AnSWErS to tHE StudEnt’S Book ExErCiSES€¦ · · 2010-09-14Higher WJEC GCSE Mathematics © 2010, Hodder Education Answers to the Student’s ... Answers to the Student’s Book

Answers to the Student’s Book exercises 193Higher WJEC GCSE Mathematics © 2010, Hodder Education

2 6 cm 3 a) 625 b) 1600 c) 1225 d) 1024 e) 1.44 4 a) 1728 b) 15.625 c) 226.981 d) 27 000 e) 157.464 5 a) 620 b) 623.98 c) 635.83 d) 660.70 e) 6124 6 a) 8 b) 8.78 c) 20 d) 21.15 e) 21.54 7 1 and 64. 0 is also a possible answer since

02 1 03 5 0

Exercise 1.5 (page 14) 1 a) 35 b) 73 c) 34 3 52

2 a) 55 b) 107 c) 84

d) 310 e) 26

3 a) 52 b) 103 c) 83

d) 32 e) 20 5 1 4 a) 54 b) 1011

c) 82 d) 34

5 a) 25 b) 32 c) 54 d) 73

Exercise 1.6 (page 17) 1 a) 1 _ 3 b) 1 _ 6 c) 1 __ 49

d) 1 ___ 100 e) 1

___ 640

2 a) 16 b) 9 c) 52 d) 67 e) 1000 3 a) 1 1 _ 4 b) 2 2 _ 3 c) 5 _ 8

d) 3 __ 10 e) 12 1 _ 2

4 a) 0.4 b) 2 c) 0.008 d) 6.25 e) 0.3125

Mixed exercise 1 (page 18) 1 a) 3 3 52 b) 22 3 5 3 7 c) 22 3 3 3 5 3 7 2 a) 24 5 23 3 3, 60 5 22 3 3 3 5, HCF 5 12,

LCM 5 120 b) 100 5 22 3 52, 150 5 2 3 3 3 52, HCF 5 50,

LCM 5 300 c) 81 5 34, 135 5 33 3 5, HCF 5 27, LCM 5 405 3 a) 212 b) 216 c) 8 d) 3 4 a) 16, 64 b) 36, 216 c) 100, 1000 5 a) 68 b) 614 6 a) 5 b) 3 7 a) 21.16, 97.336 b) 441, 9261 c) 8.41, 24.389 8 a) 69.43, 4.46 b) 611.14, 4.99 c) 631.40, 9.95

Chapter 1Exercise 1.1 (page 3) 1 2 3 3 2 2 3 5 3 3 3 5 4 3 3 7 5 2 3 2 3 2 3 2 3 2 5 25

6 2 3 2 3 3 3 3 5 22 3 32

7 2 3 2 3 5 3 7 5 22 3 5 3 7 8 2 3 5 3 5 3 5 5 2 3 53

9 3 3 3 3 5 3 7 5 32 3 5 3 710 2 3 2 3 3 3 5 3 7 5 22 3 3 3 5 3 7

Exercise 1.2 (page 7) 1 4 5 22; 6 5 2 3 3; HCF 5 2;

LCM 5 22 3 3 5 12 2 12 5 22 3 3; 16 5 24;

HCF 5 22 5 4; LCM 5 3 3 24 5 48 3 10 5 2 3 5; 15 5 3 3 5; HCF 5 5;

LCM 5 2 3 3 3 5 5 30 4 32 5 25; 40 5 23 3 5;

HCF 5 23 5 8; LCM 5 25 3 5 5 160 5 35 5 5 3 7; 45 5 32 3 5; HCF 5 5;

LCM 5 32 3 5 3 7 5 315 6 27 5 33; 63 5 32 3 7;

HCF 5 32 5 9; LCM 5 33 3 7 5 189 7 20 5 22 3 5; 50 5 2 3 52;

HCF 5 2 3 5 5 10; LCM 5 22 3 52 5 100

8 48 5 24 3 3; 84 5 22 3 3 3 7; HCF 5 22 3 3 5 12; LCM 5 24 3 3 3 7 5 336

9 50 5 2 3 52; 64 5 26; HCF 5 2; LCM 5 26 3 52 5 1600

10 42 5 2 3 3 3 7; 49 5 72; HCF 5 7; LCM 5 2 3 3 3 72 5 294

Exercise 1.3 (page 10) 1 112 2 220 3 130 4 254 5 228 6 216 7 118 8 24 9 15 10 28 11 13 12 1213 29 14 27 15 24 16 21217 221 18 26 19 29 20 220

Exercise 1.4 (page 12) 1 a) 49 b) 121 c) 66 d) 612 e) 8 f) 1000 g) 4 h) 1

AnSWErS to tHE StudEnt’S Book ExErCiSES

Answers to the Student’s Book exercises Higher WJEC GCSE Mathematics © 2010, Hodder Education194

2 a) Terminating, 5 is a factor of 10 b) Recurring, 3 is not a factor of 10 c) Recurring, 9 is not a factor of 10 d) Terminating, 16 has only factors which are

factors of 10 e) Recurring, 7 is not a factor of 10 3 a) 0.714 285 7… b) Six

Exercise 2.5 (page 36) 1 7.7 2 7.9 3 3.6 4 2.2 5 7.1 6 8.1 7 5.7 8 3.8 9 12.2 10 16.5 11 0.9 12 5.8

Exercise 2.6 (page 39) 1 a) 1.2 b) 3.5 c) 1.8 d) 7.2 e) 0.24 f) 0.48 g) 12 h) 35 i) 0.06 j) 0.08 k) 0.49 l) 0.09 2 a) 40 b) 4 c) 4 d) 9 e) 20 f) 20 g) 30 h) 50 i) 80 j) 30 k) 3.1 l) 122 3 a) 6.3 b) 14.26 c) 35.99 d) 19.44 e) 113.4 f) 388.8 g) 49.84 h) 26.16 i) 5.08 j) 1.872 k) 3.822 l) 9.968 4 a) 49 b) 17 c) 24 d) 32 e) 53 f) 78 g) 5.7 h) 4.9 i) 2.9 j) 2.7 k) 3.7 l) 4.5

Exercise 2.7 (page 42) 1 a) 1.13 b) 1.2(0) c) 1.68 d) 1.08 e) 1.02 f) 1.175 g) 2(.00) h) 2.5(0) 2 a) 0.86 b) 0.8(0) c) 0.55 d) 0.93 e) 0.97 f) 0.77 g) 0.14 h) 0.835 3 £4.78 4 £29.40 5 £2917 6 £34 696 7 £8736 8 £159 627 9 £2980.80

Mixed exercise 2 (page 44) 1 a) 30, 5 _ 6 b) 21, 1 _ 3 c) 40, 13

__ 20

2 a) 1 2 _ 5 b) 1 2 __ 21 c) 11 __ 24 d) 5 _ 6 e) 13

__ 24

f) 5 5 __ 12 g) 3 7 __ 20 h) 8 3 _ 8 i) 6 1 __ 18 j) 1 13 __ 20

k) 2 _ 5 l) 10 __ 21 m) 15

__ 16 n) 2 1 __ 10 o) 9 __ 20

p) 3 2 _ 3 q) 1 13 __ 21 r) 3 3 _ 4 s) 17

__ 44 t) 9 3 _ 7

3 a) 0.125 b) 0.222 c) 0.714 d) 0.273 4 a) 9.7 b) 5.3 c) 8.7 d) 4.6 5 a) 2 b) 0.07 c) 0.72 d) 10.8 e) 9.18 f) 18.72 6 a) 30 b) 8 c) 3 d) 80 e) 79 f) 8.7 7 a) 1 1 __ 66 b) 11

__ 40 c) 3 1 _ 7 d) 3 1 _ 9 8 £5263 9 £29.52

9 a) 57 b) 103 c) 82

d) 26 e) 33

10 a) 1 _ 5 b) 1 _ 8 c) 8

d) 10 e) 0.625 or 5 _ 8

Chapter 2Exercise 2.1 (page 24)

1 a) 9, 7 _ 9 b) 24, 7 _ 8 c) 40, 3 _ 8

2 a) 7 _ 9 b) 7 __ 11 c) 1 _ 3

d) 5 __ 13 e) 5 _ 6 f) 1 1 _ 8

g) 1 _ 3 h) 1 3 _ 5 i) 3 1 _ 2

j) 2 1 _ 4 k) 4 1 _ 9 l) 2 6 _ 7

3 a) 7 _ 8 b) 7 _ 9 c) 7 __ 12

d) 1 _ 4 e) 1 3 __ 10 f) 1 13 __ 28

g) 13 __ 18 h) 1 1 _ 2 i) 1 13

__ 18

j) 23 __ 30 k) 13

__ 36 l) 39 __ 40

4 a) 5 7 __ 10 b) 3 1 _ 8 c) 4 11 __ 14

d) 3 1 __ 12 e) 7 3 _ 8 f) 4 7 __ 12

g) 7 11 __ 18 h) 1 __ 26 i) 6 1 __ 21

j) 5 13 __ 20 k) 1 11

__ 14 l) 5 _ 6

Exercise 2.2 (page 29)

1 a) 19 __ 4 b) 17

__ 3 c) 13 __ 2

d) 21 __ 8 e) 23

__ 7 f) 17 __ 12

g) 17 __ 6 h) 62

__ 11

2 a) 2 2 _ 5 b) 4 1 _ 2 c) 2 __ 15

d) 4 3 _ 8 e) 5 __ 21 f) 9

3 a) 3 __ 16 b) 4 __ 27 c) 5 __ 24

d) 1 3 _ 8 e) 1 3 _ 5 f) 15 __ 28

g) 4 __ 27 h) 7 _ 8 i) 20 __ 27

j) 1 5 _ 9 k) 5 1 _ 3 l) 7 __ 32

4 a) 7 7 __ 10 b) 2 1 _ 7 c) 1 4 _ 7

d) 17 __ 40 e) 8 2 _ 5 f) 1 9 __ 14

g) 10 h) 3 1 _ 3 i) 7 5 _ 7

j) 2 2 _ 5 k) 18 1 _ 2 l) 1 10 __ 39


1 a) 13 __ 21 b) 7 __ 20 c) 5 __ 22

d) 1 7 __ 15 e) 5 13 __ 14 f) 1 11

__ 12

g) 12 6 _ 7 h) 3 9 __ 44

2 a) 2 _ 5 b) 5 _ 8 c) 7 __ 15

d) 7 __ 16 e) 1 _ 9

3 a) 2 1 __ 15 b) 7 11 __ 17 c) 4 13

__ 15

d) 4 4 _ 5 e) 8 3 __ 13

4 a) 20 5 _ 6 cm b) 24 3 _ 4 cm2

Exercise 2.4 (page 34) 1 a) 0.8 b) 0.375 c) 0.182 d) 0.111 e) 0.45


4 Cost per nail: 50 pack 5 2.5p, 144 pack 5 2.64p

Nails per penny: 50 pack 5 0.4 nails, 144 pack 5 0.38 nails

50 pack is better value 5 Cost per roll: 12 pack 5 14.92p, 50 pack 5 14.4p Rolls per penny: 12 pack 5 0.067 rolls,

50 pack 5 0.069 rolls (answers rounded to 3 decimal places)

50 pack is better value 6 Cost per ml: 80 ml tube 5 3.49p,

150 ml tube 5 3.33p Millilitres per penny: 80 ml tube 5 0.29 ml,

150 ml tube 5 0.3 ml 150 ml tube is better value 7 Cost per litre: 3 litres 5 66.33p,

2 litres 5 67.5p, 1 litre 5 57p Litres per pound: 3 litres 5 1.51 litres,

2 litres 5 1.48 litres, 1 litre 5 1.75 litres 1 litre bottle is the best value 8 Cost per gram: 750 g 5 0.239p,

1.4 kg 5 0.229p, 2 kg 5 0.245p (answers rounded to 3 decimal places)

Grams per penny: 750 g 5 4.19 g, 1.4 kg 5 4.38 g, 2 kg 5 4.09 g

1.4 kg packet is the best value

Mixed exercise 3 (page 58) 1 a) 10 : 7 b) 5 : 12 c) 3 : 1 d) 9 : 20 e) 3 : 5 2 a) 1 : 4 b) 1 : 2.4 c) 1 : 50 d) 1 : 250 000 e) 1 : 0.4 3 a) 60 cm b) 5 cm 4 250 g 5 a) 60 g b) 250 g 6 a) 500 ml b) 240 ml

In questions 7 and 8 either cost per unit or number of units per penny can be used. Both are given in each answer. Where necessary, answers are rounded to 2 decimal places, unless otherwise indicated. 7 Cost per litre: 5 litres 5 £3.70, 2 litres 5 £3.50

Litres per pound: 5 litres 5 0.270 litres, 2 litres 5 0.286 litres2 litres is better value

8 Cost per millilitre: pint 5 0.076p, litre 5 0.075p (answers rounded to 3 decimal places)Millilitres per penny: pint 5 13.2 ml, litre 513.3 mlLitres are better value

Chapter 4Exercise 4.1 (page 61) 1 a) 26 b) 1.5 c) 58 d) 200 e) 80.4 f) 20.1 g) 661 h) 10.1 i) 283 j) 275

Chapter 3Exercise 3.1 (page 47) 1 a) 2 : 1 b) 1 : 3 c) 5 : 1 d) 1 : 3 : 5 e) 3 : 6 : 4 2 a) 1 : 20 b) 3 : 20 c) 4 : 1 d) 16 : 3 e) 3 : 20 3 5 : 6 4 5 : 8 : 10 5 4 : 2 : 3

Exercise 3.2 (page 48) 1 a) 1 : 3 b) 1 : 5 c) 1 : 2.5 d) 1 : 1.75 e) 1 : 7.5 f) 1 : 125 g) 1 : 0.2 h) 1 : 500 000 2 1 : 250 000 3 1 : 6

Exercise 3.3 (page 52) 1 a) 12 cm b) 3.5 cm 2 a) 24 babies b) 9 helpers 3 a) 6 litres b) 4 litres 4 140 mm or 14 cm 5 a) 28 miles b) 10 inches 6 a) 30 ml b) 10 teaspoons 7 a) 100 ml b) 36 ml 8 a) 150 g b) 48 g 9 £12010 a) 500 g b) 180 g

Exercise 3.4 (page 54) 1 Dave £8, Sam £12 2 a) 15 litres b) 25 litres 3 30 kg sand 4 a) 50 ml b) 250 ml 5 Amit £320, Bree £800, Chris £480 6 1560 7 £292 8 a) 400 g b) 80 g

Exercise 3.5 (page 57)In all these questions either cost per unit or number of units per penny can be used. Both are given in each answer. Where necessary, answers are rounded to 2 decimal places, unless otherwise indicated. 1 Cost per gram: 420 g bag 5 0.38p,

325 g bag 5 0.34p Grams per penny: 420 g bag 5 2.64 g,

325 g bag 5 2.98 g 325 g bag is better value 2 Cost per litre: 2 litre bottle 5 42.5p,

5 litre bottle 5 35.8p Millilitres per penny: 2 litre bottle 5 23.53 ml,

5 litre bottle 5 27.93 ml 5 litre bottle is better value 3 Cost per gram: 680 g pack 5 0.47p,

1.4 kg pack 5 0.39p Grams per penny: 680 g pack 5 2.125 g,

1.4 kg pack 5 2.59 g 1.4 kg pack is better value


Exercise 4.4 (page 69) 1 a) 4900 b) 0.036 c) 0.04 d) 0.0016 e) 0.04 f) 30 000 g) 560 000 h) 1120 i) 120 300 j) 9040 k) 0.12 l) 0.06 2 a) 25 b) 0.05 c) 0.13 d) 70 e) 0.006 f) 90 g) 30 h) 35 i) 15 000 j) 4000 k) 0.013 l) 1000 3 a) 2160 b) 1.35 c) 16 d) 0.002 16 e) 1600 f) 216 4 a) 9880 b) 3.8 c) 260 d) 38 e) 0.038 f) 26 5 a) 408.51 b) 40 851 c) 1530 d) 4.0851 e) 26.7 f) 15 300

Mixed exercise 4 (page 69) 1 a) 92 b) 1.6 c) 141.9 d) 21.1 e) 441 f) 7.2 g) 158 h) 8.1 i) 383 j) 6.9 2 a) 153 b) 0.54 c) 710 d) 345 e) 1025 f) 17 g) 0.35 h) 0.82 i) 9 j) 12 3 a) 24 b) 6 c) 17 d) 29 e) 1 f) 228 g) 30 h) 29 i) 5 j) 23 4 a) 49 b) 0.81 c) 144 d) 10 000 e) 196 5 £15.41 6 a) 9 b) 4 c) 30 d) 30 e) 600 f) 6 g) 1000 h) 0.3 i) 0.6 j) 50 000 7 32 cm2 or 30 cm2

8 50 mph 9 a) 4000 b) 10 c) 200 d) 36 or 40 e) 4200 or 4000 f) 0.1 g) 30 h) 15 000 or 20 000 i) 300 j) 21 or 2010 a) 9.2 b) 4.7 c) 0.014 d) 160 000 e) 51011 a) 1480 b) 10.2 c) 0.0219 d) 21.0 e) 0.005 6212 a) 10p b) 49p c) 22p

d) 9p e) 85p f) 63p

13 a) 15 000 b) 80 c) 48 d) 0.09 e) 1539 f) 20 g) 0.16 h) 70 i) 4500 j) 12 00014 a) 4446 b) 2.34 c) 19 d) 0.004 446 e) 1900 c) 4.44615 a) 38.682 b) 38 682 c) 1260 d) 3.8682 e) 30.7 f) 1.26

2 a) 16 b) 0.9 c) 87 d) 58 e) 69.6 f) 65 g) 69 h) 277 i) 85.7 j) 996 3 a) 72 b) 52 c) 2.4 d) 160 e) 800 f) 336 g) 185 h) 520 i) 1000 j) 775 4 a) 4 b) 80 c) 13 d) 16 e) 13 f) 4 g) 0.08 h) 6 i) 20 j) 3 5 a) 4 b) 24 c) 27 d) 6 e) 22 f) 10 g) 26 h) 24 i) 24 j) 0 6 a) 28 b) 212 c) 12 d) 235 e) 40 f) 23 g) 24 h) 3 i) 22 j) 16 7 a) 36 b) 25 c) 121 d) 100 e) 169 f) 400 g) 90 000 h) 0.16 i) 0.49 j) 0.09 8 a) 64 b) 63 c) 67 d) 613 e) 615 9 a) 1 b) 125 c) 8 d) 64 000 e) 0.02710 £9.20 11 8 cm12 £15.28 13 420 ml14 a) 17 cm b) 18 cm2

15 8 and 5

Exercise 4.2 (page 65) 1 a) 8 b) 7 c) 20 d) 30 e) 500 f) 7 g) 1000 h) 0.4 i) 0.7 j) 4000 k) 4000 l) 3 2 a) 10 b) 200 c) 20 d) 800 e) 6000 f) 0.01 g) 0.6 h) 0.04 i) 1 j) 20 000 k) 0.07 l) 2000 3 a) 18 b) 180 c) 5700 d) 98 000 e) 50 f) 0.17 g) 0.039 h) 0.0061 i) 0.031 j) 0.99 4 a) 8.26 b) 69.8 c) 16 300 d) 208 e) 12 500 f) 7.10 g) 50.9 h) 0.416 i) 0.0386 j) 3.14 5 £180 or £200 6 3 7 54 cm2 or 50 cm2 8 21 cm or 20 cm 9 £2000 or £3000 10 64 cm3 or 60 cm3

11 50 mph12 a) 4000 b) 9 c) 125 or 100 d) 49 or 50 e) 24 000 or 20 000 f) 0.1 g) 0.75 or 0.8 h) 36 000 or 40 000 i) 43 or 40 j) 60 k) 3.5 or 3 or 4 l) 50 or 60

Exercise 4.3 (page 67) 1 a) 8p b) 64p c) 36p

d) 26p e) 81p f) 7p

2 a) 14p b) 80p c) 32p

d) 50p e) 4 f) 16p

3 5 : 2 4 225pcm2 5 64 2 4pcm2


8 54 mph 9 £191.11 or £191.1210 60 73611 200.8 (to 1 d.p.)12 2.2 g/cm3

13 15.2 cm

Mixed exercise 5 (page 83) 1 a) 3 b) 3.5 2 2.32 (to 2 d.p.) 3 a) The answer should be more than 72, which

is 49.b) Multiplying 2.4 by a number less than 1

should give an answer less than 2.4.c) 120 3 49 < 120 3 50 5 6000, not nearly

60 000 (or equivalent argument).d) The answer should be negative.

4 a) 900 3 30 5 27 000b) 0.62 5 0.36c) 248 4 6 5 28 or 250 4 6 < 28

5 a) 1270.26 b) 2.88 6 a) (i) 203 5 8000 (ii) 30 3 0.3 5 9

(iii) √_______

82 2 42 5 √_______

64 2 16 5 √___

48 < 7

(iv) 30 1 50 _______ 0.04

5 80 ____ 0.04

5 2000

b) (i) 9800.344(ii) 7.743(iii) 6.93 (to 2 d.p.)(iv) 2054 (to nearest integer)

7 3 8 a) 3 hours 42 minutes b) 2 hours 45 minutes c) 48 minutes d) 51 minutes 9 34.3 mph (to 1 d.p.)10 66 834 people11 £96.91 or £96.9212 2.9 cm (to 1 d.p.)

