Answers

612 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3

Answers

Chapter 1Start up

1 a 3 b −6 c 25 d −48e −8 f −10 g 0 h 9i 3 j −32 k 2 l −2

2 a b c d e 1

3 a 7400 b 158 c 0.3 d 0.0301e 0.535 f 0.0072

4 a 48% b 66 % c 175%

d 10.8% e 8.45%

5 a b 1 c d e 2

6 a $8.70 b $24 c 14 kg d 32 Le $15 f $28

7 a 1 : 1 b 2 : 1 c 1 : 2d 1 : 30 e 13 : 12 f 5 : 1g 1 : 10 h 1 : 2 i 1 000 000 : 1j 3 : 2

8 a 7 : 5 b 3 : 8 c 27 : 40d 18 : 13 e 3 : 2 f 3 : 1g 4 : 7 h 4 : 3

9 a 9 : 8 b 21 : 16 c 4 : 5d 5 : 6 e 25 : 1 f 1 : 4g 5 : 6 h 10 : 3

10 a 7 : 5 b 1 : 5 c 20 : 1 d 2 : 1511 a 4 : 3 : 6 b 6 : 3 : 8 c 17 : 5 : 1312 a 3 : 8 b 3 : 4 c 9 : 80

d 12 : 35 e 16 : 5 f 9 : 50

Exercise 1-011 a b − c d

e − f

2 a 2 b 1 c −4 d -1

e 1 f 2

3 a b = 1 c − = −1

d − e f −

4 a 29, 30, 31, 32, 33, 34b 33, 34, 35c 13, 14, 15, 16, 17d 49, 50, 51, 52, 53, 54, 55

5 a b c d e

6 a 1 b 3 c 6 d 4

e 2 f 1 g 2 h 6

7 a b c 2 d 6 e 22

f 3 g 7 h 17 i 8 j 8

8 a 6 b c 1 d 1 e

f g h 2 i j 2

9 a b c 4 d 9

e 12 f − g h 2

10 Any number greater than and less than

, such as

11 56 12 2 times

13 2 minutes or 2 minutes 11 seconds

(to the nearest second)

14 15 15 4 cans 16 $180

Skillbank 1A2 a 3.5 b 2.4 c 0.12 d 0.36

e 0.8 f 0.027 g 0.2 h 8.8i 0.24 j 0.012 k 1.8 l 0.028

4 a 66.3 b 6630 c 6.63 d 0.663e 6.63 f 663 g 0.663 h 663i 6630 j 66.3 k 0.663 l 0.0663

Exercise 1-021 a 56.37 b 0.1 c 636.0 d 3.765

e 19 f 2.47 g 8.00 h 244.00i 0.7 j 24.371 k 20 l 3.092

2 a 3.90 b 12.6 c 36.843d 0.6154 e 245.6615 f 7.37g 8.1 h 54.87 i 2.9861

3 a 56 000 b 19.71 c 1 600 000d 30 000 e 0.0079 f 1.00g 3.01 h 0.31 i 8 000 000j 0.0520 k 4 l 81

4 a 3 b 3 c 2 d 4 e 2f 2 g 4 h 5 i 2

5 a 4000 b 2.72 c 0.04729d 0.94 e 3 360 000 f 0.002

6 1 7 43 5008 a 6091.2 cents b $60.90

c i 2 ii 39 B 10 B 11 A

Exercise 1-031 a , recurring b 1.075, terminating

c 0.32, terminating d , recurring

e 1.375, terminating

f , recurring g 4.6, terminating

h , recurring i , recurring

j 0.85, terminating

2 a b c 5 d e

f 3 g h i 3 j

3 a 0.79 b 0.33 c 0.666 d

e f 0.55

4 a b c d e

f g h i j 1

Skillbank 1B2 a 0.5 b 90 c 8 d 0.9

e 90 f 0.7 g 8 h 30i 0.08 j 3.5 k 4 l 0.3

4 a 16 b 160 c 1.6 d 1.6e 1.6 f 16 g 160 h 0.16i 0.016 j 160 k 0.16 l 0.0016

Exercise 1-041 a 1 b 3 c d e 4

f g h i 1 j

2 a 0.37 b 0.08 c 0.042d 1.15 e 0.0004

3 a 32.5% b 1.5% c 66 %

d 87 % e 180% f 325%

g 133 % h 65% i 60%

j 68%4 a $24 b $161 c 36 kg

d $5100 e $35.70 f $2.985 a $50.40 b $1500 c 13.5 kg

d 1.827 cm e 97.2 kg f $1940.136 a $101.20 b $20.91 c 90.2 kg

d $921.38 e 12.9 L f 132 m7 a $1406 b $12588 $37.50 9 $70 157.50

10 $20.05 11 $831.25

12 a % b 8% c 76%

d 30% e % f 30%

g 7.5% h % i 7.1%13 28.6% 14 30% 15 10%16 40.6% 17 14.8%18 a 600 b 900 c 500

d e f 68

g h i 10219 a $75 b $75020 $16.32 21 19 44322 a $4347 b $24 15323 $275 24 80 25 $500 26 $560

Exercise 1-051 a 5.6 × 103 b 7.2 × 107

c 7 × 10−2 d 1 × 10−9

e 7.128 × 102 f 4 × 103

g 2.78 × 10−4 h 5 × 10−1

i 9 × 108

2 C 3 D4 a 37 000 b 0.0987

c 0.000 000 8 d 15 760 000e 0.3 f 80 700g 0.000 046 1 h 1 280 000i 0.030 61 j 99 100 000 000k 0.001 l 0.000 000 021

5 a 1.21 × 1013 b 1.44 × 10−15

c 1.76 × 1017 d 2.37 × 10−4

e 4.19 × 10−9 f 2.92 × 1015

g 8.23 × 1023 h 1.21 × 1012

i 1.96 × 10−5 j 9.13 × 10−4

6 a Maximum 2.06 × 108 km,

minimum 2.49 × 108 km

b 2.275 × 108 km7 a 1.5 × 104 b 7 × 106 c 3.5 × 105

d 7.61 × 107

8 a 1.9 × 10−3 kg b 1.66 × 10−24 g

c 1 × 10−6 m d 1.66 242 × 108 km2

e 3 × 108 f 2.817 × 10−15 m

37--- 3

5--- 17

40------ 9

16------ 2

5---

23---

725------ 9

20------ 21

25------ 1

40------ 21

50------

114

------ 138

------ 1003

--------- 214

------

52--- 15

8------

17--- 1

2--- 1

4--- 2

3---

19--- 4

7---

17--- 5

3--- 2

3--- 3

2--- 1

2---

18--- 4

11------ 7

10------

1518------ 12

15------ 3

5--- 4

3--- 8

11------

740------ 1

4--- 1

2--- 4

5---

920------ 5

6--- 23

24------ 1

10------

1528------ 1

5--- 2

5--- 1

2---

13--- 6

7--- 1

2--- 3

50------ 2

5---

23--- 9

10------ 3

4--- 19

21------ 5

48------

12--- 14

45------ 1

7--- 5

8--- 19

25------

1720------ 3

8--- 8

9--- 1

3---

35--- 7

10------ 39

100--------- 16

37------

1520------

1620------ 31

40------

47---

211------

23---

0.7̇

0.7̇2̇

0.3̇

0.46̇ 5.6̇

1750------ 13

20------ 3

10------ 1

250--------- 4

5---

625------ 41

200--------- 1

400--------- 5

8--- 7

20------

1340------

29---

79--- 38

99------ 875

999--------- 22

45------ 1

11------

17198--------- 35

198--------- 2

225--------- 541

990--------- 37

99------

720------ 9

20------ 7

25------ 1

200---------

33400--------- 1

3--- 3

8--- 1

2--- 11

200---------

23---

12---

13---

58.3̇

33.3̇

16.6̇

666.6̇ 545.4̇5̇ 47---

53.3̇ 32.6̇ 67---

ANSWERS 613

Exercise 1-061 a 5 : 6 b 7 : 6 c 9 : 13

d 2 : 11 e 16 : 11 f 21 : 55g 9 : 8 h 1 : 12 i 10 : 1j 3 : 2 k 5 : 6 l 50 : 1

m 50 : 3 n 32 : 15 o 20 : 1p 9 : 4

2 a 1 : 2 b 2 : 3 c 3 : 40d 1 : 10 e 1 : 7 f 1 : 10g 1 : 8 h 8 : 1 i 15 : 1j 15 : 1 k 500 : 3 l 3 : 8

m 1 : 50 000 n 9 : 4 o 1 : 3003 a 3 : 2 : 7 b 6 : 1 : 4

c 7 : 12 : 4 d 3 : 4 : 2e 8 : 1 : 6 f 9 : 8 : 12 : 12g 15 : 4 : 10 h 4 : 7 : 5i 12 : 7 : 10

4 a 48 b 48 c 5 d 7e 72 f 50 g 625 h 36i 9 j 55 k 39 l 4

m 4.8 n 28 o 37.5 p 175q 55 r 20

Exercise 1-071 10802 a 32 years b 38 : 18 = 19 : 93 $45 000 4 $146.705 a 8 cups b 9 cups c 10 cups

d 12 cups

6 a = 48% b 60 white, 40 red

7 a 43 : 1 : 6 b 2%c 45 kg plastic, 270 kg glassd 8.3%

8 a 1000b i 444.44 ii 750

9 972 tickets10 84 STD calls11 75 students12 1.24 kg of nickel and 2.17 kg of copper13 96 kg

14 30 m3 gravel, 22.5 m3 sand, 7.5 m3 cement15 a i 12 : 12 = 1 : 1 ii 18 : 6 = 3 : 1

b 18 g

Exercise 1-081 a 1 : 500 b 1 : 100 c 2000 : 1

d 1 : 50 000 e 1 : 10 f 1 : 5g 250 : 1 h 1000 : 1 i 1 : 200 000j 800 : 1 k 1 : 8 l 1 : 120 000

m 500 : 1 n 1 : 937.5 o 1 : 12 5002 a 1.5 m b 2.4 m c 1.15 m

d 2.75 m e 1.9 m3 C 4 825 m5 a 12 m b 4 m6 a 3.5 mm b 2.65 mm c 2.25 mm

d 1.4 mm e 0.4 mm7 A 8 D 9 B

10 1 : 2 000 000 11 1 : 25 000 00012 a i 20 km ii 37.5 km

b 9.7 km c 41 km d 2.5 km13 a 1 : 20 000 000 b 4000 km14 a 1 : 100 b 80 cm15 a 1 : 1000 b 1 : 2000

Exercise 1-091 a 90 km/h b $36/h

c 13 km/L d 80c/kge 18 mm/h f 76 cents/call

g $34/m h 1250 parts/hi $8.25/bottle j 25 km/day

k 42 words/min l 20 g/m2

m $0.90/min n 6.67 m/so 333.33 mL/s o 11 km/L

2 a $22.50/hour b $787.50c 120 hours

3 a $11.50/kg b $69 c 3.478 kg4 Sydney5 a 2000 L/h b 4000 L6 a 25 min b 4 h 35 min7 a 65 km b 162.5 km c 487.5 km8 85 km/h9 a 12 hours b 70.9 km/h

10 a 250 g b 1500 kg11 a i 255 km ii 495 km

b 5.5 hours c midnight

12 a 1 909 090.9 km2 b 930 769 23113 a $2360 NZ b $A4594

c i $76.18 US ii $129 Fiii 28 154 yen iv 298 200 baht

v 27 727.92 peso vi 610 000d $172e i $1.31 ii $2.38 iii $2.34

iv $5.79 v $819.67 vi $12.43

Exercise 1-101 a 14 L/100 k b 7.5 L/100 km

c 11 L/100 km d 8 L/100 kme 12.25 L/100 km f 12.8 L/100 kmg 12.5 L/100 km h 10 L/100 km

i 4 L/100 km

2 a 500 km b 50 km c 150 kmd 25 km e 47.9 km f 8.3 km

3 a 67.6 L b 39.52 L c 152.88 Ld 6.864 L e 0.26 L f 78 L

4 a 61.5 L b $67.595 a 1 birth/1000 people

b 2.5 births/1000 peoplec 13 births/1000 peopled 1.2 deaths/1000 peoplee 34 births/1000 peoplef 4.5 births/1000 people

6 a 240 000 b 130 000c 5.5 per 1000

7 John uses 324 L, Ken uses 144 L

Exercise 1-111 a 1167 m/min b 5000 m/min

c 22.2 m/s d 26.4 m/se 30.56 m/s f 1000 m/ming 2400 m/min h 1.8 km/hi 1.5 km/h j 252 km/hk 7.2 km/h l 36 km/h

2 a 50 g/cm b 125 kg/h

c 80 mL/g d 0.04 g/m2

e 8.64 kg/day f 54 L/h

g 11.52 t/day h 0.000 010 8 s/mm3

i 200 g/cm2 j 1.25 beats/sk 8 m/mL l 300 mL/s

3 a 4500 m b 9 km/h4 454.55 km 5 66.67 m6 2480 seconds7 a 15 m b 20 m c 27.5 m8 15 per 1000 9 Wonder Gal

10 240 km/h

Power plus1 a Teacher to check

b i 20 100 ii 500 500 iii (1 + N)

2 a i 1 ii 1 iii 2

iv 2 v 4 vi 49

b i + + = 1

ii + + + + + = 3

iii + + + + + + + …

+ = 7

iv 22 v 2475

3 3, 37, 1114 a 3 b 5 c 7 d 9 e 11

f 13 g 15 h 17 i 195 25196 a i 15.5% increase

ii 4% decreaseiii 8% decreaseiv 17.2% decrease

17 16 %

18 a , , , ,

b Depends on the values assigned to the pronumerals, e.g. if x = 3, y = 2, w = 1

then � ; if x = 6, y = 2, w = 1,

then �

9 10 M210 a Timmy b h

Chapter 1 Review1 a − b 2 c d 1

e 3 f − g −3 h 13

i 1 j 3 k 1 l 3

2 a $200 b 60 kg c 90 d

3 a 23.75 b 2.303 c 0.6d −0.8516 e 12.43 f 4.859

4 a 38 920 b 39 000 c 40 0005 9 461 000 000 000 km

6 a , recurring

b 0.575, terminating

c , recurring

d 0.95, terminating

e , recurring

7 a 0.84 b c

8 a $24.96 b $52.80 c 12 kgd 6.12 m e $132.60 f $10.50

9 a 62.5 b $84.80 c $432.32

10 a 1.27 × 105 b 7.01 × 10−2

c 7 × 10−6 d 3.7 × 103

11 a 0.000 000 43 b 87 530c 0.000 005

12 a 8.3 × 1011 b 2.0 × 1013

c 5.1 × 1012 d 1.1 × 10−10

13 a 2 : 5 b 4 : 3 c 1 : 10

1225------

23---

N2----

12---

12--- 1

2--- 1

2---

12--- 1

3--- 2

3--- 1

2---

12--- 1

3--- 2

3--- 1

4--- 2

4--- 3

4---

12--- 1

3--- 2

3--- 1

4--- 2

4--- 3

4--- 1

5---

56--- 1

2---

12---

23---

wx---- w

y---- y

x-- x

y-- x

w----

32--- 2

1---

62--- 2

1---

d720---------

115------ 3

10------ 9

35------ 19

20------

18--- 25

32------ 1

4--- 1

3---

23--- 2

5--- 3

8--- 3

4---

17120---------

0.2̇7̇

1.5̇71428̇

3.6̇2131------ 5

8---


d 9 : 2 e 3 : 8 f 1 : 3g 2 : 1 h 1 : 2 i 4 : 9

14 a 6 : 4 : 3 b 7 : 2 : 3 c 1 : 6 : 2d 5 : 9 : 6 e 3 : 8 : 6 f 1 : 2 : 4

15 a 4 : 5 = 12 : 15 = 32 : 40

b = =

16 a 180 b 30 c 48 d $16 250e i 240 : 360 ii 100 : 200 : 300

iii 120 : 60 : 240 : 18017 a 1 : 400 b 20 : 1

c 1 : 250 d 1 : 10 000 00018 a 93.33 km b 126.67 km

c 133.33 km d 266.67 km19 a i 36 km/h ii $5.98/kg

b i 36.60 ii 40 220 yeniii 15.25 iv $18.65

20 a i 12 L/100 km ii 17.2 L/100 kmb i 10.5 births/1000 people

ii 6.4 deaths/1000 people21 a 1.5 kg/min b 0.3456 kL/day

c 12.2 L/100 km d 10.53 km/Le 33.33 m/s f 18 km/h

Chapter 2Start up

1 a 115 b 15.7 c 37.7 d 69.9e 201 f 76.1 g 103 h 905i 48.5

2 a 1681 mm2 b 1650 mm2

c 1248 mm2 d 4750 mm2

e 2310 mm2 f 1800 mm2

g 182 mm2 h 770 mm2

i 680 mm2

3 a y = 29 b k = 29c 19

4 a 33 cm, 85 cm2 b 88 cm, 616 cm2

c 396 cm, 12 469 cm2

d 581 cm, 26 880 cm2

5 a 4.913 m; 1.508 m2

b 38.562 cm; 88.357 cm2

c 12.727 m; 8.727 m2

6 a 105 m3 b 480 m3 c 640 m3

Exercise 2-011 a 282 m2 b 204 m2 c 165 m2

2 a 1036 cm2 b 1020 mm2

c 204 m2 d 390 cm2

e 672 cm2 f 5672 mm2

3 a 57 m2 b 22 m c 269 m2

d 61 m2 e 20 m2 f 40 m2

4 a 657.1 cm2 b 2858.8 cm2

c 8314.2 cm2 d 2793.5 cm2

5 a 26.14 m2 b 29 m2

6 35 403 cm2

Exercise 2-021 a 275 m2 b 564 mm2 c 87.4 cm2

2 a 166.4 m2 b 3456 mm2 c 743.1 cm2

3 a 843 cm2 b 1592 cm2 c 3116 cm2

4 a 432 cm2 b 2150 cm2 c 173 cm2

5 a 144 m2 b 150 m2 c 137.8 m2

6 44 436 m2

7 a 1344 mm2 b 180 cm2 c 343.4 m2

8 a 42 m b A = 735 m2

c 35 m d 28 m

Exercise 2-031 a 101 cm2 b 628 cm2 c 2419 cm2

2 a 392.7 mm2 b 62.8 m2

c 192.4 cm2

3 a 90π b 224π c 577π4 a 942 cm2 b 393 cm2 c 1017 cm2

d 402 cm2

5 5525 cm2

6 a b 30.16 cm c 4.8 cm

d 8 cm e 193 cm2

7 a 17.3 cm b 16.5 cm

8 a 7.1 cm b 5.6 cm c 360 cm2

9 a 24.4 cm b 573.1 cm2

10 a 3.99 cm b 27.9 cm c 27.6 cm11 a 6.9 cm b 85.3 cm c 85 cm12 a r = 6.31, l = 11.2, h = 9.25

b r = 10.9, l = 12.8, h = 6.71c r = 13.1, l = 17.0, h = 10.9

Exercise 2-041 a 2827.43 mm3 b 380.13 m3

c 366.44 cm3

2 a 432π m2 b 192π m2 c 768π m2

3 a 628 m2 b 314 m2 c 236 m2

d 1257 m2

4 5.1 × 108 km2

5 a 716.3 m2 b 72 L6 a 6.91 cm b 7.98 cm c 9.77 cm7 a Teacher to check

b i 3800 cm2 ii 8.5 m2

c 6.5 cm

Exercise 2-051 a 857.7 cm2 b 270.9 cm2 c 1186.0 cm2

d 5969.0 cm2 e 250.6 cm2 f 628.3 cm2

g 282.7 cm2 h 652.9 cm2 i 501.6 cm2

j 3769.9 cm2 k 1148.8 cm2 l 3017.7 cm2

m 6615.4 cm2 n 3908.4 cm2 o 328 cm2

Exercise 2-061 a 64.8 m3 b 1534.5 m3

c 13.6 m3 d 135.7 m3

e 107.5 m3 f 146.3 m3

2 a 1539.4 m3 b 14 431.7 cm3

c 226.4 m3 d 4.7 m3

3 a 23 562 cm3 b 4825 cm3

c 5027 cm3 d 1989 cm3

e 536 cm3 f 12 900 cm3

g 33 117 cm3 h 2639 cm3

i 794 cm3

4 a 20 slices b 138 cm2 c 0.3 m2

5 Triangular prism 36.4 m2, half cylinder

28.8 m2.The triangular tent has the greater surface area.

6 2073.5 cm2 or 0.21 m2

7 a 1508 cm3 b 75.4 cm3

8 1028.3 cm3

9 a 29 040 cm3 b 24 Lc 0.0238 kL

10 4948 cm3 11 8.7 m2 12 10.7 cm

13 36 mm 14 3.2 m 15 3375 cm3

16 64 m2

Exercise 2-071 a 149.3 cm3 b 240 cm3

c 120 cm3 d 336 m3

e 1200 cm3 f 106.7 m3

2 a 950.7 cm3 b 41.1 m3

c 2400 mm3 d 840 cm3

e 14 842.7 mm3 f 21.9 m3

g 7883.3 cm3 h 8064 mm3

3 a i 24 cm ii 3200 cm3

iii 3200 mL

b i 20 m ii 19 200 m3

iii 19 200 kL

c i 40 mm ii 4320 mm3

iii 4.32 mL

d i 60 mm ii 28 160 mm3

iii 28.16 mL

e i 7.7 m ii 133.056 m3

iii 133.056 kL

f i 84 cm2 ii 564 480 cm3

iii 564 480 mL or 56.448 L

4 a Volume 343 cm3, capacity 343 mL

b Volume 240 cm3, capacity 240 mL

c Volume 1152 cm3, capacity 1152 mL

d Volume 6100 cm3, capacity 6100 mL

e Volume 6048 cm3, capacity 6048 mL

f Volume 2500 cm3, capacity 2500 mL

5 a 33.75 m3 b 18.98 tonnes

6 a 946 729 m3 b 0.411 m3

7 27 m 8 24 cm2 9 19 mm10 4.9 cm

Exercise 2-081 a 209 m3 b 872 cm3 c 1272 mm3

d 616 cm3 e 393 cm3 f 2545 mm3

2 a 12 566.4 cm3 b 3141.6 cm3

c 18 849.6 cm3 d 4712.4 cm3

3 a Perpendicular height 6.9 cm,

volume 115.6 cm3

b Perpendicular height 27.2 cm,

volume 13786.1 cm3

c Perpendicular height 12.4 m,

volume 378.6 cm3

d Perpendicular height 3.5 m,

volume 2.3 m3 e Perpendicular height 244.6 m,

volume 296 103.1 m3 f Perpendicular height 71.9 cm,

volume 129 674.2 cm3

4 a i Teacher to check. ii 40 mm

iii 3392.9 mm3

b i Teacher to check. ii 12 cm

iii 314.2 cm3

c i Teacher to check. ii 3 m

iii 8.0 m3

d i Teacher to check. ii 72 mm

iii 33 250.6 mm3

5 a 9 cm b 4 cm c 15 cm d 2 cm6 a 5.7 m b 9.7 cm c 9.2 cm

d 8.5 m7 2 cm 8 5.23 mm

23--- 12

18------ 14

21------

35---

ANSWERS 615

Exercise 2-091 a 14 137 mm3 b 697 m3

c 660 cm3 d 3619 m3

e 1072 cm3 f 8578 mm3

2 a 33 510.3 cm3 b 97.7 m3

c 179.6 cm3 d 0.9 m3

e 356.8 cm3 f 4.1 cm3

3 a 4 mm b 7 cm c 3 m

4 a 11 500 mm3 b 5150 cm3

c 409 000 mm3 d 1.07 m3

e 38.8 m3 f 51 000 mm3

5 1.1 × 1012 km3

Exercise 2-101 a cylinder and cone

b 58.6 m3 c 58.6 kL

2 a Volume 31 416 cm3, capacity 3.142 L

b Volume 616 cm3, capacity 0.616 L

c Volume 302 cm3, capacity 0.302 L

3 a Volume 1950 cm3, capacity 1.95 L

b Volume 22 988 cm3, capacity 22 988 L

c Volume 1309 cm3, capacity 1.309 L

d Volume 455 cm3, capacity 0.455 L

e Volume 25 656 cm3, capacity 25.656 L

f Volume 1527 cm3, capacity 1.527 L

4 a Tank A 28.27 m3, Tank B 56.55 m3

b 28.27 kL5 a 12 balls b 60 balls

c 31 416 cm3 d 48%

6 a 1963 cm3 b 0.55 cm3/min

7 a 250 m3 b 210 kL c $205.80

8 1.447 × 1015 km3

Exercise 2-111 a 1 : 16 b 9 : 16 c 25 : 4

d 4 : 492 a 9 : 1 b 9 : 25 c 81 : 25

d 4 : 93 a 3 : 5 b 1 : 10 c 8 : 5

d 4 : 94 a 7 : 6 b 7 : 95 a 1 : 11.56 b 1 : 39.3046 x = 3.5 7 a 1 : 11.56

b 1 : 39.3018 a i4 : 25ii 8 : 125

b i 1 : 4 ii 1 : 8c i 49 : 16 ii 343 : 64

9 a i 3 : 2 ii 9 : 4b i 5 : 3 ii 125 : 27c i Increased 8 times

ii of original length

10 a 162 cm2 b 154 mL

c 25 : 49 d 363.41 cm2

e Increases 9 times

f Decreases by

11 a V1 = 72 cm3 b V2 = 4V1 c V3 = 2V1

d V4 = 9V1 e V5 = V1

12 350% increase

Power plus1, 2, 3, 4, 5, 6 Teacher to check.

7 a b

Chapter review1 a 5236 cm2 b 277.7 m2

c 104.3 m2 d 14 294.4 cm2

e 5871.2 cm2 f 4427.8 cm2

2 a 960 cm2 b 7776 cm2 c 1356 cm2

3 a 704 mm2 b 3270 mm2

c 2490 mm2

4 a 452 m2 b 681 m2 c 5890 m2

5 a 3318 cm2 b 2592 cm2

c 3436 cm2 d 1268 cm2

e 16 416 cm2 f 3016 cm2

6 a 11 cm3 b 20 160 cm3

c 10 472 cm3

7 a 183 m3 b $21.96

8 a 323 m3 b 540 cm3 c 348 cm3

9 a i 1340.4 m3 ii 10 262.5 m3

iii 31.7 m3

b i 10.7 cm ii 6.9 cm

10 a i V = 180 000 mm3, C = 180 mL

ii V = 46.0 m3, C = 46.0 kLb i 12 mm ii 3 m

11 a 360 498 mm3 b 145 125 mm3

c 455 cm3 d 1152 m3

e 3054 cm3 f 12 667 m3

12 a 250 cm2

b i Increases 27 times

ii Decreases by

Chapter 3Start up

1 a m2 + 10m + 21 b k2 + 7k + 10

c y2 − 3y − 4 d w2 + 4w − 21

e n2 − 5n + 6 f 6d2 + 11d + 3

g 4 − 17p − 15p2 h 3a2 + 17af + 10f 2

i 3x2 − 7vx + 2v2 j ac + ad + bc+ bd

k 10e2 + 15eg + 6e + 9g

l 6h2 − 7h − 52 a i a2 + 2ab + b2 ii a2 − 2ab + b2

b i x2 + 8x + 16 ii y2 − 6y + 9iii 4k2 + 4k + 1 iv 9m2 − 24m + 16v 9k2 − 12kf + 4f 2

vi a2 + 4ab + 4b2

3 a a2 − b2

b i d2 − 9 ii 9a2 − 16iii 16w2 − y2 iv 25h2 − 9e2

v 1 − 9y2 vi 81d2 − 16w2

4 a 26 b 5−2 c 1010 d 70 e 3−5

f 157 g h 74 i 32

5 4 and 56 a 1.93 b 1.78 c 3.66

d 4.33 e 207.06 f 7.37g 5623.41 h 26.75

Exercise 3-011 a −1.8 b 0.7 c 0.4 d −3.5

e −2.5 f 2.6 g 1.6 h 1.9

2 a R b I c R d R e If R g R h R i I j Rk I l R m R n I o I

3 A, C, D

4 a , 5, b π, ,

5 a 1 , , b 2 , ,

6 b i 57 mmii Approximately 1.4 (= 56 ÷ 40)

7 a Teacher to check.b i Construct a right-angled triangle with

the two shorter sides equal to 10 units.

