IIT JAM Mathematics 2023 Past Year Paper | IIT JAM Past Year Papers and Model Test Paper (All Branches) PDF Download

 Page 1


JAM 2023 MATHEMATICS- MA
SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q. 10 carry one mark each.
Q. 1 LetG be a finite group. Then G is necessarily a cyclic group if the order of G is
(A) 4
(B) 7
(C) 6
(D) 10
Q. 2 Letv
1
,...,v
9
be the column vectors of a non-zero9×9 real matrixA. Leta
1
,...,a
9
? R,
not all zero, be such that
P
9
i=1
a
i
v
i
=0. Then the systemAx =
P
9
i=1
v
i
has
(A) no solution
(B) a unique solution
(C) more than one but only finitely many solutions
(D) infinitely many solutions
MA 2 / 27
Page 2


JAM 2023 MATHEMATICS- MA
SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q. 10 carry one mark each.
Q. 1 LetG be a finite group. Then G is necessarily a cyclic group if the order of G is
(A) 4
(B) 7
(C) 6
(D) 10
Q. 2 Letv
1
,...,v
9
be the column vectors of a non-zero9×9 real matrixA. Leta
1
,...,a
9
? R,
not all zero, be such that
P
9
i=1
a
i
v
i
=0. Then the systemAx =
P
9
i=1
v
i
has
(A) no solution
(B) a unique solution
(C) more than one but only finitely many solutions
(D) infinitely many solutions
MA 2 / 27
JAM 2023 MATHEMATICS- MA
Q. 3 Which of the following is a subspace of the real vector spaceR
3
?
(A) {(x,y,z)?R
3
: (y+z)
2
+(2x- 3y)
2
= 0}
(B) {(x,y,z)?R
3
:y?Q}
(C) {(x,y,z)?R
3
:yz = 0}
(D) {(x,y,z)?R
3
:x+2y- 3z+1 = 0}
Q. 4 Consider the initial value problem
dy
dx
+ay = 0,
y(0) = 1,
wherea ?R. Then
(A) there is ana such thaty(1) = 0
(B) there is a uniquea such that lim
x?8
y(x) = 0
(C) there is NOa such thaty(2) = 1
(D) there is a uniquea such thaty(1) = 2
MA 3 / 27
Page 3


JAM 2023 MATHEMATICS- MA
SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q. 10 carry one mark each.
Q. 1 LetG be a finite group. Then G is necessarily a cyclic group if the order of G is
(A) 4
(B) 7
(C) 6
(D) 10
Q. 2 Letv
1
,...,v
9
be the column vectors of a non-zero9×9 real matrixA. Leta
1
,...,a
9
? R,
not all zero, be such that
P
9
i=1
a
i
v
i
=0. Then the systemAx =
P
9
i=1
v
i
has
(A) no solution
(B) a unique solution
(C) more than one but only finitely many solutions
(D) infinitely many solutions
MA 2 / 27
JAM 2023 MATHEMATICS- MA
Q. 3 Which of the following is a subspace of the real vector spaceR
3
?
(A) {(x,y,z)?R
3
: (y+z)
2
+(2x- 3y)
2
= 0}
(B) {(x,y,z)?R
3
:y?Q}
(C) {(x,y,z)?R
3
:yz = 0}
(D) {(x,y,z)?R
3
:x+2y- 3z+1 = 0}
Q. 4 Consider the initial value problem
dy
dx
+ay = 0,
y(0) = 1,
wherea ?R. Then
(A) there is ana such thaty(1) = 0
(B) there is a uniquea such that lim
x?8
y(x) = 0
(C) there is NOa such thaty(2) = 1
(D) there is a uniquea such thaty(1) = 2
MA 3 / 27
JAM 2023 MATHEMATICS- MA
Q. 5 Letp(x) = x
57
+3x
10
- 21x
3
+x
2
+21 and
q(x) = p(x)+
57
X
j=1
p
(j)
(x) for allx?R,
wherep
(j)
(x) denotes thej
th
derivative ofp(x). Then the functionq admits
(A) NEITHER a global maximum NOR a global minimum onR
(B) a global maximum but NOT a global minimum onR
(C) a global minimum but NOT a global maximum onR
(D) a global minimum and a global maximum onR
Q. 6 The limit
lim
a?0
?
?
?
?
a
R
0
sin(x
2
)dx
a
R
0
(ln(x+1))
2
dx
?
?
?
?
is
(A) 0
(B) 1
(C)
p e
(D) non-existent
MA 4 / 27
Page 4


