ENGAA 2017 Section 1, Part B (Mathematics questions)

The diagram shows the outline of a keyhole consisting of three straight sides and an arc from a circle.

The sides $PQ$ and $RS$ are both 18 mm in length and when extended meet at the centre of the circle $O$ forming an angle of $\frac{\pi }{6}$ radians.

The longer arc from $Q$ to $R$ has length $22\pi$ mm.

What is the area, in ${\text{mm}}^2$, of the keyhole as shaded in the diagram?

It is given that $y=8^p$ and $z={\left(\frac{1}{2}\right)}^{2q}$ where $p$ and $q$ are real numbers.

Which of the following expressions is a simplification of ${\log }_2\left(\frac{y^3}{z^2}\right)$?

In triangle $PQR$$$PQ=4x\text{ cm}$$$$QR=(8-3x)\text{ cm}$$$$\angle PQR=60^{\circ }$$What is the maximum value of the area, in ${\text{cm}}^2$, of triangle $PQR$?

The curve $y=\sin x$ is stretched by a scale factor of $\frac{1}{2}$ parallel to the $x$-axis and then translated by $\frac{\pi }{4}$ in the negative $x$-direction.

What is the equation of the new curve?