511 answers

A n s w e r s

511

answers

CHAPTER 1 Number systems: the

Real Number System

Exercise 1A — Classification of

numbers

1

2

Exercise 1B — Recurring decimals

1 c d e f g j k l n o p q r t u v w x y z 2

7 Irene. It can also be written as .

Exercise 1C — Real and complex

numbers

1

2

3

9 a 2i b 3i c −5i d −10i

Exercise 1D — Surds: a subset of

irrational numbers

1 b d f g h i l m o q r s t w z

Exercise 1E — Simplifying surds

1

2

3

Answers

a Q b Q c Q d I

e I f Q g Q h I

i Q j Q k Q l Q

m I n Q o I p Q

q Q r I s I t I

u Q v I w I x Q

y I z Q

a Q b Q c Q d Q

e Q f I g I h Q

i I j Undefined k I l I

m I n Q o Q p Q

q I r I s Q t Q

u I v Q w Q x Indeterminate

y I z Q

3 B 4 E 5 C 6 D

a b c d

e f g h

i j k 2 l

m n 1 o 3 p

q r s t 3

u v w x

y 1 z

3 E 4 D 5 C 6 E

a Z+ b Q c Q d Q

e Z+ f Q g Z+ h Z+

i Z+ j Z− k Z− l Q

m Z− n Z+ o Q p Q

q Z+ r Z− s Z+ t Q

u Z− v Q w Z+ x Z+

y Z− z Q

a Z+ b I c I d Q

e Z− f I g Z− h I

i Q j Q k Z+ l Z−

m Q n Q o Z+ p I

q Z− r I s I t Z−

u I v Z+ w Q x I

y Z+ z I

2 9

--- 7

9

--- 8

9

--- 5

9 ---4

9

--- 1

6

--- 17

45

--- 19 45 ---31

45

--- 32 45

--- 28 45

--- 53 99 ---4

33

--- 34 99

--- 367 495

--- 361 999 ---427

999

--- 868 1665

--- 323 999

--- 152 333 ---13

18

--- 157 300

--- 1237 1980

--- 5611 9000 ---268

999

--- 2 13

---0.02

a Q b I c Q d Z−

e Z+ f Z g Z+ h Q

i Z+ j I k Z+ l Z

m I n Q o I p Q

q Z+ r Z− s Q t Q

u Z− v I w Z− x Z+

y Z z I

4 C 5 C 6 B 7 D

2 A 4 E 5 B 6 C

7 A perfect square and cube 8 m= 4

a b c d

e f g h

i j k l

m n o p

q r s t

u v w x

y z

a b c d

e f g h

i j k l

m n o p 5

q r s t

u v w x

y z

a 4a b 9ab c

d e f

g h i

j k l

m n o

p q r

s t u

v w x

y z

4 E 5 C 6 D 7 C

2 3 3 2 2 6 2 14

3 3 5 3 5 5 3 11

3 6 2 15 4 7 7 2

2 17 5 6 6 5 13 2

2 22 3 15 9 2 10 2

7 5 8 5 8 7 7 15

9 5 7 6

4 2 15 2 24 10 24 7

36 5 10 17 21 6 40 2

30 3

– 18 7 –28 5 18 30

64 3 10 2 2

2 2 6 2 3 2 2

1 3

--- 15 _{20 5} 3

2

--- 7 7

2 --- 11 8 3 –3₂--- 5

6a 2

3ab 6 3a 10b 4a 3ab

13a2 _{2 5a}2_{b 6 13ab 2ab}

2a2_b3 _{3ab 2ab}2 _{17ab 4x}3 _5y

5x3_y2 _{5 24x y 20xy 5x}

14xy 7xy 54c3_d2 _{2cd 9c}2_d2 _14d 18c3_d4 _{5cd 28c}5_d5 _{6 22ef} 2

3

---e2f3 30 _7e5_f5 _2ef 3 4 ---e6f2 7f 1

9

---xy4 _6xy 1 3 ---x5_y6 ₃

1A

➔

512

A n s w e r s

answers

Exercise 1F — Addition and subtraction

of surds

1

2

3

Exercise 1G — Multiplication of surds

1

2

Exercise 1H — Division of surds

1

2

7

8

9

Exercise 1I — The Distributive Law

1

2 a b c d e f

a b c

d e f

g h i

j k l

m n o

p

a b

c d

e f

g h

i j

k l

m 15 − 10 + 10 n 0

o p

q r

s t 0

u v

w x

a b

c d

e f

g h

i j

k l

4 D 5 E 6 A 7 E 8 B

9 a cm b cm

c cm d m

e m f m

a b c d

e f g 10 h

i j k 27 l

m n o p 126

q 120 r 144 s t

u v w 2 x

y z

7 5 17 2 8 3

19 7 15 5+5 3 15 2+7 6

4 11 5 13 13 2

10 7–11 5 –3 6 –7 2+5 6 17 3–18 7 5 xy 8 x+3 y

x–5 y+7 xy

10( 2– 3) 2 2

5( 5+ 6) –6 6+2 3

7 3 10 2

4 5 5 5

14 3+3 2 11–4 11

3 6+6 3 17 2

10 15

8 11+22

– 39 3

12 30–16 15 2 5

– –5 2–2 30+2 15 12 ab+7 3ab

7 2

--- 2+2 3 –3 2

15 2 58--- 3

34 a–6 2a 52 a–29 3a

6 6ab 32a+2 6a+8a 2

a 2a a+2 2a

3a a+a2 _3a (_a2+_a) _ab

4ab ab+3a2_{b b 3 ab}(_2a+₁)

6ab 2a

– +4a2_b3 _{3a –2a b}

12 2 (6 6+8 3)

