GATE 2023 Question Paper MA Mathematics

The Graduate Aptitude Test in Engineering

GATE 2023
Question Paper
GATE is an entrance examination conducted in India that
primarily tests the comprehensive understanding of
undergraduate subjects in engineering and sciences for
admission into technical postgraduate programs.

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General Aptitude (GA)
Q.1 – Q.5 Carry ONE mark Each

Q.1 The village was nestled in a green spot, _______ the ocean and the hills.

(A) through

(B) in

(C) at

(D) between

Q.2 Disagree : Protest : : Agree : _______

(By word meaning)

(A) Refuse

(B) Pretext

(C) Recommend

(D) Refute

Page 1 of 53

Q.3 A ‘frabjous’ number is defined as a 3 digit number with all digits odd, and no two
adjacent digits being the same. For example, 137 is a frabjous number, while 133 is
not. How many such frabjous numbers exist?

(A) 125

(B) 720

(C) 60

(D) 80

Q.4 Which one among the following statements must be TRUE about the mean and the
median of the scores of all candidates appearing for GATE 2023?

(A) The median is at least as large as the mean.

(B) The mean is at least as large as the median.

(C) At most half the candidates have a score that is larger than the median.

(D) At most half the candidates have a score that is larger than the mean.

Page 2 of 53

Q.5 In the given diagram, ovals are marked at different heights (h) of a hill. Which one
of the following options P, Q, R, and S depicts the top view of the hill?

(A) P

(B) Q

(C) R

(D) S

Page 3 of 53

Q.6 – Q.10 Carry TWO marks Each

Q.6 Residency is a famous housing complex with many well-established individuals
among its residents. A recent survey conducted among the residents of the complex
revealed that all of those residents who are well established in their respective fields
happen to be academicians. The survey also revealed that most of these
academicians are authors of some best-selling books.

Based only on the information provided above, which one of the following
statements can be logically inferred with certainty?

(A) Some residents of the complex who are well established in their fields are also
authors of some best-selling books.

(B) All academicians residing in the complex are well established in their fields.

(C) Some authors of best-selling books are residents of the complex who are well
established in their fields.

(D) Some academicians residing in the complex are well established in their fields.

Page 4 of 53

Q.7 Ankita has to climb 5 stairs starting at the ground, while respecting the following
rules:

1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.

Let 𝐹(𝑁) denote the number of possible ways in which Ankita can reach the 𝑁 𝑡ℎ
stair. For example, 𝐹(1) = 1, 𝐹(2) = 2, 𝐹(3) = 3.

The value of 𝐹(5) is _______.

(A) 8

(B) 7

(C) 6

(D) 5

Page 5 of 53

Q.8 The information contained in DNA is used to synthesize proteins that are necessary
for the functioning of life. DNA is composed of four nucleotides: Adenine (A),
Thymine (T), Cytosine (C), and Guanine (G). The information contained in DNA
can then be thought of as a sequence of these four nucleotides: A, T, C, and G. DNA
has coding and non-coding regions. Coding regions—where the sequence of these
nucleotides are read in groups of three to produce individual amino
acids—constitute only about 2% of human DNA. For example, the triplet of
nucleotides CCG codes for the amino acid glycine, while the triplet GGA codes for
the amino acid proline. Multiple amino acids are then assembled to form a protein.

Based only on the information provided above, which of the following statements
can be logically inferred with certainty?

(i) The majority of human DNA has no role in the synthesis of proteins.
(ii) The function of about 98% of human DNA is not understood.

(A) only (i)

(B) only (ii)

(C) both (i) and (ii)

(D) neither (i) nor (ii)

Page 6 of 53

Q.9 Which one of the given figures P, Q, R and S represents the graph of the following
function?

𝑓(𝑥) = | |𝑥 + 2| − |𝑥 − 1| |

(A) P

(B) Q

(C) R

(D) S

Page 7 of 53

Q.10 An opaque cylinder (shown below) is suspended in the path of a parallel beam of
light, such that its shadow is cast on a screen oriented perpendicular to the direction
of the light beam. The cylinder can be reoriented in any direction within the light
beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT
possible?

(A) P

(B) Q

(C) R

(D) S

Page 8 of 53

USEFUL DATA
ℕ Set of all positive integers
ℤ Set of all integers
ℚ Set of all rational numbers
ℝ Set of all real numbers
ℂ Set of all complex numbers
ℝ𝑛 {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∶ 𝑥𝑖 ∈ ℝ, 𝑖 = 1,2, … , 𝑛}
ℝ×ℝ Cartesian product of ℝ with ℝ

Page 9 of 53

Q.11 – Q.35 Carry ONE mark Each
Q.11 Let 𝑓, 𝑔 ∶ ℝ2 → ℝ be defined by
3
𝑓(𝑥, 𝑦) = 𝑥 2 − 2 𝑥𝑦 2 and 𝑔(𝑥, 𝑦) = 4𝑥 4 − 5𝑥 2 𝑦 + 𝑦 2

for all (𝑥, 𝑦) ∈ ℝ2 .
Consider the following statements:
𝑃: 𝑓 has a saddle point at (0,0).
𝑄: 𝑔 has a saddle point at (0,0).

Then

(A) both 𝑃 and 𝑄 are TRUE

(B) 𝑃 is FALSE but 𝑄 is TRUE

(C) 𝑃 is TRUE but 𝑄 is FALSE

(D) both 𝑃 and 𝑄 are FALSE

Page 10 of 53

Q.12 Let ℝ3 be a topological space with the usual topology and ℚ denote the set of
rational numbers. Define the subspaces 𝑋, 𝑌, 𝑍 and 𝑊 of ℝ3 as follows:
𝑋 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 ∶ |𝑥| + |𝑦| + |𝑧| ∈ ℚ}
𝑌 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 ∶ 𝑥𝑦𝑧 = 1}
𝑍 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 ∶ 𝑥 2 + 𝑦 2 + 𝑧 2 = 1}
𝑊 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 ∶ 𝑥𝑦𝑧 = 0 }

Which of the following statements is correct?

