The diagram shows the curve with equation y = 4 cos2x+8 sinx for 0 ≤ x ≤ π. The maximum points on the curve are denoted by A and B, and the shaded region is bounded by the line segment AB and the curve. [Figure X.X]

Mark scheme

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

1. [5] (a) Differentiate to obtain -8sin2x+cos x
2. [5] (a) Attempt to solve equation of form k₁ sin 2x + k₂ cosx = 0 using correct identity
3. [5] (a) Obtain at least sin x = ½
4. [5] (a) Attempt to find both coordinates of at least one maximum point
5. [5] (a) Obtain (¼π, 6) and (5/6π, 6) and no others between 0 and π
6. [6] (b) Integrate to obtain expression of form k₁ sin 2x + k₂ cos x
7. [6] (b) Obtain correct 2sin2x−8cos x
8. [6] (b) Apply their x-limits correctly to evaluate area under curve between A and B
9. [6] (b) Obtain 6√3 or exact unsimplified equivalent
10. [6] (b) Carry out correct process to find area of shaded region
11. [6] (b) Obtain 6(⅕π-¼π)−6√3, and hence 4π − 6√3