2026 CCHMS P2 Q7

In the diagram, \(OABC\) is a quadrilateral where the diagonals \(OB\) and \(AC\) intersect at \(X\).

\(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OC}=\mathbf b\) and \(\overrightarrow{OB}=2\mathbf a+\mathbf b\). The lines \(BA\) and \(CO\) are produced to meet at \(P\).

1. Find \(\overrightarrow{CB}\), giving your answer in terms of \(\mathbf a\) and/or \(\mathbf b\).
   Hence, explain why \(ABCO\) is a trapezium.[3]
2. Giving your answers in terms of \(\mathbf a\) and/or \(\mathbf b\), find
   1. \(\overrightarrow{OX}\),[1]
   2. \(\overrightarrow{AB}\).[1]
3. Find the ratio of area of triangle \(OAX\) : area of triangle \(OAC\).[1]
4. Using vector method, find \(\overrightarrow{OP}\) and show that \(O\) is the midpoint of \(PC\).[4]