Engineering Mathematics & Analytical Skills Flashcards

The eigenvalues of a real symmetric matrix are always:

Answer: Real

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

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

Answer: x = 1 and x = 2/3

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.

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

Answer: 2/3

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

The Laplace transform of t·e^(at) is:

Answer: 1/(s-a)²

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

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

Answer: Newton-Raphson method

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

For a standard normal distribution, approximately what percentage of data lies within two standard deviations of the mean?

Answer: 95%

The empirical rule states approximately 95% of data lies within ±2σ of the mean.

The curl of a gradient field ∇f is:

Answer: Zero vector

For any scalar field f with continuous second-order partial derivatives, ∇×(∇f) = 0.