JAM 2024 Question Paper Mathematics (MA)

JOINT ADMISSION TEST FOR MASTERS

2024

IIT

JAM
2024
Question Papers | Answer Key

The Joint Admission Test for Masters is a
common admission test conducted every
year for admission into Master of Science
and other post-graduate science
programs at Indian Institutes of
Technology

JAM 2024 Mathematics (MA)

Special Instructions / Useful Data

ℤ𝑛 = {0̅, 1̅, … , ̅̅̅̅̅̅̅
𝑛 − 1} denotes the additive group of integers modulo 𝑛
ℝ = the set of all real numbers
ℕ = the set of all positive integers
ℤ = the set of all integers
ℂ = the set of all complex numbers
ℚ = the set of all rational numbers
gcd(𝑟, 𝑛) = the greatest common divisor of the integers 𝑟 and 𝑛
𝑆𝑛 = the symmetric group of all permutations of {1,2, … , 𝑛}
𝐴𝑛 = the group of all even permutations in 𝑆𝑛
𝑀𝑛 (ℂ) = the set of all 𝑛 × 𝑛 matrices with entries from ℂ
𝑀𝑛 (ℝ) = the set of all 𝑛 × 𝑛 matrices with entries from ℝ
𝑀𝑇 = the transpose of the matrix 𝑀
𝐼𝑛 = the 𝑛 × 𝑛 identity matrix
𝑃𝑛 (𝑥) = the real vector space of polynomials, in the variable 𝑥 with real coefficients and having
degree at most 𝑛, together with the zero polynomial. These polynomials are regarded as
functions from ℝ to ℝ
𝑛 𝑛 𝑛!
( ) = the binomial coefficient defined as ( ) = 𝑘!(𝑛−𝑘)!
𝑘 𝑘
𝑓 ∘ 𝑔 = the composite function defined by (𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥))
𝐴 ∖ 𝐵 = the complement of the set 𝐵 in the set 𝐴, that is, {𝑥 ∈ 𝐴: 𝑥 ∉ 𝐵}
log 𝑥 = the logarithm of 𝑥 to the base 𝑒 for a positive number 𝑥
ℝ𝑛 = the 𝑛-dimensional Euclidean space
𝐴 × 𝐵 = the Cartesian product of the sets 𝐴 and 𝐵
𝑀−1 = the inverse of an invertible matrix 𝑀

MA 1/43

JAM 2024 Mathematics (MA)

Section A: Q.1 – Q.10 Carry ONE mark each.

Q.1 Let 𝑦𝑐 : ℝ → (0, ∞) be the solution of the Bernoulli’s equation

𝑑𝑦
− 𝑦 + 𝑦 3 = 0, 𝑦(0) = 𝑐 > 0.
𝑑𝑥
Then, for every 𝑐 > 0, which one of the following is true?

(A) lim 𝑦𝑐 (𝑥) = 0
𝑥→∞

(B) lim 𝑦𝑐 (𝑥) = 1
𝑥→∞

(C) lim 𝑦𝑐 (𝑥) = 𝑒
𝑥→∞

(D) lim 𝑦𝑐 (𝑥) does not exist
𝑥→∞

MA 2/43

JAM 2024 Mathematics (MA)

Q.2 For a twice continuously differentiable function 𝑔: ℝ → ℝ, define
1 𝑦
𝑢𝑔 (𝑥, 𝑦) = ∫ 𝑔(𝑥 + 𝑡)𝑑𝑡 for (𝑥, 𝑦) ∈ ℝ2 , 𝑦 > 0.
𝑦 −𝑦

Which one of the following holds for all such 𝑔?

𝜕 2 𝑢𝑔 2 𝜕𝑢𝑔 𝜕 2 𝑢𝑔
(A) = +
𝜕𝑥 2 𝑦 𝜕𝑦 𝜕𝑦 2

𝜕 2 𝑢𝑔 1 𝜕𝑢𝑔 𝜕 2 𝑢𝑔
(B) = +
𝜕𝑥 2 𝑦 𝜕𝑦 𝜕𝑦 2

𝜕 2 𝑢𝑔 2 𝜕𝑢𝑔 𝜕 2 𝑢𝑔
(C) = −
𝜕𝑥 2 𝑦 𝜕𝑦 𝜕𝑦 2

𝜕 2 𝑢𝑔 1 𝜕𝑢𝑔 𝜕 2 𝑢𝑔
(D) = −
𝜕𝑥 2 𝑦 𝜕𝑦 𝜕𝑦 2

MA 3/43

JAM 2024 Mathematics (MA)

Q.3 Let 𝑦(𝑥) be the solution of the differential equation
𝑑𝑦 𝜋 𝜋
= 1 + 𝑦 sec 𝑥 for 𝑥 ∈ (− , )
𝑑𝑥 2 2
𝜋
that satisfies 𝑦(0) = 0. Then, the value of 𝑦 ( 6 ) equals

3
(A) √3 log ( )
2

√3 3
(B) ( ) log ( )
2 2

√3
(C) ( ) log 3
2

(D) √3 log 3

MA 4/43

JAM 2024 Mathematics (MA)

Q.4 Let ℱ be the family of curves given by
𝑥 2 + 2ℎ𝑥𝑦 + 𝑦 2 = 1, −1<ℎ <1.
Then, the differential equation for the family of orthogonal trajectories to ℱ is

