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Answers to Chapter 3
Exercise 3A 1 a 7 � 7 (cos � � i sin �)
b �5i � 5 ( cos ( � � __ 2 ) � i sin ( � � __
2 ) )
c √__ 3 � i � 2 ( cos � __
6 � i sin � __
6 )
d 2 � 2i � 2 √__ 2 ( cos � __
4 � i sin � __
4 )
e 1 � i � √__ 2 ( cos ( � � __
4 ) � i sin ( � � __
4 ) )
f �8 � 8(cos � � i sin �) g 3–4i � 5(cos(�0.93c) � i sin(�0.93c)) h �8 � 6i � 10(cos(2.50c) � i sin(2.50c)) i 2 � √
__ 3 i � √
__ 7 (cos(�0.71c) � i sin(�0.71c))
2 a 5i
b √
__ 3 ___
4 � 1 __
4 i
c �3 √__ 3 � 3i
d � 3 __ 2 � 3 √
__ 3 ____
2 i
e 2 � 2i
f 2 √__ 3 � 2i
3 a �3 � 3e�i
b 6i � 6 e �i __ 2
c �2 √__ 3 � 2i � 4 e
�5�i _____ 6
d �8 � i � √___
65 e3.02i
e 2 � 5i � √___
29 e�1.19i
f �2 √__ 3 � 2 √
__ 3 i � 2 √
__ 6 e
3�i ____ 4
g 2 √__ 2 e
�i __ 4
h 8 e � �i __
6
i 2 e � �i __
5
4 a 1 __ 2 �
√__ 3 ___
2 i
b �4 c 3 � 3i d 4 √
__ 3 � 4i
e �3i
f � √__ 3 _____
2 � 1 __
2 i
g �1 h �3 � 3i i �4 � 4 √
__ 3 i
5 a cos ( � 10� ____ 13
) � i sin ( � 10� ____ 13
)
b 4 ( cos ( � 3� ___ 5 ) � i sin ( � 3� ___
5 ) )
c 5 ( cos ( 7� ___ 8 ) � i sin ( 7� ___
8 ) )
Exercise 3B 1 a cos 5� � i sin 5� b �1 c 3 � 3 √
__ 3 i d 3 � 3i
e � √__ 3 � i f �5 √
__ 3 � 5i
g cos 3� � i sin 3� h 3 � 3i
2 a cos 3� � i sin 3� b 2 � 2i
c � 3 __ 4 i
d cos(�5�) � i sin(�5�) or cos 5� � i sin 5�
3 a z � 6 √__ 3 ( cos 5� ___
6 � i sin 5� ___
6 )
b w � √__ 3 ( cos 7� ___
12 � i sin 7� ___
12 )
c 18 cos ( � 7� ___ 12
) � i sin ( � 7� ___ 12
)
d 6 ( cos � __ 4 � i sin � __
4 )
Exercise 3C 1 a cos 6� � i sin 6� b cos 12� � i sin 12�
c � √
__ 3 ___
2 � 1 __
2 i d � 1 __
2 �
√__ 3 ___
2 i
e 1 f i g cos � � i sin � h cos 2� � i sin 2� i cos 6� � i sin 6� j cos � � i sin � k cos 11� � i sin 11� l cos 5� � i sin 5� 2 1 3 a (1 � i)5 � �4 � 4i b (�2 � 2i)8 � 4096 c (1 � i)6 � 8i d (1 � √
__ 3 i)6 � 64
e ( 3 __ 2 � 1 __
2 √
__ 3 i ) 9 � 81 √
__ 3 i
f (�2 √__ 3 � 2i)5 � 512 √
__ 3 � 512i
4 (3 � √__ 3 i)5 � �432 � 144 √
__ 3 i
Exercise 3D 6 a 32 cos6 � � cos 6� � 6 cos 4� � 15 cos 2� � 10
b ∫ 0 � __ 6
cos6 � d� � 5� ___ 96
� 9 ___ 64
√__ 3
7 a sin4 � � 4 cos3 � sin � � 4 cos � sin3 �
b tan4 � � 4 tan � � 4 tan3 � ___________________ 1 � 6 tan2 � � tan4 �
c x � 0.20, 1.50, �5.03, �0.67 (2 d.p.)
