Complex numbers: notes and practice questions

Consider the complex numbers $w_1 = 1 + k\text{i}$ and $w_2 = k - \text{i}$, where $k \in \mathbb{R}$ and $k \neq 0$.

(a) Find an expression for $w_1 w_2$ in terms of $k$.

(b) Hence, given that $\text{arg}(w_1 w_2) = \frac{\pi}{6}$, find all possible values of $k$.

Consider the complex numbers $w = 2 + ki$ and $z = 1 - 2i$, where $k \in \mathbb{R}$.

(a) Find an expression for $wz$ in terms of $k$.

(b) Hence, given that $\arg(wz) = -\frac{\pi}{4}$, find the value of $k$.

In a study of wave propagation, a mathematical model uses the function $g(x) = \frac{e^x - e^{-x}}{2}$, where $x \in \mathbb{R}$, to describe a certain physical quantity. This function is also known as the hyperbolic sine function, $\sinh x$.

Verify that $y = g(x)$ satisfies the differential equation $\frac{d^2y}{dx^2} = y$.

Another related function, the hyperbolic cosine, is defined as $f(x) = \frac{e^x + e^{-x}}{2}$, also known as $\cosh x$. Show that $(\cosh x)^2 - (\sinh x)^2 = 1$.

The functions $\cosh x$ and $\sinh x$ can be extended to complex numbers. Using Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$, where $\theta \in \mathbb{R}$, express $\cosh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Similarly, express $\sinh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Hence, show that $(\cosh(i\theta) )^2 + (\sinh(i\theta) )^2 = \cos(2\theta)$.

In a design project, a component's profile is described by a hyperbola with parametric equations $x = A \cosh t$ and $y = B \sinh t$, where $A, B$ are positive constants and $t \in \mathbb{R}$.

Given that the component's profile passes through the point $(6, 0)$ and has asymptotes $y = \pm \frac{4}{3}x$, find the values of $A$ and $B$.

(a) Find the binomial expansion of $(\cos \theta + i \sin \theta)^4$. Give your answer in the form $a + bi$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

(b) By using De Moivre's theorem and your answer to part (a), show that $\cos 4\theta = 8 \cos^4\theta - 8 \cos^2\theta + 1$.

(c) Hence, find the four distinct roots of the equation $8x^4 - 8x^2 + 1 = 0$, expressing them in the form $\cos(\alpha)$ where $0 < \alpha < \pi$.

(d) By considering the roots of the equation in part (c), or otherwise, find the exact value of $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

Consider the complex number $w = 1 - i$.

By finding the modulus and argument of $w$, show that $w = \sqrt{2}e^{-i\frac{\pi}{4}}$.

Find the smallest positive integer $n$ such that $w^n$ is a real number.

Find the value of $w^n$ when $n$ takes the value found in part (b)(i).

Consider the equation $z^3 - 2z + 4 = 0$, where $z \in \mathbb{C}$.

Given that $w=1-i$ is a root of this equation, find the other roots.

By using a suitable transformation from $z$ to $v$, or otherwise, find the roots of the equation $4v^3 - 2v^2 + 1 = 0$, where $v \in \mathbb{C}$.

Consider the equation $z^2 = -4z^*$, where $z \in \mathbb{C}, z \neq 0$.

By expressing $z$ in the form $a + bi$, find the roots of the equation.

In an alternating current (AC) circuit, the impedance of two components connected in series are given by the complex numbers $Z_1 = 4\left(\cos\frac{2\pi}{3} + \text{i}\sin\frac{2\pi}{3}\right)$ ohms and $Z_2 = 2\left(\cos\frac{m\pi}{6} - \text{i}\sin\frac{m\pi}{6}\right)$ ohms, where $m \in \mathbb{Z}^+$ is a positive integer parameter related to the second component's properties.

The total impedance of the series circuit is given by the product $Z_{total} = Z_1 Z_2$.

(a) Find the modulus of $Z_{total}$.

(b) Find the argument of $Z_{total}$ in terms of $m$.

(c.i) Suppose that the total impedance $Z_{total}$ is purely resistive (i.e., $Z_{total} \in \mathbb{R}$). Find the minimum positive integer value of $m$.

(c.ii) For the value of $m$ found in part (c.i), find the value of $Z_{total}$.

Find the binomial expansion of $(\cos \theta + i \sin \theta)^4$. Give your answer in the form $a + bi$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

By using De Moivre's theorem and your answer to part (a), show that $\cos 4\theta = 8\cos^4\theta - 8\cos^2\theta + 1$.

