GATE MA topic:real-analysis PYQs & Answers — Page 4

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topic:real-analysis

Questions 31–40 of 43. Imported answers require independent review unless marked reviewed.

Topic practice 43 questionsPage 4 of 5

The value of

is (rounded off to three decimal places)

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then which one of the following statements is TRUE?

Your answerChoose one option.

Both the sequence

and the series

are convergent

Both the sequence

and the series

are divergent

The sequence

is convergent but the series

is divergent

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Let

and only finitely many

's are non-zero

and

be a metric on

defined by

(

is the set of all real numbers and

is the set of all natural numbers)

Consider the following statements:

is a complete metric space.

: The set

is compact, where

is the zero element of

Which of the above statements is/are TRUE?

Your answerChoose one option.

Both

and

only

Neither

nor

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Consider the following statements:

I. The set

is uncountable.

II. The set

is uncountable.

III. The set

is uncountable.

IV. For any infinite set, there exists a bijection from the set to one of its proper subsets.

(

is the set of all rational numbers,

is the set of all integers and

is the set of all natural numbers)

Which of the above statements are TRUE?

Your answerChoose one option.

I and IV only

II and IV only

II and III only

I, II and IV only

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Let

(the set of all natural numbers). Then

is equal to _____.

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Consider the sequence

of functions, where

and

is the derivative of

with respect to

(

is the set of all real numbers,

is the set of all natural numbers.)

Then which one of the following statements is TRUE?

Your answerChoose one option.

does NOT converge uniformly on

converges uniformly on any closed interval which does NOT contain 1

converges point-wise to a continuous function on

converges uniformly on any closed interval which does NOT contain 0

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Let

be the set of all rational numbers in the interval

be the Cantor (ternary) set, and let

. Then the Lebesgue measure of the set

equals ________.

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for

, then

_________.

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For

, let

. Then

_________.

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Let

be given by

. Then the sequence

Your answerChoose one option.

converges uniformly on

does NOT converge uniformly on

but has a subsequence that converges uniformly on

does NOT converge pointwise on

converges pointwise on

but does NOT have a subsequence that converges uniformly on

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GATE MA topic:real-analysis PYQs

Question 55

Question 5

Question 7

Question 8

Question 14

Question 38

Question 16

Question 50

Question 51

Question 5