$\text{ Solve the equation }{z}^3=1\text{, giving your answers in Cartesian form. }$

$\text{ On the Argand diagram below, the point }\mathrm{A}\text{ represents the complex number }4\mathrm{i}\text{ and the point }\mathrm{B}\text{ represents the complex number }-5+\mathrm{i}\text{. The shape }\mathrm{ABCD}\text{ is a square. }$



Determine the complex number represented by:
1. the point C ;
2. the point D .

  Consider the complex number emi8 r qgo3l4/wf vv:ayl4f56 $z=\frac{w_1}{w_2}$ where $w_1=\sqrt{2}+\sqrt{6}\mathrm{i}$ and $w_2=3+\sqrt{3}\mathrm{i}$ .
1. Express $w_1$ and $w_2$ in modulus-argument form and write down
1. the modulus of z ;$\frac{\sqrt{a}}{b}$ a =    b =   
2. the argument of z .$\frac{\pi }{a}$ a =   
2. Find the smallest positive integer value of n such that $z^n$ is a real number. n =   

  In this question give all angles v8ks-z xv)ru(; 9vqz fin radians.
Let $z=1+2\mathrm{i}$ and $w=4+\mathrm{i}$ .
1. Find z+w . a+bi a =    b =   
2. Find:
1. $|z+w|$ ;≈   
2. $\arg (z+w)$ .≈   
3. Find $\theta$, the angle shown on the diagram below.≈   

Let $z=2+\mathrm{i}$ and w=1-2 $\mathrm{i}$ .
1. Find z w .
2. Illustrate z, w and z w on the same Argand diagram.
3. Let $\theta$ be the angle between z w and w . Find $\theta$, giving your answer in radians.

A circle of radius 3 and centre (0,3) is drawn on an Argand diagram. xfoo47qx u hx4* 1iw;nThe tangent to the circle from the point B(0,9) muqoxw4 oxi1 *;7n4hfx eets the circle at the point A as shown. Let $w=\overrightarrow{\mathrm{OA}}$

The complex numbers w an ies7x38pkg* td z satisfy the equations

$\begin{array}{rl}\frac{z}{w}& =\mathrm{i}\\ w^{\ast }+2z& =4+5\mathrm{i}.\end{array}$

Find w and z in the form a+b i where a, b $\in \mathbb{Z}$ .

  Consider the equation df r.rw .*nlajm+1ur * $\frac{3z}{5-z^{\ast }}=\mathrm{i}$ , where $z=x+\mathrm{i}$ y and $x,y\in \mathbb{R}$ .
Find the value of x and the value of y . x =    y =   

  $\text{ Points }\mathrm{A}\text{ and }\mathrm{B}\text{ represent the complex numbers }{z}_1=\sqrt{3}-\mathrm{i}\text{ and }{z}_2=-3-3\mathrm{i}\text{ as shown on the Argand diagram below. }$



1. Find the angle A O B .$\frac{a\pi }{b}$ a =    b =   
2. Find the argument of $z_1{z}_2$ .$-\frac{a\pi }{b}$ a =    b =   
3. Given that the real powers of $p{z}_1{z}_2$ , for p>0 , all lie on a unit circle centred at the origin, find the exact value of p .$\frac{\sqrt{a}}{b}$ a =    b =   

The complex numbers z and w correspond to the points A and B as showbe(* 8 jo * sqikn 5t*g7/3rw.qrjnkgmn on the diagram bel ogqr.g*ki s5nqj wj*37b/ mkern *t(8ow.


1. Find the exact value of |z-w| .
2. 1. Find the exact perimeter of triangle A O B .
2. Find the exact area of triangle A O B .

Let $z=r{e}^{1\frac{4}{3}}wherer\in {\mathbb{R}}^{+}$ .
1. For $r=\sqrt{2}$ ,
1. express $z^2$ and $z^3$ in the form $+b\mathrm{i}$ where $a,b\in \mathbb{R}$ ;
2. draw $z^2$ and $z^3$ on the following Argand diagram.

$\text{ 2. Given that the integer powers of }w=(3-3\mathrm{i})z\text{ lie on a unit circle centred at the origin, find the value of }r\text{. }$

Let $z=r{e}^{\mathrm{i}\frac{\pi }{6}}$ where $r\in {\mathbb{R}}^{+}$ .
1. For $r=\sqrt{3}$ ,
1. express $z^2$ and $z^3$ in the form $a+b\mathrm{i}$ where $a,b\in \mathbb{R}$ ;
2. draw $z^2$ and $z^3$ on the following Argand diagram.

2. Given that the integer powers of $w=\frac{z}{6+2\mathrm{i}}$ lie on a unit circle centred at the origin, find the value of r .