Exercise 3E 1 a z � 1, i, �1, �i
b z � √
__ 3 ___
2 � 1 __
2 i, �
√__ 3 ___
2 � 1 __
2 i, �i
c z � 3, � 3 __ 2 � 3 √
__ 3 ____
2 i, � 3 __
2 � 3 √
__ 3 ____
2 i
d z � 2 � 2i, �2 � 2i, 2 � 2i, �2 � 2i e z � 1 � i, �1 � i, 1 � i, �1 � i f z � √
__ 3 � i, 2i, � √
__ 3 � i
Answers to Chapter 3
2
ANSWERS
2 a z � cos 0 � i sin 0, cos 2� ___ 7 � i sin 2� ___
7
cos 4� ___ 7 � i sin 4� ___
7 , cos 6� ___
7 � i sin 6� ___
7
cos ( � 2� ___ 7 ) � i sin ( � 2� ___
7 ) ,
cos ( � 4� ___ 7 ) � i sin ( � 4� ___
7 )
cos ( � 6� ___ 7 ) � i sin ( � 6� ___
7 )
b z � 2 ( cos ( � � __ 8 ) � i sin ( � � __
8 ) ) ,
2 ( cos ( 3� ___ 8 ) � i sin ( 3� ___
8 ) )
2 ( cos ( 7� ___ 8 ) � i sin ( 7� ___
8 ) ) ,
2 ( cos ( � 5� ___ 8 ) � i sin ( � 5� ___
8 ) )
c z � 2 ( cos � __ 5 � i sin � __
5 ) , 2 ( cos 3� ___
5 � i sin 3� ___
5 ) ,
2(cos � � i sin �), 2 ( cos ( � � __ 5 ) � i sin ( � � __
5 ) ) ,
2 ( cos ( � 3� ___ 5 ) � i sin ( � 3� ___
5 ) )
d z � √__ 2 ( cos � ___
12 � i sin � ___
12 ) , √
__ 2 ( cos 3� ___
4 � i sin 3� ___
4 ) ,
√__ 2 ( cos ( � 7� ___
12 ) � i sin ( � 7� _____
12 ) )
e z � √__ 2 ( cos ( � � ___
12 ) � i sin ( � � ___
12 ) ) ,
√__ 2 ( cos ( 5� ___
12 ) � i sin ( 5� ___
12 ) ) ,
√__ 2 ( cos ( 11� ____
12 ) � i sin ( 11� ____
12 ) ) ,
√__ 2 ( cos ( � 7� ___
12 ) � i sin ( � 7� ___
12 ) )
f z � 4 ( cos ( � 5� ___ 18
) � i sin ( � 5� ___ 18
) ) , 4 ( cos ( 7� ___
18 ) � i sin ( 7� ___
18 ) ) ,
4 ( cos ( � 17� ____ 18
) � i sin ( � 17� ____ 18
) ) 3 a z � 5
1 _ 4 e0.23i, 5
1 _ 4 e1.80i, 5
1 _ 4 e�1.34i, 5
1 _ 4 e�2.91i
b z � √__ 3 e�0.29i, √
__ 3 e1.80i, √
__ 3 e�2.39i
c z � √__ 2 e0.57i, z � √
__ 2 e2.14i, z � √
__ 2 e�1.00i, z � √
__ 2 e�2.57i
4 a z � � 1 __ 2 �
√__ 3 ___
2 i, �2, 1 __
2 �
√__ 3 ___
2 i
c centre (�1, 0) radius 1
5 a z � 1, cos ( 2� ___ 5 ) � i sin ( 2� ___
5 ) , cos ( 4� ___
5 ) � i sin ( 4� ___
5 ) ,
