A-Level H2 Mathematics 2 — Questions and Answers
Question 1: Find the stationary points of y = x³ - 3x.
- (0,0) only
- (1,-2) and (-1,2) (Correct answer)
- (1,0) and (-1,0)
- (3,0) and (-3,0)
Correct answer: (1,-2) and (-1,2)
dy/dx = 3x² - 3 = 0 → x² = 1 → x = ±1. At x=1: y = 1-3 = -2. At x=-1: y = -1+3 = 2. Points: (1,-2) and (-1,2).
Question 2: What is the value of ∫₀² x² dx?
- 4/3
- 8/3 (Correct answer)
- 3
- 4
Correct answer: 8/3
∫₀² x² dx = [x³/3]₀² = 8/3 - 0 = 8/3.
Question 3: If A = [[2,1],[3,4]], what is det(A)?
- 5 (Correct answer)
- 8
- 11
- 6
Correct answer: 5
det(A) = (2)(4) - (1)(3) = 8 - 3 = 5.
Question 4: Express 2sin(x) + 2cos(x) in the form R sin(x + α), where R > 0 and 0 < α < π/2.
- 2√2 sin(x + π/4) (Correct answer)
- 2 sin(x + π/4)
- √2 sin(x + π/4)
- 2√2 sin(x + π/3)
Correct answer: 2√2 sin(x + π/4)
R = √(2² + 2²) = √8 = 2√2. tan(α) = 2/2 = 1, so α = π/4. Result: 2√2 sin(x + π/4).
Question 5: Find the general solution of dy/dx = 2xy.
- y = Ce^(x²) (Correct answer)
- y = e^(x²) + C
- y = 2Cx
- y = Ce^(2x)
Correct answer: y = Ce^(x²)
Separating variables: dy/y = 2x dx. Integrating: ln|y| = x² + k. So y = Ce^(x²) where C = e^k.
Question 6: What is the gradient of the curve xy + y² = 6 at the point (1, 2)?
- −2/5 (Correct answer)
- 2/5
- −5/2
- 5/2
Correct answer: −2/5
Implicit differentiation: y + x(dy/dx) + 2y(dy/dx) = 0. At (1,2): 2 + (dy/dx)(1 + 4) = 0. dy/dx = -2/5.
A-Level H2 Mathematics 2
Find the stationary points of y = x³ - 3x.