Complex numbers (cartesian form, complex plane, imaginary solutions to quadratic equations): notes and practice questions

In a complex signal processing system, an input signal $z$ is transformed into an output signal $w$ using the relationship $w = -iz + (3 + 2i)$, where $w, z \in \mathbb{C}$.

(a.i) Calculate $w$ when $z = 4 + i$.

(a.ii) Calculate $w$ when $z = 5$.

(b) Describe the two geometric transformations that map point $z$ to point $w$ on the Argand diagram, stating the order in which they are applied.

(c) Find the complex number $z$ such that the output signal $w = 1 - i$.

A specialized navigation chart uses complex numbers to represent locations. The central hub is at the origin $(0,0)$. Two important landmarks, 'Anchor Point' ($z_A$) and 'Beacon Tower' ($z_B$), are located at $z_A = 5 - 2i$ and $z_B = 4e^{i\frac{\pi}{3}}$ respectively.

(a) (i) Find the modulus of $z_A$.

(a) (ii) Find the argument of $z_A$, giving your answer in radians.

(b) Find the area of the triangular region formed by the central hub, Anchor Point ($z_A$), and Beacon Tower ($z_B$).

A signal generator produces a sequence of complex voltage signals, $V_n$, where $n$ is the signal number. The first signal is $V_1 = 10$ volts. Each subsequent signal is generated by multiplying the previous signal by a complex factor $k = \frac{1}{2} + \frac{1}{2}i$.

Write down the value of $V_2$.

Write down the value of $V_3$.

The engineering team claims that the sequence of signal magnitudes $|V_1|, |V_2|, |V_3|, \dots$ also forms a geometric sequence.

Show that this claim is correct, stating the exact value of the common ratio for this sequence.

Hence, find the sum of the infinite sequence of magnitudes $|V_1|, |V_2|, |V_3|, \dots$.

The stability of a new energy shield is modeled by a complex number $w$, which satisfies the equation $w^2 - 2kw + (k^2 + 9) = 0$, where $k$ is a control parameter and $k \in \mathbb{R}$.

(a) When $k = -2$, find the value of $w$ which satisfies $\frac{\pi}{2} < \arg w < \pi$. Give your answer in the form $a + bi$.

The stability index of the shield is given by $|w|^2$. The control parameter $k$ can vary in the interval $[-4, 4]$.

(b) Find the value(s) of $k$ in the interval $[-4, 4]$ for which the stability index is maximized.

Two complex numbers are $P = k + 3i$ and $Q = m + 2i$, where $k, m \in \mathbb{R}$.

(a) Find an expression for the real part of $P^2$, in terms of $k$.

(b) Given that $P^2 + 2Q = 1 + 8i$, find the value of $k$ and of $m$.

Let $ω = \frac{-1 + \text{i}\sqrt{3}}{2}$.

Show that $ω^2 + ω + 1 = 0$.

Hence, find the value of $(1+ω)^9$.

A matrix $B$ is defined by $B = \begin{pmatrix} -1 & 1 \\ -1 & 0 \end{pmatrix}$.

Show that $B^2 + B + I = O$, where $I$ is the $2 \times 2$ identity matrix and $O$ is the $2 \times 2$ zero matrix.

Hence, or otherwise, deduce that $B^3 = I$.

Find an expression for $B^{-1}$ in the form $aB + bI$, where $a, b \in \mathbb{Z}$.