AnSWErS to tHE HoMEWork Book · PDF fileAnswers to the Homework Book exercises 248 Higher WJEC GCSE Mathematics © 2010, Hodder Education 3 a) 63 b) 96 c) 154 d) 132 e) 40 f) 133 g)

Answers to the Homework Book exercises Higher WJEC GCSE Mathematics © 2010, Hodder Education246

4 a) 56 b) 105 c) 82 d) 33

5 a) 26 b) 30 or 1 c) 54 d) 71 or 7 6 a) (i) 36 5 22 3 32 (ii) 49 5 72

(iii) 64 5 26 (iv) 100 5 22 3 52

(v) 324 5 22 3 34

b) All the powers are even 7 50 5 2 3 52; 2 8 a) (i) 12 5 22 3 3 (ii) 27 5 33

(iii) 60 5 22 3 3 3 5 (iv) 75 5 3 3 52

(v) 112 5 24 3 7 b) (i) 33 (ii) 33 (iii) 315 (iv) 33 (v) 37

Exercise 1.6H 1 a) 1 _ 4 b) 1 _ 9 c) 1 __ 65

d) 1 __ 10 or 0.1 e) 2 _ 9 2 a) 6 b) 10 c) 25 d) 71 e) 7.5 3 a) 1 2 _ 3 b) 2 1 _ 4 c) 5 __ 12

d) 3 __ 16 e) 33 1 _ 3 4 a) 0.04 b) 5 c) 0.156 25 d) 0.0016 e) 6.25

2 Fractions, decimals and percentages

Exercise 2.1H 1 a) 8, 7 _ 8 b) 99, 7 __ 11 c) 60, 1 _ 6

2 a) 5 _ 7 b) 11 __ 15 c) 5 __ 11

d) 3 __ 17 e) 10 __ 16 5 5 _ 8 f) 11

__ 9 5 1 2 _ 9

g) 2 __ 12 5 1 _ 6 h) 13 __ 11 5 1 2 __ 11 i) 5 5 _ 7

j) 3 4 _ 6 5 3 2 _ 3 k) 5 5 __ 13 l) 2 6 _ 8 5 2 3 _ 4

3 a) 5 _ 9 b) 10 __ 12 5 5 _ 6 c) 13

__ 20

d) 7 __ 16 e) 1 5 __ 24 f) 1 19 __ 30

g) 11 __ 24 h) 1 4 __ 20 5 1 1 _ 5 i) 1 13

__ 55

j) 1 17 __ 60 k) 17

__ 24 l) 19 __ 60

4 a) 7 7 __ 12 b) 5 2 _ 9 c) 5 5 _ 8

d) 3 9 __ 16 e) 6 1 __ 12 f) 4 13 __ 18

g) 8 13 __ 40 h) 10

__ 33 i) 8 1 __ 28

j) 3 7 __ 12 k) 4 11 __ 18 l) 1 _ 2

1 Integers, powers and rootsExercise 1.1H 1 2 3 7 2 24

3 22 3 7 4 5 3 7 5 2 3 3 3 7 6 72

7 22 3 33 8 22 3 3 3 13 9 32 3 52 10 23 3 53

Exercise 1.2H 1 6 5 2 3 3, 8 5 23, HCF 5 2, LCM 5 24 2 8 5 23, 18 5 2 3 32, HCF 5 2, LCM 5 72 3 15 5 3 3 5, 25 5 52, HCF 5 5, LCM 5 75 4 36 5 22 3 32, 48 5 24 3 3, HCF 5 12,

LCM 5 144 5 25 5 52, 55 5 5 3 11, HCF 5 5, LCM 5 275 6 33 5 3 3 11, 55 5 5 3 11, HCF 5 11, LCM 5 165 7 54 5 2 3 33, 72 5 23 3 32, HCF 5 18, LCM 5 216 8 30 5 2 3 3 3 5, 40 5 23 3 5, HCF 5 10,

LCM 5 120 9 45 5 32 3 5, 63 5 32 3 7, HCF 5 9, LCM 5 31510 24 5 23 3 3, 50 5 2 3 52, HCF 5 2, LCM 5 600

Exercise 1.3H 1 6 2 240 3 12 4 224 5 235 6 221 7 20 8 24 9 25 10 25 11 4 12 513 25 14 28 15 23 16 2217 218 18 23 19 6 20 215

Exercise 1.4H 1 a) 1 b) 169 c) 68 d) 614 e) 27 f) 125 g) 2 h) 3 2 64 cm3

3 a) 400 b) 1764 c) 26.01 d) 3600 e) 0.81 4 a) 343 b) 42.875 c) 830.584 d) 8000 e) 1 000 000 5 a) 630 b) 68.66 c) 616.85 d) 6178 e) 6201 6 a) 9 b) 5.24 c) 2.1 d) 16.92 e) 46.42 7 8.94 cm

Exercise 1.5H 1 a) 26 b) 74 c) 22 3 34 3 53

2 a) 26 b) 38 c) 45 d) 57 3 a) 53 b) 76 c) 22 d) 34

AnSWErS to tHE HoMEWork Book ExErCISES

Answers to the Homework Book exercises 247Higher WJEC GCSE Mathematics © 2010, Hodder Education

4 a) 59 b) 29.25 c) 26 d) 52 e) 47 f) 63 g) 17.6 h) 10.8 i) 316 j) 5.7 k) 2.7 l) 6.3

Exercise 2.7H 1 a) 1.17 b) 1.3 c) 1.73 d) 1.06 e) 1.01 f) 1.125 g) 2.6 2 a) 0.87 b) 0.6 c) 0.65 d) 0.92 e) 0.96 f) 0.73 g) 0.845 3 £700 4 £14 790 5 £392 6 £4258 7 £10 352 (to the nearest pound) 8 33.8 million 9 Jane should choose 3% interest every 6 months.

She will receive £13.66 more (to the nearest penny).

10 £34.46 (to the nearest penny)

3 ratio and proportionExercise 3.1H 1 a) 4 : 3 b) 2 : 5 c) 7 : 11 d) 1 : 3 : 4 e) 3 : 5 : 4 2 a) 2 : 5 b) 1 : 5 c) 12 : 5 d) 5 : 2 e) 1 : 6 3 2 : 1 4 4 : 7 : 9 5 1 : 2 : 5

Exercise 3.2H 1 a) 1 : 5 b) 1 : 6 c) 1 : 4.5 d) 1 : 2.25 e) 1 : 0.6 f) 1 : 20 g) 1 : 24 h) 1 : 0.35 i) 1 : 250 000 2 1 : 250 000 3 1 : 6

Exercise 3.3H 1 a) 75 cm b) 9 cm 2 a) 6 buckets b) 5 buckets 3 a) 2.5 kg b) 900 g 4 24 cm 5 a) 50 km b) 16.5 cm 6 480 g 7 a) 125 ml b) 80 ml 8 a) 10 cm b) 3 cm 9 a) 200 ml b) 750 ml10 a) 6000 votes b) 3000 votes

Exercise 3.4H 1 Paula £15, Tariq £25 2 a) 4 litres b) 6 litres 3 Copper 150 g, iron 200 g, nickel 100 g 4 3 hours 30 minutes 5 £520 6 £2.50 7 72 8 a) 3 : 2 b) Iain 126 000 euros, Stephen 84 000 euros

Exercise 2.2H

1 a) 23 __ 5 b) 25

__ 4 c) 25 __ 7 d) 14

__ 9

e) 29 __ 6 f) 73

__ 10 g) 39 __ 8

2 a) 2 1 _ 7 b) 3 1 _ 3 c) 3 __ 20

d) 2 8 __ 11 e) 1 __ 18 f) 24

3 a) 10 __ 21 b) 5 __ 48 c) 14

__ 45

d) 5 _ 6 e) 1 1 _ 8 f) 20 __ 99

g) 3 __ 28 h) 7 __ 10 i) 7 __ 32

j) 27 __ 28 k) 1 17

__ 25 l) 1 _ 9

4 a) 8 4 _ 9 b) 11 __ 15 c) 1 4 _ 7

d) 4 __ 11 e) 16 1 _ 2 f) 1 1 _ 3

g) 5 1 _ 9 h) 1 19 __ 44 i) 18 1 _ 3

j) 3 1 _ 5 k) 13 2 _ 9 l) 2 23 __ 26

Exercise 2.3H

1 a) 11 __ 12 b) 19

__ 56 c) 5 __ 24 d) 1 1 __ 20

e) 4 11 __ 20 f) 4 2 __ 21 g) 22 2 _ 5 h) 3 19

__ 21

2 a) 8 __ 25 b) 4 _ 7 c) 3 _ 8

d) 9 __ 14 e) 1 _ 9

3 a) 1 2 _ 3 b) 11 7 __ 13 c) 4 10 __ 19

d) 3 5 _ 9 e) 17 13 __ 17

4 a) 15 7 __ 10 cm b) 12 3 _ 5 cm2

Exercise 2.4H 1 a) 0.875 b) 0.07 c) 0.556 d) 0.182 2 a) Terminating. 4 is a multiple of only 2, a factor

of 10. b) Recurring. 6 is not a factor of 10. c) Recurring. 11 is not a factor of 10. d) Terminating. 25 is a multiple of only 5,

a factor of 10. e) Terminating. 32 is a multiple of only 2,

a factor of 10. 3 a) 0.029 702 970 … b) 4 digits

Exercise 2.5H 1 9.5 2 7.9 3 3.5 4 3.5 5 6.3 6 9.2 7 6.3 8 1.9 9 11.3 10 11.5 11 0.9 12 3.4

Exercise 2.6H 1 a) 2 b) 4.8 c) 2.8 d) 5.4 e) 0.21 f) 0.36 g) 35 h) 32 i) 0.07 j) 0.08 k) 0.64 l) 0.04 2 a) 20 b) 24 c) 4 d) 18 e) 30 f) 40 g) 30 h) 50 i) 80 j) 160 k) 3.1 l) 95 3 a) 5.04 b) 15.08 c) 34.83 d) 20.8 e) 125.8 f) 243.2 g) 22.04 h) 36.48 i) 20.16 j) 2.403 k) 10.14 l) 17.424


3 a) 63 b) 96 c) 154 d) 132 e) 40 f) 133 g) 240 h) 200 i) 252 j) 882 4 a) 6 b) 6 c) 500 d) 23 e) 12 f) 6 g) 6 h) 0.16 i) 80 j) 4 5 a) 4 b) 4 c) 2 d) 212 e) 25 f) 26 g) 8 h) 20 i) 5 j) 24 6 a) 224 b) 220 c) 18 d) 40 e) 2400 f) 22 g) 23 h) 5 i) 7 j) 28 7 a) 49 b) 81 c) 144 d) 196 e) 1 f) 900 g) 40 000 h) 0.25 i) 0.64 j) 0.04 8 a) 66 b) 62 c) 68 d) 611 e) 612 9 a) 27 b) 64 c) 1000 d) 125 000 e) 0.00810 a) 19.2 cm b) 23 cm2

11 £32.1912 11 cm13 £10.8014 9 and 515 568 g

Exercise 4.2H 1 a) 10 b) 40 c) 7 d) 300 e) 1000 f) 0.8 g) 0.6 h) 0.05 i) 2000 j) 0.01 2 a) 8 b) 20 c) 700 d) 8000 e) 100 f) 0.7 g) 0.005 h) 0.02 i) 400 j) 20 000 3 a) 29 b) 150 c) 7800 d) 47 000 e) 21 f) 0.19 g) 0.049 h) 0.0036 i) 0.051 j) 0.90 4 a) 7.39 b) 24.8 c) 28 500 d) 309 e) 16 400 f) 3.92 g) 60.7 h) 0.914 i) 0.004 16 j) 2.24 5 £6 or £8 6 £20 000 7 50 mph 8 5 cm 9 £120 or £10010 80 cm11 58 cm2 or 60 cm2

12 a) 4200 or 4000 b) 7 c) 1200 or 1000 d) 400 e) 480 000 or 500 000 f) 0.3 g) 0.6 h) 320 000 or 300 000 i) 4000 j) 70

Exercise 3.5H 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: 80 g bag 5 1.24p, 200 g bag 5 1.10p

Grams per penny: 80 g bag 5 0.808 g, 200 g bag 5 0.913 g (to 3 d.p)

200 g bag is better value 2 Cost per litre: 2 litre bottle 5 64.5p,

3 litre bottle 5 66.3p Litres per pound: 2 litre bottle 5 1.55 litre,

3 litre bottle 51.51 litre 2 litre bottle is the better value 3 Cost per gram: 200 g tub 5 0.48p,

450 g tub 5 0.47p Grams per penny: 200 g tub 5 2.11 g,

450 g tub 5 2.14 g 450g tub is the better value 4 Cost per tub: 4 tubs 5 19.75p, 12 tubs 5 18.25p Tubs per penny: 4 tubs 5 0.051,

12 tubs 5 0.055 (to 3 d.p.) Pack of 12 tubs is the better value 5 Cost per kilogram: 1.8 kg pack 5 £1.39,

1.5 kg pack 5 £1.33 Kilograms per pound: 1.8 kg pack 5 0.72 kg,

1.5 kg pack 5 0.75 kg 1.5 kg pack is the better value 6 Cost per millilitre: 75 ml bottle 5 1.59p,

200 ml bottle 5 1.45p Millilitres per penny: 75 ml bottle 5 0.63 ml,

200 ml bottle 5 0.69 ml 200 ml bottle is the better value 7 Cost per can: 12 can pack 5 35.8p,

20 can pack5 36.25p Cans per pound: 12 can pack 5 2.79 cans,

20 can pack 5 2.76 cans 12 can pack is the better value 8 Cost per gram: 750 g pack 5 0.42p,

1.5 kg pack 5 0.40p, 2.5 kg pack 5 0.28p Grams per penny: 750 g pack 5 2.38 g,

1.5 kg pack 5 2.50 g, 2.5 kg pack 5 3.58 g 2.5 kg pack is the best value

4 Mental methodsExercise 4.1H 1 a) 33 b) 62 c) 5.8 d) 201 e) 212 f) 18.4 g) 335.3 h) 539 i) 1331 j) 9170 2 a) 7 b) 27 c) 9.6 d) 11 e) 43 f) 114 g) 0.5 h) 167 i) 226 j) 1047


6 a) 30.04 b) 265.20 7 a) 0.862 b) 233.558 8 a) 3.9 b) 50 9 a) (i) 203 5 8000

(ii) 200 3 0.3 5 60

(iii) √________

102 2 52 5 √________

100 2 25 5 √___

75 < 9

(iv) 500 1 40 ________ 0.02

5 90 ____ 0.02

5 9000 _____ 2 5 4500

b) (i) 9528.128 (ii) 58.59 (iii) 10.0 (to 3 s.f.) (iv) 4140 (to 3 s.f.)

