$$\begin{gathered} Given the quadratic equation {\mathcal{x}}^2-\mathcal{x}-3=0 with roots {\alpha }_1 , {\alpha }_2 \\ \left(i\right) Show that {\mathcal{x}}^4=7\mathcal{x}+12. \\ \left(ii\right) Hence or otherwise find a quadratic equation with roots {\alpha }_1^4 {\alpha }_2^4. \end{gathered}$$

$$\begin{gathered} The polynomial equation {\mathcal{x}}^3-3{\mathcal{x}}^2+\mathcal{x}-5=0 has roots \alpha , \beta and \gamma . Find the value of {\alpha }^{-1}+{\beta }^{-1}+{\gamma }^{-1}. \\ Find \int \mathcal{x}{\tan }^{-1}\mathcal{x}d\mathcal{x} \end{gathered}$$

$If \alpha , \beta and \gamma are roots of {\mathcal{x}}^3-4\mathcal{x}-1=0 find the polynomial equation with roots 2\alpha , 2\beta and 2\gamma (answer is expanded form)$

$$\begin{gathered} Given that (\mathcal{x}+\mathcal{i}) is a factor of P\left(\mathcal{x}\right)= {\mathcal{x}}^4+3{\mathcal{x}}^3+ {\mathcal{x}}^2 +3\mathcal{x}+5 \\ factorise P\left(\mathcal{x}\right) over the complex field. \\ Given that the equation {\mathcal{x}}^4- 5{\mathcal{x}}^3- 9{\mathcal{x}}^2+81\mathcal{x}-108 =0 \\ has a root of multiplicity 3, find all the roots of this equation. \\ If \alpha ,\beta , \gamma find the equation whose roots are {\mathcal{x}}^3- 3{\mathcal{x}}^2 + 2\mathcal{x}-1=0 \\ find the equation whose roots are \\ \left(i\right) \frac{1}{\alpha },\frac{1}{\beta },\frac{1}{\gamma } \end{gathered}$$

$In the Argand diagram below,P represents the complex number \mathcal{z}.$

$Which of the following Argand diagram shows the point Q representing \mathcal{z}+\overline{\mathcal{z}}?$

$$\begin{gathered} In the figure above the length of AC is twice the length of AB. \\ Explain why \overrightarrow{AB} represents the complex number -4+6\mathcal{i}. \\ Explain why \overrightarrow{AC} represents the complex number -10\sqrt{2}+2\sqrt{2}\mathcal{i}. \\ Find the complex number C represents. \end{gathered}$$

$$\begin{gathered} A polynomial p\left(\mathcal{x}\right)={\mathcal{x}}^n+a{\mathcal{x}}^2-2 has a factor of ( \mathcal{x}-1) and leaves a \\ remainder of -6 on division by (\mathcal{x}+2). \\ Find: \\ the value of a \\ the value of n \\ the zeros of p\left(\mathcal{x}\right). \\ Find the values of a and b so that \\ p\left(\mathcal{x}\right)=2{\mathcal{x}}^3-(2a+1){\mathcal{x}}^2+(2+b)\mathcal{x}-1 has a double root at \mathcal{x}=1. \\ If l,m,n are the roots of the equation {\mathcal{x}}^3-2\mathcal{x}+5=0, \\ find the cubic equation whose roots are 2l, 2m, 2n. \\ find the value of l^3+m^3+n^3 \\ Find all the values of k for which the polynomial equation 3{\mathcal{x}}^4-4{\mathcal{x}}^3+k=0 has no real roots. \end{gathered}$$

$$\begin{gathered} Solve , graphically or otherwise ,\frac{1}{\mathcal{x}-1}<|3\mathcal{x}-5| \\ The area enclosed by the curve \mathcal{Y}=\frac{1}{{\mathcal{x}}^2},\mathcal{Y}=\frac{1}{\mathcal{x}} and \mathcal{x}=\frac{1}{10} is rotated about the \mathcal{Y} axis. \\ use the method of cylinders shells to find the volume of the solid formed. \\ \left(i\right) Use the formula A=\frac{1}{2}ab\sin C to show that the area of a regular pentagon of side D units \\ is given by \\ A=\frac{5}{2}{D}^2\frac{{\sin }^2{54}^{°}}{\sin {72}^{°}} \\ \left(ii\right) The area enclosed by \mathcal{Y}={\mathcal{x}}^2 and \mathcal{Y}=3 is the base of a certain solid. Cross sections of the \\ solid, parallel to the \mathcal{x}-axis are regular pentagons with one side on the base. Find the volume of solid. \end{gathered}$$

