Download Worksheet 2.1

Maths Quest Maths C Year 11 for Queensland
WorkSHEET 2.1
Chapter 2 Number systems: complex numbers WorkSHEET 2.1
1
Number systems: complex numbers
Name: ___________________________
1
Plot the following complex numbers on the
same fully labelled Argand diagram.
(a) A = 2 + 3i
(b) B = –5 –i
(c) C = 1 – 4i
(d) D = –4 + 2i
(e) E = 6i
(f) F = –4
2
Perform the following operations using
z = 2 – 6i
w = –1 + 2i
x=4+i
(a)
(a)
2z = 2(2 – 6i) = 4 – 12i
(b)
w – z = –1 + 2i – (2 – 6i)
= –1 + 2i – 2 + 6i
= –3 + 8i
(c)
3x + 2w = 3(4 + i) + 2(–1 + 2i)
= 12 + 3i –2 + 4i
= 10 + 7i
(d)
x × w = (4 + i)(–1 + 2i)
= –4 –i + 8i – 2
= –6 + 7i
(e)
z × w = (2 – 6i)(–1 + 2i)
= –2 + 6i + 4i + 12
= 10 + 10i
2z
(b)
w–z
(c)
3x + 2w
(d)
6
x×w
(e) z × w
Plot the answers on the same Argand diagram.
5
Maths Quest Maths C Year 11 for Queensland
3
Chapter 2 Number systems: complex numbers WorkSHEET 2.1
Find the modulus of each of the following
complex numbers:
(a)
z = 2 – 5i
(b)
w = –3 + 6i
(c)
y = –6 + i
(d)
x=4
(a)
z  2 2  (5) 2
 4  25
 29
(b)
w  (3) 2  6 2
 9  36

45
3 5
(c)
y  (6) 2  12
 36  1
 37
(d)
|x| = 4
2
4
Maths Quest Maths C Year 11 for Queensland
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Chapter 2 Number systems: complex numbers WorkSHEET 2.1
Sketch the following complex numbers then
find the argument and principal argument of
each:
(a)
z = 1 – 4i
(b)
w = –2 – 7i
(c)
y = 3 + 4i
(d)
x = –5i
3
4
(a)
4
arg( z )  tan 1 

 1 
 1.33
Arg( z )  1.33
(b)
7
arg( w)  tan 1 

2
 1.29
Arg( w)  1.29  
 1.85
(c)
4
arg( y )  tan 1  
3
 0.64
Arg( y )  0.64
(d)
arg( x) 
3
2
Arg( x)   

2
Maths Quest Maths C Year 11 for Queensland
5
Chapter 2 Number systems: complex numbers WorkSHEET 2.1
Sketch the following complex numbers and
express them in polar form:
(a)
z = 2 + 3i
(b)
w = –4 –2i
(c)
y = –3 + i
(d)
x = 5 – 5i
(a)
4
4
z  2 2  3 2  4  9  13
3
arg( z )  tan 1    0.98
2
z  13 cis 0.98
(b)
w  (4) 2  (2) 2  16  4
 20  2 5
2
arg( w)  tan 1 
  0.46
4
Arg( w)  0.46    2.68
w  2 5 cis (2.68)
(c)
y  (3) 2  12  9  1  10
 1 
arg( y )  tan 1 
  0.32
  3
Arg( y )  0.32    2.82
y  10 cis 2.82
(d)
x  5 2  (5) 2  25  25
 50  5 2
 5 
arg( x)  tan 1 

 5  4
Maths Quest Maths C Year 11 for Queensland
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Chapter 2 Number systems: complex numbers WorkSHEET 2.1
Use common trigonometric ratios in their
relevant triangles (but no formal calculator
work) to express the following in polar form:
(a)
z = 2 – 2i
(b)
w =  1  3i
(c)
y = 4 3  4i
(d)
x =  5  5i
(a)
5
4
 
z  2 2 cis   
 4
(b)
 2 
w  2 cis  

 3 
(c)
y  8 cis

6
(d)
x  10 cis
3
4