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

4

Answer

1/2

Answer

-1/2

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

y = Ae^(2x) + Be^(-2x)

Answer

y = (A + Bx)e^(2x)

Answer

y = A cos(4x) + B sin(4x)

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

5cos(θ) + 5i sin(θ)

Answer

cos(θ/5) + i sin(θ/5)

Answer

cos²(5θ) - sin²(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

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

Answer

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

Answer

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

Question 5

What is lim(x→0) of sin(x)/x?

Accepted Answer

1

Answer

0

Answer

∞

Answer

undefined

Question 6

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

Accepted Answer

32

Answer

17

Answer

28

Answer

45

Question 7

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

Accepted Answer

1 + x + x²/2 + x³/6

Answer

1 + x + x² + x³

Answer

x + x²/2 + x³/6

Answer

1 + x + 2x² + 6x³