Complex Numbers & Argand Diagrams (Edexcel A Level Further Maths: Core Pure): Exam Questions

In an Argand diagram, the points $A$ and $B$ are represented by the complex numbers $-3+2\mathrm{i}$ and $5-4\mathrm{i}$ respectively. The points $A$ and $B$ are the end points of a diameter of a circle $C$.

Find the equation of $C,$ giving your answer in the form 

$|z-a|=b$      $a\in \mathrm{ℂ}$, $b\in \mathrm{ℝ}$

The circle $D$, with equation $|z-2-3\mathrm{i}|=2$, intersects $C$ at the points representing the complex numbers $z_1$ and $z_2$

Find the complex numbers $z_1$ and $z_2$

$\mathrm{f}\left(z\right)=z^4+a{z}^3+6z^2+bz+65$

where $a$ and $b$ are real constants.

Given that $z=3+2\mathrm{i}$ is a root of the equation $\mathrm{f}\left(z\right)=0$, show the roots of $\mathrm{f}\left(z\right)=0$  on a single Argand diagram.

$\mathrm{f}\left(z\right)=z^4-4z^3+a{z}^2+bz+26$

where $a$ and $b$ are real constants.

Given that $3+2\mathrm{i}$ is a root of the equation $\mathrm{f}\left(z\right)=0$,

find the value of $a$ and the value of $b$.

Hence find the other three roots of the equation $\mathrm{f}\left(z\right)=0$.

Show all four roots of $\mathrm{f}\left(z\right)=0$ on a single Argand diagram.

The locus $C$ is given by

$|z-2\text{i}|=2$

The locus $D$ is given by

$\arg z=\frac{\pi }{6}$

Sketch, on the same Argand diagram, the locus $C$ and the locus $D$.

The set of points $A$ is defined by

$A=\left\{z:|z-2\text{i}|\le 2\right\}\cap \left\{z:\frac{\pi }{6}\le \arg z\le \frac{\pi }{2}\right\}$

Show, by shading on your Argand diagram, the set of points $A$.