$Sketch the graph of the function \mathcal{y}=\frac{{\mathcal{x}}^2-\mathcal{x}+1}{{(\mathcal{x}-1)}^2}, clearly showing the coordinates of any points of intersection with the \mathcal{x} \mathcal{y} and axes, the coordinates of any turning points and the equations of any asymptotes. There is no need to investigate points of inflexion.$

$$\begin{gathered} If \alpha , \beta and \gamma are the roots of {\mathcal{x}}^3+2{\mathcal{x}}^2-3\mathcal{x}-4=0 \\ \left(i\right) Evaluate {\alpha }^2+{\beta }^2+{\gamma }^2 \\ \left(ii\right) From the equation whose roots are \beta \gamma , \alpha \gamma , \alpha \beta . \end{gathered}$$

$Find \int \frac{1+\mathcal{x}}{4+{\mathcal{x}}^2}dx$

$By completing the square find \int \frac{1}{\sqrt{6-{\mathcal{x}}^2-\mathcal{x}}}d\mathcal{x}$

$Which of the following is the equation of the circle below?$

$$\begin{gathered} \left(A\right) (\mathcal{z}+2)(\overline{\mathcal{z}}+2)=4 \\ \left(B\right) (\mathcal{z}-2)(\overline{\mathcal{z}}-2)=4 \\ \left(C\right) (\mathcal{z}+2\mathcal{i})(\overline{\mathcal{z}}-2\mathcal{i})=4 \\ \left(D\right) (\mathcal{z}+2)(\overline{\mathcal{z}}-2)=4 \end{gathered}$$

$$\begin{gathered} \left(i\right) Factorise the polynomial {\mathcal{z}}^3 -1 over the rational field. \\ \left(ii\right) If w is a complex root of 1, show that 1+\mathcal{w}+{\mathcal{w}}^2=0. \\ \left(iii\right) Hence, or otherwise, simplify \left(1+{\mathcal{w}}^2\right)\left(1+{\mathcal{w}}^4\right)\left(1+{\mathcal{w}}^8\right)\left(1+{\mathcal{w}}^{10}\right). \\ Prove that if a\ne c there are always two real values of k which \\ will makea{\mathcal{x}}^2+2b\mathcal{x}+c+k\left({\mathcal{x}}^2+1\right) a perfect square. \\ The points , p\left(cp,\frac{c}{p}\right) and , Q\left(cp,\frac{c}{p}\right) are two variable points on the \\ hyperbola \mathcal{xy}=c^2 which move so that the points P , Q and S\left(c\sqrt{2},c\sqrt{2}\right) \\ are always collinear. The \tan gents to the hyperbola at \\ P and Q meet at the point R \\ \left(i\right)Show that the equation of the chord PQ is \mathcal{x}+ pq\mathcal{y} =c\left(p+q\right) \\ \left(ii\right) Hence show that p+q =\sqrt{2}\left(1+pq\right). \\ Show that R is the point\left(\frac{2cpq}{p+q},\frac{2c}{p+q}\right) . You may assume that the \\ \tan gent at any point , T \left(ct,\frac{c}{t}\right) has equation {\mathcal{x}}^2+t^2\mathcal{y}=2ct \\ (Do NOT prove this) \\ \left(iv\right) Hence find the equation of the locus of R . \end{gathered}$$

$Given that {\mathcal{z}}_1=5+2\mathcal{i} and {\mathcal{z}}_2=3-4\mathcal{i}, find the value of Re\left(\frac{{\mathcal{z}}_1}{{\mathcal{z}}_2}\right) in \mathcal{x}+\mathcal{iy} form.$

$$\begin{gathered} \left(i\right) Show that the square roots of -35+12\mathcal{i} are \pm (1+6\mathcal{i}). \\ \left(ii\right) Hence solve {\mathcal{z}}^2-(5+4\mathcal{i})\mathcal{z}+11+7\mathcal{i}=0 \end{gathered}$$

