GATE MA kind:question PYQs & Answers — Page 48

Questions 471–480 of 650. Imported answers require independent review unless marked reviewed.

Let

and

be a function whose all second order partial derivatives exist and are continuous. If

satisfies the Laplace equation

for all

, then which one of the following statements is TRUE in

(

is the set of all real numbers, and

)

Your answerChoose one option.

is solenoidal but not irrotational

is irrotational but not solenoidal

is both solenoidal and irrotational

is neither solenoidal nor irrotational

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Let

and only finitely many

's are non-zero

and

be a metric on

defined by

(

is the set of all real numbers and

is the set of all natural numbers)

Consider the following statements:

is a complete metric space.

: The set

is compact, where

is the zero element of

Which of the above statements is/are TRUE?

Your answerChoose one option.

Both

and

only

Neither

nor

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Consider the following statements:

I. The set

is uncountable.

II. The set

is uncountable.

III. The set

is uncountable.

IV. For any infinite set, there exists a bijection from the set to one of its proper subsets.

(

is the set of all rational numbers,

is the set of all integers and

is the set of all natural numbers)

Which of the above statements are TRUE?

Your answerChoose one option.

I and IV only

II and IV only

II and III only

I, II and IV only

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Let

be defined by

(

is the set of all real numbers and

)

Which one of the following statements is TRUE?

Your answerChoose one option.

has a local maximum at origin

has a local minimum at origin

has a saddle point at origin

The origin is not a critical point of

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Let

be any sequence of real numbers such that

. If the radius of convergence of

, then which one of the following statements is necessarily TRUE?

Your answerChoose one option.

is infinite

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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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