GATE MA 2025 Paper 1 — page 16

Questions 46–48 of 65. Imported answers require review unless explicitly marked reviewed.

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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2025 · Paper 1

Question 46

Question 47

Question 48