2016 NYJC Promo Q9

Relative to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\), \(\overrightarrow{OC}=\mathbf{c}\) and \(\mu \mathbf{a}=\left( \mu +1 \right)\mathbf{b}-\mathbf{c}\), where \(\mu \in {{\mathbb{R}}^{+}}\) .

It is given that \(OAB\) is an equilateral triangle.

1. Show that \(\mathbf{a}\cdot \mathbf{b}=\frac{1}{2}{{\left| \mathbf{b} \right|}^{2}}\).[2]
2. The point \(D\) lies on \(OB\) produced such that \(OD=\lambda OB\) and \(OD\) is perpendicular to \(CD\). Find the value of \(\lambda \) in terms of \(\mu \).[4]
3. Hence by using the vector product, find the area of \(\Delta OCD\), leaving your answer in the form of \(k{{\left| \mathbf{b} \right|}^{2}}\), where \(k\) is to be expressed in terms of \(\mu \).[4]