JAM 2025 Question Paper Mathematics (MA)

JOINT ADMISSION TEST FOR MASTERS

2025

IIT

JAM
2025
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 2025 Confidential 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 will be more than one choices 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-mark 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. A Scribble Pad will be provided for rough work.

BT 1/49

JAM 2025 Confidential Mathematics (MA)

Special Instructions / Useful Data

ℕ = The set of all natural numbers
ℤ = The set of all integers
ℤ𝑛 = {0̅, 1̅, … , ̅̅̅̅̅̅̅
𝑛 − 1}, the group of integers modulo 𝑛, under addition modulo 𝑛, for 𝑛 ∈ ℕ
ℝ = The set of all real numbers
ℝ𝑛 = The 𝑛 −dimensional Euclidean space
ln x = The natural logarithm of x (to the base e)
𝑆𝑛 = The symmetric group of all permutations on {1,2, . . . , 𝑛}
𝑖𝑑 = The identity element in 𝑆𝑛
(𝑎𝑛 ) = The infinite sequence 𝑎1 , 𝑎2 , 𝑎3 , . . .
𝑓 ∘ 𝑔 = The composition of 𝑓 and 𝑔, defined by (𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥))
𝑓 ′ (𝑥) = The first derivative of 𝑓 at the point 𝑥
𝑓 ′′ (𝑥) = The second derivative of 𝑓 at the point 𝑥
span 𝑆 = The linear span of the subset 𝑆 of a vector space
𝑃𝑛 (ℝ) = The real vector space of real polynomials of degree less than or equal to 𝑛,
together with the zero polynomial. These polynomials can be regarded as functions
from ℝ to ℝ
ker(𝑇) = The kernel of the linear transformation 𝑇
𝑀 = (𝑚𝑖𝑗 ) = Matrix of appropriate order with the entry/element in the 𝑖 𝑡ℎ row and
𝑗 𝑡ℎ column denoted by 𝑚𝑖𝑗 , 𝑚𝑖𝑗 ∈ ℝ
gcd(𝑚, 𝑛) = The greatest common divisor of the natural numbers 𝑚 and 𝑛
det(𝑀) = The determinant of the matrix 𝑀
𝜕𝑓
= The partial derivative of 𝑓 with respect to 𝑥
𝜕𝑥
𝜕𝑓
= The partial derivative of 𝑓 with respect to 𝑦
𝜕𝑦

BT 2/49

JAM 2025 Confidential Mathematics (MA)

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

Q.1 The sum of the infinite series
∞
𝑛+1
𝜋 2𝑛+1
∑(−1)
22𝑛+1 (2𝑛)!
𝑛=1

is equal to

(A) − π

(B) 𝜋
4

(C) 𝜋
2

(D) − 𝜋
4

BT 3/49

JAM 2025 Confidential Mathematics (MA)

Q.2 For which one of the following choices of 𝑁(𝑥, 𝑦), is the equation
(𝑒 𝑥 sin 𝑦 − 2𝑦 sin 𝑥) d𝑥 + 𝑁(𝑥, 𝑦) d𝑦 = 0

an exact differential equation?

(A) 𝑁(𝑥, 𝑦) = 𝑒 𝑥 sin 𝑦 + 2 cos 𝑥

(B) 𝑁(𝑥, y) = 𝑒 𝑥 cos 𝑦 + 2 cos 𝑥

(C) 𝑁(𝑥, 𝑦) = 𝑒 𝑥 cos 𝑦 + 2 sin 𝑥

(D) 𝑁(𝑥, 𝑦) = 𝑒 𝑥 sin 𝑦 + 2 sin 𝑥

BT 4/49

JAM 2025 Confidential Mathematics (MA)

Q.3 Let 𝑓 , 𝑔 : ℝ → ℝ be two functions defined by

1
𝑓(𝑥) = { 𝑥 |𝑥| |sin | if 𝑥≠0
𝑥
0 if 𝑥 =0

and

2
1 1
𝑔(𝑥) = { 𝑥 sin 𝑥
+ 𝑥 cos
𝑥
if 𝑥≠0
0 if 𝑥 = 0.

Then, which one of the following is TRUE?

(A) 𝑓 is differentiable at 𝑥 = 0, and 𝑔 is NOT differentiable at 𝑥 = 0

(B) 𝑓 is NOT differentiable at 𝑥 = 0, and 𝑔 is differentiable at 𝑥 = 0

(C) 𝑓 is differentiable at 𝑥 = 0, and 𝑔 is differentiable at 𝑥 = 0

(D) 𝑓 is NOT differentiable at 𝑥 = 0, and 𝑔 is NOT differentiable at 𝑥 = 0

BT 5/49

JAM 2025 Confidential Mathematics (MA)

Q.4 Let 𝑓, 𝑔 : ℝ → ℝ be two functions defined by

1⁄8
1
𝑓(𝑥) = { |𝑥| |sin | cos 𝑥 if 𝑥≠0
𝑥
0 if 𝑥 = 0

and

𝑥
1
𝑔(𝑥) = { 𝑒 cos if 𝑥≠0
𝑥
1 if 𝑥 = 0 .