18–2 3+2 5

( ) 3π 5

18 2+2 5

( ) 21 11

14 55 42 2 6

4 3 6 2 5 15

3 7 4 10 30 3

10 33 96 6 180 5

120 3 360 3

2 6 6 23

---4 3 --- 5 2

5

--- 6 _{3 3}

a b

c d

e f

g h

i j

3 a 98 cm2 _{b 75}π _cm2

c m2 d m2

e m2 _{f m}2

4 E 5 C 6 D 7 A 8

a b c 2 d

e f g 4 h

i j k l

m n 1 o 1 p

q 1 _{r s} t 2

u v w x

a b c d

e f

3 B 4 E 5 A 6 C

a m b cm c m

d m e cm f cm

a b c

a b 126 L

a b

c d

e f

g h

i j

x2_{y y x}2_y3 _x

3a4_b2 _{2ab 5abc}2 _2abc

6a5_b2 _{2b 6a}3_b4

3x2_y2 _{10xy 15x}6_y2 ₂ 9

2

---a2_b4 _5ab 1

2

---a3_b2 _2ab

20 11 6 6

45π+96 10

( ) 72 15

15360 2

5 7 2 3

6 15 35

---3 4

--- 4 2 3

--- 5 2

--- 5 6

2 3 15--- 2 6

4 5

--- _{3 3 2 17} 2

5 ---x

y

-- 1

x3_y2

--- 2

x3_y4

--- 6x xy

2xy 3y 2x 2

3y

--- 4 a 3

--- 3b

2 _2b

2a a

---2 ---2 3m3_{n m}

--- 15 2m2_n2

---4 13 4 6 7 11

3 7 5 13 15---₂ 5

4 5 5 3

3

--- 2 2

35 2

21+6 3 3 10–7 5

2 5– 10 6+ 10

126 2–14 3 10 21–4 6 72+14 30 –30 15+80 6

24 2

– +12 60 3+24 5

3 10+9 2–5 5–15 35

– –11 4 – –40 3

24 3–18 30–8 10+60 112–140 3+24 6–90 2 2 55–2 22–4 15+4 6

A n s w e r s

513

answers

g h i j 3

4

9 a b

Exercise 1J — Rationalising

denominators

1

2

7

Exercise 1K — Rationalising

denominators using conjugate surds

1 a

b

c

d

e

f

g

h

i

j

k

l

m

n

o

p

q

r

s

t

u

v

w

x

a b

c d

e f

g h

i j

a −46 b 18 c 11

d 50 e 6 f −2

g 10 h −5 i 7

j 51 k 26 l −1

m 7 n 17 o 63

p 44 q 53 r 343

s 76 t 17 u x−y

v 2x− 3y w 9x− 16y x 4x3− 25y

y xy(49x− 9y) z xy(81x− 25y)

5 A 6 C 7 E 8 D

a b c d

e f g h

i j k l

m n o

a b

c d

e f

g h

i j

k

3 B 4 D 5 C 6 A

a b c

10 35+14 14–15 10–42

180–30 3–18 6+9 2

15x+26 xy+8y

4x+2 5xy–10y

27+10 2 16+4 15

18+6 5 53+10 6

35+12 6 53+12 10

104+60 3 14–6 5

10–2 21 37–8 10

57–12 15 59–12 15+9 5–6 3

5 2 2

--- 7 3

3

--- 4 11

11

--- 4 6

3

---2 ---21 7

--- 10

2

--- 2 15

5

--- 3 35

5

---5 6 6

--- 4 15

15

--- 5 7

14

--- 8 15

15

---8 21 49

--- 8 105

7

--- 10 3

---2+2 3 10–2 33

6

---12 5–5 6

10

--- 9 10 5

---3 10+6 14

4

--- 5 6 3

---3 22–4 10

6

--- 21– 15 3

---14–5 2

6

--- 12– 10

16

---6 15–25

70

---21 7

---± 15

3

---± 2 6

3 ---±

5–2

3+ 6

3

---2 ---2+ 5

3

---2 6+ 7

17

---8 11+4 13

31

---2 ---21– 35

14

---15 ---15–20 6

13

---9 11+9

20

---5 14+2 10–25 7–10 5

155

---12 2–17

19–4 21

5

---9 2+ 154

4

---20 2+9 10+4 30

– –9 6

2

---3 12

---3 ---3+2 6

18

---10 3+15 6+9 2+27

( )

–

42

---12 3–4+3 6– 2

52

---60 2+10 30–6 10–5 6

35

---115+31 21

148

---71–12 33

17

---18 2+10 6–9 3–15

102+48 6

95

---9 154

– +132+42 2–8 77

50

---7 3+9

3

---

1F

➔

514

A n s w e r s

answers

y

z

2 a

b

c d e

f g h

i

j

k

l 3 4 a

b

c

d e 44

f g h 5 Yes

6 a

b

Exercise 1L — Further properties of

real numbers — modulus

1

6 a

b c R and y≥ 0

Exercise 1M — Solving equations using

absolute values

1 a x= ±5 b x= 4 or x= −6

c x= 1 or x= − d x= or x= −2

e x= − f x=±9

2 a x= 2 b No solutions

c No solutions d x= 5

3 a x= 6 or x= 1 b x= 10 or x=

c x= 3 or x=− d x= 1 or x= 3

Exercise 1N — Solving inequations

1 a 1 < x < 2

b x > 2 or x < −2

c 1 < x < 2

d 2 < x < 3

2 a x < 0 or x > 2

b x < 1 or x > 1

c x < 3 or x > 4

d x > 2 or x < 2

a −2 b

7 D 8 B 9 C 10 A

21 5–6 14–5 70 20–

27

---6

– +6 2+ 10–2 5

2

---9 2+8

14

---9 7–13 3

120

---16 210–12 14

77

---6–7 2

45+15 14+9 10+6 35

( )

–

5

---66+24 6

5

---5–4 14

959+281 77+182 7+6 11

629

---3+7 65–16 11

28

---41+6 30

( )

– 12

---230+257 3–137 5–80 15

431

---–6+2 15+3 10–5 6

6

---14 19

--- 6 35

19

---210 2–120

41

---200 2–126

41

---99 238–50 400 2

1681

---99 120–50 460 2

1681

---103–90 2

7 2+4

295 2–382

49

---11+ 5

6

---42 5+28 7

17

---a 19 b c 0.75 d

e 8 f 2a g 12 h

i 3.21 j 0 k − l 4

m10 n 10 o a2b2 p −16

q 27 r 15 s −72 t −54

u v −8 w −11 x 30

y −3a z −6cd

2 C 3 E 4 D 5 B

x –2 –1 –0 –1 2 3 4 5 6

y –8 –6 –4 –2 0 2 4 6 8

|x| –8 –6 –4 –2 0 2 4 6 8

1 4

--- 15

1 2

---2 3

---1 2

---y = |2x – 4| y y = 2x – 4

x 2 –4

4

1 2

--- 1

2

--- 2

3

---1 2

---1 3

--- 2

5

---1 5

--- 1

2

--- 1

4

---0 1 2 3 4

0 –1 –2

–3 1 2 3

1 2

---0

–1 1 1–2 3 4 2

1

0 1 2 3 4

0

–1 1 2 3 4

1 3

---0 1 1– 2 3

3 1 1

2

---1 2 3 4 1– 5 2 4 1

2

A n s w e r s

515

answers

3 a −4 < x < 4

b −5 < x < −3

c 2 < x < 3

d −9 < x < 11

4 a x < −1 or x > −

b x < 1 or x > 1

c x < 3 or x > 9

d x < −3 or x > −

e 1 < x < 7, x≠ 3

f 3 < x < 5 , x≠ 4

Investigation — Real numbers —

application and modelling

1 −1 − <x<

2 3 x≠−3

4 + , 3 + 2 5 , m= 3, k= 4

6 g.c.d. = 225

Chapter review

1 A

2 a Irrational, since equal to recurring and non-terminating decimal

b Rational, since can be expressed as a whole number

c Rational, since given in a rational form

d Rational, since it is a recurring decimal

e Irrational, since equal to recurring and non-terminating decimal

3A 4D 5E

6

7 B

8 9 E

10 a , , ,

b , ,

11B 12C

13 14 A

15

16

17 D

18

19A 20E

21 22 A

23 D

24 B

25 a

b

c 26 C

27

28

29 E

30 C

31

32 m2

33 A

34

35 C 36C 37C 38A

39 D 40B

Modelling and application

1

2

3

CHAPTER 2 Number systems:

