Engineering Mathematics & Analytical Skills Flashcards

For matrix A = [[2,1],[1,2]], the spectral radius (largest eigenvalue magnitude) is:

Answer: 3

Eigenvalues satisfy (2-λ)²-1=0, giving λ=1 and λ=3; the spectral radius is 3.

A function f(x,y) has a saddle point at (a,b) if the second-order test gives D = f_xx·f_yy - (f_xy)² equal to:

Answer: D < 0

When D < 0, the critical point is a saddle point (neither a maximum nor minimum).

The Z-transform of the unit step sequence u[n] (u[n]=1 for n≥0) is:

Answer: z/(z-1)

Z{u[n]} = Σ z^(-n) for n=0 to ∞ = 1/(1-z⁻¹) = z/(z-1) for |z|>1.

In a Gaussian elimination, partial pivoting is used to:

Answer: Reduce round-off errors by choosing the largest pivot

Partial pivoting selects the largest available element as pivot to improve numerical stability and reduce round-off errors.

The mean and variance of a Binomial distribution B(n,p) are, respectively:

Answer: np and np(1-p)

For B(n,p), mean = np and variance = npq where q = 1-p.

The inverse Fourier transform of a rectangular pulse in the frequency domain produces which time-domain signal?

Answer: A sinc function

A rectangular pulse in the frequency domain transforms to a sinc function (sin(πt)/(πt)) in the time domain.

Which of the following is a necessary condition for a maximum of f(x) at x=c?

Answer: f'(c) = 0

f'(c) = 0 is a necessary condition (stationary point), though not sufficient to confirm a maximum.