Then, which one of the following is TRUE?

(A) 𝑓 is continuous at 𝑥 = 0, and 𝑔 is NOT continuous at 𝑥 = 0

(B) 𝑓 is NOT continuous at 𝑥 = 0, and 𝑔 is continuous at 𝑥 = 0

(C) 𝑓 is continuous at 𝑥 = 0, and 𝑔 is continuous at 𝑥 = 0

(D) 𝑓 is NOT continuous at 𝑥 = 0, and 𝑔 is NOT continuous at 𝑥 = 0

BT 6/49

JAM 2025 Confidential Mathematics (MA)

Q.5 Which one of the following is the general solution of the differential equation
𝑑2𝑦 𝑑𝑦
2
−8 + 16𝑦 = 2𝑒 4𝑥 ?
𝑑𝑥 𝑑𝑥

(A) α1 𝑒 4𝑥 + α2 𝑥𝑒 4𝑥 + 𝑥 2 𝑒 4𝑥 , where α1 , α2 ∈ ℝ

(B) α1 𝑒 4𝑥 + α2 𝑥𝑒 4𝑥 + 2𝑥 2 𝑒 4𝑥 , where α1 , α2 ∈ ℝ

(C) α1 𝑒 − 4𝑥 + α2 𝑒 4𝑥 + 2𝑥 2 𝑒 4𝑥 , where α1 , α2 ∈ ℝ

(D) α1 𝑥𝑒 − 4𝑥 + α2 𝑥 2 𝑒 − 4𝑥 + 𝑥 2 𝑒 4𝑥 , where α1 , α2 ∈ ℝ

Q.6 Define T: ℝ3 → ℝ3 by
T(𝑥, 𝑦, 𝑧) = (𝑥 + 𝑧, 2𝑥 + 3𝑦 + 5𝑧, 2𝑦 + 2𝑧), for all (𝑥, 𝑦, 𝑧) ∈ ℝ3 .

Then, which one of the following is TRUE?

(A) T is one-one and T is NOT onto

(B) T is NOT one-one and T is onto

(C) T is one-one and T is onto

(D) T is NOT one-one and T is NOT onto

BT 7/49

JAM 2025 Confidential Mathematics (MA)

Q.7 2 0 −1
Let 𝑀 = ( 4 1 −4 ) for some real number 𝑥.
2 0 𝑥

1
If 0 is an eigenvalue of 𝑀, then (𝑀4 + 𝑀) (0) is equal to
1

(A) 1
(0)
1

(B) 2
(0)
2

(C) 5
(0)
5

(D) 17
(0)
17

BT 8/49

JAM 2025 Confidential Mathematics (MA)

Q.8 Let T ∶ 𝑃2 (ℝ) → 𝑃2 (ℝ) be the linear transformation defined by
T(𝑝(𝑥)) = 𝑝(𝑥 + 1), for all 𝑝(𝑥) ∈ 𝑃2 (ℝ). If 𝑀 is the matrix representation of
T with respect to the ordered basis {1, 𝑥, 𝑥 2 } of 𝑃2 (ℝ), then which one of the
following is TRUE?

(A) The determinant of 𝑀 is 2

(B) The rank of 𝑀 is 2

(C) 1 is the only eigenvalue of 𝑀

(D) The nullity of 𝑀 is 2

Q.9 Let 𝐺 be a finite abelian group of order 10. Let 𝑥0 be an element of order 2 in 𝐺.

If 𝑋 = {𝑥 ∈ 𝐺 : 𝑥 3 = 𝑥0 }, then which one of the following is TRUE?

(A) 𝑋 has exactly one element

(B) 𝑋 has exactly two elements

(C) 𝑋 has exactly three elements

(D) 𝑋 is an empty set

BT 9/49

JAM 2025 Confidential Mathematics (MA)

Q.10 The value of
1 1
3
∫ (∫ 3𝑒 𝑥 𝑑𝑥 ) 𝑑𝑦
0 √𝑦
is equal to

(A) 𝑒 − 1

(B) 𝑒−1
2

(C) √𝑒 − 1

(D) √𝑒−1
2

BT 10/49

JAM 2025 Confidential Mathematics (MA)

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

Q.11 Let 𝒞 denote the family of curves described by 𝑦𝑥 2 = λ , for λ ∈ (0, ∞) and
lying in the first quadrant of the 𝑥𝑦 plane. Let 𝒪 denote the family of orthogonal
trajectories of 𝒞.

Which one of the following curves is a member of 𝒪, and passes through the
point (2, 1) ?