(A) 𝑋 is homeomorphic to 𝑌

(B) 𝑍 is homeomorphic to 𝑊

(C) 𝑌 is homeomorphic to 𝑊

(D) 𝑋 is NOT homeomorphic to 𝑊

Page 11 of 53

Q.13 Let 𝑃(𝑥) = 1 + 𝑒 2𝜋𝑖𝑥 + 2 𝑒 3𝜋𝑖𝑥 , 𝑥 ∈ ℝ, 𝑖 = √−1. Then
𝑁−1
1
lim ∑ 𝑃( 𝑘√2 )
𝑁→∞ 𝑁
𝑘=0

is equal to

(A) 0

(B) 1

(C) 3

(D) 4

Page 12 of 53

Q.14 Let 𝑇 ∶ ℝ3 → ℝ3 be a linear transformation satisfying
𝑇(1, 0, 0) = (0, 1, 1), 𝑇(1, 1, 0) = (1, 0, 1) and 𝑇(1, 1, 1) = (1, 1, 2).

Then

(A) 𝑇 is one-one but 𝑇 is NOT onto

(B) 𝑇 is one-one and onto

(C) 𝑇 is NEITHER one-one NOR onto

(D) 𝑇 is NOT one-one but 𝑇 is onto

Page 13 of 53

Q.15 Let 𝔻 = { 𝑧 ∈ ℂ ∶ |𝑧| < 1} and 𝑓: 𝔻 → ℂ be defined by
𝑧 5 𝑧 7 𝑧 9 𝑧11
𝑓(𝑧) = 𝑧 – 25𝑧 3 + − + −
5! 7! 9! 11!
Consider the following statements:
𝑃: 𝑓 has three zeros (counting multiplicity) in 𝔻 .
1
𝑄: 𝑓 has one zero in 𝕌 = { 𝑧 ∈ ℂ ∶ 2 < |𝑧| < 1}.

Then

(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

Page 14 of 53

Q.16 Let 𝒩 ⊆ ℝ be a non-measurable set with respect to the Lebesgue measure on ℝ.

Consider the following statements:
𝑃: If 𝑀 = { 𝑥 ∈ 𝒩 ∶ 𝑥 is irrational }, then 𝑀 is Lebesgue measurable.
𝑄: The boundary of 𝒩 has positive Lebesgue outer measure.

Then

(A) both 𝑃 and 𝑄 are TRUE

(B) 𝑃 is FALSE and 𝑄 is TRUE

(C) 𝑃 is TRUE and 𝑄 is FALSE

(D) both 𝑃 and 𝑄 are FALSE

Page 15 of 53

Q.17 1
For 𝑘 ∈ ℕ, let 𝐸𝑘 be a measurable subset of [0,1] with Lebesgue measure 𝑘 2 .
Define
∞ ∞
𝐸 = ∩∞ ∞
𝑛=1 ∪𝑘=𝑛 𝐸𝑘 and 𝐹 = ∪𝑛=1 ∩𝑘=𝑛 𝐸𝑘

Consider the following statements:

𝑃: Lebesgue measure of 𝐸 is equal to zero.

𝑄: Lebesgue measure of 𝐹 is equal to zero.

Then

(A) both 𝑃 and 𝑄 are TRUE

(B) both 𝑃 and 𝑄 are FALSE

(C) 𝑃 is TRUE but 𝑄 is FALSE

(D) 𝑄 is TRUE but 𝑃 is FALSE

Page 16 of 53

Q.18 Consider ℝ2 with the usual Euclidean metric. Let
1
𝑋 = {(𝑥, 𝑥 sin 𝑥) ∈ ℝ2 ∶ 𝑥 ∈ (0,1]} ⋃ {(0, 𝑦) ∈ ℝ2 : −∞ < 𝑦 < ∞} and

1
𝑌 = {(𝑥, sin 𝑥) ∈ ℝ2 : 𝑥 ∈ (0,1]} ⋃ {(0, 𝑦) ∈ ℝ2 : −∞ < 𝑦 < ∞}.

Consider the following statements:

𝑃: 𝑋 is a connected subset of ℝ2 .

𝑄: 𝑌 is a connected subset of ℝ2 .

Then

(A) both 𝑃 and 𝑄 are TRUE

(B) 𝑃 is FALSE and 𝑄 is TRUE

(C) 𝑃 is TRUE and 𝑄 is FALSE

(D) both 𝑃 and 𝑄 are FALSE

Page 17 of 53

Q.19 4 −3
Let 𝑀 = [ ].
1 0
Consider the following statements:

𝑃: 𝑀8 + 𝑀12 is diagonalizable.

𝑄: 𝑀7 + 𝑀9 is diagonalizable.

Which of the following statements is correct?

(A) 𝑃 is TRUE and 𝑄 is FALSE

(B) 𝑃 is FALSE and 𝑄 is TRUE

(C) Both 𝑃 and 𝑄 are FALSE

(D) Both 𝑃 and 𝑄 are TRUE

Page 18 of 53

Q.20 Let 𝐶[0, 1] = { 𝑓 ∶ [0, 1] → ℝ ∶ 𝑓 is continuous}.
Consider the metric space (𝐶[0,1], 𝑑∞ ), where
𝑑∞ (𝑓, 𝑔) = sup{ |𝑓(𝑥) − 𝑔(𝑥)| ∶ 𝑥 ∈ [0, 1] } for 𝑓, 𝑔 ∈ 𝐶[0,1].
Let 𝑓0 (𝑥) = 0 for all 𝑥 ∈ [0,1] and
1
𝑋 = {𝑓 ∈ (𝐶[0, 1], 𝑑∞ ) ∶ 𝑑∞ (𝑓0 , 𝑓) ≥ } .
2
Let 𝑓1 , 𝑓2 ∈ 𝐶[0, 1] be defined by 𝑓1 (𝑥) = 𝑥 and 𝑓2 (𝑥) = 1 − 𝑥 for all 𝑥 ∈ [0,1].
Consider the following statements:
𝑃: 𝑓1 is in the interior of 𝑋.
𝑄: 𝑓2 is in the interior of 𝑋.

Which of the following statements is correct?