𝑑𝑦
(A) (𝑥 2 𝑦 − 𝑦 3 + 𝑦) − (𝑥𝑦 2 − 𝑥 3 + 𝑥) = 0
𝑑𝑥

𝑑𝑦
(B) (𝑥 2 𝑦 − 𝑦 3 + 𝑦) + (𝑥𝑦 2 − 𝑥 3 + 𝑥) = 0
𝑑𝑥

𝑑𝑦
(C) (𝑥 2 𝑦 + 𝑦 3 + 𝑦) − (𝑥𝑦 2 + 𝑥 3 + 𝑥) = 0
𝑑𝑥

𝑑𝑦
(D) (𝑥 2 𝑦 + 𝑦 3 + 𝑦) + (𝑥𝑦 2 + 𝑥 3 + 𝑥) = 0
𝑑𝑥

Q.5 Let 𝐺 be a group of order 39 such that it has exactly one subgroup of order 3
and exactly one subgroup of order 13. Then, which one of the following
statements is TRUE?

(A) 𝐺 is necessarily cyclic

(B) 𝐺 is abelian but need not be cyclic

(C) 𝐺 need not be abelian

(D) 𝐺 has 13 elements of order 13

MA 5/43

JAM 2024 Mathematics (MA)

Q.6 For a positive integer 𝑛, let 𝑈(𝑛) = {𝑟̅ ∈ ℤ𝑛 ∶ gcd(𝑟, 𝑛) = 1} be the group
under multiplication modulo 𝑛. Then, which one of the following statements is
TRUE?

(A) 𝑈(5) is isomorphic to 𝑈(8)

(B) 𝑈(10) is isomorphic to 𝑈(12)

(C) 𝑈(8) is isomorphic to 𝑈(10)

(D) 𝑈(8) is isomorphic to 𝑈(12)

Q.7 Which one of the following is TRUE for the symmetric group 𝑆13 ?

(A) 𝑆13 has an element of order 42

(B) 𝑆13 has no element of order 35

(C) 𝑆13 has an element of order 27

(D) 𝑆13 has no element of order 60

MA 6/43

JAM 2024 Mathematics (MA)

Q.8 Let 𝐺 be a finite group containing a non-identity element which is conjugate to
its inverse. Then, which one of the following is TRUE?

(A) The order of 𝐺 is necessarily even

(B) The order of 𝐺 is not necessarily even

(C) 𝐺 is necessarily cyclic

(D) 𝐺 is necessarily abelian but need not be cyclic

Q.9 Consider the following statements.

P: If a system of linear equations 𝐴𝑥 = 𝑏 has a unique solution, where 𝐴 is an
𝑚 × 𝑛 matrix and 𝑏 is an 𝑚 × 1 matrix, then 𝑚 = 𝑛.

Q: For a subspace 𝑊 of a nonzero vector space 𝑉, whenever 𝑢 ∈ 𝑉 ∖ 𝑊 and
𝑣 ∈ 𝑉 ∖ 𝑊, then 𝑢 + 𝑣 ∈ 𝑉 ∖ 𝑊.

Which one of the following holds?

(A) Both P and Q are true

(B) P is true but Q is false

(C) P is false but Q is true

(D) Both P and Q are false

MA 7/43

JAM 2024 Mathematics (MA)

Q.10 Let 𝑔: ℝ → ℝ be a continuous function. Which one of the following is the
solution of the differential equation
𝑑2𝑦
+ 𝑦 = 𝑔(𝑥) for 𝑥 ∈ ℝ,
𝑑𝑥 2
satisfying the conditions 𝑦(0) = 0, 𝑦 ′ (0) = 1 ?

𝑥
(A) 𝑦(𝑥) = sin 𝑥 − ∫ sin(𝑥 − 𝑡) 𝑔(𝑡)𝑑𝑡
0

𝑥
(B) 𝑦(𝑥) = sin 𝑥 + ∫ sin(𝑥 − 𝑡) 𝑔(𝑡)𝑑𝑡
0

𝑥
(C) 𝑦(𝑥) = sin 𝑥 − ∫ cos(𝑥 − 𝑡) 𝑔(𝑡)𝑑𝑡
0

𝑥
(D) 𝑦(𝑥) = sin 𝑥 + ∫ cos(𝑥 − 𝑡) 𝑔(𝑡)𝑑𝑡
0

MA 8/43

JAM 2024 Mathematics (MA)

Section A: Q.11 – Q.30 Carry TWO marks each.

Q.11 Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does
not have an element of order greater than or equal to 6 ?

(A) The alternating group 𝐴6

(B) The alternating group 𝐴5

(C) 𝑆6

(D) 𝑆5

Q.12 Consider the group 𝐺 = {𝐴 ∈ 𝑀2 (ℝ): 𝐴𝐴𝑇 = 𝐼2 } with respect to matrix
multiplication. Let
𝑍(𝐺) = {𝐴 ∈ 𝐺 ∶ 𝐴𝐵 = 𝐵𝐴, for all 𝐵 ∈ 𝐺}.
Then, the cardinality of 𝑍(𝐺) is

(A) 1

(B) 2

(C) 4

(D) Infinite

MA 9/43

JAM 2024 Mathematics (MA)

Q.13 Let 𝑉 be a nonzero subspace of the complex vector space 𝑀7 (ℂ) such that every
nonzero matrix in 𝑉 is invertible. Then, the dimension of 𝑉 over ℂ is

(A) 1

(B) 2

(C) 7

(D) 49

MA 10/43

JAM 2024 Mathematics (MA)

Q.14 For 𝑛 ∈ ℕ, let
1 𝑛3 + cos(3𝑛 )
𝑎𝑛 = and 𝑏𝑛 = .
(3𝑛 + 2)(3𝑛 + 4) 3𝑛 + 𝑛 3
Then, which one of the following is TRUE?