(A) 𝑥2
𝑦= , 𝑥 > 0, 𝑦 > 0
4

(B) 𝑥 2 − 2 𝑦 2 = 2 , 𝑥 > 0, 𝑦 > 0

(C) 𝑥 − 𝑦 = 1 , 𝑥 > 0, 𝑦 > 0

(D) 2𝑥 − 𝑦 2 = 3 , 𝑥 > 0, 𝑦 > 0

BT 11/49

JAM 2025 Confidential Mathematics (MA)

Q.12 Let 𝜑 : (0, ∞) → ℝ be the solution of the differential equation
𝑑𝑦
𝑥 = ( ln 𝑦 − ln 𝑥 )𝑦 ,
𝑑𝑥

satisfying 𝜑(1) = 𝑒 2 . Then, the value of 𝜑(2) is equal to

(A) 𝑒 2

(B) 2𝑒 3

(C) 3𝑒 2

(D) 6𝑒 3

BT 12/49

JAM 2025 Confidential Mathematics (MA)

Q.13 Let 𝑋 = {𝑥 ∈ 𝑆4 : 𝑥 3 = 𝑖𝑑} and 𝑌 = {𝑥 ∈ 𝑆4 : 𝑥 2 ≠ 𝑖𝑑}.

If 𝑚 and 𝑛 denote the number of elements in 𝑋 and 𝑌, respectively, then which
one of the following is TRUE?

(A) 𝑚 is even and 𝑛 is even

(B) 𝑚 is odd and 𝑛 is even

(C) 𝑚 is even and 𝑛 is odd

(D) 𝑚 is odd and 𝑛 is odd

BT 13/49

JAM 2025 Confidential Mathematics (MA)

Q.14 Let 𝜑 : ℝ → ℝ be the solution of the differential equation
𝑑𝑦
= (𝑦 − 1) (𝑦 − 3) ,
𝑑𝑥

satisfying 𝜑(0) = 2 . Then, which one of the following is TRUE?

(A) lim 𝜑(𝑥) = 0
𝑥→∞

(B) lim 𝜑(𝑥) = 1
𝑥→ln √2

(C) lim 𝜑(𝑥) = 3
𝑥→ − ∞

(D) lim 1 𝜑(𝑥) = 6
𝑥→ ln
√2

BT 14/49

JAM 2025 Confidential Mathematics (MA)

Q.15 6 2 −6 8
Let 𝑀 = (5 3 −9 8) . Consider the system 𝑆 of linear equations given
3 1 −2 4
by

6𝑥1 + 2𝑥2 − 6𝑥3 + 8𝑥4 = 8
5𝑥1 + 3𝑥2 − 9𝑥3 + 8𝑥4 = 16
3𝑥1 + 𝑥2 − 2𝑥3 + 4𝑥4 = 32

where 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 are unknowns.

Then, which one of the following is TRUE?

(A) The rank of 𝑀 is 3, and the system 𝑆 has a solution

(B) The rank of 𝑀 is 3, and the system 𝑆 does NOT have a solution

(C) The rank of 𝑀 is 2, and the system 𝑆 has a solution

(D) The rank of 𝑀 is 2, and the system 𝑆 does NOT have a solution

BT 15/49

JAM 2025 Confidential Mathematics (MA)

Q.16 −2 0 0
Let 𝑀 = ( 3 2 3 ) for some real number 𝑥. Suppose that −2 and 3
4 −1 𝑥
0 0
3
are eigenvalues of 𝑀. If 𝑀 (1) = (125) , then which one of the following
1 125
is TRUE?

(A) 𝑥 = 5, and the matrix 𝑀2 + 𝑀 is invertible

(B) 𝑥 ≠ 5, and the matrix 𝑀2 + 𝑀 is invertible

(C) 𝑥 = 5, and the matrix 𝑀2 + 𝑀 is NOT invertible

(D) 𝑥 ≠ 5, and the matrix 𝑀2 + 𝑀 is NOT invertible

Q.17 Let 𝑓(𝑥) = 10𝑥 2 + 𝑒 𝑥 − sin(2𝑥) − cos 𝑥 , 𝑥 ∈ ℝ . The number of points at
which the function 𝑓 has a local minimum is

(A) 0

(B) 1

(C) 2

(D) greater than or equal to 3

BT 16/49

JAM 2025 Confidential Mathematics (MA)

Q.18 For 𝑛 ∈ ℕ , define 𝑥𝑛 and 𝑦𝑛 by

𝑛
1 1
𝑥𝑛 = (−1)𝑛 cos and 𝑦𝑛 = ∑ .
𝑛 𝑛+𝑘
𝑘=1

Then, which one of the following is TRUE?

(A) ∑∞ ∞
𝑛=1 𝑥𝑛 converges, and ∑𝑛=1 𝑦𝑛 does NOT converge

(B) ∑∞ ∞
𝑛=1 𝑥𝑛 does NOT converge, and ∑𝑛=1 𝑦𝑛 converges

(C) ∑∞ ∞
𝑛=1 𝑥𝑛 converges, and ∑𝑛=1 𝑦𝑛 converges

(D) ∑∞ ∞
𝑛=1 𝑥𝑛 does NOT converge, and ∑𝑛=1 𝑦𝑛 does NOT converge

BT 17/49

JAM 2025 Confidential Mathematics (MA)

Q.19 5
Let 𝑥1 = . For 𝑛 ∈ ℕ, define
2
1
𝑥𝑛+1 = (𝑥𝑛2 + 6) .
5

Then, which one of the following is TRUE?