Exercise 5.3H 1 a) 8.65 hours b) 5.35 hours c) 0.55 hours d) 0.05 hours 2 a) 4 hours and 48 minutes b) 5 hours and 51 minutes c) 27 minutes d) 36 minutes 3 a) 3 mph b) 52.5 miles c) 4 minutes 4 23 hours and 40 minutes 5 2.48 kg 6 £31 500 7 625 g or 0.625 kg 8 1320 people/km2 (to 3 s.f.) 9 £26.6310 3.71%11 3100 g (to 3 s.f.)12 37.8 litres13 a) 36 minutes b) 63.3 mph (to 1 d.p.)

6 Further percentagesExercise 6.1H 1 a) £5700.75 b) £637.61 c) £63 756.68 2 a) £48.23 b) £21.30 c) £8.75 3 £2575.67 4 £14.87 5 £855.36

Exercise 6.2H 1 a) £810 b) £360 c) £44.41 d) £241.90 2 a) £1760 b) £252.63 c) £250.90 d) £371.43 3 75 tonnes 4 56 500 5 £8112.50 6 £8800 7 2.48 million 8 £11 000 9 a) £87.87 b) £14.99 (or £15)10 £882.63 11 £172.80 12 475 g

Exercise 4.3H 1 a) 12p b) 49p c) 144p

d) 7.6 p e) 121p

2 a) 19p b) 45p c) 48p

d) 126p e) 9 3 30p 4 9 : 4 5 144p 2 25

Exercise 4.4H 1 a) 36 b) 0.009 c) 0.036 d) 0.0025 e) 0.09 f) 400 000 g) 150 000 h) 1530 i) 40 600 j) 9100 2 a) 15 b) 0.03 c) 0.18 d) 7 e) 0.0008 f) 80 g) 20 h) 250 i) 5000 j) 400 3 a) 75 600 b) 0.45 c) 168 d) 0.007 56 e) 4500 4 a) 1800 b) 18 c) 7020 d) 0.39 e) 7.02 5 a) 3480 b) 75 168 000 c) 75.168 d) 21.6 e) 751.68

5 Solving problemsExercise 5.1H 1 2.77 2 9.48 3 4.03 4 9.66 5 7.8 6 7.84 7 0.69 8 2.91 9 15.26 10 0.33 11 2.34 12 0.05

Exercise 5.2H 1 Check student’s answers. Possible answers

include the following.a) Positive 4 negative 5 negativeb) Multiplying by number greater than

1 should make answer largerc) 5 3 300 5 1500 and answer should be less

than thisd) 72 5 49 and answer should be more than this

2 a) 9 3 60 5 540b) Negative 3 negative 5 positivec) 0.22 5 0.04d) 5 3 1 5 5 so last digit should be 5

In questions 3 and 4, estimates may vary from those given. Check the validity of other estimates. 3 a) 90 3 100 5 9000 b) 0.62 5 0.36 c) 220 1 5 5 215 4 a) 3 3 £20 5 £60 b) 40 3 £6 5 £240 c) 5 3 £8 1 2 3 £2 5 £44 5 a) 83.7 3 235 5 19 669.5

b) 71.22 5 5069.44c) 120.28 4 9.7 5 12.4 or 120.28 4 12.4 5 9.7d) 147.32 2 68.2 5 79.12, 79.12 4 4.6 5 17.2 or

79.12 4 17.2 5 4.6


8 Using a calculatorExercise 8.1H 1 a) 1.19 b) 0.135 c) 0.380 2 a) 0.407 b) 0.0162 c) 600 3 a) 1.87 b) 0.386 c) 3.97 4 a) 0.252 b) 2.95 c) 21.78 5 a) 81.9° b) 27.6° c) 35.3° 6 a) 6.36 3 1010 b) 58 c) 1.42 3 1018

7 a) 0.002 36 b) 1.33 c) 0.002 47 8 a) 0.312 b) 0.356 c) 0.291 9 a) 1.89 b) 7.57 c) 2.6310 a) 1.30 3 10215

b) 2200 c) 2.67 3 1012

Exercise 8.2H 1 a) a 5 30 000 3 1.18t

b) (i) £68 632.73 (ii) £218 627.78 2 a) c 5 250 000 3 0.94t

b) (i) 195 187.24 scuds (ii) 143 248.70 scuds c) 15 months 3 a) 60 000 b) (i) 7500 (ii) 58 c) 15 years 4 a) £500 b) 4% c) Number of years d) (i) £584.93 (ii) £740.12 5 a) P 5 38 000 000 3 1.03t

b) (i) 44 052 414 (ii) 629 970 109 c) 24 years 6 a) m 5 0.90t

b) (i) 656 g (ii) 430 g c) 7 years 7 a)

0

5

10

15

20

25

30

y

x1 2 3

b) (i) 5.2 (ii) 2.5 8 a)

5

10

15

y

x�3�4 0�2 �1

b) (i) 5.7 (ii) 23.3

7 Indices and standard formExercise 7.1H 1 a) x

1 _ 5 b) x 2

1 _ 2 c) x

3 _ 4

2 a) 1 __ 25 b) 5 c) 1 d) 1 _ 5 e) 125 f) 9 g) 10 h) 10 i) 64 j) 1 k) 4 l) 1 m) 64 n) 1 _ 2 o) 128 p) 12 q) 25

__ 4 3 a) 400 b) 343

___ 10 c) 125 d) 43 e) 4 f) 2 __ 25

Exercise 7.2H 1 a) 789.05 b) 0.193 49 c) 1.2299 d) 0.388 90 e) 13 f) 1.5597 g) 3.1054 h) 27 2 a) 488.67 b) 475.53 c) 3255.0 d) 5.0174 e) 193.69 f) 20.142 53 g) 145.93 h) 163.84

Exercise 7.3H 1 a) 26 b) 22 c) 222

d) 23 e) 2n f) 24n

2 a) 73 b) 3 4 _ 3

c) 224 d) 222

e) Not possible f) 323

g) 35n

3 a) 22 3 3 3 5 b) 26 3 3 c) 23 3 3 3 52 d) 23 3 34

4 a) 33 3 53 b) 5 9 _ 2 3 2

1 _ 2

c) 23n 3 5n d) 23n11 3 5n11

Exercise 7.4H 1 a) 6 3 104 b) 8.4 3 103

c) 8.63 3 105 d) 7.25 3 107

e) 9.02 3 106 f) 7.8 3 10(1)

g) 5.2 3 106

2 a) 8 3 1022 b) 9.6 3 1023

c) 3.08 3 1024 d) 6.3 3 1025

e) 4.8 3 1026 f) 2.3 3 1028

3 a) 3000 b) 46 000 c) 0.000 02 d) 720 000 e) 0.000 19 f) 5 780 000 g) 287 000 000 h) 0.000 005 13 i) 0.002 07 j) 72 800 000

Exercise 7.5H 1 a) 1.8 3 1010 b) 2 3 103

c) 3.6 3 104 d) 6 3 1027

e) 4 3 104 f) 7.98 3 105

g) 5.2 3 1023

2 a) 1.188 3 1012 b) 1.495 3 109

c) 5.893 3 107 d) 2.35 3 106

e) 9.36 3 108 f) 7.15 3 104

g) 5.835 3 105


3 a) 6 __ 40 b) 7 _ 8 e) 11 __ 80

4 a) 0.15 b) 0.875 c) 0.1 6 .

d) 0.02 6 . e) 0.1375

5 a) 4 __ 25 b) 61 ___ 200 c) 5 _ 8 d) 51

___ 125

6 a) 1 _ 3 b) 8 _ 9 c) 7 __ 45 d) 13

__ 18

7 a) 7 __ 11 b) 47 __ 99 c) 5 __ 33 d) 38

__ 99

8 a) 34 ___ 111 b) 46

___ 111 c) 8 __ 99 d) 1 __ 37

Exercise 9.2H 1 a) 4 √

__ 7 b) 7 √

__ 5 c) 3 √

__ 3

d) 15 e) 6 √___

10 f) 18 2 a) 3 √

__ 2 b) 15 √

__ 2 c) 10 √

__ 3

d) 12 √__

5 e) 6 √__

3 f) 25 √__

2 3 a) 7 √

__ 2 b) 2 √

__ 3 c) 5 √

__ 3

d) 12 √__

3 e) 6 √__

7 f) 100 4 a) 2 1 5 √

__ 2 b) 6 1 √

__ 6

c) 60 1 8 √__

5 d) 18 1 3 √___

10 5 a) 59 1 10 √

__ 7 , I b) 21 2 5 √

__ 3 , I

c) 22 2 11 √__

5 , I d) 1 2 3 √__

5 , I e) 215, R f) 50 2 16 √

___ 13 , I

6 a) 18 2 6 √__

5 b) 16 2 3 √__

5 c) 7 √

__ 5 d) 14 2 2 √

__ 5

7 a) 28 1 8 √__

3 b) 26 1 7 √__

3 c) 4 √__

3 d) 37 e) 61 1 28 √

__ 3 f) 61 2 28 √

__ 3

8 a) 0.75 b) 23.5

9 a) 5 √__

3 b) 4 √__

5 ____ 5 c) 3 √

__ 2 ____

5

d) √

__ 3 ___

7 e) 3 √

__ 6 ____

5 f) 4 √

___ 15 _____

11

10 a) 2 1 2 √__

5 b) 1 1 4 √__

3 ________ 4

c) 5 1 3 √__

2 c) √

___ 15 1 6 √

__ 5 __________

3

10 Algebraic manipulation 1Exercise 10.1H 1 21a 1 42b 2 10c 1 15d 3 12e 2 20f 4 21g 2 6h 5 12i 1 6j 2 9k 6 15m 2 6n 1 9p 7 24r 2 18s 2 12t 8 32r 1 16s 1 8t 9 12u 1 20v 10 24w 1 18x11 10y 1 2z 12 12y 1 8z13 15v 1 10 14 21 1 12w15 5 2 15a 16 24g 2 15

Exercise 10.2H 1 a) 18a 1 23 b) 26b 1 18 c) 26 1 40c 2 a) 23x 1 4 b) 27y 2 2 c) 18 1 11z 3 a) 26s 1 27t b) 18v 1 19w c) 24x 1 36y d) 16v 1 11w 4 a) 18n 1 5p b) 22q 1 3r c) 36d 1 4e d) 27f 1 4g e) 8h 1 6j f) 14k 2 9m

Exercise 8.3H 1 a) upper bound 5 8.55 m,

lower bound 5 8.45 mb) upper bound 5 57.5 kg,

lower bound 5 56.5 kgc) upper bound 5 0.755 m,

lower bound 5 0.745 md) upper bound 5 12.955 s,

lower bound 5 12.945 se) upper bound 5 500.5 ml,

lower bound 5 499.5 ml 2 42.5 m < length , 43.5 m 3 a) Yes. For example, if the unit measures

between 60 cm and 59.5 cm and the gap measures between 60 cm and 60.5 cm.

b) No. For example, if the unit measures between 60 cm and 60.5 cm and the gap measures between 60 cm and 59.5 cm.

4 a) upper bound 5 15.5 kg, lower bound 5 14.5 kg

b) upper bound 5 70 cm, lower bound 5 50 cm

c) upper bound 5 4050 s, lower bound 5 3950 s

Exercise 8.4H 1 upper bound 5 38 cm, lower bound 5 34 cm 2 upper bound 5 20.625 cm2,

lower bound 5 14.625 cm2

3 upper bound 5 2 kg, lower bound 5 1.8 kg 4 a) upper bound 5 29.20 s

b) upper bound 5 1.06 s 5 a) upper bound 5 44.5475,

lower bound 5 43.1775b) upper bound 5 1.7129,

lower bound 5 1.6602 6 upper bound 5 86 mph,

lower bound 5 80.8 mph 7 upper bound 5 417.1625 g,

lower bound 5 409.3425 g 8 upper bound 5 3.42, lower bound 5 3 9 upper bound 5 0.409, lower bound 5 0.36510 upper bound 5 1.147, lower bound 51.06511 a) 14.5 kg, 15.5 kg

b) (i) 1450 kg, 1550 kg(ii) 96

12 a) 495 cm3, 505 cm3

b) 30 3 505 , 15 450

9 Decimals and surdsExercise 9.1H 1 b) 7 __ 30 c) 4 _ 7 e) 11

___ 120

2 a) 0.125 b) 0.2 3 . c) 0.5

.71 42 8

.

d) 0.3125 e) 0.091 6 .