complex numbers

Exercise 2A — Introduction to complex

numbers

1

a b c

a Z− b Z+ c Q d I

0 –4

–5 –3–2–1 1 2 3 4 5

0 –4

–5

–6 –3 –2 –1

5 2

1

0 3 4

10 0

–5

–10 5

1 3

--- 2 3

---0 –1 –2

–3 1_– 1 2

3

–1 2_–

3

–

1 4

--- 3 4

---2 1

0 1_– 3

4

1 3_–

4

1

3 4 5 6 7 8 9 10 2

1 3

---1 –1

–3 –2 0

–4 1_– 2

3

_

2 3

---5 3

1 2 4

0 2_– 6 7 8

3

1

1 5

--- 1 3

---6

4 5

3 1– 7

3

5

1 – 5

3

2 2

√2 0 –1 –2.4–2

–3 1 2

a– b

( ) a2–b

a2–b

--- 1 2

---7 5 2 1

4

---7 1

1 1

3 1

1 1

2 1

13 ---+ ---+ ---+

---+

---+

62 99

--- 337 900

--- 157 165

---2m 20

m

--- 3m 3_8m

25m m

16 --- 20

m

---a b

a b

a m b cm

c m d 22 cm

a 27 b

a b c 3a

a −11 b −3

a amps b amps

c amps d amps

a cm b cm3

c cm3 _{d cm}3

a cm3b cm c cm

a 3i b 5i c 7i d

e f g i h i

72x3_y4 _2xy 1

4 ---x2_y5 _xy

–

25 3 3ab ab

5 (17–4 6)

26–4 2

( )

720 2

23 6–48 cm

2323–594 14 50

---2277–606 14 50

---51–12 14–18 7+27 2

5 7 4

---x 5y 2

--- x2_{y 2}

3 7– 3 40

---3 2

---5 10 3

--- 5 102

6

---135 38 19

--- 5 34 2

---2 37 18π 37

54π 3 18π( 37+3 3)

360 10 3 10π

π

--- 2 15π

π

---3 i

11 i 7i 23

---6 5

---

1K

➔

516

A n s w e r s

answers

2

3

4 z= −2 − 3i, w= 7 + 3i 5

6 4 − i

7 a E b C c C d E

8 Solution will vary.

Investigation — Plotting complex

numbers

1

2 zw= −17 + 7i (see figure above)

3 Length of z= , length of w= ,

length of zw=

γ= 11.31°, β= 146.31°, α= 157.62°

Length of zw= length of z× length of w; α=γ+β

Exercise 2B — Basic operations using

complex numbers

1

2

3

4

5

8

9

10

11 a b

c d

e f

g h

12

Exercise 2C — Conjugates and division

of complex numbers

1

2 a

b a 9, 5 b 5, −4 c −3, −8

d −6, 11 e 27, 0 f 0, 2

g –5, 1 h 0, –17

a −1 +i b 1 +i c 1 −i d 0 + 0i

e −1 + 2i f −1 +i g 1 + 0i h 1 – 2i

a −5 b 15 c 0 d −6

e 2 f 0 g −9 h 2

a b

c d

e f

a 4 −i b 1 − 14i c −4 − 2i d 9 − 13i e −12 + 4i f −9 − 5i a −12 + 3i b −19 − 8i c 12 + 23i d −25 + 3i e −50 − 48i f −41 − 28i a 7 − 23i b 4 + 45i c −50 − 13i d 63 − 37i e −85 − 132i f 176 − 61i a 111 + 33i b 31 − 8i c 22 − 48i d 61 e −53 f 32 − 126i

Im (z)

0 Re (z) zw = –17 + 7i

w = 5 + i z = –3 + 2i

α β

γ

13 26

338

Im(z)

1 2 3 0

1 3 + i

Re(z)

1 2 3 4 –1

–2 –3 –4

–5 4 – 5i Im(z)

Re(z)

–1 –2–1 0

–2 –3 –4 –5 –6 –2 – 6i

Im(z) Re(z)

0 _1234567 2

1

7 + 3i 3

Im(z)

Re(z)

1 0 _{2 3} –1

–2 5 – 2i Im(z)

Re(z)

0 2 1

–8 –8 + i3

Im(z)

Re(z)

6 14 + 52i 7 −3

a −8 b −5 c −9 d 35

e −30 f −115

a x= 5, y= −2 b ,

c x= 1, y= 5 d x= −2, y= −3

a E b B c C

a 7 − 10i b 5 + 9i c 3 − 12i

d e 5 − 2i f

x 21

41

---= y 16

41 ---– =

0 4

3 3 + 4i Im(z)

Re(z)

0

–2 11

11 – 2i Im(z)

Re(z)

0

–2 6

6 – 2i Im(z)

Re(z)

0

–4 2

2 – 4i Im(z)

Re(z)

0 ₁₀

Im(z)

Re(z)

0

–10 –10i Im(z)

Re(z)

0

–24

32

32 – 24i Im(z)

Re(z)

0 16

–88 –88 + 16i

Im(z)

Re(z)

–3 –2 –1 1 2 3

–3 –2 –1 1 0 2 3

i2_z iz, i5_z

i3_{z, –iz} z, i4_{z, –i}2_z Im(z)

Re(z)

7+3i –6+ 11 i

Im(z)

Re(z) = 3 – i = 3 + i z

z

= –1 + 3i

= –1 – 3i Im(z) z

z

A n s w e r s

517

answers

c

3 Answers will vary.

4

5 (10 + 24i)

6 + i

7

8

12

17

22 x= 2, y= 1; x= −2, y= −1

23 a i 13 ii 5

c i − i ii − i

d −8 + 16i

e −2 + 10i

f

Exercise 2D — Radians and coterminal

angles

1

2 a b c d e

3 a 210° b 225° c 240° d 300°

4

History of mathematics

1 Probability 2 He was a foreigner.

3 Tutoring students and writing books

4 Newton

5 De Moivre predicted it.

Exercise 2E — Complex numbers in

polar form

1 a b

2 a 13 b 3 c

d e f 5

3 a i ii

b i ii

c i ii 10

d i ii

e i ii

a b c

d e f

a 0 − i b 0 − i c − + i

d − i e + i

f

a b c d − e −

9 + i 10 –29 11 −33 + 58i

a D b C c B

13, 14 and 15 Solution will vary. 16 −16

a −12 + 11i b −30 − 19i c 0

18 Solution will vary. 19 a –4, 16, –64

20 x= −1, y= 21 a= − , b=

= – 4 –5i = – 4 + 5i Im (z) z

z

Re (z)

