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