JEE Main 2015 Solved Paper PDF Download

 Page 1


Time : 3 hrs. M.M. : 360
Answers & Solutions
for for for for for
JEE (MAIN)-2015
Important Instructions :
1. The test is of 3 hours duration.
2. The Test Booklet consists of 90 questions. The maximum marks are 360.
3. There are three parts in the question paper A, B, C consisting of Physics, Chemistry and
Mathematics having 30 questions in each part of equal weightage. Each question is allotted 4 (four)
marks for each correct response.
4. Candidates will be awarded marks as stated above in Instructions No. 3 for correct response of each
question. ¼ (one-fourth) marks will be deducted for indicating incorrect response of each question.
No deduction from the total score will be made if no response is indicated for an item in the answer
sheet.
5. There is only one correct response for each question. Filling up more than one response in each
question will be treated as wrong response and marks for wrong response will be deducted accordingly
as per instruction 4 above.
6. Use Blue/Black Ball Point Pen only for writing particulars/marking responses on Side-1 and Side-2
of the Answer Sheet. Use of pencil is strictly prohibited.
7. No candidate is allowed to carry any textual material, printed or written, bits of papers, pager, mobile
phone, any electronic device, etc. except the Admit Card inside the examination room/hall.
8. The CODE for this Booklet is A. Make sure that the CODE printed on Side-2 of the Answer Sheet
and also tally the serial number of the Test Booklet and Answer Sheet are the same as that on this
booklet. In case of discrepancy, the candidate should immediately report the matter to the Invigilator
for replacement of both the Test Booklet and the Answer Sheet.
(Physics, Chemistry and Mathematics)
Page 2


Time : 3 hrs. M.M. : 360
Answers & Solutions
for for for for for
JEE (MAIN)-2015
Important Instructions :
1. The test is of 3 hours duration.
2. The Test Booklet consists of 90 questions. The maximum marks are 360.
3. There are three parts in the question paper A, B, C consisting of Physics, Chemistry and
Mathematics having 30 questions in each part of equal weightage. Each question is allotted 4 (four)
marks for each correct response.
4. Candidates will be awarded marks as stated above in Instructions No. 3 for correct response of each
question. ¼ (one-fourth) marks will be deducted for indicating incorrect response of each question.
No deduction from the total score will be made if no response is indicated for an item in the answer
sheet.
5. There is only one correct response for each question. Filling up more than one response in each
question will be treated as wrong response and marks for wrong response will be deducted accordingly
as per instruction 4 above.
6. Use Blue/Black Ball Point Pen only for writing particulars/marking responses on Side-1 and Side-2
of the Answer Sheet. Use of pencil is strictly prohibited.
7. No candidate is allowed to carry any textual material, printed or written, bits of papers, pager, mobile
phone, any electronic device, etc. except the Admit Card inside the examination room/hall.
8. The CODE for this Booklet is A. Make sure that the CODE printed on Side-2 of the Answer Sheet
and also tally the serial number of the Test Booklet and Answer Sheet are the same as that on this
booklet. In case of discrepancy, the candidate should immediately report the matter to the Invigilator
for replacement of both the Test Booklet and the Answer Sheet.
(Physics, Chemistry and Mathematics)
JEE (MAIN)-2015
2
1. Two stones are thrown up simultaneously from the
edge of a cliff 240 m high with initial speed of
10 m/s and 40 m/s respectively. Which of the
following graph best represents the time variation of
relative position of the second stone with respect to
the first?
(Assume stones do not rebound after hitting the
ground and neglect air resistance, take g = 10 m/s
2
)
(The figures are schematic and not drawn to scale)
(1)
(y
2 
– y
1
) m
240
8
t (s)
12 t ?
(2)
(y
2 
– y
1
) m
240
t (s)
12
(3)
(y
2 
– y
1
) m
240
8
t (s)
12
(4)
(y
2 
– y
1
) m
240
8
t (s)
12
Answer (3)
Sol. Till both are in air (From t = 0 to t = 8 sec)
?x = x
2
 – x
1
 = 30t
??x ? t
When second stone hits ground and first stone is in
air ?x decreases.
