The diagram shows points \(O,A,B,C,D\) and \(X\). The position vectors of \(A\), \(B\), and \(C\) relative to \(O\) are

\(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OB}=\mathbf b\), and \(\overrightarrow{OC}=\dfrac32\mathbf b\). The vector \(\overrightarrow{CD}=3\mathbf a\).

(i) If \(\overrightarrow{OX}=\lambda\overrightarrow{OD}\), express \(\overrightarrow{OX}\) in terms of \(\lambda\), \(\mathbf a\), and \(\mathbf b\).

(ii) If \(\overrightarrow{AX}=\mu\overrightarrow{AB}\), express \(\overrightarrow{OX}\) in terms of \(\mu\), \(\mathbf a\), and \(\mathbf b\).

(iii) Use your two expressions for \(\overrightarrow{OX}\) to find \(\lambda\) and \(\mu\).

(iv) Find \(\dfrac{AX}{XB}\).

(v) Find \(\dfrac{OX}{XD}\).