GATE 2026 Question Paper Mathematics (MA)

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GATE 2026
Question Paper ·
Mathematics ( )
EXAM YEAR TYPE SUBJECT COURSE

GATE 2026 Question Paper Mathematics ( ) MA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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

Q.1 Suresh said, “I did it yesterday.”

Which one of the following options is the correct form of this sentence in indirect
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speech?

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(A) a Suresh said that I did it yesterday.

(B) Suresh says I did it yesterday.

(C) Suresh says that he did it the day before.

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(D) Suresh said that he had done it the day before.

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Mathematics (MA)

Q.2 To continue the sequence of tiles shown, the tile indicated by the question mark
should be

(A)

(B)

(C)

(D)

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Q.3 Consider an art gallery whose walkways are shown as lines in the diagram. A
black dot represents a junction of two walkways. A guard may be placed at a
junction to watch over the walkways that join at that junction. The minimum
number of guards needed to watch all the walkways is ________.

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(A) 2
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(B) 3
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(C) 4 a
(D) 5

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Mathematics (MA)

Q.4 The 2nd of June is a Thursday in a certain year. Which day of the week is the
3rd of July in that year?

(A) Thursday

(B) Friday

(C) Saturday

(D) Sunday

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Q.5 A coin with heads facing up is shown as and a coin with tails facing up is
shown as .

Six coins are placed in the Starting Arrangement, as shown in the figure below.
A “step” is defined as interchanging a pair of adjacent coins without flipping
them. The minimum number of steps needed to go from the Starting Arrangement
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to the Final Arrangement, as shown in the figure, is ________.
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(A) 3

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(B) 6

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(C) 9
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(D) 12

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Mathematics (MA)

Q.6 – Q.10 Carry TWO marks Each

Q.6 Exacerbate : Mitigate :: ___________

Choose the option with the correct pair of words to fill the blank.

(A) Aggravate : Alleviate

(B) Alleviate : Precipitate

(C) Aggravate : Precipitate

(D) Emancipate : Exonerate

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Q.7 A paper shown in Panel I is folded along the dashed lines (- - -) to construct a
cube. The shaded regions shown in Panel I appear on the outer surface of the cube.
Referring to cubes shown in Panel II, which one of the options is correct?

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(A) Only (i) can correspond to the unfolded cube in Panel I.

(B) Only (ii) can correspond to the unfolded cube in Panel I.

(C) Both (i) and (ii) can correspond to the unfolded cube in Panel I.
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(D) Neither (i) nor (ii) can correspond to the unfolded cube in Panel I.

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Mathematics (MA)

Q.8 In a population, patients who have high cholesterol also have high blood-pressure
(BP). Some patients with high BP also have diabetes. There are no patients who
have both high cholesterol and diabetes. Furthermore,

1. the total number of patients with at least one of these conditions is 75,
2. the number of patients with high cholesterol is 10,
3. the number of patients with high BP is 45, and
4. the number of patients with only high BP and no other conditions is 20.

Then the number of patients who have both diabetes and high BP is _______

(A) 0

(B) 15

(C) 20

(D) 10

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Q.9 Four people P, Q, R, and S, of different ages, make the following observations.

P – I am younger than S.

Q – I am neither the youngest nor the oldest.

R – P is older than me. m
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Based on these observations, the youngest person is ______.
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(A) P

(B) Q

(C) R
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(D) S
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Mathematics (MA)

Q.10 Circles 𝐶1 , 𝐶2 , and 𝐶3 , with centers 𝑂1, 𝑂2, and 𝑂3 , and radii 𝑟1 , 𝑟2 , and 𝑟3 ,
respectively, touch each other as shown in the following figure. Given 𝑟1 = 2 cm,
𝑟2 = 1 cm and the angle ∠𝑂1 𝑂3 𝑂2 is 90°, 𝑟3 = _____ cm.

(A) 1
(−3 + √17 )
2

(B) 1
(3 + √17 )
2

(C) 1
(−2 + √17 )
2

(D) 1
(−3 + 2√17 )
2

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Q.11 – Q.35 Carry ONE mark Each

Q.11 Let 𝑀3 (ℝ) be the vector space of all 3 × 3 real matrices over ℝ under usual
matrix addition and scalar multiplication. Let 𝑇3 (ℝ) be the set of all 3 × 3 real
upper triangular matrices. Which one of the following is TRUE?

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(A) The s em space 𝑀 (ℝ)⁄𝑇 (ℝ) is isomorphic to the vector space of all 3 × 3 glas
3

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quotient
3
g
a real skew symmetric matrices over ℝ under usual matrix addition and scalar
multiplication.

(B) 𝑀3 (ℝ)
The quotient space ⁄𝑇 (ℝ) is isomorphic to the vector space of all 3 × 3
3

real symmetric matrices over ℝ under usual matrix addition and scalar

m
multiplication.

