$$\begin{gathered} Find \frac{d}{d\mathcal{x}}\left(\mathcal{x}{\tan }^{-1}\mathcal{x}\right) \\ Evaluate {\int }_1^{\sqrt{2}}\frac{d\mathcal{x}}{\sqrt{4-{\mathcal{x}}^2}} \\ Use the substitution u=\mathcal{x}+2 to evaluate {\int }_{-2}^2\mathcal{x}\sqrt{\mathcal{x}+2} d\mathcal{x}. \end{gathered}$$

$The diagram below shows the graph of \mathcal{y}=\cos \frac{1}{2}\mathcal{x} for 0\le \mathcal{x}\le \mathrm{\pi }.$

$$\begin{gathered} The shaded area is rotated about the \mathcal{x}-axis.Find the volume of the solid formed. \\ i) Show by means of a sketch, that the curves \mathcal{y}={\mathcal{x}}^2 and \mathcal{y}=\frac{1}{2}\ln \mathcal{x} meet at a \sin gle point. \\ ii) By taking 0.5 as a first approximation to the root of {\mathcal{x}}^2+\frac{1}{2}\ln \mathcal{x}=0, use Newton\text{'}s method \\ once to find a better approximation of where the two curves meet.Give your answer correct \\ to 2 decimal places. \end{gathered}$$

$$\begin{gathered} A particle is moving in a straight line and its position \mathcal{x} is given by the equation \mathcal{x}=2{\sin }^{2}t \\ \left(i\right) Express the acceleration of the particle in terms of \mathcal{x} in the form \stackrel{..}{\mathcal{x}}=-n^2(\mathcal{x}-a) \\ \left(ii\right) Hence state the period and centre of motion. \\ \left(iii\right) What are the extremities of the motion? \end{gathered}$$

$$\begin{gathered} A particle P moves in a straight line so that its dis\tan ce from O after t secs is \\ \mathcal{x} metres.The acceleration of P from O is given by the equation \frac{d^2\mathcal{x}}{d{t}^2}=10\mathcal{x}-2{\mathcal{x}}^3.The \\ particle is at rest 1 metre to the right of O. \\ i) Find v^2 in term of \mathcal{x} where v is the velocity of the particle. \\ ii) Hence find all position where the particle is at rest. \end{gathered}$$

$$\begin{gathered} A particle moves such that \stackrel{..}{\mathcal{x}}=\mathcal{x}–1. Initially, \mathcal{x}=2 and \mathcal{v}=1 \\ \left(i\right) Show that \mathcal{v}=\mathcal{x}-1 \\ \left(ii\right) Find \mathcal{x} as a function of t. \end{gathered}$$

$Water is poured into a hemispherical bowl of radius 5cm at the rate of 22c{m}^3/sec.$

$the point that dividies the interval joining (-2,3) to B(5,4) externally in the ratio of 2:3 is$

$$\begin{gathered} The interval AB between A(2, -1) and B(-6,3) is divided internally by the \\ point P in the ratio 1:3. The correct coordinate of P is given by: \end{gathered}$$

$Which of the following is not true about the function y =|x^2 -9| + 2 ?$

$$\begin{gathered} The volume , V of a sphere of radius r is increa\sin g at a cons\tan t rate of \\ 200m{m}^3 per second \\ i) Find \frac{dr}{dt} in terms of r. \\ ii) Determine the rate of increaase of the surface area .S of the sphere \\ when the radius is 50 mm, (note V=\frac{4}{3}{\mathrm{\pi r}}^3,\mathrm{S} =4{\mathrm{\pi r}}^2 \end{gathered}$$