JEE  >  JEE Advance 2022 Question Paper - 2 with Solutions

# JEE Advance 2022 Question Paper - 2 with Solutions | JEE Main & Advanced Mock Test Series

``` Page 1

[Paper-2]

PHYSICS

Question: A particle of mass 1 kg is subjected to a force which depends on the position as
( )
2
ˆˆ
F k xi yj kgms
-
=-+
??
with k =1 kgs
-2
. At time t = 0, the particle’s position
1
ˆˆ
2
2
r i jm
??
= +
??
??
?
and its velocity
1
2
ˆ ˆˆ
22 v i j k ms
p
-
??
= -+ +
??
??
?
. Let vx and vy denote the x and
the y components of the particle’s velocity, respectively. Ignore gravity. When z = 0.5 m, the
value of
( )
yx
xv yv - is _____ m
2
s
-1
.
Solution:
( ) ( )
( )
x
2
2
ma
,
sin
m
1
m
and cos
1
0,
2
and 2 m / s
1
sin
2
and 2 co
1

n
/
s
ta
2
5
and
2
x x x
x
x x
xx
x
x
x
xx
x
vA t
A
F
A
att x
d
v
So A
A
x
x
So a x
dt
= +
= =
=- =
= = -
?= +
=
=
= -
=
-
?=-
=
??
?
f
f
f
f
f
?

Similarly
2
2
y y
F y ma
dy
y
dt
=-=
?=-

Page 2

[Paper-2]

PHYSICS

Question: A particle of mass 1 kg is subjected to a force which depends on the position as
( )
2
ˆˆ
F k xi yj kgms
-
=-+
??
with k =1 kgs
-2
. At time t = 0, the particle’s position
1
ˆˆ
2
2
r i jm
??
= +
??
??
?
and its velocity
1
2
ˆ ˆˆ
22 v i j k ms
p
-
??
= -+ +
??
??
?
. Let vx and vy denote the x and
the y components of the particle’s velocity, respectively. Ignore gravity. When z = 0.5 m, the
value of
( )
yx
xv yv - is _____ m
2
s
-1
.
Solution:
( ) ( )
( )
x
2
2
ma
,
sin
m
1
m
and cos
1
0,
2
and 2 m / s
1
sin
2
and 2 co
1

n
/
s
ta
2
5
and
2
x x x
x
x x
xx
x
x
x
xx
x
vA t
A
F
A
att x
d
v
So A
A
x
x
So a x
dt
= +
= =
=- =
= = -
?= +
=
=
= -
=
-
?=-
=
??
?
f
f
f
f
f
?

Similarly
2
2
y y
F y ma
dy
y
dt
=-=
?=-

( )
, sin( )
0 2 m
and 2 m / s
So 2 sin   and
2 cos
and 2
4
yy
yy y
y
yy
yy
y y
So y A t
v A t
att y
v
A
A
A
= +
= = +
= =
=
=
=
?= =
?f
? ? ?f
f
f
p
f

( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
(
( )
( )
55
So, sin 2cos 2sin cos
22
5
2 sin cos sin cos
2
10 sin
10 sin cos cos sin
y x x y y x
x y y x
xy
x y xy
xv yv t t t t
t t t t
n
?f ?f ?f ?f
?f ?f ?f ?f
ff
f f ff
- = + × + - + × +
= × + + - +× +
= -
= -

11 2 1
10 3
52 5 2
?? ??
= × - - × =
?? ??
?? ??

Question: In a radioactive decay chain reaction,
230
90
Th nucleus decays into
214
84
Po nucleus. The
ratio of the number of a to number of
-
ß particles emitted in this process is_______.
Solution:
Let number of a particles are n
a
and
-
ß particles are n
ß
so
( )
4 230 214
4
84 90 2
2
So 2
n
n
nn
n
n
n
a
a
ßa
ß
a
ß
= -
?=
= --
=
=

Question: Two resistances
1
R X = ? and
2
1 R = ?are connected to a wire AB of uniform
resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0.2
mm at A to l mm at B. A galvanometer (G) connected to the center of the wire, 50 cm from
Page 3

[Paper-2]

PHYSICS

Question: A particle of mass 1 kg is subjected to a force which depends on the position as
( )
2
ˆˆ
F k xi yj kgms
-
=-+
??
with k =1 kgs
-2
. At time t = 0, the particle’s position
1
ˆˆ
2
2
r i jm
??
= +
??
??
?
and its velocity
1
2
ˆ ˆˆ
22 v i j k ms
p
-
??
= -+ +
??
??
?
. Let vx and vy denote the x and
