$Solve the inequality \frac{3x+2}{x-2}>1.$

$$\begin{gathered} Use Newton’s method to find a better approximation to the root of \\ {\log }_{e}x-\sin x=0, given that the roots is near x=2. (x is a radian) \\ Give your answer to 2 decimal places. \end{gathered}$$

$$\begin{gathered} Find \frac{d}{d\mathcal{x}}(1+{\mathcal{x}}^2) {\tan }^{-1}\mathcal{x}. \\ \left(A\right) 2\mathcal{x} {\tan }^{-1}\mathcal{x} \\ \left(B\right)\hspace{0.17em}\frac{2\mathcal{x}}{1+{\mathcal{x}}^2} \\ \left(C\right) \frac{1+{\mathcal{x}}^2}{\sqrt{1-{\mathcal{x}}^2}} \\ \left(D\right)\hspace{0.17em}1+2\mathcal{x} {\tan }^{-1}\mathcal{x} \end{gathered}$$

$$\begin{gathered} \int \frac{d\mathcal{x}}{4+9{\mathcal{x}}^2}= \\ \left(A\right) \frac{1}{3} {\tan }^{-1}\left(\frac{2\mathcal{x}}{3}\right) \\ \left(B\right) \frac{1}{6} {\tan }^{-1}\left(\frac{3\mathcal{x}}{2}\right) \\ \left(C\right) \frac{1}{6} {\tan }^{-1}\left(\frac{2\mathcal{x}}{3}\right) \\ \left(D\right) \frac{1}{3} {\tan }^{-1}\left(\frac{3\mathcal{x}}{2}\right) \end{gathered}$$

$a) Evaluate {\int }_0^1\frac{dx}{\sqrt{2-x^2}}.$

$b) Solve \frac{2}{3-x}\le 1$

$$\begin{gathered} Tap water at 24°C is placed in a fridge-freezer maintained at a temperature of -11°C. \\ After t minutes the rate of change of temperature T of the water is given by \\ \frac{dT}{dt}=-k(T+11) \\ \left(i\right) Show that T=A{e}^{-kt}-11 is a solution of the above equation, what A is a cons\tan t. \\ \left(ii\right) Find the values of A. \\ \left(iii\right) After 15 minutes the temperature of the water falls to 10°C. \\ Find, correct to the nearest minute, the time taken for the water to start freezing. \\ (Freezing point of water is 0°C). \end{gathered}$$

$$\begin{gathered} Differentiate {\log }_e({\mathcal{x}}^3+1) with respect to \mathcal{x}. \\ Evaluat {\int }_0^3\frac{d\mathcal{x}}{9+{\mathcal{x}}^2} \end{gathered}$$

$$\begin{gathered} li{m}_{\mathcal{x}\to 0}\frac{\tan 3\mathcal{x}}{2\mathcal{x}} is equal to : \\ \left(A\right) 0 \\ \left(B\right) 1 \\ \left(C\right) \frac{3 }{2} \\ \left(D\right) 2 \end{gathered}$$

$$\begin{gathered} Find \int \frac{\mathcal{x}}{\sqrt{1+\mathcal{x}}}d\mathcal{x} u\sin g the substitution \mathcal{x}=u^2-1 \\ Evaluate{\int }_0^{\frac{\mathrm{\pi }}{6}}{\cos }^{2}3\mathcal{x} d\mathcal{x} \end{gathered}$$