(A) 𝑃 is TRUE and 𝑄 is FALSE

(B) 𝑃 is FALSE and 𝑄 is TRUE

(C) Both 𝑃 and 𝑄 are FALSE

(D) Both 𝑃 and 𝑄 are TRUE

Page 19 of 53

Q.21 Consider the metrics 𝜌1 and 𝜌2 on ℝ, defined by

0, 𝑖𝑓 𝑥 = 𝑦
𝜌1 (𝑥, 𝑦) = |𝑥 − 𝑦| and 𝜌2 (𝑥, 𝑦) = {
1, 𝑖𝑓 𝑥 ≠ 𝑦
1
Let 𝑋 = {𝑛 ∈ ℕ ∶ 𝑛 ≥ 3} and 𝑌 = { 𝑛 + 𝑛 ∶ 𝑛 ∈ ℕ}.

2, 𝑖𝑓 𝑥 ∈ 𝑋
Define 𝑓: 𝑋 ∪ 𝑌 → ℝ by 𝑓(𝑥) = {
3, 𝑖𝑓 𝑥 ∈ 𝑌

Consider the following statements:

𝑃: The function 𝑓: (𝑋 ∪ 𝑌, 𝜌1 ) → (ℝ, 𝜌1 ) is uniformly continuous.

𝑄: The function 𝑓: (𝑋 ∪ 𝑌, 𝜌2 ) → (ℝ, 𝜌1 ) is uniformly continuous.

Then

(A) 𝑃 is TRUE and 𝑄 is FALSE

(B) 𝑃 is FALSE and 𝑄 is TRUE

(C) both 𝑃 and 𝑄 are FALSE

(D) both 𝑃 and 𝑄 are TRUE

Page 20 of 53

Q.22 Let 𝑇 ∶ ℝ4 → ℝ4 be a linear transformation and the null space of 𝑇 be the
subspace of ℝ4 given by
{ (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∈ ℝ4 ∶ 4𝑥1 + 3𝑥2 + 2𝑥3 + 𝑥4 = 0}.
If 𝑅𝑎𝑛𝑘(𝑇 − 3𝐼) = 3, where 𝐼 is the identity map on ℝ4 , then the minimal
polynomial of 𝑇 is

(A) 𝑥(𝑥 − 3)

(B) 𝑥(𝑥 − 3)3

(C) 𝑥 3 (𝑥 − 3)

(D) 𝑥 2 (𝑥 − 3)2

Page 21 of 53

Let 𝐶[0,1] denote the set of all real valued continuous functions defined on [0,1]
Q.23
and ‖𝑓‖∞ = sup{|𝑓(𝑥)| ∶ 𝑥 ∈ [0,1]} for all 𝑓 ∈ 𝐶[0,1]. Let
𝑋 = { 𝑓 ∈ 𝐶[0,1] ∶ 𝑓(0) = 𝑓(1) = 0 }.
1
Define 𝐹 ∶ (𝐶[0,1], ‖⋅‖∞ ) → ℝ by 𝐹(𝑓) = ∫0 𝑓(𝑡)𝑑𝑡 for all 𝑓 ∈ 𝐶[0,1].
Denote 𝑆𝑋 = {𝑓 ∈ 𝑋 ∶ ‖𝑓‖∞ = 1}.

Then the set {𝑓 ∈ 𝑋 ∶ 𝐹(𝑓) = ‖𝐹‖} ∩ 𝑆𝑋 has

(A) NO element

(B) exactly one element

(C) exactly two elements

(D) an infinite number of elements

Page 22 of 53

Q.24 Let 𝑋 and 𝑌 be two topological spaces. A continuous map 𝑓 ∶ 𝑋 → 𝑌 is said to be
proper if 𝑓 −1 (𝐾) is compact in 𝑋 for every compact subset 𝐾 of 𝑌, where 𝑓 −1 (𝐾)
is defined by 𝑓 −1 (𝐾) = {𝑥 ∈ 𝑋 ∶ 𝑓(𝑥) ∈ 𝐾} .
Consider ℝ with the usual topology. If ℝ ∖ {0} has the subspace topology induced
from ℝ and ℝ × ℝ has the product topology, then which of the following maps is
proper?

(A) 𝑓: ℝ ∖ {0} → ℝ defined by 𝑓(𝑥) = 𝑥

(B) 𝑓: ℝ × ℝ → ℝ × ℝ defined by 𝑓(𝑥, 𝑦) = (𝑥 + 𝑦, 𝑦)

(C) 𝑓: ℝ × ℝ → ℝ defined by 𝑓(𝑥, 𝑦) = 𝑥

(D) 𝑓: ℝ × ℝ → ℝ defined by 𝑓(𝑥, 𝑦) = 𝑥 2 − 𝑦 2

Q.25 Consider the following Linear Programming Problem P:
Minimize 3𝑥1 + 4𝑥2
subject to 𝑥1 − 𝑥2 ≤ 1,
𝑥1 + 𝑥2 ≥ 3,
𝑥1 ≥ 0, 𝑥2 ≥ 0.

The optimal value of the problem P is _____________.

Page 23 of 53

Q.26 Let 𝑢(𝑥, 𝑡) be the solution of
𝜕 2𝑢 1 𝜕 2𝑢
− = 0, 𝑥 ∈ (−∞, ∞), 𝑡 > 0,
𝜕𝑥 2 𝑐 2 𝜕𝑡 2
𝑢(𝑥, 0) = sin 𝑥, 𝑥 ∈ (−∞, ∞),
𝜕𝑢
(𝑥, 0) = cos 𝑥, 𝑥 ∈ (−∞, ∞),
𝜕𝑡
for some positive real number 𝑐.
Let the domain of dependence of the solution 𝑢 at the point 𝑃(3,2) be the line
segment on the 𝑥-axis with end points 𝑄 and 𝑅.

If the area of the triangle 𝑃𝑄𝑅 is 8 square units, then the value of 𝑐 2 is ___________.

Q.27 Let
∞
𝑧
= ∑ 𝑎𝑛 𝑧 𝑛 , 𝑎𝑛 ∈ ℝ
1 − 𝑧 − 𝑧2
𝑛=0

for all 𝑧 in some neighbourhood of 0 in ℂ.

Then the value of 𝑎6 + 𝑎5 is equal to _________.

