Chapter 5: Complex numbers Test B - Weebly

© John Wiley & Sons Australia, Ltd 1

Chapter 5: Complex numbers Test B Name: _____________________ Simple familiar 1 Using the imaginary number , write the

expression for .

2

2 Determine the real component of .

3

3 If and , write the expression for .

3

4 Draw and on an Argand diagram.

2

i80-

80 80 1

16 5

4 5

i

i

- = ´ -

= ´ ´

=

( )2 57 2 5i i- -( )( ) ( )

( )( )( )

( )

2 5 2 5

4

2

Re 7 2 5 Re 7 4 5

Re 7 4 1 5

Re 7 4 5 1

Re 11 511

i i i i

i i

i

i

- - = - -

= - ´- -

= + - -

= -

=1 4 2z i= - 2 3 5z i= - +

1 2z z( )( )

( )

1 2

2

4 2 3 5

12 20 6 1012 26 10 12 26

z z i i

i i iii

= - - +

= - + + -

= - + - -

= - +4 5z i= - - z

Maths Quest 12 Specialist Mathematics Units 3 & 4 for Queensland Chapter 5: Complex Numbers Test B


5 Express in polar form.

5

6 Express in the

form where .

3

7 Prove the identity using polar

arithmetic.

4

2 3 2z i= - + 1

1

tan

2tan2 3

6

656

yx

q

p

pp

p

-

-

æ ö= ç ÷è øæ ö= ç ÷-è ø

= -

= -

=

( ) ( )2 22 3 2

454cis6

z

z p

= - +

=

æ ö= ç ÷è ø

3 42cis 5 cis4 5p pæ ö æ ö´ -ç ÷ ç ÷

è ø è øcisr q ( , ]q p pÎ -

3 42cis 5 cis4 5

3 42 5cis4 5

2 5cis20

p p

p p

p

æ ö æ ö´ -ç ÷ ç ÷è ø è ø

æ ö= + -ç ÷è øæ ö= -ç ÷è ø

11

2 2

zzz z

=( ) ( )

( ) ( )

( )( )

( )

1 1 1 2 2 2

1 1 2 2

1

2

1 1

2 2

11 2

2

1

2

Let cis and cis

arg and arg

LHS=

ciscis

cis

RHSLHS = RHS

z z z z

z z

zz

zz

zz

zz

q q

q q

qq

q q

= =

\ = =

=

= -

=

=\



8 If and , show that

4

9 Use De Moivre’s theorem to simplify

.

4

6 2v i= + 3w i= -v w v w- = -

( ) ( )

( ) ( )

6 2 , 6 2 , 3 , 3

LHS

6 2 3

3 33 3

RHS6 2 3

3 3LHS = RHS

v i v i w i w i

v w

i i

ii

v wi ii

= + = - = - = +

= -

= + - -

= += -= -

= - - +

= -

3

2

29cis3

4cis4

p

p

æ öæ öç ÷ç ÷è øè ø

æ öæ öç ÷ç ÷è øè ø

33

22

2 29cis 9 cis 33 3

33 4 cis 24cis 44729 3cis 216 2729 cis16 2

p p

pp

pp

p

æ öæ ö æ ö´ç ÷ç ÷ ç ÷è øè ø è ø=æ öæ öæ ö ´ç ÷ç ÷ç ÷ è øè øè øæ ö= -ç ÷è øæ ö= ç ÷è ø



10 Sketch and describe the region of the complex planed defined by

.

The equation represents a circle with a centre at C = (4, –5) and radius, r = 4.

5

11 Do that last question again as an INEQUALITY!

12 Use De Moivre’s theorem to solve

for .

5

{ }: 4 5 4z z i- + =

{ }

( ) ( )

( ) ( )( ) ( )

2 2

2 2

: 4 5 4

Substitute 4 5 4

4 5 4

4 5 4

4 5 16

z z i

z x yix yi i

x y i

x y

x y

- + =

= +

+ - + =

- + + =

- + + =

- + + =

3 125cis3

z pæ ö= ç ÷è ø

z3

3

1

2

3

125cis3

23125cis3 3

25cis9 3

0, 5cis971, 5cis952, 5cis9

z

kz

kz

k z

k z

k z

p

pp

p p

p

p

p

æ ö= ç ÷è øæ öç ÷

= +ç ÷ç ÷è ø

æ ö= +ç ÷è ø

æ ö= = ç ÷è øæ ö= = ç ÷è ø-æ ö= = ç ÷

è ø



13 Solve over by completing the square.

4

2 12 40 0z z- + = C

( )( )( ) ( )( )( )