(A) (𝑥𝑛 ) is an increasing sequence, and (𝑥𝑛 ) is NOT a bounded sequence

(B) (𝑥𝑛 ) is NOT an increasing sequence, and (𝑥𝑛 ) is NOT a bounded sequence

(C) (𝑥𝑛 ) is NOT a decreasing sequence, and (𝑥𝑛 ) is a bounded sequence

(D) (𝑥𝑛 ) is a decreasing sequence, and (𝑥𝑛 ) is a bounded sequence

BT 18/49

JAM 2025 Confidential Mathematics (MA)

Q.20 1
Let 𝑥1 = 2 and 𝑥𝑛+1 = 2 + for all 𝑛 ∈ ℕ .
2𝑥𝑛

Then, which one of the following is TRUE?

(A) 𝑥 4
𝑛+1 ≥ for all 𝑛 ∈ ℕ, and (𝑥𝑛 ) is a Cauchy sequence
𝑥𝑛

(B) 𝑥 4
𝑛+1 < for some 𝑛 ∈ ℕ, and (𝑥𝑛 ) is a Cauchy sequence
𝑥𝑛

(C) 𝑥 4
𝑛+1 ≥ for all 𝑛 ∈ ℕ, and (𝑥𝑛 ) is NOT a Cauchy sequence
𝑥𝑛

(D) 𝑥 4
𝑛+1 < for some 𝑛 ∈ ℕ, and (𝑥𝑛 ) is NOT a Cauchy sequence
𝑥𝑛

BT 19/49

JAM 2025 Confidential Mathematics (MA)

Q.21 For 𝑛 ∈ ℕ , define 𝑥𝑛 and 𝑦𝑛 by

3𝑛 1
𝑥𝑛 = (−1)𝑛 3
and 𝑦𝑛 = (4𝑛 + (−1)𝑛 3𝑛 ) ⁄𝑛 .
𝑛

Then, which one of the following is TRUE?

(A) (𝑥𝑛 ) has a convergent subsequence, and NO subsequence of (𝑦𝑛 ) is convergent

(B) NO subsequence of (𝑥𝑛 ) is convergent, and (𝑦𝑛 ) has a convergent subsequence

(C) (𝑥𝑛 ) has a convergent subsequence, and (𝑦𝑛 ) has a convergent subsequence

(D) NO subsequence of (𝑥𝑛 ) is convergent, and NO subsequence of (𝑦𝑛 ) is
convergent

BT 20/49

JAM 2025 Confidential Mathematics (MA)

Let 𝑀 = (𝑚𝑖𝑗 ) be a 3 × 3 real, invertible matrix and σ ∈ 𝑆3 be the
Q.22
permutation defined by σ(1) = 2, σ(2) = 3 and σ(3) = 1. The matrix
𝑀σ = (𝑛𝑖𝑗 ) is defined by 𝑛𝑖𝑗 = 𝑚𝑖𝜎(𝑗) for all 𝑖, 𝑗 ∈ {1, 2, 3}.

Then, which one of the following is TRUE?

(A) det(𝑀) = det(𝑀σ ), and nullity of the matrix 𝑀 − 𝑀σ is 0

(B) det(𝑀) = − det(𝑀σ ), and nullity of the matrix 𝑀 − 𝑀σ is 1

(C) det(𝑀) = det(𝑀σ ), and nullity of the matrix 𝑀 − 𝑀σ is 1

(D) det(𝑀) = − det(𝑀σ ), and nullity of the matrix 𝑀 − 𝑀σ is 0

BT 21/49

JAM 2025 Confidential Mathematics (MA)

Q.23 Let ℝ/ℤ denote the quotient group, where ℤ is considered as a subgroup of the
additive group of real numbers ℝ.

Let 𝑚 denote the number of injective (one-one) group homomorphisms from ℤ3
to ℝ/ℤ and 𝑛 denote the number of group homomorphisms from ℝ/ℤ to ℤ3 .

Then, which one of the following is TRUE?

(A) 𝑚 = 2 and 𝑛 = 1

(B) 𝑚 = 3 and 𝑛 = 3

(C) 𝑚 = 2 and 𝑛 = 3

(D) 𝑚 = 1 and 𝑛 = 1

BT 22/49

JAM 2025 Confidential Mathematics (MA)

Q.24 Let 𝑓1 , 𝑓2 , 𝑓3 be nonzero linear transformations from ℝ4 to ℝ and

ker(𝑓1 ) ⊆ ker(𝑓2 ) ∩ ker(𝑓3 ) .

Let 𝑇 ∶ ℝ4 → ℝ3 be the linear transformation defined by
𝑇(𝑣) = (𝑓1 (𝑣), 𝑓2 (𝑣), 𝑓3 (𝑣)) , for all 𝑣 ∈ ℝ4 .

