$$\begin{gathered} The eccentricity of the ellipse \frac{\mathcal{x}}{k}+\frac{\mathcal{y}}{k-1}=1 k>1 is equal to \\ A) \frac{\sqrt{2k-1}}{k} \\ B) \frac{1}{\sqrt{k}} \\ C) \sqrt{\frac{2k-1}{k}} \\ D) \frac{\sqrt{2k^2-2k+1}}{k} \end{gathered}$$

$$\begin{gathered} The equations of the directrices of the hyperbola \frac{{\mathcal{x}}^2}{144}-\frac{{\mathcal{y}}^2}{25}=1 are : \\ A) \mathcal{x}\pm \frac{13}{144} \\ B) \mathcal{x}\pm \frac{13}{25} \\ C) \mathcal{x}\pm \frac{25}{13} \\ D) \mathcal{x}\pm \frac{144}{13} \end{gathered}$$

$Use the substitution u=\cos \mathcal{x} to find \int {\cos }^2\mathcal{x}{\sin }^5\mathcal{x}d\mathcal{x}$

$$\begin{gathered} Find \int \frac{e^{\tan \mathcal{x}}}{{\cos }^2\mathcal{x}}d\mathcal{x} \\ Use partial fractions to evaluate \\ {\int }_0^1\frac{5dt}{(2t+1)(2-t)} \\ Hence,and by using the substitution t=\tan \frac{\theta }{2} evaluate \\ {\int }_0^{\frac{\pi }{2}}\frac{d\theta }{3\sin \theta +4\cos \theta } \\ By using the table of standard integrals and manipulation,find \\ {\int }_0^1\frac{d\mathcal{x}}{\sqrt{4{\mathcal{x}}^2+36}} \\ If, I=\int e^{\mathcal{x}}\sin \mathcal{x}d\mathcal{x},find I, \\ By completing the square find \\ \int \frac{d\mathcal{x}}{\sqrt{1-4\mathcal{x}-{\mathcal{x}}^2}} \end{gathered}$$

$Use integration by parts to evaluate {\int }_1^e{\mathcal{x}}^4{\log }_e\mathcal{x}d\mathcal{x}$

$$\begin{gathered} The gradient of the curve {\mathcal{x}}^2\mathcal{y}-{\mathcal{xy}}^2+6=0 at point P(2, 3) is equal to \\ A) –5 \\ B) 38 \\ C) 98 \\ D) 1 \end{gathered}$$

$$\begin{gathered} Express \frac{3\mathcal{x}}{(\mathcal{x}+1)(\mathcal{x}+2)(\mathcal{x}+3)} in partial fraction and hence prove that \\ {\int }_0^1\frac{3\mathcal{x}}{(\mathcal{x}+1)(\mathcal{x}+2)(\mathcal{x}+3)}d\mathcal{x}=\ln 2 \end{gathered}$$

$$\begin{gathered} \mathrm{Where} \mathrm{there} \mathrm{are} \mathrm{vertical} \mathrm{tangents} \mathrm{on} \mathrm{a} \mathrm{curve} \mathcal{y}=\mathrm{f}\left(\mathcal{x}\right) \mathrm{then}, \mathrm{in} \mathrm{expression} \mathrm{for} \frac{\mathrm{d}\mathcal{y}}{\mathrm{d}\mathcal{x}}. \\ \mathrm{A}. \mathrm{The} \mathrm{numerators} \mathrm{equals} \mathrm{zero} \\ \mathrm{B}. \mathrm{The} \mathrm{denominator} \mathrm{equals} \mathrm{zero} \\ \mathrm{C}. \mathrm{Both} \mathrm{the} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \mathrm{equal} \mathrm{zero} \\ \mathrm{D}. \mathrm{None} \mathrm{of} \mathrm{the} \mathrm{above} \end{gathered}$$

$$\begin{gathered} The ellipse E cartesian equation \frac{x^2}{25}+\frac{y^2}{16}=1 \\ Sketch the curve and write down the: \\ \left(\alpha \right) eccentricity; \\ \left(\beta \right) coordinates of the foci \mathcal{S} and \mathcal{S}\text{'}; \\ \left(\gamma \right) equation of the directerices. \\ \left(\alpha \right) Show that the point P on E can be represented by the coordinates \\ (5\cos \theta ,4\sin \theta ). \\ \left(\beta \right) Prove that PS+PS\text{'} is independent of the position of P on the curve. \\ Show that the \tan gent at the points P(cp,\frac{c}{p}) and Q(cq,\frac{c}{q}) on the \\ rec\tan gular hyperbola xy=c^2 meet at the point (\frac{2cpq}{q+p},\frac{2c}{p+q}) \end{gathered}$$

$$\begin{gathered} \mathrm{The} \mathrm{definite} \mathrm{integral} {\int }_0^3\sqrt{9-{\mathcal{x}}^2}\mathrm{d}\mathcal{x} \\ \mathrm{A}. \mathrm{Could} \mathrm{be} \mathrm{evaluated} \mathrm{by} \mathrm{the} \mathrm{substitution} \mathcal{x}=3\mathrm{tan\theta } \\ \mathrm{B}. \mathrm{Could} \mathrm{be} \mathrm{evaluated} \mathrm{by} \mathrm{the} \mathrm{substitution} \mathcal{x}=3\mathrm{sec\theta } \\ \mathrm{C}. \mathrm{Could} \mathrm{be} \mathrm{evaluated} \mathrm{by} \mathrm{the} \mathrm{substitution} \mathcal{x}=3\mathrm{cosec\theta } \\ \mathrm{D}. \mathrm{Equals} \frac{9\mathrm{\pi }}{4} \end{gathered}$$

