Answers

540

N EW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

Answers

Chapter 1

Start up

1 a

18

b

18

c

−

2

d

6

e

64

f

23

2 a

$44.95

b

$17.80

c

$8.05

3 a b c

1

d

1

4 a

20%

b

70%

c

8%

d

150%

e

35.5%

5 a

0.3

b

0.15

c

1.75

d

0.65

e

0.025

6 a

$8.40

b

16

c

28

d

12

e

29

f

10

Exercise 1-01

1 a

6, 12, 20, 30, 42

b

The sum of the first

two

even numbers is

2

×

3, the sum of the first

three

even numbers is

3

×

4, … .

c i

72

ii

110

iii

342

iv

2550

2 a

4, 9, 16, 25, 36, 49, 64

b

The sum of the first

n

odd numbers is

n

2

.

c i

100

ii

225

iii

1600

iv

10 000

3 a

1, 3, 6, 10, 15, 21, 28, 36, 45, 55

b

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

c

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

d

1, 36

e

1, 64

f

A square number

g

A triangular number

4 a

The pattern begins with 1. Each number that follows is the sum of the two preceding numbers.

b

8, 13, 21, 34, 55, 89, 144, 233, 377, 610

c i

every 4th number

ii

every 5th number

iii

every 6th number

iv

every 7th number

v

every 8th number

5 a

0

b

0

c

0

d

The sum of any ten consecutive Fibonacci numbers is divisible by 11.

6 a

The 12th number (

=

144)

b

The 6th number (

=

8)

7

Check with teacher.

8

3 and 5; 5 and 7; 11 and 13; 17 and 19; 29 and 31; 41 and 43; 59 and 61; 71 and 73

9 a

28

=

1

+

2

+

4

+

7

+

14

b

496

=

1

+

2

+

4

+

8

+

16

+

31

+

62

+

124

+

248

10 a i

1001

ii

1 000 001

b i

999

ii

999 999

c

2002

11 a

99

b

848

c

44 044

12 a

b

c

35, 51

d

1, 6, 15, 28, 45

Skillbank 1A

2 a

360

b

5700

c

9420

d

86 200

e

320

f

3200

g

18 000

h

16 000

i

3000

j

9600

k

4500

l

400

m

4400

n

6000

o

1500

p

20 000

q

120 000

r

3500

s

140 000

t

45 000

Exercise 1-02

1 a b c d e f

2 a

3

b

1

c

2

d

7

e

6 f 7

3 a 14 b 4 c 15 d 63 e 33f 9 g 6 h 6 i 10 j 9

4 a b − c d e − f −

5 a 31, 32, 33 or 34 b 10, 11c 17, 18, 19, 20, 21, 22 or 23d 41, 42, 43, 44, 45, 46 or 47

6 a B, D b A, C D c Cd C e B f A, D

7 a 5 b 8, 9, 10 or 11c 10, 11 or 12 d 6, 7, 8 or 9

Exercise 1-031 a � b � c � d �

e � f � g � h �

2 a , , b , , c , , ,

3 a , , b , , c , , ,

Exercise 1-041 a b c d e f

2 a 1 b 1 c 1 d 1 e 1

3 a − b c − d e −

720------ 2

5--- 1

8--- 1

5---

1 3 6 10 15 21

1 4 9 16 25

32--- 19

5------ 8

3--- 47

8------ 31

4------ 19

8------

12--- 4

5--- 1

4--- 1

3--- 3

11------ 5

6---

34--- 3

5--- 9

11------ 1

3--- 9

11------ 2

3---

23--- 4

5--- 5

6--- 5

7--- 3

4--- 4

5--- 1

3--- 1

2--- 4

7--- 3

5---

34--- 2

3--- 3

5--- 5

7--- 2

3--- 5

8--- 4

5--- 2

3--- 1

2--- 5

11------

35--- 2

3--- 1

3--- 5

6--- 6

7--- 1

2---

512------ 1

14------ 7

15------ 13

24------ 3

40------

18--- 11

30------ 2

15------ 1

3--- 1

15------

15_NC_Maths9_Stages_5.2/5.3_ans Page 540 Friday, February 6, 2004 1:55 PM

ANSWERS 541

4 a 5 b 5 c 11 d 4 e 6

f 4 g h 10 i 10 j 1

5 a b 3 c 2 m d e

Exercise 1-051 a b c d e

2 a 4 b 4 c 4 d 15 e 3

3 a 48 b 3 m c $9.50 d 12 kg

e $59.50 f 12.5 km g $4.50 h 27.5m

4 a 1 b 9 c 4 d 9 e 3

f 28 g 8 h i 8 j 3

5 a 9 b 4 h = 4 h 7 min c $312.50

Exercise 1-061 a = 1 b − c d e −

2 a 1 b 1 c d e

3 a 8 b 3 c 1 d 48 e 2

f 2 g 4 h 1 i j 1

4 a 25 b $1200 c 70 d 84 kg e 16

Exercise 1-071 a b c d

e f g h

2 a b c d e

f g h i

3 a b c d e

4 a b

5 6

Skillbank 1B2 a 5600 b 11 700 c 390 d 7700

e 6400 f 35 200 g 1300 h 9300i 7930

4 a 630 b 3900 c 16 000 d 3800e 1570 f 2700 g 68 000 h 840i 37 800

Exercise 1-081 a 39.06 b 100.5 c 7.378 d 0.0

e 0.05 f 15 g 246 h 2.2i 43.000

2 a 13.2 b 1.2322 c 33.02d 0.066 e 3.9210 f 35.7g 1.304 h 1.22 i 3.0j 11 k 13.7571 l 4.6m 1322.222 n 1.23

Exercise 1-091 a 38 b 9000 c 68.4

d 3 e 16.0 f 64 000g 1 800 000 h 389 800 i 200 000

2 a 0.06 b 0.70 c 0.85d 0.000 02 e 0.0876 f 0.039g 0.800 h 0.000 040 00 i 1

3 a 3 b 2 c 2 d 5e 1 f 1 g 3 h 5i 2 j 2 k 3 l 6

4 a 7.5 b 5710 c 400 d 0.007 660e 0.507 f 11 000 g 1860 h 0.0008i 60 000 000

5 a $28 000 000, two significant figures.b $30 000 000, one significant figure.

6 a 5 b 14 400 c 4 100 0007 a 180 b −5 c 6.16 d −4.176

e 3.8627 f 0.179 g 0.74 092 h 0.7435i 0.06 j 0.67 k 33 300 l 0.6565

8 a 5000 b 5000 c 5000 d 5002

Exercise 1-101 a 20 + 30 = 50 b 200 + 100 = 300

c 50 − 20 = 30 d 760 − 20 = 740e 930 − 130 = 800 f 6 × 20 = 120g 50 × 8 = 400 h 80 × 10 = 800i 90 ÷ 10 = 9 j 750 ÷ 5 = 150k 20 ÷ 10 = 2 l 900 ÷ 90 = 10

2 a = 30 b 700 000 ÷ 2 = 350 000

c 40 ÷ 20 = 2 d 800 ÷ 40 × 4 = 80

e = 10 f = 4

g (100 + 150) ÷ 50 = 5h 10 + 5 × 10 = 60i 100 × 9 + 100 × 7 = 1600

j = 6

k 82 = 64 l 52 ÷ 25 = 13 a B b C c A d B e D4 a A b C c D d B

Exercise 1-111 a 0.625, T b 0.4

.28 571

., R c 0.7

., R

d 0.83., R e 1.2, T f 6.8, T

g 0.416., R h 4.3

., R i 0.4

.5., R

j 0.35, T

2 a b c d

e f g h

1115------ 7

20------ 1

20------ 7

20------ 1

6---

715------ 7

10------ 17

20------ 5

7--- 3

4---

16--- 1

12------ 1

8--- 1

12------ 4

15------

625------ 1

4--- 7

16------ 12

55------ 1

2---

45--- 1

2--- 2

3--- 3

4---

34---

115------ 1

2--- 1

4--- 11

12------ 1

16------

47--- 5

6--- 3

7--- 1

7---

18--- 1

2---

43--- 1

3--- 1

7--- 5

12------ 8

27------ 2

5---

15--- 1

6--- 3

5--- 1

5--- 5

16------

14--- 1

3--- 11

25------ 13

21------

1225------ 34

35------ 93

100--------- 29

51------

13--- 3

10------ 2

5--- 3

4---

25--- 1

2--- 3

5--- 11

17------

112------ 1

10------ 3

10------ 3

10------ 1

3---

15--- 1

20------ 3

20------ 2

3---

710------ 7

16------ 3

7--- 5

8--- 3

7---

819------ 11

19------

15--- 13

20------

6 10×2

---------------

100 300 100–( )50

-----------------------------

30 60+25 10–------------------

15--- 7

10------ 1

4--- 9

20------

1225------ 1

200--------- 1

40------ 61

200---------


542 NEW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

i j 2 k 1 l 3

m 2 n o 4 p 10

3 a b c 0.69 d 0.8

4 a , , 0.678, 0.68,

b 0.34, , , , 0.305


e f g h

i j k l

m 1 n 2 o 1

2 a b

Exercise 1-131 a b c d e

f g h 1 i 3 j 1

k 1 l m 1 n 6 o 1

2 a b c d e

f g h i j

k l m n o

3 a 0.55 b 1.33 c 0.46 d 0.05e 0.024 f 2 g 0.095 h 0.0525i 0.12 j 0.003 k 0.125 l 0.2525m 0.083

.n 0.109 o 3.5

Exercise 1-141 a 37.5% b 45% c 75% d 250%

e 85% f 63.6.3.% g 32% h 20%

i 60% j 125% k 260% l 400%m 233.3

.% n 66.6

.% o 62.5%

2 a 50% b 186% c 35% d 170%e 0.8% f 239% g 20% h 25%

3 a 31% b 80% c 65% d 87.5%e 56% f 17.5% g 72% h 61%i 83.3

.% j 130% k 260% l 566.6

.%

m 487.5% n 352% o 72.5%

4 a 0.57 b c 18%

5 a b 40% c 130%

6 a , 55%, , 0.59

b , 0.79, , 72%

c 77%, 0.764, , , 7.95%

Exercise 1-151 a 14 b $1144 c $3.75

d 19.8 kg e 2.4 m f 24 Lg 156 g h 190 km

2 a $2.25 b 0.84 m c $29.18d 4 L e 81 kg f $78.10g $40 h 0.9 km i 16 t

3 a 28% b 66 2404 24 5 $35 200 6 105 g7 a 11 250 b 86 2508 $17 325 9 4.62 g

Exercise 1-161 a 72 kg b 44.8 km c 82.8

d $2460 e 85.6 kg f 63.84 L2 a 82.8 L b $568.75 c $110

d 2.6 km e $106.50 f $866.883 $835.764 a $73 b $4385 $102 6 $15 895 7 $259.608 $3.44 9 $59 020.80

10 a $1020 b $714 c $11411 a $408 b $391.6812 $84.15

Exercise 1-171 a 80% b 95% c 70%

d 6.6.% e 6% f 33.3

.%

2 a 33.3.% b 60% c 14.375%

d 37.5% e 20.83.% f 14.4%

3 History (history 70%, science 68%)4 a $350 b 70%5 a $800 b 11.4%6 1.11%7 a $52 b 11.3%8 a $150 b 42.9%9 39% 10 23% 11 6.2% 12 64.2%

13 a $102 b $78 c 43.3%14 a $165 b 120%15 a $4800 b 75%

Exercise 1-181 a 108 b 110 c 64

d 130 e 256 f 1202 $8000 3 $2250 4 $640 5 406 $32 0007 a 60 kg b 57 kg8 a $625 b $93.759 a 409 b 660 c 3200 mm d 800

e $36 f 384 g 518 kg h 72i 132 j 50 m k 20 l 58 km

10 200 11 $41 690.6512 $485 13 $3667 14 1200 g

3340------ 4

5--- 9

20------ 1

20------

13200--------- 21

100 000------------------- 5

8--- 1

100---------

1120------ 5

8---

711------ 6

9--- 7

10------

13--- 13

40------ 8

25------

29--- 2

3--- 7

45------ 7

18------

163225--------- 65

99------ 31

99------ 103

165---------

17111--------- 832

999--------- 1

30------ 25

33------

7399------ 1

45------ 26

165---------

13--- 2

3---

720------ 18

25------ 19

20------ 2

25------ 2

5---

45--- 9

50------ 1

5--- 1

4--- 1

20------

25--- 12

25------ 11

20------ 3

4--- 1

2---

18--- 1

40------ 1

12------ 17

400--------- 21

200---------

1180------ 1

3--- 11

200--------- 1

6--- 3

8---

23--- 1

200--------- 1

15------ 3

200--------- 5

12------

35---

1225------

12--- 4

7---

1720------ 3

4---

1925------ 11

40------


ANSWERS 543

Power plus1 a + b + c + +

d + e + + +

f +

2 + , + + , + + , + + +

3 a + = and 162 + 632 = 652

b + = and 242 + 1432 = 1452

+ = and 122 + 352 = 372

4 Check with teacher.5 Irrational numbers are a, d, e and f.6 a The square of any number is always positive.

b The cube of a negative number is negative.9 One possible answer is

+ + + + + + .

10 a i 24 ii 120 iii 3 628 800iv 5040 v 5040

b i Yes ii 362 880c i 30 ii 696 iii 144 iv 8 v 210

11 a = 2

b ≈ 1.4, ≈ 1.7, ≈ 2.2

≈ 2.4, ≈ 2.6, ≈ 2.8

Chapter 1 review1 a = = b 29, 30 or 31

c 24, 25, 26, 27, 28, 29, 30, 31 or 32d 6, 7 or 8

2 a b 5 c −2 d 2

e 9 f − g h 1

i 6

3 a 4.94 b 0.2 c 15.093 d 0.00004 a 2641 b 1.587 c −0.90285 a 420 ÷ 60 + 20 × 5 = 107

b = 80

6 a b c7 D

8 , 0.48, 43%, ,

9 a 84 b 88610 $236.7311 a 22.2% b 1.44% loss12 259

Chapter 2Start up

1 a 8y b 9y − 9 c 10abd 9a + 5y e 9ab f 26 + xyg 3y2 + 2y h 20ay i 6x + 5j 20abc2 − 3ab2c k 6m2 − 5ml −5p2q + 4pq2

2 a 18xy b −6w2 c 12apd 50m2 e 8y3 f 15a2c

3 a 4a b 4y c 4ab

d e 6a f 5bc

4 a 36 b −36 c 15 d 2 e −4.5 f 05 a 1, 2, 3, 4, 6, 12 b 1, 3, 9, 27

c 1, 3, 9, x, 3x, 9xd 1, 2, 4, 8, 16, ab, 2ab, 4ab, 8ab, 16ab, a, 2a,

4a, 8a, 16a, b, 2b, 4b, 8b, 16b6 a 2 b 10 c 3 d x e 8a f 4

g 6m h 2 i 4

7 a 1 b c d e

f g 1 h

Exercise 2-011 a 13y b 10x c 13 + 4y

d 8a e 0 f 18yg 9y3 + 2y2 h 14n i −11yj 8ac k 5xy l 10e2f

2 a 10y + 6 b 2a + 10 c 5m + 5yd 16 + 6x e 10ac + 2a f 8xy + 5g 2y + 5 h 4ay + 8y i −6 − 6x

3 a 6m + 10 b y + 3g c p + 7hd 15 + k e 2 − 10p f 15y − 9kg 6t − 7y h 11 − 7n i −2a − fj 11 + 11w k 8e − 5t l −ab − 4qm 9c2 − 6c n 2x2 − 5x3 o ab − 2mp 2hy2 − 7h q y − 11x r −9a2 − 16y

4 a 2a b 7m2 + 2m c −6kd 2u e 3d − 5d2 f 3m + 3tg 13k + 2y h 13 − 3r i −14b − 3uj 5e + 12 k 8w l 9t − 4gm 7y2 − 20 n 8mn o 4 − fp 4 − 11f q 4ab − 2b + 5ar −1 − 12l s 6 t −4n + 2u 8p3 + 8p v 2s w 2 + 3cdx 10k − 3m − m2

5 a 6p + 3 b 90h c 18c + 12ad 5x + 8 e 7p + 12 f 7x + y + 3

6 a 2m b 8n c 16n + 6m7 a u + 4t b 2u + 3t c 10u + 14t8 a a units b (a − 2) units

c (2a + 2c − 2) units

12--- 1

10------ 1

3--- 1

9--- 1

4--- 1

8--- 1

40------

14--- 1

8--- 1

4--- 1

8--- 1

16------ 1

32------

12--- 1

5---

12--- 1

6--- 1

4--- 1

3--- 1

12------ 1

2--- 1

8--- 1

24------ 1

2--- 1

12------ 1

16------ 1

48------

17--- 1

9--- 16

63------

111------ 1

13------ 24

143---------

15--- 1

7--- 12

35------

12--- 1

3--- 1

12------ 1

18------ 1

72------ 1

108--------- 1

216---------

4

2 3 5

6 7 8

510------ 15

30------ 1

2---

112------ 13

20------ 1

12------

1124------ 1

18------ 7

25------

110------

100 40×50

---------------------

745------ 817

999--------- 251

990---------

12--- 3

7--- 19

50------

ay2------

415------ 17

30------ 13

16------ 2

5--- 3

8---

1635------ 2

15------ 3

4---



Exercise 2-021 a 15p b 18x c 24m

d 8hy e 6yq f 10y2

g −12w h −8nd i −24p2

j 24ab k −60xyk l 30r2

m 24c2f n −36t2 o 12hk2

p 8c2e2 q 6a2d2 r −15m3n2

2 a 3y b 15 c 4 d 3ke 3y f 6n g −4a h 5ai a j 2x k 5m l 7c

m 4w n o p

q r s t

u v − w − x

3 a 10 b 18a c −3xy d −8ye −5y f −9m g m2 h −19di 1 j −22m2 k −4r2 l 5ym 11c n −2x2 o −g2

4 a 6ad units2 b 12py units2

c 10ck units2

5 a 8m3 units3 b 60dty units3

c 12kw2 units3

Exercise 2-031 a 4 b 0 c 2

2 a 2 b −9 c 6 d −2

3 a 1 b −10 c 0 d 6

4 a −6 b −6 c 9 d

5 a 4 b −2 c −2.56 a 4 b 5 c −15 d −5

e −17 f −1 g 0 h 2i 1 j 20 k 21 l −1

7 a 15 b −64 c −3 d −11e 128 f −40 g 15 h −15

i 121 j −1 k −20 l 2

8 a 14.4 b 91 c 90 d 31.5 e 20.16

Exercise 2-041 a 5t − 15 b 4 − 12b c 12k + 20

d 2d + 2e e mn + mp f 5 − 10wg 2c2 + cd h 32 − 28f i 3k2 − 6kwj a2 − 3a k 14p − 21m l 12n2 + 20ny

2 a −3s − 6 b −20f − 15c −4m − 2c d −12d − 20ee −2w2 + 5w f −35 + 7ag −21b + 14q h −16d − 24i −4r2 + 20ry j −9 + 15mk −2p2 − p3 l −6v + 14vn − 6vr

3 a −4 − 3m b −5x − 2 c −3f − 5d −8 − y e −1 − p f −6n + 1g −1 + 2a h −1 + e i −10 + 5q

4 a 10k + 15 b 3 − 9y c 4m + 14nd −6 − 3x e 43 − 10d f 24p − 4g −w − 8 h 8t + 26 i 11v + 37j 18 − 16h k 6m + 12 l 26 − 7dm e + 16 n 14c + 23 o −10w − 1

5 a 13m2 + 8m b 3w2

c 3r2 − 6r d 2q2 + 13qe 4v2 + 23v f 2a2 + 13ag −3t2 + 18t h −m2 − 7mi −11e2 j 26d − 18k 9h + 1 l −14r2 + 4r + 4m 2k2 − 6k − 5 n 7f 2 + 10fo −12b2 + 19b − 8

6 a 5w + 5 b 6k − 8 c −3r + 4d 3 − 6m e 5h − 6 f 3x2 − xg 5 − 4s h 7e − 1 i 9q − 1j −5e + 24 k 9 − 6b l 10d − 10em 2y − 2 n 14g + 2 o t2 + 2th + h2

7 a 11d + 7b i 7.05n − 0.7 ii $1409.30c i 70N − 500 ii $200

Skillbank 22 a 783 b 4455 c 684 d 8811

e 1592 f 2786 g 74 925 h 2772i 8055 j 9405 k 5445 l 3582

Exercise 2-051 a 3(3m + 4) b 3(t + 2) c 4(w − 3)

d 6(k − 4) e 5(4c + 3) f 9(k − 3)g 8(2 − 3w) h 6(2e + 3) i 5(3 + 4p)j 2(5 + 6t) k 6(4y − 5) l 2(q − 11)m 9(2g + 3) n 8(2w − 1) o 3(3v + 11)p 7(4 − 3t)

2 a h(k − 1) b y(w + 1) c m(n − 3)d p(6 + r) e a(b + c) f 4(a + 3b)g x(x + 1) h 5(1 − q) i 6(2y + 3x)j y(2 + y) k d(d + 8) l 3e(e + 2)m 2m(6m − 5) n a(a + c) o 5n(n − 4)p g(7 + 9g)

3 a 4x(4x − 3y) b 2pr(4p + 3) c 4m(mh2 − 1)d 7k(2h + 3y) e 7v(4w − 3v) f 6ab(3c + 4)g 2mn(2n − 3) h 2r(4t − 9r) i 3y(3a2 + 5y)j 2wy(10 + 9y) k 15(1 − 2b2) l 4xh(2w + x)m 3(3e + 4 + 2e2) n 3h(5k − h + 3)o 3(3ef − 4e + 2f 2)

4 a −4(p + 2) b −9(k − 3) c −3(g − 2)d −6(3w − 2) e −2(3x − 2) f −4(3m + 4)g −5(4e + 3) h −3(6n + 7) i −x(x + 7)j −8(w2 + 3h) k −a(p − q) l −8x(3y + 2)

5 a d(d + h) b 2p(p − 10) c −4(y + 4)d t(r − 1) e −4kt(4kt − 1) f −10a(b − 2)

g −3(3 + w) h (p + q) i x(x − y)

j k(3 − t) k −3xy(3y − 4) l −6m(4m − 1)

a3--- x

2--- b

3---

x2y------ b

5--- 3

4m------- 5

3ab----------

a5bc--------- 7xw

3y---------- 2a

3bc--------- 8p

9m-------

13--- 2

5---

25---

15---

12--- 1

4---

14---

15_NC_Maths9_Stages_5.2/5.3_ans 44 Friday, February 6, 2004 1:55 PM

ANSWERS 545

6 a 4(2a + 3y + 1) b 5(4xy − 2x2 + y)c tm(t + m + 1) d (a + 6)(5 + h)e (m + 2)(7y − p) f (a + b)(c − d)g −10p(p + q)h −(d + 3)(w − 4) or (d + 3)(4 − w)i 5a(x − y)(1 + 2y)

Exercise 2-061 a b k c d e

f g h i j

k l m n o

p

2 a b c d

e f g h

i j k l

m n o p

3 a b c d

e f g h

i j k l

4 a + b − c

d e − f

g h − i

j k l

m n o

p q r

s t u


e f g h

i j k l

m n 1 o

2 a = 2 b c 4 d

e f g h

i = 7 j k 12 l = −1

m n 6 o x

3 a b c d

e f g h

i j k = 1 l

m n o p

Power plus1 a i 11, 13, 15 ii 2n + 1 iii 41

b i −9, −13, −17 ii 11 − 4n iii −69c i 40, 53, 68 ii n2 + 4 iii 404

2 3r

3 a 40m + 50n b =

4

5 a k + 5, k − 5 b 4k + 10, 4k − 106 a 2b

b Possible answers include 4a, 5b; 20a, b; 10ab, 2; 5, 4ab

7 a 26 b 225 c 45 d −3e − f 34 g 16 h 13

8 a b c

d e

f

Chapter 2 review1 a 4x b −3km c 11ky

d 2p + 2d e 6w − 5 f 3x2 − 3xg 7m + 10 h 3w + 4f i 5d + 9j −2y2 + 3 − y k 10 + 3p l 2xy − 7m 10y2 − 6y n −a2b + 5ab − 2bo −2t2 − 8k

2 a 18y b 30k c 3ad d −6re 15xw f 8xy2 g 6a h 3

i 4p j k 2d l

3 a 7 b −17 c 200 d 14e 96 f 3 g 10 h −30

4 a 15t − 20 b 7 + 21x c −8 − 10kd −3 + 6y e 12p2 − 4p f −8m + 6m2

g −4 + 9w h −3 − 2n i −5a2 + 15at5 a 11y + 4 b 12 − 12p c 3g + 3

d 30m − 14 e 35 − 12v f 20q + 19g 7 + 10y h 13r2 + 5r i 12d − 23j −m2 + 7m k 2a2 + 2ab − 4b2 l 17e − 18e2

m 17h − 13 n 2kx6 a 2(m − 4) b xy(x − y) c −3(g + 9)

d 4(3x + 2 − 4a) e 4w(w − 3) f k(5 + hk)g −4(2a + 1) h −5y(1 + 5a)

3w4

------- m2----

3------−x 9

q---

1d--- 3r

w------ 3

c--- 4t s+

3-------------- y

h---

1a--- 10

e 1+------------ p 3+

z------------- u

4g------ 1

3 f------

e2---

7x12------ 4s

21------

15---------- −2h

14------------

−5m

9w20------- 33t

20-------- 11p

15---------- 5r

6------

13c10--------- 5d 2r–

10------------------- 9h 10a+

15----------------------- 25 24w+

30-----------------------

21 8a+28

------------------- 3a 2b+6

-------------------- 5e24------ 9 20k–

15------------------

173p------- 5

2k------

4m-------−3

3y---------−11

95m------- 19

3x------ 22

5a------ k 8+

4c------------

a 4b–4m

----------------8y------−5 49

15d--------- 8a c–

4w---------------

14k3

--------- 7m4

------- 13y5

--------- 5m3

------- 8x 10+15

-------------------

7y 5+12

---------------- 23h4

--------- 14k5

--------- 9x 13–20

-------------------

2 m 8+15

------------------- − 9e2------ 7t

4----- 3 k 31+

70--------------------- −

7m 11–12

-------------------- k 5+6

------------ 7x 31+20

-------------------

12 m 17+35

------------------------- − 7 k 33+6

--------------------- − 14m 49+36

-----------------------

13 x 14+6

------------------------ − 59x 40+60

---------------------- 101w 24+70

--------------------------

13x 23–12

---------------------- 19 r 11+30

------------------------ − 13y 21+6

----------------------

w6---- st

20------ 5

2hk--------- 12

mn-------

5l3 f------ 2

3v2-------- 6

x2----- ac

bd------

23--- d2

36------ d

g--- 12

5k2--------

3u5

------ 23d------

52--- 1

2--- m

2n------ q

3---

3d10------ 20

27------ 3

5--- 9

b---

152------ 1

2--- h2

k2-----

8---------−11 3

8---

5u9

------

6p--- 12

35------ 3

20y--------- 2s

35------

5xz

------ c3b------ 25b

3--------- 4t

27------

23n------ 50p2t

7-------------- 5

4--- 1

4--- 1

6h2---------

t2

k2----- 2yh

9--------- 3a f 2

25------------- 2h3

3---------

40m 50n+100

--------------------------- 4m 5n+10

---------------------

15r 2–6

------------------

34---

13---

10e 1+------------ 7

6v------ 15 3r–

20------------------

143x 3–--------------- 11

2x 2–---------------

36w2 2w+27 w 1+( )---------------------------

2y3

------ 3a4d------


546

N EW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

7 a b c d

−

e f g h

i j k

+

l

−

m n o

8 a b c d

1

e

2

f

1

g h

i j

Chapter 3

Start up

1 a

−

12

km

b

40

gh

c

8

x

2

y

3

d

108

m

3

n

3

e

30

a

3

w

4

f

−

18

t

3

2 a

5

x

+

10

b

4y − 12 c −n − 3d 2x2 − 8x e 12g + 3g2 f −12m + 8m2

3 a 8x b −2x c −8xd y2 − 25 e x2 + 4x − 12 f 3n2 − 5n − 2g 2a − 3 h 13x − x2 i 5x − 3