PART–A : PHYSICS
2. The period of oscillation of a simple pendulum is
2
L
T
g
??
. Measured value of L is 20.0 cm known
to 1 mm accuracy and time for 100 oscillations of
the pendulum is found to be 90 s using a wrist
watch of 1 s resolution. The accuracy in the
determination of g is
(1) 2% (2) 3%
(3) 1% (4) 5%
Answer (2)
Sol.
2
2
4.
l
g
T
??
?
100 100 2 100
g lT
gl T
? ??
?? ? ? ?
= 
100 2. 100
lt
lt
??
?? ?
= 
0.1 1
100 2 100
20.0 90
??? ?
= 
100 200 1 20
3%
200 90 2 9
?? ? ?
3.
A B
F
Given in the figure are two blocks A and B of
weight 20 N and 100 N,  respectively. These are
being pressed against a wall by a force F as shown.
If the coefficient of friction between the blocks is 0.1
and between block B and the wall is 0.15, the
frictional force applied by the wall on block B is
(1) 100 N (2) 80 N
(3) 120 N (4) 150 N
Answer (3)
Sol.
A B
20 N 100 N
f
s
F
N
Clearly f
s
 = 120 N (for vertical equilibrium of the
system)
Page 3


Time : 3 hrs. M.M. : 360
Answers & Solutions
for for for for for
JEE (MAIN)-2015
Important Instructions :
1. The test is of 3 hours duration.
2. The Test Booklet consists of 90 questions. The maximum marks are 360.
3. There are three parts in the question paper A, B, C consisting of Physics, Chemistry and
Mathematics having 30 questions in each part of equal weightage. Each question is allotted 4 (four)
marks for each correct response.
4. Candidates will be awarded marks as stated above in Instructions No. 3 for correct response of each
question. ¼ (one-fourth) marks will be deducted for indicating incorrect response of each question.
No deduction from the total score will be made if no response is indicated for an item in the answer
sheet.
5. There is only one correct response for each question. Filling up more than one response in each
question will be treated as wrong response and marks for wrong response will be deducted accordingly
as per instruction 4 above.
6. Use Blue/Black Ball Point Pen only for writing particulars/marking responses on Side-1 and Side-2
of the Answer Sheet. Use of pencil is strictly prohibited.
7. No candidate is allowed to carry any textual material, printed or written, bits of papers, pager, mobile
phone, any electronic device, etc. except the Admit Card inside the examination room/hall.
8. The CODE for this Booklet is A. Make sure that the CODE printed on Side-2 of the Answer Sheet
and also tally the serial number of the Test Booklet and Answer Sheet are the same as that on this
booklet. In case of discrepancy, the candidate should immediately report the matter to the Invigilator
for replacement of both the Test Booklet and the Answer Sheet.
(Physics, Chemistry and Mathematics)
JEE (MAIN)-2015
2
1. Two stones are thrown up simultaneously from the
edge of a cliff 240 m high with initial speed of
10 m/s and 40 m/s respectively. Which of the
following graph best represents the time variation of
relative position of the second stone with respect to
the first?
(Assume stones do not rebound after hitting the
ground and neglect air resistance, take g = 10 m/s
2
)
(The figures are schematic and not drawn to scale)
(1)
(y
2 
– y
1
) m
240
8
t (s)
12 t ?
(2)
(y
2 
– y
1
) m
240
t (s)
12
(3)
(y
2 
– y
1
) m
240
8
t (s)
12
(4)
(y
2 
– y
1
) m
240
8
t (s)
12
Answer (3)
Sol. Till both are in air (From t = 0 to t = 8 sec)
?x = x
2
 – x
1
 = 30t
??x ? t
When second stone hits ground and first stone is in
air ?x decreases.
PART–A : PHYSICS
2. The period of oscillation of a simple pendulum is
2
L
T
g
??