the y components of the particle’s velocity, respectively. Ignore gravity. When z = 0.5 m, the
value of
( )
yx
xv yv - is _____ m
2
s
-1
.
Solution:
( ) ( )
( )
x
2
2
ma
,
sin
m
1
m
and cos
1
0,
2
and 2 m / s
1
sin
2
and 2 co
1

n
/
s
ta
2
5
and
2
x x x
x
x x
xx
x
x
x
xx
x
vA t
A
F
A
att x
d
v
So A
A
x
x
So a x
dt
= +
= =
=- =
= = -
?= +
=
=
= -
=
-
?=-
=
??
?
f
f
f
f
f
?

Similarly
2
2
y y
F y ma
dy
y
dt
=-=
?=-

( )
, sin( )
0 2 m
and 2 m / s
So 2 sin   and
2 cos
and 2
4
yy
yy y
y
yy
yy
y y
So y A t
v A t
att y
v
A
A
A
= +
= = +
= =
=
=
=
?= =
?f
? ? ?f
f
f
p
f

( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
(
( )
( )
55
So, sin 2cos 2sin cos
22
5
2 sin cos sin cos
2
10 sin
10 sin cos cos sin
y x x y y x
x y y x
xy
x y xy
xv yv t t t t
t t t t
n
?f ?f ?f ?f
?f ?f ?f ?f
ff
f f ff
- = + × + - + × +
= × + + - +× +
= -
= -

11 2 1
10 3
52 5 2
?? ??
= × - - × =
?? ??
?? ??

Question: In a radioactive decay chain reaction,
230
90
Th nucleus decays into
214
84
Po nucleus. The
ratio of the number of a to number of
-
ß particles emitted in this process is_______.
Solution:
Let number of a particles are n
a
and
-
ß particles are n
ß
so
( )
4 230 214
4
84 90 2
2
So 2
n
n
nn
n
n
n
a
a
ßa
ß
a
ß
= -
?=
= --
=
=

Question: Two resistances
1
R X = ? and
2
1 R = ?are connected to a wire AB of uniform
resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0.2
mm at A to l mm at B. A galvanometer (G) connected to the center of the wire, 50 cm from

each end along its axis, shows zero deflection when A and B are connected to a battery. The
value of X is ______

Solution:
Resistance of frustum shaped conductor shown is

l
R
ab
?
p
=
For the shown conductor in the diagram

Thus, the resistance of left half is
6
0.5 10
0.2 0.6
P
?
p
××
=
××

And the resistance of right half is
6
0.5 10
0.6 1
Q
?
p
××
=
××

For Wheatstone to be balanced
Page 4

[Paper-2]

PHYSICS

Question: A particle of mass 1 kg is subjected to a force which depends on the position as
( )
2
ˆˆ
F k xi yj kgms
-
=-+
??
with k =1 kgs
-2
. At time t = 0, the particle’s position
1
ˆˆ
2
2
r i jm
??
= +
??
??
?
and its velocity
1
2
ˆ ˆˆ
22 v i j k ms
p
-
??
= -+ +
??
??
?
. Let vx and vy denote the x and
the y components of the particle’s velocity, respectively. Ignore gravity. When z = 0.5 m, the
value of
( )
yx
xv yv - is _____ m
2
s
-1
.
Solution:
( ) ( )
( )
x
2
2
ma
,
sin
m
1
m
and cos
1
0,
2
and 2 m / s
1
sin
2
and 2 co
1

n
/
s
ta
2
5
and
2
x x x
x
x x
xx
x
x
x
xx
x
vA t
A
F
A
att x
d
v
So A
A
x
x
So a x
dt
= +
= =
=- =
= = -
?= +
=
=
= -
=
-
?=-
=
??
?
f
f
f
f
f
?

Similarly
2
2
y y
F y ma
dy
y
dt
=-=
?=-

( )
, sin( )
0 2 m
and 2 m / s
So 2 sin   and
2 cos
and 2
4
yy
yy y
y
yy
yy
y y
So y A t
v A t
att y
v
A
A
A
= +
= = +
= =
=
=
=
?= =
?f
? ? ?f
f
f
p
f

( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
(
( )
( )
55
So, sin 2cos 2sin cos
22
5
2 sin cos sin cos
2
10 sin
10 sin cos cos sin
y x x y y x
x y y x
xy
x y xy
xv yv t t t t
t t t t
n
?f ?f ?f ?f
?f ?f ?f ?f
ff
f f ff
- = + × + - + × +
= × + + - +× +
= -
= -

11 2 1
10 3
52 5 2
?? ??
= × - - × =
?? ??
?? ??

Question: In a radioactive decay chain reaction,
230