4 a 3(5k + 3) b −6(2d − 3) c −5x(x − 2)d 4h(2g − 3) e −xy(x + 4) f 9ay(a − 3y)

Exercise 3-011 a 80 + 10b + 8a + ab b 3y + xy + 36 + 12x

c k2 + 8k + 152 a x2 − 3x − 54 b w2 + 4w − 5

c n2 − 12n + 35 d 12k − 27 − k2

3 a dg + d + 4g + 4 b k2 + k − 12c 2l2 − 9l − 5 d 2x2 + 13x + 15e m2 − 2m − 15 f 6gb + 12g − 2b − 8g 12y2 + 11y + 2 h h2 − 5h + 6i 12w2 − 11w + 2

Exercise 3-021 a x2 + 5x + 6 b km + 4k + m + 4

c 6 − y − y2 d 8e2 + 19e + 6e 2g2 + 9g + 7 f 16a2 − 42a + 5g −3p2 + 4p + 4 h 6 − 7b + 2b2

i 4m2 − 8m − 5 j 16t2 − 252 a a2 + 7a + 12 b x2 + 11x + 30

c m2 + 11m + 24 d e2 + 8e + 12e g2 + 14g + 45 f h2 + 22h + 120g s2 + 3s − 10 h w2 − 3w − 18i k2 + 2k − 15 j p2 + 2p − 3k n2 + n − 56 l v2 − v − 30m c2 + c − 6 n l2 + 2l − 8o p2 + 3p − 18 p t2 + t − 2q y2 + 6y − 40 r u2 + 5u − 84s d2 − 7d + 10 t b2 − 14b + 48u r2 − 6r + 8 v z2 − 16z + 55w u2 − 11u + 24 x n2 − 12n + 27

3 a 3y2 + 8y + 5 b 16c2 − 20c − 6c 2m2 − 9m − 5 d 8d2 − 2d − 15e 3p2 − 8p − 60 f 25y2 + 5y − 2g 3g2 + 23g + 30 h 15 − 22q + 8q2

i −2w2 + 5w − 2 j 6t2 − 13t + 5k 25a2 + 15a + 2 l −6h2 + 5h + 6m 7b2 − 57b − 54 n 9e2 − 25o 10d2 + 13d + 4

4 a l = 100 + x; w = 75 + xb A = (100 + x)(75 + x) = 7500 + 175x + x2

c 175x + x2 d 17.51 cm2

5 a l = 4 + x; w = 3 + y b A = (4 + x)(3 + y)c 12 + 4y + 3x + xy d 4y + 3x + xy

Exercise 3-031 The missing terms are:

a 20x b m2 c t2 d +8he − 18k f + 80f g 4d2; 12d h 36a2; 1

2 a m2 + 18m + 81 b u2 + 6u + 9c y2 − 12y + 36 d 64 + 16k + k2

e 25 − 10h + h2 f 49 + 14k + k2

g f 2 + 40f + 400 h q2 − 22q + 121i 100 + 20t + t2 j x2 − 2xw + w2

k a2 + 2ag + g2 l 4m2 − 12m + 9m 25x2 − 60x + 36 n 81a2 + 36a + 4o 9e2 − 24e + 16 p 25 + 70b + 49b2

q 16 − 40p + 25p2 r 121 − 44c + 4c2

s 100g2 + 60g + 9 t 9k2 + 66k + 121u 25 + 20v + 4v2 v 81d2 − 72d + 16w 1 − 16e + 64e2 x 144 + 120n + 25n2

3 a m4 + 6m2 + 9 b 1 − 6f 2 + 9f 4

c n2 − 2nw2 + w4 d 49h2 + 28hk + 4k2

e 64a2 − 48ay + 9y2 f 16a2 − 24ab + 9b2

g 49p2 + 28pn + 4n2 h 64 + 16pq + p2q2

i x2y2 + 2xyw + w2 j x4 − 4x2 + 4k 16 + 8k2 + k4 l x6 + 4x3 + 4

4 a 1 + + b t2 − 2 +

c k2 + 2 + d + 6 + w2

e − 2 + f − 2a + b2

5 a (m + 2)2 b (p + 10)2 c (a − 5)2

d (w − 4)2 e (g + 1)2 f (x − 7)2

g (2y + 2)2 h (5h − 4)2 i (4b + 8)2

j (n + 2m)2 k (3a − 5b)2 l (10g + 3h)2

6 a x2 + 2xy + y2 b 2xy7 a 2 × 10 × 20 = 400 b 800 c 1200

d 1000 e 1200 f 5008 a 441 b 2025 c 841

d 3481 e 10 404 f 96049 b 100 − 40x + 4x2

c 10x − 2x2

e 10 × 10 − 4 × x × x = 100 − 4x2

5h6

------ k2--- 2

m---- 3w

20-------

5k6------

2d------−3 3x 13+

4------------------- m 2+

12--------------

y 22+10

--------------- 293 c 5+( )--------------------- 5d

3------ 17m

4---------- 39g

7--------- 4k

3------

21c 19+20

---------------------- 23h 11–15

---------------------- 10

---------------------------−11w + 11

20mt------ 3h

k2------ 21

d------ 1

4---

12--- 2d

5------ p

3---

10a3

--------- 13---

2y--- 1

y2----- 1

t2----

1k2----- 9

w2------

16a2------ a2

16------ a2

b2-----


ANSWERS 547

Exercise 3-041 a m2 − 25 b c2 − 100 c a2 − 144

d 36 − y2 e 64 − m2 f p2 − 1g 25 − e2 h v2 − 121 i w2 − 9j x2 − 100 k q2 − 49 l 81 − g2

m b2 − 4 n 225 − r2 o d2 − 1692 a 4h2 − 9 b 25r2 − 16 c 25b2 − 64

d 16p2 − 49 e 9 − 64k2 f 49x2 − 25g 4 − 81m2 h 81k2 − 16l2 i 49n2 − 64m2

j 16g2 − 25h2 k 49u2 − 9w2 l 121a2 − 9b2

m 25a2b2 − 4 n t2 − o − 4

p 1 − q x2y2 − t2 r − 4

3 a length = x + l, width = x − l b x2 − l2

4 a p − 1 b p + 1 c (p − 1)(p + 1) d 75 8996 a 399 b 2499 c 8099 d 63967 a 29 b 47 c 3000 d 400

e 240 f 1968 a l = x + y, b = x − y

Exercise 3-051 a m2 b y2 + 18 c −k2 − 36

d d2 + 21 e 2e2 + e f 48mg 27t − 10 h 2f 2 − 8 i 12h + 18j 5xy − 6x + 3y + 9k bc − ab l 18a2 − 6a

2 a 60a2 b n2 + 4n + 4 c −2t2 − 8td −8mn e x − 2 f 1 − 2b2

g am − 2m2 + bm − ac + 2mc − bch 11l2 + 30l − 25 i 19e + 256j 2r3 − 36r + 24

3 a 3a2 + 15a + 20 b 3y2 + 12y + 14c 3x2 + 9 d 34n2 − 34 e −4b2

Skillbank 3A2 a 24 b 51 c 32 d 110

e 9 f 31 g 32 h 72i 45 j 9 k 118 l 40

4 a 28 b 28 c 9 d 27e 27 f 40 g 150 h 35i 216 j 24 k 588 l 36

Exercise 3-061 a 6y b 6a c 5 d 3

e m f 24c g 3pr h 8wi 9mn3 j 4e k 15ab l 4

2 a 3 b 6x + y c 10 d h; 8e p − 2q f c − 2t + 3a g 5h x2 + 7 i 4m2 j 3q − 1 k 4c − 8t + al 4(9y2 − 4 − 3y)

3 a −9 b r c +3 d −7; 3ze g − 4 f l + 2m g ac; 3a h −4di −3d2 j 4 + x

4 a 3(t + 12) b 4(w − 3) c 6(k − 4)d 4(5w − h) e h(g + h) f x(y − w)g 3e(1 + 2e) h 2m(6m − 5) i 2(b2 − c)j a(a + c) k 5n(n − 4) l 4x(4x − 7y)m 4(3cf + 4) n −4(y + 4) o −t(r + 1)p −4kl(4kl − 1) q −10a(b − 2) r −3(3 + e)

s x(5 + 12y) t (p + g) u g(7 + 9g)

v x(x − y) w −6m(4m − 1) x k(3 − t)

5 a 4(2a + 3y + 1) b 5(4yx − 2x2 + y)c tm(t + m + 1) d (a + 6)(5 + h)e (m + 2)(7y − p) f (a + b)(c − d)g −10p(p + q) h 5a(x − y)(1 + 2ay)i (d + 3)(4 − w) j −5g(u + 3)(1 + 2g)

Exercise 3-071 a (m + n)(5 + t) b (r + 3)(y − 4)

c (3g − 4)(1 − 4g) d (x2 + 1)(5x + 2)e (3 − 4t)(d − t) f (e + 2)(4e + 1)g (3 − 2x + y)(w + 3) h (k + 3)(k − 3)i (7 − 3a)(1 − w) j (3b − 1)(b2 + 2)

2 a (b + d)(4a + 5c) b (y + t)(2x − 5w)c (3c + 4d)(3a + 2b) d (10 + x)(x2 + 3)e 3(a + b)(a + c) f 6(t + p)(r − 3w)g (7 + d)(2e − 3) h (h − 2)(k − h)i (3m + p)(n − 2) j (9 + q)(p2 − 3)k ( f − 10)(g − h) l 3(l + n)(3k − 4m)m (2 − p)(p − c) n (l − 3)(l2 + m2)o (a + y)(x + 1 − k) p (a − b + 3q)(p − 2q)

Exercise 3-081 a (w − 3)(w + 3) b (y − 6)(y + 6)

c (k − 1)(k + 1) d (m − 11)(m + 11)e (p − 8)(p + 8) f (t − 10)(t + 10)g (2e − f )(2e + f ) h (a − 3b)(a + 3b)i (4y − 1)(4y + 1) j (2 − b)(2 + b)k (5 − e)(5 + e) l (1 − 4x)(1 + 4x)m (k − l)(k + l) n (7 − 4m)(7 + 4m)o (b − 11d)(b + 11d) p (6c − 5k)(6c + 5k)q (4 − 9h)(4 + 9h) r (5a − 8m)(5a + 8m)s (10 − 7n)(10 + 7n) t (11p − 12q)(11p + 12q)

u ( − 5c)( + 5c) v (2t − )(2t + )

w (5h − )(5h + ) x (1 − mn)(1 + mn)

2 a 2(a − b)(a + b) b 7(k − 2)(k + 2)c 3(1 − 5u)(1 + 5u) d x(x − 7)(x + 7)e k(1 − 4k)(1 + 4k) f 2(5q − 1)(5q + 1)g 3(d − 2v)(d + 2v) h 5t3(t − 5)(t + 5)i 2(ab − 1)(ab + 1) j x2(y − w)(y + w)k 12(4f − 3g)(4f + 3g)

l 5(3d − )(3d + ) or (6d − 1)(6d + 1)

m 2(x − 2a)(x + 2a) n 25(2 − w)(2 + w)

o 5( − 4e)( + 4e) or (1 − 8e)(1 + 8e)

p (6c − 5)(6c + 5) or (3c − 2 )(3c + 2 )

1t2---- w2

9------

1r2----- a2

b2-----

12---

14--- 1

4---

12--- 1

2--- 1

3--- 1

3---

32--- 3

2---

12--- 1

2--- 5

4---

12--- 1

2--- 5

4---

14--- 1

2--- 1

2---



3 a ( − )( + ) b (x − )(x + )

c ( − )( + ) d 2( − )( + )

e ( + )( − ) f (t − 3)(t + 3)(t2 + 9)

g (10 − n2)(10 + n2) h 25(c2 − 2)(c2 + 2)i (2k − 1)(2k + 1)(4k2 + 1)j x(y2 − 2)(y2 + 2) k (a + b − x)(a + b + x)l (x + y − 2c)(x + y + 2c)m (a − c)(a + 2b + c) n (x − a + c)(x + a − c)

o −(p + 3q)(3p − q) p (x − )(x + )

q (x − 1)(x + 1)(x2 + 1) r 4ab

Exercise 3-091 a −1, −6 b −3, 4 c −5, 3

d 3, 4 e −5, −4 f −2, 7g −2, 5 h −5, 5 i −2, 1j −9, 2

2 a 10, 10 b −4, −4 c −11, 2d 2, 10 e −4, −2 f −9, 3g −2, 5 h −9, 9 i −5, −1

3 a (m + 3)(m + 4) b (k + 7)(k + 5)c (d + 4)(d + 1) d (t + 5)(t + 2)e (x + 5)(x + 4) f (t + 1)(t + 5)g (e + 2)(e + 3) h (h + 2)(h + 2)i (n + 10)(n + 1) j (a + 6)(a + 5)k (d + 4)(d + 6) l (y + 4)(y + 11)

4 a (x + 4)(x − 1) b (t + 8)(t − 3)c (m + 5)(m − 3) d (a + 2)(a − 1)e (k + 7)(k − 2) f (w + 6)(w − 2)g ( f + 15)(f − 2) h (y + 4)(y − 2)i (b + 4)(b − 3) j (p + 12)(p − 4)k (c + 9)(c − 1) l (w + 13)(w − 9)

5 a (y − 3)(y + 1) b (r − 7)(r + 2)c (h − 4)(h + 1) d (w − 9)(w + 2)e (e − 9)(e + 3) f (a − 6)(a + 2)g (p − 5)(p + 1) h (u − 7)(u + 6)i (y − 10)(y + 1) j (x − 7)(x + 4)k (d − 9)(d + 5) l (h − 11)(h + 2)

6 a (m − 4)(m − 1) b (w − 2)(w − 4)c (k − 3)(k − 4) d (p − 6)(p − 4)e (n − 2)(n − 1) f (a − 1)(a − 1)g (t − 7)(t − 1) h (x − 7)(x − 5)i (g − 5)(g − 4) j (d − 3)(d − 3)k (t − 16)(t − 3) l (e − 7)(e − 9)

7 a 3(m + 1)(m + 2) b 2(y + 2)(y − 1)c 5(t − 10)(t + 8) d 5e2(e + 8)(e − 3)e x(x − 11)(x + 10) f 4(b − 7)(b + 6)g 4(w + 4)(w − 3) h 3a(a − 4)(a + 1)i 2(e + 5)(e + 4) j 6(h − 3)(h − 3)k 8(p + 2)(p + 2) l 2r(r + 3)(r + 8)m −(t + 8)(t − 3) n −(u − 7)(u + 6)o −(x − 7)(x + 4) p −(b + 4)(b − 3)q −(k − 3)(k − 4) r −(x − 5)(x − 7)

s −3( f + 15)(f − 2) t −5(a − 6)(a + 2)u 3(y + 6)(y − 8)

8 a (h − 6)(h + 3) b (t − 5)(t + 5)c (w + 1)(w + 7) d (k + 3)(k − 15)e (v − 2)(v + 10) f 3(c + 5)(c + 5)g (q − 1)(q − 5) h (a − 3)(a − 1)i −(t + 9)(t − 2) j (x − 8)(x − 8)k 8(u − 4)(u + 1) l (b + 5)(b + 6)m (y − 6)(y − 6) n 5(r − 2)(r + 1)o 4(l − 4)(l + 2) p (g − 20)(g − 4) q −2(d − 6)(d + 9) r −(n + 2)(n − 13)s 6(x + 8)(x − 2) t 10(t − 1)(t − 1)u −4(k − 8)(k − 5)

Exercise 3-101 a (m + 5)(2m + 1) b (d + 3)(4d + 1)

c (k + 3)(5k + 2) d (2p + 5)(3p + 2)e (w + 15)(2w + 1) f (e + 3)(4e + 3)g (2f + 3)(4f + 1) h (d + 1)(3d + 2)i (b + 1)(2b + 7) j (y + 1)(5y + 11)k (4g + 3)(2g + 5) l (3a + 7)(2a + 3)

2 a (y − 4)(2y − 3) b (2k − 3)(5k − 2)c (2e − 3)(3e − 2) d (b − 3)(4b − 1)e (w − 3)(6w − 5) f (2t + 5)(4t + 3)g (3x − 2)(3x − 2) h (3f − 4)(4f − 3)i (2h − 9)(2h − 9) j (3n − 4)(5n − 1)k (6p − 5)(2p − 3) l (5m − 1)(3m − 4)

3 a (y + 1)(5y − 11) b (4d − 5)(d + 1)c (2m + 3)(m − 3) d (3t − 10)(t + 3)e (6h − 7)(h + 1) f (y − 4)(2y + 3)g (2a + 1)(4a − 3) h (5u − 4)(3u + 1)i (3c + 1)(3c − 5) j (6p − 5)(3p + 2)k (3w − 4)(2w + 1) l (5e + 1)(3e − 1)

4 a (5m + 7)(m − 1) b (3g − 4)(2g + 3)c (3p − 2)(p + 2) d (7w − 1)(w + 1)e (5y − 1)(y + 3) f (3n − 2)(n + 4)g (4b − 3)(b + 3) h (4m − 1)(2m + 3)i (3x + 8)(x − 2) j (3q − 5)(q + 3)k (2a − 3)(2a + 5) l (6y − 5)(y + 4)

5 a 2(t + 2)(3t − 1) b 3(g + 4)(2g − 3)c 4(2e − 3)(3e + 1) d 2(a − 2)(4a + 3)e 4(t + 2)(3t − 1) f −1(5q + 3)(5q − 2)g −2(2m − 1)(3m − 2) h −1(3h + 4)(4h − 5)i 6(2c + 3)(2c + 1) j −3(z + 1)(2z − 5)k 2(2d − 3)(3d + 5) l −2(x − 3)(3x − 2)

6 a (a + 1)(2a + 3) b (2m − 5)(6m − 1)c (4x − 1)(x + 3) d (k − 2)(7k + 1)e (u + 5)(8u + 1) f (e − 3)(6e − 1)g (w − 1)(7w − 1) h (h − 3)(4h + 5)i (4x − 3)(2x + 1) j (r + 5)(5r + 1)k (d − 7)(2d − 1) l (3n + 1)(2n − 3)m −(3m − 2)(3m + 4) n −(5c − 3)(c + 1)o (3g + 2)(5g + 3) p −(4q + 3)(2q − 5)q (x − 2)(3x − 7) r −(3d − 4)(d + 4)s −(3v + 1)(2v − 5) t −3(y − 4)(y + 3)u 3(k + 1)(4k − 3)

p4--- x

5--- p

4--- x

5--- 1

3--- 1

3---

v7--- u

9--- v

7--- u

9--- y

3--- m

11------ y

3--- m

11------

4a7

------ 5b2

------ 4a7

------ 5b2

------

14--- y

3--- y

3---


ANSWERS 549

Exercise 3-111 (m − 8)(m − 8) 2 3d(d − 1)3 (d − 3)(3d + 5) 4 (3 + h)(k − 5)5 (5y + 8)(5y − 8) 6 4(5f + 4)(5f − 4)7 q(q + 3 − 3p) 8 (3 − g)(g + 1)9 4(2b + 5)(3b − 2) 10 (5r − 1)(5r + 1)

11 (l2 + 1)(l + 1) 12 (2x − 5)2

13 −(5d − 4)(d + 1) 14 (b − 1)2(b + 1)15 2(2 − v)(2 + v) 16 m(n + 3)(n + p)17 2(w − 6)(w − 6) 18 (6h + 1)(6h + 1)19 (3r − 8t)(5r + 3t) 20 (2d + 1)2

21 9(g − 2k)(g + 2k) 22 e(e − 5)(e + 2)23 5[(p + q)2 − 15(p − q)2]24 7(2x − 1)(2x + 1) 25 (a − b)(a + b + 4)26 (c − 2)2(c + 2) 27 (3a − 1)(2a + 5)28 (t + 7)(t − 5) 29 2(3p + 2)(3p + 2)30 −(6a − 1)(4a + 1) 31 9(x + 2)(x − 3)32 (a − 3)(2ab − 3) 33 2(a + 3)(a + 3)34 (5u − 1)(5u − 1) 35 (k − 3)(4k + 7)36 3(4 − w)(4 + w) 37 3(1 − 3s)(1 + 3s)38 (k − 4)(k + 4)(k + 4) 39 5y(y2 − 2y + 3)40 mn(m − 2)(m + 2) 41 −2(a − 2)(a + 2)42 4(2c − 3)(4c + 1)

Skillbank 3B2 a 11 b 40 c 14 d 32

e 23 f 7.5 g 11.25 h 58i 57 j 18 k 15 l 42.9m 40 n 135 o 11 p 14q 135 r 54 s 123 t 56

Exercise 3-121 x + y 2 3

4 5 6

7 8 −1 9 w − 4

10 11 12

13 − 14 15 3(c − 1)

16 17 18

19 20 21

22 23 24

25 26 27

28 or

29 30

Exercise 3-131 a 6m b c

d 6k e f

g h i

j

2 a b 4t c

d e f

g h

Exercise 3-141 a b

c d

e f

g h

2 a b

c d

e f

g h

i

Power plus1 a a2 + b2 + 2ab + ac + bc

b x2 + xy − 5x − 2y + 6c p3 − 6p2 − 7p + 60d xyz − xy − yz − xz + x + y + z − 1

2 a w2t2 − 6rwt + 9r2 b 1 − 6c2 + 9c4

c x6 + 2x3 + 4 d 1 + +

3 a 16a4 − 9a2 b x2y2 − t2 c −

d − 4 e d4 − d2 f 1 −

4 a (n + 2m)2 b (x − y)2 c (5x − 4y)2

d 5(a − 3b)(a − 3b) e (c2 + 1)2

f −(t2 − 1)(t2 + 8) = −(t − 1)(t + 1)(t2 + 8)

5 a ( − )( + ) b (x − 1)(x + 1)(x2 + 1)

c 25(c2 − 2)(c2 + 2) d (a + b − c)(a + b + c)

e ( − )( + ) f 4pq

6 a 7h2 + 28 b 8r2 − 204 c 385v2 − 385

d e f

7 a both positive

1m n–------------- 1

2 t r–( )-------------------

b c–a

------------ 37--- a

b---

2 d 2e+( )t

------------------------

5d t+----------- 1

l m–------------ -2

a b+------------

23--- k 5+

k 5–------------

a 1+m n+-------------- y 4+

2------------ k 1+

k 4+------------

4a 5c+a c–

------------------- e 6–2

------------ 4 h–4 h+------------

3 d 5–( )d 5+

--------------------- s 2+s 3–------------ m 3–

m 4+--------------

3d 2–d 5+---------------- k 1–

3k 2+---------------- 2n 3–

n 3+----------------

-1 2c 1–( )3c 1–

-------------------------- 1 2c–3c 1–---------------

2x y+x y+

---------------- a 4+2 p 2+( )----------------------

2524------ 1

2---

10h 1+------------ 3

2 a b–( )---------------------

-r5 r t+( )------------------- 4m

m 1–------------- x 3+

x x 1–( )--------------------

4p---

r2 r 4–( )-------------------- 3

7---

23 x 2–( )--------------------- 10

3------ d 1+( ) d 3–( )

6------------------------------------

f 3+4 f 3–( )--------------------- 3

f 2–-------------

7m 10+m m 1+( ) m 2+( )-------------------------------------------- 2w 20–

w w 3+( ) w 5+( )-------------------------------------------

4b 7–b 1–( ) b 2+( ) b 3–( )

------------------------------------------------------ 2 3k 2+( )k k 3+( ) k 1+( )---------------------------------------

4w 5–2 w 3–( ) w 2+( ) w 1–( )------------------------------------------------------------ 2n 3–

n 2n 1+( ) n 1+( )-------------------------------------------

p2 2p 4+ +p 2+( ) p 5–( )

------------------------------------- 11a 9a2

–2a 3–( ) a 4+( ) a 4–( )

---------------------------------------------------------

2g 5+3 g 2–( ) g 2+( )--------------------------------------- 3q 1–

q 1+( ) q 1–( )------------------------------------

f 12–f f 3–( ) f 3+( )---------------------------------------- -k 2–

k k 1+( ) k 1–( )---------------------------------------

5h 12+4h h 1+( )------------------------- -4d 1–

d 2+( ) d 1+( )------------------------------------

42 5r–4 r 6–( ) r 6+( )-------------------------------------- 3d 6– d2+

d d 2+( ) d 2–( )----------------------------------------

9k 5– k2–k 4–( ) k 1+( ) k 1–( )

-----------------------------------------------------

2y--- 1

y2-----

x2

4----- y2

9-----

a2

b2----- 1

r2-----

x4--- y

5--- x

4--- y

5---

4a7

------ 5b2

------ 4a7

------ 5b2

------

d 1+d d 6+( )--------------------- 1

k--- -3

u 1+( ) u 6+( )------------------------------------



b The larger of p, q is negative; the smaller of p, q is positive.

c The larger of p, q is positive; the smaller of p, q is negative.

d both negative

Chapter 3 review1 a 35 + 12b + b2 b 2d2 + 13d + 15

c y2 − 8y + 15 d 3m2 − 5m + 22 a m2 − m − 6 b p2 + 10p + 21

c s2 − 11s + 24 d 15 − 2k − k2

e 12 − 7g + g2 f 6f 2 + 11f + 3g 20a2 − 19a + 3 h 20r2 − 85rw + 20w2

i 12h2 − 14h − 10 j 6a2 − 3ab + 9b2

k 36 + 7m − 15m2 l −6x2 + 21x + 453 a n2 + 10n + 25 b u2 − 6u + 9

c 81 + 18x + x2 d 36 − 12k + k2

e 9y2 + 6y + 1 f 4x2 − 20x + 25g 16 + 24a + 9a2 h 36 − 84b + 49b2

i 16a2 + 40ab + 25b2 j 9g2 − 24gh + 16h2

k 1 − 8h + 16h2 l 4x2 + 28x + 494 a n2 − 36 b b2 − 1 c 100 − m2

d 49 − y2 e 9r2 − 1 f 16w2 − 25g 64x2 − 9y2 h 36p2 − 25q2 i 1 − 64w2

5 a 3x2 + 9x + 4 b 7y + 18c −6a − 2 d 8p2 + 18q2

6 a 4(t + 6) b x(x + 6)c 3y(1 + 2y) d pq(p − q)e −(y + 16) f −y(x + 1)g −5g(1 + 2g2) h w(3 − w + 4w2)i (x + 4)(x − 3) j 5(4ab − 2a2 + b)