. Measured value of L is 20.0 cm known
to 1 mm accuracy and time for 100 oscillations of
the pendulum is found to be 90 s using a wrist
watch of 1 s resolution. The accuracy in the
determination of g is
(1) 2% (2) 3%
(3) 1% (4) 5%
Answer (2)
Sol.
2
2
4.
l
g
T
??
?
100 100 2 100
g lT
gl T
? ??
?? ? ? ?
= 
100 2. 100
lt
lt
??
?? ?
= 
0.1 1
100 2 100
20.0 90
??? ?
= 
100 200 1 20
3%
200 90 2 9
?? ? ?
3.
A B
F
Given in the figure are two blocks A and B of
weight 20 N and 100 N,  respectively. These are
being pressed against a wall by a force F as shown.
If the coefficient of friction between the blocks is 0.1
and between block B and the wall is 0.15, the
frictional force applied by the wall on block B is
(1) 100 N (2) 80 N
(3) 120 N (4) 150 N
Answer (3)
Sol.
A B
20 N 100 N
f
s
F
N
Clearly f
s
 = 120 N (for vertical equilibrium of the
system)
3
JEE (MAIN)-2015
4. A particle of mass m moving in the x direction with
speed 2v is hit by another particle of mass 2m
moving in the y direction wth speed v. If the
collision is perfectly inelastic, the percentage loss in
the energy during the collision is close to
(1) 44% (2) 50%
(3) 56% (4) 62%
Answer (3)
Sol. m
2v
2m
v
v'
?
22
'
3
mv
v
m
=
KE loss = ?? ?? ?
2
2
11
22
22
mv mv
??
2
2
122 5
3
23 3
mv
mmv
m
??
?? ?
??
??
??
Required % = 
??
?
2
22
5
3
100 56%
2
mv
mv mv
5. Distance of the centre of mass of a solid uniform
cone from its vertex is z
0
. If the radius of its base is
R and its height is h then z
0
 is equal to
(1)
2
4
h
R
(2)
3
4
h
(3)
5
8
h
(4)
2
3
8
h
R
Answer (2)
Sol. ?? ?
2
.. dm r dy
y
R
?
h
r
????
??
??
? ?
?
2
0
CM
2
1
3
h
rdyy ydm
y
dm
Rh
?
3
4
h
6. From a solid sphere of mass M and radius R a cube
of maximum possible volume is cut. Moment of
inertia of cube about an axis passing through its
center and perpendicular to one of its faces is
(1)
2
32 2
MR
?
(2)
2
16 2
MR
?
(3)
2
4
93
MR
?
(4)
2
4
33
MR
?
Answer (3)
Sol.
23 dRa ??
?
2
3
aR ?
3
3
4
3
3
2
2
3
R
M
M
R
?
???
?
??
??
??
?
2
'
3
M
M ?
?
2
2
'2 41
636 3
Ma M
IR ?? ? ?
?
2
4
93
MR
I ?
?
7. From a solid sphere of mass M and radius R, a
spherical portion of radius 
2
R
 is removed, as
shown in the figure. Taking gravitational potential
V = 0 at r = ?, the potential at the centre of the
cavity thus formed is
(G = gravitational constant)
(1)
?
2
GM
R
(2)
?GM
R
(3)
?2
3
GM
R
(4)
?2GM
R
Page 4


Time : 3 hrs. M.M. : 360
Answers & Solutions
for for for for for
JEE (MAIN)-2015
Important Instructions :
1. The test is of 3 hours duration.
2. The Test Booklet consists of 90 questions. The maximum marks are 360.
3. There are three parts in the question paper A, B, C consisting of Physics, Chemistry and
Mathematics having 30 questions in each part of equal weightage. Each question is allotted 4 (four)
marks for each correct response.
4. Candidates will be awarded marks as stated above in Instructions No. 3 for correct response of each
question. ¼ (one-fourth) marks will be deducted for indicating incorrect response of each question.
No deduction from the total score will be made if no response is indicated for an item in the answer
sheet.
