A-Level H2 Mathematics 2

Question 1

Find the stationary points of y = x³ - 3x.

Accepted Answer

(1,-2) and (-1,2)

Answer

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?

Accepted Answer

8/3

Answer

∫₀² x² dx = [x³/3]₀² = 8/3 - 0 = 8/3.

Question 3

If A = [[2,1],[3,4]], what is det(A)?

Accepted Answer

5

Answer

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.

Accepted Answer

2√2 sin(x + π/4)

Answer

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.

Accepted Answer

y = Ce^(x²)

Answer

Separating variables: dy/y = 2x dx. Integrating: ln|y| = x² + k. So y = Ce^(x²) where C = e^k.

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