$The diagram below shows the hyperbola \frac{{\mathcal{x}}^2 }{a^2}- \frac{{\mathcal{y}}^2}{b^2} = 1 where a >b >0. The points P(a sec \theta ,b \tan \theta ) and Q(asec \theta ,b\tan \theta ) lie on the hyperbola and the chord PQ subtends a right angle at the origin.$

$$\begin{gathered} Which of the following is correct? \\ A) \sin \theta \sin \alpha =-\frac{a^2}{b^2} \\ B) \sin \theta \sin \alpha =\frac{a^2}{b^2} \\ C) \tan \theta \tan \alpha =-\frac{a^2}{b^2} \\ D) \tan \theta \tan \alpha =\frac{a^2}{b^2} \end{gathered}$$

$The graphs of \mathcal{y} = f\left(\mathcal{x}\right) and \mathcal{y} = g\left(\mathcal{x}\right) are shown below.$

$$\begin{gathered} Which of the following best describes the relationship between f\left(\mathcal{x}\right) and g\left(\mathcal{x}\right) ? \\ A) \left|f\left(\mathcal{x}\right)\right|=g(\mathcal{x}\left) \\ B\right) g\left(\mathcal{x}\right)=\ln \left[f\left(\mathcal{x}\right)\right] \\ C) f(\mathcal{x})=\pm \ln \left|g\left(\mathcal{x}\right)\right| \\ D) g\left(\mathcal{x}\right)=\pm \sqrt{f\left(\mathcal{x}\right)} \end{gathered}$$

$$\begin{gathered} Consider the hyperbola with the equation \frac{{\mathcal{x}}^2}{16}-\frac{{\mathcal{Y}}^2}{9}=1 \\ Find the coordinates of the foci and \mathcal{x}-intercepts of the hyperbola. \\ Find the equations of the directrices and the asymptotes of the hyperbola. \\ What are the parametric equations of this hyperbola? \end{gathered}$$

$$\begin{gathered} Given that a, b and c are real positive numbers \\ \left(i\right) Prove that a^2+b^2+c^2\ge ab+bc+ac \\ \left(ii\right) Show that \\ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{9}{a+b+c} \\ \left(iii\right) Given that a^2+b^2+c^2= 9, prove that \\ \frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ac}\ge \frac{3}{2} \\ If z and w are complex numbers such that \left|z\right| = \left|w\right|, show that \\ \frac{1}{2}\left(z+w\right).\frac{1}{2}\stackrel{⏜}{\left(z+w\right)}\frac{1}{2}\left(z+w\right)\frac{1}{2}\stackrel{⏜}{\left(z+w\right)}=z\stackrel{⏜}{z} \\ A particle of mass 𝑚 kg is projected vertically upwards with a speed of Um{s}^{-1} \\ At time 𝑡 seconds the particle has vertical height 𝑥 metres above the \\ point of projection, speed vm{s}^{-1} and acceleration am{s}^{-2}. The particle moves under \\ gravity in a medium where the resis\tan ce to motion has magnitude \\ \frac{m}{g}{v}^{2}Newtons where gm{s}^{-1} is the acceleration due to gravity. \\ \left(i\right) Show thata=-\frac{1}{g}\left(g^2+v^2\right). \\ \left(ii\right) Show that \\ v=g\left(\frac{U-g\tan t}{g+U\tan t}\right) \\ \left(iii\right) Find the time taken for the particle to reach its maximum height \\ \left(iv\right) Express 𝑥 in terms of 𝑡. \end{gathered}$$

$$\begin{gathered} Evaluate {\int }_0^1\frac{e^{\mathcal{x}}}{1+e^{\mathcal{x}}}d\mathcal{x} \\ By completing the square, find \int \frac{d\mathcal{x}}{\sqrt{{\mathcal{x}}^2-6\mathcal{x}+10}}. \\ Use integration by parts to find \int \mathcal{x}{\log }_e\mathcal{x}d\mathcal{x}. \\ Use the substitution u=l-\mathcal{x} to evaluat {\int }_{-1}^0\frac{2+\mathcal{x}}{\sqrt{1-\mathcal{x}}}d\mathcal{x}. \\ Find real numbers a and b such that \\ \frac{4{\mathcal{x}}^2-5\mathcal{x}-1}{(\mathcal{x}-3)({\mathcal{x}}^2+1)}=\frac{a}{\mathcal{x}-3}+\frac{b\mathcal{x}+1}{{\mathcal{x}}^2+1}. \\ Hence find \int \frac{4{\mathcal{x}}^2-5\mathcal{x}-1}{(\mathcal{x}-3)({\mathcal{x}}^2+1)}d\mathcal{x}. \end{gathered}$$