Chapter 6Exercise 6.1 (page 88) 1 a) £696.73 b) £18 310.68 c) £3124.13 2 a) £91.75 b) £399.19 c) £512.33 3 £18 575 4 £17.78 5 2340 rabbits


1 a) £92 b) £47 c) £192.40 d) £358.60 2 a) £696 b) £123 c) £221.33 d) £392.15 3 £813.95 4 £75.05 5 £12 500 6 £759 or £760 or £758.97 7 £34 440 or £34 439 or £34 439.25

Chapter 5Exercise 5.1 (page 73) 1 5 2 3.84 3 4.5 4 3.45 5 9.2 6 62.41 7 1.20 8 2.68 9 4.4810 2.65 11 1.78 12 1.85

Exercise 5.2 (page 76) 1 Check students’ answers. Possible answers

include the following. a) positive 3 negative 5 negative b) Multiplying by number less than 1 should

make answer smaller c) 3 3 60 5 180 and answer should be more

than this d) √

___

64 5 8 and answer should be less than this 2 Check students’ answers. Possible answers

include the following. a) Answer is too large, since 0.9 is near 1 b) negative 4 negative 5 positive c) 62 5 36 and answer should be less than this d) 8 1 3 5 11 so last digit should be 1 3 a) 1000 3 20 5 20 000 b) 0.4 3 0.4 5 0.16 c) 220 4 5 5 24 4 a) 7 3 £9 5 £63 b) 30 3 £15 5 £450 or 30 3 £10 5 £300 c) 3 3 £6 1 3 3 £2 5 £24 5 a) 62 067.5 4 81.4 5 762.5

or 62 067.5 4 762.5 5 81.4 b) √

_______

1466.89 5 38.3 c) 17.6 3 3.8 5 66.88 d) 278.93 1 18.2 5 297.13, 297.13 4 4.3 5 69.1

or 297.13 4 69.1 5 4.3 6 a) 468.38 b) 6.80 7 a) 12.647 b) 896.874 8 a) 61.1 b) 4.8 9 a) (i) 403 5 64 000 (ii) 20 3 0.2 5 4 (iii) √

_______

72 2 42 5 √_______

49 2 16 5 √___

33 < 6

(iv) 10 1 30 _______ 0.08

.5 40 ____ 0.08

5 4000 _____ 8 5 500

b) (i) 60 236.288 (ii) 3.496 (iii) 5.93 (to 2 d.p.) (iv) 516 (to nearest whole number)

Exercise 5.3 (page 82) 1 a) 2 hours and 51 minutes b) 9 minutes 2 a) 1.45 hours b) 0.9 hours 3 3 hours and 15 minutes 4 1 hour and 48 minutes 5 5 6 £31.32 and £20.88 7 700 ml or 0.7 litre


Exercise 7.4 (page 104) 1 a) 7 3 103 b) 8.4 3 104

c) 5.63 3 102 d) 6.5 3 106

e) 7.23 3 105 f) 2.7 3 101

g) 8 3 106 h) 3.92 3 107

2 a) 3 3 1023 b) 5.6 3 1022

c) 3.8 3 1024 d) 6.3 3 1026

e) 8.2 3 1025 f) 3.8 3 1027

3 a) 50 000 b) 370 000 c) 0.0007 d) 6 900 000 e) 0.0061 f) 47 300 g) 27 900 000 h) 0.000 048 3 i) 0.0103 j) 989 000 000 k) 0.000 002 61 l) 370

Exercise 7.5 (page 106) 1 a) 8 3 107 b) 1.2 3 1011

c) 5.6 3 106 d) 4 3 105

e) 5.2 3 107 f) 5 3 104

g) 6.4 3 104 h) 5.83 3 105

2 a) 2.356 3 1013 b) 1.5 3 105

c) 1.45 3 1011 d) 1.073 3 1026

e) 3.25 3 104 f) 8.019 3 105

g) 3.692 3 106 h) 5.202 3 1023

Mixed exercise 7 (page 108) 1 a) 3 b) 9 c) 1 _ 9

d) 1 _ 4 e) 100 000

2 a) 2 _ 5 b) 3 _ 4

c) 109 d) 7 _ 6 or 1 1 _ 6 3 a) 17.58 b) 2.074 c) 0.069 25 d) 3.869 e) 13.57 4 a) 64.39 b) 0.028 97 c) 3.166 d) 21.158 5 a) 72 b) 522 c) 222

d) 1 1 2 _ 3 e) 3 2 1 _ 2

6 a) 26 3 33 b) 25 3 32 c) 23 3 324

d) 53 3 2–2 3 322 e) 3 1 _ 3 3 5

1 _ 3

7 a) 1.65 3 104 b) 8.69 3 1024

c) 5.3 3 107 d) 8.3 3 1028

8 a) 530 000 b) 0.006 32 c) 726 000 000 d) 0.000 001 28 9 a) 8 3 108 b) 2.4 3 106 c) 2.25 3 104

d) 2.5 3 109 e) 8 3 104 f) 2.1 3 1027

g) 4.6 3 104 h) 7.6 3 10–4 i) 5.96 3 107

j) 7.7 3 1024

10 a) 6.81 3 108 b) 2.88 3 10210 c) 8.75 3 107

d) 8.36 3 1025 e) 7.26 3 105 f) 2.04 3 1024

Chapter 8Exercise 8.1 (page 112) 1 a) 0.207 b) 0.429 c) 0.211 2 a) 9.30 b) 4.68 3 1024 c) 0.293 3 a) 2.37 b) 0.977 c) 2.42

8 1540 votes 9 £42510 a) 79p b) £1.2411 a) £56 b) £54.7512 £84

Mixed exercise 6 (page 93) 1 1679 bacteria 2 331 people 3 a) £724.50 b) £27.31 c) £346.75 d) £903.23 e) £75.60 4 £520 5 9216 copies 6 234 seats 7 a) £5705.83 b) £5824.56 c) £6738.28 8 a) £2163.20 b) £2576.41 9 a) £112.50 b) £173.8010 40 miles per gallon

Chapter 7Exercise 7.1 (page 99) 1 a) x

1 _ 2 b) x24 c) x

5 _ 2

2 a) 1 _ 4 b) 2 c) 1

d) 1 __ 16 e) 8 f) 1000

g) 16 h) 1 _ 9 i) 1

j) 16 k) 2 l) 1 _ 8

m) 16 n) 8 o) 1 _ 2

3 a) 18 b) 243 c) 1 _ 5 d) 18 e) 16 f) 72 g) 36

__ 5 5 7 1 _ 5 h) 26

Exercise 7.2 (page 101) 1 a) 74.088 b) 0.073 116 c) 65.944 d) 0.029 401 e) 7 f) 1.5187 g) 2.8439 h) 27 2 a) 225.10 b) 389.33 c) 278.92 d) 2.8730 e) 67.349 f) 0.18 303 g) 768.20 h) 0

Exercise 7.3 (page 102) 1 a) 34 b) 321 c) 3

3 _ 2 d) 32

e) 32n f) 39n g) 311n

2 a) 521 b) 53

c) 54 d) 55

e) Cannot be simplified f) 57n

3 a) 2 3 32 b) 23 3 32

c) 2 1 _ 3 3 3

2 _ 3 d) 22 3 322

e) 33 3 221

4 a) 3 3 52 b) 25 3 32

c) 22 3 53 d) 23 3 34 3 5


10 a) Least 5 14.5 kg, Greatest 5 15.5 kg b) (i) Least 5 1450 kg, Greatest 5 1550 kg (ii) 1550 2 3 3 15.5 5 1503.5 . 1500

97 blocks is too many. 1550 2 4 3 15.5 5 1488 , 1500 96 blocks is under 1500 kg. At most 96 blocks.

Mixed exercise 8 (page 120) 1 a) 28.7 b) 1760 c) 3.23 2 a) 2.80 b) 55.6 c) 11.3 3 a) 2.59 b) 0.009 53 c) 1.47 4 a) 1.64 3 1012

b) 2.07 3 10–4

5 a) £3000 b) 5% c) Number of years d) (i) £3646.52 (ii) £5387.57 6 a) v 5 12 000 3 0.87t

b) (i) £7902.04 (ii) £3938.54 c) 5 years 7 a) UB 5 1.75 kg, LB 5 1.65 kg b) UB 5 255 kg, LB 5 245 kg 8 UB 5 21.2 m, LB 5 20.8 m 9 UB 5 64.3 km/h, LB 5 61.4 km/h

Chapter 9Exercise 9.1 (page 124) 1 a) 4 __ 15 and c) 4 __ 35

2 a) 0.2 6 . b) 0.15 c) 0.1 1

. 4 285 7

.

d) 0.072 e) 0.6875 3 a) 2 _ 5 , b) 17

__ 20 , c) 38 ___ 125 and e) 3 _ 8

4 a) 0.4 b) 0.85 c) 0.304 d) 0.3 8

. e) 0.375

5 a) 3 __ 25 b) 41 ___ 200 c) 3 _ 8 d) 5 __ 16

6 a) 4 _ 9 b) 7 _ 9 c) 11 __ 90 d) 43

__ 90

7 a) 35 __ 99 b) 6 __ 11 c) 4 __ 33 d) 17

__ 99

8 a) 4 __ 37 b) 41 ___ 333 c) 7 __ 99 d) 31

___ 333

Exercise 9.2 (page 128) 1 a) 6 √

__ 3 b) 9 √

__ 5 c) 5 √

__ 5

d) 12 e) 5 √___

21 f) 12 2 a) 5 √

__ 2 b) 10 √

__ 5 c) 24 √

__ 2

d) 15 √__

6 e) 3 √___

10 f) 20 √__

5 3 a) 7 √

__ 3 b) 5 √

__ 5 c) 5 √

__ 2

d) 12 √__

5 e) 24 f) 150 √__

2 4 a) 3 1 2 √

__ 3 b) 10 1 2 √

___ 10

c) 12 √__

7 1 28 √__

2 d) 20 1 10 √__

3 5 a) 32 1 9 √

__ 3 b) 13 2 2 √

__ 2 c) 8

d) 12 1 5 √__

6 e) 10 f) 44 2 23 √__

7 6 a) 8 1 20 √

__ 3 b) 5 2 2 √

__ 3

c) 29 √__

3 d) 299 1 √__

3

4 a) 0.722 b) 1.26 c) 20.649 5 a) 33.7° b) 65.5° c) 69.6° 6 a) 4.67 3 108 b) 6.57 3 107

c) 3.6 3 1028

7 a) 37.8 b) 0.0308 c) 2.95 8 a) 0.070 b) 0.151 c) 0.719 9 a) 1.51 b) 2.47 c) 31.1°10 a) 2.21 3 1011 b) 0.08 c) 4.06 3 1017

Exercise 8.2 (page 114) 1 a) a 5 5000 3 1.03t

b) (i) £5627.54 (ii) £9030.56 2 a) v 5 9000 3 0.88t

b) (i) £6133.25 (ii) £3236.71 c) 5 years 3 a) 1 000 000 b) (i) 31 250 (ii) 244 c) 20 hours 4 a) £2000 b) 8% c) Number of years invested d) (i) £2938.66 (ii) £6344.34 5 a) P 5 60 000 000 3 1.05t

b) (i) 76 576 894 (ii) 6 182 080 587 c) 14.2 years 6 a) m 5 50 3 0.90t

b) (i) 36.5 g (ii) 17.43 g c) 6.6 years

Exercise 8.3 (page 117) 1 a) UB 5 4.75 m, LB 5 4.65 m b) UB 5 37.5 kg, LB 5 36.5 kg c) UB 5 1.955 m, LB 5 1.945 m d) UB 5 28.455 seconds, LB 5 28.445 seconds e) UB 5 330.5 ml, LB 5 329.5 ml 2 2.995 m < length , 3.005 m 3 No. UB for card 5 12.55 cm,

LB for envelope 5 12.5 cm 4 a) UB 5 65 g, LB 5 55 g b) UB 5 210 m, LB 5 190 m c) UB 5 1365 seconds, LB 5 1355 seconds

Exercise 8.4 (page 119) 1 UB 5 26.5 cm, LB 5 23.5 cm 2 UB 5 536.25 cm2, LB 5 490.25 cm2

3 UB 5 3 kg, LB 5 2.8 kg 4 UB 5 1 cm, LB 5 0 cm 5 a) UB 5 13.8, LB 5 13.6 b) UB 5 3.6, LB 5 3.4 6 UB 5 7.88 m/s, LB 5 7.7 4 m/s 7 UB 5 908 people/square mile,

LB 5 892 people/square mile 8 UB 5 84.5, LB 5 20.2 9 UB 5 0.405 LB 5 0.384


Exercise 10.3 (page 137) 1 a) 5(2x 1 3) b) 2(x 1 3) c) 4(2x 2 3) d) 4(x 2 5) 2 a) 7(2 1 x) b) 4(2 1 3x) c) 5(3 2 2x) d) 3(3 2 4x) 3 a) x(3x 1 5) b) 5x(x 1 4) c) 4x(3x 2 2) d) 2x(3x 2 4)

Exercise 10.4 (page 139) 1 a) a2 1 10a 1 21 b) b2 1 11b 1 28 c) 6 1 5c 1 c2

2 a) 9d2 1 3d 2 20 b) 8e2 2 2e 2 15 c) 6 1 5f 2 56f2

3 a) 4g2 2 20g 1 21 b) 4h2 2 28h 1 49 c) 6j2 2 37j 1 56 4 a) 10k2 1 23k 2 42 b) 6 2 11m 2 72m2

c) 6 2 n 2 15n2

5 a) 4 2 4p 2 63p2 b) 6r2 2 37r 1 56 c) 4s2 2 20s 1 21

Exercise 10.5 (page 141) 1 a) 34 b) 73 c) 105

2 a) x5 b) y4 c) z7

3 a) m2n4 b) f 4g5 c) p3r4

4 a) 56k3 b) 120y3 c) 8d3

Mixed exercise 10 (page 142) 1 a) 24a 1 16b b) 20a 1 15b c) 36a 2 60b d) 9a 2 18b e) 12x 1 15y f) 18x 2 12y g) 20x 2 12y h) 8x 1 2y i) 15f 2 20g j) 6j 1 15k k) 7r 1 14s l) 12v 2 4w 2 a) 12x 1 11 b) 20x 1 27 c) 7x 1 12 d) 16y 1 5 e) 15y 1 7 f) 21y 1 4 g) 3a 1 6 h) 6m 1 1 i) 6p 1 18 j) 14t 1 24 k) 2j 1 31 l) 46 3 a) 4(x 1 2) b) 6(x 1 2) c) 3(3x 2 2) d) 6(2x 2 3) e) 2(3 2 5x) f) 5(2 2 3x) g) 8(3 1 x) h) 4(4x 1 3) i) 2(3x 1 4) j) 4(8x 2 3) k) 4(5 2 4x) l) 5(3 1 4x) m) x(2 2 x) n) y(3 2 7y) o) z(5z 1 2) 4 a) a2 1 9a 1 20 b) a2 1 5a 1 6 c) 12 1 7a 1 a2 d) x2 1 7x 2 8 e) x2 1 4x 2 45 f) x2 2 3x 1 2 g) 3x2 1 31x 1 36 h) 2y2 1 y 2 21 i) 14 2 25p 1 6p2 j) 6p2 1 8p 2 8 k) 4t2 2 23t 1 15 l) 6a2 1 a 2 15 5 a) 46 b) 54 c) 25 d) a7

e) j3 f) t6 g) v3w3 h) d3e6

i) x3y5 j) 60p3

7 a) 9 2 15 √__

2 b) 10 1 3 √__

2 c) 6 d) 241 e) 59 1 30 √

__ 2 f) 59 2 30 √

__ 2

8 37

9 a) 5 √__

2 b) √

___ 10 ____

5 c) 2 √

___ 10 _____

15

d) 7 √__

2 ____ 10

e) √

__ 6 ___

3 f) 12 √

___ 10 ______

35

10 a) 3 1 3 √__

2 b) 1 1 3 √__

5 ________ 2

c) 4 √__

3 1 √__

6 _________ 2

d) 5 √__

6 1 6 √__

2 __________ 6

Mixed exercise 9 (page 130) 1 a) 5 _ 6 , c) 5 __ 18 and e) 11

__ 15

2 a) 0.8 3 . b) 0.625 c) 0.2 7

.

d) 0.44 e) 0.7 3 .

3 a) 18 __ 25 b) 13

__ 40 c) 5 _ 8 d) 24 ___ 125

4 a) 4 _ 9 b) 49 __ 90 c) 6 __ 11 d) 56

___ 111

5 a) 4 √__

2 b) 10 √__

3 c) 15 √__

2

d) 6 √__

3 e) 2 √___

15 f) 15 √__

3

6 a) 7 √__

2 b) 4 √__

5 c) 7 √__

3

d) 60 e) 8 √__

3 f) 12 √__

5

7 a) 5 √__

2 1 2 b) 3 √___

15 1 30

c) 2 √___

11 1 22 √__

2 d) 9 √__

2 1 18

8 a) 17 1 11 √__

3 b) 2 1 3 √__

2 c) √__

5 2 11

d) 17 1 7 √__

7 e) 4 f) 46 2 21 √__

3

9 a) 42 2 12 √__

3 b) 35 √__

3 1 30 c) 4 √__

3

d) 37 e) 61 1 28 √__

3 f) 61 2 28 √__

3

10 a) 4 √__

2 b) 2 √___

15 _____ 5 c)

√___

10 ____ 8

d) 2 √__

5 1 5 ________ 5 e) 4 √

__ 3 1 1 f) 2 √

__ 7 1

√__

3 ___ 3

Chapter 10Exercise 10.1 (page 133) 1 20a 1 30b 2 6c 1 21d 3 15e 2 40f 4 28g 2 21h 5 10u 1 15v 6 30w 1 18x 7 21y 1 7z 8 16v 1 40 9 12 1 42w 10 12 2 32a11 8g 2 6 12 35 2 20b13 6i 1 8j 2 10k 14 20m 2 12n 1 8p15 12r 2 18s 2 24t

Exercise 10.2 (page 135) 1 a) 20a 1 38 b) 27b 1 53 c) 12 1 37c d) 32 1 22a 2 a) 18s 1 43t b) 14v 1 49w c) 27x 1 77y d) 38v 1 27w 3 a) 21x 1 8 b) 14y 1 1 c) 22 1 3z d) 36 1 10x 4 a) 16n 1 9p b) 21q 1 13r c) 23d 1 6e d) 14f 1 g e) 2h 1 11j f) 6k 1 3m


3 a) (3x 2 2)(x 2 4) b) (5x 2 4)(x 2 3) c) (3x 2 5)(x 2 7) d) (2x 2 5)(x 2 8) e) (2x 2 3)(x 2 4) f) (4x 2 3)(x 2 2) g) (2x 2 5)(x 2 8) h) (3x 2 2)(x 2 1) i) (3x 2 4)(x 2 1)

Exercise 11.6 (page 152) 1 (2x 1 3)(3x 1 2) 2 (4x 1 3)(5x 1 1) 3 (9x 1 2)(3x 1 2) 4 (2x 2 3)(5x 1 2) 5 (3x 2 2)(5x 1 1) 6 (5x 2 2)(10x 2 3) 7 (10x 1 3)(2x 1 7) 8 3(10x 2 4)(x 2 1) 9 (20x 1 3)(x 1 2) 10 (8x 2 3)(x 2 2)


1 3 _____ x 2 2

2 6 _____ x 1 2

3 x 2 2 _____ x 2 1

4 x 2 4 _____ x 2 5

5 x 1 1 _____ x 1 3

6 3(x 1 2)

________ x 1 4

7 3x 1 2 ______ 2x 2 3

8 2x 2 3 ______ x 2 1

9 3x ______ 2x 2 1

10 5(x 1 3)

________ x 2 3

11 x 1 2 12 x 1 2 ______ 2x 2 3

Mixed exercise 11 (page 155)

1 a) x 1 2 _____ 3 2 x

b) 2(2x 1 3)

_________ 5 1 3x

c) 2(3x 2 2)

_________ 5(2x 1 3)

d) 2 2 3x ______ 3x 1 2

e) 3(3 1 5x)

_________ 5(x 2 2)

f) 4 ___ 3x

g) 3 ______ 2x 1 3

h) 5x ___ 4

2 a) (x 1 1)(x 1 4) b) (x 1 3)(x 1 4) c) (x 1 2)(x 1 7) d) (x 1 1)(x 1 7) e) (x 1 2)(x 1 5) f) (x 1 1)(x 1 5) g) (x 1 2)(x 1 15) h) (x 1 2)(x 1 18) 3 a) (x 1 9)(x 2 9) b) (x 1 8)(x 2 8) c) (x 1 13)(x 2 13) d) (x 1 15)(x 2 15) e) 3(x 1 4)(x 2 4) f) 5(x 1 3)(x 2 3) g) 7(x 1 7)(x 2 7) h) 10(x 1 10)(x 2 10)

