If $\alpha , \beta and \gamma$ are the roots of ${\mathcal{x}}^3-4{\mathcal{x}}^2+5\mathcal{x}+3=0$ evaluate ${\alpha }^3+{\beta }^3+{\gamma }^3$.

$$\begin{gathered} The region enclosed by \mathcal{Y}={\mathcal{x}}^3 , \mathcal{Y}=0 and \mathcal{x}=2 is rotated around the y-axis to produce a \\ solid. What is the volume of this solid? \end{gathered}$$

$$\begin{gathered} What is the angle at which a road must be banked so that a car may round a curve with \\ a radius of 100 metres at 90 km/h without sliding? Assume that the road is smooth and \\ gravity to be 9.8 2m{s}^{-2} . \end{gathered}$$

$$\begin{gathered} A particle of mass m falls from rest under gravity and the resis\tan ce to its motion is mk{v}^2,where v is its speed and k is a positive cons\tan t. \\ Which of the following is the correct expression for square of the velocity where \mathcal{x} is the dis\tan ce fallen? \end{gathered}$$

$$\begin{gathered} \mathrm{The} \mathrm{foci} \mathrm{of} \mathrm{the} \mathrm{hyperbola} \mathcal{xY}=8 \mathrm{are} \\ \left(\mathrm{A}\right) \left(\pm 4,\pm 4\right) \\ \left(\mathrm{B}\right) \left(\pm 2\sqrt{2},\pm 2\sqrt{2}\right) \\ \left(\mathrm{C}\right) \left(\pm 8\sqrt{2},\pm 8\sqrt{2}\right) \\ \left(\mathrm{D}\right) (\pm 4\sqrt{2},\pm 4\sqrt{2}) \end{gathered}$$

$$\begin{gathered} \mathrm{Consider} \mathrm{a} \mathrm{polynomial} \mathrm{P}\left(\mathcal{x}\right)\mathrm{of} \mathrm{degree} 3. \mathrm{You} \mathrm{are} \mathrm{given} 2 \mathrm{numbers} \mathrm{a} \mathrm{and} \mathrm{b} \mathrm{such} \mathrm{that} \\ .\mathrm{a}<\mathrm{b} \\ .\mathrm{P}\left(\mathrm{a}\right)>\mathrm{P}\left(\mathrm{b}\right)>0 \\ .\mathrm{P}\text{'}\left(\mathrm{a}\right)=\mathrm{P}\text{'}\left(\mathrm{b}\right)=0 \\ \mathrm{The} \mathrm{polynomial} \mathrm{has} \\ \left(\mathrm{A}\right) 3 \mathrm{real} \mathrm{zeros} \\ \left(\mathrm{B}\right) 1 \mathrm{real} \mathrm{zero} \mathrm{\gamma such} \mathrm{that} \mathcal{Y}<\mathrm{a} \\ \left(\mathrm{C}\right) 1 \mathrm{real} \mathrm{zero} \mathrm{\gamma such} \mathrm{that} \mathrm{a}< \mathcal{Y}< \mathrm{b} \\ \left(\mathrm{D}\right) 1 \mathrm{real} \mathrm{zero} \mathrm{\gamma } \mathrm{such} \mathrm{that} \mathcal{Y}>\mathrm{b} \end{gathered}$$

$$\begin{gathered} \mathrm{Given} \mathrm{that} {\mathcal{x}}^2+ {\mathcal{Y}}^2+\mathcal{x} \mathcal{Y}=2, \mathrm{which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{true}? \\ \left(\mathrm{A}\right) \frac{d \mathcal{Y}}{d\mathcal{x}}=\frac{2\mathcal{x}+ \mathcal{Y}}{2 \mathcal{Y}+\mathcal{x}} \\ \left(\mathrm{B}\right) \frac{d \mathcal{Y}}{d\mathcal{x}}=-\frac{2\mathcal{x}+ \mathcal{Y}}{2 \mathcal{Y}+\mathcal{x}} \\ \left(\mathrm{C}\right)\frac{d \mathcal{Y}}{d\mathcal{x}}=\frac{2\mathcal{x}- \mathcal{Y}}{2 \mathcal{Y}+\mathcal{x}} \\ \left(\mathrm{D}\right) \frac{d \mathcal{Y}}{d\mathcal{x}}=\frac{-2\mathcal{x}+ \mathcal{Y}}{2 \mathcal{Y}+\mathcal{x}} \end{gathered}$$

$$\begin{gathered} A particle of mass m kg is set in motion, with speed u m{s}^{-1} and moves in a straight line \\ before coming to rest. At time t seconds the particle has displacement x metres from its starting \\ point O, velocity v m{s}^{-1} and acceleration a m{s}^{-2}. \\ The resul\tan t force acting on the particle directly opposes its motion and has magnitude m (1+v) Newtons. \\ Show that a=-(1+v) \\ Find expressions for, \\ \mathcal{x} in terms of v \\ v in terms of t \\ \mathcal{x} in terms of t \\ Show that \mathcal{x}+v+t=u \\ Find the dis\tan ce travelled and time taken by the particle in coming to rest. \end{gathered}$$

$$\begin{gathered} A person is s\tan ding on the outer edge of a circular disc that is spinning. His relative position on \\ the disc remains unchanged. Which description below best describes the situation? \end{gathered}$$

$$\begin{gathered} \mathrm{Find} \mathrm{real} \mathrm{constants} \mathrm{A}, \mathrm{B} \mathrm{and} \mathrm{C} \mathrm{such} \mathrm{that} \\ \frac{4\mathcal{x}-2}{\left({\mathcal{x}}^2-1\right)(\mathcal{x}-2)}\equiv \frac{A\mathcal{x}+\mathrm{B}}{\left({\mathcal{x}}^2-1\right)}+\frac{\mathrm{C}}{(\mathcal{x}-2)} \end{gathered}$$