7 a (a + 3)(5a + 2) b (5m − 3)(n + 2)c (b − 1)(a − 1) d (x + y2)(xy − 5)e (p2 + 3)(10 + p) f 6(t + p)(r − 3w)

8 a (w − 3)(w + 3) b (m − n)(m + n)c (6 − r)(6 + r) d (4a − 3)(4a + 3)e (9x − y)(9x + y) f (7a − 5b)(7a + 5b)g (2n − 5m)(2n + 5m) h 7(h − 2)(h + 2)i 3(q − 3)(q + 3) j h(1 − 8h)(1 + 8h)k 2(5m − 4)(5m + 4) l 6(2 − 5d)(2 + 5d)

9 a (m − 12)(m + 1) b (y − 3)(y − 1)c (t − 13)(t + 5) d (p + q)(p − 3)e 3(n + 1)(n + 2) f 4(y + 2)(y − 1)g −(x − 6)(x + 3) h −5(b + 4)(b − 3)i −(2k − 1)(3k − 4)

10 a (3x + 2)(2x − 1) b (5s + 3)(s − 2)c (4w − 1)(2w − 3) d 3(e + 2)2

e 3(2y − 3)(y + 4) f 4(3a + 1)(2a − 3)g −3(3x − 2)2 h −2(4t − 3)(2t − 5)i 2(3g − 1)(2g − 7)

11 a (8y − 5)(8y + 5) b (x − 8)2

c 16(r − 3) d (g − 5)(3 + h)e (a − 1)(a2 − 1) f 3b(b − 1)(b + 1)g (4y + 7)(y − 3) h 2(3q + 2)2

i −(6p − 1)(4p + 1) j (5w − 1)2

12 a m − 2 b c d

13 a b c d

14 a b

c d

e f

g h

Mixed revision 11 a $161.59 b 79.12 kg2 a $259.50 b $87 c $48.98

d $30.15 e $14.10 f $163.20

3 a b c 4 d 6

4 a i 4 ii iii iv

b i 2.18 ii 0.058 iii 0.004 iv 1.735 a 8 + 7 = 15 b 5 × 10−2 = 0.05

c 8 × 9 + = 72 d 1300 − 13 = 1287

6 a 29 b 12

7 a 52 + 10 × 30 = 325 b = 5

c = 9 d 5 × 50 = 250e 100 ÷ 25 + 150 = 154

f = 6

8 a 100 b 2.21 c 1.499 d 19

9 a i ii iii iv v

b C10 36

11 a b c

12 a b c

13 1.04 g14 a $280 b 103.7%

15 16 $748.61

17 33.4 18 $166719 a 3 + 11d b 10x + 2 c 5 − 7y

d 22w − 2 e 1 − 16k f 26t − 2320 a 3(8 − 5m) b 4(9i − 7mk)

c xy(x − 5y) d h(h − 1)e 3(16 − 5kd) f 3y(3 + 5y − 2n)

21 a b c

22 a b c

3y 5+4

---------------- x 1–3x 2+---------------- w 5+

w 5–-------------

95--- 2 y 3–( )

y2-------------------- 1

2 d 2–( )--------------------- 5

4---

x 4–x x 1+( ) x 1–( )--------------------------------------- 2

3 x 3–( )---------------------

10 x2 2x––2 x 2–( ) x 2+( )--------------------------------------- k2 3k– 10+

2 k 3+( ) k 3–( )---------------------------------------

5x 9–x 3–( ) x 2+( ) x 3+( )

------------------------------------------------------ 4d2 3d– 3+d d 1–( ) d 2–( )---------------------------------------

m 18+m 3–( ) m 1+( ) m 4+( )

---------------------------------------------------------- h2 3h 6–+h h 2+( ) h 2–( )---------------------------------------

1235------ 5

24------ 3

8--- 3

4---

320------ 22

25------ 49

300--------- 1

16------

14--- 1

4---

13---

1 100×20

------------------

81

500 200–50

------------------------

6096------ 16

28------ 8

5--- 10

9------ 6

15------

1160------ 7

80------ 4

25------

1433------ 59

90------ 56

333---------

1396------

10a2b2

3c------------------ 12x

y--------- 3m2

4n2----------

7m3

------- 6d11------ 13t

20--------


ANSWERS 551

d e f

23 a − or b − or

c d

e f − or

24 a b c

25 a 20m − 2 b x2 − 8x + 10 c 28v − 12v2 − 14d 9dh − 3d e −pr − 7p f 20n − 13

26 a d2 − 18d + 81 b 25a2 + 30a + 9c 81 − 126h + 49h2

27 a 25y2 − 9 b m2 − 64 c 49 − 9h2

28 a k2 + 18k + 77 b w2 − 11w + 28c x2 − 6x − 7 d a2 − 12a − 45e 12g2 + 17g − 5 f 6h2 + 7h − 24

29 a 25k2 b 24m c 6x − 18 d 11d − 5530 a (5k − 1)(k + 3) b (2e − 3)(2e + 7)

c (3y − 8)(y + 3) d (4k + 1)(3k − 5)e (h − 3)(4h + 5) f (4w − 5)(2w − 7)

31 a b c

32 a (y − 9)(y + 7) b (k − 5)(k − 2)c (d + 5)(d − 11) d (w − 7)(w + 2)e (g + 3)(g + 2) f (m − 9)(m − 8)

33 a (k − 8)(k + 8) b (9 − 5d)(9 + 5d)c 4(3y − 2x)(3y + 2x)

34 a −3(5m + 9) b 3(2d2 − 6 − 9d)c 7gh(g + 3)

35 a b

36 a (m + 3)(m − 5n) b (2y − 3)(ky − 1)

37 a b c

38 a b

39 a 2(2m − 5)(2m + 5) b (3g + 5)2

c (2p − 3q)(2p − q)

Chapter 4Start up

1 a 110° (vertically opposite angles)b 27° (vertically opposite angles)c 53° (angle sum of a right angle)d 18° (angle sum of a right angle)e 41° (angles on a straight line)f 105° (angles on a straight line)

2 a p = 55, v = 125 b a = 35, c = 75c x = 75 d b = 90, h = 45, q = 115e w = 55 f c = 30, y = 150

3 a 3k + 66 = 180 (co-interior angles)3k = 114k = 38

b 5m = 80 (alternate angles)m = 16

c 4w = 120 (corresponding angles)w = 30

d 3c = 75 (alternate angles)c = 25

e 5m + 30 + 70 = 180 (co-interior angles)5m + 100 = 180

5m = 80m = 16

f 4k + 60 = 180 (vertically opposite angles, then co-interior angles)

4k = 120k = 30

Skillbank 42 a 6 b 12 c 10 d 6

e 24 f 24 g 24 h 18i 32 j 28 k 36 l 26

4 a 6 b 9 c 7.5 d 10.5e 6 f 22.5 g 17.5 h 14i 12

6 a 3.75 b 8.5 c 13.5 d 5.2e 12.6 f 30

Exercise 4-011 a d = 70 b k = 70 c y = 24 d k = 60

e w = 33 f c = 222 a m = 115 b m = 85 c m = 68 d m = 130

e m = 135 f m = 120 g m = 40 h m = 112i m = 121

3 a 125° b 110° c 67° d 50°e 60° f 56°

4 a A and D b D and C c B5 a d = 16 b a = 20 c g = 73

d y = 20 e k = 28 f e = 108, y = 36g h = 35 h m = 27, t = 24, v = 50i h = 80, n = 65, r = 50, w = 130j t = 35 k h = 30 l m = 29, w = 64

6 a 52.5° (vertically opposite angles, then angle sum of an isosceles triangle)

b 105° (angles on a straight line, then angles in an isosceles triangle, then angles on a straight line)

c 110° (angle sum of isosceles triangle, then corresponding angles, then angles on a straight line)

d 115° (angles on a straight line, then angle sum of isosceles triangle, then angles on a straight line)

e 36° (angle sum of isosceles triangle, then co-interior angles)

f 60° (angle sum of a triangle, then alternate angles)

10y 3m–6

----------------------- 72x------ 15 6k+

6h-------------------

13a3

--------- 5d2

------ 26a 15d–6

------------------------- 13e3

--------- 8w5

------- 65e 24w–15

--------------------------

10m 11+12

----------------------- 12x 1+10

-------------------

4a 23–15

------------------- 9y5

------ 7p3

------- 27y 35p–15

-------------------------

2b2dc

------------ 3k4c------ 3m2

10n2------------

1k 3–------------ 1

x 3–------------ y 5+

6------------

9k 8+k 1–( ) k 2+( ) k 3–( )

----------------------------------------------------- n2 1+n n 1–( ) n 1+( )---------------------------------------

d 5+d 7+------------- e 3–

e 5+------------ 3m 2–

5m 4+-----------------

5m3

------- k 4+( ) k 1–( )5

-----------------------------------



7 a

b a = 65 b = 65 c = 50c

8 a ∠ P = 30°, ∠ Q = 60°, ∠ R = 90°b ∠ P = 48°, ∠ Q = 42°, ∠ R = 90°

9 20°, 60°, 100°. Yes, rule is m + n + 3n = m + 4n = 180 which has more than one possible solution (e.g. m = 80, n = 25 gives 25°, 75°, 80°).

10 a Isosceles, missing angle is 70°.b Isosceles, missing angles are both 56°.

Exercise 4-02

2 a Trapezium, parallelogramb Trapeziumc Square, rhombus d Square, rectanglee Square, rhombus, parallelogram, rectanglef Square g Square, rhombush Kitei Square, rhombus, parallelogram, rectanglej Rectangle, square k Rectangle, rhombusl Parallelogram, square, rectangle, rhombusm Parallelogram, rectangle, square, rhombusn Square, rhombus, kite

3 A, C, D, E, G4 a 74° b 30° c 221° d 57°

e 30° f 81°5 a k = 70 (opposite angles of parallelogram are

equal)b a = 45 (angles of square bisected by diagonals)c p = 55 (co-interior angles, opposite sides of

rhombus parallel)d w = 112 (opposite angles of parallelogram are

equal, then angles on a straight line)e t = 55 (angles of a rectangle equal 90°)f n = 67 (corresponding angles, opposite sides of

a rhombus are parallel)g a = 55 (co-interior angles)h c = 9 (diagonals bisect angles of a square)i r = 110 (angles between unequal sides of a kite

are equal)6 a ∠ PQR = 72° (alternate angles, RW � PV)

b ∠ TQU = ∠ TUQ = 53° (angles in isosceles triangle)∠ QUP = 53° (alternate angles, QT � PU)∠ UPQ = ∠ PQU = 63.5° (angles in isosceles triangle)∴ ∠ PQR = 63.5° (alternate angles, RQ � PU)

c (30° + ∠ QRT) + 110° = 180° (co-interior angles, QT � PR)∠ QRT = 40°∴ ∠ PQR = 40° (alternate angles, PQ � RT)

d ∠ PRQ = 35° (vertically opposite angles)∴ ∠ PQR = 72.5° (angles in isosceles ∆PQR, diagonals of rectangle are equal and bisect each other)

e ∠ QRP = 75° (angles on a straight line)∠ QPR = 75° (angles in isosceles triangle)∴ ∠ PQR = 30° (angle sum of triangle)

f ∠ TPQ = 60° (alternate angles, RT � PQ)∴ ∠ PQR = 30° (angle sum of triangle, diagonals of rhombus meet at 90°)

7 a n = 61 (angle sum of isosceles triangle)m = 122 (angle sum of right angle, then angle sum of isosceles triangle)

b p = 90 (diagonals of a rhombus bisect each other at right angles)q = 65 (alternate angles and angle sum of a triangle)

Properties

One pair of opposite sides parallel

✓

Opposite sides parallel ✓ ✓ ✓ ✓

Opposite sides equal ✓ ✓ ✓ ✓

All sides equal ✓ ✓

Two pairs of adjacent sides equal

✓

Diagonals equal ✓ ✓

Diagonals bisected ✓ ✓ ✓ ✓

Diagonals are perpendicular

✓ ✓ ✓

Diagonals bisect angles ✓ ✓

Opposite angles are equal

✓ ✓ ✓ ✓

One pair opposite angles equal

✓

All angles 90° ✓ ✓

Axes of symmetry 0 0 2 4 2 1

Order of rotational symmetry

✓ ✓ ✓ ✓

6 cm

40°

70° 70°

6 cm100°

40° 40°or

d°

Tra

pez

ium

Par

alle

logr

am

Rec

tan

gle

Sq

uar

e

Rh

ombu

s

Kit

e


ANSWERS 553

c t = 34 (alternate angles)v = 34 (angle sum of a triangle)w = 112 (opposite angles of parallelogram are equal)

d 2g + 3g = 180 (co-interior angles)5g = 180g = 36

e y = 90 (co-interior anglesk = 85 (vertically opposite angles)

f a = 44 (angle sum of isosceles triangle)b = 68 (alternate angles)c = 56 (angle sum of isosceles triangle)

8 a ∠ N = 180 − ∠ R (co-interior angles, KL � NM)∴ ∠ M = 180 − (180 − ∠ K) (co-interior angles,

KL � NM)= 180 − 180 + ∠ K= ∠ K

∴ ∠ L = 180 − ∠ K = ∠ N (co-interior angles,KL � NM)

b equal9 a Similar to 8 a

b i Triangle is isoscelesii Alternate angles iii Alternate angles

c From b ∠ GFD = ∠ EFD (both equal ∠ GDF)and ∠ GDF = ∠ FDE (both equal ∠ GFD)

∴ ∠ F and ∠ D are bisected by FD.10 ∠ CBA = c (alternate angles, DC � AB)

and ∠ CBA = a (alternate angles, DA � CB)∴ a = c

Exercise 4-031 a 1800° b 2340° c 3240° d 1980°

e 900° f 2160° g 1260° h 3060°i 4140°

2 a 14 b 11 c 24 d 17e 8 f 21 g 30 h 13i 16

3 a 5 b 540° c x = 36d 36°, 72°, 108°, 144°, 180°

4 173°

Exercise 4-041 a Equilateral b Square2 a 135° b 144° c 150° d 156°3 a 162° b 156° c 166.2° d 120°

e 168° f 144°4 a 22 b 163.6°5 a 157.5° b 162° c 161.1° d 165°

e 150° f 170°6 a 24 b 36

Exercise 4-051 a 15 b 10 c 9 d 36

e 20 f 62 a 45° b 36° c 24°

3 a 12 b 72 c 20 d 9e 5 f 24

4 a 120° b 170° c 128.6° d 175°e 162° f 157.5°

5 10

Power plus1 a 135° b 82.5° c 127.5°2 a Construction b 70.71 mm3 a 900° b 2520°4 Proof5 b

c d = n(n − 3)

Chapter 4 review1 a e = 75 b x = 24

c t = 107.5 d h = 19, x = 20.5e d = 6 f a = 18, c = 53

2 a True b False3 a Square, parallelogram, rectangle, rhombus,

trapeziumb Square, rectangle c Square, rhombusd Square, parallelogram, rectangle, rhombuse Square, rhombusf Rhombus, square, kite

4 a True b False c True d False5 a x = 65 b m = 60, y = 120

c g = 60 d a = 55, k = 756 Proof7 152.3°8 24

Chapter 5Start up

1 a $56 b $2 c $140 d $28e $6 f $98 g $10 h $84

2 a 10 b 2 c 3 d 100e 20 f 13 g 63 h 32

3 a $9 b $15 c $3 d $12e $30 f $10.50 g $52.50 h $90

4 a $384.62 b $6250 c $3150d $1566.89 e $1830 f $14 911.22g $764.09 h $1208.13 i $6848

5 a $55 b $88 c $165 d $13.20e $7.48 f $23.32 g $50.60 h $41.47

Number of sides

3 4 5 6 7 8 9 10

Number of diagonals

0 2 5 9 14 20 27 35

12---



Exercise 5-011 a $621.04 b $1242.09 c $2700.50

d $17.252 a $502.80 b $719.95 c $556 d $9903 Dana earns $321.86 per week

Darren earns $320.60 per week∴ Dana by $1.26

4 Fortnightly income of $10755 a $457.20 b $21 945.60 c $1828.806 a Zane, by $0.55 b Mitch, by $1.40

c Philip, by $0.30 d Fiona, by $2.60e Robert, by $0.90

7 a $331.20 b $15 897.60 c $1324.808 a i $42 000 ii $804.91

b i $72 000 ii $1379.84c i $103 680 ii $1986.97d i $57 600 ii $1103.87

9 a $187 b $935 c $4051.6710 a $108 b $119.70 c $92.30 d $10411 a $3226.67 b $742.05 c $17.6712 $19.25 13 $724.20 14 $2584.6215 $716.45

Exercise 5-021 a $550.40 b $678.40 c $742.40

d $566.40 e $720 f $707.202 a $556.80 b $1165 c $911.40

d $1040 e $1560 f $897.903 a 43.5 hours b 46 hours c 46 hours

d 21 hours e 12.75 hoursf 15 hours4 a $19.60 b $19.59 c $17.14

d $13.65 e $16.73 f $8.275 a $613.20 b $394.20 c $503.70

d $671.60 e $580.35 f $744.606 a $94.80 b $110.60 c $110.60

d $268.60 e $233.05 f $191.58

Skillbank 52 a $19.50 b $7.50 c $87.40

d $20.20 e $3.76 f $21.67g 93c h 90c i 5cj 10c k $152.76 l $8.27m $3.15 n 11c o 43cp $2431.76

4 a $124 b $490 c $1.72d $7.72 e $100.40 f $2550g $192.40 h $7.60 i $1.50j $1.63 k $5.08 l $64

6 a $100 b $2.50 c 60cd $1.35 e $1.84 f $3.62g 4c h 1c i 48cj $1.93 k 40c l $42.88

Exercise 5-031 a $1440 b $1215 c $4232 $10 9793 a Perry $700, Amy $750

b Perry $1330, Amy $1800c Perry $1600, Amy $2250d Perry $2275, Amy $3375

4 a $82 000 b $68305 a $165 b $550 c $1287

d $1430 e $2736.25 f $3201g $5486.25 h $9460 i $11 000

6 $13487 a $6000 b $8500 c $95908 $45549 a $8300 b $9820 c $11 400

10 a $18 222.22 b $34 888.89 c $62 222.2211 a $29 000 b $51 000 c $83 00012 a $74 285.71 b $99 142.86 c $134 285.7113 a $163.80 b $90 c $188

d $307.80 e $97.1314 a i $79.90 ii $124.10 iii $172.55

iv $362.95b 540 toysc i 236 ii 424 iii 559 iv 730

15 a $136.36 b $142.10 c $182.35d $457.10 e $1090.25

16 a $734 b $122.33 c $14.4017 a i $51.60 ii $86 iii $111.80

iv $172 v $223.60 vi $374.10b 1380 callsc i 140 ii 419 iii 698 iv 1675

18 a i $4.80 ii $9.60 iii $17.60 iv $36.40v $39.20 vi $57.20vii $60 viii $66.40

b $180.6019 a i $564 735 ii $10 860.29

b i $41 975.80 ii $807.23c i $409 410.75 ii $7873.28d i $217 426 ii $4181.27e i $115 833.50 ii $2227.57f i $470 510.75 ii $9048.28

Exercise 5-041 a $2014 b $3116 c $2704.65

d $1961.18 e $2416.80 f $3847.50g $3897.66 h $3638.31

2 a $1023.76 b $1237.20c $1527.84 d $1542.96

3 a $1270.12 b $1315.324 a $680.40 b $558

c $691.20 d $722.305 a $18.50 per hour b $755.20

c $37 762.406 a $3384 b $4531.74 c $5099.50

d $7050 e $6679.78 f $4337.917 a $536 b $375.20 c $2519.20


ANSWERS 555

8 a $1993.10 b $697.59 c $4683.799 a $632 b $2970.40

10 $784.70 11 $97 44012 a i $891.60 ii $1046.40

b $3142.89

Exercise 5-051 a $1156.20 b $847.80 c $365.50

d $1205.10 e $438.602 a 53.5% b 61.2% c 78.6%

d 44.0% e 29.3%3 a i $638.40 ii $277.80 iii $360.60

b i $525 ii $396.90 iii $128.10c i $1071.60 ii $459.60 iii $612

4 a $2099.66 b $764.28 c $1060.385 a $353.01 b $55.07 c $268.746 a $251.70 b 24.5%7 a i $139.47 ii $89.66 iii $393.50

b i $410.53 ii $167.45 iii $772.44c i $116.05 ii $167.63 iii $432.69d i $522.34 ii $336.05 iii $967.98e i $2119.22 ii $788.71 iii $3950.40f i $2458.56 ii $685.56 iii $4735.88

Exercise 5-061 a 10.2% b $2600 c $64 d 24 weeks2 a $315

b i 25% ii 21.9%c $35

b She owns her house already.

b $90c i 7.1% ii 11.4%d 90 × 52 = $4680 saved in one year

∴ Still requires $120 to pay for trip.e Decrease magazines and entertainment.

5 a $547.50b i $252.50 ii $13 130c $90

Exercise 5-071 a Item B b Item B c Item A2 a Item A b Item A c Item A3 a Brand Z b Brand Y

c Brand X d Brand W4 D Chocolate Swirl5 Peta’s battery6 a ii b i c i d ii e i

Exercise 5-081 a $182.16 b $300.80 c $3.362 a B b D c A d B3 a 30.56% b 40.60% c 22.81%

d 20.83% e 39.20%4 a $120 b $183.33

c i $106.15 ii $148.62d $104.71

5 a $9692.55 b $11 664c $7052.83 d $12 915.36

6 a $125.80 b $119.51c $118.40 d No. 20% is better.

7 $749.528 a i $4036.69 ii $962.31 iii 19.25%

b i $612.89 ii $146.11 iii 19.25%c i $484.50 ii $115.50 iii 19.25%d i $879.37 ii $209.63 iii 19.25%

9 Store 2, you save more.10 A discount of 10% followed by one of 15% is

equivalent to an overall discount of 23.5%, which is less than a single 25% discount.

Power plus1 Brenda: $514 per fortnight

Barry: $454.80 per fortnightDifference = $59.20

2 $26 7903 75 cm x 150 cm @ $7.95 for 2 rolls.4 Option 3 is the cheapest option.5 Store 2 has the cheapest offer for 90 drawers.6 Option 1: Selling price = $945

Option 2: Selling price = $945There is no difference.

7 a $45 084 b $44 557 c $9747.10d $8964.80 e Tax debt of $782.30

8 a $6662.80 b $21 434.32c $202.30 per week d $1003.11 per fortnighte $1397.03 per month

9 a $61 830 b $357010 $212.77

Chapter 5 review1 $23 128.802 a $56.40 b $94 c $65.803 a $958.22 b $1916.44 c $4166.67

3 a Income Expenses

Earnings $494 Groceries $100Household bills $65Car $45Clothes $80Ent’ment $90Savings $114

4 a Income Expenses

Earnings $350 Board $50Fares $35Clothes $80Magazines $25Food $30Entertainment $40Savings $90



4 Brian5 a $702 b $708.40 c $8606 a $440.75 b $344 c $521.387 $176.408 a $288 b $1134.909 a $4200 b $50 400

10 a $1260 b $2250 c $11 55011 87 bags 12 $11 625 13 $4117.2014 $821.70 15 $691.4216 a 300 mL for $2.70 b 750 g for 94c17 B18 a $490.50 b $236.25

c 9.7% (to 1 dec. pl.)19 a $550 b $99020 Yes

Chapter 6Start up

1 a b c d

e f g h

2 a b c d 4

e 100 f g h

3 a 60° b 50° c 160°d 60° e 40° f x° = 45°, y° = 135°

4 a AC b w c 105 a 85 b 16 c 3.9

Exercise 6-011 a b

c

2 a R b d c B d x e Q3 a T b T c F d T e F

f i F ii T4 The hypotenuse is always the side opposite the

right angle.5 The positions of the opposite and adjacent sides

depend on which of the two angles (other than the right angle) is being considered.