Hence, show that $\theta = \frac{\pi}{8}$ and $\theta = \frac{3\pi}{8}$ are solutions of the equation $8\cos^4\theta - 8\cos^2\theta + 1 = 0$.

Hence, find the exact value of $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

By considering De Moivre's theorem, show that $\cos(4\theta) = 8\cos^4\theta - 8\cos^2\theta + 1$.

Let $w = 1+i$. Show that $w^4 = -4$.

Hence, find the four roots of the equation $z^4 = -4$ in Cartesian form.

The four roots are represented by points A, B, C, D on an Argand diagram, forming a square. Find the area of this square.

Each of the points A, B, C, D is rotated counter-clockwise about the origin by an angle of $\frac{\pi}{6}$ to form new points A', B', C', D'. These points are the roots of an equation $z^4 = K$. Find $K$ in Cartesian form.

It is given that the eight points represented by the roots of $z^4=-4$ and $z^4=K$ are all solutions of $z^n = \alpha$ for some $\alpha \in \mathbb{C}$ and $n \in \mathbb{N}$. Find the smallest positive value of $n$.

A function $f$ is defined by $f(x) = x^3 - 6x^2 + cx + d$, where $c, d \in \mathbb{R}$. The equation $f(x)=0$ has three real roots, $\alpha, \beta$ and $\gamma$.

(a) Write down the value of $\alpha + \beta + \gamma$.

A polynomial $P$ is defined by $P(z) = z^5 - 8z^4 + pz^3 + qz^2 + rz - 12$, where $p, q, r \in \mathbb{R}$. The roots of $f(x)=0$ are also roots of the equation $P(z)=0$. It is given that $z=1-i$ is a root of $P(z)=0$.

(b) Find the other non-real root of $P(z)=0$, giving a reason for your answer.

(c) Find the value of $\alpha\beta\gamma$.

(d) It is given that the roots $\alpha, \beta, \gamma$ form an arithmetic progression. Find the values of $\alpha, \beta$ and $\gamma$.

Consider the complex number $z = \frac{2}{1-i}$.

(a) Show that $z = 1+i$.

(b) Find

(i) $|z|$;

(ii) $\arg z$.

(c) Use De Moivre's theorem to find the two square roots of $z$. Give your answers in the form $re^{i\theta}$, where $r> 0$ and $-\pi < \theta\leq \pi$.

The complex number $z$ can be expressed in the form $(c + di)^2$, where $c, d \in \mathbb{R}^+$.

(d) (i) Show that $c^2 = \frac{1+\sqrt{2}}{2}$.

(ii) Find the value of $d^2$.

(e) Hence, express $\tan \frac{\pi}{8}$ in the form $p+q\sqrt{2}$, where $p, q \in \mathbb{Z}$.

The polynomial $f(z) = az^3 + bz^2 + cz + d$ has integer coefficients. One root of the equation $f(z)=0$ is $z_1 = (1+i)^4$.

(a) Show that $z_1 = -4$.

(b) Another root is $z_2 = \frac{1+3i}{1-i}$. Find $z_2$ in the form $x+yi$, where $x, y \in \mathbb{Z}$.

(c) Given that the coefficients $a, b, c, d$ are relatively prime, find the polynomial $f(z)$.

Consider the complex number $z_1 = 1 - i\sqrt{3}$.

(a) (i) Express $z_1$ in modulus-argument form.

(a) (ii) Find the smallest positive integer $n$ for which $z_1^n$ is a real number.

Consider the equation $(z+i)^5 - (z-i)^5 = 0$, where $z \in \mathbb{C}$.

(b) Show that the roots of the equation are given by $z = \cot\left(\frac{k\pi}{5}\right)$ for $k=1, 2, 3, 4$.

(c) (i) By using the binomial expansion, show that the equation in part (b) can be written as $5z^4 - 10z^2 + 1 = 0$.

(c) (ii) Let the roots of the equation in (c)(i) be $z_1, z_2, z_3, z_4$. Without finding the roots, show that $\sum_{j=1}^4 z_j^2 = 4$.

(d) Hence, find the exact value of $\cot^2\left(\frac{\pi}{5}\right)$.

Let $z_1 = 1+i\sqrt{3}$ and $z_2 = 1-i$.

(a) Find the value of $\frac{z_1^6}{z_2^{10}}$. Express your answer in the form $a+bi$, where $a, b \in \mathbb{R}$.

(b) Find the smallest positive integer $n$ such that $(\frac{z_1}{z_2})^n$ is a purely real number.