4 a) 4a2 ____

3bc2 b) 5x2

____ 3yz2

c) 3q3s2

_____ 10r2

d) 21v2 _____

10tw e)

x2z2 ____

6y2 f) 1 ___

ten

g) 8 __ x

7 _ 2 h) 4 ____

3a23

5 a) (5x 1 2)(x 1 5) b) (3x 1 7)(x 1 3) c) (3x 2 4)(x 2 6) d) (4x 2 1)(x 2 3) e) (2x 1 7)(x 2 2) f) (6x 1 7)(x 2 1) g) (3x 1 8)(x 2 4) h) (5x 1 3)(x 2 3) i) (7x 1 2)(4x 1 3) j) (20x 2 3)(x 1 2) k) (6x 1 5)(5x 2 3) l) (8x 1 3)(3x 1 2)

6 a) 2 _____ x 2 3

b) 2(x 2 3)

________ x(x 1 2)

c) x 2 1 _____ x 1 2

d) x 2 3 _____ x 2 1

e) 5x _____ x 2 2

f) 3x 1 4 ______ x 1 3

g) 2(x 2 5)

________ x 1 5

h) 2x(x 1 3)

_________ x 2 2

Chapter 11Exercise 11.1 (page 144)

1 x 2 3 ______ 2x 2 1

2 2(x 1 3)

________ 5(x 2 2)

3 4 1 x ______ 3 2 2x

4 4(3 2 x)

_________ 3(2 2 3x)

5 3x 2 1 ______ 2x 1 1

6 x 2 2 ______ 3x 2 4

7 3(3 2 2x)

_________ 2(x 1 4)

8 x(x 1 3)

________ 2x 2 1

Exercise 11.2 (page 146) 1 a) (x 1 1)(x 1 2) b) (x 1 1)(x 1 5) c) (x 1 2)(x 1 6) d) (x 1 3)(x 1 5) e) (x 1 1)(x 1 9) f) (x 1 3)(x 1 3) g) (x 1 4)(x 1 5) h) (x 1 2)(x 1 12) 2 a) (x 1 6)(x 2 1) b) (x 1 10)(x 2 1) c) (x 1 4)(x 2 2) d) (x 1 6)(x 2 2) e) (x 1 5)(x 2 3) f) (x 1 10)(x 2 2) g) (x 1 8)(x 2 3) h) (x 1 6)(x 2 4) 3 a) (x 2 2)(x 2 5) b) (x 2 1)(x 2 8) c) (x 2 1)(x 2 2) d) (x 2 2)(x 2 7) e) (x 2 2)(x 2 8) f) (x 2 2)(x 2 12) g) (x 2 5)(x 2 6) h) (x 2 2)(x 2 3)

Exercise 11.3 (page 148) 1 a) (x 1 3)(x 2 3) b) (x 1 4)(x 2 4) c) (x 1 7)(x 2 7) d) (x 1 9)(x 2 9) e) (x 1 10)(x 2 10) f) (x 1 12)(x 2 12) 2 a) 3(x 1 2)(x 2 2) b) 5(x 1 3)(x 2 3) c) 3(x 1 6)(x 2 6) d) 7(x 1 7)(x 2 7) e) 10(x 1 20)(x 2 20) f) 8(x 1 5)(x 2 5)


1 4a2c ____ 3b2

2 8x2y2

_____ 5z2

3 a2b2c3

______ 5

4 6a6b3 _____

5c3 5 9t3w2

_____ 4v3

6 3e2g

____ f

7 2a2d2 _____

bc2e 8

z ______

tv2x3y 9 8a2c2

_____ 9b2d2

10 3e2 _____

8f 2g3 11 3x3 12 9 ____

a5b4

13 6 __ x2

14 12 ___ a9

15 20x

Exercise 11.5 (page 151) 1 a) (3x 1 5)(x 1 4) b) (2x 1 3)(x 1 2) c) (3x 1 1)(x 1 4) d) (5x 1 3)(x 1 3) e) (4x 1 2)(x 1 1) f) (3x 1 5)(x 1 2) g) (2x 1 1)(x 1 2) h) (4x 1 5)(x 1 3) i) (5x 1 3)(x 1 1) 2 a) (2x 1 5)(x 2 3) b) (3x 1 7)(x 2 2) c) (5x 1 3)(x 2 4) d) (3x 1 4)(x 2 3) e) (4x 1 5)(x 2 2) f) (2x 1 3)(x 2 5) g) (4x 1 1)(x 2 2) h) (3x 1 2)(x 2 6) i) (4x 2 3)(x 1 6)


Mixed exercise 12 (page 165) 1 m 5 9 2 p 5 2 3 x 5 8 4 x 5 3 5 p 5 8 6 x 5 4 or 24 7 x 5 6 or 26 8 p 5 4 or 24 9 x 5 4 or 2410 a 5 5 or 25 11 x 5 25 12 x 5 3013 y 5 18 14 y 5 49 15 a 5 3616 x < 2

0 1 2 3 4 5

17 x > 3

0 1 2 3 4 5

18 x , 3

0 1 2 3 4 5

19 x . 4

0 1 2 3 4 5

20 x . 2

0 1 2 3 4 5

Chapter 13Exercise 13.1 (page 169) 1 x 5 20 2 x 5 4 3 x 5 2 2 _ 3 4 x 5 4 5 x 5 3 6 x 5 2 7 x 5 3 _ 4 8 x 5 4 9 x 5 24

10 x 5 2 _ 5 11 x 5 4 12 x 5 413 x 5 3 14 x 5 21 15 x 5 1

Exercise 13.2 (page 171) 1 x 5 4 2 x 5 1 _ 3 3 x 5 12.5 4 x 5 32 5 x 5 1 _ 2 6 x 5 2.74 7 x 5 4.81 8 x 5 9.18 9 x 5 3.2 10 x 5 0.283

Exercise 13.3 (page 173) 1 x . 3 2 x < 5 3 x . 4 4 x . 6 5 x .1.6 6 x < 8 7 x . 216 8 x < 12 9 x < 3 10 x < 3.85

Exercise 13.4 (page 176) 1

01

R1

2

3

4

5

6

y

2 3 4 5 6 x

x � 2y � 6

Chapter 12Exercise 12.1 (page 158) 1 x 5 7 2 x 5 0.5 3 x 5 7 4 x 5 7 5 x 5 1 6 x 5 7 7 x 5 5 8 x 5 4 9 x 5 1010 x 5 6 11 x 5 65 12 x 5 6713 y 5 68 14 m 5 69 15 m 5 6516 x 5 610 17 x 5 68 18 x 5 6419 x 5 62 20 x 5 610

Exercise 12.2 (page 159) 1 p 5 16 2 x 5 3 3 x 5 7 4 x 5 1 5 x 5 15 6 x 5 1 7 x 5 14 8 x 5 5 9 x 5 2410 x 5 10 11 x 5 10 12 a 5 113 x 5 5 1 _ 6 14 x 5 3 15 x 5 216 x 5 3 17 x 5 8 18 x 5 419 x 5 2 20 x 5 5.5

Exercise 12.3 (page 160) 1 x 5 3 2 x 5 3 3 x 5 2 4 x 5 10 5 x 5 3 6 x 5 2 7 x 5 10 8 x 5 22 9 x 5 310 x 5 2 11 x 5 2 12 x 5 113 x 5 3 14 x 5 4.5 15 x 5 416 x 5 6 17 x 5 1 18 x 5 219 x 5 23 20 x 5 7

Exercise 12.4 (page 161) 1 x 5 16 2 a 5 40 3 x 5 20 4 y 5 50 5 y 5 30 6 p 5 35 7 m 5 24 8 x 5 64 9 x 5 2710 y 5 33

Exercise 12.5 (page 164) 1 x . 13

6 8 10 12 14 16

2 x , 4

0 1 2 3 4 5

3 x , 13

6 8 10 12 14 16

4 x < 4

0 1 2 3 4 5

5 x > 3

0 1 2 3 4 5

6 x > 8

0 2 4 6 8 10

7 x , 2 8 x > 25 9 x , 2910 x . 1 11 x . 5 12 x < 913 x < 2 14 x . 6 15 x . 516 x , 0.25 17 x . 2 18 x > 319 x < 4 20 x , 4


15

0�1

�2

�1

1

2

3

4

5

6

7

8

9

10

R

y

1 2 3 4 5 6 x

4x � 3y � 24

x � 0

y � 3x � 2

Chapter 14Exercise 14.1 (page 179) 1 Three points plotted, e.g. (23, 212), (0, 0),

(3, 12), straight line drawn through them. 2 Three points plotted, e.g. (23, 0),

(0, 3), (3, 6), straight line drawn through them. 3 Three points plotted, e.g. (22, 210), (0, 24),

(4, 8), straight line drawn through them. 4 Three points plotted, e.g. (22, 210), (0, 22),

(3, 10), straight line drawn through them. 5 Three points plotted, e.g. (24, 8), (0, 24),

(2, 210), straight line drawn through them.

Exercise 14.2 (page 181) 1 Three points plotted, e.g. (22, 24), (0, 21),

(4, 5), straight line drawn through them. 2 Points (0, 3) and (7.5, 0) plotted, straight line

drawn through them. 3 Points (0, 7) and (2, 0) plotted, straight line

drawn through them. 4 Three points plotted, e.g. (23, 26), (0, 1 1 _ 2 ),

(3, 9), straight line drawn through them. 5 Points (0, 7) and (3.5, 0) plotted, straight line

drawn through them.

Exercise 14.3 (page 184) 1 a) She walked at 3 km/h for 20 minutes

(distance 5 1 km), waited for 5 minutes, travelled in a bus (or car) at 45 km/h for 12 minutes (distance 5 9 km).

b) Approximately 35 minutes c) (i) 0.25 km/minute (ii) 15 km/h d) (i) Approximately 0.75 km/minute (ii) Approximately 45 km/h

2

R

�3 �2 �1 0 1

1

�1

�2

�3

2

3

4

y

x2 3 4

y � 3

y � 2x � 3

3

�4 �2�3 �1 0 1

1

�1

�2

�3

�4

2

3

4

y

x2 3 4

y � 3

x � �2

y � 2xR

4

01

R

1

2

3

4

5

6

y

2 3 4 5 6 x

3x � 4y � 12

x � 4

y � 3

5

0�1

R

�1

1

2

3

4

5

y

1 2 3 4 5 x

3x � 5y � 15

y � x � 1

Mixed exercise 13 (page 177) 1 x 5 3 2 x 5 5 3 x 5 4 4 x 5 22 5 x 5 2 6 x 5 2 _ 3 7 x 5 2.5 8 x 5 1 _ 6 9 x 5 7.5 10 x 5 2.4911 x 5 2.78 12 x , 413 x > 1 _ 4 14 x . 4.6


Exercise 14.4 (page 192)Answers read from graphs may be approximate.

1 a) x 23 22 21 0 1 2 3

x2 9 4 1 0 1 4 9

22 22 22 22 22 22 22 22

y 5 x2 2 2 7 2 21 22 21 2 7

b)

�3 �2 �1 0 1 2 3

1

�1�2�3�4

2345678y � x2 �2

c) (i) 3.3 (ii) 2.4 to 2.5 or 22.4 to 22.5 2 a) x 21 0 1 2 3 4 5

x2 1 0 1 4 9 16 25

24x 4 0 24 28 212 216 220

y 5 x2 2 4x 5 0 23 24 23 0 5

b)

�1 1 30 2 4 5 x

y

�2�3�4�5

�1

12345 y � x2 �4x

c) (i) 0.8 to 0.9 (ii) 0.6 or 3.4 3 a) x 24 23 22 21 0 1 2 3

x2 16 9 4 1 0 1 4 9

x 24 23 22 21 0 1 2 3

23 23 23 23 23 23 23 23 23

y 5 x2 1 x 2 3 9 3 21 23 23 21 3 9

b)

�3�4 �2 �1 0 1 2 3 x

y

1

�1�2�3�4

23456789

y � x2 � x � 3

c) (i) 21.8 (ii) 22.3 or 1.3

2 Description to include:• A winning, B second, C third• A ran at a steady speed, B started off quickly

but gradually slowed down, C ran in a series of sprints and rests.

May also mention approximately when overtaking took place.

3 a) £2 b) 80p c)

10 0 20 30 Distance (km)

Ch

arg

e (

£)

40 50

5

10

15

20

25

30

d)

d) £24 e) 69p (to nearest penny) 4 a) D b) H c) F d) C 5 a) £49.50 b) £46.50 c)

20 4 6Number of boxes

Cos

t (£)

8 10 121 3 5 7 9 11

10

20

30

40

50

60

6 a)

40

0

80 120Water used (cubic metres)

Cha

rge

(£)

16020 60 100 140

2010

40

60

80

100

120

30

50

70

90

110

130140150160170180

b)

b) Less than 91 cubic metres


c) (i) 21.8 or 2.8 (ii) 22.4 or 3.4 7 a) x 23 22 21 0 1 2 3

2x2 18 8 2 0 2 8 18

25 25 25 25 25 25 25 25

y 5 2x2 2 5 13 3 23 25 23 3 13

b)

�3 �2 �1 0 1 2 3 x

y

1

�1�2�3�4�5�6

23456789

10111213

y � 2x2 � 5

c) (i) 21.6 or 1.6 (ii) 22.7 or 2.7 8 a) x 0 1 2 3 4 5

A 5 6x2 0 6 24 54 96 150

b)

x

A

A � 6x2

10 2 3 4 5

20

40

60

80

100

120

140

160

c) (i) 1.8 (ii) 3.6 to 3.7

Mixed exercise 14 (page 195) 1

x

yy � 2x � 1

�1 2 41 3

�3

�1

1

0

3

5

�2

2

4

6

7

4 a) x 22 21 0 1 2 3 4 5

x2 4 1 0 1 4 9 16 25

23x 6 3 0 23 26 29 212 215

4 4 4 4 4 4 4 4 4

y 5 x2 2 3x 1 4 14 8 4 2 2 4 8 14

b) y

123456789

1011121314

y � x2 �3x � 4

�2 �1 10 2 3 4 5 x

c) (i) 1.75 (ii) 21.4 or 4.4 5 a) x 22 21 0 1 2 3 4 5

3x 26 23 0 3 6 9 12 15

2x2 24 21 0 21 24 29 216 225

y 5 3x 2 x2 210 24 0 2 2 0 24 210

b) y

�10�9�8�7�6�5�4�3�2�1

123

y � 3x � x2

�2 �1 10 2 3 4 5 x

c) (i) 2.25 (ii) 20.5 to 20.6 or 3.5 to 3.6 6 a) x 23 22 21 0 1 2 3 4

x2 9 4 1 0 1 4 9 16

2x 3 2 1 0 21 22 23 24

25 25 25 25 25 25 25 25 25

y 5 x2 2 x 2 5 7 1 23 25 25 23 1 7

b)

�3 �2 �1 0 1 2 3 4 x

y

1

�1�2�3�4�5�6

234567

y � x2 � x � 5


8 a)

x 25 24 23 22 21 0 1 2

x2 25 16 9 4 1 0 1 4

3x 215 212 29 26 23 0 3 6

y 5 x2 1 3x 10 4 0 22 22 0 4 10

b)

x

y

�1�2�3�4�5 1 20

�4

�2

2

4

6

8

10

c) (i) 22.25 (ii) x 5 23.8 or x 5 0.8 9 a)

x 24 23 22 21 0 1 2 3

(x 1 3) 21 0 1 2 3 4 5 6

(x 2 2) 26 25 24 23 22 21 0 1

y 5 (x 1 3)(x 2 2) 6 0 24 26 26 24 0 6

b)

x

y

�1�2�3�4 1 2 3

�3

�1

1

0

3

5

�4

�5

�6

�7

�8

�2

2

4

6

c) (i) 26.25 (ii) x 5 22.5 to 22.6 or x 5 1.5 to 1.6

2

2x � y � 8 � 0

x

y

2 41 3

1

0

3

5

2

4

6

7

8

3 a) £39 b) £11 c) 8p 4 a)

1200 1400 1600600 800

Gas used (kWh)

Cos

t (£)

10000

200 400

10

20

30

40

50

b) Gas by £20 5 a) d

t

b) d

t

6 a) She stopped (e.g. waited for bus) b) 68 minutes c) 27 km/h d) (i) 1 km (ii) 10 km 7 a) Quadratic, b) Not quadratic c) Quadratic, d) Quadratic, e) Not quadratic f) Not quadratic g) Quadratic,


5 a) Line drawn through (0, 3) and (3, 9), gradient 5 2

b) Line drawn through (0, 22) and (3, 13), gradient 5 5

c) Line drawn through (0, 1) and (3, 25), gradient 5 22

d) Line drawn through (0, 0) and (3, 23), gradient 5 21

e) Line drawn through (0, 6) and (4, 0), gradient 5 21.5

6 a) 4 km/h b) 212.5 m/s

Exercise 15.2 (page 203) 1 a) y 5 3x 1 2 b) y 5 2x 1 4 c) y 5 5x 2 a) y 5 4x 1 2 b) y 5 1 _ 3 x 1 4 or 3y 5 x 1 12 c) y 5 2x 1 7 3 a) gradient 5 3, y-intercept 5 22 b) gradient 5 2, y-intercept 5 4 c) gradient 5 21, y-intercept 5 3 d) gradient 5 23, y-intercept 5 2 e) gradient 5 2.5, y-intercept 5 26 4 a) gradient 5 22, y-intercept 5 5 b) gradient 5 22, y-intercept 5 4.5 c) gradient 5 21.2, y-intercept 5 2.4 d) gradient 5 1.5, y-intercept 5 23 e) gradient 5 20.75, y-intercept 5 3 5 a) y 5 5x b) y 5 22x 1 8 c) y 5 0.5x 2 5 or 2y 5 x 210

Exercise 15.3 (page 204) 1 Graph of y 5 x 1 1 drawn through (0, 1), (3, 4)

and (5, 6) Graph of y 5 4x 2 5 drawn through (0, 25),

(3, 7) and (5, 15) x 5 2, y 5 3 2 Graph of y 5 2x drawn through (22, 24), (0, 0)

and (4, 8) Graph of y 5 8 2 2x drawn through (22, 12),

(0, 8) and (4, 0) x 5 2, y 5 4 3 Graph of y 5 3x 1 5 drawn through (23, 24),

(0, 5) and (2, 11) Graph of y 5 x 1 3 drawn through (23, 0), (0, 3)

and (2, 5) x 5 21, y 5 2 4 Graph of y 5 2x 2 7 drawn through (21, 29),

(0, 27) and (5, 3) Graph of y 5 5 2 x drawn through (21, 6), (0, 5)

and (5, 0) x 5 4, y 5 1 5 Graph of 2y 5 2x 1 1 drawn through (0, 0.5),

(3, 3.5) and (7, 7.5) Graph of x 1 2y 5 7 drawn through (0, 3.5),

(3, 2) and (7, 0) x 5 2, y 5 2.5

10 a)

x 22 21 0 1 2 3 4

x2 4 1 0 1 4 9 16

22x 4 2 0 22 24 26 28

21 21 21 21 21 21 21 21

y 5 x2 2 2x 2 1 7 2 21 22 21 2 7

b)

x�1�2 1 2 3 4

�3

�1

1

0

3

5

�4

�2

2

4

6

7

8

y

c) (i) x 5 20.4 or x 5 2.4 (ii) x 5 21.4 to 21.5 or x 5 3.4 to 3.511 a)

x 21 0 1 2 3 4 5 6

5x 25 0 5 10 15 20 25 30

2x2 21 0 21 24 29 216 225 236

y 5 5x 2 x2 26 0 4 6 6 4 0 26

b)

x

y

�1 1 2 3 4 5 6

�3

�1

1

0

3

5

�4

�5

�6

�2

2

4

6

c) (i) x 5 0.7 or x 5 4.3 (ii) 6.25

Chapter 15Exercise 15.1 (page 200) 1 a) 1.5 b) 22 c) 3 2 a) 6 b) 2 c) 25 d) 1 e) 0 3 AB 5 0.4, AC 5 2.5, BC 5 21 4 a) 2 b) 1.25 c) 20.5


b)

x

y

2 4 5 61 3

�3

�1

1

0

3

5

�4

�5

�6

�7

�2

2

4

6

7

8

9

10

11y � 3x � 7

2x � 3y � 12

x 5 3, y 5 2 6 a) x 5 4, y 5 2 b) x 5 4, y 5 1 c) x 5 4, y 5 2 d) x 5 21, y 5 3 e) x 5 3, y 5 2 f) x 5 2, y 5 21 g) x 5 21, y 5 2 h) x 5 3, y 5 1

Chapter 16Exercise 16.1 (page 217) 1 a)

0

2468

101214161820

2 4 6Time (seconds)

Vel

ocit

y (m

/s)

8 10

b) 1.8 m/s2

2 a)

�6

�4

�2

0

2

4

6

2 4 6Time(seconds)

Vel

ocit

y (m

/s)

8 10

b) 24.5 m/s

Exercise 15.4 (page 206) 1 x 5 4, y 5 1 2 x 5 2, y 5 3 3 x 51, y 5 3 4 x 5 2, y 5 3 5 x 5 5, y 5 1 6 x 5 2, y 5 1 7 x 5 3, y 5 2 8 x 5 4, y 5 3 9 x 5 3, y 5 1 10 x 5 3, y 5 21

Exercise 15.5 (page 209) 1 x 5 2, y 5 1 2 x 5 2, y 5 3 3 x 5 3, y 5 2 4 x 5 2, y 5 3 5 x 5 5, y 5 1 6 x 5 2, y 5 1 7 x 5 4, y 5 2 8 x 5 22, y 5 3 9 x 5 2.5, y 5 1.5 10 x 5 2, y 5 21

Mixed exercise 15 (page 210 ) 1 a) 1.5 b) 21 2 a) 7.5 b) 25 3 a) Gradient 5 3, y-intercept 5 22

b) Gradient 5 2.5, y-intercept 5 22c) Gradient 5 21.5, y-intercept 5 4d) Gradient 5 2, y-intercept 5 27

4 a) y 5 x 1 5 b) y 5 24x 1 8 or y 1 4x 5 8 5 a)

x

y

�1 2 41 3

�3

�1

1

0

3

5

�4

�5

�6

�7

�2

2

4

6

7

8

9

10

11

12

13y � 4x � 3

y � 3x � 1

x 5 2, y 5 5


2 a), b)

�5

�10

5

10

15

20

25

30

y

y � 4x � 5

y � x2 � 3x � 1

20 4�2�4 x

c) x 5 23, y 5 17 and x 5 2, y 5 23 d) x 5 1.5, y 5 23.25 3 a), b)

10

20

30

y

y � 3x � 7

y � x2 � 3

20 4 6�2 x

c) x 5 21, y 5 4 and x 5 4, y 5 19 4 a), b)

10

20

30

y

y � 4x � 1

y � x2 � 2x � 3

20 4 6�2 x

c) x 5 0.4, y 5 2.4 and x 5 5.6, y 5 24

Exercise 16.3 (page 221)Answers are given to 1 decimal place. 1 a)

�5

5

y

y � x

y � �2

y � 2

y � x2 � 2x � 3

20 4�2 x

b) (i) x 5 21 or 3 (ii) x 5 20.4 or 2.4 (iii) x 5 20.8 or 3.8 (iv) x 5 21.4 or 3.42 a)

10

5

y

y � 8y � 7

y � x2 � 2x � 2y � 5 � x

20 4�2 x

b) (i) x 5 21.6 or 3.6 (ii) x 5 21.3 or 2.3 (iii) x 5 21.4 or 3.4

3 The velocity increases from 0 m/s to 8 m/s in 2 seconds (acceleration 5 4 m/s2).

The velocity is constant at 8 m/s for 4 seconds. The velocity decreases from 8 m/s to 0 m/s in

4 seconds (acceleration 5 22 m/s2).

