ANSWERS TO HigHER gCSE MATHEMATiCS FOR CCEA PRACTiCE · PDF fileH gCSE M CCEA P B H S ANSWERS TO HigHER gCSE MATHEMATiCS FOR CCEA PRACTiCE BOOK Chapter 1 1 a 18, 24, 60 b 25, 60 c

Higher GCSE Mathematics for CCEa Practice Book © Hodder & Stoughton Ltd 2014

ANSWERS TO HigHER gCSE MATHEMATiCS FOR CCEA PRACTiCE BOOK

Chapter 1 1 a 18, 24, 60

b 25, 60c 24, 60 d 24e 60

2 a 4, 12 b 35c 12, 21 d 17

3 a 2, 18 b 1, 2, 5c 15, 18 d 2, 5, 17

4 a 36 b 1c 100 d 25e 121f 64g 1h 1000i 0.49j 8

5 a 7 b 3c 9d 10e 6f 1g 10h 1i 2j 0.6

6 a 28b 65

c 117d 290

7 a 43 b 74

c 39 d 96

e 27

f 42 × 53

g 32 × 84

h 23 × 64

8 a 16 b 1c 1024 d 2401e 256

9 30

10 273

11 4, 12, 20, 60

12 15 and 16

13 a 2 × 7b 2 × 2 × 2 × 2 = 24

c 2 × 2 × 7 = 22 × 7d 5 × 7e 2 × 3 × 7f 7 × 7 = 72

g 2 × 2 × 3 × 3 × 3 = 22 × 33

h 2 × 2 × 3 × 13 = 22 × 3 × 13i 3 × 3 × 5 × 5 = 32 × 52

j 2 × 2 × 2 × 53 = 23 × 53k 2 × 2 × 2 × 2 × 2 × 3 × 3 ×

3 = 25 × 33

l 2 × 3 × 3 × 5 × 7 × 11 = 2 × 32 × 5 × 7 × 11

14 a a = 2 b = 3 c = 2b x = 4 y = 1 z = 2

c m = 5 n = 2 p = 3d d = 4 e = 3

15 a 7b 2c 1d 3e 2f 3g 20h 26

16 a 336b 140c 1404d 95 400

e 3150f 39 690g 9900h 40 018

17 a 84 = 22 × 3 × 7; 154 = 2 × 7 × 11; HCF = 14; LCM = 924

b 75 = 3 × 52; 135 = 33 × 5; HCF = 15; LCM = 675

c 150 = 2 × 3 × 52 95 = 5 × 19 HCF = 5; LCM = 2850

d 645 = 3 × 5 × 43 225 = 32 × 52 HCF = 15; LCM = 9675

18 a HCF = 1; LCM = 680 b HCF = 13; LCM = 884 c HCF = 7; LCM = 1078

19 a 28b 44

20 231

21 1pm

1


Chapter 2 1 a 717

b 829c 5972 d 62 201e 4881f 16 457g 494 886h 8818

2 a 147b 626c 769d 1311e 1253f 1802g 54 166h 11 378

3 530

4 It is not possible to make exactly 1000

5 54 kilometres

6 a 3179b 6628c 3182d 9807

7 a 3600b 240c 5200d 124e 9000

8 a 18 000b 36 000c 400d 20e 2520

9 a 3854 b 4266c 33 540 d 188 448e 3 716 601

10 360

11 a 51.529 (3dp)b 27c 35

d 325e 690.95

12 8

13 23

14 200 g

15 82 miles

16 95p

17 a 83b 26

18 a 169 600b 169 600

19 a 2165b 3554

20 a 150b 135

21 a 9158b 174 900

22 a 140b 9600

Chapter 3 1 a 57.74

b 87.71c 77.356d 22.355e 21.096f 63.1784

2 a 4.8b 7.2c 112.2d 53.6e 5.04f 15.08g 34.992h 21.0273i 125.8j 2.432

3 a 2.4b 1.7c 12.5d 13.5e 30f 50

g 26h 52i 1.39j 56.38

4 6.25 m

5 £53.65

6 60 cm or 0.6 m

7 535.5 cm

8 119.2 m

9 23.5 mm or 2.35 cm

10 13.9 g

11 a 24.91 b 24 910 c 0.2491 d 2.491

12 £80.04

13 a 1.35 kgb £2.97

14 5

15 41.6 miles

16 45p

Chapter 4 1 The differences for each

country are:Northern Ireland 48.3°CEngland 64.6°CWales 56.9°CScotland 60.1°C England has the biggest difference

2 a –2 b –3 c –8 d –56 e –8 f 17 g 9 h 0 i –23j –9 k –72 l –80


m –31 n 54 o –8 p –33 q –28 r –63 s –24 t 49 u –85 v –78 w –22 x –61 y –52

3 a –4°Cb 3°C

4 a 0 b –40c 12d –24e –35f –21g 20h –4 i –5 j –5 k 16 l 5m –5 n –8 o –3p –2q –24 r –3s 6t –15

5 a –26 b 10 c 1 d 24 e –3 f –9 g 4 h 1 i –21 j –1 k 13

l 32 m 13 n 8 o –15 p –31 q 64r –64

Chapter 5

Exercise A 1 a = 3

41520

b 1521

57

=

c 12

1122

=

d 1860

310

=

e 1618

89

=

2 a 45

b 56

c 34

d 12

e 35

f 13

g 58

h 34

i 411

j 25

k 35

l 13

3 a 14

27100

310

720

b 512

35

1320

23

4 a 3 12

b 3 13

c 2 18

d 1 59

e 3 45

f 1 37

g 1 57

h 5 12

I 3 14

J 3 23

5 a 97

b 138

c 152

d 114

e 175

f 145

g 299

h 296

i 418

j 133

6 a 1 45

b 123

c 116


d 14

e 1 12

f 5

14

g 123

h 1 13

i 1 211

j 1 13

Exercise B 1 a 13

18

b 14

c 118

d 45

e 2360

2 a 7 712

b 8 59

c 5 18

d 3 916

e 8 128

f 3 712

g 4 49

h 1212

3 6 38

inches

4 a 4 718

b 7 15

c 5 2140

5 1334

m

6 5 1320

cut off, 4 720

left, not enough

7 a 34

b 4

c 10 15

d 8 14

e 1 910

f 5 110

8 a 2 89

b 2 1112

c 12

d 2 1720

e 12

f 4877

9 a 9 34

b 16 12

c 9 15

d 2 25

e 8

f 1641

g 9 13

h 3 922

i 5 58

j 2 67

k 3 717

l 14

m 6985

n 8 710

o 81155

10 a 2 13

b 2 2655

c 5 13

d 5 13

e 4 120

f 6 16

Exercise C 1 a 84

b £168c 141 milesd £204.20

2 They are the same, £90.

3 415

4 150

5 241500

6 £15

7 p = 1 q = 20 (or any pq = 20)

Exercise D 1 8

15x

2 m6

3 3340

w

4 730

e

5 5 420

c d+

6 2 14 535

( )m e−

7 10 3

12x y+

8 y x+ 4

12

9 56

2xy

10 x x( )3 212+


11 23

12 x15

13 1 14

14 310

cd

15 x32

16 ny2

17 2342

x

18 215x

19 w2

6

20 xm

2

2

Chapter 6 1 a i 80.9 ii 80.93

b i 5.1 ii 5.12c i 4.4 ii 4.40d i 0.0 ii 0.03e i 649.0 ii 649.00f i 17.0 ii 16.98

2 a 10b 40 c 7 d 300 e 1000 f 0.8 g 0.6 h 0.05 i 2000 j 0.01 k 8 l 20 m 700 n 8000 o 100 p 0.7 q 0.005 r 0.02 s 400 t 20 000

3 600 people

4 a 3 mb 300 cm

5 a 900 b 600 000c 7 d 0.0004e 19f 0.99g 7.39 h 2.72i 8.0

6 a 200 + 400 = 600b 800 ÷ 80 = 10c 20 × 50 = 1000d 7002 = 490 000

e 90

40 20980×

=

f 60 × 6 = 360g 4 × 9 = 36h 20 × 30 = 600i 70 × 50 = 3500

j 20 640

3 × =

k 10 10 100 10× = =l 0.8 × 30 = 24

m 0 6 70

67. × =

n 203 = 8000o 200 × 0.3 = 60

p 10 5 100 25 75 8 72 2− = − = ≈ .

q 50 400 02

900 02

90002

4500. .+ = = =

r 70 × 60 = 4200

s 6000 5

1200.