6 a i 13 ii 12 iii 5b i BC ii AC iii AB

c i v ii u iii wd i SR ii TR iii STe i x ii y iii zf i 41 ii 9 iii 40

7 a i angle , hyp = DF, opp = EF, adj = DEii angle φ, hyp = DF, opp = DE, adj = EF

b i angle , hyp 15, opp = 12, adj = 9ii angle φ, hyp = 15, opp = 9, adj = 12

c i angle , hyp = d, opp = f, adj = eii angle φ, hyp = d, opp = e, adj = f

d i angle , hyp = HJ, opp = HI, adj = IJii angle φ, hyp = HJ, opp = IJ, adj = HI

e i hyp = 45, opp = 36, adj = 27ii hyp = 45, opp = 27, adj = 36

f i hyp = a, opp = c, adj = bii hyp = a, opp = b, adj = c

8 a i K ii L iii M iv M v Lb i ‘opposite’ and ‘adjacent’

ii ‘opposite’ and ‘adjacent’9 a b

c

d

Exercise 6-021 a , , , b , , ,

c , , ,

2 a i 0.90 ii 0.44 iii 2.05b i 0.91 ii 0.46 iii 1.93c i 0.90 ii 0.44 iii 2.05d i ∠ G ii ∠ I

3 a T b T c Fd T e F f Tg F h T i T

Exercise 6-031 a sin = , cos = , tan =

b sin = , cos = , tan =

c sin R = , cos R = , tan R =

513------ 13

5------ 25

7------ 7

25------

1213------ 13

12------ 1

4--- 4

1---

15--- 1

3--- 1

4---

1100--------- 2

3--- 3

2---

d

ef

EF

D

r

qf

PQ

R

c

ba

B

C

A

θ

θ

θ

θ

C

A

Bθ α

Z

Y

X

P

hypotenuse = PR

Q R

F

isosceles right-angledtriangle

D E

DHXH--------- CG

XG-------- BF

XF-------- AE

XE-------- XD

XH-------- XC

XG-------- XB

XF-------- XA

XE--------

DHXD--------- CG

XC-------- BF

XB-------- XA

XE--------

θ 5573------ θ 48

73------ θ 55

48------

θ ef--- θ g

f--- θ e

g---

STRT-------- RS

RT-------- ST

RS-------


ANSWERS 557

d sin α = , cos α = , tan α =

e sin M = , cos M = , tan M =

f sin W = , cos W = , tan W =

2 a i X ii Y iii Yiv Y v X vi X

b i α ii β iii βiv α v α vi β

3 a i , , ii , ,

b i , , ii , ,

c i , , ii , ,

d i , , ii , ,

e i , , ii , ,

f i , , ii , ,

4 a i F ii F iii Tb i F ii T iii Fc i T ii F iii T

5 a i tan Y ii tan Xiii sin X or cos Y iv sin Y or cos X

b i sin φ or cos ii tan φiii sin or cos φ iv tan

c i tan W ii sin W or cos Uiii tan U iv sin U or cos W

6 a

sin A = , cos A =

b

cos B = , tan B =

c d

sin C = , tan C = cos Y = , tan Y =

e f

sin P = , tan P = =

Exercise 6-041 a 57.916

.° b 84.2° c 16.683

.°

d 24.5° e 50.883.° f 75.13

.°

g 30.50694.° h 82.67083

.°

2 a 33°45'36'' b 14°6'0'' c 78°9'0''d 55°30'0''

3 a 56° b 57° c 57°d 27° e 28° f 28°g 34° h 33° i 33°

4 a 68°40' b 68°39' c 68°40'd 18°30' e 18°31' f 18°30'

5 a 38.2° b 66.1° c 27.2°d 45.8° e 8.4° f 81.1°

6 a 34°27' b 71°5' c 5°29'd 69°27' e 41°19' f 50°13'

7 a 0.3907 b 0.4985 c 0.8258d 0.9489 e 3.9959 f 0.8311

8 a 1.78 b 9.56 c 3.79d 34.43 e 19.84 f 244.91g 32.20 h 53.21 i 31.40j 18.55 k 3.86 l 174.05

Exercise 6-051 a 41° b 64° c 60° d 81°

e 61° f 30° g 6° h 55°i 60° j 82° k 84° l 30°

2 a 64°59' b 54°35' c 36°52'd 20°10' e 78°14' f 72°0'g 27°2' h 61°39' i 86°12'j 70°32' k 35°16' l 51°3'

3 a 5.7° b 63.0° c 45.2°d 53.1° e 23.9° f 53.9°g 18.2° h 24.2° i 73.4°j 26.7° k 72.5° l 41.8°

4 a α = 5°44' cos 5°44' = 1.0, tan 5°44' = 0.1b β = 29° sin 29° = 0.48, tan 29° = 0.55c γ = 53°58' sin 53°58' = 0.809,

cos 53°58' = 0.588d = 30° sin 30° = 0.50, tan 30° = 0.58

Skillbank 62 a ≈ 2.8 b ≈ 4.2 c ≈ 5.3

d ≈ 10.5 e ≈ 8.9 f ≈ 5.6

g ≈ 3.5 h ≈ 8.1 i ≈ 8.7

j ≈ 5.4 k ≈ 6.3 l ≈ 11.2

nk--- m

k---- n

m----

3.68.5------- 7.7

8.5------- 3.6

7.7-------

YXYW--------- WX

YW--------- YX

WX---------

2425------ 7

25------ 24

7------ 7

25------ 24

25------ 7

24------

uw---- v

w---- u

v--- v

w---- u

w---- v

u---

FGHG--------- HF

HG--------- FG

FH-------- HF

HG--------- FG

HG--------- FH

FG--------

8485------ 13

85------ 84

13------ 13

85------ 84

85------ 13

84------

ba--- c

a--- b

c--- c

a--- b

a--- c

b---

QSQR-------- RS

QR-------- QS

SR------- RS

QR-------- QS

QR-------- RS

QS-------

θθ θ

13

12

5

A

513------ 12

13------

5

4

3

B

45--- 3

4---

17

8

15

C

5

3

4

Y

1517------ 15

8------ 3

5--- 4

3---

3

1

8

P

1

1M

2

sin M = , cos M = 1

2-------

83

------- 81

------- 8

θ

8 18 28

111 80 31

12 65 75

29 40 126



Exercise 6-061 a tan b cos c tan

d sin e tan f cos 2 a 27.63 b 24.44 c 8.90

d 31.17 e 38.71 f 53.023 a 66.5 b 1.9 c 17.2

d 47.4 e 31.3 f 251.04 a 17 b 58 c 9

d 7 e 115 f 2655 a 167.2 b 131.0 c 622.2

d 2.2 e 24.5 f 3.56 a 3273 mm b 228 cm c 141 m

d 671 m e 1053 cm f 17 km

7 a

AB = 12.3 m

b

MR = 10 cm

c

PQ = 64.15 m

d

XW = 5.76 m

e

HK = 3.4 m

f

BE = 30 m

8 a 933 cm b 5738 mm c 48 md 624 m e 364 m × 265 m

f 1.6 km g 0.47 m h 5.1 mi 89 m j 930 mm

Exercise 6-071 a 102.3 b 62.9 c 96.4

d 21.5 e 245.7 f 378.32 a 89 b 1188 c 45 d 1924 a 27.5 b 109.4 c 293.5

d 5.7 e 88.1 f 20.85 a 638 cm b 1203 cm c 414 mm

d 7 m e 4 km f 1044 cm

6 a

b

c

d

e

f

7 a 782 cm b 12.1 m c 4.9 kmd 6.88 m e 3.84 m f 77.3 mg 1698 mm h 146 m

Exercise 6-081 a 34° b 51° c 60°

d 29° e 67° f 54°

θ θ θθ θ θ

14.8 m 56°

A B

C

19 cm27°14'

M N

R

43.5 m

55.86°T P

Q

8.34 m43.7°

X W

Y

13.9 m76°

A K

H

3.5 m

83°15'

B E

I

12 m

75.2°L W

K

KW = 12 m

4.95 m

36°

C D

E

CD = 3.60 m

230 mm45°35'

W L

P

LP = 329 mm

18.4 cm

19°47'M T

H

HM = 54.4 mm

42.1 m

84°

F G

W

WG = 402.8 m

25.34 m

18.3°D N

R

DR = 80.70 m


ANSWERS 559

2 a 69°52' b 27°2' c 55°9'd 22°33' e 62°22' f 78°28'

3 a 36.7° b 25.2° c 36.9°d 31.5° e 26.8° f 48.2°

4 a 12.3° b 68.0° c 26.6°d 27.2° e 30.0° f 88.1°

5 a

b

c

d

e

f

6 a 11° b 37° c 6° d 40°e 71° f 35° g 11° h 50°i 25° j 42°

Exercise 6-091 a i ii = 20°

b i ii = 30°

c i ii = 54°

2 a i ii = 49°

b i ii = 28°

c i ii = 43°

3 a 12 671 cm b 177 mc 2224 m d 180 m

4 a 14° b 9° c 50° d 19°5 a 14.5 m b 970 m c 1256 cm

d 57 m e i 379 m ii 273 km/h

Exercise 6-101 a 243° b 290° c 040°

d 115° e 210° f 140°g 312° h 253° i 065°

2 a 000° b 090° c 180°d 270° e 038° f 125°g 330° h 225° i 072°j 187°

3 a

b

10 cm∠ W = 39°

Y X

W

8 cm

19.5 m ∠ F = 10°45'

G H

F

3.7 m

35 m

∠ M = 38.9°H T

M

45 m

9.5 cm

∠ V = 62°S V

T

8.4 cm

5 m

∠ W = 77.7°

B K

W

23 m

3.9 m

∠ A = 68°13'C A

E4.2 m

70°

θ

θ

30°

θ

θ

54°

θ

θ

41°θ θ

62°

θ θ

43°

θ θ

40°

W E

N

S

60°

W E

N

S



c

d

e

f

g

h

4 a

b

c

d

10°W E

N

S

35°W E

N

S

10°

W E

N

S

3°W E

N

S

9°

W E

N

S

32°

W E

N

S

42°

T

P

W E

N

S

80°

M

P

W E

N

S

65°

P

W E

N

S

X

10°

P

W E

N

S

K


ANSWERS 561

e

f

5 a 240° b 280° c 140° d 32° e 355°6 a 45° b 90° c 135°7

8 a

b

c

9 a 230° b 270° c 160° d 340°e 050° f 090°

Exercise 6-111 a 21 km b 260°2 a = 37° b 163 km c 143°3 a 63 km b 29 km c 025°4 a 12 km b 035°5 a 7.16 km b 252°6 a 19 km b 020°7 45.7 km8 a 14 km, 13 509 m b 321°9 a 3.7 km b 071.1°

10 a 2122 km b 330°11 a 15 km b 26 km12 a 261 km b 168 km13 6.558 km14 a 322 nautical miles b 276°45''

Power plus1 a i 0.342020143 …, 0.342020143 …

ii 0.731353701 …, 0.731353701 …iii 0.819152044 …, 0.819152044 …iv 0.996194698 …, 0.996194698 …v 0.5, 0.5

vi 0.788010753 …, 0.788010753 …b Each pair of trignometric ratios has the same

value.c The pairs of angles are complementary.d sin 60° = 0.8660e i T as 75° + 15° = 90° ii F

iii F iv T as 30° + 60° = 90°v T as 65° + 25° = 90°

vi T as 12° + 78° = 90°

35°

R

P

W E

N

S

5°Q

P

W E

N

S

45°

W E

N

S

W E

N

S

A

B

60°

180°

40 km

50 km

W E

N

S

W E

N

S

280°

80°

150 km

250 km

W E

N

S

W E

N

S

210°

20 k

m

E

N

S

W

N

S

15 km

θ



2 a 1932 m b 31°3 a β = 72° b γ = 22.5° c α = 15°4 a 24°12'26'' b 63°17'3'' c 26°42'57''5 a 5° b 0.7 m6 smaller building 21 m

taller building 30 m7 a x° = 50° y° = 40°

b i 085° ii 265°8 a 0.86 km b 2.35 km9 a 4 km b 343°

10 RP = 75 mRQ = 60 m

11 a i ii iii 1

b i ii iii 1

c i 1 ii 1

Chapter 6 review1 a U: O = 15, A = 20, H = 25

V: O = 20, A = 15, H = 25b U: O = VW, A = UW, H = VU

V: O = UW, A = VW, H = VUc U: O = u, A = v, H = w

V: O = v, A = u, H = w2 a i sin ii cos iii tan

b sin X = , cos X = , tan X =

sin Y = , cos Y = , tan Y =

c cos β = , tan β =

3 a i 39° ii 45° iii 66° iv 12° v 28°b i 12°9' ii 12°8' iii 12°9'

iv 12°31' v 12°40'c i 73.5° ii 18°15' iii 46.4°

iv 14°21' v 50°33'd i 65.0 ii 64.1 iii 210.8

4 a 60° b 42° c 43°5 a 14.67 b 72.50 c 0.886 a 99 b 21 c 237 a 32°35' b 28°4' c 37°49'

8 a

Height of tower = 195 m

b

Angle of depression = 29°

9 a i 060° ii 240°b i 320° ii 140°

10 a 5.2 km b i 1281 km ii 024°

Mixed revision 21 a 32 b 168.75°2 a C b A3 B, D4 a i 24 ii 120 b i 45 ii 85 a x = 22.5 (co-interior angles in a parallelogram)

b w = 35 (diagonals of rhombus meet at 90°, angle sum of a triangle)

c p = 93 (angle sum of a kite)6 a Item B b Item B7 a $496.41 b $1218.75 c $320.328 John (by 50 cents)9 a $752.40 b $526.68 c $3536.28

10 D11 a i 53° ii 78° iii 59°

b i 77°53' ii 69°33' iii 56°19'c i 80.0° ii 35.5° iii 28.0°

12 a 7.32 b 194.95 c 10.1813 a 11.9 b 8.2 c 14.314 a 39.3° b 51.9° c 44.4°15 a 17.9 b 11.7 c 32.416 a 2.7 m b 2.2 m17 212 km 18 75.7 m 19 50 m20 40.7°

21 a

b

35--- 4

5---

45--- 3

5---

5665------ 33

65------ 56

33------

3365------ 56

65------ 33

56------

7785------ 36

77------

tower

400 m

26°

100 m

55 m

θ

θ

10°

260°

M

A

70°

P

A


ANSWERS 563

c

d

Chapter 7Start up

1 a 16 b 32 c 9 d 125 e 100 000f 49 g 64 h 262 144 i 20 736

2 a 64 b 100 c 64 d 81e 16 f 1 000 000

3 a 52 b 45 c 63 d 36 e y2

f m5 g a3 h x6 i d4

4 a 10 × 10 × 10 b 8 × 8c 1 × 1 × 1 × 1 × 1 d 2 × 2 × 2 × 2e 3 f k × kg w × w × w × w h d × d × d × d × di p j c × c × c

5 a 20 b 17 c 32 d 15 e 256 a 2 b 3 c −2 d −6 e 10 f −3

Exercise 7-011 a i 3 ii 7 iii three to the power of seven

b i 7 ii 3 iii seven to the power of threec i k ii 4 iii k to the power of fourd i 4 ii k iii four to the power of ke i a ii n iii a to the power of n

2 a 54 b 102 c 83 d 322

e 95 f 123 g 14 h 61

3 a a4 b m2 c y4 d q6 e p3 f w1

4 a 25 × 32 b 34 × 73 c 56 × 82

d 63k2 e x3y2 f 52n3

5 a 6 × 6 × 6 × 6 b 10 × 10 × 10c 6 × 6 × 6 × 6 × 10 × 10 × 10d p × p × p × p e 5 × p × p × p × pf 5 × 5 × p × p × p × pg p × p × p × p × q × q × q × q × q

h 5 × p × p × p × p × q × q × q × q × qi 5 × 5 × p × p × p × p × q × q × q × q × qj a × b × b × bk a × b × b × b × c × cl a × a × a × a × b × c × cm m × m × m × n × n × n × nn 2 × y × y × y × d × do 4 × 4 × a × a × a × mp w × w × w × w × y × y × v × v × v

6 a 16 b 27 c 25 d 64 e 128f 125 g 169 h 512 i 1296 j 343k 1024 l 243 m 78 125 n 270 000o 9216 p 30 375

7 a 2751.261 b 0.021 c 0.026d −610.352 e 1.611 f 2.733g −0.132 h 0.032

8 a 3 b 4 c 3 d 2e 12 f 4 g 6 h 4

9 a 3 000 000 b −160 000 c 4100d 30 e 1.2 f 2700

10 a 256 b 108 c 1728d 180 e 3600 f 0.216

11 a −1, 1, −1, 1, −1, 1, …b i 1 ii −1c i 1 ii −1d i 0 ii −1

12 a i 36 ii 4356 iii 443 556iv 44 435 556

b i 44 444 355 556ii 444 444 443 555 555 556

Exercise 7-021 a 105 b 105 c 37 d 75 e 88 f 59

g 610 h 412 i 1120 j 24 k 312 l 78

2 a x5 b g8 c w8 d b13 e p20 f r2

g y6 h m8

3 a 6p7 b 12y12 c 18m9 d 5h11

e 24q11 f 10a7 g 30n16 h 30b9

i 3e10 j 50p5q k 16a4b5 l 20w9y5

m 10a3b4c3 n 10p5q9 o 20g4h6

4 a F b F c F d T e T f Fg T h F i F j T

5 a 256 b 200 c 102 400d 30 375 e 729 f 1000g 100 000 h 1 024 000

6 a x9 b x3y7 c 60n3 d 20mn2

e 100p4q3 f a8b5 g 4a + b h 22x + 1

i 33y j (p + q)5 k (x − y)3 l (a + 3)n + 1

Exercise 7-031 a 56 b 99 c 224 d 71

e 100 f 84 g 2010 h 219

i 660 j 84 k 36 l 27

2 a h16 b y6 c a8 d b1

15°

165°

T

A

304°

34°

W

A



e m0 f n6 g t9 h w24

i e20 j d4 k p5 l w18

3 a 2y12 b 5w6 c 8r6 d 30xe 5m8 f 2g6 g 2h9 h y4

i 3g56 j a4b k 9p4q2 l 20fg2

m 10mn n 3x4y7 o 11e3f 8 p 5km2

4 a F b F c T d F e T f Fg F h F

5 a 32 b 128 c 3.375 d 125e 1 f 40.96 g 1 h 64i 1600 j 512 k 0.1

.48

.l 81

6 a x5 b y6 c a3 d 4a − b e 2f 3y g 2m4 h 4n12 i p7

Exercise 7-041 a 46 b 516 c 312 d 228 e 22 f 912

g 100 h 620 i 515 j 250 k 35 l 70

m 220 n 134 o 416

2 a e8 b t25 c y21 d c5

e m35 f y16 g h0 h p18

i w4 j x10 k n24 l d9

m k50 n d12 o a64

3 a 16d4 b 25m2 c 16y10 d 81x8

e 3125m30 f 8w15 g 10 000d20 h 27e21

i 2b4 j 36d12 k 243f 20 l 1024c30

m 81h20 n 36k2 o 64w6

4 a b c d

e f g h

i j k l

5 a m30 b 53t3 c 28 d −x3

e y36 f 44w20 g −25d5 h 2100

i −33p6 j 52m6 k 35f 25 l m8

6 a 64 b 81 c 1 000 000 d 25e −8 f −4096 g 6561 h −15 625

i j k l

7 a l18m30 b x10y20 c

d e −w3e6k9 f

g h w16a8k32 i −243m10n5

j 16p8w12 k 1 l 1

m n − o 64k9m3

p 64k8y10 q 243a15d5f 20 r 1296d20p8

s − t

8 a 52x b 52x c (x + 1)6

d b4 e b20 f 324n12

g 4n4 h 27x7 i 20y2

Exercise 7-051 a 1 b 1 c 1 d 1 e 1 f 1

g 1 h 1 i 1 j 1 k 1 l 12 a 1 b 1 c 1 d 1 e 1 f 1

g 1 h 13 a 2 b 3 c −1 d 3 e 1 f 2

g 3 h 6 i 4 j 9 k 5 l 7

m 5 n −3 o 0 p 1 q 0 r 6

s 3 t 27 u 5m4 v 12 w 1 x 28


e f g h

i j k l

2 a b c d

e f g h

i j k l

m n o

3 a b c d

e f g h

4 a b c d

e f g h

5 a m−1 b w−1 c 8−1 d 9−1 e 2−2

f n−4 g 3−4 h 10−3 i e−3 j t −2

k 2a−1 l 4t −2 m 2w −5 n 5d −1 o (2y) −1

p (7e) −1 q (3a2) −1 r 5(3m4) −1 s (8p3) −1

t 2(3k) −6

6 a 3 b 16 c = 2

d = 15 e = 1 f = 1

g 100 000 h

7 a b c 4 d e

f g h i j −

k l

8 a y3 b e4 c 1 d n e 12g2

f 30a g h i j 4q3

k l m n o

9 a b c d

e m7n4 f g h

e5

32------ x2

49------ 27m3

8------------- 25h

2

36------------

f 8

81------ n40

p16-------- w10

t15-------- a4m4

c4-------------

4k6

25-------- 9r8

c4-------- a8b4

d20----------- 125c6

27x9--------------

94--- 4

25------ 125

8--------- 9

16------

a28

d21--------

m8

n4------ 8y9

x6--------

p10q15

t20----------------

d6e15 f 3

64-------------------- m10n15

32y25-----------------

243a5y20

b10----------------------- k2 p6

9q8-----------

12---

152----- 1

37----- 1

4---182-----

1104-------- 1

m----1h3----- 1

w2------

1204-------- 1---------

−111k8----- 1

c6-----

4d--- 3

x5----- 2

d3

----- 4m2------

a

b2----- m2

n4------ w

y2----- 4a

c------

3p2------ 15k

w4--------- 12y2

m3----------- m2

a4------

y3

d3----- 4x

y3------ 1

vm2----------

12m------- 1

xy------ 1

4h( )2-------------- 1

5k( )3-------------

13h( )2

-------------- 14k( )3

------------- 12c( )4

------------- 18y------

19--- 1

64------ 1

6--- 1

49------

111------ 1

32------ 1

16------ 1

100---------

94--- 1

4---

1258

--------- 58--- 3

2--- 1

2--- 4

3--- x1

3---

45---

w4---- n

m---- 5

4--- 3

k---

9x2----- 64

a6------ 9

16------ 25

4d2--------- m15

h10---------

64a6c9----------- p9

125d 6---------------

10x

------ 60e4------ 4

p3------

r3

4----- 1

4t----- 1

h4----- 1

b3----- 25

x2------

1y--- 5

p2q4------------ 1

m4n6------------- p2

w2------

-8a2b2----------- 2

xy4-------- 9

m11---------


ANSWERS 565

i j k l

m n o p

q r

10 a b 27 c d

e f

Skillbank 72 a 625 b 3025 c 2025 d 7225

e 13 225 f 56.25 g 9025 h 38 025i 2.25 j 4225 k 24 025 l 60 025

4 a 441 b 10 201 c 961 d 8281 e 26.01f 3721 g 40 401 h 1.21

6 a 3481 b 4761 c 7921 d 361 e 11 881f 24.01 g 6241 h 141.61

8 a 144 b 169 c 324d 361 e 121 f 2.56

Exercise 7-07

1 a b c d

e f g h

2 a b c d

e f g h3 a 8 b 7 c 10 d 25

e 0.2 f 0.5 g −4 h 32i −2 j −9 k 14 l 30

4 a b c d

e f g h

5 a 1.59 b 2.83 c 4.64 d 31.62e −3.68 f 33.33 g 1.90 h −0.20

6 a n b c

d 6m e 10g f

g pq2 h 3a3b3 i 2mn2

j 5 k l

7 a m6 b b8 c e6

d p4q12 e 125q15 f 4h4

g 4m2n8 h 32x10y5

Exercise 7-08

1 a b c y3 d x4

e f g h

2 a 8 b 2 c 32 d 9e 100 000 f 256 g 216 h 27

i 100 j k l

m n o

3 a 1.97 b 1.52 c 132.96 d 0.30e 0.03 f 5.28 g 0.02 h 2.41

4 a 2p b c d

e 32d10 f 256n16 g 125q15 h 100e2d4

Exercise 7-091 a 211 b 1511 c 43 d 712

e 91 f 32 g 46 h 88

i 719 j 28 k 71 l 149

2 a x7 b w17 c m5 d k2

e m8 f y24 g a2 h pi t13 j d4 k q7 l 6b7

m 20d13 n 6c4 o 8e6 p m3 a 1 b 4 c 1 d 7

e 4 f 2 g 1 h 4

4 a 9m10 b a5w9 c x5y5

d p4q11 e m7n5 f 6c5d8

g 20w8m5 h 24a3b6 i 48y17

j 2v k m2n l a8c4

m 4l3d2 n 3x3y3 o 8g2h6

5 a 7 b 64 c 16 d 1 e 25 f 31g 3 h 1 i −8 j 4 k 2 l 4

6 a b c d

e f g h

7 a b c d

e f g h

i j k l

m n o

8 a b c d

e f g h

9 a 4−3 b 2−1 c 10−3 d 7−4

e w−1 f k−4 g d −7 h 3−3

10 a b 1 c 4 d 1

e 1 f 16 777 216 g 27

h 3125 i 2 j 1 k

l

1p5q3------------ -2

h5-----

1

a4k10------------- 5

2x5y4--------------

20tr

-------- a

2b6--------- 1

4h------ 4v13

w3-----------

1a11------- c

d14--------

12--- 2

5--- 4

3---

14--- 1

9---

812---

3512---

1013---

1513---

2012---

51213---

10013---

7212---

m12---

w13---

8k( )13---

ab( )12---

9y( )13---

xy( )12---

18 f( )12---

10mn( )13---

37 83 d 20

4p3 100h3 3c7

w3

f12---

x43---

14h35---g

35---

5a12--- 9

a13---

-----

k15---

d54---

n53---

e32---

n-14---

a-73---

12--- 1

3--- 1

256---------

18000------------ 1

64------ 1

625---------

2m113------

4d25---

9m203------

512---

d12---

3y( )12---

1013---

4p( )13---

xy( )12---

100013---

3m2n( )13---

178----- 1

210------- 1

151-------- 1

y3-----

15x------ 1

92----- 1

103-------- 1

ab------

4y8----- 1

3a2--------- 10

d------ p3

q5------

m

w3------ c2

e3----- 8t3

m4-------

14--- 1

3--- 1

20------ 1

125---------

1512--------- 1

10 000---------------- 1

1296------------ 1

243---------

18---

19---

16---



11 a 16a6 b m10n15 c d

e f 4c2 g h

i j k 2m2 l

m 256k12n8 n 9g4 o 32x10 p

Exercise 7-101 a 1.52 × 108 km b 1.3 × 105 kg

c 2.9 × 10−7 cm d 8 × 10−5 me 3 × 108 m/s f 4 × 1013 kmg 2 × 10−24 g h 1.2 × 10−4 mi 6.1 × 102 cm j 1 × 10−10 mk 7.7 × 1020 m l 1 × 10−6 sm 2.2 × 106 light years

2 a 2.4 × 103 b 7.86 × 105 c 5.5 × 107

d 9.5 × 101 e 7.8 × 100 f 3.48 × 108

g 5.967 × 104 h 1.5 × 101 i 3 × 108

j 8 × 101 k 7.63 × 102 l 2.47 × 101

m 4.563 × 102 n 8.007 × 100 o 9.0576 × 103

p 1.302 × 102

3 a 3.5 × 10−2 b 7.6 × 10−5 c 8 × 10−1

d 7.13 × 10−2 e 3 × 10−6 f 9.13 × 10−1

g 7.146 × 10−6 h 9 × 10−3 i 1 × 10−6

j 8.9 × 10−1 k 7.8 × 10−8 l 1 × 10−1

4 a 2.5 × 104 b 4.4 × 106 c 1.85 × 108

d 7 × 100 e 4 × 10−1 f 2.7 × 10−2

g 8.75 × 10− 4 h 6.7 × 100 i 2.0345 × 1010

j 9 × 10−10 k 7.3 × 101 l 6 × 10−2

m 5.52 × 10−1 n 2.299 × 103 o 3.5 × 106

p 5.637 × 102 q 1 × 10−4 r 7.03 × 100

s 2.7 × 108 t 4.004 × 102 u 5 × 101

5 a 5700 b 570 c 57 d 5.7e 0.57 f 0.057 g 0.0057 h 800i 80 j 8 k 0.8 l 0.08

6 a 600 000 b 7100c 302 000 000 d 3.14e 0.000 06 f 0.0071g 0.000 000 0302 h 0.000 000 000 59i 1 100 000 000 000 j 0.0004k 5000 l 0.000 476m 0.803 n 63 200o 0.016 p 0.000 000 22q 9 000 000 r 0.111

7 a 2 × 100 b 9 × 101 c 7 × 102

d 4 × 103 e 5 × 106 f 3 × 10−1

g 7 × 10−2 h 5 × 10−6 i 1.5 × 10−1

j 1.5 × 103 k 3 × 105 l 6 × 10−3

8 a 1 b 0 c −6 d 10 e 2 f −1g 0 h −4 i 5

Exercise 7-111 a 8 × 108 b 2.7 × 105 c 1.3 × 107

d 6.3 × 10−7 e 9.3 × 109 f 9.3 × 102

g 3.04 × 100 h 4.5 × 10−5 i 2 × 10−15

j 9.7 × 10−2

2 a 6 × 102, 6 × 103, 6 × 105

b 7.3 × 109, 5.5 × 109, 3.8 × 109

c 3 × 10−6, 3 × 10−5, 3 × 10−4

d 9.5 × 10−3, 6.4 × 10−3, 4.1 × 10−3

e 4.9 × 10−4, 3.5 × 100, 5.3 × 104

f 6.9 × 10−1, 4.3 × 10−4, 2.1 × 10−8

g 6.3 × 102, 9.76 × 101, 8 × 10−4, 5 × 10−9

3 a China, Indonesia, Japan, Vietnam, South Africa, Australia, Cambodia, Lebanon, New Zealand, Tonga

b Tonga, Lebanon, Cambodia, New Zealand,Vietnam, Japan, South Africa, Indonesia,Australia, China

Exercise 7-121 a 6 × 108 b 2 × 105 c 8 × 1015

d 3 × 106 e 2.4 × 1016 f 5 × 104

g 1.024 × 1018 h 1.8 × 107 i 4 × 10−6

j 3 × 10−12 k 4 × 1015 l 7 × 109

2 a 2.856 × 1016 b 8.3094 × 1015

c 4.53 × 10−9 d 3.15 × 10−10

e 3.01 × 10−8 f 6.3 × 109

g 1.024 × 1019 h 2 × 10−3

i 3.81 × 1013 j 1.14 × 1017

3 a 4.25 × 1013 b 2.15 × 1035

c 7.04 × 1030 d 4.68 × 105

e 8.12 × 1013 f 2.27 × 1036

g 5.45 × 1050 h 6.46 × 108

i 1.81 × 10−4 j 9.50 × 102

4 a 1.2 × 109 b 3.2 × 1013 c 3 × 1015

d 4 × 1016 e 7 × 1019 f 1 × 104

g 3 × 10−4 h 2 × 109

5 a 4 × 1033

b No, the power of ten is incorrect. Manal’s answer is ten times larger, which makes it incorrect.

