$$\begin{gathered} Let I_n={\int }_0^1{\mathcal{x}}^n{e}^{\mathcal{x}}d\mathcal{x} \\ i) Evaluate I_0 \\ ii) Show I_n=e-n{I}_{n-1} for n\ge 1 \\ iii) Hence evaluate I_3={\int }_0^1{\mathcal{x}}^n{e}^{\mathcal{x}}d\mathcal{x} \end{gathered}$$

$$\begin{gathered} Find \\ \int \frac{1}{\sqrt{{\mathcal{x}}^2+9}}d\mathcal{x} \\ \int {\cos }^3\mathcal{x}d\mathcal{x} \\ \int \frac{2\mathcal{x}+5}{{\mathcal{x}}^2+4\mathcal{x}+13}d\mathcal{x} \\ U\sin g the t method to evaluate {\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{4}{3+5\cos \mathcal{x}}d\mathcal{x} \\ Given that I_n={\int }_0^3\frac{{\mathcal{x}}^n}{\sqrt{4-\mathcal{x}}}d\mathcal{x} \\ show that I_n=\frac{2}{2n+1}(4n{I}_{n-1}-3^n). \\ Use your result to evaluate {\int }_0^3\frac{{\mathcal{x}}^2}{\sqrt{4-\mathcal{x}}}d\mathcal{x} \end{gathered}$$

$$\begin{gathered} Consider the quadratic equation {\mathcal{x}}^2-\mathcal{x}+k=0 where k is a real number. \\ The equation has 2 distinct positive roots \alpha and \beta . \\ \left(i\right) Show 0<k<\frac{1}{4} \\ \left(ii\right) Show that \frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}>8 \end{gathered}$$

$$\begin{gathered} Given I={\int }_{-1}^1\frac{{\mathcal{x}}^2{e}^{\mathcal{x}}}{e^{\mathcal{x}}+1}d\mathcal{x} and J={\int }_{-1}^1\frac{{\mathcal{x}}^2}{e^{\mathcal{x}}+1}d\mathcal{x} \\ \left(i\right) Use substitution u=-\mathcal{x} in I to show I=J. \\ \left(ii\right) Hence evaluate I and J. \end{gathered}$$

$$\begin{gathered} On a hyperbola the dis\tan ce between its vertices is 6 units and the dis\tan ce \\ between its focii is 10 units. \\ Find \\ the dis\tan ce between its directrices \\ the acute angle between its asymptotes. \\ Points P and Q are the end points of a focal chord of the ellipse \frac{{\mathcal{x}}^2}{a^2}+\frac{{\mathcal{y}}^2}{b^2}=1 \\ If the parametres at points P and Q are \theta and \phi , show that the eccentricity e \\ is given by \\ \frac{\sin (\theta -\phi )}{\sin \theta -\sin \phi } \end{gathered}$$

$$\begin{gathered} Find \\ \int e^{\mathcal{x} }\cos \mathcal{x} d\mathcal{x} \\ \int \frac{1}{{\mathcal{x}}^2\sqrt{{\mathcal{x}}^2+4}}d\mathcal{x} \\ \int \frac{\sqrt{1+\mathcal{x}}}{\sqrt{1=\mathcal{x}}}d\mathcal{x} \end{gathered}$$

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

$Find \int {\sin }^4\mathcal{x} d\mathcal{x}$

$Find \int \frac{e^{-2\mathcal{x}}}{e^{-\mathcal{x}}+1}d\mathcal{x}$

$$\begin{gathered} Given that {\int }_1^6\frac{1}{\sqrt{{\mathcal{x}}^2+2\mathcal{x}}}d\mathcal{x}=\ln (a+\sqrt{b}), find the values of a and b where \\ a and b are rational. \end{gathered}$$

