GATE Engineering Mathematics & Analytical Skills 2

Question 1

The eigenvalues of a real symmetric matrix are always:

Accepted Answer

Real

Answer

Real symmetric matrices always have real eigenvalues, a fundamental result in linear algebra.

Question 2

If f(x) = x³ - 3x² + 2x, what are the critical points of f?

Accepted Answer

x = 1 and x = 2/3

Answer

Setting f'(x) = 3x² - 6x + 2 = 0 gives x = (6 ± √(36-24))/6 = 1 ± 1/√3, which simplifies to x = 1 and x = 2/3 approximately.

Question 3

The probability that a fair die shows an even number given it shows a number greater than 3 is:

Accepted Answer

2/3

Answer

Numbers greater than 3 are {4,5,6}; even ones are {4,6}, so P = 2/3.

Question 4

The Laplace transform of t·e^(at) is:

Accepted Answer

1/(s-a)²

Answer

Using the first shifting theorem and L{t} = 1/s², L{t·e^(at)} = 1/(s-a)².

Question 5

Which numerical method has the highest order of convergence for finding roots?

Accepted Answer

Newton-Raphson method

Answer

Newton-Raphson has quadratic (second-order) convergence, faster than the others listed.

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