Answer only one of the following two alternatives. EITHER The diagram shows part of the curve y = 9x² – x³, which meets the x-axis at the origin O and at the point A. The line y – 2x + 18 = 0 passes through A and meets the y-axis at the point B. [Figure 11.1]

Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

1. [M1] (i) for differentiation (dy/dx = 18x - 3x²)
2. [M1] (i) for equating to zero and attempt to solve (0 = 3x(6-x))
3. [M1] (i) for finding y (When x = 6, y = 108)
4. [A1] (i) for showing y <= 108
5. [M1] (ii) for attempt to integrate curve (∫ (9x² - x³) dx = 3x³ - x⁴/4)
6. [DM1] (ii) for correct application of limits for curve area ([3x³ - x⁴/4] from 0 to 9, or from 0 to 6) (Area under curve from 0 to 6 = 546.75)
7. [A1] (ii) for calculation of curve area (546.75 or 486)
8. [B1] (ii) for B(0, -18) and A(9,0) for the line (or area of triangle = 81)
9. [A1] (ii) for total area = 628
10. OR
11. [M1] (ii) for ∫ (equation of line - equation of curve) dx (∫ (2x - 18 - (9x² - x³)) dx)
12. [A3,2,1,0] (ii) for correct integration (∫ (x³ - 9x² + 2x - 18) dx = x⁴/4 - 3x³ + x² - 18x)
13. [DM1] (ii) for correct application of limits (from 0 to 9)
14. [A1] (ii) for result 628