2025 SJI PRELIMS P2 Q8

In triangle \(ABC\), \(AY:AB=1:3\), \(X\) is the midpoint of \(AZ\), \(Z\) is the midpoint of \(CB\), \(\overrightarrow{CZ}=\mathbf a\), and \(\overrightarrow{AY}=\mathbf b\).

1. Express \(\overrightarrow{AC}\) in terms of \(\mathbf a\) and \(\mathbf b\).[1]
2. Express \(\overrightarrow{XZ}\).[1]
3. Express \(\overrightarrow{CX}\).[1]
4. Express \(\overrightarrow{CY}\).[1]
5. Hence explain why \(C,X,Y\) are collinear.[2]
6. Find \(\dfrac{\text{area of triangle}\hspace{0.5em}ACY}{\text{area of triangle}\hspace{0.5em}ACZ}\).[2]