90
Th nucleus decays into
214
84
Po nucleus. The
ratio of the number of a to number of
-
ß particles emitted in this process is_______.
Solution:
Let number of a particles are n
a
and
-
ß particles are n
ß
so
( )
4 230 214
4
84 90 2
2
So 2
n
n
nn
n
n
n
a
a
ßa
ß
a
ß
= -
?=
= --
=
=

Question: Two resistances
1
R X = ? and
2
1 R = ?are connected to a wire AB of uniform
resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0.2
mm at A to l mm at B. A galvanometer (G) connected to the center of the wire, 50 cm from

each end along its axis, shows zero deflection when A and B are connected to a battery. The
value of X is ______

Solution:
Resistance of frustum shaped conductor shown is

l
R
ab
?
p
=
For the shown conductor in the diagram

Thus, the resistance of left half is
6
0.5 10
0.2 0.6
P
?
p
××
=
××

And the resistance of right half is
6
0.5 10
0.6 1
Q
?
p
××
=
××

For Wheatstone to be balanced

12
66
0.2 0.6 1 0.6 1
0.5 10 0.5 10
RR
PQ
X
=
×× × ×× ×
=
×× ××
pp
? ?

5 X ?=
Question: In a particular system of units, a physical quantity can be expressed in terms of the
electric charge e, electron mass ???? e, Planck’s constant h, and Coulomb’s constant
0
1
,
4
k
pe
=
where
0
e is the permittivity of vacuum. In terms of these physical constants, the dimension of
the magnetic field is
[ ] [ ] [ ] [ ] [ ] .
e
B e m hk
a ß? d
= The value of a ß? d + + + is ______
Solution:
[ ] [ ] [ ] [ ] [ ]
[ ] [ ]
2
1 1 2 1 3 42
e
B e m hk
M T I IT M MLT MLT I
-
- - --
=
??
? ?? ? =
? ?? ?
??
a ß? d
?d
aß

So, 1 ß? d + + =  …(i)
23 0 += ?d   …(ii)
42 a? d -- =-  …(iii)
On solving
So, 4 a ß? d + + + =
Question: Consider a configuration of n identical units, each consisting of three layers. The
first layer is a column of air of height
1
,
3
h cm = and the second and third layers are of equal
thickness
3 1
,
2
d cm
-
= and refractive indices
12
3
and 3,
2
= = µµ respectively.  A light
source O is placed on the top of the first unit, as shown in the figure. A ray of light from O is
incident on the second layer of the first unit at an angle of ???? = 60° to the normal. For a specific
value of n, the ray of light emerges from the bottom of the configuration at a distance
8
,
3
l cm = as shown in the figure. The value of n is ____.
Page 5

[Paper-2]

PHYSICS

Question: A particle of mass 1 kg is subjected to a force which depends on the position as
( )
2
ˆˆ
F k xi yj kgms
-
=-+
??
with k =1 kgs
-2
. At time t = 0, the particle’s position
1
ˆˆ
2
2
r i jm
??
= +
??
??
?
and its velocity
1
2
ˆ ˆˆ
22 v i j k ms
p
-
??
= -+ +
??
??
?
. Let vx and vy denote the x and
the y components of the particle’s velocity, respectively. Ignore gravity. When z = 0.5 m, the
value of
( )
yx
xv yv - is _____ m
2
s
-1
.
Solution:
( ) ( )
( )
x
2
2
ma
,
sin
m
1
m
and cos
1
0,
2
and 2 m / s
1
sin
2
and 2 co
1

n
/
s
ta
2
5
and
2
x x x
x
x x
xx
x
x
x
xx
x
vA t
A
F
A
att x
d
v
So A
A
x
x
So a x
dt
= +
= =
=- =
= = -
?= +
=
=
= -
=
-
?=-
=
??
?
f
f
f
f
f
?

Similarly
2
2
y y
F y ma
dy
y
dt
=-=
?=-

( )
, sin( )
0 2 m
and 2 m / s
So 2 sin   and
2 cos
and 2
4
yy
yy y
y
yy
yy
y y
So y A t