$$\begin{gathered} In this question, neglect air resis\tan ce and assume that acceleration due to gravity g=10m{s}^{-2}. \\ A ball is thrown with initial speed 8m{s}^{-1} at an angle of {60}^{°} to the horizontal. The ball lands on the \\ top of brick wall adn it becomes off again. \\ \left(i\right) Derive all the equations of motion of the ball. \\ \left(ii\right) If the ball hits the wall after \frac{\sqrt{27}}{3} seconds, show that the wall is 1.8 meters high and 12\frac{\sqrt{3}}{5} \\ meters horizontally from the origin. \\ \left(iii\right) Also show that the ball strieks the top of the wall at an angle of \alpha , where \tan \alpha =\frac{\sqrt{3}}{2} \\ Now assume that the ball bounces forward off the wall with the same angle \alpha and assume that the \\ speed of projection off the wall is equal to the speed of impact onto the wall. \\ \left(iv\right) Show that the speed of projection off the wall is 2\sqrt{7} m/s. \\ \left(v\right) Write down a new set of equation of motion of the ball from the moment it bounces off the \\ wall. \\ \left(vi\right) Calculate where the ball first reaches the ground measured from the original poiny of projection. \end{gathered}$$

$$\begin{gathered} Ice cream begins to melt at 0^{°}C. When a frozen ice cream thaws, its temperature slowly rises until it increases \\ {0}^{°}C. An ice cream manufacturer has developed a new wrapper that slows down the rate of thawing after an \\ ice cream is taken from the shop freezer. Ice creams are stored at -2^{°}C in the shop freezer. Temperature measurements \\ recorded show that it takes 400 seconds for the temperature of an ice cream to rise from -{20}^{°}C to -4^{°}C \\ \left(i\right) Assuming a cons\tan t rate of thawing, estimate when the temperature of the ice cream will reach 0^{°}C. \\ In fact, the rate of thawing of the ice cream is not cons\tan t and the temperature, {\theta }^{°} C of the ice creaam, t seconds \\ after being taken out of the shop freezer is given by \ln (60-\theta )=c-kt, wher k and c are cons\tan ts. \\ \left(ii\right) Find \frac{d\mathcal{Y}}{d\mathcal{x}}and hence show that the rate of thawing is proportional to the amount by which the temperature i \\ below {60}^{°}C. \\ \left(iii\right) Find the values of c and k and hence ind an improved estimate of the time before the ice crea, begins to melt. \end{gathered}$$

$$\begin{gathered} The point P({\mathcal{x}}_1,{\mathcal{y}}_1) is one point of intersection of the rec\tan gular hyperbolas \\ {\mathcal{x}}^2+{\mathcal{y}}^2=2 and \mathcal{xy}=1. \\ The \tan gent at P to the first hyperbola cuts its asymptotes at A and C, \\ and the \tan gent at P to the second hyperbola cuts its asymptotes at B \\ and D. \\ Draw a sketch illustrating the given information and show that the equation \\ of AC is {\mathcal{xx}}_1-{\mathcal{yy}}_1=2 \\ Show that the coordinates of B and D are (0,2{\mathcal{y}}_1) and (2{\mathcal{x}}_1,0) \\ Prove that A,B,C,D are the vertices of a square. \end{gathered}$$

$$\begin{gathered} Two circle intersect at A and B as shown, such that the smaller circle passes through \\ the center O of the larger circle. \\ The \tan gent XA to the smaller circle at A cuts the larger circle at T. \\ Show that △ABT is isosceles. \end{gathered}$$

$$\begin{gathered} The circle {\mathcal{x}}^2+{\mathcal{y}}^2+2g\mathcal{x}+2f\mathcal{y}+h=0 cuts the rec\tan gular by hyperbola \mathcal{x}=ct, \\ \mathcal{y}=\frac{c}{t} in four distinct points P,Q,R and S. \\ Show that the parameters at these four points are the roots of the equation \\ {c}^2{t}^4+3gc{t}^3+h{t}^2+2fct+c^2=0 \\ If the midpoint of the chord joining two of these points is the centre of the circle, \\ show that the midpoint of the chord joining the other two points is the origin. \end{gathered}$$