cos ( � 2� ___ 5 ) � i sin ( � 2� ___
5 ) ,
cos ( � 4� ___ 5 ) � i sin ( � 4� ___
5 )
6 a r � 4, � � � 2� ___ 3
b z � √__ 2 e
� �i __ 6 , √
__ 2 e
�i __ 3 , √
__ 2 e
5�i ____ 6 , √
__ 2 e
� 2�i ____ 3
7 a r � √__ 8 , � � � __
6
b z � 4 e 2�i ____ 9 , z � 4 e
8�i ____ 9 , z � 4 e
� 4�i ____ 9
Exercise 3F 1 a
x6
6
O
y
x2 � y2 � 36
b
x10
10
O
y
x2 � y2 � 100
c
x51
2
�2
3O
y
(x � 3)2 � y2 � 4
d
x�3 3
(0, �3)
O
y
x2 � ( y � 3)2 � 9
e
x�5 5
(0, 4)
(0, 9)
(0, �1)O
y
x2 � ( y � 4)2 � 25
ANSWERS
3
f
x
�1
1
�2 (�1, 0) O
y
(x � 1)2 � y2 � 1
g
x�4
�4
6
6
(1, 1)
y
(x � 1)2 � ( y � 1)2 � 25
h
xO�7
�8
�3 1
(�3, �4)
y
(x � 3)2 � ( y � 4)2 � 16
i
xO
�1
�6
�11
5 10
(5, �6)
y
(x � 5)2 � ( y � 6)2 � 25
j
xO�1
�6
2
5
(�3, 2)
y
(x � 3)2 � ( y � 2)2 � 9
k
xO 7�1
�2
6
3
(3, 2)
y
(x � 3)2 � ( y � 2)2 � 16
l
xO
6�1
5
2(3, 2)
y
(x � 3)2 � ( y � 2)2 � 9
2 a
O x
y
π3
b
O x
y
π4
3
(�3, 0)
c
O x
y
π2
4
ANSWERS
d
O x
y
π4
�4
(�2, �2)
e
O x
y
3π4
2
(1, 1)
f
O x
y
π
g
O x
y
2π3
(1, �3)
h
O x
y
π2
(3, �4)
i
O x
y
3π2
(0, 4)
�4
3 a
O x
x � 4
y
2 6
Equation: x � 4
b
O x
x � �2
y
�8 4
Equation: x � �2
c
O x
y � �3
y
�6
Equation: y � �3
ANSWERS
5
d
O x
y � 2.5
y
�3
8
Equation: y � 2.5
e
Ox
y � �x
y
(2, 2)
(�2, �2)
Equation: y � �x
f
O x
y
y �
�4
�1
�6
�72
Equation: y � � 7 __ 2
g
O x
y
7
(7, 5)(�3, 5)
�3
x � 2
Equation: x � 2
h
O x
y
2
(0, �6)
(8, �2)
(�4, 2)
y � 3x � 6
Equation: y � 3x � 6
i
O x
y
(0, )
(�3, 0)
(0, 6)
y � � x �
94
( , 0)92
12
94
Equation: y � � 1 _ 2 x � 9 _ 4
j
x
y
O
(�6, �1)
(10, 5)
(0, 11)
y � �4x � 11
( , 0)114
Equation: y � �4x � 11
6
ANSWERS
k
x
y
O(�7, �2)
(4, 3)
y � �
( , 0)�1411
(0, )�145
145
�11x5
y � � 11x ____ 5 � 14 ___
5
l
x
y
y � x � 4O
(�1, 6)
(0, 4)
(2, 3)
(�4, 0)
Equation: y � x � 4
4 a The Cartesian equation of the locus of z � z* � 0 is y � 0.
b The Cartesian equation of the locus of z � z* � 0 is x � 0.