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(C) 𝑀3 (ℝ)
s em
⁄𝑇 (ℝ) is isomorphic to the vector space of all 3 × 3
la
The quotient space
3

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a over ℝ under usual matrix addition and scalar
real lower triangular matrices
multiplication.

(D) 𝑀3 (ℝ)
The quotient space ⁄𝑇 (ℝ) is isomorphic to the vector space of all 3 × 3
3

real matrices with trace zero over ℝ under usual matrix addition and scalar
multiplication.
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Mathematics (MA)

Q.12 Let 𝐼 be the integral defined as follows:
1 𝑦 2 1
𝐼 = ∫0 ∫0√ 𝑑𝑥 𝑑𝑦 + ∫1 ∫√𝑦−1 𝑑𝑥 𝑑𝑦

If the order of the integration is changed, then which one of the following is the
correct expression for 𝐼?

(A) 1/2 𝑥 1 −𝑥 2 +1
∫0 ∫−𝑥 𝑑𝑦 𝑑𝑥 + ∫1/2 ∫−𝑥 2 𝑑𝑦 𝑑𝑥

(B) 1 𝑥 2 +1
∫0 ∫𝑥 2 𝑑𝑦 𝑑𝑥

(C) 1/2 𝑥 2 +1 1 √𝑥+1
∫0 ∫𝑥 2 𝑑𝑦 𝑑𝑥 + ∫1/2 ∫√𝑥 𝑑𝑦 𝑑𝑥

(D) 1 𝑥 2 −1
∫0 ∫𝑥 2 𝑑𝑦 𝑑𝑥

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Q.13 Let 𝐶 denote the Cantor set and 𝑓: [0,1] → ℝ be defined as follows:

𝑥 2026 for 𝑥 ∈ 𝐶
1
cos(𝜋𝑥) for 𝑥 ∈ [ 0, ] \𝐶
𝑓(𝑥) = m
.co
2
m 1
.co { sin(𝜋𝑥) m
for 𝑥 ∈ [ , 1 ] \𝐶
2
e
The s emof the Lebesgue integral of 𝑓(𝑥) over the interval [0,1] is equal to glas
la
value

g a
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(A) 2
𝜋

(B) 1
m
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𝜋

e m
(C) 3
las
𝜋
ag
(D) 0

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Mathematics (MA)

Q.14 Let 𝐺 be a group of order 595. Which one of the following is TRUE?

(A) 𝐺 cannot have a proper normal subgroup.

(B) 𝐺 must have a proper normal subgroup.

(C) 𝐺 cannot have an element of order 17.

(D) The number of Sylow 5-subgroups is 17.

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Q.15 Let 𝑢(𝑥, 𝑡) be the solution of the initial value problem for the heat equation on the
real line:

𝜕𝑢 𝜕 2𝑢
=𝑘 , −∞ < 𝑥 < ∞, 𝑡 > 0, 𝑘∈ℝ
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𝜕𝑡 𝜕𝑥 2
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with the initial condition
s e m
s em 𝑢(𝑥, 0) = 𝑒 −𝑎|𝑥|
, 𝑎 > 0.
g l a
l a a
agtoIf 𝑥,𝑢̂(𝑤,then𝑡) 𝑢̂(𝑤,
∞ 𝑖𝑤𝑥
= ∫ 𝑢(𝑥, 𝑡)𝑒
−∞
𝑑𝑥 is the Fourier transform of 𝑢(𝑥, 𝑡) with respect
𝑡) is equal to

(A) 2𝑎 −𝑘𝑤 𝑡 2
𝑎2 +𝑤 2𝑒
m
𝑎
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e
(B) −𝑘𝑤 𝑡 2
2𝑒 s
la
𝑎2 +𝑤

ag
(C) 2𝑎 −𝑘𝑎 𝑡 2
𝑎2 +𝑤 2𝑒

(D) −𝑤2
1
𝑒 4𝑘𝑡
√4 𝜋𝑘𝑡

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Mathematics (MA)

Q.16 Let 𝑓: ℝ2 → ℝ2 be defined as

𝑓(𝑥, 𝑦) = (𝑒 𝑥 cos 𝑦 , 𝑒 𝑥 sin 𝑦) .

Which one of the following is TRUE?

(A) 𝑓 is one-to-one.

(B) The Jacobian of 𝑓 is negative.

(C) 𝑓 is locally invertible.

(D) 𝑓 is invertible on ℝ2 .

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Q.17 Let 𝑓: ℂ → ℂ be defined by 𝑓(𝑧) = |𝑧|2 − 5𝑧̅ + 2.

Which one of the following is TRUE?

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(A)
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𝑓 is differentiable at 𝑧 = 5𝑖.
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(B)
g ladifferentiable at 𝑧 = 5.
𝑓 is ag
a
(C) 𝑓 is differentiable at 𝑧 = −5𝑖.

(D) 𝑓 is differentiable at 𝑧 = −5.

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Mathematics (MA)

Q.18 Let 𝐷 = {𝑧 ∈ ℂ | |𝑧| < 1} denote the unit disc in the complex plane ℂ.

