Mathematics & Engineering Fundamentals Flashcards

The Laplace transform of f(t) = t·e^(2t) is:

Answer: 1/(s-2)^2

Using the shifting theorem, L{t·e^(at)} = 1/(s-a)^2, so with a=2 the result is 1/(s-2)^2.

A matrix A is singular if and only if:

Answer: det(A) = 0

A singular matrix has a zero determinant, meaning it has no inverse and its rows/columns are linearly dependent.

Which numerical method has the fastest convergence rate for finding roots?

Answer: Newton-Raphson

Newton-Raphson converges quadratically (order 2), making it faster than bisection and false position near the root.

The curl of a conservative vector field F is:

Answer: Zero everywhere

A conservative field can be expressed as the gradient of a scalar potential, and the curl of any gradient is identically zero.

For a second-order linear ODE with constant coefficients, if the characteristic equation has repeated real roots r, the general solution is:

Answer: (C1 + C2·t)·e^(rt)

Repeated roots require the second independent solution to be multiplied by t to avoid linear dependence.

In probability, if P(A) = 0.4 and P(B) = 0.3 and events A and B are independent, what is P(A ∩ B)?

Answer: 0.12

For independent events, P(A ∩ B) = P(A)·P(B) = 0.4 × 0.3 = 0.12.

The Taylor series expansion of e^x about x = 0 truncated after the cubic term is:

Answer: 1 + x + x²/2 + x³/6

The Maclaurin series for e^x is Σ(x^n/n!), so the first four terms are 1 + x + x²/2! + x³/3! = 1 + x + x²/2 + x³/6.