The curve $C _ { 1 }$ has polar equation $r ^ { 2 } = 2 \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.\\
(i) The point on $C _ { 1 }$ furthest from the line $\theta = \frac { 1 } { 2 } \pi$ is denoted by $P$. Show that, at $P$,

$$2 \theta \tan \theta = 1$$

and verify that this equation has a root between 0.6 and 0.7 .\\

The curve $C _ { 2 }$ has polar equation $r ^ { 2 } = \theta \sec ^ { 2 } \theta$, for $0 \leqslant \theta < \frac { 1 } { 2 } \pi$. The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the pole, denoted by $O$, and at another point $Q$.\\
(ii) Find the exact value of $\theta$ at $Q$.\\

(iii) The diagram below shows the curve $C _ { 2 }$. Sketch $C _ { 1 }$ on this diagram.\\
(iv) Find, in exact form, the area of the region $O P Q$ enclosed by $C _ { 1 }$ and $C _ { 2 }$.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q11 EITHER}}