IIT JAM Mathematics 2019 Past Year Paper | Past Year Papers of IIT JAM Mathematics PDF Download

 Page 1


JAM 2019  MATHEMATICS - MA 
MA 1/13 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
  
Page 2


JAM 2019  MATHEMATICS - MA 
MA 1/13 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
  
JAM 2019  MATHEMATICS - MA 
MA 2/13 
Notation 
 
N  set of all natural numbers 1, 2, 3, ? 
 
R  set of all real numbers 
 
?? ?? ×?? ( R)  real vector space of all matrices of size ?? × ?? with entries in R 
 
Ø   empty set 
 
?? \ ??   set of all elements from the set ?? which are not in the set  ?? 
 
Z
??   group of all congruence classes of integers modulo ?? 
 
?? ^ , ?? ^ , ?? ^
  unit vectors having the directions of the positive ?? , ?? and ?? axes of a three dimensional 
rectangular coordinate system, respectively 
 
?? ??  group of all permutations of the set { 1, 2, 3, ? , ?? } 
 
ln  logarithm to the base ?? 
 
log  logarithm to the base 10 
 
?  ?? ^
?? ????
+ ?? ^
?? ????
+ ?? ^
?? ????
    
 
det( ?? )              determinant of a square matrix ?? 
 
 
 
 
  
Page 3


JAM 2019  MATHEMATICS - MA 
MA 1/13 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
  
JAM 2019  MATHEMATICS - MA 
MA 2/13 
Notation 
 
N  set of all natural numbers 1, 2, 3, ? 
 
R  set of all real numbers 
 
?? ?? ×?? ( R)  real vector space of all matrices of size ?? × ?? with entries in R 
 
Ø   empty set 
 
?? \ ??   set of all elements from the set ?? which are not in the set  ?? 
 
Z
??   group of all congruence classes of integers modulo ?? 
 
?? ^ , ?? ^ , ?? ^
  unit vectors having the directions of the positive ?? , ?? and ?? axes of a three dimensional 
rectangular coordinate system, respectively 
 
?? ??  group of all permutations of the set { 1, 2, 3, ? , ?? } 
 
ln  logarithm to the base ?? 
 
log  logarithm to the base 10 
 
?  ?? ^
?? ????
+ ?? ^
?? ????
+ ?? ^
?? ????
    
 
det( ?? )              determinant of a square matrix ?? 
 
 
 
 
  
JAM 2019  MATHEMATICS - MA 
MA 3/13 
SECTION – A 
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q. 1 – Q.10 carry one mark each. 
Q.1  Let ?? 1
= ?? 1
= 0, and for each ?? = 2, let ?? ?? and ?? ?? be real numbers given by 
 
?? ?? = ?
( -1)
?? ?? ( log( ?? ) )
??  and  ?? ?? = ?
1
( log( ?? ) )
?? ?? ?? =2
  
?? ?? =2
. 
 
Then which one of the following is TRUE about the sequences { ?? ?? } and  { ?? ?? }?  
(A)  Both  { ?? ?? }  and  { ?? ?? } are divergent 
(B)  { ?? ?? }  is convergent and  { ?? ?? }  is divergent 
(C)  { ?? ?? }  is divergent and  { ?? ?? }  is convergent 
(D)  Both  { ?? ?? } and  { ?? ?? }  are convergent 
 
 
Q.2  Let  ?? ? ?? ?? ×?? ( R) . Let  ??  be the subspace of  ?? ?? ×?? ( R)  defined by  
 
?? = { ?? ? ?? ?? ×?? ( R) : ???? = 0}. 
 
Then the dimension of  ?? is 
(A)  ???? - rank( ?? ) (B)  ???? - ?? rank( ?? ) 
(C)  ?? ( ?? - rank( ?? ) ) (D)  ?? ( ?? - rank( ?? ) ) 
 
 
Q.3  Let ?? : R ? R be a twice differentiable function. Define ?? : R
3
? R by  
 
?? ( ?? , ?? , ?? ) = ?? ( ?? 2
+ ?? 2
- 2?? 2
) . 
 
