Complex PDF

MATHEMATICS (EXTENSION 2)
Notes and exercises for Topic 1:

With references to
Complex numbers
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Initial version by H. Lam, August 2012
Various corrections by students & members of the Department of Mathematics.
Gentle reminder
For a thorough understanding of the topic, every blank space/example
question in this handout is to be completed!
Additional questions from the selected texts will be completed at the
discretion of your teacher.
Remember to copy the question into your exercise book!
Contents
1 A new number system 5

1.1 Review of number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The imaginary numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Basic operations with complex numbers . . . . . . . . . . . . . . . . . . . . . 9
1.4.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.3 Complex conjugate pairs . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Properties of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5.1 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5.2 Solutions to equations . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Further arithmetic & algebra of complex numbers 14

2.1 Vector representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.4 Scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Modulus/argument of a complex number . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Natural ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 (Principal) Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.4 Triangle inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Mod-arg & polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Application of De Moivres Theorem 31

3.1 Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Locus problems 37
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Past HSC questions 45

5.1 2001 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 2002 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 2003 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 2004 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3
4 Contents
5.5 2005 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.6 2006 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.7 2007 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.8 2008 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.9 2009 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.10 2010 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.11 2011 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.12 2012 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
References 60
NORTH SYDNEY BOYS HIGH SCHOOL

Section 1
A new number system
1.1 Review of number systems
. . . . . . . . . . . . . . . . . . numbers. N = {1, 2, 3 }
Example 1
Solve x + 1 = 5 and x + 3 = 0 over N. Answer: x = 4, no solution
. . . . . . . . . . . . . . . . . . . Z = { , 3, 2, 1, 0, 1, 2, 3, }
Example 2
Solve x + 3 = 0 and 2x + 4 = 7 over Z. Answer: x = 3, no solution

p
. . . . . . . . . . . . . . . . . . . . numbers. Q = : p, q Z, q 6= 0
q
Example 3
Solve 2x + 4 = 7 and x2 2 = 0 over Q. Answer: x = 76 , no solution
. . . . . . . . . . numbers. R
Example 4

Solve x2 2 = 0 and x2 + 5 = 0 over R. Answer: x = 2, no solution
NZQR
5
6 A new number system Rotation

5 4
log3 8
11 4 R
e7 23 7
Q
9 3
1.1 Z 5 2
8 10
2 1
5
3 N 6
2 7 75
5.45 4 5 4 9
2
6 5 73

1+ 50.4453
6 16
2 1
2
cosh 7
sin 1.25
1.2 Rotation
From x = 1, go to x = 1 by rotating radians in the usual direction.
Multiply 1 by 1 to obtain 1 corresponds to rotating by . . radians.

Stop halfway whilst rotating? Quarter of way whilst rotating?
2 1 0 1 2

A new number system The imaginary numbers 7
1.3 The imaginary numbers

Definition 1
Imaginary number The imaginary number i to be the quantity to multiply with
a real number when rotating anti-clockwise by 2 about x = 0.
Jump off the real number line.
Definition 2
The imaginary number i has property such that
i i = i2 = 1
Why?
Definition 3
The set of all imaginary numbers, called the complex numbers, is defined to be
C = {z : z = x + iy; x, y R}
Example 5
Find the values of i2 , i3 , i4 and i5 .
i2 = . . . . . . . . . . i3 = . . . . . . . . . . i4 = . . . . . . . . . . i5 = . . . . . . . . . .

8 A new number system The imaginary numbers
Definition 4
Complex number A complex number z has real and imaginary parts and is defined
by z = x + iy.
The real part of z: Re(z) = x. The imaginary part of z: Im(z) = y.
Treat real and imaginary parts as . . . . . . . . . . . . . . . . . . . . . . . . . . . . of a complex number.
z = x + iy is known as . . . . . . . . . . . . . . . . . . . . . . form.
Plot on A . . . . . . . . . . . . . d . . . . . . . . . . . . . . . . , similar to plotting points coordinate geometry.
Example 6
On the following diagram, plot the location of:
z1 = 3 + 4i. z2 = 2 i. z3 = 1 3i. z4 = 21 + 32 i.
Im
4
Re
4 3 2 1 1 2 3 4
1

A new number system Basic operations with complex numbers 9
1.4 Basic operations with complex numbers

Example 7
Find the value of

1. 2+ 3 54 3 2. 2+ 3 54 3
1.4.1 Addition
Operations similar to surds (group rational parts with rational parts, irrational parts
with irrational parts).
Group . . . . . . . . parts with . . . . . . . . parts
Group . . . . . . . . . . . . . . . . . . . . . . . parts with . . . . . . . . . . . . . . . . . . . . . . . parts.
1.4.2 Multiplication
Use distributive law.
Beware that i2 = . . . . , which becomes . . . . . . . .
Example 8
If z1 = 2 + 3i and z2 = 1 + 5i, find the value of
(a) z1 + z2 (b) z1 z2 (c) 3z1 (d) 3iz1 (e) z1 z2
Example 9
Find z C such that Re(z) = 2 and z 2 is imaginary.