$$\begin{gathered} Prove that if x and y are positive numbers then \\ {\left(\mathcal{x}+\mathcal{y}\right)}^2\ge 4\mathcal{xy} \\ \left(ii\right) Deduce that if a b c , , and d are positive numbers then \\ \frac{1}{4}{\left(a+b+c+b\right)}^2\ge ac+ad+bc+bd \\ Scientists use a pressure gauge which measures depth as it \sin ks \\ towards the ocean floor. The gauge of mass 2 kg is released from \\ rest at the ocean’s surface. As it \sin ks in a vertical line, the water \\ exerts a resis\tan ce to its motion of 4v Newtons, where v m{s}^{-1} is \\ the velocity of the gauge. \\ Let x be the displacement of the ball measured vertically downwards \\ from the ocean’s surface, t be the time in seconds elapsed after \\ the gauge is released, and g be the cons\tan t acceleration due to gravity. \\ \left(i\right) Show that \frac{d^2\mathcal{x}}{d{\mathcal{x}}^2}=g-2v \\ \left(ii\right) Hence show that t=\frac{1}{2}{\log }_e\left(\frac{g}{g-2v}\right) \\ \left(iii\right)Show that v=\frac{g}{2}\left(1-e^{-2t}\right) \\ \left(iv\right) Write down the limiting \left(terminal\right) velocity of the gauge. \\ \left(v\right) At a particular location, the gauge takes 180 seconds \\ to hit the ocean floor. U\sin g 10m{s}^{-1} , calculate the depth \\ of the ocean at that location, giving your answer correct \\ to the nearest metre. \end{gathered}$$

$Show that if the polynomial f\left(\mathcal{x}\right)={\mathcal{x}}^3+p\mathcal{x}+q has a multiple root, then 4p^3+27q^2=0$

$$\begin{gathered} Find the five roots of the equation {\mathcal{z}}^5=1 . Give the roots in modulus-argument form. \\ Show that z^5-1 can be factorised in the form : \\ {\mathcal{z}}^5-1=({\mathcal{z}}^2-2\mathcal{z}\cos \frac{2\pi }{5}+1)({\mathcal{z}}^2-2\mathcal{z}\cos \frac{4\pi }{5}+1) \\ Hence show that \cos \frac{2\pi }{5}+\cos \frac{4\pi }{5}=-\frac{1}{2} \end{gathered}$$

$$\begin{gathered} In the triangle ABC , AD is the perpendicular from A to BC . E is any point on AD and the circle drawn \\ with AE as diameter cuts AC at F and AB at G \end{gathered}$$

$Prove B G F C , , and are concyclic$

$$\begin{gathered} If a + b + c = 1, \\ Prove a^2 +b^2 \ge 2ab \\ Prove \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \ge 9 . \\ Prove (1- a)(1 - b)(1- c) \ge 8abc . \end{gathered}$$

$$\begin{gathered} Explain why the domain of the function, f\left(\mathcal{x}\right)=\sqrt{2-\sqrt{\mathcal{x}}} is 0 \le \mathcal{x}\le 4 \\ Show that f\left(\mathcal{x}\right) is a decrea\sin g function and hence find its range. \\ U\sin g the substitution, u=2-\sqrt{\mathcal{x}} or otherwise, \\ find the area bounded by the curve and the x and y axes. \end{gathered}$$

$$\begin{gathered} Let I_n={\int }_0^{\frac{\mathrm{\pi }}{4}}{\tan }^n\mathcal{x} d\mathcal{x} where n is an integer and n\ge 3. \\ Show that I_n+I_{n-2}=\frac{1}{n-1} \end{gathered}$$

$$\begin{gathered} A body mass of 1 kg falls vertically downwards, from rest, \\ in a medium which exerts a resis\tan ce to its motion of \frac{1}{100} v^2 Newtons \\ (where v metres per second is the speed of the body when it has fallen a dis\tan ce of x metres). \\ Show (on a diagram) that the equation of motion of the body is \stackrel{..}{\mathcal{x}}=g-\frac{1}{100} v^2 \\ where g is the acceleration due to gravity. \\ Show that the terminal speed V_T is given by V_T=10\sqrt{g} \\ Prove that v^2={\left(V_T\right)}^2(1-e^{-\frac{\mathcal{x}}{50}}) \end{gathered}$$

