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 \(0\le \theta \le 2\pi \).

1. Sketch \(C\) and state the cartesian equations of its lines of symmetry.[2]
2. 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]
3. Show that the area enclosed by the axes and the part of \(C\) in the first quadrant is given by
   \(\displaystyle\displaystyle \displaystyle\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{\left( 27{{\cos }^{2}}\theta -27\cos \theta \cos 3\theta +6{{\cos }^{2}}3\theta \right)\,\mathrm{d}\theta }\),
   where the values of \({{\theta }_{1}}\) and \({{\theta }_{2}}\) should be stated.[3]
4. Hence find the exact total area enclosed by \(C\).[5]