Page 24 of 53

Q.28 Let 𝑝(𝑥) = 𝑥 3 − 2𝑥 + 2. If 𝑞(𝑥) is the interpolating polynomial of degree less than
or equal to 4 for the data
𝑥 −2 −1 0 1 3
,
𝑞(𝑥) 𝑝(−2) 𝑝(−1) 2.5 𝑝(1) 𝑝(3)

𝑑4𝑞
then the value of at 𝑥 = 0 is ______________.
𝑑𝑥 4

Q.29 2
For a fixed 𝑐 ∈ ℝ, let 𝛼 = ∫0 (9𝑥 2 − 5𝑐𝑥 4 )𝑑𝑥.
2
If the value of ∫0 (9𝑥 2 − 5𝑐𝑥 4 )𝑑𝑥 obtained by using the Trapezoidal rule is equal
to 𝛼, then the value of 𝑐 is __________ (rounded off to 2 decimal places).

Page 25 of 53

Q.30 If for some 𝛼 ∈ ℝ,
𝜋
4 𝑥 𝛼 sec 𝜃
1 4 1
∫ ∫ 2 2
𝑑𝑦𝑑𝑥 = ∫ ∫ 𝑑𝑟𝑑𝜃,
1 −𝑥 𝑥 + 𝑦 𝑟
𝜋
− sec 𝜃
4

then the value of 𝛼 equals _______.

Q.31 Let 𝑆 be the portion of the plane 𝑧 = 2𝑥 + 2𝑦 − 100 which lies inside the cylinder
𝑥 2 + 𝑦 2 = 1. If the surface area of 𝑆 is 𝛼𝜋, then the value of 𝛼 is equal to
___________.

Page 26 of 53

Q.32 Let
1
𝐿2 [−1, 1] = {𝑓: [−1, 1] → ℝ ∶ 𝑓 is Lebesgue measurable and ∫−1|𝑓(𝑥)|2 𝑑𝑥 < ∞}
1
1 2
and the norm ‖𝑓‖2 = (∫−1|𝑓(𝑥)|2 𝑑𝑥) for 𝑓 ∈ 𝐿2 [−1,1].

Let 𝐹: (𝐿2 [−1, 1], ‖⋅‖2 ) → ℝ be defined by
1
𝐹(𝑓) = ∫−1 𝑓(𝑥)𝑥 2 𝑑𝑥 for all 𝑓 ∈ 𝐿2 [−1,1] .

If ‖𝐹‖ denotes the norm of the linear functional 𝐹, then 5‖𝐹‖2 is equal to
_________.

Q.33 Let 𝑦(𝑡) be the solution of the initial value problem

𝑡, 0 ≤ 𝑡 ≤ 2,
𝑦 ′′ + 4𝑦 = { and 𝑦(0) = 𝑦 ′ (0) = 0.
2, 2 < 𝑡 < ∞,

𝜋 4
If 𝛼 = 𝑦 ( 2 ), then the value of 𝜋 𝛼 is ________(rounded off to 2 decimal places).

Page 27 of 53

Q.34 Consider ℝ4 with the inner product < 𝑥, 𝑦 > = ∑4𝑖=1 𝑥𝑖 𝑦𝑖 , for 𝑥 = (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 )
and 𝑦 = (𝑦1 , 𝑦2 , 𝑦3 , 𝑦4 ).

Let 𝑀 = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∈ ℝ4 ∶ 𝑥1 = 𝑥3 } and 𝑀⊥ denote the orthogonal
complement of 𝑀. The dimension of 𝑀⊥ is equal to ________.

Q.35 3 −1 −2 1 0 0
Let 𝑀 = [ 0 2 4 ] and 𝐼 = [0 1 0]. If 6𝑀 −1 = 𝑀2 − 6𝑀 + 𝛼𝐼 for
0 0 1 0 0 1
some 𝛼 ∈ ℝ, then the value of 𝛼 is equal to ___________.

Page 28 of 53

Q.36 – Q.65 Carry TWO marks Each
Q.36 Let 𝐺𝐿2 (ℂ) denote the group of 2 × 2 invertible complex matrices with usual
matrix multiplication. For 𝑆, 𝑇 ∈ 𝐺𝐿2 (ℂ), < 𝑆, 𝑇 > denotes the subgroup
0 −1
generated by 𝑆 and 𝑇. Let 𝑆 = [ ] ∈ 𝐺𝐿2 (ℂ) and 𝐺1 , 𝐺2 , 𝐺3 be three
1 0
subgroups of 𝐺𝐿2 (ℂ) given by
𝑖 0
𝐺1 = < 𝑆, 𝑇1 >, where 𝑇1 = [ ],
0 𝑖
𝑖 0
𝐺2 = < 𝑆, 𝑇2 >, where 𝑇2 = [ ],
0 −𝑖
0 1
𝐺3 = < 𝑆, 𝑇3 >, where 𝑇3 = [ ].
1 0
Let 𝑍(𝐺𝑖 ) denote the center of 𝐺𝑖 for 𝑖 = 1, 2, 3.

Which of the following statements is correct?

(A) 𝐺1 is isomorphic to 𝐺3

(B) 𝑍(𝐺1 ) is isomorphic to 𝑍(𝐺2 )

(C) 1 0
𝑍(𝐺3 ) = { [ ] }
0 1

(D) 𝑍(𝐺2 ) is isomorphic to 𝑍(𝐺3 )

Page 29 of 53

Q.37 Let ℓ2 = {(𝑥1 , 𝑥2 , 𝑥3 , … ) ∶ 𝑥𝑛 ∈ ℝ for all 𝑛 ∈ ℕ and ∑∞ 2
𝑛=1 𝑥𝑛 < ∞}.
1
For a sequence (𝑥1 , 𝑥2 , 𝑥3 , … ) ∈ ℓ2 , define ‖(𝑥1 , 𝑥2 , 𝑥3 , … )‖2 = (∑∞ 2
𝑛=1 𝑥𝑛 )2 .

