1
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 3ei
b 6i 6 e i __ 2
c 2 √__ 3 2i 4 e
5i _____ 6
d 8 i √
___ 65 e3.02i
e 2 5i √___
29 e1.19i
f 2 √__ 3 2 √
__ 3 i 2 √
__ 6 e
3i ____ 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 ) )
6 sin 1 __ 2i
(ei ei)
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 22
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 1 sin 3 3 sin 4 sin3 2 sin 5 16 sin5 20 sin3 5 sin 3 cos 7 64 cos7 112 cos5 56 cos3 7 cos5 4 cos4 1 _ 8 (cos 4 4 cos 2 3)
5 sin5 1 __ 16 (sin 5 5 sin 3 10 sin ) 6 a 32 cos6 cos 6 6 cos 4 15 cos 2 10
b ∫ 0 __ 6 cos6 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 e1.34i, 5
1 _ 4 e2.91i
b z √__ 3 e0.29i, √
__ 3 e1.80i, √
__ 3 e2.39i
c z √__ 2 e0.57i, z √
__ 2 e2.14i, z √
__ 2 e1.00i, z √
__ 2 e2.57i
4 a z 1 __ 2
√__ 3 ___
2 i, 2, 1 __
2
√__ 3 ___
2 i
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
5i ____ 6 , √
__ 2 e
2i ____ 3
7 a r √__ 8 , __
6
b z 4 e 2i ____ 9 , z 4 e
8i ____ 9 , z 4 e
4i ____ 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)
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
x
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 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
�6
�2 �8i
10
ANSWERS
e
xO
y
�1
�2
�3
3
2
1
�1�2�3 321
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
ANSWERS
11
5
O x
yiii arg(z � 4 � 2i) �
R
|z � 2| � |z � 6 � 8i|
π2
ii arg(z � (4 � 2i)) � 0
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 The locus is a circle, radius 6, centre ( 2 5 )
4 A circle, centre (1, 3) radius 8
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)
12
ANSWERS
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, with u 0 c w lies on the negative real axis, with v 0
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 _ 4 ) 2 v 2 1 __ 16
15 k 416 Centre (1, 1) and radius √
__ 2
Exercise 3I 1 cos 7x i sin 7x 2 a 16 b 256
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 ___ 20 , √
__ 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
ANSWERS
13
7 a ( x 4 __ 3 ) 2 y2 25 ___
9
b z satisfies 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
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
15 Centre (2, 4) radius 2 √__ 5
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