
 Hz 
6. Two thin metallic spherical shells of radii r
1
 and r
2
 
(r
1
 < r
2
) are placed with their centres coinciding.  
A material of thermal conductivity K is filled in 
the space between the shells. The inner shell is 
maintained at temperature ?
1
 and the outer shell at 
temperature ?
2
( ?
1
< ?
2
). The rate at which heat 
flows radially through the material is :- 
 (1) 
? ? ? ?
?
1 2 2 1
21
4 Krr ( )
rr
 (2) 
? ? ? ?
?
1 2 2 1
21
rr ( )
rr
 
 (3) 
? ? ?
?
21
21
K( )
rr
 (4) 
? ? ? ?
?
2 1 2 1
12
K( )(r r )
4 rr
 
 Official Ans. by NTA (1) 
Sol. 
 
r
1 
r
 
dr
 
r
2 
 
 Thermal resistance of spherical sheet of thickness 
dr and radius r is  
 
2
dr
dR
K(4 r )
?
?
 
 
2
1
r
2
r
dr
R
K(4 r )
?
?
?
 
 
21
1 2 1 2
rr 1 1 1 1
R
4 K r r 4 K r r
? ? ? ? ?
? ? ?
? ? ? ?
??
? ? ? ?
 
 Thermal current (i) = 
21
R
? ? ?
 
 
12
21
21
4 Kr r
i ( )
rr
?
? ? ? ?
?
 
Page 3


 
    1 
(Held On Tuesday 31
st
 August, 2021)                     TIME : 3 : 00 PM  to  6 : 00 PM 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. Four identical hollow cylindrical columns of mild 
steel support a big structure of mass 50 × 10
3
 kg, 
The inner and outer radii of each column are 50 cm 
and 100 cm respectively. Assuming uniform local 
distribution, calculate the compression strain of 
each column. [Use Y = 2.0 × 10
11
 Pa, g = 9.8 m/s
2
] 
 (1) 3.60 × 10
–8
 (2) 2.60 × 10
–7 
 (3) 1.87 × 10
–3
 (4) 7.07 × 10
–4 
 Official Ans. by NTA (2) 
Sol. Force on each column =  
mg
4
 
 Strain = 
mg
4AY
 
            
3
11
50 10 9.8
4 (1 0.25) 2 10
??
?
? ? ? ? ?
 
          = 2.6 × 10
–7
 
2. A current of 1.5 A is flowing through a triangle, of 
side 9 cm each. The magnetic field at the centroid 
of the triangle is : 
 (Assume that the current is flowing in the 
clockwise direction.) 
 (1) 3 × 10
–7
 T, outside the plane of triangle  
 (2) 23 × 10
–7
 T, outside the plane of triangle  
 (3) 23 × 10
–5
 T, inside the plane of triangle  
 (4) 3 × 10
–5
 T, inside the plane of triangle  
 Official Ans. by NTA (4) 
Sol. 
 
A 
? 
i 
i 
60° 
r 
 
 
0
i
B 3 (sin60 sin60 )
4r
???
? ? ? ?
??
?
??
 
 
/2
tan 60
r
?? 
 Where r = 
2
9 10
M
23
?
?
 
 ? B = 3 × 10
–5
 T 
 Current is flowing in clockwise direction so, B is 
inside plane of triangle by right hand rule. 
3. A system consists of two identical spheres each of 
mass 1.5 kg and radius 50 cm at the end of light 
rod. The distance between the centres of the two 
spheres is 5 m. What will be the moment of inertia 
of the system about an axis perpendicular to the 
rod passing through its midpoint ? 
 (1) 18.75 kgm
2
  
 (2) 1.905 × 10
5
 kgm
2 
 (3) 19.05 kgm
2 
 (4) 1.875 × 10
5
 kgm
2 
 Official Ans. by NTA (3) 
Sol. 
 
r 
C
1 
r 
C
2 
A 
5m 
d d 
 
 M = 1.5 kg,  r = 0.5 m,  d = 
5
2
m 
 
22
2
I 2 Mr Md
5
??
??
??
??
 
 = 19.05 kgm
2
 
 
2 
 
 
4. Statement I : 
 Two forces 
? ?
? PQ and 
? ?
? PQ where ? PQ , 
when act at an angle ?
1
 to each other, the 
magnitude of their resultant is ?
22
3(P Q ), when 
they act at an angle ?
2
, the magnitude of their 
resultant becomes ?
22
2(P Q ) . This is possible 
only when ?
1
 < ?
2
. 
 Statement II : 
 In the situation given above. 
 ?
1
 = 60° and ?
2
 = 90° 
 In the light of the above statements, choose the 
most appropriate answer from the options given 
below :- 
 (1) Statement-I is false but Statement-II is true 
 (2) Both Statement-I and Statement-II are true 
 (3) Statement-I is true but Statement-II is false 
 (4) Both Statement-I and Statement-II are false. 
 Official Ans. by NTA (2) 
Sol. A P Q ?? 
 B P – Q ? PQ ? 
 
22
| A| | B| P Q ? ? ? 
 
