GATE MA purpose:exam-prep PYQs & Answers — Page 37

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Questions 361–370 of 650. Imported answers require independent review unless marked reviewed.

Topic practice 650 questionsPage 37 of 65

Let

denote the ring of integers. Consider the subring

of the field

of complex numbers. Consider the following statements: P:

is an irreducible element. Q:

is a prime element. 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 second-order partial differential equation (PDE)

Consider the following statements: P: The PDE is parabolic on the ellipse

. Q: The PDE is hyperbolic inside the ellipse

. 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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is the solution for the Cauchy problem

then

is equal to

Your answerChoose one option.

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Let

be the solution of the initial value problem

obtained by the method of Laplace transform. Then

Your answerChoose one option.

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The critical point of the differential equation

is a

Your answerChoose one option.

node and is asymptotically stable

spiral point and is asymptotically stable

node and is unstable

saddle point and is unstable

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The initial value problem

where

, is solved by the following Euler method

with step-size

. Then

, provided

Your answerChoose one option.

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Consider the Linear Programming Problem

subject to

where

and

are real numbers (

Let

be a feasible solution of

such that

and let all feasible solutions

satisfy

. Then, which one of the following statements is NOT true?

Your answerChoose one option.

has an optimal solution

The feasible region of

is a bounded set

is a feasible solution of the dual of

, then

The dual of

has at least one feasible solution

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Let

be the Hilbert space of real valued square integrable functions on

equipped with the norm

. Consider the subspace

. For

, define

. Then

Your answerChoose one option.

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Let

be the Banach space of real valued continuous functions on

equipped with the supremum norm. Define

Let

denote the range space of

. Consider the following statements: P:

is a bounded linear operator. Q:

exists and is bounded. 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 the sequence space equipped with the norm

. Consider the subspace

and the linear transformation

given by

for

. Then

Your answerChoose one option.

is closed but NOT bounded

is bounded

is neither closed nor bounded

exists and is an open map

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