O-Level Additional Mathematics 2 — Questions and Answers
Question 1: Find the range of values of k for which the equation x² + kx + 9 = 0 has no real roots.
- k > 6 or k < -6
- -6 < k < 6 (Correct answer)
- k = 6 or k = -6
- All values of k
Correct answer: -6 < k < 6
For no real roots, the discriminant must be negative: b² - 4ac < 0. k² - 4(1)(9) < 0, k² - 36 < 0, k² < 36, -6 < k < 6.
Question 2: What is the exact value of cos 60°?
- √3/2
- 1/2 (Correct answer)
- √2/2
- 0
Correct answer: 1/2
cos 60° = 1/2 is a standard trigonometric value derived from the equilateral triangle (60-60-60) bisected to form a 30-60-90 triangle.
Question 3: What is the equation of the tangent to the curve y = x² at the point (2, 4)?
- y = 2x
- y = 4x - 4 (Correct answer)
- y = 4x + 4
- y = x + 2
Correct answer: y = 4x - 4
dy/dx = 2x. At x = 2, gradient = 4. Using y - y₁ = m(x - x₁): y - 4 = 4(x - 2), y = 4x - 4.
Question 4: Simplify: (2ⁿ⁺¹ × 4ⁿ) / 8ⁿ
- 2ⁿ⁺¹
- 2 (Correct answer)
- 2ⁿ
- 4
Correct answer: 2
Express all as powers of 2: (2ⁿ⁺¹ × 2²ⁿ) / 2³ⁿ = 2ⁿ⁺¹⁺²ⁿ / 2³ⁿ = 2³ⁿ⁺¹ / 2³ⁿ = 2³ⁿ⁺¹⁻³ⁿ = 2¹ = 2.
Question 5: What is the area under the curve y = x² between x = 0 and x = 3?
- 9 (Correct answer)
- 3
- 27
- 27/3
Correct answer: 9
Area = ∫₀³ x² dx = [x³/3]₀³ = 27/3 - 0 = 9.
Question 6: If the roots of 2x² - 5x + 1 = 0 are α and β, what is α + β?
- 5
- 5/2 (Correct answer)
- -5/2
- 1/2
Correct answer: 5/2
For ax² + bx + c = 0, the sum of roots = -b/a. Here, α + β = -(-5)/2 = 5/2.
O-Level Additional Mathematics 2
Find the range of values of k for which the equation x² + kx + 9 = 0 has no real roots.