22
| A B| 2(P Q )(1 cos ) ? ? ? ? ? 
 For 
22
| A B| 3(P Q ) ? ? ? 
 ?
1
 = 60° 
 For 
22
| A B| 2(P Q ) ? ? ? 
 ?
2
 = 90° 
5. A free electron of 2.6 eV energy collides with a  
H
+
 ion. This results in the formation of a hydrogen 
atom in the first excited state and a photon is 
released. Find the frequency of the emitted photon. 
(h = 6.6 × 10
–34
 Js) 
 (1) 1.45 × 10
16
 MHz 
 (2) 0.19 × 10
15
 MHz 
 (3) 1.45 × 10
9
 MHz 
 (4) 9.0 × 10
27
 MHz 
 Official Ans. by NTA (3) 
Sol. For every large distance P.E. = 0 
 & total energy = 2.6 + 0 = 2.6 eV 
 Finally in first excited state of H atom total energy  
= –3.4 eV 
 Loss in total energy  = 2.6 – (–3.4) 
                = 6eV  
 It is emitted as photon 
 
1240
206 nm
6
? ? ? 
 
8
9
3 10
f
206 10
?
?
?
?
 = 1.45 × 10
15
 Hz 
      = 1.45 × 10
9
 Hz 
6. Two thin metallic spherical shells of radii r
1
 and r
2
 
(r
1
 < r
2
) are placed with their centres coinciding.  
A material of thermal conductivity K is filled in 
the space between the shells. The inner shell is 
maintained at temperature ?
1
 and the outer shell at 
temperature ?
2
( ?
1
< ?
2
). The rate at which heat 
flows radially through the material is :- 
 (1) 
? ? ? ?
?
1 2 2 1
21
4 Krr ( )
rr
 (2) 
? ? ? ?
?
1 2 2 1
21
rr ( )
rr
 
 (3) 
? ? ?
?
21
21
K( )
rr
 (4) 
? ? ? ?
?
2 1 2 1
12
K( )(r r )
4 rr
 
 Official Ans. by NTA (1) 
Sol. 
 
r
1 
r
 
dr
 
r
2 
 
 Thermal resistance of spherical sheet of thickness 
dr and radius r is  
 
2
dr
dR
K(4 r )
?
?
 
 
2
1
r
2
r
dr
R
K(4 r )
?
?
?
 
 
21
1 2 1 2
rr 1 1 1 1
R
4 K r r 4 K r r
? ? ? ? ?
? ? ?
? ? ? ?
??
? ? ? ?
 
 Thermal current (i) = 
21
R
? ? ?
 
 
12
21
21
4 Kr r
i ( )
rr
?
? ? ? ?
?
 
 
 
3 
 
?  
7. If V
A
 and V
B
 are the input voltages (either 5V  
or 0V) and V
o
 is the output voltage then the two 
gates represented in the following circuit (A) and 
(B) are:- 
 
D
1 
D
2 
V
A 
V
B 
R=1k ?
 
V
o 
V=5V
 
R
C
 = 1 k ? 
V
o 
?=150 
npn 
R
B
=100 k ? 
V
B
 
(A)
 
(B)
 
(1) AND and OR Gate 
 (2) OR and NOT Gate 
 (3) NAND and NOR Gate 
 (4) AND and NOT Gate  
 Official Ans. by NTA (2) 
Sol. V
A
 = 5V ? ? ? ? ?? ?? ?
 V
A
 = 0 V ? ? ? ?? ?? ?
 V
B
 = 5 V ? ? ? ? ? ?? ?
 V
B
 = 0 V ? ? ? ? ? ?? ?
 If A = B = 0, there is no potential anywhere here  
 V
0
 = 0 
 If A = 1, B = 0, Diode D
1
  is forward biased, here 
V
0
 = 5V 
 If A = 0, B = 1, Diode D
2
 is forward biased hence 
V
0
 = 5V 
 If A = 1, B = 1, Both diodes are forward biased 
hence V
0
 = 5V 
 Truth table for I
st 
 
A B Output  
 0 0 0 
 0 1 1 
 1 0 1 
 1 1 1 
 ? Given circuit is OR gate 
 For II
nd
 circuit   
 V
B
 = 5V, A = 1 
 V
B
 = 0V, A = 0 
 When A = 0, E–B junction is unbiased there is no 
current through it  
 ? V
0
 = 1 
 When A = 1, E–B junction is forward biased 
     V
0
 = 0 
 ? Hence this circuit is not gate. 
8. Consider two separate ideal gases of electrons and 
protons having same number of particles. The 
temperature of both the gases are same. The ratio 
of the uncertainty in determining the position of an 
electron to that of a proton is proportional to :- 
 (1) 
??
??
??
3/2
p
e
m
m
 (2) 
e
p
m
m
 (3) 
p
e
m
m
 (4) 
p
e
m
m
 
 Official Ans. by NTA (3) 
Sol. ?x.?p = 
h
4 ?
 
 
h
x
4 m v
??
??
 