The hypotenuse is units.

ii Construct a right-angled triangle with the two shorter sides equal to a units.

Exercise 3-021 a 2 b 5 c 27 d 250

e 0.09 f 28 g 45 h 50

2 a 2 b 3 c 2 d 3

e 9 f 3 g 4 h 10

i 4 j 3 k 12 l 6

m 5 n 7 o 4 p 11

q 9 r 7 s 5 t 16

u 3 v 10 w 4 x 3

y 4

3 a 25 b 6 c 12

d 56 e f

g h 6 i 18

j k 3 l 3

m 40 n 15 o 14

p q 2 r 6

s 4 t 6 u 12

v 5 w 3 x 49

y 3

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

Exercise 3-031 a b − c 4

d 6 e −5 f 45

g 15 h −10 i 140

j −30 k 36 l − 60

m −112 n 24 o 80

p 90 q −396 r 160

s 216 t −96 u 36

v − 60 w 252 x 72

2 a b − c 3

d 2 e − or −

f 21 g 1 h 8

1

23-------

2764------

34---

2 H4----

14---

4196

------

−4 −3 −2 −1 0 1 2 3 4

− 153 −145--- 2 5

9---

411------12 74%

π2--- 187%−

17 26 10 11

47--- π

2--- 2 7

9--- 203 2.6̇

10 2

2 3 6 6

3 5 3 2

6 7 2 3

3 3 2 2

2 5 5 2

35 7 7 11

10

2 2 3

2 10 3

73

------- 6 17

5 52

---------- 2 3

10 3 17

133

---------- 5 5

5 2 10

10 6 7

30

14 35 3

2

30 21

2 2

6

6 5

2 6 5

10 3 14

5 3 6

2 72

------- 12--- 7


i 5 j 5 k 4

l 2 m 2

n − or − o 1

p 10 q 4 r −21

s 12 t 2 u

3 a 2 b 4 c

d e 14 f 2

Exercise 3-041 a 7 b −5 c 7

d 4 e 0 f

g 8 h 0 i 2

j 10 k 6 l −14

2 a 5 − 8 b 13 + 3

c −9 + 5 d 7 + 8

e −3 − 2 f − − 8

g 13 − h − 6 + 11

i − 6 j −33 a C b D c B d A e C

4 a 6 b 3 c −2 d −

e 5 f 7 g − h 8

i 6 j 8 k 3 l 9

5 a 11 b −

c − 6 d 30

e 5 f 41

g 5 h 29

i −15 j 0

k 6 + 2 l 12 + 3

m 3 − 6 n −10

o 4 p −5

q 17 − 10 r −10 − 7

s 30 − 15 t − 4

u 8 v −13 + 10

w 9 − x −

Exercise 3-051 a − b 6 −

c −2 + 7 d 3 − 15

e 12 + 6 f − 4 +

g 42 − 8 h 5 − 75

i 24 − 3

2 a 10 + − 6 − 3

b 7 + 2 − − 2

c 28 + 21 + 8 + 2

d 20 + e 109 + 10

f 72 − 23 g 16 + 54

h −16 − i − − 1

j 45 − 5

3 a 8 − 2 b 9 + 2

c 21 − 4 d 49 + 12

e 77 + 30 f 179 − 20

g 110 + 60 h 35 + 20

i 159 − 30 j 73 − 12

4 a 5 b 22 c 67 d −57e 1 f 166 g 2 h −35

5 a 88 − 30 b −10 + 21

c 29 + 5 d 73 + 40

e 83 f 85 − 3

g 29 h 92 − 12

i − 6 j − 4

k 4 l −12 + 2

m −9 n 30 + 4

Exercise 3-061 a b c d 4

e f g h

i j k l

m = 1 − n

o or − 1

p or 2 −

q r 1 +

s t

Skillbank 32 a b c d

e f g h

i 5 : 9 j 5 : 9 k 9 : 20 l 4 : 5

m 9 : 7 n 4 : 3 o p

q r s t

3 a b c d e f

Exercise 3-071 a y8 b d4 c k15

d 1 e 9e f 12k9

g 4g3 h x3 i 2a4

j 20p3q7 k 3m2n l l14m13

m 48b17 n 4g2 o 1

p 3 q 6c9 r 36t5

s 5w2 t 5g4 u wt3

v −27e6 w fh3 xy 8

2 a 2 b 8 c −1 d −2e −4 f 3 g 15 h 24

3 a b c d e

f g h i j

4 a 12−1 b 5y−4 c (4k)−2

d h−4 or 3−1h−4 or e 2p−3

5 a 3 b 6 c 2 d

e f g h 7

i j k l

6 a 5 b 5 c 8 d 1 e 2f 10 g 3 h 25 i 6 j 3k 2 l 150 m 25 n 1 o 3

p 1 q r 27 s 4 t 8

u 1 v 1 w 1 x 24

7 a 3 b 3 c 4 d 2

e 2 f 4

Exercise 3-081 a 36 b (2d) c 8k

d (15w) e 5p f 74

g (9ab) h a

2 a b c

d e 14 f 4

g 7 h

3 a d b y c y

d h3 e p− f x−

g a h (5y4)

4 a b c 6 d 32

e f g h

5 a 125 b c 8

d e 27 f 100 000

g h 256 i

j 625 k 243 l

m 27 n 1000 o

6 a 1.7 b 0.23 c 2.4 d 0.0098e 0.046 f 0.30 g 15 h 2.3

Exercise 3-091 a y12 b k9 c m d w9

e a15 f m8 g d4 h x3

i p19 j c11 k 5k5 l 12q10

m 2 n 81e2 o 4mn6 p t8w7

q r 2x3 s 2d11 t 16a6b7

u 2c2h2 a 9 b 50 c 16 d 2

e 1 f 64 g 24 h 81

i 2 j 2 k −27 l

3 a 8 b (2m) c 20 d k

e y f (9k) g c h (5b)

4 a b c

3 2 3

6 14

24

------- 14--- 2

2

23---

6 30

245------ 3

7 2 5 7–

5 10

15 3

5 10 3

5 10 2

3 2 15 2

7 5 6 3

11 3 13 7

7 5

2 3 5 7

6 5 10 11

2 3 2 2

3 5

3 3

7 2

6 2

3

2 3 3 6

2 5 2

6 2

5 7 10 3

5 3 a

y n k

d 2a 2

15 10 3 6

21 10

6 11 55

7 5

6

10 5 2

7 21 3

6 2 3

10 77

6 10

35 15

21

15 14

5 10

6 7

2 3

6 35

7 2

35 3

77

5

30

15

15

33

------- 5 77

---------- 4 55

---------- 2

7 520------- 3 2

4---------- 10

8----------

146

---------- 5 66

---------- 772

---------- 4 155

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

2 2–2

---------------- 22

------- 5 2 6+4

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

35 5–5

------------------- 355

----------

22 11–11

---------------------- 1111

----------

2 3 2–2

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

1 2 7+2

-------------------- 6 5 3–3

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

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------ 1

4--- 5

24------ 2

25------

5 p2

3---------

142----- 1

65----- 1

8--- 1

53----- 1

27-----

1x2----- 1

5w------- 3

g4----- 4d2

y5--------- 15

n3------

13--- h 4–

3-------

14--- 10

27------ w

4----

9m6------ 125

k6--------- 5

a2b3----------- 1

9---

259d2--------- w4

m8 x12-------------- 16c10

a4------------- 64n3

27d6------------

14--- 1

16------

13--- 1

2---

12--- 1

16------

13-- 1

2-- 1

2--

13-- 1

3-- 1

2--

13-- 3

2--

5 603 2x3

8v a3 15w

t3 c8

------

14-- 5

3-- 1

2--

14-- 3

5--

72-- 1

3--

k5 1

d34---------- w25 p53

1

x4------- 1

8 a34------------- 1

64c35---------------- 10 k37

12---

18---

125------ 1

2---

1128---------

1216---------

2a3c------

12---

12-- 1

4-- 1

3-- 1

5--

43-- 3

5-- 7

4-- 8

3--

123----- 1

4--- 1

k2-----

ANSWERS 617

d e or

f g h

i 93 j k

l

5 a b c 2 d

e 2 f 4 g 4 h

i j 1 k 32 l

m 3 n 4 o p 27

Powerplus1 a Yes, since has been multiplied

by 1 .

b i ii Yes

2 a − 2 b

c d

3 a 1189 mm × 841 mm b m

4 s = or s = 5 =

6 (2 + 2 ) m 7

Chapter 3 Review1 a I b R c R d I e R

f R g R h R i N j I

2 a 6 b 7 c 5

d 8 e 15 f 14

g 48 h 15 i 28

j k l 2

m 4 n 6 o 6

p 3

3 a b 2 c 4

d e 2 f

g 4 h 1 i

j k

4 a −2 b 5

c 7 d 3 − 3

e 2 + 6 f 3

5 a 13 b 5 − 7

c 14 + 17 d 32 − 9

e 38 − 24 f 8

6 a −12 + 9 b − 10

c 7 − 27 d 23 − 8

e 77 + 10 f 43

g 9 h 70 + 38

7 a b c

d

8 a 30m3 b 3 c 4k6

d 4mn e 81y8 f 8

9 a d b d c w d y

10 a 32 b 10 000 cd e 4

Mixed Revision 11 202 a 17.0 b 14.72 c 2.0643 a 3 : 7 b 5 : 3 c 3 : 2

d 3 : 4 e 10 : 3 f 1 : 4g 12 : 1 h 4 : 1 i 1 : 180

4 40 320 km/h

5 a 6 b 6 c 2 d 12

6 a 673.9 b 0.0010

7 a 1.7 × 107 b 6 × 10−6 c 3.574 × 10−3

8 a 75 km/h b 70c/kgc $7.65/m d 85 words/min

9 a , recurring b 0.85, terminatingc 0.335, terminatingd 0.0019, terminatinge , recurring

10 a 95.2 L b 8.288 L c 154.56 L11 a $12 520 b $28 17012 a 3.6 m b 2.56 m

13 a 145.25 b $A204.82

14 a 7.18 × 1015 b 1.06 × 1020

15 20.6%

16 a b c 1

17 a 2145 mm3 b 8181 mm3

c 7069 mm3

18 a 6362 mm3 b 90 mm3

c 30 708 cm3

19 a 1294.43 m2 b 868 m2

c 1555.63 m2

20 a 15 927.9 cm3 b 8143.0 cm2

21 a 580 cm2 b 7.96 m c 7 cm

22 a 46 m3 b $49.22

23 a 22.4 m2 b 192.8 m2

c 303 m2

24 a 18 473 mm3 b 2413 mm3

c 1105 mm3

25 a 800 mm2 b 8.9 cm c 4904 cm2

26 a 445 cm2 b 61 m2

27 a 320 cm3 b 746.7 cm3

c 512 cm3

28 a i 1 : 5.76 ii 1 : 13.824b 64 : 125 c 1.86 md i 3.375 : 1 ii 45 cm

29 14 mm 30 4.89 m 31 10 cm

32 a 10 cm b 400 cm3

33 a 6 b 12 c 9 d 1

34 a 3 b 2 c −6d −12 e 2 f 13

g 14 h −2 i −18

35 a b c −10

d 60 e − f

g 15 h i

36 a 118 b y7 c 3k7m2

d 12 e f 35m4

37 a k b (3m5) c 2w

d 2y

38 a b c

d 27 e f

g h

39 a −2 b 15

c 9 d 7

e 0 f 32 − 6

g −14 h 9 − 6

i −940 a Irrational b Rational c Rational

d Neither e Rational f Irrational

41 a 1 b 2 c 12

42 a 5 b 24 − 12

c 4 + 10 d −7 − 7

e 33 + 4 f 54 + 22

g 8 − 2 h 163 + 8

i 5 j 259

k −20 l −18x − 6

43 a b c

d 3 e f

g h i

j 1 k l

44 a 5 − b 5

c 2 + d 7 − 2

e 5 − 2 f 2 − 7

45 (9 − ) cm2

Chapter 4Start up

1 a y = 5 b x = −3 c m = −24

d a = 7 e y = 2 f x = 1

2 a y < 50 b x � −13 c x > −8

d x < −1 e y < −10 f m � 10

3 a a2 + 13a + 30 b 2y2 − 3y + 1

c x2 + 9x + 20 d y2 − y − 6

e k2 + 2k − 15 f m2 − 4m + 4

g 25y2 + 30y + 9 h 9a2 − 24a + 26

i a2 + 14a + 514 a (4 − m)(4 + m) b (d − 11)(d + 11)

c 2y(7 − y) d 5p(2p + 5)e 5(x − 8)(x + 8) f 2(3w − 5)(3w + 5)

5 a (k + 1)(k + 4) b (y − 8)(y − 2)c (m − 8)(m + 7) d (u + 13)(u − 5)e (w − 7)(w − 3) f (x − 6)(x + 4)

6 a (y + 2)2 b (d − 3)2

c (n − 6)2 d (p + 9)2

e (2w − 3)2 f (8q + 5)2

7 a (3a + 1)(a + 3) b (5x + 2)(x − 3)c (2y − 5)(3y + 8) d (3t − 1)(5t + 4)e (5v + 3)(v − 7) f (2y + 5)(4y + 7)g (3h − 4)(5h − 1) h (4p − 3)(3p + 5)

i (4d + 5)2

3a5----- 1

25g2------------ 1

5g( )2--------------

p3

h5----- 1

d2 f 3------------ 7a2

x4---------

25m4------ 9

h3-----

27k6

y12-----------

19--- 1

4--- 1

15------

12--- 1

36------

45--- 1

2--- 1

216---------

38--- 17

27------ 1

32------

1

7 2–---------------------

7 2+

7 2+--------------------- 1=⎝ ⎠

⎛ ⎞

7 2+5

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

10 2 15 3+17

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

13 5 5+11

----------------------- 8 3 6–2

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

3 22

----------

D

3------- 3D

3------------ 1

2------- 2

2-------

3 43--- 10 3

3-------------

2 2 11

2 6 7

2 5 3

6 503

------ 5 2

11 2

6

21 10 3

55 6 14

6 66

-------

3 22

---------- 23---

3 2

5 7 2

5 7

2 2 5

2 3 5 7

2 3 11

2 10 5

35 7

6

5

3 1010

------------- 2 2 1+3

-------------------- 34

-------

5 66

----------

15-- 2

3-- 7

4-- 3

2--

12---

1343---------

25--- 3

8--- 3

4---

0.5̇4̇

0.64̇

6599------ 7

55------ 313

990---------

2 5 2

5 5 2

7 3 10

2 6 43--- 14

2 2 18---

22

------- 3 32

----------

56---

75-- 1

2-- 5

4--

13--

y23 32m53 24

p3 454 1025

1

3125 c5---------------------- 4n( )56

2 5

3 2

5 3

6 3 10

y

916------ 5

6---

2

7 2

14 10

15 30

6 wx

55

------- 7 33

---------- 10

5 115

---------- 2 23

----------

302

---------- 6 2 355

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

3 14 7–21

--------------------------- 6 15+3

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

10

5 2 10

10 10

12--- 5

34---

12---


8 a 5.6 b 13.54

Exercise 4-011 a y = 2 b a = 5 c x = 8

d a = 1 e a = −5 f a = −2g w = 6 h x = 7.5 i y = −2

2 a y = 10 b a = −2.5 c y = 5d a = −3 e y = 8 f x = 1g y = −1 h x = −10 i m = 1.5

3 a x = 16 b m = 6 c y = 4d y = 8 e y = −7 f x = 11g m = −8 h a = −1 i y = −5

j a = k a = l a =

4 a y = 15 b a = 9 c m = 7d k = 57 e n = −35 f y = −7g x = 31 h y = 46 i m = 18j x = −29 k x = 24 l m = 10

5 a m = 2 b x = 4 c w =

d x = 3 e y = 3 f a = −8

g p = 9 h y = 3 i y =

j w = 10 k w = 50 l w = 9

m a = n y = 60 o a = 1

6 a x = b y = 15 c m = 19

d x = 10 e x = 10 f y = 10

g a = -41 h a = 7 i y = -14

j a = 5 k w = 11 l y = 2

Exercise 4-021 $7.50, $182 18 mm, 36 mm, 36 mm3 57 mm, 19 mm 4 29 cm, 13 cm5 61, 62, 636 Vatha is 22, Chris is 14.7 213, 214, 215, 216 8 269 Scott is 11 and his mother is 34

10 85 10-cent coins, 117 20-cent coins11 612 The son is 3 and the father is 27.13 25°, 50°, 105° 14 72 L15 8 teachers, 120 students

Skillbank 4A2 a 160 b 70 c 240 d 900

e 2600 f 900 g 220 h 300i 180 j 770 k 18 l 34

m 46 n 26 o 18 p 12q 40 r 8 s 104 t 24u 44 v 135 w 600 x 80

Exercise 4-031 a 52 b 172 a i 36 km/h ii 86.4 km/h

iii 180 km/hb m/s

3 434 a I = 300 b P = 781.255 a 27°C b 0°C c 100°C

d 39°C6 a 11.2 b 9 c 17.3

7 a 15.1 m b 31.8 cm8 a 21.0 b 105.84 kg

9 a 137.3 cm3 b 4.9 m10 a 80 km/h b km/h

c 325 km d 1 hours

11 a $97.50 b 620 km

12 a 410.0 cm2 b 73.9 m2

Exercise 4-041 a y = b y = c y =

d y = e y = f y =

g y = h y =

i y = − 3 or y =

j y = ± k y = ±

l y = kx2 m y = ±

n y = T 2c − k o y =

2 a r = b b = ±

c a = d h =

e r = f s =

g R = ± h l =

i n = j T =

Exercise 4-051 a x � 9

b x � 5

c y � − 4

d m � −8

e a �

f w � −10

g a � 5

h y � 20

i a � 3

j a � −1

k w � −3

l a � − 6

2 a x � 1 b m � 6 c y � -8

d r � e a � 3 f w � 0

g a � 4 h a � 6 i m � 3

j m � 3 k d � 2 l m � −2

3 a x � 3 b y � −8 c k � −11d m � 0 e p � −6 f p � −4g x � −3 h k � −12 i y � 4

j t � −2 k x � 12 l h � −18

m w � −11 n m � 29 o h � −9p x � −8 q p � −4 r y � −6

4 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

5 a m � −10 b y � −2

c k � d x �

6 a 22, 23, 24b length = 23.2 cm, width = 5.8 cmc 12 cm, 12 cm, 21 cm d 3, 5 and 17

Exercise 4-061 a x = 3, y = −1 b x = 1, y = 3

c x = 7, y = 3 d x = 2, y = 3e x = 1, y = 4 f x = 5, y = 4

1113------ 4

3--- 9

10------

25--- −3

4-----

35---

23--- −5

6-----

35---

611------ 11

13------

715------ 4

7---

67--- 2

9--- 3

5---

914------ 1

5---

514------ 1

2---

30.5̇

55.5̇12---

5 x–2

------------ k m–p

------------- 5 2x–d

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

P 8–k

------------- 5m3

------- K D–M

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

4d 5–8

---------------- 2c k+a

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

20m3

---------- 20m 9–3

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

t2 d2– w 5–x

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

a2 p–m

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

5n d–n

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

IPn------- c2 a2–

2 s ut–( )t2

---------------------- 2 Aa b+------------

3V4π-------3

v2 u2–2a

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

A πr2+π

------------------- A πr2–πr

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

s 360+180

------------------ DS----

0 3 6 9 12 15

0 1 2 3 4 5 6

−6 −4 −2 0 2 4

−12 −10 −8 −6 −4 −2 0

12---

−1 0 1 2 3 4

−12 −10 −8 −6 −4 −2 0 2

0 1 2 3 4 5 6 7

0 5 10 15 20 25

0 1 2 3 4 5

−3 −2 −1 0 1 2

−4 −3 −2 −1 0 1 2

−10 −8 −6 −4 −2 0

12--- 2

5---

12---

25--- 3

5--- 1

2---

25---

−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 −312---

13--- 11

3------

ANSWERS 619

2 a x = 2, y = 2 b x = 1, y = 4c x = 2, y = 1

3 a x = −2, y = 2 b x = 5, y = 10c x = 1, y = 4 d x = 1, y = 3e x = 2, y = 8 f x = 1, y = 1

Exercise 4-071 a x = −2, y = −3 b x = 0, y = 1

c x = 4, y = −32 a x = 1, y = 2 b x = −3, y = −3

c x = 3, y = 2 d x = 0, y = 1e x = 3, y = −1 f x = −2, y = 9g x = 5, y = − 4 h x = 3, y = 2

3 b The lines are parallel, so they do not intersect.

Exercise 4-081 a x = −2, y = −5 b x = 8, y = 2

c x = 2, y = 2 d x = 24, y = 8

e x = , y = 2 f x = , y = 2

2 a x = 1, y = 3 b x = 5, y = 2c x = 1, y = 6 d x = 4, y = 1e x = 4, y = 1 f x = 3, y = 2g x = 9, y = 3 h x = -13, y = 6

i x = -8, y = -3 j x = 3 , y = −1

Exercise 4-091 a a = 2, c = −3 b e = 4, g = − 4

c x = 1, y = −3 d n = 1 , p = −2

e x = 3, y = 2 f w = −1, y = −2g x = 5, y = −10 h x = 5, y = −5

i c = 1 , e = 1

2 a x = 6, y = 1 b a = 3, c = 2c h = −2, p = 3 d x = − 4, y = 6e x = 3, w = 6.5 f p = 1, w = 5

g x = 3, y = −3 h x = −3, y =

i x = 6, y =

3 a x = 10, y = 3 b a = 1, c = 3c x = 5, w = 5 d a = 3, g = 4e x = 6, y = −7 f x = 2, y = −2g p = 2, q = 0 h m = −1, n = 1i x = 4, y = 7 j x = 1, y = − 4k a = 2, c = −8 l h = −2, d = −2

Skillbank 4B3 a 13 b 2460 c 0.13 d 24.6

e 24.6 f 1300 g 1.3 h 2460i 2.46 j 13 k 130 l 246

6 a 14 076 b 1407.6 c 140.76d 1407.6 e 140.76 f 140.76g 140 760 h 14.076 i 1407.6j 14 076 k 14 076 l 1.4076

Exercise 4-101 120 women 2 280 children3 b 110 adults4 210 children 5 22 inkjet, 38 laser6 10 400 adults, 4600 children7 4 videos 8 11 Supreme pizzas9 Jenni is 19 years old

10 Adam is 12 years old11 a Pie = $2.20 b Hotdog = $1.8012 b 115 × 50-cent coins

Exercise 4-111 a m = ±12 b x = ±20 c y = ±15

d k = ±13 e y = ±1 f w = ±4g x = ±2 h t = ±4 i a = ±4j k = ±6 k w = ±10 l d = ±12

m k = ±1 n w = ±5 o x = ±

p m = ±6 q y = ±1 r p = ±3s k = ±2 t y = ±10 u x = ±9

2 a m = +2 b a = ±9 c m = ±5.29d m = ±1.94 e k = ±0.58 f x = ±7.58g k = ±9.80 h k = ±9.49 i y = ±0.35j w = ±7.07 k a = ±9.24 l y = ±6.20

3 Because the square of a positive number or a negative number is always a positive number.

4 a, c and f

5 a x = 1 or x = −3 b a = 0 or a = −5

c m = −5 or m = 6 d k = or k = −1

e p = −7.5 or p = 4.5 f x = 21 or x = −19

g m = −2 or m = 14 h y = or y = 2

i x = −32 or x = 33

Exercise 4-121 a m = −7 or −3 b d = 3 or 7

c y = −5 or 3 d k = 0 or 3e t = −7 or 0 f p = 0 or 3

g w = 0 or h n = − or 3

i a = or j x = − or −1

k e = l f =

m c = − or − n h = −1 or

o e = or 1

2 a m = 0 or −2 b y = 0 or 3c f = −5 or 5 d p = 4 or −4e x = 3 or −3 f g = −1 or −2g t = −3 or 6 h u = −2 or −24i n = 7 j w = −11 or 6k p = 4 or 6 l k = 3 or 4

m d = 6 or −3 n y = 5 or −3o k = −9 or 3 p a = 3 or −2q c = −5 or 3 r r = 11 or −3s y = 2 t d = −6u m = 1

3 a k = −3 or − b g = −1 or −1

c d = −1 or d t = −2 or −1

e m = or f y = −1 or 3

g x = or −2 h a = 2

i u = or 3 j q = −4 or 1

k w = −1 or 1 l c = −4 or 3

4 a x = −2 or 3 b t = −2 or

c u = − or −5 d m = or 1

e p = −4 or 7 f e = −1 or 5

g t = or 5 h d = − or

i h = ±5 j f = 0 or

k w = or 3 l a = −2 or

5 8

Exercise 4-131 a x2 + 2x + 1 = (x + 1)2

b p2 − 6p + 9 = (p − 3)2

c m2 − 8m + 16 = (m − 4)2

d k2 + 4k + 4 = (k + 2)2

e y2 + 7y + = (3 + )2

f w2 − 3w + = (w − )2

g x2 + x + = (x + )2

h h2 − 5h + = (h − )2

i a2 + a + = (a + )2

j v2 − v + = (v − )2

2 a −3 + , −3 −

b 5 + , 5 −

c −1 + , −1 −

d 1 + , 1 −

e + , − or ,

f ,

g ,

h ,

i ,

j ,

k 2 + , 2 −

l ,

3 a h = −1 ± b r = 1 ±

c m = −3 ± d w = 2 ±

e a = 5 + f x =

g p = h c =

i f = j y =

k x = l e =

m k = n u =

o b = −2 ±

4 a x = −0.88 or −5.12 b m = 4.40 or −0.40c g = 0.72 or −1.39 d h = 1.27 or −2.77e w = −1.27 or −0.47 f y = 1.14 or −1.47g p = −2 or 1.33 h e = 1.13 or −0.88i n = 1 or −2.5

Exercise 4-141 a x = 36, −3 b n = −2 , 1

c k = , 1 d p = , −2

e y = , 2 f x = −3 ±

g a = h m =

i c = j n =

34--- 3

4--- 2

3--- 1

3---

13---

12---

12---

12---

12---

12---

23---

23--- 1

3---

23---

23--- 1

2---

12--- 3

5--- 1

3--- 1

2---

52--- 1

2---

13--- 1

4--- 1

2---

57---

25--- 1

2---

−23----- 1

5---

56--- −2

3----- 1

4---

13--- 1

2--- 1

2---

−45----- 1

3---

25---

12--- 1

2--- 1

2---

18--- 1

7---

32--- 7

3--- 1

2---

12---

16--- 1

3---

494

------ 72---

94--- 3

2---

14--- 1

2---

254

------ 52---

72--- 49

16------ 7

4---

53--- 25

36------ 5

6---

7 7

5 5

10 10

2 2

12--- 5 1

2--- 5 1 2 5+

2-------------------- 1 2 5–

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

−2 3 3+3

------------------------- −2 3 3–3

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

−2 42+2

------------------------ −2 42–2

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

6 82+2

-------------------- 6 82–2

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

−2 7+3

--------------------- −2 7–3

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

3 71+2

-------------------- 3 71–2

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

5 5

3 4 2+4

-------------------- 3 4 4–4

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

6 2

7 3

30 7 61±2

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

−1 21±2

----------------------- 9 73±2

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

−5 17±2

----------------------- 3 17±2

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

3 5±2

----------------- −5 17±2

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

−7 61±2

----------------------- 1 21±2

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

2

12---

56--- −2

3-----

−15----- 7

1 17±4

-------------------- −1 22±3

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

3 41±8

-------------------- 3 21±6

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


k d = -2 ± l h =

m g = n w =

o v = p

2 a 8.89, 0.11 b 1.41, −1.41c 5.37, 0.37 d 3.19, 0.31e 0.85, −2.35 f 0.30, −1.13g 2.39, 0.28 h 3.83, −1.83i 1.62, −0.62 j 4, 9k 1.48, −1.48 l 2.31, 0.69m 8.09, −3.09 n 4.27, 7.73o 3.31, −1.81

