2025 CGS PRELIMS P2 Q9

In the diagram, \(\overrightarrow{AD}=\mathbf a\), \(\overrightarrow{AB}=3\mathbf b\) and \(\overrightarrow{AD}=\dfrac14\overrightarrow{AC}\). \(E\) is the midpoint of \(BC\). \(F\) is a point on \(AE\) such that \(AF=\dfrac23FE\).

1. Express and simplify your answers in terms of \(\mathbf a\) and \(\mathbf b\):
   1. \(\overrightarrow{BC}\),[1]
   2. \(\overrightarrow{AF}\),[2]
   3. \(\overrightarrow{BF}\).[1]
2. Explain if \(BF\) produced will meet \(AC\) at \(D\).[3]
3. Find the value of
   1. \(\dfrac{\text{area of}\hspace{0.5em}\triangle ABF}{\text{area of}\hspace{0.5em}\triangle ABE}\),[1]
   2. \(\dfrac{\text{area of}\hspace{0.5em}\triangle BEF}{\text{area of}\hspace{0.5em}\triangle ABC}\).[2]