The diagram shows the curve y = cos x √sin 2x for 0 ≤ x ≤ π/2. The curve has a maximum point at M, where x = a. [Figure 11.1]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [B1] d/dx(√sin 2x) = 2cos2x / (2√sin 2x) Accept k cos 2x / √sin 2x SOI
2. [*M1] Use correct product rule
3. [A1] Obtain correct derivative in any form E.g. dy/dx = -sin x √sin 2x + cos x * (2cos 2x / (2√sin 2x))
4. [DM1] Equate the derivative to zero and obtain a horizontal equation E.g. -sin a √sin 2a + cos a cos 2a / √sin 2a = 0
5. [DM1] Use correct trig formulae to obtain an equation in one trig function E.g. cos 3a = 0 or sin² a = 1/4.
6. [A1] Obtain a = π/6 Exact answer in radians only.
7. [M1*] Use [π] ∫ y² dx
8. [DM1] Use double angle formula to obtain a form that can be integrated directly, e.g. [π] ∫ cos²x sin 2x dx = [π] ∫ 2cos³x sin x dx Or [π] ∫ sin 4x + sin 2x dx, or [π] ∫ 2(u-u³)du by substituting u = sin x.
9. [A1] Obtain -[π] × (1/4)cos 4x - (1/2)cos 2x, OE.
10. [DM1] Use correct limits correctly [- [π] x (1/4)cos 4x - (1/2)cos 2x] from 0 to π/2 = -[π] ((1/4)cos(2π) - (1/2)cos(π)) - ((1/4)cos(0) - (1/2)cos(0))
11. [A1] Obtain final answer π/2 If the factor of π is missing throughout, allow the first 4 marks and A0 here.