Let 𝑓: 𝐷 → 𝐷 be an analytic function which satisfies 𝑓(0) = 0. Then which one of
the following is a possible value of 𝑓′(0)?

(A) 5
𝑖
2

(B) 𝑖
10

(C) 3
2

(D) 5
− 𝑖
2

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Q.19 Let 𝑋 be the space of all continuously differentiable real valued functions on
[0,1]. Define the following norms on 𝑋:

𝑝1 (𝑥) = sup{|𝑥(𝑡)| ∶ 𝑡 ∈ [0,1]}

𝑑
𝑝2 (𝑥) = sup {| 𝑥(𝑡)| ∶ 𝑡 ∈ [0,1]}
m
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𝑑𝑡
m
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s e m
s e l a
a ag
Which one of the following is TRUE?

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a
(A) (𝑋, 𝑝1 ) is a Banach space.

(B) (𝑋, 𝑝2 ) is a Banach space.

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(C) (𝑋, 𝑝3 ) is not a Banach space.
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(D) (𝑋, 𝑝 ) is a Banach space.
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Mathematics (MA)

Q.20 Let 𝑋 be a set with at least two elements. Let 𝑑1 , 𝑑2 and 𝑑3 be metrics on 𝑋.
Which one of the following is NOT a metric on 𝑋?

(A) 𝑑(𝑥, 𝑦) ≔ min(𝑑1 (𝑥, 𝑦), 3) for all 𝑥, 𝑦 ∈ 𝑋.

(B) 𝑑(𝑥, 𝑦) ≔ max(𝑑2 (𝑥, 𝑦), 3) for all 𝑥, 𝑦 ∈ 𝑋.

(C) 10 𝑑3 (𝑥,𝑦)
𝑑(𝑥, 𝑦) ≔ for all 𝑥, 𝑦 ∈ 𝑋.
1+𝑑3 (𝑥,𝑦)

1
(D) 𝑑(𝑥, 𝑦) ≔ (𝑑1 (𝑥, 𝑦) + 𝑑2 (𝑥, 𝑦) + 𝑑3 (𝑥, 𝑦)) for all 𝑥, 𝑦 ∈ 𝑋.
3

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Q.21 Let 𝑃 be a 3 × 3 real symmetric positive definite matrix. Which of the following
statements is/are TRUE?

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(A)
m
𝑄 ⊤ 𝑃𝑄 is positive definite for all nonzero 𝑄 ∈ ℝ3×3 .
.co
.co s e m
s em
𝑄 ⊤ 𝑃𝑄 is positive semidefinite for all 𝑄 ∈ ℝ3×3 . l a
ag
(B)

g la
(C) a 𝑄 𝑃𝑄 is positive definite if 𝑄 ∈ ℝ
⊤ 3×3
is nonsingular.

(D) 𝑄 ⊤ 𝑃𝑄 is not positive semidefinite for some 𝑄 ∈ ℝ3×3 .

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Mathematics (MA)

Q.22 Let 𝑓: ℝ2 → ℝ be a function defined as follows:

𝑥 √𝑥 2 + 𝑦 2
for 𝑥 ≠ 0
𝑓(𝑥, 𝑦) = { |𝑥|
0 for 𝑥 = 0
Which of the following statements is/are TRUE?

(A) 𝑓(𝑥, 𝑦) is continuous at (0,0).

(B) For any 𝑢 ∈ ℝ2 , the directional derivative of 𝑓 at (0,0) in the direction of 𝑢 exists.

(C) 𝜕𝑓 𝜕𝑓
(0,0) = 1 = (0,0).
𝜕𝑥 𝜕𝑦

(D) 𝑓(𝑥, 𝑦) is differentiable at (0,0).

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Q.23 Let ℤ11 be the ring of integers modulo 11. Let ℤ11 [𝑥] be the ring of all
polynomials with coefficients in ℤ11 . Which of the following statements is/are
TRUE?

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(A)
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The polynomial 𝑥 2 + 𝑥 + 4 is irreducible over ℤ11 .
s e m
s e l a
(B)
g
ℤ a[𝑥]
l ⁄〈𝑥 + 𝑥 + 4〉 is a field.
11 ag
a 2

(C) The ideal 〈𝑥 2 + 𝑥 + 4〉 is maximal.

(D) The ideal 〈𝑥 2 + 𝑥 + 4〉 is not a prime ideal.

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Mathematics (MA)

Q.24 Consider ℝ with the topology

𝜏 = {𝐴 ⊆ ℝ ∶ 𝐴𝑐 is finite} ∪ {𝜙}.

Which of the following statements is/are TRUE?

(A) (ℝ, 𝜏) is a Hausdorff space.

(B) Any finite subset of (ℝ, 𝜏) is closed.

(C) (ℝ, 𝜏) is compact.

(D) (ℝ, 𝜏) is connected.