∞ ∞

(A) ∑ 𝑎𝑛 is convergent but ∑ 𝑏𝑛 is divergent
𝑛=1 𝑛=1

∞ ∞

(B) ∑ 𝑎𝑛 is divergent but ∑ 𝑏𝑛 is convergent
𝑛=1 𝑛=1

∞ ∞

(C) Both ∑ 𝑎𝑛 and ∑ 𝑏𝑛 are divergent
𝑛=1 𝑛=1

∞ ∞

(D) Both ∑ 𝑎𝑛 and ∑ 𝑏𝑛 are convergent
𝑛=1 𝑛=1

MA 11/43

JAM 2024 Mathematics (MA)

Q.15 1
√3
−1
Let 𝑎 = √2 . Consider the following two statements.
1
√6
[ 0 ]

P: The matrix 𝐼4 − 𝑎𝑎𝑇 is invertible.

Q: The matrix 𝐼4 − 2𝑎𝑎𝑇 is invertible.

Then, which one of the following holds?

(A) P is false but Q is true

(B) P is true but Q is false

(C) Both P and Q are true

(D) Both P and Q are false

MA 12/43

JAM 2024 Mathematics (MA)

Q.16 Let 𝐴 be a 6 × 5 matrix with entries in ℝ and 𝐵 be a 5 × 4 matrix with entries
in ℝ. Consider the following two statements.

P: For all such nonzero matrices 𝐴 and 𝐵, there is a nonzero matrix 𝑍 such that
𝐴𝑍𝐵 is the 6 × 4 zero matrix.

Q: For all such nonzero matrices 𝐴 and 𝐵, there is a nonzero matrix 𝑌 such that
𝐵𝑌𝐴 is the 5 × 5 zero matrix.

Which one of the following holds?

(A) Both P and Q are true

(B) P is true but Q is false

(C) P is false but Q is true

(D) Both P and Q are false

MA 13/43

JAM 2024 Mathematics (MA)

Q.17 Let 𝑃11 (𝑥) be the real vector space of polynomials, in the variable 𝑥 with real
coefficients and having degree at most 11, together with the zero polynomial.
Let

𝐸 = {𝑠0 (𝑥), 𝑠1 (𝑥), … , 𝑠11 (𝑥)}, 𝐹 = {𝑟0 (𝑥), 𝑟1 (𝑥), … , 𝑟11 (𝑥)}
be subsets of 𝑃11 (𝑥) having 12 elements each and satisfying

𝑠0 (3) = 𝑠1 (3) = ⋯ = 𝑠11 (3) = 0 , 𝑟0 (4) = 𝑟1 (4) = ⋯ = 𝑟11 (4) = 1 .

Then, which one of the following is TRUE?

Any such 𝐸 is not necessarily linearly dependent and any such 𝐹 is not
(A)
necessarily linearly dependent

Any such 𝐸 is necessarily linearly dependent but any such 𝐹 is not necessarily
(B)
linearly dependent

Any such 𝐸 is not necessarily linearly dependent but any such 𝐹 is necessarily
(C)
linearly dependent

Any such 𝐸 is necessarily linearly dependent and any such 𝐹 is necessarily
(D)
linearly dependent

MA 14/43

JAM 2024 Mathematics (MA)

Q.18 For the differential equation
𝑦(8𝑥 − 9𝑦)𝑑𝑥 + 2𝑥(𝑥 − 3𝑦)𝑑𝑦 = 0 ,

which one of the following statements is TRUE?

(A) The differential equation is not exact and has 𝑥 2 as an integrating factor

(B) The differential equation is exact and homogeneous

The differential equation is not exact and does not have 𝑥 2 as an integrating
(C)
factor

(D) The differential equation is not homogeneous and has 𝑥 2 as an integrating factor

MA 15/43

JAM 2024 Mathematics (MA)

Q.19 For 𝑥 ∈ ℝ, let ⌊𝑥⌋ denote the greatest integer less than or equal to 𝑥.

For 𝑥, 𝑦 ∈ ℝ, define
𝑥 if 𝑥 ≤ 𝑦,
min{𝑥, 𝑦} = {
𝑦 otherwise.

Let 𝑓: [−2𝜋, 2𝜋] → ℝ be defined by

𝑓(𝑥) = sin(min{𝑥, 𝑥 − ⌊𝑥⌋}) for 𝑥 ∈ [−2𝜋, 2𝜋].

Consider the set 𝑆 = {𝑥 ∈ [−2𝜋, 2𝜋]: 𝑓 is discontinuous at 𝑥}.

Which one of the following statements is TRUE?

(A) 𝑆 has 13 elements

(B) 𝑆 has 7 elements

(C) 𝑆 is an infinite set

(D) 𝑆 has 6 elements

MA 16/43

JAM 2024 Mathematics (MA)

Q.20 Define the sequences {𝑎𝑛 }∞ ∞
𝑛=3 and {𝑏𝑛 }𝑛=3 as
1
(1+ )
𝑎𝑛 = (log 𝑛 + log log 𝑛)log 𝑛 and 𝑏𝑛 = 𝑛 log 𝑛 .
Which one of the following is TRUE?

