Maths Quest Maths C Year 12 for Queensland 2e 1

WorkSHEET 1.1 Complex numbers Name: ___________________________ 1 Using the imaginary number i, write down

expressions for: (a) (b)

2 Simplify in the form of .

3 (a) Represent on an Argand diagram.

(b) Evaluate

(a)

*** Note: there should be a line from the Origin to the “dots”. ***

81-

5-

ii

98181 (a) 2

±==-

i

i

5

55 (b) 2

±=

=-

2352 7653 +-+- iiiiiyx + ( ) ( ) ( ) ( )( )

( ) ( )( ) ( )( )

3 5 6 7

2 4 4 2 4 2

2 5 3 2

2 5 3 2

1 2 1 5 1 1 3 1 1 22 5 3 2

3

i i i i

i i i i i i i i i

i i ii i i

- + - +

= - + - +

= - - + - - - +

= - - - + += -

iwiz 34 and 32 +=-=

.34 wz - ( ) ( )

iii

iiwz

214912128

34332434 (b)

--=---=

+--=-

Maths Quest Maths C Year 12 for Queensland 2e 2

4 For determine: (a)

(b) (c) Im

(a)

(b)

(c)

,23 and 21,35 iuiwiz -=-=+=

zwu

( )2Re z

( ) ( )wzuz 2Re-

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

iiii

iiiiii

iiizwu

431914212233

237112363105

232135

2

2

-=+--=

--=--+-=

--+=

( )

( ) 16Re3016

9302535

2

2

22

=

+=++=

+=

zi

iiiz

( )( )

( )( )

( ) ( )77761ReIm276

60303216213016

21610915

3523

2

2

2

2

-=--=-

-=-+-=

-+=

-=--+=

+-=

wzuzi

iiiiiwz

iiii

iiuz

Maths Quest Maths C Year 12 for Queensland 2e 3

5 Find the values of x and y if:

** AS an alternative method, simply divide both sides by (2+3i) and then rationalise the right hand side of the equation.

6 Express

( )( ) iyixi 22332 +=++( )( )

( ) ( ) iiyxyxiyiixiyxiiyxi

2232332223332222332

2

+=++-+=+++

+=++

[ ] [ ][ ] [ ][ ] [ ]

[ ]

2 3 23 [1]3 2 2 [2]

1 2 : 4 6 46 [3]

2 3 : 9 6 6 [4]

3 4 : 13 524

Substitute 4 into [2]3 4 2 2

12 2 22 10

5Solution 4, 5

x yx y

x y

x y

xxxyyyyx y

- =+ =

´ - =

´ + =

+ =

==

+ =

+ == -= -= = -

. form in the 2334 yixii

+-+

i

ii

iiiii

ii

ii

1317

13613176

4969812

2323

2334

2334

2

2

+=

+=

-+++

=

++

´-+

=-+

Maths Quest Maths C Year 12 for Queensland 2e 4

7 Write

8 Find the modulus and argument of

. form in the 3235

23 yix

ii

ii

+--

-++

( ) ( )

i

i

ii

ii

iii

iiii

iiiii

ii

ii

ii

ii

ii

6558

6546558465

4595139165

919571313919

57

94961510

4236

3232

3235

22

23

3235

23

2

2

2

2

--=

--=

---=

+--=

+-

-=

---+

--

-+-=

++

´--

---

´++

=

--

-++

.77 iz -= ( )

( )

4Argument

27 Modulus4

4

quadrant)(4th 1tan177tan

27249

984949

77

1

22

p

p

pq

qq

q

-=

=

-=

-=

-=

-=

-=

=´=

=+=

-+=

-

z

Maths Quest Maths C Year 12 for Queensland 2e 5

9 Express

10 Express in cartesian form.

form.polar in 721 iz -=

( ) ( )