3KT
v
m
? 
 
p
e
pe
m
x
xm
?
?
?
 
9. A bob of mass 'm' suspended by a thread of length l 
undergoes simple harmonic oscillations with time 
period T. If the bob is immersed in a liquid that has 
density 
1
4
 times that of the bob and the length of 
the thread is increased by 1/3
rd
 of the original 
length, then the time period of the simple harmonic 
oscillations will be :- 
 (1) T (2) 
3
2
T (3) 
3
4
T (4) 
4
3
T 
 Official Ans. by NTA (4) 
Sol. T 2 / g ?? 
 When bob is immersed in liquid 
 mg
eff
 = mg – Buoyant force  
 mg
eff
 = mg – v ?g ( ? = density of liquid) 
         = mg – v 
4
?
g 
         = mg – 
mg
4
 = 
3mg
4
 
 ? ?g
eff
 = 
3g
4
 
 
1
1
eff
T2
g
??           
l
4
33
? ? ? ,   
eff
3g
4
? 
 By solving  
 T
1
 = 
4
2 / g
3
? 
 T
1
 = 
4T
3
 
Page 4


 
    1 
(Held On Tuesday 31
st
 August, 2021)                     TIME : 3 : 00 PM  to  6 : 00 PM 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. Four identical hollow cylindrical columns of mild 
steel support a big structure of mass 50 × 10
3
 kg, 
The inner and outer radii of each column are 50 cm 
and 100 cm respectively. Assuming uniform local 
distribution, calculate the compression strain of 
each column. [Use Y = 2.0 × 10
11
 Pa, g = 9.8 m/s
2
] 
 (1) 3.60 × 10
–8
 (2) 2.60 × 10
–7 
 (3) 1.87 × 10
–3
 (4) 7.07 × 10
–4 
 Official Ans. by NTA (2) 
Sol. Force on each column =  
mg
4
 
 Strain = 
mg
4AY
 
            
3
11
50 10 9.8
4 (1 0.25) 2 10
??
?
? ? ? ? ?
 
          = 2.6 × 10
–7
 
2. A current of 1.5 A is flowing through a triangle, of 
side 9 cm each. The magnetic field at the centroid 
of the triangle is : 
 (Assume that the current is flowing in the 
clockwise direction.) 
 (1) 3 × 10
–7
 T, outside the plane of triangle  
 (2) 23 × 10
–7
 T, outside the plane of triangle  
 (3) 23 × 10
–5
 T, inside the plane of triangle  
 (4) 3 × 10
–5
 T, inside the plane of triangle  
 Official Ans. by NTA (4) 
Sol. 
 
A 
? 
i 
i 
60° 
r 
 
 
0
i
B 3 (sin60 sin60 )
4r
???
? ? ? ?
??
?
??
 
 
/2
tan 60
r
?? 
 Where r = 
2
9 10
M
23
?
?
 
 ? B = 3 × 10
–5
 T 
 Current is flowing in clockwise direction so, B is 
inside plane of triangle by right hand rule. 
3. A system consists of two identical spheres each of 
mass 1.5 kg and radius 50 cm at the end of light 
rod. The distance between the centres of the two 
spheres is 5 m. What will be the moment of inertia 
of the system about an axis perpendicular to the 
rod passing through its midpoint ? 
 (1) 18.75 kgm
2
  
 (2) 1.905 × 10
5
 kgm
2 
 (3) 19.05 kgm
2 
 (4) 1.875 × 10
5
 kgm
2 
 Official Ans. by NTA (3) 
Sol. 
 
r 
C
1 
r 
C
2 
A 
5m 
d d 
 
 M = 1.5 kg,  r = 0.5 m,  d = 
5
2
m 
 
22
2
I 2 Mr Md
5
??
??
??
??
 
 = 19.05 kgm
2
 
 
2 
 
 
4. Statement I : 
 Two forces 
? ?
? PQ and 
? ?
? PQ where ? PQ , 
when act at an angle ?
1
 to each other, the 
magnitude of their resultant is ?
22
3(P Q ), when 
they act at an angle ?
2
, the magnitude of their 
resultant becomes ?
22
2(P Q ) . This is possible 
only when ?
1
 < ?
2
. 
 Statement II : 
 In the situation given above. 
 ?
1
 = 60° and ?
2
 = 90° 
 In the light of the above statements, choose the 
most appropriate answer from the options given 
below :- 
 (1) Statement-I is false but Statement-II is true 
 (2) Both Statement-I and Statement-II are true 
 (3) Statement-I is true but Statement-II is false 
 (4) Both Statement-I and Statement-II are false. 
 Official Ans. by NTA (2) 
Sol. A P Q ?? 
 B P – Q ? PQ ? 
 
22
| A| | B| P Q ? ? ? 
 