∞ ∞
1 1
(A) ∑ is convergent but ∑ is divergent
𝑎𝑛 𝑏𝑛
𝑛=3 𝑛=3

∞ ∞
1 1
(B) ∑ is divergent but ∑ is convergent
𝑎𝑛 𝑏𝑛
𝑛=3 𝑛=3

∞ ∞
1 1
(C) Both ∑ and ∑ are divergent
𝑎𝑛 𝑏𝑛
𝑛=3 𝑛=3

∞ ∞
1 1
(D) Both ∑ and ∑ are convergent
𝑎𝑛 𝑏𝑛
𝑛=3 𝑛=3

MA 17/43

JAM 2024 Mathematics (MA)

Q.21 For 𝑝, 𝑞, 𝑟 ∈ ℝ, 𝑟 ≠ 0 and 𝑛 ∈ ℕ, let

2
𝑛
𝑛 𝑛𝑞
𝑛𝑛 𝑛+2
𝑎𝑛 = 𝑝 𝑛 ( ) and 𝑏𝑛 = ( √ ).
𝑛+2 𝑛! 𝑟 𝑛 𝑛

Then, which one of the following statements is TRUE?

∞

(A) If 1 < 𝑝 < 𝑒 2 and 𝑞 > 1, then ∑ 𝑎𝑛 is convergent
𝑛=1

∞
2 4
(B) If e < 𝑝 < 𝑒 and 𝑞 > 1, then ∑ 𝑎𝑛 is convergent
𝑛=1

∞

(C) If 1 < 𝑟 < 𝑒, then ∑ 𝑏𝑛 is convergent
𝑛=1

∞
1
(D) If < 𝑟 < 1, then ∑ 𝑏𝑛 is convergent
e
𝑛=1

MA 18/43

JAM 2024 Mathematics (MA)

Q.22 Let 𝑃7 (𝑥) be the real vector space of polynomials, in the variable 𝑥 with real
coefficients and having degree at most 7, together with the zero polynomial.
Let 𝑇: 𝑃7 (𝑥) → 𝑃7 (𝑥) be the linear transformation defined by

𝑑𝑓(𝑥)
𝑇(𝑓(𝑥)) = 𝑓(𝑥) + .
𝑑𝑥
Then, which one of the following is TRUE?

(A) 𝑇 is not a surjective linear transformation

(B) There exists 𝑘 ∈ ℕ such that 𝑇 𝑘 is the zero linear transformation

(C) 1 and 2 are the eigenvalues of 𝑇

There exists 𝑟 ∈ ℕ such that (𝑇 − 𝐼)𝑟 is the zero linear transformation, where 𝐼
(D)
is the identity map on 𝑃7 (𝑥)

MA 19/43

JAM 2024 Mathematics (MA)

Q.23 For 𝛼 ∈ ℝ, let 𝑦𝛼 (𝑥) be the solution of the differential equation
𝑑𝑦 1
+ 2𝑦 = for 𝑥 ∈ ℝ
𝑑𝑥 1 + 𝑥2
satisfying 𝑦(0) = 𝛼. Then, which one of the following is TRUE?

(A) lim 𝑦𝛼 (𝑥) = 0 for every 𝛼 ∈ ℝ
𝑥→∞

(B) lim 𝑦𝛼 (𝑥) = 1 for every 𝛼 ∈ ℝ
𝑥→∞

There exists an α ∈ ℝ such that lim 𝑦𝛼 (𝑥) exists but its value is different from
𝑥→∞
(C)
0 and 1

(D) There is an α ∈ ℝ for which 𝑥→∞
lim 𝑦𝛼 (𝑥) does not exist

MA 20/43

JAM 2024 Mathematics (MA)

Q.24 Consider the following two statements.

P: There exist functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ such that 𝑓 is continuous at
𝑥 = 1 and 𝑔 is discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1.

Q: There exist functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ such that both 𝑓 and 𝑔 are
discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1.

Which one of the following holds?

(A) Both P and Q are true

(B) Both P and Q are false

(C) P is true but Q is false

(D) P is false but Q is true

MA 21/43

JAM 2024 Mathematics (MA)

Q.25 Let 𝑓: ℝ → ℝ be defined by
(𝑥 2 + 1)2
𝑓(𝑥) = 4 for 𝑥 ∈ ℝ.
𝑥 + 𝑥2 + 1
Then, which one of the following is TRUE?

𝑓 has exactly two points of local maxima and exactly three points of local
(A)
minima

𝑓 has exactly three points of local maxima and exactly two points of local
(B)
minima

𝑓 has exactly one point of local maximum and exactly two points of local
(C)
minima

𝑓 has exactly two points of local maxima and exactly one point of local
(D)
minimum

MA 22/43

JAM 2024 Mathematics (MA)

Q.26 Let 𝑓: ℝ → ℝ be a solution of the differential equation
𝑑2𝑦 𝑑𝑦
− 2 + 𝑦 = 2𝑒 𝑥 for 𝑥 ∈ ℝ .
𝑑𝑥 2 𝑑𝑥
Consider the following statements.

P: If 𝑓(𝑥) > 0 for all 𝑥 ∈ ℝ, then 𝑓 ′ (𝑥) > 0 for all 𝑥 ∈ ℝ .

