GATE 2025 Question Paper Mathematics (MA)

GATE
2025
Question Paper | Answer Key
Graduate Aptitude Test in Engineering
(GATE) is a prestigious national-level exam
that assesses candidates for
comprehensive understanding in various
undergraduate-level subjects in
Engineering, Technology, Science,
Architecture, and Humanities.

General Aptitude
Q.1 – Q.5 Carry ONE mark Each

Q.1 Ravi had ______ younger brother who taught at ______ university. He was widely
regarded as ______ honorable man.

Select the option with the correct sequence of articles to fill in the blanks.

(A) a; a; an

(B) the; an; a

(C) a; an; a

(D) an; an; a

Organizing Institute: IIT Roorkee Page 1 of 49

Q.2 The CEO’s decision to downsize the workforce was considered myopic because it
sacrificed long-term stability to accommodate short-term gains.

Select the most appropriate option that can replace the word “myopic” without
changing the meaning of the sentence.

(A) visionary

(B) shortsighted

(C) progressive

(D) innovative

Organizing Institute: IIT Roorkee Page 2 of 49

Q.3 The average marks obtained by a class in an examination were calculated as 30.8.
However, while checking the marks entered, the teacher found that the marks of one
student were entered incorrectly as 24 instead of 42. After correcting the marks, the
average becomes 31.4. How many students does the class have?

(A) 25

(B) 28

(C) 30

(D) 32

Organizing Institute: IIT Roorkee Page 3 of 49

Q.4 Consider the relationships among P, Q, R, S, and T:

• P is the brother of Q.
• S is the daughter of Q.
• T is the sister of S.
• R is the mother of Q.

The following statements are made based on the relationships given above.

(1) R is the grandmother of S.

(2) P is the uncle of S and T.

(3) R has only one son.

(4) Q has only one daughter.

Which one of the following options is correct?

(A) Both (1) and (2) are true.

(B) Both (1) and (3) are true.

(C) Only (3) is true.

(D) Only (4) is true.

Organizing Institute: IIT Roorkee Page 4 of 49

Q.5 According to the map shown in the figure, which one of the following statements is
correct?

Note: The figure shown is representative.

1st Main Road
Library Canteen

Physics Lab Hospital

5th Cross Road

Hostels Chemistry
Lab
N
Classrooms
W E

S

(A) The library is located to the northwest of the canteen.

(B) The hospital is located to the east of the chemistry lab.

(C) The chemistry lab is to the southeast of physics lab.

(D) The classrooms and canteen are next to each other.

Organizing Institute: IIT Roorkee Page 5 of 49

Q.6 – Q.10 Carry TWO marks Each

Q.6 “I put the brown paper in my pocket along with the chalks, and possibly other things.
I suppose every one must have reflected how primeval and how poetical are the
things that one carries in one’s pocket: the pocket-knife, for instance the type of all
human tools, the infant of the sword. Once I planned to write a book of poems
entirely about the things in my pocket. But I found it would be too long: and the age
of the great epics is past.”

(From G.K. Chesterton’s “A Piece of Chalk”)

Based only on the information provided in the above passage, which one of the
following statements is true?

(A) The author of the passage carries a mirror in his pocket to reflect upon things.

(B) The author of the passage had decided to write a poem on epics.

(C) The pocket-knife is described as the infant of the sword.

(D) Epics are described as too inconvenient to write.

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Q.7 In the diagram, the lines QR and ST are parallel to each other. The shortest distance
between these two lines is half the shortest distance between the point P and line
QR. What is the ratio of the area of the triangle PST to the area of the trapezium
SQRT?

Note: The figure shown is representative.

P

S T

Q R

(A) 1
3

(B) 1
4

(C) 2
5

(D) 1
2

Organizing Institute: IIT Roorkee Page 7 of 49

Q.8 A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled
thrice. What is the probability of rolling ‘6’ exactly once?

(A) 75
216

(B) 1
6

(C) 1
18

(D) 25
216

Organizing Institute: IIT Roorkee Page 8 of 49

Q.9 A square paper, shown in figure (I), is folded along the dotted lines as shown in the
figures (II) and (III). Then a few cuts are made as shown in figure (IV). Which one
of the following patterns will be obtained when the paper is unfolded?

Note: The figures shown are representative.

(I) (II) (III) (IV)

(A)

(B)

(C)

(D)

Organizing Institute: IIT Roorkee Page 9 of 49

Q.10 A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops
of any flavor. The order in which the scoops are purchased is inconsequential.
If one wants to purchase 3 scoops of ice-cream, in how many ways can one make
that purchase?

(A) 4

(B) 20

(C) 24

(D) 48

Organizing Institute: IIT Roorkee Page 10 of 49

Q.11 – Q.35 Carry ONE mark Each

Q.11 Let
𝑤1 1
𝑆 = {𝑤 = [𝑤2 ] ∈ ℝ3 ∶ [ −3 ] [𝑤1 𝑤2 𝑤3 ] is diagonalizable and ‖𝑤‖ = 1},
𝑤3 2
1
where ‖𝑤‖ = (𝑤12 + 𝑤22 + 𝑤32 )2 . Then, which one of the following is TRUE?

(A) 𝑆 is compact and connected

(B) 𝑆 is neither compact nor connected

(C) 𝑆 is compact but not connected

(D) 𝑆 is connected but not compact

Q.12 Given that the Laplace transforms of 𝐽0 (𝑥), 𝐽0′ (𝑥) and 𝐽0′′ (𝑥) exist, where 𝐽0 (𝑥) is
the Bessel function. Let 𝑌 = 𝑌(𝑠) be the Laplace transform of the Bessel function
𝐽0 (𝑥). Then, which one of the following is TRUE?

(A) 𝑑𝑌 2𝑠𝑌
+ 2 = 0, 𝑠>0
𝑑𝑠 𝑠 +1

(B) 𝑑𝑌 2𝑠𝑌
− 2 = 0, 𝑠>0
𝑑𝑠 𝑠 +1

(C) 𝑑𝑌 𝑠𝑌
− 2 = 0, 𝑠>0
𝑑𝑠 𝑠 +1

(D) 𝑑𝑌 𝑠𝑌
+ 2 = 0, 𝑠>0
𝑑𝑠 𝑠 +1

Organizing Institute: IIT Roorkee Page 11 of 49

Q.13 3
To find a real root of the equation 𝑥 3 + 4𝑥 2 − 10 = 0 in the interval (1, 2 ) by
using the fixed-point iteration scheme, consider the following two statements:

10
S1: The iteration scheme 𝑥𝑘+1 = √4+𝑥 , 𝑘 = 0, 1, 2, …, converges for any initial
𝑘
3
guess 𝑥0 ∈ (1, 2 ).
1
S2: The iteration scheme 𝑥𝑘+1 = √10 − 𝑥𝑘3 , 𝑘 = 0, 1, 2, …, diverges for some
2
3
initial guess 𝑥0 ∈ (1, 2 ).