 Then   
?? 2
?? ?? ?? 2
+
?? 2
?? ?? ?? 2
+
?? 2
?? ?? ?? 2
   is equal to 
(A)  4( ?? 2
+ ?? 2
- 4?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
) 
(B)  4( ?? 2
+ ?? 2
+ 4?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
) 
(C)  4( ?? 2
+ ?? 2
- 2?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
) 
(D)  4( ?? 2
+ ?? 2
+ 4?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
)+ 8?? '
( ?? 2
+ ?? 2
- 2?? 2
) 
 
 
Q.4  Let { ?? ?? }
?? =0
8
  and  { ?? ?? }
?? =0
8
 be sequences of positive real numbers such that  ?? ?? ?? < ?? ?? < ?? 2
?? ?? for 
all ?? = 2. If the radius of convergence of the power series  ? ?? ?? ?? ?? 8
?? =0
 is 4, then the power series   
? ?? ?? ?? ?? 8
?? =0
 
(A) converges for all ?? with  |?? | < 2 
(B) converges for all ?? with  |?? | > 2 
(C) does not converge for any ?? with  |?? | > 2 
(D) does not converge for any ?? with  |?? | < 2 
 
Q.5  
Let ?? be the set of all limit points of the set  {
?? v2
+
v2
?? : ?? ? N}. Let Q
+
 be the set of all positive 
rational numbers. Then 
(A)  Q
+
? ?? (B)  ?? ? Q
+
 
(C)  ?? n ( R \ Q
+
) ? Ø (D)  ?? n Q
+
? Ø 
 
Page 4


JAM 2019  MATHEMATICS - MA 
MA 1/13 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
  
JAM 2019  MATHEMATICS - MA 
MA 2/13 
Notation 
 
N  set of all natural numbers 1, 2, 3, ? 
 
R  set of all real numbers 
 
?? ?? ×?? ( R)  real vector space of all matrices of size ?? × ?? with entries in R 
 
Ø   empty set 
 
?? \ ??   set of all elements from the set ?? which are not in the set  ?? 
 
Z
??   group of all congruence classes of integers modulo ?? 
 
?? ^ , ?? ^ , ?? ^
  unit vectors having the directions of the positive ?? , ?? and ?? axes of a three dimensional 
rectangular coordinate system, respectively 
 
?? ??  group of all permutations of the set { 1, 2, 3, ? , ?? } 
 
ln  logarithm to the base ?? 
 
log  logarithm to the base 10 
 
?  ?? ^
?? ????
+ ?? ^
?? ????
+ ?? ^
?? ????
    
 
det( ?? )              determinant of a square matrix ?? 
 
 
 
 
  
JAM 2019  MATHEMATICS - MA 
MA 3/13 
SECTION – A 
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q. 1 – Q.10 carry one mark each. 
Q.1  Let ?? 1
= ?? 1
= 0, and for each ?? = 2, let ?? ?? and ?? ?? be real numbers given by 
 
?? ?? = ?
( -1)
?? ?? ( log( ?? ) )
??  and  ?? ?? = ?
1
( log( ?? ) )
?? ?? ?? =2
  
?? ?? =2
. 
 
Then which one of the following is TRUE about the sequences { ?? ?? } and  { ?? ?? }?  
(A)  Both  { ?? ?? }  and  { ?? ?? } are divergent 
(B)  { ?? ?? }  is convergent and  { ?? ?? }  is divergent 
(C)  { ?? ?? }  is divergent and  { ?? ?? }  is convergent 
(D)  Both  { ?? ?? } and  { ?? ?? }  are convergent 
 
 
Q.2  Let  ?? ? ?? ?? ×?? ( R) . Let  ??  be the subspace of  ?? ?? ×?? ( R)  defined by  
 
?? = { ?? ? ?? ?? ×?? ( R) : ???? = 0}. 
 
Then the dimension of  ?? is 
(A)  ???? - rank( ?? ) (B)  ???? - ?? rank( ?? ) 
(C)  ?? ( ?? - rank( ?? ) ) (D)  ?? ( ?? - rank( ?? ) ) 
 
 
Q.3  Let ?? : R ? R be a twice differentiable function. Define ?? : R
3
? R by  
 
?? ( ?? , ?? , ?? ) = ?? ( ?? 2
+ ?? 2
- 2?? 2
) . 
 