Q: If 𝑓 ′ (𝑥) > 0 for all 𝑥 ∈ ℝ, then 𝑓(𝑥) > 0 for all 𝑥 ∈ ℝ .

Then, which one of the following holds?

(A) P is true but Q is false

(B) P is false but Q is true

(C) Both P and Q are true

(D) Both P and Q are false

MA 23/43

JAM 2024 Mathematics (MA)

Q.27 For 𝑎 > 𝑏 > 0, consider

𝐷 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 : 𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 𝑎2 and 𝑥 2 + 𝑦 2 ≥ 𝑏 2 }.

Then, the surface area of the boundary of the solid 𝐷 is

(A) 4𝜋(𝑎 + 𝑏)√𝑎2 − 𝑏 2

(B) 4𝜋 (𝑎2 − 𝑏√𝑎2 − 𝑏 2 )

(C) 4𝜋(𝑎 − 𝑏)√𝑎2 − 𝑏 2

(D) 4𝜋 (𝑎2 + 𝑏√𝑎2 − 𝑏 2 )

MA 24/43

JAM 2024 Mathematics (MA)

Q.28 For 𝑛 ≥ 3, let a regular 𝑛-sided polygon 𝑃𝑛 be circumscribed by a circle of
radius 𝑅𝑛 and let 𝑟𝑛 be the radius of the circle inscribed in 𝑃𝑛 . Then
2
𝑅𝑛 𝑛
lim ( )
𝑛→∞ 𝑟𝑛

equals

2
(A) 𝑒 (𝜋 )

𝜋2
(B) (2)
𝑒

𝜋2
(C) ( )
𝑒 3

2
(D) 𝑒 (2𝜋 )

MA 25/43

JAM 2024 Mathematics (MA)

Q.29 Let 𝐿1 denote the line 𝑦 = 3𝑥 + 2 and 𝐿2 denote the line 𝑦 = 4𝑥 + 3. Suppose
that 𝑓: ℝ → ℝ is a four times continuously differentiable function such that the
line 𝐿1 intersects the curve 𝑦 = 𝑓(𝑥) at exactly three distinct points and the line
𝐿2 intersects the curve 𝑦 = 𝑓(𝑥) at exactly four distinct points. Then, which one
of the following is TRUE?

𝑑𝑓
(A) does not attain the value 3 on ℝ
𝑑𝑥

𝑑2 𝑓
(B) vanishes at most once on ℝ
𝑑𝑥 2

𝑑3 𝑓
(C) vanishes at least once on ℝ
𝑑𝑥 3

𝑑𝑓 7
(D) does not attain the value on ℝ
𝑑𝑥 2

MA 26/43

JAM 2024 Mathematics (MA)

Q.30 Define the function 𝑓: ℝ2 → ℝ by

𝑓(𝑥, 𝑦) = 12𝑥𝑦 𝑒 −(2𝑥+3𝑦−2) .
If (𝑎, 𝑏) is the point of local maximum of 𝑓, then 𝑓(𝑎, 𝑏) equals

(A) 2

(B) 6

(C) 12

(D) 0

MA 27/43

JAM 2024 Mathematics (MA)

Section B: Q.31 – Q.40 Carry TWO marks each.

Q.31 Let {𝑎𝑛 }∞
𝑛=1 be a sequence of real numbers.

Then, which of the following statements is/are always TRUE?

∞ ∞

(A) If ∑ 𝑎𝑛 converges absolutely, then ∑ 𝑎𝑛2 converges absolutely
𝑛=1 𝑛=1

∞ ∞

(B) If ∑ 𝑎𝑛 converges absolutely, then ∑ 𝑎𝑛3 converges absolutely
𝑛=1 𝑛=1

∞ ∞

(C) If ∑ 𝑎𝑛 converges, then ∑ 𝑎𝑛2 converges
𝑛=1 𝑛=1

∞ ∞

(D) If ∑ 𝑎𝑛 converges, then ∑ 𝑎𝑛3 converges
𝑛=1 𝑛=1

MA 28/43

JAM 2024 Mathematics (MA)

Q.32 Which of the following statements is/are TRUE?

∞
1
(A) ∑ 𝑛 log (1 + ) is convergent
𝑛3
𝑛=1

∞
1
(B) ∑ (1 − cos ( )) log 𝑛 is convergent
𝑛
𝑛=1

∞
1
(C) ∑ 𝑛2 log (1 + ) is convergent
𝑛3
𝑛=1

∞
1
(D) ∑ (1 − cos ( )) log 𝑛 is convergent
𝑛=1
√𝑛

MA 29/43

JAM 2024 Mathematics (MA)

Q.33 Which of the following statements is/are TRUE?

The additive group of real numbers is isomorphic to the multiplicative group of
(A)
positive real numbers

The multiplicative group of nonzero real numbers is isomorphic to the
(B)
multiplicative group of nonzero complex numbers

The additive group of real numbers is isomorphic to the multiplicative group of
(C)
nonzero complex numbers

The additive group of real numbers is isomorphic to the additive group of
(D)
rational numbers

Q.34 Let 𝑓: (1, ∞) → (0, ∞) be a continuous function such that for every 𝑛 ∈ ℕ,
𝑓(𝑛) is the smallest prime factor of 𝑛. Then, which of the following options
is/are CORRECT?