5. There is only one correct response for each question. Filling up more than one response in each
question will be treated as wrong response and marks for wrong response will be deducted accordingly
as per instruction 4 above.
6. Use Blue/Black Ball Point Pen only for writing particulars/marking responses on Side-1 and Side-2
of the Answer Sheet. Use of pencil is strictly prohibited.
7. No candidate is allowed to carry any textual material, printed or written, bits of papers, pager, mobile
phone, any electronic device, etc. except the Admit Card inside the examination room/hall.
8. The CODE for this Booklet is A. Make sure that the CODE printed on Side-2 of the Answer Sheet
and also tally the serial number of the Test Booklet and Answer Sheet are the same as that on this
booklet. In case of discrepancy, the candidate should immediately report the matter to the Invigilator
for replacement of both the Test Booklet and the Answer Sheet.
(Physics, Chemistry and Mathematics)
JEE (MAIN)-2015
2
1. Two stones are thrown up simultaneously from the
edge of a cliff 240 m high with initial speed of
10 m/s and 40 m/s respectively. Which of the
following graph best represents the time variation of
relative position of the second stone with respect to
the first?
(Assume stones do not rebound after hitting the
ground and neglect air resistance, take g = 10 m/s
2
)
(The figures are schematic and not drawn to scale)
(1)
(y
2 
– y
1
) m
240
8
t (s)
12 t ?
(2)
(y
2 
– y
1
) m
240
t (s)
12
(3)
(y
2 
– y
1
) m
240
8
t (s)
12
(4)
(y
2 
– y
1
) m
240
8
t (s)
12
Answer (3)
Sol. Till both are in air (From t = 0 to t = 8 sec)
?x = x
2
 – x
1
 = 30t
??x ? t
When second stone hits ground and first stone is in
air ?x decreases.
PART–A : PHYSICS
2. The period of oscillation of a simple pendulum is
2
L
T
g
??
. Measured value of L is 20.0 cm known
to 1 mm accuracy and time for 100 oscillations of
the pendulum is found to be 90 s using a wrist
watch of 1 s resolution. The accuracy in the
determination of g is
(1) 2% (2) 3%
(3) 1% (4) 5%
Answer (2)
Sol.
2
2
4.
l
g
T
??
?
100 100 2 100
g lT
gl T
? ??
?? ? ? ?
= 
100 2. 100
lt
lt
??
?? ?
= 
0.1 1
100 2 100
20.0 90
??? ?
= 
100 200 1 20
3%
200 90 2 9
?? ? ?
3.
A B
F
Given in the figure are two blocks A and B of
weight 20 N and 100 N,  respectively. These are
being pressed against a wall by a force F as shown.
If the coefficient of friction between the blocks is 0.1
and between block B and the wall is 0.15, the
frictional force applied by the wall on block B is
(1) 100 N (2) 80 N
(3) 120 N (4) 150 N
Answer (3)
Sol.
A B
20 N 100 N
f
s
F
N
Clearly f
s
 = 120 N (for vertical equilibrium of the
system)
3
JEE (MAIN)-2015
4. A particle of mass m moving in the x direction with
speed 2v is hit by another particle of mass 2m
moving in the y direction wth speed v. If the
collision is perfectly inelastic, the percentage loss in
the energy during the collision is close to
(1) 44% (2) 50%
(3) 56% (4) 62%
Answer (3)
Sol. m
2v
2m
v
v'
?
22
'
3
mv
v
m
=
KE loss = ?? ?? ?
2
2
11
22
22
mv mv
??
2
2
122 5
3
23 3
mv
mmv
m
??
?? ?
??
??
??
Required % = 
??
?
2
22
5
3
100 56%
2
mv
mv mv
5. Distance of the centre of mass of a solid uniform
cone from its vertex is z
0
. If the radius of its base is
R and its height is h then z
0
 is equal to
(1)
2
4
h
R
(2)
3
4
h
(3)
5
8
h
(4)
2
3
8
h
R
Answer (2)
Sol. ?? ?