10 A new number system Basic operations with complex numbers
1.4.3 Complex conjugate pairs

Definition 5
If z = x + iy, then its complex conjugate is denoted z such that
z = x iy

Analogous to conjugate surds, where the conjugate of a + b c is . . . . . . . . . . . . . . . .
Geometrically,
Im
b
z = x + iy
Re
Example 10
If z1 = 2 + i and z2 = 1 3i, evaluate in Cartesian form:
(a) z1 + z2 (c) z2 (e) z1 z2

1
(b) z2 z1 (d) z1 (f)
z2

A new number system Basic operations with complex numbers 11
Example 11
Find the square roots of 3 + 4i in Cartesian form.
Further exercises
Lee (2006, Ex 2.3)

12 A new number system Properties of complex numbers
1.5 Properties of complex numbers

1.5.1 Equality
Definition 6
Two complex numbers z1 and z2 are equal iff the real and imaginary parts are equal.
Proof
Let z1 = a + ib, z2 = c + id
If z1 = z2 , then . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , or
....................................................
1.5.2 Solutions to equations

Example 12
Solve z 2 + 1 = 0 for z C. Answer: z = i
Example 13
Solve z 2 + 2z + 10 = 0 for z C. Answer: z = 1 3i

A new number system Properties of complex numbers 13
Example 14
Solve 2z 2 + (1 i)z + (1 i) = 0 for z C. Answer: z = i, z = 12 12 i
Observations
Equations with . . . . . . . . coefficients will have . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

roots.
Equations with . . . . . . . . . . . . . . . . . . . coefficients do not necessarily have . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . roots.
History
Girolamo Cardano (
Further exercises
Fitzpatrick (1991, Ex 31(a), (b), (c))
Lee (2006, Ex 2.1, 2.2)
Arnold and Arnold (2000, Ex 2.1)

Section 2
Further arithmetic & algebra of complex

numbers
2.1 Vector representation

2.1.1 Equivalence
Definition 7
Two vectors p and q on the Argand diagram are equal iff both
Modulus ( . . . . . . . . . . . . . . . . . . . . . . . . ), and
Argument ( . . . . . . . . . . . . . . . . . . . . )
are equal.
The starting point ( . . . . . . . ) is irrelevant for a vector.
14
Further arithmetic & algebra of complex numbers Vector representation 15
2.1.2 Addition
Place vectors, head-to-tail.
Im
p q
p+q
Re
O
q
Parallelogram of vectors when adding two vectors.
Example 15
If z = 3 + 2i and w = 2 + 4i, draw z + w on the Argand diagram.
Im Im
6 6
4 4
2 2
Re Re
3 2 1 1 2 3 3 2 1 1 2 3
2 2
2.1.3 Subtraction
For z1 z2 , add z2 to z1 .
Im
p + (q)
p pq
Re
O p+q
q

16 Further arithmetic & algebra of complex numbers Vector representation
Alternatively, what vector get you to z1 from z2 ?

Quick & easy parallelogram of vectors:
1. z1 + z2 starts from tails of z1 & z2 ,
2. z2 z1 starts from head of z1 , goes to head of z2 .
Example 16
If z = 3 2i and w = 2 5i, draw z w on the Argand diagram
Im Im
3 3
2 2
1 1
Re Re
11 1 2 3 11 1 2 3
2 2
3 3
4 4
5 5
2.1.4 Scalar multiplication

For kz1 where k R, stretch z1 by factor of k.
If k < 0, direction of new vector is opposite to original vector.
Example 17
If z = 1 + 2i, draw 3z and 2z on separate Argand diagrams.
Im Im
6 6
5 5
4 4
3 3
2 b
z 2 b
z
1 1
Re Re
3 2 1 1 2 3 4 3 2 1 1 2 3 4
2 2
3 3
4 4

Further arithmetic & algebra of complex numbers Vector representation 17
Example 18

[2011 HSC Q2] On the Argand diagram, the complex numbers 0, 1 + i 3, 3 + i,
and z form a rhombus. Im
z b

1+i 3
b
3+i
b
Re
O
(i) Find z in the form a + ib, where a and b are real numbers.
(ii) An interior angle, , of the rhombus is marked on the diagram. Find the value
of .
5
Answer: z = ( 3 + 1) + i( 3 + 1), = 6

18 Further arithmetic & algebra of complex numbers Modulus/argument of a complex number
2.2 Modulus/argument of a complex number

2.2.1 Natural ordering
N, Z, Q, R are well ordered :
22 5 5
6 ... 1 7
... 4
... 7
However:
6 + 4i . . . . . . 3 + 2i 3 3i . . . . . . 2 + i
Natural ordering does not exist with complex numbers.
2.2.2 Modulus
Definition 8
The modulus of a complex number, denoted |z| (where z = x + iy) is the magnitude
( . . . . . . . . . . . . . . ) of the vector from O to z on the Argand diagram.
Im
z = x + iy
b
|
|z
Re
x
i.e. p
|z| = x2 + y 2

Further arithmetic & algebra of complex numbers Modulus/argument of a complex number 19
2.2.3 (Principal) Argument

