Physics - 2020 Past Year Paper | IIT JAM Past Year Papers and Model Test Paper (All Branches) PDF Download

 Page 1


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2020 
SECTION – A 
0 x =
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q1 – Q10 carry one mark each. 
Q1. Which one of the following functions has a discontinuity in the second derivative at 
, where x is a real variable?  
 (a) ( )
3
fx x =  (b) ( ) f x xx = (c) ( ) ( )
cos fx x = (d) ( )
2
fx x = 
Ans. : (b)  
Solution: 
( )
2
2
2 dx
xx
dx x
=  
Assuming a function to be real. 
 Plots:  
 
Q2. A collimated beam of laser light of wavelength 514nm is normally incident on a smooth 
glass slab placed in air. Given the refractive indices of glass and air are 1.47 and 1.0, 
respectively, the percentage of light intensity reflected back is  
 (a) 0   (b) 4.0  (c) 3.6   (d) 4.2 
Ans. : (c)  
Solution: 
2
12
12
1 1.47
0.036 0.036 100 3.6%
1 1.47
nn
R
nn
? ? - - ? ?
= = = = ×=
? ?? ?
++
? ?
? ?
 
Q3. Two stationary point particles with equal and opposite charges are at some fixed distance 
from each other. The points having zero electric potential lie on:  
 (a) A sphere   (b) A plane   (c) A cylinder            (d) Two parallel planes  
Ans. : (b)  
Solution: All points on the plane are at equal distance from charges, so potential will be 
zero. 
y
2
1
1 - 2 - 1 2
x
1 -
2 -
( ) from 2 to 2 x -
Page 2


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2020 
SECTION – A 
0 x =
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q1 – Q10 carry one mark each. 
Q1. Which one of the following functions has a discontinuity in the second derivative at 
, where x is a real variable?  
 (a) ( )
3
fx x =  (b) ( ) f x xx = (c) ( ) ( )
cos fx x = (d) ( )
2
fx x = 
Ans. : (b)  
Solution: 
( )
2
2
2 dx
xx
dx x
=  
Assuming a function to be real. 
 Plots:  
 
Q2. A collimated beam of laser light of wavelength 514nm is normally incident on a smooth 
glass slab placed in air. Given the refractive indices of glass and air are 1.47 and 1.0, 
respectively, the percentage of light intensity reflected back is  
 (a) 0   (b) 4.0  (c) 3.6   (d) 4.2 
Ans. : (c)  
Solution: 
2
12
12
1 1.47
0.036 0.036 100 3.6%
1 1.47
nn
R
nn
? ? - - ? ?
= = = = ×=
? ?? ?
++
? ?
? ?
 
Q3. Two stationary point particles with equal and opposite charges are at some fixed distance 
from each other. The points having zero electric potential lie on:  
 (a) A sphere   (b) A plane   (c) A cylinder            (d) Two parallel planes  
Ans. : (b)  
Solution: All points on the plane are at equal distance from charges, so potential will be 
zero. 
y
2
1
1 - 2 - 1 2
x
1 -
2 -
( ) from 2 to 2 x -
 
 
 
  
 
 
 
 
 
 
 
   
 
 
Q4. For a system undergoing a first order phase transition at a temperature 
c
T , which one of 
the following graphs best describes the variation of entropy ( S) as a function of 
temperature (T )?  
(a) 
 
(b) 
 
(c) 
 
(d) 
 
Ans. : (d)  
Solution: 
P
g
S
T
? ? ?
= -
? ?
?
? ?
 specific entropy. In first order Phase transition first order 
derivative of specific Gibbs free energy is discontinuous that is specific entropy.    
Q5. In a photoelectric effect experiment, a monochromatic light source emitting photon with 
energy greater than the work function of the metal under test is used. If the power of the 
light source is doubled, which one of the following statements is correct?  
 (a) The number of emitted photoelectrons remains the same  
 (b) The stopping potential remains the same  
 (c) The number of emitted photoelectrons decreases  
 (d) The stopping potential doubles  
Ans.: (b)  
Solution: Stopping potential is not defined at power of light source 
Q6. The figure below shows a cubic unit cell with lattice constant a. The shaded 
crystallographic plane intersects the x -axis at 0.5a. The Miller indices of the shaded 
plane are 
 
