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Complex Numbers

Different Forms, Roots, De Moivre’s Theorem, Argand Diagram, Geometric Applications…

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Question 1

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easy

[Maximum mark: 6]

Solve the equation $z^3 = 1$, giving your answers in Cartesian form.

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Question 2

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easy

[Maximum mark: 6]

Consider the complex number $z = \dfrac{w_1}{w_2}$ where $w_1 = \sqrt{2} + \sqrt{6}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $w_2 = 3 + \sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

1. Express $w_1$ and $w_2$ in modulus-argument form and write down

1. the modulus of $z$;

2. the argument of $z$. [4]

2. Find the smallest positive integer value of $n$ such that $z^n$ is a real number. [2]

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Question 3

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medium

[Maximum mark: 7]

The complex numbers $w$ and $z$ satisfy the equations

\begin{aligned} \dfrac{z}{w} &= {\mathrm{\hspace{0.05em}i}\mkern 1mu}, \\[6pt] w^\ast + 2z &= 4 + 5{\mathrm{\hspace{0.05em}i}\mkern 1mu}.\end{aligned}

Find $w$ and $z$ in the form $a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{Z}$.

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Question 4

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medium

[Maximum mark: 6]

1. Verify that $1+2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $-1-2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ are the second roots of $-3+4{\mathrm{\hspace{0.05em}i}\mkern 1mu}$. [2]

2. Hence, find two distinct roots of the equation $z^2 - 3z + (3-{\mathrm{\hspace{0.05em}i}\mkern 1mu}) = 0$, $z \in \mathbb{C}$,
giving your answers in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$. [4]

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Question 5

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medium

[Maximum mark: 6]

Consider the equation $\dfrac{3z}{5-z^{*}}=\text{i}$, where $z=x+\text{i}y$ and $x$, $y \in \mathbb{R}$.
Find the value of $x$ and the value of $y$.

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Question 6

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medium

[Maximum mark: 7]

Consider the complex numbers $u = 1 + 2 {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $v = 2 + {\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

1. Given that $\dfrac{1}{u} + \dfrac{1}{v} = \dfrac{6\sqrt{2}}{w}$, express $w$ in the form $a + b {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a,b \in \mathbb{R}$. [4]

2. Find $w^\ast$ and express it in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$. [3]

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Question 7

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medium

[Maximum mark: 9]

1. Find three distinct roots of the equation $z^3 + 64 = 0$, $z \in \mathbb{C}$, giving your answers in modulus-argument form. [6]

The roots are represented by the vertices of a triangle in an Argand diagram.

1. Show that the area of the triangle is $12\sqrt{3}$. [3]

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Question 8

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medium

[Maximum mark: 5]

Consider $z=\cos\theta+{\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin\theta$ where $z \in \mathbb{C}$, $z\neq 0$.

Show that $\dfrac{1}{2z^2}+\left(\dfrac{1}{2z^2}\right)^\ast = \cos(2\theta)$.

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Question 9

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medium

[Maximum mark: 7]

Consider the equation $2z^4 + az^3 + bz^2 +cz + d = 0$, where $a, b, c, d \in \mathbb{R}$ and $z \in \mathbb{C}$. Two of the roots of the equation are $\log_2 10$ and ${\mathrm{\hspace{0.05em}i}\mkern 1mu}\sqrt{5}$ and the sum of all the roots is $4 + \log_25$.

Show that $15a + d + 90 = 0$.

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Question 10

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medium

[Maximum mark: 7]

Consider the complex numbers

$z = 3\left(\cos\dfrac{\pi}{6}-{\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin\dfrac{\pi}{6}\right) \enskip\text{ and }\enskip w = 5\left(\cos\dfrac{k\pi}{12}+{\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin\dfrac{k\pi}{12}\right)$

where $k\in\mathbb{Z}^{+}$.

