N2016 P2 Q3

A curve \(D\) has parametric equations \[x=t-\cos t,\qquad y=1-\cos t,\qquad\text{for}\hspace{0.5em}0\le t\le2\pi.\]

1. Sketch the graph of \(D\). Give in exact form the coordinates of the points where \(D\) meets the \(x\)-axis, and also give in exact form the coordinates of the maximum point on the curve.[4]
2. Find, in terms of \(a\), the area under \(D\) for \(0\le t\le a\), where \(a\) is a positive constant less than \(2\pi\).[3]

The normal to \(D\) at the point where \(t=\frac12\pi\) cuts the \(x\)-axis at \(E\) and the \(y\)-axis at \(F\).

3. Find the exact area of triangle \(OEF\), where \(O\) is the origin.[4]