$$\begin{gathered} \left(i\right) Sketch the graph of y=2 \sin x for 0 \le x \le \mathrm{\pi }. \\ \left(\mathrm{ii}\right) \mathrm{On} \mathrm{the} \mathrm{same} \mathrm{set} \mathrm{of} \mathrm{axes} \mathrm{sketch} \mathrm{x}-3\mathrm{y}=0 \\ \left(\mathrm{iii}\right) \mathrm{Amusing} \mathrm{the} \mathrm{approximate} \mathrm{point} \mathrm{of} \mathrm{intersection} \mathrm{of} \mathrm{the} \mathrm{two} \mathrm{graphs} \mathrm{is} \mathrm{at} \mathrm{x}=2.7, \mathrm{use} \mathrm{one} \\ \mathrm{application} \mathrm{of} \mathrm{Newton}\text{'}\mathrm{s} \mathrm{method} \mathrm{to} \mathrm{find} \mathrm{a} \mathrm{second} \mathrm{approximation}. (\mathrm{Answer} \mathrm{correct} \mathrm{to} 3 \\ \mathrm{decimal} \mathrm{places}). \end{gathered}$$

$Evaluate \underset{\mathcal{x}\to 0}{\lim }\left(\frac{\sin 3\mathcal{x}}{2\mathcal{x}}\right).$

$Evaluate {\int }_0^{\frac{\mathrm{\pi }}{3}}{\sin }^2\mathcal{x}d\mathcal{x}$

$$\begin{gathered} The radius of a spherical balloon is increa\sin g at the rate of 2cm/sec. Find the rate at \\ which the volume of the balloon is increa\sin g when the radius is 10cm \left(in terms of \mathrm{\pi }\right). \end{gathered}$$

$$\begin{gathered} A cylindrical \tan k has diameter 4 metres and height 2.25 metres. Water is flowing into the \tan k at a rate of \frac{2\mathrm{\pi }}{5}{m}^3/min. \\ There is a tap at a point T at the base of the \tan k. When the tap is opened, water leaves the \tan k at a rate of \frac{\mathrm{\pi }}{5}\sqrt{h} m^3/min \\ where h is the height of the water in metres. \\ \left(i\right) Show that at time t minutes after the tap has opened, the volume of water in the \tan k satisfies the differential equation \\ \frac{dv}{dt}=\frac{\mathrm{\pi }(2-\sqrt{\mathrm{h}})}{5} \\ \left(ii\right) Show that at time t minutes after the tap has opened, the height of the water in the \tan k satisfies the differential equation \\ \frac{dh}{dt}=\frac{(2-\sqrt{\mathrm{h}})}{20} \\ \left(iii\right) When the tap is opened the height of the water is 0.16 metres. The time taken to fill the \tan k to a height of 2.25 metres \\ can be calculated u\sin g t={\int }_{0.16}^{2.25}\frac{20}{(2-\sqrt{\mathrm{h}})}dh (DO NOT PROVE THIS) \\ U\sin g the substitution H={(2-x)}^2 , where 0<x<2 , find the time taken to fill the \tan k, correct to the nearest minute. \end{gathered}$$

$$\begin{gathered} A stone is thrown at an angle of a to the horizontal. The position of the stone \\ at time t seconds is given by \mathcal{x}=Vt \cos a and y=Vt \sin a-\frac{1}{2}g{t}^2 where g m/s^2 \\ is the acceleration due to gravity and V m/s is the initial velocity of projection. \\ At what time does the stone reach its maximum height? \\ A) \frac{V\sin \alpha }{g} \\ B) \frac{g\sin \alpha }{V} \\ C) \frac{V^2{\sin }^2 \alpha }{2g} \\ D) \frac{g{\sin }^2 \alpha }{2V^2} \end{gathered}$$

$$\begin{gathered} What is the derivative of {\cos }^{-1} \left(\frac{\mathcal{x}}{3}\right)? \\ A) \frac{-1}{3\sqrt{9-{\mathcal{x}}^2}} \\ B) \frac{1}{3\sqrt{9-{\mathcal{x}}^2}} \\ C) \frac{-1}{\sqrt{9-{\mathcal{x}}^2}} \\ D)\frac{1}{\sqrt{9-{\mathcal{x}}^2}} \end{gathered}$$

$$\begin{gathered} Let f\left(x\right)=5-\sqrt{x}. \\ \left(i\right) Sketch the inverse function y=f^{\text{'}}\left(x\right). \\ \left(ii\right) Find the equation of the inverse function y=f^{\text{'}}\left(x\right). \end{gathered}$$

$The area enclosed by the curves \mathcal{y} =\sqrt{4-{\mathcal{x}}^2} and \mathcal{y} ={\mathcal{x}}^2-2 is shaded in the diagram below.$

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$$\begin{gathered} Which expression could be used to calculate this area? \\ A\left)2{\int }_0^1\left({\left(4-{\mathcal{x}}^2\right)}^{\frac{1}{2}}-{\mathcal{x}}^2+2\right)d\mathcal{x} \\ B\right){\int }_2^{-2}\left({\left(4-{\mathcal{x}}^2\right)}^{\frac{1}{2}}-{\mathcal{x}}^2-2\right)d\mathcal{x} \\ C\left)2{\int }_0^{\sqrt{3}}\left({\left(4-{\mathcal{x}}^2\right)}^{\frac{1}{2}}-{\mathcal{x}}^2+2\right)d\mathcal{x} \\ D\right){\int }_{-\sqrt{3}}^{\sqrt{3}}\left({\left(4-{\mathcal{x}}^2\right)}^{\frac{1}{2}}-{\mathcal{x}}^2-2\right)d\mathcal{x} \end{gathered}$$

$$\begin{gathered} The diagram show the graph of \mathcal{y}=f\left(\mathcal{x}\right),sketch the graph of \\ \mathcal{y}=\frac{1}{f\left(\mathcal{x}\right)}show all impor\tan t features. \end{gathered}$$

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