JAM 2018 Mathematics - MA

Paper Specific 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
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor.
No choices will be shown for these 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 Scientific 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.

Useful information

ℕ set of all natural numbers {1, 2, 3, … }

ℤ set of all integers {0, ±1, ±2, … }
ℚ set of all rational numbers
ℝ set of all real numbers
ℂ set of all complex numbers
ℝ -dimensional Euclidean space {( , , … , ) | ∈ ℝ, 1 ≤ ≤ }
group of all permutations of distinct symbols
ℤ group of congruence classes of integers modulo
̂, ̂, unit vectors having the directions of the positive , and axes of a three
dimensional rectangular coordinate system
∇ ̂ + ̂ +
× (ℝ) real vector space of all matrices of order × with entries in ℝ
sup supremum
inf infimum

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JAM 2018 Mathematics - MA

SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q.10 carry one mark each.
Q.1 Which one of the following is TRUE?

(A) ℤ is cyclic if and only if is prime

(B) Every proper subgroup of ℤ is cyclic
(C) Every proper subgroup of is cyclic
(D) If every proper subgroup of a group is cyclic, then the group is cyclic

Q.2 Let = , where = 1 , = 1 and = + , ∈ ℕ. Then lim is

→

√ √ √ √
(A) (B) (C) (D)

Q.3 If { , , } is a linearly independent set of vectors in a vector space over ℝ, then which one of
the following sets is also linearly independent?

(A) { + − , 2 + +3 , 5 +4 }
(B) { − , − , − }
(C) { + − , + − , + − , + + }
(D) { + , +2 , +3 }

Q.4 Let be a positive real number. If is a continuous and even function defined on the interval
( )
[− , , then is equal to

( )
(A) ( ) (B) 2

( )
(C) 2 ( ) (D) 2

Q.5 The tangent plane to the surface = +3 at (1, 1, 2) is given by

(A) −3 + =0 (B) +3 −2 =0
(C) 2 + 4 − 3 = 0 (D) 3 − 7 + 2 = 0

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Q.6 In ℝ , the cosine of the acute angle between the surfaces + + − 9 = 0 and
− − + 3 = 0 at the point (2, 1, 2) is

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

√ √ √ √

Q.7 Let : ℝ → ℝ be a scalar field, : ℝ → ℝ be a vector field and let ∈ ℝ be a constant vector.
If represents the position vector ̂+ ̂+ , then which one of the following is FALSE?

(A) ( )= ( )× + ( )

(B) ( ( )) = + +

(C) ( × ) = 2 | |

(D) = 0 , for ≠0
| |

/ / /
Q.8 In ℝ , the family of trajectories orthogonal to the family of asteroids + = is
given by

/ / / / / /
(A) + = (B) − =
/ / / / / /
(C) − = (D) − =

Q.9 Consider the vector space over ℝ of polynomial functions of degree less than or equal to 3
defined on ℝ. Let ∶ → be defined by ( )( ) = ( ) − ′( ). Then the rank of is

(A) 1 (B) 2 (C) 3 (D) 4

Q.10 Let = 1 + + + ⋯+ for ∈ ℕ. Then which one of the following is TRUE for the
! ! !

sequence { }

(A) { } converges in ℚ
(B) { } is a Cauchy sequence but does not converge in ℚ
(C) the subsequence { } is convergent in ℝ, only when is even natural number
(D) { } is not a Cauchy sequence

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Q. 11 – Q. 30 carry two marks each.

Q.11 ( )
2+ ,
Let = , ∈ ℕ.
1+ ,

Then which one of the following is TRUE?

(A) sup { | ∈ ℕ} = 3 and inf { | ∈ ℕ} = 1

(B) lim inf ( ) = lim sup ( ) =

(C) sup { | ∈ ℕ} = 2 and inf { | ∈ ℕ} = 1

(D) lim inf ( ) = 1 and lim sup ( )=3

Q.12 Let , , ∈ ℝ . Which of the following values of , , do NOT result in the convergence of the
series

?
(log )

(A) | | < 1, ∈ ℝ, ∈ ℝ (B) = 1, > 1, ∈ℝ

(C) = 1, ≥ 0, < 1 (D) = −1, ≥ 0, > 0

Q.13 Let = + , ∈ ℕ. Then the sum of the series ∑ (−1) is

!

(A) −1 (B) (C) 1 − (D) 1 +

Q.14 Let ( )
= and let = ∑ , where ∈ ℕ ∪ {0}. Then which one of the
√

following is TRUE?