1. Express $z$ and $w$ in modulus-argument form and write down
1. the modulus of $zw$.

2. the argument of $zw$ in terms of $k$.[3]

Suppose that $zw\in \mathbb{Z}$.

1. Find the minimum value of $k$.

2. For the value of $k$ found in part (i), find the value of $zw$.[4]

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Question 11

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medium

[Maximum mark: 12]

Consider the complex numbers $z_1 = 3 \mathop{\mathrm{cis}}(\ang{120})$ and $z_2 = 2 + 2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

1. Calculate $\dfrac{z_1}{z_2}$ giving your answer both in modulus-argument form and

Cartesian form. [7]

2. Use your results from part (a) to find the exact value of $\sin \ang{15}\cdot\,\sin \ang{45} \cdot\,\sin \ang{75}$,

giving your answer in the form $\dfrac{\sqrt{a}}{b}$ where $a, b \in \mathbb{Z}^+$. [5]

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Question 12

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medium

[Maximum mark: 18]

1. Express $-4 + 4\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r > 0$ and $- \pi < \theta \leq \pi$. [5]

Let the roots of the equation $z^3 = -4 + 4\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ be $z_1$, $z_2$ and $z_3$.

1. Find $z_1$, $z_2$ and $z_3$ expressing your answers in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r > 0$ and $-\pi < \theta \leq \pi$. [5]

On an Argand diagram, $z_1$, $z_2$ and $z_3$ are represented by the points A, B and C, respectively.

1. Find the area of the triangle ABC. [4]

2. By considering the sum of the roots $z_1$, $z_2$ and $z_3$, show that

$\cos\Big(\dfrac{2\pi}{9}\Big) + \cos\Big(\dfrac{4\pi}{9}\Big) + \cos\Big(\dfrac{8\pi}{9}\Big) = 0$

[4]

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Question 13

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medium

[Maximum mark: 8]

Find the six distinct roots of the equation $z^6+(1-8{\mathrm{\hspace{0.05em}i}\mkern 1mu})z^3-8{\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0$, $z \in \mathbb{C}$, giving your answers in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$.

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Question 14

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medium

[Maximum mark: 7]

Consider the polynomial equation $z^4 + 10z^3 + 50z^2 + 130z + 169 = 0$ where $z \in \mathbb{C}$.

Two of the roots of this equation are $a + {\mathrm{\hspace{0.05em}i}\mkern 1mu}b$ and $b + {\mathrm{\hspace{0.05em}i}\mkern 1mu}a$ where $a,b \in \mathbb{Z}$.

1. Write down the other two roots in terms of $a$ and $b$. [1]

2. Find the possible values of $a$. [6]

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Question 15

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medium

[Maximum mark: 7]

Two distinct roots for the polynomial equation $z^4 - 12z^3 + 57z^2 - 120z + 100 =0$ are $a - 1 + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $a + 1 - 2b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $z \in \mathbb{C}$ and $a,b \in \mathbb{Z}$.

1. Write down the other two roots in terms of $a$ and $b$. [1]

2. Find the value of $a$. [2]

3. Hence, or otherwise, find the four roots of the polynomial. [4]

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Question 16

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hard

[Maximum mark: 15]

Consider $w = \dfrac{z - 1}{z + {\mathrm{\hspace{0.05em}i}\mkern 1mu}}$ where $z = x + {\mathrm{\hspace{0.05em}i}\mkern 1mu}y$ and ${\mathrm{\hspace{0.05em}i}\mkern 1mu}= \sqrt{-1}$.

