Complex Numbers

[Maximum mark: 6]

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

[Maximum mark: 6]

In this question give all angles in radians.

Let $z = 1 + 2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $w = 4 + {\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

1. Find $z+w$. [1]
   

2. Find:

   1. $|z+w|$;

   2. $\arg(z+w)$. [3]
      

3. Find $\theta$, the angle shown on the diagram below. [2]
   

   

[Maximum mark: 7]

The complex numbers $w$ and $z$ satisfy the equations

Find $w$ and $z$ in the form $a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{Z}$.

[Maximum mark: 8]

Let $w = 2e^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\frac{2\pi}{3}}$.

1. 1. Write $w$, $w^2$ and $w^3$ in the form $a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in\mathbb{R}$.

   2. Draw $w$, $w^2$ and $w^3$ on an Argand diagram. [6]
      

2. Find the smallest integer $k > 3$ such that $w^k$ is a real number. [2]
   

[Maximum mark: 9]

1. Find three distinct roots of the equation $z^3 + 64 = 0$, $z \in \mathbb{C}$, giving your answers in modulus-argument form. [6]
   

The roots are represented by the vertices of a triangle in an Argand diagram.

2. Show that the area of the triangle is $12\sqrt{3}$. [3]
   

[Maximum mark: 15]

Consider $w = \dfrac{z - 1}{z + {\mathrm{\hspace{0.05em}i}\mkern 1mu}}$ where $z = x + {\mathrm{\hspace{0.05em}i}\mkern 1mu}y$ and ${\mathrm{\hspace{0.05em}i}\mkern 1mu}= \sqrt{-1}$.

1. If $z = {\mathrm{\hspace{0.05em}i}\mkern 1mu}$,

   1. write $w$ in the form $r\mathop{\mathrm{cis}}\theta$;

   2. find the value of $w^{14}$. [5]
      

2. Show that in general,

   $$w = \dfrac{(x^2 - x + y^2 + y) + {\mathrm{\hspace{0.05em}i}\mkern 1mu}(y - x + 1)}{x^2 + (y + 1)^2}$$

   [4]
   

3. Find condition under which $\mathrm{Re}(w) = 1$. [2]
   

4. State condition under which $w$ is:

   1. real;

   2. purely imaginary. [2]
      

5. Find the modulus of $z$ given that $\arg w = \dfrac{\pi}{4}$. [2]
   

[Maximum mark: 22]

1. Solve $2\sin(x+\ang{120}) = \sqrt{3}\cos(x + \ang{60})$, for $x \in [0,\ang{180}]$. [5]
   

2. Show that $\sin \ang{75} + \cos \ang{75} = \dfrac{\sqrt{6}}{2}$. [3]
   

3. Let $z = \sin 4\theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}(1 - \cos 4\theta)$, for $z \in \mathbb{C}$, $\theta \in [0,\ang{90}]$.

   1. Find the modulus and argument of $z$ in terms of $\theta$.

   2. Hence find the fourth roots of $z$ in modulus-argument form. [14]