2+i

5

--- 3–i 10

--- 4+3i 25

---5–4i

41

--- –3–2i 13

--- 3+ 2i 5

---1 2 --- 1

2

---7 25 --- 26

25

---14 29 --- 23

29

--- 43

53 --- 18

53

---2 5– 6

7

--- 2 2+ 15 7 ---i

+

23 10

--- 9

10

--- 17

5

--- 16

5

--- 14

5

---17 2 --- 9

2

---2

± 1

2

--- 1

2

---2 13 --- 3

13

--- 1

5 --- 2

5

---4 – +7i

65

---2 0 π π

_– 6 π _– 2 π –

4 π

7 _— 4π 3 — 2 π 5 — 4 π 7 — 6 π 5 — 6π 3 — 4π

Im (z)

Re (z)

π

4

--- π

3

--- 3π

4

--- 3π 2

--- 5π 6

---(b) (a) (d)

(e) (f)

(c) Im (z)

Re (z)

4 8

0

z = 4 + 8i Im (z)

Re (z)

z = 4 5

65

3 5 5

0 4

–1 z – w

Im (z)

Re (z)

17

0 6

1 u + z Im (z)

Re (z)

37

0

–8 6

w – u Im (z)

Re (z)

0

–2 7

w + z Im (z)

Re (z)

53

0

–7

9

z + w – u Im (z)

Re (z)

130

2A

➔

518

A n s w e r s

answers

f i ii 10

4 a b 42.5 square units

5 a b 24 square units

6 a 0.588 b c d 2.034

e f 5.253 g h π

i j 0 or 2π

7 a b c d

e f g h

8

9

10

Investigation — Multiplication in polar

form

1 a 2 − 2 + 2( + 1)i

b 4 cis

2 a 2 − 2 − 2( + 1)i

b 4 cis

Exercise 2F — Basic operations on

complex numbers in polar form

1

2

3

4

5

6 a − b − i c − + i

d e 0.171 – 0.046i f 16

7 8 1

9 a B b C c E

10 11 16 − 16i

12 −64 + 64i 13

14 a ±(3 + 2i) b ±(3 − 2i)

c d ±(2 −i)

a b c

d e

f g

a b

c d

e cis f

a b (1 +i)

c d

e (1 +i) f −8i g

11 C 12 B 13 D 14 E 15 D

0 6

8

z2

Im(z)

Re(z)

–4 –2 2 4 6 8 10 4

6 z2 z3

z₄ z₁ 2

Im(z)

Re(z)

–4 –2 2 4 6 8 2

0 4 6 8 10 12

z w

u

Im(z)

Re(z)

π 6

--- 7π

4

---4π 3

--- π

2

---3π 2

---π 2

---– π

6

--- π

8

---– 3π

4

---5π 6

---– 6π

7

--- 2π

5

--- 11π

12

---3 2 π

4 ---–

, 5 2 3π

4

---, 2 2π

3 ---– ,

8 π 6

---, 149,–2.182

2 10 1.893, 4 π 3 ---,

2 cis 3π 4

--- 2 2 cis π

6

---10 cis 3π 4 ---–

 

  2 5 cis π

3 ---–    

2π 3 ---–

 

  2

4 --- cis 3π

4

---1

– + 3i 3 2

2

---5 2

--- (− 3+i) 2–2 3 i

14 2

--- – 3

a b c

d e

a b

c d

e

a b

c

a b

c d

e

a i ii

b i 16 cis π ii −16

c i 9 cis π ii −9

d i ii

3 3

2 7π

12

---3 3

2 7π

12 ---–

 

 

6 cis 3π 4

--- 20 cis π 3

--- 6 5 cis π

4 ---–    

6 cis 2π 3

--- 2 7 cis π

6 ---–    

3 2

– +3 2 i 10+10 3 i

3 10–3 10 i 6

2

---– 3 2

2 --- i

+

21– 7 i

4 2 cis 5π 12

--- 8 3 cis π

2 ---–    

8 2 cis π

12 ---–

 

 

3 cis π 2

--- 4 cis 11π

12

---2 cis 3π 10 ---–

 

  2 2 cis 3π

14 ---–

 

 

3 2

4 --- cis 7π

12

---3 ---3 cis π 4

--- 3 6

2 --- 3 6

2 ---i

+

32 cis 3π 4

--- –16 2 + 16 2i

1 4

--- 1

8

--- 1

4

--- 1

4

---3 64 --- – 1

64 ---i

64 3

– –64i

8 9 --- π

6 ---– ,

2 5 --- – π

120 ---,

1+ 2+ –1+ 2i

( )

±

A n s w e r s

519

answers

Investigation — Complex numbers:

applications

2 a iii z2=r2 cis 2θ

iii z3=r3 cis 3θ iii z4_=r4 _{cis 4θ} iv z5=r5 cis 5θ iv z6_=r6 _{cis 6θ} vi z7=r7 cis 7θ b GP, a=r r=r GP,T_n=arn – 1

c Geometric progression

3

z₁= 1 = cis 0

z₂=−1 + i= cis

z₃=−1 − i= cis

Chapter review

1 B 2 B 3 −1 − 2i

4 C 5 E 6 C

7

8

9 C 10 A 11 A

12 13 D 14 B

15 16 E 17 A

18 B 19 D 20 C

CHAPTER 3 Matrices

Exercise 3A — Operations with

matrices

1

2

3

4

5

6

7 Different orders

8

9

10

11

12

Exercise 3B — Multiplying matrices

1 a A (2 × 2), B (2 × 2), C (3 × 2), D (1 × 2),

E (2 × 3), I (2 × 2)

b CA, DB, AE, AI, IA, IB, A2, EC

c (3 × 2), (1 × 2), (2 × 3), (2 × 2), (2 × 2), (2 × 2), (2 × 2), (2 × 2)

d

2

a −48 b a= −549, b= 296

a 11 −i b c

Matrix Order 2, 1 element 1, 3 element

A 2 × 2 8 —

B 3 × 1 5 —

C 1 × 4 — 10

D 2 × 3 4 4

E 3 × 3 1 2

a b c d

0 z₁

z₃ z₂

π 2— 3

π –2_—– 3

Im (z)

Re (z)

3 2π

3

---3 2π

3 ---–

 