=

t 60005

= 1200

u 202 = 400v 60 × 8000 = 480 000

w 620

= 0.3

x 3050

= 0.6

y 8000 × 40 = 320 000

z 900 409 × = 4000

7 a 6 × 7 = 42 b 30 ÷ 6 = 5c 50 × 30 = 1500 d 8000 ÷ 200 = 40e 300 × 0.3 = 90

f 20 × 20 = 400g 900 ÷ 200 = 4.5 h 5 × 6 × 10 = 300

i 700 0 84 2

70.×× =

j 9 60 5 0 3

270. .×− =

8 20 × 40p = 800p = £8

9 £20 × 4000 = £80 000

10 4 × 8 = 32 cm²

11 a 30 ÷ 5 = 6 cmb Estimate is smaller than

the actual length because the numerator is reduced and the denominator is increased.

12 3 ÷ 0.2 = 15

13 Number of tiles = 5 × 6 ÷ 0.3 = 100

14 £600 ÷ 5 = £120

15 6000 77≈

16 3 × 4² = 48 cm²

Chapter 7

Exercise A 1 18

2 10

3 27

4 11

5 19

6 30

7 10

8 2.11

9 20

10 3 5

12

11 2

12 20

13 26 ÷ 2 + 4 = 17

14 6 × 2 + 3 = 15

15 18 + 6 ÷ 2 = 21


16 19 – 2 × 3 = 13

17 5 × 3 + 4 × 5 = 35

18 6 + 5 – 1 × 2 = 9

19 3 × 8 ÷ 2 – 1 = 11

20 10 – 2 – 6 + 3 = 5

Exercise B 1 7.75

2 32.768

3 10

4 1.65

5 1.6

6 5.1975

7 100

8 12

9 6 1324

10 3403

Exercise C 1 –1.7

2 –21.24

3 0.5625

4 –1.67

5 1.37

6 2.45

7 41.67

8 0.39

9 7.03

10 39.54

11 104.8576

12 0.872

13 49.215

14 5.840

15 7.963

16 22.09

17 46.53

18 490.912

19 4.347

20 2.626

21 3.21

22 11.6

23 8.33

24 12.5

25 0.510

26 26.88 or 26.9 cm2

27 38.8129 or 38.8 cm2

28 £78.98

Chapter 8

Exercise A 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 : 6f 2 : 3g 6 : 7h 63 : 44

3 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 : 25 000

4 4 : 7 : 9

5 1 : 2 : 5

6 Paula £15, Tom £25

7 £520 : £650 : £780

8 Sue £14 000, Jane £21 000, Christine £35 000

9 1.5 litres

10 Copper 150 g, iron 200 g, nickel 100 g

11 £2.50

12 a 6000 votes b 3000 votes

13 £12 800

Exercise B 1 1 : 250 000

2 a 2.5 kg b 900 g

3 a 72 b 40

4 a 600 m b 38 cm

5 a 12 cm

b 6.3 cm

6 a 6 cm b 180 cm

7 a 150 gb 13 people

8 a 2.6 kmb 12.5 cm

9 24

10 780

Chapter 9

Exercise A

1 a 920

0.45

b 34

0.75

c 325

0.12

d 1110

1.1

e 9100

0.09

f 740

0.175

2 a 47% 47100

b 82% 4150


c 4% 125

d 42 12

1740

%

e 135% 2720

f 33 13

13

%

3 a 0.19 19%b 0.18 18%c 0.65 65% d 0.03 3%e 1.75 175%

f 0. 2̇ 7̇ 27.2̇ 7̇ %

4 a 29100

, 0.3, 31%, 0.32, 13

b 0.25, 0.27, 28%, 310

, 0.35

c 920

, 0.46, 56%, 64100

, 0.65

d 23

, 1725

, 710

, 72%, 0.84

e 16.9%, 725

, 0.29, 30%, 13

f 0.605, 66%, 23

, 0.67, 79

5 Man. Utd. by 5%

6 0.344

7 62 12

%

8 3200 000

9 a £2.16 b £6 c £108

10 294

11 In the first week of the sale, the price would be £360. In the second week of the sale, the price would be £324. In the third week of the sale, the price would be £291.60. So the selling price of the television was £291.60.

12 a 2.52 mb £1071 c £3.24

13 a £660 b £690 c £612 d £657

14 a £300

b £328 c £376 d £356

15 £52.80

16 £15 525

17 £23.76

18 583.6996mm

Exercise B 1 a 2

25

b 1925

c 28%d 12%e 65%f 24%

2 a 14% b 12% c 60% d 27% e 24% f 16.7% g 17.1% h 22.2% i 30% j 39.2%

3 39.7%

4 57.4%

5 12.9%

6 1623

%

7 4%

8 10.25%

Exercise C 1 £24 520

2 £45

3 £1440

4 £13 500

5 28 million

6 £6

7 £42

8 £75

9 £72 000

10 £90

Exercise D 1 a Cheaper Sounds by £88.80

b Cheaper Sounds by £62

2 380

3 S.I. by £328.76

4 £6959

5 6 years

6 4.5%: £2604.525%: £2552.56 4.5% for 6 years is better.

7 Anglo Bank: £5692.15Bonus Bank: £5705.83Bonus Bank pays more.by £13.68

8 6 years

9 £6400

10 20%

Chapter 10 1 a 1

3

b 16

c 1640

d − 110

e − 1317

2 a 18b 52c 1000d 2 1

2e −2

3 a 114

b 2 23

c 58


d 310

e 1212

4 a 0.4b 5c 4

5d 6.25e 5

16

5 a ba

b 1y

c p2

d − 32x

e nn2 1+

6 59

7 a 58

b 85

c 481

d 0.0723e 500

8 a 0

b 415

c false e.g. 5 > 3 but 15

< 13

d 2e 5

8

Chapter 11

Exercise A 1 a 4 × 104

b 4.8 × 103

c 7.37 × 105

d 2.5 × 10e 8 × 106

f 1.234 × 104

g 6 × 106

h 7.89 × 107

I 3.6 × 106

J 9.3 × 107

k 5 × 105

l 2 × 1012

2 a 7 × 10–3

b 2.04 × 10–1

c 4.5 × 10–5

d 7.07 × 10–3

e 1 × 10–7

f 7.936 × 10–2

g 1.0008 × 10–1

h 6.47 × 10–4

i 3 × 10–3

j 2.9 × 10–3

k 5.2 × 10–11

l 7.3 × 10–5

3 a 30 000b 5 700 000c 0.875d 0.00402e 773f 0.0000012g 80 300 000 000h 548 000i 0.0999j 0.000386

4 1 × 1012

5 19

6 1.26 × 10–3, 1.62 × 10−1, 6.12 × 100, 216 × 10−1, 2.16 × 102

Exercise B 1 3.400412 × 106

2 5.47 × 10−2

3 8.2825 × 105

4 6.087 × 10−2

5 6.66 × 106

6 4.12 × 10−2

7 3.2 × 1011

8 2 × 105

9 3 × 10−5

10 3 × 103

11 3.4 × 10−3

12 3.0256 × 104

13 3.2 × 1012

14 48 mm or 4.8 cm

15 2.48 × 10−17 g

16 4.88 × 103 mm

Exercise C 1 a 4.32 × 108

b 2.1 × 102

c 2.924 × 10–1

d 5.785 × 10–5

e 7.3 × 106

f 9.2 × 10–11

g 2.89 × 10–4

h 7.744 × 1013

i 3.1884 × 10–2

j 7.3081 × 108

k 1.815 × 10–5

l −4.44 × 10–3

2 a 6.48 × 1011

b 3.46 × 104

c 1.73 × 10–7

d 3.39 or 3.39 × 100

e 3.20 × 1010 f 2.28 × 106

3 a 1.472 × 1011

b 3.9 × 107

c 1.08 × 10–6

d 2.3 × 102

e 1.312 × 101

f 5.4756 × 1010

4 a 3.16 × 109

b 3.17 × 10–10

5 1.67 × 10−24g

6 5.9 × 1012 miles

Chapter 12 1 a Rational, terminating

decimalb Rational, terminating

decimalc Not rational (irrational),

π is irrationald Rational, terminating

decimale Rational, 144 = 12f Not rational (irrational),

66 = 3.899 538 434… (non-recurring, non-terminating)

g Not rational (irrational), 3 is irrational

h Rational, fractioni Rational, recurring decimal


2 a Rational 1425

b Rational 5699

c Rational 311

d Irrational

e Rational 5

f Rational 491

g Rational 11

h Irrational

i Rational 103999

j Irrational k Irrational

3 a 7299 = 8

11

b 2245

c 34111

d 41333

e 1 311

f 1 718

g 606990

101165

=

h 5190

1730

=

i 31599990

117370

=

4 a π, , , ...2 3 5b e.g. 2 5and 2 +( )