 Then   
?? 2
?? ?? ?? 2
+
?? 2
?? ?? ?? 2
+
?? 2
?? ?? ?? 2
   is equal to 
(A)  4( ?? 2
+ ?? 2
- 4?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
) 
(B)  4( ?? 2
+ ?? 2
+ 4?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
) 
(C)  4( ?? 2
+ ?? 2
- 2?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
) 
(D)  4( ?? 2
+ ?? 2
+ 4?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
)+ 8?? '
( ?? 2
+ ?? 2
- 2?? 2
) 
 
 
Q.4  Let { ?? ?? }
?? =0
8
  and  { ?? ?? }
?? =0
8
 be sequences of positive real numbers such that  ?? ?? ?? < ?? ?? < ?? 2
?? ?? for 
all ?? = 2. If the radius of convergence of the power series  ? ?? ?? ?? ?? 8
?? =0
 is 4, then the power series   
? ?? ?? ?? ?? 8
?? =0
 
(A) converges for all ?? with  |?? | < 2 
(B) converges for all ?? with  |?? | > 2 
(C) does not converge for any ?? with  |?? | > 2 
(D) does not converge for any ?? with  |?? | < 2 
 
Q.5  
Let ?? be the set of all limit points of the set  {
?? v2
+
v2
?? : ?? ? N}. Let Q
+
 be the set of all positive 
rational numbers. Then 
(A)  Q
+
? ?? (B)  ?? ? Q
+
 
(C)  ?? n ( R \ Q
+
) ? Ø (D)  ?? n Q
+
? Ø 
 
JAM 2019  MATHEMATICS - MA 
MA 4/13 
 
Q.6  
If ?? h
?? ?? is an integrating factor of the differential equation 
 
?? ( 1 + ???? ) ???? + ?? ( 1 - ???? ) ???? = 0, 
 
then the ordered pair ( h, ?? ) is equal to 
(A)  ( -2, -2) (B)  ( -2, -1) (C)   ( -1, -2) (D)   ( -1, -1) 
 
 
Q.7  
If  ?? ( ?? ) = ?? ?? 2?? + ?? ???? 
, ?? ? 2, is a solution of the differential equation  
 
?? 2
?? ?? ?? 2
+
????
????
- 6?? = 0 
 
satisfying  
????
????
( 0) = 5, then ?? ( 0) is equal to 
(A)  1 (B)  4 (C)  5  (D)   9 
 
 
Q.8  
The equation of the tangent plane to the surface  ?? 2
?? + v8 - ?? 2
- ?? 4
= 6  at the point ( 2, 0, 1) 
is 
(A)  2?? + ?? = 5 (B)  3?? + 4?? = 10 
(C)  3?? - ?? = 10 (D)  7?? - 4?? = 10 
 
 
Q.9   The value of the integral 
 
? ? ?? sin ( ?? ( 1 - ?? )
2
)???? ???? 1-?? 2
?? =0
1
?? =0
 
 
is  
(A)   
1
2?? 
 
(B)  2?? 
(C)   
?? 2
 
(D)   
2
?? 
 
 
Q.10  
The area of the surface generated by rotating the curve ?? = ?? 3
, 0 = ?? = 1, about the ?? -axis, is 
(A)  
?? 27
10
3/2 
 
(B)  
4?? 3
( 10
3/2
- 1) 
(C)   
?? 27
( 10
3/2
- 1) 
(D)   
4?? 3
10
3/2
 
 
 
 
Q. 11 – Q. 30 carry two marks each. 
Q.11  Let  ??  and  ??  be subgroups of  Z
144
. If the order of ?? is 24 and the order of  ?? is 36, then the 
order of the subgroup  ?? n ?? is 
(A)  3 (B)  4 (C)   6 (D)   12 
 
 
Q.12  
Let ?? be a 4 × 4 matrix with entries from the set of rational numbers.  If v2 + ?? ,  with ?? = v-1,  is 
a root of the characteristic polynomial of ?? and ?? is the 4 × 4 identity matrix, then 
(A)  ?? 4
= 4?? 2
+ 9?? (B)  ?? 4
= 4?? 2
- 9?? (C)   ?? 4
= 2?? 2
- 9?? (D)  ?? 4
= 2?? 2
+ 9??  
 
Page 5


JAM 2019  MATHEMATICS - MA 
MA 1/13 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
  
JAM 2019  MATHEMATICS - MA 
MA 2/13 
Notation 
 
N  set of all natural numbers 1, 2, 3, ? 
 
R  set of all real numbers 
 
?? ?? ×?? ( R)  real vector space of all matrices of size ?? × ?? with entries in R 
 
Ø   empty set 
 
?? \ ??   set of all elements from the set ?? which are not in the set  ?? 
 
Z
??   group of all congruence classes of integers modulo ?? 
 
