IIT JAM Mathematics 2021 Past Year Paper | IIT JAM Past Year Papers and Model Test Paper (All Branches) PDF Download

 Page 1


JAM 2021 MATHEMATICS - MA
SECTION–A
MULTIPLECHOICEQUESTIONS(MCQ)
Q.1–Q.10carryonemarkeach.
Q. 1 Let 0 <  < 1 be a real number. The number of differentiable functionsy : [0;1]! [0;1),
having continuous derivative on[0;1] and satisfying
y
0
(t) = (y(t))

; t2 [0;1];
y(0) = 0;
is
(A) exactly one. (B) exactly two.
(C) ?nite but more than two. (D) in?nite.
Q. 2 LetP :R!R be a continuous function such thatP(x) > 0 for allx2R. Lety be a twice
differentiable function onR satisfyingy
00
(x)+P(x)y
0
(x)y(x) = 0 for allx2R. Suppose
that there exist two real numbersa;b(a<b) such thaty(a) =y(b) = 0. Then
(A)y(x) = 0 for allx2 [a;b]. (B)y(x)> 0 for allx2 (a;b).
(C)y(x)< 0 for allx2 (a;b). (D)y(x) changes sign on(a;b).
Q. 3 Letf :R!R be a continuous function satisfyingf(x) =f(x+1) for allx2R. Then
(A)f is not necessarily bounded above.
(B) there exists a uniquex
0
2R such thatf(x
0
+) =f(x
0
).
(C) there is nox
0
2R such thatf(x
0
+) =f(x
0
).
(D) there exist in?nitely manyx
0
2R such thatf(x
0
+) =f(x
0
).
MA 1 / 17
Page 2


JAM 2021 MATHEMATICS - MA
SECTION–A
MULTIPLECHOICEQUESTIONS(MCQ)
Q.1–Q.10carryonemarkeach.
Q. 1 Let 0 <  < 1 be a real number. The number of differentiable functionsy : [0;1]! [0;1),
having continuous derivative on[0;1] and satisfying
y
0
(t) = (y(t))

; t2 [0;1];
y(0) = 0;
is
(A) exactly one. (B) exactly two.
(C) ?nite but more than two. (D) in?nite.
Q. 2 LetP :R!R be a continuous function such thatP(x) > 0 for allx2R. Lety be a twice
differentiable function onR satisfyingy
00
(x)+P(x)y
0
(x)y(x) = 0 for allx2R. Suppose
that there exist two real numbersa;b(a<b) such thaty(a) =y(b) = 0. Then
(A)y(x) = 0 for allx2 [a;b]. (B)y(x)> 0 for allx2 (a;b).
(C)y(x)< 0 for allx2 (a;b). (D)y(x) changes sign on(a;b).
Q. 3 Letf :R!R be a continuous function satisfyingf(x) =f(x+1) for allx2R. Then
(A)f is not necessarily bounded above.
(B) there exists a uniquex
0
2R such thatf(x
0
+) =f(x
0
).
(C) there is nox
0
2R such thatf(x
0
+) =f(x
0
).
(D) there exist in?nitely manyx
0
2R such thatf(x
0
+) =f(x
0
).
MA 1 / 17
JAM 2021 MATHEMATICS - MA
Q. 4 Letf :R!R be a continuous function such that for allx2R,
Z
1
0
f(xt)dt = 0: ()
Then
(A) f must be identically0 on the whole ofR.
(B) there is anf satisfying() that is identically0 on(0;1) but not identically0 on the whole
ofR.
(C) there is anf satisfying() that takes both positive and negative values.
(D) there is anf satisfying() that is0 at in?nitely many points, but is not identically zero.
Q. 5 Letp andt be positive real numbers. LetD
t
be the closed disc of radiust centered at (0;0),
i.e.,D
t
=f(x;y)2R
2
:x
2
+y
2
t
2
g. De?ne
I(p;t) =
ZZ
Dt
dxdy
(p
2
+x
2
+y
2
)
p
:
Thenlim
t!1
I(p;t) is ?nite
(A) only ifp> 1. (B) only ifp = 1.
(C) only ifp< 1. (D) for no value ofp.
Q. 6 How many elements of the groupZ
50
have order10?
(A) 10 (B) 4 (C) 5 (D) 8
MA 2 / 17
Page 3