2

2

2

2 2

2 2

1 2

0 12 400 12 36 36 40

0 6 4

0 6 4

0 6 2

0 6 2 6 2Using the Null factor law

6 2 , 6 2

z zz z

z

z i

z i

z i z i

z i z i

= - +

= - + - +

= - +

= - -

= - -

= - + - -

= - = +


Chapter 5: Complex numbers Test B Name: _____________________ Complex familiar 14 Calculate the values of and that

satisfy the following equation.

.

Equating the real and imaginary components with to create equations [1] and [2]:

Simultaneously solve

The values are, and .

6

15 Solve for if .

6

x y

( )( )6 2 34 42i x yi i- + = +

( )( )( )

26 2 6 6 2 2

6 2 2 6

i x yi x yi xi yi

x y i x y

- + = + - -

= + + - +

34 42i+

6 2 34 [1]2 6 42 [2]x yx y+ =

- + =

1

6 2 342 6 42

6 2 342 6 42

38

xy

xy

xy

-

é ù é ù é ù=ê ú ê ú ê ú-ë û ë û ë û

é ù é ù é ù=ê ú ê ú ê ú-ë û ë û ë û

é ù é ù=ê ú ê ú

ë û ë û

3x = 8y =z 4 23 70 0z z- - =

( )( )

4 2

2

2

2 2

2

1 2 3 4

3 70 0Let

3 70 010 7 0

10 , 710 , 7

10 , 7

10 , 7

10, 10, 7 , 7

z zz a

a aa aa az z

z z

z z i

z z z i z i

- - =

=

- - =

- + =

= = -

= = -

= ± = ± -

= ± = ±

= = - = = -



16 Determine all solutions to the equation, , over .

Express the solutions in polar form.

7

17 If is a zero of , use

polynomial long division to calculate all the roots for .

8

3 8z = - C( )

( )( )

( )( )

( )

3

13

1

1

1

2

2

1

3

Express 8 as cis :

8 8 cis

8cis

8cis 2

22cis3 3

2 00, 2cis3 3

2cis3

2 11, 2cis3 332cis3

2cis

2 22, 2cis3 352cis3

r

z

z k

kz

k z

z

k z

z

z

k z

z

q

p

p

p p

p p

p p

p

p p

p

p

p p

p

-

- = -

=

= +

æ ö= +ç ÷è ø

´ ´æ ö= = +ç ÷è øæ ö= ç ÷è ø

´ ´æ ö= = +ç ÷è øæ ö= ç ÷è ø

=

´ ´æ ö= = +ç ÷è øæ= çè

3 2cis3

z p

ö÷ø

-æ ö= ç ÷è ø

2z =( ) 3 26 9 50P z z z z= + + -

( )P z

2

3 2

3 2

2

2

8 25

2 6 9 50( 2 )

8 9 50( 8 16 )

25 50( 25 50)

0

z z

z z z zz z

z zz z

zz

+ +

- + + -

- - ¯ ¯

+ -

- - ¯

-- -

( )( ) ( )( )( )

2

2

2

2 2

1 2 3

0 8 250 8 16 16 25

0 4 9

0 4 3

0 4 3 4 32 , 4 3 , 4 3

z zz z

z

z i

z i z iz z i z i

= + +

= + + - +

= + +

= + -

= + + + -

= = - - = - +



Chapter 5: Complex numbers Test B Name: _____________________ Complex unfamiliar 18 Determine the four roots of

given,

is a quadratic factor of .

9

( ) 4 3 24 16 3 196 637P z z z z z= + + - -2 4 13z z+ + ( )P z

( ) ( )( ) ( )( ) ( )

( )

( )

2

2

2

2 4 3 2

4 3 2

2

2

4 13 is a factor of .