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Q.25 Consider the following partial differential equation:

𝜕 2 𝑓(𝑥, 𝑦) 𝜕 2 𝑓(𝑥, 𝑦)
𝑎 + 𝑏 = 8𝑓(𝑥, 𝑦)
𝜕𝑥 2 𝜕𝑦 2

where 𝑎 and 𝑏 are distinct positive real numbers.
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The combination(s) of the values of the real parameters 𝜉 and 𝜂 for which
𝑓(𝑥, 𝑦) = 𝑒 2𝜉𝑥+𝜂𝑦 is a solution of the given partial differential equation, is/are e m
e m l as
l as ag
ag
(A) 1 2
𝜉= , 𝜂=
√𝑎 √𝑏

(B) 𝜉= 0, 𝜂= 0

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(C) 1 1
𝜉= , 𝜂=
.co
√2𝑎 √2𝑏

e m
(D)
𝜉 = 0, 𝜂 = 2 √
2
las
𝑏
ag

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Mathematics (MA)

Q.26 Consider the following assignment problem where 𝑋, 𝑌, 𝑍 are tasks, 𝑃, 𝑄, 𝑅 are
agents and the cost matrix is given by

X Y Z

P 4 2 8

Q 2 3 7

R 3 1 6

Which of the following statements is/are TRUE for an optimal assignment?

(A) The cost is 9.

(B) Agent P is assigned to task Y.

(C) Agent Q is assigned to task X.

(D) Agent R is assigned to task Y.

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Q.27 Consider the following differential equation:

𝑑2𝑦 𝑑𝑦
2𝑥(𝑥 − 2)2 2
+ 3𝑥 + (𝑥 − 2)𝑦 = 0
𝑑𝑥 𝑑𝑥
Which of the following statements is/are TRUE ?
m
m .co
m .co s e m
s e l a
(A)
g la point 𝑥 = 2 is not a singular point.
The ag
a
(B) The point 𝑥 = 2 is a singular point.

(C) The point 𝑥 = 2 is a regular singular point.

(D) The point 𝑥 = 2 is not a regular singular point.
m
.co
s em
la
Let 𝑇: ℝ → ℝ be the lineargtransformation
a
3 3 3
Q.28 which reflects every vector in ℝ
3 3×3
through a two-dimensional subspace of ℝ . Let 𝑃 ∈ ℝ be the matrix
1 0 0
representation of 𝑇 using the basis { (0) , (1) , (0) }.
0 0 1
Then the value of

2 × trace(𝑃) − 3 × determinant(𝑃) is equal to . (answer in integer)
m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q.29 Let 𝑥 3 − 𝑥 + 1 ∈ ℤ3 [𝑥], where ℤ3 [𝑥] is the ring of all polynomials with

ℤ3 [𝑥]
coefficients in ℤ3 . Then the degree of the field extension ⁄〈𝑥 3 − 𝑥 + 1〉 of
ℤ3 is equal to . (answer in integer)

Q.30 𝑑𝑦
Consider the differential equation = 𝑥 + 𝑦 with the initial condition 𝑦(0) = 1.
𝑑𝑥
Using the modified Euler’s method, the second approximation to 𝑦(ℎ), where
ℎ = 0.05 (step size), is equal to . (rounded off to TWO decimal places)

Q.31 Let
1
𝑛2 + (sin 𝑒 𝑥 )𝑛
𝛼 = lim ∫ 𝑑𝑥 .
𝑛→∞ 7𝑛2 + 𝑥 8
0

The value of 14𝛼 is equal to . (answer in integer)

Q.32 𝑥
Let 𝑋 = (ℝ3
, ‖ ⋅ ‖1 ) where ‖(𝑦)‖ = |𝑥| + |𝑦| + |𝑧| and 𝑇: 𝑋 → 𝑋 be a linear
𝑧 1
transformation defined as
𝑥 1 1 3 𝑥
𝑇 (𝑦) = (2 2 −2) (𝑦) .
𝑧 1 3 −3 𝑧
Then the operator norm of 𝑇 is equal to . (answer in integer)

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a
Q.33 Consider the following Linear Programming Problem (LPP):

maximize 𝑧 = 10𝑥1 + 20𝑥2

subject to

m
.co
𝑥1 ≤ 36
m
m .co 𝑥2 ≥ 42
s e m
s e l a
ag
𝑥1 + 𝑥2 ≥ 48

g l a
a 5𝑥1 + 𝑥2 ≤ 150

𝑥1 , 𝑥2 ≥ 0

The maximum value of 𝑧 in the above LPP is equal to . (answer in
integer)

o m
c
The number of distinct topologies on the. set {1, 2, 3} consisting of exactly four
em in integer)
Q.34
elements is equal to
la s
. (answer

a g

Q.35 For 𝑎𝑛 ∈ ℂ, 𝑛 = 0,1,2, …, the power series
∞

∑ 𝑎𝑛 (𝑧 − 2)𝑛
𝑛=0
m
m converges at 𝑧 = 5 and diverges at 𝑧 = −1. Then the radius of convergence of
.co
m .co this power series is equal to . (answer in integer)
s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q.36 – Q.65 Carry TWO marks Each