4

�2

0

2

4

2

Time (seconds)

Vel

ocit

y (m

/s)

4 6 8 10

5 a)

x

y

�20

�10

20

10

5

�15

�5

15

3 4 5 6 2 1 �2 �1 0

b) Gradient is approximately 21 c) x 5 2.5

6 a)

x

y

20

25

30

35

10

5

15

1.5 2 2.5 3 3.5 4 1 0.5 0

b) 62 square units of area c) The answer in b) is slightly greater than the

actual area as all the trapeziums are formed by going above the curve.

7 a) Approximate acceleration is 20.075 m/s2

b) Approximate total distance in the first 30 seconds is 93 metres.

8 Velocity approximately 6 km/h

Exercise 16.2 (page 219) 1 a)–b)

�5

5

0

10

15

20

y

y � 9 � 2xy � x2 � 5x � 5

1 2 3 4 5 6�2 �1 x

c) x 5 21, y 5 11 and x 5 4, y 5 1 d) x 5 2.5, y 5 21.25


b)

0

10 000

20 000

30 000

40 000

50 000

1970 1980 1990Year

2000 2010

n

n � 50 000 � 0.8y

c) n 5 50 000 3 0.8y where y is the number of years from the start (1970)

3 a) The table of values is shown.x 24 23 22 21 0 1 2

y 16 8 4 2 1 0.5 0.25

10

20

y

5

15y � 2�x

20�2�4 x

b) (i) y 5 0.7 (ii) x 5 23.32 4 a) n 5 2 3 3y22005

b) The missing values for n are 162 and 486. 5 a)

5

10

15

20

25

30

35

y

y � 2x

20�1 4 51 3�2 x

b) (i) y 5 9.2 (ii) x 5 3.3 6 a) The table of values is shown.x 24 23 22 21 0 1 2 3 4

y 20.75 21 21.5 23 — 3 1.5 1 0.75

�2

�4

2

4

y

y �

20 4�2�4 x

3x

b) y 5 1.7

3 a)

�5

�10

5

y

y � �5

y � �1

y � 2x2 � 3x � 9

20�2�4 x

b) (i) x 5 22.9 or 1.4 (ii) x 5 22.4 or 0.94 a)

�5

5

10

15

20

25

30y

y � 2x

y � 5

y � x2 � 5x � 3

20 4 6 8�2 x

b) (i) x 5 0.7 or 4.3 (ii) x 5 20.4 or 5.4 (iii) x 5 0.5 or 6.5 5 y 5 24 6 y 5 2x2 1 4x 2 3

Exercise 16.4 (page 227) 1 a)

�20

20

40

60

y

y � 4x

y � x3

20 4�2�4 x

�40

�60

b) x 5 22, 0 or 2 c) y

�15

�10

�5

5

10

15

x�3 �2 �1 1 32

y � x3 � 4x

0

2 a) Year Number of birds (n)

1970 50 000

1980 40 000

1990 32 000

2000 25 600

2010 20 480


4 a)

x

y

�2

�4

�6

�8

�10

�12

1 2 3 4 5 6 70

y � x2 � 7x

y � �2x � 1

y � �9

b) (i) x 5 1.7 and x 5 5.3 (ii) x 5 0.2 and x 5 4.8

5 a) y

x4 5 6 7 8321�2 �1

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

�20

y � 2

y � x2 � 4x � 3

b) x 5 0.3 and x 5 3.7

6 a)

x

y

�30

�20

�10

30

20

10

321�2�3 �1 0

y � 1y � x

y � x3 � 1

b) (i) x 5 1.3 (ii) x 5 1.3

7 a)

x2

y

y � ax2 � b

2 4 6 8

10 12 14 16 18 20 22 24 26 28 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0

b) a 5 21 6 0.2, b 5 30 6 2

Mixed exercise 16 (page 229) 1 Velocity 5 m/s at start, slowing down. Stops at

t 5 4. (Negative acceleration until t 5 6, then positive acceleration.) Then moves in opposite direction. Stops again when t 5 9.

2 a), b) y

x

16

14

12

10

8

6

4

2

�2

�4

�6

�8

�10

�12

4321�2 �1 0

7x � 2y � 11

y � x2 � 5x � 3

c) (21, 9) and (2.5, 23.25)

3 a) y

x

8

7

6

5

4

3

2

1

�14321�1 0

y � x2 � 2x

y � x � 1

b) x 5 20.3 and x 5 3.3


8 a) 14 __ 18 or 7 _ 9 b) 27 weeks

9 a) 8 _ 6 or 4 _ 3 b) 13.5 hours

10 a) 15 __ 9 or 5 _ 3 b) 12 days

Exercise 17.3 (page 238) 1 a) Direct b) y 5 3 __ 40 x

2 a) Direct b) y 5 7x c) 70

3 a) Direct b) y 5 4 _ 3 x

4 a) Direct b) y 5 2 _ 5 x c) 60

5 a) Direct b) y 5 11 __ 3 x

6 a) Inverse b) y 5 240 ____ x c) 16

7 a) Inverse b) y 5 150 ____ x

8 a) Inverse b) y 5 1440 _____ x c) 7.2

9 a) Inverse b) y 5 360 ____ x

10 a) Inverse b) y 5 1800 _____ x c) 11.25

Exercise 17.4 (page 241) 1 a) y ~ x2 b) y 5 x2

2 a) y ~ x2 b) y 5 2x2 c) 128 3 a) y ~ x2 b) y 5 1 _ 3 x

2

4 a) y ~ x2 b) y 5 0.4x2 c) 490 5 a) y ~ x2 b) y 5 1.2x2

6 a) y ~ 1 __ x2 b) y 5 225 ____

x2 c) 1

7 a) y ~ 1 __ x2 b) y 5 1250 _____

x2

8 a) y ~ 1 __ x2

b) y 5 20 ___ x2 c) 0.2

9 a) y ~ 1 __ x2 b) y 5 800 ____

x2

10 a) y ~ 1 __ x2 b) y 5 256 ____

x2 c) 1

Mixed exercise 17 (page 244) 1 a) 161 revolutions b) 988 metres c) 16 miles d) 25 grams e) 9.8 litres 2 a) 10 hours b) 17.5 hours c) 9 days d) 9 days e) 10 days 3 a) (i) y ~ x (ii) y 5 4x b) (i) y ~ x (ii) y 5 15

__ 4 x (iii) 120

c) (i) y ~ x (ii) y 5 2 _ 3 x

d) (i) y ~ x (ii) y 5 1 _ 5 x (iii) 55

e) (i) y ~ 1 __ x (ii) y 5 12 ___

x

f) (i) y ~ 1 __ x (ii) y 5 24 ___

x (iii) 2.4

4 a) h 5 0.05 s2

b) 0.6125 m c) 9.5 m/s

7 a) x 22.5 22 21.5 21 20.5 0 0.5 1

y 32 16 8 4 2 1 0.5 0.25

b)

10�1�2 x

y

10

20

30

y � 25

y � 4�x

c) (i) y 5 12.1 (ii) x 5 22.3 8 Gradient at x 5 21 should be around 11,

gradient at x 5 2 should be around 38. 9 (2)50 square units, the negative indicates that

the area is below the x-axis.10 6.7 km/hr11 262.5 m12 Approximately y 5 10x2 2 613 y 5 5 3 3x

Chapter 17Exercise 17.1 (page 234) 1 a) 8 _ 3 b) 440 metres

2 a) 12 __ 27 or 4 _ 9 b) 96 miles

3 a) 175 ___ 50 or 7 _ 2 b) $315

4 a) 5 _ 7 b) 20 rungs

5 a) 15 __ 27 or 5 _ 9 b) 195 g

6 a) 7 _ 4 b) 45.5 hours

7 a) 40 __ 15 or 8 _ 3 b) 16 hours

8 a) 5 __ 12 b) £37.50

9 a) 14 __ 18 or 7 _ 9 b) £343

10 a) 0.9 __ 2.5 or 9 __ 25 b) 178.2 kg

Exercise 17.2 (page 236) 1 a) 48

__ 32 or 3 _ 2 b) 12 minutes

2 a) 12 __ 8 or 3 _ 2 b) 4 weeks

3 a) 3 _ 4 b) 24 hours

4 a) 40 __ 50 or 4 _ 5 b) 55 minutes

5 a) 20 __ 12 or 5 _ 3 b) 9 days

6 a) 2 _ 3 b) 9 hours

7 a) 3 _ 4 b) 16 kilometres per hour


5 a) y 5 5 _ 2 x 2 10 or 2y 5 5x 2 20

b) y 5 2 _ 5 x 1 8 _ 5 or 5y 5 2x 1 8 c) 3x 1 y 5 9 6 AB: x 1 3y 5 14

BC: y 5 2x – 7 AC: 5x 1 y 5 14

7 a) y 5 7x – 34 b) 2y 5 5x – 20 c) 5x 1 4y 5 6 8 a) x 1 2y 5 4 b) 3x 1 2y 5 2 c) 2y 5 5x – 11 9 (i)

01

1

2

3

4

5

6

7

8

y

2 3 4 5 6 7 8 x

a) (ii) 4x 1 3y 5 24 b) (ii) (4.5, 0) c) (ii) 3x 5 4y10 a) y 5 2x 1 4 b) (3, 10)

Chapter 19Exercise 19.1 (page 258) 1 x 5 22 or 0 2 x 5 1 or 5 3 x 5 23 or 4 4 x 5 22 or 27 5 x 5 0 or 3 6 x 5 64 7 x 5 210 or 6 8 x 5 1 1 _ 2 or 7

9 x 5 24 or 1 _ 2 10 x 5 2 1 _ 3 or 2 1 _ 2

11 x 5 22 1 _ 3 or 4 1 _ 2 12 x 5 2 1 _ 4 or 0

Exercise 19.2 (page 260) 1 x 5 21 or 22 2 x 5 2 or 4 3 x 5 24 or 1 4 x 5 21 or 5 5 x 5 24 or 3 6 x 5 27 or 0 7 x 5 3 or 5 8 x 5 23 or 7 9 x 5 0 or 4 10 x 5 6611 x 5 28 or 22 12 x 5 23 or 813 x 5 26 or 1 14 x 5 2 or 915 x 5 62 1 _ 2 16 x 5 210 or 217 x 5 0 or 1 _ 3 18 x 5 23 or 1019 x 5 25 or 0 20 x 5 4 or 521 x 5 21 or 2 1 _ 3 22 x 5 1 _ 2 or 3

23 x 5 22 1 _ 2 or 1 24 x 5 22 or 2 1 _ 3

25 x 5 21 1 _ 2 or 2 26 x 5 2 _ 3 or 3

27 x 5 2 1 _ 5 or 5 28 x 5 2 2 _ 3 or 1 _ 2

29 x 5 5 (repeated) 30 x 5 2 1 _ 3 or 3 _ 5

31 x 5 210 or 10 32 x 5 24 1 _ 2 or 4 1 _ 2

33 x 5 2 2 _ 3 or 3 _ 2

5 a) y 5 24 ___ x

b) 48

6 a) d 5 250 ____ s2

b) (i) 2.5 m (ii) 31.6 m/s

Chapter 18Exercise 18.1 (page 248) 1 2y 5 3x 1 4 2 y 5 3x 1 5 3 8x 1 3y 5 24 4 3x 1 8y 5 24 5 3y 5 2x 1 5 6 3x 1 4y 5 9 7 y 5 x 1 3 8 y 5 2x 2 1 9 3x 1 y 5 2 10 2x 1 3y 5 9

Exercise 18.2 (page 253) 1 a) 1 _ 2 b) 2 1 _ 4 c) 4 _ 5 2 y 5 3x 1 2 3 3x 1 2y 5 6 4 Check students’ answers. 5 Check students’ graphs. a) 2 1 _ 3 b) 3 6 x 1 3y 5 16 7 3y 5 2x 1 9 8 a) y 5 4x 1 3 and 4x 2 y 5 5 b) 2y 2 3x 5 5 and 6y 1 4x 5 1 9 3y 5 x 1 410 a) y 5 2x 1 2 b) x 1 2y 5 4 c) (0, 2)

Mixed exercise 18 (page 255) 1 a)

01

1

2

3

4

5

y

2 3 4 5 x

2x � 3y � 9

b)

0

y

x

5x � 2y � 10 � 0

�2 �1 �1

�2

�3

�4

�5

2 a) y 5 1 _ 2 x 1 2 or 2y 5 x 1 4

b) y 5 2 15 __ 8 x 1 15 or 15x 1 8y 5 120

c) y 5 5 _ 3 x 2 1 or 3y 5 5x – 3 3 a) y 5 3x 1 2 b) y 5 22x 1 6 or 2x 1 y 5 6 c) y 5 1 _ 2 x 1 5 or 2y 5 x 1 10

4 a) 4 b) 2 _ 5 c) 2 2 _ 3 d) 26


4 a) x 5 0 or 7 b) x 5 1 or 24 c) x 5 1 or 7 d) x 5 4 or 5 e) x 5 1 _ 3 or 2 1 _ 2 5 a) (x 2 1)2 1 6 b) (x 1 4)2 2 26 c) (x 2 2)2 2 13 d) (x 2 2.5)2 2 5.25 6 a) x 5 5.24 or 0.76 b) x 5 20.53 or 29.47 c) x 5 4.65 or –0.65 d) x 5 0.30 or –3.30 7 a) x 5 3.41 or 0.59 b) x 5 20.59 or 28.41 c) x 5 7.53 or 20.53 d) x 5 0.63 or 22.63 8 b) and c) have no solutions. For both, b2 2 4ac is

negative (27 in each case). 9 (x 1 1)(x 2 7) 5 33, x 5 10 or x 5 24 (reject),

dimensions are 3 cm and 11 cm.10 x(2x 2 3) 5 77, x 5 25.5 or 711 a) 2x(x 1 3) 1 2x(x 1 2) 1 (x 1 3)(x 1 2) 2

242 5 0 Students assuming the cuboid is solid rather than open will end up with

2x2 1 6x 1 2x2 1 4x 1 x2 1 5x 1 6 2 242 → 5x2 1 15x 2 236 5 0

b) 5.53, 28.53 c) 5.53, 8.53, 7.53 cm12 a) h2 5 x2 1 (x 1 4)2

5 x2 1 x2 1 8x 1 16 5 2x2 1 8x 1 16

b) 1 _ 2 x(x 1 4) 5 45.6 x2 1 4x 2 91.2 5 0 7.8 cm

Chapter 20Exercise 20.1 (page 276) 1 x 5 1, y 5 2 2 x 5 4, y 5 3 3 x 5 2, y 5 4 4 x 5 3, y 5 21 5 x 5 21, y 5 5 6 x 5 2 1 _ 2 , y 5 3 7 x 5 22, y 5 23 8 x 5 4, y 5 1 9 x 5 5, y 5 3 10 x 5 3, y 5 2411 x 5 22, y 5 24 12 x 5 3 1 _ 2 , y 5 4 1 _ 2

13 x 5 4, y 5 2 _ 3 14 x 5 2 1 _ 2 , y 5 1 _ 3

15 x 5 23, y 5 5 16 x 5 3, y 5 1 1 _ 2 17 x 5 22, y 5 25 18 x 5 2, y 5 319 x 5 5, y 5 21 20 x 5 4, y 5 2321 x 5 2 1 _ 2 , y 5 1 1 _ 2

Exercise 20.2 (page 279) 1 x 5 2, y 5 1 2 x 5 3, y 5 21 3 x 5 22, y 5 5 4 x 5 2, y 5 23 5 x 5 2, y 5 4 6 x 5 3, y 5 24


1 a) x 5 3, y 5 21 b) x 5 1 1 _ 2 , y 5 22 c) x 5 23, y 5 1 d) x 5 4, y 5 3 2 a) x 5 21, y 5 2 b) x 5 24, y 5 215 3 a) 3x 1 2y 5 155, 4x 1 y 5 140 b) Pen 25p, exercise book 40p.

Exercise 19.3 (page 262) 1 x 5 23 or 4 2 x 5 22 or 5 3 x 5 22 or 7 4 x 5 1 or 6 5 x 5 0 or 6 6 x 5 25 or 1 7 x 5 22 or 4 8 x 5 22 or 1 _ 2 9 x 5 23 or 5 10 x 5 24 or 1

Exercise 19.4 (page 264) 1 a) (i) (x 1 4)2 2 16 (ii) (x 2 5)2 2 25 (iii) (x 1 6)2 2 36 (iv) (x 1 0.5)2 2 0.25 b) (i) (x 1 4)2 2 19 (ii) (x 2 5)2 1 6 (iii) (x 1 6)2 2 41 (iv) (x 1 0.5)2 1 1.75 2 a) (x 1 1)2 2 4 b) (x 1 2)2 2 5 c) (x 2 3)2 1 3 d) (x 1 5)2 2 31 e) (x 2 10)2 2 150 f) (x 1 6)2 2 37 g) (x 1 4)2 1 3 h) (x 2 1.5)2 2 5.25 i) (x 1 2.5)2 1 3.75

Exercise 19.5 (page 267) 1 a) x 5 6.16 or 20.16 b) x 5 0.41 or 22.41 c) x 5 20.71 or 211.29 d) x 5 0.12 or 28.12 2 a) (x 2 2)2 2 14 5 0, x 5 5.74 or 21.74 b) (x 1 4)2 2 5 5 0, x 5 21.76 or 26.24 c) (x 2 1)2 2 8 5 0, x 5 3.83 or 21.83 d) (x 2 5)2 2 6 5 0, x 5 7.45 or 2.55 e) (x 2 6)2 2 15 5 0, x 5 9.87 or 2.13 f) (x 1 2)2 2 10 5 0, x 5 1.16 or 25.16 g) (x 1 4)2 2 3 5 0, x 5 22.27 or 25.73 h) (x 1 10)2 2 50 5 0, x 5 22.93 or 217.07 i) (x 2 1.5)2 2 13.25 5 0, x 5 5.14 or 22.14

Exercise 19.6 (page 270) 1 x 5 20.68 or 27.32 2 x 5 4.30 or 0.70 3 x 5 0.65 or 27.65 4 x 5 2.30 or 21.30 5 x 5 20.56 or 24.44 6 No real solutions 7 x 5 1.12 or 23.12 8 x 5 20.13 or 22.54 9 x 5 2.53 or 20.53 10 x 5 3.28 or 1.2211 x 5 0.23 or 20.73 12 x 5 0.48 or 22.0813 x 5 2.21 or 0.79 14 No real solutions15 x 5 2.92 or 23.42 16 x 5 0.22 or 20.6517 x 5 11.05 or 29.05 18 x 5 20.32 or 21.8819 No real solutions 20 x 5 2.91 or 20.9121 x 5 21.26 or 28.74

Mixed exercise 19 (page 271) 1 a) x 5 0 or 6 b) x 5 23 or 7 c) x 5 22 1 _ 2 or 4 d) x 5 21 1 _ 3 or 2 1 _ 5 2 a) x 5 0 or 29 b) x 5 7 or 27 c) x 5 24 or 21 d) x 5 28 or 1 e) x 5 2 or 7 3 a) x 5 0 or 4 b) x 5 2 1 _ 2 or 22 1 _ 2

c) x 5 2 1 _ 2 or 3 d) x 5 1 _ 3 or 2

e) x 5 21 1 _ 3 or 3


9 a) 1, 4, 9, 16, 25b) (i) Each term is 3 more than the equivalent

term in the sequence of square numbers.(ii) n2 1 3

c) 10 00310 a) Each term is 3 times the equivalent term in

the sequence of square numbers. b) 3n2 c) 1200

Mixed exercise 21 (page 289) 1 a) 300 b) Tn 5 3n 2 a)

b) 64. This is the sequence of square numbers; the 8th pattern has 8 3 8 stars.