 

29 6 5

1 17

--- 12( –14i)

7 2 cis 3π

4 ---–

 

 

3 3 0 9

7

– 3

8 5

6–3

2

– 8

3 –6

2

– –1

a b c

d e

a C b D c E d A e B

a b c

d e f

a b c d

a b

a b

a True b True c False d True

a b No

4

– 6

8 14

9

– 6

12 12

11 6 4

– 20

9 9 0 27

7 18 12

– –4

2 0 14 4 0 0 6 0 18

0 8 0 0 10 16 0 12 0

2 8 14 4 10 16 6 12 18

3 4 21 6 5 8 9 6 27

4 0 28 8 0 0 12 0 36

1

– 4–7

2

– 5 8

3

– 6–9

2 0

3–1

2

– 0

6

– 2

1

– 0 1

2 3 –1

2 –6

1

– –12

2

– –1

0 1 3 1 1 0 2 2 3 2 0 1 1 2 1 0

0 0 1 1 0 0 1 2 1 1 0 3 1 2 3 0

82 54 75 68 91 82

15 14 104 7 10 52

13 7 5 1 31 18 26 12 4 4 4 17 15 16

14 8 5 1 35 19 29 13 4 4 5 18 19 16

20 14 44 22 4 5

2–2 –418–8

8

– –8 6

2–3

4 5

2–3

4 5

1 1 1 0

8

– –21

28 13

14 15

24

– –30

10 20 5

– 10

8 26 4

– 12

2E

➔

3B

520

A n s w e r s

answers

3

4 a i ii iii

b All are I c Multiplicative inverses

5

6

7 a b

c Southport 120, Broadbeach 99, Lions 74,

Eagles 70

8 a b Shop A = $820, Shop B = $345

History of mathematics

1 Matrix theory and number theory

2 Computer development

3 Cross of Honour 4 Caltech

Investigation — Matrix powers

1 a 2A=

b A2= Not conformable

c Cannot multiply A×A if A is a 3 × 2 matrix

2 a 2A=

b A2=

A2=

3 a 2_A=

b A2_{= Not conformable}

c Cannot multiply A×A if A is a 2 × 3 matrix

4 a If a matrix is to be raised to a power it must be a

square matrix.

Exercise 3C — Powers of a matrix

1 a b c

2 a b c

3 a b

4 a

Investigation — Applications of

matrices

Patio Office Bank Hotel

1 a iii Venue: V= [ 5 7 3 4 ]

iii Type and number:

Hanging Indoor Fern Camellia Geranium basket plant Palm

T=

iii Cost: C=

b iii Quantity:

Hanging Indoor Fern Camellia Geramium basket plant Palm

Q= [ 22 22 12 39 49 18 ]

iii TC= $2192

iii CD=

a b c d

e f g h

i j

a A b C c D d B

a

b Sharks have a total of 32 points. Dolphins

have a total of 31 points.

4 3

0–9

2–7

24 9

0 0 0 0

3 2 8

– 5

2 0

0–3

10–4

24

– –9

31 0 0 31

10–11

16 1

0 0 0 0

3 2 8

– 5

1 0 0 1

1 0 0 1

1 0 0 1

3 1 0

32 31

18 12 14 15 10 14 9 16

6 1

10 25 12

x y z w

p q

x y z w

p q

x y z w

p q

x y

z w

x y

z w

x y

z w

x2+yz xy+yw

zx +zw zy+w2

x y z

w p q

x y z

w p q

x y z

w p q

4–2

0 0

8–4

0 0

16–8

0 0

1 0 0

0 1 0

0 0 1

1 0 0

0 1 0

0 0 1

1 0 0

0 1 0

0 0 1

1 0 0

8 9 0

0 0 1

1 0 0 26 27 0 0 0 1

1

– –1–1

1 1 2

1

– –1–1

2 1 1 2 0 0

0 0 1 3 2 1

0 3 0 0 5 1

3 2 0 2 5 2

22 15

8 12 10 18

163.80 147.60 203.40 370.80

Fern Camellia Geranium Hanging basket Indoor plant Palm Patio

Office Bank Hotel

Patio Office Bank Hotel

A n s w e r s

521

answers

2 a $5018 b $1 661 420

Eggs Flour Sugar Shortening Milk

3 a I=

S₁= S₂ = O =

Eggs Flour Sugar Shortening Milk

b TI= [ 260 780 198.75 142.70 95 ]

c S₁ quote = $13 679.90

S₂ quote = $16 585.50

S₁ cheaper by $2905.60

d SP=

e Total takings: $25 992.20

Exercise 3D — Multiplicative inverse

and solving matrix equations

1 a b

2

3

4

5

6 Answers will vary.

7

8 a D− det = 0 b E− det = 0

c F− Not a square matrix

9

10

11

12

Exercise 3E — The transpose of a

matrix

Answers will vary.

Exercise 3F — Applications of matrices

1

2 a and b Answers will vary. c det = 0

d i ii

e In i there are parallel lines; in ii there is only one

line.

3 a E b B 4 a C b D

5 16, 4 6 15, 10 7 Anh 8 $51 070

Exercise 3G — Dominance matrices

1 Mair, Ann, Janine

2 20 points to Hamilton, 15 to Leslie, 10 to

Cunningham, 5 to Barnes

3 a M=

b W= 5, C= 4, I= 3, S= 2, G= 1 based on

V = V₁ + V₂ and original results.

a 5 b 12 c −2 d −8 e 7 f 14

a b c

d e f

a C b E c D d A

a b c

d e f

1 4 0.25 0.25 1

0 3 – 0.25 0

4 3 2 1 1

1 1 – 0.33 0

0 2 3 1 1

10 8 10 12 12

12 10 12 15 12

15 150

45 65 35

113.05 51.30 205.20 41.73 133.00

AB 6 1 0 0 1

= A–1 1

6 ---B

= B–1 1

6 ---A =

MN 2 1 0

0 1 M–1 ,

– 1

2

---N, N–1

– 1

2 ---M –

= = =

1 5 --- 10–3

5

– 2

1 12

--- 0 3

4

– –2

1 2

--- 1 6

0–2

1 8

--- –1 3

4–4

1 7

--- –5 1

3–2

1 14 --- 4 1

6

– 2

1 2

--- 1 6

0–2

1 4

--- 1 2

2

– 0

12 2

2

– –1

1 8

--- 1 2

2

– –12

1 8 --- –112

4 0

1 8

--- 1 2

2

– –12

a b

a b c

d e f

g h

a b

a x= −2, y= 1 b x= 1, y= 2 c x= −2, y= 3 d x= 7, y= 4

a (5, −1) b (3, 0) c (10, 2)

d (0, 0) e (4, 4) f (−2, −3)

