GATE Engineering Mathematics & Analytical Skills 3

Question 1

The rank of a matrix A is defined as:

Accepted Answer

The maximum number of linearly independent rows (or columns)

Answer

Rank is the dimension of the column space (or row space) of a matrix, equal to the max number of linearly independent rows or columns.

Question 2

If ∫₀^∞ e^(-st) f(t) dt = F(s), then ∫₀^∞ e^(-st) f'(t) dt equals:

Accepted Answer

sF(s) - f(0)

Answer

Using integration by parts, the Laplace transform of a derivative is L{f'(t)} = sF(s) - f(0).

Question 3

A bag contains 3 red and 5 blue balls. Two balls are drawn without replacement. What is the probability both are red?

Accepted Answer

3/28

Answer

P = (3/8) × (2/7) = 6/56 = 3/28.

Question 4

The solution to the differential equation dy/dx + y = e^(-x) with y(0)=0 is:

Accepted Answer

y = xe^(-x)

Answer

The integrating factor is e^x; solving gives y = xe^(-x) after applying the initial condition.

Question 5

Which of the following is NOT a property of a Poisson distribution?

Accepted Answer

The distribution is symmetric about the mean

Answer

Poisson distributions are right-skewed (not symmetric), though they approach symmetry as λ → ∞.

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