Exercise 4-151 a −5, −9 b 20, −7 c 5 ±

d −2 ± e −4, 2 f no solution

g 2, −2 h i

j k 3 , −5 l

m 1 , −2 n o

p 3.5 q r no solution

2 0, 1 or 2 solutions

Exercise 4-161 a 58 m × 38 m b 6.5 m × 13 m

c 17 cm × 12 cm d x = 62 a 24 and 25 b 42 and 44

c 15 and 273 a 1275 b 3240 c 1965

d 36 e 444 15 and 175 a 1800 m b 1080 m

c 19 s (approximately) d 13.4 s6 a $9200 b $5550

c 471 (truncate)

Exercise 4-171 a 16a2p2 + 7 b 2at − 4a2t2

c 11 + 6h + h2 d 3k2 − 6k − 4

e u2 − 3u + 4 f k2 − 4k + 5

2 a −1, b − , 1 c − , −1

d , 2 e −1 , 1

3 a x = ±2 or ±3 b x = ±2 or ±2

c m = ± or ±2 d y = ± or ±3

e k = ± or ±2

4 a y = ±3 b m = ±4c no solutions

5 a m = 1 or 2 b m = −1 or 10c x = ±2.8 or ±1.2 d x = ±0.9 or ±1.4e x = −0.894 or 1.26

Exercise 4-181 a x = 3, y = 9 and x = 2, y = 4

b x = 0, y = 0 and x = 4, y = 32c x = 4, y = 26 and x = −2, y = 14

d x = −1, y = 5 and x = 1 , y = 7

e x = −2, y = 18 and x = , y = 11

f x = −2, y = −22 and x = 5, y = 20g x = 7, y = 39 h no solution

i no solution j

Power plus1 a x = b x =

c x = −8 d x = ±2 x � 10 3 Teacher to check.4 p = 4, q = 3 5 b x = 1

6 Intersects twice

Chapter 4 review1 a w = −3 b y = −2 c p = 0

d m = 2.5

2 a m = 4 b w = 13 c m =

d m = 8

3 a y = b m = 1

4 a 92, 93, 94, 95b Grace is 13, Jane is 16

5 a i 160 mm3 ii 300 m2

iii 1.5 cmb 120 m

6 a = − 6 or a =

7 a y � 5

b y � −5

c y � −1

d m � −2

8

∴ x = −2, y = 69 a x = −3, y = −11 b x = 3, y = 3

10 a x = 2, y = 11 b x = −4, y = 12c x = 1, y = 1 d x = −1, y = 1

11 a m = 5, c = −9.5 b x = −5, y = −2c x = 9, y = −4 d k = −3, w = −10

12 a 900 childrenb 17 DVDs and 13 videos

13 a y = ±11 b m = ±5c m = 6 or −2 d y = 1.5 or -4.5

14 a m = −2 or 2 b x = 0 or 10c p = ±6 d m = -6 or 8

e m = − or 2 f x = −3 or 5

15 a y = -4 ± b x = 3 ±

c y = d k =

16 a m = −5.5 or 0.54 b y = −1.2 or 4.7c k = 2.8 or −2.5 d m = 1.8 or −2.2e y = 2.0 or 4.0 f m = −1.9 or 2.4

17 a 32 and 34 b 14 m × 22 m

18 a y = or b m = ± or ±2

c y = ±219 a x = 4, y = −20 and x = −3, y = 29

b x = 0, y = 0 and x = 3, y = 6

Chapter 5Start up

1 a m = 63 (vertically opposite angles)b x = 51 (angles on a straight line)c y = 57 (angles at a point)d k = 318 (angles at a point)e w = 89 (vertically opposite angles)f a = 19 (angles at a point)

2 a d = 127 (alternate angles)b 4m = 88 (corresponding angles)

m = 22c 2h + 66 = 180 (co-interior angles)

2h = 114h = 57

d a = 72 (alternate angles, corresponding angles)

e k = 104 (vertically opposite angles)m = 86 (angles on a straight line, corresponding angles)

f p = 81 (corresponding angles)w = 99 (angles on a straight lines, alternate angles)

3 a h = 46 (angle sum of a triangle)b m + 35 = 124 (exterior angle of a

triangle)m = 89

c 10h = 180 (angle sum of a triangle)h = 18

d 5y + 137 + 83 = 360 (angle sum of a quadrilateral)5y = 140y = 28

e 3k + 105 = 180 (co-interior angles)3k = 75k = 25

f a + 27 + 197 + 34 = 360° (angle sum of a quadrilateral)a = 102

4 a No, corresponding angles are not equal.b Yes, co-interior angles have a sum of 180°.c No, alternate angles (or corresponding

angles) are not equal.

Exercise 5-011 a y = 50 (angle sum of an isosceles

triangle)b m = 108 (angle sum of an isosceles

triangle)c h = 120 (exterior angle of an equilateral

triangle)d x = 45 (angle sum of a right-angled

isosceles triangle)e a = 135 (exterior angle of a right-angled

isosceles triangle)f 5d = 180 (angle sum of an isosceles

triangle)d = 36

g y + 70 + 70 = 180 (angle sum of an isosceles triangle)y = 40

5 9 17±4

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

−1 6±5

-------------------- 1 7±3

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

−2 14±5

----------------------- 2 7±3

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

7

11

−1 97±6

----------------------- 2 2 10±3

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

3 37±2

-------------------- 12--- -3 11±

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

13--- 4 46±

3-------------------- 5 3 5±

4--------------------

−7 33±4

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

13--- 4

5--- 2

5--- 1

2--- 1

2---

45--- 1

3--- 1

2---

2

1

3------- 1

2---

2

5-------

15--- 1

5---

13---

3 41±2

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

1113------ 1

3---

k2 2y–

35--- 1

11------ −1

9-----

13---

57--- 1

2---

2 Ah

------- 2 A bh–h

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

0 1 2 3 4 5

−5 −4 −3 −2 −1 0

−2 −1 0 1 2 315---

−3 −2 −1

x −3 −2 −1 0 1

y 7 6 5 4 3

x −3 −2 −1 0 1

y 4 6 8 10 12

53---

26 29

−3 3 3±2

----------------------- 6 6±3

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

−12----- 1

2--- 2

ANSWERS 621

h 2p = 130 (exterior angle of an isosceles triangle)p = 65

i h = 84 (exterior angle of an isosceles triangle)

2 a y = 53 (rhombus, co-interior angles)b k = 27 (opposite angles of a

parallelogram are equal)c w = 35 (angles in a square)d h = 109 (co-interior angles in a

trapezium)e f = 25 (angle sum of a right-angled

triangle in a rhombus)f a = 46 (opposite angles of a

parallelogram are equal, angles on a straight line)

3 a t = 70 (angle sum of an isosceles triangle)

b m = 58 (alternate angles)y = 61 (angle sum of an isosceles triangle)

c d + 74 = 120 (exterior angle equals sum of interior opposite angles)d = 46

d m = 115 (corresponding angles)k = 115 (vertically opposite angles)w = 65 (co-interior angles)h = 65 (corresponding angles)

e c = 94 (angle sum of isosceles triangle, and vertically opposite angles)

f w = 24 (alternate angles and angle sum of an isosceles triangle)a = 102 (angle sum of straight line or alternate angles)

g ∠DXA = 90° (diagonals of a rhombus meet at right angles)∴ p = 56 (angle sum of ∆DXA)

h h = 70 (corresponding angles, angle sum of a triangle)

i w = 93 (isosceles triangle, and angle sum of a quadrilateral)

4 a q = 70 (alternate angles)p = 70 (angle sum of a straight line)w = 40 (angle sum of ∆XYZ)

b ∆XYZ is isosceles since p = q = 70.5 a T b T c F d F e T

f F g T h T i F j Fk T l T m F n T

6 ∠LMK = ∠LKM = 45° (angle sum of a right-angled isosceles triangle)∠PMN = 135° (angles on a straight line)2x + 135 = 180 (angle sum of isosceles triangle MNP)∴ x = 22.5

7 Let ∠XYP = x°, ∠TWP = y°∠PYW = x° (YP bisects ∠XWY)and ∠PWY = y° (WP bisects ∠TWY)2x + 2y = 180 (co-interior angles, YX || WT)∴ x + y = 90But x° + y° + ∠YPW = 180° (angle sum of ∆YPW)90° + ∠YPW = 180°∴ ∠YPW = 90°

8 ∠DBF = ∠DCE (corresponding angles, BF || CE)∠BFD = ∠DEC (corresponding angles, BF || CE)But ∠DBF = ∠BFD (equal angles of isosceles ∆BDF)∴ ∠DCE = ∠DEC

∴ ∆CDE is an isosceles triangle (two angles equal)

9 ∠NKL + 93° = 147° (exterior angle of ∆NKL)∠NKL = 54∠NKH = 54 (NK bisects ∠NHK)∠HKL = 108°∠KHL + 108° = 147 (exterior angle of ∆HKL)∴ ∠KHL = 39°

10 a ∠CED = ∠CDE = 42° (equal angles in isosceles ∆CDE)∠DCE = 96 (angle sum of ∆CDE)∴ k = 96

b ∠BCE = ∠CED = 42° (alternate angles, CB || DE)∠BEC = 42° (equal angles in isosceles ∆BCE)∠ABE = ∠BCE + ∠BEC (exterior angle of ∆BCE)= 84°∴ m = 84

11 a y° = 60 (angle in equilateral triangle)x = 120 (co-interior angle in a rhombus)

b ∠ACB = 60° (angle in equilateral triangle)x + x = 60 (exterior angle in isosceles triangle)∴ x = 30∠B = 60° (angle in equivalent triangle)∴ y = 90 (angle sum of ∆ABD)

c 2x + 40 = 180 (angle sum of isosceles triangle)∴ x = 70y + y = 40 (exterior angle of isosceles triangle)∴ y = 20

d ∠HJK = ∠JHK = 36 (equal angles in isosceles triangle)∠IJK = 72° (co-interior angles on parallel lines IJ, HK)∴ x = 72 − 36 = 36∴ y = 180 − x − x (equal angles in isosceles triangle)y = 108°

12 ∠CDB = ∠CBD (equal angles of isosceles ∆CDB)∠EAB = ∠CBD (corresponding angles, AE || BD)∠EBD = ∠CDB (alternate angles, BE||CD)and ∠EBD = ∠AEB (alternate angles, AE || BD)∴ ∠CBD = ∠AEBBut ∠CDB = ∠CBD = ∠EAB∴ ∠AEB = ∠EAB∴ ∆ABE is an isosceles triangle (two equal angles)

13 ∠LMN = ∠LNM (equal angles of isosceles ∆LMN)∠KLN = ∠LMN + ∠LNM (exterior angle of triangle)= 2 × ∠LMN

∠KLP = × (2 × ∠KMN)

(LP bisects ∠KLN)∠KLP = ∠LMN∴ LP || MN (corresponding angles are equal)

14 Let ∠B = ∠A = x° (equal angles of isosceles ∆ABC)

∠ACB = 180 − 2x° (angle sum of ∆ABC)∠DCE = 180 − 2x° (vertically opposite angles)

∠D = ∠E = [180 − (180 − 2x)] (angle

sum of isosceles ∆DCE)∠D = ∠E = x°∠B = ∠D∴ AB || DE (alternate angles are equal)

15 ∠YUX = ∠UYX = ∠UXY = 60° (angles in equilateral ∆UXY)∠UXW = 120° (angles on a straight line)∠XWU + ∠WUX + 120° = 180° (angle sum of ∆WXU)∠XUW = ∠XWU = 30° (∆WXU is isosceles)∴ ∠WUY = ∠XUW + ∠YUX= 30° + 60°= 90°

16 ∠WTP = ∠P and ∠YTQ = ∠Q (alternate angles, WY || PQ)∴ Angle sum ∆PQT = ∠P + ∠PTQ + ∠Q= ∠WTP + ∠PTQ + ∠YTQ (angles on a straight line)= 180°

17 ∠BAD + ∠DAH + ∠ BAC + ∠CAF = 180° (angles on a straight line)But ∠BAD = ∠DAH (AD bisects ∠HAB)and ∠BAC = ∠CAF (AC bisects ∠FAB)2 ∠BAD + 2 ∠BAC = 180°∠BAD + ∠BAC = 90°∴ ∠CAD = 90°

Exercise 5-021 a 3240° b 1440° c 2340°

d 3960° e 5040° f 1800°2 a 28 b 12 c 7 d 40 e 14 f 93 a 108° b 140° c 150°

4 a 168° b 172° c 128 ° d 174°

5 a 18 b 24 c 8 d 45 e 126 a 60° b 30° c 12°7 a 6 b 24 c 45 d 8 e 158 24

Skillbank 5A2 a 70% b 66% c 45% d 88%

e 75% f 75% g 80% h 80%i 55% j 35% k 30% l 80%m 90% n 45% o 52%

4 a 25% b 68% c 17%d 60% e 10% f 33.3%g 59% h 70.2% i 84%j 70% k 42.8% l 5.5%

m 91% n 78.25% o 31.4%

Exercise 5-031 a No b Yes, SAS c No

d Yes, SSS e No f Yes, RHSg Yes, AAS h Yes, SAS i Yes, AAS

2 a d = 31 b k = 25c y = 12, w = 25 d a = 33, p = 9e p = 109 f k = 11

3 a AB = CB (given)EB = CB (given)∠ABE = ∠CBD (vertically opposite angles)∴ ∆ABE ≡ ∆CBD (SAS)

12---

12---

47---


b QW = PT (given)TW is common.∠QTW = ∠PWT = 90° (given)∴ ∆QTW ≡ ∆PWT (RHS)

c ∠HCD = ∠GEF (corresponding angles, CH || EG)∠HDC = ∠GHE (corresponding angles, DH || FG)CH = EG (given)∴ ∆CDH ≡ ∆EFG (AAS)

d UX = WX (given)XY is common.∠UXY = ∠WXY (YX bisects ∠UXW)∴ ∆UXY ≡ ∆WXY (SAS)

e AB = CB (equal sides of a square)AY = CX (given)∠A = ∠C = 90° (angles in a square)∴ ∆ABY ≡ ∆CBX (SAS)

f LM = NP (given)LP = NM (given)PM is common.∴ ∆LMP ≡ ∆NPM (SSS)

g OA = OC (equal radii of small circle)OB = OD (equal radii of large circle)∠AOB = ∠COD (vertically opposite angles)∴ ∆AOB ≡ ∆COD (SAS)

h FH = FG (given)∠F is common.∠HNF = ∠GMF = 90° (HN ⊥ FG, GM ⊥ FH)∴ ∆FHN ≡ ∆FGM (AAS)

4 a i PQ = PR (∆PQR is isosceles)QA = RB (given)∠Q = ∠R (equal angles of an isosceles triangle)∴ ∆PQA ≡ ∆PRB (SAS)

ii PA = PB (matching sides of congruent triangles)∴ ∆PAB is isosceles (two sides of the triangle are equal)

b i TP = XP (given)AP = CP (given)∠TPA = ∠XPC (vertically opposite angles)∴ ∆TAP ≡ ∆XCP (SAS)

ii ∠T = ∠X (matching angles of congruent triangles)∴ TA || XC (alternate angles proved equal)

c i ∠ADB = ∠CBD (alternate angles, AD||CB)∠ABD = ∠CDB (alternate angles, AB ||| CD)BD is common.∴ ∆ABD ≡ ∆CDB (AAS)

ii ∴ AD = CB and AB = CD (matching sides of congruent triangles)

d i AD = AC (equal radii of circle, centre A)BD = BC (equal radii of circle, centre B)AB is common.∴ ∆ADB ≡ ∆ACB (SSS)

ii ∠DAC = ∠CAB (matching angles of congruent triangles)∴ AB bisects ∠DAC

e i ∠HEF = ∠GFE (given)EH = FG (given)EF is common.

∴ ∆HEF ≡ ∆GFE (SAS)ii ∠EHF = ∠FGE (matching angles of

congruent triangles)f i OT = ON (equal radii)

OL = OM (equal radii)LT = MN (given)∴ ∆LOT ≡ ∆MON (SSS)

ii ∠LOT = ∠MON (matching angles of congruent triangles)

g i OC = OE (equal radii)OD is common.∠ODC = ∠ODE = 90° (OD⊥CE)∴ ∆OCD ≡ ∆OED (RHS)

ii CD = ED (matching sides of congruent triangles)∴ OD bisects CE

h i AB = AD (given)CB = CD (given)AC is common.∴ ∆ABC ≡ ∆ADC (SSS)

ii ∴ ∠BCA = ∠DCA (matching angles of congruent triangles)

iii Since ∠BCA = ∠BCYand ∠DCA = ∠DCY,∠BCY = ∠DCY (∠BCA = ∠DCA, proved in ii)CB = CD (given)CY is common.∴ ∆BCY ≡ ∆DCY (SAS)

iv BY = DY (matching angles of congruent triangles)

Exercise 5-041 a TX = TY (given)

XW = WY (given)TW is common.∴ ∆TXW ≡ ∆TYW (SSS)

b ∠X = ∠Y (matching angles of congruent triangles)

2 a PL = ML (equal sides of a rhombus)PN = MN (equal sides of a rhombus)NL is common.∴ ∆LMN ≡ ∆LPN (SSS)

b MN = ML (equal sides of a rhombus)PN = LP (equal sides of a rhombus)PM is common.∴ ∆PMN ≡ ∆PML (SSS)

c ∠PNL = ∠MNL and ∠PLN = ∠MLN (matching angles of isosceles triangles LMN and NPL)∴ LN bisects ∠PNM and ∠PLM.Similarly, by considering matching angles in ∆PMN and ∆MLP, PM bisects ∠NPL and ∠NML.∴ Angles of a rhombus bisected by the diagonals.

3 a ∠ABD = ∠CDB (alternate angles, AB || CD)∠ADB = ∠CBD (alternate angles, AD || CB)BD is common.∴ ∆ABD ≡ ∆CDB (AAS)

b AB = CD (matching sides of congruent triangles)and AD = CB∴ Opposite sides of a parallelogram are equal.

4 a i AC = BC (equal sides of an equilateral triangle)AX = BX (given)

CX is common.∴ ∆AXC ≡ ∆BXC (SSS)

ii ∠A = ∠B (matching angles of congruent triangles)

b i AB = CB (equal sides of an equilateral triangle)AY = CY (given)BY is common.∴ ∆AYB ≡ ∆CYB (SSS)

ii ∠A = ∠C (matching angles of congruent triangles)

c ∴ ∠A = ∠B = ∠CBut ∠A + ∠B + ∠C = 180°∴ ∠A = ∠B = ∠C = 60°

5 a ∠XWY = ∠VYW (alternate angles, WX || VY)∠XYW = ∠VWY (alternate angles, WV || XY)WY is common.∴ ∆WXY ≡ ∆YVW (AAS)

b ∠WXV = ∠YVX (alternate angles, WX || VY)∠WVX = ∠YXV (alternate angles, WV || XY)XV is common.∴ ∆VWX ≡ ∆XYV (AAS)

c ∠V = ∠X (matching angles in congruent triangles WXY and YVW)∠W = ∠Y (matching angles in congruent triangles VWX and XYV)∴ Opposite angles of a parallelogram are equal.

6 a ∠XDE = ∠XFG (alternate angles, DE || FG)∠XED = ∠XGF (alternate angles, DE || FG)DE = FG (opposite sides of a parallelogram are equal)∴ ∆DEX ≡ ∆FGX (AAS)

b DX = FX and EX = GX (matching sides of congruent triangles)∴ Diagonals of a parallelogram bisect each other (X is the midpoint of both diagonals).

7 a XW = XY (given)∠XTW = ∠XTY = 90° (XT⊥WY)XT is common.∴ ∆WXT ≡ ∆XYT (RHS)

b ∴ WT = YT (matching sides of congruent triangles)

8 a AB = AC (given)AX is common.BX = CX (AX bisects BC)∴ ∆AXB ≡ ∆AXC (SSS)

b ∴ ∠AXB = ∠AXC (matching angles of congruent triangles)

c ∠AXB + ∠AXC = 180° (angles on a line)∠AXB = ∠AXC = 90°∴ AX ⊥ BC

9 a ∠P = ∠R (given)∠TXP = ∠TXR = 90° (TX⊥PR)TX is common.∴ ∆PXT ≡ ∆RXT (AAS)

b TP = TR (matching sides of congruent triangles)∴ Sides opposite the equal angles are equal.

10 a FC = FE (equal sides of a rhombus)FB is common.

ANSWERS 623

∠CFB = ∠EFB (diagonals of a rhombus bisect the angles)∴ ∆CBF ≡ ∆EBF (SAS)

b CB = EB (matching sides of congruent triangles)∴ Diagonal CE is bisected at B.

c Prove ∆FBC ≡ ∆DBC (SAS)FB = DB (matching sides of congruent triangles)∴ Diagonal DF is bisected at B.

d i Matching angles of congruent triangles CBF and EBF.

ii ∠CBF = ∠EBFBut ∠CBF + ∠EBF = 180° (angles on a straight line)∠CBF = ∠EFB = 90°∴ FB ⊥ CEand FD ⊥ CE

e The diagonals of a rhombus bisect each other at right angles.

11 a In ∆LMN and ∆LPN,LM = LP (given)NM = NP (given)LN is common.∴ ∆LMN ≡ ∆LPN (SSS)i ∴ ∠LMN = ∠LPN (matching angles

of congruent triangles)ii ∠MLN = ∠PLN and ∠MNL = ∠PNL

(matching angles of congruent triangles)∴ ∠MLP and ∠MNP are bisected by the diagonal LN.

b LM = LP (given)LT is common.∠MLT = ∠PLT (LN bisects ∠MLP (proved in a)∆LMT ≡ ∆LPT (SAS)MT = PT (matching sides of congruent triangles)∴ Diagonal MP is bisected.Also, ∠LTM = ∠LTP (matching angles of congruent triangles)and ∠LTM + ∠LTP = 180° (angles on a straight line)∠LTM = ∠LTP = 90°LT ⊥ MPand LN ⊥ MP∴ Diagonal MP is bisected at right angles by diagonal LN.

Exercise 5-051 a ∠A = ∠C and ∠B = ∠D

Now ∠A + ∠C + ∠B + ∠D = 360° (angle sum of a quadrilateral)∴ 2∠A + 2∠B = 360° (∠C = ∠A, ∠D = ∠B)∴ ∠A + ∠B = 180°Note: A pair of co-interior angles have a sum of 180°∴ AD || BCAlso, from ∠A + ∠B + ∠C + ∠D = 360°2∠A + 2∠D = 360° (∠C = ∠A, ∠D = ∠B)∠A + ∠D = 180°AB || DC (co-interior angles have a sum of 180°)Opposite sides are parallel.∴ ABCD is a parallelogram.

b Draw the diagonal PM.In ∠LMP and ∆NPM,LM = NP (given)PM is common.∠LMP = ∠NPM (alternate angles, LM || NP)∆LMP ≡ ∆NPM (SAS)∠LPM = ∠NMP (matching angles of congruent triangles)LP || NM (alternate angles proved equal)∴ LMNP is a parallelogram (opposite sides are parallel).

c In ∆LMF and ∆GMH,LM = GM (given)FM = HM (given)∠LMF = ∠GMH (vertically opposite angles)∆LMF ≡ ∆GMH (SAS)∠FLM = ∠HGM (matching angles of congruent triangles)∴ FL || GH (alternate angles proved equal).Similarly, ∠FGM = ∠HLM (matching angles of congruent triangles FGM and HLM)FG || HLOpposite sides are parallel∴ FGHL is a parallelogram.

d Join the diagonal PR.In ∆PQR and ∆RTP,PQ = RT (given)QR = TP (given)PR is common.∆PQR ≡ ∆RTP (SSS)∠QPR = ∠TRP and ∠QRP = ∠TPR (matching angles of congruent triangles)PQ || TR and PT || QR (alternate angles proved equal)∴ PQRT is a parallelogram.However, since the sides of PQRT are equal, PQRT is a rhombus.

e ∆FHC ≡ ∆FHE ≡ ∆DHE ≡ ∆DHC (SAS)FC = FE = DE = DC (matching sides of congruent triangles)Also, ∠CFH = ∠EDH and ∠CDH = ∠EFH (matching angles of congruent triangles)CF || DE and CD || FE (alternate angles equal)∴ CDEF is a rhombus (opposite sides parallel and all sides equal).

f Since WY = XV, and diagonals bisect each other, TW = TV = TY = TXThen ∆TWV ≡ ∆TXY (SAS), and ∆TVY ≡ ∆TWX (SAS)∠VWT = ∠XYT and ∠TVY = ∠TWX (matching angles of congruent triangles)WV || XY and VY || WX (alternate angles equal)∴ WXYV is a parallelogram.Also ∆WXV ≡ ∆YVX ≡ ∆WVY ≡ ∆WXY (AAS)∠W = ∠X = ∠Y = ∠V (matching angles of congruent triangles)Since the angle sum of WXYV = 360°∠W = ∠X = ∠Y = ∠V = 90°.∴ WXYV is a rectangle.

g Since ∠A + ∠D = 180° and ∠A + ∠B = 180°AB || DC and AD || BC (co-interior angles have a sum of 180°)ABCD is a parallelogram with right angles∴ ABCD is a rectangle.

h Since the sides are equal, TWNE is a rhombus (proved in part d).Since ∠M = 90°, TWME is a square (a square is a rhombus with a right angle).

i Since the angles of the quadrilateral are right angles, GHKL is a rectangle (proved in part g).If GL = GH,GL = GH = LK = KH (opposite sides of a rectangle are equal)∴ GHKL is a square (all sides are equal, all angles are 90°).

j The diagonals bisect each other at right angles, so MNPT is a rhombus.∴ MN = NP = PT = MT∆MNT ≡ ∆NPT ≡ ∆PTM ≡ ∆MNP (SSS, since TN = PM)∠M = ∠N = ∠P = ∠T = 90° (angle sum of a quadrilateral and matching angles of congruent triangles)∴ MNPT is a square.

2 a i BX = DY (given)∠B = ∠D (opposite angles of a parallelogram)AB = CD (opposite sides of a parallelogram)∴ ∆ABX ≡ ∆CDY (SAS)

ii AX = CY∴ XC = AY as BX = YD and BC = AD (opposite sides of a parallelogram)∴ AXCY is a parallelogram as pairs of opposite sides are equal

b i AE = EB (given)∠DAE = ∠CEB (corresponding angles, AD || EC)AD = EC (sides of a rhombus)∴ ∆DAE ≡ ∆CEB (SAS)

ii ED = BC (matching sides in congruent triangles above)AE = DC and AE = EB∴ DC = EB∴ BCDE is a parallelogram because opposite pairs of sides are equal

c i AP = CR (given)∠A = ∠C (opposite angles of a parallelogram)AS = CQ (given)∴ ∆APS ≡ ∆CRQ (SAS)∴ PS = RQPB = RD (given AP = CR and opposite sides of a parallelogram)∠B = ∠D (opposite angles of a parallelogram)BQ = DS (given CQ = AS)∴ ∆PBQ ≡ ∆RDS (SAS)∴ PQ = RS

ii PQRS is a parallelogram because pairs of opposite sides are equal

d AC and DB are the diagonals.DO = BO (equal radii small circle)AO = CO (equal radii large circle)∴ ABCD is a parallelogram because the diagonals bisect each other.


e SQ and PR are the diagonals.PO = RO (equal radii small circle)SO = QO (equal radii large circle)PR ⊥ SQ (given)∴ PQRS is a rhombus because the diagonals bisect each other at right angles.

f Since WD = WE = GY = YF, and DZ = ZG = FX = EX, WZ = WX = YX = ZY (by Pythagoras’ theorem)∴ i WXYZ is a parallelogram (opposite sides equal).∴ ii WXYZ is a rhombus (all sides equal).