?? ^ , ?? ^ , ?? ^
  unit vectors having the directions of the positive ?? , ?? and ?? axes of a three dimensional 
rectangular coordinate system, respectively 
 
?? ??  group of all permutations of the set { 1, 2, 3, ? , ?? } 
 
ln  logarithm to the base ?? 
 
log  logarithm to the base 10 
 
?  ?? ^
?? ????
+ ?? ^
?? ????
+ ?? ^
?? ????
    
 
det( ?? )              determinant of a square matrix ?? 
 
 
 
 
  
JAM 2019  MATHEMATICS - MA 
MA 3/13 
SECTION – A 
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q. 1 – Q.10 carry one mark each. 
Q.1  Let ?? 1
= ?? 1
= 0, and for each ?? = 2, let ?? ?? and ?? ?? be real numbers given by 
 
?? ?? = ?
( -1)
?? ?? ( log( ?? ) )
??  and  ?? ?? = ?
1
( log( ?? ) )
?? ?? ?? =2
  
?? ?? =2
. 
 
Then which one of the following is TRUE about the sequences { ?? ?? } and  { ?? ?? }?  
(A)  Both  { ?? ?? }  and  { ?? ?? } are divergent 
(B)  { ?? ?? }  is convergent and  { ?? ?? }  is divergent 
(C)  { ?? ?? }  is divergent and  { ?? ?? }  is convergent 
(D)  Both  { ?? ?? } and  { ?? ?? }  are convergent 
 
 
Q.2  Let  ?? ? ?? ?? ×?? ( R) . Let  ??  be the subspace of  ?? ?? ×?? ( R)  defined by  
 
?? = { ?? ? ?? ?? ×?? ( R) : ???? = 0}. 
 
Then the dimension of  ?? is 
(A)  ???? - rank( ?? ) (B)  ???? - ?? rank( ?? ) 
(C)  ?? ( ?? - rank( ?? ) ) (D)  ?? ( ?? - rank( ?? ) ) 
 
 
Q.3  Let ?? : R ? R be a twice differentiable function. Define ?? : R
3
? R by  
 
?? ( ?? , ?? , ?? ) = ?? ( ?? 2
+ ?? 2
- 2?? 2
) . 
 
 Then   
?? 2
?? ?? ?? 2
+
?? 2
?? ?? ?? 2
+
?? 2
?? ?? ?? 2
   is equal to 
(A)  4( ?? 2
+ ?? 2
- 4?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
) 
(B)  4( ?? 2
+ ?? 2
+ 4?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
) 
(C)  4( ?? 2
+ ?? 2
- 2?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
) 
(D)  4( ?? 2
+ ?? 2
+ 4?? 2
) ?? ''
( ?? 2
+ ?? 2
- 2?? 2
)+ 8?? '
( ?? 2
+ ?? 2
- 2?? 2
) 
 
 
Q.4  Let { ?? ?? }
?? =0
8
  and  { ?? ?? }
?? =0
8
 be sequences of positive real numbers such that  ?? ?? ?? < ?? ?? < ?? 2
?? ?? for 
all ?? = 2. If the radius of convergence of the power series  ? ?? ?? ?? ?? 8
?? =0
 is 4, then the power series   
? ?? ?? ?? ?? 8
?? =0
 
(A) converges for all ?? with  |?? | < 2 
(B) converges for all ?? with  |?? | > 2 
(C) does not converge for any ?? with  |?? | > 2 
(D) does not converge for any ?? with  |?? | < 2 
 
Q.5  
Let ?? be the set of all limit points of the set  {
?? v2
+
v2
?? : ?? ? N}. Let Q
+
 be the set of all positive 
rational numbers. Then 
(A)  Q
+
? ?? (B)  ?? ? Q
+
 
(C)  ?? n ( R \ Q
+
) ? Ø (D)  ?? n Q
+
? Ø 
 
JAM 2019  MATHEMATICS - MA 
MA 4/13 
 
Q.6  
If ?? h
?? ?? is an integrating factor of the differential equation 
 
?? ( 1 + ???? ) ???? + ?? ( 1 - ???? ) ???? = 0, 
 
then the ordered pair ( h, ?? ) is equal to 
(A)  ( -2, -2) (B)  ( -2, -1) (C)   ( -1, -2) (D)   ( -1, -1) 
 
 
Q.7  
If  ?? ( ?? ) = ?? ?? 2?? + ?? ???? 
, ?? ? 2, is a solution of the differential equation  
 