Exercise 11.5H 1 (3x 1 2)(x 1 5) 2 (x 1 3)(2x 1 1) 3 (2x 1 7)(x 2 3) 4 (5x 1 1)(x 2 4) 5 (2x 2 5)(x 1 3) 6 (3x 2 4)(x 1 5) 7 (2x 2 3)(x 2 4) 8 (5x 2 4)(x 2 3) 9 (3x 1 1)(2x 1 5) 10 (2x 2 5)(5x 1 2)11 (3x 2 2)(4x 2 3) 12 (7x 2 3)(3x 1 4)

Exercise 11.6H

1 1 _____ x 2 2

2 x 1 3 3 x 2 3 _____ x 1 3

4 x 1 2 _____ x 2 2

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

______________ (x 2 1)(x 2 6)

6 2(x 1 2)

________ x 1 3

7 x 1 2 _____ x 1 1

8 x 2 3 _____ x 2 4

9 4 _____ x 1 4

10 5 ___ 2x

11 x 1 4 _____ x 2 3

12 x 1 2 _____ x 1 4

13 3x _____ x 2 3

14 x 1 4 ______ 2x 2 3

15 3(x 2 4)

________ 4(x 1 4)

16 3x(x 1 4)

_________ x 2 3

12 Equations and inequalities 1Exercise 12.1H 1 x 5 5 2 x 5 3 3 x 5 6 4 x 5 3 5 x 5 4 6 x 5 5 7 x 5 2 8 x 5 1.5 9 x 5 8.510 x 5 21 11 x 5 ±4 12 x 5 ±313 y 5 ±9 14 x 5 3 15 x 5 21.5

Exercise 12.2H 1 x 5 8 2 x 5 3 3 x 5 7 4 x 5 2 5 x 5 5 6 x 5 2 7 x 5 8 8 x 5 3 9 x 5 1110 x 5 1.5 11 x 5 3 12 x 5 313 x 5 5.5 14 x 5 4.5 15 x 5 3.5

Exercise 12.3H 1 x 5 3 2 x 5 4 3 x 5 2 4 x 5 10 5 x 5 3 6 x 5 1 7 x 5 1 8 x 5 3 9 x 5 310 x 5 3 11 x 5 4.5 12 x 5 513 x 5 2 14 x 5 2 15 x 5 23

Exercise 12.4H 1 x 5 14 2 x 5 15 3 x 5 12 4 x 5 30 5 x 5 6 6 x 5 15 7 x 5 96 8 x 5 8 9 x 5 7 10 x 5 24

Exercise 10.3H 1 a) 4(2x 1 5) b) 3(x 1 2) c) 3(3x 2 4) d) 5(x 2 6) 2 a) 8(2 1 x) b) 3(3 1 5x) c) 4(3 2 4x) d) 4(2 2 3x) 3 a) 4x(x 1 4) b) 6x(x 1 5) c) 4x(2x 2 5) d) 3x(3x 2 5)

Exercise 10.4H 1 a) a2 1 8a 1 15 b) b2 1 6b 1 8 c) 12 1 7c 1 c2

2 a) 8d2 1 6d 2 9 b) 15e2 1 2e 2 8 c) 6 1 f 2 40f 2

3 a) 15g2 2 28g 1 12 b) 12h2 2 43h 1 35 c) 15j2 2 58j 1 48 4 a) 12k2 1 13k 2 35 b) 6 1 5m 2 56m2

c) 8 2 14n 2 15n2

5 a) 6 1 p 2 35p2 b) 10r2 2 37r 1 30 c) 12s2 2 35s 1 18

Exercise 10.5H 1 a) 75 b) 35 c) 26

2 a) d7 b) m6 c) t7

3 a) a4b2 b) c4d5 c) r3s2t4

4 a) 36xyz b) 24abc c) 120r2s2t

11 Algebraic manipulation 2Exercise 11.1H

1 7(x 2 2)

________ 4(x 1 2)

2 x 2 2 ______ 3 2 4x

3 2(3 1 2x)

_________ 3x 2 2

4 2(3 2 x)

________ 4 1 3x

5 2x 1 1 ______ x 2 3

6 3x 2 4 ______ 5x 2 6

7 4 2 3x _________ x(2x 1 3)

8 x 1 2 ______ 3 2 4x

Exercise 11.2H 1 (x 1 1)(x 1 6) 2 (x 1 3)(x 1 5) 3 (x 1 3)(x 1 6) 4 (x 1 1)(x 1 13) 5 (x 1 5)(x 1 5) 6 (x 1 3)(x 1 8) 7 (x 1 5)(x 1 8) 8 (x 1 3)(x 1 16)

Exercise 11.3H 1 (x 1 1)(x 2 1) 2 (x 1 6)(x 2 6) 3 (x 1 11)(x 2 11) 4 (x 1 14)(x 2 14) 5 2(x 1 4)(x 2 4) 6 4(x 1 5)(x 2 5) 7 6(x 1 3)(x 2 3) 8 9(x 1 8)(x 2 8)

Exercise 11.4H

1 10b2 _____

3a2c2 2

9z2 _____

20xy3 3

5p3

___ 2s3

4 5k2 ______

3h3m3 5 8t4w2 _____

9v2 6 8a3c4

_____ 15b2

7 3ab4c3 ______

2d4 8

w5x4z2 ______

y11


3

0

1

2

3

4

5

6

7

8

9

y

x1

R

2 3

y � 2x

y � 3x

4

0

1

2

3

4

5

6

7

8

y

x2

R

4 6 7 81 3 5

3x � 4y � 24

y � x

5

0

1

2

3

4

5

6

7

8

y

x2

R

4 6 7 81 3 5

8x � 3y � 24

5x � 6y � 30

14 Graphs 1Exercise 14.1H 1 Three points plotted, e.g. (23, 29), (0, 0), (3, 9),

straight line drawn through them. 2 Three points plotted, e.g. (24, 22), (0, 2), (2, 4),

straight line drawn through them. 3 Three points plotted, e.g. (23, 210), (0, 2),

(3, 14), straight line drawn through them. 4 Three points plotted, e.g. (21, 27), (0, 25),

(5, 5), straight line drawn through them. 5 Three points plotted, e.g. (24, 4), (0, 24),

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

Exercise 14.2H 1 Three points plotted, e.g. (23, 0), (0, 2), (3, 4),

straight line drawn through them. 2 Straight line joining (0, 2) and (5, 0). 3 Straight line joining (0, 7.5) and (5, 0).

Exercise 12.5H 1 x . 3

0 1 2 3 4 5

2 x , 2 0 1 2 3 4 5

3 x > 3 0 1 2 3 4 5

4 x < 2.5 0 1 2 3 4 5

5 x > 2 0 1 2 3 4 5

6 x > 4 7 x , 3 8 x > 3 9 x , 24 10 x < 5 11 x . 612 x . 2.5 13 x < 5 14 x . 315 x > 2

13 Equations and inequalities 2Exercise 13.1H 1 x 5 3 2 x 5 3 3 x 5 2.5 4 x 5 6 5 x 5 3 6 x 5 8 7 x 5 2 8 x 5 1.5 9 x 5 2210 x 5 1 _ 5 11 x 5 2 12 x 5 113 x 5 22 14 x 5 1 _ 3 15 x 5 1

Exercise 13.2H 1 x 5 0.5 2 x 5 4 3 x 5 2.5 4 x 5 10 5 x 5 0.5 6 x 5 5.43 7 x 5 10.8 8 x 5 12.88 9 x 5 5.01 10 x 5 0.593

Exercise 13.3H 1 x . 3 2 x > 2.5 3 x , 4 4 x . 1.5 5 x , 28 6 x > 40 7 x , 1.6 8 x > 0.653

Exercise 13.4H 1

0

1

2

3

4

5

6

y

2x � 3y � 12

x2

R

4 61 3 5

2

�2 �1 0

�4

4

8

y

x1

R

2

y � 3x � 2


6 a) 9 miles b) 54 min c)

0

Dis

tanc

e (i

n m

iles)

from

hom

e

10

20

30

1:00 2:00 3:00

Time (pm)

4:00

d) answers between 3.39 and 3.42

Exercise 14.4H 1 a) Quadratic, <-shaped b) Not quadratic c) Quadratic, <-shaped d) Quadratic >-shaped e) Not quadratic f) Not quadratic g) Quadratic, <-shaped h) Quadratic >-shaped 2 a)

x 23 22 21 0 1 2 3

x2 9 4 1 0 1 4 9

y 5 2x2 18 8 2 0 2 8 18

b)

�3 �2 �1 0

123456789

101112131415161718

y

y � 2x2

x1 2 3

c) (i) y 5 6.5 (ii) x 5 22.5 or x 5 2.5 3 a)

x 24 23 22 21 0 1 2 3

x2 16 9 4 1 0 1 4 9

y 5 x2 1 x 12 6 2 0 0 2 6 12

4 Three points plotted, e.g. (22, 29), (0, 24), (4, 6), straight line drawn through them.

5 Straight line joining (0, 6) and (8, 0).

Exercise 14.3H 1 a) 16 gallons b) 20 minutes c) 4 gallons/minute d) 8 gallons/minute e) Simon lets out 9 gallons of water over

1.5 minutes, then adds another 9 gallons over 2 minutes

f) 6 gallons/minute 2 a) £600 b) £1.20 c)

0 1000

1000

2000

3000

4000

5000

6000

2000 3000 4000 5000 6000

Number of programmes

Cha

rge

in p

ound

s (£

)

d) £3800 e) £1.09 (to the nearest penny) 3 a) Accept 20–21 minutes b) 50 kilometres c) 22 kilometres d) Accept 2.5–3 minutes e) Between about 16 and 18 minutes 4 a) E b) C c) B d) G 5 a) £5 b) £57.50 c)

0 50

10

20

30

40

50

60

100 150 200 250 300

Total number of minutes for calls per month

Plan APlan B

Tota

l cos

t (£)

d) 185 minutes e) Plan A by £5


b)

�3�4�5 �2 �1 0

1

�1�2�3�4�5�6�7

23456789

1011

y

x1 2 3

y � x2 � 2x � 5

c) (i) y 5 25.8 (ii) x 5 23.5 to 23.4 or x 5 1.4 to 1.5 6 a)

x 23 22 21 0 1 2 3

8 8 8 8 8 8 8 8

2x2 29 24 21 0 21 24 29

y 5 8 2 x2 21 4 7 8 7 4 21

b)

�3 �2 �1 0

1

�1

2345678y

y � 8 � x2

x1 2 3

c) (i) y 5 7.75 (ii) x 5 23.2 to 23.1 or x 5 3.1 to 3.2 7 a)

x 23 22 21 0 1 2 3 4

x 2 2 25 24 23 22 21 0 1 2

x 1 1 22 21 0 1 2 3 4 5

y 5 (x 2 2)(x 1 1) 10 4 0 22 22 0 4 10

b)

�3�4 �2 �1 0

1

�1

23456789

101112

y

y � x2 � x

x1 2 3

c) (i) y 5 4.2 (ii) x 5 23.4 or x 5 2.4 4 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

2 2 2 2 2 2 2 2 2

y 5 x2 2 x 1 2 14 8 4 2 2 4 18 14

b)

�3 �2 �1 0

123456789

1011121314

y

y � x2 � x � 2

x1 2 3 4

c) (i) y 5 1.8 (ii) x 5 21.6 or x 5 2.6 5 a)

x 25 24 23 22 21 0 1 2 3

x2 25 16 9 4 1 0 1 4 9

2x 210 28 26 24 22 0 2 4 6

25 25 25 25 25 25 25 25 25 25

y 5 x2 1 2x 2 5 10 3 22 25 26 25 22 3 10


d) Line drawn through (0, 21) and (3, 27), gradient 5 22

e) Line drawn through (0, 3) and (4, 0), gradient 5 20.75

6 a) 4 m/s b) 216 km/h

Exercise 15.2H 1 a) y 5 2x 1 6 b) y 5 23x 1 5 c) y 5 x 2 a) y 5 2x 1 3 b) y 5 4x c) y 5 21.5x 2 3 or 3x 1 2y 1 6 5 0 3 a) gradient 5 4, y-intercept 5 22

b) gradient 5 3, y-intercept 5 5c) gradient 5 22, y-intercept 5 1d) gradient 5 21, y-intercept 5 2e) gradient 5 23.5, y-intercept 5 28

4 a) gradient 5 23, y-intercept 5 7b) gradient 5 23, y-intercept 5 2.5c) gradient 5 22, y-intercept 5 3d) gradient 5 2.5, y-intercept 5 24e) gradient 5 21.5, y-intercept 5 2.25

5 a) y 5 2.5x 2 2 b) y 5 22x 1 30 c) y 5 3x 1 20

Exercise 15.3H 1 Graph of y 5 3x drawn through (21, 23) and

(4, 12) Graph of y 5 4x 2 2 drawn through (21, 26) and (4, 14) x 5 2, y 5 6

2 Graph of y 5 2x 1 3 drawn through (22, 21) and (3, 9) Graph of y 5 4x 1 1 drawn through (22, 27) and (3, 13) x 5 1, y 5 5