Then, which one of the following is correct?

(A) S1 is TRUE and S2 is FALSE

(B) S2 is TRUE and S1 is FALSE

(C) both S1 and S2 are TRUE

(D) neither S1 nor S2 is TRUE

Organizing Institute: IIT Roorkee Page 12 of 49

Q.14 For the linear programming problem:

Maximize 𝑍 = 2𝑥1 + 4𝑥2 + 4𝑥3 − 3𝑥4

Subject to 𝛼𝑥1 + 𝑥2 + 𝑥3 = 4,
𝑥1 + 𝛽𝑥2 + 𝑥4 = 8,
𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ≥ 0,

consider the following two statements:

S1: If 𝛼 = 2 and 𝛽 = 1, then (𝑥1 , 𝑥2 )𝑇 forms an optimal basis.

S2: If 𝛼 = 1 and 𝛽 = 4, then (𝑥3 , 𝑥2 )𝑇 forms an optimal basis.

Then, which one of the following is correct?

(A) S1 is TRUE and S2 is FALSE

(B) S2 is TRUE and S1 is FALSE

(C) both S1 and S2 are TRUE

(D) neither S1 nor S2 is TRUE

Organizing Institute: IIT Roorkee Page 13 of 49

Q.15 Consider the following subsets of the Euclidean space ℝ4 :

𝑆 = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∈ ℝ4 ∶ 𝑥12 + 𝑥22 + 𝑥32 − 𝑥42 = 0 },

𝑇 = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∈ ℝ4 ∶ 𝑥12 + 𝑥22 + 𝑥32 − 𝑥42 = 1 }, and

𝑈 = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∈ ℝ4 ∶ 𝑥12 + 𝑥22 + 𝑥32 − 𝑥42 = −1 }.

Then, which one of the following is TRUE?

(A) 𝑆 is connected, but 𝑇 and 𝑈 are not connected

(B) 𝑇 and 𝑈 are connected, but 𝑆 is not connected

(C) 𝑆 and 𝑈 are connected, but 𝑇 is not connected

(D) 𝑆 and 𝑇 are connected, but 𝑈 is not connected

Consider the system of ordinary differential equations
Q.16
𝑑𝑋
= 𝑀𝑋,
𝑑𝑡

where 𝑀 is a 6 × 6 skew-symmetric matrix with entries in ℝ. Then, for this system,
the origin is a stable critical point for

(A) any such matrix 𝑀

(B) only such matrices 𝑀 whose rank is 2

(C) only such matrices 𝑀 whose rank is 4

(D) only such matrices 𝑀 whose rank is 6

Organizing Institute: IIT Roorkee Page 14 of 49

Q.17 Let 𝑋 = { 𝑓 ∈ 𝐶[0, 1] ∶ 𝑓(0) = 0 = 𝑓(1)} with the norm ‖𝑓‖∞ = sup |𝑓(𝑡)|,
0≤𝑡≤1
where 𝐶[0, 1] is the space of all real-valued continuous functions on [0, 1].
1
1 2
Let 𝑌 = 𝐶[0, 1] with the norm ‖𝑓‖2 = (∫0 |𝑓(𝑡)|2 𝑑𝑡) . Let 𝑈𝑋 and 𝑈𝑌 be the
closed unit balls in 𝑋 and 𝑌 centred at the origin, respectively. Consider 𝑇: 𝑋 → ℝ
and 𝑆: 𝑌 → ℝ given by
1 1
𝑇𝑓 = ∫0 𝑓(𝑡) 𝑑𝑡 and 𝑆𝑓 = ∫0 𝑓(𝑡) 𝑑𝑡.

Consider the following statements:

S1: sup |𝑇𝑓| is attained at a point of 𝑈𝑋 .
𝑓∈𝑈𝑋

S2: sup |𝑆𝑓| is attained at a point of 𝑈𝑌 .
𝑓∈𝑈𝑌

Then, which one of the following is correct?

(A) S1 is TRUE and S2 is FALSE

(B) S2 is TRUE and S1 is FALSE

(C) both S1 and S2 are TRUE

(D) neither S1 nor S2 is TRUE

Organizing Institute: IIT Roorkee Page 15 of 49

Q.18 Let 𝑔(𝑥, 𝑦) = 𝑓(𝑥, 𝑦)𝑒 2𝑥+3𝑦 be defined in ℝ2 , where 𝑓(𝑥, 𝑦) is a continuously
differentiable non-zero homogeneous function of degree 4. Then,
𝜕𝑔 𝜕𝑔
𝑥 𝜕𝑥 + 𝑦 𝜕𝑦 = 0 holds for

(A) all points (𝑥, 𝑦) in ℝ2

(B) all points (𝑥, 𝑦) on the line given by 2𝑥 + 3𝑦 + 4 = 0

(C) all points (𝑥, 𝑦) in the region of ℝ2 except on the line given by 2𝑥 + 3𝑦 + 4 = 0

(D) all points (𝑥, 𝑦) on the line given by 2𝑥 + 3𝑦 = 0

Q.19 The partial differential equation

𝜕2 𝑢 𝜕2 𝑢 𝜕2 𝑢 𝜕𝑢 𝜕𝑢
(1 + 𝑥 2 ) 2
+ 2𝑥(1 − 𝑦 2 ) + (1 − 𝑦 2 ) +𝑥 + (1 − 𝑦 2 ) =0
𝜕𝑥 𝜕𝑦𝜕𝑥 𝜕𝑦2 𝜕𝑥 𝜕𝑦

is

(A) elliptic in the region {(𝑥, 𝑦) ∈ ℝ2 : |𝑦| ≤ 1}

(B) hyperbolic in the region {(𝑥, 𝑦) ∈ ℝ2 : |𝑦| > 1}

(C) elliptic in the region {(𝑥, 𝑦) ∈ ℝ2 : |𝑦| > 1}

(D) hyperbolic in the region {(𝑥, 𝑦) ∈ ℝ2 : |𝑦| < 1}

Organizing Institute: IIT Roorkee Page 16 of 49

Q.20 Let 𝑢(𝑥, 𝑡) be the solution of the following initial-boundary value problem

𝜕𝑢 𝜕2 𝑢
− = 0, 𝑥 ∈ (0, 𝜋), 𝑡 > 0,
𝜕𝑡 𝜕𝑥2

𝑢(0, 𝑡) = 𝑢(𝜋, 𝑡) = 0, 𝑢(𝑥, 0) = sin 4𝑥 cos 3𝑥 .
𝜋
Then, for each 𝑡 > 0, the value of 𝑢 ( 4 , 𝑡) is

