GATE MA Real Analysis PYQs & Answers — Page 3

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Questions 21–30 of 43. Imported answers require independent review unless marked reviewed.

Topic practice 43 questionsPage 3 of 5

Let

. Then, which of the following statements is TRUE?

Your answerChoose one option.

is not equicontinuous on

is uniformly convergent on

is equicontinuous on

is uniformly bounded and has a subsequence converging uniformly on

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Let

be given by

. Consider the following statements:

for all

has a fixed point.

Then

Your answerChoose one option.

both P and Q are TRUE

P is TRUE and Q is FALSE

P is FALSE and Q is TRUE

both P and Q are FALSE

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Consider the following statements:

is a metric on

Then

Your answerChoose one option.

both P and Q are TRUE

P is TRUE and Q is FALSE

P is FALSE and Q is TRUE

both P and Q are FALSE

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Let

be given by

for

. Consider the following statements: P:

is equicontinuous on

. Q:

does NOT converge uniformly on

. Then

Your answerChoose one option.

both P and Q are TRUE

P is TRUE and Q is FALSE

P is FALSE and Q is TRUE

both P and Q are FALSE

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For each

, consider the decimal representation

. Define

is rational and

is irrational, where

is the number of zeroes immediately after the decimal point up to the first nonzero digit in the decimal representation of

. Then the Lebesgue integral

__________.

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For

, define the sequence

Then the sequence

converges to __________ (correct up to two decimal places)

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Let

denote the power set of

, equipped with the metric

, where

and

denote the characteristic functions of the subsets

and

, respectively, of

. The set

in the metric space

Your answerChoose one option.

bounded but not totally bounded

totally bounded but not compact

compact

not bounded

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Let

be defined by

, where

is the characteristic function of the interval

. For

, let

. Then

Your answerChoose one option.

is open

is not measurable

is closed

is measurable

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For

, let

be functions defined by

and

. Then

Your answerChoose one option.

converges uniformly but

does not converge uniformly

converges uniformly but

does not converge uniformly

both

and

converge uniformly

neither

nor

converge uniformly

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Suppose that

and

are metric spaces with metrics induced by the Euclidean metric of

. Let

and

be the open unit balls around

and

, respectively. Consider the following statements: I: The closure of

is compact. II: The closure of

is compact. Then

Your answerChoose one option.

both I and II are true

I is true but II is false

I is false but II is true

both I and II are false

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GATE MA Real Analysis PYQs

Question 40

Question 10

Question 11

Question 36

Question 54

Question 21

Question 30

Question 31

Question 32

Question 35