[Figure 11.1] In the triangle OAB, OA = a_and OB = b. The mid-point of the line OB is X, and the mid-point of the line AB is Y. The lines OY and AX intersect at the point Z. It is given that AZ = λAX and OZ = μOY where λ and μ are rational numbers.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [3] (a) B1 Allow unsimplified
2. [3] (a) M1 for OZ = a + λ × their (b/2 - a)
3. [3] (a) Allow unsimplified Mark final answer
4. [2] (b) B1 Allow unsimplified
5. [2] (b) B1 FT on their OY , allow unsimplified Mark final answer
6. [3] (c) M1 For equating their final answer for (a) and their final answer for (b) and attempt to equate like vectors at least once to obtain a scalar equation
7. [3] (c) M1 dep for solving their simultaneous equations to obtain at least one unknown. Each equation must be in terms of λ and μ
8. [3] (c) A1 for both.
9. [1] (d) B1 Must be from correct work