Skillbank 5B2 a b c d e

f g h i j

k l m n o

4 a b c d e

f g h i j

k l m n o

Exercise 5-061 a Yes b No c Yes d Yes

e Yes f Yes2 a 16.2 b or 6

c 10.5 d 10e 5.625 f or 5

3 a 18 b c 31.5

d 7 e 30 f

g h 16.5

4 a A and C,

b A and B, . Also A and D, 1 . Also B

and D, 2.4.5 15.8 6 6.3 7 21.35 m8 a T b T c T d T

e F f T g T h F

Exercise 5-071 a RHS, b AA c SAS,

d SSS, e SAS, f AA

g SAS h SSS, i RHS,

2 a ∠X is common.∠XLM = ∠XYW (corresponding angles)∴ ∆XLM ||| ∆XYW (AA)

b = =

= =

∴ =

and ∠DCE = ∠BCA (vertically opposite)∴ ∆ABC ||| ∆EDC (SAS)

c ∠P is common.∠PLM = ∠PTW = 90° (given)∴ ∆PLM ||| ∆PTW (AA)

d = =

= =

∴ =

and ∠K = ∠T = 90°∴ ∆KGX ||| ∆TYX (RHS)

e ∠GMK = ∠LHK (given)∠K is common.∴ ∆GMK ||| ∆LHK (AA)

f ∠Y is common.∠WZY = ∠ZXY = 90° (given)∴ ∆WYZ ||| ∆ZYX (AA)

3 a i ∠K is common.∠FGK = ∠MHK (corresponding angles)∴ ∆GKF ||| ∆HKM (AA)

ii m = 6 or 6.86

b i ∠F = ∠C (alternate angles)∠G = ∠D (alternate angles)∴ ∆CDE ||| ∆FGE

ii k = 11.25c i ∠F = ∠E (given)

∠FCB = ∠ECD (vertically opposite angles)∴ ∆BCF ||| ∆DCE (AA)

ii x = 7.5d i ∠A is common.

∠AXM = ∠AYN = 90° (given)∴ ∆AXM ||| ∆AYN (AA)

ii AX = 12e i ∠T is common.

∠TFR = ∠TPD = 90° (given)∴ ∆TFR ||| ∆TPD (AA)

ii TR = 14.4f i ∠E is common.

∠ECF = ∠EBA (corresponding angles, FC || AB)∴ ∆EFC ||| ∆EAB (AA)

ii ∠AEB = ∠FAD (alternate angles, AD || BE)∠B = ∠D (opposite angles of a parallelogram)∴ ∆EAB || ∆AFD (AA)

iii ∠ECF = ∠ADF (alternate angles, AD || EC)∠EFC = ∠AFD (vertically opposite angles)∴ ∆EFC ||| ∆AFD (AA)

iv AB = 14.4 cm4 a ∠T = 90 − ∠P (in ∆PWT)

∴ ∠TWN = 180 − [90 + (90 − ∠P)] (angle sum of ∆TWN)= ∠PIn ∆PWN and ∆WTN,∠WPN = ∠P = ∠TWN (proved)∠PNW = ∠WNT = 90° (WN⊥PT)∴ ∆PWN ||| ∆WTN (AA)

b WN = 6, WP = 2 , WT = 3

5 a ∠FEG = ∠EHF∠F is common.∴ ∆EFG ||| ∆HFE (AA)

b FH = 25 cm

Exercise 5-081 ∠ABC = 180° − (∠CAB + ∠ACB)

(angle sum of ∆ABC)

∠CBD = 180° − ∠ABC (angle on a straight line)∴ ∠CBD = 180° − [180° − (∠CAB + ∠ACB)]= 180° − 180° + (∠CAB + ∠ACB)∴ ∠CBD = ∠CAB + ∠ACB

2 a XA = YA (equal radii of circle, centre A)XB = YB (equal radii of circle, centre B)AB is common.∴ ∆AXB ≡ ∆AYB (SSS)∴ ∠XAC = ∠YAC (matching angles of congruent triangles)

b XA = YA (equal radii)AC is common.∠XAC = ∠YAC (proved in part a)∴ ∆XAC ≡ ∆YAC (SAS)

c ∴ XC = YC (matching sides of congruent triangles)Also, ∠XCA + ∠YCA = 180° (angles on a straight line)and since ∠XCA = ∠YCA (matching angles of congruent triangles)∠XCA = ∠YCA = 90°∴ XY ⊥ AB

3 a ∠P is common.∠PQR = ∠PMN (corresponding angles, QR || MN)∆PQR ||| ∆PMN (AA)

b = (ratio of matching sides)

Since Q is the midpoint of PM. =

=

∴ PR = RN4 a ∠A is common

∠ADC = ∠ACB = 90° (given)∆ADC ||| ∆ACB (AA)

= (matching sides are in the

same ratio)

∴ AC2 = AB × ADb ∠B is common.

∠BDC = ∠BCA = 90° (given)∆BDC ||| ∆BCA (AA)

= (matching sides in the same

ratio)

∴ BC2 = AB × DB

c AC2 + BC2 = AB × AD + AB × DB= AB × (AD + DB)= AB × AB

= AB2

5 In ∆DEG: DE2 = EG2 + DG2

DG2 = DE2 − EG2

In ∆DFG: DG2 = DF2 − GF2

DE2 − EG2 = DF2 - GF2

∴ DE2 + GF2 = DF2 + EG2

6 a no b yes c yes7 a 53 cm b 12 m

c 5 cm d 96 m

8 a In ∆AXB: AB2 = AX2 + BX2 (1)∆AXD: AD2 = AX2 + DX2 (2)∆CXD: CD2 = CX2 + DX2 (3)∆BXC: BC2 = BX2 + CX2 (4)Adding (1) and (3): AB2 + CD2

= AX2 + BX2 + CX2 + DX2

= (AX2 + DX2) + (BX2 + CX2)

= AD2 + BC2 (using (2) and (4))∴ AB2 + CD2 = BC2 + AD2

34--- 7

25------ 3

10------ 7

50------ 3

50------

1720------ 8

25------ 49

100--------- 14

25------ 9

10------

1825------ 13

20------ 1

5--- 6

25------ 53

100---------

1925------ 1

10------ 4

5--- 9

20------ 22

25------

1425------ 3

4--- 31

100--------- 17

25------ 1

20------

35--- 27

50------ 3

50------ 49

100--------- 41

50------

6.6̇ 23---

5.0̇9̇ 111------

7.3̇17--- 22.53̇

10.26̇58---

58--- 1

2---

32--- 3

2---

45--- 8

15------

45--- 3

5---

ACEC-------- 10

7.5------- 4

3---

BCDC--------- 8

6--- 4

3---

ACEC-------- BC

DC---------

KXTX-------- 7.5

18------- 0.416̇

GXYX-------- 12

28.8---------- 0.416̇

KXTX-------- GX

YX--------

67---

13 13

PRPN-------- PQ

PM---------

PQPM--------- 1

2---

PRPN-------- 1

2---

ACAB-------- AD

AC--------

BCAB-------- DB

BC--------

23

ANSWERS 625

9 a Since 52 = 32 + 42,∠D is a right angle.In ∆ABC and ∆EDC:∠B = ∠D = 90°∠ACB = ∠ECD (vertically opposite angles)∴ ∆ABC ||| ∆EDC (AA)

b m = 16 or

Power plus1 Teacher to check proofs.2 X and Y are midpoints of BC and AY.

Medians AX and BY meet at B.Draw CP to T, so that CP = PT.Prove ∆CYP ||| ∆CAT (SAS)∴ YP || AT∴ PB || ATSimilarly, prove PA || BT∴ APBT is a parallelogram∴ W is the midpoint of AB (the diagonals of a parallelogram bisect each other).

Chapter review1 a ∠BCE = 110° (corresponding angles,

BD || CE)∴ c = 110 (corresponding angles, BC || DE)

b 4w = 114 + w (exterior angles of a triangle)∴ w = 38

c 2y + 36 = 180 (angle sum of an isosceles triangle)y = 72

2 a Let ∠PTR = ∠PRT = x (equal angles of an isosceles ∆PRT)∠TRM = 180 − x (angles on a straight line)∠MTR = ∠RMT = [180 − (180 − x)] ÷ 2 (angle sum of isosceles ∆MRT)

∠MTR = x

∠MTR = × ∠PTR

∴ ∠PTR = 2 × ∠MTRb ∠DBA = 120° (exterior angle of

equilateral ∆BCD)∴ ∠ADB = ∠BAD = 30° (angle sum of isosceles ∆ABD)∴ ∠ADC = ∠ADB + ∠BDC= 30° + 60°= 90°∴ AD ⊥ CD

3 a Teacher to check b 724 a SAS b AAS c SAS

5 a No, the angle in the second triangle is not included.

b Yes, RHSc No, sides are not opposite the same

angle.6 a i KM = PN (sides of a square)

MX = NX (sides of an equilateral triangle)∠KMX = ∠PNX = 30° (angle of a square and angle of an equilateral triangle)∴ ∆KMX ≡ ∆PNX (SAS)

ii XK = XP (matching sides of congruent triangles)∴ ∆KPX is isosceles (two equal sides)

b i DF = EG (given)EF = DG (given)DE is common.∴ ∆DEF ≡ ∆EDG (SSS)

ii ∴ ∠DEG = ∠EDF (matching angles of congruent triangles)∴ ∆DEY is isosceles (two equal angles)

iii DY = EY (equal sides of isosceles ∆DEY)and DF = EG (given)By subtraction: YF = YG∴ ∆FGY is isosceles.

iv Let ∠DYE = ∠GYF = x (vertically opposite angles)

∠EDY = (180° − x°) (angle sum of

isosceles ∆DEY)

and ∠GFY = (180° − x°) (angle

sum of isosceles ∆FGY)∠EDY = ∠GFY∴ DE || GF (alternate angles proved equal).

7 a Opposite angles of a parallelogram are equal.

b BC = AD (opposite sides of a parallelogram are equal)BC = DX (given)∴ AD = DX

c ∠DAX = ∠BCY (opposite angles in parallelogram)Since ∆DAX, ∆BCY are isosceles:∠DXA = ∠BYC (equal to ∠DAX, ∠BCY)DX = BY∴ ∆DAX = ∆BCY (AAS)

d AX = CY (matching sides of congruent triangles)AB = CD (opposite sides of parallelogram)AB − AX = CD − CY∴ BX = DYAlso, BY = DX∴ BXDY is a parallelogram (opposite sides are equal).

8

a 2α + 2θ + 2β + 2φ = 360° (angle sum of quadrilateral)α + θ + β + φ = 180° (1)

b In ∆ABD, β + α + 2φ = 180°φ = 180 − (α + β + φ) (2)From (1): θ = 180 − (α + β + θ)∴ θ = φ

c Similarly, α = βABCD is a parallelogram (opposite angles equal)Also, ∆ABC is isosceles (θ = φ)AB = BC

d ∴ ABCD is a rhombus (a parallelogram with two adjacent sides equal).

9 a 12 b 6 c 4.2 d 5.5

10 10.6 11 11

12 In ∆ACD and ∆ABC:

= = 0.4

and = = 0.4

Also ∠DAC = ∠CAB (included angles equal)∴ ∆ACD ||| ∆ABC (SAS)

13 a ∠CDA = ∠BAD = 90° (angles in a square)∠WAB = 90° and ∠YDC = 90° (angles on a straight line)Let ∠AWX = ∠AWB = x°∠WYX = ∠DYC = (90 − x°) (angle sum of ∆WYX)∠YCD = x° (angle sum of ∆CYD)In ∆WBA and ∆CYD:∠WAB = ∠YDC (proved above)∠AWB = ∠YCD (proved)∴ ∆WBA ||| ∆CYD (AA)

b = (matching pairs of sides in

similar triangles)WA × DY = AB × CDBut AB = CD = AD (equal sides of a square)∴ WA × DY = AD × AD

and AD2 = WA × DY

14 a

JK2 = JX2 + KX2

= ( JL)2 + ( KM)2

JK2 = JL2 + KM2

∴ JK2 + KM2 = 4JK2

23--- 16.6̇

C

X

BT

A

Y

P

W

12---

12---

12---

12---

A

B

D Cθθ

φφ

β

αα

β

67---

37---

ADAC-------- 3.2

8------- } (matching

pairs of sides in same ratio)

ACAB-------- 8

20------

WACD--------- AB

DY--------

M L

J K

X

12--- 1

2---

14--- 1

4---


b Prove ∆NMA = ∆PQB (SAS)Hence show AC = BC (∆ABC is isosceles)Prove NC = PC∴ ∆NPC is isosceles.

Chapter 6Start up

1 a $100 b $3.25 c $42d $1425 e $100 000 f $0.38

2 a $2.52 b $13.96 c $81.50d $8.91 e $215.63 f $0.31

3 a 0.07 b 0.11 c 0.2d 0.085 e 0.105 f 0.02g 0.045 h 0.0675 i 0.1225j 0.032

4 a $60 b $189 c $119.10d $58.21 e $4.20 f $179.29

5 a $73.5 b $4060.80 c $963.20d $4795.50 e $564.08 f $1027.04

6 a $837 b $1610 c $18 619d $538.13 e $35.52 f $2229.98

7 a 8.3% b 7.1% c 0.3%8 a 1152 b 83.2 c 600 d 0.069 a i 24 ii 60 iii 120

b i 52 ii 156 iii 26010 a 10 b 20 c 2 d 411 a 596.26 b 2388.10 c 3425.78

d 357.99

Exercise 6-011 a $120 b $1600 c $7500

d $8 e $3645 f $602 a $78.75 b $3750 c $117 000

d $156 0003 a $11 200 b $1575 c $22 800

d $7400 e $9690 f $9943.75g $618 h $70 000

4 a $1.25 b $80 c $53.10d $0.85 e $1.15 f $4.40g $322.75 h $2.05

5 a $106 600 b $429 c $25.70d $131.73 e $640.15 f $3856.44

6 a 2 years b 39 weeks c 137 daysd 5 years e 4.5% p.a.f i $3150 ii 13.125% p.a.

g 11 years 156 days h 2 years

i 1.9% p.a. j 9.75% p.a.

Skillbank 6A2 a 0.75 b 0.7 c 0.6

d 0.07 e 0.45 f 0.98g 0.4 h 0.85 i 0.264j 0.15 k 0.648 l 0.44

m 0.725 n 0.54 o 0.2624 a 0.61 b 0.04 c 0.29

d 0.9 e 0.085 f 0.271g 0.87 h 0.124 i 0.05j 0.1825 k 0.2 l 0.401

m 0.007 n 0.0628 o 0.3005

Exercise 6-021 a $84 b $1284 c $89.88

d $1373.88 e $173.882 a $270 b $282.15 c $6552.15

d $294.84 e $6846.99 f $846.993 a i $563.08 ii $63.08

b i $2823.33 ii $1233.33c i $3846.79 ii $446.79d i $32 756.04 ii $4756.04e i $7956.75 ii $456.75f i $861.13 ii $21.13g i $255 256.31 ii $55 256.31h i $20 528.77 ii $2278.77i i $41 787.70 ii $3337.70j i $11 099.89 ii $2199.89

4 a $47.29 b $223.73 c $90.83d $1314.50 e $1939.39 f $199.78

Exercise 6-031 a $7919.61 b $12 298.74

c $13 433.47 d $1592.04e $4032.59 f $19 233.59g $30 387.65 h $5418.33i $68 135.36 j $185 196.18

2 a $42.31 b $275.56 c $1337.90d $2039.04 e $4311.43 f $10 324.28g $100 902.81 h $25 625.00 i $25 464.10j $217 461.14

3 a $849.27, $49.27b $13 488.50, $3488.50c $52 751.13, $17 251.13d $53 366.91, $11 366.91e $19 473.43, $2973.43f $21 832.76, $9832.76g $4448.55, $948.55h $1901.76, $101.76

4 a $2910.78 b $2934.27 c $2946.20d $2954.22 e $2957.32

Exercise 6-041 a $2148.79 b $9586.40 c $3411.52

d $935.92 e $1616.28 f $12 750.402 a $307.60 b $18.12 c $5597.21

d $104.09 e $8922.06 f $7155.04g $480.45 h $777.97

Skillbank 6B2 a $88 b $800 c 900

d $60 e $24 000 f $800g $140 h 48 i $550 000j 45 k $22 500 l $75, $66

Exercise 6-051 $1589.25 2 $17 977.643 a i $8215.83 ii 60.0%

b i $4416.17 ii 52.2%c i $16 120 ii 58.6%d i $470.21 ii 45.2%e i $403.03 ii 46.3%

4 a i 90% ii 72.9% iii 53.1%iv 47.8%

b 6–7 years5 a i $7120 ii $5696 iii $2916.35

b 32.77%6 a $11 984.47 b $10 666.18

c $8448.687 a i $10 540 ii $8959

iii $6472.88b 8–9 yearsc 23.2%

8 Yes, since the depreciated value is 0.48 (or 48%) of the original value.

Exercise 6-061 $731 2 $409 3 $78 4 $1255 $162.706 a i $299.80 ii $1199.20

iii $139.11 iv $1338.31v $55.76

b i $119.25 ii $675.75iii $54.06 iv $729.81v $28.07

c i $247.50 ii $1402.50iii $561 iv $1963.50v $40.91

d i $1281 ii $2989iii $807.03 iv $3796.03v $105.45

e i $64 ii $576iii $69.12 iv $645.12v $12.41

7 a $217.50 b $1232.50 c $295.80d $1528.30 e $63.68 f $1745.80

8 a $1100 b $1440 c $340 d 30.9%9 a $2080 b $8320 c $13 200

d $4880 e 14.7%10 a $990 b $190 c 23.75% p.a.11 a $2599 b $3576 c $677

d 10.4% p.a.12 $140.45

Exercise 6-071 a $22 080 b $9580 c 19.2%2 a $9120 b $3120 c 26%3 a $33 280 b $8280 c 6.6%4 a i $209.60 ii $12 576 iii $4576

b i $322.20 ii $38 664 iii $20 664c i $1096 ii $263 040 iii $183 040d i $580.31 ii $6963.72 iii $713.72e i $40.96 ii $983.04 iii $183.04

5 a $33.20 b $29 100 c $68 400d $3540

Exercise 6-081 a Teacher to check.

b $2000 + $1800 + $1580 = $5380c $6000 d $620

2 a i $14 700 ii $14 376b $2376 c $2400d $24

3 a 0.005 b $1500c $297 500 d $1487.50e $294 987.50 f $2987.50g $3000

4 a $40 b $3940c i $39.40 ii $3879.40d i $38.79 ii $3818.19e $118.19 f $120 g $1.81

Exercise 6-091 a $10 000 b $6485.13

c i $729.75 ii $47.58d i 16.99% p.a. ii 16.99% p.a.e July 8, 2004f The first $34.90 is a purchase charge, the

second $34.90 is a credit (or refund)2 a i $896.60 ii $44.83

b i $926 ii $46.30c i $320.60 ii $16.03d i $708 ii $35.40e i $1649 ii $82.45f i $250 ii $12.50

12---

ANSWERS 627

g i $3738 ii $186.90h i $711.50 ii $35.58

3 a $10 b $2.57 c $81.25d $8.06 e $0.43 f $10.85g $56.85 h $22.25

Power plus1 4 years, 61 days 2 $4444.443 a $541.57, $41.57

b $12 838.71, $2838.714 $63 367.49 5 $5839.78 6 3 924 8727 a 17.67 years b 17.67 years

c No (when compounded annually it doesn’t double till the end of the 18th year)

Chapter 6 review1 $14402 a $45 b $208.33 c $495.19

d $67.593 6.25% p.a. 4 1.85 years5 a $2249.73 b $9274.196 a $215.87 b $2063.297 a $2497.04 b $391.048 $16 638.94 9 65.61% 10 $440

11 a i $320 ii $88.80b i $176 ii $48.84

12 23.8%13 a $2600 b $3200 c $600 d 9.23%14 a $15 120 b $5120 c 10.2% 15 a $8.04 b $36.50

Mixed Revision 21 a 8 b 1 c −9 d 2

e −1 f −1

2 a m = 2, n = 1 b k = 3, h = −2

c y = 1 d =

3 20 mL4 a x = ±4.5 b m = ±2.2 c a = ±1.75 87, 88, 89 6 $357 a 0, 3 b no solutions c 1, 3

d , −2 e , 5 f −1, 2

g 7, 1 h − , i −2, 6

8 a b c

d

9 a x = 4, y = 16 and x = −1, y = 1

b x = 2 + , y = 10 + 2 ;

and x = 2 − , y = 10 − 2

10 −2 + , −2 − 11 r = ±

12 a 1.9 or −1.4 b −0.4, −2.6

13 a 4w2 − 5w − 1071 = 0b 17 m × 63 m

14 a x = 1, y = 4 b x = 2, y = 1

c x = − , y = −3

15 a x � 1

b x � 6

c x � −3

16 x = ±2, x = ±317 a ii x = 3, y = −3 b ii x = 4, y = 218 a x = 1, y = 2 b x = 1, y = 5

c x = −1, y = 619 a m = 35 b d = 25 c w = −3020 a w = 64 (angle sum of an isosceles

triangle)b m = 42 (opposite angles of a

parallelogram, angles on a straight line)c x = 48 (angles at centre is 90° since the

shape is a rhombus, angle sum of a triangle)

d h = 23 (the shape is a rectangle, angle sum of right-angled triangle)

e a = 108 (equal angles of an isosceles triangle, angles on a straight line)

f y = 52 (angle sum of an isosceles triangle)

21 160°22 a ∠CDW = ∠ABW (alternate angles,

CD || AB)∠DWC = ∠BWA (vertically opposite angles)∴ ∆CDW ||| ∆ABW (AA)

b 4 ≈ 4.7

23 a 12 b 6

24 10.6 25 165°26 a ∠LPM = ∠NMP, ∠PLN = ∠MNL

b Opposite sides of a rectangle are equalc AASd Matching sides of congruent triangles

LPT and MTNe Rectangle, bisect

27 a PT = RQ (given)RT = PQ (given)PR is common.∆PRT ≡ ∆RPQ (SSS)

b ∠PRT = ∠RPQ (matching angles of congruent triangles)TR || QP (alternate angles proved equal in part b)

c PQRT is a parallelogam because it has one pair of opposite sides equal and parallel.

28 CO = DO (equal radii of circle)∠CEO = ∠DFO (CE⊥AB, DF⊥AB)∠COE = ∠DOF (vertically opposite angles)∴ ∆CEO ≡ ∆DFO (AAS)

29 a ∠P is common.∠PTW = ∠PRQ (corresponding angles, TW || RQ)∴ ∆PWT ||| ∆PQR (AA)

b = (matching sides of similar

triangles)But T is the midpoint of PR.

=

=

∴ TW = PQ

30 a m = 12.375 b 10

31 a No b Yes, SAS c No32 a $1400 b $264 c $1.56 d $7.5433 a $7941.60 b $903.86 c $21 216

d $6080 e $206034 a i $20 400 ii $17 340

iii $14 739b 61.4%

35 a $15 900 b $3900 c 6.5% p.a.36 D 37 B 38 45 months39 $232.33 40 $10.1241 a $8.70 b $878.7042 $6843 a $373.50 b $2116.50 c $465.63

d $2582.13 e $107.59 f $2955.66

Chapter 7Start up

1 a i (3, 2) ii (6, 1)iii (−1, − 6) iv (−5, −4)

b i 6 units ii 2 unitsiii 6 units

c 4.2 units, isoscelesd (−2, 2) e (6, 6)

f i ii −

2 a

b

c

d

3 a N b X c P d P e N f X4 a −2 b 8 c −1 d 5 e 64 f 64

5 a − = − b − = − c 8.25

6 y = 4 7 x = −28 a x = 7 b y = −3 c (7, −3)

Exercise 7-011 a i (3 , 1 ) ii iii

b i (8, ) ii iii

c i (−2 , 3 ) ii iii −

d i (2 , −2 ) ii iii

e i ( , −7) ii iii −

f i (2 , 1) ii iii −

g i (−1 , − 6 ) ii iii 1

35--- 1

4---

23--- 1

2---

3941------ −36

41--------

34--- 1

3--- 1

2---

12--- 4

5---

−1 5±2

-------------------- 1 21±5

-------------------- −5 17±4

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

3 3±3

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

10 10

10 10

2 2 πR2 A–π

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

23---

−2 −1 0 1 2

−1 0 1 32 54 7614---

−5 −4 −3 −2 −1 0

57---

67---

TWRQ--------- PT

PR--------

PTPR-------- 1

2---

TWRQ--------- 1

2---

12---

522------

13--- 2

3---

x 0 1 2 3

y −3 −2 −1 0

x −2 −1 0 1

y −4 −1 2 5

x −1 0 1 2

y 3 1 −1 −3

x −2 0 2 4

y −4 −1 2 5

28--- 1

4--- 2

8--- 1

4---

12--- 1

2--- 26 1

5---

12--- 41 5

4---

12--- 1

2--- 34 3

5---

12--- 1

2--- 50 1

7---

12--- 29 2

5---

12--- 85 2

9---

12--- 1

2--- 2


h i (4, −3) ii iii −2

i i (1, 2) ii iii −4

2 a AB = , BC = 1, AC = ; scalene

b XW = , WV = , XV = ;

scalene

c PQ = , QR = , PR = ;

scalene

d MN = 5, NT = , MT = 5; isosceles

e AF = 13, FL = , AL = 13; isosceles

f DX = 10, XP = , DP = 10; isosceles

3 a AB = , BC = , CD = , AD

=

b 25.5 units

c mAB = , mBC = − , mCD = , mAD = −

d A parallelogram, because opposite sides are equal

4 a LM = , MN = , NP = ,

LP =

b mLM = − , mMN = , mNP = − , mLP =

c LN = = MP

d A rectangle, because opposite sides are equal and diagonals are equal

e 68 units2

5 a WX = , XY = , WY =

b i 104 ii 104c They are the same.d A right-angled triangle (using

Pythagoras’ theorem)

6 a The length of each side is units.

b DF = EG =

c A square, because all sides are equal and the diagonals are also equal

7 a The length of each side is units.

b mHJ = , mJK = , mKL = , mHL =

c HK = , JL =

d A rhombus, because all sides are equal and the diagonals are not equal

Skillbank 7A2 a y = 3x + 4 b y = 2x + 10

c b = 5a − 2 d n = 8m − 6e p = 10k − 3 f z = r + 9g h = 4 − 2d h w = 9t − 9

Exercise 7-021 a No b No c Yes d Yes e Yes

f Yes g No h No i Yes j Yes2 b (2, 2)3 They intersect at one point because (1, 2)

lies on each of the three lines.

