GATE MA year:2019 PYQs & Answers — Page 3

Questions 21–30 of 65. Imported answers require independent review unless marked reviewed.

Let

be the co-countable topology on

(the set of real numbers) and

be the co-finite topology on

Consider the following statements:

I. In

, the sequence

converges to 0.

II. In

, the sequence

converges to 0.

III. In

, there is no sequence of rational numbers which converges to

IV. In

, there is no sequence of rational numbers which converges to

Which of the above statements are TRUE?

Your answerChoose one option.

I and II only

II and III only

III and IV only

I and IV only

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Let

and

be normed linear spaces, and let

be any bijective linear map with closed graph. Then which one of the following statements is TRUE?

Your answerChoose one option.

The graph of

is equal to $X\times Y

is continuous

The graph of

is closed

is continuous

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Let

be a function defined by

and

. (

is the set of all real numbers and

.) Which one of the following statements is TRUE?

Your answerChoose one option.

is injective

is the continuous inverse of

, defined in some neighbourhood of

, such that

, then the Jacobian of

is the continuous inverse of

, defined in some neighbourhood of

, such that

, then the Jacobian of

is surjective

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Let

(the set of all natural numbers). Then

is equal to _____.

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If the differential equation

is solved using the Euler’s method with step-size

, then

is equal to ____ (round off to 2 places of decimal).

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Let

be any polynomial function of degree at most 2 over

(the set of all real numbers). If the constants

and

are such that

then

is equal to _____ (round off to 2 places of decimal).

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Let

denote the value of the line integral

, where

, a circle of radius 2 with centre at origin of the

-plane is traversed once in the anti-clockwise direction. Then

is equal to _____.

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The temperature

at any point

is inversely proportional to the square of the distance of

from the origin. If the value of the temperature

at the point

, then the rate of change of

at the point

in the direction of

is equal to _____ (round off to 2 places of decimal).

(

is the set of all real numbers,

and

denotes

excluding the origin.)

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Let

be a continuous function defined on

such that

for all

. If the area bounded by

and

, where

, then

is equal to _____ (round off to 1 place of decimal).

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If the characteristic polynomial and minimal polynomial of a square matrix

are

and

, respectively, then the rank of the matrix

is _____, where

is the identity matrix of appropriate order.

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GATE MA year:2019 PYQs

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20