JAM 2021 MATHEMATICS - MA
SECTION–A
MULTIPLECHOICEQUESTIONS(MCQ)
Q.1–Q.10carryonemarkeach.
Q. 1 Let 0 <  < 1 be a real number. The number of differentiable functionsy : [0;1]! [0;1),
having continuous derivative on[0;1] and satisfying
y
0
(t) = (y(t))

; t2 [0;1];
y(0) = 0;
is
(A) exactly one. (B) exactly two.
(C) ?nite but more than two. (D) in?nite.
Q. 2 LetP :R!R be a continuous function such thatP(x) > 0 for allx2R. Lety be a twice
differentiable function onR satisfyingy
00
(x)+P(x)y
0
(x)y(x) = 0 for allx2R. Suppose
that there exist two real numbersa;b(a<b) such thaty(a) =y(b) = 0. Then
(A)y(x) = 0 for allx2 [a;b]. (B)y(x)> 0 for allx2 (a;b).
(C)y(x)< 0 for allx2 (a;b). (D)y(x) changes sign on(a;b).
Q. 3 Letf :R!R be a continuous function satisfyingf(x) =f(x+1) for allx2R. Then
(A)f is not necessarily bounded above.
(B) there exists a uniquex
0
2R such thatf(x
0
+) =f(x
0
).
(C) there is nox
0
2R such thatf(x
0
+) =f(x
0
).
(D) there exist in?nitely manyx
0
2R such thatf(x
0
+) =f(x
0
).
MA 1 / 17
JAM 2021 MATHEMATICS - MA
Q. 4 Letf :R!R be a continuous function such that for allx2R,
Z
1
0
f(xt)dt = 0: ()
Then
(A) f must be identically0 on the whole ofR.
(B) there is anf satisfying() that is identically0 on(0;1) but not identically0 on the whole
ofR.
(C) there is anf satisfying() that takes both positive and negative values.
(D) there is anf satisfying() that is0 at in?nitely many points, but is not identically zero.
Q. 5 Letp andt be positive real numbers. LetD
t
be the closed disc of radiust centered at (0;0),
i.e.,D
t
=f(x;y)2R
2
:x
2
+y
2
t
2
g. De?ne
I(p;t) =
ZZ
Dt
dxdy
(p
2
+x
2
+y
2
)
p
:
Thenlim
t!1
I(p;t) is ?nite
(A) only ifp> 1. (B) only ifp = 1.
(C) only ifp< 1. (D) for no value ofp.
Q. 6 How many elements of the groupZ
50
have order10?
(A) 10 (B) 4 (C) 5 (D) 8
MA 2 / 17
JAM 2021 MATHEMATICS - MA
Q. 7 For everyn2 N, letf
n
: R! R be a function. From the given choices, pick the statement
that is the negation of
“For everyx2 R and for every real number  > 0, there exists an integer N > 0 such that
P
p
i=1
jf
N+i
(x)j< for every integerp> 0.”
(A) For everyx2R and for every real number> 0, there does not exist any integerN > 0
such that
P
p
i=1
jf
N+i
(x)j< for every integerp> 0.
(B) For everyx2R and for every real number> 0, there exists an integerN > 0 such that
P
p
i=1
jf
N+i
(x)j for some integerp> 0.
(C) There existsx2R and there exists a real number> 0 such that for every integerN > 0,
there exists an integerp> 0 for which the inequality
P
p
i=1
jf
N+i
(x)j holds.
(D) There existsx2R and there exists a real number> 0 such that for every integerN > 0
and for every integerp> 0 the inequality
P
p
i=1
jf
N+i
(x)j holds.
Q. 8 Which one of the following subsets ofR has a non-empty interior?
(A) The set of all irrational numbers inR.
(B) The setfa2R : sin(a) = 1g.
(C) The setfb2R :x
2
+bx+1 = 0 has distinct rootsg.
(D) The set of all rational numbers inR.
Q. 9 For an integerk 0, letP
k
denote the vector space of all real polynomials in one variable of
degree less than or equal tok. De?ne a linear transformationT :P
2
!P
3
by
Tf(x) =f
00
(x)+xf(x):
Which one of the following polynomials is not in the range ofT ?
(A)x+x
2
(B)x
2
+x
3
+2 (C)x+x
3
+2 (D)x+1
MA 3 / 17
Page 4


