GATE MA format:mcq PYQs & Answers — Page 29

Questions 281–290 of 366. Imported answers require independent review unless marked reviewed.

Let

be non-zero and analytic at all points in

. If

for

, then the residue of

is _____.

(

is the set of all complex numbers,

is the set of all integers and

denotes the set of all complex numbers excluding integers)

Your answerChoose one option.

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The general integral of the partial differential equation

is given by

, where

is a continuously differentiable function.

(

is the set of all real numbers and

)

Then which one of the following is TRUE?

Your answerChoose one option.

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Consider the following statements:

I. If

denotes the additive group of rational numbers and

is a non-trivial homomorphism, then

is an isomorphism.

II. Any quotient group of a cyclic group is cyclic.

III. If every subgroup of a group

is a normal subgroup, then

is abelian.

IV. Every group of order 33 is cyclic.

Which of the above statements are TRUE?

Your answerChoose one option.

II and IV only

II and III only

I, II and IV only

I, III and IV only

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A solution of the Dirichlet problem

is given by

Your answerChoose one option.

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Consider the subspace

of the normed linear space

. If

is a bounded linear functional on

, defined by

, then which one of the following sets is equal to

(

is the set of all complex numbers,

and

)

Your answerChoose one option.

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Consider the following statements:

I. The ring

is a unique factorization domain.

II. The ring

is a principal ideal domain.

III. In the polynomial ring

, the ideal generated by

is a maximal ideal.

IV. In the polynomial ring

, the ideal generated by

is a prime ideal.

(

denotes the set of all integers,

denotes the set of all integers modulo

, for any positive integer

)

Which of the above statements are TRUE?

Your answerChoose one option.

I, II and III only

I and II only

I, II and IV only

II and III only

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Let

be a

real symmetric matrix with eigenvalues

and

with the respective eigenvectors

and

Consider the following statements:

II. The vector

satisfies

III. For any

has a solution.

IV. The trace of the matrix

is 8.

(

denotes the transpose of the vector

)

Which of the above statements are TRUE?

Your answerChoose one option.

I, II and III only

I and II only

II and IV only

III and IV only

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Consider the region

in the complex plane. The transformation

maps the region

onto the region

(the set of all complex numbers). Then the area of the region

is equal to

Your answerChoose one option.

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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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Consider the boundary value problem (BVP)

with the boundary conditions

(

is a non-zero real number).

Then which one of the following statements is TRUE?

Your answerChoose one option.

For

, the BVP has infinitely many solutions

For

, the BVP has a unique solution

For

, the BVP has a solution

such that

for all

For

, the BVP has a solution

such that

for all

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GATE MA format:mcq PYQs

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