2025 GESS PRELIMS P2 Q8

In triangle \(OAB\), the point \(C\) on \(OA\) is such that \(3OC=2OA\). \(E\) and \(D\) are the midpoints of \(CD\) and \(OB\) respectively, and \(\overrightarrow{AF}=m\overrightarrow{AB}\). \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\).

1. Express, as simply as possible, in terms of a and b,
   1. \(\overrightarrow{AB}\),[1]
   2. \(\overrightarrow{CE}\).[2]
2. 1. Write down \(\overrightarrow{OF}\) in terms of \(\mathbf a\), \(\mathbf b\) and \(m\).[2]
   2. \(\overrightarrow{OE}=\dfrac35(1-m)\mathbf a+\dfrac35m\mathbf b\). Show that \(O\), \(E\) and \(F\) lie on a straight line.[2]
3. Find, as a fraction in its simplest form,
   1. \(\dfrac{\text{area of triangle}\hspace{0.5em}AEC}{\text{area of triangle}\hspace{0.5em}OEC}\),[1]
   2. \(\dfrac{\text{area of triangle}\hspace{0.5em}OCD}{\text{area of triangle}\hspace{0.5em}OAB}\).[1]