ESAT ESAT Further Mathematics 3

Question 1

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

Accepted Answer

-2

Answer

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

Question 2

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

Accepted Answer

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

Answer

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

Question 3

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

Accepted Answer

cos(5θ) + i sin(5θ)

Answer

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).

Question 4

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

Accepted Answer

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

Answer

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.

Question 5

What is lim(x→0) of sin(x)/x?

Accepted Answer

1

Answer

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

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