  Consider the complex nu5 8u( ln4+uc ;:5qdrj,hdhxbil; isxbmbers $u=1+2\mathrm{i}andv=2+\mathrm{i}$ .
1. Given that $\frac{1}{u}+\frac{1}{v}=\frac{6\sqrt{2}}{w}$ , express w in the form $a+b\mathrm{i}$ where $a,b\in \mathbb{R}$ .$a\sqrt{b}+c\sqrt{d}\mathrm{i}$ a =    b =    c =    d =   
2. Find $w^{\ast }$ and express it in the form r $e^{\mathrm{i}\theta }$ .$a{e}^{-i\frac{\pi }{b}}$ a =    b =   

1. Find three distinctxmt2kl/. bl wc1g:v y kr 7+r,(vp6f1ic.grul roots of the equation $z^3+64=0$, z $\in \mathbb{C}$ , giving your answers in modulus-argument form.

The roots are represented by the vertices of a triangle in an Argand diagram.
2. Show that the area of the triangle is 12 $\sqrt{3}$ .

$\text{ Let }z=2\text{ cis }2\theta \text{ where }0<\theta <{45}^{\circ }\text{. Find the modulus and argument of }z+2\text{, expressing your answers in terms of }\theta \text{. }$

Let $w=2e^{\mathrm{i}\frac{2\pi }{3}}$ .
1. 1. Write w, $w^2$ and $w^3$ in the form $a+b\mathrm{i}$ where a, $b\in \mathbb{R}$ .
2. Draw w, $w^2$ and $w^3$ on an Argand diagram.
2. Find the smallest integer k>3 such that $w^k$ is a real number.

Consider the equation fpgrd/;-k o ;f2 $z^4+a{z}^3+b{z}^2+cz+d=0$ , where a, b, c, $d\in \mathbb{R}$ and $z\in \mathbb{C}$ . Two of the roots of the equation are ${\log }_2$ 10 and $\mathrm{i}\sqrt{5}$ and the sum of all the roots is $4+{\log }_{2}5$ .
Show that 15 a+d+90=0 .

Consider the complex numbepy y0e*6k3ukfj***:g )mzdgurok e5; vj fmt6rs $z_1=3\mathrm{cis}\left({120}^{\circ }\right)$ and $z_2=2+2\mathrm{i}$ .
1. Calculate $\frac{z_1}{z_2}$ giving your answer both in modulus-argument form and Cartesian form.
2. Use your results from part (a) to find the exact value of $\sin {15}^{\circ }\cdot \sin {45}^{\circ }\cdot \sin {75}^{\circ }$ , giving your answer in the form $\frac{\sqrt{a}}{b}wherea,b\in {\mathbb{Z}}^{+}$ .

1. Express $-4+4\sqrt{3}\mathrm{i}$ in the form r $e^{\mathrm{i}\theta }$ , where r>0 and $-\pi <\theta \le \pi$ .

Let the roots of the equation $z^3=-4+4\sqrt{3}\mathrm{i}$ be $z_1$, $z_2$ and $z_3$ .
2. Find $z_1$, $z_2$ and $z_3$ expressing your answers in the form $r{e}^{\mathrm{i}\theta }$ , where r>0 and $-\pi <\theta \le \pi$ .

On an Argand diagram, $z_1$, $z_2$ and $z_3$ are represented by the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ , respectively.
3. Find the area of the triangle $\mathrm{ABC}$ .
4. By considering the sum of the roots $z_1,z_2$ and $z_3$ , show that

$\cos \left(\frac{2\pi }{9}\right)+\cos \left(\frac{4\pi }{9}\right)+\cos \left(\frac{8\pi }{9}\right)=0$

Consider $w=\frac{z-1}{z+\mathrm{i}}$ where $z=x+\mathrm{i}$ y and $\mathrm{i}=\sqrt{-1}$
1. If $z=\mathrm{i}$ ,
1. write w in the form $r\mathrm{cis}\theta$ ;
2. find the value of $w^{14}$ .
2. Show that in general,
$w=\frac{(x^2-x+y^2+y)+\mathrm{i}(y-x+1)}{x^2+(y+1{)}^2}$

3. Find condition under which $\mathrm{Re}(w)=1$ .
4. State condition under which w is:
1. real;
2. purely imaginary.
5. Find the modulus of z given that $\arg w=\frac{\pi }{4}$ .

On an Argand diagram, the compw ;ns4k tu/ng2lex numbers $z_1=2+2\sqrt{3}\mathrm{i},z_2=1-\mathrm{i}$ and $z_3=z_1{z}_2$ are represented by the vertices of a triangle. The exact area of the triangle can be expressed in the form $p+\sqrt{q}$ . Find the value of p and of q .

1. 1. Expand $(\cos \theta +\mathrm{i}\sin \theta {)}^4$ by using the binomial theorem.
2. Hence use de Moivre's theorem to prove that

$\cos 4\theta ={\cos }^4\theta -6{\cos }^2\theta {\sin }^2\theta +{\sin }^4\theta$

3. State a similar expression for $\sin 4\theta$ in terms of $\cos \theta$and $\sin \theta$ .

Let $z=r(\cos \alpha +\mathrm{i}\sin \alpha )$ , where $\alpha$ is measured in degrees, be the solution of $z^4-\mathrm{i}=0$ which has the smallest positive argument.
2. Find the modulus and argument of z .
3. Use (a) (ii) and your answer from (b) to show that $8{\cos }^4\alpha -8{\cos }^2\alpha +1=0$ .
4. Hence express $\cos {22.5}^{\circ }$ in the form $\frac{\sqrt{a+b\sqrt{c}}}{d}$ where a, b, c, d $\in \mathbb{Z}$ .