$Evaluate {\int }_1^{\frac{\mathrm{\pi }}{4}}\cos \log \mathcal{x} d\mathcal{x}$

$$\begin{gathered} \left(i\right) Find the four solutions of {Ƶ}^4+1=0, writing them in the form \mathcal{x}+\mathcal{iy} \\ \left(ii\right) Hence, or otherwise, write {Ƶ}^4+1 as the product of two quadratic factors with real coefficients. \end{gathered}$$

$$\begin{gathered} which of the following graph is the locus of the point P representing the complex number \\ \mathcal{z} moving in an argand digram such that |\mathcal{z}-2i|=2+Im\mathcal{z}? \\ \left(A\right) a circle \\ \left(B\right)a parabola \\ \left(C\right)a hyperbola \\ \left(D\right)a straight line \end{gathered}$$

6. $What are the values of real numbers p and q such that 1+\mathcal{i}\sqrt{2} is a root of the equation{\mathcal{z}}^3+p\mathcal{z}+q=0?$

(a) $$\begin{gathered} Let \mathcal{z}=3-\mathcal{i} and w=2+\mathcal{i}. Express the following in the form \mathcal{x}+\mathcal{iy}, where \mathcal{x} and \mathcal{y} are real numbers: \\ \left(i\right) \frac{\mathcal{z}}{w} \\ \left(ii\right) \overline{-2\mathcal{iz}} \end{gathered}$$

(b) $$\begin{gathered} Let \mathcal{z}=\frac{1}{2}+\frac{\sqrt{3}}{2}\mathcal{i} \\ \left(i\right) Express \mathcal{z} in modulus-argument form. \\ \left(ii\right) Show that {\mathcal{z}}^6=1 \\ \left(iii\right) Hence, or otherwise, graph all roots of {\mathcal{z}}^6-1=0 on an Argand diagram \end{gathered}$$

(b) $$\begin{gathered} If w is a complex root of the equation {\mathcal{z}}^3=1 \\ \left(i\right) Show that 1+w+w^2=0 \\ \left(ii\right) Find the value of \left(1+2w+3w^2\right)\left(1+2w^2+3w\right) \end{gathered}$$

$By considering the binormal expansion of {(1+\mathcal{i})}^n Show that 1-\left(\begin{array}{c}n\\ 2\end{array}\right)+\left(\begin{array}{c}n\\ 4\end{array}\right)-\left(\begin{array}{c}n\\ 6\end{array}\right)+......=2^{\frac{n}{2}}\cos \frac{n\mathrm{\pi }}{4}$

$$\begin{gathered} In the diagram, MAN is the common \tan gent to two circles touching \\ internally at A. B and C are two points on the larger circle such that \\ BC is a \tan gent to the smaller circle with point of contact D. \\ AB and AC cut the smaller circle at E and F respectively \\ Copy the diagram. \\ Show that AD bisects \angle BAC. \end{gathered}$$

$$\begin{gathered} A particle of mass m kg is dropped from rest in a medium where there \\ the motion has magnitude \frac{1}{40}m{v}^2 when the speed of the particle is \\ vm{s}^{-1} after t seconds the particle has fallen x metres. \\ The acceleration due to gravity is 10m{s}^{-2}. \\ Explain why x=\frac{1}{40}(400-v^2) \\ Find an expression for t in terms of v by integration. \\ Show that v=20(1-\frac{2}{1+e\text{'}}) \\ Consider the polynomial P\left(\mathcal{x}\right)={\mathcal{x}}^4-2{\mathcal{x}}^3-{\mathcal{x}}^2+6\mathcal{x}-6 over the complex \\ field. \\ Given that P(l-i)=0,find all f our solutions to P\left(\mathcal{x}\right)=0 \\ In the diagram below, F is a focus of the hyperbola. \frac{{\mathcal{x}}^2}{a^2}-\frac{{\mathcal{Y}}^2}{b^2}=1 \\ with eccentricity e. \\ This branch of the hyperbola cuts the x axis at A where AF = h. P \\ is the point on the hyperbola vertically above F and the normal at P \\ cuts the x axis at B making an acute angle B with the X axis. \\ Show that \tan \theta =\frac{1}{e} \\ Show that PF=h(e+1) \\ A bowl is formed by rotating the hyperbola above through one revolution \\ about the x axis. The bowl is then placed on a horizontal table with point \\ A on the table. A particle P of mass m is set in motion around the inside \\ of the bowl, travelling with cons\tan t angular velocity OJ in a horizontal circle \\ with centre F. \\ Show that {\omega }^2=\frac{g}{he(e+1)} \\ N is the normal reaction force between the particle P and the bowl. Show that \\ if the hyperbola used to form the bowl is a rec\tan gular hyperbola, then N = mg\sqrt{\frac{3}{2}} \end{gathered}$$

