The diagram shows part of the curve y = 3x - x³ and the lines y = 3x and 2y = 27 - 3x. The curve and the line y = 3x meet the x-axis at O and the curve and the line 2y = 27 – 3x meet the x-axis at A. [Figure 11 with the curve, lines, and shaded region]

Mark scheme

Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.

1. [B1+B1] 3x²/2 - 2x⁴/5 (+c) isw
2. [B1] (9, 0) oe Not just x = 9
3. [B1] Substitute (3, 9) into both lines 3 x 3 = 9 and (27-3×3)/2 = 9 Or solves simultaneously (6x = 27 – 3x oe) to get x = 3, y = 9
4. [M1] [Area AOB =] ½ × 9 × 9 oe (or 40.5) Uses their (ii). May split into 2 triangles (13.5 and 27). May integrate. Must be a complete method.
5. [M1] their 3(9)²/2 - 2(9)⁴/5 [- 0] (= 24.3) lower limit may be omitted but must be correct if seen
6. [M1] must be from genuine attempts at area of triangle and area under curve their 81/2 - their 243/10
7. [A1] 16.2