Question 1a

Marks: 4

Consider the complex numbers $z_{1} = \sqrt{3} + 2 i$ and $z_{2} = i - 3 \sqrt{3}$ .

(a)

Find

(i)

u = z_{1} z_{2}

(ii)

v = \frac{z_{1}}{z_{2}}

Key Concepts

Complex Addition, Subtraction & Multiplication
Complex Conjugation & Division

Question 1b

Marks: 3

The complex numbers $u$ and $v$ are represented by the points $A$ and $B$ respectively on an Argand diagram with origin $O$ .

(b)

Determine whether the angle made by

OA

with the positive horizontal axis is greater than or less than the angle made by

OB

with the positive horizontal axis. Give a reason for your answer.

Key Concepts

Modulus & Argument
Argand Diagrams

Question 2a

Marks: 4

Consider the complex number $z = - a + \frac{3}{4} i$ .

(a)

Write down, in terms of

a

,

(i)

Re (z^{2})

(ii)

Im (z^{3})

Key Concepts

Complex Addition, Subtraction & Multiplication
Cartesian Form

Question 2b

Marks: 4

(b)

In the case where

a = 2

, find the modulus and argument of

z^{3}

.

Key Concepts

Modulus & Argument
Cartesian Form

Question 3a

Marks: 3

Consider the complex numbers $z_{1} = i - \frac{1}{2}$ and $z_{2} = \frac{1}{2} - \frac{3}{i}$ .

(a)

Express

z_{2}

in the form

a + b i

, where

a, b \in ℝ

.

Key Concepts

Cartesian Form
Complex Conjugation & Division

Question 3b

Marks: 6

(b)

Find

(i)

{z_{1}}^{*} z_{2}

(ii)

\frac{z_{2}}{z_{1}}

(iii)

|\frac{z_{2}}{z_{1}}|

, giving your answer as an exact value.

Key Concepts

Complex Addition, Subtraction & Multiplication
Complex Conjugation & Division

Question 4

Marks: 6

Consider a general complex number $z = x + i y$ , where $x, y \in ℝ$ , $z \in ℂ$ and $z \neq 0$ .

Show that

(i)

Re (\frac{1}{z} + \frac{1}{z^{*}}) = \frac{2 x}{x^{2} + y^{2}}

(ii)

Im (\frac{1}{z} + \frac{1}{z^{*}}) = 0

(iii)

z z^{*} = {|z|}^{2}

Key Concepts

Cartesian Form
Complex Conjugation & Division

Question 5a

Marks: 4

Consider the equation $z w - w + i z + 1 = 0$ , where $w, z \in ℂ$ , $w = x + i y$ .

(a)

Find an expression in terms of

x

and

y

for

Re (z)

.

Key Concepts

Cartesian Form
Complex Conjugation & Division

Question 5b

Marks: 4

(b)

Find in terms of

x

given that

z

is purely real.

Key Concepts

Complex Conjugation & Division
Solving Equations Analytically

Question 6a

Marks: 5

Consider the complex numbers $z_{1} = \frac{3 - i}{1 - 2 i}$ and $z_{2} = - 3 i + 1$ .

(a)

Find the modulus of

\frac{z_{1}}{{z_{2}}^{*}}

giving your answer as an exact value.

Key Concepts

Modulus & Argument
Complex Conjugation & Division

Question 6b

Marks: 2

(b)

The argument of

\frac{z_{1}}{{z_{2}}^{*}}

is given as

θ = \tan^{- 1} x

, where

0 < θ < 2 π

. Find the value of

x

.

Key Concepts

Modulus & Argument

Question 7a

Marks: 3

Consider the complex numbers $z = \frac{v}{w}, v = 1 - p i and w = 3 i - 2$

(a)

Express

z

in the form

a + b i

, where

a, b, p \in ℝ .

.

Key Concepts

Cartesian Form
Complex Conjugation & Division

Question 7b

Marks: 4

(b)

In the case where

z

is purely imaginary, represent

v, w

and

z

on an Argand diagram.

Key Concepts

Cartesian Form
Argand Diagrams

Question 8a

Marks: 4

Consider the complex numbers $z = \frac{a - 3 i}{2 + i}, w = a + b i$ and $\frac{z}{w} = 1 + 2 i$ where $a, b \in ℝ .$

(a)

Find the values of

a

and

b

.

Key Concepts

Complex Conjugation & Division

Question 8b

Marks: 2

(b)

Find the modulus of

\frac{w}{z}

, giving your answer as an exact value.

Key Concepts

Modulus & Argument

Question 8c

Marks: 2

(c)

Find the argument of

\frac{w}{z}

, giving your answer in the range

- π \leq \arg \frac{w}{z} \leq π

.

Key Concepts

Modulus & Argument

Question 9

Marks: 7

Consider the complex numbers $a - w = 2 z - i$ and $w - 2 z = b i - 1$ .

Find the values of $a$ and $b$ such that $Re (w) = Im (z)$ and $Re (w) = Re (z) + 1$ .

Key Concepts

Cartesian Form
Solving Equations Analytically

Question 10a

Marks: 3

Consider the complex numbers $z_{1} = 5 + p i$ , $z_{2} = a + b i$ and $\frac{z_{1}}{z_{2}} = - 1 + i$ , where $z \in ℂ$
and $a, b \in ℝ$ .

(a)

Find the values of

a

and

b

in terms of

p

.

Key Concepts

Complex Conjugation & Division

Question 10b

Marks: 3

(b)

Given that

|z_{2}| = \sqrt{73}

, find the possible values of

p

.

Key Concepts

Modulus & Argument

Question 10c

Marks: 2

(c)

Given additionally that

\arg (z_{2}) = 2.78

radians correct to 2 decimal places, determine the exact value of

Im (z_{2})

.

Key Concepts

Cartesian Form

DP IB Maths: AA HL

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1.8 Complex Numbers

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