2022 ASRJC Promo Q10

A curve \(C\) has parametric equations

\(x=2{{\sin }^{2}}t\), \(y=2t-\sin \left( 2t \right)\), \(0 < t < \frac{\pi }{2}\).

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=\tan t\).[3]
2. The normal to the curve at the point where \(t=a\) meets the \(y\)-axis at the point \(A\), where \(a\in \mathbb{R}\). Show that the \(y\)-coordinate of \(A\) is \(\lambda a\), where \(\lambda \) is a constant to be found.[3]
3. What can be said about the values of the gradients to the curve \(C\) as \(t\to 0\) and as \(t\to \frac{\pi }{2}\). Hence sketch \(C\), clearly showing the features of the curve when \(t\to 0\) and \(t\to \frac{\pi }{2}\).[4]