0 8

1

– –2

1 8 --- –2–8

1 0

1 2

--- –31–22

24 18

1 2

--- –5 5

14–8

1 6 --- –62

6

– 4

1 2

--- 18 23

12

– –16

1 30

--- 78 103

24

– –34

1 0 0 1

1 15 --- 1 –5

2 5

1 15

--- –132–114

186 162

2 1 –

2 3

x y

0 _{2 3}

–3 –2 –1

4 – 3

Both lines

x y

0 2

–3

0 0 0 1 1

1 0 1 1 0

1 0 0 0 0

0 0 1 0 1

0 1 1 0 0

1 2

---

3B

➔

3G

Eggs

Flour

Sugar

Shortening

Milk

Sugar rolls

Bread

Cakes

Pastries

Buns Sugar rolls

Bread loaves

Cakes

Pastries

Buns

Sugar rolls

Bread loaves

Cakes

Pastries

Buns

522

A n s w e r s

answers

Chapter review

Modelling and application

a 1: Despatch for Deluxe model takes 1 hour. b 14: Packaging at Plant 1 has a wage rate of $14 per

hour.

c 3 × 3, 3 × 2, 3 × 2

d

e The total costs for the Standard model at Plants 1 and 2 f The assembly costs for each model at Plant 1 g i $529.50 ii $514.00

CHAPTER 4 An introduction to

groups

Exercise 4A — Groups

1 a 12, 20, 28, 36 … b 4, 10, 16, 22

2 a 0, 1, 2 b 0, 1, 2, … 8

c 0, 1, … 10

3 a b

c

History of mathematics

1 He tutored students.

2 Abelian groups are those that have the property of commutativity.

Exercise 4B — The terminology of

groups

1 a = 2 and 2 is not an element of the set of

whole numbers. b None.

2 103 = =

Not a whole number, ∴ not closed.

3 IE = 1 Inverse of b= 2 −a 4 IE = 0 a_° 0 =a+ 0 −a0 5 IE =b= 4a× ( )2_=a

6 Assuming this operation has an identity then let

=a

a+b=a2_b a=a2b−b

But a≠a2b−b therefore the operation has no identity.

7 Let (0, 1) = (a, b) = IE, therefore (0, 1) _° (c, d) = (0 +c, 1 ×d) = (c, d) 8 Let (a+b)2=a where b= IE

Take the square root of both sides: a+b= ± If a is negative then ∉R. Since an identity must be applicable to all elements of the set, there is no IE for a_°b.

Exercise 4C — Properties of groups

1 a [R, +] It is closed, associative, IE+ = 0,

inverse = −a, therefore it is a group. b It is Abelian.

2 a Closed, associative, no IE since

0 ∉ {even numbers}, there is an inverse; therefore not a group.

b Closed, associative, no IE since

1 ∉ {even numbers}, no inverse; therefore not a group.

3 a 12_{+ 1}3 _{is not closed; not a group.}

b 12_{× 1}3 _{is closed, and associative, IE× = 1, there is} an inverse; so it is a group.

5

Closed, associative, IE× = 1 and there is an inverse; therefore it is a group.

6 a

b Under addition: not closed, associative, no IE+ since 0 ∉ 22n, no inverse (always +ve); not a group.

c Under multiplication: closed, associative, IE× = 1 is not present as no 20 (0 ∉ {even numbers}), inverse is 2–2n; not a group.

7

It is closed and associative, IE× = 1; inverse does not exist since there are no 1s in the first row or column. This is not a group; therefore, it is not Abelian, even though the commutative law does apply.

1 D 2 b 3 C 4 B 5 a, f, g, i

6 D 7 C 8 C 9 A 10

11 B 12 a b (5, −1)

9 3 6

– 0

1 10 --- 1 2

3

– 4

433.25 420.50 529.50 514.00 605.25 587.50

+ 0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

1 2 3 4 5 0

2 3 4 5 0 1

3 4 5 0 1 2

4 5 0 1 2 3

5 0 1 2 3 4

× 0 1 2 3

0 1 2 3

0 0 0 0

0 1 2 3

0 2 0 2

0 3 2 1

× 0 1 2 3 4

0 1 2 3 4

0 0 0 0 0

0 1 2 3 4

0 2 4 1 3

0 3 1 4 2

0 4 3 2 1

3+2 2 --- 1

2

--- 1

2

---1+9 10

1 2

--- 1

2

---a+b ab

---a a

× 1 2 3 4

1 2 3 4

1 2 3 4

2 4 1 3

3 1 4 2

4 3 2 1

+ 2 4 8 16 …

2 4 8 16 . . .

4 6 10 18 . . .

6 8 12 20 . . .

10 12 16 24 . . .

18 20 24 32 . . .

… … … …

× 0 1 2

0 1 2

0 0 0

0 1 2

0 2 1

A n s w e r s

523

answers

8 a Yes b No, not closed

c No, no inverse for b d Yes

9 a

b _{It is closed, associative, IE°} = 5, no inverse; so not a group.

10

Closed, associative, IE_°= N, there is an inverse, N appears in every row and column.

Investigation — Application of groups

Not closed ∴ Not a group

P_2° P₂= P_2° P₃=

P_3° P₂= P_3° P₃= = P₁

History of mathematics

1 He worked towards having women accepted at Cambridge University.

2 Abstract algebra, group algebra, n-dimensional geometry, matrices and determinants

Exercise 4D — Further examples of

groups — transformations

1

a

b IE = R₀, Inverse exists for all elements. It is an Abelian group because the table is symmetrical about the leading diagonal.

2

a

b Does not form a group.

3 Not Abelian.

4 a

b

c

5 … H H H H … R_V … H H H H …

R_H R₁₈₀ … H H H H …

… H H H H … R₀ … H H H H …

6

° 5 10 20

5 10 20

5 10 20

10 10 20

20 20 20

° N L R A

N L R A

N L R A

L A N R

R N A L

A R L N

° P1 P2 P3

P₁ P₂ P₃

P₁ P₂ P₃

P₂ P₃

P₁

1 2 3 4 5

3 4 5 1 2

 

  1 2 3 4 5

5 4 3 2 1

 

 

1 2 3 4 5

2 1 5 4 3

 

  1 2 3 4 5

1 2 3 4 5

 

 

R120

R0 R240

A

B C

B

C A

C

A B

° R0 R120 R240

R₀ R₁₂₀ R₂₄₀

R₀ R₁₂₀ R₂₄₀

R₁₂₀ R₂₄₀ R₀

R₂₄₀ R₀ R₁₂₀

RV B

C

RL RR

A

° R0 RV RL RR

R₀ R_V R_L R_R

R₀ R_V R_L R_R

R_V R₀ — —

R_L — R₀ —

R_R — — R₀

4 3

2 1

3 4 _R

V

R_H

R₁₈₀1 2

2 1

3 4

2

3 4

1 4

1 2

3

R₀

1

2 4 3

1 4

4 3

2

3 3 2

1 4

2 1

R_L R_R

2 1

3 4

R₀ R₁₈₀

R_V

R₀ R_H

R₁₈₀ 2

4 4

1 3

2 4

3 1

2

2 3 1

4 2

1 3

4 1

3

4A

➔

524

A n s w e r s

answers

7 a b Not a group

8 a Closed: addition of 2 × 2 matrices results in a 2 × 2 matrix.

