IB Mathematics AA HL  Popular Quizzes
Complex Numbers
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Question 1
[Maximum mark: 6]
Solve the equation $z^3 = 1$, giving your answers in Cartesian form.
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Question 2
[Maximum mark: 6]
In this question give all angles in radians.
Let $z = 1 + 2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $w = 4 + {\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

Find $z+w$. [1]

Find:

$z+w$;

$\arg(z+w)$. [3]


Find $\theta$, the angle shown on the diagram below. [2]
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Question 3
[Maximum mark: 7]
The complex numbers $w$ and $z$ satisfy the equations
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
[Maximum mark: 8]
Let $w = 2e^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\frac{2\pi}{3}}$.


Write $w$, $w^2$ and $w^3$ in the form $a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in\mathbb{R}$.

Draw $w$, $w^2$ and $w^3$ on an Argand diagram. [6]


Find the smallest integer $k > 3$ such that $w^k$ is a real number. [2]
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Question 5
[Maximum mark: 9]

Find three distinct roots of the equation $z^3 + 64 = 0$, $z \in \mathbb{C}$, giving your answers in modulusargument form. [6]
The roots are represented by the vertices of a triangle in an Argand diagram.
 Show that the area of the triangle is $12\sqrt{3}$.
[3]
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Question 6
[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 ${\mathrm{\hspace{0.05em}i}\mkern 1mu}= \sqrt{1}$.

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

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

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


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]

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

State condition under which $w$ is:

real;

purely imaginary. [2]


Find the modulus of $z$ given that $\arg w = \dfrac{\pi}{4}$. [2]
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Question 7
[Maximum mark: 22]

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

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

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

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

Hence find the fourth roots of $z$ in modulusargument form. [14]

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