JAM 2021 MATHEMATICS - MA
SECTION–A
MULTIPLECHOICEQUESTIONS(MCQ)
Q.1–Q.10carryonemarkeach.
Q. 1 Let 0 <  < 1 be a real number. The number of differentiable functionsy : [0;1]! [0;1),
having continuous derivative on[0;1] and satisfying
y
0
(t) = (y(t))

; t2 [0;1];
y(0) = 0;
is
(A) exactly one. (B) exactly two.
(C) ?nite but more than two. (D) in?nite.
Q. 2 LetP :R!R be a continuous function such thatP(x) > 0 for allx2R. Lety be a twice
differentiable function onR satisfyingy
00
(x)+P(x)y
0
(x)y(x) = 0 for allx2R. Suppose
that there exist two real numbersa;b(a<b) such thaty(a) =y(b) = 0. Then
(A)y(x) = 0 for allx2 [a;b]. (B)y(x)> 0 for allx2 (a;b).
(C)y(x)< 0 for allx2 (a;b). (D)y(x) changes sign on(a;b).
Q. 3 Letf :R!R be a continuous function satisfyingf(x) =f(x+1) for allx2R. Then
(A)f is not necessarily bounded above.
(B) there exists a uniquex
0
2R such thatf(x
0
+) =f(x
0
).
(C) there is nox
0
2R such thatf(x
0
+) =f(x
0
).
(D) there exist in?nitely manyx
0
2R such thatf(x
0
+) =f(x
0
).
MA 1 / 17
JAM 2021 MATHEMATICS - MA
Q. 4 Letf :R!R be a continuous function such that for allx2R,
Z
1
0
f(xt)dt = 0: ()
Then
(A) f must be identically0 on the whole ofR.
(B) there is anf satisfying() that is identically0 on(0;1) but not identically0 on the whole
ofR.
(C) there is anf satisfying() that takes both positive and negative values.
(D) there is anf satisfying() that is0 at in?nitely many points, but is not identically zero.
Q. 5 Letp andt be positive real numbers. LetD
t
be the closed disc of radiust centered at (0;0),
i.e.,D
t
=f(x;y)2R
2
:x
2
+y
2
t
2
g. De?ne
I(p;t) =
ZZ
Dt
dxdy
(p
2
+x
2
+y
2
)
p
:
Thenlim
t!1
I(p;t) is ?nite
(A) only ifp> 1. (B) only ifp = 1.
(C) only ifp< 1. (D) for no value ofp.
Q. 6 How many elements of the groupZ
50
have order10?
(A) 10 (B) 4 (C) 5 (D) 8
MA 2 / 17
JAM 2021 MATHEMATICS - MA
Q. 7 For everyn2 N, letf
n
: R! R be a function. From the given choices, pick the statement
that is the negation of
“For everyx2 R and for every real number  > 0, there exists an integer N > 0 such that
P
p
i=1
jf
N+i
(x)j< for every integerp> 0.”
(A) For everyx2R and for every real number> 0, there does not exist any integerN > 0
such that
P
p
i=1
jf
N+i
(x)j< for every integerp> 0.
(B) For everyx2R and for every real number> 0, there exists an integerN > 0 such that
P
p
i=1
jf
N+i
(x)j for some integerp> 0.
(C) There existsx2R and there exists a real number> 0 such that for every integerN > 0,
there exists an integerp> 0 for which the inequality
P
p
i=1
jf
N+i
(x)j holds.
(D) There existsx2R and there exists a real number> 0 such that for every integerN > 0
and for every integerp> 0 the inequality
P
p
i=1
jf
N+i
(x)j holds.
Q. 8 Which one of the following subsets ofR has a non-empty interior?
(A) The set of all irrational numbers inR.
(B) The setfa2R : sin(a) = 1g.
(C) The setfb2R :x
2
+bx+1 = 0 has distinct rootsg.
(D) The set of all rational numbers inR.
Q. 9 For an integerk 0, letP
k
denote the vector space of all real polynomials in one variable of
degree less than or equal tok. De?ne a linear transformationT :P
2
!P
3
by
Tf(x) =f
00
(x)+xf(x):
Which one of the following polynomials is not in the range ofT ?
(A)x+x
2
(B)x
2
+x
3
+2 (C)x+x
3
+2 (D)x+1
MA 3 / 17
JAM 2021 MATHEMATICS - MA
Q. 10 Letn > 1 be an integer. Consider the following two statements for an arbitrarynn matrix
A with complex entries.
I. IfA
k
=I
n
for some integerk 1, then all the eigenvalues ofA arek
th
roots of unity.
II. If, for some integerk 1, all the eigenvalues ofA arek
th
roots of unity, thenA
k
=I
n
.
Then
(A) both I and II are TRUE. (B) I is TRUE but II is FALSE.
(C) I is FALSE but II is TRUE. (D) neither I nor II is TRUE.
MA 4 / 17
Page 5


