A-Level H2 Mathematics 5

Question 1

A random variable X ~ N(50, 16). Find P(X > 54).

Accepted Answer

0.1587

Answer

Z = (54-50)/4 = 1. P(X>54) = P(Z>1) = 1 - Φ(1) = 1 - 0.8413 = 0.1587.

Question 2

Find the inverse of the matrix [[3,1],[5,2]].

Accepted Answer

[[2,-1],[-5,3]]

Answer

det = 3×2 - 1×5 = 1. Inverse = (1/1)[[2,-1],[-5,3]] = [[2,-1],[-5,3]].

Question 3

For X ~ B(10, 0.3), find P(X = 3).

Accepted Answer

0.2668

Answer

P(X=3) = ¹⁰C₃ × (0.3)³ × (0.7)⁷ = 120 × 0.027 × 0.0823... = 120 × 0.02223... ≈ 0.2668.

Question 4

Find the exact value of ∫₀^(π/2) sin(x) dx.

Accepted Answer

1

Answer

∫₀^(π/2) sin(x) dx = [-cos(x)]₀^(π/2) = -cos(π/2) + cos(0) = 0 + 1 = 1.

Question 5

Differentiate y = ln(3x² + 1).

Accepted Answer

6x/(3x²+1)

Answer

Using chain rule: dy/dx = 1/(3x²+1) × 6x = 6x/(3x²+1).

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