7 The curve \(C _ { 1 }\) has polar equation \(r = \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
- The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\),
$$2 \theta \tan \theta - 1 = 0$$
and verify that this equation has a root between 0.6 and 0.7 .
The curve \(C _ { 2 }\) has polar equation \(r = \theta \sin \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\). - Find the polar coordinates of \(Q\), giving your answers in exact form.
- Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
- Find, in terms of \(\pi\), the area of the region bounded by the arc \(O Q\) of \(C _ { 1 }\) and the arc \(O Q\) of \(C _ { 2 }\). [7]
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