JAM 2021 MATHEMATICS - MA
SECTION–A
MULTIPLECHOICEQUESTIONS(MCQ)
Q.1–Q.10carryonemarkeach.
Q. 1 Let 0 <  < 1 be a real number. The number of differentiable functionsy : [0;1]! [0;1),
having continuous derivative on[0;1] and satisfying
y
0
(t) = (y(t))

; t2 [0;1];
y(0) = 0;
is
(A) exactly one. (B) exactly two.
(C) ?nite but more than two. (D) in?nite.
Q. 2 LetP :R!R be a continuous function such thatP(x) > 0 for allx2R. Lety be a twice
differentiable function onR satisfyingy
00
(x)+P(x)y
0
(x)y(x) = 0 for allx2R. Suppose
that there exist two real numbersa;b(a<b) such thaty(a) =y(b) = 0. Then
(A)y(x) = 0 for allx2 [a;b]. (B)y(x)> 0 for allx2 (a;b).
(C)y(x)< 0 for allx2 (a;b). (D)y(x) changes sign on(a;b).
Q. 3 Letf :R!R be a continuous function satisfyingf(x) =f(x+1) for allx2R. Then
(A)f is not necessarily bounded above.
(B) there exists a uniquex
0
2R such thatf(x
0
+) =f(x
0
).
(C) there is nox
0
2R such thatf(x
0
+) =f(x
0
).
(D) there exist in?nitely manyx
0
2R such thatf(x
0
+) =f(x
0
).
MA 1 / 17
JAM 2021 MATHEMATICS - MA
Q. 4 Letf :R!R be a continuous function such that for allx2R,
Z
1
0
f(xt)dt = 0: ()
Then
(A) f must be identically0 on the whole ofR.
(B) there is anf satisfying() that is identically0 on(0;1) but not identically0 on the whole
ofR.
(C) there is anf satisfying() that takes both positive and negative values.
(D) there is anf satisfying() that is0 at in?nitely many points, but is not identically zero.
Q. 5 Letp andt be positive real numbers. LetD
t
be the closed disc of radiust centered at (0;0),
i.e.,D
t
=f(x;y)2R
2
:x
2
+y
2
t
2
g. De?ne
I(p;t) =
ZZ
Dt
dxdy
(p
2
+x
2
+y
2
)
p
:
Thenlim
t!1
I(p;t) is ?nite
(A) only ifp> 1. (B) only ifp = 1.
(C) only ifp< 1. (D) for no value ofp.
Q. 6 How many elements of the groupZ
50
have order10?
(A) 10 (B) 4 (C) 5 (D) 8
MA 2 / 17
JAM 2021 MATHEMATICS - MA
Q. 7 For everyn2 N, letf
n
: R! R be a function. From the given choices, pick the statement
that is the negation of
“For everyx2 R and for every real number  > 0, there exists an integer N > 0 such that
P
p
i=1
jf
N+i
(x)j< for every integerp> 0.”
(A) For everyx2R and for every real number> 0, there does not exist any integerN > 0
such that
P
p
i=1
jf
N+i
(x)j< for every integerp> 0.
(B) For everyx2R and for every real number> 0, there exists an integerN > 0 such that
P
p
i=1
jf
N+i
(x)j for some integerp> 0.