$$\begin{gathered} A particle of mass m is thrown vertically upwards with initial velocity U in a medium with \\ resistive force R= mkv where v is the velocity of the particle at time t and k is a cons\tan t. \\ The equation of the motion of the particle is then \frac{dv}{dt}=-g-kv where g is the acceleration \\ due to gravity (Do not prove this). \\ Use \frac{dv}{dt}=v\frac{dv}{d\mathcal{x}} to show that the vertical displacement \mathcal{x} \\ from the point of projection of the particle is given by \\ \mathcal{x} =\frac{1}{k}(u-v)-\frac{g}{k^2}{\log }_e\left(\frac{g+KU}{g+KV}\right). \\ Hence find an expression for H the maximum height reached by the particle. \\ Find an expression for the time taken for the particle to reach its maximum height. \\ The hyperbola H has equation \mathcal{xy} =16 . The points p\left(4p\left(\frac{4}{p}\right)\right) for p > 0 and \\ Q\left(4q,\frac{4}{q}\right) for q>0 are two distinct arbitrary points on H. \\ Show that the equation of the \tan gent at P is \mathcal{x}+p^2\mathcal{y}=8p \\ Find the coordinates of T, the point of intersection of \\ the \tan gents at P and Q. \\ The equation of the chord pas\sin g through PQ \\ is given by pq\mathcal{x}+\mathcal{y}=4(p+q)(Do not prove this). \\ If chord PQ passes through the point N (0,8) find the Cartesian \\ equation of the locus of T. \end{gathered}$$

$\mathrm{Find} \mathrm{all} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} 18{\mathrm{x}}^3 + 3{\mathrm{x}}^2 - 28𝑥 + 12 = 0, \mathrm{given} \mathrm{that} \mathrm{two} \mathrm{of} \mathrm{the} \mathrm{roots} \mathrm{are} \mathrm{equal}.$

$$\begin{gathered} If a > 0 and b > 0, prove that: \\ \frac{1}{a}+\frac{1}{b}\ge \frac{4}{a+b} \\ \frac{1}{a^2}+\frac{1}{b^2}\ge \frac{8}{{(a+b)}^2} \end{gathered}$$

$$\begin{gathered} Draw a one third page sketch the graph of \mathcal{y}=\frac{{\mathcal{x}}^3}{{\mathcal{x}}^2-4}, \\ indicating the coordinates of all stationary points and all asymptotes. \\ For what values of k will {\mathcal{x}}^3-k{\mathcal{x}}^2+4k=0 have exactly one real root. \end{gathered}$$

$$\begin{gathered} The diagram above shows the horizontal square base of a solid. \\ Vertical cross-sections of the solid perpendicular to the \mathcal{x}-axis \\ are right-angled isosceles triangles with hypotenuse in the base. \\ Find, as a function of \mathcal{x}, the area of a typical cross-section s\tan ding \\ on the interval PQ. \\ Find the volume of the solid \\ If U_1 = 8 and U_2 = 20 and U_n =4U_{n-2} for n \ge 3 , \\ prove by mathematical induction that U_n=(n+3)2^n for n\ge 1 \end{gathered}$$