IIT JAM 2016 Question Paper Mathematics (MA)

JAM 2016 MATHEMATICS - MA

Notation

The set of all natural numbers 1,2,3, …

The set of all integers

ℚ The set of all rational numbers

The set of all real numbers

The group of permutations of distinct symbols

0, 1, 2, … , 1 with addition and multiplication modulo

empty set

Transpose of

√ 1

, ̂, unit vectors having the directions of the positive , and axes of a three dimensional rectangular

coordinate system

̂ ̂

Identity matrix of order

ln logarithm with base

MA 2/13

JAM 2016 MATHEMATICS - MA

SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q.10 carry one mark each.
Q.1 The sequence of real numbers given by

sin sin sin
2 2 ⋯ 2
1⋅2 2⋅3 ⋅ 1
is
(A) a divergent sequence
(B) an oscillatory sequence
(C) not a Cauchy sequence
(D) a Cauchy sequence

Q.2 Let be the vector space (over of all polynomials of degree 3 with real coefficients. Consider
the linear transformation : → defined by

.

Then the matrix representation of with respect to the ordered basis 1, , , satisfies
(A) 0 (B) 0
(C) 0 (D) 0

Q.3 Let : 1, 1 → be a continuous function. Then the integral

sin

is equivalent to
(A) (B)

sin cos
2 2

(C) (D)

cos sin

Q.4 Let be an element of the permutation group . Then the maximum possible order of is
(A) 5 (B) 6 (C) 10 (D) 15

Q.5 Let be a strictly monotonic continuous real valued function defined on , such that
and . Then which one of the following is TRUE?
(A) There exists exactly one ∈ , such that
(B) There exist exactly two points , ∈ , such that , 1,2
(C) There exists no ∈ , such that
(D) There exist infinitely many points ∈ , such that

MA 3/13

JAM 2016 MATHEMATICS - MA

Q.6
The value of lim is
, → ,

(A) 0 (B) (C) (D)

Q.7 Let ̂ ̂ and | |. If ln and , 0, satisfy
2 0, then is

(A) (B) (C) 2 (D)

Q.8 The nonzero value of for which the differential equation

3 3 0, 0,

becomes exact is
(A) 3 (B) 2 (C) 2 (D) 3

Q.9 One of the points which lies on the solution curve of the differential equation

0,

with the given condition 0 1, is
(A) 1, 2 (B) 2, 1 (C) 2, 1 (D) 1, 2

Q.10 Let be a closed subset of , a compact subset of such that ∩ . Then ∩ is
(A) closed but not compact
(B) not closed
(C) compact
(D) neither closed nor compact

Q. 11 – Q. 30 carry two marks each.
Q.11 Let be the series

1
2 1 2
and be the series
3 4
3 2

of real numbers. Then which one of the following is TRUE?
(A) Both the series and are convergent
(B) is convergent and is divergent
(C) is divergent and is convergent
(D) Both the series and are divergent

MA 4/13

JAM 2016 MATHEMATICS - MA

Q.12 Let be a sequence of positive real numbers satisfying

4 3
, 1, 1.
81

Then all the terms of the sequence lie in

(A) , (B) 0, 1 (C) 1, 2 (D) 1, 3

Q.13 1 4 16
The largest eigenvalue of the matrix 4 16 1 is
16 1 4

(A) 16 (B) 21

(C) 48 (D) 64

Q.14 The value of the integral

2 !
1 , ∈
2 !
is

(A) (B)
! !

! !
(C) (D)

Q.15 If the triple integral over the region bounded by the planes

2 4, 0, 0, 0

is given by

,

,

then the function , is

(A) (B) (C) (D)

Q.16 The surface area of the portion of the plane 2 2 within the cylinder 3 is
√ √ √ √
(A) (B) (C) (D)

MA 5/13

JAM 2016 MATHEMATICS - MA

Q.17 Let : → be defined by

if 0
, .
0 if 0

Then the value of at the point 0, 0 is

(A) 0 (B) 1 (C) 2 (D) 4

Q.18 The function , 3 4 3 12 1 has a saddle point at
(A) 0, 0 (B) 0, 2 (C) 1, 1 (D) 2, 1

Q.19 Consider the vector field ̂ ̂ , where ∈ , ̂ ̂ and | |. If the
absolute value of the line integral ∮ ⋅ along the closed curve : (oriented
counter clockwise) is 2 , then is
(A) 2 (B) 1 (C) 1 (D) 2

Q.20 Let be the surface of the cone bounded by the planes 0 and 3. Further,
let be the closed curve forming the boundary of the surface . A vector field is such that
̂ .̂ The absolute value of the line integral ∮ ⋅ , where
̂ ̂ and | |, is
(A) 0 (B) 9 (C) 15 (D) 18

Q.21 Let be the solution of the differential equation

; 1 0, 0.