÷øö

çèæ -=

-=

-=

÷÷ø

öççè

æ-=

-=

-=

==

+=

-+=

-=

-

6cis72

6

6

quadrant)(4th 3

1tan

3121

7tan

7228

721721

721

1

22

p

p

pq

qq

q

z

r

iz

45cis2 p

i

i

i

i

--=

÷ø

öçè

æ--=

÷øö

çèæ --=

÷øö

çèæ +=

121

212

4sin

4cos2

45sin

45cos2

45cis2

pp

pp

p

Maths Quest Maths C Year 12 for Queensland 2e 6

11 Determine𝑧!"where𝑧 = −3 + 3𝑖.

𝑧!" =1𝑧

so we need to rationalize the denominator of;

1−3 + 3𝑖

=1

−3 + 3𝑖 ×−3 − 3𝑖−3 − 3𝑖

=−3 − 3𝑖9 − 9𝑖#

=−3 − 3𝑖9 + 9

∴ 𝑧!" =−318 −

3𝑖18

12 If𝛼 = $%,showthat&'$(

(= 𝛼(𝜆 − 𝑖). Take LHS;

𝐿𝐻𝑆 =𝛼 + 𝛽𝑖𝑖

rationalise the denominator

=𝛼 + 𝛽𝑖𝑖 ×

𝑖𝑖

=𝛼𝑖 + 𝛽𝑖#

𝑖#

=𝛼𝑖 − 𝛽−1

= −𝛼𝑖 + 𝛽

= 𝛽 − 𝛼𝑖

since 𝛼 = $

%, then 𝛽 = 𝛼𝜆

and by substitution we have;

= 𝛼𝜆 − 𝛼𝑖

= 𝛼(𝜆 − 𝑖)

𝑄𝐸𝐷

Maths Quest Maths C Year 12 for Queensland 2e 7

13 Calculate𝑧),where𝑧 = √3 + 1𝑖. Convert to Polar à use diagram or rules à 𝑧 = 2 cis *

+.

By DeMoivre;

𝑧, = 𝑟, cis 𝑛𝜃 so,

𝑧) = 2) cis 5 ×𝜋6

= 32 cis5𝜋6

= 32 cos5𝜋6 + 32 sin

5𝜋6 𝑖

= 32 × −√32 + 32 ×

12 𝑖

= −16√3 + 16𝑖

*** You can check this using your calculator!!

14 Find𝑧,suchthat;

𝑅𝑒(𝑧) + 𝐼𝑚(𝑧) = 9and

𝑅𝑒(𝑧)𝐼𝑚(𝑧) = 20

Set 𝑧 = 𝑎 + 𝑏𝑖

Clearly, 𝑎 + 𝑏 = 9

and 𝑎𝑏 = 20

Simultaneously,

𝑏 = 9 − 𝑎 and

𝑎(9 − 𝑎) = 20

𝑎# − 9𝑥 + 20 = 0

(𝑎 − 5)(𝑎 − 4) = 0

𝑎 = 4, 𝑜𝑟5 When,

𝑎 = 4 → 𝑏 = 5 and when,

𝑎 = 5 → 𝑏 = 4 Hence;

𝑧 = 4 + 5𝑖 or

𝑧 = 5 + 4𝑖

Maths Quest Maths C Year 12 for Queensland 2e 8

15 Directly from the Syllabus document: Prove and use DeMoivre’s theorem for integral powers. OK, so the proof is NOT in my exam … on page 279 of the Textbook there is a mathematical Induction proof of DeMoivre … but in the Induction section of the Syllabus it only requires Induction to be used with integer proofs, so I don’t think this is needed. If we want to prepare for a proof question in the external exam, I will run through this proof for you … https://www.youtube.com/watch?v=NjYZS_XYlEQ To Use DeMoivre’s theorem, that’s simply raising complex numbers to powers, multiplying complex numbers in polar form and rooting complex numbers. One last possibility is to use DeMoivre for Proofs. Yes, that is fair game in my exam … and below are two examples of proofs requiring DeMoivre. (Proofs are about juggling as you are already given the answer, you just need to get there!