5 a
x
y
O�1
�3
3
(2, 0) 5
(x � 2)2 � y2 � 9
b
x
y
O�4 4
(0, 5)
(0, 9)
(0, 1)
x2 � ( y � 5)2 � 16
c
x
y
(3, �2)
3
1
�5
�2
6
(x � 3)2 � ( y � 2)2 � 9
6 a The Cartesian equation of the locus of z is (x � 6)2 � y2 � 9
x
y
O 3 9(6, 0)
3
�3
b
x
y
O�1915
�315
�73
( x � 19 ___ 15
) 2 � y2 � 256 ____ 225
ANSWERS
7
c
x
y
O
�13
�53
�3
x2 � ( y � 5 __ 3 ) 2 � 16 ___
9
d y
O 5 10 15 20 25 x
(x � 14)2 � ( y � 5)2 � 100
e y
2
2
4
6
8
10
4 6 8 xO
(x � 4)2 � ( y � 6)2 � 20
f
x
y
O 43
83
123
( x � 8 __ 3 ) 2 � y2 � 16 ___
9
7 a
� 32
32
π4
x
y
O
P
b
x
y
P
O�4
3
π6
c
x
y
O
π3
d
x
y
P
O
3
5
π4
8
ANSWERS
e
x
y
P
O
π3
(2 � 3i)
f
P
y
O�4
4i
8 z � �4 � 3i
9 a
x
y
O
2
2
(2, 2)
4
4
b z � (2 � √__ 3 ) � 3i
10 a, b
x
y
y � 5
O 2
(0, 2)
(0, 8)
1
arg(z � 2 � i) �
|z � 2i| � |z � 8i|
π4
π4
c z � 6 � 5i
11 a, b
x
y
O 1
(3, �2)
arg(z � 1) � �
|z � 3 � 2i| � 4
π4
π4
c a � 3 � 2 √__ 2 , b � �2 � 2 √
__ 2
12 a (x � 4)2 � ( y � 3)2 � 64
b
x
y
OY
X
4
8
3
C(4, 3)
c The maximum value of |z| is 13 and the minimum value of |z| is 3
13 a
x
y
O(�4, 0)
dminπ3
b Hence the minimum value of |z| is |z|min � 2 √__ 3
ANSWERS
9
14 b The exact radius is 2 √__ 2
15 a z � 4 � 4 √__ 3 i
b arg(z � 8) � 2� ___ 3
16 a
x
y
O�4
(�2, )2 32 3
2 3 � 2
2 3 � 2
b Minimum value of arg(z) is � __ 2
c Maximum value of arg(z) is 5� ___ 6 .
17 a
x
y
OA (�4, 0)
P
φ θ
b (�2, 2)
c 2 √__ 2
d (x � 2)2 � ( y � 2)2 � 8
e 6� � 4
Exercise 3G 1 a
x
y
O
|z| � 3
b
x
y
O
�2
2
5
�2 2
c
xO
y
�2
�4
�6
2
4
�1�2�3�4�5�6�7 21
d
xO
y
�2�6 4
�2 � 8i
e
xO
y
�1
�2
�3
3
2
1
�1�2�3 321
10
ANSWERS
f
xO
y
�2
�4
�6
�8
�10
�2�4 42
g
xO
y
�2
�4
�6
�8
�10
4 6 82�2
h
xO
y
1
1�1�3
2
x
|z| � 5
|z| � |z � 6i|
O
y
R
3
O x
y
4
O x
y
�3
3
5
O x
yiii arg(z � 4 � 2i) �
R
|z � 2| � |z � 6 � 8i|
π2
ii arg(z � (4 � 2i)) � 0
ANSWERS
11
6 a i y � �2 √__ 2 (x � 1)
ii (x � 1)2 � y2 � 9
b z � �2 √__ 2 i and z � �2 � 2 √
__ 2 i
c
O x
y
R
Exercise 3H 1 a
O u
v
2
3
4
3 4
(4, 3)
5
|w � 4 � 3i| � 1
b
O u
v
3
4
arg (w � 4 � 3i) �π2
π2
c
O
v
v � u �1
u1
�1
2 4z � 8 � 12i
3 (u � 2)2 � (v � 5)2 � 36
4 (u � 1)2 � (v � 3)2 � 64
5 a
O u
u2 � (v � 2)2 � 9
v
�1
�3
5
(0, 2)
3
b
O
v
u
(0, 1)
arg (w � i) �
π4
π4
c
O
v
v � 2u � 4
u(�2, 0)
(0, 4)
6 a A circle, centre (0, 0), radius 1 _ 2
b arg w � � � __ 4
Ow
c Circle, centre ( � 1, � 1 __ 2 ) radius √
__
5 __ 2
7 b w lies on the positive real axis, including the origin, 0
c w lies on the negative real axis, including the origin, 0
12
ANSWERS
8 a
O x
y
(0, �2)
�2 2
R
�4
|z � 2i| � 2
b the interior of a circle, radius 2, centre (�2, 3) c w is the interior of a circle, radius 8, centre (2, �4) d w represents everything outside the circle,
centre (0, 0) radius 1 _ 2 .