Then 2 is

(A) ln 2 (B) ln 2

(C) ln 2 (D) ln 2

Q.22 The general solution of the differential equation with constant coefficients

0

approaches zero as → ∞, if
(A) is negative and is positive
(B) is positive and is negative
(C) both and are positive
(D) both and are negative

MA 6/13

JAM 2016 MATHEMATICS - MA

Q.23 Let ⊂ and denote the set of points in such that every neighbourhood of contains
some points of as well as some points of complement of S. Further, let denote the closure
of . Then which one of the following is FALSE?
(A) ℚ
(B) ∖ ∂ , ⊂
(C) ∪ ∪ , , ⊂ , ∩
(D) ∩ ∖ , ⊂

Q.24 The sum of the series
1
2
is

(A) ln 2 (B) ln 2 (C) ln 2 (D) ln 2

Q.25 Let for all ∈ 1, 1 . Then which one of the following is TRUE?
| | | |

(A) Maximum value of is
(B) Minimum value of is
(C) Maximum of occurs at
(D) Minimum of occurs at 1

Q.26 cos sin
The matrix is a unitary matrix when is
sin cos
(A) 2 1 , ∈ (B) 3 1 , ∈

(C) 4 1 , ∈ (D) 5 1 , ∈

Q.27 0 1 2
Let 1 0 , ∈ ∖ 0 and a non-zero vector such that for some
2 0
∈ . Then the value of is
(A) (B) (C) 0 (D) 1

Q.28 The number of group homomorphisms from the cyclic group to the cyclic group is
(A) 7 (B) 3 (C) 2 (D) 1

Q.29 In the permutation group 5 , if is the smallest subgroup containing all the 3-cycles, then
which one of the following is TRUE?
(A) Order of is 2
(B) Index of in is 2
(C) is abelian
(D)

MA 7/13

JAM 2016 MATHEMATICS - MA

Q.30 Let : → be defined as

1 sin ln if 0.
0 if 0

Then, at 0, the function is
(A) continuous and differentiable when 0
(B) continuous and differentiable when 0
(C) continuous and differentiable when 1 0
(D) continuous and differentiable when 1

SECTION - B
MULTIPLE SELECT QUESTIONS (MSQ)
Q. 31 – Q. 40 carry two marks each.

Q.31 Let be a sequence of positive real numbers satisfying

3
2 , 1.
4

If and are the roots of the equation 2 0 and , then which of the
following statement(s) is(are) TRUE ?
(A) is monotonically decreasing
(B) is monotonically increasing
(C) lim →
(D) lim →

Q.32 The value(s) of the integral

| | cos , 1

is (are)
(A) 0 when is even
(B) 0 when is odd
(C) when is even
(D) when is odd

MA 8/13

JAM 2016 MATHEMATICS - MA

Q.33 Let : → be defined by

if 0
| |
, .
0 elsewhere

Then at the point 0, 0 , which of the following statement(s) is(are) TRUE ?
(A) is not continuous
(B) is continuous
(C) is differentiable
(D) Both first order partial derivatives of exist

Q.34 Consider the vector field ̂ ̂ on an open connected set ⊂ . Then which of the
following statement(s) is(are) TRUE ?

