2024 IGCSE Additional Mathematics October/November 0606/13 Q9

The diagram shows the trapezium \(OABC\), where \(\overrightarrow{OA}=4\mathbf a\), \(\overrightarrow{OC}=\mathbf c\), and \(\overrightarrow{CB}=2\mathbf a\). The point \(D\) lies on \(AB\) such that \(AD:DB=2:1\). The point \(X\) is the point of intersection of the lines \(OD\) and \(AC\). It is given that \(\overrightarrow{AX}=\lambda\overrightarrow{AC}\) and \(\overrightarrow{OX}=\mu\overrightarrow{OD}\).

Find in terms of \(\mathbf a\) and \(\mathbf c\)

1. \(\overrightarrow{AB}\)[1]
2. \(\overrightarrow{OD}\).[2]
3. Find \(\overrightarrow{OX}\) in terms of \(\mathbf a\), \(\mathbf c\) and \(\mu\).[1]
4. Find \(\overrightarrow{AX}\) in terms of \(\mathbf a\), \(\mathbf c\) and \(\lambda\).[2]
5. Hence find the values of \(\lambda\) and \(\mu\).[4]