5 a 3

b 1π

c 1

7 or 12 7

6 For example, 3 12 36 6× = =

7 For example, 6 62( ) =

8 a irrationalb irrationalc rational

9 Any root between 36 and 49

10 4 or 16( )Chapter 13 1 2 7

2 3 7

3 5 5

4 10 6

5 6 10

6 20 2

7 12 6

8 12 7

9 2 3

10 2

11 a 20b 5

3c 16d 9 5

e 5

3f 2 2

12 a 9 2

b 6 3

c 29 2

d 3 5

e 8 3

f 7 2

13 a 10b 2 2c 23

14 a 12 3−

b 2 3 3+

c 52 14 3+

d 29 9 3−

15 a 28 + 10 3

b 1

c 48 – 24 3d 32 + 11 12 7

16 2 6

17 a 4 cm

b 8 2 cmc 8 cm2

18 a 33

b 3 55

c 7 1010

d 2 5

e 5 36

f 26

g 66

h 355

19 13 630

20 9 22

21 a falseb truec trued truee false

Chapter 14

Exercise A 1 4a

2 2a

3 abc

4 y2

5 3p + 4r

6 2c

7 5x + 4y

8 7a + 2b


9 4a + b

10 2y + 3s

11 a3b2

12 12xy

13 10x

14 3ab + 3ac

15 6ab

16 3x + 2y

17 6a + 3b

18 a + 2b

19 2a2 + 9a

20 4p – 3q + 4

21 5p + q

22 5x2 – 3x

23 5x3 – 3x2 + 2x

24 2ab + 3ac

25 7x – 3y + 6z

26 3a2 – 5ab + 4b2

27 6a + 3b

28 a – b + c + 6

29 9ab + 6ac

30 –2a2 + ab

31 10a pence

32 7a2 + 3ab

33 3x – 6

34 a a + 3bb −3a – 3bc 3a – 6

Exercise B 1 a 6

b 2c 8d 28e 10f 16g 8h 16

i 24j 10k 10l 28m 4n 12o 48p 24q 32r 8s 2t 5u −2v 1w 4x 0.5y 32z 64

2 a 12b 42c –58d 102e 4.5

3 a 4b 11c –13d –7

4 a 3b –8c 12d 350e –6.5

5 a 16b –11c –18d –12e 97

6 a 38b 7.6c 52d –2e 438

7 3

8 –4

Chapter 15

Exercise A 1 a 24

b 22 × 32× 53

c a5

2 a 211

b 38

c 45

d 57

3 a 53

b 76

c 22

d 34

4 a 26

b 30 or 1c 54

d 71 or 7e 5–2

f 33

5 a 2a6

b a2

c 12a5

d 3a

e 14p

or p−4

f y

g k−310

h t2

i lj 6

2y or 6y−2

k 64a6

l 36p4q6

6 a 6a5

b 18x6

c 20a3b4

d Cannot be simplified e 2a4b5

f 14c3

g 16y10

h 2p2qi 4p–3

j 2p–2


k 32mn

l 43

2pq

m 2p13

n 36a5b5

o 12b4

7 a x12

b x6

c x 2

4

d 4n32

e mnf 36k4

g 13p or p–3

h 7ti 9p–4 or 9

4pj 6m3n2

8 a 26

b 23

c 2–3

d 20

e 272

f 22n+1

9 29a–6b–3

10 a 48b 64c 216

d 113

e 36f 512

Exercise B 1 a 1

5b 1c 5d 25e 32

f 12

2 a 243b 100c 16d 1

64

e 32f 1000

3 a 16

b 6c 32d 27e 1

25f 1

4 a 36

b 175

c 1 920

d 8e 61 1

2 5 a 7776

b 27

6 211

7 a − 12

b 23

c –10

Exercise C 1 4

2 4

3 2

4 2

5 −1

6 −32

7 −2

8 1

9 2

10 −16

11 −18

12 −23

13 −1

14 4

15 38

Chapter 16

Exercise A 1 21a + 42b

2 −10c − 15d

3 −12e – 20f

4 21g – 6h

5 12i + 6j – 9k

6 −15m + 6n − 9p

7 24r – 18s – 12t

8 32r + 16s + 8t

9 12u + 20v

10 −24w − 18x

11 10y + 2z

12 12y + 8z

13 −15v − 10

14 21 + 12w

15 −5 + 15a

16 24g – 15

17 2x2 + xy

18 10p2 −5p

19 a2b – ab3

20 −10k +6k2

21 2a + 1

22 5p −4

23 7x −7

24 4t2 – 10t

25 15 – 5y

26 5x2 + 2x

27 −3b −3

28 –t + 4r −6

29 21x + 29

30 i 5x + 2ii 4 1

2 x −3 or 9 6

2x −


Exercise B 1 x2 + 3x + 2

2 x2 + 7x + 12

3 x2 + x – 2

4 x2 + 2x – 15

5 x2 – 3x + 2

6 x2 – 4x – 5

7 x2 – 9

8 x2 + 4x + 4

9 x2 – 14x + 49

10 x2 – 21x + 108

11 x2 – 9x + 20

12 x2 – 4x – 21

13 4x2 – 33x + 8

14 6x2 + 16x + 8

15 3x2 – x – 10

16 8x2 – 10x – 12

17 14x2 – 53x + 14

18 6x2 – x – 15

19 9x2 + 42x + 49

20 15a2 + 26ab + 8b2

21 6m2 – 5mn –6n2

22 10p2 – 11pq +3q2

23 3a2 + 7ab –6b2

24 6x2 + xy –12y2

25 25a2 – 4b2

26 a2 – ab – 2b2

27 18p2 + 15p – 12

28 2x2 + 8

29 27y3 – 27y2 + 9y – 1

30 1 – 4x2 – 12x

31 2x2 + 4x – 6

32 a = 16 b = −4

33 4x2 = > multiple of 4

35 c = 6

36 d = 4 or −4

Chapter 17

Exercise A 1 x = 7

2 x = 6

3 x = 2

4 x = −45

5 x = 1.5

6 x = −2

7 x = 5

8 x = −2.25

9 p = −7.5

10 y = 2.5

11 a = −0.25

12 k = 0.5

13 y = 2

14 d = −3

15 x = 2.5

16 x = 1.75

Exercise B 1 x = 3

2 x = 4

3 x = 2

4 x = 10

5 x = 3

6 x = 1

7 x = 1

8 x = 3

9 x = 3

10 x = 3

11 x = 4.5

12 x = 5

13 x = 2

14 x = 2

15 x = −3

16 y = 1

17 p = − 15

18 a = 710

19 t = −4

20 y = 1.5

Exercise C 1 x = 8

2 x = 1

3 x = 7

4 x = 0

5 x = 3

6 y = 1 711

7 a = −1 18

8 t = −2

9 p = 58

10 h = 0.5

11 x = 6

12 x = 5

13 y = 13

14 x = 3 12

15 x = −20

16 x = 3

Exercise D 1 x = 4

2 x = 20

3 x = 6

4 x = 27

5 x = 35

6 x = 8

7 x = 50

8 x = 3 13

9 x = 9

10 x = −2

11 x = 1 12


12 x = 715

13 x = 15

14 x = 12

15 x = 17

16 x = 11

Exercise E 1 x = 1 4

11

2 x = −5

3 x = 2 47

4 x = − 811

5 x = 81

6 x = 9

7 x = 13

8 x = 7

9 x = 19

10 x = 1 517

11 x = − 58

12 x = 1 811

Exercise F 1 1

2 b × 7 = 21, base = 6 cm

2 n n n3 4

2 24= + =,

3 a x x− =7 34

b x = 28; he had 28m of rope.