Let 𝑆 ∶ (ℓ2 , ‖⋅‖2 ) → (ℓ2 , ‖⋅‖2 ) and 𝑇 ∶ (ℓ2 , ‖⋅‖2 ) → (ℓ2 , ‖⋅‖2 ) be defined by
0, 𝑛 = 1
𝑆(𝑥1 , 𝑥2 , 𝑥3 , … ) = (𝑦1 , 𝑦2 , 𝑦3 , … ), where 𝑦𝑛 = {
𝑥𝑛−1 , 𝑛 ≥ 2
0, 𝑛 𝑖𝑠 𝑜𝑑𝑑
𝑇(𝑥1 , 𝑥2 , 𝑥3 , … ) = (𝑦1 , 𝑦2 , 𝑦3 , … ), where 𝑦𝑛 = {
𝑥𝑛 , 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛

Then

(A) 𝑆 is a compact linear map and 𝑇 is NOT a compact linear map

(B) 𝑆 is NOT a compact linear map and 𝑇 is a compact linear map

(C) both 𝑆 and 𝑇 are compact linear maps

(D) NEITHER 𝑆 NOR 𝑇 is a compact linear map

Page 30 of 53

Q.38 Let

𝑐00 = {(𝑥1 , 𝑥2 , 𝑥3 , … ) ∶ 𝑥𝑖 ∈ ℝ, 𝑖 ∈ ℕ, 𝑥𝑖 ≠ 0 only for finitely many indices 𝑖}.

For (𝑥1 , 𝑥2 , 𝑥3 , … ) ∈ 𝑐00 , let ‖(𝑥1 , 𝑥2 , 𝑥3 , … )‖∞ = sup{|𝑥𝑖 | ∶ 𝑖 ∈ ℕ}.

Define 𝐹, 𝐺 ∶ (𝑐00 , ‖⋅‖∞ ) → (𝑐00 , ‖⋅‖∞ ) by
1 1
𝐹((𝑥1 , 𝑥2 , … , 𝑥𝑛 , … )) = ((1 + 1)𝑥1 , (2 + 2) 𝑥2 , …, (𝑛 + 𝑛) 𝑥𝑛 , … ),

𝑥1 𝑥2 𝑥𝑛
𝐺((𝑥1 , 𝑥2 , … , 𝑥𝑛 , … )) = ( , , …, ,… ),
1+1 2+1 𝑛 +
1
2 𝑛
for all (𝑥1 , 𝑥2 , … , 𝑥𝑛 , … ) ∈ 𝑐00 .

Then

(A) 𝐹 is continuous but 𝐺 is NOT continuous

(B) 𝐹 is NOT continuous but 𝐺 is continuous

(C) both 𝐹 and 𝐺 are continuous

(D) NEITHER 𝐹 NOR 𝐺 is continuous

Page 31 of 53

Q.39 Consider the Cauchy problem

𝜕𝑢 𝜕𝑢
𝑥 +𝑦 = 𝑢;
𝜕𝑥 𝜕𝑦

𝑢 = 𝑓(𝑡) on the initial curve Γ = (𝑡, 𝑡); 𝑡 > 0.

Consider the following statements:

𝑃: If 𝑓(𝑡) = 2𝑡 + 1, then there exists a unique solution to the Cauchy problem
in a neighbourhood of Γ.
𝑄: If 𝑓(𝑡) = 2𝑡 − 1, then there exist infinitely many solutions to the Cauchy
problem in a neighbourhood of Γ.

Then

(A) both 𝑃 and 𝑄 are TRUE

(B) 𝑃 is FALSE and 𝑄 is TRUE

(C) 𝑃 is TRUE and 𝑄 is FALSE

(D) both 𝑃 and 𝑄 are FALSE

Page 32 of 53

Q.40 2 −1 −2
Consider the linear system 𝑀𝑥 = 𝑏, where 𝑀 = [ ] and 𝑏 = [ ].
−4 3 5
Suppose 𝑀 = 𝐿𝑈, where 𝐿 and U are lower triangular and upper triangular square
matrices, respectively. Consider the following statements:

𝑃: If each element of the main diagonal of 𝐿 is 1, then 𝑡𝑟𝑎𝑐𝑒(𝑈) = 3.

𝑄: For any choice of the initial vector 𝑥 (0) , the Jacobi iterates 𝑥 (𝑘) , 𝑘 = 1,2,3 …
converge to the unique solution of the linear system 𝑀𝑥 = 𝑏.

Then

(A) both 𝑃 and 𝑄 are TRUE

(B) 𝑃 is FALSE and 𝑄 is TRUE

(C) 𝑃 is TRUE and 𝑄 is FALSE

(D) both 𝑃 and 𝑄 are FALSE

Page 33 of 53

Q.41 Let 𝜙 and 𝜓 be two linearly independent solutions of the ordinary differential
equation
𝑦 ′′ + (2 − cos 𝑥) 𝑦 = 0, 𝑥 ∈ℝ.
Let 𝛼, 𝛽 ∈ ℝ be such that 𝛼 < 𝛽, 𝜙(𝛼) = 𝜙(𝛽) = 0 and 𝜙(𝑥) ≠ 0 for all
𝑥 ∈ (𝛼, 𝛽).
Consider the following statements:
𝑃: 𝜙 ′ (𝛼)𝜙 ′ (𝛽) > 0.
𝑄: 𝜙(𝑥)𝜓(𝑥) ≠ 0 for all 𝑥 ∈ (𝛼, 𝛽).

Then

(A) 𝑃 is TRUE and 𝑄 is FALSE

(B) 𝑃 is FALSE and 𝑄 is TRUE

(C) both 𝑃 and 𝑄 are FALSE

(D) both 𝑃 and 𝑄 are TRUE

Page 34 of 53

Q.42 Let 𝔻 = { 𝑧 ∈ ℂ ∶ |𝑧| < 1} and 𝑓: 𝔻 → ℂ be an analytic function given by the
1
power series 𝑓(𝑧) = ∑∞ 𝑛
𝑛=0 𝑎𝑛 𝑧 , where 𝑎0 = 𝑎1 = 1 and 𝑎𝑛 = 22𝑛 for 𝑛 ≥ 2.

Consider the following statements:

𝑃: If 𝑧0 ∈ 𝔻, then 𝑓 is one-one in some neighbourhood of 𝑧0 .
1
𝑄: If 𝐸 = { 𝑧 ∈ ℂ ∶ |𝑧| ≤ 2}, then 𝑓(𝐸) is a closed subset of ℂ.

