2019 EJC P2 Q2

The diagram shows a mechanism for converting rotational motion into linear motion. The point \(P\) is on the circumference of a disc of fixed radius r which can rotate about a fixed point \(O\). The point \(Q\) can only move on the line \(OX\), and \(P\) and \(Q\) are connected by a rod of length \(2r\). As the disc rotates, the point \(Q\) is made to slide along \(OX\). At time t, angle \(POQ\) is \(\theta\) and the distance \(OQ\) is \(x\).

1. State the maximum and minimum values of \(x\).[1]
2. Show that \(x=r\left( \cos \theta +\sqrt{4-{{\sin }^{2}}\theta } \right)\).[2]
3. At a particular instant, \(\theta =\frac{\pi }{6}\) and \(\frac{\mathrm{d}\theta }{\mathrm{d}t}=0.3\). Find the numerical rate at which point \(Q\) is moving towards point \(O\) at that instant, leaving your answer in terms of \(r\).[3]