4 a 80 13

− =x x

b x = 60, she has spent £60

5 a 13

15

50 180x x x+ + − =

b x = ° ° ° °150 100 30 50; , ,

6 a x x x

x

− + + + −

+ − =

15 3 27

3 68

14

12

b x = 40; 25cm, 13cm, 13cm, 17cm

7 125, 127, 129

8 336 + 24k + 44 = 500; k = 5

9 24

Chapter 18 1 a 19, 22, 25

b 59, 52, 45c 1.0, 1.2, 1.4 d 43, 50, 57e 22, 28, 35 f 57, 65, 73g 1, 7, 0 h 46, 51, 56i 0, −10, −22j 6.25, 3.125, 1.5625

2 a 3, 6, 9, 12b 26, 24, 22, 20c 1, 10, 100, 1000d x, x + 4, x + 8, x + 12e 1000, 100, 10, 1f n n n n,

2, ,

4 8

3 a Half the previous term 1 1

234

38

, ,

b Subtract 1; 96, 95, 94c Multiply by 2; 16, 32, 64d Subtract 0.5; 4, 3.5, 3e Subtract 6; 5, −1, −7f Subtract 7; −3, −10, −17g Subtract 5; −4, −9, −14h double the previous term

64, 128, 256i Subtract 8; −5, −13, −21j Subtract 13; −10, −23, −36k Add the two previous

terms 8, 13, 21l Subtract 5; 2, −3, −8

4 a 13, 31; Add 6b 23, 59; Add 12c 76, 58; Subtract 9d 91, 75; Subtract 4e 37, 52; Add 15f 27, −1; Subtract 7

5 a 25, 36, 49b 84, 119, 160c 57, 74, 93

6 a 2n b 3n + 1 c 3n – 3 d n + 20

e 4n – 3 f 3n + 7 g 2n – 5 h 30 – 5n or −5n + 30 i 6 – 2n or −2n + 6 j 4 – n or −n + 4k – 3n

l n

n + 5

7 a 7, 8, 9, 10 b 6, 12, 18, 24 c −1, 0, 1, 2 d 5, 8, 11, 14 e −5, −3, −1, 1 f 1, 4, 7, 10g 8, 12, 16, 20 h −1, −2, −3, −4i 7, 12, 17, 22 j −1, −3, −5, −7k 2, −1, −6, −13l 7, 13, 23, 37

8 9n – 6

9 3n – 4 100th term = 296

10 14th term

11 a 51b s = 5p +1c 76d 16

12 a T = 3n + 1b K = 4r – 1 c P = 5m – 7d w = 2d + 5

13 a Squares 1, 5, 9, 13, 17, 4n – 3b 15th

14 a 4, 6, 8, 10, 12, 2n + 2b 38c 38d No nth = 25.5 not possible

15 (n + 2) (n + 3) – n(n+5) Expand brackets n² + 5n +6 − n² – 5n = 6

16 15 terms

17 1st, 3rd


Chapter 19

Exercise A 1 4(2x + 5)

2 3(x + 2)

3 3(3x – 4)

4 5(x – 6)

5 8(2 + x)

6 3(3 + 5x)

7 4(3 – 4x)

8 4(2 – 3x)

9 4x(x + 4)

10 6x(x + 5)

11 4x(2x – 5)

12 3x(3x – 5)

13 3(2x + y)

14 2(2a – 5b)

15 a(5 + 7a)

16 2x(3x – 2)

17 a(3a – b + c)

18 y(4x + 2y – 1)

19 2y(1 + 4x)

20 3a(3a + b)

21 4ab(2 – b)

22 a2(a + b)

23 x2(2 – y)

24 3x(2x2 – 5y)

25 5ab(a + 3b)

26 3a(4bc + 5b2 – ac)

27 4xyz(2x + y + 3)

28 2πr(h + r)

29 x2(3 – x2)

30 2pq(3p + 5q² + 4q)

Exercise B 1 (k + m) (a + b)

2 (p² – 2) (p + 3)

3 (a + c²) (b – c)

4 (x – z) (x + 1)

5 (2a² – t) (3a + t)

6 (m – t) (m + n)

7 (r² + 1) (r + 1)

8 (a – 8) (c + 1)

9 (c – 8) (6c – d)

10 (2a + 3d) (6c – b)

Exercise C 1 (x + 3)(x – 3)

2 (x + 8)(x – 8)

3 (x + 14)(x – 14)

4 (x + y)(x – y)

5 (5 + y)(5 – y)

6 (4 + c)(4 – c)

7 (a + 100)(a – 100)

8 (9 + s)(9 – s)

9 (x – 6)(x + 6)

10 (7 – y)(7 + y)

11 (3x – 5)(3x + 5)

12 (2y – 3)(2y + 3)

13 (1 – 8t)(1 + 8t)

14 (x – 11y)(x + 11y)

15 (9a – 4b)(9a + 4b)

16 (6a – 5b)(6a + 5b)

17 (10 – 7y)(10 + 7y)

18 2(x – 5y)(x + 5y)

19 p p+

−

1

101

10

20 (0.5a + 0.6b) (0.5a – 0.6b)

21 a 117

b 14

c 80 000

22 7x² + 6x or x(7x + 6)

Exercise D 1 (x + 3)(x + 1)

2 (x + 4)(x + 3)

3 (x – 1)(x – 4)

4 (x + 20)(x + 1)

5 (x + 9)(x + 1)

6 (x + 5)(x + 4)

7 (a – 8)(a – 1)

8 (a – 2)(a – 8)

9 (y – 6)(y – 5)

10 (x – 16)(x – 1)

11 (x + 5)(x – 3)

12 (x + 7)(x – 1)

13 (x – 7)(x + 2)

14 (x + 9)(x – 4)

15 (x + 10)(x – 1)

16 (a + 6)(a – 2)

17 (b + 8)(b – 3)

18 (c – 9)(c + 4)

19 (x + 14)(x – 1)

20 (a – 7)(a + 3)

21 (x + 2)(x + 6)

22 (2x – 1)(x – 2)

23 (2x – 3)(2x – 1)

24 (4x + 1)(x + 5)

25 (x – 3)(5x – 1)

26 (x – 4)(x + 3)

27 (2x + 3)(x – 4)

28 (3x – 1)(2x + 1)

29 (4x – 3)(2x + 1)

30 (6x + 5)(x – 2)

Exercise E 1 p(p – 1) (p + 1)

2 2(2x + 7) (x – 3)

3 4xy²(3xy + 5)

4 10(b – 1)(b +1)

5 3 (x + 1) (y +2)

6 2(4 + k) (5 – 2k)

7 cannot be factorised

8 (3c + d)(2c – d)

9 (2m + 5n)(4m + 3n)

10 (5w – 3e)(2w + 5e)


Chapter 20

Exercise A 1 F = 1.19

2 C = 26.85

3 a 140°b 162°

4 a 30 °Cb 248 °F

5 12.7

6 54.1

7 9 1027 cubic units

8 k = 2xp + mk = 850

Exercise B 1 a b = 180 – a – c

b y x c= −5

c t p c= −3

d g p f= + 22

2 hh

= −S ππ

rr

2

2

3 a ut

= −v

4 r C=2π

5 x f ed

= −

6 t p r= − − 18

7 a hb

= 2A

b n a= + 360180

c q A prp

= −

d n Ft m= −4

8 x = 4y

9 xy

= +8

3

10 m Ev

= 22

11 h Vr

= 32π

12 RP

= 100

13 a r Ah

=2π

b a P= −( )1

c c a b= −2 2

d x y= + 43

14 v as u= +( )2 2

15 t Sa

= −30 2

16 x = t2 v2

17 x p q y= + +( )2 2 2

18 v r= 43

3π

19 a u fvf v

= −

b r p qq

= +−

( )11

c x qyp q

= +

d t su a

= +2

2

e r pqt s

= +f x y

y= +

−3 4

1

g y KK

= −3

2

h vf u

u= +( )2 1

20 a equationb identityc expressiond formulae equationf identity

Chapter 21 1 16 days

2 £1.85

3 250 g butter, 350g flour

4 5 hours 20 mins

5 4560

6 30

7 a y = 4wb y = 32c w = 1.5

8 a m = 16h

b m = 0.8c h = 0.16

9 81.25

10 18G

11 64

12 a 921.6 gb 3.5 mm

13 a t is 8 times biggerb t is 27 times smaller

14 p ∝ r2

15 a P = 1000V

b P = 2

16 0.8

17 a It is 19

of its value at the Earth’s surface.

b 6400 km

18 p = 6, r = 16

19 distance needs to be halved

20 1600 m

Chapter 22 1 0 is between –2 and 3 so one

solution is between 1 and 2. x = 1.6

2 0 is between –3 and 3 so one solution is between 0 and 1. x = 0.6

3 0 is between –10 and 4 so one solution is between –3 and –2. x = –2.4

4 0 is between 2 and –2 so another solution is between 0 and 1. x = 0.4

5 1.77

6 5.8

7 4.8

8 2.69

9 2.5

10 19.6

11 3.29

Chapter 23 1 a 4

5b 3

7c 3

5 2 a pq

3


b 1y

c 35qp

3 a 5x

b x4

c 12x

d 45x

4 a p−1b 3

2

2xyx y−

c 13a

d 43

e xx + 3

f t − 1

5 a 75x −

b 22

xx +

c xx−+

15

d xx+ 4

2

e xx−+

43

f xx2 1−

g 3 12

xx−+

h 3 21

xx

−−

i x bx a

++

j k rk r+−

36

6 a 5 26

a +

b 9 1310

x +

c 8 2115

x −

d 3 11

bb b

++( )

e 8 92 3x

x x+

− +( )( )

f 7 2

12p +

g 1 212− p

h –( )x + 1121

i 7 82

xx x

−−( )

j 4 111 2

mm m

++ +( )( )

k 2 6 51 2

2x xx x

+ ++ +( )( )

l x xx x x

2 2 11 1− −+ −( )( )

m 7 312

aa a

−−( )

n 3 212

pp

+−

o 4 32 6xx+−

7 a 2 6ab

b 2 14

2x yx−

c 16

d 23pq

e m qm q

2

3−+( )

f −23

g −54

h 1i ( )( )

