GATE MA format:msq PYQs & Answers — Page 4

Questions 31–40 of 62. Imported answers require independent review unless marked reviewed.

Topic practice 62 questionsPage 4 of 7

Let

and

be field extensions which are not algebraic. Let

be algebraic over

and

. Let

be the subfield of

generated over

by the coefficients of the monic polynomial of minimal degree over

which has

as a zero. Then, which of the following is/are TRUE?

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

is a finite extension if and only if

is a finite extension

The dimension of

over

is greater than the dimension of

over

The dimension of

over

is smaller than the dimension of

over

is an algebraic extension if and only if

is an algebraic extension

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Let

endowed with the inner product

Let

be given by

, where

Which of the following is/are correct?

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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?

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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?

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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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Let

be the two fixed points of the function

, where

. For

, let the sequence

be generated by the fixed point iteration

Which one of the following is/are correct?

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converges to

for any

converges to

for any

converges to

for any

converges to

for any

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Which of the following is/are eigenvalue(s) of the Sturm–Liouville problem

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Let

be a function such that

Which of the following is/are correct?

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is continuous at

, but not differentiable at

is differentiable at

All the directional derivatives of

exist and they are equal to zero

Both the partial derivatives of

exist and they are equal to zero

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For an integer

, let

, where

. Let

. Consider the metric space

, where

Which of the following statement(s) is/are true?

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is an equi-continuous family of continuous functions

is closed in

is bounded in

is compact in

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Let

be an

-linear transformation such that 1 and 2 are the only eigenvalues of

. Suppose the dimensions of

and

are 1 and 2, respectively. Which of the following is/are possible (upper triangular) Jordan canonical form(s) of

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

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GATE MA format:msq PYQs

Question 60

Question 48

Question 49

Question 50

Question 51

Question 52

Question 53

Question 54

Question 55

Question 56