(A) 𝑒 −49𝑡
(𝑒 48𝑡 − 1)
2√2

(B) 𝑒 −49𝑡
(1 − 𝑒 48𝑡 )
2√2

(C) 𝑒 −49𝑡
(1 + 𝑒 48𝑡 )
2√2

(D) 𝑒 −49𝑡
(1 − 𝑒 48𝑡 )
4√2

Organizing Institute: IIT Roorkee Page 17 of 49

Q.21 Consider the function 𝐹: ℝ2 ⟶ ℝ2 given by

𝐹(𝑥, 𝑦) = (𝑥 3 − 3𝑥𝑦 2 − 3𝑥 , 3𝑥 2 𝑦 − 𝑦 3 − 3𝑦).

Then, for the function 𝐹, the inverse function theorem is

(A) applicable at all points of ℝ2

(B) not applicable at exactly one point of ℝ2

(C) not applicable at exactly two points of ℝ2

(D) not applicable at exactly three points of ℝ2

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Q.22 Let the functions 𝑓: ℝ2 → ℝ and 𝑔: ℝ2 → ℝ be given by

𝑓(𝑥1 , 𝑥2 ) = 𝑥12 + 𝑥22 − 2𝑥1 𝑥2 , and
𝑔(𝑥1 , 𝑥2 ) = 2𝑥12 + 2𝑥22 − 𝑥1 𝑥2 .

Consider the following statements:

S1: For every compact subset 𝐾 of ℝ, 𝑓 −1 (𝐾) is compact.

S2: For every compact subset 𝐾 of ℝ, 𝑔−1 (𝐾) is compact.

Then, which one of the following is correct?

(A) S1 is TRUE and S2 is FALSE

(B) S2 is TRUE and S1 is FALSE

(C) both S1 and S2 are TRUE

(D) neither S1 nor S2 is TRUE

Organizing Institute: IIT Roorkee Page 19 of 49

Q.23 Let 𝑝𝐴 (𝑥) denote the characteristic polynomial of a square matrix 𝐴. Then, for
which of the following invertible matrices 𝑀, the polynomial 𝑝𝑀 (𝑥) − 𝑝𝑀−1 (𝑥) is
constant?

(A) 5 7
𝑀=[ ]
2 3

(B) 3 1
𝑀=[ ]
4 2

(C) 1 3
𝑀=[ ]
2 −1

(D) 5 −8
𝑀=[ ]
2 −3

Organizing Institute: IIT Roorkee Page 20 of 49

Q.24 Consider the balanced transportation problem with three sources 𝑆1 , 𝑆2 , 𝑆3 , and four
destinations 𝐷1 , 𝐷2 , 𝐷3 , 𝐷4 , for minimizing the total transportation cost whose cost
matrix is as follows:

𝐷1 𝐷2 𝐷3 𝐷4 Supply

𝑆1 2 6 20 11 𝛼 + 10

𝑆2 12 7 4 10 𝛼 + 𝜆 + 10

𝑆3 8 14 16 11 5

Demand 𝛼+5 10 𝜆+5 𝛼+𝜆

where 𝛼, 𝜆 > 0. If the associated cost to the starting basic feasible solution obtained
by using the North-West corner rule is 290, then which of the following is/are
correct?

(A) 𝛼 2 + 𝜆2 = 100

(B) 𝛼 2 + 𝛼𝜆 = 150

(C) The optimal cost of the transportation problem is 260

(D) The optimal cost of the transportation problem is 290

Organizing Institute: IIT Roorkee Page 21 of 49

Q.25 Consider the following regions:

𝑆1 = {(𝑥1 , 𝑥2 ) ∈ ℝ2 ∶ 2𝑥1 + 𝑥2 ≤ 4, 𝑥1 + 2𝑥2 ≤ 5, 𝑥1 , 𝑥2 ≥ 0},

𝑆2 = {(𝑥1 , 𝑥2 ) ∈ ℝ2 ∶ 2𝑥1 − 𝑥2 ≤ 5, 𝑥1 + 2𝑥2 ≤ 5, 𝑥1 , 𝑥2 ≥ 0}.

Then, which of the following is/are TRUE?

(A) The maximum value of 𝑥1 + 𝑥2 is 3 on the region 𝑆2

(B) The maximum value of 𝑥1 + 𝑥2 is 5 on the region 𝑆2 − 𝑆1

(C) The maximum value of 𝑥1 + 𝑥2 is 3 on the region 𝑆1 ∩ 𝑆2

(D) The maximum value of 𝑥1 + 𝑥2 is 4 on the region 𝑆1 ∪ 𝑆2

Organizing Institute: IIT Roorkee Page 22 of 49

Q.26 Let 𝑓: ℝ2 − {(0,0)} ⟶ ℝ be a function defined by

𝑥2 − 𝑦2 1
𝑓(𝑥, 𝑦) = ( 2 2
) + 𝑥 sin ( 2 ).
𝑥 +𝑦 𝑥 + 𝑦2

Consider the following three statements:

S1: lim lim 𝑓(𝑥, 𝑦) exists.
𝑥→0 𝑦→0

S2: lim lim 𝑓(𝑥, 𝑦) exists.
𝑦→0 𝑥→0

S3: lim 𝑓(𝑥, 𝑦) exists.
(𝑥,𝑦)→(0,0)

Then, which of the following is/are correct?

(A) S2 and S3 are TRUE and S1 is FALSE

(B) S1 and S2 are TRUE and S3 is FALSE

(C) S1 and S3 are TRUE and S2 is FALSE

(D) S1, S2 and S3 all are TRUE

Q.27 Let 𝑀 be a 7 × 7 matrix with entries in ℝ and having the characteristic polynomial

𝑐𝑀 (𝑥) = (𝑥 − 1)𝛼 (𝑥 − 2)𝛽 (𝑥 − 3)2 , where 𝛼 > 𝛽.

