N2015 P1 Q11

A curve \(C\) has parametric equations

\(x={{\sin }^{3}}\theta \), \(y=3{{\sin }^{2}}\theta \cos \theta \), \(0\le \theta \le \frac{\pi }{2}\).

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=2\cot \theta -\tan \theta \).[3]
2. Show that \(C\) has a turning point when \(\tan \theta =\sqrt{k}\), where \(k\) is an integer to be determined.
   Find, in non-trigonometric form, the exact coordinates of the turning point and explain why it is a maximum.[6]
3. Show that the area of the region bounded by \(C\) and the \(x\)-axis is given by
   \(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{1}{2}\pi }{9{{\sin }^{4}}\theta {{\cos }^{2}}\theta \mathrm{ d}\theta }\).
   Use your calculator to find the area, giving your answer correct to \(3\) decimal places.[3]

The line with equation \(y=ax\), where \(a\) is a positive constant, meets \(C\) at the origin and at the point \(P\).

4. Show that \(\tan \theta =\frac{3}{a}\) at \(P\). Find the exact value of \(a\) such that the line passes through the maximum point of \(C\).[3]