2026 RI P1 Q4

With reference to the origin \(O\), three distinct points \(A\), \(B\) and \(C\) are such that \(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OB}=\mathbf b\) and \(\overrightarrow{OC}=\mathbf c\). The mid-point of \(AC\) is \(D\).

1. Express \(\overrightarrow{OD}\) in terms of \(\mathbf a\) and \(\mathbf c\).[1]
2. Given that \(|\mathbf a|=|\mathbf c|\), \(|\mathbf b|\ne0\) and \(BD\) is perpendicular to \(AC\), show that \(OB\) is also perpendicular to \(AC\) by using a suitable scalar product.[3]
3. Given further that point \(D\) lies on line segment \(OB\), \(|\mathbf a|=2\), \(|\mathbf b|=6\) and \(\angle AOC=60^\circ\), find the exact area of quadrilateral \(OABC\).[2]