( )( )a aa a+ −− +

3 43 4

j −2p

k xx− 64

l a3

6

m 2 2 12

( )xx

−+

Chapter 24

Exercise A 1 a x = 0, x = 3

b x = 2 x = –6

c x = − 14

d p p= =,12

13

1

e y y= =0 13

,

f x x= = −,12

23

2 a x = –3, x = −1b x = –5, x = 6

c x x= − =,1 213

d x x= =,35

4

e x x= = −,56

4

f x x= = −,12

53

3 a x = 0, x = –4b x = 7, x = –7


c y = 3, y = –3

d p = 0, p = −16

e h h= = −,16

16

f x x= = −,53

53

4 a x = 0, x = 3b x = 0, x = 4c x = 5, x = –7d x = 0, x = 8e x = −1, x = 4

f x x= − = −,13

3

5 a x = 18, x = 1b x = 0, x = 5

c x x= = −,34

2

d x x= = −,25

3

e p p= =0 112

,

f m m= − = −1 13

,

6 a x = −0.46, x = −6.54b x = −0.10, x = −2.10c x = 1.81, x = −1.47d x = 1.16, x = −1.82e x = 0.85, x = −3.52f x = 0.74, x = 0.15

7 a x =− ±2 14

b x = ±3 372

c x =− ±9 2 29

d x = − ±1 2 53

e x = ±2 22

f x = ±2 2 115

8 a x = −12.4, x = 0.403b x = 0.586, x = 3.41c x = 0.551, t = 5.45d x = −2.58, x = 6.58e x = 0.172, x = 5.83f x = −1.72, x = 0.387

Exercise B 1 x = 2

2 x = 4 or x = −2

3 x = 115

4 x = −0.78 or x = 1.28

5 x = −3 or x = 6

6 x = 4 or x = −3

7 n = −4 or n = 7

8 x = − 34

or x = 2

9 x = − 57

or x = 6

10 p = −0.72 or p = 1.39

11 x = −0.48 or x = 1.19

12 x = −0.19 or x = 0.11

Exercise C 1 10 m

2 17, 19

3 12 cm

4 50 km/hr

5 10p

6 25

7 48.28

8 3.2, 0.3125 (3 15 and 5

16 )

Chapter 25

Exercise A 1

x –3 –2 –1 0 1 2 3

y = 6x + 1 –17 –11 –5 1 7 13 19

2

x –2 –1 0 1 2 3 4

y = 3x – 5 –11 –8 –5 –2 1 4 7

3

x –3 –2 –1 0 1 2 3

y = 4 – 5x 19 14 9 4 –1 –6 –11


4

a x –2 0 4

y = 3x – 3 –9 –3 9

b

2

0–2

–1 1 2 3 4 x–2–3

–4–6

–8

–10

4

6

8

10

y = 3x – 3

y

c x = 2.3

5

�4 �3 �2 �1 0 1 2

3x + 4y = 12

3 4

y

x

�3

�2

�1

1

2

3

4

5

6

6

�3 �2 �1 0 1 2

2y – 5x = 15

3

y

x

�2

�1

1

2

3

4

5

6

7

8

7 a nob yesc nod yes

8 k = 5

9 a (4,5)

b 5 12

, −

c −2 61

2,

d −3 11

2,

e − −

1 51

212

,

10 a (−1, 7)b (5, 3)

c 5 412

12

,

d 5 4 12

,

e −2 3 1

2,

f − −

6 4 1

2,

11 A(−3, 7)

12 a 7.21b 6.40c 7.28d 8.06e 7.62

13 a 5.39b 8.60c 9.06d 5e 13f 18.79

14 AB = 5 20 BC = 5, AC = 5 20 BC2 = AB 2 + AC 2, 25 = 5 + 20, therefore ABC is a right-angled triangle

Exercise B 1 a 3

b 12

c −2

d 1 12

e 1

2 a 3b 2c 1

5d –4 e − 1

4f 1

6


3 a 7b – 4c 1d – 1e 2f – 4g 31

2

h − 13

4 a y = 4x + 3 y = 4x + 5b y = 3x – 2 y = 5 + 3xc y = 6 – x y = –x + 3d y = x + 4 2y = 2x + 3e 3x + y = 2 y = 5 – 3x

5 – 4

6 – 12

7 14

8 a (0, 5)b (0, 6)c (0, 0)d (0, 2)e (0, –2)

f 0 14

,−

9 a y = 5x + 1b y = 3x – 8c y = x + 2d y = 10x + 4

10 a A; y = 3x – 5b B; y = 7 – xc C; y = 2x + 3d D; y = 2

11 y = 3x + 7

12 y = 2x – 4

13 a 2y = 3x + 4b y = 3x + 5c 8x + 3y = 24d x + 3y = 9

14 y = 4x + 1

15 y = 3x – 5

16 a y = 4x + 2b y = 4x – 4c y = –3x + 5d y = 2x – 1e y = –2x + 9f y x= −4

525

5

17 y = − 14

x + 10

18 y = 3x + 8

19 y = − 25

x + 2

20 y = –2x + 6

Chapter 26

Exercise A 1

x –2 –1 0 1 2 3 4 5

x2 4 1 0 1 4 9 16 25

+3x –6 –3 0 3 6 9 12 15

–5 –5 –5 –5 –5 –5 –5 –5 –5

y = x2 + 3x – 5 –7 –7 –5 –1 5 13 23 35

1−1

2

−2−4−6−8

468

1012141618202224262830323436

−2−3 2 3 4 5 x

y

y = x2 + 3x – 5


2

0–1

–5

–10

1

5

2 3 x

y

–2–3

y = 2 + x – x2

3

–2 –1

–5

0

5

10

15y

1 2 3 x

y = 2x2 – 3x – 1

4

1−1

2

4

6

8

10

12

14

16

18

−2 2 3 4 x

y = (3 – 2x)(1 – x)

y

20

5 a

x –3 –2 –1 0 1 2 3

x3 –27 –8 –1 0 1 8 27

–5x 15 10 5 0 –5 –10 –15

x –12 2 4 0 –4 –2 12

b

–2–3 –1

–5

–10

–15

0 1 2 3 x

y15

10

5

y = x3 – 5x


6 a

x –4 –3 –2 –1 0 1 2

x3 –64 –27 –8 –1 0 1 8

4x2 64 36 16 4 0 4 16

–6 –6 –6 –6 –6 –6 –6 –6

y –6 3 2 –3 –6 –1 18

b

x

y

•

••

•

•

•

•

–3–4 –2 –1 0 1 2

2

4

6

8

10

12

14

16

18

–2

–4

–6

y = x3 + 4x2 – 6

7

x

y

•

•

•

•

•

•

–3 –2 –1 1 20

14

16

18

20

12

10

8

6

4

2

y = 3

y = x3 + 3x2

8

6

4

2

8

10

12y

y = 5x – x3

0 1 2 3 x–1

–2

–4

–6

–8

–10

–12

–2–3

Exercise B 1

1−1

1

−1

−2

−3

−4

−5

−6

−7

−8

−9

−10

2

3

4

5

6

7

8

9

10

11

12

−2−3 2 3 x

y

y = 3x


2

1−1

−2

−4

−6

−8

−10

2

4

6

8

10

−2−3−4−5 2 3 4 5 x

y

y = −10x

3

1−1

−2

−4

−6

−8

−10

2

4

6

8

10

−2−3−4 2 3 4 x

y

y = 12x

4

1−1

−2

−4

−6

−8

−10

−12

−14

2

4

6

8

10

−2−3−4 2 3 4 x

y

y = 82−x

5

–2 –1 0

5

10

15

20

25

30

35

40

1 32 4 x

y

y = 2.5x

b i 22.5 ii 2.6

6 a x y

–8 5.96–6 3.81–4 2.44–2 1.560 12 0.644 0.416 0.268 0.17


b

–8 –6 –4 –2 0 2 4 6 8 x

1

2

3

4

5

6y

y = 0.8x

c i x = 3.1 ii x = −7.2

7

10−1

2

4

6

8

10

12

14

16

18

20

22

24

26

27

y

y = 3–x

−2−3 2 3 x

8

−1

2

4

6

8

−2−3 1 32 x

y

10

y = 1

y = 2x + 1

Asymptote y = 1

Exercise C 1 a

30

1

–1

x

y

60 90 120150180210240270300330360

y = cos x

b 60°, 300°

2 a

1

–1

x

y

90 180 270 360

b 45º, 135º

3 140º, 220º

4 a 150°b 60°c 135°d 340°

5 a 240°b 330°c 150°d 270


6 a x = 5.7° 174.3°b x = 60° 300°c 108.4° x = 288.4°d x = 60° 120°e x = 135° 315°

