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

 Page 1


JAM 2020 MATHEMATICS - MA
Paper Speci?c 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 ques-
tions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. No
choices will be shown for this 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 Scienti?c 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.
MA 1 / 17
Page 2


JAM 2020 MATHEMATICS - MA
Paper Speci?c 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 ques-
tions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. No
choices will be shown for this 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 Scienti?c 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.
MA 1 / 17
JAM 2020 MATHEMATICS - MA
NOTATION
1. N =f1;2;3;


g
2. R - the set of all real numbers
3. Rnf0g - the set of all non-zero real numbers
4. C - the set of all complex numbers
5. fg - composition of the functionsf andg
6. f
0
andf
00
- ?rst and second derivatives of the functionf, respectively
7. f
(n)
-n
th
derivative off
8.r =
^
i
@
@x
+
^
j
@
@y
+
^
k
@
@z
9.
H
C
- the line integral over an oriented closed curveC
10.
^
i;
^
j;
^
k - unit vectors along the Cartisean right handed rectangular co-ordinate system
11. ^ n - unit outward normal vector
12. I - identity matrix of appropriate order
13. det(M) - determinant of the matrixM
14. M
1
- inverse of the matrixM
15. M
T
- transpose of the matrixM
16. id - identity map
17.hai - cyclic subgroup generated by an elementa of a group
18. S
n
- permutation group onn symbols
19. S
1
=fz2C :jzj = 1g
20. o(g) - order of the elementg in a group
MA 2 / 17
Page 3


JAM 2020 MATHEMATICS - MA
Paper Speci?c 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 ques-
tions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. No
choices will be shown for this 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 Scienti?c 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.
MA 1 / 17
JAM 2020 MATHEMATICS - MA
NOTATION
1. N =f1;2;3;


g
2. R - the set of all real numbers
3. Rnf0g - the set of all non-zero real numbers
4. C - the set of all complex numbers
5. fg - composition of the functionsf andg
6. f
0
andf
00
- ?rst and second derivatives of the functionf, respectively
7. f
(n)
-n
th
derivative off
8.r =
^
i
@
@x
+
^
j
@
@y
+
^
k
@
@z
9.
H
C
- the line integral over an oriented closed curveC
10.
^
i;
^
j;
^
k - unit vectors along the Cartisean right handed rectangular co-ordinate system
11. ^ n - unit outward normal vector
12. I - identity matrix of appropriate order
13. det(M) - determinant of the matrixM
14. M
1
- inverse of the matrixM
15. M
T
- transpose of the matrixM
16. id - identity map
17.hai - cyclic subgroup generated by an elementa of a group
18. S
n
- permutation group onn symbols
19. S
1
=fz2C :jzj = 1g
20. o(g) - order of the elementg in a group
MA 2 / 17
JAM 2020 MATHEMATICS - MA
SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q. 10 carry one mark each.
Q. 1 Lets
n
= 1+
(1)
n
n
;n2N. Then the sequencefs
n
g is
(A) monotonically increasing and is convergent to 1
(B) monotonically decreasing and is convergent to 1
(C) neither monotonically increasing nor monotonically decreasing but is convergent to 1
(D) divergent
Q. 2 Letf(x) = 2x
3
9x
2
+7. Which of the following is true?
(A)f is one-one in the interval[1;1]
(B)f is one-one in the interval[2;4]
(C)f is NOT one-one in the interval[4;0]
(D)f is NOT one-one in the interval[0;4]
Q. 3 Which of the following is FALSE?
(A) lim
x!1
x
e
x
= 0 (B) lim
x!0
+
1
xe
1=x
= 0
(C) lim
x!0
+
sinx
1+2x
= 0 (D) lim
x!0
+
cosx
1+2x
= 0
Q. 4 Letg :R!R be a twice differentiable function. Iff(x;y) =g(y)+xg
0
(y), then
(A)
@f
@x
+y
@
2
f
@x@y
=
@f
@y
(B)
@f
@y
+y
@
2
f
@x@y
=
@f
@x
(C)
@f
@x
+x
@
2
f
@x@y
=
@f
@y
(D)
@f
@y
+x
@
2
f
@x@y
=
@f
@x
MA 3 / 17
Page 4


