The equation of a curve is y = 1/2 k²x² - 2kx+2 and the equation of a line is y = kx+p, where k and p are constants with 0 < k < 1.

Mark scheme

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

1. [M1*] Using (5/2, 1/2) in the curve equation or equating the line and the curve and then using x = 5/2 and p = 1/2 - kx. Simplify to get a three-term quadratic in k. Condone errors in simplification.
2. [A1] k = 2/5 OE Condone inclusion of k = 6/5.
3. [DM1*] Using (5/2, 1/2) and their k in an equation in p. Either the line (as shown) or 4p² + 12p + 5 = 0 are the most likely and solving for p.
4. [A1] p = -1/2 OE Condone inclusion of p = -5/2.
5. [DM1] Equating the line and curve using their k and p and simplify to get a three-term quadratic [= 0].
6. [A1 A1] (25/2, 9/2) OE Accept x = 25/2, y = 9/2.
7. [M1*] Equate the original equations of the curve and the line and collect like terms; k and p must still be present.
8. [DM1] Use of b² – 4ac for their quadratic in x to give an expression in k and p. This expression can come from their equation in (a).
9. [A1] p < -5/2