Let rank(𝑀 − 𝐼7 ) = rank(𝑀 − 2𝐼7 ) = rank(𝑀 − 3𝐼7 ) = 5, where 𝐼7 is the 7 × 7
identity matrix. If 𝑚𝑀 (𝑥) is the minimal polynomial of 𝑀, then 𝑚𝑀 (5) is equal to
_____ (in integer)

Organizing Institute: IIT Roorkee Page 23 of 49

Q.28 Let 𝑦 = 𝑃𝑛 (𝑥) be the unique polynomial of degree 𝑛 satisfying the Legendre
differential equation

(1 − 𝑥 2 )𝑦 ′′ − 2𝑥𝑦 ′ + 𝑛(𝑛 + 1)𝑦 = 0 and 𝑦(1) = 1.
′
Then, the value of 𝑃11 (1) is equal to _____ (in integer)

Q.29 ̂ be a unit vector parallel to the tangent at the point 𝑃(1, 1, √2 ) to the curve
Let 𝒂
of intersection of the surfaces 2𝑥 2 + 3𝑦 2 − 𝑧 2 = 3 and 𝑥 2 + 𝑦 2 = 𝑧 2 .
Then, the absolute value of the directional derivative of

𝑓(𝑥, 𝑦, 𝑧) = 𝑥 2 + 2𝑦 2 − 2√11 𝑧

̂ is _____ (in integer)
at 𝑃 in the direction of 𝒂

Q.30 The volume of the region bounded by the cylinders 𝑥 2 + 𝑦 2 = 4 and 𝑥 2 + 𝑧 2 = 4
is _____ (rounded off to TWO decimal places)

Q.31 Let 𝑊 be the vector space (over ℝ) consisting of all bounded real-valued solutions
of the differential equation

𝑑4 𝑦 𝑑2𝑦
+2 2+𝑦 =0.
𝑑𝑥 4 𝑑𝑥
Then, the dimension of 𝑊 is _____ (in integer)

Organizing Institute: IIT Roorkee Page 24 of 49

Q.32 ̂ be a vector field, and let 𝑆 be the surface
Let 𝐹⃗ = (𝑦 − 𝑧)𝒊̂ + (𝑧 − 𝑥)𝒋̂ + (𝑥 − 𝑦)𝒌
2 2 2
𝑥 + 𝑦 + (𝑧 − 1) = 9, 1 ≤ 𝑧 ≤ 4. If 𝒏 ̂ denotes the unit outward normal vector
to 𝑆, then the value of

1
⃗⃗ × 𝐹⃗ ) ⋅ 𝒏
|∬(∇ ̂ 𝑑𝑆|
𝜋
𝑆

is equal to _____ (in integer)

Q.33 Consider
1 sin 𝑧
𝐼= ∮ 𝑑𝑧,
2𝜋𝑖 𝐶 1 − cos(𝑧 3 )

where 𝐶 = {𝑧 ∈ ℂ ∶ 𝑧 = 𝑥 + 𝑖𝑦, |𝑥| + |𝑦| = 1, 𝑥, 𝑦 ∈ ℝ} is oriented positively
as a simple closed curve. Then, the value of 120𝐼 is equal to _____ (in integer)

Q.34 Let 𝛼, 𝛽, 𝛾, 𝛿 ∈ ℝ be such that the quadrature formula
1
∫ 𝑓(𝑥) 𝑑𝑥 = 𝛼𝑓(−1) + 𝛽𝑓(1) + 𝛾𝑓 ′ (−1) + 𝛿𝑓′(1)
−1

is exact for all polynomials of degree less than or equal to 3.
Then, 9(𝛼 2 + 𝛽 2 + 𝛾 2 + 𝛿 2 ) is equal to _____ (in integer)

Organizing Institute: IIT Roorkee Page 25 of 49

Q.35 Let 𝑦(𝑥) be the solution of the initial value problem

𝑑𝑦
= sin(𝜋(𝑥 + 𝑦)), 𝑦(0) = 0.
𝑑𝑥
Using Euler’s method, with the step-size ℎ = 0.5, the approximate value of
𝑦(1.5) + 2𝑦(1) is equal to _____ (in integer)

Organizing Institute: IIT Roorkee Page 26 of 49

Q.36 – Q.65 Carry TWO marks Each

Q.36 Consider the linear system 𝐴𝒙 = 𝒃, where 𝐴 = [𝑎𝑖𝑗 ], 𝑖, 𝑗 = 1, 2, 3, and 𝑎𝑖𝑖 ≠ 0 for
𝑎11 0 0
𝑖 = 1, 2, 3, is a matrix with entries in ℝ. For 𝐷 = [ 0 𝑎22 0 ], let
0 0 𝑎33

1 1 −2 4
𝐷−1 𝐴 = [ 3 1 2] and 𝐷 −1 𝒃 = [ 4 ].
1 1 1 1
Consider the following two statements:

S1: The approximation of 𝒙 after one iteration of the Jacobi scheme with initial
vector 𝒙𝟎 = [1 1 1]𝑇 is 𝒙𝟏 = [5 − 1 − 1]𝑇 .

S2: There exists an initial vector 𝒙𝟎 for which Jacobi iterative scheme diverges.

Then, which one of the following is correct?

(A) S1 is TRUE and S2 is FALSE

(B) S2 is TRUE and S1 is FALSE

(C) both S1 and S2 are TRUE

(D) neither S1 nor S2 is TRUE

Organizing Institute: IIT Roorkee Page 27 of 49

Q.37 Let 𝑦(𝑥) be the solution of the differential equation

𝑥 2 𝑦 ′′ + 7𝑥𝑦 ′ + 9𝑦 = 𝑥 −3 log 𝑒 𝑥, 𝑥 > 0,

satisfying 𝑦(1) = 0 and 𝑦 ′ (1) = 0. Then, the value of 𝑦(𝑒) is equal to

(A)
1 −3
𝑒
3

(B)
1 −3
𝑒
6

(C)
2 −3
𝑒
3

(D)
1 −3
𝑒
2

Organizing Institute: IIT Roorkee Page 28 of 49

Q.38 Let 𝑦1 (𝑥) and 𝑦2 (𝑥) be the two linearly independent solutions of the differential
equation
(1 + 𝑥 2 )𝑦 ′′ − 𝑥𝑦 ′ + (cos2 𝑥)𝑦 = 0,

satisfying the initial conditions

𝑦1 (0) = 3, 𝑦1′ (0) = −1 and 𝑦2 (0) = −5, 𝑦2′ (0) = 2.