(A) Both ∑ and ∑ are convergent

(B) ∑ is convergent but ∑ is not convergent
(C) ∑ is convergent but ∑ is not convergent
(D) Neither ∑ nor ∑ is convergent

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JAM 2018 Mathematics - MA

Q.15 Suppose that , ∶ ℝ → ℝ are differentiable functions such that is strictly increasing and is
strictly decreasing. Define ( ) = ( ( )) and ( )= ( ) , ∀ ∈ ℝ. Then, for > 0, the

sign of ′( ) ( ( ) − 3) is

(A) positive (B) negative (C) dependent on (D) dependent on and

Q.16 sin , ≠0
For ∈ ℝ, let ( )= . Then which one of the following is FALSE?
0, =0

( )
(A) lim = 0
→
( )
(B) lim = 0
→
( )
(C) has infinitely many maxima and minima on the interval (0,1)
( )
(D) is continuous at = 0 but not differentiable at =0

Q.17 , ( , ) ≠ (0,0)
Let ( , ) = ( )
0, ( , ) = (0,0)
Then which one of the following is TRUE for at the point (0,0)?

(A) For = 1, is continuous but not differentiable

(B) For = , is continuous and differentiable

(C) For = , is continuous and differentiable

(D) For = , is neither continuous nor differentiable

Q.18 Let ,
∈ ℝ and let : ℝ → ℝ be a thrice differentiable function. If = ( ), where
= + and = − , then which one of the following is TRUE?

(A) − =4 ( ) (B) − = −4 ( )
(C) + = (D) + =−

Q.19 Consider the region in the plane bounded by the line = and the curve + = 1, where

≥ 0. If the region is revolved about the -axis in ℝ , then the volume of the resulting solid is

(A) (B) √ (D) √3

√ √ (C)

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Q.20 If ( , ) = (3 − 8 ) ̂ + (4 − 6 ) ̂ for ( , ) ∈ ℝ , then ∮ ⋅ , where is the

boundary of the triangular region bounded by the lines = 0, = 0 and + = 1 oriented in the
anti-clockwise direction, is

(A) (B) 3 (C) 4 (D) 5

Q.21 Let , and be finite dimensional real vector spaces, : → , : → and : → be

linear transformations. If range ( ) = nullspace ( ), nullspace ( ) = range ( ) and
rank ( ) = rank ( ), then which one of the following is TRUE?

(A) nullity of = nullity of

(B) dimension of ≠ dimension of
(C) If dimension of = 3 , dimension of = 4, then is not identically zero
(D) If dimension of = 4 , dimension of = 3 and is one-one, then is identically zero

Q.22 Let ( ) be the solution of the differential equation + = ( ), for ≥ 0, (0) = 0, where
2, 0 ≤ < 1
( ) = . Then ( )=
0, ≥ 1

(A) 2(1 − ) when 0 ≤ < 1 and 2( − 1) when ≥1

(B) 2(1 − ) when 0 ≤ < 1 and 0 hen ≥1
(C) 2(1 − ) when 0 ≤ < 1 and 2(1 − ) when ≥1
(D) 2(1 − ) when 0 ≤ < 1 and 2 when ≥1

Q.23 An integrating factor of the differential equation + + + ( + ) = 0 is

(A) (B) 3 log (C) (D) 2 log

Q.24 A particular integral of the differential equation ′′ + 3 ′ + 2 = is

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

Q.25 Let be a group satisfying the property that : →ℤ is a homomorphism implies

( ) = 0, ∀ ∈ . Then a possible group is

(A) ℤ (B) ℤ (C) ℤ (D) ℤ

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Q.26 Let be the quotient group ℚ/ ℤ. Consider the following statements.

I. Every cyclic subgroup of is finite.
II. Every finite cyclic group is isomorphic to a subgroup of .
Which one of the following holds?

(A) I is TRUE but II is FALSE (B) II is TRUE but I is FALSE

(C) both I and II are TRUE (D) neither I nor II is TRUE

Q.27 Let denote the 4 × 4 identity matrix. If the roots of the characteristic polynomial of a 4 × 4 matrix

±√
are ± , then =

(A) + (B) 2 + (C) 2 + 3 (D) 3 + 2

Q.28 Consider the group ℤ = {( , )| , ∈ ℤ} under component-wise addition. Then which of the
following is a subgroup of ℤ ?

(A) {( , ) ∈ ℤ | = 0}
(B) {( , ) ∈ ℤ | 3 + 2 = 15}
(C) {( , ) ∈ ℤ | 7 divides }
(D) {( , ) ∈ ℤ | 2 divides and 3 divides }

Q.29 Let : ℝ → ℝ be a function and let be a bounded open interval in ℝ. Define

( , ) = sup { ( ) | ∈ } − inf { ( ) | ∈ } .
Which one of the following is FALSE?

