The diagram shows the graph of y = 5 sin 2x cos²x for 0 < x ≤ ½π and its maximum point M. [Figure X.X] (a) Find the exact x-coordinate of M. (b) By using the substitution u = cosx, find the area of the region bounded by the curve, the x-axis between x = 0 and x = ¼π, and the line x = ½π.

Mark scheme

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

1. [B1] Differentiate cos^2x to obtain -2sinxcosx OE Could be stated as -sin 2x.
2. [M1] Use correct product rule With a '+' in the middle.
3. [A1] Obtain derivative -10sin2xsinxcosx+10cos2xcos^2 x OE
4. [DM1] Equate derivative to zero and obtain an equation in one trig function Trigonometry formulas used need to be correct.
5. [A1] Obtain 3 tan^2x = 1, 4 sin^2x = 1 or 4 cos^2x = 3 or cos3x=0 OE
6. [A1] Obtain x = π/6 only
7. [B1] du/dx = -sin x SOI
8. [M1] Reach an integral of the form ∫Au^3 du OE The question requires use of the substitution method.
9. [A1] Obtain -10∫u^3 du OE Ignore limits, but check order of limits if no minus sign.
10. [DM1] Substitute correct limits correctly in an expression of the form Cu^4 or Ccos^4 x u = 1 and u = √2/2. x = 0 and x = π/4. -10[u^4/4] from 1 to √2/2 or 10[u^4/4] from √2/2 to 1. Allow the correct answer from the correct integration and relevant limits to imply M1.
11. [A1] Obtain answer 15/8 or 1.875 WWW ISW