𝑦1 (𝑥) 𝑦2 (𝑥) 1
Define 𝑊(𝑥) = | |. Then, the value of 𝑊 (2) is
𝑦1′ (𝑥) 𝑦2′ (𝑥)

(A) √5
4

(B) √5
2

(C) 2
√5

(D) 4
√5

Organizing Institute: IIT Roorkee Page 29 of 49

Q.39 Let 𝐶 be the curve of intersection of the surfaces 𝑧 2 = 𝑥 2 + 𝑦 2 and
4𝑥 + 𝑧 = 7. If 𝑃 is a point on 𝐶 at a minimum distance from the 𝑥𝑦-plane, then the
distance of 𝑃 from the origin is

(A) 7
5

(B) 7√2
5

(C) 14
5

(D) 14√2
5

Organizing Institute: IIT Roorkee Page 30 of 49

Q.40 Let 𝑢(𝑥, 𝑡) be the solution of the initial-value problem

𝜕2 𝑢 𝜕2 𝑢 𝜕𝑢
2
−9 = 0, 𝑥 ∈ ℝ, 𝑡 > 0, 𝑢(𝑥, 0) = 𝑒 𝑥 , (𝑥, 0) = sin 𝑥.
𝜕𝑡 𝜕𝑥2 𝜕𝑡
𝜋 𝜋
Then, the value of 𝑢 ( 2 , 6 ) is

(A) 1 𝜋 1
(𝑒 − )
2 3

(B) 1 𝜋 1
(𝑒 + )
2 3

(C) 1 𝜋 5
(𝑒 + )
2 3

(D) 1 𝜋 5
(𝑒 − )
2 3

Organizing Institute: IIT Roorkee Page 31 of 49

Q.41 1
Let 𝑇 be the Mö bius transformation that maps the points 0, 2 and 1 conformally
onto the points −3, ∞ and 2, respectively, in the extended complex plane.
If 𝑇 maps the circle centred at 1 with radius 𝑘 onto a straight line given by the
2𝑘(𝛼+𝛽)+𝛾
equation 𝛼𝑥 + 𝛽𝑦 + 𝛾 = 0, then the value of is equal to
𝛼+𝛽−2𝑘𝛾

(A) 1
7

(B) 2
7

(C) 1
3

(D) 2
3

Organizing Institute: IIT Roorkee Page 32 of 49

Q.42 Let 𝑈 = {𝑧 ∈ ℂ ∶ Im(𝑧) > 0} and 𝐷 = {𝑧 ∈ ℂ ∶ |𝑧| < 1}, where Im(𝑧) denotes
the imaginary part of 𝑧. Let 𝑆 be the set of all bijective analytic functions
𝑓: 𝑈 → 𝐷 such that 𝑓(𝑖) = 0. Then, the value of sup |𝑓(4𝑖)| is
𝑓∈𝑆

(A) 0

(B) 1
4

(C) 1
2

(D) 3
5

Organizing Institute: IIT Roorkee Page 33 of 49

Q.43 Let Ω be a non-empty open connected subset of ℂ and 𝑓: Ω → ℂ be a non-constant
function. Let the functions 𝑓 2 : Ω → ℂ and 𝑓 3 : Ω → ℂ be defined by
2 3
𝑓 2 (𝑧) = (𝑓(𝑧)) and 𝑓 3 (𝑧) = (𝑓(𝑧)) , 𝑧 ∈ Ω.

Consider the following two statements:

S1: If 𝑓 is continuous in Ω and 𝑓 2 is analytic in Ω, then 𝑓 is analytic in Ω.

S2: If 𝑓 2 and 𝑓 3 are analytic in Ω, then 𝑓 is analytic in Ω.

Then, which one of the following is correct?

(A) S1 is TRUE and S2 is FALSE

(B) S2 is TRUE and S1 is FALSE

(C) both S1 and S2 are TRUE

(D) neither S1 nor S2 is TRUE

Organizing Institute: IIT Roorkee Page 34 of 49

Q.44 In the following, all subsets of Euclidean spaces are considered with the respective
subspace topologies.
Define an equivalence relation ~ on the sphere

𝑆 = {(𝑥1 , 𝑥2 , 𝑥3 ) ∈ ℝ3 ∶ 𝑥12 + 𝑥22 + 𝑥32 = 1}

by (𝑥1 , 𝑥2 , 𝑥3 ) ~ (𝑦1 , 𝑦2 , 𝑦3 ) if 𝑥3 = 𝑦3 , for (𝑥1 , 𝑥2 , 𝑥3 ), (𝑦1 , 𝑦2 , 𝑦3 ) ∈ 𝑆.
Let [𝑥1 , 𝑥2 , 𝑥3 ] denote the equivalence class of (𝑥1 , 𝑥2 , 𝑥3 ), and let 𝑋 denote the set
of all such equivalence classes. Let ℒ ∶ 𝑆 → 𝑋 be given by
ℒ((𝑥1 , 𝑥2 , 𝑥3 )) = [𝑥1 , 𝑥2 , 𝑥3 ]. If 𝑋 is provided with the quotient topology induced
by the map ℒ, then which one of the following is TRUE?

(A) 𝑋 is homeomorphic to {𝑥 ∈ ℝ ∶ −1 ≤ 𝑥 ≤ 1}

(B) 𝑋 is homeomorphic to {(𝑥1 , 𝑥2 ) ∈ ℝ2 ∶ 𝑥12 + 𝑥22 = 1}

(C) 𝑋 is homeomorphic to {(𝑥1 , 𝑥2 ) ∈ ℝ2 ∶ 𝑥12 + 𝑥22 ≤ 1}

(D) 𝑋 is homeomorphic to {(𝑥1 , 𝑥2 , 𝑥3 ) ∈ ℝ3 ∶ 𝑥12 + 𝑥22 = 1 and − 1 ≤ 𝑥3 ≤ 1}

Organizing Institute: IIT Roorkee Page 35 of 49

Q.45 Consider the following two spaces:

𝑋 = (𝐶[−1, 1], ∥⋅∥∞ ), the space of all real-valued continuous functions defined on
[−1, 1] equipped with the norm ∥ 𝑓 ∥∞ = sup |𝑓(𝑡)|.
𝑡∈[−1,1]

𝑌 = (𝐶[−1, 1], ∥⋅∥2 ), the space of all real-valued continuous functions defined on
1 1/2
[−1, 1] equipped with the norm ∥ 𝑓 ∥2 = (∫−1|𝑓(𝑡)|2 𝑑𝑡) .

