2020 TMJC Promo Q9

A curve \(C\) has parametric equations \(x=\cos 2\theta \) and \(y=\cos \mathrm{ec }\theta \), where \(0 < \theta \le \frac{\pi }{2}\).

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{4}\cos \mathrm{e}{{\mathrm{c}}^{3}}\theta \). What can be said about \(\frac{\mathrm{d}y}{\mathrm{d}x}\) as \(\theta \to 0\)?[3]
2. Sketch \(C\), stating the equation of the asymptote and coordinates of the point of intersection with the axis.[3]
3. Find the equation of the normal to \(C\) at the point where \(\theta =\frac{\pi }{6}\).[3]
4. Show that the point(s) of intersection of the normal found in part (iii) and the curve \(C\) satisfy the equation \({{\sin }^{3}}\theta +\frac{7}{4}\sin \theta -1=0\). Hence using a non-calculator method, show that this normal does not meet the curve again.[4]