2026 IGCSE Additional Mathematics May/June 0606/12 Q11

The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are \(\overrightarrow{OA}=\begin{pmatrix}4\\ -2\end{pmatrix}\) and \(\overrightarrow{OB}=\begin{pmatrix}-4\\ 6\end{pmatrix}\).

The point \(C\) lies on \(AB\) and is such that \(AC:CB=3:1\).

The point \(D\) lies on \(OA\) such that \(\overrightarrow{OD}=\lambda\overrightarrow{OA}\) and \(\overrightarrow{CD}=\begin{pmatrix}5\\ -5.5\end{pmatrix}\).

1. Find \(\overrightarrow{OD}\).[5]
2. Hence find the value of \(\lambda\).[1]