Let 𝑊 be the linear span over ℝ of all the Legendre polynomials. Then, which one
of the following is correct?

(A) 𝑊 is dense in 𝑋 but not in 𝑌

(B) 𝑊 is dense in 𝑌 but not in 𝑋

(C) 𝑊 is dense in both 𝑋 and 𝑌

(D) 𝑊 is dense neither in 𝑋 nor in 𝑌

Organizing Institute: IIT Roorkee Page 36 of 49

Q.46 Consider the metric spaces 𝑋 = (ℝ, 𝑑1 ) and 𝑌 = ([0, 1], 𝑑2 ) with the metrics
defined by 𝑑1 (𝑥, 𝑦) = |𝑥 − 𝑦|, 𝑥, 𝑦 ∈ ℝ and 𝑑2 (𝑥, 𝑦) = |𝑥 − 𝑦|, 𝑥, 𝑦 ∈ [0,1],
respectively. Then, which one of the following is TRUE?

(A) 1
[0, 4) is open in 𝑋 but not in 𝑌

(B) 1
[0, 4) is open in 𝑌 but not in 𝑋

(C) 1
[0, 4) is open in both 𝑋 and 𝑌

(D) 1
[0, 4) is open neither in 𝑋 nor in 𝑌

Q.47 Let 𝐾 be an algebraically closed field containing a finite field 𝐹. Let 𝐿 be the
subfield of 𝐾 consisting of elements of 𝐾 that are algebraic over 𝐹.

Consider the following statements:

S1: 𝐿 is algebraically closed.

S2: 𝐿 is infinite.

Then, which one of the following is correct?

(A) S1 is TRUE and S2 is FALSE

(B) S2 is TRUE and S1 is FALSE

(C) both S1 and S2 are TRUE

(D) neither S1 nor S2 is TRUE

Organizing Institute: IIT Roorkee Page 37 of 49

Q.48 Let 𝑀2 (ℝ) be the vector space (over ℝ) of all 2 × 2 matrices with entries in ℝ.
Consider the linear transformation 𝑇: 𝑀2 (ℝ) → 𝑀2 (ℝ) defined by 𝑇(𝑋) = 𝐴𝑋𝐵,
1 −2 6 5
where 𝐴 = [ ] and 𝐵 = [ ]. If 𝑃 is the matrix representation of
1 4 −2 −1
𝑇 with respect to the standard basis of 𝑀2 (ℝ), then which of the following is/are
TRUE?

(A) 𝑃 is an invertible matrix

(B) The trace of 𝑃 is 25

(C) The rank of (𝑃2 − 4𝐼4 ) is 4, where 𝐼4 is the 4 × 4 identity matrix

(D) The nullity of (𝑃 − 2𝐼4 ) is 0, where 𝐼4 is the 4 × 4 identity matrix

Organizing Institute: IIT Roorkee Page 38 of 49

Q.49 Consider the linear programming problem (LPP):

Maximize 𝑍 = 3𝑥1 + 5𝑥2

Subject to 𝑥1 + 𝑥3 = 4 ,
2𝑥2 + 𝑥4 = 12,
3𝑥1 + 2𝑥2 + 𝑥5 = 18,
𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 ≥ 0.

Given that 𝑥𝐵 = (𝑥3 , 𝑥2 , 𝑥1 )𝑇 forms the optimal basis of the LPP with basis matrix
𝛼 𝛽 −𝛽
−1
𝐵 and respective 𝐵 = [ 0 𝛾 0 ] . If ( 𝑝, 𝑞, 𝑟) is the optimal solution of the
0 −𝛽 𝛽
dual of the LPP, then which of the following is/are TRUE?

(A) 𝛼 + 3𝛽 + 2𝛾 = 3

(B) 𝛼 − 3𝛽 + 4𝛾 = 1

(C) 5
𝑝+𝑞+𝑟 =2

(D) 17
𝑝2 + 𝑞 2 + 𝑟 2 = 4

Organizing Institute: IIT Roorkee Page 39 of 49

Q.50 Let 0 < 𝛼 < 1. Define

|𝑓(𝑡) − 𝑓(𝑠)|
𝐶 𝛼 [0, 1] = {𝑓: [0, 1] → ℝ ∶ sup < ∞}.
𝑠≠𝑡 |𝑡 − 𝑠|𝛼
𝑠,𝑡∈[0,1]

It is given that 𝐶 𝛼 [0, 1] is a Banach space with respect to the norm ‖⋅‖𝛼 given by

|𝑓(𝑡) − 𝑓(𝑠)|
‖𝑓‖𝛼 = |𝑓(0)| + sup .
𝑠≠𝑡 |𝑡 − 𝑠|𝛼
𝑠,𝑡∈[0,1]

Let 𝐶[0, 1] be the space of all real-valued continuous functions on [0, 1] with the
norm ‖𝑓‖∞ = sup |𝑓(𝑡)|. If 𝑇: 𝐶 𝛼 [0, 1] → 𝐶[0,1] is the map 𝑇𝑓 = 𝑓,
0≤𝑡≤1
𝑓 ∈ 𝐶 𝛼 [0, 1], then which of the following is/are TRUE?

(A) 𝑇 is a compact linear map

(B) Image of 𝑇 is closed in 𝐶[0, 1]

(C) Image of 𝑇 is dense in 𝐶[0, 1]

(D) 𝑇 is not a bounded linear map

Organizing Institute: IIT Roorkee Page 40 of 49

Q.51 Let 𝑢(𝑥, 𝑡) be the solution of the initial value problem

𝜕𝑢 𝜕𝑢
+3 = 𝑢, 𝑥 ∈ ℝ, 𝑡 > 0, 𝑢(𝑥, 0) = cos 𝑥,
𝜕𝑡 𝜕𝑥

and let 𝑣(𝑥, 𝑡) be the solution of the initial value problem

𝜕𝑣 𝜕𝑣
+3 = 𝑣 2 , 𝑥 ∈ ℝ, 𝑡 > 0, 𝑣(𝑥, 0) = cos 𝑥.
𝜕𝑡 𝜕𝑥

Then, which of the following is/are TRUE?