(A) lim 𝑓(𝑥) exists
𝑥→∞

(B) lim 𝑓(𝑥) does not exist
𝑥→∞

(C) The set of solutions to the equation 𝑓(𝑥) = 2024 is finite

(D) The set of solutions to the equation 𝑓(𝑥) = 2024 is infinite

MA 30/43

JAM 2024 Mathematics (MA)

Q.35 Let
𝑆 = {(𝑥, 𝑦) ∈ ℝ2 : 𝑥 > 0, 𝑦 > 0} ,

and 𝑓: 𝑆 → ℝ be given by

1
𝑓(𝑥, 𝑦) = 2𝑥 2 + 3𝑦 2 − log 𝑥 − log 𝑦 .
6
Then, which of the following statements is/are TRUE?

(A) There is a unique point in 𝑆 at which 𝑓(𝑥, 𝑦) attains a local maximum

(B) There is a unique point in 𝑆 at which 𝑓(𝑥, 𝑦) attains a local minimum

For each point (𝑥0 , 𝑦0 ) ∈ 𝑆, the set {(𝑥, 𝑦) ∈ 𝑆: 𝑓(𝑥, 𝑦) = 𝑓(𝑥0 , 𝑦0 ) } is
(C)
bounded

For each point (𝑥0 , 𝑦0 ) ∈ 𝑆, the set {(𝑥, 𝑦) ∈ 𝑆: 𝑓(𝑥, 𝑦) = 𝑓(𝑥0 , 𝑦0 ) } is
(D)
unbounded

MA 31/43

JAM 2024 Mathematics (MA)

Q.36 The center 𝑍(𝐺) of a group 𝐺 is defined as

𝑍(𝐺) = {𝑥 ∈ 𝐺 ∶ 𝑥𝑔 = 𝑔𝑥 for all 𝑔 ∈ 𝐺}.
Let |𝐺| denote the order of 𝐺. Then, which of the following statements is/are
TRUE for any group 𝐺?

If 𝐺 is non-abelian and 𝑍(𝐺) contains more than one element, then the center of
(A)
the quotient group 𝐺/𝑍(𝐺) contains only one element

(B) If |𝐺| ≥ 2, then there exists a non-trivial homomorphism from ℤ to 𝐺

If |𝐺| ≥ 2 and 𝐺 is non-abelian, then there exists a non-identity isomorphism
(C)
from 𝐺 to itself

(D) If |𝐺| = 𝑝3 , where 𝑝 is a prime number, then 𝐺 is necessarily abelian

MA 32/43

JAM 2024 Mathematics (MA)

Q.37 For a matrix 𝑀, let Rowspace(𝑀) denote the linear span of the rows of 𝑀 and
Colspace(𝑀) denote the linear span of the columns of 𝑀. Which of the
following hold(s) for all 𝐴, 𝐵, 𝐶 ∈ 𝑀10 (ℝ) satisfying 𝐴 = 𝐵𝐶 ?

(A) Rowspace(𝐴) ⊆ Rowspace(𝐵)

(B) Rowspace(𝐴) ⊆ Rowspace(𝐶)

(C) Colspace(𝐴) ⊆ Colspace(𝐵)

(D) Colspace(𝐴) ⊆ Colspace(𝐶)

MA 33/43

JAM 2024 Mathematics (MA)

Q.38 Define 𝑓: ℝ → ℝ and 𝑔: ℝ → ℝ as follows

∞ ∞
(−1)𝑚 𝑥 2𝑚 𝑥 (−1)𝑚 𝑥 2𝑚
𝑓(𝑥) = ∑ 2𝑚 and 𝑔(𝑥) = ∑ 2𝑚 for 𝑥 ∈ ℝ.
2 (𝑚!)2 2 2 (𝑚 + 1)! 𝑚!
𝑚=0 𝑚=0

Let 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ∈ ℝ be such that 0 < 𝑥1 < 𝑥2 , 0 < 𝑥3 < 𝑥4 ,

𝑓(𝑥1 ) = 𝑓(𝑥2 ) = 0, 𝑓(𝑥) ≠ 0 when 𝑥1 < 𝑥 < 𝑥2 ,

𝑔(𝑥3 ) = 𝑔(𝑥4 ) = 0 and 𝑔(𝑥) ≠ 0 when 𝑥3 < 𝑥 < 𝑥4 .

Then, which of the following statements is/are TRUE?

(A) The function 𝑓 does not vanish anywhere in the interval (𝑥3 , 𝑥4 )

(B) The function 𝑓 vanishes exactly once in the interval (𝑥3 , 𝑥4 )

(C) The function 𝑔 does not vanish anywhere in the interval (𝑥1 , 𝑥2 )

(D) The function 𝑔 vanishes exactly once in the interval (𝑥1 , 𝑥2 )

MA 34/43

JAM 2024 Mathematics (MA)

Q.39 For 0 < 𝛼 < 4, define the sequence {𝑥𝑛 }∞
𝑛=1 of real numbers as follows:

𝑥1 = 𝛼 and 𝑥𝑛+1 + 2 = −𝑥𝑛 (𝑥𝑛 − 4) for 𝑛 ∈ ℕ.
Which of the following is/are TRUE?

