A-Level H2 Mathematics 4

Question 1

What is the modulus and argument of the complex number z = 1 + i√3?

Accepted Answer

modulus 2, argument π/3

Answer

|z| = √(1² + (√3)²) = √(1+3) = 2. arg(z) = arctan(√3/1) = π/3 (first quadrant).

Question 2

The function f(x) = x³ - 3x² + 2. Find f'(x) and determine the nature of all stationary points.

Accepted Answer

f'(x) = 3x²-6x; maximum at x=0, minimum at x=2

Answer

f'(x) = 3x²-6x = 3x(x-2). Stationary at x=0, x=2. f''(x) = 6x-6. At x=0: f''= -6 < 0 (maximum). At x=2: f'' = 6 > 0 (minimum).

Question 3

Solve for x: log₂(x) + log₂(x-6) = 4.

Accepted Answer

x = 8

Answer

log₂(x(x-6)) = 4 → x(x-6) = 16 → x²-6x-16 = 0 → (x-8)(x+2) = 0. x = 8 or x = -2. Since x > 6 (domain), x = 8.

Question 4

Find the area enclosed between y = x² and y = x + 2.

Accepted Answer

4.5 sq units

Answer

Intersection: x²=x+2 → x²-x-2=0 → (x-2)(x+1)=0 → x=-1,2. Area = ∫₋₁²(x+2-x²)dx = [x²/2+2x-x³/3]₋₁² = (2+4-8/3)-(-1/2-2+1/3) = 10/3+13/6 = 9/2.

Question 5

If P(A) = 0.4, P(B) = 0.3, and A and B are independent, find P(A∪B).

Accepted Answer

0.58

Answer

P(A∩B) = P(A)×P(B) = 0.4×0.3 = 0.12. P(A∪B) = 0.4+0.3-0.12 = 0.58.

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