2020 YIJC P2 Q3
A curve $C$ has parametric equations
$x=3\cos \theta -2\cos 3\theta $,
$y=9\sin \theta -\sin 3\theta $,
for $0\le \theta \le 2\pi $.
(i)
Sketch $C$ and state the cartesian equations of its lines of symmetry.
[2]
(ii)
Given that $\cos 3\theta =4{{\cos }^{3}}\theta -3\cos \theta $, find the exact values of $\theta $ at the points where $C$ meets the $y$-axis.
[2]
(iii)
Show that the area enclosed by the axes and the part of $C$ in the first quadrant is given by
$\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{\left( 27{{\cos }^{2}}\theta -27\cos \theta \cos 3\theta +6{{\cos }^{2}}3\theta \right)\,\text{d}\theta }$,
where the values of ${{\theta }_{1}}$ and ${{\theta }_{2}}$ should be stated.
[3]
(iv)
Hence find the exact total area enclosed by $C$.
[5]
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