3 Graph of y 5 x 1 4 drawn through (23, 1) and (3, 7) Graph of 4x 1 3y 5 12 drawn through (23, 8) and (3, 0) x 5 0, y 5 4

4 Graph of y 5 2x 1 8 drawn through (25, 22) and (1, 10) Graph of y 5 22x drawn through (25, 10) and (1, 22) x 5 22, y 5 4

5 Graph of 2y 5 3x 1 6 drawn through (0, 3) and (4, 9) Graph of 3x 1 2y 5 12 drawn through (0, 6) and (4, 0) x 5 1, y 5 4.5

b)

�3 �2 �1 0

1

�1�2

23456789

10y

y � (x � 2)(x � 1)

x1 2 3 4

c) (i) y 5 22.25 (ii) x 5 21.7 or x 5 2.7 8 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

12 2 2 2 2 2 2 2 2

y 5 x2 2 3x 1 2 12 6 2 0 0 2 6 12

b)

�2 �1 0

1

�1

23456789

101112

y

y � x2 � 3x � 2

x1 2 3 4 5

c) (i) x 5 0.4 or x 5 2.6 (ii) x 5 21.7 or x 5 4.7

15 Graphs 2Exercise 15.1H 1 a) 2 b) 22.5 c) 1.5 2 a) 3 b) 1 c) 23 d) 20.5 e) 0 3 AB 5 1.5, BC 5 21.5, AC 5 20.5 4 a) 2 b) 24 c) 5 5 a) Line drawn through (0, 1) and (3, 10),

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

gradient 5 1c) Line drawn through (0, 2) and (3, 27),

gradient 5 23


4 a)

�5

0

�10

5

10

Time (seconds)Vel

ocit

y (m

/s)

5 10 15 20

b) 6 m/s c) At t 5 0 and t 5 12.8 d) 29 m/s 5 From O to A the particle travels at a constant

acceleration of 1.6 m/s2 for 5 seconds.Between A and B it travels at a constant velocity of 8 m/s for 6 seconds.Between B and C it travels at a constant acceleration of 0.5 m/s2 for 4 seconds.Between C and D it travels at a constant acceleration of 22 m/s2 for 5 seconds.

6 a) x 5 22.1, 2.7 b) 29

Exercise 16.2H 1 a), b)

�2

0

2

4

6

y y � x2 � 3x � 2

y � x � 1

x1�1 2 3 4

c) (1, 0) and (3, 2) 2 a), b)

�2

0�1

�3

�4

2

4

1

3

5

yy � x2 � 2x � 3

y � 3 � x

x1�1�2 2 3 4

c) (22, 5) and (3, 0)

Exercise 15.4H 1 x 5 1, y 5 2 2 x 5 2, y 5 2 3 x 5 4, y 5 1 4 x 5 5, y 5 2 5 x 5 3, y 5 1 6 x 5 3, y 5 1 7 x 5 3, y 5 2 8 x 5 2, y 5 8 9 x 5 2, y 5 4 10 x 5 6, y 5 1

Exercise 15.5H 1 x 5 2, y 5 1 2 x 5 4, y 5 1 3 x 5 2, y 5 3 4 x 5 1, y 5 2 5 x 5 2, y 5 21 6 x 5 1, y 5 3 7 x 5 22, y 5 2 8 x 5 2, y 5 1 9 x 5 6, y 5 1 10 x 5 0.5, y 5 1.5

16 Graphs 3Exercise 16.1H 1 From O to A the particle moves 8 metres in 5

seconds, with increasing velocity.Between A and B it travels a further 4 metres in 5 seconds, with a constant velocity of 0.8 m/s. Between B and C it is stationary for 7 seconds.Between C and D it moves back to the start 2 a distance of 12 metres 2 in 8 seconds, with constant velocity of 21.5 m/s.

2 a)

0 1

5

10

15

20

2 3 4 5 6

Time (seconds)

Dis

tanc

e (m

)

7 8 9 10

b) Velocity 5 2.25 m/s. The object travels 18 metres in 8 seconds with a constant velocity.

3 a)

0 1

5

10

15

20

2 3 4 5 6

Time (seconds)

Vel

ocit

y (m

/s)

7 8 9 10

b) 9 m/s c) 1.5 m/s2


2 a)

0

y

y � x2 � 4x � 3

y � 2x � 4

y � 6

x�1 1 2 3 4 5�1

�2

8

7

6

5

4

3

2

1

b) (i) x 5 1, 3(ii) x 5 20.6, 4.6 [Accept 60.1] (iii) x 5 1.6, 4.4 [Accept 60.1]

3 a)

0

y

y � x2 � 2x � 15

y � �x � 11

y � �5

x�2 �1�3�4�5�6 2 41 3

�5

�10

�15

10

5

b) (i) x = 25, 3(ii) x 5 24.3, 2.3 [Accept 60.1](iii) x 5 24, 1

c) (i) y 5 x 2 18(ii) The line and the curve do not intersect

(or the equation has no roots). 4 a) y 5 5 b) y 5 3x 2 2 5 a) y 5 2x b) y 5 212 c) y 5 3 2 2x 6 a) y 5 4 b) y 5 2x c) y 5 5 2 2x

Exercise 16.4H 1 a) x 1 2 3 4 6 8 12

y 12 6 4 3 2 1.5 1

b)

x

y

y � y � x

x � 9

20 4 6 8 10 12

2

4

6

8

10

12 12x

c) x 5 1.3 d) (i) x 5 y 5 3.5 (ii) √

___ 12

3 a), b)

�2

�1

�3

�4

�5

2

4

1

0

3

y

y � x2 � 6x � 4

x1 2 3 4 5 6

4y � 3x � 12

c) (1.3, 22.0) and (5.5, 1.2) [Accept 60.5]

4 a), b)

�4

0�2

�6

�8

�10

�12

4

8

2

6

10

y

y � 10 � x � 2x2

x1�1�2�3 2 3

3y � 4x � 12

c) (21.2, 5.8) and (2.4, 0.8) [Accept 60.5]

Exercise 16.3H 1 a)

�4

0

�8

�12

4

8

y

y � x2 � 5x � 6

y � 2 � 3x

y � �10

x2�2 �1 4 6 71 3 5

b) (i) x 5 21, 6(ii) x 5 1, 4(iii) x 5 22, 4


c) y

x1 2 3 4 5

20

0

40

60

80

100

120

y � 4 � 2x

d) x 5 4.3 [Accept 60.1] 6 a 5 3, b 5 2

17 Proportion and variationExercise 17.1H 1 5 revolutions 2 25 minutes 3 8232 metres 4 900 grams 5 49 litres 6 48p 7 £125 8 70 litres 9 135 kilometres 10 255 metres

Exercise 17.2H 1 20 minutes 2 8 weeks 3 24 hours 4 12 days 5 9 hours 6 5 1 _ 4 hours 7 12 hours 8 £715 9 14 days 10 1 1 _ 2 hours

Exercise 17.3H 1 a) w 5 4f 2 b) w 5 64 c) f 5 5

2 a) y 5 3200 _____ x2 b) (i) y 5 200 (ii) x 5 10

3 a) y 5 30 ___ x b) y 5 20 c) x 5 60

4 a) L 5 144 ____ d2 b) d 5 2

5 a) y 5 4x3 b) (i) y 5 4 __ 27 (ii) x 5 10

Exercise 17.4H 1 a) y ~ x2 b) y 5 2x2

2 a) y ~ 1 __ x2 b) y 5 36 ___

x2

3 a) y ~ 1 __ x2 b) y 5 216 ____

x2

4 a) y ~ x2 b) y 5 1 _ 4 x2

5 a) y ~ 1 __ x2 b) y 5 100 ____

x2

2 a), b)

�20

0�10

�30

�40

�50

20

40

10

30

50

yy � 2x3

y � 10x

x1�1�2�3 2 3

c) (i) 2x3 2 10x 5 0(ii) x 5 22.2, 0, 2.2 [Accept 60.1]

3 a), c)y

x1 2 3 4 5 6

1

0

2

3

4

5

6

y � x � 1

y � 6x � 1

b) x 5 0.5d) (i) x2 5 7

(ii) x 5 ±2.6 [Accept 60.1] 4 a)

x�2 �1 0 1 2

y

y � 4�x

4

8

12

16

b) (i) y 5 0.5 [Accept 60.1](ii) x 5 21.7 [Accept 60.1]

5 a) a = 4, b 5 2b) x 0 1 2 3 4 5

y 4 8 16 32 64 128


9 x 5 3, 1 _ 4 10 x 5 2 1 _ 3 , 2 1 _ 2

11 x 5 0, 2 _ 3 12 x 5 1 1 _ 2 , 1 1 _ 3

Exercise 19.2H 1 x 5 24, 21 2 x 5 1, 7 3 x 5 25, 1 4 x 5 21, 2 5 x 5 24, 3 6 x 5 0, 27 7 x 5 2, 7 8 x 5 22, 5 9 x 5 0, 5 10 x 5 6411 x 5 22, 26 12 x 5 22, 1013 x 5 24 (repeated) 14 x 5 3, 515 x 5 610 16 x 5 222, 117 x 5 0, 3 18 x 5 2, 919 x 5 0, 2 1 _ 2 20 x 5 61 2 _ 3

21 x 5 2 1 _ 2 , 21 22 x 5 2 _ 3 , 1

23 x 5 22 1 _ 3 , 21 24 x 5 23, 1 _ 2

25 x 5 2 _ 3 , 22 26 x 5 1 _ 2 , 8

27 x 5 1 _ 2 (repeated) 28 x 5 2 1 _ 4 , 3

29 x 5 2 1 _ 2 , 22 30 x 5 1 2 _ 3 , 2 1 _ 2

31 x 5 1 _ 5 , 2 2 _ 3 32 x 5 1 1 _ 2 , 4 _ 9

33 will not factorise 34 x 5 1 _ 6 , 22 1 _ 2

35 x 5 2 1 _ 4 , 2 2 _ 7 36 x 5 2 3 _ 4 , 5 _ 6

Exercise 19.3H 1 x 5 2, 23 2 x 5 2, 5 3 x 5 21, 5 4 x 5 3, 7 5 x 5 0, 11 6 x 5 26, 2 7 x 5 21, 6 8 x 5 1 _ 2 , 1 9 x 5 21, 9 10 x 5 25, 1

Exercise 19.4H 1 a) (i) (x 1 1)2 2 1 (ii) (x 2 2)2 2 4 (iii) (x 2 7)2 2 49 (iv) (x 2 0.5)2 2 0.25 b) (i) (x 1 1)2 2 6 (ii) (x 2 2)2 1 3 (iii) (x 2 7)2 2 48 (iv) (x 2 0.5)2 2 7.25 2 a) (x 1 3)2 2 10 b) (x 1 4)2 2 18 c) (x 2 2)2 2 1 d) (x 2 1)2 2 4 e) (x 2 6)2 1 1 f) (x 1 5)2 2 28 g) (x 2 3)2 1 10 h) (x 1 1.5)2 2 4.25 i) (x 2 2.5)2 1 0.75

Exercise 19.5H 1 a) x 5 20.24, 4.24 b) x 5 20.35, 25.65 c) x 5 20.47, 8.47 d) x 5 24.32, 2.32 2 a) (x 2 3)2 2 14 5 0; x 5 6.74, 20.74 b) (x 1 2)2 2 3 5 0; x 5 20.27, 23.73 c) (x 2 2)2 2 7 5 0; x 5 4.65, 20.65 d) (x 1 5)2 2 20 5 0; x 5 20.53, 29.47 e) (x 1 1)2 2 9 5 0; x 5 2 or 24

[exact solutions show that the original would have factorised]

f) (x 1 3)2 2 11 5 0; x 5 0.32, 26.32 g) (x 2 4)2 2 10 5 0; x 5 7.16, 0.84 h) (x 1 7)2 2 52 5 0; x 5 0.21, 214.21 i) (x 1 1.5)2 2 8.25 5 0; x 5 1.37, 24.37

Exercise 17.5H 1 a) y ~ x b) y 5 6x c) 90

2 a) y ~ 1 __ x b) y 5 144 ____

x c) 8

3 a) y ~ x2 b) y 5 1 _ 8 x2 c) 162

4 a) y ~ 1 __ x2 b) y 5 72 ___

x2 c) 2.88

5 a) y ~ 1 __ x b) y 5 60 ___

x c) 3

6 a) y ~ x b) y 5 5 _ 4 x c) 125

7 a) y ~ 1 __ x b) y 5 140 ____

x c) 3.5

8 a) y ~ 1 __ x2 b) y 5 36 ___

x2 c) 1

9 a) y ~ x2 b) y 5 10x2 c) 100010 a) y ~ x b) y 5 5x c) 375

11 a) y ~ 1 __ x b) y 5 72 ___

x c) 7.2

12 a) y ~ x2 b) y 5 2x2 c) 2048

13 a) y ~ x b) y 5 1 _ 3 x c) 50

14 a) y ~ 1 __ x2 b) y 5 200 ____

x2 c) 0.5

18 Linear equationsExercise 18.1H 1 4y 5 x 1 4 2 y 5 2x 2 2 3 3x 1 10y 5 30 4 3y 5 5x 1 20 5 2x 1 y 5 4 6 4y 5 3x 2 4 7 y 5 2x 1 1 8 3x 1 y 5 9 9 5x 1 2y 5 5 10 3y 5 4x 2 9

Exercise 18.2H 1 a) 22 b) 3 _ 4 c) 21 2 y 5 2x 22 3 4x 1 2y 5 14 or 2x 1 y 5 7 4 a) 2 3 _ 5 b) 3y 5 5x 22 5 x 1 4y 5 8 6 3x 1 y 5 5 7 2y 5 7x 2 4 8 a) None of them b) y 5 3x 1 5 and x 1 3y 5 5 9 x 1 y 5 710 a) 2x 1 y 5 7 b) 2y 5 x 1 4 c) (4, 4) and (0, 2)

19 Quadratic equationsExercise 19.1H 1 x 5 0, 23 2 x 5 0, 4 3 x 5 62 4 x 5 5, 26 5 x 5 0, 25 6 x 5 210, 1 7 x 5 3, 7 8 x 5 29, 2 1 _ 2


21 SequencesExercise 21.1H 1 a) 199 b) Multiply the pattern number by 2 then

subtract 1 2 a) Pattern number 1 2 3 4 5

Number of matchsticks 7 9 11 13 15

b) They go up by 2 each time. c) 105 3 a)

b) 72. The rectangle will have 8 columns and 8 1 1 5 9 rows

4 a) 7 3 1 2 10 5 7 2 10 5 23 b) 4, 11, 18, 25 5 a) 10, 20, 30, 40 b) 10, 18, 26, 34 6 a) 1, 4, 9, 16, 25 b) 2, 8, 18, 32, 50 c) 5, 20, 45, 80, 125 7 3, 9, 27, 81, 243 8 12, 7, 2, 23 9 For example

and so on.10 For example

and so on.