(A) |𝑢(𝑥, 𝑡)| ≤ 𝑒 𝑡 for all 𝑥 ∈ ℝ and for all 𝑡 > 0

(B) 𝑣(𝑥, 1) is not defined for certain values of 𝑥 ∈ ℝ

(C) 𝑣(𝑥, 1) is not defined for any 𝑥 ∈ ℝ

(D) 𝑢(2𝜋, 𝜋) = −𝑒 𝜋

Organizing Institute: IIT Roorkee Page 41 of 49

Q.52 Let 𝑢(𝑥, 𝑡) be the solution of the initial-boundary value problem

𝜕𝑢 𝜕2 𝑢
=2 , 0 < 𝑥 < 1, 𝑡 > 0,
𝜕𝑡 𝜕𝑥2

𝑢(0, 𝑡) = 𝑢(1, 𝑡) = 0, 𝑢(𝑥, 0) = 2𝑥(1 − 𝑥).

Then, which of the following is/are TRUE?

(A) 1
0 ≤ 𝑢(𝑥, 𝑡) ≤ 4 for all 𝑡 ≥ 0 and 𝑥 ∈ [0, 1]

(B) 𝑢(𝑥, 𝑡) = 𝑢(1 − 𝑥, 𝑡) for all 𝑡 ≥ 0 and 𝑥 ∈ [0, 1]

(C) 1 2
∫0 (𝑢(𝑥, 𝑡)) 𝑑𝑥 is a decreasing function of 𝑡

(D) 1 2
∫0 (𝑢(𝑥, 𝑡)) 𝑑𝑥 is not a decreasing function of 𝑡

Q.53 Consider the function 𝑓: ℝ2 → ℝ2 given by

𝑓(𝑥, 𝑦) = (𝑒 2𝜋𝑥 cos 2𝜋𝑦, 𝑒 2𝜋𝑥 sin 2𝜋𝑦).

Then, which of the following is/are TRUE?

(A) If 𝐺 is open in ℝ2 , then 𝑓(𝐺) is open in ℝ2

(B) If 𝐺 is closed in ℝ2 , then 𝑓(𝐺) is closed in ℝ2

(C) If 𝐺 is dense in ℝ2 , then 𝑓(𝐺) is dense in ℝ2

(D) 𝑓 is surjective

Organizing Institute: IIT Roorkee Page 42 of 49

Q.54 Let {𝑥𝑘 }∞
𝑘=1 be an orthonormal set of vectors in a real Hilbert space 𝑋 with inner
product <⋅ , ⋅>. Let 𝑛 ∈ ℕ, and let 𝑌 be the linear span of {𝑥𝑘 }𝑛𝑘=1 over ℝ.
For 𝑥 ∈ 𝑋, let 𝑆𝑛 (𝑥) = ∑𝑛𝑘=1 < 𝑥, 𝑥𝑘 > 𝑥𝑘 . Then, which of the following is/are
TRUE?

(A) 𝑆𝑛 (𝑥) is the orthogonal projection of 𝑥 onto 𝑌

(B) 𝑆𝑛 (𝑥) is the orthogonal projection of 𝑥 onto 𝑌 ⊥

(C) (𝑥 − 𝑆𝑛 (𝑥)) is orthogonal to 𝑆𝑛 (𝑥) for all 𝑥 in 𝑋

(D) ∑𝑛𝑘=1|< 𝑥, 𝑥𝑘 >|2 = ‖𝑥‖2 for all 𝑥 in 𝑋

Q.55 Consider the sequence { 𝑓𝑛 } of continuous functions on [0, 1] defined by

𝑥 1 2
𝑓1 (𝑥) = , 𝑓𝑛+1 (𝑥) = 𝑓𝑛 (𝑥) − ( (𝑓𝑛 (𝑥)) − 𝑥) , 𝑛 = 1, 2, 3, … .
2 2
Then, which of the following is/are TRUE?

(A) The sequence { 𝑓𝑛 } converges pointwise but not uniformly on [0, 1]

(B) The sequence { 𝑓𝑛 } converges uniformly on [0, 1]

(C) 2√𝑥
√𝑥 − 𝑓𝑛 (𝑥) > 2+𝑛√𝑥 for all 𝑥 ∈ [0, 1] and 𝑛 = 1, 2, 3, …

(D) 0 ≤ 𝑓𝑛 (𝑥) ≤ √𝑥 for all 𝑥 ∈ [0, 1] and 𝑛 = 1, 2, 3, …

Organizing Institute: IIT Roorkee Page 43 of 49

Q.56 sin 𝑛𝑥
For 𝑥 ∈ (0, 𝜋), let 𝑢𝑛 (𝑥) = , 𝑛 = 1, 2, 3, … . Then, which of the
√𝑛
following is/are TRUE?

(A) ∑∞
𝑛=1 𝑢𝑛 (𝑥) converges uniformly on (0, 𝜋)

(B) ∑∞ ′
𝑛=1 𝑢𝑛 (𝑥) converges uniformly on (0, 𝜋)

(C) ∑∞
𝑛=1 𝑢𝑛 (𝑥) converges pointwise but not uniformly on (0, 𝜋)

(D) ∑∞
𝑛=1 𝑢𝑛 (𝑥) converges uniformly on every compact subset of (0, 𝜋)

Q.57 Let ℝ1 and ℝ2 be provided with the respective Euclidean topologies, and let

𝑆 1 = {(𝑥1 , 𝑥2 ) ∈ ℝ2 ∶ 𝑥12 + 𝑥22 = 1}

be assigned the subspace topology induced from ℝ2 . If 𝑓: 𝑆 1 → ℝ1 is a non-constant
continuous function, then which of the following is/are TRUE?

(A) 𝑓 maps closed sets to closed sets

(B) 𝑓 is injective

(C) 𝑓 is surjective

(D) There exists 𝜆 ∈ ℝ such that 𝑓(cos 𝜆, sin 𝜆) = 𝑓(− cos 𝜆, − sin 𝜆)

Organizing Institute: IIT Roorkee Page 44 of 49

Q.58 Let 𝑋 be an uncountable set. Let the topology on 𝑋 be defined by declaring a subset
𝑈 ⊆ 𝑋 to be open if 𝑋 − 𝑈 is either empty or finite or countable, and the empty set
to be open. Then, which of the following is/are TRUE?

