The diagram shows the curve y = sin2x(1+sin2x), for 0 ≤ x ≤ ¾π, and its minimum point M. The shaded region bounded by the curve that lies above the x-axis and the x-axis itself is denoted by R. [Figure 6.1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. [M1*] Correct use of product rule to differentiate 2cos2x(1+sin2x) + sin 2x x 2cos2x, or 2cos²x-2sin²x+8sinxcos²x-8cosxsin³x. All terms needed but could have errors in the coefficients.
2. [A1] Obtain 2cos 2x + 4cos2xsin 2x OE
3. [DM1] Equate derivative to zero and solve for 2x or x 2cos2x(1+2sin2x)=0 ⇒ x=sin⁻¹(−½) Condone if they only consider cos 2x = 0.
4. [A1] Obtain x = ⁷⁄₁₂π, y = -¼ Mark degrees as a misread. The Q asks for an exact answer.