GATE MA branch:ma PYQs & Answers — Page 24

Questions 231–240 of 650. Imported answers require independent review unless marked reviewed.

Topic practice 650 questionsPage 24 of 65

Let

denote the group of

invertible complex matrices with usual matrix multiplication. For

denotes the subgroup generated by

and

. Let

and

be three subgroups of

given by

Let

denote the center of

for

. Which of the following statements is correct?

Your answerChoose one option.

is isomorphic to

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Let

. For a sequence

, define

. Let

and

be defined by

Then

Your answerChoose one option.

is a compact linear map and

is NOT a compact linear map

is NOT a compact linear map and

is a compact linear map

both

and

are compact linear maps

NEITHER

NOR

is a compact linear map

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Let

. For

, let

. Define

for all

. Then

Your answerChoose one option.

is continuous but

is NOT continuous

is NOT continuous but

is continuous

both

and

are continuous

NEITHER

NOR

is continuous

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Consider the Cauchy problem

on the initial curve

. Consider the following statements:

: If

, then there exists a unique solution to the Cauchy problem in a neighborhood of

: If

, then there exist infinitely many solutions to the Cauchy problem in a neighborhood of

Then

Your answerChoose one option.

both

and

are TRUE

is FALSE and

is TRUE

is TRUE and

is FALSE

both

and

are FALSE

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Consider the linear system

, where

and

. Suppose

, where

and

are lower triangular and upper triangular square matrices, respectively. Consider the following statements:

: If each element of the main diagonal of

is 1, then

: For any choice of the initial vector

, the Jacobi iterates

converge to the unique solution of the linear system

Then

Your answerChoose one option.

both

and

are TRUE

is FALSE and

is TRUE

is TRUE and

is FALSE

both

and

are FALSE

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Let

and

be two linearly independent solutions of the ordinary differential equation

Let

be such that

and

for all

. Consider the following statements:

for all

Then

Your answerChoose one option.

is TRUE and

is FALSE

is FALSE and

is TRUE

both

and

are FALSE

both

and

are TRUE

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Let

and

be an analytic function given by the power series

, where

and

for

. Consider the following statements:

: If

, then

is one-one in some neighbourhood of

: If

, then

is a closed subset of

Which of the following statements is/are correct?

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

is TRUE

is FALSE

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Let

be an open connected subset of

containing

. Let

. Consider the following statements:

: There exists

such that

for all

, where

denotes the third derivative of

Then

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

is TRUE

is FALSE

is TRUE

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Let

be a topological space, where the topology

is defined as

Which of the following statements is/are correct?

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

is first countable

is Hausdorff

is separable

The closure of

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Let

, where

denotes the set of rational numbers and

denotes the set of integers. For

, let

denote the ideal generated by

Which of the following statements is/are correct?

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

is an irreducible element in

, then

is a prime ideal in

is a unique factorization domain

is a prime ideal in

is NOT a principal ideal domain

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GATE MA branch:ma PYQs

Question 36

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