2025 YCKSS PRELIMS P1 Q27

\(\overrightarrow{BC}=4\mathbf a\), \(\overrightarrow{OB}=3\mathbf b\), and \(\overrightarrow{OA}=\dfrac34\overrightarrow{BC}\). Point \(E\) lies on \(OC\) with \(OE:EC=1:2\).

1. Show that \(\overrightarrow{AE}=-\dfrac53\mathbf a+\mathbf b\).[2]
2. Point \(F\) lies on \(OB\) with \(\overrightarrow{OF}=\dfrac35\overrightarrow{OB}\). Express \(\overrightarrow{AF}\) in terms of \(\mathbf a\) and/or \(\mathbf b\).[2]
3. Determine whether A, E and F are collinear. Justify using vectors.[2]
4. The area of triangle \(AEC\) is \(30\text{ cm}^2\). Find the area of triangle \(OAE\).[1]