(A) Every compact subset of 𝑋 is closed

(B) Every closed subset of 𝑋 is compact

(C) 𝑋 is 𝑇1 (singleton subsets are closed) but not 𝑇2 (Hausdorff)

(D) 𝑋 is 𝑇2 (Hausdorff)

Organizing Institute: IIT Roorkee Page 45 of 49

Q.59 All rings considered below are assumed to be associative and commutative with
1 ≠ 0. Further, all ring homomorphisms map 1 to 1.

Consider the following statements about such a ring 𝑅:

P1: 𝑅 is isomorphic to the product of two rings 𝑅1 and 𝑅2 .
P2: ∃ 𝑟1 , 𝑟2 ∈ 𝑅 such that 𝑟12 = 𝑟1 ≠ 0 ≠ 𝑟2 = 𝑟22 , 𝑟1 𝑟2 = 0 and 𝑟1 + 𝑟2 = 1.
P3: ∃ ideals 𝐼1 , 𝐼2 ⊆ 𝑅 with 𝑅 ≠ 𝐼1 ≠ (0) ≠ 𝐼2 ≠ 𝑅 such that 𝑅 = 𝐼1 + 𝐼2 and
𝐼1 ⋂ 𝐼2 = (0).
P4: ∃ 𝑎, 𝑏 ∈ 𝑅 with 𝑎 ≠ 0 ≠ 𝑏 such that 𝑎𝑏 = 0.

Then, which of the following is/are TRUE?

(A) 𝑃1 ⟹ 𝑃2

(B) 𝑃2 ⟹ 𝑃3

(C) 𝑃3 ⟹ 𝑃4

(D) 𝑃4 ⟹ 𝑃1

Organizing Institute: IIT Roorkee Page 46 of 49

Q.60 Let 𝐸 ⊂ 𝐹 and 𝐹 ⊂ 𝐾 be field extensions which are not algebraic. Let 𝛼 ∈ 𝐾 be
algebraic over 𝐹 and 𝛼 ∉ 𝐹. Let 𝐿 be the subfield of 𝐾 generated over 𝐸 by the
coefficients of the monic polynomial of minimal degree over 𝐹 which has 𝛼 as a
zero. Then, which of the following is/are TRUE?

(A) 𝐹(𝛼) ⊃ 𝐿(𝛼) is a finite extension if and only if 𝐹 ⊃ 𝐿 is a finite extension

(B) The dimension of 𝐿(𝛼) over 𝐿 is greater than the dimension of 𝐹(𝛼) over 𝐹

(C) The dimension of 𝐿(𝛼) over 𝐿 is smaller than the dimension of 𝐹(𝛼) over 𝐹

(D) 𝐹(𝛼) ⊃ 𝐿(𝛼) is an algebraic extension if and only if 𝐹 ⊃ 𝐿 is an algebraic
extension

Q.61 Consider the inner product space of all real-valued continuous functions defined on
[−1, 1] with the inner product
1
〈𝑓, 𝑔〉 = ∫ 𝑓(𝑥)𝑔(𝑥) 𝑑𝑥.
−1

If 𝑝(𝑥) = 𝛼 + 𝛽𝑥 2 − 30𝑥 4 , 𝛼, 𝛽 ∈ ℝ is orthogonal to all the polynomials having
degree less than or equal to 3, with respect to this inner product, then 𝛼 + 5 𝛽 is
equal to _____ (in integer)

Organizing Institute: IIT Roorkee Page 47 of 49

Q.62 For 𝑋 = (𝑥1 , 𝑥2 , 𝑥3 )𝑇 ∈ ℝ3 , consider the quadratic form:

𝑄(𝑋) = 2𝑥12 + 2𝑥22 + 3𝑥32 + 4𝑥1 𝑥2 + 2𝑥1 𝑥3 + 2𝑥2 𝑥3 .

Let 𝑀 be the symmetric matrix associated with the quadratic form 𝑄(𝑋) with
respect to the standard basis of ℝ3 . Let 𝑌 = (𝑦1 , 𝑦2 , 𝑦3 )𝑇 ∈ ℝ3 be a non-zero vector,
and let

𝑌 𝑇 (𝑀 + 𝐼3 )𝑛+1 𝑌
𝑎𝑛 = , 𝑛 = 1, 2, 3, … ,
𝑌 𝑇 (𝑀 + 𝐼3 )𝑛 𝑌

where 𝐼3 is the 3 × 3 identity matrix. Then, the value of lim 𝑎𝑛 is _____
𝑛→∞
(in integer)

Q.63 Let 𝛼, 𝛽 be distinct non-zero real numbers, and let 𝑄(𝑧) be a polynomial of degree
less than 5. If the function

𝛼 6 sin 𝛽𝑧 − 𝛽 6 (𝑒 2𝛼𝑧 − 𝑄(𝑧))
𝑓(𝑧) =
𝑧6
𝛼
satisfies Morera’s theorem in ℂ\{0}, then the value of is equal to
4𝛽
____ (in integer)

Q.64 Let 𝐺 be a group with identity element 𝑒, and let 𝑔, ℎ ∈ 𝐺 be such that the following
hold:

(i) 𝑔 ≠ 𝑒, 𝑔2 = 𝑒,
(ii) ℎ ≠ 𝑒, ℎ2 ≠ 𝑒, and 𝑔ℎ𝑔−1 = ℎ2 .

Then, the least positive integer 𝑛 for which ℎ𝑛 = 𝑒 is _____ (in integer)

Organizing Institute: IIT Roorkee Page 48 of 49

Q.65 Let (ℝ2 , 𝑑1 ) and (ℝ2 , 𝑑2 ) be two metric spaces with

𝑑1 ((𝑥1 , 𝑥2 ), (𝑦1 , 𝑦2 )) = |𝑥1 − 𝑦1 | + |𝑥2 − 𝑦2 |

and

𝑑1 ((𝑥1 , 𝑥2 ), (𝑦1 , 𝑦2 ))
𝑑2 ((𝑥1 , 𝑥2 ), (𝑦1 , 𝑦2 )) = .
1 + 𝑑1 ((𝑥1 , 𝑥2 ), (𝑦1 , 𝑦2 ))
1
If the open ball centred at (0,0) with radius in (ℝ2 , 𝑑1 ) is equal to the open ball
7
1
centred at (0,0) with radius in (ℝ2 , 𝑑2 ), then the value of 𝛼
𝛼
is _____ (in integer)

Organizing Institute: IIT Roorkee Page 49 of 49

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