Q. 36 𝑦 (𝑥)
The solution 𝑦(𝑥) = ( 1 ) of the initial value problem
𝑦2 (𝑥)

𝑑𝑦 −3 4 1
=( ) 𝑦, 𝑦(0) = ( )
𝑑𝑥 −2 3 2
is equal to

(A) 1 2
3 ( ) 𝑒 𝑥 + ( ) 𝑒 −𝑥
1 1

(B) 1 2
−3 ( ) 𝑒 𝑥 + ( ) 𝑒 −𝑥
1 1

(C) 1 2
3 ( ) 𝑒 𝑥 + 2 ( ) 𝑒 −𝑥
1 1

(D) 1 2
3 ( ) 𝑒 𝑥 − ( ) 𝑒 −𝑥
1 1

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Q.37 1 0 0
Let 𝑃 = (0 9 0 ) . Define an inner product on ℝ3 with respect to 𝑃 as
0 0 16
〈𝑥, 𝑦〉𝑃 = 𝑥 ⊤ 𝑃𝑦, for all 𝑥, 𝑦 ∈ ℝ3 .

1 1
m
m
Consider the subspace 𝑉 = span { (0) , (1) }. Which one of the following sets is
.co
.co 0 1
an orthonormal basis of 𝑉 with respect to 〈𝑥, 𝑦〉𝑃 ?
m s e m
s e l a
g l a ag
a
(A) 1 0
{ (0) , (1) }
0 1

(B) 0 0
{ (1) , (1) }
m
.co
0 1

e m
(C)
1
0
las
g
1
{ (0) , ( 5 ) }
0 1 a
5

(D) 0
1 1
{ (0) , (√2) }
1
0
√2

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Mathematics (MA)

Q. 38 The value of the contour integral

(1 − 𝑧)(3cos𝑧 + 5sin𝑧)
∫ 𝑑𝑧 ,
Γ 1 − 𝑧101
7
where Γ = {𝑧 ∈ ℂ ∶ |𝑧| = }, oriented in the counter-clockwise direction, is equal
9
to

(A) 2π𝑖

(B) −2π𝑖

(C) 0

(D) 4π𝑖

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a
Q. 39 Let 𝑓(𝑧) = ∑∞ −𝑛
𝑛=1 5 cos(𝑛𝑧). On which one of the following domains, does 𝑓(𝑧)

represent an analytic function?

m
m .co
(A)
m .co
{𝑧 ∈ ℂ ∶ |Im 𝑧| < ln5}
s e m
s e l a
(B) {𝑧 a
l ∈ ℂ ∶ |Im 𝑧| > ln5} ag
ag
(C) {𝑧 ∈ ℂ ∶ |Re 𝑧| < ln5}

(D) {𝑧 ∈ ℂ ∶ |Re 𝑧| > ln5}

m
m .co
s e
g la
a

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Mathematics (MA)

Q. 40 Let δ(𝑡) be the unit impulse function defined by δ(𝑥 − 𝑥0 ) = 0 when 𝑥 ≠ 𝑥0 and
∞
∫ δ(𝑥 − 𝑥0 ) 𝑑𝑥 = 1.
−∞

Consider the following initial value problem

𝑑2 𝑦 𝑑𝑦
2 + + 2𝑦 = δ(𝑥 − 5)
𝑑𝑥 2 𝑑𝑥
𝑑𝑦
with 𝑦(0) = 0 and (0) = 0. Which one of the following is TRUE?
𝑑𝑥

(A) 10 √15
𝑦(10) = 𝑒 −5/4 sin ( 4 )
√15

(B) 2 5√15
𝑦(10) = 𝑒 −5/4 sin ( 4 )
√15

(C) 10 √15
𝑦(10) = 𝑒 5/4 sin ( 4 )
√15

(D) 2 5√15
𝑦(10) = 𝑒 5/4 sin ( 4 )
√15

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a
Q.41 Which one of the following statements is TRUE?

The set of all 2 × 2 real matrices under usual matrix addition and multiplication is m
.co
(A)
m
a Unique Factorization Domain.

m .co s e m
s e l a
ag
(B) The set of all real valued functions under usual function addition and

g l a
multiplication is a Unique Factorization Domain.
a
(C) The set of all polynomials with coefficients in ℝ under usual polynomial addition
and multiplication is a Unique Factorization Domain.

(D) The set of Riemann integrable functions in [0,1] under usual function addition and
multiplication is a Unique Factorization Domain.

m
m .co
s e
g la
a

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Let ℓ∞ = {𝑥 = (𝑥𝑛 )𝑛≥1 | 𝑥𝑛 ∈ ℝ, sup {|𝑥𝑛 |: 𝑛 = 1,2, … } < ∞} with the
Q. 42 𝑥 𝑥
supremum norm. Let 𝑇: ℓ∞ → ℓ∞ be given by 𝑇(𝑥1 , 𝑥2 , 𝑥3 , … ) = (𝑥1 , 22 , 33 , … ).
Which one of the following is TRUE?