22
| A B| 2(P Q )(1 cos ) ? ? ? ? ? 
 For 
22
| A B| 3(P Q ) ? ? ? 
 ?
1
 = 60° 
 For 
22
| A B| 2(P Q ) ? ? ? 
 ?
2
 = 90° 
5. A free electron of 2.6 eV energy collides with a  
H
+
 ion. This results in the formation of a hydrogen 
atom in the first excited state and a photon is 
released. Find the frequency of the emitted photon. 
(h = 6.6 × 10
–34
 Js) 
 (1) 1.45 × 10
16
 MHz 
 (2) 0.19 × 10
15
 MHz 
 (3) 1.45 × 10
9
 MHz 
 (4) 9.0 × 10
27
 MHz 
 Official Ans. by NTA (3) 
Sol. For every large distance P.E. = 0 
 & total energy = 2.6 + 0 = 2.6 eV 
 Finally in first excited state of H atom total energy  
= –3.4 eV 
 Loss in total energy  = 2.6 – (–3.4) 
                = 6eV  
 It is emitted as photon 
 
1240
206 nm
6
? ? ? 
 
8
9
3 10
f
206 10
?
?
?
?
 = 1.45 × 10
15
 Hz 
      = 1.45 × 10
9
 Hz 
6. Two thin metallic spherical shells of radii r
1
 and r
2
 
(r
1
 < r
2
) are placed with their centres coinciding.  
A material of thermal conductivity K is filled in 
the space between the shells. The inner shell is 
maintained at temperature ?
1
 and the outer shell at 
temperature ?
2
( ?
1
< ?
2
). The rate at which heat 
flows radially through the material is :- 
 (1) 
? ? ? ?
?
1 2 2 1
21
4 Krr ( )
rr
 (2) 
? ? ? ?
?
1 2 2 1
21
rr ( )
rr
 
 (3) 
? ? ?
?
21
21
K( )
rr
 (4) 
? ? ? ?
?
2 1 2 1
12
K( )(r r )
4 rr
 
 Official Ans. by NTA (1) 
Sol. 
 
r
1 
r
 
dr
 
r
2 
 
 Thermal resistance of spherical sheet of thickness 
dr and radius r is  
 
2
dr
dR
K(4 r )
?
?
 
 
2
1
r
2
r
dr
R
K(4 r )
?
?
?
 
 
21
1 2 1 2
rr 1 1 1 1
R
4 K r r 4 K r r
? ? ? ? ?
? ? ?
? ? ? ?
??
? ? ? ?
 
 Thermal current (i) = 
21
R
? ? ?
 
 
12
21
21
4 Kr r
i ( )
rr
?
? ? ? ?
?
 
 
 
3 
 
?  
7. If V
A
 and V
B
 are the input voltages (either 5V  
or 0V) and V
o
 is the output voltage then the two 
gates represented in the following circuit (A) and 
(B) are:- 
 
D
1 
D
2 
V
A 
V
B 
R=1k ?
 
V
o 
V=5V
 
R
C
 = 1 k ? 
V
o 
?=150 
npn 
R
B
=100 k ? 
V
B
 
(A)
 
(B)
 
(1) AND and OR Gate 
 (2) OR and NOT Gate 
 (3) NAND and NOR Gate 
 (4) AND and NOT Gate  
 Official Ans. by NTA (2) 
Sol. V
A
 = 5V ? ? ? ? ?? ?? ?
 V
A
 = 0 V ? ? ? ?? ?? ?
 V
B
 = 5 V ? ? ? ? ? ?? ?
 V
B
 = 0 V ? ? ? ? ? ?? ?
 If A = B = 0, there is no potential anywhere here  
 V
0
 = 0 
 If A = 1, B = 0, Diode D
1
  is forward biased, here 
V
0
 = 5V 
 If A = 0, B = 1, Diode D
2
 is forward biased hence 
V
0
 = 5V 
 If A = 1, B = 1, Both diodes are forward biased 
hence V
0
 = 5V 
 Truth table for I
st 
 
A B Output  
 0 0 0 
 0 1 1 
 1 0 1 
 1 1 1 
 ? Given circuit is OR gate 
 For II
nd
 circuit   
 V
B
 = 5V, A = 1 
 V
B
 = 0V, A = 0 
 When A = 0, E–B junction is unbiased there is no 
current through it  
 ? V
0
 = 1 
 When A = 1, E–B junction is forward biased 
     V
0
 = 0 
 ? Hence this circuit is not gate. 
8. Consider two separate ideal gases of electrons and 
protons having same number of particles. The 
temperature of both the gases are same. The ratio 
of the uncertainty in determining the position of an 
electron to that of a proton is proportional to :- 
 (1) 
??
??
??
3/2
p
e
m
m
 (2) 
e
p
m
m
 (3) 
p
e
m
m
 (4) 
p
e
m
m
 
 Official Ans. by NTA (3) 
Sol. ?x.?p = 
h
4 ?
 
 
h
x
4 m v
??
??
 
3KT
v
m
? 
 
p
e
pe
m
x
xm
?
?
?
 