Then, the nullity of 𝑇 is equal to

(A) 1

(B) 2

(C) 3

(D) 4

BT 23/49

JAM 2025 Confidential Mathematics (MA)

Q.25 Let 𝑥1 = 1. For 𝑛 ∈ ℕ , define
1 sin2 𝑛
𝑥𝑛+1 = ( + ) 𝑥𝑛 .
2 𝑛

Then, which one of the following is TRUE?

(A) ∑∞
𝑛=1 𝑥𝑛 converges

(B) ∑∞
𝑛=1 𝑥𝑛 does NOT converge

(C) ∑∞ 2
𝑛=1 𝑥𝑛 does NOT converge

(D) ∑∞
𝑛=1 𝑥𝑛 𝑥𝑛+1 does NOT converge

BT 24/49

JAM 2025 Confidential Mathematics (MA)

Q.26 Let 𝑥1 > 0. For 𝑛 ∈ ℕ , define 𝑥𝑛+1 = 𝑥𝑛 + 4 . If
1 1 1 1
lim ( + +⋯ + ) = ,
𝑛 → ∞ 𝑥2 𝑥3 𝑥3 𝑥4 𝑥𝑛+1 𝑥𝑛+2 24

then the value of 𝑥1 is equal to

(A) 1

(B) 2

(C) 3

(D) 8

BT 25/49

JAM 2025 Confidential Mathematics (MA)

Q.27 Let 𝑓: ℝ2 → ℝ be defined by
𝑓(𝑥, 𝑦) = 𝑒 𝑦 (𝑥 2 + 𝑦 2 ) for all (𝑥, 𝑦) ∈ ℝ2 .

Then, which one of the following is TRUE?

(A) The number of points at which 𝑓 has a local minimum is 2

(B) The number of points at which 𝑓 has a local maximum is 2

(C) The number of points at which 𝑓 has a local minimum is 1

(D) The number of points at which 𝑓 has a local maximum is 1

BT 26/49

JAM 2025 Confidential Mathematics (MA)

Q.28 Let Ω be the bounded region in ℝ3 lying in the first octant (𝑥 ≥ 0, 𝑦 ≥ 0,
𝑧 ≥ 0), and bounded by the surfaces 𝑧 = 𝑥 2 + 𝑦 2 , 𝑧 = 4 , 𝑥 = 0 and 𝑦 = 0.

Then, the volume of Ω is equal to

(A) π

(B) 2π

(C) 3π

(D) 4𝜋

BT 27/49

JAM 2025 Confidential Mathematics (MA)

Q.29 Let 𝜑 : [0, ∞) → ℝ be the continuous function satisfying

𝑥

𝜑(𝑥) = (∫ 𝜑(𝑡)𝑑𝑡) + sin 𝑥 , for all 𝑥 ∈ [0, ∞) .
0

Then, the value of lim (2𝜑(𝑥) − 𝑒 𝑥 ) is equal to
𝑥 → 𝜋⁄2

(A) 1

(B) 2

(C) 3

(D) 4

BT 28/49

JAM 2025 Confidential Mathematics (MA)

Q.30 The number of elements in the set
{𝑥 ∈ ℝ ∶ 8𝑥 2 + 𝑥 4 + 𝑥 8 = cos 𝑥}

is equal to

(A) 0

(B) 1

(C) 2

(D) greater than or equal to 3

BT 29/49

JAM 2025 Confidential Mathematics (MA)

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

Q.31 Let 𝑓: ℝ2 → ℝ be defined by

𝑥𝑦 2 + 𝑦 5
if (𝑥, 𝑦) ≠ (0, 0)
𝑓(𝑥, 𝑦) = { 𝑥 2 + 𝑦 4
0 if (𝑥, 𝑦) = (0, 0)
Then, which of the following is/are TRUE?

(A) The iterated limits lim ( lim 𝑓(𝑥, 𝑦)) and lim ( lim 𝑓(𝑥, 𝑦)) exist
𝑥 → 0 𝑦 →0 𝑦 → 0 𝑥 →0

(B) Exactly one of the partial derivatives 𝜕𝑓 and 𝜕𝑓 exists at (0, 0)
𝜕𝑥 𝜕𝑦

(C) Both the partial derivatives 𝜕𝑓 and 𝜕𝑓 exist at (0, 0)
𝜕𝑥 𝜕𝑦

(D) 𝑓 is NOT differentiable at (0, 0)

BT 30/49

JAM 2025 Confidential Mathematics (MA)

Q.32 If 𝑀, 𝑁, 𝜇, 𝑤: ℝ2 → ℝ are differentiable functions with continuous partial
derivatives, satisfying

𝜇(𝑥, 𝑦)𝑀(𝑥, 𝑦)𝑑𝑥 + 𝜇(𝑥, 𝑦)𝑁(𝑥, 𝑦)𝑑𝑦 = 𝑑𝑤

then which of the following is/are TRUE?

