A-Level A-Level Mathematics 1 — Questions and Answers
Question 1: What is the derivative of f(x) = x⁴ − 3x² + 5x − 2?
- 4x³ − 6x + 5 (Correct answer)
- 4x³ + 6x − 5
- 4x³ − 3x + 5
- x⁵ − x³ + 5x²
Correct answer: 4x³ − 6x + 5
Using the power rule: d/dx(xⁿ) = nxⁿ⁻¹. d/dx(x⁴) = 4x³, d/dx(−3x²) = −6x, d/dx(5x) = 5, d/dx(−2) = 0. So f'(x) = 4x³ − 6x + 5.
Question 2: What does ∫(3x² + 2x) dx equal?
- 6x + 2 + C
- x³ + x² + C (Correct answer)
- 3x³ + x² + C
- x³ + 2x + C
Correct answer: x³ + x² + C
Integrate term by term: ∫3x² dx = x³ and ∫2x dx = x². Add constant C. So ∫(3x² + 2x) dx = x³ + x² + C.
Question 3: What is the binomial expansion of (1 + x)ⁿ for small x?
- 1 + nx only
- 1 + nx + n(n−1)/2! · x² + n(n−1)(n−2)/3! · x³ + ... (Correct answer)
- nxⁿ
- 1 + x + x² + x³
Correct answer: 1 + nx + n(n−1)/2! · x² + n(n−1)(n−2)/3! · x³ + ...
The binomial expansion is (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + ... This is valid for |x| < 1 when n is not a positive integer. For positive integers it terminates.
Question 4: Solve: 2sin²θ − sinθ − 1 = 0 for 0° ≤ θ ≤ 360°
- θ = 30°, 150°
- θ = 90°, 210°, 330° (Correct answer)
- θ = 30°, 150°, 270°
- θ = 60°, 120°, 270°
Correct answer: θ = 90°, 210°, 330°
Let u = sinθ: 2u² − u − 1 = 0 → (2u + 1)(u − 1) = 0. So sinθ = −1/2 or sinθ = 1. sinθ = 1 → θ = 90°. sinθ = −1/2 → θ = 210°, 330°. So θ = 90°, 210°, 330°.
Question 5: What is the chain rule for differentiation?
- dy/dx = dy/du + du/dx
- dy/dx = dy/du × du/dx (Correct answer)
- dy/dx = du/dy × dx/du
- dy/dx = (dy/du) ÷ (du/dx)
Correct answer: dy/dx = dy/du × du/dx
The chain rule states: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x), or equivalently dy/dx = (dy/du)(du/dx). It is used to differentiate composite functions.
Question 6: What is the sum of an infinite geometric series with first term a and common ratio r (|r| < 1)?
- a/(1 − r) (Correct answer)
- a(1 − r)
- ar/(1 − r)
- a/(1 + r)
Correct answer: a/(1 − r)
For an infinite geometric series where |r| < 1, the sum S∞ = a/(1 − r). This converges because successive terms get smaller and smaller, approaching zero.
A-Level A-Level Mathematics 1
What is the derivative of f(x) = x⁴ − 3x² + 5x − 2?