3 a) T1 5 1 3 6 2 2 5 4 b) 10, 16, 22, 28 4 a) 7, 12, 17, 22 b) 2, 5, 10, 17 c) 88, 86, 84, 82 5 4, 8, 16, 32, 64 6 a) Tn 5 3n 1 2 b) Tn 5 6n 2 5 c) Tn 5 7 2 5n 7 a) Linear: 24, 29 b) Not linear c) Not linear d) Linear: 212, 218 8 a) 3, 9, 27, 81, 243 b) Powers of 3 9 A suitable answer is as follows.

1 3 6 10 15

10 a) 1, 4, 9, 16, 25 b) Tn 5 n2 1 4

Chapter 22Exercise 22.1 (page 292) 1 a) 90 minutes b) 210 minutes 2 4°C 3 £31 4 a) s 5 d __ t b) 60 km/h

5 490 m 6 18 7 a) 55 b) 20 8 a) 8x b) 3x2

9 a) 2 hours 10 minutes b) 1.5 kg10 511 15 miles12 a) 12

b) Number of posts, P 1 2 3 4 5 6

Number of chains, C 0 3 6 9 12 15

c) C 5 3P 2 3 or C 5 3(P 2 1) d) 8713 a) 24 cm2 b) 22.5 cm2

Chapter 21Exercise 21.1 (page 283) 1 a) Multiply the pattern number by 2 b) 200 2 a) Pattern number 1 2 3 4 5

Number of matchsticks 4 6 8 10 12

b) They go up by 2 each time; they are the 2 times table plus 2.

c) 102 3 a)

b) 36 You add 1 more each time, so T4 5 10,

T5 5 15, T6 5 21, T7 5 28, T8 5 36

4 a) 3 3 1 2 5 5 22 b) 1, 4, 7, 10 5 a) 4, 10, 16, 22 b) 5, 9, 13, 17 c) 4, 2, 0, 22 6 a) 1, 4, 9, 16, 25 b) 3, 6, 11, 18, 27 c) 1, 8, 27, 64, 125 7 2, 4, 8, 16, 32 8 5, 2, 21, 24 9 For example

and so on.10 For example

and so on.

Exercise 21.2 (page 287) 1 a) 2n 1 3 b) 3n 2 1 c) n 1 6 2 a) 20 2 3n b) 10 2 5n c) 1 2 n 3 a) Linear: 17, 20 b) Not linear c) Linear: 30, 36 d) Not linear 4 a) 6, 0, 26, 212, 218 b) 14 2 6n 5 a) Number of tickets 1 2 3 4

Cost in £ 17 32 47 62

b) 15n 1 2 c) 7 6 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 7 210 8 a) 10, 100, 1000, 10 000, 100 000 b) Powers of 10


5 a) q 5 p 2 2r b) r 5 x 2 s _____ 5

c) r 5 ms ___ pq

d) y 5 tx 2 A ______ 2t

or y 5 x __ 2 2 A __

2t

6 a) w 5 T 2 40 _______ 45

b) w 5 2.4 kg

7 a) 4 cm b) 8 cm 8 a) 30 words b) 60 words 9 a) 13 1 4 3 1 5 5 23 1 4 3 2 5 16 b) x 5 1.710 a) 33 2 32 5 18 43 2 42 5 48 b) x 5 3.411 a) x3 1 x 5 100 b) x 5 4.6

Chapter 23Exercise 23.1 (page 303) 1 r 5

pq _____

s 1 t 2 P 5 100A _________

RT 1 100

3 x 5 4y 1 15

________ 3(1 1 y)

4 p 5 rq 2 r

______ q 1 1

5 r 5 pq 1 p

_______ q 2 1

6 x 5 qy ______

q 1 p

7 t 5 2s ______ 2u 1 a

8 x 5 b 1 a _______ 3a 2 2b

9 u 5 sv _____ v 2 s

10 v 5 fu _____

u 2 f

11 f 5 5g 2 3e

_______ 2 12 b 5

c(5a 1 1) _________

15a 2 2


1 a) x 5 6 √______

y 1 4

_____ 3 b) l 5

gt2

____ 4p2

c) h 5 6 √__________

(  A ___ pr

) 2 2 r2 d) u 5 6 √________

v2 2 2as

e) r 5 6 √___

3V ___ ph

f) t 5 6 √_______

30 2 2s _______ a

2 a) 9000 cm3 b) r 5 √___

V ___ ph

c) 5.64 cm 3 a) 5.39 b) 3.68 m c) x 5 √

___________

d2 2 y2 2 z2 d) 0.71 m

Mixed exercise 23 (page 305) 1 a) a 5 2rb _____

r 2 1 b) g 5 4lp2

____ t2

c) u 5 fv _____

v 2 f d) x 5 b 2 ac ______

c 2 1

e) a 5 2 1 b2 ______

b 2 1 f) y 5 2x

g) a 5 2bc _____ b 1 1

h) x 5 6 √__

y __

3

i) x 5 1 ______ y2 2 1

j) n 5 p 2 2r 2 2

__________ p 2 2

14 a) 120 cm3 b) 216 cm3

15 a) £45 b) 43 miles

Exercise 22.2 (page 296) 1 a) b 5 a 1 c

b) x 5 4a 2 y

______ w

c) t 5 v 2 u _____ a

d) t 5 p 2 c

_____ 3

e) q 5 A 2 pr

_______ p

or q 5 A __ p 2 r

f) g 5 p 1 2f

______ 2

g) n 5 Ft 2 m _______ 4

2 u 5 sbn ____ 3v

3 h 5 2a ___ b

4 R 5 100I _____ PT

5 a) h 5 3V ___ pr2

b) 16.6 cm

6 a) 140°F b) C 5 (F 1 40) 3 5 _ 9 2 40

7 r 5 √___

3V ___ ph

8 a) a 5 v2 2 u2

_______ 2s

b) u 5 √________

v2 2 2as

Exercise 22.3 (page 298) 1 x 5 1.7 2 a) 23 2 5 3 2 5 22 33 2 5 3 3 5 12 b) x 5 2.8 3 a) 43 2 4 5 60 53 2 5 5 120 b) x 5 4.6 4 a) 3.73 2 3.7 5 47.0 3.83 2 3.8 5 51.1 b) x 5 3.77 5 x 5 2.3 6 x 5 4.33 7 22 8 31.6 9 a) x2 1 x 5 10 b) 2.7 (or 23.7)10 a) x(x 2 4) 5 621, i.e. x2 2 4x 5 621 b) 27, 23 (or 227, 223)11 7.8

Mixed exercise 22 (page 300) 1 a) 104°F b) 32°F c) 23°F 2 95p 3 a) 12 cm2 b) 21.6 cm2

4 a) 34 minutes b) 47.8 minutes


4 a) y

y � sin θ � 1

θ0

�2

�3

�190° 180° 270° 360°

b) y 5 sin 2 1 5 a) y 5 x2 2 5 b) y 5 (x 2 2)2

6

4

6

2

�2

�4

y

y � f(x)

y � f(x � 2)

y � f(x) � 2

x�1 0 1 2 3

7 4

2

�2

y y � g(x) � 1

y � g(x � 1)

x

y � g(x)

0 1�1 2 3 4

8 y 5 (x 2 1)2 1 2 9 y 5 cos 1 210 a) y 5 (x 1 2)2 1 3 b) y 5 (x 1 2)2 1 3

5 x2 1 4x 1 4 1 3 5 x2 1 4x 1 7


1

yy � �sin θ

y � sin θ

θ0

�1

90° 180° 270° 360°

b) A reflection in the x-axis 2 a)

1

y

y � sin θ

y � sin

θ0

�1

90° 180° 270° 360°

θ2

k) q 5 (x 2 np)2

_________ npy2

l) x 5 6 √______

F ______ F 1 1

m) y 5 6 √________

ns2 2 x2 n) b 5 ax 2 am ________ m 2 y

2 a) 524 units3 (to 3 s.f.)

b) r 5 3 √___

3V ___ 4p

c) 9.42 m (to 3 s.f.)

Chapter 24Exercise 24.1 (page 307) 1 a) 4 b) 21 2 a) 22 b) 17 c) 1 3 a) x 5 6 b) (i) 2x 2 9 (ii) 4x 2 5 4 a) x 5 2 b) (i) 21 2 9x (ii) 23x 2 2 5 a) x 5 3 _ 4 b) (i) 4x 1 17 (ii) 4x 1 2 6 a) x 5 63 b) (i) x2 1 2x 2 1 (ii) x2 2 1 7 a) 56 b) (i) 3x2 2 8x 1 5 (ii) 12x2 2 4x 8 a) x 5 4 or 21 b) (i) 4x2 2 6x (ii) x2 2 x 2 2


4

8

12

16

yy � x2 y � (x � 5)2

x20 4 6 8 10�4 �2

b) A translation of (  5 0 )

2 a)

�4

�8

�12

�16

�20

y

y � �x2

y � �x2 � 4

x0 2 4�4 �2

b) A translation of (  0 24

) 3 a)

4

0

8

12

16

�4

y

y � x2

y � (x � 2)2

y � (x � 2)2 � 3

x2 4�8 �6 �4 �2

b) A translation of (  22 23

)


8 a) y 5 cos 1 3 b) y 5 cos 4

9 a) y 5 2x3 2 5 b) y 5 2x3 1 510 a) y 5 2[(x 2 3)2 1 1] or y 5 2x2 2 12x 1 20 b) (3, 2)11 y

y � �(x � 3)2

y � (x � 3)2

x3

12 y

y � f(x) � 2

x

(0, 1)

Chapter 25Exercise 25.1 (page 323) 1 5x 2 1 ______

6 2 23x 1 1 ________

10

3 7x 2 26 _______ 12

4 2x 1 12 ________ 4

5 x2 1 x 1 4 __________ 3(x 2 1)

6 10 2 x 2 x2 ___________

4(x 2 2)

7 2x2 2 7x 2 21 _____________ 3(x 2 5)

8 29x 1 12 _________ 7(4x 2 1)

9 17 1 14x _________ 4(2x 2 3)


1 3x 1 5 _____________ (x 1 1)(x 1 2)

2 x2 1 6x 1 3 ___________ x(x 1 3)

3 4x2 1 5x 1 6 ____________ 3x(x 1 3)

4 2(x2 1 12)

_____________ (x 1 4)(x 2 4)

5 x 2 7 _____________ (x 2 1)(x 1 2)

6 2x2 1 2 _____________ (x 2 2)(x 2 4)

7 x2 2 2x 2 21 _____________ (x 2 3)(x 2 5)

8 4x2 2 9x 1 16 ______________ (3x 2 4)(x 1 2)

9 3(x 1 1)(2x 1 3)

_______________ (2x 1 1)(x 1 2)

10 2(x2 1 7x 2 3)

_____________ (x 1 3)(x 2 3)

11 x2 2 33x 1 24 _______________ (3x 2 2)(5x 2 4)

12 1 4(x2 2 x 2 1)

______________ (x 1 1)(5x 1 3)

Exercise 25.3 (page 327) 1 x 5 5 2 x 5 22 3 x 5 5 4 x 5 2 or x 5 6 5 x 5 5 or x 5 0 6 x 5 5.30 or x 5 1.70 7 x 5 20.86 or x 5 0.46 8 x 5 2.30 or x 5 21.30 9 x 5 20.21 or x 5 223.79

b) A one-way stretch of scale factor 2, parallel to the x-axis

3 a) A reflection in the x-axis b) A reflection in the y-axis c) A one-way stretch of scale factor 5 parallel

to the y-axis 4 y 5 cos 3

5 a) y 5 x2 1 5 b) y 5 2x2 2 5 6 a) y 5 3x 1 6 b) y 5 2x 1 2

7 a) A translation of (  0 1 )

b) A one-way stretch parallel to the y-axis of scale factor 3

c) A one-way stretch parallel to the x-axis of scale factor 0.5

d) A one-way stretch parallel to the y-axis of scale factor 5 and a one-way stretch parallel to the x-axis of scale factor 1 _ 3

8 a) y 5 x2 2 2x b) y 5 2x2 2 2x c) y 5 2(x 2 3)2 1 2(x 2 3) or

y 5 2x2 1 8x 2 15 9 a) y 5 1 _ 4 x

2 b) (2, 1) c) 1 _ 4 d) (1, 1 _ 4 )10 k 5 3

Mixed exercise 24 (page 319) 1 a) x 5 1 _ 2 b) (i) 2x 1 9 (ii) 2x 1 4 2 a) x 5 ±1 b) (i) x2 1 2x 1 7 (ii) 4x2 1 7 3 a) 24 b) (i) x2 2 6x 1 8 (ii) 4x2 2 4x 1 3 4 a) y

x

y � x2 � 4

�4

y � x2

b) Translation by (  0 24

) 5 y 5 2x 2 3 6 a) y

θ

�2

�1

0 90° 180° 270° 360°

y � sin θ � 1

7 a) Translation by (  21 0 )

b) One-way stretch with scale factor 1 _ 3 , parallel to the x-axis

c) One-way stretch with scale factor 4, parallel to the y-axis

d) Reflection in the y-axis


10 u 5 45° (Alternate angles) v 5 67° (Corresponding angles) w 5 68° (Angles on a straight line add up to

180°)

Exercise 26.4 (page 341) 1 a 5 35° (Angles in a triangle add up to 180°) 2 b 5 41° (Angles in a triangle add up to 180°) 3 c 5 43° (Angles in a triangle add up to 180°) 4 d 5 35° (Angles in an isosceles triangle) 5 e 5 60° (Angles in an equilateral triangle) 6 f 5 55° (The exterior angle of a triangle is

equal to the sum of the opposite interior angles)

7 g 5 141° (The exterior angle of a triangle is equal to the sum of the opposite interior angles)

h 5 39° (Angles on a straight line add up to 180°)

8 i 5 115° (Angles in a quadrilateral add up to 360° or angle sum of a triangle 5 180°)

9 j 5 123° (Angles in a quadrilateral add up to 360°)

k 5 57° (Angles on a straight line add up to 180°)

10 l 5 100° (Angles on a straight line add up to 180°)

m 5 84° (Angles in a quadrilateral add up to 360°)

Exercise 26.5 (page 344) 1 A: trapezium B: rhombus C: square D: kite E: rectangle F: square G: arrowhead H: parallelogram I: isosceles trapezium J: rhombus K: parallelogram 2 a) Square, rhombus b) Rectangle, parallelogram, kite, arrowhead c) Rectangle, parallelogram d) Trapezium e) Square, rhombus, kite 3 a)

0 2

2

4

6

8

y

x4 6 8

Rhombus


1 a) 5x ___ 6 b) x 2 19 ______

6

c) 2x2 1 8x 2 15 _____________ 10x

d) 13x 1 5 ________________ (4x 2 1)(3x 1 2)

2 a) 2(x 2 1)

_____________ (x 1 4)(x 1 2)

b) x2 1 4x 2 2 ___________ 3(x 1 2)

c) 4(x2 2 1)

_________ x(x 1 2)

d) 5x2 1 4x 2 19 ______________ (x 2 1)(3x 1 5)

3 a) x 5 1 b) x 5 2 c) x 5 8 d) x 5 2 or x 5 6 4 a) x 5 0.638 or x 5 21.305 b) x 5 0.422 or x 5 21.422

Chapter 26Exercise 26.1 (page 331) 1 16 cm2 2 50 cm2

3 160 m2 4 75 mm2

5 64 cm2 6 17.5 cm2

7 55 cm2 8 144 cm2

9 150 m2 10 120 m2

Exercise 26.2 (page 333) 1 24 cm2 2 72 cm2

3 64 m2 4 35 cm2

5 14 cm2 6 30 cm2

7 42 cm2 8 144 m2

9 42 cm2 10 16 cm2

Exercise 26.3 (page 337) 1 a 5 74° (Alternate angles) 2 b 5 65° (Allied angles) 3 c 5 125° (Corresponding angles) 4 d 5 65° (Alternate angles) e 5 65° (Corresponding or vertically opposite

angles) 5 f 5 110° (Alternate angles) g 5 110° (Corresponding or vertically opposite

angles) 6 h 5 55° (Allied angles) i 5 125° (Corresponding angles) j 5 125° (Alternate or vertically opposite

angles) 7 k 5 65° (Corresponding angles) l 5 65° (Alternate or vertically opposite

angles) m 5 115° (Allied angles) n 5 115° (Vertically opposite angles) 8 o 5 101° (Corresponding angles) p 5 101° (Corresponding angles) q 5 101° (Alternate angles) 9 r 5 97° (Allied angles) s 5 83° (Corresponding angles) t 5 29° (Angles in a triangle add up to 180°)


4 a) a 5 19° Angles in a triangle add up to 180°.

b) b 5 47° Base angles in an isosceles triangle are equal.

c) c 5 31° Exterior angle of a triangle is equal to the sum of the opposite, interior angles.

d 5 77° Angles on a straight line add up to 180° or Angles in a triangle add up to 180°.

5 a) a 5 126° Angles in a quadrilateral add up to 360°.

b) b 5 129° Angles on a straight line add up to 180°.

c 5 53° Angles in a quadrilateral add up to 360°.

6 a) Rectangle, parallelogram, kite, arrowhead b) Rhombus, isosceles trapezium, parallelogram c) Square, rectangle, isosceles trapezium 7 a) 900° b) Exterior angle 5 24°, interior angle 5 156° c) 36

Chapter 27Exercise 27.1 (page 352) 1 12 cm2 2 85 cm2 3 50 cm2 4 28 cm2

Exercise 27.2 (page 354) 1 a) 13 cm b) 5.83 m c) 6.24 cm d) 24 cm e) 7.64 cm f) 6.34 cm g) 30.50 m h) 9.35 m i) 8.30 cm 2 4.53 m 3 23.66 inches

Exercise 27.3 (page 357) 1 Yes 62 1 82 5 102 (This is 2 3 the 3, 4, 5 triangle) 2 No 42 1 72 ? 82

3 No 82 1 82 ? 112

4 Yes 7.5 2 1 182 5 19.52 (This is 1.5 3 the 5, 12, 13 triangle)

5 Yes 3.52 1 122 5 12.52 (This is 0.5 3 the 7, 24, 25 triangle)

6 No 52 1 102 ? 112

7 No 4.52 1 92 ? 102

8 Yes 28.82 1 122 5 31.22 (This is 2.4 3 the 5, 12, 13 triangle)

Exercise 27.4 (page 360) 1 a) (6.5, 3) b) (3, 5) c) (21.5, 6) d) (24.5, 0) e) (20.5, 25) f) (5.5, 24.5) 2 a) (1, 6) b) (4, 4) c) (5, 4.5) d) (5.5, 4.5) e) (1, 2) f) (25, 27)

b)

0 2

2

4

y

x4 6 8

Trapezium c)

0 2

2

4

y

x4 6 8

Parallelogram d)

0 2

2

4

y

x4 6 8

Kite 4 All the angles are 90° and the diagonals are

equal length. 5 Isosceles trapezium, parallelogram or rhombus

Isosceles trapezium

110°110°

70°70°

110°

110°

70°

70°

Parallelogram

Rhombus

110°

110°

70°

70°

Exercise 26.6 (page 348) 1 2340° 2 3240° 3 a) 131° b) 86°, 130°, 95°, 49° 4 a) 58° b) 90°, 100°, 123°, 105°, 122° 5 e 5 30°; i 5 150° 6 e 5 3.6°; i 5 176.4° 7 15 8 20

Mixed exercise 26 (page 349) 1 a) 9 cm2 b) 8 cm2 c) 30 cm2

2 a) 28 cm2 b) 300 cm2 c) 27 cm2

3 a) a 5 110° Corresponding angles. b 5 70° Allied angles or Angles on a

straight line.b) c 5 50° Alternate angles. d 5 50° Corresponding angles or

Opposite angles.c) e 5 36° Alternate angles. f 5 72° Angles on a straight line. g 5 72° Corresponding angles or angles

in a triangle.