Associative: matrix addition is associative.

Identity exists: is the identity element.

Inverses exist: the inverse of A is −A.

b i ii

c iClosed: multiplication of 2 × 2 matrices results in a 2 × 2 matrix.

Associative: matrix multiplication is associative.

Identity exists: is the identity element.

If ad−bc≠ 0 inverses exist,

A–1₌

9 Identity =I. Inverse is present as I is present in each row and column. Closed and associative

10 a IE+ = (Remember 0 is a complex

number.)

Inverse =

b IE× =I Inverse = where the

determinant is real. The inverse exists if the determinant ≠ 0.

11 b Yes

12 a , , ,

b Yes.

Investigation — Some applications of

group theory

1 Yes

2 29

5 1, 3, 5, 9, 13, 15, 19, 23, 25, 27, 39, 45

6

8 a Yes b No

10 a

b Yes

Chapter review

1 a Yes b No, not closed

2 a Yes

b No, 0 does not have an inverse

c Yes d Yes

3 a No, no identity b No, not associative

5 a Commutativity

b There is only element x such that p_°x=q and

x_°q=p.

c Each element has a unique inverse.

6 a Yes b No, 0–1 doesn’t exist

c No, inverses don’t always exist

d Yes

CHAPTER 5 Matrices and their

applications

Exercise 5A — Inverse matrices and

systems of linear equations

1 2 3 4

5 6 7

8 c d

History of mathematics

3 Science of determining the size and shape of the Earth.

Exercise 5B — Solving simultaneous

equations — the Gaussian elimination

method

1 x= 1, y= 1

2 a=−3, b=−1

3 x= 2, y= 1, z=−3

4 x= 1, y= 2, z= 0

5 x=−2, y=−1, z= 0

6 a b

° f1 f2 f3 f4

f₁ f₂ f₃ f₄

f₁ f₂ f₃ f₄

f₂ f₁

— —

f₃ f₄

— —

f₄ f₃

— —

0 0 0 0

1 2 3 4

1 2 1 2

1 0 0 1

1

ad–bc

--- d–b

c

– a

0 0

0 0

z1

– –z2

z2 –z1

1

z1 2

z2 2

+

--- z1–z2

z2 z1

i 0

0 i

1

– 0

0 –1

i

– 0

0 –i

1 0

0 1

e a b c d

e a b c d

e a b c d

a b c d e

b c d e a

c d e a b

d e a b c

P Q R S T V U W

P Q R S

P Q R S

Q S P R

R P S Q

S R Q P

T U V W

U W T V

V T W U

W V U T T

U V W

T U V W

V T W U

U W T V

W V U T

S R Q P

Q S P R

R P S Q

P Q R S

1 0 0 1

0 1 1

– 0

0–1

1 0

1

– 0

0–1

0 i

i0

i – 0

0 i

i 0

0–i

0–i

i

– 0

42 36

10.38 17.31

38 –

37 –

400 370

3 –

6

100.29 92.06

98.04 108.82

0.681

1.81

480.68 481.82

3

– 2

2 –1

1 2 --- 1

4

---–

1 2

---– 3₄

A n s w e r s

525

answers

c d

7 a x= 2.7, y= 0.6

b x= 2 , y= 1 , z=−

c x=−2 , y=−2 , z= 1

Exercise 5C — Introducing

determinants

1 2 2 13 3 − 4 5 a −5 b −33 c 81 d −3

e 61 f 0 g 0 h 24

i −24 j 122

Exercise 5D — Properties of

determinants

1 a −3 b −3 c Property 1

2 a 0 b Property 2

3 a 0 b Property 3

4 a 22 b −22 c Property 4

5 a −18 b −36 c Property 5

6 a 2 b 2 c Property 6

7 a 8 b Property 7

8 a 1 b Property 8

9 a −6 b −4 c 3 d −8 e 3

Exercise 5E — Inverse of a 3

¥

3 matrix

1 a iii 1 iii iii iv b iii −12 iii iii iv c iii 8 iii iii iv d iii −33 iii

iii iv 2 a b

c

3 X=

4 a b 5 x= ±

Exercise 5F — Cramer’s Rule for

solving linear equations

1 x= 2, y=−3 2 x=−1, y= 4

3 x=−2, y=−5 4 4x= 0, y= 3

5 x= 3, y=−1, z= 2

6 x=−11 , y=−5 , z=−

7 a (y−z)(v−u)

b (x−z)[(y2_{− y(x + z)] + x(x − z)}

c (1 − x)(1 − y) + y + x − 1

d (a − b)2 (2a + b)