Let $z=\cos \theta +\mathrm{i}\sin \theta$ , for $-\frac{\pi }{4}<\theta <\frac{\pi }{4}$ .
1. 1. Find $z^3$ using the binomial theorem.
2. Use de Moivre's theorem to show that $\cos 3\theta =4{\cos }^3\theta -3\cos \theta$ and $\sin 3\theta =3\sin \theta -4{\sin }^3\theta$ .
2. Hence show that $\frac{\sin 3\theta -\sin \theta }{\cos 3\theta +\cos \theta }=\tan \theta$ .
3. Given that $\sin \theta =\frac{1}{3}$ , find the exact value of $\tan 3\theta$.

Let $z=\cos \theta +\mathrm{i}\sin \theta$ , for $-\frac{\pi }{4}<\theta <\frac{\pi }{4}$ .
1. 1. Find $z^3$ using the binomial theorem.
2. Use de Moivre's theorem to show that $\cos 3\theta =4{\cos }^3\theta -3\cos \theta$ and $\sin 3\theta =3\sin \theta -4{\sin }^3\theta$ .
2. Hence show that $\frac{\sin 3\theta -\sin \theta }{\cos 3\theta +\cos \theta }=\tan \theta$ .
3. Given that $\sin \theta =\frac{1}{3}$ , find the exact value of $\tan 3\theta$.

1. Solve $2\sin (x+{120}^{\circ })=\sqrt{3}\cos (x+{60}^{\circ })$, for $x\in [0,{180}^{\circ }]$ .
2. Show that $\sin {75}^{\circ }+\cos {75}^{\circ }=\frac{\sqrt{6}}{2}$ .
3. Let $z=\sin 4\theta +\mathrm{i}(1-\cos 4\theta )$ , for $z\in \mathbb{C}$, $\theta \in [0,{90}^{\circ }]$ .
1. Find the modulus and argument of z in terms of $\theta$ .
2. Hence find the fourth roots of z in modulus-argument form.

1. Find the roots of z^{16}=1 whi 5n4- 2pemmc40hjpr each satisfy the condition $0<\arg (z)<\frac{\pi }{2}$ , expressing your answer in the form $r{e}^{\mathrm{i}\theta }$ , where r, $\theta \in {\mathbb{R}}^{+}$ .
2. Let S be the sum of the roots found in part (a).
1. Show that $\mathrm{Re}(S)=\mathrm{Im}(S)$ .
2. By writing $\frac{\pi }{8}$ as $\frac{1}{2}\cdot \frac{\pi }{4}$ , find the value of $\cos \left(\frac{\pi }{8}\right)$ in the form $\frac{\sqrt{a+\sqrt{b}}}{c}$, where a, b and c are integers to be determined.
3. Hence, or otherwise, show that $S=\frac{1}{2}(\sqrt{2+\sqrt{2}}+\sqrt{2}+\sqrt{2-\sqrt{2}})(1+\mathrm{i})$ .

1. Solve the equation zery - mgffet o1.;4vnb,* bc)$\sin (x+{90}^{\circ })=2\cos (x-{60}^{\circ })$, $0^{\circ}2. Show that $\sin {15}^{\circ }+\cos {15}^{\circ }=\frac{\sqrt{6}}{2}$ .
3. Let $z=1-\cos 4\theta -\mathrm{i}\sin 4\theta$ , for $z\in \mathbb{C}$, $0<\theta <\frac{\pi }{2}$ .
1. Find the modulus and argument of z . Express each answer in its simplest form.
2. Hence find the fourth roots of z in modulus-argument form.

1. Use de Moivre's theorem to findfi j k4+d52xhgy)vfr; the value of ${[\cos \left(\frac{\pi }{6}\right)+\mathrm{i}\sin \left(\frac{\pi }{6}\right)]}^{12}$ .
2. Use mathematical induction to prove that

$(\cos \alpha -\mathrm{i}\sin \alpha {)}^n=\cos (n\alpha )-\mathrm{i}\sin (n\alpha )\text{ for all }n\in {\mathbb{Z}}^{+}\text{. }$

eet $w=\cos \alpha +\mathrm{i}\sin \alpha$ .
3. Find an expression in terms of $\alpha$ for $w^n-{\left(w^{\ast }\right)}^n$, $n\in {\mathbb{Z}}^{+}$ , where $w^{\ast }$ is the complex conjugate of w .
4. 1. Show that $w{w}^{\ast }=1$ .
2. Write down and simplify the binomial expansion of ${(w-w^{\ast })}^3$ in terms of w and $w^{\ast }$ .
3. Hence show that $\sin (3\alpha )=3\sin \alpha -4{\sin }^3\alpha$ .
5. Hence solve $4{\sin }^3\alpha +(2\cos \alpha -3)\sin \alpha =0$ for $0\le \alpha \le \pi$ .