4 13

4 49

4 13 4 16 3 196 637

4 16 52

49 196 637

49 196 637

0

z z P z

P z z z z z

z

z z z z z z

z z z

z z

z z

a b

+ +

= + + - -

-

+ + + + - -

- + + ¯ ¯

- - -

- - - -

( ) ( )( )( ) ( ) ( )( )( )( )( )( )

( ) ( )( )( )( )( )( )( )( )

2 2

2 22

2

2 2

4 13 4 49

4 4 4 13 2 7

2 9 2 7 2 7

2 3 2 7 2 7

2 3 2 3 2 7 2 7

P z z z z

z z z

z z z

z i z z

z i z i z z

= + + -

= + + - + -

= + + + -

= + - + -

= + + + - + -

( )( )( )( )( )

1 2 3 4

Let 0,

0 2 3 2 3 2 7 2 7Using the Null factor law,

7 7z , z , z 2 3 , z 2 32 2

P z

z i z i z z

i i

=

= + + + - + -

-= = = - - = - +



19 Determine the value of if is

one solution for .

Hence, calculate all possible solutions to the equation.

9 d RÎ 2z i= +2 5 9 dz z

z- = - +

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

2

2

3 2

3 2

3 0 2 1 1 2 0 3

2

Rearrange

5 9

5 9

5 9 0Substitute the first solution

2

2 5 2 9 2 0

1 2 3 2 3 2 1 2

5 4 4 18 9 0

8 12 6 20 20 5 18 9 05 0

5

dz zzdz zz

z z z d

z i

i i i d

i i i i

i i i d

i i i i dd

d

- = - +

- + =

- + - =

= +

+ - + + + - =

+ + +

- + + + + - =

+ - - - - + + + - =- ==

( ) ( )( )( )( ) ( )

( )( )

( ) ( )( )( )( )

3 2

1 2 1

2

3 2 2

1 2 3

5 9 5 02 , 2

2 2 ,

4 5

5 9 5 4 5

Hence,5 5

55

11 2 2

The solutions to 0 are,2 , 2 , 1

z z zz i z z iP z z a z i z i a R

z a z z

z z z z a z z

a

a

aP z z z i z i

P zz i z i z

- + - == + = = -

= - - + - - Î

= - - +

- + - = - - +

- = --

=-

=

= - - + - -

=

= + = - =



20 Show that the complex equation represents a circle.

Determine its centre and radius.

8

21 Given and , calculate the value of the real constant and determine all possible roots.

It is known,

9

2 4 2 42 0zz z z+ + - =

( ) ( ) ( )

2 2

2 2

2 2

2 4 2 42 0Let,

2 4 2 42 0

2 2 4 4

zz z zz x yiz x yizz x y

x y x yi x yi

x y x yi

+ + - == += -

= +

+ + + + - - =

+ + + 4 4x yi+ -

( )( )( )

( )

( )( )

2 2

2 2

2 2

2 2

2 2

2 2

2 2

42 0

2 2 8 42 02 8 2 42

2 4 2 42

2 4 4 2 42 8

2 2 2 505022

2 25

Circle centre 2,0 , radius, 5

x y xx x y

x x y

x x y

x y

x y

x y

r

- =

+ + - =

+ + =

+ + =

+ + + = +

+ + =

+ + =

+ + =

- =

( ) 4 3 2104 576 2448P z z bz z z= + + + +

( )6 0P i =b

( ) 4 3 2104 576 2448P z z bz z z= + + + +

( )6 0P i =

( ) ( )6 is a factor 6 is a factorz i z i- \ +

( ) ( )( )( )( ) ( )( )( )( ) ( )

( )

( )

2

2 2

4 3 2 2

4 3 2

4 3 2

4 3 2

6 6

36

36 36 36

36 36 36Given,

104 576 24481, 16, 68

16 104 576 32

P z z i z i az bz c

P z z az bz c

P z az bz cz az bz c

P z az bz c a z bz c

P z z bz z za b cP z z z z z

= - + + +

= + + +

= + + + + +

= + + + + +

= + + + +

= = =

\ = + + + +

( ) ( )( )( ) ( )( )( )( )

( )

2 2

1 2 3 4

36 16 68

6 6 8 2 8 2

let 0Using the Null factor law,

6 , 6 , 8 2 , 8 2

P z z z z

P z z i z i z i z i

P z

z i z i z i z i

= + + +

= + - + - + +

=

= = - = - + = - -

Chapter 5: Complex numbers Test B - Weebly

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Transcript of Chapter 5: Complex numbers Test B - Weebly