 (a) ( ) 210  (b) 
( )
210  (c) ( ) 110  (d) ( ) 102 
x
y
a
z
S
c
T
T
S
c
T
T
S
c
T
T
S
c
T
T
Page 3


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2020 
SECTION – A 
0 x =
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q1 – Q10 carry one mark each. 
Q1. Which one of the following functions has a discontinuity in the second derivative at 
, where x is a real variable?  
 (a) ( )
3
fx x =  (b) ( ) f x xx = (c) ( ) ( )
cos fx x = (d) ( )
2
fx x = 
Ans. : (b)  
Solution: 
( )
2
2
2 dx
xx
dx x
=  
Assuming a function to be real. 
 Plots:  
 
Q2. A collimated beam of laser light of wavelength 514nm is normally incident on a smooth 
glass slab placed in air. Given the refractive indices of glass and air are 1.47 and 1.0, 
respectively, the percentage of light intensity reflected back is  
 (a) 0   (b) 4.0  (c) 3.6   (d) 4.2 
Ans. : (c)  
Solution: 
2
12
12
1 1.47
0.036 0.036 100 3.6%
1 1.47
nn
R
nn
? ? - - ? ?
= = = = ×=
? ?? ?
++
? ?
? ?
 
Q3. Two stationary point particles with equal and opposite charges are at some fixed distance 
from each other. The points having zero electric potential lie on:  
 (a) A sphere   (b) A plane   (c) A cylinder            (d) Two parallel planes  
Ans. : (b)  
Solution: All points on the plane are at equal distance from charges, so potential will be 
zero. 
y
2
1
1 - 2 - 1 2
x
1 -
2 -
( ) from 2 to 2 x -
 
 
 
  
 
 
 
 
 
 
 
   
 
 
Q4. For a system undergoing a first order phase transition at a temperature 
c
T , which one of 
the following graphs best describes the variation of entropy ( S) as a function of 
temperature (T )?  
(a) 
 
(b) 
 
(c) 
 
(d) 
 
Ans. : (d)  
Solution: 
P
g
S
T
? ? ?
= -
? ?
?
? ?
 specific entropy. In first order Phase transition first order 
derivative of specific Gibbs free energy is discontinuous that is specific entropy.    
Q5. In a photoelectric effect experiment, a monochromatic light source emitting photon with 
energy greater than the work function of the metal under test is used. If the power of the 
light source is doubled, which one of the following statements is correct?  
 (a) The number of emitted photoelectrons remains the same  
 (b) The stopping potential remains the same  
 (c) The number of emitted photoelectrons decreases  
 (d) The stopping potential doubles  
Ans.: (b)  
Solution: Stopping potential is not defined at power of light source 
Q6. The figure below shows a cubic unit cell with lattice constant a. The shaded 
crystallographic plane intersects the x -axis at 0.5a. The Miller indices of the shaded 
plane are 
 
 (a) ( ) 210  (b) 
( )
210  (c) ( ) 110  (d) ( ) 102 
x
y
a
z
S
c
T
T
S
c
T
T
S
c
T
T
S
c
T
T
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
Ans. : (a)  
Solution: (i) Intercepts along , , x yz - axis; ,,
2
a
x y az = = = 8 
 (ii)  Divide by lattice parameters  
 
/2 1
, , , 1,
2
aa
a aa
8
? 8 
 (iii) Take reciprocal, 2,1,0 
 ? The Miller indices is ( ) 21 0 
Q7. For a particle moving in a central potential, which one of the following statements is 
correct?  
 (a) The motion is restricted to a plane due to the conservation of angular momentum  
 (b) The motion is restricted to a plane due to the conservation of energy only  
 (c) The motion is restricted to a plane due to the conservation of linear momentum  
 (d) The motion is not restricted to a plane  
Ans. : (a)  
Solution: For central force problem angular momentum J
?
 is conserved and .0 rJ =
?
?
 
which ensure that motion of particle is confined in plane 
Q8. Consider the motion of a quantum particle of mass m and energy E under the influence 
of a step potential of height 
0
V . If R denotes the reflection coefficient, which one of the 
following statements is true?  
 