(A) 𝑇 is bounded but not one-to-one.

(B) 𝑇 is one-to-one but not bounded.

(C) 𝑇 is bounded and the inverse (from the range of 𝑇) exists but not bounded.

(D) 𝑇 is bounded and the inverse (from the range of 𝑇) is bounded.

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Q. 43 Let 𝑢(𝑥, 𝑡) satisfy the wave equation

∂2 𝑢 2
∂2 𝑢
= 𝑐 , − ∞ < 𝑥 < ∞, 𝑡 > 0, 𝑐>0
∂𝑡 2 ∂𝑥 2
with
m
m 1 if |𝑥| < 1
.co
.co
𝑢(𝑥, 0) = {
0 otherwise
e m
e m l as
l as and
ag
ag ∂𝑢
(𝑥, 0) = 0.
∂𝑡

By using D’Alembert’s formula, the maximum value of 𝑢(0, 𝑡) for 𝑡 > 0 is

m
(A) 1
m .co
s e
(B) 1
g la
2 a
(C) 1
4

(D) 0

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q. 44 ∂2 𝑇 ∂2 𝑇
Consider the Laplace equation + = 0, 0 < 𝑥 < 1, 0 < 𝑦 < 1, with the
∂𝑥 2 ∂𝑦 2
boundary conditions

𝑇(𝑥, 0) = 𝑥, 𝑇(0, 𝑦) = 𝑦

𝑇(𝑥, 1) = 1 + 𝑥, 𝑇(1, 𝑦) = 1 + 𝑦.
1 1
Then the value of 𝑇 ( , ) is equal to
2 3

(A) 7
6

(B) 5
6

(C) 1
6

(D) 1
2

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a
Q.45 Let 𝐸1 and 𝐸2 be subsets of a normed linear space 𝑋, and

𝐸1 + 𝐸2 = {𝑥 + 𝑦 ∶ 𝑥 ∈ 𝐸1 , 𝑦 ∈ 𝐸2 }

𝐸1 × 𝐸2 = {(𝑥, 𝑦) ∶ 𝑥 ∈ 𝐸1 , 𝑦 ∈ 𝐸2 }.

m
.co
Which one of the following is NOT TRUE?
m
.co s e m
s em l a
g a
Ifleither of 𝐸 or 𝐸 is open, then 𝐸 + 𝐸 is open.
ag
(A)
a 1 2 1 2

(B) If 𝐸1 and 𝐸2 are convex, then 𝐸1 + 𝐸2 is convex.

(C) If 𝐸1 and 𝐸2 are closed, then 𝐸1 + 𝐸2 is closed.

(D) If 𝐸1 and 𝐸2 are connected, then 𝐸1 × 𝐸2 is connected.
m
m .co
s e
g la
a

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q. 46 For a transportation problem, let 𝑐𝑖𝑗 denote the unit cost of the cell (𝑖, 𝑗). Assume
that 𝑐11 = 10, 𝑐12 = 12, 𝑐21 = 11 and 𝑐22 = 13. Let α𝑖 and β𝑗 , 𝑖, 𝑗 = 1,2,3,
represent the simplex multipliers associated with any basis corresponding to the
unit cost 𝑐𝑖𝑗 . Assume that α1 = 𝑥, α2 = 𝑥 + 1, β1 = 𝑦 and β2 = 𝑦 + 2. The
relative cost coefficient 𝑑𝑖𝑗 is the difference between the current solution and the
new improved solution.

If 𝑥 = 4, 𝑐13 = 19 and β3 = 𝑦 + 5, then which one of the following is TRUE?

(A) 𝑦 = 6 and 𝑑13 = 7

(B) 𝑦 = 6 and 𝑑13 = 4

(C) 𝑦 = 5 and 𝑑13 = 7

(D) 𝑦 = 5 and 𝑑13 = 4

Q. 47 Let 𝑋 and 𝑌 be topological spaces and 𝑓: 𝑋 → 𝑌 be a continuous and bijective
mapping. Which one of the following statements is TRUE?

(A) 𝑓 is a homeomorphism if 𝑋 and 𝑌 are compact.

(B) 𝑓 is a homeomorphism if 𝑋 is Hausdorff and 𝑌 is compact.

(C) 𝑓 is a homeomorphism if 𝑋 is compact and 𝑌 is Hausdorff.

(D) 𝑓 is a homeomorphism if 𝑋 and 𝑌 are Hausdorff.