2
.. dm r dy
y
R
?
h
r
????
??
??
? ?
?
2
0
CM
2
1
3
h
rdyy ydm
y
dm
Rh
?
3
4
h
6. From a solid sphere of mass M and radius R a cube
of maximum possible volume is cut. Moment of
inertia of cube about an axis passing through its
center and perpendicular to one of its faces is
(1)
2
32 2
MR
?
(2)
2
16 2
MR
?
(3)
2
4
93
MR
?
(4)
2
4
33
MR
?
Answer (3)
Sol.
23 dRa ??
?
2
3
aR ?
3
3
4
3
3
2
2
3
R
M
M
R
?
???
?
??
??
??
?
2
'
3
M
M ?
?
2
2
'2 41
636 3
Ma M
IR ?? ? ?
?
2
4
93
MR
I ?
?
7. From a solid sphere of mass M and radius R, a
spherical portion of radius 
2
R
 is removed, as
shown in the figure. Taking gravitational potential
V = 0 at r = ?, the potential at the centre of the
cavity thus formed is
(G = gravitational constant)
(1)
?
2
GM
R
(2)
?GM
R
(3)
?2
3
GM
R
(4)
?2GM
R
JEE (MAIN)-2015
4
Answer (2)
Sol. V = V
1
 – V
2
2
2
1
3
3
2
2
GM
R
V
R
R
??
??
??
?
?? ??
?? ??
2
3
8
2
2
M
G
V
R
??
??
??
??
??
??
??
?
GM
V
R
?
?
8. A pendulum made of a uniform wire of cross-
sectional area A has time period T.  When an
additional mass M is added to its bob, the time
period changes to T
M
. If the Young's modulus of the
material of the wire is Y then 
1
Y
 is equal to
(g = gravitational acceleration)
(1)
??
??
? ??
??
??
??
??
2
1
M
T A
TMg
(2)
??
??
? ??
??
??
??
??
2
1
M
Mg T
TA
(3)
??
??
???
??
??
??
??
2
1
M
T A
TMg
(4)
??
??
?? ?
??
??
??
??
2
1
M
TA
TMg
Answer (1)
Sol.
?? 2
l
T
g
...(1)
??
?? 2
M
ll
T
g
...(2)
? ???
?
Mgl Fl
Yl
AlAY
...(3)
??
??
??? ??
??
??
??
??
2
1
1
M
T A
YMg T
9. Consider a spherical shell of radius R at
temperature T. The black body radiation inside it
can be considered as an ideal gas of photons with
internal energy per unit volume 
4
U
uT
V
??
 and
pressure 
??
?
??
??
1
3
U
P
V
. If the shell now undergoes an
adiabatic expansion the relation between T and R is
(1) T ? e
–R
(2) T ? e
–3R
(3)
1
T
R
?
(4)
3
1
T
R
?
Answer (3)
Sol.
4
11
33
U
PkT
V
??
??
??
??
...(i)
PV = ?RT ...(ii)
4
1
3
RT
kT
V
?
?
? V ? T
–3
1
R
T
?
10. A solid body of constant heat capacity 1 J/°C is
being heated by keeping it in contact with reservoirs
in two ways :
(i) Sequentially keeping in contact with 2
reservoirs such that each reservoir supplies
same amount of heat.
(ii) Sequentially keeping in contact with 8
reservoirs such that each reservoir supplies
same amount of heat.
In both the cases body is brought from initial
temperature 100°C to final temperature 200°C.
Entropy change of the body in the two cases
respectively is
(1) ln 2, 4ln 2 (2) ln 2, ln 2
(3) ln 2, 2ln 2 (4) 2ln 2, 8ln 2
Answer (None)
Sol.
dQ dT
ds ms
TT
???
?? ?? ? ? ?
??
2
1
473
1log log
373
ee
T dT
sds ms
TT
11. Consider an ideal gas confined in an isolated closed
chamber. As the gas undergoes an adiabatic
expansion, the average time of collision between
molecules increases as V
q
, where V is the volume of
the gas. The value of q is
??
