$$\begin{gathered} A man is in a boat at point B on a lake and AD is a straight stretch of the lake\text{'}s edge. \\ B is 3 kilometres from a point A on the river bank. The man wishes to travel from \\ point B to point D. He intends to row in a straight line to point C and then walk to D. \\ He can row at 4 km/h and walk at 5 km. /h. \end{gathered}$$

$$\begin{gathered} Let the dis\tan ce AC be x kilometres and let the total time for the trip be T hours. \\ \left(i\right)Explain why T=\frac{\sqrt{{\mathcal{x}}^2+9}}{4}+\frac{6-\mathcal{x}}{5} \\ \left(ii\right)Find the value of \mathcal{x} which will enable him to complete the trip in the minimum tme. \end{gathered}$$

$Find the area of XYZ$

$$\begin{gathered} The diagram shows the area of a minor segment of a circle with \\ radius 10cm.Find the area of this minor segment. \end{gathered}$$

$Simplify \frac{2se{c}^{2}A-2}{4\tan A}$

$$\begin{gathered} What are the coordinates of the focus of the parabola defined by \\ the equation \mathcal{y}=-16{\mathcal{x}}^2 \end{gathered}$$

$$\begin{gathered} The diagram shows the graphs of the curves \mathcal{y}=\sin \mathcal{x} \\ and \mathcal{y}= \cos \mathcal{x} for 0\le \mathcal{x}\le 2\mathrm{\pi }. \\ \mathrm{i})\mathrm{Show} \mathrm{that} \mathrm{the} \mathcal{x}-\mathrm{coordinates} \mathrm{of} \mathrm{the} \mathrm{two} \mathrm{marked} \mathrm{points} \mathrm{A} \mathrm{and} \mathrm{B} \mathrm{are} \frac{\mathrm{\pi }}{4} \\ \mathrm{and} \frac{5\mathrm{\pi }}{4} \mathrm{respectively}. \\ \mathrm{ii})\mathrm{Calculate} \mathrm{the} \mathrm{shaded} \mathrm{area} \mathrm{leaving} \mathrm{your} \mathrm{answer} \mathrm{in} \mathrm{surd} \mathrm{form}. \end{gathered}$$

$$\begin{gathered} For the curve f\left(\mathcal{x}\right)=\frac{2}{3}{\mathcal{x}}^3-8\mathcal{x}+1 \\ \left(i\right) Find the coordinates of the stationary point\left(s\right) and determine their nature. \\ \left(ii\right) Determine any point\left(s\right) of inflection. \\ \left(iii\right) Sketch the curve, showing all of the above features. \\ \left(iv\right) State the domain of x for which f \left(x\right) is monotonic increa\sin g. \end{gathered}$$

$$\begin{gathered} Find the volume of the solid formed when the area bounded by \\ \mathcal{y}=\sqrt{4-\mathcal{x}}, \mathcal{y}=0 and the \mathcal{y}-axis is rotated around the \mathcal{x}-axis. \end{gathered}$$ 

$$\begin{gathered} A Year 12 Bio\log y student tested to see how much bacteria was present in a \\ variety of food samples left in the classroom. It is known that after t hours the \\ number of bacteria \left(N\right) present in a particular type of food is given by the formula N=N_o{e}^{kt} \\ i) If initially there were 20 000 bacteria present and after three hours there were 45 000 \\ bacteria present, calculate the value of k (correct to 2 decimal places). \\ ii) How long would it take for the initial number of bacteria to triple in quantity? \\ iii) What will be the rate of increase of the bacteria after 4\frac{1}{2}hours? \end{gathered}$$

$Express \frac{\sqrt{6}}{\sqrt{6}-\sqrt{5}} in the form of a+b\sqrt{c} where a,b,c are integers$

$$\begin{gathered} The diagram above was sketched by a surveyor, who measured the \\ angle of elevation of the top of a tree on the other side of a river \\ to be 7°12\text{'} from point A. \\ From point B, 100 metres directly towards the tree from A, the \\ angle of elevation of the top of the same tree was 9942\text{'}. \\ i)Show that the height of the tree can be expressed in the form \\ h=\frac{100\sin 7°12\text{'}\sin 9°42\text{'}}{\sin 2°30\text{'}} \\ ii)Calculate the height of the tree correct to three significant figures. \end{gathered}$$