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Q.48 Let 𝑃 and 𝑄 be 3 × 3 nonzero real matrices. Assume that there exists a 3 × 3 real
nonsingular matrix 𝑆 such that 𝑆 −1 𝑃𝑆 and 𝑆 −1 𝑄𝑆 are both upper triangular.
Which of the following statements is/are TRUE?

m
m .co
(A)
.co
The matrix 𝑃𝑄 − 𝑄𝑃 is invertible.
s e m
s em l a
(B)
g la matrix 𝑃𝑄 − 𝑄𝑃 is nilpotent.
The ag
a
(C) The matrix 𝐼 + 𝑃𝑄 − 𝑄𝑃 is invertible, where 𝐼 is the 3 × 3 identity matrix.

(D) If 𝑃𝑄 − 𝑄𝑃 is a nonzero matrix, then 𝑃𝑄 − 𝑄𝑃 is diagonalizable.

o m
Q. 49 Let 𝑆 = ⋃ {(𝑥, 𝑦) ∈ ℝ : (𝑥 − 𝑛) m
∞
c
+ .𝑦 = } with usual topology. Which of
2 2 2 1
𝑛=1
s e 𝑛2

la
the following statements is/are TRUE?
g
a

(A) 𝑆 is not compact.

(B) 𝑆 is compact.

m
m .co
.co
(C) 𝑆 is connected.
e m
e m l as
las (D) 𝑆 is not connected.
ag
ag

m .
.co s e m
s em l a
la
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Q. 50 𝑧 1+𝑧
Let 𝑓(𝑧) = and 𝑔(𝑧) = be two Mӧbius transformations defined on the
1−𝑧 1−𝑧

unit disc 𝐷 = {𝑧 ∈ ℂ ∶ |𝑧| < 1}. Consider the following statements:

𝑆1: 𝑓(𝐷) ⊆ 𝑔(𝐷)

𝑆2 : 𝑔(𝐷) ⊆ f(𝐷)

Which of the following statements is/are CORRECT?

(A) 𝑆1 is true.

(B) 𝑆2 is true.

(C) 𝑆2 is true and 𝑆1 is false.

(D) Neither 𝑆1 is true nor 𝑆2 is true.

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Q. 51 Let α and β be the roots of the indicial equation obtained in the method of finding
Frobenius series solution to the differential equation:

𝑑2𝑦 𝑑𝑦
2𝑥 2 − 𝑥 + (1 − 𝑥 2 )𝑦 = 0 .
𝑑𝑥 2 𝑑𝑥

m
.co
Which of the following statements is/are TRUE?
m
m .co s e m
s e l a
g a
l≠ β ag
(A)
a α

(B) α−β∈ℤ

(C) α−β ∉ℤ

α= β
m
.co
(D)

e m
las
ag

m
m .co
m .co s e m
s e g l a
g la a
a

m .
.co s e m
s em l a
la
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Q.52 Let ℚ[𝑥] be the ring of all polynomials with coefficients in ℚ under the usual
polynomial addition and multiplication. Let 𝑇: ℚ[𝑥] → ℚ[𝑥] be defined by

𝑇(𝑝(𝑥)) = 𝑝(𝑥 2 ), for all 𝑝(𝑥) ∈ ℚ[𝑥] .

Which of the following statements is/are TRUE?

(A) 𝑇 is a ring homomorphism.

(B) 𝑇 is one-to-one.

(C) 𝑇 is onto.

(D) 𝑇 is a ring isomorphism.

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Q. 53 Let 𝑋 be any normed linear space and 𝑋 ′ be the dual space of 𝑋. Which of the
following statements is/are TRUE?

m
(A)
m
If 𝑋 is separable and 𝑋 ′ is non-separable, then 𝑋 must be reflexive.
.co
.co s e m
s em
If 𝑋 is separable and 𝑋 ′ is non-separable, then 𝑋 cannot be reflexive. l a
ag
(B)

g la
(C) a If 𝑋 is reflexive, then 𝑋 is reflexive.
′

(D) If 𝑋 ′ is separable, then 𝑋 is separable.

m
.co
Let (ℤ𝑛 , +) be the group of integers modulo 𝑛. Let 𝐺 = ℤ30 ⨁ ℤ12 be the external
Q. 54
direct product of ℤ30 and ℤ12 .
e m
s
la is/are TRUE?
g
Then which of the following statements
a

(A) 𝐺 is not a cyclic group.

(B) The order of the element (14,7) in 𝐺 is 98.
m
m
c. o(C) The order of the element (14,7) in 𝐺 is 60.
m.co
m s e
s e g l a
g la (D) There is an element in 𝐺 of order 360.
a
a

m .
.co s e m
s em l a
la
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Q.55 2 1 0 0 0 0
0 2 0 0 0 0
0 0 2 1 0 0
Let 𝐽 = .
0 0 0 2 0 0
0 0 0 0 3 1
(0 0 0 0 0 3)
Then the geometric multiplicity of the eigenvalue 2 of 𝐽 is equal to .
(answer in integer)