Exercise 21.2H 1 a) 3n 1 7 b) n 2 1 c) 2n 2 5 2 a) 30 2 5n b) 6 2 2n c) 4 2 n 3 a) Not linear b) Linear: 14, 17 c) Not linear d) Linear: 28, 212 4 a) 100, 200, 300, 400, 500 b) 100n 2 1 5 a) Number of shirts 1 2 3

Cost in £ 28 53 78

b) 25n 1 3 c) 5 6 a) 1, 4, 9, 16, 25 b) n2 2 1 7 1830 8 a) 2, 4, 8, 16, 32 b) Powers of 2 9 a) 1, 8, 27, 64, 125 b) n3 1 2 c) 100210 a) 5n2 b) 200011 a) n(n 1 1) b) n(2n 1 1) c) 2n(2n 2 1)

Exercise 19.6H 1 x 5 20.38, 22.62 2 x 5 6.54, 0.46 3 x 5 2.46, 24.46 4 x 5 2.19, 23.19 5 x 5 20.63, 22.37 6 x 5 20.22, 22.28 7 No real solutions 8 x 5 20.61, 22.72 9 x 5 2.81, 20.47 10 x 5 2.27, 21.7711 x 5 20.14, 21.4612 x 5 1, 23.5 [exact solutions show that the

original would have factorised]13 x 5 2.37, 0.63 [dividing the original equation by 2

would make this easier]14 x 5 3.14, 20.64 15 No real solutions16 x 5 0.46, 21.46 17 x 5 8.73, 25.7318 x 5 20.41, 21.84 19 x 5 20.46, 21.7420 x 5 1.81, 21.47 21 No real solutions

Exercise 19.7H 1 a) 2[5(x 1 3) 1 5(x 1 3) 1 (x 1 3)2] 5 195

⇒ 2x2 1 32x 2 117 5 0 b) x 5 3.07, 219.1 c) 6.07 cm 3 6.07 cm 3 5 cm 2 a) 4x 1 x(5x 1 2) 5 128 ⇒5x2 1 6x 2 128 5 0 b) x 5 4.5, 25.7 c) 29.0 s 3 a) 2[x2 1 x(x 1 2) 1 x(x 1 2)] 5 83

⇒ 6x2 1 8x 2 83 5 0 b) x 5 3.11, 24.45 c) 3.11 cm 3 3.11 cm 3 5.11 cm 4 a) 2(2x2) 1 3(7x 1 5) 5 202

⇒ 4x2 1 21x 2 187 5 0 b) x 5 4.7, 29.9 c) cross section 44.2 cm2

rectangular face 37.9 cm2

20 Simultaneous equationsExercise 20.1H 1 x 5 3, y 5 2 2 x 5 2, y 5 21 3 x 5 13, y 5 24 4 x 5 22, y 5 23 5 x 5 6, y 5 21 6 x 5 3 1 _ 2 , y 5 2 1 _ 2 7 x 5 5, y 5 210 8 x 5 4, y 5 3 9 x 5 5, y 5 1 _ 2 10 x 5 1 1 _ 2 , y 5 311 x 5 5, y 5 22 12 x 5 22, y 5 2313 x 5 21, y 5 3 1 _ 2 14 x 5 2 1 _ 7 , y 5 2 5 _ 7

15 x 5 3, y 5 27 16 x 5 1 1 _ 3 , y 5 217 x 5 5, y 5 3 18 x 5 1, y 5 419 x 5 3, y 5 22 20 x 5 21, y 5 221 x 5 3 1 _ 2 , y 5 1 1 _ 2

Exercise 20.2H 1 x 5 3, y 5 24 2 x 5 22, y 5 8 3 x 5 22, y 5 5 4 x 5 21, y 5 10 5 x 5 4, y 5 3 6 x 5 23, y 5 25


b) x 5 4.3 6 x 5 4.4 7 x 5 3.83 8 x 5 3.79 9 a) x 1 x2 5 1000 b) x 5 31.1 (or x 5 232.1)10 x 5 8.47

23 Formulae 2Exercise 23.1H

1 r 5 C ___ 2p

2 V 5 Tk ___ P

3 x = 2b _____ a 2 2

4 x 5 y(a 1 3)

________ a 2 3

5 p 5 r 1 rq 2 t

_________ q 6 q 5 r 2 t _____ p 2 r

7 r 5 pq 1 t

______ q 1 1

8 t 5 2s _____ u 1 v

9 v 5 2s 2 tu _______ t 10 x 5 3 _______

3V 1 1

11 g 5 Pe 1 f

______ 4p 2 3

Exercise 23.2H

1 a) x 5 √______

y 2 3

_____ 2 b) t 5 2p √

__

h __ g

c) x 5 3y2 d) x 5 y2 2 a

e) x 5 √______

s2 2 y2 f) l 5 3A 2 3pr2 __________

pr

g) x 5 3y 1 15

_______ 2 h) x 5 a 1 √

__ y

2 a) 113.1 cm3 b) r 5 3 √___

3V ___ 4p

c) 4.9 cm

3 a) 7.83

b) (i) x 5 m4 ____

abc or m4

_________ 4 3 7 3 8

(ii) 10

24 FunctionsExercise 24.1H 1 a) 13 b) 8 2 a) 16 b) 1 c) 1 3 a) x 5 2 b) (i) 5x 2 13 (ii) 10x 2 3 4 a) x 5 21.5 b) (i) 4x 1 8 (ii) 4x 1 10 5 a) x 5 0.6 b) (i) 5x 1 12 (ii) 10x 2 3 6 a) x 5 62 b) (i) x2 1 6x 1 11 (ii) 4x2 1 3

22 Formulae 1Exercise 22.1H 1 £39 2 a) (i) £140 (ii) £210 b) C 5 60 1 n 3 a) (i) £180 (ii) £240 b) C 5 80 1 20h 4 a) 17 cm b) P 5 2(x 1 y) or 2x 1 2y

5 a) V 5 28 cm3 b) V 5 Ah ___ 3

6 a) 7 minutes b) T 5 w __ r

7 a) t 5 d __ s b) 3.5 h

8 C 5 26.69 cm 9 a) £2.50 b) A 5 1 1 0.5n10 a) A 5 2.5 b) B 5 7 c) C 5 8 d) D 5 12 e) E 5 47 f) F 5 0.5

Exercise 22.2H

1 a) b 5 a 2 c b) x 5 a 1 y

_____ 3

c) t 5 a 2 b _____ c d) q 5 F __

2 2 p

e) y 5 x 1 3z

______ 2 f) n 5 5P 2 3 _______

4

2 d 5 C __ p

3 a) a 5 2nA ____ 3b

b) n 5 3ab ____ 2A

4 a 5 P __ 2 2 b

5 m 5 y 2 c

_____ x

6 r 5 √___

A ___ 4p

7 a) V 5 34.4 (to 1 d.p.)

b) (i) r 5 √___

4V ___ ph

(ii) r 5 3.6 (to 1 d.p.)

8 a) h 5 S ____ 2pr

2 r

Exercise 22.3H 1 a) 10 b) 30 c) x 5 18.125 2 a) (i) 60 (ii) 120 (iii) 92.736 (iv) 99.123 (v) 95.894 625 b) x 5 4.6 3 x 5 2.2 4 a) 23 1 3 3 2 5 14

33 1 3 3 3 5 36 b) x 5 2.8 5 a) 43 2 2 3 4 5 56

53 2 2 3 5 5 115


6 y

x

3

f(x)

f(x � 3)

f(x) � 3

4

2

1

0

�1

�2

2 4 6 8 10

7 y

x

10

g(x)

g(x � 1)

g(x) � 2

8

6

4

2

10�1 2 3

8 y 5 (x 2 3)2 2 4 or y 5 x2 2 6x 1 5 9 y 5 sin x 2 110 a) y 5 (x 2 2)2 1 3(x 2 2) 1 3 b) y 5 x2 2 4x + 4 + 3x 2 6 1 3 5 x2 2 x 1 1

Exercise 24.3H 1 a)

4

xO

y

f(x) � 4

b)

x

�g(x)

O

y

7 a) 14 b) (i) 2x2 1 5x 1 2 (ii) 18x2 2 9x 8 a) x 5 21 or 22 b) (i) 3x2 1 9x 1 4 (ii) 4x2 1 10x 1 4

Exercise 24.2H 1 a)

�4 �3 �2 �1 0 1

4

�4

�8

�12

�16

y

y � �x2

y � 2 � x2

x2 3 4

b) A translation of (  0 2 )

2 a)

�4 �3 �2 �1 0 1

4

�4

8

12

16

y

x2 3

y � x2

y � x2 � 4

b) A translation of (  0 24

) 3 a)

�4 �3 �2 �1 0 1

4

�4

8

12

16

20

24

y

x2 3 4 5 6

y � x2

y � (x � 2)2 � 3

y � (x � 2)2

b) A translation of (  2 3 )

4 a) y

θ

3

2

1

0 90° 180° 270° 360°

b) y 5 cos 1 2 5 a) y 5 tan 1 4 b) y 5 tan( 2 3)


b)

x

y � g(x) � 3 �2

O1

y

c)

x

y � h(x � 2)

2O

y

Exercise 24.4H 1 a)

yy � 2 cos θ

y � cos θ

θ

2

1

�1

�2

090° 180° 270° 360°

b) A one-way stretch parallel to the y-axis with scale factor 2

2 a) y

θ

�1

1

090° 180° 270° 360°

y � sin θy � sin 2 θ

b) A one-way stretch parallel to the x-axis with scale factor 0.5

3 a) A reflection in the y-axis b) A reflection in the x-axis c) A one-way stretch parallel to the y-axis with

scale factor 3 d) A one-way stretch parallel to the x-axis with

scale factor 3

c)

x

h(4x)

O

y

d)

x

j(x � 2)

42�2

�4

O

y

2 a)

x

f(x � 6)

�6O

y

b)

x

g(x) � 6

�2

4

O

y

c)

x

y � 3 � 2x2

y � �2x2

3

O

y

3 a)

x

y � 3x2

O

y


Exercise 26.3H 1 a 5 60° Corresponding angles. 2 b 5 75° Allied angles. 3 c 5 65° Alternate angles. 4 d 5 49° Corresponding angles. e 5 49° Alternate or opposite angles. 5 f 5 60° Allied angles. g 5 120° Alternate angles or Angles on a

straight line add up to 180°. 6 h 5 100° Corresponding angles. i 5 80° Allied angles or Angles on a straight

line add up to 180°. 7 j 5 60° Alternate angles. k 5 120° Angles on a straight line add up to

180°. 8 l 5 118° Corresponding angles. m 5 62° Allied angles. n 5 118° Alternate angles. 9 o 5 40° Alternate angles. p 5 47° Allied angles.10 q 5 35° Corresponding angles. r 5 55° Angles in a triangle add up to 180°. s 5 55° Corresponding angles or Angles in a

triangle add up to 180°.

Exercise 26.4H 1 a 5 20° Angles in a triangle add up to 180°. 2 b 5 27° Angles in a triangle add up to 180°. 3 c 5 25° Angles in a triangle add up to 180°. 4 d 5 45° Angles in an isosceles triangle. 5 e 5 79° Exterior angle of a triangle equals

the sum of the opposite interior angles.

6 f 5 160° Exterior angle of a triangle equals the sum of the opposite interior angles.

7 g 5 132° Exterior angle of a triangle equals the sum of the opposite interior angles.

h 5 48° Angles on a straight line add up to 180°.

8 i 5 75° Angles in a quadrilateral add up to 360°.

9 j 5 115° Angles in a quadrilateral add up to 360°.

k 5 65° Angles on a straight line add up to 180°.

10 l 5 115° Angles on a straight line add up to 180°.

m 5 136° Angles in a quadrilateral add up to 360°.