$$\begin{gathered} If \alpha , \beta , and \gamma are the roots of the cubic equation {\mathcal{x}}^3+m\mathcal{x}+n=0 , find in terms of m and n, the values of \\ \frac{1}{\alpha } +\frac{1}{\beta }+\frac{1}{\gamma } \\ {\alpha }^2+{\beta }^2+{\gamma }^2 \\ Detennine the cubic equation whose roots are {\alpha }^2, {\beta }^2 and {\gamma }^2 \end{gathered}$$

$$\begin{gathered} Given that the equation {\mathcal{x}}^4-5{\mathcal{x}}^3-9{\mathcal{x}}^2+81\mathcal{x}-108=0 has a triple root, \\ find all the roots of the equation \end{gathered}$$

$$\begin{gathered} The solid S is generated by moving a straight edge so that it is \\ always parallel to a fixed plane P. It is constrained to pass \\ through a circle C and line segment l. The circle C, which \\ forms a base for S, has radius 𝑎 and the line segment l \\ is at a dis\tan ce 𝑑 from C. Both C and 𝒍 are \\ perpendicular to P. The perpendicular to C at \\ its centre O bisects 𝒍. \end{gathered}$$

.

$$\begin{gathered} \left(i\right) Calculate the area of the triangular cross-section EFG which \\ is parallel to P and dis\tan ce 𝑥 from the centre O of C. \\ \left(ii\right) Calculate the volume of S. \\ \left(i\right) Prove the identity: \\ {\cos }^{3}A-\frac{3}{4}\cos A=\frac{1}{4}\cos 3A \\ \left(iii\right) Show that 𝑥 = 2\surd 2\cos A satisfies the cubic equation \\ {\mathcal{x}}^3+6\mathcal{x}+2=0 given that \cos 3𝐴 = \frac{-1}{2\sqrt{2}} \\ For n=1,2,3...s_{n }and T_n are two different sequences of positive integers. \\ Given that S_n=T_1+T_2+T_3+.....T_n \\ Also s_1=6, s_2=20 and s_n=6s_{n-1}-8s_{n-2} foe n=3,4,5... \\ \left(i\right) Prove by mathematical induction that s_n=4^n+2^n,n=1,2,3.... \\ \left(ii\right) Hence or otherwise, find 𝑇𝑛,n = 1, 2, 3 \dots in simplest form \end{gathered}$$

$$\begin{gathered} If f\left(\mathcal{x}\right)=(\mathcal{x}-1)(\mathcal{x}-3) then sketch \\ \left(i\right)\mathcal{y}=\frac{1}{f\left(\mathcal{x}\right)} \\ \left(ii\right)\mathcal{y}=f\left(l\mathcal{x}l\right) \\ \left(iii\right)l\mathcal{y}l=f\left(\mathcal{x}\right) \\ \left(i\right)Find the stationary points and the asymptotes of the function \\ \mathcal{y}=\frac{{(\mathcal{x}+l)}^4}{{\mathcal{x}}^4+1} \\ \left(ii\right)Sketch this function labelling all essential features. \\ \left(iii\right)Use the graph to find the set of values of k for which{(\mathcal{x}+1)}^4=k({\mathcal{x}}^4+1) \\ has two distinct real roots. \\ Given the graph of \mathcal{y}=f\text{'}\left(\mathcal{x}\right) below, sketch the graph of \mathcal{y}= f \left(\mathcal{x}\right) .f\text{'}\left(\mathcal{x}\right) is the \\ derivative of \mathcal{y}= f \left(\mathcal{x}\right) . \end{gathered}$$

$$\begin{gathered} \left(i\right) Find \int \frac{\mathcal{x}}{\sqrt{9-16}{\mathcal{x}}^2}d\mathcal{x} \\ \left(ii\right)Find \int \frac{{\mathcal{x}}^2}{\mathcal{x}+1}d\mathcal{x} \\ \left(iii\right) Evaluate {\int }_0^{\ln 3}\mathcal{x}{e}^{\mathcal{x}}d\mathcal{x} \\ \left(i\right) Find real numbers A , B and C such that \frac{2}{\left(t+1\right){\left(t+1\right)}^2}=\frac{A}{t+1}+\frac{Bt+C}{t^2+1} \\ \left(ii\right)Hence,find{\int }_0^1\frac{2}{\left(t+1\right){\left(t+1\right)}^2}dt \\ \left(iii\right) By u\sin g the substitution t =\tan \left(\frac{\mathcal{x}}{2}\right) evaluate{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{\sin \mathcal{x}}{1+\sin \mathcal{x}-\cos \mathcal{x}}d\mathcal{x} \end{gathered}$$