Exercise 7-031 a − b = c =

d − e =

2 a L b N c N d L e N f P

3 a − b − c

4 a − b c − d −

e − f g − h m

5 a Yes b Yes c A parallelogram

6 a mPQ = mRT = b mPT = mQR = −

c PT = , QP =

d No, because − × ≠ −1

e A parallelogram

Exercise 7-041 a

b

c

d

e

f

g

h

i

j

k

l

20

272

17 20

41 5 52

26 50 16

20

208

80

41 40 41

40

45--- 1

3--- 4

5--- 1

3---

136 34 136

34

53--- 3

5--- 5

3--- 3

5---

170

32 72 104

40

80

58

37--- 7

3--- 3

7--- 7

3---

200 32

15--- 10

3------ 31

3--- 7

5--- 12

5---

23--- 20

7------ 26

7---

65--- 6

5--- 5

6---

16--- 4

5--- 20

17------ 1

k---

ba--- n

3--- 10

21------

211------ 8

3---

73 73

83--- 2

11------

20

4

y

x

y = 4 − 2x

−4 0

2

y

2x = 4y − 8

x

−6 0

6y

x

y − x = 6

40

−6

y

x

3x − 2y = 12

2.50

2.5

y

x

2x + 2y = 5

60

3

y

x

6 − x = 2y

−2 0

4

y

x

y = 4 + 2x

30

5

y

x

5x + 3y − 15 = 0

20

−6

y

x

3x − y = 6

100

−4

y

x

2x − 5y − 20 = 0

20

4

y

x

4x + 2y − 8 = 0

20−0.5

x − 4y − 2 = 0

y

x

ANSWERS 629

Exercise 7-051 a m = 6, b = −5 b m = −3, b = 8

c m = − , b = 9 d m = , b = 6

e m = − , b = 2 f m = 2, b = −

g m = 2, b = −7 h m = − 4, b = 3

i m = , b = 2 j m = , b = −

2 a y = 3x + 7 b y = −5x − 2

c y = + 8 d y =

e y = + 7 f y = − 2

g y = h y = − 4x + 2

i y = − + 1 j m = 4x +

3 a m = , b = 2, y = + 2

b m = − , b = 5, y = − + 5

c m = , b = 4, y = + 4

d m = −1, b = −5, y = −x − 5

4 a y = + 2 b y = − + 3

c y = − 2 d y = −x − 4

5 a C b B c B d C, D e D6 a B, C b A c B, D

d C, D7 a y = 5x + 6 b y = − 4x − 1

c y = − + 4 d y = − + 2

e y = − − 5 f y = − − 2

Exercise 7-061 a i m = 1, b = − 4

ii

b i m = 2, b = 5

ii

c i m = −2, b = 3

ii

d i m = − , b = 1

ii

e i m = − , b = 7

ii

f i m = 6, b = 1ii

g i m = 2, b = − 4

ii

h i m = − , b = − 4

ii

i i m = − , b =

ii

Exercise 7-071 a x + y − 2 = 0 b 3x − y + 2 = 0

c 5x − y + 8 = 0 d x − 2y + 3 = 0e x − 5y + 1 = 0 f x − 2y − 6 = 0g 8x − y + 2 = 0 h 6x − y − 3 = 0i x + y − 10 = 0 j 2x − y + 4 = 0k 8x − 2y − 1 = 0 l x − 2y − 6 = 0

m 3x − 5y + 10 = 0 n x + 2y + 10 = 0o 15x − 3y − 2 = 0 p 3x − 24y − 8 = 0q 4x − 3y − 2 = 0 r 4x − 3y + 15 = 0s 3x − 2y − 12 = 0 t 2x + 5y + 3 = 0

2 a y = −2x − 5, m = −2, b = −5b y = −x − 6, m = −1, b = − 6c y = −2x + 1, m = −2, b = 1

d y = + 2, m = b = 2

e y = 4x − 5, m = 4, b = −5f y = 2x − 6, m = 2, b = − 6

g y = − + 4, m = − b = 4

h y = − , m = b = − (or −3 )

12--- 5

4---

23--- 1

2---

54--- 1

2--- 3

4--- 1

4---

x3--- x 4+

5------------

5x2

------ 4x3

------

3x10------

5x3

------ 34---

23--- 2x

3------

27--- 2x

7------

12--- x

2---

x2--- x

3---

3x2

------

x2--- 4x

3------

x3--- 2x

3------

−2 2

11

4 6 8−4

2

4

6

−2

−8

0

−6

−4

y

y = x − 4

x

−2 2

1

2

4 6−4

2

4

6

−20

−6

−4

8

y

y = 2x + 5

x

−2 2

12

4 6−4

2

4

6

−20

−6

−4

8

y

x

12---

−2 2

−12

4 6−4

2

−20

−4

y

x

23---

−2 2

−23

4 6 8−4

2

0

4

6

8

y

x

−2 2

6

1

4 6 8−4

2

0

4

6

8

y

x

−2 24

2

4

2

−20

−4

−6

−8

4

6

y

x

32---

−2 2

−32

4−20

−4

−6

−8

y

x

54--- 1

2---

−2−4−6 2−5

4

4−2

2

4

6

8

0

−4

−6

−8

y

x

3x2

------ 32--- ,

4x3

------ 43--- ,

3x2

------ 72--- 3

2--- , 7

2--- 1

2---


i y = − − 2, m = − b = −2

j y = + 4, m = b = 4

k y = + , m = b =

l y = 2x − , m = 2, b = −

Exercise 7-081 a 2x − y + 1 = 0 b x + y + 2 = 0

c 4x − y − 20 = 0 d x − 2y − 10 = 0e 2x − 3y − 4 = 0 f x + 5y + 38 = 0g 3x + y − 4 = 0 h 4x + y + 1 = 0i 3x − 4y + 10 = 0 j 2x + y + 10 = 0

2 i and ii

3 a k: x + 2y − 7 = 0, l: 3x − y + 7 = 0b 4x + y − 20 = 0 c 5x − 7y − 30 = 0d 2x − 3y + 18 = 0 e 3x + 5y − 30 = 0

4 a x − y − 4 = 0 b 4x − 5y + 18 = 0c 5x − 6y + 23 = 0 d 8x + 3y − 10 = 0e 3x + 2y − 6 = 0 f 5x − 3y − 1 = 0g 6x + 11y + 38 = 0 h 13x − 10y + 24 = 0i x + y − 3 = 0 f 4x − 3y − 11 = 0

Exercise 7-091 a 2x − y + 4 = 0 b x + 2y − 11 = 0

c 3x + y − 13 = 0 d 2x − y − 13 = 0e 5x + 4y + 37 = 0

2 a 2x + y + 2 = 0 b x + 3y − 4 = 0c 2x − 3y − 19 = 0 d 2x + y − 2 = 0e 2x + 9y − 8 = 0

3 a x + y − 9 = 0 b 3x − 4y + 22 = 0c 2x + y − 11 = 0 d x + 2y − 16 = 0

4 a x − 3y + 3 = 0 b 3x + y − 11 = 05 a 4x + 5y − 40 = 0 b A(10, 0)

c 5x − 4y − 50 = 0 d B(0, −12.5)

6 a i m = − ii (6, 6)

iii 3x − 4y + 6 = 0

b i m = − ii (1, 2)

iii 7x − 3y − 1 = 0c i m = −2 ii (−2, 1)

iii x − 2y + 4 = 0

d i m = − ii (−2, −3)

iii 3x − y + 3 = 0

e i m = − ii (3, −1)

iii 2x − y − 7 = 0

f i m = − ii (2, 1)

iii 8x − y − 15 = 0

7 a x − 4y + 10 = 0 b x + 2y + 4 = 0c x − 4y − 2 = 0 d x − y + 1 = 0e (6, 1)

8 a i x − 2y + 4 = 0 ii (− 4, 0)

iii 2x + y − 7 = 0 iv A(0, 7), B(3 , 0)

b Area ∆AOB = 12.25 units2

Area ∆RQB = 14.25 units2

Area ∆RQB − Area ∆AOB = 2 units2

Skillbank 7B2 a 19 b $7.50 c 87.5

d $20.20 e $3.76 f 40g $0.93 h 89.6 i $270j 9.7c (10c) k $152.76 l $8.26m 31.54 n $1.01 o 42.6p $2431.76

4 a 10 b 124 c $490d $1.72 e 160.4 f $255g 79.4 h $0.76 i 8.8c (9c)j $1.45 k $768 l 64

6 a 100 b $0.60 c 2.5 d $1.35e $1.84 f $4.23 g 4c h 6.6i $0.48 j $6.95 k $0.40 l 42.9

Exercise 7-101 a i 5x + 2y − 18 = 0

ii 3x − 4y − 16 = 0

iii (0, 9) iv (0, − 4) v 26 units2

b i x − 5y + 20 = 0ii x + 2y + 6 = 0

iii (0, 4) iv (0, −3) v 35 units2

c i 3x − y − 46 = 0ii 7x + 15y + 66 = 0

iii (15 , 0) iv (−9 , 0)

v 123 units2

2 a 5x − 2y − 25 = 0 b 5x + 7y − 25 = 0

c 5 d t = −10

3 a DE = EF = FG = DG = 5 unitsb For DE and GF, m = 0

For DG and EF, m = −

c Diagonal DF, m = −

Diagonal EG, m = 2

Since − × 2 = −1 it is true that DF ⊥ EG

d Midpoint of DF = (0, 0)Midpoint of EG = (0, 0)The diagonals bisect each other because their midpoints are the same.

e Opposite sides are equal and parallel, adjacent sides are equal, diagonals bisect each other at right angles

4 a units b units c (−1, −1)

d (−1, −1) e No, since mPR × mQS ≠ −1f Rectangle, diagonals are equal and

bisect each other but not at right angles

5 a CE = units, DF = units

b and

c mCE = , mPR = −

∴ CE ⊥ DF because × − = −1

d Square, diagonals are equal and bisect each other at right angles

6 a BC = DE = units,

CD = BE = units

b mBC = , mCD = − , mDE = , mBE = −

c Midpoint of BD = ,

Midpoint of CE =

d Parallelogram, opposite sides are parallel and equal

7 a AC = BD = units

b Midpoint of AC = (1, 2), midpoint of BD = (1, 2)

c mAC = −5, mBD = , ∴ AC ⊥ BD

d The diagonals are equal and bisect each other at right angles.

8 Midpoint of KM = Midpoint of LN

=

mKM × mLN = 1 × −1 = −19 Teacher to check.

10 a mJK = − , mLM = − , mKL = − , mJM = −

b JK = LM = units, KL = JM

= units∴ JKLM is a parallelogram because opposite sides are parallel.

11 Teacher to check. 12 Trapezium

13 ST = WX = units

TW = SX = units

XS ⊥ ST because mXS = − , mST = 6∴ STWX is a rectangle because opposite sides are equal and angles are right angles.

14 a Midpoint of TU = A(4, −1)Midpoint of UV = B(0, 3)Midpoint of SV = C(−5, −1)Midpoint of ST = D(−1, −5)

b Gradient of AB = −1 = gradient of CD

Gradient of AD = = gradient of BC

AC = 9 units, BD = units∴ ABCD is a parallelogram.

15 a X(3, ) Y(−1 , 3 )

b mXY = − , mCB = −

∴ XY || CB

c XY = , CB =

∴ CB || 2.XY

16 a i mLM = ii mLM = iii mMN =

b L, M and N are collinear points.

Exercise 7-111 a C b A c B d C e D2 a B b A c C d D e C

3 a

x2--- 1

2--- ,

x3--- 1

3--- ,

x10------ 7

10------ 1

10------ , 7

10------

92--- 9

2---

0

y

xP

a

d

b

c

4x + y − 10 = 0

x − y − 5 = 0

x − 5y − 13 = 0

x + 3y + 3 = 0

(3, −2)

43---

37---

13---

12---

18---

12---

13--- 3

7---

1721------

25---

43---

12---

12---

6 5 6 5

130 130

12--- 1

2---,⎝ ⎠

⎛ ⎞ 12--- 1

2---,⎝ ⎠

⎛ ⎞

113

------ 311------

113

------ 311------

61

6556--- 4

7--- 5

6--- 4

7---

112--- 21

2---,⎝ ⎠

⎛ ⎞

112--- 21

2---,⎝ ⎠

⎛ ⎞

104

15---

212---

12---,⎝ ⎠

⎛ ⎞

13--- 1

3--- 5

2--- 5

2---

40

29

37

2 3716---

45---

65

12--- 1

2--- 1

2---

23--- 2

3---

12--- 117 117

13--- 1

3--- 1

3---

0

2

y

x

ANSWERS 631

b

c

d

e

f

g

h

i

4 a

b

c

d

e

f

g

h

i

5 a i y � −x and y �

ii y � −x and y �

b i y = − − 1 ii y � − − 1 and y < 2

c i y = 3 − 3x ii y = x + 3iii A: y > x + 3 and y � 3 − 3x

B: y < x + 3 and y � 3 − 3xC: y < x + 3 and y � 3 − 3xD: y > x + 3 and y � 3 − 3x

d i y = ii p: x = 3, q: x = − 4

iii y � and x � − 4 and x � 3 and

y � 0iv 14 units2

Exercise 7-121 a

0

1

y

x

0 3

y

x

0−4

y

x

0

−2

y

x

0 4−1

y

x

1

5

0

y

x

2

0

y

x

−2

4

0

y

x

0

yy = 2x

x

0

7

7

yy = 7 − x

x

0

1

1

y

y = 1 − x

x

0

2

(1, 6)

yy = 4x + 2

x

0

2

2

yy = 2 − x

x

0(−1, −1)

−4

yy = 3x − 4

x

0

−66

y

x − y = 6

x

0−4

2

y

x = 2y − 4

x

0

−6

12

y

12 = x − 2y

x

x2---

x2---

12--- x x

2---

13 2x–7

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

13 2x–7

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

x −3 −2 −1 0 1 2 3

y 9 4 1 0 1 4 9


b

2 a

b

3 a

b

4 a

b

5 a

b

6 a

b

Power plus1 D(−3, 5)

2 a y = x − 2

b C(9, 4), 4 is the y-coordinate3 a k = 5 b k = −2

4 a − b 3x + 2y + 2 = 0 or y = − − 1

5 B(2, −1)6 x + 5y + 9 = 0, x − y − 1 = 0, 2x + y + 3 = 0.

Point of intersection is

7

Area = (15 + 5)

= 5 × 20

= 100 units2

Chapter 7 review1 a (4, 2) b units c m = 4

2 a T(−1, 6) b Z(5, 10)

c mTZ = d mAC =

e Length of AC = units

= units

f Length of TZ = units

∴ AC || TZ and AC = 2.TZ3 a No b Yes c No

d Yes e Yes f Yes4 a Neither b Parallel

c Perpendicular d Parallele Neither f Parallel

5 a i m = −2 ii m =

b i m = 3 ii m = −

c i m = − ii m = 3

d i m = 2 ii m = −

e i m = − ii m = 4

f i m = ii m = −

6 a

b

−2−4 2 4

2

4

6

8

10

0

y

x

y = x2

x −3 −2 −1 0 1 2 3 4 5

y 15 8 3 0 −1 0 3 8 15

−2−4 2 4 6

2

4

6

8

10

0

12

14

16

y

x

x −2 −1 0 1 2

y −8 −1 0 1 8

−2−4 2 4

2

−2

−4

−6

−8

4

6

8

0

y

x

x −3 −2 −1 0 1 2 3

y 1 2 4 818--- 1

4--- 1

2---

−2−4 2 4

2

4

6

8

0

y

x

x −3 −2 −1 0 1 2 3

y 8 4 2 1 12--- 1

4--- 1

8---

−2−4 2 4

2

4

6

8

0

y

x

x −3 −2 −1 0 1 2 3

y 1 2 1411------ 2

3--- 1

3--- 1

3--- 2

3--- 4

11------

−2−4 2 40

y

x−2

−4

2

4

23---

32--- 3x

2------

− 23--- − 5

3---,⎝ ⎠

⎛ ⎞

−2−4−6−8 2 4 6 80

y

x

x = 4x = −6

−2

−4

−6

−8

2

4

6

8

y = − 3x_2

y = 6 − x_2

102

------

68

23--- 2

3---

208

2 52

52

12---

13---

13---

12---

14---

32--- 2

3---

40

y

x

x − y = 44

−4

160

y

x

x + 2y = 168

ANSWERS 633

c

d

7 a m = −3, b = 5 b m = 6, b = 10

c m = , b = − 4 d m = , b = −2

e m = − , b = f m = 2, b = −5

8 a y = 2x − 4 b y = − + 5

9 a m = −3, b = 5

b m = , b = 3

c m = , b = −3

10 m = − , b = 5

11 a 3x + y − 9 = 0 b 5x − 4y + 4 = 0c 2x + 7y − 3 = 0 d 3x − 2y − 36 = 0e x + 4y − 10 = 0 f x − 6y − 12 = 0

12 a y = + 2, m = , b = 2

b y = + , m = , b =

c , m = − , b = 6

13 a 3x − y − 10 = 0 b 2x − 3y + 26 = 0c x + 5y + 33 = 0 d x − 4y − 10 = 0

14 a 3x − 5y − 20 = 0 b x + y + 3 = 0c 2x − 5y − 16 = 0 d 8x − 5y + 22 = 0

15 a 4x − y − 19 = 0 b 3x − 2y − 18 = 0c x − 2y − 10 = 0 d 5x − 2y − 26 = 0

16 8x + 3y − 95 = 0 17 5x − 4y + 4 = 018 a x + y − 6 = 0 b Q(0, 6)

c 5x − 8y − 30 = 0 d P(0, −3.75)

e 29.25 units2

19 PN = LM = units,

PN = PL = units,

mPN = , mPL = − , ∴ PN ⊥ PL

∴ LMNP is a square because all sides are equal and PN ⊥ PL.

20 A only

21 a

b

c

22 a

b

c

Chapter 8Start up

1 a i 17.6 ii 5.0 iii 70.5b i 65° ii 70° iii 28°

2 a 37.2 m b 164.5 cm c 270.3 mm3 a 29°46′ b 55°7′ c 34°53′

Exercise 8-011 a 64.7 cm b 14.2 cm c 54.5 cm

d 18.5 cm e 5.1 cm f 17.4 cmg 48.8 cm h 59 cm i 17.5 cm

2 a 39° b 56° c 43°3 a 52°57′ b 64°37′ c 45°1′4 a 73° b 5.7 m5 10° 6 65 m 7 60° 8 3.60 m9 a 11.6 m b 11.2 m

40

y

x

3x + 4y − 12 = 0

3

80

y

x

5x − 2y = 40

−20

13--- 2

3---

52--- 1

2---

x2---

−1−2−3−4 1 2 3−1

2

3

4

5

0

1

−2

−3

−4

yy = 5 − 3x

x

−1−2−3 1

1

2

2 3−1

1

2

3

4

0

−2

−3

y

x

y = + 3x_2

12---

32---

−2−4−6 23

2

4 6 8−2

2

4

6

8

0

−4

−6

−8

y 3x − 2y = 6

x

−2−4−6 2

5

−2

4 6 8

2

4

6

8

0

y

y = − x + 5

x

2_5

25---

2x5

------ 25---

3x5

------ 65--- 3

5--- 6

5---

9x2

------ 6+ 92---

34

34

35--- 5

3---

0

6

y

x

0 2

y

x

0 1

3

y

x

0−1.5

3

y

x

0 3

3

y

x

0 3.5−2

y

x1_3


10 33.64 m 11 11 m 12 79°13 480 m 14 48 m15 a 052° b 206° c 300°

d 125° e 238° f 338°16 a 68 km b 015°17 a 58 km b 301°18 2757 km 19 367 km20 a 14.1 m b 53.0 m21 389 m

Exercise 8-021 a 43° b 16° c 87.45°

d 34.8° e 51°43′ f 72°22′2 a 0.6 b 0.75

c cos β = , cos α = , sin β =

d sin F = , sin E = , cos F =

e cos Y = , sin Y = , sin X =

f cos φ = , sin φ = , cos =

3 a 0 b 1 c 1

4 a sin = , cos = , tan = , =

so = tan

b sin = , cos = ,

tan = , =

so = tan

c sin = , cos = ,

tan = ,

= so = tan

5 a tan A = b tan Y = c tan X =

d tan P = e tan Q =

6 a sin X = b cos X = c sin X =

d cos X =

7

8 a 45° b 30° c 30°

9 a 8 b 12 c

10 11 Teacher to check.

Exercise 8-031 a P b P c N d N e P f N2 a 10° b 70° c 50° d 83° e 65° f 12°3 a − 0.89 b 0.95 c −1.17 d − 0.11

e − 0.19 f 0.26 g 0.05 h − 0.60i − 0.43

4 a −cos 38° b −tan 59.5° c sin 79°25′d −tan 22° e −cos 27.3° f −cos 59°35′g sin 85° h −tan 9.2° i −cos 45°j −tan 39°50′ k sin 4.5° l sin 75°

5 a b −1 c d − e 0

f g − h − i 0 j −

6 a 123° b 143° c 110° d 130° e 173°f 135° g 100° h 155° i 114° j 147°k 105° l 118°

7 a 34°51′, 145°9′ b 48°3′, 131°57′c 20°34′, 159°26′ d 6°12′, 173°48′e 27°2′, 152°58′ f 64°9′, 115°51′

8 a 53°8′ b 126°52′c 16°42′ d 163°18′e 53°8′, 126°52′ f 25°23′, 154°37′g 136°39′ h 136°28′i 61°3′, 118°57′ j 69°18′k 41°49′, 138°11′ l 143°8′

9 a 30.5 b 23.1 c 44.3d 11.5 e 0.61 f 69.6

10 Teacher to check. 11 Teacher to check.

Exercise 8-041 a 18.4 b 21.1 c 105.02 a a = 20.51 b b = 11.91 c c = 12.58

d d = 4.10 e e = 30.85 f f = 3.553 a 27° b 37° c 54°4 a 44.5° b 46.6° c 32.0°

d 67.3° e 18.8° f 31.8°5 a 149°7′ b 129° c 142°8′

d 135°29′ e 129°29′ f 162°13′6 a k = 6.1 cm b w = 28.7 m

c p = 8.3 m7 a 46° or 134° b 48°

c 55° or 125° d 44° or 136°e 51°

8 a 75° or 21° b 41° c 84°9 79 m 10 106°31′

11 a 113° b 1042 cm12 a 110° b 131.6 m13 235° or 205°14 d 124.7 m15 b 595 m

Skillbank 8The following are the exact answers. Check

how close your estimates were.2 a 331 b 157 c 1587 d 255

e 421 f 203 g 413 h 734i 6723 j 15 744 k 276l 72.43 (to two decimal places)

4 a 177.4967 b 416.752 c 16.6957d 5.0237 e 38.36 f 4.4066g 5.8065 h 9097.3444 i 8.1998

Exercise 8-051 a 5.6 b 13.1 c 35.82 a a = 8.30 b c = 54.52 c e = 88.41

d b = 16.33 e d = 19.44 f f = 40.723 a 70° b 33° c 109° d 131°4 a 111.8° b 108.0° c 121.2°

d 23.0° e 60.0° f 82.8°5 20.8° 6 64°40′ 7 99° 8 47 km

Exercise 8-061 a 413.4 m2 b 463.1 cm2

c 326.9 mm2 d 132.9 mm2

e 320.4 cm2 f 0.1 m2

2 a 97.4 m2 b 463.6 m2 c 246.2 m2

d 227.6 m2 e 93.5 m2 f 152.2 m2

3 a 46 cm2 b 20 cm2 c 294 cm2

d 321 cm2

4 a 225 m b 2770 m2

5 a 130° b 82 m c 766.7 m2

6 852.7 m2

7 a 1256.6 cm2 b 418.9 cm2

c 173.2 cm2 d 245.7 cm2

8 a 112° b 37 cm2 c 740 cm3

Exercise 8-071 a 10.2 m b 16.1 mm c 17.1 cm

d 13.1 m e 3.9 m f 18.2 m2 a 32° b 142° (or 38°)

c 34° d 55° e 37° f 125°3 105 m4 a 32°–23° = 55° (exterior angle of a

triangle)b 108.45 m c 89 m

5 a 15.4 b 15.4c Results are the same. When using the

sine rule:

=

becomes:

d = (since sin 90° = 1)

which is the same result when using the sine ratio.

6 7.5 km 7 486 km

Power plus1 a x = 12 b y = 16 c g = 10

2 a b

3 a 45° b 60° c 30°d 150° e 120° f 135°

4 a 0 b 1c, d and e Teacher to check.

5 Teacher to check. 6 Teacher to check.7 a 67°1′ or 112°59′ b 30° or 150°

c 20.7° or 159.3°

Chapter 8 Review1 a 10.9 m b 4.4 m c 11.5 cm2 a 64°59′ b 48°59′ c 57°12′3 a 12° b 0.114 km or 114 m4 a 27° b 38°48′ c 61.63°

5 a cos α = b tan A =

6 a 30° b 45° c 60°

7 a 24 b 48 c =

8 a −cos 17°13′ b sin 25.6°c −tan 67.19° d sin 84°43′

9 a 23°19′ b 64°28′c 25°32′ or 154°28′ d 117°2′e 114°27′ f 27°2′ or 152°58′

10 a 0.4 m b 14.8 cm c 136.4 mm11 a 81°54′ b 77°24′ or 102°36′

c 49°37′12 a 6.8 m b 112.1 mm c 7.6 cm13 a 95.7° b 55.9° c 125.1°14 a 165 cm2 b 286 m2 c 30 mm2

513------ 12

13------ 12

13------

4041------ 9

41------ 9

41------

53

------- 23--- 5

3-------

54

------- 114

---------- 114

----------

35--- 4

5--- 3

4---

35---

45---

34---

sincos------------

513------ 12

13------

512------

513------

1213------

512------

sincos------------

5 1–

2 3---------------- 5 1+

2 3-----------------

5 1–

5 1+-----------------

5 1–

2 3----------------

5 1+

2 3-----------------

5 1–

5 1+----------------- sin

cos------------

6091------ 1.3̇ 2

3---

940------ 40

3----------

1161------ 7

25------ 2

3---

1

3-------

2

3 2

35 3 m

12--- 1

2------- 1

2---

32

------- 32

------- 3 1

2-------

d90°sin

------------------ 12.856°sin

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

12.856°sin

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

3 1– 12--- 3 1–( )

6061------ 39

80------

3 48

2------- 24 2

ANSWERS 635

Chapter 9Start up

1 a quantitative, continuousb quantitative, discretec quantitative, continuousd quantitative, continuouse categoricalf quantitative, discreteg categoricalh quantitative, continuousi quantitative, discretej categorical

2 a 34 b 8 c 4 d 44%3 a i 26% ii 7% iii 5%

b i 312 or 313 ii 187 or 188iii 75

4 a

b

c 4 d 115 a 30 b 2 c 27%6 a i 5 ii 5.3 iii 5 iv 5

b i 9 ii 16.4 iii 15 iv 15c i 9 ii 8.7 iii 8.5 iv 7

Exercise 9-011 a i 5 ii 16.125 iii 15.5

iv 15b i 6 ii 3.1 iii 3 iv 2c i 6 ii 6.75 iii 6.5 iv 6, 8d i 1.6 ii 7.6 iii 7.5 iv 7.5

2 a range = 5, mean = 17.325, median = 17, mode = 17

b range = 52, mean = 28.5, median = 29, mode = 28, 29

c range = 6, mean = 5.4, median = 5, mode = 5

d range = 5, mean = 12.4, median = 13, mode = 13

3 a

= 7.42

b i 7 ii 8 iii 8

4 a

b i 36 ii 70.3 iii 70 iv 685 a i 21 ii 74 iii 74.5 iv 76

b 69, 78, 75, 90, 81, 81c i 21 ii 79 iii 79.5 iv 81d Range, no change; mean, median and

mode have increased by 56 a i 26 ii 20.5 iii 17.5 iv 16

b i 26 ii 30.5 iii 27.5 iv 267 a 120 b 154 c 348 85%9 11 or 42

10 a 12, 12; 11, 13; 10, 14; 9, 15b 17, 17c Any three scores who have a sum of 62,

such as 20, 20, 22.11 a 342 b 38

Exercise 9-021 a i no outliers

ii some clustering near 5 to 18iii symmetrical

b i outlier at 24ii clustered near 15 to 18

iii positively skewedc i no outliers

ii clustered near 13, 16iii not symmetrical, not skewed

d i no outliersii clustered near 3, 4 and 7, 8

iii symmetricale i no outliers ii no clustering

iii symmetricalf i no outliers ii clustered near 7

iii negatively skewedg i one outlier at 25

ii clustered near 18, 19iii positively skewed

h i two outliers ii clustered near 4, 6iii symmetrical

i i no outliers ii clustered near 12iii positively skewed

j i one outlier (24)ii clustered near 5, 9

iii symmetrical (if outlier is ignored)k i no outliers ii clustered near 51

iii symmetricall i no outliers

ii clustered near 20, 26iii symmetrical

2 a

b no outliersc i clustered near 69–70 and 72–74

ii not symmetrical, not skewed3 a

b no outliersc i clustered around 2–3

ii positively skewed4 a 26

b Clustering in 30s, 40s and 50sc Skewed towards lower marksd The test was difficult, or most students

did not study.5 a Yes, 7

The opposition team in this game was missing some of its better players, possibly through injury.

b Clustered around 2c Symmetrical (especially without the

outlier). The team is consistent in the number of goals it scores.

6 a No, because results of 3, 2, 1 are not far from the main body of data which begins at 5.

b Clustering occurs around 8–9.c Skewed towards the higher marksd The students practised spelling the

words before the test, or the words were very easy to spell.

Exercise 9-031 a Richard 167, James 45

b Richard 46, James 41.8c Richard 7, James 46.5d James because he is the more consistent

batsman. However, Richard who frequently scores 0, has had two very good scores.

2 a i $95 250 ii $91 500b The median because the value of the

mean has been affected by $170 000 which is much higher than the other salaries.

