Let $z=2-3i$.

A transformation of the complex plane is defined by $w=\alpha z$, where $\alpha=1+\sqrt3i$. The point $P$ represents the complex number $z=-1+2i$, and its image is the point $Q$ representing $w$.

Two alternating voltages, measured in volts, are given by
$V_1(t)=8\cos(12t+0.4),\qquad V_2(t)=11\cos(12t-0.9)$
where $t$ is measured in seconds. The total voltage is $V(t)=V_1(t)+V_2(t)$.

Three water waves have heights
$h_1(t)=4\sin(5t),\qquad h_2(t)=3\sin\left(5t+\frac{\pi}{2}\right),\qquad h_3(t)=2\sin(5t+\pi)$
where height is measured in centimetres and $t$ in seconds. Their combined height is $h(t)=h_1(t)+h_2(t)+h_3(t)$.

A complex number $z$ satisfies
$z+\overline z=8\qquad\text{and}\qquad z\overline z=25$

An input signal has complex amplitude $P=6-8i$. A filter multiplies this amplitude by the transfer factor $H=\dfrac{1+i}{2}$. The output complex amplitude is $Q=HP$.

The points $A$ and $B$ on an Argand diagram represent $z_A=3+i$ and $z_B=-1+4i$, respectively. Multiplication by a complex number $q$ maps $A$ to $B$, so that $z_B=qz_A$.

Let $z=-1+\sqrt3i$ and define $w_n=z^n$ for positive integers $n$.

Three actuators produce vibrations with the same angular frequency. Their phasors are $P_1=9e^{0.35i}$, $P_2=6e^{-0.80i}$ and $P_3=Ae^{i\theta}$, where $A>0$ and $-\pi<\theta\leq\pi$. The resultant phasor is $P=P_1+P_2+P_3$.

A sequence of points on an Argand diagram is represented by $z_0=5-2i$ and
$z_{n+1}=qe^{0i}z_n,\qquad q=0.82e^{i\pi/7}$
for $n\geq0$.

The equation
$z^2-6z+k=0$
where $k\in\mathbb R$, has two non-real roots. One root is denoted by $\alpha$, and $|\alpha|=5$.

A transformation $T$ of the complex plane is defined by $T(z)=qz$. The point represented by $2-3i$ is mapped to the point represented by $-4+7i$.

A complex-valued calculation is defined by
$Q=\frac{(3-2i)^8}{(1+i)^5}$

The horizontal displacement, in millimetres, of a platform is modelled by
$x(t)=5\cos(8t)+7\cos\left(8t+1.10\right)+4\cos\left(8t-0.65\right)$
where $t$ is measured in seconds.

A quadratic function is given by $f(x)=2x^2+px+q$, where $p,q\in\mathbb R$. The equation $f(x)=0$ has one root $-1+3i$.

Let
$z=3\operatorname{cis}\left(\frac{5\pi}{12}\right),\qquad w=4\operatorname{cis}\left(-\frac{\pi}{8}\right)$

A point in an animation is represented by $z_0=5e^{-0.2i}$. Each frame multiplies its position by $\alpha=1.2e^{0.35i}$, so that $z_n=\alpha^nz_0$ after $n$ frames.

Three synchronized seismic signals are
$s_1(t)=6\cos(4t+0.4),\quad s_2(t)=8\cos(4t+1.7),\quad s_3(t)=5\cos(4t-0.8)$
where displacement is measured in millimetres and $t$ in seconds.

Two loudspeakers produce signals
$L_1(t)=10\cos(6t+\phi),\qquad L_2(t)=10\cos(6t-\phi)$
where $0<\phi<\frac{\pi}{2}$. The measured amplitude of their combined signal is $13$. Here, $t$ is measured in seconds, so the angular frequency is $6\ \text{rad s}^{-1}$.

The vertices of a regular hexagonal plate are represented by
$z_k=3e^{k\pi i/3},\qquad k=0,1,\ldots,5$
A manufacturing process maps each vertex to $w_k=\alpha z_k$, where $\alpha=1.5e^{i\pi/4}$.

A resultant signal is measured as
$V(t)=10\cos(5t+0.4)$
One known component is $V_1(t)=8\cos(5t)$. A second component is $V_2(t)=A\cos(5t+\phi)$, where $A>0$ and $-\pi<\phi\leq\pi$. The variable $t$ is measured in seconds.

The points $A$, $B$ and $C$ on an Argand diagram represent the complex numbers $a=1+2i$, $b=6+i$ and $c=4+5i$, respectively. A fourth point $D$ is chosen so that $ABCD$ is a parallelogram, with the vertices in this order.

A logo is generated from the complex number $z_0=3e^{0.20i}$. Successive vertices are defined by
$z_{n+1}=\left(1.05e^{0.60i}\right)z_n$
The centre of the logo is the origin.

Let $z=a+bi$, where $a,b\in\mathbb R$, $b\ne0$ and $z\ne-1$. Define
$w=\frac{z-1}{z+1}$

Four speakers emit tones of the same angular frequency. Their complex amplitudes are shown in the table. The total complex amplitude is the sum of the four amplitudes.

Two alternating currents are modelled by
$I_1(t)=14\cos(50t+\alpha),\qquad I_2(t)=10\cos(50t-0.40)$
where current is measured in amperes. It is known that the resultant current has phase shift zero and a positive amplitude.

On an Argand diagram, the points $O$, $A$ and $B$ represent $0$, $a=4+i$ and $b=-2+5i$, respectively. A transformation $T(z)=qz$ maps $A$ to $B$.

Two vibration sources have phasors $P_1=12e^{0.2i}$ and $P_2=9e^{-1.1i}$. A controllable source has phasor $C=5e^{\theta i}$, where $-\pi<\theta\leq\pi$.

For $0\leq t<2\pi$, consider the quadratic equation
$x^2-2(1+2\cos t)x+(5+4\cos t)=0$