Chapter 27 1 a £60

b £12c £16

2 a AB and CD b 11:50 c 16 milesd BCe 12 mph

3 a Imran walked 1 kilometre in 20 minutes. He stopped for 5 minutes and then travelled the remaining 9 kilometres in 9 minutes. (Perhaps he got a lift or caught a bus.)

b 32 minutes c 6 minutes d 15 km/hr

4 a

2

1

3

4

5

6

7

1200 1210 1220 1230 1240

Time

Dis

tanc

e fr

om h

ome

(km

)

1250 1300 1310 1320

• •

• •

•

b 15 km/h

5 a t 0 1 2 3 4

H 0 3 4 3 0

b

2

1

3

4

Hei

ght

(met

res)

H

10 2 3 4t

H = 4t– t2

c 4 metresd 1 second, 3 seconds

6 a x 0 1 2 3 4 5 6 7 8

T –16 –15 –12 –7 0 9 20 33 48

b

–10

–20

(Tem

pera

ture

°C

)

10

20

30

40

50

y

x1 2 3 4 5 6 7 8

T = x2 – 16

c –16 °Cd 7.5 minutes

7 a

40

20

(Year)

Mas

s (g

)

60

80

100

120

y

x0 10(2010)

20 30 40 50 60

140

160

b 2030


8 a

10

20

10

40

60

80

100

(Pop

ulat

ion

Rob

bits

)

120

140

160

2 3 4 5 6 7

(Months)

8 9 10 11 12 x

y

b 32 rabbitsc 7 months

Chapter 28

Exercise A 1 a i x = 1; y = 2

ii x = 2; y = 1 iii x = −2; y = −1 iv x = −4; y = −2b They are parallel so they will never intersect

2

x

y

–2

2

0

6

4

14

12

10

8y = x + 2

y = 3x – 2

1 2 3 4 5

(2, 4)

3

y

x4321 5

y = 4x − 1

x +y = 4

20

18

16

14

12

10

8

6

4

2

0

–2

x = 1y = 3

4 y7

6

5

4

3

2

1

0

–1

–2

–3

x54321

2x + y = 7

2y = x + 3

x = 2.2, y = 2.6

5 (−2, 5) (4, 2) (6, 3)


Exercise B 1 x = −1, y = 10

2 x = 5, y = −2

3 x = 4, y = 3

4 x = −2, y = −3

5 x = 2 17 , y = −5

7

6 x = 4, y = 3

7 x = 6, y = −2

8 x = 4, y = 3

9 x = −2, y = −3

10 x = −1, y = 31 12

11 x = 5, y = 3

12 x = 1 12

, y = 3

13 x = 8, y = 41 12

14 x = −2, y = 5

15 x = −3, y = −7

16 x = 21 12 , y = −8

Exercise C 1 x + y = 40, x – y = 14,

x = 27, y = 13

2 2 = –2m + c, 1 = 4m + c,

c m

y x

= = −

= − +

1 23

16

16

1 23

,

3 2a + 4c = 27, a + 3c = 17, adult £6.50, child £3.50

4 3p + 2r = 135, 4p + 3r = 190, pen 25p, ruler 30p

5 a = 2c, 2a + 3c = 84, adult £24, child £12

6 a x + y = 46b y + 1 = 3(x + 1) y – 3x = 2c Lan 11 years old, mother 35 years old

7 4.5 hours

8 £24

Exercise D 1 a, b

y8

6

4

2

0

–2

–4

–6

x654321

y = x2 − 6x + 8

x + 2y = 8

y = 2x − 5

c (2.3, –0.5) and (5.7, 6.5)

2 (4.6, 1.7) and (0.9, 3.6)

3 a x = 1, y = 2 or x = –2 , y = 5b x = – 1 1

2, y = –2 1

4 or x = –2, y = –3

c x = 9, y = –6 or x = 1, y = 2

d x = 2.56, y = 7.56 or x = –1.56, y = 3.44

e x = 1, y = 5 or x = –3 1 12

, y = –4

f x = 2.62, y = –1.43 or x = 0.104, y = 2.34

g x = 4, y = 2 or x = –3, y = – 2 23

h x = 2.39, y = 0.706 or x = –0.218, y = 2.66

i x = 10 15

, y = 1 − 25

or x = –1, y = –1

j x = –3, y = –2 or x = 32

, y = 52

Chapter 29 1 a

–2

–4

–6

–8

2

4

6

–2–4 2 4

y = x2 – 2x – 8

x

y


b i x = 4, x = –2 ii x = 4.7, x = –2.7 iii x = 3.4, x = –1.4

2 a

1

10

20

30

–1–2–3 2 3 4

y = 3x2 – 2x

y

x

b i – 13

ii x = 2.2, x = –1.5

3 a

1

2

4

6

8

10

–1–2–3 2 3 4 5 6–4

12

y = x2 – 2x + 512

y

x0

2

b i x = 5.5, x = –1.5 ii x = 3.4, x = 0.6

4 a

2

–2

–4

–6

–8

–10

–12

–14

–16

–18

–20

4

6

8

10

–1–2–3 1 2 3 4 x

y

y = 5x – 2x2

b i y = 3.1 when x = 1.25 ii x = 2.5, x = 0 iii x = 0, x = 1.5 5 a

1

–1

–2

2

3

4

5

6

10 2 3 4 5

y = x2 – 5x + 6

x

y

–1

b i x = 2, x = 3 ii x = 0.7, x = 4.3

6 a

–1

–2

–3

–4

1

–1–2 10 2 3

2

y = x2 – x – 4

y

x

b i x = 2.6, x = –1.6 ii x = 1.8, x = –0.8


7 a

2

4

8

10

–1–2–3 1 2 3

6

y = 5 – 12

x

y = x2

y

x

b x2 + 12

x – 5 = 0 c x = 2, x = –2 1

2

8 a

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

2

1

4

3

–1–2–3 1 2 3

5

y = 2x2 + 3x – 9

y

x

b i x = 1.4, x = –2.9 ii x = 0.9, x = –2.4

9 a, b

2

–2–4–6

468

10

14161820

–1–2–3 1 2 3 4

12

y = x2 – 4x

22

y = x – 3

y

x5

c x = 0.7, x = 4.3

10 a

2

–2

–4

–6

4

6

8

10

14

16

18

20

24

26

28

–1–2–3 1 2 3 4 5 6 7

12 y = x2 – 5x + 3

22

8 x

y

b i x = 4.3, x = 0.7 ii x = 5.4, x = –0.4 iii x = 6.5, x = 0.5

11 a

–2

2

6

8

10

–1–2 1 2 3 4 5

4

y = x2 – 3x

x

y

b i x = 3.6, x = –0.6 ii x = 3.7, x = 0.3

12 y = x + 2

13 x2 – 1 = 0

14 a y = 2x – 8

b y = –4x – 2

c y = –x – 1


Chapter 30

Exercise A 1 a 3, 4

b 3, 4, 5, 6c –2, –1, 0, 1,d –7, –6e –3, –2f –5

2 a –3, –2, –1b 4, 5c 2, 3, 4, 5, 6d –2, –1, 0, 1e –2f –4, –3, –2, –1, 0, 1, 2

3 3

4 –3

5 a x 2b x –3c 1 x 4d –1 x 3

6 a 543210–1–2–3–4

b 543210–1–2–3–4

c 543210–1–2–3–4–5

d 543210–1–2–3–4–5

7 a x 2.5b x 2c x 4d x 3e x 3f x –4g x 5h x 6i x 2.5j x 5k x 3l x 2m x –1

n x 2o x –1 3

8p –1 x 7q 0 x 3

r 12

x 3

s − 25

x 0

t 6 x 10

Exercise B 1

x = –3

y5

x

4

3

2

1

–5 –4 –3 –2 –1–1–2

–3

–4

–5

10 2 3 4 5

R

2

y = x

x

y

–2

–3

–4

–5

–1

1

2

3

4

5

–1 1

0

32 4 5–2–3–4–5

3

x + y = 3

x

y5

4

3

2R 1

0–1–5 1 2 3 4 5–4 –3 –2 –1

–2

–3

–4

–5

4 y = 2x – 1

y

–3 –2 –1 x3210–1–2

–3

–4

–5

5

4

3

2

1

R

5 y

–2 –1 10 2 3 4 5 6 x–1

–2

4x + 3y = 12

6

5

4

3

2

1R

6

x76543210–3 –2–1

y = 2x + 1

y

7

6

5

4

3

2

1

–3

–2–1

y = 4

x = –2

R

7 y5

4

3

2

1

x–5 –4 –3 –2 –1 10 2 3 4 5–1–2

–3

–4

–5

x = 4

y = 3

y = –x

R

8 y7

6

5

4

3

2

1

0 x7654321

3x + y = 6

R


9

x542

y5

4

3

2

1

–1–2

–3

x = –1

x + 2y = 4

–3 –2 –1 10 3

R

10

x876543210

y8

7

6

5

4

3

2

1

–1–1–2

–2

x = 1

x + y = 5

y = 2x + 1R

11 a y 3b x –1c y x – 3d 5x + 2y 10

12 a x 2 y 1b y x + 1 y –2c y x x 3 d y 2x + 1 y 3 – x y –1

Exercise C 1 a (4, 1)

b (−1, 1)