9. A bob of mass 'm' suspended by a thread of length l 
undergoes simple harmonic oscillations with time 
period T. If the bob is immersed in a liquid that has 
density 
1
4
 times that of the bob and the length of 
the thread is increased by 1/3
rd
 of the original 
length, then the time period of the simple harmonic 
oscillations will be :- 
 (1) T (2) 
3
2
T (3) 
3
4
T (4) 
4
3
T 
 Official Ans. by NTA (4) 
Sol. T 2 / g ?? 
 When bob is immersed in liquid 
 mg
eff
 = mg – Buoyant force  
 mg
eff
 = mg – v ?g ( ? = density of liquid) 
         = mg – v 
4
?
g 
         = mg – 
mg
4
 = 
3mg
4
 
 ? ?g
eff
 = 
3g
4
 
 
1
1
eff
T2
g
??           
l
4
33
? ? ? ,   
eff
3g
4
? 
 By solving  
 T
1
 = 
4
2 / g
3
? 
 T
1
 = 
4T
3
 
 
4 
 
 
10. Statement :1 
 If three forces 
12
F,F and 
3
F are represented by 
three sides of a triangle and ? ? ?
1 2 3
F F F , then 
these three forces are concurrent forces and satisfy 
the condition for equilibrium. 
  Statement : II 
 A triangle made up of three forces 
12
F,F and 
3
F as 
its sides taken in the same order, satisfy the 
condition for translatory equilibrium. 
 In the light of the above statements, choose the 
most appropriate answer from the options given 
below: 
 (1) Statement-I is false but Statement-II is true 
 (2) Statement-I is true but Statement-II is false  
 (3) Both Statement-I and Statement-II are false 
 (4) Both Statement-I and Statement-II are true. 
 Official Ans. by NTA (4) 
Sol. 
 
 
 
 
 
 Here  1 2 3 F F F 0 ? ? ? 
                   1 2 3 F F F ? ? ? 
 Since net F0 ? (equilibrium) 
 Both statements correct 
11. If velocity [V], time [T] and force [F] are chosen 
as the base quantities, the dimensions of the mass 
will be : 
 (1) [FT
–1
 V
–1
] 
 (2) [FTV
–1
] 
 (3) [FT
2
 V]  
 (4) [FVT
–1
] 
 Official Ans. by NTA (2) 
Sol. [M] = K[F]
a
 [T]
b
 [V]
c 
 [M
1
] = [M
1
L
1
T
–2
]
a
 [T
1
]
b
 [L
1
T
–1
]
c 
 a = 1, b = 1, c = –1 
 ? ?[M] = [FTV
–1
]
 
12. The magnetic field vector of an electromagnetic 
wave is given by B = 
?
o
ˆˆ
ij
B
2
cos(kz – ?t); where 
ˆˆ
i,j represents unit vector along x and y-axis 
respectively. At t = 0 s, two electric charges q
1
 of 
4? coulomb and q
2
 of 2 ? coulomb located at 
? ??
??
??
0,0,
k
 and 
? ??
??
??
3
0,0,
k
, respectively, have the 
same velocity of 0.5 c
ˆ
i , (where c is the velocity of 
light). The ratio of the force acting on charge  
q
1
 to q
2
 is :- 
 (1) 22 : 1  
 (2) 1 :  2 
 (3) 2 : 1  
 (4) 2 : 1 
 Official Ans. by NTA (3) 
Sol. 
? ?
F q V B ??  
 1
0
ˆˆ
ij
ˆ
F 4 0.5ci B cos K. – 0
2K
?? ??
?? ??
? ? ?
?? ??
??
??
??
?? ?? ??
 
 2
0
ˆˆ
i j 3
ˆ
F 2 0.5ci B cos K. – 0
2K
?? ??
?? ??
? ? ?
?? ??
??
??
??
?? ?? ??
 
 cos ? = –1,     cos3 ? = –1 
 
1
2
F
 2
F
?? 
13. The equivalent resistance of the given circuit 
between the terminals A and B is : 
  
 
    
 
 
 
 
  
2 ? 2 ? 
5 ? 2 ? 3 ? 3 ? 
A 
B 
 
 (1) 0 ?  (2) 3 ?  
 (3) 
9
2
?  (4) 1 ? 
 Official Ans. by NTA (4) 
Page 5


 
    1 
(Held On Tuesday 31
st
 August, 2021)                     TIME : 3 : 00 PM  to  6 : 00 PM 
PHYSICS 
 
TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
 
SECTION-A 
1. Four identical hollow cylindrical columns of mild 
steel support a big structure of mass 50 × 10
3
 kg, 
The inner and outer radii of each column are 50 cm 
and 100 cm respectively. Assuming uniform local 
distribution, calculate the compression strain of 
each column. [Use Y = 2.0 × 10
11
 Pa, g = 9.8 m/s
2
] 
 (1) 3.60 × 10
–8
 (2) 2.60 × 10
–7 
 (3) 1.87 × 10
–3
 (4) 7.07 × 10
–4 
 Official Ans. by NTA (2) 
Sol. Force on each column =  
mg
4
 
 Strain = 
mg
4AY
 
            
3
11
50 10 9.8
4 (1 0.25) 2 10
??
?
? ? ? ? ?
 
