Further Complex Numbers (Cambridge International A-Level Mathematics): Exam Questions

A complex number has modulus-argument form

6 \left( \text{cos} \; 2 + i \, \text{sin} \; 2 \right)

Write this complex number in exponential form.

The complex number $w$ is given by $w = 3 + 3 i$.

(i) Find the modulus and argument of $w$, giving your answers as exact values.

(ii) Hence write $w$ in both modulus-argument form and exponential form.

The complex numbers $z_{1}$ and $z_{2}$ are given by

$z_{1} = 8 e^{-i}$

$z_{2} = 2 e^{2i}$

Find $z_{1} z_{2}$ and $\frac{z_{1}}{z_{2}}$, giving your answers in exponential form.

Express $z_{1} z_{2}$ and $\frac{z_{1}}{z_{2}}$ in modulus-argument form.

Express the numbers $3$ and $- i$ in exponential form.

The complex number $z$ is given by $z = r e^{iθ}$.

Hence express each of the following in exponential form, in terms of $r$ and $θ$:

(i) $3 z$

(ii) $- i z$

The point $z$, where |z| \neq 0, is represented on an Argand diagram.

Describe the geometrical transformation that maps $z$ to each of the following points:

(i) $3 z$

(ii) $- i z$

The complex number $z$ is given by

z = 9 \left( \text{cos} \dfrac{2 \pi}{3} + i \, \text{sin} \dfrac{2 \pi}{3} \right)

Show that 3 \left( \text{cos} \dfrac{\pi}{3} + i \, \text{sin} \dfrac{\pi}{3} \right) is a square root of $z$.

Show that 3 \left( \text{cos} \left( - \dfrac{2 \pi}{3} \right) + i \, \text{sin} \left( - \dfrac{2 \pi}{3} \right) \right) is also a square root of $z$.

Given that

- 1 = \text{cos} \left( - \pi \right) + i \, \text{sin} \left( - \pi \right)

show that

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

State how this compares with the relationship between the two square roots of a positive real number.

The complex numbers $z$ and $w$ satisfy

$z = - 3 + 3 i$

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

\left| z w \right| = 5 \left| z \right|

Find \left| w \right|.

Show that $\text{arg} z = \frac{3π}{4}$, and write down the two possible values of \text{arg} \left( z w \right).

Hence find the two possible values of $\text{arg} w$, giving your answers in the interval $- π < \text{arg} w \leq π$.

Write down the two possibilities for $w$, giving your answers in exponential form.

For each of the possible values of $w$, describe the geometrical transformation that maps $z$ to $z w$ on an Argand diagram.

The complex numbers $z_{1}$ and $z_{2}$ are given by

$z_{1}=6e^{i}\,z_{2}=3e^{-2i}$

(i) Find $z_{1} z_{2}$ and $\frac{z_{1}}{z_{2}}$, giving your answers in the form $r e^{iθ}$, where $r > 0$ and $- π < θ \leq π$.

(ii) Express your answers to part (i) in the form r \left( \text{cos} \; \theta + i \, \text{sin} \; \theta \right).

Express each of the following complex numbers in the form $r e^{iθ}$, where $r > 0$ and $- π < θ \leq π$. Give the exact values of $r$ and $θ$ in each case.

(i) 7 \left( \text{cos} \; 3 + i \, \text{sin} \; 3 \right)

(ii) $2 - 2 i$

The complex number $z$, where $z \neq 0$, 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) $4 z$

(ii) $i z$

(iii) $w z$, 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( - \dfrac{\pi}{3} \right) + i \, \text{sin} \left( - \dfrac{\pi}{3} \right) \right)

By considering points on an Argand diagram, explain why

2 \left( \text{cos} \left( - \dfrac{\pi}{3} \right) + i \, \text{sin} \left( - \dfrac{\pi}{3} \right) \right) = 2 \left( \text{cos} \left( - \dfrac{\pi}{3} + 2 \pi \right) + i \, \text{sin} \left( - \dfrac{\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 $- π < α \leq π$.

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

The complex numbers $z$ and $w$ satisfy

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

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

$z_{1} = 6 e^{4i}$

$z_{2} = 8 e^{-i}$

(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 $θ$ being given in the interval $- π < θ \leq π$.

The complex number $u$ is given by $u = - 5 \sqrt{3} - 5 i$, 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( \dfrac{- 5 \pi}{6} \right) + i \, \text{sin} \left( \dfrac{- 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 $α$, where $α$ 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 $a + b i$, 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

$e^{iπ} + 1 = 0$

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

$z_{1} = 14 e^{9i}$

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

(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 $θ$ being given in the interval $- π < θ \leq π$.

Express the following complex numbers in exponential form:

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

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

The complex number $z$, where $z \neq 0$, 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) $- 2 z$

(ii) \left| z \right|

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

The complex number $z$ is given by $z = r e^{iθ}$, where $r$ and $θ$ are real and $r \geq 0$.

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 $θ$.

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 $θ$.

The complex numbers $z$ and $w$ satisfy

$z = \sqrt{3} - i$

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

\left| \dfrac{z^{2}}{w} \right| = \dfrac{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 $z = \text{cos} θ + i \text{sin} θ$, where $θ \in ℝ$, show that

$\frac{\text{d}z}{\text{d}θ} = i z$

Express each of the following complex numbers in the form $r e^{iθ}$, where $r > 0$. Give the exact values of $r$ and $θ$ 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 $z \neq 0$, 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.