2022 TJC Promo Q10

Referred to the origin \(O\), the points \(P\) and \(Q\) are such that \(\overrightarrow{OP}=4\,\mathbf{a}\) and \(\overrightarrow{OQ}=3\,\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors. The points \(M\) and \(S\) are on \(OP\) produced such that \(M\) is the midpoint of \(P\) and \(S\). The point \(R\) is on \(OQ\) produced. (see diagram above)

It is given that \(\overrightarrow{MQ}=3\mathbf{b}-7\mathbf{a}\), \(\overrightarrow{MR}=\frac{15}{2}\mathbf{b}-7\mathbf{a}\) and \(\left| \overrightarrow{MQ} \right|=\left| \overrightarrow{MR} \right|\).

1. Using a suitable scalar product, show that \(\cos \angle POQ=\frac{3}{4}\).[5]
2. Find \(\overrightarrow{SR}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). Show that \(\overrightarrow{SR}=\lambda \overrightarrow{PQ}\) where \(\lambda \) is a constant to be found.[3]
3. Find the exact area of the triangle \(OPQ\).[2]
4. By considering the ratio of the area of the triangle \(OSR\) to the area of the triangle \(OPQ\), determine the exact area of the quadrilateral \(PQRS\).[2]