$Sketch the following on separate Argand Diagrams.$

$\left(i\right)\left|\mathcal{z}+1\right|=3$

 $\left(ii\right) arg\left(\mathcal{z}-1\right)=\frac{\mathrm{\pi }}{4}$

 $\left(iii\right) \left|\mathcal{z}-1\right|=Re \mathcal{z}$

 $\left(iv\right) Re \left(\mathcal{z}\left(\overline{\mathcal{z}}+2\right)\right)=3$

 $Find the greatest and least values of arg \mathcal{z} for which|\mathcal{z}+4\mathcal{i}|=2$

$Find \int \frac{4{\mathcal{x}}^3-2{\mathcal{x}}^2+1}{2\mathcal{x}-1}d\mathcal{x}$

 $\left(i\right) Express 1-\sqrt{3}\mathcal{i} in mod-arg form.$

$\left(ii\right) Hence find {\left(1-\sqrt{3}\mathcal{i}\right)}^6$ $in the form a+bi$.

$$\begin{gathered} PT is a common \tan gent to the circles which touch at T. PA is a \tan gent to the the smaller circle at Q. \\ \left(i\right) Prove that ∆BTP is similar to ∆TAP. \end{gathered}$$

$\left(ii\right) Hence show that T{P}^2=PA.PB$

$\left(iii\right) If PT = t, QA = a and QB = b prove that t=\frac{ab}{a-b}$.

Let $\alpha ,\beta ,\gamma$ be the roots of equation ${\mathcal{x}}^3+4{\mathcal{x}}^2-3\mathcal{x}+1=0$. The equation with roots ${\alpha }^{-1},{\beta }^{-1},{\gamma }^{-1}$ is:

A. ${\mathcal{x}}^3+4{\mathcal{x}}^2-3\mathcal{x}+1=0$

B. ${\mathcal{x}}^3-3{\mathcal{x}}^2+4\mathcal{x}+1=0$

C. ${\mathcal{x}}^3-8{\mathcal{x}}^2+12\mathcal{x}+3=0$

D. ${\mathcal{x}}^3-4{\mathcal{x}}^2+8\mathcal{x}-3=0$

Given the complex numbers ${\mathcal{z}}_1=\frac{p}{1+2i}$ and ${\mathcal{z}}_2=\frac{q}{1+2i}$ where p and q are real, find p and q if ${\mathcal{z}}_1-{\mathcal{z}}_2=4i$.

$$\begin{gathered} Consider the complex number z which satisfies \left|z\right|=1. \\ \left(i\right) U\sin g double angle trigonometric identities show that: \\ 1+\cos \alpha +\iota \sin \alpha \equiv 2 \cos \frac{1}{2}\alpha \left(\cos \frac{1}{2}\alpha +\iota \sin \frac{1}{2}\alpha \right). \\ \left(ii\right) If z=\cos \theta +i \sin \theta , -\pi <\theta \le \pi , write 1+z^2 in terms \\ of \cos \theta and \sin \theta . Hence deduce that if in an Argand diagram, \\ points A and B represent z and 1+ z^2 respectively, then A, B and O are collinear, \\ where O is the origin. State the values of \theta such that B lies on the interval OA. \end{gathered}$$

$Express 1+\mathcal{x}+{\mathcal{x}}^2+{\mathcal{x}}^3+{\mathcal{x}}^4+{\mathcal{x}}^5 as a product of real factors$

$$\begin{gathered} The area enclosed by the curve \mathcal{y} ={(\mathcal{x} -3)}^2 and the line \mathcal{y} =9 is \\ rotated about the \mathcal{y}-axis. Use the method of cylindrical shells to \\ find the exact volume of the solid formed. \end{gathered}$$

$$\begin{gathered} The base of a solid is a right angled triangle on the horizonal \mathcal{x}-\mathcal{Y} plane, bounded by the lines \mathcal{Y}=0, \mathcal{x}=4 and \mathcal{Y}=\mathcal{x} \\ Vertical cross-sections of the solid, parallel to the \mathcal{Y}-axis are semicircles with their diameters on the base o solid as shown \\ in diagram below. Find the volume of solid \end{gathered}$$

$$\begin{gathered} The Argand diagram below shows the complex number \mathcal{z} \\ Which diagram best represents the locus of P such that P=\left|\mathcal{z}\right| ? \end{gathered}$$

$Which diagram best represents \mathcal{z}={\overline{\mathcal{z}}}^2=16.$

$$\begin{gathered} The diagram shows the graph of the function \mathcal{y}=f\left(\mathcal{x}\right)? \\ Which of the following is the graph of \mathcal{y}=\left|f\left(\mathcal{x}\right)\right|? \end{gathered}$$

$$\begin{gathered} Sketch separately the following loci in an Argand plane \\ 2\left|\mathcal{z}-(1+\mathcal{i})\right|=\left|\mathcal{z}-(4+\mathcal{i}\right| \\ \left\{\mathcal{z}:0\le arg(\mathcal{z}+4+\mathcal{i})\le \frac{2\mathrm{\pi }}{3} and \left|\mathcal{z}+4+\mathcal{i}\right|\le 4\right\} \end{gathered}$$

$$\begin{gathered} If a,b,c are positive real numbers; \\ Show that a^2+b^2\ge 2ab \\ Hence prove (a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\ge 9 \end{gathered}$$

$$\begin{gathered} The points A, B, C and D lie on the circle C. From the exterior point T, a \tan gent is drawn to \\ point A on C. The line CT passes through D and TC is parallel to AB. \\ Copy or trace the diagram onto your page. \\ Prove that △ADT is similar to △ABC. \\ The line BA is produced through A to point M, which lies on a second circle. \\ The points A, D, T also lie on this second circle and the line DM crosses \\ AT at O. \\ Show that △OMA is isosceles. \\ Show that TM = BC . \end{gathered}$$

