.$$\begin{gathered} A metal tray, in the shape of a rec\tan gular prism with a square base, is \\ made out of 108square centimetres of sheet metal. The tray is open at \\ the top. Let \mathcal{x} cm be the side length of the base and h cm be the height as shown. \end{gathered}$$

$$\begin{gathered} Show that h= \frac{108-{\mathcal{x}}^2}{4\mathcal{x}} \\ Show that the volume, V of the tray is given by V=27\mathcal{x}-\frac{{\mathcal{x}}^2}{4} \\ Find the maximum volume of the tray. \end{gathered}$$

$$\begin{gathered} A hollow bowl is made by rotating part of the curve \mathcal{y} =\sqrt{\mathcal{x}+2} between \mathcal{x}=-1 and \\ \mathcal{x}= 1 around the x axis, as shown below. Find the exact volume occupied by the \\ bowl. \end{gathered}$$

$\text{In the diagram }A,B\text{ and }C\text{ are the points }(-5,3),(2,2)\text{ and }(1,-5)\text{ respectively (Diagram is not to scale). }$

$$\begin{gathered} \text{(i) Calculate the gradient of }AC\text{. } \\ \left(ii\right)Find the coordinates of X,the midpoint of AC. \\ \left(iii\right)Calculate the length of EX. \end{gathered}$$

$$\begin{gathered} \text{(iv) Hence, or otherwise, find the coordinates of }D\text{ if }X\text{ is the midpoint of ABD.} \\ \text{(v) Show that }AC\perp BD \\ \text{(vi) Explain why the quadrilateral }ABCD\text{ is a square. } \\ \text{(vii) Calculate the area of }ABCD\text{. } \end{gathered}$$

$$\begin{gathered} On a golf course, the tee is a dis\tan ce of 130 metres due west from the hole. \\ On her first shot, Kelly hits the ball 100 metres but not at the correct angle. On \\ her second shot she hits the ball 35 metres and gets it in the hole. On what \\ bearing, a, did she hit her first shot? Give your answer correct to the nearest degree \end{gathered}$$

$$\begin{gathered} A(4,2)\text{ and }B(-2,-8)are \mathrm{two} \mathrm{points} on \mathrm{the} \mathrm{number} \mathrm{plane}.\mathrm{Th}e \mathrm{point} P(\mathcal{x},\mathcal{y}) moves so that PA is always perpendicular to PB. \\ (i) Find the gradiant of PA and PB in term of \mathcal{x} and \mathcal{y}. \\ (ii) Hence, show that the equation of the locus of P is (\mathcal{x}-1{)}^2+(\mathcal{y}+3{)}^2=34 \\ (iii) Describe the locus of P geometrically. \end{gathered}$$

$$\begin{gathered} A fence 8m high is parallel to the wall of a building and 1m from the building \\ Write an expression for the length of AB in terms of \theta . \\ Find the shortest lentgh of the plank AB that can go over the fence from the level ground \\ in order to rest against the wall. \end{gathered}$$

$$\begin{gathered} The value of \alpha in the above diagram is \\ \left(A\right) 9 \\ \left(B\right) 11 \\ \left(C\right) 12 \\ \left(D\right) 12.6 \end{gathered}$$

$$\begin{gathered} ABCD is a square of side length 2 units. P is the midpoint of AD. \\ CQ is drawn perpendicular to PB and \angle APB={\mathcal{x}}^{°}. \\ Prove \angle APB=\angle QBC . \\ Hence, or otherwise show QC=\frac{4}{\sqrt{5}} units. \\ Show QD=CD. \end{gathered}$$

$$\begin{gathered} Find the largest vertical dis\tan ce between graphs \mathcal{y}=\frac{1}{2}\mathcal{x} \\ and \mathcal{y}=\sin \mathcal{x} in the interval 0\le \mathcal{x}\le 2\mathrm{\pi }. \end{gathered}$$

$$\begin{gathered} Ship X is 30 nautical miles from port P and is on a bearing of {065}^o. \\ Ship Y is 40 nautical miles from port P and is on a bearing of {125}^o. \\ Show that \angle XPY = 60°. \\ Determine the dis\tan ce between the two ships, correct to one decimal place. \\ Find the bearing of ship X from ship Y, to the nearest degree. \end{gathered}$$