(A) {𝑥𝑛 }∞
𝑛=1 converges for at least three distinct values of 𝛼 ∈ (0,1)

(B) {𝑥𝑛 }∞
𝑛=1 converges for at least three distinct values of 𝛼 ∈ (1,2)

(C) {𝑥𝑛 }∞
𝑛=1 converges for at least three distinct values of 𝛼 ∈ (2,3)

(D) {𝑥𝑛 }∞
𝑛=1 converges for at least three distinct values of 𝛼 ∈ (3,4)

Q.40 Consider
𝐺 = {𝑚 + 𝑛√2 ∶ 𝑚, 𝑛 ∈ ℤ}
as a subgroup of the additive group ℝ.
Which of the following statements is/are TRUE?

(A) 𝐺 is a cyclic subgroup of ℝ under addition

(B) 𝐺 ∩ 𝐼 is non-empty for every non-empty open interval 𝐼 ⊆ ℝ

(C) 𝐺 is a closed subset of ℝ

𝐺 is isomorphic to the group ℤ × ℤ, where the group operation in ℤ × ℤ is
(D)
defined by (𝑚1 , 𝑛1 ) + (𝑚2 , 𝑛2 ) = (𝑚1 + 𝑚2 , 𝑛1 + 𝑛2 )

MA 35/43

JAM 2024 Mathematics (MA)

Section C: Q.41 – Q.50 Carry ONE mark each.

Q.41 The area of the region

1 1
𝑅 = {(𝑥, 𝑦) ∈ ℝ2 : 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1 and ≤ 𝑥𝑦 ≤ }
4 2

is __________ (rounded off to two decimal places).

Q.42 Let 𝑦: ℝ → ℝ be the solution to the differential equation
𝑑2𝑦 𝑑𝑦
+ 2 + 5𝑦 = 1
𝑑𝑥 2 𝑑𝑥
satisfying 𝑦(0) = 0 and 𝑦 ′ (0) = 1.

Then, lim 𝑦(𝑥) equals __________ (rounded off to two decimal places).
𝑥→∞

Q.43 For 𝛼 > 0, let 𝑦𝛼 (𝑥) be the solution to the differential equation
𝑑 2 𝑦 𝑑𝑦
2 2− −𝑦 =0
𝑑𝑥 𝑑𝑥
satisfying the conditions
𝑦(0) = 1, 𝑦 ′ (0) = 𝛼.

Then, the smallest value of 𝛼 for which 𝑦𝛼 (𝑥) has no critical points in ℝ equals
___________ (rounded off to the nearest integer).

MA 36/43

JAM 2024 Mathematics (MA)

Q.44 Consider the 4 × 4 matrix
0 1 2 3
𝑀 = (1 0 1 2) .
2 1 0 1
3 2 1 0
If 𝑎𝑖,𝑗 denotes the (𝑖, 𝑗)th entry of 𝑀−1 , then 𝑎4,1 equals __________ (rounded

off to two decimal places).

Q.45 Let 𝑃12 (𝑥) be the real vector space of polynomials in the variable 𝑥 with real
coefficients and having degree at most 12, together with the zero polynomial.
Define

𝑉 = {𝑓 ∈ 𝑃12 (𝑥): 𝑓(−𝑥) = 𝑓 (𝑥) for all 𝑥 ∈ ℝ and 𝑓 (2024) = 0} .

Then, the dimension of 𝑉 is ______________

Q.46 Let

2024
𝑆 = {𝑓: ℝ → ℝ ∶ 𝑓 is a polynomial and 𝑓(𝑓(𝑥)) = (𝑓(𝑥)) for 𝑥 ∈ ℝ} .

Then, the number of elements in 𝑆 is ______________

MA 37/43

JAM 2024 Mathematics (MA)

Q.47 Let 𝑎1 = 1, 𝑏1 = 2 and 𝑐1 = 3. Consider the convergent sequences
{𝑎𝑛 }∞ ∞ ∞
𝑛=1 , {𝑏𝑛 }𝑛=1 and {𝑐𝑛 }𝑛=1
defined as follows:

𝑎𝑛 + 𝑏𝑛 𝑏𝑛 + 𝑐𝑛 𝑐𝑛 + 𝑎𝑛
𝑎𝑛+1 = , 𝑏𝑛+1 = and 𝑐𝑛+1 = for 𝑛 ≥ 1.
2 2 2
Then,
∞ ∞

∑ 𝑏𝑛 𝑐𝑛 (𝑎𝑛+1 − 𝑎𝑛 ) + ∑(𝑏𝑛+1 𝑐𝑛+1 − 𝑏𝑛 𝑐𝑛 )𝑎𝑛+1
𝑛=1 𝑛=1
equals _____________ (rounded off to two decimal places)

Q.48 Let
𝑆 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 ∶ 𝑥 2 + 𝑦 2 + 𝑧 2 = 4, (𝑥 − 1)2 + 𝑦 2 ≤ 1, 𝑧 ≥ 0}.

Then, the surface area of 𝑆 equals ___________ (rounded off to two decimal
places).

MA 38/43

JAM 2024 Mathematics (MA)

Q.49 Let 𝑃7 (𝑥) be the real vector space of polynomials in 𝑥 with degree at most 7,
together with the zero polynomial. For 𝑟 = 1, 2, … , 7, define
𝑠𝑟 (𝑥) = 𝑥(𝑥 − 1) ⋯ (𝑥 − (𝑟 − 1)) and 𝑠0 (𝑥) = 1.
Consider the fact that 𝐵 = {𝑠0 (𝑥), 𝑠1 (𝑥), … , 𝑠7 (𝑥)} is a basis of 𝑃7 (𝑥).
If
7

𝑥 5 = ∑ 𝛼5,𝑘 𝑠𝑘 (𝑥) ,
𝑘=0

where 𝛼5,𝑘 ∈ ℝ, then 𝛼5,2 equals ___________ (rounded off to two decimal
places)

Q.50 Let
0 0 0 0 −1
2 0 0 0 −4
𝑀= 0 2 0 0 0 .
0 0 2 0 3
(0 0 0 2 2)
If 𝑝(𝑥) is the characteristic polynomial of 𝑀, then 𝑝(2) − 1 equals __________

MA 39/43

JAM 2024 Mathematics (MA)

Section C: Q.51 – Q.60 Carry TWO marks each.

