9 The diagram shows the curve y = sin x cos 2x, for 0 ≤ x ≤ π, and a maximum point M, where x = a. The shaded region between the curve and the x-axis is denoted by R. [Figure 9.1] (a) Find the value of a correct to 2 decimal places. (b) Find the exact area of the region R, giving your answer in simplified form.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. *M1 Use correct product rule As far as p cos x cos 2x + q sin x sin 2x or full working (u, v, du/dx, dv/dx) shown.
2. A1 Obtain dy/dx = cos x cos 2x - 2 sin x sin 2x OE
3. DM1 Equate derivative to zero and use correct double angle formulae Allow if only have one double angle in their derivative.
4. A1 Obtain cosx(1-6sin²x) = 0 or equivalent e.g. cosx(6cos²x-5)=0, 5tan²x=1. Simplified but not necessarily factorised - like terms must be collected.
5. A1 Obtain a=0.42 Only. Accept x=0.42.
6. *M1 Use double angle formula and obtain p cos 3x + q cos x correctly e.g. from ∫[2cos²xsin x − sin x] dx .
7. A1 Obtain ± (-⅔cos³x + cosx) Correct for their integral.
8. DM1 Correct use of limits ¼π and ½π (or use double the integral from ¼π to ½π) OE [± ( (2(-1/√2)³/3 + 1/√2) - (0) ) ]
9. A1 Obtain 2√2/3 Or simplified exact equivalent. Final answer must be positive.