6 a 5.7 × 105 b 3.3 × 102 c 1.0 × 106

d 5.5 × 10−1 e 1.2 × 10−2 f 1.7 × 10−5

g 3.9 × 108 h 4.2 × 100

7 a 3.78 × 1019 atomsb i 507 seconds (to the nearest second)

ii 8 minutes 27 seconds (to the nearest second)

8 a 2.3 × 10−2 mm b 1.9 × 1014 tc 1.7 × 103 m/s = 1.7 × 100 km/sd 9.5 × 1012 km/yeare 4.3 yearsf 2.7 × 109 m/s

16k20

m4-------------- 64y15

27m6--------------

5a4

------ 49m2

100------------- 3125w

252------

125

d3--------- a2b6

c4----------- y2

16------

16a12

c8--------------


ANSWERS 567

g i 1.1 × 10−4 s ii 100 s

9 a 9.999 999 999 × 1099

b 1.000 000 000 × 10−99

Power plus1 a 512 b 256 c 625 d e

f 5 g 2 h 1600 i 1 000 000j 2 k 4 l 3200

2 a 42y b 1 c 102x d 36e 9a2 f 32a g 53n h 8x

i 2m2 j 3p2 k 2n

3 a = 2 b = 3 c = 2

d = 2 e f

g h i

j

4 a a4 b 4p4 c 2y3 d 3y5

e 10g10 f 4h8 g n4 h 4h5

5 a 9.42 × 1010 b 5.2 × 10−4

c 4 × 104 d 1.05 × 10−2

6 a 6.7 × 106 b 1.57 × 107

c 5.78 × 104 d 4 × 109

e 3.2 × 109 f 1.27 × 108

7 a a = 4, m = 1, n = 2

= 2b a = 9, m = 1, n = 2

= 3c a = 16, m = 3, n = 2

= 64d a = 625, m = 3, n = 4

= 125

8 If a = b there is an infinite number of answers. Otherwise a = 2, b = 4.

10 a 2n + 1, where n = 1, 2, 4, 8 … (doubling the power each time.

b 4 294 967 297

Chapter 7 review1 a 43 b 62 c 101

d x4 e y2 f mn3

g 42 × 73 h a2b2 i 32a3

2 a y13 b a5 c h10

d 6p7 e 9q11 f 10m8

g 15x8y h 30x3y3 i 20a4b5

3 a 416 b x6 c b11

d 2e12 e 5n6 f 8g6

g p4q h 9a4b2 i 20xy2

4 a a8 b y25 c b3

d 32x15 e 125r6 f 256w16

g a4b2 h 25a4b2 i 1000g3

j k l 243m15n5p10

5 a 1 b 1 c 1 d 1 e 1 f qg 2 h 16 i 4

6 a b c

d e f

g h i

7 a = 8 b = 4 c = −

d = q5 e = p5 f

g = 2a2 h i

8 a 625d20 b 2y5 c d

9 9.15 10 B (81) 11 D

12 a 64p3 b −4n4 c 125w12

d e −32x15 f

g 10 000e8d12 h i 4k2m

j k −243d20h5 l 2b3

13 a b −24 c 5832

14 a 5.5 × 104 b 5.5 × 10−1 c 2.5 × 105

d 2.5 × 10−4 e 8 × 100 f 8 × 10−11

15 a 8100 b 60 000 000 c 3.075d 0.0081 e 0.000 0006 f 0.030 75

16 a 3.5 × 107, 6.8 × 107, 7.5 × 107

b 9 × 10−8, 4 × 100, 3 × 103

c 5.7 × 10−7, 4.4 × 10−3, 3.1 × 10−1

17 a 2.701 × 10−13 b 4 × 105

c 1.25 × 1017 d 2.5 × 10−4

18 a 9.6 × 10−8 b 1.5 × 107 c 3.0 × 10−16

d 3.9 × 1034 e 7.4 × 1010 f 1.0 × 105

g 3.8 × 1018 h 8.0 × 10−5

Chapter 8Start up

1 a P = 40 cm b P = 88 cm c P = 112 cmA = 100 cm2 A = 384 cm2 A = 336 cm2

18--- 1

16------

164 814 325

1287 x34 m

25

k23 d

710 325

yn

4

12---

9

12---

16

32---

625

34---

m20

w25-------- 16a28

e4--------------

181----- 1

82----- 1

9m-------

1m5------ 1

y---1y2-----

1y3----- 1

3x------ 3

x---

64 643 -18-----3 1

2---

q153 p

10 2q3

4a4

xy3 8k

4x165------ p2

v4-----

1

4h4-------- 1

9m6n

2----------------

4a85---b

125------

1

125a6--------------

916------



d P = 70 cm e P = 84 cm f P = 42 cmA = 200 cm2 A = 336 cm2 A = 96 cm2

2 a 3400 mm b 9.2 m c 0.565 kmd 2.385 m e 38 mm f 45 mm

3 a i ii b i ii

c i ii d i ii

4 a 180 cm3 b 2197 cm3

c 1440 cm3 d 4500 cm3

5 a i 5 facesii two right-angled triangles and three

rectangles.iii cross section is a right

triangle

b i 8 facesii two hexagons and six rectangles

iii cross-section is a hexagon

c i 6 facesii two trapeziums and four rectangles

iii cross-section is a trapezium

6 a

b

c

d

Skillbank 82 a 895.4 b 37 c 83.1

d 4200 e 5271.6 f 1561g 31 840 h 6430 i 22.4j 48.94 k 7389 l 1142

4 a 73.34 b 0.94 c 6.52d 0.104 e 0.704 f 1.985g 0.02 h 4.159 i 12.3j 0.007 58 k 0.0849 l 0.0251

Exercise 8-011 180 min 2 27 t 3 0.6 km4 4500 mL 5 360 m 6 50 000 m2

7 230 cm 8 45 000 mg 9 9 m3

10 675 cm 11 50 min 12 47 mm

13 0.4 m3 14 125 000 m2

15 8 h 16 0.75 L 17 170 000 cm2

18 60 h 19 15 g 20 8150 m21 4.5 ha 22 300 000 g23 165 s24 14.5 m2 25 6 days 26 16 kL

27 15 cm3 28 3.8 ha 29 6 200 000 cm3

30 8 000 kg 31 48 mL 32 1.25 kg33 10 800 s 34 800 000 mg35 55 000 000 mL 36 600 000 cm37 4320 min 38 2.9 t 39 700 mm40 0.05 kL 41 0.375 t 42 5 days43 6 000 000 mm 44 240 h45 50 000 000 mg 46 500 m47 18 000 000 g48 0.08 km

Exercise 8-021 a 53 b 13.82 c 0.80

d 10.91 e 4.90 f 5.392 224 cm 3 17.3 m 4 4770 cm5 1146 m6 a 2.1 km, 1.1 km, 1.1 km

b 6.3 km7 a 4387 mm b 6652 mm8 a 47.2 m b 55.9 m9 a 1.14 m b 1.21 m

10 a 70 cm (not 71 cm) b 86 cm11 246 mm 12 14.7 cm 13 13.2 m

12--- 1

2--- 1

4--- 3

4---

13--- 2

3--- 23

36------ 13

36------


ANSWERS 569

Exercise 8-031 a 60, P = 144 b 85, P = 204

c 60, P = 132 d 12, P = 84e 65, P = 316 f 45, P = 230

2 a 270 mm b 360 mm3 16 sides 4 Length = 48 cm5 Cost = $42706 Length = 60 cm, breadth = 30 cm7 a 360 m b 6 laps8 a sides = 16.97 cm, P = 67.88 cm

b sides = 21.63 cm, P = 86.52 cmc short side = 13.42 cm

long side = 24.74 cmP = 76.32 cm

9 a 69.5 m b 64.02 m

Exercise 8-041 a 59.69 b 120.95 c 50.27 d 71.00

2 a i 90° ii iii 12.6 m iv 28.6 m

b i 180° ii iii 28.3 cm iv 46.3 cm

c i 270° ii iii 21.2 m iv 30.2 m

d i 60° ii iii 36.6 mm iv 106.6 mm

e i 280° ii iii 58.6 cm iv 82.6 cm

f i 140° ii iii 97.7 mm iv 177.7 mm

3 a 167.83 mm b 307.08 cm c 139.67 cmd 34.27 m e 33.56 m f 94.25 cmg 314.16 mm h 43.98 m i 41.13 m

4 a 182.8 cm b 72.8 mm c 121.1mmd 129.1 m e 554.2 cm f 1416.7 mmg 104.0 m h 46.6 cm i 56.5 cmj 106.5 cm k 54.8 m l 760.9 mm

5 a 31 cm b 201 cm c 251 cm6 40 laps 7 4.9 m8 a 40 074 km b 47 614 km9 a 19 cm b 19 m

10 13 mm by 1 mm, 12 mm by 2 mm, 11 mm by 3 mm, 10 mm by 4 mm, 9 mm by 5 mm, 8 mm by 6 mm, 7 mm by 7 mm

11 50π12 a P = 2w + n b P = 4(x + 2)

c P = y( + 1) d P = 2h(1 + )

e P = 4(p + q) f P = x(10 + π)13 a 399 m b 406 m c 7 m14 73.9 m

Exercise 8-051 a 96 cm2 b 45 cm2 c 84 cm2

d 300 cm2 e 540 cm2 f 180 cm2

2 a 960 mm2 b 352 mm2 c 2200 mm2

d 540 mm2 e 630 mm2 f 1824 mm2

g 540 mm2 h 324 mm2 i 180 mm2

3 a 75 m2 b 220 m2 c 392 m2

d 108 m2 e 232 m2 f 133 m2

4 a i 6 cm ii P = 24 cmb i 9 m ii P = 36 mc i 12 mm ii P = 48 mmd i 1.7 m ii P = 6.8 m

5 a 44 m2 b $1546 a 2.5 m b 31.25 m2

7 900 cm2

8 a 100 cm2 b 50 cm

Exercise 8-061 a 300 mm2 b 172.5 mm2 c 90 mm2

d 360 mm2 e 120 mm2 f 72 mm2

g 66.5 mm2 h 510 mm2 i 112 mm2

2 a 84 m2 b 800 mm2 c 52.25 m2

d 190 m2 e 54 cm2 f 3.15 m2

g 84 cm2 h 144.6 m2 i 752.5 mm2

3 a 46 m2 b 140 cm2 c 24 m2

d 168 cm2 e 100.8 cm2 f 47.04 m2

4 a 325 m2 b 141 m2 c 7.8 m2

5 a 288 cm2 b 432 cm2 c 216 cm2

Exercise 8-071 a 201.1 m2 b 84.9 m2 c 706.9 m2

d 624.6 m2

2 a 509 m2 b 79 m2 c 2313 m2

d 1056 m2 e 2566 m2 f 9 m2

g 9 m2 h 4618 m2

3 a 2228 cm2 b 293 cm2 c 952 cm2

d 442 cm2 e 185 cm2 f 71 cm2

g 370 cm2 h 942 cm2 i 52 cm2

4 a 503 cm2 b 151 cm2 c 193 cm2

d 743 cm2 e 4664 cm2 f 471 cm2

g 1383 cm2 h 11 cm2 i 314 cm2

j 1257 cm2 k 104 cm2 l 236 cm2

5 a 20 056 m2 b $18956 1.2 m2

7 a 35 m2 (to the nearest m2) b $1137.508 a 4503 m2 b $1125.759 a 10 m2 b 40%

10 a 4 b 1 : 9

Exercise 8-081 a 253.5 m2 b 4140 cm2 c 103 680 cm2

d 125.36 m2 e 9600 mm2 f 9680 cm2

2 a 408 m2 b 636 m2 c 360 m2

d 648 m2

3 a 1344 m2 b 4172 m2 c 2100 m2

d 2804 m2 e 492.7 m2 f 446.96 m2

14---

12---

34---

16---

79---

718------

π2--- π

3---



4 a 80 m2, $8400 b 171 m2

5

Shed 1 has the greater wall area.6 464 cm2

7 a 15.1 m b 151 m2 c 100 m2 d 251 m2

8 1.66 m2

9 a 3682 cm2

b The surface area has increased by 52 cm2.

Exercise 8-091 a 747.70 m2 b 3573.56 cm2

c 206.47 m2 d 15.83 m2

2 a 904.8 cm2 b 8545.1 mm2

c 35.3 m2 d 3477.7 cm2

3 a 1009 m2 b 2160 m2 c 4 m2

d 1895 m2 e 7 m2 f 14 m2

4 a 9721.7 cm2 b 14 031.4 cm2

c 14 778.1 cm2 d 4619.6 cm2

e 3827.4 cm2 f 2708.7 cm2

5 a 352 cm2 b 75.5 cm2

6 a 20 slices b 13 823 mm2 c 0.3 m2

7 2953 cm2 8 1028.32 cm2

9 5770 cm2

Exercise 8-101 a 1620 mm3 b 504.8 m3 c 10 500 mm3

d 1045 cm3 e 19 968 cm3 f 4.05 m3

2 a 9.7 m3 b 94 247.8 m3 c 7.4 m3

d 42.3 m3 e 5026.5 m3 f 216.9 m3

3 a 536 cm3 b 5027 cm3 c 167 cm3

d 12 900 cm3 e 33 117 cm3 f 3258 cm3

4 a 251.33 cm3 b 320.00 cm3 c 68.67 cm3

5 a 1508.0 cm3 b 75.4 cm3

6 13 666 cm3 7 2094.4 cm3

8 a 300 m3 b 300 kL9 h = 24 m 10 1728 cm3

Power plus1 a square: 10 cm by 10 cm, rectangle:

20 cm by 5 cmb rectangle: 6 cm by 4 cm, triangle: base 6 cm,

height 8 cmc parallelogram: base 9 cm, height 4 cm, square:

6 cm by 6 cm2 P = (20π + 8) m A = 40π m2

3 r = 33.9 cm 4 side = 17 cm5 628 m2

6 a A = (x + 2)2, P = 4(x + 2)

b A = w2π, P = w( + 1)

c A = , P = h(2 + )

7 a A = 2(pq + pr + rp) V = pqr

b A = x(y + )π V =

8 a r = 4.6 cm b 432.2 cm2

9 50 cm/h10 a seventy 4 cm cubesb 1520 cm3

11 160 cm3 12 40 mm

Chapter 8 review1 a 8000 m b 67 mm c 900 s

d 700 mm e 5760 min f 9400 kgg 250 kg h 4 h i 125 000 Lj 0.025 km k 17 kg l 0.075 kL

2 a 9.95 mm b 15.80 cm c 1.39 m3 a RQ = 15 b AG = 18.5

PQ = 394 a 142 m b 112 m c 60 m

d 36 m e 76 m f 52 m5 a 12.3 cm b 36.2 cm c 91.5 cm

d 180.0 cm e 106.5 cm f 90.8 cm6 a 468 m2 b 72 m2 c 480 m2

d 450.56 m2 e 210 m2 f 784 m2

7 a 14.1 mm2 b 177.0 mm2 c 162.6 mm2

d 326.7 mm2 e 121.1 mm2 f 73.1 mm2

roof

wall

floor

wall

roof

Shed 1

3 m

2.5 m

2.5 m

0.5 m

wall area = 28.75 m2

roof

wall

floor

wall

roof

Shed 2

3.5 m

3 m

2 m

0.5 m

wall area = 27.5 m2

π2---

h2

2----- 2

x2---

πx2y4

------------


ANSWERS 571

8 a 9360 cm2 b 2290 cm2 c 2592 cm2

d 1120 cm2 e 1268 cm2 f 3180 cm2

9 a i closed cylinder ii 942 m2

b i open cylinder ii 311 m2

c i cylindrical tube open at both endsii 170 m2

10 a 7389.0 cm2 b 1437.3 cm2 c 296.9 cm2

11 a 11 084 cm3 b 10 016 cm3 c 36 816 cm3

12 a 10.5 kL b 172.5 m2 c 3375 cm3

d i 59 000 L ii 1.44 m

Chapter 9Start up1 A(2, 4) B (6, 1) C(−4, 10)

D(−7, 0) E(5.5, −3) F(−2, −7)G(0, −2) H(12, 4.5) I(9, 9)J(−3, 0) K(−8, −4) L(0, −9)M(3, 8) N(9, −5) P(−6, 6)Q(2, 1) R(−6, −1) S(6, −7)T(0, 10) U(0, 6) V(−5, −5)W(2, −5) X(−7, −9) Y(−11, 2)Z(−1.5, 7.5)

2 a 6 units b 14 unitsc 8 units d 19 units

3 a y = 4 b y = 13

c y = 5 d y = −2

4 a x = 2 b x = 1

c x = 6 d x = −2

5 a y = 3x

b y = −2x + 1

c y = + 1

6 a B b A c C d B

7 a

b

c

12---

12---

12---

x −2 −1 0 1 2

y −6 −3 0 3 6

x −2 −1 0 1 2

y 5 3 1 −1 −3x2---

x −2 −1 0 1 2

y 1 1 1 212--- 1

2---

21 3 5423 1

2

1

3

4

2

4

6

5

3

10−− −

−

−

−

−

−

−

y

x

y = x − 2

21 3 423 1

2

1

3

5

4

6

7

10−− − −

y

x

y = 2x

21 3 423 1

2

1

3

5

4

6

7

10−− − −

y

x

y = x–2



d

e

f

8 a 7.8 m b 10.8 mm c 12.1 m 9 a 2 b 2 c −4

10 a −2 b c d e

Exercise 9-011 a

x-intercept: x = −4, y-intercept: y = 4

b

x-intercept: x = 6, y-intercept: y = 6

c

x-intercept: x = , y-intercept: y = −1

21 3 423 1

2

1

3

5

4

6

7

10−− − −

y

x

y = 2 − x

4321 5 7623 1

2

1

3

5

4

6

2

3

1−− −

−

−

−

y

x0

x + y = 5

4321 5 7623 1

2

1

3

5

4

6

2

3

1−− −

−

−

−

y

x0

x − y = −2

-34----- 2

5--- -5

4----- 2

5---

21 3246 5 3 1

2

1

3

5

4

6

2

3

10−−− − − −

−

−

−

y

x

y = x + 4

4321 5 7623 1

2

1

3

5

4

6

7

2

1−− −

−

−

y

x0

y = 6 − x

21 3 423 1

2

1

3

5

4

6

2

4

3

10−− −

−

−

−

−

y

x

y = 3x − 1

13---


ANSWERS 573

d

x-intercept: x = −1, y-intercept: y = −1

e

x-intercept: x = −4, y-intercept: y = 2

f

x-intercept: x = 3, y-intercept: y = 3

2 a

b

c

21 3 423 1

2

1

3

4

2

4

3

10−− −

−

−

−

−

y

x

y = −1 − x

21 3 423 1

2

1

3

4

2

4

3

10−− −

−

−

−

−

y

x

y = + 2x–2

4321 5 7623 1

2

1

3

5

4

6

2

3

1−− −

−

−

−

y

x0

x + y = 3

21 3 424 3 1

2

1

2

4

6

5

3

10−− − −

−

−

−

−

−

−

y

x

y = 3x − 6

4321 5 62 1

2

1

3

5

4

2

3

1− −

−

−

−

y

x0

y = 6 − x––––2

4321 5 761

2

1

3

5

4

6

7

2

1−

−

−

y

x0

x + y = 7



d

e

f

g

h

i

21 3 5 642 1

2

1

3

2

4

6

5

3

10− −

−

−

−

−

−

−

y

x

x − y = 6

8642 10 141246 2

4

2

6

10

8

12

14

4

2−− −

−

−

y

x0

x + y −12 = 0

4321 5 623 1

2

1

3

5

4

6

7

2

1−− −

−

−

y

x0

y = + 12x––3

21 3 5424 3 1

2

1

3

5

4

2

4

5

3

10−− − −

−

−

−

−

−

y

x

3x − 4y = 12

21 3 542 1

2

1

2

4

6

5

3

10− −

−

−

−

−

−

−

y

x

2x − y − 5 = 0

21 3 542 1

2

1

3

5

4

2

3

10− −

−

−

−

y

x

+ = 1y–3

x–2


ANSWERS 575

3 a

Point of intersection (−1, −1)

b

Point of intersection (4, 2)

c

Point of intersection (3, 2)

d

Point of intersection (−1, 4)

e


f


21 3 54245 3 1

2

1

3

5

4

2

4

6

5

3

10−−− − −

−

−

−

−

−

−

y

x

y = x

y =

2x +

1

4321 5 7623 1

2

1

3

5

4

6

7

2

1−− −

−

−

y

x0

y = x

− 2x + y = 6

4321 5 7623 1

2

1

3

5

4

6

7

2

1−− −

−

−

y

x0

y = x

− 1

y = 8 − 2x

21 3 54 623 1

2

1

3

5

4

2

4

6

5

3

10−− −

−

−

−

−

−

−

y

x

y = −4x

x + y = 3

21 3 54245 3 1

2

1

3

5

4

2

4

6

5

3

10−−− − −

−

−

−

−

−

−

y

x

y = x

y + x + 4 = 0

21 3 54245 3 1

2

1

3

5

4

2

4

6

5

3

10−−− − −

−

−

−

−

−

−

y

x

y = x

+ 3

y = x+1–––

2



Exercise 9-021 a No b Yes c No

d Yes e No f Nog No h Yes i No

2 a Proofb The lines intersect at (3, 2).

3 (−1, −5) satisfies both equations.4 (6, −12) satisfies both equations.5 a (6, 0) b (0, −3)

c (6, 0) satisfies y = 6 − x.d (0, −3) satisfies x = 3y + 9.

Skillbank 92 a (2, 2) b (−5, −7) c (4, −4)

d (8, 1) e (6, 4) f (−2, −5)g (1, 5) h (−8, 10) i (−4, 0)j (−4, −10) k (12, 2) l (−15, −1)

Exercise 9-031 a

b

c

d

e

21 3 54 6245 3 1

2

1

3

5

4

2

4

6

5

3

10−−− − −

−

−

−

−

−

−

y

x

x = −1 x = 2 x = 5

21 3 54 6246 5 3 1

2

1

3

5

4

6

2

4

3

10−−− − − −

−

−

−

−

y

x

y = 6

y = 4

y = −1

4321 5 7623 1

2

1

3

5

4

6

2

3

1−− −

−

−

−

y

x0

y = 3

x = 7

21 3 54 6245 3 1

2

1

3

5

4

2

4

6

5

3

10−−− − −

−

−

−

−

−

−

y

x

y = −3

y = 1

x = 21–2

21 3 54 6246 5 3 1

2

1

3

5

4

2

4

3

10−−− − − −

−

−

5−

−

−

y

x

y = 2

y = −3

x = −6 x = 4


ANSWERS 577

f

g

h

2 a x-axis b y-axis3 a y = 6 b x = 4 c x = −1 d y = −2

e y = 3 f x = −1 g y = 6 h y = −4i x = −1 j x = 5 k x = 4

4 a i y = 2 b i y = −1ii x = −5 ii x = 4

c i y = −4 d i y = −3ii x = 2 ii x = 8

e i y = 4 f i y = −1

ii x = 0 ii x = −2

g i y = 0 h i y = 4ii x = 11 ii x = −3

5 a i y = 1 or ii y = 3 or iii y = −1 ory = 11 y = 9 y = 13

b i y = 4 or ii y = 0 or iii y = −6 or

y = 8 y = 12 y = 18

Exercise 9-041 a (8, 7) b (7, 8)

c (4.5, 9) d (4.5, 4)e (2, 3) f (4, 5)g (7, 4) h (1, 4)i (−7.5, 6) j (3, 9)k (3, −7) l (1.5, 2)m (0.5, 4) n (1, 2.5)o (−0.5, 3.5)

2 a (2, 3) b (−1, 1) c (5, 5)3 a (−6, 6) b (7, −2)

c (−1.5, 2) d (3.5, 1)e (5, 11) f (10, 0)g (−4.5, 1.5) h (5, 12)

4 a (4, 4) b (6, −4) c (−4, −2)d (5, −2) e (7, −1)

Exercise 9-051 a 7.3 b 7.2 c 8.2 d 14.3 e 7.1 f 14.3

2 a units b units

c units d units

e units f units

g units h units

i units j units

k units l units

3 a 36.1 units b 32.4 units

4 a i 30.0 units2 b i 25.5 units2

ii 26.8 units ii 29.5 units

c i 25.5 units2 d i 29.0 units2

ii 29.5 units ii 26.0 units

e i 19.5 units2 ii 25.8 units5 (4, 1), (4, 3), (2, 1), (2, 3)6 (2, 6) (2, −4) (7, 1) (−3, 1)

(5, 5) (5, −3) (−1, 5) (−1, −3)(6, 4) (6, −2) (−2, 4) (−2, −2)

21 3 54 6245 3 1

2

1

3

5

4

2

4

6

5

3

10−−− − −

−

−

−

−

−

−

y

x

y = −1

y = −4

x = −11–2 x = 5

21 3 54 6245 3 1

2

1

3

5

4

2

4

6

5

3

10−−− − −

−

−

−

−

−

−

y

x

x = −2 x = 1–2

y = −11–2

y = −6

21 3 54 6245 3 1

2

1

3

5

4

2

4

6

5

3

10−−− − −

−

−

−

−

−

−

y

x

x = − 1–2

y = 11–2

x = 6

y = −6

12---

12---

12---

12---

45 72

128 104

29 74

85 101

40 73

17 90



Exercise 9-061 a N b P c P

d P e X f Ng X

2 The line is horizontal or vertical.

3 a 2 b − c −5

d −1 e f

4 a b c −6

d e f

5 a 1 b c −5 d −

e f − g − h −

i j

6 a − b 5 c d 7

e f g

Exercise 9-071 a (6.5, 4) b (1, 8)

c (6, 4) d (−1, 7)e (−13, 14) f (12, −6)g (9, 6) h (5.5, 1.5)i (14, 6) j (5, 3)

2 a 2.2 units b 8.2 unitsc 11.4 units d 8.6 unitse 7.1 units f 21.2 unitsg 5 units h 13 unitsi 15 units j 16.6 units

3 a units b units

c units d units

e units f units

g 5 units h units

i units j units

k units l units

4 a 1 b c d −1

e −1 f g h

i 1 j

5 a i (5.5, 4) b i (3.5, −0.5)

ii 12 ii −

iii 12.0 units iii 11.4 unitsc i (−4, 3) d i (−1.5, 0)

ii −2 ii −

iii 4.5 units iii 21.9 unitse i (2, 7) f i (4, −4)

ii −1 ii −

iii 11.3 units iii 21.3 unitsg i (7, −1.5) h i (3.5, 4.5)

ii ii

iii 15.1 units iii 22.8 units6 a S b S c S d I

7 a units (7.21 units)b 28.8 units

c units (10.20 units)

d unitse yesf (2, 1)g (2, 1)h Yes, since the midpoint of one diagonal is the

midpoint of the other.

