IIT JAM Mathematics 2021 Past Year Paper

# IIT JAM Mathematics 2021 Past Year Paper | Past Year Papers of IIT JAM Mathematics 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 arbitrarynn 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 arbitrarynn 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 allnn 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
```

16 docs

## FAQs on IIT JAM Mathematics 2021 Past Year Paper - Past Year Papers of IIT JAM Mathematics

 1. What is the eligibility criteria for appearing in the IIT JAM Mathematics 2021 exam?
Ans. To appear in the IIT JAM Mathematics 2021 exam, a candidate must have a bachelor's degree with Mathematics as a subject for at least two years/four semesters. The candidate should have secured at least 55% (50% for SC/ST/PwD category) in the qualifying degree.
 2. What is the exam pattern for IIT JAM Mathematics 2021?
Ans. The IIT JAM Mathematics 2021 exam consists of a total of 60 multiple-choice questions. The duration of the exam is 3 hours. The questions are divided into three sections - Section A, Section B, and Section C. Section A contains 30 multiple-choice questions, and each question carries one or two marks. Section B contains 10 multiple-select questions, and each question carries two marks. Section C contains 20 numerical answer type questions, and each question carries one or two marks.
 3. How can I prepare for the IIT JAM Mathematics 2021 exam effectively?
Ans. To prepare effectively for the IIT JAM Mathematics 2021 exam, you can follow these tips: - Understand the exam syllabus and pattern thoroughly. - Create a study schedule and allocate time for each topic accordingly. - Study from recommended textbooks and reference materials. - Solve previous year question papers and sample papers to understand the exam pattern and practice time management. - Take mock tests regularly to evaluate your preparation and identify areas of improvement. - Seek guidance from mentors or join coaching institutes for expert guidance and doubt clarification.
 4. Are there any negative marking in the IIT JAM Mathematics 2021 exam?
Ans. Yes, there is negative marking in the IIT JAM Mathematics 2021 exam. For Section A, each wrong answer to a one-mark question will result in a deduction of 1/3 mark, and each wrong answer to a two-mark question will result in a deduction of 2/3 mark. There is no negative marking for Section B and Section C.
 5. How can I apply for the IIT JAM Mathematics 2021 exam?
Ans. To apply for the IIT JAM Mathematics 2021 exam, you need to visit the official website of the organizing institute. Fill in the online application form with the required details, upload the necessary documents, and pay the application fee. The application fee can be paid online through net banking, debit card, or credit card. After successful submission of the application form, take a printout of the acknowledgment form for future reference.

## Past Year Papers of IIT JAM Mathematics

16 docs

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