7 The curve $C _ { 1 }$ has polar equation $r = \theta \cos \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.\\
(a) 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 $Q$.\\
(b) Find the polar coordinates of $Q$, giving your answers in exact form.\\

(c) Sketch $C _ { 1 }$ and $C _ { 2 }$ on the same diagram.\\
(d) 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]\\

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

\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q7 [7]}}