The diagram shows part of the curve y = 3√(4x + 1) – 2x. The curve crosses the y-axis at A and the stationary point on the curve is M.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

1. [B2,1,0] d/dx [3(4x + 1)^(1/2)] = 3 * (1/2) * (4x + 1)^(-1/2) * 4
2. [B1] d/dx [-2x] = -2
3. [B1] ∫ 3(4x + 1)^(1/2) dx = 3 * (4x + 1)^(3/2) / (3/2) / 4
4. [B1] ∫ -2x dx = -2x²/2
5. [M1] Sets their dy/dx to 0 and attempts to solve (as far as x = k)
6. [A1] x = 2
7. [A1] y = 5
8. [M1] Uses their integral and their '2' and 0 correctly
9. [M1] Either using the area of a trapezium with their 2, 3 and 5 or ∫(their x+3)dx using their '2' and 0 correctly.
10. [A1] (Shaded area = 9 – 8) = 1
11. [M1] Subtracts their line from given curve and uses their '2' and 0 correctly.
12. [A1] All integration correct and limits 2 and 0.
13. [M1] Evidence of substituting their '2' and 0 into their integral.
14. [A1] 1