          = 2.6 × 10
–7
 
2. A current of 1.5 A is flowing through a triangle, of 
side 9 cm each. The magnetic field at the centroid 
of the triangle is : 
 (Assume that the current is flowing in the 
clockwise direction.) 
 (1) 3 × 10
–7
 T, outside the plane of triangle  
 (2) 23 × 10
–7
 T, outside the plane of triangle  
 (3) 23 × 10
–5
 T, inside the plane of triangle  
 (4) 3 × 10
–5
 T, inside the plane of triangle  
 Official Ans. by NTA (4) 
Sol. 
 
A 
? 
i 
i 
60° 
r 
 
 
0
i
B 3 (sin60 sin60 )
4r
???
? ? ? ?
??
?
??
 
 
/2
tan 60
r
?? 
 Where r = 
2
9 10
M
23
?
?
 
 ? B = 3 × 10
–5
 T 
 Current is flowing in clockwise direction so, B is 
inside plane of triangle by right hand rule. 
3. A system consists of two identical spheres each of 
mass 1.5 kg and radius 50 cm at the end of light 
rod. The distance between the centres of the two 
spheres is 5 m. What will be the moment of inertia 
of the system about an axis perpendicular to the 
rod passing through its midpoint ? 
 (1) 18.75 kgm
2
  
 (2) 1.905 × 10
5
 kgm
2 
 (3) 19.05 kgm
2 
 (4) 1.875 × 10
5
 kgm
2 
 Official Ans. by NTA (3) 
Sol. 
 
r 
C
1 
r 
C
2 
A 
5m 
d d 
 
 M = 1.5 kg,  r = 0.5 m,  d = 
5
2
m 
 
22
2
I 2 Mr Md
5
??
??
??
??
 
 = 19.05 kgm
2
 
 
2 
 
 
4. Statement I : 
 Two forces 
? ?
? PQ and 
? ?
? PQ where ? PQ , 
when act at an angle ?
1
 to each other, the 
magnitude of their resultant is ?
22
3(P Q ), when 
they act at an angle ?
2
, the magnitude of their 
resultant becomes ?
22
2(P Q ) . This is possible 
only when ?
1
 < ?
2
. 
 Statement II : 
 In the situation given above. 
 ?
1
 = 60° and ?
2
 = 90° 
 In the light of the above statements, choose the 
most appropriate answer from the options given 
below :- 
 (1) Statement-I is false but Statement-II is true 
 (2) Both Statement-I and Statement-II are true 
 (3) Statement-I is true but Statement-II is false 
 (4) Both Statement-I and Statement-II are false. 
 Official Ans. by NTA (2) 
Sol. A P Q ?? 
 B P – Q ? PQ ? 
 
22
| A| | B| P Q ? ? ? 
 
22
| A B| 2(P Q )(1 cos ) ? ? ? ? ? 
 For 
22
| A B| 3(P Q ) ? ? ? 
 ?
1
 = 60° 
 For 
22
| A B| 2(P Q ) ? ? ? 
 ?
2
 = 90° 
5. A free electron of 2.6 eV energy collides with a  
H
+
 ion. This results in the formation of a hydrogen 
atom in the first excited state and a photon is 
released. Find the frequency of the emitted photon. 
(h = 6.6 × 10
–34
 Js) 
 (1) 1.45 × 10
16
 MHz 
 (2) 0.19 × 10
15
 MHz 
 (3) 1.45 × 10
9
 MHz 
 (4) 9.0 × 10
27
 MHz 
 Official Ans. by NTA (3) 
Sol. For every large distance P.E. = 0 
 & total energy = 2.6 + 0 = 2.6 eV 
 Finally in first excited state of H atom total energy  
= –3.4 eV 
 Loss in total energy  = 2.6 – (–3.4) 
                = 6eV  
 It is emitted as photon 
 
1240
206 nm
6
? ? ? 
 
8
9
3 10
f
206 10
?
?
?
?
 = 1.45 × 10
15
 Hz 
      = 1.45 × 10
9
 Hz 
6. Two thin metallic spherical shells of radii r
1
 and r
2
 
(r
1
 < r
2
) are placed with their centres coinciding.  
A material of thermal conductivity K is filled in 
the space between the shells. The inner shell is 
maintained at temperature ?
1
 and the outer shell at 
temperature ?
2
( ?
1
< ?
2
). The rate at which heat 
flows radially through the material is :- 
 (1) 
? ? ? ?
?
1 2 2 1
21
4 Krr ( )
rr
 (2) 
? ? ? ?
?
1 2 2 1
21
rr ( )
rr
 
 (3) 
? ? ?
?
21
21
K( )
rr
 (4) 
? ? ? ?
?
2 1 2 1
12
K( )(r r )
4 rr
 
 Official Ans. by NTA (1) 
Sol. 
 
r
1 
r
 
dr
 
r
2 
 
 Thermal resistance of spherical sheet of thickness 
dr and radius r is  
 
2
dr
dR
K(4 r )
?
?
 
 
2
1
r
2
r
dr
R
K(4 r )
?
?
?
 