JAM 2020 MATHEMATICS - MA
Paper Speci?c 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 ques-
tions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. No
choices will be shown for this 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 Scienti?c 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.
MA 1 / 17
JAM 2020 MATHEMATICS - MA
NOTATION
1. N =f1;2;3;


g
2. R - the set of all real numbers
3. Rnf0g - the set of all non-zero real numbers
4. C - the set of all complex numbers
5. fg - composition of the functionsf andg
6. f
0
andf
00
- ?rst and second derivatives of the functionf, respectively
7. f
(n)
-n
th
derivative off
8.r =
^
i
@
@x
+
^
j
@
@y
+
^
k
@
@z
9.
H
C
- the line integral over an oriented closed curveC
10.
^
i;
^
j;
^
k - unit vectors along the Cartisean right handed rectangular co-ordinate system
11. ^ n - unit outward normal vector
12. I - identity matrix of appropriate order
13. det(M) - determinant of the matrixM
14. M
1
- inverse of the matrixM
15. M
T
- transpose of the matrixM
16. id - identity map
17.hai - cyclic subgroup generated by an elementa of a group
18. S
n
- permutation group onn symbols
19. S
1
=fz2C :jzj = 1g
20. o(g) - order of the elementg in a group
MA 2 / 17
JAM 2020 MATHEMATICS - MA
SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q. 10 carry one mark each.
Q. 1 Lets
n
= 1+
(1)
n
n
;n2N. Then the sequencefs
n
g is
(A) monotonically increasing and is convergent to 1
(B) monotonically decreasing and is convergent to 1
(C) neither monotonically increasing nor monotonically decreasing but is convergent to 1
(D) divergent
Q. 2 Letf(x) = 2x
3
9x
2
+7. Which of the following is true?
(A)f is one-one in the interval[1;1]
(B)f is one-one in the interval[2;4]
(C)f is NOT one-one in the interval[4;0]
(D)f is NOT one-one in the interval[0;4]
Q. 3 Which of the following is FALSE?
(A) lim
x!1
x
e
x
= 0 (B) lim
x!0
+
1
xe
1=x
= 0
(C) lim
x!0
+
sinx
1+2x
= 0 (D) lim
x!0
+
cosx
1+2x
= 0
Q. 4 Letg :R!R be a twice differentiable function. Iff(x;y) =g(y)+xg
0
(y), then
(A)
@f
@x
+y
@
2
f
@x@y
=
@f
@y
(B)
@f
@y
+y
@
2
f
@x@y
=
@f
@x
(C)
@f
@x
+x
@
2
f
@x@y
=
@f
@y
(D)
@f
@y
+x
@
2
f
@x@y
=
@f
@x
MA 3 / 17
JAM 2020 MATHEMATICS - MA
Q. 5 If the equation of the tangent plane to the surface z = 16x
2
y
2
at the pointP(1;3;6) is
ax+by+cz +d = 0, then the value ofjdj is
(A) 16 (B) 26 (C) 36 (D) 46
Q. 6 If the directional derivative of the function z = y
2
e
2x
at (2;1) along the unit vector
!
b =

^
i+
^
j is zero, thenj+j equals
(A)
1
2
p
2
(B)
1
p
2
(C)
p
2 (D)2
p
2
Q. 7 If u = x
3
and v = y
2
transform the differential equation 3x
5
dx y(y
2
 x
3
)dy = 0 to
dv
du
=
u
2(uv)
, then is
(A)4 (B)2 (C)2 (D)4
Q. 8 LetT :R
2
!R
2
be the linear transformation given byT(x;y) = (x;y). Then
(A)T
2k
=T for allk 1
(B)T
2k+1
=T for allk 1
(C) the range ofT
2
is a proper subspace of the range ofT
(D) the range ofT
2
is equal to the range ofT
Q. 9 The radius of convergence of the power series
1
X
n=1

n+2
n

n
2
x
n
is
(A)e
2
(B)
1
p
e
(C)
1
e
(D)
1
e
2
MA 4 / 17
Page 5


JAM 2020 MATHEMATICS - MA
Paper Speci?c 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 ques-
tions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. No
choices will be shown for this 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 Scienti?c 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.
MA 1 / 17
JAM 2020 MATHEMATICS - MA
NOTATION
1. N =f1;2;3;