(A) 𝜇𝑤 is an integrating factor for 𝑀(𝑥, 𝑦)𝑑𝑥 + 𝑁(𝑥, 𝑦)𝑑𝑦 = 0

(B) 𝜇𝑤 2 is an integrating factor for 𝑀(𝑥, 𝑦)𝑑𝑥 + 𝑁(𝑥, 𝑦)𝑑𝑦 = 0

(C) 𝑤(𝑥, 𝑦) = 𝑤(0,0) + ∫𝑥(𝜇𝑀)(𝑠, 0)𝑑𝑠 + ∫𝑦(𝜇𝑁)(𝑥, 𝑡)𝑑𝑡 , for all (𝑥, 𝑦) ∈ ℝ2
0 0

(D) 𝑤(𝑥, 𝑦) = 𝑤(0,0) + ∫𝑥(𝜇𝑀)(𝑠, 𝑦)𝑑𝑠 + ∫𝑥(𝜇𝑁)(0, 𝑡)𝑑𝑡 , for all (𝑥, 𝑦) ∈ ℝ2
0 0

BT 31/49

JAM 2025 Confidential Mathematics (MA)

Q.33 Let 𝜑 : (−1, ∞) → (0, ∞) be the solution of the differential equation
𝑑𝑦
− 2 𝑦𝑒 𝑥 = 2𝑒 𝑥 √𝑦 ,
𝑑𝑥

satisfying 𝜑(0) = 1 .

Then, which of the following is/are TRUE?

(A) 𝜑 is an unbounded function

(B) lim 𝜑(𝑥) = (2𝑒 − 1)2
𝑥 → ln 2

(C) lim 𝜑(𝑥) = √2𝑒 − 1
𝑥 → ln 2

(D) 𝜑 is a strictly increasing function on the interval (0, ∞)

BT 32/49

JAM 2025 Confidential Mathematics (MA)

Q.34 Let 𝑓: ℝ2 → ℝ be defined by

(𝑥 2 + sin 𝑥)𝑦 2
if (𝑥, 𝑦) ≠ (0, 0)
𝑓(𝑥, 𝑦) = { 𝑥 2 + 𝑦 2
0 if (𝑥, 𝑦) = (0, 0)
Then, which of the following is/are TRUE?

(A) lim 𝑓(𝑥, 𝑦) exists and lim 𝑓(𝑥, 𝑦) = 1
(𝑥,𝑦)→(0,0) (𝑥,𝑦)→(0,0)

(B) lim 𝑓(𝑥, 𝑦) exists and lim 𝑓(𝑥, 𝑦) = 0
(𝑥,𝑦)→(0,0) (𝑥,𝑦)→(0,0)

(C) 𝑓 is differentiable at (0, 0)

(D) 𝑓 is NOT differentiable at (0, 0)

BT 33/49

JAM 2025 Confidential Mathematics (MA)

Q.35 Let
𝑢1 = (1,0,0, −1), 𝑢2 = (0,2, 0, −1), 𝑢3 = (0,0,1, −1) and 𝑢4 = (0,0,0,1)
be elements in the real vector space ℝ4 .

Then, which of the following is/are TRUE?

(A) {𝑢1 , 𝑢2 , 𝑢3 , 𝑢4 } is a linearly independent set in ℝ4

(B) {𝑢1 − 𝑢2 , 𝑢2 − 𝑢3 , 𝑢3 − 𝑢4 , 𝑢4 − 𝑢1 } is NOT a linearly independent set in ℝ4

(C) {𝑢1 , −𝑢2 , 𝑢3 , −𝑢4 } is NOT a linearly independent set in ℝ4

(D) {𝑢1 + 𝑢2 , 𝑢2 + 𝑢3 , 𝑢3 + 𝑢4 , 𝑢4 + 𝑢1 } is a linearly independent set in ℝ4

BT 34/49

JAM 2025 Confidential Mathematics (MA)

Q.36 For 𝑛 ∈ ℕ , let

𝑛
𝑘
𝑥𝑛 = ∑ .
𝑛2 + 𝑘
𝑘=1

Then, which of the following is/are TRUE?

(A) The sequence (𝑥𝑛 ) converges

(B) The series ∑∞
𝑛=1 𝑥𝑛 converges

(C) The series ∑∞
𝑛=1 𝑥𝑛 does NOT converge

(D) The series ∑∞ 𝑛
𝑛=1 𝑥𝑛 converges

BT 35/49

JAM 2025 Confidential Mathematics (MA)

Q.37 Let 𝑓: ℝ → ℝ be a twice differentiable function such that
𝑓(0) = 0, 𝑓 ′ (0) = 2 and 𝑓(1) = −3 .

Then, which of the following is/are TRUE?

(A) |𝑓 ′ (𝑥)| ≤ 2 for all 𝑥 ∈ [0, 1]

(B) |𝑓 ′ (𝑥1 )| > 2 for some 𝑥1 ∈ [0, 1]

(C) |𝑓 ′′ (𝑥)| < 10 for all 𝑥 ∈ [0, 1]

(D) |𝑓 ′′ (𝑥2 )| ≥ 10 for some 𝑥2 ∈ [0, 1]

BT 36/49

JAM 2025 Confidential Mathematics (MA)

Q.38 Let 𝑓: ℝ → ℝ be a twice differentiable function such that
𝑓(0) = 4, 𝑓(1) = −2, 𝑓(2) = 8 and 𝑓(3) = 2 .

