GATE MA topic:mathematical-sciences PYQs & Answers — Page 41

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Questions 401–410 of 650. Imported answers require independent review unless marked reviewed.

Topic practice 650 questionsPage 41 of 65

Suppose that

and

are topologies on

induced by metrics

and

, respectively, such that

. Then which of the following statements is TRUE?

Your answerChoose one option.

If a sequence converges in

then it converges in

If a sequence converges in

then it converges in

Every open ball in

is an open ball in

The map

from

is continuous

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Let

. If the function

is defined by

then

Your answerChoose one option.

is continuous at

both the first order partial derivatives of

exist at

is finite

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The initial value problem

has

Your answerChoose one option.

a unique solution if

no solution if

infinitely many solutions if

a unique solution if

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Consider the following statements:

is harmonic on

II:

has a harmonic conjugate on

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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Let

and

be defined by

Suppose

and

are analytic functions. Consider the following statements:

is independent of paths

joining

and

II:

is independent of paths

joining

and

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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Let

and let, for

If for a subset

denotes the closure of

, then

Your answerChoose one option.

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Suppose that

Then, with respect to the Euclidean metric on

Your answerChoose one option.

both

and

are disconnected

is disconnected but

is connected

is connected but

is disconnected

both

and

are connected

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If (D1) and (D2) denote the dual problems of the linear programming problems (P1) and (P2), respectively, where

(P1): minimize

subject to

(P2): minimize

subject to

then

Your answerChoose one option.

both (D1) and (D2) are infeasible

(P2) is infeasible and (D2) is feasible

(D1) is infeasible and (D2) is feasible but unbounded

(P1) is feasible but unbounded and (D1) is feasible

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and

are critical points of the function

where

, then

Your answerChoose one option.

is a point of local maxima of

is a saddle point of

is a point of local minima of

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Consider the iterative scheme

with initial point

. Then the sequence

Your answerChoose one option.

converges only if

converges for any

does not converge for any

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GATE MA topic:mathematical-sciences PYQs

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