??
??
??
P
v
C
C
(1)
? ? 35
6
(2)
? ? 35
6
(3)
? ?1
2
(4)
? ?1
2
Answer (3)
Sol.
?
?? ?
??
?
??
??
rms 2
1
3
2
v RT N
d
M V
...(i)
V
T
?? ...(ii)
Page 5


Time : 3 hrs. M.M. : 360
Answers & Solutions
for for for for for
JEE (MAIN)-2015
Important Instructions :
1. The test is of 3 hours duration.
2. The Test Booklet consists of 90 questions. The maximum marks are 360.
3. There are three parts in the question paper A, B, C consisting of Physics, Chemistry and
Mathematics having 30 questions in each part of equal weightage. Each question is allotted 4 (four)
marks for each correct response.
4. Candidates will be awarded marks as stated above in Instructions No. 3 for correct response of each
question. ¼ (one-fourth) marks will be deducted for indicating incorrect response of each question.
No deduction from the total score will be made if no response is indicated for an item in the answer
sheet.
5. There is only one correct response for each question. Filling up more than one response in each
question will be treated as wrong response and marks for wrong response will be deducted accordingly
as per instruction 4 above.
6. Use Blue/Black Ball Point Pen only for writing particulars/marking responses on Side-1 and Side-2
of the Answer Sheet. Use of pencil is strictly prohibited.
7. No candidate is allowed to carry any textual material, printed or written, bits of papers, pager, mobile
phone, any electronic device, etc. except the Admit Card inside the examination room/hall.
8. The CODE for this Booklet is A. Make sure that the CODE printed on Side-2 of the Answer Sheet
and also tally the serial number of the Test Booklet and Answer Sheet are the same as that on this
booklet. In case of discrepancy, the candidate should immediately report the matter to the Invigilator
for replacement of both the Test Booklet and the Answer Sheet.
(Physics, Chemistry and Mathematics)
JEE (MAIN)-2015
2
1. Two stones are thrown up simultaneously from the
edge of a cliff 240 m high with initial speed of
10 m/s and 40 m/s respectively. Which of the
following graph best represents the time variation of
relative position of the second stone with respect to
the first?
(Assume stones do not rebound after hitting the
ground and neglect air resistance, take g = 10 m/s
2
)
(The figures are schematic and not drawn to scale)
(1)
(y
2 
– y
1
) m
240
8
t (s)
12 t ?
(2)
(y
2 
– y
1
) m
240
t (s)
12
(3)
(y
2 
– y
1
) m
240
8
t (s)
12
(4)
(y
2 
– y
1
) m
240
8
t (s)
12
Answer (3)
Sol. Till both are in air (From t = 0 to t = 8 sec)
?x = x
2
 – x
1
 = 30t
??x ? t
When second stone hits ground and first stone is in
air ?x decreases.
PART–A : PHYSICS
2. The period of oscillation of a simple pendulum is
2
L
T
g
??
. Measured value of L is 20.0 cm known
to 1 mm accuracy and time for 100 oscillations of
the pendulum is found to be 90 s using a wrist
watch of 1 s resolution. The accuracy in the
determination of g is
(1) 2% (2) 3%
(3) 1% (4) 5%
Answer (2)
Sol.
2
2
4.
l
g
T
??
?
100 100 2 100
g lT
gl T
? ??
?? ? ? ?
= 
100 2. 100
lt
lt
??
?? ?
= 
0.1 1
100 2 100
20.0 90
??? ?
= 
100 200 1 20
3%
200 90 2 9
?? ? ?
3.
A B
F
Given in the figure are two blocks A and B of
weight 20 N and 100 N,  respectively. These are
being pressed against a wall by a force F as shown.
If the coefficient of friction between the blocks is 0.1
and between block B and the wall is 0.15, the
frictional force applied by the wall on block B is
(1) 100 N (2) 80 N
(3) 120 N (4) 150 N
Answer (3)
Sol.