Exercise 26.5H 1 A Parallelogram B Rectangle C Trapezium (isosceles) D Kite E Trapezium F Square

4 y 5 sin 4

5 a) y 5 x2 2 1 b) y 5 2x2 1 1 6 a) y 5 16x1 4 b) y 5 8x 1 1

7 a) A translation of (  0 22

) b) A one-way stretch parallel to the y-axis with

scale factor 3c) A one-way stretch parallel to the x-axis with

scale factor 2d) A one-way stretch parallel to the y-axis with

scale factor 4 and a one-way stretch parallel to the x-axis with scale factor 0.5

8 a) y 5 2x2 2 3b) y 5 x2 1 3c) y 5 4x2 1 3

9 a) y 5 1 _ 4 x2 2 x b) (2, 21)

10 a 5 3, b 5 6

25 Algebraic fractionsExercise 25.1H

1 5(x 1 1)

________ 6 2 x 1 11 ______

12 3 x 1 1 _____

6

4 5x 1 8 ______ 4 5 x

2 1 2x 1 3 ___________ 3(x 1 3)

6 x2 2 x 2 16 ___________ 2(x 2 5)

7 6x2 1 5x 2 19 _____________ 5(2x 2 1)

8 35x 1 6 _________ 4(3x 2 2)

Exercise 25.2H

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

2 x2 1 8x 2 9 ___________ 2x(x 2 3)

3 25x2 1 7x 1 10 ______________ 2x(x 1 2)

4 13x 1 1 _____________ (x 2 3)(x 1 2)

5 22x 2 17 _____________ (x 1 4)(x 2 5)

6 2(x2 2 2x 1 3)

_____________ (x 2 3)(x 1 1)

7 2x2 2 2x 2 23 _____________ (x 2 5)(x 1 4)

8 x2 1 21x 1 17 _______________ (3x 1 1)(2x 1 5)

Exercise 25.3H 1 x 5 11 2 x 5 21 3 x 5 5 4 x 5 23.4 5 x 5 1 __ 18 6 x 5 2 or x 5 12

__ 19

7 x 5 1 __ 13 8 x 5 4.54 or x 5 21.54 9 x 5 21 or x 5 2 11

__ 6

26 Properties of shapesExercise 26.1H 1 15 cm2 2 40 cm2 3 18 cm2 4 20 m2

5 60 cm2 6 160 cm2 7 96 mm2 8 10.5 cm2

9 38.5 cm2 10 96 cm2

Exercise 26.2H 1 28 cm2 2 54 cm2 3 25 cm2 4 56 m2

5 20 cm2 6 22 cm2 7 50 cm2 8 72 cm2

9 8 cm 10 9 cm


4 a) 58° b) 10°, 100°, 23°, 105°, 122° 5 e 5 20°, i 5 160° 6 e 5 15°, i 5 165° 7 30 8 45

27 Pythagoras’ theoremExercise 27.1H 1 19 cm2 2 40 cm2 3 12 cm2 4 46 cm2

Exercise 27.2H 1 a) 14.76 cm b) 15 m c) 11.62 cm d) 9.38 cm e) 14.81 cm f) 20.78 m g) 4.91 cm h) 19.14 cm 2 117 m (to the nearest metre) 3 199 cm (to the nearest centimetre) 4 10.4 cm 5 109.7 cm3

Exercise 27.3H 1 Yes 152 1 202 5 252 (This is 5 3 the 3, 4, 5

triangle) 2 No 4.22 1 7.3 2 8.72

3 No 122 1 122 152

4 Yes 15.52 1 37.22 5 40.32 (This is 3.1 3 the 5, 12, 13 triangle)

5 No 72 1 82 112

6 No 6.52 1 11.12 12.82

7 Yes 8.42 1 28.82 5 302 (This is 1.2 3 the 7, 24, 25 triangle)

8 No 13.52 1 27.22 30.22

Exercise 27.4H 1 a) (4, 5) b) (5, 20.5) c) (22.5, 6) d) (23.5, 1) e) (21.5, 25.5) 2 a) (21, 7) b) (5, 3) c) (5.5, 4.5) d) (5, 4.5) e) (22, 3.5) f) (26, 24.5)

Exercise 27.5H 1 a) (5, 0, 0) b) (5, 0, 3) c) (0, 0, 3) d) (0, 7, 0) e) (5, 7, 0) f) (0, 7, 3) 2 a) (8, 0, 0) b) (8, 10, 0) c) (0, 10, 0) d) (4, 5, 0) e) (4, 5, 7) 3 a) (i) (6, 5, 0) (ii) (6, 5, 3) (iii) (0, 5, 3) (iv) (6, 5, 1.5) (v) (3, 5, 3) b) (i) AB (ii) OCGD

G Parallelogram H Square I Rhombus J Arrowhead 2 a) Square, rectangle b) Rectangle, parallelogram, square, rhombus c) Rectangle, square, isosceles trapezium d) Rectangle, parallelogram, square, rhombus,

trapezium e) Square, rectangle, parallelogram, rhombus 3 a)

0 1

1

2

3

4

y

x2 3 4 5

Square b)

0 1

1

2

3

4

5

y

x2 3 4

Rectangle c)

0 1

1

2

3

4

5

6

7

y

x2 3

Parallelogram d)

0 1

1

2

3

4

5

y

x2 3 4 5 6 7 8

Trapezium (isosceles) 4 All sides are equal length. 5 Parallelogram.

80°

100° 80°

100°

Exercise 26.6H 1 1260° 2 1980° 3 a) 47.5° b) 87°, 130°, 143°, 95°, 132.5°, 132.5°


Exercise 28.2H 1

�3 �2 �1 0 1

1

�1

�2

�3

2

3

4

y

x2 3 4

B

D

A

C

2

�3 �2 �1 0

B

1

1

�1

�2

�3

2

3

y

x2 3D

C

A

3

�4 �2 0

�2

�4

2

4

y

x2 4

B

D

AC

4 C

5

O

6

150°A

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

8 a) A rotation of 90° clockwise (22, 2)b) A rotation of 90° anticlockwise about the

originc) A rotation of 180° about (0, 2)

28 transformationsExercise 28.1H 1

�2 0

�2

2

4

y

x2 4

A

DC

B

2

�3 �2 �1 0 1

1

�1

�2

�3

2

3

y

x2 3

D

A B

C

3 a)

b)

c)

4 a) A reflection in the line y 5 2 b) A reflection in the line y 5 x c) A reflection in the line x 5 3.5 5 a) A reflection in the line y 5 0.5 b) A reflection in the line y 5 2x c) A reflection in the line y 5 1.5


Exercise 28.4H 1 a), b)

B

A

0

2

4

6

y

x2 4 6

c) An enlargement with scale factor 3, centre the origin

2 a), b)

A

B

0

2

4

6

y

x2 4 6

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

3 a), b)

B

0

2

4

6

8

1

3

5

7

y

x2 4 6 81 3 5 7

A

c) An enlargement with scale factor 2 _ 5 , centre the origin

4 a), b)

B

0

2

4

6

1

3

5

7

y

x2 4 61 3 5 7

A

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

5 a) An enlargement with scale factor 2, centre the origin

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

c) An enlargement with scale factor 3, centre (0, 3)

d) An enlargement scale factor 1 _ 3 , centre (0, 3)

9 A rotation of 120° about the point P10 a) A rotation of 90° anticlockwise about the

originb) A reflection in the line x 5 2c) A rotation of 180° about (1, 0)d) A rotation of 90° clockwise about (1, 21)e) A reflection in the line y 5 21

Exercise 28.3H 1

�2 0

�2

2

4

6

y

x2 4 6

D

A

B

C

2

�3 �2 �1 0 1

1

�1

�2

�3

2

3

4

5

y

x2 3 4 5

C

A

B

D

3 a) A translation by vector (  5 0 )

b) A translation by vector (  3 23

)

c) A translation by vector (  21 22

)

d) A translation by vector (  26 22

)

4 a) A translation by vector (  1 22

) b) A reflection in the line x 5 21c) A rotation of 90° anticlockwise about the

origind) A translation by vector (  2

23 )

e) A rotation of 180° about the point (1, 2)

f) A translation by vector (  0 3 )

g) A rotation of 180° about the point (3, 1)

h) A translation by vector (  23 1 )


4 74.95 cm and 75.05 cm 5 7.825 mm and 7.835 mm 6 a) 14.05 cm and 14.15 cm, 6.75 cm and 6.85 cm,

12.15 cm and 12.25 cm b) 32.95 cm and 33.25 cm 7 Shortest 5 306 mm, longest 5 308 mm 8 a) 0.6 kg b) 0.8 kg

Exercise 29.3H 1 a) 78 kg b) 16 minutes c) 162 miles or 160 miles d) 5 m or 5.1 m e) 3 litres 2 a) 8, 9, 10 or 11 cm b) 45, 50, 55, 60 or 65° c) 10.4 cm2 d) 25 cm e) 1.9 cm f) 206° g) About 2 hours or 2 hours 10 minutes h) £2.33

Exercise 29.4H 1 8 km/h 2 65.7 mph 3 8.75 km 4 5 hours 20 minutes 5 558 g 6 3.6 g/cm3

7 4.4 km/h 8 350 g 9 4.3 cm3 10 1.3 kg/m3

11 6.712 a) 48 km/h b) 1529 or 1528

30 Constructions and lociExercise 30.1HDiagrams are not full size. Measurements given are correct for the suggested scale. 1

7 cmA B

2 Scale 1 cm 5 1 mile

3 cm

Home

3

6 cm6 cm

B

6 cmA C

4 Scale 1 cm 5 1 m

5 cm

6 cm

8 cm

F

5

3 cm

D 5 cm

5 cm

C

A B

6

Hedge

Zeke

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

b) A reflection in the line y 5 xc) A reflection in the line x 5 0d) A rotation of 180° about the origine) An enlargement with scale factor 3,

centre (0, 21) 7

�4 �2 0

�2

�4

2

4

y

x2 4

C

B

E

DA

8

�4 �2 0

�2

2

4

y

x2 4

C

B

AD

29 MeasuresExercise 29.1H 1 a) 250 mm b) 2400 cm c) 13.6 mm d) 151 mm e) 235 mm 2 a) 20 000 cm2 b) 300 mm2 c) 11 200 cm2

d) 5 mm2 e) 2 000 000 mm2

3 a) 80 cm2 b) 840 cm2 c) 200 m2

d) 1800 m2 e) 6.4 m2

4 a) 32 000 mm3 b) 24 000 000 cm3

c) 5200 mm3 d) 420 000 cm3

e) 20 mm3

5 a) 5.2 m3 b) 270 cm3

c) 0.000 21 m3 d) 8 400 000 000 mm3

e) 0.17 cm3

6 a) 36 000 cm3 b) 6.3 litres c) 1400 ml d) 0.061 litres e) 5.4 litres

Exercise 29.2H 1 a) 4.25 cm and 4.35 cm b) 462.5 ml and 463.5 ml c) 10.45 seconds and 10.55 seconds d) 77.5 kg and 78.5 kg e) 5.45 m2 and 5.55 m2

2 23 500 3 141.5 cm and 142.5 cm


3

5 cm

3 cm

BA

4 cm

4

2 cm

P Q

R

X40° 35°

6 cm

5 Scale 1 cm 5 5 km

3 cm

4 cmA B

2 cm

6 3 cm

3 cm

6 cm

6 cm

7

5 cm

Locus

7 cm

P Q

8 Scale 1 cm 5 5 km

D E

C5 cm

5 cm

4 cm

7 S 7 cm

5 cm

R

P Q

8

40°

40°

9 Scale 1 cm 5 4 m

4 cm

2.5 cm

2.5 cm

3 cm

10 Scale 1 cm 5 4 m

4 cm

3 cm3.125 cm

12.5 m

Exercise 30.2HDiagrams are not to scale. Measurements given are correct for the suggested scale.

1

4 cmP

6 cm

2 Scale 1 cm 5 4 m

3 cm

5 cm1 cm1 cm

1 cm1 cm


Exercise 31.4HAnswers to 4 s.f. 1 408 cm3 2 1330 cm3

3 422.6 cm3 4 798 cm3

5 1099 cm3 6 1098 cm3

Exercise 31.5HAnswers to 4 s.f. 1 914.6 cm3 2 815.1 cm3

3 1163 cm3 4 691.0 cm3

5 2244 cm3 6 4955 cm3

Exercise 31.6HAnswers to 3 s.f. 1 4460 cm3 2 14 300 cm3

3 157 000 cm3 4 21 600 mm3

5 19 700 mm3 6 5.77 mm3

7 45.4 m3 8 72.2 m3

Exercise 31.7HAnswers to 3 s.f. 1 a) 905 cm2 b) 1720 cm2

c) 2040 cm2 d) 3740 mm2

e) 452 mm2 f) 1460 mm2

g) 227 m2 h) 72.9 m2

2 a) 955 cm2 b) 4810 cm2

c) 8020 cm2 d) 8040 mm2

e) 10 300 mm2 f) 1200 mm2

g) 39.2 m2 h) 83.8 m2

Exercise 31.8H 1 Plan

Side elevation

9 Scale 1 cm 5 1 m

D C

A B

2 cm

5 cm

1.5 cm

3 cm

10

6 cm

8 cm7 cm

10 cm

LocusE F

G

31 Areas, volumes and 2-D representation

Exercise 31.1HAll answers to the same accuracy as the measurement given in the question. 1 25 cm 2 53 cm 3 123.2 cm 4 364 mm 5 16.0 m 6 19.85 m 7 44 cm 8 72 cm 9 245 mm 10 122 mm11 14 m 12 8.64 m

Exercise 31.2HAll answers to 3 s.f. 1 a) 908 cm2 b) 1660 cm2

c) 14 100 cm2 d) 5810 mm2

e) 17 200 mm2 f) 3220 cm2

g) 10 600 cm2 h) 58.1 cm2

i) 238 cm2 j) 6940 m2

k) 11.3 m2 l) 20.9 m2

2 a) 254 cm2 b) 616 cm2

c) 3630 cm2 d) 1130 mm2

e) 4780 mm2 f) 2640 cm2

g) 1660 cm2 h) 32.2 cm2 i) 45.4 cm2 j) 804 m2 k) 9.08 m2 l) 14.7 m2

Exercise 31.3HAnswers to 3 s.f. 1 138 cm2 2 178 cm2

3 119 cm2 4 66 cm2

5 200 cm2 6 190 cm2


Side elevation

Front elevation

4 Plan

Side elevation Front elevation

32 trigonometry 1All answers are given correct to 3 significant figures.

