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