 
21
1 2 1 2
rr 1 1 1 1
R
4 K r r 4 K r r
? ? ? ? ?
? ? ?
? ? ? ?
??
? ? ? ?
 
 Thermal current (i) = 
21
R
? ? ?
 
 
12
21
21
4 Kr r
i ( )
rr
?
? ? ? ?
?
 
 
 
3 
 
?  
7. If V
A
 and V
B
 are the input voltages (either 5V  
or 0V) and V
o
 is the output voltage then the two 
gates represented in the following circuit (A) and 
(B) are:- 
 
D
1 
D
2 
V
A 
V
B 
R=1k ?
 
V
o 
V=5V
 
R
C
 = 1 k ? 
V
o 
?=150 
npn 
R
B
=100 k ? 
V
B
 
(A)
 
(B)
 
(1) AND and OR Gate 
 (2) OR and NOT Gate 
 (3) NAND and NOR Gate 
 (4) AND and NOT Gate  
 Official Ans. by NTA (2) 
Sol. V
A
 = 5V ? ? ? ? ?? ?? ?
 V
A
 = 0 V ? ? ? ?? ?? ?
 V
B
 = 5 V ? ? ? ? ? ?? ?
 V
B
 = 0 V ? ? ? ? ? ?? ?
 If A = B = 0, there is no potential anywhere here  
 V
0
 = 0 
 If A = 1, B = 0, Diode D
1
  is forward biased, here 
V
0
 = 5V 
 If A = 0, B = 1, Diode D
2
 is forward biased hence 
V
0
 = 5V 
 If A = 1, B = 1, Both diodes are forward biased 
hence V
0
 = 5V 
 Truth table for I
st 
 
A B Output  
 0 0 0 
 0 1 1 
 1 0 1 
 1 1 1 
 ? Given circuit is OR gate 
 For II
nd
 circuit   
 V
B
 = 5V, A = 1 
 V
B
 = 0V, A = 0 
 When A = 0, E–B junction is unbiased there is no 
current through it  
 ? V
0
 = 1 
 When A = 1, E–B junction is forward biased 
     V
0
 = 0 
 ? Hence this circuit is not gate. 
8. Consider two separate ideal gases of electrons and 
protons having same number of particles. The 
temperature of both the gases are same. The ratio 
of the uncertainty in determining the position of an 
electron to that of a proton is proportional to :- 
 (1) 
??
??
??
3/2
p
e
m
m
 (2) 
e
p
m
m
 (3) 
p
e
m
m
 (4) 
p
e
m
m
 
 Official Ans. by NTA (3) 
Sol. ?x.?p = 
h
4 ?
 
 
h
x
4 m v
??
??
 
3KT
v
m
? 
 
p
e
pe
m
x
xm
?
?
?
 
9. A bob of mass 'm' suspended by a thread of length l 
undergoes simple harmonic oscillations with time 
period T. If the bob is immersed in a liquid that has 
density 
1
4
 times that of the bob and the length of 
the thread is increased by 1/3
rd
 of the original 
length, then the time period of the simple harmonic 
oscillations will be :- 
 (1) T (2) 
3
2
T (3) 
3
4
T (4) 
4
3
T 
 Official Ans. by NTA (4) 
Sol. T 2 / g ?? 
 When bob is immersed in liquid 
 mg
eff
 = mg – Buoyant force  
 mg
eff
 = mg – v ?g ( ? = density of liquid) 
         = mg – v 
4
?
g 
         = mg – 
mg
4
 = 
3mg
4
 
 ? ?g
eff
 = 
3g
4
 
 
1
1
eff
T2
g
??           
l
4
33
? ? ? ,   
eff
3g
4
? 
 By solving  
 T
1
 = 
4
2 / g
3
? 
 T
1
 = 
4T
3
 
 
4 
 
 
10. Statement :1 
 If three forces 
12
F,F and 
3
F are represented by 
three sides of a triangle and ? ? ?
1 2 3
F F F , then 
these three forces are concurrent forces and satisfy 
the condition for equilibrium. 
  Statement : II 
 A triangle made up of three forces 
12
F,F and 
3
F as 
its sides taken in the same order, satisfy the 
condition for translatory equilibrium. 
 In the light of the above statements, choose the 
most appropriate answer from the options given 
below: 
 (1) Statement-I is false but Statement-II is true 
 (2) Statement-I is true but Statement-II is false  
 (3) Both Statement-I and Statement-II are false 
 (4) Both Statement-I and Statement-II are true. 
 Official Ans. by NTA (4) 
Sol. 
 