$Consider the Argand diagram below$

$The inequality that represents the shaded area is:$

$\mathrm{ABC} \mathrm{is} \mathrm{an} \mathrm{isosceles} \mathrm{triangle} \mathrm{with} \mathrm{AC}=\mathrm{BC} \mathrm{and} \mathrm{AB}=\mathrm{b}. \mathrm{ABCDE} \mathrm{is} \mathrm{a} \mathrm{wedge} \mathrm{shape} \mathrm{with} \mathrm{height} \mathrm{DE}=\mathrm{h} \mathrm{and} \mathrm{length} \mathrm{CD}=\mathrm{l}. \mathrm{Triangle} \mathrm{ABC} \mathrm{and} \mathrm{line} \mathrm{DE} \mathrm{are} \mathrm{perpendicular} \mathrm{to} \mathrm{the} \mathrm{plane} \mathrm{of} \mathrm{ABE} \mathrm{as} \mathrm{shown} \mathrm{in} \mathrm{the} \mathrm{diagram}.$

$\mathrm{Consider} \mathrm{a} \mathrm{slice} \mathrm{of} \mathrm{the} \mathrm{wedge} \mathrm{height} \mathrm{h} \mathrm{and} \mathrm{depth}  \mathrm{y} \mathrm{as} \mathrm{in} \mathrm{the} \mathrm{diagram}. \mathrm{The} \mathrm{slice} \mathrm{is} \mathrm{parallel} \mathrm{to} \mathrm{the} \mathrm{plane} \mathrm{ABC} \mathrm{at} \mathrm{PQR}.$

$$\begin{gathered} \left(\mathrm{i}\right) \mathrm{Show} \mathrm{that} \mathrm{the} \mathrm{area} \mathrm{of} \mathrm{the} \mathrm{triangle} \mathrm{PQR} \mathrm{can} \mathrm{be} \mathrm{expressed} \mathrm{as} \frac{\mathrm{h}}{2}\left(\mathrm{b}-\frac{\mathrm{by}}{\mathrm{l}}\right). \\ \left(\mathrm{ii}\right) \mathrm{Hence} \mathrm{calculate} \mathrm{the} \mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{wedge} \end{gathered}$$

$$\begin{gathered} A car travels around a banked circular track of radius 90 metres at 54 km/h. \\ Draw a diagram showing all the forces acting on the car \\ Show that the car will have tendency to slip sideways if the angle at which \\ the banked track is banked is {\tan }^{-1}\left(\frac{1}{4}\right). \\ A second car of mass 1.2 tonnes travels around the same bend at 72 km/h. \\ Find the sideways frictional force exerted by the road on the wheels of the car in \\ Newtons. You may assume gravity = 10 m/s2. (Answer correct to 1decimal place) \end{gathered}$$

$$\begin{gathered} \mathrm{P}(\mathrm{acos\theta },\mathrm{bsin\theta }) \mathrm{is} \mathrm{a} \mathrm{point} \mathrm{on} \mathrm{the} \mathrm{ellipse} \frac{{\mathcal{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathcal{y}}^2}{{\mathrm{b}}^2}=1 \mathrm{where} \mathrm{a}>\mathrm{b}. \\ \mathrm{i})\mathrm{Show} \mathrm{that} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{normal} \mathrm{to} \mathrm{the} \mathrm{above} \mathrm{ellipse} \mathrm{at} \mathrm{the} \mathrm{point} \mathrm{P} \mathrm{is} \mathrm{given} \mathrm{by} \mathrm{the} \mathrm{equation} \\ \frac{\mathrm{a}\mathcal{x}}{\mathrm{cos\theta }}-\frac{\mathrm{b}\mathcal{y}}{\mathrm{sin\theta }}={\mathrm{a}}^2-{\mathrm{b}}^2 \\ \mathrm{ii}) \mathrm{The} \mathrm{normal} \mathrm{found} \mathrm{in} \mathrm{part} \mathrm{i}) \mathrm{meets} \mathrm{the} \mathrm{major} \mathrm{axis} \mathrm{of} \mathrm{the} \mathrm{ellipse} \mathrm{at} \mathrm{the} \mathrm{point} \mathrm{G}. \\ \mathrm{If} \mathrm{S} \mathrm{is} \mathrm{a} \mathrm{focus} \mathrm{of} \mathrm{the} \mathrm{ellipse} \mathrm{and} \mathrm{e} \mathrm{its} \mathrm{eccentricity}, \mathrm{show} \mathrm{that} \mathrm{SG}=\mathrm{eSP}. \end{gathered}$$

$$\begin{gathered} A hyperbola has equation 𝑥{ }^2 - 4𝑦{ }^2 = 4. \\ The dis\tan ce between its directrices is: \\ \left(A\right) \sqrt{5} \\ \left(B\right) \frac{4\sqrt{5}}{5} \\ \left(C\right) 2\sqrt{5} \\ \left(D\right) \frac{8\sqrt{5}}{5} \end{gathered}$$