2019 NYJC P1 Q2

The diagram shows a mechanism for converting rotational motion into linear motion. The point \(P,\) on the circumference of a disc of radius \(r\), rotates about a fixed point \(O\). The point \(Q\) moves along the line \(OX\), and \(P\) and \(Q\) are connected by a rod of fixed length \(3r\). As the disc rotates, the point \(Q\) is made to slide backwards and forwards along \(OX\). At time \(t\), angle \(POQ\) is \(\theta\), measured anticlockwise from \(OX\), and the distance \(OQ\) is \(x\).

1. Show that \(x=r\left( \cos \theta +\sqrt{9-{{\sin }^{2}}\theta } \right)\).[2]
2. State the maximum value of \(x\).[1]
3. Express \(x\) as a polynomial in \(\theta\) if \(\theta\) is sufficiently small for \({{\theta }^{3}}\) and higher powers of \(\theta\) are to be neglected.[3]