The diagram shows part of the curve y = 32x-4x² - 48 and the line AB. The curve and the line AB meet the x-axis at A and meet again at the point B(5, 12). The line CD extended is parallel to the y-axis and passes through the maximum point of the curve. Find the area of the shaded region. [Figure 9.1]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [M1] for factorising or solving 32x-4x²-48 = 0
2. [A1] A(2, 0) or x = 2 [x = 6]
3. [M1] Finds equation CD/ x-coordinate of C or D or maximum point : x = 4
4. [M1] for (2+6)/2 or 32-8x = 0
5. [A1] D(4, 8) or [equation AB is y =] 4x – 8
6. [M1] for y = (12-0)/(5-2) x or mAB = (12-0)/(5-2) soi
7. [B1] [ (32x^2)/2 - (4x^3)/3 - 48x ] must be seen
8. [B1] or [ 28x^2/2 - 4x^3/3 - 40x ]
9. [M1] Correct plan including correct substitution of upper and lower limits at some point e.g. [16x^2 - (4x^3)/3 - 48x] from 2 to 4 minus (1/2)×(4-2)×8
10. [M1] dep on attempt to integrate FT their 4 and their 8 if needed
11. [M1] or FT their 4 and their 4x – 8 of the form mx + c if needed
12. [M1] or FT their 4 and (32-their4)x-4x²+(-48-their(-8)) providing clear evidence of the derivation of this has been seen
13. [A1] 40/3 isw or 13.3[33....] nfww
14. [A1] dep on all previous marks awarded