8 a EH = FG = units

EF = GH = units b parallelogram or rectangle

c FG = EG = units d rectangle, since diagonals are equal.

9 a units

b > c D (on), H (inside), J (outside)

K (on), T (on)

10 a (−5, 4) b (1, 8) c d

e units f Yes g Yes11 a 10.3 cm b 520 m

Exercise 9-081 a b c 4 d e −1.25

2 a b c d e

f 4 g 3 h i j

3 a N b P c P d N e N f Pg L h N i N j P k L l L

4 a −1 b −1 c 1

5 a b c

d PQ is perpendicular to CD because m1 × m2 = −1

6 a b c d e −4 f

g h h i j k l

7 a yes b no c trapezium8 a Check that opposite sides have equal gradients

and lengths.b Check that adjacent sides have perpendicular

gradients.

14---

37--- 7

3---

35--- -1

12------

-103

-------- 53--- 1

5---

13--- 1

2---

13--- 1

2--- 1

2--- 1

2---

45--- 2

3---

12--- 2

3---

-16----- 5

2--- -7

2-----

89 18

194 41

82 82

181

10 18

40 8032--- 6

5---

53--- -6

5----- -5

7-----

43---

79---

209------

78---

-152

-------- 73---

52

104

104

50

8

58

73

80 73

23--- 2

3---

208

-13----- 1

2--- 7

3---

-12----- 1

5--- -4

3----- -10

3-------- 5

4---

-34----- -2

5----- 5

16------

-43----- -4

3----- 3

4---

-13----- -2

7----- -1

k----- 1

6--- -6

5-----

-1t----- -4

9----- q

p--- -100

7----------- 1

w----


ANSWERS 579

c Check for parallel gradients.

9 a 5 units, b 10 units,

c right-angled d 25 units2

e ZV; (6, ) VW; (11, −2)

f Yes, equal gradients.

Exercise 9-091 a m = 3, b = −2 b m = −2, b = 7

c m = 1, b = 4 d m = −1, b = 9

e m = , b = 6 f m = 1, b = 0

g m = , b = −11 h m = , b = 6

i m = , b = −8 j m = , b = 5

k m = , b = 10 l m = −3, b = 11

mm = , b = n m = , b = 1

o m = 1, b = p m = , b = 20

2 a y = 2x − 1 b y = x + 2

c y = −3x + 5 d y = x + 3

e y = −2x − 3 f y = −0.3x + 7

g y = x + 1 h y = −3x +

3 a y = x + 2 b y = x

c y = x + 5 d y = x + 3

e y = −3x − 3 f y = −x − 2

g y = 3x − 10 h y = x + 2

4 a y = x + 4 b y =

c y = − 5 d y = + 7

e y = + 3 f y = −2x − 6

5 a i 4, 1, 4, 2 b i −2, , 4,

ii y = 4x + 3, y = 4x − 6 ii y = + 1,

y =

c i 1, −1, 1, 4 d i , 3, −2, −2

ii y = x + 1, y = x − 4 ii y = 5 − 2x, y = −2x − 1

e i 5, 3, 3, −5ii 3x + 7 = y, y = 3x − 2

6 a i , −2, 1, −3 b i −1, 2, −2, 1

ii y = − 2, y = 4 − 3x ii y = 3 − x, y = x + 5

c i 1, 1, 6, − d i , 4, , 2

ii y = 6x + 2, y = 3 − ii y = 4x, y = 3 −

e i , 5, −5, 0 f i 0, 6, 3,

ii y = − 2, ii y = 3x − 7,

y = 5 − 5x y = 1 −

g i 4, 2, , h i 5, , , 0

ii y = 4x + 2, ii y = 5x + 2,

y = 3 − y = 8 −

Exercise 9-101

m = 3, b = 4

2

m = 1, b = −33

m = 2, b = 2

34--- -4

3-----

12---

34---

12--- 2

3---

-13----- -1

2-----

14---

-13----- 5

2--- 2

5---

-72----- -3

4-----

34---

-25-----

54--- 1

2---

12---

-12----- -1

2-----

25---

3x2

------

x3--- 3x–

2---------

x–3------

52--- 5

2---

5x2

------

5x2

------

23---

13---

x3---

16--- 1

4--- -1

4-----

x6--- x

4---

15--- -1

3-----

x5---

x3---

12--- -1

4----- 1

5--- -1

5-----

x4--- x

5---

21 3 423 1

2

1

3

5

4

6

7

10−− − −

y

x

y = 3x + 4

21 3 423 1

2

1

3

4

2

4

3

10−− −

−

−

−

−

y

x

y = x − 3

21 3 423 1

2

1

3

4

2

4

3

10−− −

−

−

−

−

y

x

y = 2x + 2



4

m = , b = −1

5

m = −2, b = 3

6

m = −1, b = 3

7

m = , b = 0

8

m = , b = 9

9

m = 6, b = −710

m = , b = 8

21 3 423 1

2

1

3

4

2

4

3

10−− −

−

−

−

−

y

x

y = − 1x–2

12---

21 3 423 1

2

1

3

4

2

4

3

10−− −

−

−

−

−

y

x

y = 3 − 2x

21 3 423 1

2

1

3

4

2

4

3

10−− −

−

−

−

−

y

x

y = −x + 3

21 3 423 1

2

1

3

4

2

4

3

10−− −

−

−

−

−

y

x

y = 2x––3

23---

4321 5 762 1

2

1

3

5

4

6

7

8

9

1− − −

y

x0

y = 9 − x–3

-13-----

21 3 424 3 1

2

1

3

5

4

2

4

6

5

3

10−− − −

−

−

−

7−

−

−

−

y

x

y = 6x − 7

4321 5 62 1

2

1

3

5

4

6

7

8

9

1− − −

y

x0

y = 8 − 3x––4

-34-----


ANSWERS 581

11

m = , b = 1

12

m = , b = −2

Power plus1 D (1, 3)

2 a y = x − 2 b C (9, 4)

3 a k = 5 b k = −24 a Z (0, 4) b m = −2

c m = d y = x + 4

5 AB = BC = 13 units; AC = units.6 B (2, −1)7 Method 1: For PQ, m = 5. For RQ, m =

∴ PQ ⊥ RQ

Method 2: PQ = units, QR = units

PR = unitsTesting Pythagoras’ theorem:

= + 130 = 26 + 104 is true.

8 dJD = units dKD = units

9 a D (0, −3)b For AD, m = ; for BC, m = ∴ AD ⊥ BC

c BC = 10.77 units, AD = 5.39 units

d Area = 29.0 units2

e y = − 3

Chapter 9 review1 a

b

c

d

21 3 423 1

2

1

3

4

2

4

3

10−− −

−

−

−

−

y

x

y = − + 14x––3

-43-----

21 3 5423 1

2

1

3

2

4

3

10−− −

−

−

−

−

y

x

y = − 23x––5

35---

23---

12--- 1

2---

208

-15-----

26 104

130

130( )2

26( )2

104( )2

40 40

52--- -2

5-----

5x2

------

21 3 424 3 1

2

1

2

4

5

3

10−− − −

−

−

−

−

−

y

x

x + y = −3

4321 5 7 862 1

4

2

4

6

2− −

−

−

8−

12−

−

y

x0

x − y = 8

21 3 423 1

2

1

3

4

2

10−− −

−

−

y

x

y = 1 − 2x

21 3 423 1

2

1

3

4

2

10−− −

−

−

y

x

y = + 3x–2



e

f

2 a No b Yes c Yes d No3 i (4.5, 0) ii (0, 9)4 a B b C c B

5 a

b

c

d

6 a y = −2 b x = −7 c y = −11d x = −8 e y = −3

7 a (5, 9) b (3, −5) c (−4, −1)d (6, 2) e (−3, 3) f (5.5, 1.5)

8 a units b 10 units c units

d units e units

9 a m = 0 b m is undefined.

4321 5 7 862 1

2

1

3

5

4

6

7

8

9

1− − −

y

x0

y = 7.5 − x

21 3 423 1

2

1

3

5

4

6

7

10−− − −

y

x

y = 2x + 4

21 3 4246 5 3 1

2

1

3

5

4

2

4

3

10−−− − − −

−

−

5−

−

−

y

x

x = −6

21 3 4246 5 3 1

2

1

3

5

4

2

4

3

10−−− − − −

−

−

5−

−

−

y

x

y = 4

21 3 424 3 1

2

1

3

5

4

2

4

3

10−− − −

−

−

5−

−

−

y

x

x = 2

21 3 4246 5 3 1

2

1

3

5

4

2

4

3

10−−− − − −

−

−

5−

−

−

y

x

y = −1

68 293

194 85


ANSWERS 583

10 a P b N c P

11 a b c

d e f 4

12 a midpoint =

b gradient, m =

c distance, d =

13 a (8, 6) b (7, 3) c (−4, −1.5) d (5, −8)

14 a m = −1 b m = c m = −1 d m = −3

15 a 12.8 units b 8.1 unitsc 6.4 units d 8.5 unitse 10.8 units f 10.8 unitsg 2.2 units h 10.3 units

16 a AB: m1 = , BC: m2 = , m1 × m2 = −1

b 5.831 units c 11.66 unitsd 34.0 units2

17 a b c 2

18 a Yes b Yes c Yes d No e rectangle

19 a m = −4 b m = c m = 5

b = 16 b = b = −1d m = −3 e m = −1 f m = 0

b = 0 b = 5 b = 6g m = −1 h m = i m =

b = −3 b = 7 b = 1

20 Line l: y = x − 6

Line k: y = + 5

21 a

b

c

Mixed revision 31 D

2 a 5.3 × 109 b 3.73 × 105

3 C

4 a 215 b 412 c y40 d f 14 e p16

5 A6 a 15 400 000 b 0.0003 c 0.9876

d 600 000 000

7 a 9k4 b h2y3 c w4 d 3m − 5

8 a 9y2 b 16y28 c −32d10 d9 B

10 a 81m12n8 b 125k3 c x22

11 a 1 b c 16 d 1

12 a 125 b c 8 d

13 a e2 b 5m6 c 2k2 d 9h8

14 a 1 b 2 c 5 d 3 e 3 f 5g 24 h −1

15 6.667 × 1024

16 a b c d

17 a 98 m2 b 217.5 m2 c 225 cm2

18 a 22 cm b 82 cm c 31 m19 B 20 60 cm21 a 98.5 cm b 120.8 cm

22 a 418 m3 b 452 389.3 mm2

-25----- 2

9--- -1

14------

73--- 2

11------

x1 x2+

2-----------------

y1 y2+

2-----------------( , )

y2 y1–

x2 x1–-----------------

x2 x1–( )2y2 y1–( )2+

32---

53--- -3

5-----

-12----- -1

2-----

-32-----

12---

12--- -3

5-----

12---

-5x4

--------

21 3 5423 1

2

1

3

2

4

3

10−− −

−

−

−

−

y

x

y = 3x − 2

21 3 5423 1

2

1

3

2

4

3

10−− −

−

−

−

−

y

x

y = − 34x––5

4321 5 7623 1

2

1

3

5

4

2

1−− −

−

−

y

x0

y = 4 − x–3

4m10

9------------

12---

132------ 1

9---

1312---

1213---

7e( )45---

k74---



23 a 108.6 cm2 b 485 cm2

24 a i 113.1 cm2 ii 144 cm2 b 21.5%

25 a i

ii

iii

b i −1 ii 1 i 1 ii 4 i 2 ii −1

26 a (1.5, −2.5) b (3, 1.5) c (−2.5, −1.5)

27 a − b − c −2

28 a

b

c

d

29 a 7.1 b 6.4 c 3.6

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

13---

35--- 4

3---

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

x =

1

x =

4

x =

−3

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

y = 2

y = 3

y = −1

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

y = 3

y = −2

x = 2

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

x = −4

y = 21–2

x = 1–2


ANSWERS 585

30 D31 a y = 4 b x = 6 c y = 3 or y = −3

32 a b c

33 a

b

34 a

b 22.6 units c 30 units2

35 a No b Yes c Yes36 B

37 a i m = −2, b = 1ii

b i m = , b = −2

ii

c i m = , b = 2

ii

38 a i (−1, 5.5) ii iii 5

b i (0.5, −4) ii iii

39 a m = − , b = 1, y = − x + 1

b m = 3, b = 3, y = 3x + 340 B, D

37 41 68

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

y =

3x

y = x

+ 2

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

2x + y = 4

y = x

− 1

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

A

B

D

C

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

32---

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

13---

21 3 423 1

2

1

3

4

2

4

3

10−−4− −

−

−

−

−

y

x

34---

25--- 29

13--- 1

3---



Chapter 10Start up

1 a 1.55 m at 4:00pm, Saturdayb 0.4 m at 7:00am–8:00am, Thursdayc 3:00am, Saturdayd 8:00pm–9:00pm, Thursday; 8:00am Friday;

9:00pm–10:00pm, Friday; 9:00am, Saturday; 11:00am Saturday

e 6 hoursf i 0.5 m ii 0.8–0.9 m

2 a Rugby League b AFLc i 30% ii 12.5% iii 17%d The percentage participation of each sport

3 a 12% b Sportc i $374 400 ii $486 000 d $515 160

4 a i 6 m ii 12 mb 24 m c 33 m d 17–18 m

5 a i 40% ii 35% iii 18%b Four students

6 a 8b i 3 ii no mode iii no mode

iv 5, 6c i 7 ii 0d i 10 ii 28

7 a

b 36 c 4d i 22.2

.% ii 44.4

.% iii 33.3

.%

Exercise 10-011 a quantitative b quantitative

c quantitative d categoricale quantitative f categoricalg quantitative h quantitativei categorical j quantitativek quantitative l categorical

2 a continuous b continuousc discrete d discretee discrete f continuousg discrete h discretei discrete j discretek discrete l continuous

Exercise 10-021 a

b

2 a

Number of DVDs

Tally Frequency

0 ||| 3

1 ||| 3

2 |||| | 6

3 |||| ||| 8

4 |||| |||| 9

5 |||| 5

6 || 2

Student results Frequency

3 1

4 3

5 4

6 5

7 8

8 5

9 3

10 1

30

Class results

Student results

1

0

2

3

4

5

6

7

8

43 5 6 7 8 9 10

Fre

qu

en

cy

Number of children Frequency

1 5

2 8

3 9

4 5

5 4

6 3

7 4

8 1

9 1

40


ANSWERS 587

b

c 20% d 45%

3 a

b

c i 4 ii 104 a 14 b 5 letters c 22 d 11

e

5 a 9 b 1 head c 2 heads

d

6 a 3 b 73 c 14d Morning; this is the time when teachers and

parents phone the school.

Exercise 10-031 a

b

2 No; the scores are spread too much (ranging from 12 to 48).

3 a

b 15 is a possible outlier.c Scores are grouped around 5, 6 and 7.

4 a

b 88 c 18 d 17.8%e i Yes, 32 could be considered as an outlier.

ii No

Survey results

Number of children

1

0

2

3

4

5

6

7

8

9

21 3 4 5 6 7 8 9

Fre

qu

en

cy

Number of goals Frequency

0 2

1 3

2 4

3 7

4 7

5 3

26

1234567

0 1 2 3 4 5

Goals scored

Number of goals

Fre

qu

en

cy

Number of letters Frequency

3 4

4 7

5 8

6 14

7 12

More than 7 10

55

Number of heads Frequency

0 3

1 12

2 22

3 9

4 4

50

Temperature15 17 19 21 23 25 27

1 3 5 7 9 11

0 2 4 6 8 10 12 14 16

Stem Leaf

3 2

4 0 4 8 9

5 6 7 7 8 9

6 2 4 5 7 8

7 1 2 2 3 7 7 9 9

8 1 2 2 4 8



5 a

b 7 c 20%d i No ii Yes, the 14s and 15s

6 a 20 b 2 c 9 d 45%e 10; it’s very different from the other scores.

7 a

b 95% c 5% d Yes, in the 50s8 a 54, 55 or 56 b 279 a Rockets 8 Blues 13

b Rockets 52 Blues 57c Blues; they scored more than 30 goals on

13 occasions; but the Rockets managed this on only 8 occasions.

10 a

b Greg c 57%

11 a

b i 9 Au ii 9 Alc Nod Yes, in the 60s and 70s for both classese 9 Au, as the fraction of students scoring over

60 is higher than the fraction of students scoring over 60 in 9 Al.

Exercise 10-041 a 9 b 15 c 1.8 d 132 a 5.4 b 62.1 c $1.23 d 10.4

e −0.43 a 19 b 7 c 5.5 d 1

e 8.74 a 15 b 5 and 6 c 25 d 7.1

5

6 a Mean = 5.7, median = 5, mode = 5b Mode; it can be found by observation.

7 a Range = 14, mean = 24.4, median = 24.5b No; all the scores occur only once.

8 a i 6 ii 50.7 iii 51 iv 51b i 52 ii 40.1 iii 39 iv 39

9 a 52 b 68 c 60d i The twelfth score is the middle score

(11 scores on either side).ii 63

10 a

b 54 c 67 d 67 e 67.911 a 62 b 38

c Since there are an even number of scores, there will be two middle scores. For 30 scores this will be the fifteenth and sixteenth scores.

d 38.5 e 37.512 a 446.5 mm b 0 mm c 402.5 mm

d The modal value of zero is not consistent with the rainfall for the other months.

e Median, since it is not affected by the rainfall for July and August.

13 276014 a 6 b 1 c 58 d 1.9

e 2; the median is the average of the 15th and 16th scores. There are 5 zeros and 9 ones, giving a total of 14 scores. Since there are 7 twos, the 15th and 16th scores must be twos.

Stem Leaf

11 8 9

12 1 1 5 6

13 1 5 7

14 3 4 5 6 9 9

15 0 0 1 4 4 7 7 7

16 1 3

17 2 6 6

18 1 7

Stem Leaf

4 7

5 1 2 3 3 3 4 4 5 5 5 6 7

6 0 2 3 4 8

7 1 7

Sandy Greg

7 1

7 5 3 2 5 8 8 9

8 7 7 5 5 3 3 4 5 6 6 7

5 2 0 4 0 2 5 7

4 5 4

9 Al 9 Au

7 3 8

9 8 0 4 6

5 5 4 3 5 0 4 6

9 8 7 4 4 3 2 1 0 0 6 0 0 2 4 5 7 7 8 8

8 7 4 3 2 2 1 0 7 0 1 2 3 3 4 9

1 8 0 2 3

Range Mean Median Mode

a 8 5.4 6 6, 8

b 8 15.9 16 12, 15, 18

c 4.5 51.5 51.6 –

d 11 13.1 14 14

e 38 67.3 68 –

Stem Leaf

4 0 1 5

5 1 3 5 6 8

6 3 4 5 6 7 7 7 8

7 1 2 4 7 9

8 2 7 8

9 0 2 4


ANSWERS 589

15 a

Median = 44

b

Median = 37

16 a

b Median = 16.5

Exercise 10-051 a

x̄ = 4.04

b

x̄ = 24.5

c

x̄ = 7.2

d

x̄ = 3.5

2

x̄ = 2.43 a x̄ = 1.44 b x̄ = 12.78

4

x̄ = 4.9

Stem Leaf

2 1 8

3 5 6 8

4 2 3 4 4 5 7

5 1 4

6 0 2

Stem Leaf

1 0 2

2 1 2 3 6

3 4 7

4 5 8 9

5 0 4 7

6 6

Stem Leaf

0 5 6 6 7

1 0 1 3

1 5 5 5 6 7 8 8

2 0 1 1 2 2 4

2 6 6

Scorex

Frequencyf fx

2 5 10

3 3 9

4 7 28

5 6 30

6 4 24

Σf = 25 Σfx = 101

x f fx

22 3 66

23 8 184

24 7 168

25 9 225

26 5 130

27 4 108

Σf = 36 Σfx = 881

x f fx

7.0 6 42.0

7.1 10 71.0

7.2 8 57.6

7.3 4 29.2

7.4 2 14.8

Σf = 30 Σfx = 214.6

x f fx

0 2 0

1 4 4

2 5 10

3 8 24

4 10 40

5 6 30

6 5 30

Σf = 40 Σfx = 138

x Tally f fx

0 |||| 4 0

1 |||| | 6 6

2 |||| |||| ||| 13 26

3 |||| |||| 10 30

4 ||| 3 12

5 ||| 3 15

6 | 1 6

Σf = 40 Σfx = 95

x Tally f fx

1 || 2 2

2 || 2 4

3 |||| 4 12

4 |||| 4 16

5 |||| | 6 30

6 |||| | 6 36

7 ||| 3 21

8 |||| 4 32

Σf = 31 Σfx = 153



Exercise 10-061 a 6.2 b 13.6 c 51.8 d 64.82 a 34.4 b 12.9 c 7.0 d 57.0 e 49.73 162.4 cm

Skillbank 102 a = 26

b Any three numbers whose sum is 39c Any four numbers whose sum is 88d 97% e 88 f 73 kgg $622 h 48 points i 12 years oldj 205 cm

Exercise 10-071 a 19 b 25.5 c 47.5

2 a

b 40.5 c 39–45 d 39–45

3

x̄ = 42.4

4 a

b

c x̄ = 0.126.

5 a

b

x

Class interval

x f fx

18–24 21 3 63

25–31 28 8 224

32–38 35 11 385

39–45 42 15 630

46–52 49 9 441

53–59 56 5 280

60–66 63 2 126

Σf = 53 Σfx = 2149

Class interval

x f fx

10–19 14.5 4 58.5

20–29 24.5 10 245.5

30–39 34.5 18 621.5

40–49 44.5 25 1112.5

50–59 54.5 14 763.5

60–69 64.5 8 516.5

70–79 74.5 1 74.5

Σf = 80 Σfx = 3390.5

Class interval

x f fx

0.00–0.04 0.02 3 0.06

0.05–0.09 0.07 7 0.49

0.10–0.14 0.12 9 1.08

0.15–0.19 0.17 7 1.19

0.20–0.24 0.22 2 0.44

0.25–0.29 0.27 2 0.54

Σf = 30 Σfx = 3.80

Pollution levels

Volume of sulphur dioxide(parts per million)

1

0

2

3

4

5

6

7

8

9

0.020.07 0.17 0.27

0.12 0.22

Fre

qu

en

cy

Class interval

Class centrex

f

30–39 34.5 1

40–49 44.5 4

50–59 54.5 5

60–69 64.5 8

70–79 74.5 9

80–89 84.5 2

90–99 94.5 1

30

1

2

3

4

5

6

7

8

9

34.5 44.5 54.5 64.5 74.5 84.5 94.5

Frequency histogram

Marks of students

Fre

qu

en

cy


ANSWERS 591

c

d Ordered stem-and-leaf, as it is easier to construct and it still shows the main features of the distribution.

6 a 45b 140–146; 147–153; 154–160; 161–167;

168–174; 175–181; 182–188c 164.5 cm

Exercise 10-081 a

Median = 2.5

b

Median = 48

c

Median class = 16–20

d

Median class = 30–39

e

Median = 14

f

Median class = 60–642 a

b Modal class = 0.05–0.09c Mean = 0.13d Median class = 0.10–0.14

Stem Leaf

3 3

4 1 7 7 8

5 2 2 3 6 8

6 0 2 3 5 7 8 8 9

7 0 0 1 1 2 3 3 6 6

8 7 7

9 5

x f cf

0 7 7

1 8 15

2 10 25

3 5 30

4 9 39

5 6 45

6 5 50

x f cf

45 2 2

46 6 8

47 7 15

48 10 25

49 3 28

50 3 31

Class interval

Class centre, x

fcf

1–5 3 2 2

6–10 8 7 9

11–15 13 10 19

16–20 18 8 27

21–25 23 10 37

26–30 28 3 40

Class interval

Class centre, x

f cf

0–9 4.5 3 3

10–19 14.5 7 10

20–29 24.5 11 21

30–39 34.5 19 40

40–49 44.5 7 47

50–59 54.5 2 49

x f cf

12 5 5

13 16 21

14 11 32

15 7 39

16 4 43

Class interval

Class centre, x

f cf

35–39 37 3 3

40–44 42 5 8

45–49 47 10 18

50–54 52 15 33

55–59 57 28 61

60–64 62 27 88

65–69 67 22 110

70–74 72 18 128

75–79 77 15 143

80–84 82 9 152

Class interval

x f cf fx

0.00–0.04 0.02 7 7 0.14

0.05–0.09 0.07 14 21 0.98

0.10–0.14 0.12 8 29 0.96

0.15–0.19 0.17 11 40 1.87

0.20–0.24 0.22 7 47 1.54

0.25–0.29 0.27 3 50 0.81

Σf = 50 Σfx = 6.30



3 a

b 101–110 c 110.75 km/hd 101–110 e 70%

4 a

b 60–69 c 60–69d 61.7 e 52%f i 9 ii 15

Exercise 10-91 a Median = 8

b Median = 32.5

c Median = 18.5 or median class = 15–19

2 a

Classinterval

x f cf fx

81–90 85.5 3 3 256.5

91–100 95.5 9 12 859.5

101–110 105.5 12 24 1266

111–120 115.5 4 28 462

121–130 125.5 6 34 753

131–140 135.5 4 38 542

141–150 145.5 2 40 291

Σf = 40 Σfx = 4430

Classinterval

x f cf fx

30–39 34.5 2 2 69

40–49 44.5 7 9 311.5

50–59 54.5 12 21 654

60–69 64.5 14 35 903

70–79 74.5 12 47 814

80–89 84.5 3 50 253.5

Σf = 50 Σfx = 3085

Cumulative frequency histogram and polygon

Score

5

0

10

15

20

25

30

35

40

76 8 9 10 11

Cu

mu

lativ

e f

req

ue

ncy


Score

2

0

4

6

8

10

12

14

16

18

20

3130 32 33 34 35

Cu

mu

lativ

e f

req

ue

ncy


Class centres

10

0

20

30

40

50

60

70

127 17 22 27 32

Cu

mu

lativ

e f

req

ue

ncy

Class interval

x f cf fx

16–21 18.5 1 1 18.5

22–27 24.5 9 10 220.5

28–33 30.5 5 15 152.5

34–39 36.5 11 26 401.5

40–45 42.5 6 32 255.5

46–51 48.5 2 34 97.5

Σf = 34 Σfx = 1145.5


ANSWERS 593

b

c Median class = 34–39d Mean = 33.7

3 a 25–31; 32–38; 39–45; 46–52; 53–59; 60–66b Median class = 39–45

c 25–31 d 62%

4 a

b

c Median class is 0.10–0.14d 0.11

Exercise 10-101 a 4, 6, 7.5 b 21.5, 25, 27

c 13, 15, 17.5 d 30, 39, 49e 25, 27.5, 40

2 a i 10 ii 5 iii 14.5 iv 9.5b i 28 ii 23 iii 31 iv 8c i 47.5 ii 41.5 iii 53.5 iv 12

3 a 3 b 2.5 c 17.5 d 19 e 21.5 f 1.54 a 3 b 3.5 c 3

5 a

b, c

Median = 16, interquartile range = 8


Number of sandwiches

5

0

10

15

20

25

30

35

24.5 36.5 48.518.5 30.5 42.5

Cu

mu

lativ

e f

req

ue

ncy

10

20

30

40

50

28 35 42 49 56 63

The age of employees

Class centres

Fre

qu

en

cy

Classinterval

Class centre

xf cf

0.00–0.04 0.02 6 6

0.05–0.09 0.07 18 24

0.10–0.14 0.12 11 35

0.15–0.19 0.17 10 45

0.20–0.24 0.22 4 49

0.25–0.29 0.27 1 50

10

20

30

40

50

0.02 0.07 0.12 0.17 0.22 0.27Blood alcohol levels

Cu

mu

lativ

e f

req

ue

ncy

Classinterval

x f cf

0–5 2.5 1 1

5–10 7.5 7 8

10–15 12.5 15 23

15–20 17.5 21 44

20–25 22.5 6 50

Σf = 50

10

20

30

40

50

2.5 7.5 12.5 17.5 22.5

Cu

mu

lativ

e f

req

ue

ncy

Class centres



6 a

b A: 76, B: 74.5c A: 38, B: 42d A: 12.5, B: 17.5e Group A has less spread because its

interquartile range is less than that of B.7 a Team 1: 51, Team 2: 36

b yesc Team 1: 22, Team 2: 23d Team 2; although its interquartile range is one

more than that of Team 1, its range is much less.