Length of queue Frequency

1 5

2 4

3 7

4 9

5 7

6 3

7 1

2

0

4

6

8

10

1 2 3 4 5 6 7

Frequency histogramand frequency polygon

Length of queue

Freq

uenc

y

x f xf

4 4 16

5 3 15

6 6 36

7 10 70

8 16 128

9 6 54

10 3 30

11 2 22

50 371

x

Stem Leaf

5 5

6 0 2 2 4 4 7 8 8 8

7 2 2 3 3 4 5 7

8 0 1

9 1

67 68 69 70 71 72 73 74 75

2

0

4

6

8

10

1 2 3 4 5 6 7 8 9

Frequency histogram

Number of children

Freq

uenc

y


3 a i 1.5 ii 0.5 iii 0b A combination of the mean and mode; the average number of

accidents is 1.5 per workplace although many of the workplaces had no accidents.

4 a Toyotab The data is categorical.c The range, mean and median can only be found for numerical data.

5 a Jade: range 39, mean 66, median 69Amy: range 62, mean 64.6, median 73

b Amy because she is more consistent, apart from the score of 23, which could be due to illness. Amy’s mean of 64.6 has been affected by the mark of 23.

6 a i 64 ii 63b i 70–79 (the 70s) ii 60–69 (the 60s)

Exercise 9-041 a

Σf = 40b

2 a

b 17 c 80%

d

e Median = 15

3 a

b

Number of calls f cf

0 5 5

1 6 11

2 6 17

3 6 23

4 9 32

5 6 38

6 1 39

7 1 40

5

0

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7

Cumulative frequency histogramand polygon

Number of calls/minute

Freq

uenc

y

Median = 3

Ages f cf

13 1 1

14 6 7

15 10 17

16 7 24

17 5 29

18 1 30

5

0

10

15

20

25

30

13 14 15 16 17 18


Ages

Cum

ulat

ive

freq

uenc

y

10

0

20

30

40

50

60

70

17 18 19 20 21 22

Cumulative frequency histogramand polygram

Score

Cum

ulat

ive

freq

uenc

y

Median = 20

5

0

10

15

20

25

30

0 1 2 3 4 5

Cumulative frequency histogramand polygram

Score

Cum

ulat

ive

freq

uenc

y

Median = 2.5

ANSWERS 637

4 a

Σf = 45

b

c Median = 8

Exercise 9-051 a

b

c 50–54

d

e 55.1

2 a

b i 145.4 ii 140–144

c

3 a

b Mean = 0.121c Modal class = 0.05–0.09

Number of seedlings f cf

4 4 4

5 2 6

6 4 10

7 8 18

8 10 28

9 9 37

10 4 41

11 3 44

12 1 45

10

0

20

30

40

50

4 5 6 7 8 9 10 11 12


Number of seedlings/punnet

Cum

ulat

ive

freq

uenc

y

Classinterval

Classcentre Frequency fx cf

35–39 37 5 185 5

40–44 42 4 168 9

45–49 47 6 282 15

50–54 52 2 624 27

55–59 57 8 45 35

60–64 62 3 186 38

65–69 67 5 335 43

70–74 72 6 452 49

75–79 77 0 0 49

80–84 82 0 0 49

85–89 87 1 87 50

50 2755

6

4

2

0

8

10

12

37 42 47 52 57 62 67 72 77 82 87

Frequency histogram and polygon

Marks in maths exam

Freq

uenc

y

10

0

20

30

40

50

37 42 47 52 57 62 67 72 77 82 87


Marks (class centres)

Cum

ulat

ive

freq

uenc

y

Median ≈ 54

Heights ofstudents (cm)

Classcentre, x f fx cf

130–134 132 3 396 3

135–139 137 7 959 10

140–144 142 10 1420 20

145–149 147 8 1176 28

150–154 152 6 912 34

155–159 157 4 628 38

160–164 162 2 324 40

Σf = 40 5815

10

0

20

30

40

132 137 142 147 152 157 162

Cumulative frequency histogram and polygon

Heights (class centres)

Cum

ulat

ive

freq

uenc

y

Median ≈ 144.5

Classinterval x f cf fx

0.00–0.04 0.02 10 10 0.2

0.05–0.09 0.07 12 22 0.84

0.10–0.14 0.12 9 31 1.08

0.15–0.19 0.17 9 40 1.53

0.20–0.24 0.22 6 46 1.32

0.25–0.29 0.27 4 50 1.08

Σf = 50 Σfx = 6.05


d

e Median ≈ 0.12f The mean and median are the most useful measures of location.

The mean and median are approximately equal; both are much larger than the mode.

4 a

b 105–109

c

d ≈ 99.9 km/h

5 a

b 100–109

c

d ≈ 99.7

e Both methods of grouping give similar results. Grouping of 70–79, … condenses data more than grouping of 75–79, … and so some information about actual speeds of vehicles may be lost.

Skillbank 92 a $408 b 132 c 200

d $408 e $672 f 81g $517 h $1170 i 525j 364 k $262.50 l $954.60

4 a 330 b $240 c 1600 d $470e $175 f 370 g $51 h $105i 68 j $388 k $108 l $237.50

Exercise 9-061 a Q1 = 4, Q2 = 8, Q3 = 16

b Q1 = 18.5, Q2 = 21, Q3 = 22.5

c Q1 = 6.5, Q2 = 9, Q3 = 13

d Q1 = 55, Q2 = 71, Q3 = 78

2 a 9 b 3 c 15 d 2 e 3 f 22.53 a 9 − 6 = 3 b 50 − 48 = 2 c 6 − 3 = 3

d 24 − 22 = 24 a Range for X is 10, and for Y is 10.

b Median for X is 5, and for Y is 9.c Interquartile range for X is 5 and for Y is 4.d Technician Y’s spread of calls is less, since the interquartile range

is 4 compared with 5 for X.

5 a

b Median speed for A is 73 km/h, and for B is 71 km/h.c Range for A is 30 km/h and for B is 24 km/h.d Interquartile range for A is 15 km/h and for B is 8 km/h.e Suburb B is much safer for driving because the speeds are more

clustered (interquartile range of B < half that of A). Also the median speed is lower, while the range is significantly lower.

10

0

20

30

40

50

0.020.07

0.120.17

0.220.27

Blood alcohol levels (class centres)

Cum

ulat

ive

freq

uenc

y

Classinterval

Classcentre, x

f fx cf

75–79 77 2 154 2

80–84 82 4 328 6

85–89 87 5 435 11

90–94 92 3 276 14

95–99 97 7 679 21

100–104 102 7 810 28

105–109 107 11 1177 40

110–114 112 9 1008 49

115–119 117 1 117 50

Σf = 50 4995

10

0

20

30

40

50

77 82 87 92 97 102 107 112 117


Speed (km/h) (class centres)

Cum

ulat

ive

freq

uenc

y

Median ≈ 102 km/h

x

Classinterval

Classcentre f fx cf

70–79 74.5 2 149 2

80–89 84.5 9 760.5 11

90–99 94.5 10 945 21

100–109 104.5 19 1985.5 40

110–119 114.5 10 1145 50

Σf = 50 4985

10

0

20

30

40

50

74.584.5

94.5104.5

114.5


Speeds (class centres)

Cum

ulat

ive

freq

uenc

y

Median ≈ 102 km/h

x

B A

8 5

9 8 8 7 4 3 3 3 2 0 6 0 0 1 2 3 5 5 7 8 9

9 9 6 5 5 4 4 3 3 2 2 1 1 0 0 0 7 0 0 2 2 3 3 5 5 5 6 6

2 0 0 8 0 2 3 4 5 5 5 8

9 0

ANSWERS 639

6 a

b The range for Day 1 is 36, and for Day 2 is 44.c The interquartile range for Day 1 is 13, and for Day 2 is 20.d Day 2, because the interquartile range and the range are

significantly larger than for Day 1.

7 a

b

c Median ≈ 37Interquartile range ≈ 45–21

= 23

Exercise 9-071 a 7 b 5.5 c 8 d 2.5

e

2 a

b Team 1 is more consistent because the interquartile range (the length of the box) is smaller than that of team 2.

3 a 21 hours b 16 hours c 24 hours d 8 hourse i 75% ii 25%

4 a i 9 ii 10b The median mark for Class 1 is 6.5, and for Class 2 is 5.5.c The interquartile range for Class 1 is 3, and for Class 2 is 4.d Class 1 is more consistent.e i 75% ii less than 50%

5 C 6 C7 a

b The interquartile range for the Bushrangers is 60, and for the Ghosts is 73.

c Bushrangers are more consistent. Their interquartile range is lower and the range of the Bushrangers is 152 compared to 190 for the Ghosts.

Exercise 9-081 a ≈ 6.4, σ ≈ 2.7 b ≈ 23.6, σ ≈ 2.6

2 a ≈ 13.71, σ ≈ 2.12 b ≈ 7.8, σ ≈ 1.83

c ≈ 22.73, σ ≈ 0.90 d ≈ 12.29, σ ≈ 1.19

e ≈ 4.07, σ ≈ 1.33

3 a i 4 ii 4 iii 1.09b i 4 ii 4 iii 1.41c i 4 ii 4 iii 1.63

4 a ≈ 7.0, σ ≈ 2.0 b ≈ 46.4, σ ≈ 13.4 c ≈ 49.3, σ ≈ 1.4

Exercise 9-091 a Men: ≈ 71.4, σ ≈ 6.8 Women: ≈ 77.5, σ ≈ 7.0

b Yes, the mean of women’s pulse rates is much higher, which may be due to the stresses involved in shopping (and looking after children at the same time).

2 a i ≈ 150.1, σ ≈ 9.5 ii ≈ 165.9, σ ≈ 11.2

iii ≈ 158, σ ≈ 13.0

b The boys are much taller than the girls, since their mean is much larger than the girls’ mean. The mean of the class is the average of the girls’ and boys’ means.

3 The first set, since the standard deviation is less than that of the second set.

4 a Team A: ≈ 122.9, σ ≈ 27.0

Team B: ≈ 120.9, σ ≈ 23.6

b Team B is slightly more consistent as its standard deviation is 23.6, compared with 27.0 for Team A.

5 a Vatha: ≈ 13.8, σ ≈ 0.5 Ana: ≈ 14.1, σ ≈ 0.7

b Vatha is more consistent as the standard deviation for her times is significantly lower than the standard deviation for Ana’s times.

6 a Test 1: ≈ 58.3, σ ≈ 14.5 Test 2: ≈ 67.7, σ ≈ 13.3

b The mean for Test 2 is higher than for Test 1 and the standard deviation for Test 2 is lower than the Test 1. This means the results are higher and not as spread, so the class has benefitted from the remedial work.

Exercise 9-101 a i The range for science is 47, and for English is 44%.

ii The interquartile range for science is 22, and for English is 14.iii The standard deviation for science is 14.7, and for English is 10.0.

b Mean mark for science is 63.5, and for English is 65.9.c The interquartile range, range and standard deviation for English

are all lower than those for science. Also the mean is higher for English, so student marks for English are better than those for science.

2 a Male 53, Female 46 b Range: Male 52, Female 45

Day 1 Day 2

0 8

9 8 7 5 5 5 4 2 2 1 1 2 3 4 5 5 6 6 6 7 8 8 9

9 8 7 6 4 2 2 1 1 0 2 2 4 7 8 8

8 8 5 1 0 0 3 1 2 3 5 6 7 7

6 7 0 4 0 2 3 8

5 2

Class interval Frequency Cumulative frequency

0–9 1 1

10–19 8 9

20–29 5 14

30–39 9 23

40–49 11 34

50–59 4 38

60–69 2 40

0–9

10–1

9

20–2

9

30–3

9

40–4

9

50–5

9

60–6

9

5

0

Number of SMS communications

Cumulative frequencyhistogram and polygon

Cum

ulat

ive

freq

uenc

y

10

15

20

25

30

35

40

Med

ian

Q1

Q3

4 5 6 7Number of hours

8 9 10

22 24 26 28 30 32 34 36Number of goals

38 40 42 44 46 48 50 52 54

Team 1

Team 2

40 60 80 100 120 140 160 180 200 220 240 260

Bushrangers

Ghosts

x x

x x

x x

x

x x x

x x

x x

x

x

x

x x

x x


Interquartile range: Male 22, Female 16.5Standard deviation: Male 13.84, Female 10.48

c Males spend more than females at the restaurant. Also females as a group are more consistent in their spending because the range, interquartile range and standard deviation are all significantly lower than those of the males.

3 a A 75.25, B 74.06b A 38, 12.5, 9.64

B 42, 17.5, 11.39c Group A has less spread since the range, interquartile range and

standard deviation are all less than Group B’s.4 a i Team 1 51, Team 2 36 ii Team 1 22, Team 2 23

iii Team 1 14.6, Team 2 11.9b Team 2 appears to be more consistent since its range and standard

deviation are lower than Team 1’s. However Team 1’s range and standard deviation have been affected by the score of 73.

Power plus1 a Maths, since it is 1 standard deviation above the mean while 65 for

English is less than 1 standard deviation above the mean.b Maths, since it is less than 1 standard deviation below the mean

while 58 for English is more than 1 standard deviation below the mean.

c Neither, since they are both 2 standard deviations below the mean.d Neither, since they are both 3 standard deviations above the mean.e English, since it is 4.5 standard deviations above the mean while

maths is only 4 standard deviations above the mean.

2 ≈ 8, σ ≈ 1.63

3 a 12 11 13 15 14 13 10 14 15 b ≈ 13, σ ≈ 1.63

c Mean increases by 5, σ is unchanged.

4 a 14 12 16 20 18 16 10 18 20 b ≈ 16, σ ≈ 3.27

c is doubled and so is σ.

5 a 1.94b i 2.67 ii 2.63

6 a Because it is a measure of the ‘distance’ of a score from the mean.b 0

Chapter 9 Review1 a Range = 10, mean = 17, median = 17.5, mode = 18

b 4, 9 , 9, 9

c 6, 13.45, 13.5, 14d 40, 103.7, 107, 94, 108 and 112

2 Teacher to check.

3

b Centre A: = 23.08, median = 23.5, range = 37

Centre B: = 16.63, median = 15.5, range = 28

c Yes. The mean of the waiting times of Centre B is much lower than that of Centre A. The waiting times for Centre B also have less spread than the waiting times for Centre A.

4 a

b

c 7

5 a

b Modal class 105–109

c

d 107 km/h

e = 105

6 a 6 b 2.5 c 12.57 a X 81, Y 73.5

b X 9, Y 12

x

x

x

x

13---

Centre A Centre B

9 7 5 4 0 5 7 8 8 8 9

8 6 6 5 2 1 1 2 2 3 3 4 5 6 7 7 8

6 5 5 2 1 2 0 1 2 4 5

9 8 8 5 1 0 0 3 0 2 3

1 0 4

x

x

x f cf

5 5 5

6 8 13

7 15 28

8 13 41

9 8 49

10 1 50

10

0

20

30

40

50

5 6 7 8 9 10


Score

Cum

ulat

ive

freq

uenc

yClass

intervalClass

centre x f cf

85–89 87 3 3

90–94 92 2 5

95–99 97 7 12

100–104 102 7 18

105–109 107 15 33

110–114 112 13 27

115–119 117 3 50

50

10

0

20

30

40

50

87 92 97 102 107 112 117


Speeds (km/h)

Cum

ulat

ive

freq

uenc

y

x

ANSWERS 641

c

d Group Y had greater spread since its range and interquartile range were both larger than those for group X.

8 a Girls: ≈ 55, σ ≈ 19.6

Boys: ≈ 55.8, σ ≈ 23.8

b The mean mark for boys was just higher than the mean for girls, but the spread of marks for boys was much greater.

9 a

b

c Box plot, as it shows the difference in spread between men and women and the positions of the medians

d Women: ≈ 41.65, σ ≈ 6.94

Men: ≈ 37.2, σ ≈ 8.97

e Yes. The mean of reaction times of men is much lower than that of women, but the reaction times for men have more spread than the women’s reaction times.

Mixed revision 31 a Neither b Parallel

c Parallel d Perpendicular

2

3 a i m = , b = −2 ii

b i m = −3, b = 1 ii

c i m = − , b = 2 ii

4

5 3x − y − 5 = 0

6 y = − x + , m = − , b =

7 a y = − x + 5 b y = 3x − 2 c y = − x + 6

8 a b

c

9 3 × (−1) − 3 + 6 = 0

10 AB = AC = , BC =

11 All sides have the same length of units, and diagonals AC and

BD both have length units.

12 y = −4x + 3

13 a y = x − 3 b y = − x + 1 c y = 5

14 a i (−1, − ) ii iii

b i (3 , ) ii iii −

c i (−1, ) ii iii −

15 a y = − x + 1 b y = −x − 4 c y = x − 2 d y = 2x + 2

16 C17 a 2x − y − 3 = 0 b 3x + 4y − 6 = 018 6x + 8y + 7 = 019 a x + y − 1 = 0 b 4x − y − 13 = 020 a 2x + 3y − 6 = 0 b 3x − 2y − 5 = 0

62 64 66 68 70 72 74 76 78 80 82 84 86 88

Group X

Group Y

Pulse rates

x

x

Women Men

2 6 8 9 9 9

9 9 8 7 7 6 5 5 4 2 3 0 1 1 2 2 3 7 8

8 5 3 2 0 0 4 3 4 7 8 9

7 4 1 1 5 3 5

25 30 35 40 45 50 55 60

Women

Men

x

x

23---

12---

−2 2 4

2

−20

4

y

x

−2 2 4

2

−4

−20

4

y

x

45---

−2 2 4

2

−4

−20

4

y

x

−2−4−6 2 4 6

2

−4

−20

−6

4

6a

b

c

d

y

x

12--- 3

2--- 1

2--- 3

2---

34--- 1

2---

−2−4 2 4

2

−4

−20

4

y

y = 2x + 1

x −2−4 2 4

2

−4

−20

4

y

2x + y = 3

x

−2−4 2 4

2

−4

−20

4

y

x + 2y − 4 = 0

x

40 32

29

58

12--- 3

4---

12--- 97 9

4---

12--- 1

2--- 34 5

3---

12--- 125 1

2---

12--- 2

5---


c d −2

c 4 square units (≈4.67)

21 a 3x − 2y + 4 = 0 b 5x − 2y − 17 = 022 a 39.3° b 51.9° c 44.4°

23 a b c

d e f 1

24 72 m 25 C 26 50 m 27 13°

28 a −1 b − c −

d e f −

g 1 h 0 i 029 a 20.6° or 159.4° b 118° c 95.4°30 a 5.4 m b 32.5 m c 5.5 m

31 a 41 m2 b 340 mm2

32 a 43°54′ b 95°12′33 52°, 128°34 a 116° b 43° c 41°35 019°36 a i 8 ii 6.7 iii 7 iv 7

b i 5 ii 5.0 iii 5 iv 4c i 6 ii 18.1 iii 18 iv 18d i 39 ii 117.7 iii 119 iv 134e i 4 ii 22.3 iii 22 iv 23

37 a 8 − 5 = 3 b 6 − 4 = 2c 19 − 17 = 2 d 128 − 102.5 = 25.5e 23 − 21.5 = 1.5

38 a 10 b i 6 ii 11.5 c 5.5

39 a A: ≈ 396.7, σ ≈ 34.2

B: ≈ 369.7, σ ≈ 27.8

b Brand A is the longer-lasting brand because the mean for Brand A is nearly 30 hours above that of Brand B.

40 a i No ii Clustered around 3 and 4iii Nearly symmetrical

b i No ii Clustered around 15–17iii Negatively skewed

c i Yes, 125 ii Clustered in 70–80siii Skewed

d i No ii No iii Symmetrical41 a i Range for Runner A is 3.3. Range for Runner B is 0.8.

ii Interquartile range for Runner A is 0.5 and for B is 0.55.iii The standard deviation for Runner A is 1.03 and for B is 0.29.

b A 12.55, B 12.6c Runner B is more consistent because the range and standard

deviation are much lower than for Runner A and the interquartile ranges are nearly the same.

d Although Runner B is more consistent, Runner A is no better. Runner A’s results have been affected by the outlier of 15.2 (which may be an error in timing or recording).

42 a ≈ 55.2, σ ≈ 13.6 b ≈ 50.3, σ ≈ 1.7

43 a

b and c

44 = 7.2, median = 6.5, mode = 2. The median best represents the data

because the mean is affected by the score of 20 and the mode is the lowest score.

45 a

b

c 40–49

2713------ 8

13------,⎝ ⎠

⎛ ⎞ 12---

3552------

32

------- 1

2------- 1

3-------

32

------- 1

2-------

32

------- 1

2-------

1

2------- 3

2------- 1

3-------

x

x

x x

Number of DVDs f cf

0 6 6

1 10 16

2 12 28

3 8 36

4 7 43

5 4 47

6 2 49

7 1 50

Σf = 50

0

10

20

30

40

50

0 1 2 3 4 5 6 7Number of DVDs

Median = 2


Cum

ulat

ive

freq

uenc

y

x

Marks in test x f cf fx

10–19 14.5 1 1 14.5

20–29 24.5 10 11 24.5

30–39 34.5 11 22 379.5

40–49 44.5 20 42 890

50–59 54.5 8 50 436

Σf = 50 Σfx = 1965

0

4

8

12

16

20

10–1

9

20–2

9

30–3

9

40–4

9

50–5

9

Marks in test

Frequency histogramand polygon

Freq

uenc

y

ANSWERS 643

d

e 42 f ≈ 39.8

46 4

Chapter 10Start up

1 a even chanceb likely or unlikely (depending on today’s

weather)c even chance d very unlikelye unlikely f very unlikelyg even chance h likelyi certain j impossible

2 a 1, 2, 3, 4, 5 or 6b Team A wins, Team B wins, or it is a

drawc Player A wins, Player B winsd It lands heads, or it lands tailse HH, HT, TH, TTf 1, 1; 1, 2; 1, 3; 1, 4; 1, 5; 1, 6; 2, 1; 2, 2;

2, 3; 2, 4; 2, 5; 2, 6; 3, 1; 3, 2; 3, 3; 3, 4; 3, 5; 3, 6; 4, 1; 4, 2; 4, 3; 4, 4; 4, 5; 4, 6; 5, 1; 5, 2; 5, 3; 5, 4; 5, 5; 5, 6; 6, 1; 6, 2; 6, 3; 6, 4; 6, 5; 6, 6

3 a A b B c B d B4 a C b C c A d D5 a C b D

Exercise 10-011 a 26 b Yes2 a 5

b No, since the coins are not identical in size.

3 a 5 b c i ii 0 iii d

4 a i 2, 4 ii 1, 1, 3, 5 iii 2, 3, 4, 5iv 1, 1, 2, 3

b i or ii or

5 a {Jack of hearts, Jack of diamonds, Jack of clubs, Jack of spades}

b {4 of clubs, 5 of clubs, 4 of spades, 5 of spades, 4 of hearts, 5 of hearts, 4 of diamonds, 5 of diamonds}

c Jack of hearts, Queen of hearts, King of hearts, Jack of diamonds, Queen of diamonds, King of diamonds}

6 a = b = c =

d =

7 a Drawing a blue marble (or any colour other than green or red)

b 18 a 0 b 1

9 a i ii iii 0

iv = 1

b + = 1

10 a Obtaining an odd numberb Obtaining a tailc Selecting a diamond, club or spaded Selecting a vowele Missing the bullseyef Selecting a red disc

11 a {1, 2, 3, 4, 5} b

c Obtaining a number other than 1

d {2, 3, 4, 5} e f + = 1

12 a i ii 0.01 iii 1%

b = 0.99 or 99%

13 0.714 a 40% b 6015 a Missing the bullseye b 0.4

Skillbank 102 a Mark $100, Jenni $50

b Simon $1200, Sunil $900c Lisa $160, Bree $560d William $500, Adriana $1500e Ed $2700, Pete $1800f Sharanya $900, Asam $2100g Cindy $3000, Carmen $600h Nancy $600, John $1000i Carol $550, Louis $440j Yvette $800, Andre $3200k Arden $2100, Ivan $2800l Tan $2000, Mai $1200

Exercise 10-021 a i ii iii or

b 1c i 102 or 103 ii 52 or 53

2 a 10 b to e Teacher to check.3 a Teacher to check. b 1

c and d Teacher to check.4 a Teacher to check. b 1

c and d Teacher to check.

5 a i = ii =

b 16 a 40

b i ii

iii iv =

v = vi =

c 1

7 a b = c

d

8 a b c

9 a b = c

d e

10 a b = c =

d =

11 a 1000b i 0.17 ii 0.155 iii 0.505

iv 0.486 v 0.845 vi 0.66912 a 0.2525 b 0.2325 c 0.515

d 0.7675

Exercise 10-031 a Each outcome to appear 5 times

b P(R) = P(G) = P(Y) = P(B) =

c to f Teacher to check.2 a to f Teacher to check.

g i

ii

iii Teacher to check.iv A 67 B 222 C 1111v Teacher to check.

3 a 1, 2, 3, 4, 5, 6

b i ii iii

4 a 1, 2, 3, 4, 5, 6, 7, 8

b i ii

c Yes d 3, 4, 5, 6 e or

f i 0.125 ii 0.5 iii 0.375iv 1 v 0.5 vi 0

5 a A, E, I b or 0.3

c i or 0.1 ii or 0.7

0

10

20

30

40

5010

–19

20–2

9

30–3

9

40–4

9

50–5

9

Marks in test


Cum

ulat

ive

freq

uenc

y

x

15--- 1

5--- 1

5--- 2

5---

46--- 2

3--- 2

6--- 1

3---

2652------ 1

2--- 12

52------ 3

13------ 5

52------ 1

13------

852------ 2

13------

37--- 4

7---

66---

36--- 3

6---

15---

45--- 1

5--- 4

5---

1100---------

99100---------

41100--------- 21

100--------- 38

100--------- 19

50------

55100--------- 11

20------ 45

100--------- 9

20------

340------ 11

40------

740------ 6

40------ 3

20------

540------ 1

8--- 8

40------ 1

5---

19100--------- 15

100--------- 3

20------ 57

100---------

97100---------

731------ 10

31------ 12

31------

320------ 2

20------ 1

10------ 1

20------

1120------ 3

20------

1160------ 21

60------ 7

20------ 8

60------ 2

15------

4260------ 7

10------

14---

Dice 1Dice 2 1 2 3 4 5 6

1 0 1 2 3 4 5

2 1 0 1 2 3 4

3 2 1 0 1 2 3

4 3 2 1 0 1 2

5 4 3 2 1 0 1

6 5 4 3 2 1 0

Difference Frequency Probability

0 6 =

1 10 =

2 8 =

3 6 =

4 4 =

5 2 =

Total 36 1

636------ 1

6---

1036------ 5

18------

836------ 2

9---

636------ 1

6---

436------ 1

9---

236------ 1

18------

16--- 1

2--- 5

6---

18--- 1

8---

48--- 1

2---

310------

110------ 7

10------


d + = 1

6 a = b or

7 6 red marbles

8 a b c d

e f g h

9 a b c

10

11 a b c d

e f g 0 h

12 a b c d

13 a b c d

14 a b c d

15 a 0.62 or or or 62%

b 0.07 or or 7%

16 D17 a 1, 2, 3, 4, 5, 6, 6

b i ii iii iv v

18 a b c 0 d

19 C

20 a b 50

21 27 or 28

Exercise 10-041 2

3 a Independent b Dependentc Dependent d Independente Dependent

4 a i ii b i ii

c i ii d i ii

5 Independent, because the outcome for the coin will not affect the outcome for the die.

6 Dependent, snce the number of balls from which the draw is made is decreasing.

Exercise 10-051 BA AB RB MB

BR AR RA MABC AC RC MRBM AM RM MC

2

b 36 c 11

3 a

b

4 a

b

5 a b No

6 i with replacement

a b c d e =

ii without replacement

a = b = c =

d = e =

7

b i = ii

iii = iv =

8

b i = ii =

iii = iv

v = vi

vii viii =

9

a = b =

c = d =

10 a i ii

b i = ii =

Exercise 10-061

310------ 7

10------

26--- 1

3--- 4

6--- 2

3---

14--- 1

2--- 1

13------ 2

13------

113------ 3

13------ 1

52------ 3

13------

511------ 4

11------ 9

11------

37---

13--- 1

2--- 1

2--- 1

3---

310------ 1

5--- 2

5---

13--- 1

3--- 1

6--- 5

6---

111------ 4

11------ 7

11------ 3

11------

12--- 1

2--- 1

5--- 1

10------

62100--------- 31

50------

7100---------

27--- 4

7--- 4

7--- 3

7--- 3

7---

12--- 2

3--- 5

6---

14---

12--- 1

6---

58--- 4

7--- 3

8--- 5

7---

58--- 3

7--- 3

8--- 2

7---

a Red die

1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6,3 6, 4 6, 5 6, 6

Blu

e d

ieB BB

B B BBR BRB BB

B B BBR BRB RB

R B RBR RR

B BBB R BR

B BBB R BR

B RBR B RB

3 334 34

36 367 373 434 44

46 467 473 63

64 646 667 673 73

74 746 767 77

4 343 6 36

7 373 43

4 6 467 473 63

6 4 647 673 73

7 4 746 76

H HHH

T HT

H THT

T TT

1sttoss

2ndtoss

425------ 4

25------ 4

25------ 16

25------ 5

25------ 1

5---

220------ 1

10------ 4

20------ 1

5--- 2

20------ 1

10------

1220------ 3

5--- 4

20------ 1

5---

a 2nd die

1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6,3 6, 4 6, 5 6, 6

1st

die

636------ 1

6--- 11

36------

1636------ 4

9--- 2

36------ 1

18------

a 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

336------ 1

12------ 6

36------ 1

6---

336------ 1

10------ 35

36------

1036------ 5

18------ 7

36------

136------ 15

36------ 5

12------

K spadesK clubs Q clubs

Q spadesK clubs

K spades Q spadesQ clubsK spades

Q clubs K clubsQ spadesK spades

Q spades K clubsQ clubs

212------ 1

6--- 2

12------ 1

6---

412------ 1

3--- 4

12------ 1

3---

925------ 4

25------

620------ 3

10------ 2

20------ 1

10------

PeopleCar J M L S

C CJ CM CL CS

F FJ FM FL FS

M MJ MM ML MS

H HJ HM HL HS

ANSWERS 645

2

3 a

b i ii iii iv v vi

4 a

b 2, 3, 4, 5, 6, 7, 8, 9, 10

c i ii iii iv

v vi vii viii

5

a b c d e f

6 a 50 b 300 c 750

7 a Teacher to check.

b i ii iii iv

8 a 0.277 b 0.54 c 0.4659 a 640 b 260 c i 0.333 ii 0.173 d 0.45

Power plus1 Teacher to check.

