2024 IGCSE Additional Mathematics May/June 0606/23 Q10

The diagram shows a parallelogram \(OABC\). The point \(D\) divides the line \(OC\) in the ratio \(2:3\).

\(\overrightarrow{OA}=\mathbf a\quad\text{and}\quad\overrightarrow{OC}=\mathbf c\)

The point \(P\) lies on \(AD\) such that \(\overrightarrow{OP}=\lambda\overrightarrow{OB}\) and \(\overrightarrow{AP}=\mu\overrightarrow{AD}\), where \(\lambda\) and \(\mu\) are scalars.

Find two expressions for \(\overrightarrow{OP}\), each in terms of \(\mathbf a\), \(\mathbf c\) and a scalar, and hence show that \(P\) divides both \(DA\) and \(OB\) in the ratio \(m:n\), where \(m\) and \(n\) are integers to be found.[7]