Exercise 32.1H 1 a 5 6.34 cm, b 5 5.60 cm, c 5 11.8 cm, d 5 25 cm,

e 5 35.2 m, f 5 13.4 cm, g 5 11.9 cm, h 5 18.4 cm 2 a) 30.3 cm b) 62.5 cm 3 a) 14 cm b) 24.2 cm c) 12.6 cm d) 258 cm2

Exercise 32.2H 1 a 5 2.60 cm, b 5 12.2 m, c 5 99.7 cm, d 5 10.1 m,

e 5 9.20 m, f 5 5.35 cm, g 5 42.7 cm, h 5 15.3 m 2 2.32 m 3 a) 21.1 cm b) 84.2 cm2

Exercise 32.3H 1 a 5 34.5°, b 5 32.6°, c 5 8.58°, d 5 61.5°,

e 5 46.7°, f 5 53.4°, g 5 71.8°, h 5 32.5° 2 42.9 m 3 12.0°

Front elevation

2 Plan

Side elevation

Front elevation

3 Plan


2 a)

y

θ

�1

1

0

�0.6

90° 180° 270° 360°

b) Approximately 127° and 233°

3 a)

y

θ

�1

1

0

�0.8

90° 180° 270° 360°


4 a)y

θ

�1

1

0

�0.3

90°�90°�180° 180° 270° 360°

b) Approximately 2107°, 107° and 253°

5 Approximately (37°), 143°, 397° and 503°

33 trigonometry 2All lengths and areas are given correct to 1 d.p. and all angles to the nearest degree.

Exercise 33.1H 1 a) 14.9 cm2 b) 8.6 cm2

c) 28.4 cm2 d) 73.4 cm2

2 a) 57° b) 29° c) 72° 3 a) 32.0 cm b) 7.4 cm

Exercise 33.2H 1 a) AB

∧

C 5 45°, BC∧

A 5 77°, AB 5 13.8 cm b) PR

∧

Q 5 40°, PQ∧

R 5 12°, PR 5 4.6 cm c) YX

∧

Z 5 36°, XY 5 8.4 cm, YZ 5 5.0 cm d) AB

∧

C 5 35°, AC 5 6.3 cm, BC 5 6.9 cm 2 RP

∧

Q 5 79°

Exercise 33.3H 1 a) 9.0 cm b) 68° c) 13.4 cm d) 47° 2 a) AB

∧

C 5 58°, AC∧

B 5 75.5°, CA∧

B 5 46.5° (1 d.p) b) AB

∧

C 5 51°, AC∧

B 5 71°, CA∧

B 5 58° c) QR

∧

P 5 58°, QP∧

R 5 83°, PQ∧

R 5 39° d) YX

∧

Z 5 34°, XY∧

Z 5 80°, XZ∧

Y 5 66° 3 a) 64° b) 43°

Exercise 33.4H 1 a)

y

θ

�1

1

0.7

�180° �90° 90°0 180°



4 y

θ

1

�1

090° 180° 270° 360°

5 y

θ

�1.0

�1.5

1.0

1.5

0.5

�0.5

�180° �90° 90°0 180°

6 y

θ

1

�1

�0.6

090° 180° 270° 360°

Approximately 32°, 58°, 122°, 148°, 212°, 238°, 302° and 328°

6 a)y

θ

�3

�2

�1

3

2

1

0 90° �90° �180°

y � �1.5

180° 270° 360°

b) Approximately 256°, 124° and 304°

Exercise 33.5H 1 a) amplitude 5 2, period 5 360° b) amplitude 5 1, period 5 90° c) amplitude 5 5, period 5 180° d) amplitude 5 3, period 5 450° e) amplitude 5 4, period 5 72° f) amplitude 5 7, period 5 600° 2 y

θ

2

1

�1

�2

090° 180° 270° 360°

3 y

θ

�1

1

�180° �90° 90°0 180°


8 a) Using similar triangles: 0.7 ___ 0.4

5 2.8 ___ 1.6

.

b) 1.17 m3

9 a) 9.59 cmb) Use the triangle formed by the

perpendicular height of the pyramid, the sloping sides of the pyramid and half the diagonal of the base and use Pythagoras.

Length of diagonal 5 √____________

12.82 1 12.82 5 18.10… cm

Length of sloping sides 5 √__________

9.62 1 9.05 2

5 13.19… 5 13.2 cm to 3 s.f.

10 a) 810 ____ p

b) 1800 cm2

35 Similarity and enlargementExercise 35.1H 1 a) 18 cm b) 22.5 cm 2 Triangle PRQ 3 AB and QR, AC and PR, BC and PQ 4 PQ 5 12.5 units, QR 5 10 units 5 a) Triangle AXY b) BC 5 3 cm, CY 5 2.5 cm 6 x 5 4 units, y 5 10 units 7 a) 4 cm b) 4.5 cm 8 8.75 cm

Exercise 35.2H 1

0 x

y

�6

�4

�2

2

4

6

8

�6 �4 �2 2A

B

C 4 6 8 10 12 14

2

�8 �6 �4 �2 0 2

2

A

B C

�2

�4

�6

�8

4

6

8

y

x4 6 8

3 a) (3, 3) b) 22

34 Length, area and volumeExercise 34.1H 1 Expressions a) and c) 2 Expressions b) and c) 3 Expressions a) and c) 4 a) Area b) Nonsense c) Volume d) Length 5 a) 3 b) 1 c) 2 d) 1 e) 1 f) 2

Exercise 34.2HAll lengths and areas are given correct to 1 d.p. and all angles to the nearest degree. 1 a) 4.9 cm b) 23.8 cm c) 9.3 cm 2 a) 16.5 cm2 b) 48.7 cm2 c) 16.2 cm2

3 a) 57° b) 282° c) 112° d) 54° 4 a) 6.3 cm b) 2.9 cm c) 3.7 cm 5 167°, 14.9 cm

Exercise 34.3HAll answers to 3 s.f. 1 a) 295 cm2 b) 137 cm2 c) 73.5 cm2

2 a) 589 cm3 b) 154 cm3 c) 67.9 cm3

3 a) 96 cm3 b) 126 cm3 c) 109 cm3

4 278 cm2

5 8.2 cm 6 a) 5.60 cm b) 7.10 cm c) 4.20 cm 7 a) 3.38 cm b) 9.45 cm c) 4.94 cm 8 a) 163 cm2 b) 908 cm2 c) 4070 cm2

9 a) 195 cm3 b) 2570 cm3 c) 24 400 cm3

10 a) 7.1 9 cm b) 4.64 cm c) 7.57 cm d) 9.71 cm11 8.19 cm12 6.88 cm2

13 8.47 cm14 14.5 cm15 319 cm3

16 SA 5 4 3 1 _ 2 (2x 3 2x 3 sin 60) 5 4 3 2x2 sin 60

5 4 √__

3 x2 cm2

Exercise 34.4HAll answers to 3 s.f. 1 a) 5.29 cm2 b) 99.4 cm2 c) 17.1 cm2

2 151 000 cm3

3 a) 54.0 cm2 b) 101 cm2 c) 48.0 cm2

4 98.2 cm2

5 a) 6.40 cm, 154 cm3

b) 8.90 cm, 627 cm3

c) 6.10 cm, 652 cm3

6 a) 771 cm3

b) 14.2 cm, 548 cm2

7 a) Check students’ answers. b) 3 _ 4


3 a) 38.7° b) 11.7 cm c) 14.1 cm d) 55.6° 4 9.1 cm 5 9.7 cm 6 a) 17.5 cm b) 59.2° 7 a) 37.0 m b) 52.9 m c) 60.8 m 8 a) AO2 5 32 1 32 5 18

b) 8.0 cmc) 9.0 cm

9 a) 10.0 cmb) 53.2°c) 12.5 cm

Exercise 36.2H 1 8.2 cm 2 70.3° 3 a) 13 cm b) 14.3 cm c) 24.8° d) 56.9° 4 53.0° 5 a) 58.0° b) 9.4 cm c) 10.7 cm d) 48.5° 6 VB 5 VD 5 12.8 cm, at 38.7° to the base.

VC 5 16.2 cm at 29.5° to the base. 7 a) 40.8 m2 b) 78.7° c) 10.1 m [10.06 m to 2 d.p.] 8 a) 6.8 cm b) 8.6 cm

37 Congruence and proofExercise 37.1H 1 a) (i) 28° (ii) 52° (iii) 100° (iv) 28°

b) QP 2 Triangles MLN and XYZ 3 a) The three sides of one triangle are equal to

the corresponding three sides of the other triangle.

b) Two angles and the side between them in one triangle are equal to the corresponding two angles and side in the other triangle.

c) Each triangle is right-angled and the hypotenuse and one other side of one triangle is equal to the hypotenuse and one other side in the other triangle.

4 a) Congruent: RHSb) Not congruent: the given angle (60°) is

opposite a different side.c) Not congruent: the given length (7) is

opposite a different angle. d) Congruent: as two corresponding angles are

equal the third will also be equal, ASA 5 In triangles ABD and CBD:

AB 5 BC (Given.) Angle ABD 5 angle DBC (Given.) BD is common to both triangles.

4

�8�10 �6 �4 �2 0 2

2

A

BC

�2

�4

�6

�8

4

6

8

y

x4 6 8 10

5 a) (4, 4) b) 2 1 _ 3

Exercise 35.3H 1 a) 3 b) (i) 16 cm2 (ii) 144 cm2

c) (i) 9(ii) It is the square of the linear scale factor.

d) (i) 64 cm2 (ii) 1728 cm2

e) (i) 27(ii) It is the cube of the linear scale factor.

2 a) 2.5 b) 17.5 cm c) 14 cm2

d) 87.5 cm2 e) 6.25 3 a) Area scale factor 5 9, volume scale

factor 5 27b) Area scale factor 5 16, volume scale

factor 5 64c) Area scale factor 5 64, volume scale

factor 5 512d) Area scale factor 5 1 _ 4 , volume scale

factor 5 1 _ 8 e) Area scale factor 5 25

__ 9 , volume scale factor 5 125

___ 27 4 a) 7 b) 9 c) 10 d) 50 e) 3 _ 5 5 a) 15 m b) 16 cm 6 1.26 m2

7 1.2 litres 8 2240 cm2

9 16 cm3

10 20 cm11 a) The angles are not equal b) 7.5 cm12 a) The angles are the same 57°, 78°, 45°

b) The ratios of the sides are not equal 15

__ 12 5 10 __ 8 9 _ 6

36 three-dimensional geometryAnswers are given correct to 1 d.p.

Exercise 36.1H 1 12.7 cm 2 3.9 cm


(Angles on a straight line.) So angle ZXY5 60° Similarly, angles XZY and ZYX are 60°. Therefore triangle XYZ is equilateral. (This could also be proved by showing that the three sides were equal.)

10

25°9 cm 9 cm

5 cm5 cm

25°

38 Circle theoremsExercise 38.1H 1 a 5 128° 2 b 5 30°, c 5 82°, d 5 46° 3 e 5 77°, f 5 58° 4 g 5 90°, h 5 45°, i 5 55° 5 j 5 106°, k 5 98° 6 m 5 26° 7 n 5 9 cm 8 p 5 70°, q 5 75° 9 r 5 40°10 s 5 88°, t 5 44°11 u 5 27°, v 5 153°12 w 5 152°, x 5 208°, y 5 104°13 a 5 55°, b 5 125°, c 5 58°14 d 5 43°, e 5 137°15 f 5 43°16 g 5 22.5°17 h 5 151°18 i 5 31°, j 5 59°19 k 5 17 cm 20 m 5 20°

Exercise 38.2H 1 a 5 25°, b 5 65° 2 c 5 63°, d 5 42° 3 e 5 36°, f 5 54°, g 5 72° 4 h 5 90°, i 5 34°, j 5 34° 5 k 5 24°, m 5 66° 6 n 5 77°, p 5 42°, q 5 61° 7 r 5 59°, s 5 59° 8 t 5 45° 9 u 5 51°, v 5 66°10 w 5 73°

So triangle ABD is congruent to triangle DBC. (SAS)

6 In triangles ABC and DEC: Angle ACB 5 angle DCE (Vertically opposite angles.) BC 5 CE (Given.) Angle ABC 5 angle CED (Alternate angles.) So triangle ABC is congruent to triangle DEC. (ASA)

7 a) Check students’ sketches of quadrilateral ABCD. Angle ABC should be marked equal to angle ADC, lines BC and AD should be parallel, and the line AC should be drawn.

b) In triangles ABC and CDA: Angle ABC 5 angle ADC (Given.) Angle BCA 5 angle DAC (Alternate angles.) The third angles will also be equal. AC is common to both triangles. So triangle ABC is congruent to triangle CDA. (ASA)

c) The opposite sides of the quadrilateral are equal and parallel.