4 a) A reflection in the line y 5 2 b) A reflection in the line x 5 3 c) A reflection in the line y 5 x 5 a) A reflection in the line y 5 0 b) A reflection in the line y 5 2x c) A reflection in the line y 5 x


�4 �3 �2 �10

A

C

1

1

�1

�2

�3

�4

2

3

4y

x2

B

3 4

2

�4 �3 �2 �10 1

A

C

D

B1

�1

�2

�3

�4

2

3

4y

x2 3 4

3

�4�3�2�10 1

CA

D

B

1

�1�2�3�4

2345678y

x2 3 4 5 6 7

4

C B90°

A

B� A�

5

90°O

6

C

B�

B120°

A�

A

Exercise 27.5 (page 362) 1 a) (5, 3, 0) b) (5, 0, 2) c) (0, 3, 2) d) (5, 3, 2) 2 a) (0, 6, 0) b) (6, 6, 0) c) (3, 3, 0) d) (3, 3, 5) 3 a) (8, 6, 0) b) (8, 3, 0) c) (8, 3, 4) d) (4, 6, 4)

Mixed exercise 27 (page 363) 1 a) 20 cm2 b) 23 cm2

2 a) 11.40 cm b) 7.55 cm c) 6.27 cm d) 4.99 cm 3 a) No 52 1 82 ? 92

b) Yes 92 1 122 5 152

(This is 3 3 the 3, 4, 5 triangle) c) Yes 17.5 2 1 602 5 62.52

(This is 2.5 3 the 7, 24, 25 triangle) d) No 5.22 1 8.32 ? 9.72

4 a) (3, 4) b) (4, 5 1 _ 2 ) c) (4 1 _ 2 , 4 1 _ 2 ) 5 1.93 m 6 30.3 cm2 7 15.66 m 8 a) (i) (15, 0, 0) (ii) (0, 10, 6) (iii) (15, 10, 6) b) (11, 5, 6)

Chapter 28Exercise 28.1 (page 368) 1

�4 �3 �2 �10 1

1

�1�2

�3

�4

2

A

C

B3

4y

y � 1

y � x

x2 3 4

2

�4 �3 �2 �10 1

1

�1�2

�3

�4

2

3

4y

y � �x

y � 2

x2 3 4

A

B

C

3 a) b)

c)


Exercise 28.4 (page 381) 1 a), b)

10 2 3 4 5

B

A

6

1

2

3

4

5

6y

x

c) An enlargement with scale factor 2, centre the origin

2 a), b)

10 2 3 4 5

A

6 7 8

1

2

3

4

5

6

7

8y

x

B

C

c) An enlargement with scale factor 3, centre (1, 2)

3 a), b)

10 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8y

x

A

B

c) An enlargement with scale factor 1 _ 4 , centre the origin

4 a), b)

10 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8y

x

C

A

B

c) An enlargement with scale factor 2 _ 3 , centre (2, 1)

5 a) An enlargement with scale factor 2, centre (0, 3)

b) An enlargement with scale factor 1 _ 2 , centre (0, 3)

c) An enlargement with scale factor 1 1 _ 2 , centre the origin

d) An enlargement with scale factor 2 _ 3 , centre the origin

7 a) A rotation of 90° anticlockwise about the origin

b) A rotation of 180° about (1, 0) c) A rotation of 90° clockwise about (1, 2) 8 a) A rotation of 90° clockwise about the

origin b) A rotation of 180° about (2, 1) c) A rotation of 90° anticlockwise about (1, 1) 9 A rotation of 150° clockwise about O10 a) A reflection in the line x 5 0 b) A rotation of 90° anticlockwise about (2, 4) c) A reflection in the line y 5 x d) A rotation of 90° anticlockwise about the

origin e) A reflection in the line y 5 2x


�2 �1 1

D0

A

B

C

2 3 4 5 6

�2

�1

1

2

3

4

5

6y

x

2

�2 �1 10 2 3 4 5 6

�2

�1

1

2

3

4

5

6y

x

D

C A

B

3 a) A translation by vector (  2 21

) b) A translation by vector (  21

22 )

c) A translation by vector (  22 1 )

d) A translation by vector (  24 2 )

4 a) A translation by vector (  25 2 )

b) A rotation of 90° anticlockwise about the origin

c) A rotation of 90° clockwise about (2, 2)

d) A translation by vector (  23 24

) e) A reflection in the line y 5 0 f) A reflection in the line y 5 x

g) A translation by vector (  2 26

) h) A rotation of 180° about (0, 2)


b)

3

�4 �3 �2 �1 0 1

1

�1

�2

�3

�4

2

3

4

y

x2

C

B

3 4

A

4

�4 �3 �2 �1 0 1

1

�1

�2

�3

�4

2

3

4

y

x2 3 4

A

B

C

D

5

10

1

2

3

4

5

6

7

8

9

y

x2 3 4 5 6 7 8 9

B

A

6

10

1

2

3

4

5

6

7

8

9

y

x2 3 4 5 6 7 8 9

A

B

6 a) A reflection in the line y 5 0 b) An enlargement with scale factor 2, centre

(0, 4) c) A translation by vector (  28

23 )

d) An enlargement with scale factor 2 1 _ 2 , centre the origin

e) A translation by vector (  26 4 )

f) An enlargement with scale factor 1 _ 3 , centre (5, 3)

7 a) A reflection in the line x 5 1 b) An enlargement with scale factor 2, centre

(1, 0) c) A rotation of 90° anticlockwise about the

origin d) A reflection in the line y 5 x e) An enlargement with scale factor 1 _ 2 , centre

the origin f) A translation by vector (  1

4 )

8 a) A reflection in the line y 5 2 b) A rotation of 90° clockwise or about the

origin c) A translation by vector (  2

22 )

d) A reflection in the line y 5 0 e) An enlargement with scale factor 1 _ 2 , centre

(0, 22) f) An enlargement with scale factor 1 _ 3 , centre

(2, 1) g) An enlargement with scale factor 3, centre

(2, 1)

Mixed exercise 28 (page 383) 1 y

x1

C

A

B

2 3 4�4 �3 �2 1

�4

�3

�2

�1

1

2

3

4

2 a)


8 2 3 (14.5 1 8.5) 5 46 cm and 2 3 (15.5 1 9.5) 5 50 cm

Exercise 29.3 (page 391) 1 a) 3.5 minutes b) 3 weeks c) 3 kg d) 2 m 2 a) 5.5 m b) £16 c) 665 km d) 0.1 cm2

Exercise 29.4 (page 392) 1 62.5 m/s 2 43 mph 3 7.5 miles 4 5 hours 5 278 cm3 6 60 mph 7 9 g/cm3 8 100 km/h 9 7.8 g/cm3

10 a) 8.9 g/cm3 b) 151.3 g11 19.3 kg12 a) 58.5 kg b) (i) 0.7692 m3 (ii) 769.2 m3

13 a) 60 km/h b) 64.3 km/h14 10.6 g/cm3

15 1156.3 people per square kilometre

Mixed exercise 29 (page 394) 1 a) 120 000 cm2 b) 371 mm2

c) 4200 cm2 d) 5 mm2

2 a) 3 000 000 mm2 b) 41.25 m2

c) 94 cm2 d) 600 cm2

3 a) 2130 cm3 b) 5.1 litres c) 421 000 ml d) 0.0917 litres 4 a) 26.5 cm and 27.5 cm b) 5.55 cm and 5.65 cm c) 1.225 m and 1.235 m 5 Greatest speed: greatest distance 4 least time

5 100.05 4 5.5 5 18.19 m/s 6 a) 33.5 mm and 34.5 mm b) Greatest perimeter: greatest length 3 3

5 34.5 3 3 5 103.5 mm 7 a) 1 hour 15 minutes b) 8 km/h 8 a) 60 mph b) 190 miles 9 5394 m2

Chapter 30Exercise 30.1 (page 401)Diagrams are not full size. The measurements given are correct for the suggested scale. 1

A5 cm

2 Scale 1 cm 5 20 m

5 cm

2.5 cm

7 a) A reflection in the line x 5 0 b) A rotation of 90° clockwise about the origin c) An enlargement with scale factor 3, centre

(24, 3) d) A translation by vector (  0

23 )

e) An enlargement with scale factor 1 _ 2 , centre the origin

f) A translation by vector (  23 2 )

g) A reflection in the line y 5 x 8 a) A reflection in the line x 5 0

b) A translation by vector (  1 23

) c) An enlargement with scale factor 3, centre

(0, 2) d) A rotation of 90° anticlockwise about

(2, 22) e) An enlargement with scale factor 1 _ 2 , centre

the origin f) A rotation of 180° about the origin

g) A translation by vector (  3 2 )

Chapter 29Exercise 29.1 (page 387) 1 a) 2500 cm b) 420 mm c) 236 cm d) 5100 mm 2 a) 30 000 cm2 b) 230 mm2

c) 95 200 cm2 d) 1.4 mm2

3 a) 900 cm2 b) 81.4 cm2

c) 720 m2 d) 9.4 m2

4 a) 3 200 000 cm3 b) 0.000 042 m3

c) 0.005 m3 d) 6 420 000 cm3

5 a) 2610 cm3 b) 9.5 litres c) 2400 ml d) 0.91 litre 6 All the units need to be the same;

change 50 cm to 0.5 m. 5 3 2 3 0.5 5 5 m3

7 624 cm3

Exercise 29.2 (page 389) 1 a) 56.5 m and 57.5 m b) 567.5 ml and 568.5 ml c) 23.925 seconds and 23.935 seconds 2 a) 633.5 g and 634.5 g b) 233.5 ml and 234.5 ml c) 8.25 m and 8.35 m 3 a) 28 to 32 m2

b) 15 cans (100 4 7 5 14.2) 4 a) 4.965 mm b) 4.975 mm 5 No; 595.5 mm is greater than 59.5 cm. 6 a) 14.9 cm and 15.1 cm b) 1.95 cm and 2.05 cm c) 1.95 cm and 2.05 cm 7 Maximum d: longest rod minus least depth 5

35.5 2 19.5 5 16 mm Minimum d: shortest rod minus greatest depth 5

34.5 2 20.5 5 14 mm


2 Scale 1 cm 5 1 km

7 cm

5 cm

4 cm

P Q

3

3 cm

3 cm

5 cm

5 cmA

D

B

C

4 Scale 1 cm 5 1 m

8 cm

6 cm 5 cm

E F

H G

T

5

9 cm7 cm

11 cmA B

C

Locus

6 Scale 1 cm 5 1 m

3 cm

2 cm

5 cm

7 1 cm 5 20 m

2.5 cm

7 cm

Locus

8 Scale 1 cm 5 1 km

B

C

A 7 cm

9.5 cm

5 cm

5 cm

5 cm

3 Scale 1 cm 5 10 m

6 cm

Hedge

Tree

4

60° 30°

5

A

B

5 cm

5 cm

3 cm

D

C

6 Scale 1 cm 5 1 m

1 cm

4 cm

2 cm

7 N

S

NW NE

SW

W E

SE

8

7 cm

3 cm

3 cm

9

7 cm 6 cm

8 cmA B

C

10 Scale 1 cm 5 10 m Note: You could have used a different scale.

6 cm2 cm

8 cm

Castle

Exercise 30.2 (page 404)Diagrams are not full size. The measurements given are correct for the suggested scale. 1

6 cm

3 cmA


6 Scale 1 cm 5 5 km

8 cm 6 cm

6 cm

7 Scale 1 cm 5 1 m

D C

A B5 cm

3 cm

8

70°8 cmE F

G

6 cm 5 cm

9 Scale 1 cm 5 1 m

3 cm

5 cm

5 cm

10 Scale 1 cm 5 1 m

10 cm

3 cm

3 cm8 cm

3 cm

3 cm

3 cm

3 cm

Chapter 31Exercise 31.1 (page 409)These answers have been calculated using the p button on a calculator. 1 37.7 cm (1 d.p.) 2 78.5 cm (1 d.p.) 3 282.7 cm (1 d.p.) 4 116.2 mm (1 d.p.) 5 207 mm (nearest whole number) 6 84.8 cm (1 d.p.)

9 Scale 1 cm 5 4 m

5 cm

3.5 cm

2 cm

2.5 cmH

ouse

2 cm

10 Scale 1 cm 5 5 miles

H K2 cm

3 cm

4 cm

Mixed exercise 30 (page 406)Diagrams are not full size. Measurements given are correct for the suggested scale. 1

100°

2

8 cm

3

7 cm

3 cm 3 cm

4

9 cmA B

C

5.2 cm6 cm 8 cm

Area 5 23.4 cm2

5 S R

P Q

5 cm

5 cm

8 cm


Exercise 31.6 (page 418)These answers have been calculated using the p button on a calculator. 1 a) 7037 cm3 (nearest whole number) b) 25 862 cm3 (nearest whole number) c) 113 097 cm3 (nearest whole number) d) 24 881 mm3 (nearest whole number) e) 11 781 mm3 (nearest whole number) f) 115 mm3 (nearest whole number) g) 707 m3 (nearest whole number) h) 370 m3 (nearest whole number) 2 11 045 cm3 (nearest whole number)

Exercise 31.7 (page 420)These answers have been calculated using the p button on a calculator. 1 a) 1810 cm2 (nearest whole number) b) 2281 cm2 (nearest whole number) c) 2827 cm2 (nearest whole number) d) 3958 mm2 (nearest whole number) e) 377 mm2 (nearest whole number) f) 465.6 mm2 (1 d.p.) g) 131.9 m2 (1 d.p.) h) 77.0 m2 (1 d.p.) 2 152 cm2 (nearest whole number) 3 a) 2111 cm2 (nearest whole number) b) 6993 cm2 (nearest whole number) c) 10 336 cm2 (nearest whole number) d) 3054 mm2 (nearest whole number) e) 4869 mm2 (nearest whole number) f) 1176.5 mm2 (1 d.p.) g) 190.0 m2 (1 d.p.) h) 59.7 m2 (1 d.p.)

Exercise 31.8 (page 424) 1 Plan

Front elevation

Side elevation

2 Plan

Front elevation

Side elevation

7 163.4 cm (1 d.p.) 8 14.8 cm (1 d.p.) 9 28.9 cm (1 d.p.)10 22.9 m (1 d.p.)11 9.1 m (1 d.p.)12 3.86 m (2 d.p.)

Exercise 31.2 (page 410)These answers have been calculated using the p button on a calculator. 1 a) 616 cm2 (nearest whole number) b) 2463 cm2 (nearest whole number) c) 20 106 cm2 (nearest whole number) d) 3848 mm2 (nearest whole number) e) 12 076 mm2 (nearest whole number) f) 5809 cm2 (nearest whole number) g) 9503 cm2 (nearest whole number) h) 75.4 cm2 (1 d.p.) i) 295.6 cm2 (1 d.p.) j) 36.3 m2 (1 d.p.) k) 21.2 m2 (1 d.p.) l) 4.91 m2 (2 d.p.) 2 a) 201 cm2 (nearest whole number) b) 452 cm2 (nearest whole number) c) 3848 cm2 (nearest whole number) d) 1018 mm2 (nearest whole number) e) 5281 mm2 (nearest whole number) f) 1810 cm2 (nearest whole number) g) 2290 cm2 (nearest whole number) h) 15.2 cm2 (1 d.p.) i) 75.4 cm2 (1 d.p.) j) 11.3 m2 (1 d.p.) k) 6.2 m2 (1 d.p.) l) 4.30 m2 (2 d.p.)

Exercise 31.3 (page 412) 1 160 cm2

2 126 cm2

3 340 cm2

4 227 cm2

5 108 cm2

6 212 cm2

Exercise 31.4 (page 414) 1 a) 515 cm3 b) 1376 cm3

c) 219.6 cm3 d) 385.5 cm3

e) 192 cm3 f) 780 cm3

2 18 000 cm3

Exercise 31.5 (page 416) 1 a) 1233 cm3 b) 17 334 cm3

c) 2092.8 cm3 d) 396.8 cm3

e) 2253.42 cm3 f) 528.82 cm3

2 119 000 mm3


6 a) 20 308.2 cm3 (1 d.p.) b) 1578.46 mm3 (2 d.p.) c) 5617.5 cm3 (1 d.p.) d) 0.8874 m3 (4 d.p.) 7 a) 2601 mm3 (nearest whole number) b) 3268.5 mm3 (1 d.p.) c) 106 437 cm3 (nearest whole number) d) 181 cm3 (nearest whole number) e) 23 091 m3 (nearest whole number) f) 27.5 m3 (1 d.p.) 8 a) 1659 m2 (nearest whole number) b) 2432 cm2 (nearest whole number) c) 16.0 cm2 (1 d.p.) d) 672 mm2 (nearest whole number) 9 a) 5202 mm2 (nearest whole number) b) 113 m2 (nearest whole number) c) 174 cm2 (nearest whole number) d) 546 cm2 (nearest whole number)10 a) Plan

Side elevation

Front elevation

b) Plan

Side elevation

Front elevation

c) Plan

Side elevation

Front elevation

d) Plan

Side elevation

Front elevation

3 Plan

Front elevation

Side elevation

4 Plan

Front elevation

Side elevation

5 Plan

Front elevation

Side elevation

6 Plan

Front elevation

Side elevation

Mixed exercise 31 (page 425)Where necessary, the p button on a calculator has been used to work out these answers, which are given to 3 s.f. (unless stated otherwise). 1 a) 44.6 cm b) 93.3 cm c) 204 cm d) 101 mm 2 a) 127 cm2 b) 24.5 m2

c) 238 mm2 d) 69.4 mm2

e) 125 cm2 f) 72.1 m2

3 Check students’ diagrams. 4 a) 89.54 cm2 (2 d.p.) b) 75 cm2

c) 176.5 cm2 (1 d.p.) d) 178.23 cm2 (2 d.p.) 5 a) 208 cm3 b) 156 cm3

c) 169.32 cm3 (2 d.p.) d) 192 cm3

e) 257.544 cm3 (3 d.p.) f) 655.2 cm3 (1 d.p.)

229Higher WJEC GCSE Mathematics © 2010, Hodder Education Answers to the Student’s Book exercises

2 a) 47° b) 70° c) 25° 3 a) 31.6 cm b) 19.0 cm

Exercise 33.2 (page 451) 1 a) a 5 7.1 cm b) angle C 5 51° c) angle B 5 39° d) r 5 16.1 cm e) p 5 7.0 cm f) angle Y 5 33° 2 a) angle R 5 56°, angle P 5 51°, QR 5 7.1 cm b) angle Y 5 20°, angle X 5 37°, YZ 5 16.8 cm c) angle B 5 38°, AB 5 6.9 cm, AC 5 4.3 cm (1 d.p) d) angle C 5 81°, AB 5 9.3 cm, BC 5 5.8 cm (1 d.p) e) angle A 5 49°, angle B 5 59°, AC 5 7.8 cm (1 d.p) f) angle Q 5 57°, PQ 5 15.0 cm, QR 5 11.5 cm (1 d.p)

Exercise 33.3 (page 454) 1 a) a 5 7.1 cm b) angle A 5 25° c) b 5 6.3 cm d) angle C 5 57° 2 a) angle A 5 43°, angle B 5 54°, AB 5 7.3 cm (1 d.p) b) angle A 5 41°, angle B 5 61°, angle C 5 78° c) angle A 5 60°, angle C 5 74°, AC 5 5.4 cm (1 d.p) d) angle A 5 50°, angle B 5 69°, angle C 5 61° e) angle P 5 49°, angle R 5 33°, PR 5 10.2 cm (1 d.p) f) angle P 5 54°, angle Q 5 26°, angle R 5 100° g) angle P 5 66°, angle Q 5 61°, angle R 5 53° h) angle X 5 27°, angle Y 5 38°, XY 5 18.6 cm (1 d.p) i) angle X 5 49°, angle Y 5 82°, angle Z 5 49° j) angle X 5 16°, angle Y 5 45°, XY 5 16.9 cm (1 d.p) 3 a) 19.5 cm b) 56° 4 a) 54° b) 80° c) 46°


0

�1

θ

y

90° 180° 270° 360° 450° 540°

1

0.5

y � sin θ

b) 5 30°, 150°, 390° and 510° 2 a)

0

�1

�0.5θ

y

90° 180° 270° 360° �180° �90°

1 y � cos θ

b) 5 2120°, 120° and 240°

Chapter 32All answers are given correct to 3 significant figures.

