Let $z$ be a non-zero complex number such that $|z|=\sqrt{2}$ and $0<\arg z<\frac{\pi}{2}$. It is given that the principal argument of $(1+i)z^2$ is $\frac{5\pi}{6}$.

The complex numbers $w$ and $z$ satisfy

where $z\ne0$.

On an Argand diagram, the point $A$ represents the complex number $z=-3+4{\text{i}}$. The point $B$ represents the complex number $w$, where

Let $z=-4+6{\text{i}}$.

Let

Two complex impedances are given by $Z_1=4+3{\text{i}}$ and $Z_2=6-2{\text{i}}$. The combined impedance $Z$ is defined by

Let $z=x+y{\text{i}}$, where $x,y\in\mathbb{R}$. The point representing $z$ on an Argand diagram satisfies both

and

Consider the equation $z^3=8i$, where $z\in\mathbb{C}$.

Let $w=16\operatorname{cis}\left(\frac{2\pi}{3}\right)$.

A monic quartic polynomial $P(z)$ has real coefficients. Two of its roots are $1+2i$ and $3-i$.

Let $z=\cos\theta+i\sin\theta$, where $\theta\in\mathbb{R}$.

The five roots of $z^5=1$ are $z_k=\operatorname{cis}\left(\frac{2\pi k}{5}\right)$, where $k=0,1,2,3,4$.

Consider the equation

The polynomial

has real coefficients. Two of its roots are $2+3{\text{i}}$ and $-1+2{\text{i}}$.

Let $z=x+y{\text{i}}$, where $x,y\in\mathbb{R}$. The complex number $z$ satisfies

Let $z=x+yi$, where $x,y\in\mathbb{R}$, and define $w=(1-i)z+2$.

Let $u=\sqrt{3}+i$, $v=1-i$ and $q=\dfrac{u}{v}$.

The transformation $T$ of the complex plane is defined by

$T(z)=(1+\sqrt{3}{\text{i}})z+2-{\text{i}}$.

The points $A$, $B$ and $C$ represent the complex numbers $1+2{\text{i}}$, $4+{\text{i}}$ and $2-2{\text{i}}$ respectively.

Consider the equation

$z^4=81\left(\cos\frac{2\pi}{3}+{\text{i}}\sin\frac{2\pi}{3}\right)$,

where $z\in\mathbb{C}$.

A monic quartic polynomial $P(z)$ has real coefficients. One root of $P(z)=0$ is $\frac32+2{\text{i}}$. Another root is $-1$, and the constant term of $P(z)$ is $-25$.

Consider the equation
$(z-1+i)^4=16\operatorname{cis}\left(\frac{2\pi}{3}\right),\qquad z\in\mathbb{C}$

A monic quartic polynomial $P(z)$ has real coefficients. One root is $1+3i$. The coefficient of $z^3$ is $-6$ and the constant term is $-50$.

Let $0<\theta<\dfrac{\pi}{2}$ and let
$z=1+\cos2\theta+i\sin2\theta$

A transformation of the complex plane is defined by $w=az+b$, where $a,b\in\mathbb{C}$ and $a\ne0$. The transformation maps $0$ to $1+i$ and maps $2$ to $3+5i$.

Let $z=x+yi$, where $x,y\in\mathbb{R}$. The point representing $z$ in the Argand diagram satisfies
$|z-2|=2|z+1|$

Let $a\in\mathbb{R}$ and consider the quadratic equation
$z^2-2az+a^2+16=0$

A complex number $z=x+y{\text{i}}$ is represented by a point $P$ on an Argand diagram. The point $P$ satisfies

$|z-(2+3{\text{i}})|=2$

and

$\arg(z+1-{\text{i}})=0.700$.

Angles are in radians.

In an alternating-current circuit, two impedances are connected in parallel. At angular frequency $\omega>0$ they are modelled by

$Z_1=6+0.5\omega{\text{i}},\qquad Z_2=10-\frac{60}{\omega}{\text{i}}$.

The combined impedance $Z$ satisfies

$\frac{1}{Z}=\frac{1}{Z_1}+\frac{1}{Z_2}$.

Consider the equation

$(z-2+{\text{i}})^3=64\left(\cos\frac{5\pi}{6}+{\text{i}}\sin\frac{5\pi}{6}\right)$.

Let $z=x+y{\text{i}}$, where $x,y\in\mathbb{R}$. The complex number $z$ satisfies

$z^2+2z^*=11+8{\text{i}}$.

The complex number $u=\sqrt{3}+i$ is used to generate a sequence of points $P_0,P_1,P_2,\ldots$ on an Argand diagram. The point $P_n$ represents the complex number $z_n=u^n$, where $n\in\mathbb{Z}_{\ge 0}$.

Consider the equation

where $0\le \phi<2\pi$. The five roots are represented by points on an Argand diagram.

For $t\ge 0$, define

In an engineering interpretation, $|H(t)|$ represents a gain and $\arg H(t)$ represents a phase shift.