8 a) Check students’ sketches of isosceles triangle ABC. Lines AB and AC should be equal, and there should be a line joining the midpoint of BC to A.

b) Call the midpoint X. In triangles ABX and ACX: AB 5 AC (Isosceles.) BX 5 XC (X is midpoint of BC.) AX is common to both triangles. So triangle ABX is congruent to triangle ACX. (SSS) So angle BAX 5 angle CAX (Corresponding angles.) So AX bisects the angle at A. Angle AXB5 angle AXC (Corresponding angles.) Angle AXB 1 angle AXC 5180° (Angles on a straight line.) So angle AXB 5 angle AXC 5 90°

9 a) Check students’ sketches of equilateral triangle ABC. The midpoints of AB, BC and AC should be labelled X, Y and Z respectively, and joined to form triangle XYZ.

b) In triangle AXZ: AX 5 AZ (Both half a side of an equilateral triangle.) Angle XAZ 5 60° (Angle in equilateral triangle.) So triangle AXZ is equilateral. Similarly, triangles BXY and CYZ are equilateral. So angles AXZ and BXY are 60°. Also AXZ 1 ZXY 1 BXY 5 180°


3 a)

10 0

10

0

20

30

40

50

60

70

80

90

20 30 40 50 60 70 Age (years)

Freq

uenc

y

1989

2009

b) Answers should include the following comments.• In 1985 there were more younger

people / fewer older people.• In 2005 there were fewer younger

people / more middle-aged and older people.• For 1985 the modal group is 0 to 10, for

2005 it is 20 to 30.

Exercise 39.3H 1 a), e)

Wei

ght (

kg)

Amount of fertiliser (g/m3)

d)

10 0

40

30

50

60

70

80

90

20 30 40 50 60 70 80

b) Strong positive correlation c) 63.5 kg d) About 87 kg 2 a), e)

Mile

age

(tho

usan

ds)

Price (£000s)10

10

0

20

30

40

50

60

70

80

90

100

2 3 4 5 6 7 8 9

d)

b) Strong negative correlation c) 55 000

f) (i) About £7900(ii) 66 000 miles

39 Statistical diagramsExercise 39.1H 1 a)

20

5

0

10

15

20

25

4 6 8 10 12 14Age (years)

Freq

uenc

y

b) 4 < a , 6 2 a)

6560

10

0

20

30

40

50

70 75 80 85 90Weight (kg)

Freq

uenc

y

b) 75 < w , 80 c) 75 < w , 80 3 a) 5 b) 40 c) 30% d) 8 < t , 9

e) Time (t mins) Frequency

5 < t , 6 3

6 < t , 7 9

7 < t , 8 10

8 < t , 9 13

9 < t , 10 5

Exercise 39.2H 1

43

5

0

10

15

5 6 7 8 9Height (cm)

Freq

uenc

y

2

50

10

0

20

30

40

10 15 20 25 30Time (minutes)

Freq

uenc

y


41 Probability 1Exercise 41.1H 1 0.8 2 5 _ 6 3 0.85 4 7 __ 10

5 0.92 6 14 __ 31

Exercise 41.2H 1 0.1 2 7 __ 19 3 0.45 4 4 __ 31 5 0.17 6 2 __ 20 5 1 __ 10

Exercise 41.3H 1 8 2 4 3 12 4 8 or 9 5 a) 15 b) 45 6 240 yellow, 60 blue, 100 green

Exercise 41.4H 1 a) Score 1 2 3 4 5 6

Relative frequency

0.15 0.17 0.18 0.16 0.17 0.18

b) Yes. For a fair dice each score should have probability 1 _ 6 5 0.166 … or 1.67. All the relative frequencies for Pete’s dice are close to this value.

2 a) Result Win Lose Draw

Relative frequency

0.64 0.22 0.14

b) No. Only a small number of trials. 3 a) Flavour Relative frequency

Plain 0.28

Salt & Vinegar 0.22

Cheese & Onion 0.38

Other 0.13

b) Large number of trials. 4 a) 0.84 b) 0.16 5 a) Score 1 2 3 4 5

Relative frequency

0.07 0.17 0.31 0.30 0.15

b) No. For a fair five-sided spinner, each score should have a probability of 1 _ 5 5 0.2.

6 a) Colour Yellow Green White Blue

Number 32 24 18 6

b) The relative frequency values must have been found after a large number of trials.

42 Probability 2Exercise 42.1H 1 a) 8 __ 10 b) 5 __ 10 2 a) 0.6 b) 0.65 3 a) 0.53 b) 0.47 c) 0.75 d) 0.39 e) 0.47 4 8 __ 52 5 1 _ 4 6 1

___ 216

3 a), e)

Hei

ght o

f dau

ghte

r (c

m)

Height of father (cm)

165160

165

160

170

175

180

170 175 180 185 190

d)

b) Strong positive correlationc) 169.4 cmf) About 176 cm

40 Statistical calculationsExercise 40.1H 1 a) (i) 3 (ii) 3 (iii) 5 (iv) 3.34 b) (i) 100 (ii) 101 (iii) 6 (iv) 100.875 c) (i) 1 (ii) 1 (iii) 6 (iv) 1.64 d) (i) 1 (ii) 2 (iii) 7 (iv) 2.09 e) (i) 49 (ii) 50 (iii) 5 (iv) 50.2 f) (i) 144 (ii) 144 (iii) 7 (iv) 144.2 2 £8.20 3 1.3275

Exercise 40.2H 1 a) (i) 20 (ii) 4.97 b) (i) 75 (ii) 17.4 c) (i) 25 (ii) 72.2 d) (i) 24 (ii) 7.42 2 a) 125 < x , 150 b) 150 < x , 175 c) 160.5 3 398.1

Exercise 40.3H 1 a) (i) 50 cm (ii) (131.3) 131 cm b) (i) 25 grams (ii) (36.4) 36 grams c) (i) 80 mm (ii) (111.7) 112 mm d) (i) 10 minutes (ii) (59.56) 60 minutes 2 a) (i) 0.3 < x , 0.6 (ii) (0.87) 0.9 m b) (i) 40 < x , 50 (ii) (44.2) 44 grams c) (i) 26 < x , 28 (ii) (25.9) 26 minutes d) (i) 35 < x , 40 (ii) (35.9) 36 mph 3 a) 1500 < x , 2000 b) 1500 < x , 2000 c) £1562.50 4 41.4 seconds 5 148 pence


5 a)Firstlight

Secondlight

Thirdlight

Outcome Probability

Go

0.6

0.4

0.7

0.3

GoGo

Go

Go

Go

0.8

0.2

Stop

Stop

Stop

Stop

Stop

GGG

GSG

SGG

SSG

GGS

GSS

SGS

SSS

0.336

0.144

0.224

0.096

0.084

0.036

0.056

0.024

0.8

0.2

Go

0.8

0.2

Stop

0.8

0.2

Stop

0.7

0.3

b) (i) 0.024 (ii) 0.452 (iii) 0.212

Exercise 42.3H 1 a)

First bead Second bead Outcome Probability

Black

White

Black

White

BB

BW

WB

WW

1242

Black

White

1242

1242

642

36

36

26

46

47

37

b) (i) 6 __ 42 5 1 _ 7 (ii) 24 __ 42 5 4 _ 7

2 a) 24 _____ 132 600 5 1

___ 5525

b) 1728 _____ 132 600 5 72

___ 5525

3 a) First day Second day Outcome Probability

Snow

Not snow

Snow

Not snow

SS

SN

NS

NN

0.14

0.06

0.08

0.72

Snow

Not snow

0.2

0.8

0.7

0.3

0.1

0.9

b) (i) 0.14 (ii) 0.72 (iii) 0.28 4 0.28 5 a) 2730

____ 12 144 5 455 ___ 2024

b) 3240 ____ 12 144 5 135

___ 506

c) 8400 ____ 12 144 5 175

___ 253

43 Planning and collectingExercise 43.1H 1 a) Primary b) Secondary c) Secondary d) Primary 2 a) There are not enough categories: for

example, red and grey should be added. b) Check students’ new questions.

7 a) 0.64 b) 0.04 8 a) 4

___ 121 b) 16 ___ 121 c) 49

___ 121 9 a) 0.16 b) 0.36 c) 0.4810 a) 0.970 299 b) 0.000 001 c) 0.000 297

Exercise 42.2H 1 a)

First spin Second spin Outcome Probability

Four

Not a four

Four

Not a four

FF

FN

NF

NN

116

Four

Not a four

316

316

916

34

14

34

14

14

34

b) 1 __ 16 c) 6 __ 16 5 3 _ 8 2 a)

First day Second day Outcome Probability

Late

Not late

Late

Not late

LL

LN

NL

NN

0.3 � 0.3 � 0.09

0.3 � 0.7 � 0.21

0.7 � 0.3 � 0.21

0.7 � 0.7 � 0.49

Late

Not late

0.3

0.7

0.3

0.7

0.3

0.7

b) (i) 0.49 (ii) 0.42 3 a)

First disc Second disc Outcome Probability

�blue

red

blue

red

BB

BR

RB

RR

blue

red

58

38

38

38 � 9

64

�38

58 � 15

64

�58

38 � 15

64

�58

58 � 25

6458

38

38

58

b) (i) 25 __ 64 (ii) 34

__ 64 5 17 __ 32 (iii) 39

__ 64

4 a) Outcome Probability

WW

WD

WL

DW

DD

DL

LW

LD

LL

0.36

0.18

0.06

0.18

0.09

0.03

0.06

0.03

0.01

First game Second game

W

D

L

W

D

L

W

0.6

0.1

0.1

0.1

0.1

0.3

0.3

0.3

0.3

0.6

0.6

0.6

D

L

W

D

L

b) (i) 0.36 (ii) 0.48 (iii) 0.46


3 a) Age (y years) Cumulative frequency

y , 0 0

y , 11 32

y , 19 57

y , 30 110

y , 50 193

y , 70 238

y , 96 259

b)

0 10 20 30 40 50 60 70 80 90 100

Age (y years)

Cum

ulat

ive

freq

uenc

y

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

c) 152d) 33e) Median 35 years, lower quartile 22 years,

upper quartile 51 years

4 a) That no girl has a foot longer than 28 cm.b) Two comments from the following, or other

valid comparisons.• The median shows that the boys’ feet are

longer than the girls’ feet, on average.• The interquartile range shows that the

boys’ feet vary more in length than the girls’ feet.

5

20 4 6 8 10Distance (miles)

Cum

ulat

ive

freq

uenc

y

10

20

30

40

6 a) 110 b) 42

3 a) He has not stated whether 1 is ‘Not at all’ or ‘Very much’.

b) Check students’ new questions. 4 a) Possible answers include the following.

• Thefirst30inthequeuemaybethosekeenest on lunch.

• Thefirst30inthequeuemayallbefromone year group.

b) For example, select two people at random from each tutor group.

5 a) This is a leading question. b) Check students’ new question. A suitable

question would be What is your opinion of the number of salads on the menu?h Too fewh About righth Too many

6 Check students’ questions.

44 representing and interpreting data

Exercise 44.1H 1 Median 115 cm, lower quartile 85 cm, upper

quartile 140 cm, interquartile range 55 cm

2 a) Mass (m grams) Cumulative frequency

m < 0 0

m < 20 4

m < 40 18

m < 60 43

m < 80 90

m < 100 126

m < 120 140

b)

20 40 60 80Mass (m g)

Cum

ulat

ive

freq

uenc

y

100 1200

20

40

60

80

100

120

140

c) Median approximately 72 g, interquartile range approximately 34 g


2 a) You are age 49 up till your 50th birthday, which is 50 years after you were born.

b)

100 20 30 40 50 60 70 80 90Age (years)

Freq

uenc

y de

nsit

y (p

eopl

e pe

r ye

ar)

2

4

6

8

10

1

3

5

7

9

3 a) Waiting time (w months) Frequency

0 < w , 1 20

1 < w , 2 32

2 < w , 3 40

3 < w , 6 135

6 < w , 12 288

12 < w , 18 126

b) Mean waiting time 5 8.2 months to 1 d.p. 4 a) 12 girls and 8 boys

b) Comparisons such as:• More of the boys than the girls spent

10–20 hours on the coursework.• The times for the boys were more evenly

spread than those for the girls.• There were 30 girls and 30 boys

represented. 5 a) 92

b) 188 cm to the nearest centimetre

7 a)

0 5 10 15 20 25

Type A

Type B

30Time (t hours)

Cum

ulat

ive

freq

uenc

y

20

40

60

80

100

120

140

160

180

200

220

b) Both types have approximately the same range and interquartile range. However, the type A batteries last longer on average, and so are more reliable in that sense.

Exercise 44.2H 1

200 40 60 80Amount spent (£)

100 120 140 160

Freq

uenc

y de

nsit

y (p

eopl

e pe

r £)

0.2

0.4

0.6

0.8

1.0

1.2

Documents

AnSWErS to tHE HoMEWork Book · PDF fileAnswers to the Homework Book exercises 248 Higher WJEC GCSE Mathematics © 2010, Hodder Education 3 a) 63 b) 96 c) 154 d) 132 e) 40 f) 133 g)