2 a b

3 a p3 b (1 − p)p2 = p2 − p3

c (1 − p)3 d 3p2 − 2p2q

4 a ≈ 0.191 b ≈ 0.417 c ≈ 0.139

Chapter 10 review1 a 5 b Yes c i ii iii

2 a 9b No, as there are two 3s, 3 is twice as likely.

c i ii iii iv

3 a Red, yellow, blue, green

b c Yellow, blue, green d e + = 1

4 a 70%5 a 0 is 0.117, 1 is 0.383, 2 is 0.41, 3 is 0.09

b i 0.383 ii 0.41 iii 0.883

6 a b c

7 a b c

8 a b c

9 a b c d e f

10

11 a i ii =

b i = ii =

c i ii =

d i = ii =

12 a b c d

13 a 0.1 or b 0.2 or c 0.2 or

14

a b i ii

1 H12 H2

H3 H34 H45 H56 H61 T12 T2

T3 T34 T45 T56 T6

Coin Die

1st 2nd 3rd Samplecoin coin coin space

H HHHH

T HHTH

H HTHT

T HTTH THH

HT THT

TH TTH

TT TTT

18--- 3

8--- 1

8--- 1

2--- 1

8--- 1

2---

Normaldie

Tetra-hedral die

1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

124------ 1

6--- 1

6--- 1

6---

14--- 1

8--- 23

24------ 23

24------

H HHHHH

T HHHTH

H HHTHT

T HHT TH

H HTHHH

T HTHTT

H HT THT

T HT T TH THHH

HT THHT

HH THTH

TT THT T

TH T THH

HT T THT

TH T T TH

TT T T T T

1stcoin

2ndcoin

3rdcoin

4thcoin

Samplespace

116------ 1

4--- 3

8--- 1

4--- 1

16------ 15

16------

18--- 1

8--- 3

8--- 1

8---

16--- 25

216---------

22115--------- 48

115--------- 16

115---------

15--- 1

5--- 2

5---

19--- 2

9--- 1

9--- 5

9---

14--- 3

4--- 1

4--- 3

4---

23--- 1

3--- 5

6---

712------ 1

4--- 5

6---

25--- 3

5--- 3

5---

152------ 2

13------ 12

13------ 1

26------ 1

13------ 3

4---

19---

2564------ 20

56------ 5

14------

1064------ 5

32------ 10

56------ 5

28------

964------ 6

56------ 3

28------

1064------ 5

32------ 10

56------ 5

28------

14--- 1

6--- 1

12------ 1

6---

110------ 1

5--- 1

5---

B B B B BB

G B B BGB

B B BGBG

G B BGGB

B BGB BB

G BGBGG

B BGGBG

G BGGGB GB B B

BG GB BG

BB GBGB

GG GBGG

GB GGB B

BG GGBG

GB GGGB

GG GGGG

12--- 3

8--- 15

16------


Chapter 11Start up

1 a i −2 ii 13 iii 23 iv −12b i −1 ii 95 iii −3 iv 29

c i 4 ii −2 iii iv −8

d i −2 ii −10 iii 5 iv −10

e i 1 ii iii 81 iv

2 a

b

c

d

3 a m = 2, b = 2, y = 2x + 2b m = −2, b = 2, y = −2x + 2

c m = , b = −1, y = x − 1

Exercise 11-011 a i 160 m/min ii 100 m/min

b i 160 ii 100c The steepness of the graph.

d Zaid 11 min, Nooreen 9 min

2 a No, because the slope of the graph changes.

b During the 3rd hour (from t = 2 to t = 3), 0 km/h

3 a i 12.5 km/h ii 25 km/hb i 12.5 ii −25c i Yes ii No, opposite in signd Car A is moving away from the starting

point (positive gradient) while car B is moving towards the starting point (negative gradient).

4 a The cyclist leaves the starting point, travels at a speed of 20 km/h for 1 h; stops for 1 h; and continues for another hour at a speed of 10 km/h. At D, the

cyclist stops for h, then cycles back

towards the starting point at a speed of 30 km/h for 1 h.

b No, since the gradients of the intervals are all different.

c The cyclist is moving back towards the starting point.

d i 10 km/h ii From C to D.5 a Kate 133 m/min, Colleen 114 m/min

b Kate, in 9 minutes c 1.5 mind 690 m e 110 mf The graph shows the distance they move

down the slope and this increases as more time passes.

Exercise 11-021 a The person starts the journey fast (the

graph is steep), then slows down (the graph becomes less steep) before increasing speed again (the graph becomes steeper).

b The person’s speed is fast initially, then slows down and stops (the graph is horizontal).

c The person starts the journey at a high speed and then gradually slows down to a stop.

2 a H b D c A–B d F e E f C

3

4 a

b

c

d

5 a i C ii B iii A

b i Is the steepest (has the greatest gradient) and must be the fastest (Jade).

ii Is the least steep (the smallest gradient) and must be the slowest (Cameron).

iii The slope of this graph is between the other two (Kiet).

c Jade stopped to talk to a friend. (Other answers possible.)

d This person speeds up slightly and maintains speed for a while, slowing down gradually to a stop.

6 No, because it does not account for variable speeds when leaving and arriving at home.

7 a C b D c E d F e B f A

Skillbank 11A2 a 12:25pm b 1:10am

c 10:50am d 10:55pme 0610 hours f 0010 hoursg 9:10am h 3:15ami 1100 hours j 2305 hoursk 12:20am l 11:35am

4 a 6:05pm b 6:40amc 12:10pm d 2:50ame 1245 hours f 0355 hoursg 10:50pm h 12:15pmi 1545 hours j 0400 hoursk 1:35pm l 7:20am

Exercise 11-031 a i Independent variable is time.

Dependent variable is temperature.ii Teacher to check.

b i Independent variable is time. Dependent variable is height of tide.

ii Teacher to check.c i Independent variable is distance.

Dependent variable is volume of petrol.

ii Teacher to check.d i Independent variable is age.

Dependent variable is height.ii Teacher to check.

2 a i Independent variable is number of persons. Dependent variable is cost.

ii Cost per personiii 1

b i Independent variable is time. Dependent variable is speed.

ii Rate of change of speed (acceleration)

iii −3c i Independent variable is quantity.

Dependent variable is profit.ii Profit/item

iii

d i Independent variable is time. Dependent variable is water level.

ii cm/min

iii −3

3 Teacher to check.

85---

13--- 1

9---

n −2 −1 0 1 4

c 1 2 1614--- 1

2---

B 1 4 10 32 80

L 40 10 4 1 14--- 1

2---

x −3 −1 0 1 2

y −27 −1 0 1 8

x −3 0 1 4 9

y 9 0 1 16 81

12--- 1

2---

14---

12---

40

20

0

60

80

100

120

2 4 6 8 10 12

Time

Dis

tanc

e

Time

Dis

tanc

e

Time

Dis

tanc

e

Time

Dis

tanc

e

13---

19---

ANSWERS 647

4 a

b

c

d

e

f

5 a

b

c

d

e

f

g

h Same as g

i

j

k

l

Exercise 11-041 a viii b i c iii d vi e ix

f iv g ii h v i vii2 a C b B c A d B e A

f B g A3 a B b A c C d B e A4 a B b H c A d F e C5 a C b E c F d D e A f B6 a i C ii A iii E iv F

b i A ii D iii F iv C7 a E b F c B d C e A f D

8 a

b

H

t

H

t

H

t

H

t

H

t

H

t

Time

Dis

tanc

e

Time

Dis

tanc

e

Time

Dis

tanc

e

Time

Dis

tanc

e

Time

Litr

es

Distance

Litr

es

Time

Dis

tanc

e

Time

Wat

er le

vel

Time

Hei

ght

Time

Dis

tanc

e fr

om s

hop

Time

Soun

d le

vel

Time

Spee

d

Time

Spee

d


c d

e f

9

10

Skillbank 11B2 a 8 h 30 min b 5 h 40 min c 3 h 25 min

d 8 h 25 min e 11 h 25 min f 1 h 40 ming 5 h 10 min h 5 h 45 min i 7 h 55 minj 7 h 25 min

Exercise 11-051 a b

c d

e f

Exercise 11-061 a y = 3x2 b y = 3x2 c y = 0.5x2

d y = −2x2 e y = −3x2 f y = −10x2

2 a i

b i

c i

d i

e i

Time

Spee

d

Time

Spee

d

Time

Spee

d

Time

Spee

d

Time

Spee

d

Time

Spee

d

Time

Spee

d

a b c

Distance from home

Spe

ed

30

−6

y

x

40

12

y

x

20

−8

y

x

0

3

y

x−11_

2

0

y

x1_3

1_2

0 5

y

x−21_

2

0

y

x

y = 4x2

y = x2

ii (0, 0)iii y = 36

y = −2x2

y

x

y = x2

0

ii (0, 0)iii y = −18

0

y

x

y = x2

y = x2

1_3

ii (0, 0)iii y = 3

y = 6x2

0

y

x

y = x2

ii (0, 0)iii y = 54

y

x

y = x2

0

y = − x2 1_2

ii (0, 0)

iii y = −4 12---

ANSWERS 649

f i

g i

h i

i i

Exercise 11-071 a b

c d

e f

g h

i j

k l

Axis of symmetry is the y-axis (or x = 0) for all parabolas.2 a vi b ix c i d xi e x f iii

g ii h xii i viii j v k vii l iv

3 a y = −x2 b y = x2 c y = −x2 +

d y = x2 + 3 e y = −x2 + 9 f y = x2 − 84 a i narrower ii moved down iii −1

b i wider ii moved up iii 3c i wider ii moved up iii 4d i wider ii moved down iii −4e i the same ii moved down iii −3

f i narrower ii moved down iii −

5 a ii b vii c iv d x e v f ixg i h xi i viii j xii k vi l iii

6 a ii y = x2

b ii y = x2 − 1

c ii y = −x2

d ii y = −x2 + 37 b 80 m c 43.2 m

d Approximately 4.1 seconds.8 a to c Teacher to check.

d i parabola ii 100 m2 iii 10 m × 10 m

y = 2x2

0

y

x

y = x2

ii (0, 0)iii y = 18

y

x

y = x2

0

y = −3x2

ii (0, 0)iii y = −27

0

y

x

y = 3x2

y = x2

ii (0, 0)iii y = 27

0

y

x

y = x2

y = x2

1_4

ii (0, 0)

iii y = 2 14---

y

x

V = (0, 10)10

0

V = (0, 10)y

x0

10

y

x0

−10V = (0, −10)

y

x0

V = (0, −10)−10

y

x

V = (0, 6)6

0 V = (0, −6)

y

x

−6

0

V = (0, 6)y

x0

6

y

x0

−6 V = (0, −6)

V = (0, 10)y

x0

10

V = (0, −3)

y

x

−3

0

y

x0

−5 V = (0, −5)

y

x

V = (0, 2)2

0

12---

12---


Exercise 11-08(Diagrams not to scale.)

1 a

b

c

d

e

f

g

h

i

Exercise 11-091 a x = 3 b x = 0 c x = 2

d x = −1 e x = 1 f x = 3

2 a i x = 3 ii (3, −1)b i x = 3 ii (3, 0)c i x = 3 ii (3, 1)d i x = 4 ii (4, 25)e i x = 4 ii (4, −9)f i x = −4 ii (−4, −80)g i x = 4 ii (4, 80)

h i x = − ii (− , − )

j i x = − ii (− , )

3 a (−2, −3) b (1, 6)

c (− , − ) d (3, −9)

e (1, 1) f (− , )

4 a i −4, 10ii −40

iii x = 3iv (3, −49)v Up

b i 0, 3ii 0

iii x =

iv ( , − )

v Up

c i Noneii 4

iii x = −

iv (− , − )

v Up

d i Approx. (−0.7, 6.7)ii 5

iii x = 3iv (3, 14)v Down

e i Approx. (−4.2, 1.2)ii 21

iii x = −

iv (− , 30)

v Down

f ii 3iii x = 4iv (4, −13)v Up

g i Noneii 4

iii x = −0.7iv (−0.7, 1.55)v Up

h i 0, 4ii 0

iii x = 2iv (2, 8)v Down

i i , 2

ii −6

iii x =

iv ( , )

v Down

y

x0 4

y

x0−2

−3

−15

−5 x

y

0

y

x0

10

2 1_22

− 2_3

y

x0−1

−2

−4 −2

8

x

y

0

x

y

0 5

5

−1

1_31−5

−20

x

y

0

1_2−2

0 x

y

2

12---

12---

14--- 1

4--- 11

4---

16--- 1

6--- 11

4---

212--- 101

4---

112--- 31

4---

0 10

−40

−4 x

y

(3, −49)

0 x

y

31_4

1_2(1 , −2 )

112---

112--- 21

4---

0

4_78

_34(− , 2 )

x

y

34---

34--- 27

8---

x

y

0

5

(3, 14)

0

21

1_2(−1 , 30)

x

y11

2---

112---

3

(4, −13)

0 x

y

4

(−0.7, 1.55)

0 x

y

4

(2, 8)

0 x

y

2

−6

0 x

y

1_21

1_8

3_4(1 , )

112---

134---

134--- 1

8---

ANSWERS 651

Exercise 11-101 a y = 3x3 is narrower than y = x3

b y = 3x3 is narrower than y = 2x3

c y = 0.5x3 is narrower than y = x3

d y = −2x3 is narrower than y = −x3

e y = −3x3 is narrower than y = − x3

f y = −10x3 is narrower than y = − x3

2 a i

ii y = 108

b i

ii y = −54

c i

ii y = 9

d i

ii y = 162

e i

ii y = −13.5

f i

ii y = 54

g i

ii y = −81

h i

ii y = 81

i i

ii y = 6.753 a i narrower ii moved down 1

b i wider ii moved up 3c i wider ii moved up 4d i wider ii moved down 4e i same ii moved down 3

f i narrower ii moved down

2 a i b v c vii d iii e vif iv g viii h ix i ii

Exercise 11-111 a

b

c

d

e

f

g

h

14---

15---

110------

0 x

y

0 x

y

0 x

y

0 x

y

0 x

y

0 x

y

0 x

y

0 x

y

0 x

y

12---

x

y

0

(1, 4)

x

y

0

(1, −2)

x

y

0

(4, 5)

x

y

0

(1, 5)

x

y

0(2, −4)

x

y

0

(−4, 4)

x

y

0

(3, 4)

x

y

0

(−1, 1)


Exercise 11-121 a

b 1

c For y = ax where a = 2, 3 or 5, as a increases the graph increases more rapidly as x becomes larger.

2 a

b i y = 3−x ii y = a−x

3 a

b

c y = −2x and y = −3x are reflections of

y = 2x and y = 3x in the x-axis.

d y = a−x

4 a increasing b decreasingc increasing d decreasing

5 a

b

c

d

e

f

g

h

Exercise 11-131 a Centre (0, 0), r = 3

b Centre (0, 0), r = 10c Centre (0, 0), r = 7d Centre (0, 0), r = e Centre (0, 0), r = 3f Centre (0, 0), r = 16

g Centre (0, 0), r =

h Centre (0, 0), r = i Centre (0, 0), r = 5

2 a x2 + y2 = 144 b x2 + y2 = 25

c x2 + y2 = d x2 + y2 = 7

3 (Diagrams not to scale.)

a

b

c

d

e

0 x

y

4

3

2

1

−2 −1 1 2

y = 2xy

= 5

x

y =

3x

0 x

y

4

3

2

1

−2 −1 1 2

y = 3xy = 3−x

0 x

y

2

1

−1

−2

−2−3 −1 1

y = 2x

y = −2x

0 x

y

2

1

−1

−2

−2−3 −1 1

y = 3x

y = −3x

x

y

(1, 2)

1

x

y

(1, −4)−1

x

y

(−1, 3)1

x

y

(−1, 2) −1

x

y

(−1, 4)

1

x

y

(−1, −5)

−1

x

y

(−1, 2)

1

x

y

(−1, −10)

−1

20

12---

6

19---

x

y

−4

4

4

−4

x

y

1

1

−1

−1

x

y

− 1–2

− 1–2

1–2

1–2

x

y8

8

− 8

− 8

x

y

5

5

−5

−5

ANSWERS 653

f

Exercise 11-141 a H b P c L d E e L

f L g H h L i P j Lk P l E m Q n L o C

2 a vii b xii c x d xi e viiif i g iv h ii i vi j iiik v l ix

3 a

b

c

d

e

f

g

h

i

Power plus1 i

ii For y = 5x + 1 asymptote is y = 1.

For y = 5x asymptote is y = 0.

For y = 5x − 2 asymptote is y = −2.2 a Centre (1, −2), r = 6

b Centre (−4, −3), r =

3 i

ii The asymptote for each graph is y = 0, the x-axis.

4 a

b

5 a i

ii

iii

x

y

11

11

−11

−11

x

y

(2, 5)

−30

x

y

(1, −1)0

x

y

(2, 25)

1

0

x

y

(−2, 11)

0

3

x

y(5, 3)

−47

0

x

y

(2, 3)

0

x

y

(5, 3)

53

0

x

y

4

20

x

y

12

12

−12

−12

−1

0

1

2

x

y

y = 1

y = −2

y = 0

y = 5x + 1y = 5x

y = 5x − 2

2

0

y = 5 × 2x

y = 2x

1

0.5

5

x

y

y = × 2x1–2

x

y

−5 2 3 40

40

x

y

30 5−11–2

−45

x

y

0−4

−4

−2−2

2

2

4

4

x

y

0−4

−4

−2−2

2

2

4

4

x

y

0−4−2

−2 2

2

4

4

6


b i x = 0, y = 1 ii x = 0, y = −2iii x = 0, y = 3

6 a

b

c

7 a i

ii

iii

b i x = 1, y = 0 ii x = −1, y = 0iii x = −3, y = 0

8 a b

c

9 a

b

c

Chapter 11 review1 a Ben, his graph is steeper

b Ben 20 km/h, Anita 16 km/hc 1 h 22.5 min

2

3 a i Time ii Distanceb Teacher to check.

4 a ii b iii c i

5 a i

ii

iii

6

x

y

40

4

−4

Semi-circleCentre (0, 0)

Radius 4

x

y

5

4

−5 0

−5


Radius 5

x

y

0 2


Radius 2

2

− 2

x

y

0−2

−2

2

2

x

y

0−2

−2

2

2

x

y

0−2

−2

2

−4

x

y

2

2

0

−2

x

y

1

1

0−1

x

y

0

−21–2

−21–2

21–2

x

y

(1, 1)

0

x

y

−2

2

20

x

y

−3

−3

3

0

Time

Dis

tanc

e

40

−8

y

x

40

8

y

x

4

0−8

y

x

y

x

y = x2

y = 4x2

0

y = − x2 1__10

0

ac

ANSWERS 655

7 a

Both have the same vertex (0, −1) and axis of symmetry x = 0. y = 4x2 − 1 is narrower than y = x2 − 1. Both are concave up.

b

Both have the same vertex (0, 3) and axis of symmetry x = 0. y = 3 − 2x2 is narrower than y = 3 − x2. Both are concave down.

8 a

x-intercepts: −1, 1y-intercept: 1

b

x-intercepts: −2, 10y-intercept: −20

c

x-intercepts: 0, 2y-intercept: 0

9 a i x = 1 ii (1, −7)

b i x = 1 ii (1 , 14 )

c i x = −4 ii (−4, −40)

10 a

b

11 a

b

c

12 a

b

c

13 a

b

c

d

14 a Centre (0, 0), radius 10 b Centre

(0, 0), radius

c Centre (0, 0), radius 715 a ix b vii c vi d iieif

ivg viii h iii i v j xikxl

xii

Mixed Revision 41 Teacher to check.

2 a = b = c =

y = 4x2 − 1

−1

y

x

y = x2 − 11–4

14---

0

3

y = 3 − x21–2

y = 3 − 2x2

x

y

12---

−1 1

1

x

y

−2

−20

10 x

y

x

y

21–2

12---

12--- 1

2--- 3

4---

x

yy = 2x3

y = x3

x

y

y = x3

y = − x31–2

x

y

(2, 23)

−10

x

y

(2, −38)

−1

2

0

x

y

(2, −1)

−5

0

x

y

0

(1, 3)

x

y

0

(−2, 4)

x

y

0

(5, 8)

x

y

(1, 4)

1

0

x

y

(−1, 4)1

0

x

y

(1, −4)

−10

x

y

(−1, −4)

−10

5

46--- 2

3--- 3

6--- 1

2--- 3

6--- 1

2---


3 a b i ii

4 a

b The theoretical probability is the expected probability, whereas the probabilities in the table are based on an experiment. The experimental probability will approach 0.5 as the number of trials is increased.

5 a b

6 0.8

7 a b c

8 B

9 a b c d e

f g h i j

10 a b c d

11 a i ii

b

c i ii iii iv

12

13 a b c

14 B

15 a

b

c

16 C

17

18 a Naim; Naim’s graph is steeper than Jenna’s.b Naim is stationary. c 6 d 6 km/h

19 a C

20 a i

b i

c i

d i

Numberof trials

10 20 30 50 80 100 200 400 1000

Numberof heads

6 10 15 29 41 51 109 214 511

P(H) 0.6 0.5 0.5 0.58 0.5125 0.51 0.545 0.535 0.511

45--- 2

3--- 1

3---

112------ 1

6---

126------ 3

13------ 12

13------

136------ 4

9--- 5

6--- 5

36------ 35

36------

14--- 1

2--- 1

4--- 11

36------ 5

6---

28121--------- 56

121--------- 49

121--------- 105

121---------

38--- 5

8---

2564------

27512--------- 135

512--------- 125

512--------- 387

512---------

12---

18--- 3

8--- 7

8---

−2 2 51 3−3

2

1

3

−2

−4

0

−5

−3

y

x4

−2 21 3−3

2

1

3

−2

−4

0

−5

−3

y

x

−1 2 51 3

2

1

3

4

5

−2

0

−3

y

x

4

1−1−2−3−4 2 3 4

2

−1−2−3−4

4

1

3

0

y

x

y

x

(1, 4)30

ii (0, 3)

y

x−20

ii (0, −2)

y

x

2

0

ii (0, 2)

y

x

11−1

0

ii (0, 1)

ANSWERS 657

e i

f i

21 y = x2 + 3 is the graph of y = x2 moved up 3 units (along the y-axis)

22 Speed on the journey away from home increases then decreases to a stop. The journey begins again towards home, increasing before decreasing and finally stopping without reaching home.

23 x-intercepts are −2, 7.y-intercept is −14.

24 a i x = − ii (− , − )

b i x = ii

c i x = 0 ii (0, 10)

d i x = ii

25 a

b

26 a i ii

iii 9

b i ii − , 5

iii −5

27 a

b

28 a Independent variable is distance, dependent variable is amount of fuel.

b Independent variable is air leaking out, dependent variable is diameter.

c Independent variable is number of people, dependent variable is cost.

d Independent variable is temperature, dependent variable is height.

29 a vii b vi c v d xii e iiif x g ii h iv i xi j ik ix l viii

30 a Centre (0, 0), r = 6 unitsb Centre (0, 0), r = 1 unit

c Centre (0, 0), r = unit

General revision1 a k = b y = 10 c m =

d k = e x = ±3 f y = ±5

g x = −11, 2 h y = −1 or y =

i d = − or d = 1

2 C3 a 4 b 2.5 c 27.54 a $5618 b $935.88 c $1597.98

5 a

b

c

6 a 2x − y + 7 = 0 b 2x + 3y + 5 = 0c 3x + 7y − 2 = 0 d x + 2y − 9 = 0

7 a i $851.20 ii $4742.40b $1200 c 69%

8 a 7.2 m3 b 78.5 cm3

c 5747.0 cm3

9 a x = 2, y = 2 b m = 1, p = −1

c a = , c = −

10 a = 63.1, σ = 10.3

b = 3.6, σ = 1.6

c = 50.3, σ = 12.2

11 a b

c

12 a i ii iii

b i −172 ii

iii

13 a 270b i 0.15 ii 0.19 iii 0.47

iv 0.47 v 0.7214 a y ≥ 1,

b x < ,

c x < −3,

15 a 19.1 m b 10.6 m c 4.8 cm16 a In ∆ABD and ∆ACD:

AB = AC (given)AD is common∠ADB = ∠ADC = 90° (AD ⊥ BC)∴ ∆ABD ≡ ∆ACD (RHS)

b i ∴ BD = CD (matching sides of congruent triangles)∴ AD bisects BC

ii ∴ ∠BAD = ∠CAD (matching sides of congruent triangles)∴ AD bisects ∠BAC

17 a 396 mm2 b 476 mm2 c 792 mm2

d 3750 mm2 e 785 mm2 f 6792 mm2

18 a x = b x =

19 a b c d

20 a 81°45′ b 37°45′ c 142°49′21 a 360 498 mm3, 25 700 mm2

b 145 125 mm3, 17 604 mm2

22 a i , m = ii , m = −

y

x

(2, 6)4

0

ii (0, 4)

y

x

(1, −7)

−50

ii (0, −5)

x

y

7−2−14

32--- 3

2--- 1

4---

34---

34--- 11

8---,⎝ ⎠

⎛ ⎞

54--- 11

4--- −11

8---,⎝ ⎠

⎛ ⎞

y

x0−1

−4

11_3

( , -4 )1_6

1_12

y

x021_

2

(1 , 3 )1_4

1_8

1 12--- 0,⎝ ⎠

⎛ ⎞ 112---

2 13--- −211

3---,⎝ ⎠

⎛ ⎞ 13---

x

y(1, 3)

1

0

x

y

(−1, −6)

−10

13---

612--- 71

3---

337---

23--- 1

2---

25--- 1

3---

x

y

110

−33

−3

(4, −49)

x

y

(2, −2)0

x

y

2

4

0

11823------ 111

23------

x

x

x

9 2 7 2

7 3 15 2–

3 55

---------- 3 2+2

----------------- 2 5–3

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

1 9 3+

98 24 10+

−2 −1 0 1 2 3

112---

−3 −1−2 0 1 2 3

−5 −3−4 −2 −1 0 1

-7 41±2

---------------------- -1 34±3

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

14--- 5

32------ 169

256--------- 169

256---------

13 23--- 40 1

3---


iii , m = iv , m = −

b i 5, (2 , 1) ii 9, (2 , 1)

c A parallelogram.