Then, which of the following is/are TRUE?

(A) |𝑓 ′ (𝑥)| < 5 for all 𝑥 ∈ [0, 1]

(B) |𝑓 ′ (𝑥1 )| ≥ 5 for some 𝑥1 ∈ [0, 1]

(C) 𝑓 ′ (𝑥2 ) = 0 for some 𝑥2 ∈ [0, 3]

(D) 𝑓 ′′ (𝑥3 ) = 0 for some 𝑥3 ∈ [0, 3]

BT 37/49

JAM 2025 Confidential Mathematics (MA)

Q.39 For 𝑛 ∈ ℕ, consider the set 𝑈(𝑛) = {𝑥̅ ∈ ℤ𝑛 ∶ gcd(𝑥, 𝑛) = 1} 𝑎s a group
under multiplication modulo 𝑛 .

Then, which of the following is/are TRUE?

(A) 𝑈(8) is a cyclic group

(B) 𝑈(5) is a cyclic group

(C) 𝑈(12) is a cyclic group

(D) 𝑈(9) is a cyclic group

BT 38/49

JAM 2025 Confidential Mathematics (MA)

Q.40 Consider the following subspaces of the real vector space ℝ3 :

𝑉1 = span {(1, 2, 3), (1, 1, 0)},

𝑉2 = span {(1, −1, 0)},

𝑉3 = span {(1, 1, 1)},

𝑉4 = span {(1, 3, 6)} and

𝑉5 = span {(1, 0, −3)}.

Then, which of the following is/are TRUE?

(A) 𝑉1 ∪ 𝑉2 is a subspace of ℝ3

(B) 𝑉1 ∪ 𝑉3 is a subspace of ℝ3

(C) 𝑉1 ∪ 𝑉4 is a subspace of ℝ3

(D) 𝑉1 ∪ 𝑉5 is a subspace of ℝ3

BT 39/49

JAM 2025 Confidential Mathematics (MA)

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

Q.41 The radius of convergence of the power series

∞ 1 𝑛
(𝑥 + )
4
∑
(−2) 𝑛2
𝑛
𝑛=1

1
about 𝑥 = − 4 , is equal to _______________ (rounded off to two decimal

places).

Q.42 The value of

1 1 1
lim 8𝑛 (𝑒 (2𝑛) − 1) (sin + |cos |)
𝑛→∞ 2𝑛 2𝑛

is equal to _________________ (rounded off to two decimal places).

Q.43 Let 𝛼 be the real number such that
(1 − cos 𝑥)(22+𝑥 − 4)
lim = 𝛼 ln 2 .
𝑥→0 𝑥3

Then, the value of 𝛼 is equal to _____________ (rounded off to two decimal
places).

BT 40/49

JAM 2025 Confidential Mathematics (MA)

Q.44 Let 𝜑: ℝ → ℝ be the solution of the differential equation

𝑑2𝑦 𝑑𝑦
4 2
+ 16 + 25𝑦 = 0
𝑑𝑥 𝑑𝑥

1
satisfying 𝜑(0) = 1 and 𝜑′(0) = − 2 .

Then, the value of lim 𝑒 2𝑥 𝜑(𝑥) is equal to _____________ (rounded off to
𝑥→𝜋⁄6

two decimal places).

Q.45 Let 𝑆 be the surface area of the portion of the plane 𝑧 = 𝑥 + 𝑦 + 3, which lies
inside the cylinder 𝑥 2 + 𝑦 2 = 1.

𝑆 2
Then, the value of ( ) is equal to ____________ (rounded off to two decimal
𝜋
places).

Q.46 Consider the following subspaces of ℝ4 :

𝑉1 = {(𝑥, 𝑦, 𝑧, 𝑤) ∈ ℝ4 ∶ 𝑥 + 𝑦 + 2𝑤 = 0},

𝑉2 = {(𝑥, 𝑦, 𝑧, 𝑤) ∈ ℝ4 ∶ 2𝑦 + 𝑧 + 𝑤 = 0},

𝑉3 = {(𝑥, 𝑦, 𝑧, 𝑤) ∈ ℝ4 ∶ 𝑥 + 3𝑦 + 𝑧 + 3𝑤 = 0}.

Then, the dimension of the subspace 𝑉1 ∩ 𝑉2 ∩ 𝑉3 is equal to __________.

BT 41/49

JAM 2025 Confidential Mathematics (MA)

Q.47 Consider the real vector space ℝ3 . Let 𝑇 ∶ ℝ3 → ℝ be a linear transformation
such that

𝑇(1, 1, 1) = 0, 𝑇(1, −1, 1) = 0 and 𝑇(0, 0, 1) = 16.