2 a (5, –1)b (−1, −1)

Chapter 31

Exercise A 1 ABD = ACD, angles in the

same segment are the same size when they touch the circumference.

2 MCO = 90°; the angle between a tangent and a radius is a right angle.

3 PQR + PSR = 180°; opposite angles in a cyclic quadrilateral add up to 180°

4 FPG = 90°; the angle at the circumference of a semicircle is a right angle.

5 Let PQO = x then POQ = 180 – 2x (isosceles triangle) Let RQO = y then ROQ = 180 – 2y (isosceles triangle) POR = 360 – (180 –2x) – (180 – 2y) = 2x + 2y = 2PQR

6 a = 150° (angle at centre = 2 × angle at circumference)

7 b = 65° (angle at centre = 2 × angle at circumference)

c = 25° (angle sum of a triangle = 180° and base angles of an isosceles triangle are equal)

8 d = 35° (base angles of an isosceles triangle are equal)

e = 70° (angle at centre = 2 × angle at circumference or exterior angle of a triangle

= sum of interior opposite angles)

9 f = 128° (angle at centre = 2 × angle at circumference)

10 g = 44° (angles in the same segment)

11 h = 90° (angle in a semicircle)i = 38° (angles in the same segment)j = 52° (angle sum of a triangle)

12 k = 26° (angle at centre = 2 × angle at circumference and angles in the same segment)

13 l = 31° (angles in the same segment)

m = 59° (angle at centre = 2 × angle at circumference and isosceles triangle)

Exercise B 1 b = a = 110° (angles at centre

= 2 × angle at circumference and angles in the same segment)c = 70° (opposite angles of a cyclic quadrilateral)

2 d = 106° (opposite angles of a cyclic quadrilateral)e = 98° (opposite angles of a cyclic quadrilateral)

3 f = 62° (angles in the same segment) g = 32° (angle BAC = angle BDC = 48°, angles in the same segment; angle sum of a triangle)

4 h = 43° (angle in a semicircle and angle sum of a triangle)i = 137° (opposite angles of a cyclic quadrilateral)

5 j = 36°, k = 72°

6 L = 123°, m = 62°

Exercise C 1 a = 55° (angle between radius

and tangent and angle sum of a triangle)

2 b = 40° (perpendicular from centre to chord and angle sum of a triangle)

c = 50° (angle between radius and tangent and angle sum of a triangle)

3 d = 53° (isosceles triangle and angle between radius and tangent)

e = 53° (angle in a semicircle and angle sum of a triangle)

4 f = 24° (angle at centre = 2 × angle at circumference)g = 66° (angle between radius and tangent)

5 n = 9 cm (perpendicular from chord to centre and Pythagoras)

6 OM = 12 cm (perpendicular from chord to centre and Pythagoras)

ON = 5 cm (perpendicular from chord to centre and Pythagoras)k = 17 cm

7 180° – 2x


Exercise D 1 a = 50° (alternate segment

theorem)b = 80° (alternate segment theorem)

2 c = 65° (alternate segment theorem)d = 55° (alternate segment theorem)

3 e = 25° (angle between radius and tangent)f = 65° (alternate segment theorem)

4 g = 63° (alternate segment theorem)h = 42° (alternate segment theorem)

5 j = 77° (opposite angles of a cyclic quadrilateral)k = 42° (alternate segment theorem)i = 61° (angles on a straight line)

6 l = 51° (alternate segment theorem)m = 66° (angles on a straight line)

Exercise E 1 a = 40° (angle in a semicircle

and exterior angle of a triangle)

2 b = 151° (angle at centre = 2 × angle at circumference and angles on a straight line)

3 c = 55° (angle sum of a triangle and base angles of an isosceles triangles give angle at centre as 110°; angle at centre = 2 × angle at circumference)

4 d = 152° (angle at the centre = 2 × angle at circumference)e = 208° (angles at a point)

f = 104° (angle at the centre = 2 × angle at circumference

5 g = 132° (angles at a point)

6 h = 28° (exterior angle of a triangle and isosceles triangle)

i = 112° (angle at centre = 2 × angle at circumference)

7 j = 22 12

° (3j + j = 90°, angle in a semicircle and

angle sum of a triangle)

8 k = 43° (isosceles triangle and angles in the same segment)

9 L = 65º, m = 65º, n = 50° (alternate segment theroem, angles, in a triangle)

10 p = 80°, q = 65°, r = 35° (opposite angles in cyclic quadrilateral, alternate segment theorem, angles on a straight line)

11 (i) 90°(ii) 2x(iii) 180 − x(iv) 180 − 2x

Chapter 32 1 a = 94 º, b = 68 º c = 111º

d = 54 º e = 75 º

2 w = 112 º, x = 113 º, y = 77 º, z = 67 º

3 p = 34 º, q = 126 º, r = 141 º, s = 146 º

4 e = 20 º, i = 160 º

5 e = 15 º, i = 165 º

6 30

7 45

8 nonagon

9 36º

10 1260º

11 1980º

12 a 47.5 ºb 87 º, 130 º, 143 º, 95 º,

132.5 º, 132.5 º

13 a 58 ºb 10 º, 100 º, 23 º, 105 º, 122 º

14 x = 59º

15 16 sides

16 2340º

17 ext = 16º, 360/16 ≠ integer number of sides ; no

18 A = 30º, D = 60º, C = 90º

19 360x

20 a 13 sidesb 79º

21 octagons 360 902– = 135º

so ext = 45º. 36045

8= sides

Chapter 33

Exercise A 1 p2 = q2 + r2

2 b2 = a2 + c2

3 52 = 32 + 42

4 19 cm2

5 12 cm2

6 6.71 cm

7 8.80 cm

8 9.48 cm

9 14.76 cm

10 15 cm

11 11.62 cm

12 9.38 cm

13 20.78 m

14 14.20 cm

15 6.51 cm

16 117 m (to the nearest metre)

17 199 cm (to the nearest centimetre)

18 9.54 m

19 7.81 cm

20 170 km

21 a Yes 15² + 20² = 25²b No 12² + 12² ≠ 15²


Exercise B 1

O

O

A

A

1

2

θ

θH

H

2

O

O

A

A

1

2

θ

θH

H

3 sinθ = ab

4 tanθ = dc

5 cosθ = ef

6 sinθ = hg

7 a = 6.34 cm

8 b = 5.60 cm

9 c = 11.8 cm

10 d = 25.0 cm

11 35.2 m

12 19.1 m

13 24.3 cm

14 31.3 cm

15 a = 34.5°

16 b = 32.6°

17 c = 8.58°

18 d = 61.5°

19 a = 2.60 cm

20 b = 99.7 cm

21 c = 10.1 m

22 d = 9.20 m

23 e = 46.7°

24 f = 53.4°

25 42.7 cm

26 32.5°

Exercise C 1 a w = 30.3 cm

b h = 62.5 cm

2 a a = 14 cmb b = 24.25 cmc c = 12.61 cmd 258 cm²

3 a 180 kmb 126 km

4 145.3 cm³

5 l = 2.32 m

6 a a = 21.06 cmb 84.24 cm²

7 16.8 km

8 13.1 cm

9 33.7°

10 120°

11 x = 108.7°

12 34.7 miles north, 197.0 miles east

Exercise D 1 a 7.21 cm

b 33.7°c 7.81 cm d 22.6°

2 a 17.0 cm b 55.6°c 13.7 cm d 64.1°

3 a 4.47 cm b 48.2°c 9.22 cm d 29.0°

4 64.4°

5 5.29 cm

6 15.3°

7 a 23.4° b 49.3°

8 62.1°

9 a 8.22 cm b 10.1 cm

10 a 22.9 cmb i 12.6° ii 60.8°


Chapter 34

Exercise A 1 Angles are the same but 6 ÷ 4 = 1.5 and

10 ÷ 7 = 1.43 or equivalent explanation

2 In first triangle, 3rd angle = 180 – (72 + 66) = 42° So both triangles have angles 42°, 66° and 72°, so the triangles are similar.