(A) ( , )≤ ( , ) if ⊂
(B) If is a bounded function in and ⊃ ⊃ ⋯⊃ ⊃ ⋯ such that the length of the
interval tends to 0 as → ∞, then lim ( , )=0
→

(C) If is discontinuous at a point ∈ , then ( , )≠0

(D) If is continuous at a point ∈ , then for any given > 0 there exists an interval ⊂
such that ( , )<

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JAM 2018 Mathematics - MA

Q.30 For > , let ( )= , ( ) = log (1 + 2 ) and ( ) = 2 . Then which one of the

following is TRUE?

√
(A) ( )< ( )< ( ) for 0 < <

(B) ( )< ( )< ( ) for >0

( ) √
(C) ( )+ ( )< for >

(D) ( )< ( )< ( ) for >0

SECTION - B
MULTIPLE SELECT QUESTIONS (MSQ)
Q. 31 – Q. 40 carry two marks each.
Q.31
Let : ℝ\ {0} → ℝ be defined by ( ) = + . On which of the following interval(s) is

one-one?

(A) (−∞, −1) (B) (0, 1) (C) (0, 2) (D) (0, ∞)

/
Q.32 The solution(s) of the differential equation = (sin 2 ) satisfying (0) = 0 is (are)

(A) ( )=0
(B) ( )=− sin

(C) ( )= sin (D) ( )= cos

Q.33 Suppose , , ℎ are permutations of the set { , , , } , where

interchanges and but fixes and ,
g interchanges and but fixes and ,
ℎ interchanges and but fixes and .
Which of the following permutations interchange(s) and but fix(es) and ?

(A) ∘ ∘ℎ∘ ∘ (B) ∘ℎ∘ ∘ℎ∘ (C) ∘ ∘ℎ∘ ∘ (D) ℎ∘ ∘ ∘ ∘ ℎ

Q.34 Let and be two non-empty disjoint subsets of ℝ. Which of the following is (are) FALSE?
(A) If and are compact, then ∪ is also compact
(B) If and are not connected, then ∪ is also not connected
(C) If ∪ and are closed, then is closed
(D) If ∪ and are open, then is open

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JAM 2018 Mathematics - MA

Q.35 Let ℂ∗ = ℂ \ {0} denote the group of non-zero complex numbers under multiplication. Suppose
= { ∈ ℂ | = 1}, ∈ ℕ. Which of the following is (are) subgroup(s) of ℂ∗ ?

(A) ⋃ (B) ⋃ (C) ⋃ (D) ⋃

Q.36 Suppose , , ∈ ℝ. Consider the following system of linear equations.

+ + = , + + = , + + = . If this system has at least one solution, then
which of the following statements is (are) TRUE?

(A) If = 1 then =1 (B) If = 1 then =

(C) If ≠ 1 then =1 (D) If = 1 then =1

Q.37 Let , ∈ ℕ, < , ∈ × (ℝ), ∈ × (ℝ). Then which of the following is (are)
NOT possible?

(A) ( )=
(B) ( )=
(C) ( )=

(D) ( ) = , the smallest integer larger than or equal to

Q.38 If ( , , ) = (2 + 3 ) ̂ + (3 + 2 ) ̂ + (3 + 2 ) for ( , , ) ∈ ℝ , then which among

the following is (are) TRUE?

(A) ∇ × =0

(B) ∮ ⋅ = 0 along any simple closed curve

(C) There exists a scalar function : ℝ → ℝ such that ∇ ⋅ = + +

(D) ∇ ⋅ =0

Q.39 Which of the following subsets of ℝ is (are) connected?

(A) { ∈ ℝ | + > 4} (B) { ∈ℝ| + < 4}

(C) { ∈ ℝ | | | < | − 4|} (D) { ∈ ℝ | | | > | − 4|}

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JAM 2018 Mathematics - MA

Q.40 Let be a subset of ℝ such that 2018 is an interior point of . Which of the following is (are)
TRUE?

(A) contains an interval

(B) There is a sequence in which does not converge to 2018
(C) There is an element ∈ , ≠ 2018 such that is also an interior point of
(D) There is a point ∈ , such that | − 2018| = 0.002018

SECTION – C
NUMERICAL ANSWER TYPE (NAT)
Q. 41 – Q. 50 carry one mark each.
Q.41 The order of the element (1 2 3) (2 4 5) (4 5 6) in the group is _________________

Q.42 Let ( , , ) = 3 +3 for ( , , ) ∈ ℝ . Then the absolute value of the directional derivative

of in the direction of the line = = , at the point (1, −2, 1) is _______________

Q.43 Let ( ) = ∑ (−1) ( − 1) for 0 < < 2. Then the value of is _____________

Q.44 Let : ℝ → ℝ be given by

( − )
, ( , ) ≠ (0,0)
( , ) = +
0, ( , ) = (0,0)

Then − at the point (0,0) is _____________

Q.45
Let ( , ) = sin + cos for ( , ) ∈ ℝ , > 0, >0.