9
uO
v l
u �
12
14
10 a u � 2
b
uO
v
�2
R
�2 2
2
12 a v � 0
b
uO
v
v � 0
R
c u2 � ( v � 5 _ 3 ) 2 � 16
__ 9
13 Circle centre ( 9 _ 2 , � 3 _ 8 ) radius � 15 __ 8
14 a u2 � ( v � 1 _ 2 ) 2 � 1 _ 4
b ( u � 1 _ 8 ) 2 � v 2 � 1 __ 64
15 k � 416 Centre (�1, �1) and radius √
__ 2
Exercise 3I 1 cos 7x � i sin 7x 2 a 16 b 256
3 b ( z2 � 1 __ z2 )
3 � 2 cos 6� � 6 cos 2�
c a � 1 _ 4 , b � 3 _ 4
d k � 3 __ 16
4 c cos2 ( 3� ___
10 ) � 5 � √
__ 5 _______
8 , cos2 ( 7�
___ 10
) � 5 � √__ 5 _______
8 ,
cos2 ( 9� ___
10 ) � 5 � √
__ 5 _______
8
5 a 4 √__ 2 ( cos ( � � __
4 ) � i sin ( � � __
4 ) )
b z � √__ 2 e � � i
__ 20 , √__ 2 e
7� i ___ 20 , √
__ 2 e
3� i ___ 4 , √
__ 2 e � 9� i
___ 20 , √__ 2 e � 17� i
____ 20
c
x
Z2
Z1
Z4
Z5
Z3
y
O
6 a i y � x � 2 ii (x � 2)2 � y2 � 8 b �2i, 4 � 2i c, d
x
R
(1, 1)
(0, �2)
(3, �1)
(2, 0)
(4, 2)
|z � 3 � i| � |z � 1 � i|
|z � 2| � 2 2
y
O2 � 2 2
2 � 2 2
7 a ( x � 4 __ 3 ) 2 � y2 � 25 ___
9
b z satisfi es both when z � 1.64 � 1.64i (2 d.p.)
c
|z � 2| � |2z � 2|
R
arg z � π
arg z �
� ( , 0) 3
( , )
O x
y
13
π4
43
4 � 346
4 � 346
ANSWERS
13
8 a (x � 1)2 � ( y � 1)2 � 1
b
O
y
x
2
1(�1, 1)
�1�2
|z � 1 �i| � 1
c |z|min � √__ 2 � 1
|z|max � √__ 2 � 1
d |z � 1|min � √__ 5 � 1
|z � 1|max � √__ 5 � 1
9 a
O x
y
(0, 6)
(4, 2)
arg ( ) �z � 4 � 2iz � 6i
π2
b 2 √__ 2
10 a
O x
y
(2, �4)
arg (z � 2 � 4i) � π4
π4
b 3 √__ 2
11 a 1 � (u � 1)2 � v 2
b
O
R
(u � 1)2 � v2 � 1
u
v
(1, 0) 2
�1
1
12 a
O
y
x
�2
�2 2
�6
(0, �4)
|z � 4i| � 2
b max value of |z| is 6 c Image is a circle, centre (0, �8) radius 4 d Image is a circle, centre (4, 0) radius 2 e Image is a circle, centre (�4, 0) radius 2 f Image is a circle, centre (0, 4) radius 213 a v � 2u � 2
b Circle centre ( � 1 __ 2 , � 1 __
2 ) , radius 1 __
2 √
___ 10
14 a � 2, b � �8, c � �15
15 Centre (�2, �4) radius 2 √__ 5 , k � 2
16 a a � 17 ___ 5 , b � � 13 ___
5 � c
b z � 3 � 4 __ 5 √
___ 10
17 a v � u � 1
c
O
v
u
(0, �1)
C
(1, 0)
( , � )
v � u � 1
u2 � v2 � u � v � 0
l
12
12
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