The diagram shows the curve y = sin 2x cos²x for 0 ≤ x ≤ ½π, and its maximum point M. [Figure 10.1]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2021). The mark or mark type is shown in brackets.

1. [B1] State or imply du = cos x dx
2. [M1] Using double angle formula for sin2x and Pythagoras, express integral in terms of u and du.
3. [A1 OE] Obtain integral ∫2(u-u³)du
4. [M1] Use limits u = 0 and u = 1 in an integral of the form au² + bu⁴, where ab ≠ 0
5. [A1] Obtain answer ½
6. [M1] Use product rule
7. [A1] Obtain correct derivative in any form
8. [*M1] Equate derivative to zero and use a double angle formula
9. [DM1] Obtain an equation in one trig variable
10. [A1] Obtain 4sin²x=1, 4cos²x=3 or 3tan²x=1
11. [A1] Obtain answer x = π/6