GATE MA topic:mathematical-sciences PYQs & Answers — Page 12

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Questions 111–120 of 650. Imported answers require independent review unless marked reviewed.

Topic practice 650 questionsPage 12 of 65

Consider the metric spaces

and

with the metrics defined by

and

, respectively. Then, which one of the following is TRUE?

Your answerChoose one option.

is open in

but not in

is open in

but not in

is open in both

and

is open neither in

nor in

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Let

be an algebraically closed field containing a finite field

. Let

be the subfield of

consisting of elements of

that are algebraic over

Consider the following statements:

S1:

is algebraically closed.

S2:

is infinite.

Then, which one of the following is correct?

Your answerChoose one option.

S1 is TRUE and S2 is FALSE

S2 is TRUE and S1 is FALSE

both S1 and S2 are TRUE

neither S1 nor S2 is TRUE

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Let

be the vector space (over

) of all

matrices with entries in

. Consider the linear transformation

defined by

, where

and

. If

is the matrix representation of

with respect to the standard basis of

, then which of the following is/are TRUE?

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

is an invertible matrix

The trace of

is 25

The rank of

is 4, where

is the

identity matrix

The nullity of

is 0, where

is the

identity matrix

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Consider the linear programming problem (LPP):

Maximize

Subject to

Given that

forms the optimal basis of the LPP with basis matrix

and respective

. If

is the optimal solution of the dual of the LPP, then which of the following is/are TRUE?

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

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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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GATE MA topic:mathematical-sciences PYQs

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