A B
20 N 100 N
f
s
F
N
Clearly f
s
 = 120 N (for vertical equilibrium of the
system)
3
JEE (MAIN)-2015
4. A particle of mass m moving in the x direction with
speed 2v is hit by another particle of mass 2m
moving in the y direction wth speed v. If the
collision is perfectly inelastic, the percentage loss in
the energy during the collision is close to
(1) 44% (2) 50%
(3) 56% (4) 62%
Answer (3)
Sol. m
2v
2m
v
v'
?
22
'
3
mv
v
m
=
KE loss = ?? ?? ?
2
2
11
22
22
mv mv
??
2
2
122 5
3
23 3
mv
mmv
m
??
?? ?
??
??
??
Required % = 
??
?
2
22
5
3
100 56%
2
mv
mv mv
5. Distance of the centre of mass of a solid uniform
cone from its vertex is z
0
. If the radius of its base is
R and its height is h then z
0
 is equal to
(1)
2
4
h
R
(2)
3
4
h
(3)
5
8
h
(4)
2
3
8
h
R
Answer (2)
Sol. ?? ?
2
.. dm r dy
y
R
?
h
r
????
??
??
? ?
?
2
0
CM
2
1
3
h
rdyy ydm
y
dm
Rh
?
3
4
h
6. From a solid sphere of mass M and radius R a cube
of maximum possible volume is cut. Moment of
inertia of cube about an axis passing through its
center and perpendicular to one of its faces is
(1)
2
32 2
MR
?
(2)
2
16 2
MR
?
(3)
2
4
93
MR
?
(4)
2
4
33
MR
?
Answer (3)
Sol.
23 dRa ??
?
2
3
aR ?
3
3
4
3
3
2
2
3
R
M
M
R
?
???
?
??
??
??
?
2
'
3
M
M ?
?
2
2
'2 41
636 3
Ma M
IR ?? ? ?
?
2
4
93
MR
I ?
?
7. From a solid sphere of mass M and radius R, a
spherical portion of radius 
2
R
 is removed, as
shown in the figure. Taking gravitational potential
V = 0 at r = ?, the potential at the centre of the
cavity thus formed is
(G = gravitational constant)
(1)
?
2
GM
R
(2)
?GM
R
(3)
?2
3
GM
R
(4)
?2GM
R
JEE (MAIN)-2015
4
Answer (2)
Sol. V = V
1
 – V
2
2
2
1
3
3
2
2
GM
R
V
R
R
??
??
??
?
?? ??
?? ??
2
3
8
2
2
M
G
V
R
??
??
??
??
??
??
??
?
GM
V
R
?
?
8. A pendulum made of a uniform wire of cross-
sectional area A has time period T.  When an
additional mass M is added to its bob, the time
period changes to T
M
. If the Young's modulus of the
material of the wire is Y then 
1
Y
 is equal to
(g = gravitational acceleration)
(1)
??
??
? ??
??
??
??
??
2
1
M
T A
TMg
(2)
??
??
? ??
??
??
??
??
2
1
M
Mg T
TA
(3)
??
??
???
??
??
??
??
2
1
M
T A
TMg
(4)
??
??
?? ?
??
??
??
??
2
1
M
TA
TMg
Answer (1)
Sol.
?? 2
l
T
g
...(1)
??
?? 2
M
ll
T
g
...(2)
? ???
?
Mgl Fl
Yl
AlAY
...(3)
??
??
??? ??
??
??
??
??
2
1
1
M
T A
YMg T
9. Consider a spherical shell of radius R at
temperature T. The black body radiation inside it
can be considered as an ideal gas of photons with
internal energy per unit volume 
4
U
uT
V
??
 and
pressure 
??
?
??
??
1
3
U
P
V
. If the shell now undergoes an
adiabatic expansion the relation between T and R is
(1) T ? e
–R
(2) T ? e
–3R
(3)
1
T
R
?
(4)
3
1
T
R
?
