N2019 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 \(0\le \theta \le 2\pi .\)

1. Sketch \(C\) and state the Cartesian equation of its line of symmetry.[2]
2. Find the values of \(\theta \) at the points where \(C\) meets the \(x\)-axis.[2]
3. Show that the area enclosed by the \(x\)-axis, and the part of \(C\) above the \(x\)-axis, is given by
   \(\displaystyle\displaystyle \displaystyle\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{{{a}^{2}}\left( 4{{\sin }^{2}}\theta -6\sin \theta \sin 2\theta +2{{\sin }^{2}}2\theta \right)}\,\mathrm{d}\theta ,\)
   where \({{\theta }_{1}}\) and \({{\theta }_{2}}\) should be stated.[3]
4. Hence find, in terms of \(a,\) the exact total area enclosed by \(C.\)[5]