7 The curve $C$ has polar equation $r ^ { 2 } = ( \pi - \theta ) \tan ^ { - 1 } ( \pi - \theta )$, for $0 \leqslant \theta \leqslant \pi$.\\
(a) Sketch $C$ and state the polar coordinates of the point of $C$ furthest from the pole.\\
(b) Using the substitution $u = \pi - \theta$, or otherwise, find the area of the region enclosed by $C$ and the initial line.\\

(c) Show that, at the point of $C$ furthest from the initial line,

$$2 ( \pi - \theta ) \tan ^ { - 1 } ( \pi - \theta ) \cot \theta - \frac { \pi - \theta } { 1 + ( \pi - \theta ) ^ { 2 } } - \tan ^ { - 1 } ( \pi - \theta ) = 0$$

and verify that this equation has a root for $\theta$ between 1.2 and 1.3.\\

If you use the following page to complete the answer to any question, the question number must be clearly shown.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q7}}