 (a) If 
0
4
,1
3
E V R = =    (b) 
0
4
,0
3
E V R = = 
 (c) 
0
1
,1
2
E VR = =    (d) 
0
1
, 0.5
2
E VR = = 
Ans. : (c)  
Solution: 
0
2
V
E =   
0
E V ?< , so 1 R = .        Therefore, option (c) is correct 
Q9. The Boolean function 
( ) ( )
PQ P Q Q Q + + is equivalent to:  
 (a) P   (b) P   (c) PQ  (d) PQ 
Ans. : (b)  
x
0
V
Page 4


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2020 
SECTION – A 
0 x =
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q1 – Q10 carry one mark each. 
Q1. Which one of the following functions has a discontinuity in the second derivative at 
, where x is a real variable?  
 (a) ( )
3
fx x =  (b) ( ) f x xx = (c) ( ) ( )
cos fx x = (d) ( )
2
fx x = 
Ans. : (b)  
Solution: 
( )
2
2
2 dx
xx
dx x
=  
Assuming a function to be real. 
 Plots:  
 
Q2. A collimated beam of laser light of wavelength 514nm is normally incident on a smooth 
glass slab placed in air. Given the refractive indices of glass and air are 1.47 and 1.0, 
respectively, the percentage of light intensity reflected back is  
 (a) 0   (b) 4.0  (c) 3.6   (d) 4.2 
Ans. : (c)  
Solution: 
2
12
12
1 1.47
0.036 0.036 100 3.6%
1 1.47
nn
R
nn
? ? - - ? ?
= = = = ×=
? ?? ?
++
? ?
? ?
 
Q3. Two stationary point particles with equal and opposite charges are at some fixed distance 
from each other. The points having zero electric potential lie on:  
 (a) A sphere   (b) A plane   (c) A cylinder            (d) Two parallel planes  
Ans. : (b)  
Solution: All points on the plane are at equal distance from charges, so potential will be 
zero. 
y
2
1
1 - 2 - 1 2
x
1 -
2 -
( ) from 2 to 2 x -
 
 
 
  
 
 
 
 
 
 
 
   
 
 
Q4. For a system undergoing a first order phase transition at a temperature 
c
T , which one of 
the following graphs best describes the variation of entropy ( S) as a function of 
temperature (T )?  
(a) 
 
(b) 
 
(c) 
 
(d) 
 
Ans. : (d)  
Solution: 
P
g
S
T
? ? ?
= -
? ?
?
? ?
 specific entropy. In first order Phase transition first order 
derivative of specific Gibbs free energy is discontinuous that is specific entropy.    
Q5. In a photoelectric effect experiment, a monochromatic light source emitting photon with 
energy greater than the work function of the metal under test is used. If the power of the 
light source is doubled, which one of the following statements is correct?  
 (a) The number of emitted photoelectrons remains the same  
 (b) The stopping potential remains the same  
 (c) The number of emitted photoelectrons decreases  
 (d) The stopping potential doubles  
Ans.: (b)  
Solution: Stopping potential is not defined at power of light source 
Q6. The figure below shows a cubic unit cell with lattice constant a. The shaded 
crystallographic plane intersects the x -axis at 0.5a. The Miller indices of the shaded 
plane are 
 
 (a) ( ) 210  (b) 
( )
210  (c) ( ) 110  (d) ( ) 102 
x
y
a
z
S
c
T
T
S
c
T
T
S
c
T
T
S
c
T
T
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
Ans. : (a)  
Solution: (i) Intercepts along , , x yz - axis; ,,
2
a
x y az = = = 8 
 (ii)  Divide by lattice parameters  
 
/2 1
, , , 1,
2
aa
a aa
8
? 8 
 (iii) Take reciprocal, 2,1,0 
 ? The Miller indices is ( ) 21 0 
Q7. For a particle moving in a central potential, which one of the following statements is 
correct?  
 (a) The motion is restricted to a plane due to the conservation of angular momentum  
 (b) The motion is restricted to a plane due to the conservation of energy only  
 (c) The motion is restricted to a plane due to the conservation of linear momentum  
 (d) The motion is not restricted to a plane  
Ans. : (a)  
Solution: For central force problem angular momentum J
?
 is conserved and .0 rJ =
?
?
 
which ensure that motion of particle is confined in plane 
Q8. Consider the motion of a quantum particle of mass m and energy E under the influence 
of a step potential of height 
0
V . If R denotes the reflection coefficient, which one of the 
following statements is true?  
 
