GATE MA topic:mathematical-sciences PYQs & Answers — Page 13

Questions 121–130 of 650. Imported answers require independent review unless marked reviewed.

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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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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Consider the inner product space of all real-valued continuous functions defined on

with the inner product

. If

is orthogonal to all the polynomials having degree less than or equal to

, with respect to this inner product, then

is equal to _____ (in integer)

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For

, consider the quadratic form:

Let

be the symmetric matrix associated with the quadratic form

with respect to the standard basis of

. Let

be a non-zero vector, and let

where

is the

identity matrix. Then, the value of

is _____ (in integer)

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Let

be distinct non-zero real numbers, and let

be a polynomial of degree less than 5. If the function

satisfies Morera’s theorem in

, then the value of

is equal to _____ (in integer)

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Let

be a group with identity element

, and let

be such that the following hold:

(i)

(ii)

, and

Then, the least positive integer

for which

is _____ (in integer)

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Let

and

be two metric spaces with

and

If the open ball centred at

with radius

is equal to the open ball centred at

with radius

, then the value of

is _____ (in integer)

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GATE MA topic:mathematical-sciences PYQs

Question 56

Question 57

Question 58

Question 59

Question 60

Question 61

Question 62

Question 63

Question 64

Question 65