(A) Divergence of is zero on
(B) The line integral of is independent of path in
(C) can be expressed as a gradient of a scalar function on
(D) The line integral of is zero around any piecewise smooth closed path in

Q.35 Consider the differential equation
sin 2 2 2 cos , 1 √2.
4
Then which of the following statement(s) is(are) TRUE?
(A) The solution is unbounded when → 0
(B) The solution is unbounded when →
(C) The solution is bounded when → 0
(D) The solution is bounded when →

Q.36 Which of the following statement(s) is(are) TRUE?
(A) There exists a connected set in which is not compact
(B) Arbitrary union of closed intervals in need not be compact
(C) Arbitrary union of closed intervals in is always closed
(D) Every bounded infinite subset of has a limit point in itself

Q.37 Let 1 for all ∈ . Then which of the following statement(s) is(are)
TRUE?
(A) The equation 0 has exactly one solution in
(B) is strictly increasing for all ∈
(C) The equation 0 has exactly two solutions in
(D) is strictly decreasing for all ∈

MA 9/13

JAM 2016 MATHEMATICS - MA

Q.38 Let be a finite group and denotes its order. Then which of the following statement(s) is(are)
TRUE?
(A) is abelian if where and are distinct primes
(B) is abelian if every non identity element of is of order 2
(C) is abelian if the quotient group is cyclic, where is the center of
(D) is abelian if , where is prime

Q.39
Consider the set ∈ , , , ∈ . For which of the

following choice(s) the set becomes a two dimensional subspace of over ?
(A) 0, 1, 0
(B) 0, 1, 1
(C) 1, 0, 0
(D) 1, 1, 0

Q.40 Let , ∈ . Then which of the following statement(s) is(are) TRUE?

(A) is closed
(B) is not open
(C) is connected
(D) 0 is a limit point of

SECTION – C
NUMERICAL ANSWER TYPE (NAT)
Q. 41 – Q. 50 carry one mark each.

Q.41 Let be a sequence of real numbers given by

1
2 1 sin , ∈ .
2

Then the least upper bound of the sequence is ____________

Q.42 Let be a sequence of real numbers, where

/
, 1, 0.
Then
/
lim …
→
is ____________

MA 10/13

JAM 2016 MATHEMATICS - MA

Q.43
Let ∈ be a non-zero vector and . Then the dimension of the vector space

∈ over is ____________

Q.44 Let be a real valued function defined by

, 2 ln , 0, 0.

Then the value of at any point , , where 0, 0, is ____________

Q.45 Let √ ̂ be a vector field for all , with 0 and ̂. Then the
value of the line integral ⋅ from 0, 0 to 1, 1 along the path
: , , 0 1 is ____________

Q.46 If : 1, ∞ → defined by is expressed as

2 1 2
2 ,
3 9 1

where lies between 2 and , then the value of is ____________

Q.47 Let , and be linearly independent solutions of the differential equation

6 11 6 0.

If the Wronskian , , is of the form for some constant , then the value of
is____________

Q.48 The radius of convergence of the power series

4
2 is ______________
1

MA 11/13

JAM 2016 MATHEMATICS - MA

Q.49 Let : 0, ∞ → be a continuous function such that

2 4 sin 2 2 cos 2 .
2

Then the value of is ____________

Q.50 Let be a cyclic group of order 12. Then the number of non-isomorphic subgroups of
is ____________

Q. 51 – Q. 60 carry two marks each.

Q.51
1
The value of lim 8 is equal to ____________
→

Q.52 Let be the region enclosed by 4 1 and 1. Then the value of

| | is ____________

Q.53 Let

1 1
1 1 , 1, , , ∈ and ∈ .
1 1

Then has infinitely many solutions if is ____________

Q.54 Let be the boundary of the region enclosed by , 2, and 0. Then the value
of the line integral

,
where is traversed in the counter clockwise direction, is ____________

MA 12/13

JAM 2016 MATHEMATICS - MA

Q.55 Let S be the closed surface forming the boundary of the region V bounded by 3,
0, 6. A vector field is defined over V with ⋅ 2 1. Then the value of

1
⋅ ,

where is the unit outward drawn normal to the surface , is ______________,

Q.56 Let be the solution of the differential equation

5 6 0, 0 1, 1.

Then attains its maximum value at ____________

Q.57 The value of the double integral
sin

is ____________

Q.58 Let denote the group of all 2 2 invertible matrices over under usual matrix multiplication.
2 3
Then the order of the matrix in is ____________
1 2

Q.59 1 2
1 2 0
Let , 1 0, the null space of and the range space of .
1 5 2
3 1
Then the dimension of ∩ over is ____________

Q.60 The maximum value of , 2 subject to the constraint
1 0 is ____________

END OF THE QUESTION PAPER

MA 13/13