23

24 a 32 b 4 c d

Chapter 12Start up

1 a SSS b SAS c RHSd AAS e SAS f AAS

2 a SSS b AA c RHSd SSS e SAS or AA f AA

Exercise 12-01 1–5 Teacher to check.6 a UC = 4.5 m

(line from the centre is the perpendicular bisector of the chord and the chords are equal because they are the same distance from centre)

b DE = 12 m(sides opposite equal angles)

c ∠UVO = 58°(angle sum of isosceles ∆; chords of equal length subtend equal angles)

d PQ = 30 mm(Pythagoras: the line from the centre is the perpendicular bisector of the chord)

e OM = 21 cm(Pythagoras: the line from the centre is the perpendicular bisector of the chord)

f OD =

(Pythagoras: the line from the centre is the perpendicular bisector of the chord)

7 a i 52 cm ii 96 cmb i 58 cm ii 8 cmc 18.4 km

8 a 77 cm b 34 cm, 20 cm

c i AB = 30 cm ii area = 540 cm2

d i 52 cm ii area = 1920 cm2

Exercise 12-02 1–3 Teacher to check.4 a 45

b 112c 120d 232 e 40f 74g 63h 104i 90 (angle in semi-circle theorem)j 48k 36l 30

m 23n 9o 45p 63 (opposite angles of a cyclic

quadrilateral are supplementary)q 75r 88

5 a x = 75 (angle at centre theorem)y = 33 (angles at circumference

theorem)z = 72 (angle sum triangle)

b x = 108 (angle at centre theorem)y = 126 (opposite angles of a cyclic

quadrilateral theorem)z = 252 (angle sum at a point or angle at

centre)c x = 70 (straight line)

y = 110 (exterior angle cyclic quadrilateral theorem)

z = 70 (straight line)d x = 96 (angle at centre theorem)

y = 42 (base angles of isosceles ∆)z = 264 (angle sum at a point)

e x = 140 (base angles of isosceles ∆)y = 70 (angle at centre theorem)z = 35 (angle sum of isosceles ∆, and

by subtraction)f x = 62 (angle in semi-circle theorem)

y = 118 (opposite angles of cyclic quadrilateral theorem)

z = 31 (base angles of isosceles ∆)6 a iii WXYZ is a cyclic quadrilateral

because ∠W + ∠Y = 180° and∠X = ∠Z = 180°i.e. opposite angles are supplementary

b Draw the circumcircle of ∆BCD. We show that A lies on the circle. Join OB, OD and let A′ be a point on the circumference.

If ∠A = x, then∠BCD = 180° − x (given)

∴ Reflex ∠BOD = 2 × (180° − x)= 360° − 2x

∴ ∠BOD = 2x∴ ∠A′ = x (angle at centre and

circumference of a circle)∴ ∠A = ∠A′ (both equal x)

∴ A lies on the circle∴ A, B, C, D are concyclic.

7 In ∆PBC:

sin P =

but ∠P = ∠A (angles at the circumference standing on the same arc are equal)

∴ sin A =

∴ = 2R

Similarly, construction diameters from A and B,

= 2R and = 2R

∴ = = = 2R

where R is the radius of the circumcircle of ∆ABC

Exercise 12-03 1–3 Teacher to check.4 a a = 56 (the angle between the radius

and the tangent is a right angle)

b b = 21 (radius is perpendicular to a tangent, and Pythagoras’ theorem)

c c = 134 (a tangent is perpendicular to the radius; angle sum of a quadrilateral)

d g = 67 (alternate segment theorem)5 a 15 b 5 c 9 d 7 e 20 f 46 a x = 7 cm

b i XP = 10 cm ii AB = 24 cm

Exercise 12-041 ∠R = ∠Q

∠P = ∠S

∠RYP = ∠QYS (vertically opposite angles)

∴ ∆PYR ||| ∆SYQ

∴ = (ratio of sides opposite equal angles)

∴ PY × YQ = RY × YS2 ∠ADC = ∠BEC (opposite angles of a

parallelogram equal)∠ADC = ∠CBE (exterior angle of

cyclic quadrilateral)∴ ∆CBE is isosceles as ∠BEC = ∠CBE (base angles equal)

3 ∠TZY = ∠X (alternate segment theorem)

∠X = ∠Y (base angles of isosceles ∆ZXY (XZ = YZ) )

∴ ∠TZY = ∠Ynow XY || ST as alternate angles ∠TZY and ∠Y are equal

4 Construction: Draw a perpendicular from O to meet DG at P. Since the perpendicular from the centre to a chord bisects the chord:

DP = GP andEP = FP

∴ DE = DP − ED= GP − FP= FG

5 ∠THJ = ∠HIJ (alternate segment theorem)

∠THJ = ∠HPI (equal alternate angles)∴ ∠HIJ = ∠HPIIn ∆HIP and ∆HJI

∠HIJ = ∠HPI (proved above)∠IHJ = ∠IHP (common angles)

∴ ∆HIP ||| ∆HJI (equiangular)∴ ∠HIP = ∠HJI (third pair of equal

angles in similar triangles)

6 In ∆UVX and ∆UWX,∠UXV = 90° = ∠UXW (angle in a semi-

circle, straight line)UV = UW (given)

13 23--- 40 1

3---

12--- 1

2---

x

y

20

−11–3

19--- 1

32------

18 2

⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫

(the angle at the centre; twice the angle at the circumference standing on the same arc)

⎭⎪⎬⎪⎫ (angles at the circumference of

a circle standing on the same arc are equal)

⎭⎪⎬⎪⎫ (angle in semi-circle,

angle sum triangle)

⎭⎬⎫ (exterior angle of cyclic

quadrilateral theorem)

B

C

D

O

A

A′

a2R-------

a2R-------

aAsin

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

bBsin

------------- cCsin

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

aAsin

------------- bBsin

------------- cCsin

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

⎭⎪⎪⎬⎪⎪⎫ (angles at the

circumference standing on the same arc are equal)

PYYS-------- RY

QY--------

ANSWERS 659

UX is common∴ ∆UVX ≡ ∆UWX (RHS)∴ VX = VW (matching sides in congruent

triangles)∴ circle bisects base of triangle

7 Let ∠QRP = x (alternate segment theorem)

∴ ∠SRP = x (PR bisects ∠QRS)∴ ∠PSQ = x = ∠QRP (angles at the

circumference standing on the same arc)

∠PQS = x = ∠SRP (angles at the circumference standing on the same arc)

∴ ∠PQS = ∠PSQ = x∴ ∆SPQ is isosceles because base angles PQS and PSQ are equal

8 Now AP2 = YP × PX (intersecting tangent and secant theorem)

and BP2 = YP × PX (intersecting tangent and secant theorem)

∴ AP2 = ΒP2

∴ AP = ΒP 9 Join APX and BQX

In ∆XPQ and ∆XAB∠X is common

∠XPQ = ∠XAB (corresponding angles)∠XQP = ∠XBA (corresponding angles)

∴ ∆XPQ ||| ∆XAB (AAA)

∴ = (ratio of sides opposite equal angles in similar ∆s)

but XP = XA (radius half diameter)

∴ =

∴ =

∴ PQ = AB

10 Let ∠PTX = a∴ ∠R = a (alternate segment

theorem)now ∠QTY = a (vertically opposite

angles)∴ ∠S = a (alternate segment

theorem)∴ ∠R = ∠S = a

∴ PR || SQ because alternate angles R and S are equal.

Chapter 13Start up

1 a 13 b 3 c 122 a (x − 4)(x + 4) b x(x − 4)(x + 4)

c 3(x − 3)(x + 3) d 3x(x − 3)(x + 3)e (x − 5)(x + 3) f (x + 8)(x − 3)g (x − 2)(2x + 5) h x(x − 10)(x + 7)i (x − 5)(x + 5)(x − 2)(x + 2)

3 a

b

c

d

e

f

4 a x = − b x = − or x = 2

c x = 0 or x = 10 d x = 0 or x =

e x = −1 or x = −5 f x = −10 or x = 12

g x = −2 or x = − h x = − or x = 2

i x = −3 or x =

Exercise 13-011 a

b

c

d

e

f

g

XPXA-------- PQ

AB--------

12---

12--- XA

XA----------- PQ

AB--------

12--- PQ

AB--------

12---

0

y

x

y = x + 2

2

−2

0

y

x

y = x2

0

y

xy = −x2

0

y

x

y = x3

0

y

x

(2, −4)

y = −x3

0

y

x

x2 + y2 = 11

1

−1

−1

52--- 5

2---

35---

32--- 5

4---

32---

0

y

x

(−1, −3)−2

0

y

x

1−1

0

y

x

(1, 2)

0

y

x

(2, −5)

3

0

y

x

(1, −3)

0

y

x

(2, −6)

2

0

y

x

(2, 11)

−5


h

i

j

k

l

m

n

o

2 a i −3, −2, 2 ii −12

iii

b i −1, 0, 2 ii 0

iii

c i −1, 1, 3 ii 3

iii

d i −1, 4, 5 ii −20

iii

e i −3, −2, 1 ii 6

iii

f i 0, 3, 6 ii 0iii

g i −3, 2, 5 ii −30

iii

h i −2, 1, 3 ii −12

iii

3 a i −2, 1, 2 ii 4

iii

b i −1, 0, 3 ii 0

iii

c i −1, 1, 3 ii −3

iii

d i −6, 0, 1 ii 0

iii

e i − 4, −1, 1 ii 4

0

y

x

(1, 1)

4

0

y

x

(2, 8)

4

0

y

x

(−3, 7)

−2

0

y

x

(2, 1)

3

(−1, 3)

0

y

x−1

−40

y

x

(3, 5)

0

y

x

4

(−2, 8)

0

y

x(1, −1)

−3

1−1−2−3 2 30

−12

y

x

1−1 20

y

x

1−1 2 30

3

y

x

1 2 3 4 5 60

−20

y

x

1−1−2−3 20

6

y

x

1−1 2 3 4 5 60

y

x

0

y

x−3

−30

2 5

0

y

x−2

−12

1 3

0

y

x−2 1 2

4

0

y

x−1

3

0

y

x−1 31

0

y

x−61

ANSWERS 661

iii

f i −2, 0, 3 ii 0

iii

g i −6, −1, 2 ii − 4

iii

h i − 4, 1, 4 ii 48

iii

4 a x-intercepts are −2, 1 and y-intercept is 6

b x-intercept is −1y-intercept is 1

c x-intercepts are −2 and 3y-intercept is 18

d x-intercepts are , −2

y-intercept is 4

Exercise 13-021 a v b viii c ix d iii e vii

f i g vi h iv i ii

2 a

b

c

d

e

f

13 a Move up 4 unitsb Move right 5 unitsc Move left 3 unitsd Move up 4 unitse Move right 3 unitsf Move left 2 units

Exercise 13-031 a i (−3, 4) ii 5

b i (5, −12) ii 13c i (−2, 4) ii 5d i (3, 1) ii 1e i (9, 12) ii 15f i (1, −3) ii 2g i (−6, −1) ii 1h i (−5, −8) ii 4i i (0, 0) ii 11j i (−2, 1) ii 1k i (2, 0) ii 8

l i (4, −3) ii

m i (3, 4) ii 5

n i (0, −1) ii

2 a x2 + y2 = 16

b (x − 1)2 + (y + 2)2 = 9

c x2 + (y + 11)2 = 4

d (x + 3)2 + (y − 2)2 = 100

e (x + 1)2 + (y + 1)2 = 1

f (x − 7)2 + y2 = 81

g (x + 6)2 + (y − 2)2 = 5

h (x + 1)2 + y2 = 8

3 a

0

y

x−4−1 1

4

0

y

x−2 3

y

x−6

−4−1 1

y

x−4 4

48

1

112---

y

x−2

6

1 2

y

x−1

1

y

x3−2

12---

y

x−2

4

1

0

y

x

4

2

y = (x − 2)2

y = x2

y = 3x2 + 1

0

y

x

y = 3x21

0

y

x

2y = −x3

y = −x3 + 2

0

y

x

y = −2x3y = −2(x + 2)4

−2

(−1, −1)

0

1

1

y

x

y = x4 y = (x − 1)4

0

y

xy = −x5 y = −x5 − 2

−2

212---

13---

0

1

2y

x

(1, 0)


b

c

d

e

f

4 a (−3, 1), r = 5 b (4, 2), r = 7c (−2, 5), r = 6 d (−10, 6), r = 1e (2, − 4), r = 5 f (6, −3), r = 4

g (−3, 10), r = 9 h (4, −1), r =

Exercise 13-041 a (0, 0) and (1, 1) b (1, 1) (3, 9)

c (3, −3), (2, −4) d (2, 0) (−1, 3)e (3, 0), (0, 3) f (1, 3), (−1, −3)g (1, 3), (3, 1) h (− 4, −3), (3, 4)i (−8, −1), (1, 8) j (−3, 9), (1, 1)k (2, 3), (−6, −1) l (12, 5), (−5, −12)

2 a ( , − )

b The algebraic method is more accurate.

Exercise 13-051 a Yes, not monic b No

c Yes, monic d Yes, not monice No f Yes, monic

g Yes, not monic h Noi Yes, monic j Nok Yes, monic l Yes, not monic

2 a i 5 ii 9 iii 1b i 5 ii −6 iii 3c i 2 ii 11 iii −10d i 1 ii −6 iii 0e i 5 ii 7 iii 3f i 0 ii 9 iii 9g i 6 ii 1 iii −11

h i 1 ii − iii 22

i i 3 ii iii 0

j i 4 ii iii 8

3 a 17 b −1 c −7

d e −

4 a 1 b −3 c 0

d e 14 f

g 56 h −7

Exercise 13-061 a 9x3 + 8x2 + 6x − 2

b 3x4 + 2x3 − 3x2 −2x

c − 4x2 − 3x − 2

d −x4 − 3x3 − 5x2 + 4

e x6 + 2x5 + 11x4 + 10x3 + 25x2

f 6x4 + x3 − 2x2 + 2x + 11

g 7x6 + x5 − 3x4 − x3 + 2x2

h 7x4 − 16x3 − 48x2 + 6x + 4

i 6x3 − 8x2 + 4x + 6

j − 4x4 − 8x3 + 5x2 + x − 2

2 a x2 + 11x − 1 b −x2 − 3x − 5

c x2 + 3x + 5 d 2x2 + 26x − 5

e 4x3 + 25x2 − 13x − 6

3 a −x2 + 7x + 3 b x2 + 4x − 15

c −3x2 + 11x + 6 d 3x2 − 15x + 3

e 3x2 − 7x − 15 f 2x2 − 15x + 9

g − 4x3 + 9x2 + 24x − 54

h 8x3 − 62x2 + 99x

4 a x2 − 7x + 6 b

c x = 6, x = 1

5 a x3 + 4x2 − 5x − 20

b

c x = −5, x = 0, x = 1

Exercise 13-071 a x2 + 7x + 4 = (x + 2)(x + 5) − 6

b x2 − 6x + 2 = (x − 3)(x − 3) − 7

c 4x2 + 3x + 10 = (x − 1)(4x + 7) + 17

d 8x2 + 9x + 11 = (2x + 1)(4x + ) +

e x3 + 6x2 + 5x − 4

= (x − 3)(x2 + 9x + 32) + 92

f 4x3 + 2x2 + x

= (x + 4)(4x2 − 14x + 57) − 228

g 2x3 − x2 + 5x + 3

= (x + 6)(2x2 − 13x + 83) − 495

h 3x3 − x2 + 11

= (x + 2)(3x2 − 7x + 14) − 17

i x5 − x4 + 8x3 + 2x2 − x − 1

= (x + 1)(x4 − 2x3 + 10x2 − 8x + 7) − 8

j x4 − x2 − 10

= (x + 3)(x3 − 3x2 + 8x − 24) + 62

2 a (3x − 1), R = 3 b (x + 7), R = 14

c (3x3 + 14x2 − 2x + 21), R = 42d (4x + 6), R = 17

3 a (2x − 1)(3x + 2)

b (2x − 1)(x2 + x + 1)c (2x − 1)(4x + 7)

d (2x − 1)(3x2 + 2x + 1)

e (2x − 1)(x3 − 3x2 − 4x + 2)

f (2x − 1)(x3 − x + 3)

g (2x − 1)(3x2 + 1)

h (2x − 1)(4 − 3x − x5)

Exercise 13-081 a 5 b −181 c −1 d 179

e −7 f 1709 g −16 h −12 a 54 b 2 c 14 d −2 e −12

f 174 g 0 h 6 i −1 j 115

Exercise 13-091 a B, C b B, C c A

d A, B, C e A, B2 Teacher to check.3 a x(x + 2)(x + 4)

b x(x − 2)(x + 1)c (x − 1)(x + 1)(x + 2)d (x − 2)(2x − 1)(x + 4)e (x − 1)(x − 2)(x − 3)f (x − 2)(x + 8)(x − 5)g (x − 6)(x + 1)(3x − 1)h (x − 2)(3x + 1)(2x − 1)

i x2(2x − 1)(x − 2)j (x − 3)(2x + 1)(2x + 3)

4 a x = − 4, , 3 b x = − 4

c x = −2, , 3 d x = ±5

e x = ±4, −3, 2 f x = −7, 0, 2g x = −3, −2, 3 h x = −2, −1, 4

i x = − 4, −1, 5 j x = − , 1, 3

k x = , , 1 l x = 3

Exercise 13-101 a

b

20

3

y

x

(2, 3)

2

0

y

x

(−1, −1)2

5

0

y

x

(3, 4)

4−3 50

y

x(1, 0)

−20

−2

y

x

2

(−2, −2)

2 3

225--- 31

5---

54---

13---

2 2

7 2 5– 58---

13 5 3– 4764------

24 21 2–

2 3 8–

212--- 81

2---

12---

52---

12---

14--- 2

3---

y

x−4

−2

y

x−2

−2

−1 1

ANSWERS 663

c

d

e

f

g

h

2 a

b

c

d

e

f

g

h

i

j

3 a

b

c

d

e

f

y

x2

8

−4 1_2

y

x1

20

y

x

−20

−4−51_2

y

x−2 1 6

−12

y

x0−2 1 4

y

x−3

−36

−2 2 3

−4

−32

2 x

y

0

−18

3−2x

y

0

−1

−4

2x

y

0

−1 3 x

y

0

−8

−2 1 2 x

y

0

4x

y

0

−4

−1 2x

y

0

−64−1 4 x

y

0

−1−2 1 2 x

y

0

4

−64

−2 2 4 x

y

0

−24

−2 34 x

y

0

1

−3

−3x

y

0

x

y

0−5 3

x

y

0

4

1 2−1−2

y

x−1

−64

4

y

x−2

−48


Exercise 13-111 a i

ii

iii

iv

b i

ii

iii

iv

c i

ii

iii

iv

d i

ii

iii

iv

2 Teacher to check.3 Teacher to check.

Chapter 14Start-up

1 a

b

c

0

y

x

2

−2

y = P(x)

y = −P(x)

0

y

x

2

y = P(−x)y = P(x)

0

y

x

2

y = P(x) − 3

y = P(x)

0

y

x

2

4 y = P(x)

y = 2P(x)

0

y

x

y = P(x)

y = −P(x)

−2−2

2

−1 1

0

y

x

y = P(−x) y = P(x)

−2−2

−1 1

0

y

x

y = P(x)

y = P(x) − 3

−2−2

−5

−1 1

0

y

x

y = P(x)

y = 2P(x)

−2−2

−4

−1 1

0

y

x

y = −P(x)

y = P(x)

−1 2

0

y

x

y = P(−x) y = P(x)

−1−2 1 2

0

y

x

y = P(x)

y = P(x − 3)

−1

−3

2

0

y

x

y = 2P(x)

y = P(x)−1 2

0

y

x

y = −P(x)

y = P(x)

31

9

−9

0

y

x

y = P(−x) y = P(x)

31

−9−1−3

−12

0

y

x

y = P(x) − 3

y = P(x)

31

−9

y = 2P(x)

0

y

x

y = P(x)

31

−9

−18

0

y

x

0

y

x

−3

0

y

x−3

ANSWERS 665

2 a

b

c

3 a

b

c

4 a

b

c

5 a x = b x =

c x =

6 a 2.47 b 3.83 c 1.58 d 3.31

Exercise 14-011 a Yes b No c No d Yes

e No f Yes g No h Yes2 a Yes b Yes c No d No

e No f Yes g No h Yesi Yes j No k Yes l Yes

Exercise 14-021 a i 6 ii −2 iii 0

b i −2 ii 2 iii 1c i 12 ii 0 iii 0

d i ii −1 iii 1

e i 27 ii iii 1

f i 3 ii 1 iii

g i ii iii −1

h i −2 ii 6 iii 12 a i 16 ii −14 iii 6x + 16 iv x = 2

b i 4 ii − iii −2 iv x =

c i 5 ii 0 iii

iv t = −2 or t =

3 a i 11 ii 15 iii 10b 9 − 2x c −2 d x = 16 e x = 3

4 a i 11 ii 4 iii 20

b Teacher to check. c x = 1 ±

5 a b x = 7.5

c has no value. d x = −

6 a i − 4 ii 4b −8 c x = 1 or x = 3

d Teacher to check. e has no value.

7 a 8 b 11 c 40

d 3k4 − 2k2 + 3

8 a −26 b −7 c 1 + y3 d −2y3

Exercise 14-031 a i −3 � x � 3 ii 0 � y � 3

b i − 4 � x � 0 ii 0 � y � 4c i x � 0 ii y � 5d i x � −1 ii y � 0e i x � −1 ii y � −3f i No restrictions ii y = −3g i No restrictions ii y > 0h i No restrictions ii y = 3i i No restrictions ii y > −2j i No restrictions ii −1 � y � 1k i No restrictions ii y � − 4l i No restrictions ii 0 < y � 4

2 a

i All x ii All y

b

i All x ii y ≤ 1

c

i All x ii y > 0

d

i x ≠ 0 ii y ≠ 0

e

i All x ii All y

0

y

x

(2, 8)

0

y

x

(1, 4)

0

y

x−11

0

y

x

(1, 1)

0

y

x−1

1

0

y

x−2

1_2

0

y

x

(1, 2)1

0

y

x

(−1, 3)

1

0

y

x

(1, −4)

−1

12--- y 1

2---+ 3

y--- 1–

y 3–±

17---

13---

3

72--- 1

2---

12--- 5

2---

12---

32---

3

2

−1 12---

30---

0

y

x

(1, 4)

1

0

y

x

1

1−1

0

y

x

1

(1, 2)

0

y

x

(1, 2)

0

y

x

1

1–2


f

i All x ii All y

Exercise 14-041 a

b i

ii

iii

iv

c i y = f(x) is moved up 1 unitii y = f(x) is moved down 3 units

iii y = f(x) is moved 2 units to the rightiv y = f(x) is moved 1 unit to the left

2 a

b i ii

iii iv

v vi

3 a

b i ii

iii

4 a b

c

5 a b

c

Exercise 14-051 b, c, h

2 a i f −1(x) = 3x

ii

b i y = 8 − 4x

ii

c i f −1(x) =

ii

d i f −1(x) =

ii

0

y

x

(2, 8)

0

y

x

(1, 2)

0

1

y

x

(1, 3)

0

−3

y

x(1, −1)

20

−4

y

x

−1 0

2

y

x

0

y

x

0

y

x

−3 0

2

y

x

−10

y

x3

0

y

x

0

y

x−10

y

x(−2, −1)

3

0

y

x

0

y

x1

0

y

x

2

0

y

x

(−2, 1)

−3

−2

y

x0

y

x0

1

y

x0

−3

3

y

x0 −2

y

x0

0

y

x

y

x0

y = 3x

y = x_3

y

x0

y = 2 − x_4

y = 8 − 4x

x3

y

x0

1

1−1

−1

y = x3

y = x3

x 5+2

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

y

x0

y = x + 5__2

y = 2x − 5

ANSWERS 667

e i f −1(x) =

ii

f i y =

ii

3 a Teacher to check.b The graph of f(x) = 2 − x is itself

symmetrical about the line y = x.

4 a b No.

c f −1(x) =

d f −1(x) = −

5 a

b f −1(x) = y = −

c x � 2

Exercise 14-061 a 2 b 3 c 2 d 4 e 5

f 3 g 3 h 2 i 6 j 8k 6 l 3

2 a log5 25 = 2 b log4 64 = 3

c log10 10 000 = 4 d log25 5 =

e log2 = − 4 f log3 = −2

g log8 4 = h log10 0.01 = −2

i log4 = j log16 4 =

k log9 27 = l log6 = −

3 a 125 = 53 b 10 = 101

c 27 = d = 23.5

e 64 = 26 f = 3−4

g = 5−3 h =

i 10 = j =

k 2 = l = 100−1

Exercise 14-071 a 7 b 3 c 2 d −1

e f −2 g − 4 h − 4

2 a 1 b 2 c 1 d 3

e 3 f 2 g −1 h

i 3 j 2 k −1 l − 43 a logx 30 b logx 5 c logx 8

d logx 2 e logx 40 f logx 10

g logx h logx i logx 12

j logx 2

4 a 1.2042 b 2.6021 c 3.6021d 0.30105 e −0.3979 f 2.2042g 0.3979 h 0.80105

5 a a + b b a + b + cc 2b + c d c − (a + b)e 2c + a + b f −a

g h −(a + c)

i (a + c) − 2b j (b − c)

6 a logm x + logm y + logm z

b logm a +logm b − logm c

c logm a − (logm b + logm c) or

logm a − logm b − logm c

d logm a + logm (x + y)

e (logm a + logm x)

f −(logm x + logm y) or −logm x − logmy

g 2 logm x − logm y

h logm x − 3 logm y

i − (logm x − logm y) or

logm y − logm x

Exercise 14-081–5 Teacher to check.

6 a i 1.3010 ii 2.7973 iii 3.7345iv 0.9138 v 0.3979 vi −0.1192

b, c, d Teacher to check.

Exercise 14-091 a k = 10 b m = 8 c d = 10

d x = 2.5 e y = −4.5 f a = 3.5g k = 1.5 h n = −1.5 i d = 2.75

2 a x = 1.425 b x = 2.227c x = 2.519 d x = −0.943e x = 0.428 f x = −0.661g x = 7.555 h x = −0.107i x = −1.121 j x = 1.011k y = −0.975 l k = −2.069

3 a x = 2 b x = c x =

d x = − e x = f −

g x = h x = −2

4 a x = 8 b x = 1000 c x =

d x = e x = f x = 32

g x = h x = i x =

j x = k x = 128 l x =

5 a x = 2 b x = c x =

d x = 0.1 e x = 256 f x = 2g x = 3.915 h x = 23.04 i x = 3.846j x = 6.687 k x = 1.682 l x = 19.180

6 a 11.89 ≈ 12 years b 22.43 ≈ 23 months7 a A ≈ 106 g b t = 20 days

c t ≈ 58 days d Teacher to check.

2 2x–3

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

y

x0

y = 2 − 2x_____3

y = 1 − 3x__2

2x 3+2

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

y

x0

y = 2x + 3_____2

y = 2x − 3_____2

0

y

x

−3

x 3+

−3

−3

0

y

x

y = x2 − 3, x � 0

(x + 3)y =

x 3+

0

y

x

−3

−3

y = x2 − 3, x � 0

(x + 3)y = −

0

y

x

2

2 x–

0

y

x2

y = − (2 − x)

12---

116------ 1

9---

23---

2 14--- 1

2---

32--- 1

6------- 1

2---

36

8 2

181------

1125--------- 2 8

16---

10012---

5 5 532---

813---

1100---------

12---

12---

14--- 1

5---

b c+2

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

13---

12---

12---

12---

12--- 1

2---

53--- 5

4---

12--- 7

2--- 13

6------

54---

125------

164------ 81

16------

11000------------ 16 2 1

10----------

18--- 1

25------

15--- 1

2---

Answers

Documents

cumulative

alternate

cab acb

vertically

wider ii moved

shorter sides

cyclic quadrilateral

angled isosceles