Complex Numbers

[Maximum mark: 5]

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

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

   

[Maximum mark: 7]

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

1. Find $zw$. [2]
   

2. Illustrate $z$, $w$ and $zw$ on the same Argand diagram. [3]
   

3. Let $\theta$ be the angle between $zw$ and $w$. Find $\theta$, giving your answer in radians.[2]
   

[Maximum mark: 6]

The complex numbers $z$ and $w$ correspond to the points A and B as shown on the diagram below.

1. Find the exact value of $|z - w|$. [2]
   

2. 1. Find the exact perimeter of triangle AOB.

   2. Find the exact area of triangle AOB. [4]
      

[Maximum mark: 6]

Let $z = re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\frac{\pi}{6}}$ where $r \in \mathbb{R}^+$.

1. For $r = \sqrt{3}$,

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

   2. draw $z^2$ and $z^3$ on the following Argand diagram. [4]
      

      

2. Given that the integer powers of $w = \dfrac{z}{6+2{\mathrm{\hspace{0.05em}i}\mkern 1mu}}$ lie on a unit circle centred

   at the origin, find the value of $r$. [2]