Exercise 32.1 (page 433) 1 a 5 3.50 cm b 5 6.76 cm

c 5 8.18 cm d 5 8.57 cm e 5 2.01 cm f 5 1.33 cm g 5 3.41 m h 5 2.00 m

2 2.05 m 3 a) 6.88 cm b) 68.8 cm2

Exercise 32.2 (page 436) 1 a 5 9.24 cm b 5 13.4 cm

c 5 10.5 cm d 5 11.4 m e 5 23.0 cm f 5 10.7 cm g 5 7.71 m h 5 26.7 m

2 9.53 km 3 a) 1.89 m b) 10.3 m3

Exercise 32.3 (page 440) 1 a 5 47.2° b 5 9.46°

c 5 45.6° d 5 69.2° e 5 52.8° f 5 42.7° g 5 39.6° h 5 56.7°

2 69.8° 3 36.9° 4 32.3°

Mixed exercise 32 (page 442) 1 a 5 9.80 cm b 5 13.9 m c 5 9.79 m d 5 14.5 cm e 5 5.57 m f 5 73.6 m 2 a 5 52.0° b 5 39.4° 3 a) 8.09 m b) 50.8° 4 a) 75.7°, safe b) 6.09 m c) 3.39 m 5 a)

120 km70°

N

b) (i) 113 km (ii) 41.0 km 6 70.0° 7 8.28 cm 8 a) 51.3° b) 93.7 cm2

9 35.3 m 10 4.34 m

Chapter 33Exercise 33.1 (page 447)All lengths and areas are given correct to 3 s.f. and all angles to the nearest degree. 1 a) 15.0 cm2 b) 33.2 cm2

c) 16.7 cm2 d) 50.0 cm2

e) 21.6 cm2 f) 24.3 cm2

g) 59.7 cm2 h) 53.1 cm2


3 a) y

θ0 90° 180°�180° �90°

�1

1y � sin θ

b) 5 44° and 136° approximately 4 a) y

θ 90° 180° 270° 360° 450° 540°

�1

1

0

y � cos θ

b) 5 114°, 246° and 474° approximately

5 5 (53°), 127°, 413° and 487° 6 a)

0

�2

�3

�1

θ

y

180° 360°90° 270°

2

3

1

y � tan θ

b) 5 50° and 230° approximately

Exercise 33.5 (page 462) 1 a) 3, 360° b) 1, 72° c) 3, 90° d) 4, 720° e) 5, 120° f) 2, 450° 2

0

�2

�3

�1

y

y � 3 cos θ2

3

1

θ30° 60° 90° 120° 150° 180°

3

0

�1

y

y � sin 2 θ1

θ30° 60° 90° 120° 150° 180°

4

0

�1

y

y � cos 2 θ 1

θ 45° 90° 135° 180° 225° 270° 315° 360°

5

0

�2

�3

�1

y

y � 2.5 sin θ 2

3

1

θ 90° 180° 270° 360°


d) (i)

0

�1

y

y � tan θ

2

1

60° �60° �120° �180° 120° 180° θ

(ii) Approximately 2117° and 63° e) (i) 3, 360° (ii) 1, 72° (iii) 3, 90° (iv) 4, 720° (v) 5, 540° (vi) 2, 450° (vii) 5, 120° (viii) 2, 600° f) (i)

0

�3

�2

�1

y

y � 3 sin θ

2

1

3

30° 60° 90° 120° 150° 180° θ

(ii)

0

�1

y y � cos 4θ 1

θ 60° 120° 180°

(iii)

0

�3

�2

�1

y y � 3 cos 2θ

2

1

3

θ 60° 120° 180° 240° 300° 360°

6

0

�1

y

y � cos 3 θ1

θ30° 60° 90° 120° 150° 180°

The solutions of cos 3 5 20.5 are 40°, 80° and 160°

Mixed exercise 33 (page 464) 1 a) 17.1 cm2 b) 30.9 cm2

c) 46.8 cm2 d) 16.3 cm2

2 a) 24° b) 31° c) 73° 3 a) 7.95 cm b) 16.2 cm 4 a) 8.1 cm b) 29° c) 53° d) 15.9 cm 5 a) PR

∧

Q 5 52°, QP∧

R 5 55°, QR 5 12.4 cm b) XY

∧

Z 5 53°, ZX∧

Y 5 19°, YZ 5 9.7 cm c) AC

∧

B 5 49°, AC 5 8.8 cm, AB 5 6.7 cm d) PQ

∧

R 5 57°, QR 5 7.2 cm, PR 5 9.0 cm 6 Side XY is longest at 8.7 cm 7 a) 6.7 cm b) 9.2 cm c) 48° d) 51° 8 a) CA

∧

B 5 36°, CB∧

A 5 63°, AB 5 6.3 cm b) CA

∧

B 5 41°, AB∧

C 5 51°, BC∧

A 5 88° c) QP

∧

R 5 66°, PQ∧

R 5 47°, PQ 5 5.6 cm d) YX

∧

Z 5 53°, XY∧

Z 5 68°, XZ∧

Y 5 59° 9 82°, 60°, 38°10 a) 60° b) 67°11 a) (i)

0

�1

y

y � sin θ 1

θ 60° 120° 180° 240° 300° 360°

(ii) Approximately 44° and 136°

b) (i)

0

�1

y

y � cos θ 1

60° �60° �120° �180° 120° 180° θ

(ii) Approximately 2107° and 107° c) 233°, 487°, 593°


Exercise 34.4 (page 479) 1 a) 2.39 cm2 b) 28.4 cm2 c) 46.6 cm2

2 1390 m3

3 a) 166 cm2 b) 67.1 cm2 c) 255 cm2

4 990 cm3

5 a) 7.3 cm, 224 cm3 b) 4.5 cm, 61.7 cm3

c) 10.0 cm, 776 cm3

6 a) 409 cm3

b) 10.7 cm, 382 cm2

7 a) Check students’ answers. b) 2.56 litres 8 1740 cm3

9 4.8 cm, 174 cm3

10 785 cm3


1 a) √___

6a2 , b) 4bc ____ d

and d) 2pa ____ 5

2 a) Length b) Volume c) None d) Area 3 a) Arc length 6.7 cm, area 20.8 cm2

b) Arc length 34.9 cm, area 136 cm2

4 a) 115° b) 302° 5 439.5 cm2

6 3.1 cm 7 38 700 cm3 (to 3 s.f.) 8 8.4(69…) cm

Chapter 35Exercise 35.1 (page 484) 1 A, B and C are similar. D and E are similar. 2 XY 5 6 cm, YZ 5 9 cm 3 AC 5 4.8 cm, AX 5 4.5 cm, XB 5 7.5 cm 4 42 m 5 x 5 2.7 cm, y 5 3 cm 6 x 5 2.143 cm, y 5 10 cm 7 27 cm 8 a) Angle BAC 5 Angle CDE

(Alternate angles)Angle ACB 5 Angle DCE(Vertically opposite angles)Angle ABC 5 Angle CED (Alternate angles)Corresponding angles are equal so triangle ABC is similar to triangle CDE.

b) AC 5 13.5 cm, BC 5 27 cm


�12�16 �8 �4 0

2

�2�4�6�8

4C� D�

A�

A

B

CD

B�

y

x4

(iv)

0

�1

y

y � 0.5 sin 3θ

1

θ 60° 120° 180° 240° 300° 360°

g)

0

�1

y y � sin 5θ 1

θ 60° �60° �120° �180° 120° 180°

Values of are approximately 2139°, 2113°, 267°, 241°, 5°, 31°, 77°, 103°, 149° and 175°

Chapter 34Unless stated otherwise, lengths are given correct to 1 d.p. and areas and volumes to 3 s.f.

Exercise 34.1 (page 469) 1 a) and b) 2 a) and c) 3 b) and c) 4 a) Length b) Volume c) Area 5 a) Volume b) Area c) Nonsense d) Length

Exercise 34.2 (page 472) 1 a) 6.1 cm b) 12.9 cm c) 42.1 cm 2 a) 16.5 cm2 b) 40.5 cm2 c) 221 cm2

3 a) 52° b) 103° c) 91° d) 257° 4 a) 9.1 cm b) 6.8 cm c) 8.6 cm 5 a) 4.9 cm b) 5.7 cm c) 9.0 cm d) 6.5 cm 6 187°, 62.6 cm2

7 208 cm (to 3 s.f.)

Exercise 34.3 (page 475) 1 a) 47.1 cm2 b) 107 cm2 c) 51.1 cm2

2 a) 37.7 cm3 b) 126 cm3 c) 39.3 cm3

3 a) 32 cm3 b) 240 cm3 c) 106 cm3

4 158 cm2

5 12 cm 6 a) 3.6 cm b) 3.1 cm c) 6.2 cm 7 a) 2.6 cm b) 5.7 cm c) 9.9 cm 8 a) 327 cm2 b) 845 cm2 c) 1260 cm2

9 a) 556 cm3 b) 2310 cm3 c) 4190 cm3

10 a) 8.5 cm b) 7.3 cm c) 6.6 cm d) 12.4 cm


Chapter 36All answers which are not exact are given correct to 1 decimal place.

Exercise 36.1 (page 496) 1 9.9 cm 2 9.7 cm 3 5.3 cm 4 8.5 cm 5 8.6 cm 6 a) 12.0 cm b) (i) 26.6° (ii) 14.4° 7 a) 11.0 cm b) 35.5° 8 a) 10.9 m b) 68.9° c) 11.7 m

Mixed exercise 36 (page 498) 1 9.6 cm 2 9.9 cm 3 a) 61.7° b) 13.7 cm 4 a) 38.4 m b) 39.1 m c) 44.5° 5 a) 30.3° b) 13 cm c) 14.8 cm d) 28.3° 6 a) 4.3 m b) 79.2° 7 a) 8.7 cm b) 30.7° 8 a) 3.4 cm b) 47.0°

Chapter 37Exercise 37.1 (page 502) 1 CX 5 CY (Given.) Angle ACB is common AC 5 BC (Given.) Therefore triangle CXB is congruent to triangle

CYA (SAS). 2 DZ 5 DV 1 VZ VC 5 VZ 1 ZC DV 5 ZC (Given.) So DZ 5 VC Triangle DXC is isosceles.

DX 5 XC (Given.) So angle BDZ 5 angle ACV Angle XAB 5 angle ACV

(Alternate angles.) Angle XBA 5 angle BDZ

(Alternate angles.) So triangle AXB is isosceles. So AX 5 XB So DB 5 AC Therefore triangle DBZ is congruent to triangle

CAV (SAS). 3 a) a 5 20 cm, b 5 12 cm b) x 5 27°, y 5 23°

2 a) (1, 8) b) 23 3

0

2

4

6

8

10

12

A B

D C

2 4 6 8 10 12 14 16

y

x

B� A�

C� D�

4 a) (0, 3) b) 22 5 a) (2, 3) b) 20.5

Exercise 35.3 (page 490) 1 a) 1.6 b) 6.4 cm c) 10 cm2

d) 25.6 cm2

e) The ratio of the areas is the square of the ratio of the lengths.

2 a) 4, 8 b) 25, 125 c) 100, 1000 d) 0.25, 0.125 3 a) 6, 216 b) 8, 512 c) 7.07, 353.6 d) 0.316, 0.0316 4 422 ml 5 a) 3.6 m b) 5.76 litres 6 a) 15 b) 225 c) 4.52 m3

7 15.1 cm 8 0.0226 m2 or 226 cm2

9 3.30 : 2.08 : 3.7810 77.44 cm2

Mixed exercise 35 (page 492) 1 a 5 3 cm b 5 4.5 cm c 5 10 cm d 5 7.5 cm e 5 2.5 cm 2 a) 96 cm2

b) 16 cm 3

B� A�

C� D�

A B

D C12345678

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

y

x

A9(14, 2.5) B9(6.5, 2.5) C9(6.5, 7) D9(14, 7) 4 a) (0, 4) b) 22 5 10 cm2

6 a) 650 m2

b) 0.5 litre or 500 ml 7 a) 1.26 : 1.14 or 1.10 : 1 b) 61.9 cm2


8 Angle ACB 5 angle ABC 5 70° (Base angles of an isosceles triangle are equal.)

Angle ACD 5 40° (Alternate angles.) So angle BCD 5 110° So angle DBC 5 angle BDC 5 35°

(Base angles of an isosceles triangle are equal.) 9 Angle BCD 5 122° (Alternate angles.) Angle DCE 5 58° (Angles on a straight line add up to 180°.) Angle BCY 5 58° (Vertically opposite angles are equal.) Angle ABC 5 58° (Angles on a straight line add up to 180°.) Angle BAD 5 89° (Angle sum of triangle BAE.)10 Angle BDC 5 80° (Base angles of an isosceles triangle are equal.) So angle DBC 5 20° (Angles in a triangle add up to 180°.) So angle ABC 5 40° (BD bisects angle ABC.) So angle BAC 5 60° (Angle sum of triangle BAC.)11 Angle BRQ 5 125° (Angles in a quadrilateral add up to 360°.) Angle BRP 5 55° (Angles on straight line add up to 180°.) So a 5 145° (Exterior angle of triangle is equal to the sum of

the opposite, interior angles.)


1 x 5 22° (Alternate angles.) y 5 26° (Base angles of an isosceles triangle

(PRS) are equal and Angles in a triangle add up to 180°.)

z 5 48° (Alternate angles.) 2 x 5 70° (Angles in a triangle add up to 180°.) y 5 60° (Alternate angles.) z 5 130° (Vertically opposite angles.) 3 Angle DCB 5 80° (Angles in a quadrilateral add up to 360°.) So angle ACB 5 angle ACD 5 40° (Given that CA bisects angle DCB.) Angle DAC 5 10° (Angles in a triangle add up to 180°.) So angle CAB 5 70° (80° – 10°.) Since CAB 5 ABC, triangle CAB is isosceles. 4 a 5 87° (Allied angles.) b 5 74° (Alternate angles.) 5 x 5 100° (Angles in a quadrilateral add up to 360°.)

4 a) (i) Angle QPX 5 angle XRS (Alternate angles.)

Angle PQX 5 angle RSX (Alternate angles.)

PQ 5 SR (Given.) So triangle PQX is congruent to triangle

RSX (ASA). (ii) Angle QRX 5 angle SPX

(Alternate angles.) Angle RQX 5 angle PSX

(Alternate angles.) QR 5 SP (Given.) So triangle RQX is congruent to triangle

SPX (ASA). b) SX 5 XQ -

(Since triangles PQX and RSX are congruent.)PX 5 XR (Since triangles RQX and SPX are congruent.)So X is the midpoint of SQ and PR.SR 5 RQ (PQRS is a rhombus.)Angle RSX 5 angle RQX (Triangle QSR is isosceles.)XR is common to both triangle SXR and triangle QXR.So triangle SXR is congruent to triangle QXR (SAS).So angle SXR 5 angle QXR.Angle SXR 5 angle QXR 5 90° (Since SQ is a straight line.)

5 A and B 6 AP 5 PB 5 BQ 5 QC 5 CR 5 RD 5 DS 5 AS (Because P, Q, R and S are the midpoints of the

sides of the square ABCD) So all the triangles are isosceles and congruent

and all the marked angles are 45°.

A P

R

QS

B

D C

So the interior angles of the quadrilateral PQRS are 90°

(Angles on a straight line.) All the sides, PS, SR, RQ and QP, are equal. So PQRS is a square. 7 Angle DCB 5 angle DCA

(CD bisects angle ACB.) Angle DCA 5 angle CAE

(Alternate angles.) Angle DCB 5 angle AEC

(Corresponding angles.) So angle AEC 5 angle CAE So triangle ACE is isosceles and AC 5 CE.


4 h 5 66° 5 i 5 50° j 5 56° k 5 34° 6 m 5 49° n 5 41° 7 p 5 45° q 5 60° 8 r 5 20 cm 9 s 5 40° t 5 80° u 5 50°10 v 5 95° w 5 126°11 x 5 42°12 y 5 30°13 z 5 57°14 a 5 44° b 5 46°15 c 5 30°16 d 5 90° e 5 40° f 5 32°17 g 5 120°18 h 5 10 cm19 i 5 98° j 5 120°20 k 5 18°

Exercise 38.2 (page 517) 1 a 5 142° 2 b 5 70° c 5 50° 3 d 5 50° e 5 50° 4 f 5 90° g 5 35° h 5 55° i 5 35° 5 j 5 36° k 5 108° 6 m 5 74° n 5 32° 7 p 5 70° q 5 40° 8 r 5 45° 9 s 5 61° t 5 61° u 5 58°10 v 5 85° w 5 71°

Mixed exercise 38 (page 519) 1 a 5 102° b 5 39° 2 c 5 13° 3 d 5 81° e 5 90° 4 f 5 25° g 5 75° 5 h 5 62° i 5 28° j 5 29° 6 k 5 5.7 cm m 5 38.9° 7 n 5 83° p 5 52° 8 q 5 122° r 5 244° 9 s 5 55° t 5 35° u 5 35°10 v 5 64° w 5 52°

Chapter 39Exercise 39.1 (page 521) 1

0

5

15 20 25 30 35 40 45 50

10

15

20

Age (years)

Freq

uenc

y

6 A

C

D BE

AD 5 AB, DE 5 DB and AE is common to triangles ADE and ABE.

So triangles ADE and ABE are congruent (SSS). So angle DEA 5 angle BEA Angle DEA 5 angle BEA 5 90°. (Since DB is a straight line.) CD 5 CB, DE 5 EB and CE is common to

triangles CDE and CBE. So triangles CDE and CBE are congruent (SSS). So angle CED 5 angle CEB Angle CED 5 angle CEB 5 90° (Since DB is a straight line.) This proves that the diagonals of a kite cross at

right angles. 7

E

A

B

C D

CE 5 ED, angle CEA 5 angle DEA and EA is common to triangles AEC and AED.

So triangles AEC and AED are congruent (SAS).

So AC 5 DA CE 5 ED, angle CEB 5 angle DEB and EB is

common to triangles CEB and DEB. So triangles CEB and DEB are congruent (SAS). So CB 5 BD AE 5 EB, angle AED 5 angle BED and ED is

common to triangles AED and BED. So triangles AED and BED are congruent

(SAS). So DA 5 BD So AC 5 CB 5 BD 5 DA

Chapter 38Exercise 38.1 (page 513) 1 a 5 20° b 5 40° c 5 60° 2 d 5 104° e 5 85° 3 f 5 110° g 5 35°


3

5

0

10Freq

uenc

y

15

20

172168 176 180

Height (cm)

184 188 192

25

4 a)

5

0

10 Freq

uenc

y

15

20

10 0 20 30 No. of words

40 50 60 70

25

30

Book 1

Book 2

b) Possible answers include:Book 1 has a wider variety of sentence lengths.Book 2 has more shorter sentences.

Exercise 39.3 (page 529) 1 a), e)

1

0

2

Num

ber

of b

ad p

each

es

3

4

2 0 4 6 Delivery time (hours)

8 10 12 14 16 18

5

6

d)

b) Strong positive correlation c) 10.4 hours f) About 3 2 a), e)

10

0

20 Pap

er 2

30

40

10 0 20 30 Paper 1

40

d)

b) Strong positive correlation c) 29.0 f) About 32 3 a), e)

10

0

20

Pet

rol c

onsu

mpt

ion

(mpg

)

30

40

1 0 2 3 Engine size (litres)

4

50

5

d)

b) Strong negative correlation c) 32.9 mpg f) About 30 mpg

2 a)

0

10

50 70 90 110 130 150 170

20

30

Rainfall (cm)

Freq

uenc

y

b) 70 < r , 90 c) 90 < r , 110 3 a)

0

10

60 70 80 90 100 110

20

30

Weight (kg)

Freq

uenc

y

b) 80 < w , 90 c) 80 < w , 90

4 Lifetime of lightbulb (h hours)

Frequency

300 < h , 400 3

400 < h , 500 9

500 < h , 600 12

600 < h , 700 15

700 < h , 800 13

800 < h , 900 19

900 < h , 1000 13

1000 < h , 1100 10

1100 < h , 1200 6


5

0

10

Freq

uenc

y

15

20

60 12 18Weight (kg)

24 30

2

20

0

40

Freq

uenc

y

60

80

30 15 45 60 Time (minutes)

75 90 105

100

120

140


Chapter 40Exercise 40.1 (page 536) 1 a) (i) 3 (ii) 3 (iii) 5 (iv) 3.456 b) (i) 50 (ii) 50 (iii) 5 (iv) 49.632 c) (i) 1 (ii) 1 (iii) 5 (iv) 1.443 d) (i) 2 (ii) 2 (iii) 4 (iv) 1.728 2 a) 1.96 b) 1.04 c) 2.987 d) 4.456 3 a) 4.51 b) 27.04 c) 12.75 d) 3.1625 4 £1.07 5 7.52

Exercise 40.2 (page 542) 1 a) (i) 50 (ii) 13.6 b) (i) 20 (ii) 7.22 c) (i) 40 (ii) 14.1 d) (i) 20 (ii) 6.55 2 a) (i) 0–4 (ii) 20 (iii) 6.98 b) (i) 0–9 (ii) 40 (iii) 15.5 c) (i) 250–299 (ii) 250 (iii) 164.7 d) (i) 30–44 (ii) 45 (iii) 31.5 3 a) (i) 3–5 (ii) 3.62 b) (i) 4–7 (ii) 7.3 c) (i) 30–39 (ii) 32.25 d) (i) 75–99 (ii) 75 4 30.3 5 179.5

Exercise 40.3 (page 546) 1 a) (i) 50 cm (ii) 31.46 cm b) (i) 40 g (ii) 25.11 g c) (i) 16 cm (ii) 68.56 cm d) (i) 50 mm (ii) 16.04 mm 2 a) (i) 0 < x , 3 (ii) 9.85 days b) (i) 120 < x , 130 (ii) 121.36 g c) (i) 22 < x , 26 (ii) 21.2 cm d) (i) 50 < x , 55 (ii) 54.83 miniutes 3 a) 250 < x , 300 b) 250 < x , 300 c) £270.00 4 18.3 g 5 165.71 pence

Mixed exercise 40 (page 548) 1 a) (i) 6 (ii) 7 (iii) 4.488 b) (i) 50 (ii) 7 (iii) 50.27 c) (i) 1 (ii) 6 (iii) 1.42 d) (i) 6 (ii) 5 (iii) 6.36 e) (i) 2 (ii) 5 (iii) 3.47 f) (i) 18 (ii) 5 (iii) 17.54 2 £11.36 3 2.71 (2 d.p.) 4 a) (i) 20 cm (ii) 22.09 cm b) (i) 100 km/h (ii) 30.11 km/h c) (i) 50 seconds (ii) 59.33 seconds d) (i) 25 kg (ii) 55.8 kg 5 a) 10 < x , 20 b) 10 < x , 20 c) 19.31 minutes 6 £24 888.89 (or £24.89 thousand)

4 a)

10

0

20 M

ark

30

40

400 500 Circumference of head (mm)

50

600

b) No correlation c) No d) Don’t know the ages of the people involved.

