2025 MGS(S) PRELIMS P2 Q5

In triangle \(ABC\), \(\overrightarrow{AB}=5\mathbf b\) and \(\overrightarrow{AC}=2\mathbf a\). It is given that \(BD:DA=1:4\), \(\overrightarrow{BN}=\dfrac13\overrightarrow{BM}\), and \(M\) is the midpoint of \(AC\).

1. Write each as simply as possible in terms of (mathbf a) and (mathbf b):
   1. \(\overrightarrow{AD}\),[1]
   2. \(\overrightarrow{BM}\),[1]
   3. \(\overrightarrow{CD}\).[1]
2. Show that \(\overrightarrow{CN}=\dfrac53(2\mathbf b-\mathbf a)\).[2]
3. Explain why \(C\), \(N\) and \(D\) lie on a straight line.[2]