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

$z_1=6e^i$

$z_2=3e^{-2i}$

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

(ii) Express your answers to part (i) as complex numbers in modulus-argument form.

Express the following complex numbers in exponential form:

(i) $7(\cos 3+i\sin 3)$

(ii) $2-2i$

Given the point $z$ on an Argand diagram, where $z\ne 0$ is a complex number, describe the geometrical transformations that will map $z$ to each of the following points:

(i) $4z$

(ii) $iz$

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

Let $z=r(\cos \theta +i\sin \theta )$ be a square root of the complex number $1-\sqrt{3}i$.

By first expressing $1-\sqrt{3}i$ in modulus-argument form, show that

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

Use the geometry of complex numbers to explain why

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

Use your answers to parts (a) and (b) to find the two square roots of the complex number  $1-\sqrt{3}i$,  giving your answers in modulus-argument form.

Express the square roots in the form  $a+bi$  where $a$ and $b$ are real numbers.

$z=5-5i,Re\left(zw\right)=0,\left|zw\right|=4\left|z\right|$

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

Express the following complex numbers in exponential form:

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

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

$z_1=6e^{4i}$

$z_2=8e^{-i}$

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

(ii) Express your answers to part (i) as complex numbers in modulus-argument form. In each case the modulus and argument should be given as exact values, with the argument $\theta$ being given in the interval $-\pi <\theta \le \pi$.

Given the point $z$ on an Argand diagram, where $z\ne 0$ is a complex number, describe the geometrical transformations that will map $z$ to each of the following points:

(i) $-2z$

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

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

Let $z=r(\cos \theta +i\sin \theta )$ be a square root of the complex number $-5\sqrt{3}-5i$.

Show that

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

Use the geometry of complex numbers to explain why

$\cos \alpha +i\sin \alpha =\cos (\alpha +2\pi )+i\sin (\alpha +2\pi )$

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 the complex number $-5\sqrt{3}-5i$. Give your answers both in modulus-argument form and in the form $a+bi$ where $a$ and $b$ are real numbers.

$z=3+3\sqrt{3}i,Re\left(z^{2}w\right)=0,\left|z^{2}w\right|=2\left|z\right|$

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

By considering the exponential and modulus-argument forms of a complex number, prove Euler’s identity

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

Express the following complex numbers in exponential form:

(i) $-5(\cos 2-i\sin 2)$

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

$z_1=14e^{9i}$

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

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

(ii) Express your answers to part (i) as complex numbers in modulus-argument form.
In each case the modulus and argument should be given as exact values, with the argument $\theta$ being given in the interval  $-\pi <\theta \le \pi$.

Given the points 1 and $z$ on an Argand diagram, where $z\ne 0$ is a complex number, explain how to find each of the following points by geometrical construction.  In each case provide a sketch to illustrate your answer.

$z^2$

$(2-i)z$

Let $z=r{e}^{i\theta }$ be a general complex number, where $r,\theta \in \mathrm{ℝ}$ and $r\ge 0$.

Use the geometry of complex numbers to explain why

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

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 $\theta$.

$z=\sqrt{3}-i,\mathrm{Im}\left(\frac{{\mathrm{z}}^2}{\mathrm{w}}\right)=0,\left|\frac{z^2}{w}\right|=\frac{1}{2}\left|z\right|$

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, differentiation and integration.

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

$\frac{dz}{d\theta }=iz$

Utilising your knowledge of differential equations, explain briefly why the result of part (a) supports the validity of Euler’s relation

$e^{i\theta }=\cos \theta +\mathrm{i}\sin \theta$