Mixed exercise 39 (page 531) 1 a)

Time (t mins)

0 < t , 5 5 < t , 10 10 < t , 15 15 < t , 20

Frequency 6 7 5 2

b)

0 5

2

4

Freq

uenc

y

6

8

y

x10 15Time (minutes)

20

c) 5 < t , 10 d) 5 < t , 10

2 a)

04030

5

10

Freq

uenc

y

15

20

y

x50 60Marks

70 80 90

b) Suitable comments include the following.The marks range from 30 to 90.There are few low and high marks.Most of the marks are around 50 to 60.

c) 50 < m , 60 d) 70 e) 2 _ 7

3 a), c)

0 1 2 3 4 5

Age of rabbit (months)

6 7 8 9

200

400

600

800

1000

1200

1400

Ave

rage

wei

ght (

g)

1600

1800

b) Strong positive correlation d) (i) About 800 g (ii) About 1700 g e) No. A rabbit’s weight cannot keep increasing.


Mixed exercise 41 (page 559) 1 7 _ 9 2 0.35 3 0.13 4 a) 7 __ 20 b) 5 red, 8 white, 7 blue 5 12 days 6 a) 10

__ 20 5 1 _ 2 , which is the probability of getting heads with a fair coin.

b) She is not necessarily correct. She should carry out more trials.

7 a) About 67 b) (i) 39

___ 400 (ii) 279 ___ 400 (iii) 121

___ 400

Chapter 42Exercise 42.1 (page 564) 1 a) 15

__ 23 b) 18 __ 23

2 a) 0.85 b) 0.65 3 a) 0.53 b) 0.168 c) 0.412 d) 0.83 e) 0.56 4 The events are not mutually exclusive: 3 _ 5 1 1 _ 2 . 1.

5 1 _ 4

6 1 ___ 125

7 a) 0.12 b) 0.42

8 a) 1 ___ 121 b) 9

___ 121 c) 16 ___ 121

9 a) 1 __ 16 b) 9 __ 16 c) 6 __ 16 5 3 _ 8 10 a) 0.216 b) 0.064 c) 0.432

Chapter 41Exercise 41.1 (page 552) 1 0.9 2 5 _ 6

3 0.15 4 3 __ 10

5 0.57 6 17 __ 49

Exercise 41.2 (page 553) 1 0.7 2 6 __ 20 5 3 __ 10

3 0.35 4 4 __ 11

5 0.11 6 1 __ 20

Exercise 41.3 (page 556) 1 4 2 12 3 17 4 £300 5 a) 50 b) 150 6 50 yellow, 75 blue, 125 green

Exercise 41.4 (page 557) 1 a) Score 1 2 3 4 5 6

Relative frequency

0.14 0.09 0.17 0.22 0.11 0.27

b) 1 _ 6 or 0.17 (to 2 d.p.) c) Yes, the dice is biased. The relative frequency

for 2 seems too low and the relative frequencies for 4 and 6 seem too high.

2 Small number of trials. Cannot be predicted from these past events.

3 a) Party Relative frequency

Labour 0.3

Conservative 0.47

Lib. Dem. 0.17

Other 0.06

b) 0.3 4 a) 10

__ 20 5 0.5 which is the probability of tossing a coin and getting a head.

b) 102 ___ 300 5 0.34 which is very different from 0.5.

c) Emma. More trials give a more reliable indication.

5 a) Score 1 2 3 4

Relative frequency 0.27 0.23 0.26 0.24

b) Yes. All values are approximately equal to 0.25.

c) Either 225, 225, 225, 225 (if the spinner is fair) or

Score 1 2 3 4

Relative frequency 240 204 237 219

6 a) 200 is a large number of trials.b) Travel by … Bus Car Bike Walk

Estimate of probability

0.245 0.24 0.115 0.4



Six

First dice Second dice

Not a six

Six

Not a six

Six

Not a six

Outcome

NS

NN

SS

SN

Probability

� �16

16

136

� �16

56

536

� �56

16

536

� �56

56

16

56

16

56

16

56

2536

b) (i) 1 __ 36 (ii) 10 __ 36 5 5 __ 18

2 a)

0.2

Late

First day Second day

Not late0.8

0.2Late

Not late0.8

0.2Late

Not late

Outcome

NL

NN

LL

LN

Probability

0.8 � 0.2 � 0.16

0.8 � 0.8 � 0.64

0.2 � 0.2 � 0.04

0.2 � 0.8 � 0.160.8

b) (i) 0.64 (ii) 0.32

3 a)

Black

First disc Second disc

White

Black

White

Black

White

Outcome

WB

WW

BB

BW

Probability

� �710

710

49100

� �710

310

21100

� �310

710

21100

� �310

310

310

310

710

710

710

310

9100

b) (i) 49 ___ 100 (ii) 58

___ 100 5 29 __ 50 (iii) 51

___ 100

4 a)

W

0.5

0.5

0.5

0.50.2

0.2

0.2

0.2

0.3 0.3

0.3

0.3

D

First match Second match

L

Outcome Probability

W

D

L

W

D

L

W

D

L

DW

DD

DL

WW

WD

WL

LW

LD

LL

0.3 � 0.5 � 0.15

0.3 � 0.3 � 0.09

0.3 � 0.2 � 0.06

0.5 � 0.5 � 0.25

0.5 � 0.3 � 0.15

0.5 � 0.2 � 0.10

0.2 � 0.5 � 0.10

0.2 � 0.3 � 0.06

0.2 � 0.2 � 0.04

b) (i) 0.25 (ii) 0.50 (iii) 0.38


Mixed exercise 42 (page 573) 1 a) 0.85 b) 0.40 2 a) 17

__ 25 b) 15 __ 25 5 3 _ 5

3 a) 0.5 b) 0.9 4 a) 1 __ 25 b) 4 __ 25 c) 6 __ 25

5 a) 1 __ 42 b) 6 __ 42 5 1 _ 7 c) 30 __ 42 5 5 _ 7

6 1 ___ 216

5 a)

H

First toss Second toss Third toss

T

Outcome Probability

12

12

H

T

12

12

H

T

12

12 H

T12

12

12

12

12

12

12

12

12

12

18

12

H

T

H

T

H

T

TTH

TTT

THH

THT

HTH

HTT

HHH � � �

12

12

12

18� � �

12

12

12

18� � �

12

12

12

18� � �

12

12

12

18� � �

12

12

12

18� � �

12

12

12

18� � �

12

12

12

18� � �

HHT

b) (i) 1 _ 8 (ii) 3 _ 8 (iii) 4 _ 8 5 1 _ 2


Yellow

First ball Second ball

Red

Yellow

Red

Yellow

Red

Outcome

RY

RR

YY

YR

Probability

� �711

610

42110

� �711

410

28110

� �411

710

28110

� �411

411

410

310

711

610

710

310

12110

b) (i) 42 ___ 110 5 21

__ 55 (ii) 56 ___ 110 5 28

__ 55

2 a) 99 ___ 2495 b) 899

___ 2495

3 a)

0.7

Judith wins

First game Second game

Suki wins0.3

0.5Judith wins

Suki wins0.5

0.7Judith wins

Suki wins

Outcome

SJ

SS

JJ

JS

Probability

0.3 � 0.5 � 0.15

0.3 � 0.5 � 0.15

0.7 � 0.7 � 0.49

0.7 � 0.3 � 0.210.3

b) (i) 0.49 (ii) 0.15 (iii) 0.36 4 a) 0.42 b) 0.34

5 a) 12 144 ____ 24 360 5 506

____ 1015 b) 12 216 ____ 24 360 5 509

____ 1015

c) 2160 ____ 24 360 5 18

___ 203


Chapter 43Exercise 43.1 (page 579) 1 a) Primary b) Secondary c) Primary d) Secondary 2 Possible answers include the following. a) Those not in the phone book have no chance

of being included. b) Those working on Saturday would not be

included. 3 Possible answers include the following. Make the categories exclusive

(in Paul’s first draft, for example, 1 hour could go in two categories); have more categories, for example, split the groups into classes of 30 minutes rather than 1 hour.

4 Possible faults include the following. a) The person’s favourite sport might be one

that isn’t listed. b) This question is vague: it needs categories,

such as hours spent. c) This is a leading question. Check students’ new questions.

7 a)

Black

First disc Second disc

White

Black

White

Black

White

Outcome

WB

WW

BB

BW

Probability

425

625

625

35

25

35

25

35

25

925

b) (i) 4 __ 25 (ii) 13 __ 25 (iii) 12

__ 25

8 a)

0.7

Red

First plant Second plant

Not red0.3

0.7Red

Not red0.3

0.7Red

Not red

Outcome

NR

NN

RR

RN

Probability

0.21

0.09

0.49

0.210.3

b) (i) 0.09 (ii) 0.42 9 a) 32

__ 72 5 4 _ 9 b) 40 __ 72 5 5 _ 9

10 a) 0.14 b) 0.14

5 a) Possible answers include the following. Q1 is too vague. Q2 is an open question and the answers

would be very difficult to analyse. b) Check students’ rewrites of Q1. It should

have better categories. 6 Check students’ questions.

Mixed exercise 43 (page 580) 1 Secondary. Jan did not collect the data herself. 2 a) Possible answers included the following.

Ask people to draw a 70° angle without a protractor.Show people three drawn angles and ask them to estimate what size the angles are.

b) Possible answers include the following.Choose 30 students from Year 7, 30 from Year 9 and 30 from Year 11.Choose a random sample of students from a mixed-age group.

c) Check students’ data collection sheets. The sheets should include the age or year-group of the student as well as data on the angle(s) drawn/estimated.

3 Check students’ answers.


5

200

100

20304050

Cum

ulat

ive

freq

uenc

y

60708090

40 60 80Earnings (£w)

100

6 a)

250

Cum

ulat

ive

freq

uenc

y500

7501000

Time (t hours)

12501500

Key

AB

00

50

100

150

200

250

300

350

400

450

500

b) Both types have about the same average lifetime but type B is more reliable since its interquartile range is smaller.


0 40 80

Earnings (£w)

20 60 160140120100

0.5

Freq

uenc

y de

nsit

y

1

Chapter 44Exercise 44.1 (page 585) 1 Q2 Q1 Q3 IQR

Girls 156 cm 148 cm 163 cm 15 cm

Boys 160 cm 150 cm 169 cm 19 cm

2 a) Mass (m grams) Cumulative frequency

m < 0 0

m < 50 16

m < 100 38

m < 150 81

m < 200 143

m < 250 183

m < 300 196

m < 350 200

b)

5000

20406080

100

Cum

ulat

ive

freq

uenc

y

120140160180200

100 150 200Mass (m grams)

250 300 350

c) Median 5 165 g approx., IQR 5 95 g approx.

3 a) Age (y years) Cumulative frequency

y , 11 0

y , 16 7

y , 19 17

y , 25 32

y , 35 52

y , 50 64

y , 65 71

b)

20100

1020304050

Cum

ulat

ive

freq

uenc

y

607080

30 40 50Age (y years)

60 70

c) 44 d) 16 e) Median 5 26 years, LQ 5 19.5 years,

UQ 5 36.5 years 4 Possible answers The median shows that the boys’ heads are larger

than the girls’ heads, on average. The interquartile range shows that the girls’ head

sizes are slightly more varied than the boys’. Boys and girls both have the same range of head

sizes.


b)

120 130 140 150 160 170 180 190 0

20

40

60

80

100

120

140

160

180

200

Height (h cm)

Cum

ulat

ive

freq

uenc

y

c) Median 5 155 cm, IQR 5 19 cm

2 a) 12 b) 35 c) Approximately 9.5 km 3 Age (years) Under 10 10–19 20–29 30–49 50–89

Frequency 240 1460 2080 4950 6120

Frequency density (people per year)

24 146 208 247.5 153

Age (years)0 10 20 30 40 50 60 70 80 90

Freq

uenc

y de

nsit

y

0

50

100

150

200

250

4 Distance (km) Frequency

0 , x < 10 6

10 , x < 15 7

15 , x < 20 9

20 , x < 25 12

25 , x < 30 5

30 , x < 40 7

Mean 5 20.2 km (to 1 d.p.)

5 Suitable comments include the following. The second group did not run as far as the first,

on average. The distances for the second group were not as

spread out as those for the first (comparatively more ran between 10 and 25 km).

The estimate of the range is the same for both groups (both distributions go from 0 to 40 km).

6 a) 21.1 b) 75 and 85 c) 16

2 a) You are aged 19 up until your 20th birthday, which is 20 years after you were born.

b) Age (years)

Under 10

10–19 20–29 30–49 50–89

Frequency 24 46 81 252 288

Frequency density

2.4 4.6 8.1 12.6 7.2

0 20 40

Age (years)

60 80 100

5

Freq

uenc

y de

nsit

y(p

eopl

e pe

r ye

ar)

10

15

3 a) Waiting time (w minutes)

Frequency

0 , w < 20 50

20 , w < 40 82

40 , w < 60 130

60 , w < 90 153

90 , w < 120 42

120 , w < 180 24

b) 60.2 minutes to 1 d.p. 4 a) 5 girls and 18 boys b) Possible answers:

In general, boys spent less time on the phone.The times for the girls were more evenly spread than those for the boys.There were 50 girls and 50 boys represented.

5 a) 100 b) 81.2 kg to 1 d.p.

Mixed exercise 44 (page 596) 1 a) Height (h cm) Cumulative

frequency

h < 120 0

h < 130 5

h < 140 32

h < 150 71

h < 160 133

h < 170 178

h < 180 196

h < 190 200


20 £11 53021 £700.56 (compared with £733.92)22 Cycling, by 3 minutes (walking 15 minutes,

cycling 12 minutes)23 96%24 a) Cheapest option is to order meals for 51

people: 48 3 7.75 5 £372, 51 3 7.25 5 £369.75.

b) Total cost 5 basic cost (44) 1 cost of ingredients (430) 5 £474. Total charge 5 £725. 53% profit.

c) 625 a) £72, 11 1 _ 4 hours

b) It would take Barbara 12 hours to earn £72 so very similar in terms of time taken. The money she earns is guaranteed. Alan has the potential to collect more money if more people are in.

26 £39127 £551.69 28 a) 1: 36p 2: 90p 3: £5.70 4: £3.82

b) Combine 4 parcels at 1.3 kg (4 3 5 5 £20 to 1 parcel at 5.2 kg costing £11.02. Saving £8.98.

29 a) 5700 b) C – smallest volume of box with sufficient

dimensions c) Flour 450 g Fruit 1650 g Eggs 6 or 7

30 a) Diesel £983.99 Petrol £1297.76 b) £300031 a) Various combinations are possible – G or F

to share with A, F to share with C or E and B to share with E

b) £704.76 c) 79%

32 a) 108 b) 96 slabs or 99 slabs

Chapter 46 (page 614)

1 6 2 9 1 8 1 7 1 6 1 5 1 43 1 21 (9 1 8 1 7 1 65 1 4 1 3 1 2 1 1 uses 7)

2 Rachel, Jane, Kim, Shaun 3 4 4 95 3 76 1 4 3 1 5 7224 5 Rounding of decimals 6 123 and 124 7 63 and 64 (Zen looks at 62 and 65 with 63 and 64

missing.) 8 1 _ 8 9 90110 6511 1 hour 59 minutes12 201713 About 23p14 x 1 5 (2x 2 3, x 1 2, x 1 5, 2x 1 1, x 2 3)

Chapter 45 (page 600)

1 a) 45 b) Grass seed by £45.75

c) (i) small box: 50 p/m² large box: 40 p/m² (ii) 2 large boxes and 1 small box

2 a) £3.75 b) £13.75 3 a) £130 b) No. The total amount collected on Sunday is

not known. 4 a) e.g. no timescale; options are exhaustive. b) e.g. not a representative sample of the

population, rugby is probably played on a Saturday afternoon and the people who watch it are unlikely to be in the supermarket at this time.

5 a) Yes – it is better to change £300 straight into pesos. Converting £300 gives 6075 pesos compared with 6075 pesos from changing $615.

b) Mexican trip cheapest by £12. 6 Any answer so long as it is justified.

Travelling by bus is the cheapest at £32 in total but they would have to catch the bus at 07:50.Travelling by car costs £63 but they don’t have to leave until 11a.m.Travelling by train is the most expensive at a total of £79, but they don’t have to catch the train until 10:52.

7 Hotel £1740 – the most expensive but includes use of leisure facilities. Guest House £1140 – includes evening meal (but not flexible) and breakfast. Cottage £880 – all meals must be paid for and prepared.

8 £10 016 9 a) 1008 b) £95010 a) Electric superstore b) £6.8011 a) £16 and £64 b) 840 ml12 48 mph13 3456, £411.26, £424.82, £21.24, £446.05, £416.6814 15 days15 £258416 a) £975 b) £1372.9617 £195618 a) Player Mean Range

A 50 8

B 33 106

C 50 99

b) Any, so long as reasoning is justified, e.g. Player A as the most consistent.

19 a) 616, 15.52% b) Each dimension must be a multiple of 6

and multiply together to give a volume of 133 056 cm3, e.g. 66 cm 3 48 cm 3 42 cm

c) 30 cm


47 3.99 m48 4.8 m49 9 km

50 2x ___ t km/h

51 a) 21.21 cmb) Pencil or box has no thickness or accuracy of

measurement52 6.25 square units53 79.8 cm2

54 1015.78 cm3

55 1 _ 4 56 a) 1.154 cm b) 0.577 cm2

57 4.7 cm2

58 Bearing 115.8°59 5 people60 24.8 mm61 11, 44, 47, 50, 7362 3563 88%64 5 __ 16

65 1 _ 2 , equal probability each time it is thrown66 a) 0.6 b) 0.2867 46.8 metres68 65 cm69 0.45

Hard (page 627) 1 54 kg wheat, 53 kg barley, 30 kg barley,

62 kg wheat, 49 kg oats, 50 kg wheat 2 18 3 69 cm2

4 e.g. Terms 4a 1 7, 4a 1 8, 4a 1 9, 4a 1 10, 4a 1 11 Sum 20a 1 45 5 5(4a 1 9) 5 Yes 6 AR 5 7.2 cm 7 a) 16 sin y 1 8 sin 60

b) (i) y 5 0 or equivalent in words, 4 √__

3 cm2 or equivalent

(ii) y 5 90° or equivalent in words, 16 1 4 √

__ 3 cm2 or equivalent

15 23 (because 0 is in the sequence and the numbers decrease by three each time)

16 1 _ 4 1 1 __ 72 17 £341018 3619 361.12 metres20 9 metres by 3 metres21 14 days22 28023 3024 99%25 Low energy bulb by £16.80

26 a) 6x2 ___

16 or 3x2

___ 8 b) y 5 3 _ 4 x

27 0.2 and 0.528 26 hens and 12 pigs29 Tile 135 mm, gap 35 mm30 Second column as they are all the multiples of 831 No, unless the 250 ml glass holds just a little more

than 250 ml32 12033 90 minutes (The wife takes 72 minutes, but has to

wait for her husband.)34 10 units35 2 days 16 hours or 2 2 _ 3 days36 1 : 837 11.5 km/h38 59.48 m2

39 Not possible as sum of angles 5 720°, 3 3 140 5 420°, remaining 3 angles total 300° so 3 angles , 90° not possible

40 72.8%41 (x 1 3)(x 1 5) m2 or x2 1 8x 1 15 m²

42 a) x(y 2 2x)2 b) 4x2 ___

y2 3 10043 5.29 cm44 6.3 cm45 x2(1 1 √

__ 3 ) shown by working out perpendicular

height of triangle and adding the area of the square base to the area of 4 of these triangles

46 4x2 2 px2 _________

4 ( 5

x2(4 2 p) _________

4 )

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AnSWErS to tHE StudEnt’S Book ExErCiSES€¦ · · 2010-09-14Higher WJEC GCSE Mathematics © 2010, Hodder Education Answers to the Student’s ... Answers to the Student’s Book