JAM 2023 MATHEMATICS- MA
SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q. 10 carry one mark each.
Q. 1 LetG be a finite group. Then G is necessarily a cyclic group if the order of G is
(A) 4
(B) 7
(C) 6
(D) 10
Q. 2 Letv
1
,...,v
9
be the column vectors of a non-zero9×9 real matrixA. Leta
1
,...,a
9
? R,
not all zero, be such that
P
9
i=1
a
i
v
i
=0. Then the systemAx =
P
9
i=1
v
i
has
(A) no solution
(B) a unique solution
(C) more than one but only finitely many solutions
(D) infinitely many solutions
MA 2 / 27
JAM 2023 MATHEMATICS- MA
Q. 3 Which of the following is a subspace of the real vector spaceR
3
?
(A) {(x,y,z)?R
3
: (y+z)
2
+(2x- 3y)
2
= 0}
(B) {(x,y,z)?R
3
:y?Q}
(C) {(x,y,z)?R
3
:yz = 0}
(D) {(x,y,z)?R
3
:x+2y- 3z+1 = 0}
Q. 4 Consider the initial value problem
dy
dx
+ay = 0,
y(0) = 1,
wherea ?R. Then
(A) there is ana such thaty(1) = 0
(B) there is a uniquea such that lim
x?8
y(x) = 0
(C) there is NOa such thaty(2) = 1
(D) there is a uniquea such thaty(1) = 2
MA 3 / 27
JAM 2023 MATHEMATICS- MA
Q. 5 Letp(x) = x
57
+3x
10
- 21x
3
+x
2
+21 and
q(x) = p(x)+
57
X
j=1
p
(j)
(x) for allx?R,
wherep
(j)
(x) denotes thej
th
derivative ofp(x). Then the functionq admits
(A) NEITHER a global maximum NOR a global minimum onR
(B) a global maximum but NOT a global minimum onR
(C) a global minimum but NOT a global maximum onR
(D) a global minimum and a global maximum onR
Q. 6 The limit
lim
a?0
?
?
?
?
a
R
0
sin(x
2
)dx
a
R
0
(ln(x+1))
2
dx
?
?
?
?
is
(A) 0
(B) 1
(C)
p e
(D) non-existent
MA 4 / 27
JAM 2023 MATHEMATICS- MA
Q. 7 The value of
Z
1
0
Z
1-x
0
cos(x
3
+y
2
)dydx- Z
1
0
Z
1-y
0
cos(x
3
+y
2
)dxdy
is
(A) 0
(B)
cos(1)
2
(C)
sin(1)
2
(D) cos

1
2

- sin

1
2

Q. 8 Let f : R
2
? R
2
be defined by f(x,y) = (e
x
cos(y),e
x
sin(y)). Then the number of
points inR
2
that do NOT lie in the range of f is
(A) 0
(B) 1
(C) 2
(D) infinite
MA 5 / 27
Page 5


JAM 2023 MATHEMATICS- MA
SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q. 10 carry one mark each.
Q. 1 LetG be a finite group. Then G is necessarily a cyclic group if the order of G is
(A) 4
(B) 7
(C) 6
(D) 10
Q. 2 Letv
1
,...,v
9
be the column vectors of a non-zero9×9 real matrixA. Leta
1
,...,a
9
? R,
not all zero, be such that
P
9
i=1
a
i
v
i
=0. Then the systemAx =
P
9
i=1
v
i
has
(A) no solution
(B) a unique solution
(C) more than one but only finitely many solutions
(D) infinitely many solutions
MA 2 / 27
JAM 2023 MATHEMATICS- MA
Q. 3 Which of the following is a subspace of the real vector spaceR
3
?
(A) {(x,y,z)?R
3
: (y+z)
2
+(2x- 3y)
2
= 0}
(B) {(x,y,z)?R
3
:y?Q}
(C) {(x,y,z)?R
3
:yz = 0}
(D) {(x,y,z)?R
3
:x+2y- 3z+1 = 0}
Q. 4 Consider the initial value problem
dy
dx
+ay = 0,
y(0) = 1,
wherea ?R. Then
(A) there is ana such thaty(1) = 0
(B) there is a uniquea such that lim
x?8
y(x) = 0
(C) there is NOa such thaty(2) = 1
(D) there is a uniquea such thaty(1) = 2
MA 3 / 27
JAM 2023 MATHEMATICS- MA
Q. 5 Letp(x) = x
57
+3x
10
- 21x
3
+x
2
+21 and
q(x) = p(x)+
57
X
j=1
p
(j)
(x) for allx?R,
wherep
(j)
(x) denotes thej
th
derivative ofp(x). Then the functionq admits
(A) NEITHER a global maximum NOR a global minimum onR
(B) a global maximum but NOT a global minimum onR
(C) a global minimum but NOT a global maximum onR
(D) a global minimum and a global maximum onR
Q. 6 The limit
lim
a?0
?
?
?
?
a
R
0
sin(x
2
)dx
a
R
0
(ln(x+1))
2
dx
?
?
?
?
is
(A) 0
(B) 1
(C)
p e
(D) non-existent
MA 4 / 27
JAM 2023 MATHEMATICS- MA
Q. 7 The value of
Z
1
0
Z
1-x
0
cos(x
3
+y
2
)dydx- Z
1
0
Z
1-y
0
cos(x
3
+y
2
)dxdy
is
(A) 0
(B)
cos(1)
2
(C)
sin(1)
2
(D) cos

1
2

- sin

1
2

Q. 8 Let f : R
2
? R
2
be defined by f(x,y) = (e
x
cos(y),e
x
sin(y)). Then the number of
points inR
2
that do NOT lie in the range of f is
(A) 0
(B) 1
(C) 2
(D) infinite
MA 5 / 27
JAM 2023 MATHEMATICS- MA
Q. 9 Leta
n
=

1+
1
n

n
andb
n
=ncos

n!p 2
10

forn?N. Then
(A) (a
n
) is convergent and(b
n
) is bounded
(B) (a
n
) is NOT convergent and (b
n
) is bounded
(C) (a
n
) is convergent and(b
n
) is unbounded
(D) (a
n
) is NOT convergent and (b
n
) is unbounded
Q. 10 Let(a
n
) be a sequence of real numbers defined by
a
n
=
?
?
?
?
?
1 ifn is prime
-1 ifn is not prime.
Letb
n
=
a
n
n
forn?N. Then
(A) both(a
n
) and(b
n
) are convergent
(B) (a
n
) is convergent but(b
n
) is NOT convergent
(C) (a
n
) is NOT convergent but (b
n
) is convergent
(D) both(a
n
) and(b
n
) are NOT convergent
MA 6 / 27