Q.56 Let α = ∬𝑆 𝐹⃗ ∙ 𝑛̂ 𝑑𝑆, where 𝐹⃗ = (2𝑥 + 3𝑧)𝑖̂ + (𝑥𝑧 − 𝑦)𝑗̂ + (𝑦 2 + 2𝑧)𝑘̂ and 𝑆 is
the sphere with centre at (3, −1, 2) and radius 9. Here, 𝑛̂ is the unit normal drawn
outward and 𝑖̂, 𝑗̂, 𝑘̂ are unit vectors.
1
Then the value of α is equal to . (answer in integer)
36 𝜋

Q.57 Consider the problem of maximizing

2 0 0 𝑥1
𝑧 = (𝑥1 𝑥2 𝑥3 ) (0 2 1) (𝑥2 )
0 1 2 𝑥3
subject to
𝑥1 𝑥1
(𝑥1 𝑥2 𝑥3 ) (𝑥2 ) = 1, 𝑥
where ( 2 ) ∈ ℝ3 .
𝑥3 𝑥3

Then the maximum value of 𝑧 is . (answer in integer)

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Q.58 Let 𝑃3 (ℝ) be the vector space of all polynomials of degree at most three with the
real coefficients under usual polynomial addition and scalar multiplication. Let
𝑇: 𝑃3 (ℝ) → ℝ2 be the linear transformation defined as

𝑇(𝑝) = (𝑝(1), 𝑝′ (1))

m
.co
for all 𝑝 ∈ 𝑃3 (ℝ), where 𝑝′ is the derivative of 𝑝. Then the nullity of 𝑇 is equal to
m
.co m
. (answer in integer)

m s e
s e l a
Q. 59
g l a
The number of zeros of the complex polynomial 𝑧 6 + 5𝑧 3 + 4𝑧 + 11 in the
annulus {𝑧 ∈ ℂ ∶ 1 < |𝑧| < 3} is equal to ag
a
. (answer in integer)

Q. 60 Let 𝐿2 [0, π] denote the space of all real valued Lebesgue square integrable
functions on [0, π]. Let 𝑇: 𝐿2 [0, π] → 𝐿2 [0, π] be defined as follows:
𝜋 𝜋
𝑇(𝑓(𝑥)) = sin𝑥 ∫ 𝑓(𝑡)cos𝑡 𝑑𝑡 + cos𝑥 ∫ 𝑓(𝑡)sin𝑡 𝑑𝑡 .
0 0

Then the value of
4
‖𝑇‖ is equal to
o m. (answer in integer)
.c
𝜋

s em
g la
a𝑦 < 1} be the open unit disc and ∂Ω be its
Q. 61 Let Ω = {(𝑥, 𝑦) ∈ ℝ : 𝑥 + 2 2 2

boundary. If 𝑢(𝑥, 𝑦) is the solution of the following Dirichlet problem

∂2 𝑢 ∂2 𝑢
+ = 0 in Ω
∂𝑥 2 ∂𝑦 2

𝑢(𝑥, 𝑦) = 𝑥 2 − 𝑦 2 on ∂Ω,
m
m .co
1 1
then the value of 4 (𝑢 ( , 0) − 𝑢 (0, )) is
.co
. (answer in integer)
2 2
m
e m Q. 62 Let 𝑋 = {5, 6, 7, 8, 9, 10} be equipped with the topology lase
l as ag
ag τ = {𝜙, 𝑋, {5,6,7}, {8,9,10}}.

Then the number of subsets of 𝑋 which are neither open nor closed is .

(answer in integer)

m .
.co s e m
s em l a
la
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g
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ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 47 of 48

Mathematics (MA)

Q. 63 Let α, β ∈ ℝ. If (4, 0, 2, β) is an optimal solution of the Linear Programming
Problem:

minimize 𝑥1 + 3𝑥2 + 2𝑥3 − α𝑥4

subject to
4𝑥1 + 𝑥2 + 𝑥3 = 18

−3𝑥1 + 2𝑥3 + 𝑥4 = 2

𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ≥ 0,

then the maximum value of 22(α + β) is equal to . (answer in integer)

Q. 64 Let 𝐷 = {(𝑥, 𝑦) ∈ ℝ2 ∶ 0 ≤ 𝑥 ≤ 2, 0 ≤ 𝑦 ≤ 2} and let 𝑓(𝑡) denote the smallest
integer greater than or equal to 𝑡. Then the value of the integral

∬ 𝑓(𝑥 + 𝑦)𝑑𝑥𝑑𝑦
𝐷

is . (answer in integer)

Q. 65 If Jacobi method is used to solve the following system of linear equations

1 2 1 𝑥1 2
(0 2 2) (𝑥2 ) = (2)
1 1 1 𝑥3 2
(𝑖)
0 𝑥1
with the initial guess 𝑥 = (0) and 𝑥 = (𝑥2(𝑖) ) , 𝑖 = 1,2,3 … , denotes the
(0) (𝑖)

0 𝑥
(𝑖)
3
𝑡ℎ (2) (2) (2)
𝑖 iterate, then the value of | 𝑥1 + 𝑥2 + 𝑥3 | is equal to . (answer in
integer)

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