3.

Complex Numbers, EXT2 N2 2013 HSC 5 MC

EXT 2: Complex Numbers (Ext2), N2 Using Complex Numbers (Ext2)
Geometrical Implications of Complex Numbers Which region on the Argand diagram is defined by

Teacher: Sally Gorman

Exam Equivalent Time: 120 minutes (based on HSC allocation of 1.5 minutes approx.
per mark)

Questions
1. Complex Numbers, EXT2 N2 2017 HSC 1 MC

The complex number is chosen so that form the vertices of the regular polygon
shown.

Which polynomial equation has all of these complex numbers as roots?

4. Complex Numbers, EXT2 N2 2015 HSC 9 MC
A.
The complex number satisfies
B.
What is the greatest distance that can be from the point on the Argand diagram?
C.
A.
D.
B.

C.
2. Complex Numbers, EXT2 N2 2017 HSC 3 MC
D.
Which complex number lies in the region ?

A.
B.
C.
D.
5. Complex Numbers, EXT2 N2 2016 HSC 5 MC 7. Complex Numbers, EXT2 N2 2011 HSC 2b
On the Argand diagram, the complex numbers and form a rhombus.
Multiplying a non-zero complex number by results in a rotation about the origin on an

Argand diagram.
What is the rotation?

A. Clockwise by

B. Clockwise by

C. Anticlockwise by

D. Anticlockwise by

i. Find in the form , where and are real numbers. (1 mark)

6. Complex Numbers, EXT2 N2 2018 HSC 7 MC
ii. An interior angle, , of the rhombus is marked on the diagram.
Which diagram best represents the solutions to the equation ? Find the value of (2 marks)

A. B.
8. Complex Numbers, EXT2 N2 2009 HSC 2d
Sketch the region in the complex plane where the inequalities and
hold simultaneously. (2 marks)

9. Complex Numbers, EXT2 N2 2010 HSC 2c

C. D.
Sketch the region in the complex plane where the inequalities and
hold simultaneously. (2 marks)

10. Complex Numbers, EXT2 N2 2012 HSC 11b

Shade the region on the Argand diagram where the two inequalities
and
both hold. (2 marks)

11. Complex Numbers, EXT2 N2 2017 HSC 11c

Sketch the region in the Argand diagram where

. (2 marks)
12. Complex Numbers, EXT2 N2 2018 HSC 11d 16. Complex Numbers, EXT2 N2 2019 HSC 12a
The points , and on the Argand diagram represent the complex numbers , and Sketch the region defined by and . (2 marks)
respectively.
The points , , and form a square as shown on the diagram.
17. Complex Numbers, EXT2 N2 2004 HSC 2c
Sketch the region in the complex plane where the inequalities
and
hold simultaneously. (3 marks)

18. Complex Numbers, EXT2 N2 2005 HSC 2c

Sketch the region on the Argand diagram where the inequalities
and
hold simultaneously. (3 marks)

19. Complex Numbers, EXT2 N2 2007 HSC 2d

It is given that .
i. Find . (1 mark)

ii. Find . (1 mark)

iii. Find . (1 mark)

13. Complex Numbers, EXT2 N2 2007 HSC 2c

The point on the Argand diagram represents the complex number , where satisfies

Give a geometrical description of the locus of as varies. (3 marks)

14. Complex Numbers, EXT2 N2 2013 HSC 11e The points and on the Argand diagram represent the complex numbers and
respectively.
Sketch the region on the Argand diagram defined by (3 marks)
The triangles and are equilateral with unit sides, so

Let
15. Complex Numbers, EXT2 N2 2014 HSC 11c
i. Explain why (1 mark)
Sketch the region in the Argand diagram where and . (3

marks) ii. Show that (1 mark)

iii. Show that and are the roots of (2 marks)

20. Complex Numbers, EXT2 N2 2010 HSC 2d 22. Complex Numbers, EXT2 N2 2012 HSC 12d

Let where . On the Argand diagram the points and correspond to the distinct complex numbers
and respectively. Let be a point corresponding to a third complex number .
On the Argand diagram the point represents , the point represents and the point Points and are positioned so that and , labelled in an anti-
represents . clockwise direction, are right-angled and isosceles with right angles at and , respectively.
The complex numbers and correspond to and , respectively.

Copy or trace the diagram into your writing booklet.

i. Explain why the parallelogram is a rhombus. (1 mark)

ii. Show that . (1 mark)

i. Explain why . (1 mark)

iii. Show that . (2 marks) ii. Find the locus of the midpoint of as varies. (2 marks)

iv. By considering the real part of , or otherwise deduce that

. (1 mark)

21. Complex Numbers, EXT2 N2 2011 HSC 4a

Let and be real numbers with . Let be a complex number such that

i. Prove that (2 marks)

ii. Hence, describe the locus of all complex numbers such that (1
mark)
23. Complex Numbers, EXT2 N2 2020 SPEC2 2 24. Complex Numbers, EXT2 N2 2013 HSC 15a
Two complex numbers, and , are defined as and . The Argand diagram shows complex numbers and with arguments and respectively,
where . The area of the triangle formed by and is .
a. Express the relation in the cartesian form , where .
(3 marks)

b. Plot the points that represent and and the relation on the Argand diagram
below. (2 marks)

Im(z)

