A-Level H2 Mathematics 4 — Questions and Answers
Question 1: What is the modulus and argument of the complex number z = 1 + i√3?
- modulus 2, argument π/3 (Correct answer)
- modulus √2, argument π/4
- modulus 2, argument π/6
- modulus √3, argument π/3
Correct answer: modulus 2, argument π/3
|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.
- f'(x) = 3x²-6x; maximum at x=0, minimum at x=2 (Correct answer)
- f'(x) = 3x²-6x; minimum at x=0, maximum at x=2
- f'(x) = 3x-6; only one stationary point
- f'(x) = x²-6x; minimum at x=2
Correct answer: f'(x) = 3x²-6x; maximum at x=0, minimum at x=2
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.
- x = 8 (Correct answer)
- x = 2
- x = 16
- x = 4
Correct answer: x = 8
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.
- 4.5 sq units (Correct answer)
- 9/2 sq units
- 3 sq units
- 5 sq units
Correct answer: 4.5 sq units
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).
- 0.7
- 0.58 (Correct answer)
- 0.12
- 0.65
Correct answer: 0.58
P(A∩B) = P(A)×P(B) = 0.4×0.3 = 0.12. P(A∪B) = 0.4+0.3-0.12 = 0.58.
Question 6: What is the Maclaurin series for sin(x) up to the x³ term?
- 1 - x²/2 + x⁴/24
- x - x³/6 (Correct answer)
- x + x³/6
- x - x²/2 + x³/6
Correct answer: x - x³/6
The Maclaurin series for sin(x) = x - x³/3! + x⁵/5! - ... = x - x³/6 + ... up to the x³ term.
A-Level H2 Mathematics 4
What is the modulus and argument of the complex number z = 1 + i√3?