Engineering Mathematics & Analytical Skills Flashcards

The rank of a matrix A is defined as:

Answer: The maximum number of linearly independent rows (or columns)

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.

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

Answer: sF(s) - f(0)

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

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

Answer: 3/28

P = (3/8) × (2/7) = 6/56 = 3/28.

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

Answer: y = xe^(-x)

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

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

Answer: The distribution is symmetric about the mean

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

Using Simpson's 1/3 rule, ∫₀^2 x² dx with n=2 subintervals gives:

Answer: 2.667

Simpson's 1/3: (h/3)[f(0)+4f(1)+f(2)] = (1/3)[0+4+4] = 8/3 ≈ 2.667, matching the exact value.

The divergence theorem relates a surface integral to:

Answer: A volume integral over the enclosed region

Gauss's divergence theorem states ∯F·dA = ∭(∇·F)dV, converting a closed surface integral to a volume integral.