2025 SST PRELIMS P2 Q8

\(ABCD\) is a parallelogram. Point \(E\) lies on \(CD\) produced such that \(CD:CE=1:2\). \(M\) is the midpoint of \(AD\). Point \(N\) lies on \(BC\) such that \(BN:BC=1:4\). \(\overrightarrow{BN}=\mathbf a\) and \(\overrightarrow{CD}=\mathbf b\).

1. Express as simply as possible in terms of \(\mathbf a\) and/or \(\mathbf b\):
   1. \(\overrightarrow{AM}\),[1]
   2. \(\overrightarrow{BM}\),[1]
   3. \(\overrightarrow{BE}\).[1]
2. Explain why \(B,M,E\) lie on a straight line.[2]
3. Find the numerical value of
   1. \(\dfrac{\text{area of triangle}\hspace{0.5em}AMB}{\text{area of triangle}\hspace{0.5em}DCN}\),[1]
   2. \(\dfrac{\text{area of triangle}\hspace{0.5em}EDM}{\text{area of}\hspace{0.5em}DCBM}\).[2]