Question 1a

Marks: 3

Consider $w = \frac{z_{1}}{z_{2}}$ , where $z_{1} = 2 + 2 \sqrt{3} i$ and $z_{2} = 2 + 2 i$ .

a)

Express

w

in the form

w = a + b i

.

Key Concepts

Complex Conjugation & Division
Cartesian Form

Question 1b

Marks: 4

b)

Write the complex numbers

z_{1}

and

z_{2}

in the form

r e^{i θ}, r \geq 0, - π < θ < π

.

Key Concepts

Modulus & Argument
Exponential (Euler's) Form

Question 1c

Marks: 3

c)

Express

w

in the form

r e^{i θ}, r \geq 0, - π < θ < π .

Key Concepts

Exponential (Euler's) Form

Question 2

Marks: 6

Solve the equation $z^{3} = 27 i$ , giving your answers in the form $a + b i$ .

Key Concepts

Modulus-Argument (Polar) Form
Roots of Complex Numbers

Question 3a

Marks: 4

Let $z_{1} = 6 cis (\frac{π}{6})$ and $z_{2} = 3 \sqrt{2} e^{i (\frac{π}{4})} .$

a)

Giving your answers in the form

r cis θ,

find

(i)

z_{1} z_{2}

(ii)

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

Key Concepts

Modulus-Argument (Polar) Form
Exponential (Euler's) Form

Question 3b

Marks: 2

b)

Write

z_{1}

and

z_{2}

in the form

a + b i .

Key Concepts

Conversion of Forms

Question 3c

Marks: 2

c)

Find

z_{1} + z_{2},

giving your answer in the form

a + b i .

Key Concepts

Modulus-Argument (Polar) Form
Conversion of Forms

Question 3d

Marks: 2

It is given that $z_{1}^{*}$ and $z_{2}^{*}$ are the complex conjugates of $z_{1}$ and $z_{2}$ respectively.

d)

Find

z_{1}^{*} + z_{2}^{*},

giving your answer in the form

a + b i .

Key Concepts

Complex Conjugation & Division
Complex Addition, Subtraction & Multiplication

Question 4a

Marks: 2

Let $z_{1} = 2 cis (\frac{π}{3})$ and $z_{2} = 2 + 2 i$ .

a)

Express

(i)

z_{1}

in the form

a + b i

(ii)

z_{2}

in the form

r cis θ

Key Concepts

Conversion of Forms

Question 4b

Marks: 2

b)

Find

w_{1} = z_{1} + z_{2},

giving your answer in the form

a + b i

.

Key Concepts

Complex Addition, Subtraction & Multiplication

Question 4c

Marks: 3

c)

Find

w_{2} = z_{1} z_{2},

giving your answer in the form

r cis θ .

Key Concepts

Modulus-Argument (Polar) Form

Question 4d

Marks: 2

d)

Sketch

w_{1}

and

w_{2}

on a single Argand diagram.

Key Concepts

Argand Diagrams

Question 5a

Marks: 3

It is given that that $z_{1} = 2 e^{i (\frac{π}{3})}$ and $z_{2} = 3 cis (\frac{nπ}{12}), n \in ℤ^{+} .$

a)

Find the value of

z_{1} z_{2}

for

n = 3 .

Key Concepts

Modulus-Argument (Polar) Form

Question 5b

Marks: 3

b)

Find the least value of

n

such that

z_{1} z_{2} \in ℝ^{+} .

Key Concepts

Modulus-Argument (Polar) Form
Argand Diagrams

Question 6a

Marks: 5

Consider the complex number $w = \frac{z_{1}}{z_{2}}$ where $z_{1} = 3 - \sqrt{3} i$ and $z_{2} = 2 c i s (\frac{2 π}{3}) .$

a)

Express

w

in the form

r c i s θ .

Key Concepts

Conversion of Forms
Modulus-Argument (Polar) Form

Question 6b

Marks: 3

b)

Sketch

z_{1}, z_{2}

and

w

on the Argand diagram below.

q6b_1-9_ib-maths-aa-hl

Key Concepts

Argand Diagrams

Question 6c

Marks: 2

c)

Find the smallest positive integer value of

n

such that

w^{n}

is a real number.

Key Concepts

Modulus-Argument (Polar) Form
Argand Diagrams

Question 7a

Marks: 4

Consider the complex number $z = - 1 + \sqrt{3} i$ .

(a)

Express

z

in the form $r cis θ$ , where

r > 0

and

- π < θ \leq π

.

Key Concepts

Modulus & Argument
Modulus-Argument (Polar) Form

Question 7b

Marks: 4

(b)

Find the three roots of the equation

z^{3} = - 1 + \sqrt{3} i

, expressing your answers in the form

r cis θ

, where

r > 0

and

- π < θ \leq π

.

Key Concepts

Roots of Complex Numbers
De Moivre's Theorem

Question 8a

Marks: 4

Consider the equation $z^{4} - 1 = 15$ , where $z \in ℂ$ .

(a)

Find the four distinct roots of the equation, giving your answers in the form

a + b i

, where

a, b \in ℝ .

Key Concepts

Complex Roots of Polynomials
Cartesian Form

Question 8b

Marks: 2

(b)

Represent the roots found in part (a) on the Argand diagram below.

Key Concepts

Argand Diagrams

Question 8c

Marks: 2

(c)

Find the area of the polygon whose vertices are represented by the four roots on the Argand diagram.

Key Concepts

Argand Diagrams

Question 9a

Marks: 4

Consider the complex numbers $w = 3 (\cos \frac{π}{3} - isin \frac{π}{3})$ and $z = 3 - \sqrt{3} i$ .

(a)

Write

w

and

z

in the form

r cis θ

, where

r > 0

and

- π < θ \leq π

.

Key Concepts

Modulus-Argument (Polar) Form
Modulus & Argument

Question 9b

Marks: 2

(b)

Find the modulus and argument of

z w

.

Key Concepts

Modulus-Argument (Polar) Form

Question 9c

Marks: 2

(c)

Write down the value of

z w

.

Key Concepts

Modulus-Argument (Polar) Form

Question 10a

Marks: 2

Let $z = 12 + 16 i$ , where $a, b \in ℝ$ .

a)

Verify that

4 + 2 i

and

- 4 - 2 i

are the second roots of

z

.

Question 10b

Marks: 4

b)

Hence, or otherwise, find two distinct roots of the equation

w^{2} + 4 w + (1 - 4 i) = 0

, where

w \in ℂ

. Give your answer in the form

a + b i

, where

a, b \in ℝ

.

Question 11a

Marks: 1

The complex numbers $ω_{1} = 3$ and $ω_{2} = 2 - 2 i$ are roots of the cubic equation $ω^{3} + p ω^{2} + q ω + r = 0,$ where $p, q, r \in ℝ .$

a)

Write down the third root,

w_{3}

, of the equation.

Question 11b

Marks: 4

b)

Find the values of

p, q

and

r

.

Question 11c

Marks: 4

c)

Express

w_{1},

w_{2}

and

w_{3}

in the form

r c i s θ

.

DP IB Maths: AA HL

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1.9 Further Complex Numbers

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