9758/2019/P1/Q10

A curve $C$ has parametric equations

$x=a(2 \cos \theta-\cos 2 \theta),$
$y=a(2 \sin \theta-\sin 2 \theta),$

for $0 \leqslant \theta \leqslant 2 \pi$

(i) Sketch $C$ and state the Cartesian equation of its line of symmetry.

[2]

(ii) Find the values of $\theta$ at the points where $C$ meets the $x$-axis.

[2]

(iii) Show that the area enclosed by the $x$-axis, and the part of $C$ above the $x$-axis, is given by

$\int_{\theta_{1}}^{\theta_{2}} a^{2}\left(4 \sin ^{2} \theta-6 \sin \theta \sin 2 \theta+2 \sin ^{2} 2 \theta\right) d \theta,$

where $\theta_{1}$ and $\theta_{2}$ should be stated.

[3]

(iv) Hence find, in terms of $a$, the exact total area enclosed by $C$.

[5]