Then (1,1) + (1,1) = ________________

Q.46 Let : [0, ∞) → [0, ∞) be continuous on [0, ∞) and differentiable on (0, ∞) . If

( ) = ( ) , then (6) = ___________

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Q.47 Let ( ( ) ) ( )
= + . Then the radius of convergence of the power series ∑

about = 0 is __________

Q.48 Let be the group of even permutations of 6 distinct symbols. Then the number of elements of
order 6 in is __________________

Q.49 Let be the real vector space of all 5 × 2 matrices such that the sum of the entries in each row is
zero. Let be the real vector space of all 5 × 2 matrices such that the sum of the entries in each
column is zero. Then the dimension of the space ∩ is ______________

Q.50 The coefficient of in the power series expansion of about = 0 is

____________ (correct up to three decimal places).

Q. 51 – Q. 60 carry two marks each.

Q.51 Let = (−1) , = + + ⋯+ and = ( + +⋯+ )/ , where , ∈ ℕ.
Then lim is _______________ (correct up to one decimal place).
→

Q.52 Let : ℝ → ℝ be such that is continuous on ℝ and (0) = 1, (0) = 0 and (0) = −1.

Then lim is ________________ (correct up to three decimal places).

→

Q.53 Suppose , , are positive real numbers such that + 2 + 3 = 1. If is the maximum value of
, then the value of is ________________

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Q.54 If the volume of the solid in ℝ bounded by the surfaces

= −1, = 1, = −1, = 1, = 2, + = 2
is − , then = ________________

Q.55 /
If = /
, then the value of 2 sin + 1 is ______________
√

Q.56 The value of the integral

is ______________ (correct up to three decimal places).

Q.57 Suppose ∈ × (ℝ) is a matrix of rank 2. Let : × (ℝ) → × (ℝ) be the linear
transformation defined by ( )= . Then the rank of is _______________

Q.58 The area of the parametrized surface

= {((2 + cos ) cos , (2 + cos ) sin , sin ) ∈ ℝ | 0 ≤ ≤ , 0≤ ≤ }

is ____________ (correct up to two decimal places).

Q.59 If ( ) is the solution to the differential equation = + , for > 0, satisfying (0) = 1,

then the value of √2 is _____________ (correct up to two decimal places).

Q.60 If ( ) = ( ) sec is the solution of − (2 tan ) +5 =0,− < < , satisfying

(0) = 0 and (0) = √6 , then is _______________ (correct up to two decimal places).

√

END OF THE QUESTION PAPER

MA 12/12
 Paper Code : MA
Question
Q No. Section Key/Range (KY)
Type (QT)
1 MCQ A B

2 MCQ A D

3 MCQ A D

4 MCQ A A

5 MCQ A B

6 MCQ A C

7 MCQ A C

8 MCQ A B

9 MCQ A C

10 MCQ A B

11 MCQ A A

12 MCQ A C

13 MCQ A D

14 MCQ A B

15 MCQ A A

16 MCQ A D

17 MCQ A C

18 MCQ A A

19 MCQ A C

20 MCQ A B

21 MCQ A C

22 MCQ A A

23 MCQ A C
 Paper Code : MA
Question
Q No. Section Key/Range (KY)
Type (QT)
24 MCQ A B

25 MCQ A A

26 MCQ A C

27 MCQ A C

28 MCQ A D

29 MCQ A B

30 MCQ A C

31 MSQ B B

32 MSQ B A,B,C

33 MSQ B A,D

34 MSQ B B,C,D

35 MSQ B B,C,D

36 MSQ B A,B

37 MSQ B A,D

38 MSQ B A,B,C

39 MSQ B B,C,D

40 MSQ B A,B,C

41 NAT C 4 to 4

42 NAT C 6.5 to 7.5

43 NAT C 1 to 1

44 NAT C 1 to 1

45 NAT C 3 to 3

46 NAT C 9 to 9
 Paper Code : MA
Question
Q No. Section Key/Range (KY)
Type (QT)
47 NAT C 2 to 2

48 NAT C 0 to 0

49 NAT C 4 to 4

50 NAT C -0.130 to -0.120

51 NAT C 0.4 to 0.6

52 NAT C 0.350 to 0.380

53 NAT C 1140 to 1160

54 NAT C 5.99 to 6.01

55 NAT C 2.9 to 3.1

56 NAT C 0.230 to 0.250

57 NAT C 6 to 6

58 NAT C 6.30 to 6.70

59 NAT C -2.80 to -2.70

60 NAT C 0.5 to 0.5