GATE MA purpose:exam-prep PYQs & Answers — Page 57

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Questions 561–570 of 650. Imported answers require independent review unless marked reviewed.

Topic practice 650 questionsPage 57 of 65

For an odd prime

, consider the ring

. Then the element 2 in

Your answerChoose one option.

a unit

a square

a prime

irreducible

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Consider the following two statements:

P: The matrix

has infinitely many LU factorizations, where

is lower triangular with each diagonal entry 1 and

is upper triangular.

Q: The matrix

has no LU factorization, where

is lower triangular with each diagonal entry 1 and

is upper triangular.

Then which one of the following options is correct?

Your answerChoose one option.

P is TRUE and Q is FALSE

Both P and Q are TRUE

P is FALSE and Q is TRUE

Both P and Q are FALSE

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If the characteristic curves of the partial differential equation

are

and

, where

and

are constants, then

Your answerChoose one option.

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Let

be a continuous map from a Hausdorff topological space

to a metric space

. Consider the following two statements:

is a closed map and the inverse image

is compact for each

Q: For every compact subset

, the inverse image

is a compact subset of

Which one of the following is true?

Your answerChoose one option.

Q implies P but P does NOT imply Q

P implies Q but Q does NOT imply P

P and Q are equivalent

neither P implies Q nor Q implies P

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Let

denote

endowed with the usual topology. Let

denote

endowed with the co-finite topology. If

is the product topological space

, then

Your answerChoose one option.

the topology of

is the same as the topology of $Z

the topology of

is strictly coarser (weaker) than that of

the topology of

is strictly coarser (weaker) than that of

the topology of

cannot be compared with that of

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Consider

with the usual topology for

. Each of the following options gives topological spaces

and

with respective induced topologies. In which option is

homeomorphic to

Your answerChoose one option.

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Let

be a sequence of independent Poisson(

) variables and let

. Then the limiting distribution of

is the normal distribution with zero mean and variance given by

Your answerChoose one option.

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Let

be independent and identically distributed random variables with probability density function given by

Also, let

. Then the maximum likelihood estimator of

Your answerChoose one option.

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Consider the Linear Programming Problem (LP):

Maximize

Subject to

where

is a constant. If

is the only optimal solution, then

Your answerChoose one option.

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Let

be the vector space of all

real matrices over the field

. Define the linear transformation

, where

denotes the transpose of the matrix

. Then the trace of

equals _________.

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