Which of the following statements is/are correct?

(A) 𝑃 is TRUE

(B) 𝑄 is TRUE

(C) 𝑄 is FALSE

(D) 𝑃 is FALSE

Page 35 of 53

Q.43 1
Let Ω be an open connected subset of ℂ containing 𝑈 = { 𝑧 ∈ ℂ ∶ |𝑧| ≤ 2}.

Let 𝔍 = { 𝑓 ∶ Ω → ℂ ∶ 𝑓 is analytic and sup |𝑓(𝑧) − 𝑓(𝑤)| = 1 }.
𝑧,𝑤 ∈ 𝑈

Consider the following statements:
𝑃: There exists 𝑓 ∈ ℑ such that |𝑓 ′ (0)| ≥ 2.
𝑄: |𝑓 (3) (0)| ≤ 48 for all 𝑓 ∈ ℑ, where 𝑓 (3) denotes the third derivative of 𝑓.

Then

(A) 𝑃 is TRUE

(B) 𝑄 is FALSE

(C) 𝑃 is FALSE

(D) 𝑄 is TRUE

Page 36 of 53

Q.44 Let (ℝ, 𝜏) be a topological space, where the topology 𝜏 is defined as

𝜏 = {𝑈 ⊆ ℝ ∶ 𝑈 = ∅ 𝑜𝑟 1 ∈ 𝑈}.

Which of the following statements is/are correct?

(A) (ℝ, 𝜏) is first countable

(B) (ℝ, 𝜏) is Hausdorff

(C) (ℝ, 𝜏) is separable

(D) The closure of (1,5) is [1,5]

Page 37 of 53

Q.45 Let ℛ = {𝑝(𝑥) ∈ ℚ[𝑥] ∶ 𝑝(0) ∈ ℤ}, where ℚ denotes the set of rational numbers
and ℤ denotes the set of integers. For 𝑎 ∈ ℛ, let ⟨𝑎⟩ denote the ideal generated by
𝑎 in ℛ.

Which of the following statements is/are correct?

(A) If 𝑝(𝑥) is an irreducible element in ℛ, then ⟨𝑝(𝑥)⟩ is a prime ideal in ℛ

(B) ℛ is a unique factorization domain

(C) ⟨𝑥⟩ is a prime ideal in ℛ

(D) ℛ is NOT a principal ideal domain

Page 38 of 53

Q.46 Consider the rings

ℤ2 [𝑥]
𝒮1 = ℤ[𝑥]⁄⟨2, 𝑥 3 ⟩ and 𝒮2 = ⁄⟨𝑥 2 ⟩

where ⟨2, 𝑥 3 ⟩ denotes the ideal generated by {2, 𝑥 3 } in ℤ[𝑥] and ⟨𝑥 2 ⟩ denotes the
ideal generated by 𝑥 2 in ℤ2 [𝑥].

Which of the following statements is/are correct?

(A) Every prime ideal of 𝒮1 is a maximal ideal

(B) 𝒮2 has exactly one maximal ideal

(C) Every element of 𝒮1 is either nilpotent or a unit

(D) There exists an element in 𝒮2 which is NEITHER nilpotent NOR a unit

Page 39 of 53

Q.47 Consider the sequence of Lebesgue measurable functions 𝑓𝑛 : ℝ → ℝ given by

2 (𝑥
1
𝑓𝑛 (𝑥) = {𝑛 − 𝑛), 𝑖𝑓 𝑥 ∈ [𝑛, 𝑛 + ]
𝑛2
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

For a measurable subset 𝐸 of ℝ, denote 𝑚(𝐸) to be the Lebesgue measure of 𝐸.

Which of the following statements is/are correct?

(A) sup |𝑓𝑛 (𝑥)| → 0 𝑎𝑠 𝑛 → ∞
𝑥∈ℝ

(B) ∫ℝ|𝑓𝑛 (𝑥)|𝑑𝑥 → 0 𝑎𝑠 𝑛 → ∞

(C) 1
𝑚({𝑥 ∈ ℝ ∶ |𝑓𝑛 (𝑥)| > 2}) → 0 𝑎𝑠 𝑛 → ∞

(D) 𝑚({𝑥 ∈ ℝ ∶ |𝑓𝑛 (𝑥)| > 0}) → 0 𝑎𝑠 𝑛 → ∞

Page 40 of 53

Q.48 Define the characteristic function 𝜒𝐸 of a subset 𝐸 in ℝ by

1, 𝑖𝑓 𝑥 ∈ 𝐸
𝜒𝐸 (𝑥) = {
0, 𝑖𝑓 𝑥 ∉ 𝐸

For 1 ≤ 𝑝 < 2, let
1
𝐿𝑝 [0, 1] = {𝑓: [0, 1] → ℝ ∶ 𝑓 is Lebesgue measurable and ∫0 |𝑓(𝑥)|𝑝 𝑑𝑥 < ∞}.

Let 𝑓: [0, 1] → ℝ be defined by
∞
2𝑛
𝑓(𝑥) = ∑ 3 𝜒[ 1 , 1 ] (𝑥).
𝑛 2𝑛+1 2𝑛
𝑛=1

Consider the following two statements:

𝑃: 𝑓 ∈ 𝐿𝑝 [0, 1] for every 𝑝 ∈ (1, 2).

𝑄: 𝑓 ∈ 𝐿1 [0, 1].

Then

(A) 𝑃 is TRUE

(B) 𝑄 is TRUE

(C) 𝑄 is FALSE

(D) 𝑃 is FALSE

Page 41 of 53

Q.49 Let 𝑥(𝑡), 𝑦(𝑡), 𝑡 ∈ ℝ, be two functions satisfying the following system of
differential equations:

𝑥 ′ (𝑡) = 𝑦(𝑡),

𝑦 ′ (𝑡) = 𝑥(𝑡),

and 𝑥(0) = 𝛼, 𝑦(0) = 𝛽, where 𝛼, 𝛽 are real numbers.

Which of the following statements is/are correct?