1. If $z = {\mathrm{\hspace{0.05em}i}\mkern 1mu}$,

1. write $w$ in the form $r\mathop{\mathrm{cis}}\theta$;

2. find the value of $w^{14}$. [5]

2. Show that in general,

$w = \dfrac{(x^2 - x + y^2 + y) + {\mathrm{\hspace{0.05em}i}\mkern 1mu}(y - x + 1)}{x^2 + (y + 1)^2}$

[4]

3. Find condition under which $\mathrm{Re}(w) = 1$. [2]

4. State condition under which $w$ is:

1. real;

2. purely imaginary. [2]

5. Find the modulus of $z$ given that $\arg w = \dfrac{\pi}{4}$. [2]

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Question 17

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hard

[Maximum mark: 19]

1. Expand $(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)^4$ by using the binomial theorem.

2. Hence use de Moivre's theorem to prove that

\begin{aligned} \cos 4\theta = \cos^4 \theta - 6\cos^2 \theta\sin^2 \theta + \sin^4 \theta. \\ \end{aligned}
3. State a similar expression for $\sin 4 \theta$ in terms of $\cos \theta$ and $\sin \theta$. [6]

Let $z = r(\cos \alpha + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha)$, where $\alpha$ is measured in degrees, be the solution
of $z^4 - {\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0$ which has the smallest positive argument.

1. Find the modulus and argument of $z$. [4]

2. Use (a) (ii) and your answer from (b) to show that $8\cos^4\alpha - 8\cos^2 \alpha + 1 = 0$. [4]

3. Hence express $\cos \ang{22.5}$ in the form $\dfrac{\sqrt{a + b\sqrt{c}}}{d}$ where $a,b,c,d \in \mathbb{Z}$. [5]

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Question 18

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hard

[Maximum mark: 19]

Let $z = \cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta$, for $-\dfrac{\pi}{4} < \theta < \dfrac{\pi}{4}$.

1. Find $z^3$ using the binomial theorem.

2. Use de Moivre's theorem to show that $\cos 3\theta = 4\cos^3\theta - 3\cos \theta$ and $\sin 3\theta = 3\sin\theta-4\sin^3\theta$. [8]

1. Hence show that $\dfrac{\sin 3\theta - \sin \theta}{\cos 3\theta + \cos \theta} = \tan \theta$. [6]

2. Given that $\sin \theta = \dfrac{1}{3}$, find the exact value of $\tan 3\theta$. [5]

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Question 19

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hard

[Maximum mark: 22]

1. Solve $2\sin(x+\ang{120}) = \sqrt{3}\cos(x + \ang{60})$, for $x \in [0,\ang{180}]$. [5]

2. Show that $\sin \ang{75} + \cos \ang{75} = \dfrac{\sqrt{6}}{2}$. [3]

3. Let $z = \sin 4\theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}(1 - \cos 4\theta)$, for $z \in \mathbb{C}$, $\theta \in [0,\ang{90}]$.

1. Find the modulus and argument of $z$ in terms of $\theta$.

2. Hence find the fourth roots of $z$ in modulus-argument form. [14]

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Question 20

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hard

[Maximum mark: 17]

1. Solve the equation $z^3 = 27$, $z \in \mathbb{C}$, giving your answer in the form
$z = r(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)$ and in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a,b \in \mathbb{R}$. [6]

2. Consider the complex numbers $z_1 = -1 + {\mathrm{\hspace{0.05em}i}\mkern 1mu} \text{ and } z_2 =\dfrac{1}{\sqrt{2}}\bigg[\mathrm{cos}\bigg(\dfrac{\pi}{3}\bigg)+\mathrm{i}\,\mathrm{sin}\bigg(\dfrac{\pi}{3}\bigg)\bigg]$ .