Exercise 10-111 a 6 b 4.5 c 8 d 3.5

e

2 a

b

c

3 a 27.5 h b 26 h c 30 h d 4 he 50%

4 a i 21.5°C ii 12°Cb Alice Springs: 11°C, Hobart: 5°Cc

5 a Xb i 12 ii 10c Y: smaller range d X: 10, Y: 9e X: 4,Y: 5f Not enough information given to make a valid

decision; the interquartile range and the range only differ by 1.

g 25%h 25%

6 a lowest score 12, Q1 23.5, median 33, Q3 40.5, highest score 57

b lowest score 1, Q1 2, median 3, Q3 4, highest score 7

Power plus1 See answer below.2 a i 940

A B

8 5 1 9

9 7 6 2 6 1 4 4

9 7 7 5 2 2 7 2 3 4 5 9

2 1 0 8 1 1 2 6

6 1 9 0 3

1 2 3 4 5 6

Number of hours worked

7 8 9 10 11

1 2 3 4 5 6 7 8 9 10 11

5 6 7 8 9 10 11 12 13 14 15

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

8 10 12 14 16 18 20 22 24 26 28

Alice Springs

Hobart

10 15 20 25 30 35 40 45 50 55 60

1 2 3 4 5 6 7

Power plus1 a

b Tide height does not jump – there are gradual rises and falls.

0

0.5

1.0

1.5

6:00amM/n Noon NoonM/n M/n6:00pm 6:00am 6:00pm3:00am 9:00am 3:00pm 9:00pm 3:00am 9:00am 3:00pm 9:00pm

Time

Tid

e (

m)


ANSWERS 595

ii Year 7 19.1%; Year 8 20.2%; Year 9 16.0%; Year 10 21.3%; Year 11 13.8%; Year 12 9.6%

iii Year 7 19; Year 8 20; Year 9 16; Year 10 21; Year 11 14; Year 12 10 Students are selected from each year in proportion to the size of the Year group.

iv A stratified sampleb i STR ii STR iii SYS iv STR

v SYS vi SYS vii STR viii SYS3 69 4 100

5 a

b Median = 171

6 a i

ii

b Median = 18c i Modal class = 10–19

ii Modal class = 15–19d Stem-and-leaf plot using intervals of 5; it gives

a better indication of how the seedlings are spread.

7 a i No ii No b 17°C

Chapter 10 review1 a Continuous b Discrete

c Categorical or discrete d Categorical

2 a

b

c 3 d 2.3e

Stem Leaf

16 0 1 3 3 4 4 4 4 4

16 5 7 8 9

17 0 1 1 1 2 2 2 4 4

17 5 6 7 9 9

18 0 1 2

Stem Leaf

0 3 5 6 7 8 9 9

1 0 1 2 4 4 5 5 6 7 7 7 8 8 8 9

2 0 2 2 3 3 4 4 5 5 5 7 8 8

3 0 0 1 3 4

Stem Leaf

0 3

0 5 6 7 8 9 9

1 0 1 2 4 4

1 5 5 6 7 7 7 8 8 8 9

2 0 2 2 3 3 4 4

2 5 5 5 7 8 8

3 0 0 1 3 4

x f cf fx

0 5 5 0

1 3 8 3

2 7 15 14

3 10 25 30

4 3 28 12

5 2 30 10

Σf = 30 Σfx = 69

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5

Frequency histogram and polygon

Children per house

Fre

qu

en

cy

5

10

15

20

25

30

0 1 2 3 4 5


Children per house

Cu

mu

lativ

e f

req

ue

ncy

Median = 2.5



3 a

b 0 is an outlier since it is separated from the main body of the data.

c Yes, the scores are clustered about 5 and 6.4 a 51 b 66 c 67.55 a 9.8 b 50.1 c 19.76 a 34.5 b 17.6

7 a

b 71.7c 65–69

d

e 70–748 a 5 b 30–39

9 a

b

c 161–170d i 38% ii 28%

10 a 67.5 b 54 c 77 d 2311 a 40 b 10 c 25% d 75%

Chapter 11Start up

1 a 20 b 14 c 16 d 9

e 54 f 2 g h 2

2 a 7n b n + 11 c n2 d

e 5(n + 8) f n − 20 g 6n − 9 h n + 23 a 60 b 38 c 70 d 600

e 35 f 724 a 1 b 5 c 7 d −6

e 0 f 13 g 2 h 125 a 4k + 10 b 3n + 6 c 4k + 3

d y − 30 e 3 − 2x f 13m − 7

g m + 20 h 2x i

Exercise 11-011 a Yes b No c Yes d No

e No f Yes g Yes h Noi Yes j Yes k No l Yes

2 LHS = RHS in each case.3 LHS = RHS in each case.4 a B b C c B d D e B

Exercise 11-021 a y = 6 b k = 18 c a = 10

d y = 4 e x = 9 f p = 11g t = 6 h h = 4 i c = 4

2 a w = 5 b p = 3 c y = 6d a = 2 e g = 1 f x = 2g n = −2 h a = −1 i y = 0

3 a m = 3 b y = 7 c x = 5d p = 4 e n = −4 f x = −6g w = −8 h x = 2 i a = 3

0 1 3 5 7 92 4 6 8 10

Class interval

x f cf fx

60–64 62 1 1 62

65–69 67 12 13 804

70–74 72 10 23 720

75–79 77 8 31 616

80–84 82 2 33 164

Σf = 33 Σfx = 2366

1

2

3

4

5

6

7

8

9

10

11

12

62 67 72 77 82

Frequency histogram and polygon

Class centres

Fre

qu

en

cy

Class interval

x f cf

131–140 135.5 2 2

141–150 145.5 7 9

151–160 155.5 10 19

161–170 165.5 17 36

171–180 175.5 11 47

181–190 185.5 3 50

10

20

30

40

50

135.5 145.5 155.5 165.5 175.5 185.5Class centres

Cu

mu

lativ

e f

req

ue

ncy

215------ 5

8--- 11

12------

n3---

x10------


ANSWERS 597

Exercise 11-031 a m = 4 b n = 6 c m = 75

d c = −10 e n = 10 f n = −2g n = 6

2 a y = 7 b y = 3 c w = 10d w = 6 e x = 1 f x = 3g y = 4 h x = −1 i c = 2

3 a y = 3 b x = 2 c a = 5

d y = −3 e m = 8 f n = 6

g a = h x = 5 i n = 7

j y = 5 k c = 10 l m = 10

Exercise 11-041 a m = 6 b a = 9 c y = 8

d w = 50 e y = −9 f a = −7

g a = 14.4 h w = −30 i p = 6

j x = 7.5 k x = −52 a m = 5 b w = 15 c a = 1

d x = 26 e a = −3 f w = −53

g x = −3 h x = 3 i p = −100

j m = 21 k x = −1 l a = 1m a = 18 n x = 2 o x = −3p x = 5 q h = 20 r y = 27

Exercise 11-051 a a = 7 b m = 2 c h = 5

d k = 5 e y = 3 f a =

g a = 5 h q = i p = 5

j y = −3 k t = 9 l e =

m x = n y = − o v = −2

p f = q x = r b = 2

s d = −10 t n = −9

2 a m = 2 b y = 2 c x = −11

d a = 0 e w = −30 f h = 25g p = 1 h v = −7 i e = 17j n = 1 k q = −27 l c = 12

m d = −1 n b = 1 o e = −1

p g = 2 q t = −20 r l = −45

s x = −9 t a = 5 u m = −5

v x = 9

Skillbank 113 a 13 b 2460 c 0.13 d 24.6

e 24.6 f 1300 g 1.3 h 2.46i 2.46 j 13 k 130 l 246

6 a 14 076 b 1407.6 c 140.76 d 1407.6e 140.76 f 140.76 g 140 760 h 14.076i 1407.6 j 14 076 k 14 076 l 1.4076

Exercise 11-061 a x = 17 b y = 11 c p = −2

d a = 9 e x = 48 f w = 3g w = 3 h x = 4 i x = −5

2 a x = −9 b y = 22 c p = 11

d w = −22 e x = 5 f m = 24

g y = 5 h x = 5 i x = 0

3 a e = 9 b n = 30 c x = 12

d v = 20 e k = 4 f w = 20

g a = 6 h c = −40 i d = −8

4 a b = b r = 1 c a = −12

d p = e w = 1 f x = −

g a = 28 h x = − i m =

j x = k y = 5 l x = 11

m b = 103 n y = 15 o n = −10

5 a a = − b q = −20 c c =

d g = 2 e x = −1 f a = 3

g h = − h n = 22 i e = 2

j b = k p = 7 l u = 9

Exercise 11-071 a Length = 40 cm, width = 10 cm

b Length = 36 cm, width = 6 cmc Length = 27 cm, width = 20 cmd Length = 26.5 cm, width = 17 cm

2 a Franco is 50, Dean is 10 and Helen is 2 years old.

b Dario is 14 years old.c Phy is 20 years old.d Janice, 45 years; Paul, 51 years; Brett, 17

years; Amanda, 12 years old3 a 26, 27, 28 b 54, 56, 58

c 23, 25, 27 d 24, 30, 36, 424 a I have 78 $2 coins.

b I have 20 $1 coins and 100 $2 coins.c Narin has 240 50-cent coins.

5 a 21 m b 315 m2

6 x = 76°, 55°, 49°7 $1.808 a 11 b 12

14--- 2

5---

13---

13--- 5

7--- 1

4---

25--- 2

3---

12---

12---

23--- 2

3---

12---

49---

23--- 1

5--- 1

2---

313------

23--- 1

4--- 1

5---

58--- 11

12------

23---

25--- 1

3--- 2

3---

13--- 1

9--- 6

7---

15---

12---

34---

15---

35---

1217------

67--- 4

7---

1415------ 4

5--- 6

7---

310------ 1

11------ 3

14------

13--- 13

15------ 19

29------

2895------ 3

4--- 5

7---

58--- 1

4---

1011------ 1

5---

67---

1315------ 1

2--- 11

16------

1929------ 3

13------ 1

5---



9 Father, 170 cm; son, 140 cm; sister, 70 cm10 12 more weeks11 $7.6012 He needs 49 $20 notes.13 She travels for 1.2 hours at 15 km/h.14 Daughter is 20 years old now.15 15 and 4216 75 litres

Exercise 11-081 a y = 28 b y = 11 c y = 52.5

d y = −40 e b = 25 f b = −1g b = 34 h b = −40 i x = 3j x = 16

2 a F = 68 b F = 95 c F = 14d F = 36.5 e C = 26.7 f C = 37.8g C = −23.3 h C = 28.9

3 a V = 120 b V = −50 c c = 8d c = 8

4 a V = 187 b u = 2 c a = 13d t = 29

5 a A = 160 b A = 72 c h = 5d h = 20 e x = 7 f y = 3

6 a $540 b $4500 c 6 hours d 9 hours7 a $200 b $825 c 45 people8 a 65°C b 91.25°C c 4 hours9 a 182 cm b 39 cm

10 a 2.9 mL b 1 year 8 months

Exercise 11-091 a x = d − c b x = c x =

d x = pm − y e x = m(y − p) f x =

g x = h x = i x =

j x = k x = − y l x =

2 a y = b y =

c y = d y =

e y = (Q − P)2 f y =

g y = h y =

i y = j y =

k y = x − t2 l y =

m y = − 2m n y =

o y =

3 a h = b r = c A =

d R = e k = f C =

g d = h b = − a

4 a a = b a = c a =

d a = e a = f a =

g a = h a = i a =

j a =

5 a D = 3 b M = 620 c V = ; V =

6 a C = 45.2448 b r = c r = 20.5

7 a $290 b d = c 76 km

8 a V = 4.6 × 102 b t =

c t = 5

9 a r = 1 − b r =

10 a S = −620 b l = − a c l = 20

11 y =

Exercise 11-101 a N � 8 b p � 3 c k � −5 d x � 0

e T � −3 f w � 5 g y � m h 6 � d2 a m + 3 � 8 b Q − 4 � −2 c 4A � 2

d � 10 e a + b � ab

f y + 3 � 5 g gw � 15h 12 � 5 + p

3 a 0 � N � 7 b −3 � x � 5c 5 � p � 12 d y � 6 � ke −1 � y � 6

4 a H � 1.3 m b A � 18 c A � 18d 20 � A � 40 e $80 � A � $120

5 a greater than or equal to 5b less than or equal to 15c greater than 3 but less than 10d greater than or equal to 1 but less than or equal

to 76 a

b

c

d

e

y b–m

------------ p y+a

-------------

cv n+m

---------------

kmr------- m

v---- c by–

a---------------

a b c–+y

---------------------- 2Ah

------- m n+d

--------------

xp k–------------ m an–

a-----------------

ad m k–( )---------------------- ab m p–( )

b a–--------------------------

r2 x2–

A2 r2– Mn-----

r2

m2------ kx

x k–------------

kw a–-------------

3mc

------- x 1 r–( )1 r+

--------------------

c2 b–2a

--------------

Vlb----- A

π--- 3v

H------

100IPT

----------- C 2–1.8

------------- 5 F 32–( )9

-------------------------

T a–n 1–( )

----------------- 2Ah

-------

4 t–4

----------- 6 t–m 4+-------------- x 2d–

dm----------------

-wh3w 2–----------------- 5 k u–( )

1 5t–-------------------- 1 g–

2------------

-5b5 r–------------ A πh–

π----------------- u 1 p–( )

pk w–---------------------

b M N+( )N M–

-------------------------

12--- M

D----- 1

6---

C2π------

C 80–4.2

----------------

2Vc

-------

aS--- 29

32------

2Sn

------

dkk d–------------

m3----

−1 0 1

−1 0 1 2

0 2 4 6

0 1 2 3 4

−2 0−4−6

15_NC_Maths9_Stages_5.2/5.3_ans 8 Friday, February 6, 2004 1:55 PM

ANSWERS 599

f

7 a x � −3 b x � 1 c x � 4

d x � e x � 0 f x � −6

g −2 � x � 3 h − � x � 2

i −12 � x � −4 j −2 � x � 3

Exercise 11-111 a A, C, D b C c B, C, D

d C, D e B, D f C, Dg A, C, D

2 a x � 3 b d � 13 c k � 9d m � 10 e h � 4 f y � 4g w � 2 h b � −9 i p � −4

Exercise 11-121 a x � 7

b d � 8

c y � 4

d k � 1

e y � 4

f m � −2

g w � 32

h k � −15

i m � 1

j p � 18

k t � −9

l n � −2

m a � −1

n x � 0

o h � −20

p f � 4

q c � −10

r x � 16

2 a y � 2 b k � 4 c w � 10d h � 32 e h � 15 f t � −3g p � −4 h a � 3 i c � −1j x � 7 k x �16 l t � 1

m h � 24 n y � 0 o x � −1

3 a x � 1

b x � 1

c x � −3

d x � 0

e x � −3

f x � 1

g x � 4

−1 0 1 2 3 4

12---

12--- 1

2---

4 5 6 7 8 9 10

5 6 7 8 9 10 11

−2 0 2 4 6 8 10

−2 −1 0 1 2 3 4

−2 0 2 4 6 8 10

−8 −6 −4 −2 0 2 4

26 28 30 32 34 36 38

−30 −25 −20 −15 −10 −5 0

12---

−1 0 1 2 3 4

12 14 16 18 20 22 24

−18 −15 −12 −9 −6 −3 0

12---

−5 −4 −3 −2 −1 0

−4 −3 −2 −1 0 1 2

−6 −4 −2 0 2 4 6

−50 −40 −30 −20 −10 0 10

12---

2 3 4 5 6 7

−25 −20 −15 −10 −5 0 5

4 8 12 16 20 24 28

35---

−2 −1 0 1 2 3 4

−2 −1 0 1 2 3 4

−6 −5 −4 −3 −2 −1 0

−3 −2 −1 0 1 2 3

−12 −9 −6 −3 0 3 6

23---

0 1 2 3

12---

2 3 4 5 6 7



h x � 1

i x � −6

j x � −3

k x � 5

l x � 10

4 a x � −3 b h � 6 c m � 3

d p � 2 e t � 1 f x � −2

g a � h m � 4 i w �

j x � 7

Exercise 11-131 a x � 3 b y � −8 c k � −11

d m � 0 e p � −6 f p � −42 a x � −3

b k � −12

c y � 4

d t � −2

e x � 12

f h � −18

3 a w � −1

b y � −2

c x � −7

d x � 4

e a � 1

f x � −2

g k � −4

h m � 3

i d � −5

4 a w � −11 b m � 29 c h � −9d x � −8 e p � −4 f y � −6

5 a y � 16

b y � −4

c y � 7

d a � 1

e m � 4

f y � 4

g m �

h y � 36

i a � −10

j m � −7

12---

−1 0 1 2 3 4

−15 −12 −9 −6 −3 0 3

12---

−6 −5 −4 −3 −2 −1

−10 −5 0 5 10 15 20

−5 0 5 10 15 20 25

23--- 1

5---

13---

6------−5 1

2---

3------−2

−12 −9 −6 −3 0 3 6

−30 −24 −18 −12 −6 0 6

−2 0 2 4 6 8 10

25---

−3 −2

−6 0 6 12 18 24 30

−36 −30 −24 −18 −12 −6 0

−4 −3 −2 −1 0 1 2

−5 −4 −3 −2 −1 0 1

−10 −9 −8 −7 −6 −5 −4

−2 0 2 4 6 8 10

−2 −1 0 1 2 3 4

14---

−3 −2 −1

−10 −8 −6 −4 −2 0 2

14---

2 3 4

12---

−8 −7 −6 −5 −4 −3

4 8 12 16 20 24 28

−10 −8 −6 −4 −2 0 2

1 3 5 7 9 11 13

−2 −1 0 1 2 3 4

−2 0 2 4 6 8 10

−2 0 2 4 6 8 10

58---

0 1

24 28 32 36 40 44 48

23---

−10−11 −9 −8

−10 −9 −8 −7 −6 −5 −4


ANSWERS 601

k y � −3

l a �

m x � −10

n x � 5

o m � −1

p a � 32

q x � −34

r x � −8

Power plus1 See answer below.

∴ m = 262 Backtracking cannot be used with variables on

both sides of the equation.

3 a m = −3 b e = 1 c x = 1

d m = 12

4 She has 26 emus.5 The daughter is 15 years old.

6 a z = 1 b y = −

c Division by zero is impossible.

d z =

7 a x = b x = c x =

8 a x � −1

b x � 0

c y � −1

d x � 5

9 a p ≠ 6 b p � −5 c p � −5d p � 4

Chapter 11 review1 LHS = RHS in each case.2 a x = 9 b a = −53 a y = 4 b x = 12 c m = 1

4 a y = −7 b p = −3 c x = −2

d y = 1 e m = −10 f x = 18

g x = 3 h n = 4 i x = −15

5 a N = 8 b 42, 44, 46 c 273 students6 a P = 72 b t = 30 c D = 4.3

7 a w = b w = c w =

8 C9 C, D

10 a y � 4 b w � 2 c x � 5

d x � 1 e n � −6 f m � −6

g y �

11 a x � −6

b x � 3

c x � −7

d x � −9

−6 −5 −4 −3 −2 −1 0

18---

−1 0 1

−25 −20 −15 −10 −5 0 5

2 3 4 5 6 7 8

−4 −3 −2 −1 0 1 2

20 24 28 32 36 40 44

−40 −38 −36 −34 −32 −30 −28

−20 −16 −12 −8 −4 0 4

910------ 3

4--- 8

9---

517------

15--- 1

50------

2ta-----

v

p2------ k 1 w–( )

w 1+( )--------------------- ms

s m–-------------

−4 −3 −2 −1 0 1 2

−3 −2 −1 0 1 2 3

14---

10 211–4

511------

5 5 65––11

23---

12--- 1

3---

2731------ 8

13------

v xt–k

-------------- mt p–m

---------------- yxy 1–---------------

13--- 4

7---

34---

−15 −12 −9 −6 −3 0 3

−6 −3 0 3 6 9 12

−10 −9 −8 −7 −6 −5 −4

−18 −15 −12 −9 −6 −3 0

Power plus

1

∴ m = 26

m m − 10 −3(m − 10)

5

=

−3(m − 10)− 417 − 3(m − 10)−−−−−−−−−−−−4

× −3

÷ −3

÷ 4

× 4

− 10

+ 10

+ 17

− 17−12−4826 16



Chapter 12Start up

1 A and K, B and I, C and L, D and J, E and R, F and M, G and N, H and Q, P and S

2 a i ∠ P and ∠ N, ∠ T and ∠ L, ∠ Q and ∠ Mii PT and NL, TQ and LM, PQ and NM

b i ∠ A and ∠ K, ∠ D and ∠ F, ∠ C and ∠ G, ∠ B and ∠ H

ii AB and KH, BC and HG, CD and GF, AD and KF

3 a A and B b A and Cc B and C d A and B

Exercise 12-011 a BC and HF, AB and HG

b ∠ B and ∠ H, ∠ C and ∠ F2 a i ∠ X and ∠ P, ∠ Z and ∠ L, ∠ W and ∠ N

ii XW and PN, WZ and NL, XZ and PLb i ∠ F and ∠ P, ∠ E and ∠ T, ∠ D and ∠ X,

∠ G and ∠ Rii FE and PT, ED and TX, DG and XR,

GF and RPc i ∠ K and ∠ S, ∠ J and ∠ R, ∠ M and ∠ P,

∠ H and ∠ Qii KJ and SR, JM and RP, MH and PQ,

HK and QSd i ∠ B and ∠ K, ∠ C and ∠ M, ∠ D and ∠ L

ii BC and KM, CD and ML, BD and KL3 corresponding, matching, angles, congruent,

equal4 a lengths of sides of both rectangles are 20 mm

and 40 mmb i Perimeter X = 12 cm, perimeter Y = 12 cm

ii Area A = 8 cm2, area Y = 8 cm2

The perimeters and areas of congruent rectangles are equal.

5 a LN = HF, LM = HG, MN = GF and∠ L = ∠ H, ∠ M = ∠ G, ∠ N = ∠ F

b i Perimeter ∆LMN = 104 mm,perimeter ∆HGF = 104 mm

ii Area ∆LMN = 20 square units,area ∆HGF = 20 square units

The perimeters and areas of congruent triangles are equal.

6 a ED = RS, DC = ST, CB = TP, BA = PQ, AE = QRand ∠ E = ∠ R, ∠ D = ∠ S, ∠ C = ∠ T, ∠ B = ∠ P, ∠ A = ∠ Q

b i Perimeter ABCDE = 102 mm, perimeter PQRST = 102 mm

ii Area ABCDE = 27 square units, area PQRST = 27 square units

The perimeter and areas of the congruent figures are equal.

7 perimeter, area, congruent, equal8 a Perimeter X = 12 cm, perimeter Y = 12 cm

b No, since length of X ≠ length of Y and width of X ≠ width of Y.

9 a Area X = 12 cm2, area Y = 12 cm2

b No, since length of X ≠ length of Y and width of X ≠ width of Y.

10 a

Perimeter = 9 cm but the triangles are not congruent

b

Area = 3 cm2 but the triangles are not congruent.

