GATE MA year:2025 PYQs & Answers — Page 5

Questions 41–50 of 65. Imported answers require independent review unless marked reviewed.

Let

be the Möbius transformation that maps the points

and

conformally onto the points

and

, respectively, in the extended complex plane. If

maps the circle centred at

with radius

onto a straight line given by the equation

, then the value of

is equal to

Your answerChoose one option.

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Let

and

, where

denotes the imaginary part of

. Let

be the set of all bijective analytic functions

such that

. Then, the value of

Your answerChoose one option.

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Let

be a non-empty open connected subset of

and

be a non-constant function. Let the functions

and

be defined by

Consider the following two statements:

S1: If

is continuous in

and

is analytic in

, then

is analytic in

S2: If

and

are analytic in

, then

is analytic in

Then, which one of the following is correct?

Your answerChoose one option.

S1 is TRUE and S2 is FALSE

S2 is TRUE and S1 is FALSE

both S1 and S2 are TRUE

neither S1 nor S2 is TRUE

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

on the sphere

, for

. Let

denote the equivalence class of

, and let

denote the set of all such equivalence classes. Let

be given by

. If

is provided with the quotient topology induced by the map

, then which one of the following is TRUE?

Your answerChoose one option.

is homeomorphic to

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Consider the following two spaces:

, the space of all real-valued continuous functions defined on

equipped with the norm

, the space of all real-valued continuous functions defined on

equipped with the norm

Let

be the linear span over

of all the Legendre polynomials. Then, which one of the following is correct?

Your answerChoose one option.

is dense in

but not in

is dense in

but not in

is dense in both

and

is dense neither in

nor in

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Consider the metric spaces

and

with the metrics defined by

and

, respectively. Then, which one of the following is TRUE?

Your answerChoose one option.

is open in

but not in

is open in

but not in

is open in both

and

is open neither in

nor in

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Let

be an algebraically closed field containing a finite field

. Let

be the subfield of

consisting of elements of

that are algebraic over

Consider the following statements:

S1:

is algebraically closed.

S2:

is infinite.

Then, which one of the following is correct?

Your answerChoose one option.

S1 is TRUE and S2 is FALSE

S2 is TRUE and S1 is FALSE

both S1 and S2 are TRUE

neither S1 nor S2 is TRUE

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Let

be the vector space (over

) of all

matrices with entries in

. Consider the linear transformation

defined by

, where

and

. If

is the matrix representation of

with respect to the standard basis of

, then which of the following is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

is an invertible matrix

The trace of

is 25

The rank of

is 4, where

is the

identity matrix

The nullity of

is 0, where

is the

identity matrix

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Consider the linear programming problem (LPP):

Maximize

Subject to

Given that

forms the optimal basis of the LPP with basis matrix

and respective

. If

is the optimal solution of the dual of the LPP, then which of the following is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

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Let

. Define

It is given that

is a Banach space with respect to the norm

Let

be the space of all real-valued continuous functions on

with the norm

. If

is the map

, then which of the following is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

is a compact linear map

Image of

is closed in

Image of

is dense in

is not a bounded linear map

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GATE MA year:2025 PYQs

Question 41

Question 42

Question 43

Question 44

Question 45

Question 46

Question 47

Question 48

Question 49

Question 50