 (a) If 
0
4
,1
3
E V R = =    (b) 
0
4
,0
3
E V R = = 
 (c) 
0
1
,1
2
E VR = =    (d) 
0
1
, 0.5
2
E VR = = 
Ans. : (c)  
Solution: 
0
2
V
E =   
0
E V ?< , so 1 R = .        Therefore, option (c) is correct 
Q9. The Boolean function 
( ) ( )
PQ P Q Q Q + + is equivalent to:  
 (a) P   (b) P   (c) PQ  (d) PQ 
Ans. : (b)  
x
0
V
 
 
 
  
 
 
 
 
 
 
 
   
 
 
Solution: 
( ) ( )
Y PQ P Q Q Q = ++
( ) ( )
.1 Y PQ PQ P PQ PQ ? = + + = + + 
( )
( )
11 Y P Q PQ P Q P ? = + + = + = 
Q10. Three point charges each carrying a charge q are placed on the vertices of an equilateral 
triangle of side L . The electrostatic potential energy of the configuration is:  
 (a) 
2
0
1
4
q
L pe
  (b) 
2
0
2
4
q
L pe
  (c) 
2
0
3
4
q
L pe
  (d) 
2
0
1 q
L pe
 
Ans. 10: (c) 
Solution:   Work done in placing charges 
1
0 W = , 
2
2
0
1
4
q
W
L pe
= and  
2
3
00
1 12
44
qq q
Wq
LL L pe pe
??
= + =
??
??
 
Also Total work done  
( )
22
1 23
00
13
1 2
44
qq
WW W
LL pe pe
= + + = +=  
Q11 - Q30 carry two marks each.  
Q11. Which of the following statements is correct? 
 Given 
( )
!
!!
n
n
m mn m
??
=
??
-
??
 is the binomial coefficient  
 (a) 
22 44
cos cos cos sin cos sin ...
24
nn n
nn
n? ? ?? ??
--
? ? ? ?
=- + -
? ? ? ?
? ? ? ?
 
 (b) 
1 33
sin cos sin cos sin ...
1 3
n n
nn
n? ?? ? ?
- -
? ? ? ?
= ++
? ? ? ?
? ? ? ?
 
 (c) 
22 44
cos cos cos sin cos sin ....
2 4
nn n
n n
n? ? ?? ??
--
? ? ? ?
=+++
? ? ? ?
? ? ? ?
 
 (d) 
22 44
sin cos cos sin cos sin ....
24
nn n
nn
n? ? ?? ??
--
? ? ? ?
=-+ -
? ? ? ?
? ? ? ?
 
Ans. : (a)  
Solution: Let’s invoke De Moivre’s theorem here. We have  
    ( ) ( ) ( ) ( ) ( )
cos sin cos sin
n
ni n i ? ? ?? += + 
Since n is a positive integer, the binomial theorem holds, 
                                         ( ) ( ) ( )
cos sin
n
i ?? + 
q
q q
•
• •
L L
L
1
2 3
Page 5


 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
IIT-JAM - 2020 
SECTION – A 
0 x =
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q1 – Q10 carry one mark each. 
Q1. Which one of the following functions has a discontinuity in the second derivative at 
, where x is a real variable?  
 (a) ( )
3
fx x =  (b) ( ) f x xx = (c) ( ) ( )
cos fx x = (d) ( )
2
fx x = 
Ans. : (b)  
Solution: 
( )
2
2
2 dx
xx
dx x
=  
Assuming a function to be real. 
 Plots:  
 
Q2. A collimated beam of laser light of wavelength 514nm is normally incident on a smooth 
glass slab placed in air. Given the refractive indices of glass and air are 1.47 and 1.0, 
respectively, the percentage of light intensity reflected back is  
 (a) 0   (b) 4.0  (c) 3.6   (d) 4.2 
Ans. : (c)  
Solution: 
2
12
12
1 1.47
0.036 0.036 100 3.6%
1 1.47
nn
R
nn
? ? - - ? ?
= = = = ×=
? ?? ?
++
? ?
? ?
 