Definition 9
The argument of a complex number, measured in . . . . . . . . . . . . . . . . . , is denoted
arg(z)
(where z = x + iy) is the ( . . . . . . . . . . . . . . . . . . . . ) that the vector from O to z makes

with the positive real axis on the Argand diagram, with angles increasing in the
anticlockwise direction.
Im
z = x + iy
b
arg(z)
Re
i.e. y
arg(z) = tan1
x
Duplicate argument(s)?
Example 19
Evaluate arg(z), where z = 1 + i.
Definition 10
The principal argument of a complex number, denoted
Arg(z)
lies within the domain < Arg(z) .
Remarks
The principal argument is generally quoted henceforth.

y
Be aware of the quadrant which z lies. Inputting tan1 x
on the calculator may give
an erroneous result.
The complex number z = 0 + 0i has no argument defined.

Example 20
Find the modulus and principal argument of the following:

(a) 2 + 2i Answer: modulus: 2 2, argument
4

(b) 1 i 3 Answer: modulus: 2, argument 2
3
Example 21
[2011 HSC Q2] Let w = 2 3i and z = 3 + 4i.
(a) Find w + z. Answer: 5 + i

(b) Find |w|. Answer: 13
w
(c) Express in the form a + ib, where a, b R. 6
Answer: 25 17
25
i
z

Further arithmetic & algebra of complex numbers Modulus/argument of a complex number 21
2.2.4 Triangle inequality

Definition 11
For every complex number z1 and z2 ,
|z1 + z2 | |z1 | + |z2 |
1
Proof
Let p and q (with P and Q being the head of the arrow) represent the complex numbers
z1 and z2 respectively, p + q with R being the head of the arrow.
On the Argand diagram:
Im
Re
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iff O, P and Q are collinear (which implies

OP k OQ k OR)
Conclusion: z1 = kz2 , where k R as vectors are parallel.
Otherwise, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hence, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Never attempt to prove this algebraically!

Example 22
If z1 = 3 + 4i and |z2 | = 13, find the greatest value of |z1 + z2 |. If |z1 + z2 | is at its
greatest value, find the value of z2 in Cartesian form.
39 52
Answer: |z1 + z2 | = 18 at its greatest; z2 = 5
+ 5
i
Further exercises
Lee (2006, Ex 2.6) Arnold and Arnold (2000, Ex 2.3)
Patel (2004, Ex 4K)

Further arithmetic & algebra of complex numbers Mod-arg & polar form 23
2.3 Modulus-argument & polar form

Definition 12
The modulus-argument form of a complex number z is
Im
z = x + iy
b
z = x + iy (Cartesian form)
y = |z| sin
|
|z
= |z| cos + i |z| sin (Mod-arg form)
= |z| (cos + i sin )
= r (cos + i sin )
Re
where Arg(z) = . x = |z| cos
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . often abbreviated2 to z = r cis .
Better to abbreviate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . to z = rei (for reasons that

will be made obvious later)
Definition 13
Polar form: Eulers formula
ei = cos + i sin
where e 2.71828
(Leonhard Euler, 1707-1783.

http://en.wikipedia.org/wiki/Leonhard Euler)
(All index laws in R also apply to the complex exponential).
2
cis does very little to assist your understanding of the rules for multiplying complex numbers!

24 Further arithmetic & algebra of complex numbers Mod-arg & polar form
Example 23

Write z = 2 cos 3
4
+ i sin 3
4
in Cartesian form. Answer: z = 1 + i
Example 24
3
Write z = 2 2i in polar form. Answer: z = 2 2ei 4
Example 25
10
Write z = in polar form, and hence write in simplest Cartesian form.
3+i
i

5 3
Answer: z = 5e 6 = 2
52 i
Example 26
2i
Write z = 6e 3 in Cartesian form. Answer: 3 3i 3

Example 27

Evaluate the product (1 + i) 1 i 3 in Cartesian form and polar form, to show

1+ 3
that cos = .
12 2 2
Further exercises
Patel (2004, Ex 4C, Q1-10) Arnold and Arnold (2000, Ex 2.2,
Q1-4)

2.3.1 Properties
Multiplication of z1 and z2 : moduli . . . . . . . . . . . . . . . . . . . , arguments . . . . . . . . , i.e.
z1 z2 = r1 r2 (cos (1 + 2 ) + i sin (1 + 2 )) = r1 r2 ei(1 +2 )
Proof Let z1 = r1 (cos 1 + i sin 1 ) and z2 = r2 (cos 2 + i sin 2 )
z1 z2 =
Proof (via index laws and complex exponential) Let z1 = r1 ei1 and z2 = r2 ei2 .
z1 z2 =
Conjugates: if z = r (cos + i sin ), then
z = r (cos i sin ) = rei
Proof Let z = r (cos + i sin ).
Further exercises
Patel (2004, Ex 4C, Q11 onwards)
Lee (2006, Ex 2.5)

Powers:
Definition 14
De Moivres Theorem
(cos + i sin )n = cos(n) + i sin(n)
(Abraham De Moivre, 16671754.

http://en.wikipedia.org/wiki/Abraham de Moivre)
Proof (via complex exponential)
Proof (by induction for later in the course)