1 2 3
Then, the value of 𝑇 (2 , 3 , 4) is equal to ____________________ (rounded

off to two decimal places).

Q.48 Let 𝑇 denote the triangle in the 𝑥𝑦 plane bounded by the 𝑥 axis and the lines
𝑦 = 𝑥 and 𝑥 = 1. The value of the double integral (over 𝑇 )

∬(5 − 𝑦)𝑑𝑥𝑑𝑦
𝑇

is equal to _________ (rounded off to two decimal places).

Q.49 Let 𝑇 , 𝑆 ∶ 𝑃4 (ℝ) → 𝑃4 (ℝ) be the linear transformations defined by
𝑇(𝑝(𝑥)) = 𝑥𝑝′ (𝑥) and 𝑆(𝑝(𝑥)) = (𝑥 + 1)𝑝′ (𝑥)

for all 𝑝(𝑥) ∈ 𝑃4 (ℝ).

Then, the nullity of the composition 𝑆 ∘ 𝑇 is _____________.

Q.50 Let 𝑓: ℝ2 → ℝ be defined by

(𝑥 2 − 𝑦 2 )𝑥𝑦
if (𝑥, 𝑦) ≠ (0, 0)
𝑓(𝑥, 𝑦) = { 𝑥 2 + 𝑦 2
0 if (𝑥, 𝑦) = (0, 0)
𝜕𝑓 𝜕𝑓
Then, the value of 𝜕𝑦 (1, 0) − 𝜕𝑥 (0, 2) is equal to ____________ (rounded

off to two decimal places).

BT 42/49

JAM 2025 Confidential Mathematics (MA)

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

Q.51 Let 𝑓: ℝ → ℝ be a continuous function satisfying

𝜋⁄4 𝑥
∫ (sin(𝑥) 𝑓(𝑥) + cos(𝑥) ∫ 𝑓(𝑡)𝑑𝑡) 𝑑𝑥 = √2 .
0 0

𝜋⁄4
Then, the value of ∫0 𝑓(𝑥) 𝑑𝑥 is equal to _________________ (rounded off
to two decimal places).

Q.52 Let σ ∈ 𝑆4 be the permutation defined by σ(1) = 2, σ(2) = 3, σ(3) = 1 and
σ(4) = 4. The number of elements in the set
{τ ∈ 𝑆4 ∶ τστ−1 = σ}
is equal to _____________.

Q.53 Let 𝑓(𝑥) = 2𝑥 − sin 𝑥, for all 𝑥 ∈ ℝ. Let 𝑘 ∈ ℕ be such that

𝑘
1 𝑥
lim ( ∑ 𝑖 2 𝑓 ( )) = 45.
𝑥 →0 𝑥 𝑖
𝑖=1

Then, the value of 𝑘 is equal to ______________.

Q.54 The value of the infinite series

∞
3 2(𝑛−1)
∑𝑛 ( )
4
𝑛=1

is equal to _______________ (rounded off to two decimal places)

BT 43/49

JAM 2025 Confidential Mathematics (MA)

Q.55 Let 𝜑: (0, ∞) → ℝ be the solution of the differential equation

𝑑2𝑦 𝑑𝑦
𝑥2 2
−𝑥 + 𝑦 = 6𝑥 ln 𝑥
𝑑𝑥 𝑑𝑥

satisfying 𝜑(1) = −3 and 𝜑(𝑒) = 0 .

Then, the value of |𝜑′(1)| is equal to ______________ (rounded off to two
decimal places).

Q.56 Let 𝜑: ℝ → ℝ be the solution of the differential equation

𝑑𝑦
+ 2𝑥𝑦 = 2 + 4𝑥 2
𝑑𝑥

satisfying 𝜑(0) = 0.

Then, the value of 𝜑(2) is equal to _________________ (rounded off to two
decimal places).

Q.57 1
Let Ω be the solid bounded by the planes 𝑧 = 0, 𝑦 = 0 , 𝑥 = 2 , 2𝑦 = 𝑥 and

2𝑥 + 𝑦 + 𝑧 = 4.

If 𝑉 is the volume of Ω, then the value of 64 𝑉 is equal to ____________
(rounded off to two decimal places).

Q.58 Let the subspace 𝐻 of 𝑃3 (ℝ) be defined as

𝐻 = {𝑝(𝑥) ∈ 𝑃3 (ℝ) ∶ 𝑥𝑝′ (𝑥) = 3𝑝(𝑥)}.

Then, the dimension of 𝐻 is equal to _________________.

BT 44/49

JAM 2025 Confidential Mathematics (MA)

Q.59 Let 𝐺 be an abelian group of order 35. Let 𝑚 denote the number of elements of
order 5 in 𝐺, and let 𝑛 denote the number of elements of order 7 in 𝐺.

Then, the value of 𝑚 + 𝑛 is equal to ______________.

Q.60 The number of surjective (onto) group homomorphisms from 𝑆4 to ℤ6 is equal
to ________________.

BT 45/49

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