2025 AISS PRELIMS P1 Q27

(OAC) is a triangle and (B) is a point on (OC). (\overrightarrow{OA}=\mathbf a), (\overrightarrow{OB}=\mathbf b) and (OB:BC=2:3). (X) is the point on (AB) such that (AX:XB=1:2).

1. Express (\overrightarrow{AC}) in terms of (\mathbf a) and (\mathbf b), as simply as possible.[2]
2. Express (\overrightarrow{XB}) in terms of (\mathbf a) and (\mathbf b), as simply as possible.[2]
3. (Y) is the point on (OC) such that (AXYC) is a trapezium. Find, in terms of (\mathbf a) and (\mathbf b), (\overrightarrow{XY}).[2]