The diagram shows part of the curve y = x³ + 4x² – 5x + 5 and the line y = 5. The curve and the line intersect at the points A, B and C. The points D and E are on the x-axis and the lines AE and CD are parallel to the y-axis. [Figure 11.1]

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

1. [B2] x⁴/4 + 4x³/3 - 5x²/2 + 5x [+c] isw
2. [B1] for any 3 correct terms
3. [B1] x³ + 4x² – 5x + 5 = 5 and rearrange to x(x² + 4x - 5) = 0 oe soi
4. [M1] Solves their x² + 4x-5[=0] soi
5. [A1] x=-5, x=1 soi
6. [A1] OEAB = 25, OBCD=5
7. [M1] Correct or correct FT substitution of 0, their -5 seen in [x⁴/4 + 4x³/3 - 5x²/2 + 5x] (their-5 / 0)
8. [M1] Correct or correct FT substitution of their 1, 0 seen in [x⁴/4 + 4x³/3 - 5x²/2 + 5x] (their1 / 0)
9. [M1] for the strategy needed to combine the areas; may be in steps; 97.916-25+5-4.083
10. [A1] 886/12 oe or 73 5/6 oe or 73.83 rot to 3 or more sig figs all method steps must be seen; not from wrong working
11. [SC3] for ∫(x³ + 4x² – 5x) dx - ∫(x³ + 4x² – 5x) dx oe with calculation
12. [SC2] for ∫(x³ + 4x² – 5x) dx - ∫(x³ + 4x² – 5x) dx oe with calculation