Q.51
For 𝛼 ∈ (−2𝜋, 0), consider the differential equation

𝑑2𝑦
2
𝑑𝑦
𝑥 + 𝛼𝑥 + 𝑦 = 0 for 𝑥 > 0.
𝑑𝑥 2 𝑑𝑥
Let 𝐷 be the set of all 𝛼 ∈ (−2𝜋, 0) for which all corresponding real solutions
to the above differential equation approach zero as 𝑥 → 0+ . Then, the number of
elements in 𝐷 ∩ ℤ equals ___________

Q.52 The value of

𝜋𝑡
2
1 −1 sin2 5𝑥
lim (log (𝑡 + 2 )) ∫ 𝑑𝑥
𝑡→∞ 𝑡 𝑥
1
( )

equals ___________ (rounded off to two decimal places).

MA 40/43

JAM 2024 Mathematics (MA)

Q.53 Let 𝑇 be the planar region enclosed by the square with vertices at the points
(0,1), (1,0), (0, −1) and (−1,0). Then, the value of

2

∬ (cos(𝜋(𝑥 − 𝑦)) − cos(𝜋(𝑥 + 𝑦))) 𝑑𝑥 𝑑𝑦
𝑇
equals ____________ (rounded off to two decimal places).

Q.54 Let
𝑆 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 : 𝑥 2 + 𝑦 2 + 𝑧 2 < 1}.
Then, the value of

1
∭ ( (𝑥 − 2𝑦 + 𝑧)2 + (2𝑥 − 𝑦 − 𝑧)2 + (𝑥 − 𝑦 + 2𝑧)2 ) 𝑑𝑥𝑑𝑦𝑑𝑧
𝜋 𝑆

equals ________ (rounded off to two decimal places).

Q.55 For 𝑛 ∈ ℕ, if
1 22 𝑛2
𝑎𝑛 = 3 + + ⋯+ 3
𝑛 + 1 𝑛3 + 2 𝑛 +𝑛
then the sequence {𝑎𝑛 }∞
𝑛=1 converges to ____________ (rounded off to two
decimal places)

MA 41/43

JAM 2024 Mathematics (MA)

Q.56 Consider the function 𝑓: ℝ → ℝ given by 𝑓(𝑥) = 𝑥 3 − 4𝑥 2 + 4𝑥 − 6.

For 𝑐 ∈ ℝ, let

𝑆(𝑐) = { 𝑥 ∈ ℝ ∶ 𝑓(𝑥) = 𝑐 }

and |𝑆(𝑐)| denote the number of elements in 𝑆(𝑐). Then, the value of
|𝑆(−7)| + |𝑆(−5)| + |𝑆(3)|
equals _________

Q.57 Let 𝑐 > 0 be such that
𝑐
2
∫ 𝑒 𝑠 𝑑𝑠 = 3
0
Then, the value of
𝑐 𝑐
2 2
∫ (∫ 𝑒 𝑥 +𝑦 𝑑𝑦) 𝑑𝑥
0 𝑥

equals __________(rounded off to one decimal place).

MA 42/43

JAM 2024 Mathematics (MA)

Q.58 For 𝑘 ∈ ℕ, let 0 = 𝑡0 < 𝑡1 < ⋯ < 𝑡𝑘 < 𝑡𝑘+1 = 1. A function 𝑓: [0,1] → ℝ is
said to be piecewise linear with nodes 𝑡1 , … , 𝑡𝑘 , if for each 𝑗 = 1, 2, … , 𝑘 + 1,
there exist 𝑎𝑗 ∈ ℝ, 𝑏𝑗 ∈ ℝ such that

𝑓(𝑡) = 𝑎𝑗 + 𝑏𝑗 𝑡 for 𝑡𝑗−1 < 𝑡 < 𝑡𝑗 .
Let 𝑉 be the real vector space of all real valued continuous piecewise linear
1 1 3
functions on [0,1] with nodes , , . Then, the dimension of 𝑉 equals
4 2 4
______________

Q.59 For 𝑛 ∈ ℕ, let
𝑛
1 𝑛! 𝑛𝑘
𝑎𝑛 = 𝑛−1 ∑
𝑛 𝑘! (𝑛 − 𝑘)! 𝑘 + 1
𝑘=0

and 𝛽 = lim 𝑎𝑛 . Then, the value of log 𝛽 equals ___________ (rounded
𝑛→∞
off to two decimal places).

Q.60 𝜋 𝜋
Define the function 𝑓: (−1,1) → (− , ) by
2 2

𝑓(𝑥) = sin−1 𝑥
2
Let 𝑎6 denote the coefficient of 𝑥 6 in the Taylor series of (𝑓(𝑥)) about

𝑥 = 0. Then, the value of 9𝑎6 equals ____________ (rounded off to two
decimal places).

MA 43/43

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