Example 28
Simplify the following, expressing the answer in polar form:
3
(a) cos 3 + i sin 3 . Answer: 1
3
(b) 2 cos 3 4
+ i sin 3
4
. Answer: 8ei 4

8
(c) 2 cos 6 i sin 6 . Answer: 16ei
2
3
1 4 1 5
(d) 2
cos 3
5
+ i sin 35
3 cos 8 i sin 8 Answer: 243 i 39
16
e 40

Example 29
If |z1 | = 3, Arg(z1 ) = 2, |z2 | = 2 and Arg(z2 ) = 3, find the modulus and
2z1 2
argument of . 9
Answer: |z| = 20 , Arg(z) = 2 5
5z2 3
Example 30
[1988 4U HSC Q4(a)]

(a) Express z = 2 i 2 in modulus-argument form.

Answer: z = 2 cos 4
i sin 4
(b) Hence write z 22 in the form a + ib, a, b R. Answer: z 22 = 222 i

Example 31

(a) If z1 = 1 + i and z2 = 3 i, find the moduli and principal arguments of z1 ,
z1
z2 and .
z1
Answer: z1 = 2 exp i , z2 = 2 exp i , z = 1 exp 5i

4 6 12
.
z2 2 2
1+i
(b) If z = , find the smallest positive integer n such that z n is real, and
3i
evaluate z n for this integer n. 1
Answer: n = 12, z 12 = 64
Further exercises
Patel (2004, Ex 4D)
Lee (2006, Ex 2.9)
Arnold and Arnold (2000, Ex 2.2, Q6 onwards)

Section 3
Application of De Moivres Theorem
3.1 Roots of complex numbers
To find n-th roots of complex numbers, use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and polar form.
Example 32
Find the cube roots of unity, i.e. solve z 3 = 1.
Solution (via De Moivres Theorem)

3 roots
z }| {
z 3 = 1 (cos 2k + i sin 2k), where k = 0, 1, 2.
1 1
Hence, z = 1 3 (cos 2k + i sin 2k) 3 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fix up out of range arguments (change to principal argument):
Solution (via polar form)
31
32 Application of De Moivres Theorem Roots of complex numbers
Example 33
[2011 HSC Q2] Find, in modulus-argument form, all solutions of z 3 = 8.
2 2

Answer: z = 2, 2 cos 3
i sin 3
Example 34

Solve for z: z 4 = 8 8 3i, and plot the solutions on the Argand diagram.
5 2
Answer: z = 2ei 3 , 2ei 6 , 2ei 3 , 2ei 6

Application of De Moivres Theorem Roots of complex numbers 33
Example 35

Find the fourth roots of z = 1 + i 3 in modulus-argument form.
1 1 1 1
7 7 11 11 5 5

Answer: 2 4 cos 12
+ i sin 12
, 2 4 cos 12
+ i sin 12
, 2 4 cos 12
i sin 12
, 2 4 cos 12
i sin 12

34 Application of De Moivres Theorem Roots of complex numbers
Example 36
(a) Find the five fifth roots of unity and plot them on the unit circle.
(b) If is a non-real fifth root of unity, show that 1 + + 2 + 3 + 4 = 0.
(c) Hence or otherwise, factorise z 5 1 completely over R.
2 4
Answer: (z 1) z 2 2z cos + 1 z 2 2z cos

5 5
+1
Further exercises
Patel (2004, Ex 4D, 4E, 4I)
Lee (2006, Ex 2.10)

Application of De Moivres Theorem Trigonometric identities 35
3.2 Trigonometric identities

Example 37
(a) [2003 Ext 2 HSC Q2(d)] Using De Moivres theorem, find an expression for
cos 5 in terms of cos .
Answer: cos 5 = 16 cos5 20 cos3 + 5 cos
(b) Hence solve 16x4 20x2 + 5 = 0 for x. Answer: x = cos

10
, cos 3
10
, cos 7
10
, cos 9
10
.
Solution
(a) By De Moivres Theorem, cos 5 + i sin 5 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Expand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . via binomial theorem:
Equate real parts & simplify:
(b)

36 Application of De Moivres Theorem Trigonometric identities
Example 38
2k
(a) Find the five fifth-roots of 1. Answer: ei( 5 + 5 ) , where k [0, 4]
(b) If is a non-real fifth root of 1 with the smallest positive argument, show
that 1 + 2 3 + 4 = 0.
(c) Find the exact values of cos 5 and cos 3
5
.
1
3 1

Answer: cos 5
= 4
1 + 5 , cos 5
= 4
1 5
Further exercises
Lee (2006, Ex 2.11 (skip Q6(iii)))
Patel (1990, Self Testing Ex 4.7 p.109)

Section 4
Locus problems
4.1 Summary
Equation Diagram
|z| = r
Derivation of Cartesian equation:
Description: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|z | = r
Derivation of Cartesian equation:
Description: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
38 Locus problems Summary
|z z1 | = |z z2 |
Description: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.....................................................
Arg(z z1 ) = , R
Description: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.....................................................
Arg(z z1 ) Arg(z z2 ) = , 0 < <
Origin: circle geometry theorem Angle at the
circumference subtended by the same arc/chord
Description: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.....................................................
|z z1 | + |z z2 | = k, k R.
Description: . . . . . . . . . . . . . . .
Will return after Conic Sections topic is completed.

