GATE MA format:msq PYQs & Answers — Page 3

Questions 21–30 of 62. Imported answers require independent review unless marked reviewed.

Topic practice 62 questionsPage 3 of 7

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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Let

be the solution of the initial value problem

and let

be the solution of the initial value problem

Then, which of the following is/are TRUE?

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

for all

and for all

is not defined for certain values of

is not defined for any

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Let

be the solution of the initial-boundary value problem

Then, which of the following is/are TRUE?

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

for all

and

for all

and

is a decreasing function of

is not a decreasing function of

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Consider the function

given by

Then, which of the following is/are TRUE?

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

is open in

, then

is open in

is closed in

, then

is closed in

is dense in

, then

is dense in

is surjective

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Let

be an orthonormal set of vectors in a real Hilbert space

with inner product

. Let

, and let

be the linear span of

over

. For

, let

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

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

is the orthogonal projection of

onto

is the orthogonal projection of

onto

is orthogonal to

for all

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Consider the sequence

of continuous functions on

defined by

Then, which of the following is/are TRUE?

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

The sequence

converges pointwise but not uniformly on

The sequence

converges uniformly on

for all

and

for all

and

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For

, let

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

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

converges uniformly on

converges pointwise but not uniformly on

converges uniformly on every compact subset of

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Let

and

be provided with the respective Euclidean topologies, and let

be assigned the subspace topology induced from

. If

is a non-constant continuous function, then which of the following is/are TRUE?

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

maps closed sets to closed sets

is injective

is surjective

There exists

such that

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Let

be an uncountable set. Let the topology on

be defined by declaring a subset

to be open if

is either empty or finite or countable, and the empty set to be open. Then, which of the following is/are TRUE?

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

Every compact subset of

is closed

Every closed subset of

is compact

(singleton subsets are closed) but not

(Hausdorff)

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All rings considered below are assumed to be associative and commute with

. Further, all ring homomorphisms map

. Consider the following statements about such a ring

P1:

is isomorphic to the product of two rings

and

P2:

such that

and

P3:

ideals

with

such that

and

P4:

with

such that

Then, which of the following is/are TRUE?

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

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GATE MA format:msq PYQs

Question 50

Question 51

Question 52

Question 53

Question 54

Question 55

Question 56

Question 57

Question 58

Question 59