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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
?
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JEE Mains 31 August 2021 Question Paper Shift 2 Notes
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The content of JEE Mains 31 August 2021 Question Paper Shift 2 is prepared as per the latest JEE syllabus.
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