Locus problems Examples 39
4.2 Examples
Example 39
For the following:
i. Describe the locus of z.
ii. Draw a sketch.
iii. Give the Cartesian equation of the locus.
(a) |z| = 2 (c) |z + 2| = 1

(b) zz = 16 (d) |z + 2 + 3i| = 2
Answer: (a) x2 + y 2 = 4 (b) x2 + y 2 = 16 (c) (x + 2)2 + y 2 = 1 (d) (x + 2)2 + (y + 3)2 = 4

40 Locus problems Examples
Example 40
Draw a sketch of locus of z if
2
(a) Arg(z) = 3
(b) 0 Arg(z) 3
(c) Arg(z 2 + 3i) = 4

Example 41
[H] Draw a sketch of the locus of z if |z 3| + |z + 3| = 12, and find its Cartesian
2 2
equation. Answer: x36 + y27 = 1

Example 42
Sketch the region in the Argand diagram defined simultaneously by
6 Re [(2 3i)z] < 12 and Re(z) Im(z) > 0
Example 43
z is a complex number which simultaneously satisfies

2 |z + 3| 3 and 0 Arg(z + 3)
3
Find the area and perimeter of the region in the Argand diagram determined by these
restrictions on z. Answer: A = 5
6
units2 , P = 2 + 5
3
units

Example 44
Sketch the curve in the Argand diagram determined by Arg(z 1) = Arg(z + 1) + 4 .
Find its Cartesian equation. Answer: x2 + (y 1)2 = 2, y > 0
Example 45
z satisfies |z i| = Im(z) + 1. Sketch the locus of the point P representing z in the
Argand diagram and write down its Cartesian equation. Answer: x2 = 4y

Example 46
Find the locus of w if
z2+i
w=
z+2i

given |z| = 1. Answer: Circle C 32 , 0 , r = 5

2
Further exercises
Patel (1990, Self Testing Ex 4.9, p.127)
Arnold and Arnold (2000, Ex. 2.5)
Fitzpatrick (1991, Ex 31(f))
Lee (2006, Ex 2.7, 2.8)

Section 5
Past HSC questions
HSC problems that can be attempted without theory from other parts of the Extension 2
course.
5.1 2001 Extension 2 HSC

Question 2
1
(a) Let z = 2 + 3i and w = 1 + i. Find zw and in the form x + iy. 2
w

(b) i. Express 1 + 3i in modulus-argument form. 2
10
ii. Hence evaluate 1 + 3i in the form x + iy. 2
(c) Sketch the region in the complex plane where the inequalities 3

|z + 1 2i| 3 and arg z
3 4
both hold.
(d) Find all solutions of the equation z 4 = 1. 3

Give your answers in modulus-argument form.
(e) In the diagram the vertices of a triangle ABC are represented by the complex numbers
z1 , z2 and z3 , respectively. The triangle is isosceles and right-angled at B.
y
D b
A b
b
C
B
x
O
i. Explain why (z1 z2 )2 = (z3 z2 )2 . 2

ii. Suppose D is the point such that ABCD is a square. Find the complex 1
number, expressed in terms of z1 , z2 and z3 , that represents D.
45
46 Past HSC questions 2001 Extension 2 HSC
Question 7
1
(a) Suppose that z = (cos + i sin ) where is real.
2
i. Find |z|. 1
ii. Show that the imaginary part of the geometric series 3
1
1 + z + z2 + z3 + =
1z
2 sin
is .
5 4 cos
iii. Find an expression for 2
1 1 1
1+ cos + 2 cos 2 + 3 cos 3 +
2 2 2
in terms of cos .
(b) Consider the equation x3 3x 1 = 0.

p
i. Let x = where p and q are integers having no common divisors other 4
q
than +1 and 1. Suppose that x is a root of ax3 3x + b = 0, where
a and b are integers.
Explain why p divides b and why q divides a. Deduce that x3 3x1 = 0

does not have a rational root.

ii. Suppose that r, s and
d are rational numbers and that d is irrational. 4
Assume that r + s d is a root of x3 3x 1 = 0.

Show that 3r 2 s + s3 d 3s = 0 and show that r s d must also be a
root of x3 3x 1 = 0.
3
Deduce from this result and part (i),
that no root of x 3x 1 = 0
can be expressed in the form r + s d with r, s and d rational.

iii. Show that one root of x3 3x 1 = 0 is 2 cos . 1
9
You may assume the identity cos 3 = 4 cos3 3 cos .