The point $P\left(asec\theta ,b\tan \theta \right)$ lies on the hyperbola $\frac{{\mathcal{x}}^2}{a^2}-\frac{{\mathcal{y}}^2}{b^2}=1$ whose center is O and focii S' and S.

i) Show that $SP=a\left(esec\theta -1\right)$. you may assume $S\text{'}P=a\left(esec\theta +1\right)$

ii) Perpendicular are drawn from S' and S to meet the tangent at P at M and N respectively.

Prove that $\sin \angle \sin S\text{'}PM$ and deduce that the tangent at P bisects the angle S'PS.

Note: You may assume the equation of the tangent at P is $\frac{\left(sec\theta \right)\mathcal{x}}{a}-\frac{\left(\tan \theta \right)\mathcal{y}}{b}=1$

$$\begin{gathered} A particle of unit mass moves in a straight line so that at time t its displacement from a \\ fixed origin is \mathcal{x} m and its speed is v m/sec. \\ The acceleration of the particle has magnitude 4-v^2 m/se{c}^2 \\ \left(i\right) Given that initially v=0 and \mathcal{x}=0 find and expression for v in terms of t. \\ \left(ii\right) Find also an expression for \mathcal{x} in terms of v and discuss whether the \\ particle again comes to rest, if so where, and if not describe what does happen. \end{gathered}$$

$the diagram show the graph of \mathcal{y}=f\left(\mathcal{x}\right).$

.

$$\begin{gathered} Draw separate one-third page sketches of the graphs of the following: \\ \mathcal{y}=\frac{1}{f\left(\mathcal{x}\right)} \\ \left|\mathcal{y}\right|=f\left(\mathcal{x}\right) \\ \mathcal{y}={\left[f\left(\mathcal{x}\right)\right]}^2 \\ \mathcal{y}=\sqrt{f\left(\mathcal{x}\right)} \\ \mathcal{y}=\mathcal{x}\left(f\left(\mathcal{x}\right)\right) \end{gathered}$$

$$\begin{gathered} In the diagram above, AB is a fixed chord of a circle and C is a variable point on the major arc AB. \\ The angle bisector of <CAB and <ABC meet the circle again at P and Q respectively. \\ Let <CAB=2\alpha , <ABC=2\beta , <BCA=2\gamma . \\ i) Show that <PCQ=\alpha +\beta +2\gamma . \\ ii) Hence explain why the dis\tan ce PQ is cons\tan t. \\ iii) Use the \sin e rule to show that \frac{AB}{PQ}=2\sin \gamma . \\ i)Use DeMoivre\text{'}s Theorem to show that when n is a positive integer, \\ {(1+i\tan \theta )}^n+{(1-i \tan \theta )}^n=\frac{2 \cos n\theta }{\cos } \end{gathered}$$

$Find in modulus-argument form all complex numbers \mathcal{z} such that {\mathcal{z}}^3+1= 0 , and plot them on the Argand diagram.$

TP is a tangent to the circle, centre O, and TQ bisects ∠OTP

Suppose that ∠ = QTP𝑥 . Give reasons why:

$$\begin{gathered} \angle TOP=\frac{\mathrm{\pi }}{2}- 2x \\ \angle TBP=\frac{\mathrm{\pi }}{4}- x \\ Find the size of \angle TQP \end{gathered}$$

$$\begin{gathered} Sketch (on separate diagrams) the region in the Argand diagram containing the points \mathcal{z} for which: \\ 0\le arg\left(\mathcal{z}-\mathcal{i}\right)\le \frac{\mathrm{\pi }}{6} \\ \left|\mathcal{z}-2\mathcal{i}\right|<2 \end{gathered}$$

$$\begin{gathered} i)Prove that \frac{\cos \theta -\cos (\theta +2\alpha )}{2\sin \alpha }=\sin (\theta +\alpha \left) \\ ii\right)Hence use mathematical induction to prove that \\ \sin \theta +\sin 3\theta +\sin 5\theta +.......\sin (2n-1)\theta =\frac{1-\cos 2n\theta }{2\sin \theta } \end{gathered}$$