2025 NASS PRELIMS P2 Q5

\(ABCD\) is a trapezium with \(AB\parallel DC\), \(\overrightarrow{AB}=16\mathbf p\), and \(\overrightarrow{AD}=5\mathbf q\). \(P\) lies on \(BD\) such that \(DP:DB=1:4\), and \(4AB=3DC\).

1. Express in terms of (mathbf p) and (mathbf q):
   1. \(\overrightarrow{DB}\),[1]
   2. \(\overrightarrow{AP}\).[2]
2. Show that \(BC\parallel AP\).[2]
3. Find the ratio area of triangle (APB) : area of triangle (BCD).[3]
4. Find the ratio area of triangle (APD) : area of trapezium (ABCD).[2]