v A t
att y
v
A
A
A
= +
= = +
= =
=
=
=
?= =
?f
? ? ?f
f
f
p
f

( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
(
( )
( )
55
So, sin 2cos 2sin cos
22
5
2 sin cos sin cos
2
10 sin
10 sin cos cos sin
y x x y y x
x y y x
xy
x y xy
xv yv t t t t
t t t t
n
?f ?f ?f ?f
?f ?f ?f ?f
ff
f f ff
- = + × + - + × +
= × + + - +× +
= -
= -

11 2 1
10 3
52 5 2
?? ??
= × - - × =
?? ??
?? ??

Question: In a radioactive decay chain reaction,
230
90
Th nucleus decays into
214
84
Po nucleus. The
ratio of the number of a to number of
-
ß particles emitted in this process is_______.
Solution:
Let number of a particles are n
a
and
-
ß particles are n
ß
so
( )
4 230 214
4
84 90 2
2
So 2
n
n
nn
n
n
n
a
a
ßa
ß
a
ß
= -
?=
= --
=
=

Question: Two resistances
1
R X = ? and
2
1 R = ?are connected to a wire AB of uniform
resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0.2
mm at A to l mm at B. A galvanometer (G) connected to the center of the wire, 50 cm from

each end along its axis, shows zero deflection when A and B are connected to a battery. The
value of X is ______

Solution:
Resistance of frustum shaped conductor shown is

l
R
ab
?
p
=
For the shown conductor in the diagram

Thus, the resistance of left half is
6
0.5 10
0.2 0.6
P
?
p
××
=
××

And the resistance of right half is
6
0.5 10
0.6 1
Q
?
p
××
=
××

For Wheatstone to be balanced

12
66
0.2 0.6 1 0.6 1
0.5 10 0.5 10
RR
PQ
X
=
×× × ×× ×
=
×× ××
pp
? ?

5 X ?=
Question: In a particular system of units, a physical quantity can be expressed in terms of the
electric charge e, electron mass ???? e, Planck’s constant h, and Coulomb’s constant
0
1
,
4
k
pe
=
where
0
e is the permittivity of vacuum. In terms of these physical constants, the dimension of
the magnetic field is
[ ] [ ] [ ] [ ] [ ] .
e
B e m hk
a ß? d
= The value of a ß? d + + + is ______
Solution:
[ ] [ ] [ ] [ ] [ ]
[ ] [ ]
2
1 1 2 1 3 42
e
B e m hk
M T I IT M MLT MLT I
-
- - --
=
??
? ?? ? =
? ?? ?
??
a ß? d
?d
aß

So, 1 ß? d + + =  …(i)
23 0 += ?d   …(ii)
42 a? d -- =-  …(iii)
On solving
So, 4 a ß? d + + + =
Question: Consider a configuration of n identical units, each consisting of three layers. The
first layer is a column of air of height
1
,
3
h cm = and the second and third layers are of equal
thickness
3 1
,
2
d cm
-
= and refractive indices
12
3
and 3,
2
= = µµ respectively.  A light
source O is placed on the top of the first unit, as shown in the figure. A ray of light from O is
incident on the second layer of the first unit at an angle of ???? = 60° to the normal. For a specific
value of n, the ray of light emerges from the bottom of the configuration at a distance
8
,
3
l cm = as shown in the figure. The value of n is ____.

Solution:

o
1
11
tan 60
3 3
x x cm = =
and,
2
3
1 sin 60 sin
2
o
×= × ?
o
2
2
45
xd
? ?=
? =

and,
3
31
3 sin
2 2
×= × ?
o
3
30 ? = ?
3
3
d
x ?=
( )
12 3
3 1
11
1
2 33
2
3
xx x
cm
-
??
? + + = + +
??
??
=

```

## JEE Main & Advanced Mock Test Series

2 videos|327 docs|203 tests

## JEE Main & Advanced Mock Test Series

2 videos|327 docs|203 tests

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