?? 2
?? ?? ?? 2
+
????
????
- 6?? = 0 
 
satisfying  
????
????
( 0) = 5, then ?? ( 0) is equal to 
(A)  1 (B)  4 (C)  5  (D)   9 
 
 
Q.8  
The equation of the tangent plane to the surface  ?? 2
?? + v8 - ?? 2
- ?? 4
= 6  at the point ( 2, 0, 1) 
is 
(A)  2?? + ?? = 5 (B)  3?? + 4?? = 10 
(C)  3?? - ?? = 10 (D)  7?? - 4?? = 10 
 
 
Q.9   The value of the integral 
 
? ? ?? sin ( ?? ( 1 - ?? )
2
)???? ???? 1-?? 2
?? =0
1
?? =0
 
 
is  
(A)   
1
2?? 
 
(B)  2?? 
(C)   
?? 2
 
(D)   
2
?? 
 
 
Q.10  
The area of the surface generated by rotating the curve ?? = ?? 3
, 0 = ?? = 1, about the ?? -axis, is 
(A)  
?? 27
10
3/2 
 
(B)  
4?? 3
( 10
3/2
- 1) 
(C)   
?? 27
( 10
3/2
- 1) 
(D)   
4?? 3
10
3/2
 
 
 
 
Q. 11 – Q. 30 carry two marks each. 
Q.11  Let  ??  and  ??  be subgroups of  Z
144
. If the order of ?? is 24 and the order of  ?? is 36, then the 
order of the subgroup  ?? n ?? is 
(A)  3 (B)  4 (C)   6 (D)   12 
 
 
Q.12  
Let ?? be a 4 × 4 matrix with entries from the set of rational numbers.  If v2 + ?? ,  with ?? = v-1,  is 
a root of the characteristic polynomial of ?? and ?? is the 4 × 4 identity matrix, then 
(A)  ?? 4
= 4?? 2
+ 9?? (B)  ?? 4
= 4?? 2
- 9?? (C)   ?? 4
= 2?? 2
- 9?? (D)  ?? 4
= 2?? 2
+ 9??  
 
JAM 2019  MATHEMATICS - MA 
MA 5/13 
 
Q.13  
 The set  {
?? 1+?? : -1 < ?? < 1}, as a subset of R, is 
(A)  connected and compact 
(B)  connected but not compact 
(C)  not connected but compact 
(D)  neither connected nor compact 
 
 
Q.14  
The set  {
1
?? +
1
?? : ?? , ?? ? N} ? { 0}, as a subset of  R, is 
(A)  compact and open (B)  compact but not open 
(C)  not compact but open (D)  neither compact nor open 
 
 
Q.15  
 For -1 < ?? < 1, the sum of the power series  1 + ? ( -1)
?? -1
 ?? 2
?? ?? -1 8
?? =2 
  is 
(A)  
1-?? ( 1+?? )
3
 
(B)  
1+?? 2
( 1+?? )
4
 
(C)  
1-?? ( 1+?? )
2
 
(D)  
1+?? 2
( 1+?? )
3
 
 
 
Q.16   Let  ?? ( ?? ) = ( ln ?? )
2
 , ?? > 0. Then 
(A)  lim
?? ?8
?? ( ?? )
??  does not exist 
(B)  lim
?? ?8
?? '
( ?? )= 2 
(C)  lim
?? ?8
( ?? ( ?? + 1)- ?? ( ?? ) ) = 0 
(D)  lim
?? ?8
( ?? ( ?? + 1)- ?? ( ?? ) )  does not exist 
 
 
Q.17  Let ?? : R ? R be a differentiable function such that ?? '
( ?? )> ?? ( ?? ) for all ?? ? R , and ?? ( 0) = 1. 
Then ?? ( 1) lies in the interval 
(A)  ( 0, ?? -1
) 
(B)  ( ?? -1
, v?? ) (C)  ( v?? , ?? ) 
(D)  ( ?? , 8) 
 
 
Q.18  For which one of the following values of  ?? , the equation  
 
2?? 3
+ 3?? 2
- 12?? - ?? = 0 
 
  has three distinct real roots? 
(A)  16 (B)  20 (C)  26  (D)  31 
 
 
Q.19  Which one of the following series is divergent? 
(A) ?
1
?? sin
2
1
?? 
8
?? =1
  (B) ? 
1
?? log ?? 
8
?? =1
 
(C)  ? 
1
?? 2
sin
1
?? 
8
?? =1
 (D)  ? 
1
?? tan
1
?? 
8
?? =1