Re(z)
–5 O 5 Show that (3 marks)

25. Complex Numbers, EXT2 N2 2017 HSC 13e

The points and on the Argand diagram represent the complex numbers and
–5 respectively. The points form a square as shown on the diagram.

c. State a geometrical interpretation of the graph of in relation to the points that

represent and . (1 mark)

d. i. Sketch the ray given by on the Argand diagram in part b. (1 mark)

ii. In Cartesian form, write down the function that describes the ray . (1 mark)

By using vectors, or otherwise, show that . (2 marks)

26. Complex Numbers, EXT2 N2 2016 HSC 16b

i. The complex numbers and form the vertices of an equilateral triangle in the Argand
diagram.
Show that (2 marks)

ii. Give an example of non-zero complex numbers and , so that and form the vertices of
an equilateral triangle in the Argand diagram. (1 mark)
27. Complex Numbers, EXT2 N2 2019 HSC 16b 29. Complex Numbers, EXT2 N2 2016 HSC 16a
Let , where and are real numbers. i. The complex numbers and , where
and , satisfy
The roots of are .

It is given that and .

i. Show that . (3 marks) By considering the real and imaginary parts of , or otherwise, show that and
form the vertices of an equilateral triangle in the Argand diagram. (3 marks)
ii. The diagram shows the position of .
ii. Hence, or otherwise, show that if the three non-zero complex numbers and satisfy

AND

then they form the vertices of an equilateral triangle in the Argand diagram. (2 marks)

Copyright © 2004-20 The State of New South Wales (Board of Studies, Teaching and Educational Standards NSW)

Copy or trace the diagram into your writing booklet.

On the diagram, accurately show all possible positions of . (2 marks)

28. Complex Numbers, EXT2 N2 2011 HSC 6c

On an Argand diagram, sketch the region described by the inequality

(2 marks)
Worked Solutions 4. Complex Numbers, EXT2 N2 2015 HSC 9 MC

1. Complex Numbers, EXT2 N2 2017 HSC 1 MC

2. Complex Numbers, EXT2 N2 2017 HSC 3 MC

i
B
3
2
A
1 5. Complex Numbers, EXT2 N2 2016 HSC 5 MC
C
_ 2 _1 Re
1 2 3 4
_1 D
_2
_3

6. Complex Numbers, EXT2 N2 2018 HSC 7 MC

3. Complex Numbers, EXT2 N2 2013 HSC 5 MC
7. Complex Numbers, EXT2 N2 2011 HSC 2b 9. Complex Numbers, EXT2 N2 2010 HSC 2c

i.

ii.

8. Complex Numbers, EXT2 N2 2009 HSC 2d

10. Complex Numbers, EXT2 N2 2012 HSC 11b
11. Complex Numbers, EXT2 N2 2017 HSC 11c 13. Complex Numbers, EXT2 N2 2007 HSC 2c

12. Complex Numbers, EXT2 N2 2018 HSC 11d

i.

ii.

iii.
14. Complex Numbers, EXT2 N2 2013 HSC 11e 15. Complex Numbers, EXT2 N2 2014 HSC 11c

16. Complex Numbers, EXT2 N2 2019 HSC 12a

17. Complex Numbers, EXT2 N2 2004 HSC 2c 18. Complex Numbers, EXT2 N2 2005 HSC 2c

_
_ _1 _1
2 2
19. Complex Numbers, EXT2 N2 2007 HSC 2d

i.

ii.

iii.
20. Complex Numbers, EXT2 N2 2010 HSC 2d
i.

♦♦ Mean mark part (iv) 44%.

21. Complex Numbers, EXT2 N2 2011 HSC 4a

i.

ii.

iii.

ii.
♦ Mean mark part (ii) 44%.

♦♦ Mean mark part (iii) 26%.

iv.
22. Complex Numbers, EXT2 N2 2012 HSC 12d 23. Complex Numbers, EXT2 N2 2020 SPEC2 2
a.
i.

ii.
b. Im(z)

Re(z)
–5 O 5
u

–5
|z - u| = |z - v|

c.

d.i.
 Im(z)
24. Complex Numbers, EXT2 N2 2013 HSC 15a

♦♦ Mean mark 29%.

5 STRATEGY: The angles shown in
the graphic should alert students
Arg(z - u) that the mod-arg approach is likely
to be easier than the form.

π Re(z)
–5 4
O 5

–5

d.ii.

25. Complex Numbers, EXT2 N2 2017 HSC 13e

♦ Mean mark 49%.

26. Complex Numbers, EXT2 N2 2016 HSC 16b 27. Complex Numbers, EXT2 N2 2019 HSC 16b
i. i.

♦♦ Mean mark 23%.

♦♦ Mean mark part (i) 26%.

ii.

ii.

♦♦ Mean mark 33%.

♦♦♦ Mean mark part (ii) 10%.

 29. Complex Numbers, EXT2 N2 2016 HSC 16a

i.

♦♦♦ Mean mark 25%.

STRATEGY: A clear graphical
image simplifies both parts of this
question significantly.

28. Complex Numbers, EXT2 N2 2011 HSC 6c

♦♦♦ Mean mark 18%.

MARKER'S COMMENT:
Substituting
immediately was common and
caused major algebraic problems.
ii.

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♦♦♦ Mean mark 5%.