ESAT Further Mathematics Flashcards

For the matrix M with rows [1, 2] and [3, 4], which entry appears in row 1, column 1 of M^(-1)?

Answer: -2

det(M) = 1·4 - 2·3 = -2; M^(-1) = (1/-2)·[[4,-2],[-3,1]], so row 1, col 1 = 4/(-2) = -2.

What is the general solution to the differential equation d²y/dx² + 4y = 0?

Answer: y = A cos(2x) + B sin(2x)

The characteristic equation r² + 4 = 0 has roots r = ±2i, giving the general solution y = A cos(2x) + B sin(2x).

According to De Moivre's theorem, (cos θ + i sin θ)^5 equals which of the following?

Answer: cos(5θ) + i sin(5θ)

De Moivre's theorem states (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ), so with n = 5 the result is cos(5θ) + i sin(5θ).

What is the partial fraction decomposition of (5x - 1) / [(x+1)(x-2)]?

Answer: 2/(x+1) + 3/(x-2)

Writing A/(x+1) + B/(x-2): substituting x = -1 gives -6 = -3A so A = 2; substituting x = 2 gives 9 = 3B so B = 3.

What is lim(x→0) of sin(x)/x?

Answer: 1

lim(x→0) sin(x)/x = 1 is a fundamental limit, provable via L'Hôpital's rule or the squeeze theorem.

For vectors a = (1, 2, 3) and b = (4, 5, 6), what is the dot product a · b?

Answer: 32

a · b = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32.

What is the Maclaurin series for e^x up to and including the x³ term?

Answer: 1 + x + x²/2 + x³/6

The Maclaurin series for e^x is Σ xⁿ/n!, so 1 + x + x²/2! + x³/3! = 1 + x + x²/2 + x³/6.