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The diagram shows a sketch of the curve $C$ with polar equation $r = 4 \cos ^ { 2 } \theta$ for $- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.\\
(i) Explain briefly how you can tell from this form of the equation that $C$ is symmetrical about the line $\theta = 0$ and that the tangent to $C$ at the pole $O$ is perpendicular to the line $\theta = 0$.\\
(ii) The equation of $C$ may be expressed in the form $r = k ( 1 + \cos 2 \theta )$. State the value of $k$ and use this form to show that the area of the region enclosed by $C$ is given by

$$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } ( 3 + 4 \cos 2 \theta + \cos 4 \theta ) d \theta ,$$

and hence find this area.\\
(iii) The length of $C$ is denoted by $L$. Show that

$$L = 8 \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos \theta \sqrt { 1 + 3 \sin ^ { 2 } \theta } \mathrm {~d} \theta$$

and use the substitution $\sinh x = \sqrt { 3 } \sin \theta$ to determine $L$ in an exact form.

\hfill \mbox{\textit{Pre-U Pre-U 9795  Q12}}