Q3. Two stationary point particles with equal and opposite charges are at some fixed distance 
from each other. The points having zero electric potential lie on:  
 (a) A sphere   (b) A plane   (c) A cylinder            (d) Two parallel planes  
Ans. : (b)  
Solution: All points on the plane are at equal distance from charges, so potential will be 
zero. 
y
2
1
1 - 2 - 1 2
x
1 -
2 -
( ) from 2 to 2 x -
 
 
 
  
 
 
 
 
 
 
 
   
 
 
Q4. For a system undergoing a first order phase transition at a temperature 
c
T , which one of 
the following graphs best describes the variation of entropy ( S) as a function of 
temperature (T )?  
(a) 
 
(b) 
 
(c) 
 
(d) 
 
Ans. : (d)  
Solution: 
P
g
S
T
? ? ?
= -
? ?
?
? ?
 specific entropy. In first order Phase transition first order 
derivative of specific Gibbs free energy is discontinuous that is specific entropy.    
Q5. In a photoelectric effect experiment, a monochromatic light source emitting photon with 
energy greater than the work function of the metal under test is used. If the power of the 
light source is doubled, which one of the following statements is correct?  
 (a) The number of emitted photoelectrons remains the same  
 (b) The stopping potential remains the same  
 (c) The number of emitted photoelectrons decreases  
 (d) The stopping potential doubles  
Ans.: (b)  
Solution: Stopping potential is not defined at power of light source 
Q6. The figure below shows a cubic unit cell with lattice constant a. The shaded 
crystallographic plane intersects the x -axis at 0.5a. The Miller indices of the shaded 
plane are 
 
 (a) ( ) 210  (b) 
( )
210  (c) ( ) 110  (d) ( ) 102 
x
y
a
z
S
c
T
T
S
c
T
T
S
c
T
T
S
c
T
T
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
Ans. : (a)  
Solution: (i) Intercepts along , , x yz - axis; ,,
2
a
x y az = = = 8 
 (ii)  Divide by lattice parameters  
 
/2 1
, , , 1,
2
aa
a aa
8
? 8 
 (iii) Take reciprocal, 2,1,0 
 ? The Miller indices is ( ) 21 0 
Q7. For a particle moving in a central potential, which one of the following statements is 
correct?  
 (a) The motion is restricted to a plane due to the conservation of angular momentum  
 (b) The motion is restricted to a plane due to the conservation of energy only  
 (c) The motion is restricted to a plane due to the conservation of linear momentum  
 (d) The motion is not restricted to a plane  
Ans. : (a)  
Solution: For central force problem angular momentum J
?
 is conserved and .0 rJ =
?
?
 
which ensure that motion of particle is confined in plane 
Q8. Consider the motion of a quantum particle of mass m and energy E under the influence 
of a step potential of height 
0
V . If R denotes the reflection coefficient, which one of the 
following statements is true?  
 
 (a) If 
0
4
,1
3
E V R = =    (b) 
0
4
,0
3
E V R = = 
 (c) 
0
1
,1
2
E VR = =    (d) 
0
1
, 0.5
2
E VR = = 
Ans. : (c)  
Solution: 
0
2
V
E =   
0
E V ?< , so 1 R = .        Therefore, option (c) is correct 
Q9. The Boolean function 
( ) ( )
PQ P Q Q Q + + is equivalent to:  
 (a) P   (b) P   (c) PQ  (d) PQ 
Ans. : (b)  
x
0
V
 
 
 
  
 
 
 
 
 
 
 
   
 
 
Solution: 
( ) ( )
Y PQ P Q Q Q = ++
( ) ( )
.1 Y PQ PQ P PQ PQ ? = + + = + + 
( )
( )
11 Y P Q PQ P Q P ? = + + = + = 
Q10. Three point charges each carrying a charge q are placed on the vertices of an equilateral 
triangle of side L . The electrostatic potential energy of the configuration is:  
 (a) 
2
0
1
4
q
L pe
  (b) 
2
0
2
4
q
L pe
  (c) 
2
0
3
4
q
L pe
  (d) 
2
0
1 q
L pe
 
Ans. 10: (c) 
Solution:   Work done in placing charges 
1
0 W = , 
2
2
0
1
4
q
W
L pe
= and  
2
3
00
1 12
44
qq q
Wq
LL L pe pe
??
= + =
??
??
 