Answer (3)
Sol.
4
11
33
U
PkT
V
??
??
??
??
...(i)
PV = ?RT ...(ii)
4
1
3
RT
kT
V
?
?
? V ? T
–3
1
R
T
?
10. A solid body of constant heat capacity 1 J/°C is
being heated by keeping it in contact with reservoirs
in two ways :
(i) Sequentially keeping in contact with 2
reservoirs such that each reservoir supplies
same amount of heat.
(ii) Sequentially keeping in contact with 8
reservoirs such that each reservoir supplies
same amount of heat.
In both the cases body is brought from initial
temperature 100°C to final temperature 200°C.
Entropy change of the body in the two cases
respectively is
(1) ln 2, 4ln 2 (2) ln 2, ln 2
(3) ln 2, 2ln 2 (4) 2ln 2, 8ln 2
Answer (None)
Sol.
dQ dT
ds ms
TT
???
?? ?? ? ? ?
??
2
1
473
1log log
373
ee
T dT
sds ms
TT
11. Consider an ideal gas confined in an isolated closed
chamber. As the gas undergoes an adiabatic
expansion, the average time of collision between
molecules increases as V
q
, where V is the volume of
the gas. The value of q is
??
??
??
??
P
v
C
C
(1)
? ? 35
6
(2)
? ? 35
6
(3)
? ?1
2
(4)
? ?1
2
Answer (3)
Sol.
?
?? ?
??
?
??
??
rms 2
1
3
2
v RT N
d
M V
...(i)
V
T
?? ...(ii)
5
JEE (MAIN)-2015
TV
? – 1
 = k ...(iii)
?
? ?
??
1
2
V
12. For a simple pendulum, a graph is plotted between
its kinetic energy (KE) and potential energy (PE)
against its displacement d. Which one of the
following represents these correctly?
(Graphs are schematic and not drawn to scale)
(1)
E
KE
d
PE
(2)
E
KE
d
PE
(3)
E
KE
d
PE
(4)
E
PE
KE
Answer (2)
Sol.
?? ?
22 2
1
KE ( )
2
mA d
??
22
1
PE
2
md
At d = ± A,
PE = maximum while KE = 0.
13. A train is moving on a straight track with speed
20 ms
–1
. It is blowing its whistle at the frequency of
1000 Hz. The percentage change in the frequency
heard by a person standing near the track as the
train passes him is (speed of sound = 320 ms
–1
)
close to
(1) 6% (2) 12%
(3) 18% (4) 24%
Answer (2)
Sol.
??
??
?? ??
??
??
??
??
??
1
320 320
Hz
320 20 300
s
v
ff f f
vv
??
???
??
?
??
2
320
Hz
340
s
v
ff f
vv
??? ? ?
?? ? ?
??? ? ??? ? 221
11
100 1 100
fff
ff
??
???
??
??
300
100 1 12%
340
14. A long cylindrical shell carries positive surface
charge ? in the upper half and negative surface
charge – ? in the lower half. The electric field lines
around the cylinder will look like figure given in
(figures are schematic and not drawn to scale)
(1)
+
+
++
+
+
–
–
–
–
–
–
–
–
(2)
+
+
+
+
+
+
–
–
–
–
–
–
–
–
(3)
+
+
++
+
+
–
–
–
–
–
–
–
–
(4)
Answer (1)
Sol. The field line should resemble that of a dipole.
15. A uniformly charged solid sphere of radius R has
potential V
0 
(measured with respect to ?) on its
surface. For this sphere the equipotential surfaces
with potentials 
000 0
353
,, and 
24 4 4
VVV V
 have radius
R
1
, R
2
, R
3
 and R
4
 respectively. Then
(1) R
1 
= 0 and R
2
 > (R
4
 – R
3
)
(2) R
1 
? 0 and (R
2
 – R
1
) > (R
4
 – R
3
)
(3) R
1 
= 0 and R
2
 < (R
4
 – R
3
)
(4) 2R
 
< R
4