GATE MA discipline:mathematics-and-statistics PYQs & Answers — Page 51

Questions 501–510 of 650. Imported answers require independent review unless marked reviewed.

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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Consider the ordered square

, the set

with the dictionary order topology. Let the general element of

be denoted by

, where

. Then the closure of the subset

Your answerChoose one option.

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Let

be the vector space of all polynomials of degree at most 2 over

(the set of real numbers). Let a linear transformation

be defined by

Consider the following statements.

I. The null space of

II. The range space of

is spanned by the set

III.

IV. If

is the matrix representation of

with respect to the standard basis

, then the trace of the matrix

is 3.

Which of the above statements are TRUE?

Your answerChoose one option.

I and II only

I, III and IV only

I, II and IV only

II and IV only

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Let

and

be two topologies defined on

(the set of all natural numbers), where

is the topology generated by

and

is the discrete topology on

. Consider the following statements:

I. In

, every infinite subset has a limit point.

II. The function

defined by

is a continuous function.

Which of the above statements is/are TRUE?

Your answerChoose one option.

Both I and II

I only

II only

Neither I nor II

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Let

. Consider the following statements:

II.

where

and

(

is the set of all real numbers)

Which of the above statements is/are TRUE?

Your answerChoose one option.

Both I and II

I only

II only

Neither I nor II

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Consider the differential equation

is the Laplace transform of

, then the value of

is ______ (round off to 2 places of decimal). (Here, the inverse trigonometric functions assume principal values only)

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Let

be the region in the

-plane bounded by the curves

and

Then the value of the integral

is equal to ______.

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GATE MA discipline:mathematics-and-statistics PYQs

Question 36

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