Also Total work done  
( )
22
1 23
00
13
1 2
44
qq
WW W
LL pe pe
= + + = +=  
Q11 - Q30 carry two marks each.  
Q11. Which of the following statements is correct? 
 Given 
( )
!
!!
n
n
m mn m
??
=
??
-
??
 is the binomial coefficient  
 (a) 
22 44
cos cos cos sin cos sin ...
24
nn n
nn
n? ? ?? ??
--
? ? ? ?
=- + -
? ? ? ?
? ? ? ?
 
 (b) 
1 33
sin cos sin cos sin ...
1 3
n n
nn
n? ?? ? ?
- -
? ? ? ?
= ++
? ? ? ?
? ? ? ?
 
 (c) 
22 44
cos cos cos sin cos sin ....
2 4
nn n
n n
n? ? ?? ??
--
? ? ? ?
=+++
? ? ? ?
? ? ? ?
 
 (d) 
22 44
sin cos cos sin cos sin ....
24
nn n
nn
n? ? ?? ??
--
? ? ? ?
=-+ -
? ? ? ?
? ? ? ?
 
Ans. : (a)  
Solution: Let’s invoke De Moivre’s theorem here. We have  
    ( ) ( ) ( ) ( ) ( )
cos sin cos sin
n
ni n i ? ? ?? += + 
Since n is a positive integer, the binomial theorem holds, 
                                         ( ) ( ) ( )
cos sin
n
i ?? + 
q
q q
•
• •
L L
L
1
2 3
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
Hence, by expanding, we have  
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( )
1 2 22
1
cos sin cos cos sin cos sin ....
12
n
nn n
nn
i ni i ? ? ? ?? ? ?
--
-
+ = + ·+ · +
·
 
Since 
2 3 45
1, , 1, ,.... i i ii i i =- =-= = , we have  
( ) ( ) ( ) ( )
( )
( ) ( )
22
1
cos sin cos cos sin
12
n
nn
nn
i ? ? ? ? ?
-
? -
+= - ·
?
·
?
 
   
( ) ( ) ( )
( ) ( )
44
12 3
cos sin ....
1 234
n
nn n n
??
-
? -- -
+ ·+
?
· ··
?
 
   ( ) ( )
( ) ( ) ( )
( ) ( )
1 33
12 3
cos sin cos sin ....
12 3
nn
nn n n
in ?? ? ?
--
? ? -- -
+ ·- · +
? ?
··
? ?
 
 By equating the real and imaginary parts, we obtain  
  ( ) ( )
( )
( ) ( )
22
1
cos cos cos sin
12
nn
nn
n? ? ? ?
-
-
=- ·
·
 
  
( ) ( ) ( )
( ) ( )
44
12 3
cos sin ....
1 234
n
nn n n
??
-
-- -
+ ·+
· ··
 
  Also, ( ) ( ) ( )
( ) ( )
( ) ( )
1 33
12
sin cos sin cos sin .....
12 3
n n
nn n
nn ? ?? ? ?
- -
--
= ·- · +
··
 
Q12. The volume integral
2
3
2
ˆ
r
R
v
r
e dr
r
??
-
??
??
??
?·
??
??
?
??
, where V is the volume of a sphere of radius R 
centered at the origin, is equal to  
 (a) 4 p   (b) 0   (c) 
3
4
3
R p  (d) 1 
Ans. 12: (a) 
Solution: 
         
2
3
2
ˆ
r
R
v
r
J e dr
r
??
-
??
??
??
= ?·
??
??
?
??
( )
22
0
33
4 44
r
RR
v
e r dr e pd p p
?? ??
--
?? ??
?? ??
??
?? = ×= =
??
??
?
 
Q13. 
0
lim
x
x
x
?+
 is equal to  
 (a) 0   (b) 8   (c) e   (d) 1 
Ans. : (d)  
Solution: 
x
yx =
ln
ln ln
1/
x
yx x
x
? = = 
 
2
0
1/
lim ln 0, ln 0
1/
x
x
y xy
x
?
= =- = =
-
, So, 1 y = is the answer. 
0
1 ye ?= =