11 a F b F c T d T

Exercise 12-021 a RHS b SSS c SAS d AAS

e SSS f SAS g RHS h AASi RHS j SAS k AAS l SSS

2 a ∆ABC ≡ ∆GHI (AAS)b ∆MNO ≡ ∆YXW (RHS)c ∆UVT ≡ ∆CDE (SSS)d ∆PQT ≡ ∆ADF (AAS)e ∆KHZ ≡ ∆MTY (SAS)f ∆BDE ≡ ∆FIJ (RHS)g ∆MNP ≡ ∆RTS (SSS)h ∆VTM ≡ ∆DCE (SAS)

3 a X and Y; AAS b Y and Z; SASc X and Z; AAS d X and Z; SASe X and Z; SSS f X and Y; AASg X and Y; RHS h Y and Z; RHS

4 a No b Yes, RHS c Nod No e Yes, SAS f No

5 a A right angle in each triangle; or the third sides equal, or angles in corresponding positions equal

b Another side or angle in each triangle to be equal

c The hypotenuse or angles in corresponding positions to be equal

d Corresponding sides in each triangle to be equal

3 cm

4 cm

2 cm 3 cm

3 cm

3 cm

3 cm

2 cm2 cm

3 cm


ANSWERS 603

Exercise 12-031 a d = 4.5 b m = 48, k = 9.4

c m = 38 d d = 10e y = 19, k = 67 f x = 110

2 a DF = 13 cm b 133 k = 15.34 a ∠ X = ∠ Z = 90° (given)

XY = ZY (given)WY is common.∴ ∆XYW ≡ ∆ZYW (SAS)

b ∠ ABC = ∠ DCB (given)AB = DC (given)BC is common.∴ ∆ABC ≡ ∆BCD (SAS)

c EG = EH (given)∠ D = ∠ F (alternate angles, HD � FG)∠ H = ∠ G (alternate angles HD � FG)∴ ∆DEH = ∆FEG (AAS)

d BC = DC (given)AB = AD (given)AC is common.∴ ∆ABC ≡ ∆ADC (SSS)

e DG = FE (given)∠ DGE = ∠ FEG = 90° (given)EG is common.∴ ∆DEG ≡ ∆FGE (SAS)

f PQ = LQ (given)NQ = MQ (given)∠ PQN = ∠ LQM (vertically opposite angles)∴ ∆PQN ≡ ∆LQM (SAS)

g ∠ DEG = ∠ EDF (given)GE = FD (given)ED is common.∴ ∆DEG ≡ ∆EDG (SAS)

h WX = YV (given)WV = YX (given)WY is common.∴ ∆WXY ≡ ∆YVW (SSS)

i PQ = QR (given)∠ VPQ = ∠ TQR (corresponding angles, PW � QT)∠ VQP = ∠ TRQ (corresponding angles, RW � QV)∴ ∆PVQ ≡ ∆QRT (AAS)

j AC = CE (given)BC = CD (given)∠ ACB = ∠ ECD (vertically opposite angles)∴ ∆ABC ≡ ∆CDE (SAS)

k XC = YB (given)∠ BXC = ∠ CYB (CX ⊥ AB, BY ⊥ AC)BC is common.∴ ∆BCX ≡ ∆CBY (RHS)

l ∠ FNM = ∠ TMN (alternate angles, FN � TM)∠ FMN = ∠ TNM (alternate angles, TN � FM)NM is common.∴ ∆FNM ≡ ∆TMN (AAS)

5 a AB is common.AD = BC (opposite sides of a rectangle are equal)∠ A = ∠ B = 90° (angles of a rectangle)∴ ∆ABD ≡ ∆BAC (SAS)

b XW = XZ (equal sides of isosceles triangle)WY = ZY (Y is midpoint of WZ)XY is common.∴ ∆WYX ≡ ∆ZYX (SSS)

c EF = GH (opposite sides of a parallelogram are equal)EH = GF (as above)HF is common.∴ ∆EFH ≡ ∆GHF (SSS)

d OA = OC (equal radii)OB = OD (equal radii)AB = CD (given)∴ ∆AOB ≡ ∆COD (SSS)

e PQ = PR (equal sides of isosceles triangle)∠ Q = ∠ R (equal angles of isosceles triangle)QT = RM (given)∴ ∆PQT ≡ ∆PRM (SAS)

f DG = DE (equal sides of a square)∠ G = ∠ E = 90° (angles in a square)GK = GF − KF; but GF = EF (sides of a

square)and KF = LF (given)

GK = EF − LF = LE∴ ∆DKG ≡ ∆DLE (SAS)

g ∠ A = ∠ C (opposite angles of a parallelogram are equal)AD = CB (opposite sides of a parallelogram are equal)AM = AB − MB; but AB = CD (opposite sides of a parallelogram).MB = ND (given); AM = CD − ND = NC∴ ∆AMD ≡ ∆CNB (SAS)

h OA = OB (equal radii)∠ OCA = ∠ OCB = 90° (OC ⊥ AB)OC is common.∴ ∆OAC ≡ ∆OBC (RHS)

6 a i PT = MS (givenPM = ST (given)MT is common.∴ ∆TPM ≡ ∆MST (SSS)

ii ∠ PMT = ∠ STM (matching sides of congruent triangles)

iii PM � ST (alternate angles proved equal in ii)b i ∠ XYB = ∠ ABY (alternate angles, XY � AB)

∠ XYZ = 180° − ∠ XYB (angles on a straight line)

= 180° − ∠ ABY (∠ XYB = ∠ ABY)= ∠ ABC

ii YZ = BC (given)∠ XYZ = ∠ ABC (proved in i)∠ XZY = ∠ ACB (alternate angles, AC � XZ)∴ ∆XYZ ≡ ∆ABC (AAS)



c i FE is common.∠ E = ∠ F = 90° (angles in a rectangle)ED = FC (opposite sides of a rectangle are equal)∴ ∆FED ≡ ∆EFC (SAS)

ii FD = EC (matching sides of congruent triangles)

d TK = LK (TL is bisected)∠ T = ∠ L (alternate angles, TS � PL)∠ S = ∠ P (as above)∴ ∆TSK ≡ ∆PLK (AAS)

e i LM = LK (equal sides of isosceles triangles)∠ M = ∠ K (equal angles of isosceles triangle)MN = KP (given)∴ ∆LMN ≡ ∆LPK (SAS)

ii LN = LP (matching sides of congruent triangles)∴ ∆LNP is isosceles (two sides of triangles are equal)

f i PQ = PR (given)TQ = TR (given)TP is common.∴ ∆PQT ≡ ∆PRT (SSS)

ii ∠ QPT = ∠ RPT (matching sides of congruent triangles)∴ DPT bisects ∠ QPR.

Exercise 12-041 a SSS b ∠ DCB, ∠ CDB, ∠ CBD

c i CD ii CB iii equaliv bisected, BD

2 a SAS b YWc The diagonals are equal

3 b c AAS

d matching, congruente f AAS

g Matching angles of congruent triangles ADC and CBA

h Angles, parallelogram, equal4 a YZ b AAS

c XY, matching, congruent, sides, equal, equal5 a SSS b ∠ SQT, ∠ STQ

c ∠ SQT, bisected, diagonald rhombus, bisected, diagonal

6 a ∠ PLT and ∠ MNT, ∠ LPT and ∠ NMTb Opposite sides of a rectangle are equal.

c AASd Matching sides of congruent trianglese rectangle, bisect

7 a AD is a side of ∆ABD and a side of ∆ACDb SSSc Matching angles of congruent trianglesd ∠ Ce angles, equal, equal

8 a SSSb Yes, matching angles of congruent trianglesd SSSe Yes, matching angles of congruent trianglesf ∠ L = ∠ M = ∠ Ng 60°h equal, 60°

9 a Draw AD ⊥ BC.In ∆ABD and ∆ACD:AD is common.∠ B = ∠ C (given)∠ ADB = ∠ ADC = 90° (supplementary angles)∴ ∆ABD ≡ ∆ACD (AAS)∴ AB = AC (matching sides of congruent triangles)∴ sides opposite equal angles are equal.

b i In ∆GHX and ∆GKX:GH = GK (∆GHK is equilateral)HX = KX (X is midpoint of HK)GX is common.∴ ∆GHX ≡ ∆GKX (SSS)∴ ∠ H = ∠ K (matching sides of congruent triangles)

ii ∆KHY ≡ ∆KGY (SSS) (proof similar to i)∴ ∠ H = ∠ G (matching sides of congruent triangles)

iii Since ∠ H = ∠ K and ∠ H = ∠ G∠ H = ∠ K = ∠ GBut ∠ H + ∠ K + ∠ G = 180°∴ ∠ H = ∠ K = ∠ G = 60°

c i QR = QT (∆QRT is isosceles)∠ R = ∠ T (given)RP = TP (RT is bisected)∴ ∆QPR ≡ ∆QPT (SAS)

ii ∠ QPR = ∠ QPR (matching angles of congruent triangles)But ∠ QPT + ∠ QPR = 180° (angles on a line)∴ ∠ QPT = ∠ QPR = 90°∴ QP ⊥ RT.

10 a i ∠ ZYW = ∠ XWY (alternate angles, ZY � WX)

A B

D C

A B

D C

A

B D C

G

H X K


ANSWERS 605

∠ ZWY = ∠ XYW (alternate angles, WZ � XY)WY is common.∴ ∆ZYW ≡ ∆XWY (AAS)

ii ∠ Z = ∠ X (matching sides of congruent triangles)

b i ∠ YZX = ∠ WXZ (alternate angles, WX � ZY)∠ YXZ = ∠ WZX (alternate angles, WZ � XY)ZX is common.∴ ∆ZXY ≡ ∆ZXW (AAS)

ii ∠ Y = ∠ W (matching angles of congruent triangles)

c opposite, parallelogram, equal.11 a FG = FL (given)

HG = HL (given)FH is common.∴ ∆FGL ≡ ∆FLH (SSS)

b ∠ GFH = ∠ LFHand ∠ GHF = ∠ LHF (matching angles of congruent triangles)∴ FH bisects ∠ GFL and ∠ GHL.

12 a i DG = FE (opposite sides of rhombus are equal)∠ DGT = ∠ FET (alternate angles, DG � FE)∠ GDT = ∠ EFT (alternate angles, DG � FE)∴ ∆DGT ≡ ∆FET (AAS)

ii GT = TE and DT = TF (matching sides of congruent triangles)

b i DE = DG (sides of rhombus are equal)DT is common.GT = TE (proved in a)∴ ∆DET ≡ ∆DGT (SSS)

ii ∠ DTG = ∠ DTE (matching angles of congruent triangles)But ∠ DTG + ∠ DTE = 180°∴ ∠ DTG = 90°

c diagonals, rhombus, right13 a AB = CD (given)

AD = BC (given)AC is common.∴ ∆ACD ≡ ∆ACB (SSS)

b ∠ DCA = ∠ BAC (matching angles of congruent triangles)∴ AB � CD (alternate angles are equal)∠ DAC = ∠ ACB (matching angles of congruent triangles)∴ AD � BC (alternate angles are equal)

Skillbank 122 a b c d

e f g h

i 5 : 9 j 5 : 9 k 9 : 20 l 4 : 5

m9 : 7 n 4 : 3 o p

q r s t

3 a b c

d e f

Exercise 12-051 a 1.5 b 3 c d

e 2 f g or 1 h

i 2 j

2 a

b

c

d

e

f

Exercise 12-061 a, c, e, h2 a i ∠ A and ∠ F, ∠ C and ∠ D, ∠ B and ∠ E

ii AC and FD, AB and FE, BC and EDb i ∠ P and ∠ V, ∠ L and ∠ Y, ∠ N and ∠ X,

∠ M and ∠ W

23--- 4

5--- 5

7--- 1

2---

14--- 1

6--- 5

6--- 2

5---

35--- 4

35------

14--- 4

9--- 5

32------ 1

4---

1740------ 2

3--- 16

25------

14--- 5

24------ 2

25------

12--- 1

3---

12--- 5

3--- 2

3--- 2

3---

23---

AA′

A

A′

A

A′

A A′

AA′

A A′



ii PL and VY, LN and YX, NM and WX, MP and WV

c i ∠ T and ∠ J, ∠ P and ∠ G, ∠ R and ∠ Hii TP and JG, PR and GH, TR and JH

d i ∠ A and ∠ B, ∠ X and ∠ K, ∠ T and ∠ G, ∠ P and ∠ F

ii AX and BK, XT and KG, TP and GF, PA and FB

e i ∠ K and ∠ V, ∠ H and ∠ U, ∠ G and ∠ Y, ∠ F and ∠ X, ∠ E and ∠ W

ii KH and VU, HG and UY, GF and YX, FE and XW, EK and WV

f i ∠ J and ∠ Y, ∠ K and ∠ M, ∠ L and ∠ Nii JK and YM, KL and MN, LJ and NY

Exercise 12-071 a Yes, since = = and matching angles are

equal.

b No, since ≠

c Yes, since = = =

d No, since ≠

e Yes, since both triangles are equilateral triangles.

f Yes, both figures are squares.

g Yes, since = = = =

h Yes, since matching angles are equal.

2 a i ii y = 10

b i ii y = 10

c i ii y = 12

d i ii y = 12.5

e i ii y = 14.4

f i ii y = 26

3 a B and C, scale factor = ;

A and D, scale factor = 2

b A and C, scale factor = 1.5

c A and D, scale factor =

d A and C, scale factor =

4 a PN = 4.8 cm b AC = 9.6 cmMN = 6.4 cm TQ = 11.7 cmWZ = 3.125 cm

c GF = 10.5 cm d CD = 24 mmKH = 4 cm LM = 12 mmHI = 6.6 cm PN = 16 mm

e JH = 6.3 mm f BE = CD = 14.3 mmXW = 7 mm LT = 20 mm

BC = 25 mmLM = 35 mm

g GF = 36 mm h XY = 6 cmGD = 12 mm DE = 7.3 cmKL = 18 mmKN = LM = 54 mm

i PQ = 15.6 mmRQ = 12.4 mmBC = 20.6 mm

5 a = b m = 4

6 a = b d = 5.6

Exercise 12-081 a Corresponding angles in triangles are equal.

b Ratios of hypotenuse to one side are equal to 4 : 3 in both triangles.

c Corresponding pairs of sides in triangles are in the same ratio (1 : 2) and included angles are equal to 110°.

d Corresponding sides in triangles are in the same ratio (3 : 2).

e Corresponding sides in triangles are in the same ratio (5 : 4).

f Corresponding angles in triangles are equal.g Corresponding sides in triangles are in the

same ratio (2 : 1).h Corresponding pairs of sides in triangles are in

the same ratio (2 : 3) and included angles are equal to 90°.

i Ratio of hypotenuse to one side is 15 : 7 in both triangles.

2 a ∠ A is common.∠ ABE = ∠ ACD (corresponding angles, BE � CD)∴ ∆ABE ∆ACD (corresponding angles of two triangles are equal)

b ∠ P is common∠ PQN = ∠ PML∴ ∆PQN ∆PML (corresponding angles are equal)

c ∠ WYX = ∠ ZYV (vertically opposite angles)

= = or WY : ZY = XY : VY = 2 : 3)

∴ ∆VZY ∆XYW (pairs of corresponding sides are in same ratio and included angles are equal)

d ∠ K = ∠ I = 90° (given)∠ G is common∴ ∆GKH ∆GIJ (corresponding angles are equal)

e Both triangles are right-angled (∠ B = ∠ ECD = 90°)DE : EC = AC : CB = 5 : 3∴ ∆ABC ∆DCE (ratios of hypotenuse to one side are equal in both right-angled triangles)

f VZ = 24, WX = 12; VZ : WX = ZY : WY = VY : XY = 2 : 1

1530------ 10

20------ 1

2---

1218------ 36

30------

3045------ 38

57------ 20

30------ 2

3---

23--- 2

4---

313---

5----- 8

12------ 6

9--- 4

6--- 2

3---

12---

23---

43---

53---

85---

103------ 2

3---

12---

53---

43---

830------ 4

15------

820------ 2

5---

≡≡

WYZY--------- XY

VY-------- 2

3---

≡≡

≡


ANSWERS 607

∴ ∆VZY ∆XWY (corresponding sides are in the same ratio)

3 a ∠ BCA = ∠ DCE (vertically opposite angles)∠ BAC = ∠ DEC (alternate angles, AB � DE)∴ ∆ABC ∆EDC (corresponding angles are equal)

Scale factor =

b Triangles are similar since corresponding angles are equal;

Scale factor =

c Triangles are similar since pair of corresponding sides are in the same ratio (4 : 3) and their included angles are equal (110°).

d Triangles are similar since ratio of hypotenuse to one side is equal for both right-angled triangles (6 : 4 = 10.5 : 7).

4 a In ∆FGH and ∆FKG:∠ F is common.∠ FGH = ∠ FKG = 90°∴ ∆FGH ∆FKG (matching angles of triangles are equal)In ∆FGH and ∆GKH:∠ H is common.∠ FGH = ∠ GKH = 90°∴∆ FGH ∆GKH (matching angles of triangles are equal)∴ ∆FGH, ∆GKH and ∆FKG are similar.

b = = and = =

∠ ABC = ∠ AED (included angles)∴ ∆ABC ∆ADE(two pairs of matching sides in the same ratio and their included angles are equal)

c = = 0.57, = = 0.57

and ∠ CED = ∠ FEG (vertically opposite angles)∴ ∆CED ∆FEG(two pairs of matching sides in the same ratio and included angles equal)

d = =

= =

= =

∴ ∆PTN ∆RWT(3 pairs of matching sides in the same ratio)

e i ∠ GFH = 90° − ∠ EFD (∠ DFG = 90°)and ∠ EDF = 90° − ∠ EFD (angle sum of ∆EDF)∴ ∠ GFH = ∠ EDF

ii ∠ GFH = ∠ EDF (proved in i)∠ EFD = ∠ FGH = 90° (DE ⊥ FH, FG ⊥ HG)∴ ∆FGH ∆DEF (matching angles are equal)

f ∠ CDY = ∠ YDW (given)

= and =

= 0.75 = 0.75

∴ ∆CDY ∆YDW(two pairs of matching sides in the same ratio and included angles equal)

g = (OA = OB, OC = OD, equal radii)

∠ AOB = ∠ DOC (vertically opposite angles)∴ ∆AOB ∆DOC(two pairs of matching sides in the same ratio and included angles equal)

Exercise 12-091 a m = 2 b k = 11.25 c y = 162 a 10 b 12.5 c 26.73 a ∠ A is common

∠ ABE = ∠ ACD (corresponding angles)∴ ∆ABE ∆ACDx = 5.6

b ∠ L is common∠ LMQ = ∠ LNP = 90° (given)∴ ∆LMQ ∆LNP∴ x = 5.25

c ∠ Z is common∠ ZYV = ∠ ZWX = 75° (given)∴ ∆ZVY ∆ZXW

x = 2

d PR : LN = QR : MN (corresponding pairs of sides in same ratio)∠ R = ∠ N (included angles equal)∴ ∆PQR ∆LMN

∴ x = 8

e ∠ BCA = ∠ ECD (vertically opposite angles)∠ ABC = ∠ DEC (alternate angles, AB � DE)∴ ∆ABC ∆DEC∴ x = 6.5

f ∠ K is common ∠ FGK = ∠ HLK (given)∴ ∆FGK ∆HLK

∴ x = 6

4 a i ∠ LPM = ∠ RPQ (vertically opposite angles)∠ LMP = ∠ RQP (alternate angles, LM � RQ)∴ ∆LPM ∆RPQ

ii m = 25, k = 14.4b i ∠ ABE = ∠ DBC (vertically opposite angles)

AB : CB = EB : DB = 3:2∴ ∆ABE ∆BCD

ii ∠ EAB = ∠ BCD (corresponding angles of similar triangles are equal)∴ EA � CD (pair of alternate angles are equal)

≡≡

53---

52---

≡≡

ABAE-------- 8

12------ 2

3--- BC

DE-------- 6

9--- 2

3---

≡

CEFE-------- 4

7--- DE

GE-------- 5

8.75----------

≡

PTRW--------- 4

12------ 1

3---

PNRT-------- 3

9--- 1

3---

TNWT--------- 5

15------ 1

3---

≡≡

CDDY-------- 6.75

9---------- DY

DW---------- 9

12------

≡

OAOC--------- OB

OD---------

≡

23---

≡≡

≡

14---

≡

13---

≡≡

23---

≡≡



c i ∠ X is common∠ XWV = ∠ XST (corresponding angles, WV � ST)∴ ∆XWV ∆XTS

ii x = 5 (or 5.4.5.)

d i ∠ HJI = ∠ GIF (alternate angles, HJ � IF)∠ HIJ = ∠ FGI (alternate angles, HI � GF)∴ ∆HIJ ∆GFI

ii HJ = 4.3; JI = 8.65 a ∠ A is common

∠ ABE = ∠ ACD (corresponding angles, BE � CD)∴ ∆ABE ∆ACD

b k = 5.5

6 10 m 7 17.5 m 8 27 m 9 29.1 m

10 height = 0.32 m; width = 0.47 m11 a let ∠ GDE = a°

∴ ∠ DGE = 90 − a°and ∠ EGF = a°In triangles DEG and GEF:∠ DEG = ∠ GEF = 90° (given)∠ GDE = ∠ EGF = a° (proved above)∴ ∆DEG ∆GEFIn triangles DEG and DGF∠ D is common∠ DEG = ∠ DGF − 90° (given)∴ ∆DEG ∆DFG

∆DFG ∆DFG (both similar to ∆DEG)∴ three triangles are similar

b GE = 2.4 cm

Power plus1 a = , = (T and L are midpoints of

PM, PN)∠ P is common.∴ ∆PTL ∆PMN(two pairs of matching sides in the same ratio and included angles equal)

b Since ∆PTL ∆PMN,∠ PTL = ∠ PMN (matching angles of similar triangles)∴ TL � MN (corresponding angles equal).

c = (pairs of matching sides of similar

triangles)

But =

∴ =

∴ TL = MN

2 a ∠ WZX = ∠ XYZ (given)

∠ WXZ = ∠ XZY (alternate angles, WX � ZY)∴ ∆WXZ ∆XZY (matching angles of triangles are equal)

b 12.23 a AB = AD (AB = AD)

CB = CD (CB = CD)AC is common.∴ ∆ABC ≡ ∆ADC (SSS)

b AB = AD (given)AE is common.∠ BAE = ∠ DAE (matching angles of congruent triangles ∆ABC and ∆ADC)∴ ∆AEB ≡ ∆AED (SAS)

c ∠ AEB = ∠ AED (matching angles of congruent triangles)But ∠ AEB + AED = 180°∴ ∠ AEB = ∠ AED = 90°∴ diagonals meet at right angles.

4

Consider quadrilateral ABCD where diagonals AC and BD bisect each other.In ∆AXB and ∆DXC;AX = CX (diagonals are bisected)BX = DX∠ AXB = ∠ CXD (vertically opposite angles)∴ ∆AXB ≡ ∆DXC (SAS)∴ AB = CD (matching sides of congruent triangles)Similarly, ∆AXD ≡ ∆CXB∴ AD = CB (matching sides of congruent triangles)∴ ABCD is a parallelogram (opposite sides equal).

5 In ∆AXN and ∆CXMAX = CX (X is midpoint of AC)∠ XAN = ∠ XCM (alternate angles, AB � CD)∠ ANX = ∠ CMX (alternate angles, AB � CD)∴ ∆AXN ≡ ∆CXM (AAS)∴ AN = CM (matching sides of congruent triangles).

Chapter 12 review1 a SAS b SSS c AAS d RHS2 ∆ABC ≡ ∆HKG (SAS)3 a i AAS ii d = 12

b i RHS ii d = 124 a AC = 10

b c = 8, d = 6, x = 10

≡

511------

≡≡

23---

(90−a)�

(90−

a)�

D E

G

Fa�

a�

≡≡

≡

PTPM--------- 1

2--- PL

PN-------- 1

2---

≡≡

PTPM--------- TL

MN----------

PTPM--------- 1

2---

TLMN---------- 1

2---

12---

≡

AB

DC

X


ANSWERS 609

5 ∠ EGF = ∠ IGH (vertically opposite angles)∠ E = ∠ I (given)FG = HG (given)∴ EFG ≡ ∆GHI (AAS)

6 AD = BC (given)∠ DAB = ∠ CBA (given)AB is common∴ ∆ABD ≡ ∆ABC (SAS)

7 a 10 b 5

8 a ∠ BCA = ∠ XCY (vertically opposite angles)∠ ABC = ∠ YXC (alternate angles, AB � YX)∴ ∆ABC � ∆YXC

b AC = 6.49 a 6.2 m b 32.3 m

Mixed revision 41 a Mode = 6, mean = 5.16

., median = 5.5,

range = 6b Mode = 46, 48, mean = 48.5

., median = 48,

range = 62 a Discrete b categorical c continuous

d continuous3 a 29.9 b 12.7 c 49.74 a 4 b 2.5 c 15.5 d 35 A, C6 a A: 53, B: 78

b A: 20, B: 30c Bowler A; range and interquartile range are

both lower than those of bowler B.7 a 10 b i 6 ii 11.5

c 5.5 d 25% e 308 C

9 a y = b y = c y =

10 a 20 b 27

11 a v = 14 b m = 46 c k = −9 d h = 8

12 a 3 b length = 26 cm, width = 16 cmc 75 $1 coins, 123 $2 coins

13 a k = 7 b y = c h = d x =

14 A 15 C16 a 19, 20, 21, 22 b 60, 62, 64

c 375 students, 263 adults

17 h = 1

18 a x � −1,

b x �

c x � 2

19 C

20 a x � 1 b x � c x � −1d −1 � x � 2

21 y =

22 In ∆PQT and ∆VQT:PQ = VQ (given)∠ PQT = ∠ VQT (given)QT is common.∴ ∆ PQT ≡ ∆VQT (SAS)

23 a In ∆ABC and ∆ADC:AB = AD (given)CB = CD (given)AC is common.∴ ∆ ABC ≡ ∆ADC (SSS)

b ∴ ∠ BAC = ∠ DAC (matching angles of congruent triangles)

c In ∆ABX and ∆ADX:AB = AD (given)∠ BAX = ∠ BAC = ∠ DAC = ∠ DAX (proved in b)AC is common.∴ ∆ ABX ≡ ∆ADX (SAS)∴ DX = BX (matching sides of congruent triangles)∴ AC bisects BD.

24 a SAS b RHS c AAd SSS e AA f SAS

25 a Yes, SAS b No c Nod No e No f Yes, RHS

26 a In ∆ABE and ∆ACD:∠ A is common.∠ ABE = ∠ ACD (corresponding angles, BE � CD)∴ ∆ ABE ∆ACD (AA)

b i 6 or 6.77 ii 8 or 8.46

27 a In ∆FED and ∆DCF:FE = FC (sides of a rhombusDE = DC are equal)FD is common.∴ ∆ FED ≡ ∆DCF (SSS)

b ∴ ∠ E = ∠ C (matching angles of congruent triangles)

28 a b 8 m

General revision1 a $1882 a 20.3 b 4.7 c 6.73 C

4 a b 0 c

d e f

57---

m at–k

---------------- T a–p

------------- ar 2a–1 r+

-------------------

47---

13--- 3

5--- 2

13------ 1

17------

1114------

−2 −1 0 1 2

-25-----

−2 −1 0 1 2 −2 ––5

−2 −1 0 1 2 3

12---

127------

≡

1013------ 6

13------

403------

2k3p------- -m

15------

13a 1–12

------------------- 3m2x------- 6bd

5----------



5

b m = 1, b = −4; m = −1, b = 2c (3, −1)

6 a 6p2 − p − 15 b 16m2 − 24mn + 9n2

c 4k2 − 97 B8 a $1188.19 b $5166.679 a (2h + 3)(2h − 5) b 5(m − 7)(m + 7)

c ( f + 9)2

10 a 3 b −1 c d −4

11 A12 C 13 2 14 D15 a 53° b 29° c 51° d 35°16 603.1917 a Construction b 61 mm18 $818 19 $981.10

20 a i In ∆DTC and ∆BTA:DT = BT

(diagonals bisect each other)CT = AT∠ DTC = ∠ BTA (vertically opposite angles)∴ ∆ DTC ≡ ∆BTA

ii ∠ CDT = ∠ ABT (matching angles of congruent triangles)

iii DC � AB (alternate angles proved equal in ii)b i In ∆DTA and ∆BTC:

DT = BT(diagonals bisect each other)

AT = CT∠ DTA = ∠ BTC (vertically opposite angles)∴ ∆ DTA ≡ ∆BTC

ii ∠ ADT = ∠ CBT (matching angles of congruent triangles)

iii AD � BC (alternate angles proved equal in ii)c ABCD is a parallelogram (opposite sides

proved equal in a and b).

21 a 9xy3(2x − 3y) b 4b(4a − 6b + 3ab)22 a (y − 4)(y + 2) b (x − 1)(x − 2)(x + 2)23 a x − 2 b c

24 a 2 b 2 c 4w2 d 1025 B 26 A, D 27 $1104.68 28 13°30'29 a 25 b 13.2

30 t = ±

31 $482.1332 1.58 × 107 33 0.63 m2

34 172

35 a b

36 55.4 m

−1−2−3−4 3 4210

y

x

1

−1−2−3−4

2

3

4x + y = 2

y = x − 4

17--- 1

4--- 2

11------ 13

23------

x 5–x 5+------------ x 2+

2x 7–---------------

2 my k–( )m

-------------------------

3k 12+k k 1–( ) k 2+( )--------------------------------------- 5

m 3+--------------


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