Past HSC questions 2002 Extension 2 HSC 47

Question 2
(a) Let z = 1 + 2i and w = 1 + i. Find, in the form x + iy,
i. zw. 1
1
ii. . 1
w
(b) On an Argand diagram, shade in the region where the inequalities 3
0 Re(z) 2 and |z 1 + i| 2
both hold.
(c) It is given that 2 + i is a root of
P (z) = z 3 + rz 2 + sz + 2
where r and s are real numbers.

i. State why 2 i is also a root of P (z). 1
ii. Factorise P (z) over the real numbers. 2
(d) Prove by induction that, for all integers n 1, 3
(cos i sin )n = cos (n) i sin (n)
(e) Let z = 2 (cos + i sin ).

i. Find 1 z. 1
1 1 2 cos
ii. Show that the real part of is 2
1z 5 4 cos
1
iii. Express the imaginary part of in terms of . 1
1z


Question 2
(a) Let z = 2 + i and w = 1 i. Find, in the form x + iy,
i. zw. 1
4
ii. . 1
z
(b) Let = 1 + i.
i. Express in modulus-argument form. 2
ii. Show that is a root of the equation z 4 + 4 = 0. 1
iii. Hence, or otherwise, find a real quadratic factor of the polynomial z 4 +4. 2

|z 1 i| < 2 and 0 < arg(z 1 i) <
4
hold simultaneously.
(d) By applying De Moivres theorem and by also expanding (cos + i sin )5 , 3

express cos 5 as a polynomial in cos .
(e) Suppose that the complex number z lies on the unit circle, and 2
0 arg(z) 2 .
Prove that 2 arg(z + 1) = arg(z).


Question 2
(a) Let z = 1 + 2i and w = 3 i. Find, in the form x + iy,
i. zw. 1

10
ii. . 1
z

(b) Let = 1 + i 3 and = 1 + i.

i. Find in the form x + iy. 1

ii. Express in modulus-argument form. 2
iii. Given that has the modulus-argument form 1

= 2 cos + i sin
4 4

find the modulus-argument form of .

iv. Hence find the exact value of sin . 1
12
|z + z| 1 and |z i| 1
(d) The diagram shows two distinct points A and B that represent the complex
numbers z and w respectively. The points A and B lie on the circle of radius
r centred at O. The point C representing the complex number z + w also lies
on this circle.
C
b
B b
b A
b b
i. Using the fact that C lies on the circle, show geometrically that 2
AOB = 2 3
.
ii. Hence show that z 3 = w 3 . 2
iii. Show that z 2 + w 2 + zw = 0. 1

Question 7
(b) Let be a real number and suppose that z is a complex number such that
1
z+= 2 cos
z
i. By reducing the above equation to a quadratic equation in z, solve for 3
z and use De Moivres theorem to show that
1
zn + = 2 cos n
zn
1
ii. Let w = z + . Prove that 2
z

3 2 1 2 1 3 1
w + w 2w 2 = z + + z + 2 + z + 3
z z z
iii. Hence, or otherwise, find all solutions of 3
cos + cos 2 + cos 3 = 0
in the range 0 2.

Question 2
(a) Let z = 3 + i and w = 1 i. Find, in the form x + iy,
i. 2z + iw. 1
ii. zw. 1
6
iii. . 1
w

(b) Let = 1 i 3.
i. Express in modulus-argument form. 2
ii. Express 5 in modulus-argument form. 2
iii. Hence express 5 in the form x + iy. 1
|z z| < 2 and |z 1| 1

(d) Let be the line in the complex plane that passes through the origin and
makes an angle with the positive real axis, where 0 < < 2 .

b
Q
b
P
The point P represents the complex number z1 , where 0 < arg(z1 ) < . The
point P is reflected in the line to produce the point Q, which represents
the complex number z2 . Hence |z1 | = |z2 |.
i. Explain why arg(z1 ) + arg(z2 ) = 2. 2
ii. Deduce that z1 z2 = |z1 |2 (cos 2 + i sin 2). 1

iii. Let = and let R be the point that represents the complex number 1
zz. 4
1 2
Describe the locus of R as z1 varies.

Question 6
(b) Let n be an integer greater than 2. Suppose is an n-th root of unity and
6= 1.
i. By expanding the left, show that 2

1 + 2 + 3 2 + 4 3 + + n n1 ( 1) = n
1 z 1
ii. Using the identity = , or otherwise, prove that 1
z2 1 z z 1
1 cos i sin
=
cos 2 + i sin 2 1 2i sin
provided that sin 6= 0.
2 2 1
iii. Hence, if = cos + i sin , find the real part of . 1
n n 1
2 4 6 8 5
iv. Deduce that 1 + 2 cos + 3 cos + 4 cos + 5 cos = . 1
5 5 5 5 2
v. By expressing the left hand side of the equation in part (iv) in terms of 3
2
cos and cos , find the exact value, in surd form, of cos .
5 5 5


Question 2
(a) Let z = 3 + i and w = 2 5i. Find, in the form x + iy,
i. z 2 . 1
ii. zw. 1
w
iii. . 1
z

(b) i. Express 3 i in modulus-argument form. 2
7
ii. Express 3 i in modulus-argument form. 2
7
iii. Hence express 3 i in the form x + iy. 1
(c) Find, in modulus-argument form, all solutions of z 3 = 1. 2
(d) The equation |z 1 3i| + |z 9 3i| = 10 corresponds to an ellipse in the

Argand diagram.
i. Write down the complex number corresponding to the centre of the 1
ellipse.
ii. Sketch the ellipse, and state the lengths of the major and minor axes. 3
iii. Write down the range of values of arg(z) for complex numbers z 1
corresponding to points on the ellipse.

