GATE MA area:engineering-mathematics PYQs & Answers — Page 12

Questions 111–120 of 403. Imported answers require independent review unless marked reviewed.

Let

be given by

. Let

be the

zero matrix. The space

may be identified with

in the usual manner. Which one of the following is correct?

Your answerChoose one option.

is a point of local minimum of

is a point of local maximum of

is a saddle point of

is not a critical point of

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Consider the following statements.

I. There exists a proper subgroup

such that

is a finite group.

II. There exists a subgroup

such that

is isomorphic to

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 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

. 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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Let

be two non-zero, non-identity

-linear transformations. Assume

. Which of the following is/are TRUE?

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

is necessarily invertible

and

are similar if

and

are similar if

has only 0 and 1 as eigenvalues

is necessarily diagonalizable

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GATE MA area:engineering-mathematics PYQs

Question 39

Question 40

Question 41

Question 42

Question 46

Question 47

Question 48

Question 49

Question 50

Question 51