The equation of a curve is y = 4x^(1/2) - x. The curve has a maximum point when x = a and crosses the x-axis at the point with coordinates (b, 0), where b > 0. The shaded region is bounded by the curve, the line x = a and the x-axis (see diagram). [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. [B1] dy/dx = 2x^(-1/2) - 1.
2. [M1] Sets their dy/dx = 0 and solves for x (or √x).
3. [A1] a = 4.
4. [B1] (Alternative) Recognizes y = 4√x - (√x)² as quadratic in √x.
5. [M1] (Alternative) Finds maximum by completing the square or derivative w.r.t √x.
6. [A1] (Alternative) a = 4.
7. [B1] b = 16 (seen or implied).
8. [B2] Integrates 4x^(1/2) - x to (8/3)x^(3/2) - (1/2)x² [+C] (B1 for one term correct).
9. [M1] Applies limits of integration (their a and b, where a<b, from 4x^(1/2)-x=0 and dy/dx=0) to their integrated expression and subtracts.
10. [DB1] (88/3) or (29 1/3) (must be exact).