(A) If 𝛼 = 1, 𝛽 = −1, then |𝑥(𝑡)| + |𝑦(𝑡)| → 0 𝑎𝑠 𝑡 → ∞

(B) If 𝛼 = 1, 𝛽 = 1, then |𝑥(𝑡)| + |𝑦(𝑡)| → 0 𝑎𝑠 𝑡 → ∞

(C) If 𝛼 = 1.01, 𝛽 = −1, then |𝑥(𝑡)| + |𝑦(𝑡)| → 0 𝑎𝑠 𝑡 → ∞

(D) If 𝛼 = 1, 𝛽 = 1.01, then |𝑥(𝑡)| + |𝑦(𝑡)| → 0 𝑎𝑠 𝑡 → ∞

Page 42 of 53

Q.50 For ℎ > 0, and 𝛼, 𝛽, 𝛾 ∈ ℝ, let

𝛼 𝑓(𝑎 − ℎ) + 𝛽𝑓(𝑎) + 𝛾𝑓(𝑎 + 2ℎ)
𝐷ℎ 𝑓(𝑎) =
6ℎ
be a three-point formula to approximate 𝑓 ′ (𝑎) for any differentiable function
𝑓: ℝ → ℝ and 𝑎 ∈ ℝ.

If 𝐷ℎ 𝑓(𝑎) = 𝑓 ′ (𝑎) for every polynomial 𝑓 of degree less than or equal to 2 and
for all 𝑎 ∈ ℝ, then

(A) 𝛼 + 2𝛾 = −2

(B) 𝛼 + 2𝛽 − 2𝛾 = 0

(C) 𝛼 + 2𝛾 = 2

(D) 𝛼 + 2𝛽 − 2𝛾 = 1

Page 43 of 53

Q.51 Let 𝑓 be a twice continuously differentiable function on [𝑎, 𝑏] such that 𝑓 ′ (𝑥) < 0
and 𝑓 ′′ (𝑥) < 0 for all 𝑥 ∈ (𝑎, 𝑏). Let 𝑓(𝜁) = 0 for some 𝜁 ∈ (𝑎, 𝑏). The Newton-
Raphson method to compute 𝜁 is given by

𝑓(𝑥𝑘 )
𝑥𝑘+1 = 𝑥𝑘 − , 𝑘 = 0, 1, 2, …
𝑓 ′ (𝑥𝑘 )

for an initial guess 𝑥0 .

If 𝑥𝑘 ∈ (𝜁, 𝑏) for some 𝑘 ≥ 0, then which of the following statements is/are
correct?

(A) 𝑥𝑘+1 > 𝜁

(B) 𝑥𝑘+1 < 𝜁

(C) 𝑥𝑘+1 < 𝑥𝑘

(D) 𝑓 ′ (𝜂)
For every 𝜂 ∈ (𝜁, 𝑥𝑘 ), 𝑓′ (𝑥 ) > 1
𝑘

Page 44 of 53

Q.52 Let 𝑓: ℝ2 → ℝ be defined by

2𝑥 2 𝑦
, (𝑥, 𝑦) ≠ (0, 0)
𝑓(𝑥, 𝑦) = { 𝑥 2 + 𝑦 2
0, (𝑥, 𝑦) = (0, 0)

Then

(A) 1 1 1
the directional derivative of 𝑓 at (0,0) in the direction of ( , ) is
√2 √2 √2

(B) the directional derivative of 𝑓 at (0,0) in the direction of (0, 1 ) is 1

(C) the directional derivative of 𝑓 at (0,0) in the direction of (1, 0) is 0

(D) 𝑓 is NOT differentiable at (0,0)

Page 45 of 53

Q.53 Let 𝐶[0,1] = { 𝑓: [0, 1] → ℝ ∶ 𝑓 is continuous} and

𝑑∞ (𝑓, 𝑔) = sup{ |𝑓(𝑥) − 𝑔(𝑥)|: 𝑥 ∈ [0, 1]} for 𝑓, 𝑔 ∈ 𝐶[0,1].

For each 𝑛 ∈ ℕ, define 𝑓𝑛 : [0,1] → ℝ by 𝑓𝑛 (𝑥) = 𝑥 𝑛 for all 𝑥 ∈ [0, 1].

Let 𝑃 = {𝑓𝑛 : 𝑛 ∈ ℕ}.

Which of the following statements is/are correct?

(A) 𝑃 is totally bounded in (𝐶[0, 1], 𝑑∞ )

(B) 𝑃 is bounded in (𝐶[0, 1], 𝑑∞ )

(C) 𝑃 is closed in (𝐶[0, 1], 𝑑∞ )

(D) 𝑃 is open in (𝐶[0, 1], 𝑑∞ )

Page 46 of 53

Q.54 Let G be an abelian group and Φ ∶ 𝐺 → (ℤ, +) be a surjective group
homomorphism. Let 1 = Φ(𝑎) for some 𝑎 ∈ 𝐺.
Consider the following statements:
𝑃: For every 𝑔 ∈ 𝐺, there exists an 𝑛 ∈ ℤ such that 𝑔𝑎𝑛 ∈ ker(Φ).

𝑄: Let 𝑒 be the identity of 𝐺 and < 𝑎 > be the subgroup generated by 𝑎. Then
𝐺 = ker(Φ) < 𝑎 > and ker(Φ) ∩ < 𝑎 > = {𝑒}.

Which of the following statements is/are correct?

(A) 𝑃 is TRUE

(B) 𝑃 is FALSE

(C) 𝑄 is TRUE

(D) 𝑄 is FALSE

Page 47 of 53

Q.55 Let 𝐶 be the curve of intersection of the cylinder 𝑥 2 + 𝑦 2 = 4 and the plane

𝑧 − 2 = 0. Suppose 𝐶 is oriented in the counterclockwise direction around the

𝑧-axis, when viewed from above. If

|∫(sin 𝑥 + 𝑒 𝑥 ) 𝑑𝑥 + 4𝑥 𝑑𝑦 + 𝑒 𝑧 cos2 𝑧 𝑑𝑧| = 𝛼𝜋,
𝐶

then the value of 𝛼 equals ____________.

