GATE MA topic:mathematical-sciences PYQs & Answers — Page 18

Questions 171–180 of 650. Imported answers require independent review unless marked reviewed.

Let

be the space

with the quotient topology induced from the usual topology on

. Consider the following statements.

is compact.

II.

is connected for any

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Let

denote the standard inner product on

. Let

be a set of unit vectors such that

is a non-positive integer for all

. Define

to be the number of pairs

, such that

. The maximum possible value of

is equal to

Your answerChoose one option.

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Let

. Which one of the following numerical integration rules gives the exact value of the integral

Your answerChoose one option.

The Simpson’s rule

The trapezoid rule

The composite Simpson’s rule by dividing

into 4 equal subintervals

The composite trapezoid rule by dividing

into 3 equal subintervals

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Consider the initial value problem (IVP)

I. IVP has a unique solution on

II. Every solution of IVP is bounded on its maximal interval of existence.

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Let

be a

non-diagonalizable real matrix with a real eigenvalue

and

be an eigenvector of

corresponding to

. Which one of the following is the general solution of the system

of first-order linear differential equations?

Your answerChoose one option.

, where

and

is a

real column vector such that

, where

and

is a

real column vector such that

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Let

. If the following second-order linear partial differential equation

is transformed to

for some

, via the co-ordinate transform

and

, then which one of the following is correct?

Your answerChoose one option.

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Let

for

. Let

. For

, define

. Consider the following statements.

I. There exists a continuous linear functional

such that

II. There exists a continuous linear functional

such that

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Let

endowed with the inner product

Let

be given by

, where

Which of the following is/are correct?

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

is a compact operator

The operator norm of

is 1

is a self-adjoint operator

is closed

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Let

be the normed space

, where

Let

and

be given by

for all

. Recall that a Hahn–Banach extension of

is a continuous linear functional

such that

and

, where

and

are the norms of

and

, respectively. Which of the following is/are true?

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

is a Hahn–Banach extension of

admits infinitely many Hahn–Banach extensions to

admits exactly two distinct Hahn–Banach extensions to

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Let

be a basis for a topology

. Which of the following is/are correct?

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

Every

with

is an open set in

Every

with

is compact in

is a first-countable space

is a second-countable space

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GATE MA topic:mathematical-sciences PYQs

Question 41

Question 42

Question 43

Question 44

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Question 47

Question 48

Question 49

Question 50