Answer only one of the following two alternatives. EITHER A curve is such that y = 5x² / (1 + x²)

Mark scheme

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

1. [M1] M1 for attempt to differentiate a quotient -1 each error
2. [A2,1,0]
3. [A1]
4. [B1] Stationary point at (0, 0)
5. [M1] M1 for a correct attempt to determine the nature of the turning point (allow change of sign method) – just finding the second derivative is not enough.
6. [A1] When x = 0, d²y/dx² is +ve, minimum
7. [B1] B1 for ∫ x/(1+x²)² dx = -1/(2(1+x²)) (+c)
8. [B1] B1 for limits
9. [M1] M1 for correct use of limits in an attempt at integration
10. [A1]
11. [M1] M1 for attempt to differentiate a quotient -1 each error
12. [A2,1,0]
13. [A1] Answer given
14. [M1] M1 for use of conditions once
15. [M1] M1 for use of conditions a second time and attempt to solve resulting equations
16. [A1]
17. [B1] B1 for correct x
18. [B1] B1 for y = -B/2
19. [M1] M1 for a correct attempt to determine the nature of the turning point (allow change of sign method) – just finding the second derivative is not enough.
20. [A1] A1 for a correct conclusion from completely correct work.