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

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



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

  Consider the complex n q-xz x sai;::1mmu(aac(9rcqumber $z=\frac{w_{1}}{w_{2}}$ 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 angle6 gmgl/qpm 5:hsk7qq2s in 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 Arg /pge ,by+9xzgand diagram. The tangent to the circgp bxe 9g+,z/yle from the point B(0,9) meets the circle at the point A as shown. Let $w=\overrightarrow{\mathrm{OA}}$

The complex numbers w and 9dc)re-ec o d8z satisfy the equations

$\begin{aligned}
\frac{z}{w} & =\mathrm{i} \\
w^{*}+2 z & =4+5 \mathrm{i} .
\end{aligned}$

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

  Consider the equatio88rz8 a0pcmpehu0bn x- m8;kmn $\frac{3 z}{5-z^{*}}=\mathrm{i}$ , 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 cox25qt)hl.if1 h1 mk fh ke.dh5rrespond to the points A and B as shown on the diagram xh5kmf . ldehh5th1.ik)2 f1qbelow.


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 $z=r e^{1 \frac{4}{3}} where r \in \mathbb{R}^{+}$ .
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 nu7vjwq ffx+ 1*mmbers $u=1+2 \mathrm{i} and v=2+\mathrm{i}$ .
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 $w^{*}$ and express it in the form r $e^{\mathrm{i} \theta}$ .$a e^{-i \frac{\pi}{b}}$ a =    b =   

1. Find three distinct roots of the equ:c:v zdzsolsmb7 -l1; ation $ 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 $\sqrt{3}$ .

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

Let $w=2 e^{\mathrm{i} \frac{2 \pi}{3}}$ .
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 azve bc yn,(sxn/ +u5*2 eb+mx 2 $z^{4}+a z^{3}+b z^{2}+c z+d=0$ , 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 numbers 01u2e,ow yk56xlu)pdyk,y sb $z_{1}=3 \operatorname{cis}\left(120^{\circ}\right)$ 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 $\frac{\sqrt{a}}{b} where a, b \in \mathbb{Z}^{+}$ .

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

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 $-\pi<\theta \leq \pi$ .

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 $r \operatorname{cis} \theta$ ;
2. find the value of $w^{14}$ .
2. Show that in general,
$w=\frac{\left(x^{2}-x+y^{2}+y\right)+\mathrm{i}(y-x+1)}{x^{2}+(y+1)^{2}}$

3. Find condition under which $\operatorname{Re}(w)=1$ .
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:pcs-3c7 py5xuxia)z , the complex 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 \left(x+120^{\circ}\right)=\sqrt{3} \cos \left(x+60^{\circ}\right)$, for $x \in\left[0,180^{\circ}\right]$ .
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 $z \in \mathbb{C}$, $\theta \in\left[0,90^{\circ}\right]$ .
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 which sati/ 7 g6yox3b9cagqcs 8bz o/0orsfy 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 $\operatorname{Re}(S)=\operatorname{Im}(S)$ .
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 equatio/cms (d-s9 pm f)vs+axn $\sin \left(x+90^{\circ}\right)=2 \cos \left(x-60^{\circ}\right)$, $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 find the3ljey tj(mhlj6q 3-b 8 value of $\left[\cos \left(\frac{\pi}{6}\right)+\mathrm{i} \sin \left(\frac{\pi}{6}\right)\right]^{12}$ .
2. Use mathematical induction to prove that

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

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