Let ℓ2 = {(𝑥1 , 𝑥2 , 𝑥3 , … ) ∶ 𝑥𝑛 ∈ ℝ 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ∈ ℕ 𝑎𝑛𝑑 ∑∞ 2
𝑛=1 𝑥𝑛 < ∞}.
Q.56
For a sequence (𝑥1 , 𝑥2 , 𝑥3 , … ) ∈ ℓ2 , define
1
∞ 2
‖(𝑥1 , 𝑥2 , 𝑥3 , … )‖2 = ( ∑ 𝑥𝑛2 )
𝑛=1
𝑥
Consider the subspace 𝑀 = {(𝑥1 , 𝑥2 , 𝑥3 , … ) ∈ ℓ2 ∶ ∑∞ 𝑛
𝑛=1 𝑛 = 0} . 4

Let 𝑀 denote the orthogonal complement of 𝑀 in the Hilbert space (ℓ2 , ‖⋅‖2 ).
⊥

1 1 1
Consider (1, 2 , 3 , 4 , … ) ∈ ℓ2 .
1 1 1
If the orthogonal projection of (1, 2 , 3 , 4 , … ) onto 𝑀⊥ is given by
1 1 1 1
𝛼 (∑∞
𝑛=1 𝑛4𝑛 ) (4 , 42 , 43 , … ) for some 𝛼 ∈ ℝ, then 𝛼 equals _________.

Page 48 of 53

Q.57 Consider the transportation problem between five sources and four destinations as
given in the cost table below. The supply and demand at each of the source and
destination are also provided:

DESTINATIONS Supply
P Q R S
1 13 8 12 9 20

SOURCES
2 10 7 5 20 10
3 3 19 5 12 50
4 4 9 7 15 30
5 14 0 1 7 40
Demand 60 10 20 60

Let 𝐶𝑁 and 𝐶𝐿 be the total cost of the initial basic feasible solution obtained from
the North-West corner method and the Least-Cost method, respectively. Then
𝐶𝑁 − 𝐶𝐿 equals _________.

Q.58 Let 𝜎 ∈ 𝑆8 , where 𝑆8 is the permutation group on 8 elements. Suppose 𝜎 is the
product of 𝜎1 and 𝜎2 , where 𝜎1 is a 4-cycle and 𝜎2 is a 3-cycle in 𝑆8 . If 𝜎1 and 𝜎2
are disjoint cycles, then the number of elements in 𝑆8 which are conjugate to 𝜎 is
_________.

Page 49 of 53

Q.59 Let 𝐴 be a 3 × 3 real matrix with det(𝐴 + 𝑖 𝐼) = 0, where 𝑖 = √−1 and 𝐼 is the
3 × 3 identity matrix. If det(𝐴) = 3, then the trace of 𝐴2 is __________.

Q.60 Let 𝐴 = [𝑎𝑖𝑗 ] be a 3 × 3 real matrix such that

1 1 0 0 −1 −1
𝐴 [2] = 2 [2] , 𝐴 [1] = 2 [1] and 𝐴 [ 1 ] = 4 [ 1 ] .
1 1 1 1 0 0

If 𝑚 is the degree of the minimal polynomial of 𝐴, then 𝑎11 + 𝑎21 + 𝑎31 + 𝑚
equals __________.

Page 50 of 53

Q.61 Let Ω be the disk 𝑥 2 + 𝑦 2 < 4 in ℝ2 with boundary 𝜕Ω. If 𝑢(𝑥, 𝑦) is the solution
of the Dirichlet problem

𝜕 2𝑢 𝜕 2𝑢
+ = 0, (𝑥, 𝑦) ∈ Ω,
𝜕𝑥 2 𝜕𝑦 2

𝑢(𝑥, 𝑦) = 1 + 2 𝑥 2 , (𝑥, 𝑦) ∈ 𝜕Ω,

then the value of 𝑢(0,1) is _____________.

Q.62 For every 𝑘 ∈ ℕ ∪ {0}, let 𝑦𝑘 (𝑥) be a polynomial of degree 𝑘 with 𝑦𝑘 (1) = 5.
Further, let 𝑦𝑘 (𝑥) satisfy the Legendre equation

(1 − 𝑥 2 )𝑦 ′′ − 2𝑥𝑦 ′ + 𝑘(𝑘 + 1)𝑦 = 0.

If
1 𝑛 1 𝑛
1 2 2
∫ ∑(𝑦𝑘 (𝑥) − 𝑦𝑘−1 (𝑥)) 𝑑𝑥 − ∫ ∑(𝑦𝑘 (𝑥)) 𝑑𝑥 = 24,
2
−1 𝑘=1 −1 𝑘=1

for some positive integer 𝑛, then the value of 𝑛 is ____________.

Page 51 of 53

Q.63 Consider the ordinary differential equation (ODE)

4 (ln 𝑥) 𝑦 ′′ + 3 𝑦 ′ + 𝑦 = 0, 𝑥 > 1.
If 𝑟1 and 𝑟2 are the roots of the indicial equation of the above ODE at the regular
singular point 𝑥 = 1, then |𝑟1 − 𝑟2 | is equal to _________ (rounded off to 2 decimal
places).

Q.64 Let 𝑢(𝑥, 𝑡) be the solution of the non-homogeneous wave equation

𝜕 2𝑢 𝜕 2𝑢
− = sin 𝑥 sin(2𝑡), 0 < 𝑥 < 𝜋, 𝑡 > 0
𝜕𝑥 2 𝜕𝑡 2
𝜕𝑢
𝑢(𝑥, 0) = 0, 𝑎𝑛𝑑 (𝑥, 0) = 0, 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 𝜋,
𝜕𝑡
𝑢(0, 𝑡) = 0, 𝑢(𝜋, 𝑡) = 0, 𝑓𝑜𝑟 𝑡 ≥ 0.
𝜋 3𝜋
Then the value of 𝑢 ( 2 , 2 ) is ____________ (rounded off to 2 decimal places).

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Q.65 Consider the Linear Programming Problem P:

Maximize 3𝑥1 + 2𝑥2 + 5𝑥3

subject to

𝑥1 + 2𝑥2 + 𝑥3 ≤ 44,

𝑥1 + 2𝑥3 ≤ 48,

𝑥1 + 4𝑥2 ≤ 52,

𝑥1 ≥ 0, 𝑥2 ≥ 0, 𝑥3 ≥ 0.

The optimal value of the problem P is equal to ____________.

END OF QUESTION PAPER

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