(C) There existsx2R and there exists a real number> 0 such that for every integerN > 0,
there exists an integerp> 0 for which the inequality
P
p
i=1
jf
N+i
(x)j holds.
(D) There existsx2R and there exists a real number> 0 such that for every integerN > 0
and for every integerp> 0 the inequality
P
p
i=1
jf
N+i
(x)j holds.
Q. 8 Which one of the following subsets ofR has a non-empty interior?
(A) The set of all irrational numbers inR.
(B) The setfa2R : sin(a) = 1g.
(C) The setfb2R :x
2
+bx+1 = 0 has distinct rootsg.
(D) The set of all rational numbers inR.
Q. 9 For an integerk 0, letP
k
denote the vector space of all real polynomials in one variable of
degree less than or equal tok. De?ne a linear transformationT :P
2
!P
3
by
Tf(x) =f
00
(x)+xf(x):
Which one of the following polynomials is not in the range ofT ?
(A)x+x
2
(B)x
2
+x
3
+2 (C)x+x
3
+2 (D)x+1
MA 3 / 17
JAM 2021 MATHEMATICS - MA
Q. 10 Letn > 1 be an integer. Consider the following two statements for an arbitrarynn matrix
A with complex entries.
I. IfA
k
=I
n
for some integerk 1, then all the eigenvalues ofA arek
th
roots of unity.
II. If, for some integerk 1, all the eigenvalues ofA arek
th
roots of unity, thenA
k
=I
n
.
Then
(A) both I and II are TRUE. (B) I is TRUE but II is FALSE.
(C) I is FALSE but II is TRUE. (D) neither I nor II is TRUE.
MA 4 / 17
JAM 2021 MATHEMATICS - MA
Q.11–Q.30carrytwomarkseach.
Q. 11 LetM
n
(R) be the real vector space of allnn matrices with real entries,n 2.
LetA2 M
n
(R). Consider the subspaceW ofM
n
(R) spanned byfI
n
;A;A
2
;:::g. Then the
dimension ofW overR is necessarily
(A)1. (B)n
2
. (C)n. (D) at mostn.
Q. 12 Lety be the solution of
(1+x)y
00
(x)+y
0
(x)
1
1+x
y(x) = 0; x2 (1;1);
y(0) = 1; y
0
(0) = 0:
Then
(A)y is bounded on(0;1). (B)y is bounded on(1;0].
(C)y(x) 2 on(1;1). (D)y attains its minimum atx = 0.
Q. 13 Consider the surfaceS =f(x;y;xy)2R
3
: x
2
+y
2
 1g. Let
~
F = y
^
i+x
^
j +
^
k. If ^ n is the
continuous unit normal ?eld to the surfaceS with positivez-component, then
ZZ
S
~
F
 ^ ndS
equals
(A)

4
: (B)

2
: (C). (D)2.
Q. 14 Consider the following statements.
I. The group(Q;+) has no proper subgroup of ?nite index.
II. The group(Cnf0g;
) has no proper subgroup of ?nite index.
Which one of the following statements is true?
(A) Both I and II are TRUE. (B) I is TRUE but II is FALSE.
(C) II is TRUE but I is FALSE. (D) Neither I nor II is TRUE.
MA 5 / 17