8 a a = 1, 2 b a = c a = 0, 3

d a = 0, 1, 2 e x = 3, 4 f x =−1, 3

9

10 a −189 b c 12 a ′A =

a A′ =

13 a 1 + i b 0 c −4 − 7i

Investigtion — Applications of

determinants

1 y = −5x + 14

21₃--- 2

3

---– –11₃

---1

– 0 1

1 3

--- 1

3

--- 1

3

---–

1 2

--- 1

4

---– 0

0 1₂--- 0

1

– –1₂--- 1

1 5

--- 1

5

--- 3

5

---1 7

--- 3

7

--- 3

7

---1 2

--- 3

4

---3 –2

1

– 1

3 –1

2

– 1

3 –1

2

– 1

9 –6

5

– 2

9 –5

6

– 2

3 4

---– ₁₂---5 1 2

--- 1

6

---–

8 2 –4

12

– 2 8

8–2 –4

8 –12 8

2 2 –2

4

– 8 –4

1 –11₂--- 1

1 4

--- 1

4

--- 1

4

---–

1 2

---– 1 –1₂

---13 –5 11

12

– –3 0

4

– –1 –11

13 –12 –4

5

– –3 –1

11 0 –11

13 33

---– 12₃₃--- 4

33

---5 33

--- 1

11

--- 1

33

---1 3

---– 0 1₃

---3

– –1 5

1 0 –1

6 2 –9

0.16

– 0.08 0.44

0.52 0.24 –0.68

0.28 0.36 –0.52

1.5

– 0.5 0.5

0.85

– 0.45 0.25

0.6 –0.2 0

1.5

– 1 1.25

0.5

– 0 –0.25

1.5

– 1 0.75

19 2 – 15 2

3 4 –

1 0 3

3

1 3

--- 1

3

--- 1

3

---1 2

---±

1 3

--- 5

6

--- ₁2

3

---–

1 6

---– ₁₂---7 2

3

---–

0 1

2

---– 1

0.3 0.05 0.03

0.15 –0.06 0.07

0.24

– 0.03 0.13

1 2

---1 0

0.25 –0.5 0.75

0.5 0 –0.5

0.25

– 0.5 0.25

0.25 0.5 –0.25

0.5

– 0 0.5

0.75 –0.5 0.25

4D

➔

5F

526

A n s w e r s

answers

Chapter review

1 x=−1, y= 8, z= 9

2 a b

3 a −2 b 10 c −2

4 a iii −2 iii

iii iv

b iii −12 iii

iii iv

5 x= , y=−2 , z=

6 a y= and x=

where ax+by=u and cx+dy=v

b x= 2, y=−1

CHAPTER 6 Transformations

using matrices

Exercise 6A — Geometric

transformations and matrix algebra

1 a (0, 0) b (0, −26) c (0, 4) d (51, 7) 2 a (2, −5) b (4, −9) c (5, 0) d (−2, −4) 3 a A′(4, 2), B′(7, 7), C′(11, 4)

b A′(4, 0), B′(7, 5), C′(11, 2) c A′(0, −2), B′(3, 3), C′(7, 0) d A′(0, 0), B′(3, 5), C′(7, 2) 5 y′= 2x′+ 7

6 a y′=−(x′− 3)2− 1 b y′= (x′− 5)2− 5

c y′= (x′− 1)(x′+ 4) d x′2− 4x′+y′2− 4y′+ 4 = 0 e (y′− 2)2+ (x′+ 4)2+ 6y′− 12 = 0

7 a b c

Exercise 6B — Linear transformations

1 a c

2 a iii A′(0, −1), B′(4, 2), C′(−5, −2) iii A′(−1, 2), B′(−2, 0), C′(3, −1) iii A′(2, −4), B′(4, 0), C′(−6, 2) iv A′(1, −2), B′(2, 0), C′(−3, 1) b iii

iii

iii

iv No change

3 a A′(8, −7) b B′(−1, 5) c C′(2, −7)

4 A′(7, −6), B′(−1, 2), C′(15, −10)

5 a b

c d

0.2 –0.4 0.2 0.6

1 –1 0 5 –7 1 2

– 3 0

4 –3 2

– 1

4 –2 3

– 1

2

– 1

1.5 –0.5

7

– 4 5

5–8 –7 1–4 1

7

– 5 1

4–8 –4 5–7 1

1 12

---7 –5 –1 4

– 8 4

5

– 7 –1

17 27

--- 2

27

--- 2

9

---a u

c v

a b

c d

---u b

v d

a b

c d

---–3.5 –1.5 y' = 2x' + 7

7

3

0 x

y

y = 2x + 3

2 –

3 –

1 –

6

4 –

7 –

1 1 2 1

2 –3

3 2

x y

B'(4, 2)

C'(–5, –2)

A'

B(2, 0)

A(1, –2) C(–3, 1)

0

x B(2, 0)

A(1, –2) B'(–2, 0) C'(3, –1)

A'(–1, 2) y

C(–3, 1)

0

C'(–6, 2)

B'(4, 0)

A'(2, –4) x y

A B C

0

–3

–2 2 4 6 8 x

y

B

C 5 4 3 2 1

A

A' C' B'

–2 –3 –4 –5 –6 –7

A'(7, –6)

C'(15, –10) B'(–1, 2)

B(–4, 1)

A(4, 1) C(0, 5)

–10 y

x 5

–5

–4 –1 45 7 10 15

1 3

---– –12₃

---11₃--- 1 6

---–

1.5 1 1.5 1

1 1

1.75 2.5

1 3

---– 1.5

22₃

---– 2

A n s w e r s

527

answers

Exercise 6C — Linear transformations

and group theory

1 a P′(31, 18)

b det A= 1 (non-singular)

d iii y′= x′ ii y′= x′+

iii 10x′2− 34x′y′+ 29y′2= 2

2 a 17x′2_{− 26x′y′+ 10y′}2_{= 9} b 10y′2+x′2− 2x′y′= 81

c 13x′2+ 10x′y′+ 2y′2_{= 9}

3 a y′=x′ b y′= x′ c y′= x′

5 a c i

a c ii

b i y′= x′+ ii y′= 4

6 a x′2− 6x′y′+ 9y′2− 2x′+ 5y′= 0

b 11y′= 4x′− 5

Exercise 6D — Rotations

1 a b c d

2 a (1 − , + ) b (0, −4)

c ( , ) d ( − , − − )

e (−3, 2) f ( − , + )

3 a iii y′= +

a iii y′= +

a iii y′= −

b iii

b iii, iii

4 No change, rotation about the centre of the circle.

Exercise 6E — Reflections

1 a b c

d and then

e f

2 a iii (−3, −1) ii (−4, 2) iii (1, −3)

iv (2, 4) v (−3, 0) vi (2, −1)

b iii (3, 1) ii (4, −2) iii (−1, 3)

iv (−2, −4) v (3, 0) vi (−2, 1)

c iii (1, −3) ii (−2, −4) iii (3, 1)

iv (−4, 2) v (0, −3) vi (1, 2)

d iii ( − , + ) iii (2 + , 2 − 1)

iii ( − , + )

iv (−1 + 2 , − − 2)iv ( , )

4 7

--- 10

17

--- 2

17

---3 2

--- 1

2

---x y

y = 2x – 1

1 0.2

–1

y' = 0.3x' + 0.2

0

x y

y = –x + 4

4

4

y' = 4

0

3 10

--- 1

5

---0 –1

1 0

1

– 0

0 –1

0 1

1

– 0

1 0

0 1

3 2

--- 3 1

2

---3 2 2 --- 9 2

2

--- 1

2

--- 3 3

2 --- 3

2 --- 3

2

---3 2 --- 1

2 --- 1

2

--- 3

2

---x′

2

---- 2

4

---x

3

--- 1

3

---x

3

--- 1

3

---x y

y = –3x + 1

1 – 3

1

y' = x' – √ 2

√ 2 √ 2 0 –

y

y = –3x + 1 1

1 x

1 – 3

y' = x'–1– 3 – 3

y' = x'+1– 3 – 3 1

– 3

–

1

– 0

0 1

1 0

0 –1

0 1

1 0

1 0

0 –1

0 4

1 2

---– ---₂3

3 2 --- 1

2

---0 –1 1

– 0

x y

m_{x = 0} (i)'

(v)'

(ii)'

(iv)'

(iii)'