Question 2
(a) Let z = 4 + i and w = z. Find, in the form x + iy,
i. w. 1
ii. w z. 1
z
iii. . 1
w
(b) i. Write 1 + i in the form r (cos + i sin ). 2

ii. Hence, or otherwise, find (1 + i)17 in the form a + ib, where a and b are 3
integers.
(c) The point P on the Argand diagram represents the complex number z, where 3
z satisfies
1 1
+ =1
z z
Give a geometrical description of the locus of P as z varies.

(d) The points P , Q and R on the Argand diagram represent the complex
numbers z1 , z2 and a respectively.
The triangles OP R and OQR are equilateral with unit sides, so

|z1 | = |z2 | = a = 1.
Let = cos 3 + i sin 3 .

Im
Q(z2 )
R(a)
Re
P (z1 )
i. Explain why z2 = a. 1
ii. Show that z1 z2 = a2 . 1
iii. Show that z1 and z2 are the roots of z 2 az + a2 = 0. 2
Question 8
(b) i. Let n be a positive integer. Show that if z 2 6= 1, then 2
n
2 4 2n2 z z n
1+z +z ++z = 1
z n1
zz
ii. By substituting z = cos + i sin , where sin 6= 0 in to part (i), show 3
that
1 + cos 2 + + cos(2n 2) + i [sin 2 + + sin(2n 2)]

sin n
= [cos(n 1) + i sin(n 1)]
sin

iii. Suppose = . Using part (ii), show that 3
2n
2 (n 1)
sin + sin + + sin = cot
n n n 2n


Question 2
(a) Find real numbers a and b such that (1 + 2i)(1 3i)= a + ib. 2

1+i 3
(b) i. Write in the form x + iy, where x and y are real. 2
1+i

ii. By expression
both 1 + i 3 and 1 + i in modulus-argument form, write 3
1+i 3
in modulus-argument form.
1+i

iii. Hence find cos in surd form. 1
12
!12
1+i 3
iv. By using the result of part (ii), or otherwise, calculate . 1
1+i
(c) The point P on the Argand diagram represents the complex number 3
z = x + iy, which satisfies
z2 + z2 = 8
Find the equation of the locus of P in terms of x and y. What type of curve
is the locus?
(d) The point P on the Argand diagram represents the complex number z.
The points Q and R represent the points z and z respectively, where
= cos 2
3
+ i sin 2
3
. The point M is the midpoint of QR.
Im
Re
O
b
S
i. Find the complex number representing M in terms of z. 2
ii. The point S is chosen so that P QSR is a parallelogram. 2
Find the complex number represented by S.


Question 2
(a) Write i9 in the form a + ib where a and b are real. 1
2 + 3i
(b) Write in the form a + ib where a and b are real. 1
2+i
(c) The points P and Q on the Argand diagram represent the complex numbers
z and w respectively.
Im
b
P (z)
b
Q(w)
Re
Copy the diagram into your writing booklet, and mark on it the following
points:
i. the point R representing iz 1
ii. the point S representing w 1
iii. The point T representing z + w. 1
(d) Sketch the region in the complex plane where the inequalities |z 1| 2 and 2
4 arg(z 1) 4 hold simultaneously.
(e) i. Find all the 5th roots of 1 in modulus-argument form. 2

ii. Sketch the 5th roots of 1 on an Argand diagram. 1
(f) i. Find the square roots of 3 + 4i. 3

ii. Hence, or otherwise, solve the equation 2
z 2 + iz 1 i = 0


Question 2
(a) Let z = 5 i.
i. Find z 2 in the form x + iy. 1
ii. Find z + 2z in the form x + iy. 1
i
iii. Find in the form x + iy. 2
z

(b) i. Express 3 i in modulus-argument form. 2
6
ii. Show that 3 i is a real number. 2
(c) Sketch the region in the complex plane where the inequalities 1 |z| 2 2
and 0 z + z 3 hold simultaneously.

(d) Let z = cos + i sin where 0 < < .
2
On the Argand diagram the point A represents z, the point B represents z 2
and the point C represents z + z 2 .
Im b
C
b
B
b A

b
Re
O
Copy or trace the diagram into your writing booklet.

i. Explain why the parallelogram OACB is a rhombus. 1
3
ii. Show that arg (z + z 2 ) = 1
2

iii. Show that |z + z 2 | = 2 cos . 2
2
iv. By considering the real part of z + z 2 , or otherwise, deduce that 1
3
cos + cos 2 = 2 cos cos
2 2


Question 2
(a) See Example 21 on page 20
(b) See Example 18 on page 17.
(c) See Example 33 on page 32.
(d) i. Use the binomial theorem to expand (cos + i sin )3 . 1

ii. Use De Moivres theorem and your result from part (i) to prove that 3
1 3
cos3 = cos 3 + cos
4 4
iii. Hence, or otherwise, find the smallest positive solution of 2
4 cos3 3 cos = 1


1. Let z = 5 i and w = 2 + 3i. 1
What is the value of 2z + w?
(A) 12 + i (B) 12 + 2i (C) 12 4i (D) 12 5i
2. The complex number z is shown on the Argand diagram below. 1

y
b
z
x
O
Which of the following best represents iz?