16 UseDeMoivretoprove;

cos 2𝜃 = cos# 𝜃 − sin# 𝜃**Yesit’sinyourformulasheet,butwhyisitarule…well,here‘stheproofitworks?

Too wide … see below

Maths Quest Maths C Year 12 for Queensland 2e 9

17 De Moivre states;

(𝑟 cis 𝜃), = 𝑟, cis 𝑛𝜃 The proof … Consider;

(cis 𝜃)# = (cos 𝜃 + 𝑖 sin 𝜃)#

= cos# 𝜃 + 2 cos 𝜃 sin 𝜃 𝑖 − sin# 𝜃

= cos# 𝜃 − sin# 𝜃 + 2 sin 𝜃 cos 𝜃 𝑖 By DeMoivre;

(cis 𝜃)# = cis 2𝜃

= cos 2𝜃 + 𝑖 sin 2𝜃 Hence we can say;

𝐜𝐨𝐬 𝟐𝜽 + (sin 2𝜃)𝑖 = 𝐜𝐨𝐬𝟐 𝜽 − 𝐬𝐢𝐧𝟐 𝜽 + (2 sin 𝜃 cos 𝜃)𝑖 By equating Real components we have;

cos 2𝜃 = cos# 𝜃 − sin# 𝜃

𝑄𝐸𝐷 ** you can see the same working can prove … sin 2𝜃 = 2 sin 𝜃 cos 𝜃

18 UseDeMoivretoprove;

sin 3𝑥 = 3 sin 𝑥 − 4 sin. 𝑥**Yesit’sinyourformulasheet,butwhyisitarule…well,here‘stheproofitworks?

Too wide … see below

Maths Quest Maths C Year 12 for Queensland 2e 10

19 De Moivre states;

(𝑟 cis 𝜃), = 𝑟, cis 𝑛𝜃 The proof … Consider

(cis 𝑥). = (cos 𝑥 + 𝑖 sin 𝑥).

= cos. 𝑥 + 3 cos# 𝑥 𝑖 sin 𝑥 + 3 cos 𝑥 𝑖# sin# 𝑥 +𝑖. sin. 𝜃

= cos. 𝑥 − 3 cos 𝑥 sin# 𝑥 + (3 cos# 𝑥 sin 𝑥 − sin. 𝜃)𝑖 ** you can see we aren’t quite there yet, so a bit more juggling before heading to the second part of the proof ** Use the Pythagorean identity: sin# 𝑥 + cos# 𝑥 = 1

= cos. 𝑥 − 3 cos 𝑥 sin# 𝑥 + (3(1 − sin# 𝑥)sin 𝑥 − sin. 𝑥)𝑖

= cos. 𝑥 − 3 cos 𝑥 sin# 𝑥 +(3 sin 𝑥 − 3 sin. 𝑥 −sin. 𝑥)𝑖

= cos. 𝑥 − 3 cos 𝑥 sin# 𝑥 + (3 sin 𝑥 − 4 sin. 𝑥)𝑖 By DeMoivre;

(cis 𝑥). = cis 3𝑥

= cos 3𝑥 + 𝑖 sin 3𝑥 Hence we can say;

cos 3𝑥 + (𝐬𝐢𝐧𝟑𝒙)𝑖 = cos. 𝑥 − 3 cos 𝑥 sin# 𝑥 + (𝟑 𝐬𝐢𝐧 𝒙 − 𝟒 𝐬𝐢𝐧𝟑 𝒙)𝑖 By equating Imaginary components we have;

sin 3𝑥 = 3 sin 𝑥 − 4 sin. 𝑥

𝑄𝐸𝐷 ** you can see the same working can prove … cos 3𝜃 = 4 cos. 𝜃 − 3 cos 𝜃

20 CAUTION with Proofs … you can’t fool a confirmer. Because the answer is given to you, you need FULL working, with FULL justification. If you skip steps, it will look like you are hiding the fact you don’t know the steps in the proof. There really aren’t that many lines above, so please do them all

Maths Quest Maths C Year 12 for Queensland 2e 11

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