g
2. R - the set of all real numbers
3. Rnf0g - the set of all non-zero real numbers
4. C - the set of all complex numbers
5. fg - composition of the functionsf andg
6. f
0
andf
00
- ?rst and second derivatives of the functionf, respectively
7. f
(n)
-n
th
derivative off
8.r =
^
i
@
@x
+
^
j
@
@y
+
^
k
@
@z
9.
H
C
- the line integral over an oriented closed curveC
10.
^
i;
^
j;
^
k - unit vectors along the Cartisean right handed rectangular co-ordinate system
11. ^ n - unit outward normal vector
12. I - identity matrix of appropriate order
13. det(M) - determinant of the matrixM
14. M
1
- inverse of the matrixM
15. M
T
- transpose of the matrixM
16. id - identity map
17.hai - cyclic subgroup generated by an elementa of a group
18. S
n
- permutation group onn symbols
19. S
1
=fz2C :jzj = 1g
20. o(g) - order of the elementg in a group
MA 2 / 17
JAM 2020 MATHEMATICS - MA
SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q. 10 carry one mark each.
Q. 1 Lets
n
= 1+
(1)
n
n
;n2N. Then the sequencefs
n
g is
(A) monotonically increasing and is convergent to 1
(B) monotonically decreasing and is convergent to 1
(C) neither monotonically increasing nor monotonically decreasing but is convergent to 1
(D) divergent
Q. 2 Letf(x) = 2x
3
9x
2
+7. Which of the following is true?
(A)f is one-one in the interval[1;1]
(B)f is one-one in the interval[2;4]
(C)f is NOT one-one in the interval[4;0]
(D)f is NOT one-one in the interval[0;4]
Q. 3 Which of the following is FALSE?
(A) lim
x!1
x
e
x
= 0 (B) lim
x!0
+
1
xe
1=x
= 0
(C) lim
x!0
+
sinx
1+2x
= 0 (D) lim
x!0
+
cosx
1+2x
= 0
Q. 4 Letg :R!R be a twice differentiable function. Iff(x;y) =g(y)+xg
0
(y), then
(A)
@f
@x
+y
@
2
f
@x@y
=
@f
@y
(B)
@f
@y
+y
@
2
f
@x@y
=
@f
@x
(C)
@f
@x
+x
@
2
f
@x@y
=
@f
@y
(D)
@f
@y
+x
@
2
f
@x@y
=
@f
@x
MA 3 / 17
JAM 2020 MATHEMATICS - MA
Q. 5 If the equation of the tangent plane to the surface z = 16x
2
y
2
at the pointP(1;3;6) is
ax+by+cz +d = 0, then the value ofjdj is
(A) 16 (B) 26 (C) 36 (D) 46
Q. 6 If the directional derivative of the function z = y
2
e
2x
at (2;1) along the unit vector
!
b =

^
i+
^
j is zero, thenj+j equals
(A)
1
2
p
2
(B)
1
p
2
(C)
p
2 (D)2
p
2
Q. 7 If u = x
3
and v = y
2
transform the differential equation 3x
5
dx y(y
2
 x
3
)dy = 0 to
dv
du
=
u
2(uv)
, then is
(A)4 (B)2 (C)2 (D)4
Q. 8 LetT :R
2
!R
2
be the linear transformation given byT(x;y) = (x;y). Then
(A)T
2k
=T for allk 1
(B)T
2k+1
=T for allk 1
(C) the range ofT
2
is a proper subspace of the range ofT
(D) the range ofT
2
is equal to the range ofT
Q. 9 The radius of convergence of the power series
1
X
n=1

n+2
n

n
2
x
n
is
(A)e
2
(B)
1
p
e
(C)
1
e
(D)
1
e
2
MA 4 / 17
JAM 2020 MATHEMATICS - MA
Q. 10 Consider the following group under matrix multiplication:
H =
8
>
<
>
:
2
6
4
1 p q
0 1 r
0 0 1
3
7
5
:p;q;r2R
9
>
=
>
;
:
Then the center of the group is isomorphic to
(A)(Rnf0g;) (B)(R;+)
(C)(R
2
;+) (D)(R;+)(Rnf0g;)
Q. 11 – Q. 30 carry two marks each.
Q. 11 Letfa
n
g be a sequence of positive real numbers. Suppose that l = lim
n!1
a
n+1
a
n
. Which of the
following is true?
(A) Ifl = 1, then lim
n!1
a
n
= 1 (B) Ifl = 1, then lim
n!1
a
n
= 0
(C) Ifl< 1, then lim
n!1
a
n
= 1 (D) Ifl< 1, then lim
n!1
a
n
= 0
Q. 12 De?nes
1
=> 0 ands
n+1
=
r
1+s
2
n
1+
; n 1. Which of the following is true?
(A) Ifs
2
n
<
1

, thenfs
n
g is monotonically increasing and lim
n!1
s
n
=
1
p

(B) Ifs
2
n
<
1

, thenfs
n
g is monotonically decreasing and lim
n!1
s
n
=
1

(C) Ifs
2
n
>
1

, thenfs
n
g is monotonically increasing and lim
n!1
s
n
=
1
p

(D) Ifs
2
n
>
1

, thenfs
n
g is monotonically decreasing and lim
n!1
s
n
=
1

MA 5 / 17