H2 Mathematics Flashcards

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

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).

What is the value of ∫₀² x² dx?

Answer: 8/3

∫₀² x² dx = [x³/3]₀² = 8/3 - 0 = 8/3.

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

Answer: 5

det(A) = (2)(4) - (1)(3) = 8 - 3 = 5.

Express 2sin(x) + 2cos(x) in the form R sin(x + α), where R > 0 and 0 < α < π/2.

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).

Find the general solution of dy/dx = 2xy.

Answer: y = Ce^(x²)

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

What is the gradient of the curve xy + y² = 6 at the point (1, 2)?

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.