Further Complex Numbers (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

The complex number $z$, where $z\ne 0$, is represented by a point on an Argand diagram.

In each of the following cases, describe fully the geometrical transformation, or combination of transformations, that maps the point representing $z$ to the point representing the given complex number.

(i) $4z$

(ii) $iz$

(iii) $wz$, where $w$ is a non-zero complex number

The complex number $u$ is given by $u=1-\sqrt{3}i$, and $z=r\left(\text{cos}\theta +i \text{sin}\theta \right)$ is a square root of $u$.

By first expressing $u$ in modulus-argument form, show that

$r^2\left(\text{cos}2\theta +i \text{sin}2\theta \right)=2\left(\text{cos}\left(-\frac{\pi }{3}\right)+i \text{sin}\left(-\frac{\pi }{3}\right)\right)$

By considering points on an Argand diagram, explain why

$2\left(\text{cos}\left(-\frac{\pi }{3}\right)+i \text{sin}\left(-\frac{\pi }{3}\right)\right)=2\left(\text{cos}\left(-\frac{\pi }{3}+2\pi \right)+i \text{sin}\left(-\frac{\pi }{3}+2\pi \right)\right)$

(i) Use your answers to parts (a) and (b) to find the two square roots of $u$, giving your answers in the form $R\left(\text{cos}\alpha +i \text{sin}\alpha \right)$, where $R>0$ and $-\pi <\alpha \le \pi$.

(ii) Express these square roots in the form $a+bi$, where $a$ and $b$ are real constants.

The complex numbers $z$ and $w$ satisfy

$\begin{array}{rcl}z& =& 5-5i\\ \text{Re}\left(zw\right)& =& 0\\ \left|zw\right|& =& 4\left|z\right|\end{array}$

By considering the position of $z$ in an Argand diagram and using geometrical reasoning, find the two possibilities for $w$, giving your answers in exponential form.

The complex numbers $z_1$ and $z_2$ are given by

$z_1=6e^{4i}$

$z_2=8e^{-i}$

(i) Find $z_1{z}_2$ and $\frac{z_1}{z_2}$, giving your answers in exponential form.

(ii) Express your answers to part (i) in the form $r\left(\text{cos}\theta +i \text{sin}\theta \right)$. In each case the modulus and argument should be given as exact values, with the argument $\theta$ being given in the interval $-\pi <\theta \le \pi$.

The complex number $u$ is given by $u=-5\sqrt{3}-5i$, and $z=r\left(\text{cos}\theta +i \text{sin}\theta \right)$ is a square root of $u$.

Show that

$r^2\left(\text{cos}2\theta +i \text{sin}2\theta \right)=10\left(\text{cos}\left(\frac{-5\pi }{6}\right)+i \text{sin}\left(\frac{-5\pi }{6}\right)\right)$

By considering points on an Argand diagram, explain why

$\text{cos}\alpha +i \text{sin}\alpha =\text{cos}\left(\alpha +2\pi \right)+i \text{sin}\left(\alpha +2\pi \right)$

for any value of $\alpha$, where $\alpha$ is a real number.

Use your answers to parts (a) and (b) to find the two square roots of $u$. Give your answers both in modulus-argument form and in the form $a+bi$, where $a$ and $b$ are real constants.

The complex numbers $z$ and $w$ satisfy

$z=3+3\sqrt{3}i$

$\text{Re}\left(z^{2}w\right)=0$

$\left|z^{2}w\right|=2\left|z\right|$

Use geometrical reasoning to find the two possibilities for $w$, giving your answers in exponential form.

The result

$e^{i\pi }+1=0$

is known as Euler's identity.

By considering the exponential and modulus-argument forms of a complex number, show that this result is true.

The complex numbers $z_1$ and $z_2$ are given by

$z_1=14e^{9i}$

$z_2=10e^{-2i}$

(i) Find $z_1{z}_2$ and $\frac{z_2}{z_1}$, giving your answers in exponential form.

(ii) Express your answers to part (i) in the form $r\left(\text{cos}\theta +i \text{sin}\theta \right)$. In each case the modulus and argument should be given as exact values, with the argument $\theta$ being given in the interval $-\pi <\theta \le \pi$.

Express the following complex numbers in exponential form:

(i) $3\left(2 \text{cos}2-2i \text{sin}\left(-2\right)\right)$

(ii) $-2+2\sqrt{3}i$

The complex number $z$, where $z\ne 0$, is represented by a point on an Argand diagram.

In each of the following cases, describe fully the geometrical transformation, or combination of transformations, that maps the point representing $z$ to the point representing the given complex number.

(i) $-2z$

(ii) $\left|z\right|$

(iii) $\frac{z}{w}$, where $w$ is a non-zero complex number

The complex number $z$ is given by $z=r{e}^{i\theta }$, where $r$ and $\theta$ are real and $r\ge 0$.

By considering points on an Argand diagram, explain why

$r{e}^{i\theta }=r{e}^{i\left(\theta +2\pi \right)}$

for any value of $\theta$.

Hence use the properties of complex numbers to determine the two distinct square roots of $z$, giving your answers in exponential form in terms of $r$ and $\theta$.

The complex numbers $z$ and $w$ satisfy

$z=\sqrt{3}-i$

$\text{Im}\left(\frac{z^2}{w}\right)=0$

$\left|\frac{z^2}{w}\right|=\frac{1}{2}\left|z\right|$

Use geometrical reasoning to find the two possibilities for $w$, giving your answers in exponential form.

Note: You may assume throughout this question that $i=\sqrt{-1}$ behaves exactly the same as any other constant for purposes of algebraic manipulation and differentiation.

For a complex number $z=\text{cos}\theta +i \text{sin}\theta$, where $\theta \in \mathrm{ℝ}$, show that

$\frac{\text{d}z}{\text{d}\theta }=iz$

Express each of the following complex numbers in the form $r{e}^{i\theta }$, where $r>0$. Give the exact values of $r$ and $\theta$ in each case.

(i) $-5\left(\text{cos}2-i \text{sin}2\right)$

(ii) $\left(\sqrt{2}-\sqrt{6}\right)-\left(\sqrt{2}+\sqrt{6}\right)i$

The points representing the complex numbers 1 and $z$, where $z\ne 0$, are marked on an Argand diagram.

Explain how to find the point representing $z^2$ by geometrical construction, and provide a sketch to illustrate your answer.

Explain how to find the point representing $\left(2-i\right)z$ by geometrical construction, and provide a sketch to illustrate your answer.