1. Write $z_1$ in the form $r(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)$.

2. Calculate $z_1z_2$ and write in the form $a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a,b \in \mathbb{R}$.

3. Hence find the value of $\tan\left(\dfrac{\pi}{12}\right)$ in the form $c + d\sqrt{3}$ where $c,d \in \mathbb{Z}$.

4. Find the smallest $p \in \mathbb{Q}^+$ such that $(z_1z_2)^p$ is a positive real number. [11]

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Question 21

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hard

[Maximum mark: 16]

1. Find the roots of $z^{16} = 1$ which satisfy the condition $0 < \arg(z) < \dfrac{\pi}{2}$,

expressing your answer in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r, \theta \in \mathbb{R}^+$. [5]

2. Let $S$ be the sum of the roots found in part (a).

1. Show that $\mathrm{Re}(S) = \mathrm{Im}(S)$.

2. By writing $\dfrac{\pi}{8}$ as $\dfrac{1}{2}\cdot\dfrac{\pi}{4}$, find the value of $\cos \Big(\dfrac{\pi}{8}\Big)$ in the form $\dfrac{\sqrt{a + \sqrt{b}}}{c}$,

where $a, b$ and $c$ are integers to be determined.

3. Hence, or otherwise, show that $S = \dfrac{1}{2}\big(\hspace{-0.1em}\sqrt{2 + \sqrt{2}} + \sqrt{2} + \sqrt{2 - \sqrt{2}}\hspace{0.1em}\big)\big(1+{\mathrm{\hspace{0.05em}i}\mkern 1mu}\big)$. [11]

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Question 22

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hard

[Maximum mark: 18]

1. Verify that $1+{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $-1-{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ are the second roots of $2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

2. Find two distinct roots of the equation $z^2 - 2z + (1-2i) = 0$, $z \in \mathbb{C}$, giving your answers in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$. [5]

1. Find six distinct roots of the equation $z^6+(8-{\mathrm{\hspace{0.05em}i}\mkern 1mu})z^3-8{\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0$, $z \in \mathbb{C}$, giving your answers in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$. [8]

On an Argand diagram, $z_1 = \sqrt{3}+{\mathrm{\hspace{0.05em}i}\mkern 1mu}$, $z_2 = -\sqrt{3} +{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $z_3 = -2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ are represented by the points A, B and C, respectively.

1. Verify that $z_1$, $z_2$ and $z_3$ are the roots of the equation $z^3 - 8{\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0$, $z \in \mathbb{C}$.

2. Show that the area of the triangle ABC is $3\sqrt{3}$. [5]

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Question 23

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hard

[Maximum mark: 20]

1. Solve the equation $\sin (x + \ang{90}) = 2\cos(x - \ang{60})$, $\ang{0} < x < \ang{360}$. [5]

2. Show that $\sin \ang{15} + \cos \ang{15} = \dfrac{\sqrt{6}}{2}$. [4]

3. Let $z = 1 - \cos 4\theta - {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin 4\theta$, for $z \in \mathbb{C}$, $0 < \theta < \dfrac{\pi}{2}$.

1. Find the modulus and argument of $z$. Express each answer
in its simplest form.

2. Hence find the fourth roots of $z$ in modulus-argument form. [11]

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Question 24

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hard

[Maximum mark: 21]

1. Use de Moivre's theorem to find the value of $\left[\cos\left(\dfrac{\pi}{6}\right) + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \left(\dfrac{\pi}{6}\right)\right]^{12}$. [2]

2. Use mathematical induction to prove that

$\hspace{3.5em} (\cos \alpha - {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha)^n = \cos (n\alpha) - {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin (n\alpha) \hspace{1em} \text{for all } n \in \mathbb{Z}^+.$

[6]

Let $w = \cos \alpha + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha$.

1. Find an expression in terms of $\alpha$ for $w^n - (w^\ast)^n$, $n \in \mathbb{Z}^+$, where $w^\ast$ is the complex conjugate of $w$. [2]

1. Show that $ww^\ast = 1$.

2. Write down and simplify the binomial expansion of $(w - w^\ast)^3$ in terms of $w$ and $w^\ast$.

3. Hence show that $\sin (3\alpha) = 3\sin \alpha - 4 \sin^3 \alpha$. [5]

2. Hence solve $4\sin^3\alpha + (2 \cos \alpha - 3) \sin \alpha = 0$ for $0 \leq \alpha \leq \pi$. [6]

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