2025 IGCSE Additional Mathematics May/June 0606/13 Q10

The diagram shows four points, \(O\), \(A\), \(B\) and \(C\).

\(A\), \(B\) and \(C\) lie in a straight line and are such that \(\dfrac{AB}{AC}=\dfrac13\).

\(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\).

1. Find \(\overrightarrow{OC}\) in terms of \(\mathbf a\) and \(\mathbf b\).
   Simplify your answer.[3]
2. The line \(OA\) is extended to the point \(D\) such that \(OA:AD=2:7\).
   Point \(E\) lies on \(CD\) such that \(\overrightarrow{OE}=\lambda\mathbf b\).
   Find the value of \(\lambda\).[5]