$The area enclosed by the curve \mathcal{y}={(\mathcal{x}-2)}^2 and the line \mathcal{y}=4 is rotated around the \mathcal{y}-axis. Use the method of cylindrical shells to find the volume formed.$

$$\begin{gathered} On the same number plane diagram sketch the curves. \\ \mathcal{y}=\left|\mathcal{x}\right|-2 and \mathcal{y}=4+3\mathcal{x}-{\mathcal{x}}^2 \\ Hence or otherwise solve the inequality \frac{\left|\mathcal{x}\right|-2}{4+3\mathcal{x}-{\mathcal{x}}^2}>0. \end{gathered}$$

$$\begin{gathered} A man ascending in a hot air balloon throws a set of car keys to his wife who is on the ground. The keys are projected at a cons\tan t velocity of V m{s}^{-1 }at an \\ angle of \theta to the horizontal, 0°<\theta <90° , and from a point \frac{V^2{\sin }^2\theta }{g} m vertically above the ground. \\ The edge of a dam closest to where the balloon took off, lies \frac{V^2(1+\sqrt{3})}{4g} m horizontally from the point of projection. The dam is \frac{V^2}{2g}m wide. \\ The position of the keys at time t seconds after they are projected is given by: \end{gathered}$$

$\mathcal{x}=Vt\cos \theta , \mathcal{y}=\frac{-g{t}^2}{2}+Vt\sin \theta +\frac{V^2{\sin }^2\theta }{g}$

$$\begin{gathered} \left(i\right) Show that the Cartesian equation of the path of the keys is given by : \\ \left(ii\right) Show that the horizontal range of the keys on the ground is given by: \\ \mathcal{y}=\frac{-g{\mathcal{x}}^{2}se{c}^2\theta }{2V^2}+\mathcal{x}\tan \theta +\frac{V^2{\sin }^2\theta }{g} \\ \mathcal{x}=\frac{V^2(1+\sqrt{3})\sin 2\theta }{2g} \\ \left(iii\right) Find the values of \theta for which the keys will NOT land in the dam. \end{gathered}$$

$$\begin{gathered} The directrices of the hyperbola \frac{{\mathcal{y}}^2}{9}-\frac{{\mathcal{x}}^2}{16} = 1 are \\ A) \mathcal{x}=\pm \frac{9}{5} \\ B) \mathcal{y}=\pm \frac{9}{5} \\ C) \mathcal{y}=\pm 5 \\ D) \mathcal{x}=\pm 5 \end{gathered}$$

$$\begin{gathered} What are the solutions to the quadratic equation (\mathcal{x}- 2 - i)(\mathcal{x} + 3 + 2i) = 0? \\ \left(A\right) \mathcal{x}= 2 - i or -3 + 2i \\ \left(B\right) \mathcal{x}= -2 - i or 3 + 2i \\ \left(C\right) \mathcal{x}= 2 + i or -3 - 2i \\ \left(D\right) \mathcal{x}= -2 + i or 3 - 2i \end{gathered}$$

$$\begin{gathered} If \alpha ,\beta and \gamma are the roots of 4{\mathcal{x}}^3-6{\mathcal{x}}^2+11\mathcal{x}-5=0 then the polynomial equation with roots and \frac{1}{\alpha },\frac{1}{\beta }and \frac{1}{\gamma } is \\ A) 12{\mathcal{x}}^3+9{\mathcal{x}}^2+16\mathcal{x}+2=0 \\ B) 3{\mathcal{x}}^3-7{\mathcal{x}}^2+18\mathcal{x}+11=0 \\ C) 5{\mathcal{x}}^3-11{\mathcal{x}}^2+6\mathcal{x}-4=0 \\ D) 2{\mathcal{x}}^3-3{\mathcal{x}}^2+22\mathcal{x}+10=0 \end{gathered}$$

$$\begin{gathered} Which of the following is a primitive of \frac{e^{\sqrt{\mathcal{x}}}}{\sqrt{\mathcal{x}}}? \\ \left(A\right) 2e^{\sqrt{\mathcal{x}}} \\ \left(B\right) {\left(e^{\sqrt{\mathcal{x}}}\right)}^2 \\ \left(C\right) \ln \left(e^{-\sqrt{\mathcal{x}}}\right) \\ \left(D\right) \sqrt{\mathcal{x}}{e}^{\sqrt{\mathcal{x}}} \end{gathered}$$