Use this diagram to determine the roots of the corresponding equation of the form $\left(z-r\right)\left(z^2-2az+a^2+16\right)=0$ for $z\in \mathrm{ℂ}$.

Express $z_1$ in terms of $z_2$.

Use the result from part (c)(ii) to show that $a^2-3b=0$.

Sketch the graph of $x^2-y^2=1$, stating the coordinates of any axis intercepts and the equation of each asymptote.

Find the values of $x$ that satisfy the equation $\left|p\right|=\left|q\right|$.

The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.

Show that $(\omega -3{\omega }^2)({\omega }^2-3\omega )=13$.

Hence, using the part (g)(i) and part (f) results, or otherwise, prove your suggested result to part (e).

Deduce from part (d)(i) that the complex roots of the equation $\left(z-r\right)\left(z^2-2az+a^2+b^2\right)=0$ can be expressed as $a\pm \text{i}\sqrt{g\text{'}\left(a\right)}$.

State the coordinates of ${\text{C}}_2$.

Given that $\text{Re}\left(z_1{z_2}^{\ast }\right)=0$, show that ${\text{Z}}_1{\text{OZ}}_2$ is a right-angled triangle.

Hence show that ${z_1}^2+{z_2}^2=z_1{z}_2$.

Consider the equation $z^2+az+12=0$, where $z\in \mathrm{ℂ}$ and $a\in \mathrm{ℝ}$.

Given that $0<\alpha -\theta <\pi$, deduce that only one equilateral triangle ${\text{Z}}_1{\text{OZ}}_2$ can be formed from the point $\text{O}$ and the roots of this equation.

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

$\frac{w}{z}=2\text{i}$

${\text{z}}^{\ast }-3w=5+5\text{i}$.

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

Express $z^{n-1}+ z^{n-2}+ \dots +z+1$ as a product of linear factors over the set $\mathrm{ℂ}$.

Show that $(\omega -1)({\omega }^2+\omega +1)={\omega }^3-1$.

Show that ${\text{P}}_0{\text{P}}_1\times {\text{P}}_0{\text{P}}_2=3$.

Hence, deduce that ${\omega }^2+\omega +1=0$.

Suggest a value for ${\text{P}}_0{\text{P}}_1\times {\text{P}}_0{\text{P}}_2\times \dots \times {\text{P}}_0{\text{P}}_{n-1}$.

Show that ${\text{P}}_0{\text{P}}_1\times {\text{P}}_0{\text{P}}_2\times {\text{P}}_0{\text{P}}_3=4$.

Hence, write down an expression for ${\text{P}}_0{\text{P}}_{n-1}$ in terms of $n$ and $\omega$.

Plot the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ on an Argand diagram.

By using de Moivre’s theorem, show that $\alpha =\frac{1}{1-{\mathrm{e}}^{\mathrm{i}{\displaystyle \frac{\mathrm{\pi }}{6}}}}$ is a root of this equation.

Show that ${\left(f\left(t\right)\right)}^2+{\left(g\left(t\right)\right)}^2=f\left(2t\right)$.

$f\left(\text{i}u\right)$.

The hyperbola with equation $x^2-y^2=1$ can be rotated to coincide with the curve defined by $xy=k, k\in \mathrm{ℝ}$.

Find the possible values of $k$.

Find an expression for $z_1{z}_2$ in terms of $b$.

Show that $z_1{z_2}^{\ast }=r_1{r}_2{\text{e}}^{\text{i}\left(\alpha -\theta \right)}$ where ${z_2}^{\ast }$ is the complex conjugate of $z_2$.

Determine the value of $\text{Re}\left(\alpha \right)$.

Verify that $u=f\left(t\right)$ satisfies the differential equation $\frac{d^{2}u}{d{t}^2}=u$.

Show that ${\left(f\left(t\right)\right)}^2-{\left(g\left(t\right)\right)}^2={\left(f\left(\text{i}u\right)\right)}^2-{\left(g\left(\text{i}u\right)\right)}^2$.

Solve the inequality .

Determine the value of

(i)     $1+\omega +{\omega }^2$;

(ii)     $1+\omega \text{*}+(\omega \text{*}{)}^2$.

Write down expressions for ${\text{P}}_0{\text{P}}_2$ and ${\text{P}}_0{\text{P}}_3$ in terms of $\omega$.

Consider the equation $\frac{2z}{3-z*}=\text{i}$, where $z=x+\text{i}y$ and $x, y\in \mathrm{ℝ}$.

Find the value of $x$ and the value of $y$.

Use the binomial theorem to expand ${\left(\cos \theta +\mathrm{i} \sin \theta \right)}^4$. Give your answer in the form $a+b\mathrm{i}$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

Consider $z=\cos \theta +\text{i} \sin \theta$ where $z\in \mathrm{ℂ}, z\ne 1$.

Show that $\text{Re}\left(\frac{1+z}{1-z}\right)=0$.

By factorizing $z^4-1$, or otherwise, deduce that ${\omega }^3+{\omega }^2+\omega +1=0$.

Verify that ${\omega }_1=1+{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{6}}$ is a root of this equation.

Find ${\omega }_2$ and ${\omega }_3$, expressing these in the form $a+{\mathrm{e}}^{\mathrm{i}\theta }$, where $a\in \mathrm{ℝ}$ and $\theta >0$.

Find $\mathrm{AC}$.

$g\left(\text{i}u\right)$.

Hence find, and simplify, an expression for ${\left(f\left(\text{i}u\right)\right)}^2+{\left(g\left(\text{i}u\right)\right)}^2$.