 
 
 
 
 Here  1 2 3 F F F 0 ? ? ? 
                   1 2 3 F F F ? ? ? 
 Since net F0 ? (equilibrium) 
 Both statements correct 
11. If velocity [V], time [T] and force [F] are chosen 
as the base quantities, the dimensions of the mass 
will be : 
 (1) [FT
–1
 V
–1
] 
 (2) [FTV
–1
] 
 (3) [FT
2
 V]  
 (4) [FVT
–1
] 
 Official Ans. by NTA (2) 
Sol. [M] = K[F]
a
 [T]
b
 [V]
c 
 [M
1
] = [M
1
L
1
T
–2
]
a
 [T
1
]
b
 [L
1
T
–1
]
c 
 a = 1, b = 1, c = –1 
 ? ?[M] = [FTV
–1
]
 
12. The magnetic field vector of an electromagnetic 
wave is given by B = 
?
o
ˆˆ
ij
B
2
cos(kz – ?t); where 
ˆˆ
i,j represents unit vector along x and y-axis 
respectively. At t = 0 s, two electric charges q
1
 of 
4? coulomb and q
2
 of 2 ? coulomb located at 
? ??
??
??
0,0,
k
 and 
? ??
??
??
3
0,0,
k
, respectively, have the 
same velocity of 0.5 c
ˆ
i , (where c is the velocity of 
light). The ratio of the force acting on charge  
q
1
 to q
2
 is :- 
 (1) 22 : 1  
 (2) 1 :  2 
 (3) 2 : 1  
 (4) 2 : 1 
 Official Ans. by NTA (3) 
Sol. 
? ?
F q V B ??  
 1
0
ˆˆ
ij
ˆ
F 4 0.5ci B cos K. – 0
2K
?? ??
?? ??
? ? ?
?? ??
??
??
??
?? ?? ??
 
 2
0
ˆˆ
i j 3
ˆ
F 2 0.5ci B cos K. – 0
2K
?? ??
?? ??
? ? ?
?? ??
??
??
??
?? ?? ??
 
 cos ? = –1,     cos3 ? = –1 
 
1
2
F
 2
F
?? 
13. The equivalent resistance of the given circuit 
between the terminals A and B is : 
  
 
    
 
 
 
 
  
2 ? 2 ? 
5 ? 2 ? 3 ? 3 ? 
A 
B 
 
 (1) 0 ?  (2) 3 ?  
 (3) 
9
2
?  (4) 1 ? 
 Official Ans. by NTA (4) 
 
 
5 
 
?  Sol. 
 
 
    
 
 
 
 
  
2 ? 2 ? 
5 ? 2 ? 3 ? 3 ? 
V 
i=0 
   
 
 
 
 
2 ? 
 
2 ? 2 ? 3 ? 3 ? 
Parallel Parallel 
 
 
 
A 
 
 
 
2 ? 
3/2 ? 
 
 
1 ? 
? ?
B 
A 
B 
A 
B 
 
Series 
 
 
eq
3 3 / 2 9 / 2
R1
3 3 / 2 9 / 2
?
? ? ? ?
?
 
14. Choose the incorrect statement : 
(a) The electric lines of force entering into a 
Gaussian surface provide negative flux. 
(b) A charge 'q' is placed at the centre of a cube. The 
flux through all the faces will be the same. 
(c) In a uniform electric field net flux through a 
closed Gaussian surface containing no net 
charge, is zero. 
(d) When electric field is parallel to a Gaussian 
surface, it provides a finite non-zero flux.  
 Choose the most appropriate answer from the 
options given below  
 (1) (c) and (d) only (2) (b) and (d) only 
 (3) (d) only  (4) (a) and (c) only 
 Official Ans. by NTA (3) 
Sol. Since E.A EAcos ? ? ? ? 
  
90°
 
 
 
 
 ? = 90° 
 ? ? ?= 0  
15. A mixture of hydrogen and oxygen has volume 
500 cm
3
, temperature 300 K, pressure 400 kPa and 
mass 0.76 g. The ratio of masses of oxygen to 
hydrogen will be :- 
 (1) 3 : 8  (2) 3 : 16 
 (3) 16 : 3  (4) 8 : 3 
 Official Ans. by NTA (3) 
Sol. PV = nRT 
 400 × 10
3
 × 500 × 10
–6 
=
 
n 
25
3
??
??
??
 (300)
  
 
2
n
25
? 
 n = n
1
 + n
2
 
 
12
MM 2
25 2 32
?? 
 Also M
1
 + M
2
 = 0.76 gm 
 
2
1
M 16
M3
? 
16. A block moving horizontally on a smooth surface 
with a speed of 40 m/s splits into two parts with 
masses in the ratio of 1:2. If the smaller part moves 
at 60 m/s in the same direction, then the fractional 
change in kinetic energy is :- 
 (1) 
1
3
 (2) 
2
3
 (3) 
1
8
 (4) 
1
4
  
 Official Ans. by NTA (3) 
Sol. 
 
3M
 
V
0 
2M
 
V
2 
M
 
V
1   
 3MV
0
 = 2MV
2
 + MV
1
 
 3V
0
 = 2V
2
 + V
1
 
 120 = 2V
2
 + 60 ? ?V
2
 = 30 m/s 
 
2 2 2
1 2 0
2
0
1 1 1
MV 2MV 3MV
K.E.
2 2 2
1
K.E.
3MV
2
??
?
? 
            
2 2 2
1 2 0
2
0
V 2V 3V
3V
??
? 
          
3600 1800 4800
4800
??
? 
                   
1
8
?