(A) (C)
y y
b
iz
x x
O O
iz
(B) (D)
y y
iz b
x x
O O
iz

Question 11

2 5+i
(a) Express in the form x + iy, where x and y are real. 2
51
(b) Sketch the region in the complex plane where the inequalities 2
|z + 2| 2 and |z i| 1
both hold.

(d) i. Write z = 3 i in modulus-argument form. 2
ii. Hence express z 9 in the form x + iy, where x and y are real. 1
Question 12
(d) On the Argand diagram the points A1 and A2 correspond to the distinct
complex numbers u1 and u2 respectively. Let P be a point corresponding to
a third complex number z.
Points B1 and B2 are positioned so that A1 P B1 and A2 B2 P , labelled in

an anti-clockwise direction, are right-angled and isosceles with right angles
at A1 and A2 respectively. The complex numbers w1 and w2 correspond to
B1 and B2 respectively.
y
B1 (w1 )
b
B2 (w2 )
b
P (z)
b
A1 (u1)
b
A2
i. Explain why w1 = u1 + i(z u1 ). 1

ii. Find the locus of the midpoint of B1 B2 as P varies. 2

References
Arnold, D., & Arnold, G. (2000). Cambridge Mathematics 4 Unit (2nd ed.). Cambridge University
Press.
Fitzpatrick, J. B. (1991). New Senior Mathematics Four Unit Course for Years 12. Rigby
Heinemann.
Lee, T. (2006). Advanced Mathematics: A complete HSC Mathematics Extension 2 Course (2nd
ed.). Terry Lee Enterprise.
Patel, S. K. (1990). Excel 4 Unit Maths. Pascal Press.
Patel, S. K. (2004). Maths Extension 2 (2nd ed.). Pascal Press.
60

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MATHEMATICS (EXTENSION 2)

Notes and exercises for Topic 1:

This document is available

1 A new number system 5

2 Further arithmetic & algebra of complex numbers 14

3 Application of De Moivres Theorem 31

5 Past HSC questions 45

5.5 2005 Extension 2 HSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

NORTH SYDNEY BOYS HIGH SCHOOL

A new number system

1.1 Review of number systems

Multiply 1 by 1 to obtain 1 corresponds to rotating by . . radians.

NORTH SYDNEY BOYS HIGH SCHOOL

1.3 The imaginary numbers

Jump off the real number line.

NORTH SYDNEY BOYS HIGH SCHOOL

Treat real and imaginary parts as . . . . . . . . . . . . . . . . . . . . . . . . . . . . of a complex number.

Plot on A . . . . . . . . . . . . . d . . . . . . . . . . . . . . . . , similar to plotting points coordinate geometry.

NORTH SYDNEY BOYS HIGH SCHOOL

1.4 Basic operations with complex numbers

Group . . . . . . . . parts with . . . . . . . . parts

Group . . . . . . . . . . . . . . . . . . . . . . . parts with . . . . . . . . . . . . . . . . . . . . . . . parts.

Beware that i2 = . . . . , which becomes . . . . . . . .

(a) z1 + z2 (b) z1 z2 (c) 3z1 (d) 3iz1 (e) z1 z2

NORTH SYDNEY BOYS HIGH SCHOOL

1.4.3 Complex conjugate pairs

(a) z1 + z2 (c) z2 (e) z1 z2

NORTH SYDNEY BOYS HIGH SCHOOL

NORTH SYDNEY BOYS HIGH SCHOOL

1.5 Properties of complex numbers

1.5.2 Solutions to equations

NORTH SYDNEY BOYS HIGH SCHOOL

Equations with . . . . . . . . coefficients will have . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Equations with . . . . . . . . . . . . . . . . . . . coefficients do not necessarily have . . . . . . . . . . . . . . . . . . .

NORTH SYDNEY BOYS HIGH SCHOOL

Further arithmetic & algebra of complex

2.1 Vector representation

The starting point ( . . . . . . . ) is irrelevant for a vector.

Parallelogram of vectors when adding two vectors.

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Alternatively, what vector get you to z1 from z2 ?

2.1.4 Scalar multiplication

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NORTH SYDNEY BOYS HIGH SCHOOL

2.2 Modulus/argument of a complex number

Natural ordering does not exist with complex numbers.

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2.2.3 (Principal) Argument

The argument of a complex number, measured in . . . . . . . . . . . . . . . . . , is denoted

(where z = x + iy) is the ( . . . . . . . . . . . . . . . . . . . . ) that the vector from O to z makes

lies within the domain < Arg(z) .

The principal argument is generally quoted henceforth.

NORTH SYDNEY BOYS HIGH SCHOOL

NORTH SYDNEY BOYS HIGH SCHOOL

2.2.4 Triangle inequality

|z1 + z2 | |z1 | + |z2 |

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iff O, P and Q are collinear (which implies

NORTH SYDNEY BOYS HIGH SCHOOL