3 A and B; length is 1 12

× width or equivalent statement

(scale factor = 1.25) C and E; length is 1 1

3 × width or equivalent statement

(scale factor = 0.75)

4 equilateral triangles, regular polygons, circles

5 x = 4.8 cm

6 x = 4.2 cm

7 ST = 4.8 cm, SU = 4.2 cm

8 AB = 3 cm, EF = 8.75 cm

9 a PSR = 85°b AB = 10.5 cm, SR = 10 cm

10 a similar, angles are 50°, 60° and 70° in each triangle

b not similar, angles are 65°, 40° and 75° in first triangle angles are 65°, 50° and 65° in second triangle

c not similar, Corresponding sides are not in the same ratio:

35

412

513

≠ ≠

d similar, they are equilateral trianglese not similar, corresponding sides are not in the

same ratio:5

12615

≠

Exercise B 1 BAC and DAE are the same in each triangle

ABC = ADE since they are corresponding ACB = AED since they are corresponding The triangles ABC and ADE are similar since all the angles in each triangle are equal.

2 MON = POQ since they are vertically opposite NMO = OQP since they are alternate ONM = OPQ since they are alternate The triangles MON and POQ are similar since all the angles in each triangle are equal.

3 Using Pythagoras’ theorem, GH = 18cm Using Pythagoras’ theorem, HI = 24cm

GFGH

= =10 818

35

.

FHHI

= =14 824

35

.

GHGI

= =1 83 0

35

The triangles FGH and GHI are similar since all the corresponding sides are in the same ratio

4 KT = 8 cm TL = 8 cm Using Pythagoras’ Theorem KL = = ( )128 8 2cm cm

Using Pythagoras’ Theorem JK = = ( )128 8 2cm cm

KTTL

JKKL

KLKT

JLJK

KLTL

= =

= = = =

=

1 1

2 28 28

168 2

8 228

168 2

2 2= = =JLKL

The triangles JKL and KTL are similar since all the corresponding sides are in the same ratio.

5 x = 7.5 cm

6 y = 7.2 cm

7 p = 11.2 cm q = 15 cm

8 m = 5.6 cm n = 5.25 cm

9 c = 5.25 cm d = 17.6 cm

10 364.5 cm2

11 172.8 cm2

12 0.4 m3

13 11.7 cm

14 a 75 cm b 1.25 litresc 24 000 kg

15 a 4.75 cm b 256 cm²c 2432 cm³

16 a 3.2 cm b 3.73 cm²c 14.18 cm³

17 3024 cm²


Chapter 35

Exercise A 1 10.9 cm

2 4.79 cm

3 8.36 cm

4 17.0 cm

5 55.4°

6 21.9°

7 22.8°

8 24.0°

9 2.96 cm

10 25.2 cm

11 12.2 cm

12 20.7 cm

13 86.6°

14 101.6°

15 44.6°

16 109.5°

17 a 13.1 cm²b 163.8 cm²

18 59.1 cm²

19 4.9 cm

20 66.0°

Exercise B 1 a 63.6°

b 55.4°c 3.95 cm

2 a 6.18 cmb 50°c 6.09 cm

3 a BC = 250.4 mb BT = 29.4 m

4 a 4.37 cmb 82.8°

5 A = 134.4° B = 29.0° C = 16.6°

6 a 15.7 cmb 5.64 cmc 121.6 cm2

7 4.52 km

8 50.74 cm²

9 a 54.0°b 87.0°c 11.1 cm

10 AC = 3.44 m, BC = 3.18 m

11 5 385 40 55.

sin sin sin,

°=

°=

°PR PQ

PR = 3.42km, PQRR = 5.3km, PQ = 4.36km,

total = 13.08km

12 a 6.84 cmb 12.1 cm

13 Area = 16 800 m², perimeter = 533 m

14 79.9°

Chapter 36

Exercise A 1 a 12 cm

b 8 cm²

2 a 36 cmb 66 cm²

3 a 30xb 36x²

4 a 10x + 10b 5x2 − 3x + 15

5 a 10x + 12yb 2x2 + 18 xy

6 a 23.41 cmb 23.5 cm²

7 320 m²

8 26 cm²

9 1.4 m

10 4.5 m

11 1−57

m


Exercise B 1 a 12π cm

b 50π m

2 a 42 cm b 0.8 m

3 a 2.5 cmb 8 m

4 a 25.1 cm b 53.4 cm c 123.2 cm

5 a 490.1 mmb 245.0 mm c 27.6 m

6 a 4.46 cm b 0.51 m

7 a 1.34 mb 1.59 cm

8 30.8 cm

9 6.17 cm

10 4.46 m

11 21.4 cm

12 9.06 m

13 a 908 cm2

b 58.1 cm2

14 a 254 cm2

b 10.4 m2

c 3 530 000 mm2

15 3.09 cm

16 7.57 m

17 38.8 cm

18 30.2 m2

19 101 cm2

20 4.52 cm2

21 35.0 cm

22 a 42.8 m2

b 28.6 m

23 9.05 litres

24 76.4 cm2

Exercise C 1 125 cm3

2 a 48.36 m3

b 182.5 m3

3 a 43 200 cm3

b 0.0432 m3

4 4 cm

5 12 500

6 190 cm2

7 324 cm2

8 144 cm3

9 384 m2

10 108 m3

11 408 m3

12 1330 cm3

13 422.55 cm3

14 798 cm3

15 1099 cm3

16 1098.24 cm3

17 a 160 000 cm3

b 25 155 cm2

18 1.285 m

Exercise D 1 62.8 cm3

2 3.18 cm

3 2.54 cm

4 180.96 cm2

5 314.16 cm2

6 50 cm

7 34.14 cm2

8 3.20 cm

9 6.30 cm

10 3451 litres

Chapter 37

Exercise A 1

a i 51.1 cm² ii 70.7 cm² iii 39.3 cm³b i 65.9 cm² ii 108.9 cm² iii 61.6 cm³c i 106.0 cm² ii 169.6 cm² iii 127.2 cm³d i 357.6 cm² ii 704.0 cm² iii 311.7 cm³e i 549.8 cm² ii 703.7 cm² iii 1231.5 cm³

2 8.95 cm

3 677.72 cm2

4 816.74 cm2

5 2412.74 cm3

6 a 980.18 cm3b 1060.98 cm³c 770.15 cm³

7 a 659.73 cm²b 682.44 cm²c 502.86 cm²

8 a 314.16 cm²b 523.60 cm³

9 a 190.85 cm²b 190.85 cm²

10 46.69 cm²

11 94.03 cm3

12 888.30 cm3

13 579.46 cm3

14 a 12 cm³b 57 cm³c 48 cm³

15 6.45 × 6.45 × 6.45

16 2.02 mm


17 3.39 cm

18 a 805.86 cm3

b 440.33 cm2

19 1.5 cm

20 a 1490 cm³b 503 cm2

Exercise B 1 a 6.03 cm

b 41.4 cmc 25.9 cm d 13.7 cme 47.6 cm

2 a 14.5 cm2

b 161 cm2

c 123 cm2 d 55.5 cm2

e 295 cm2

3 a 19.5 cm b 19.3 cmc 58.8 cm

4 a 57° b 244°c 74° d 65°e 108° f 213°

5 a 10.7 cm b 6.2 cmc 8.3 cm d 3.6 cm

6 a 49.75 cm2

b 34.44 cm

7 a 3.49 cm2

b 11.48 cm

8 a 5.29 cm2

b 11.41 cm

Exercise C 1 length

2 area

3 area

4 area

5 none

6 area

7 none

8 length

9 none

10 volume

11 length

12 area

13 none

14 area

15 none

16 none

17 area

18 volume

Chapter 38

Exercise A 1 a B

A B

40º

7 cm

5 cm

b 4.5 cm

2 a

40° 50°

6 cm

B

A B

b 4.6 cm

3

A C5 cm

3 cm 4.5 cm

B

b <ABC = 81° <BAC = 63° <ACB = 36°


4 a

6.5 cm

7.4 cm

52 °

BA

C

b 6.1 cm

5 a E

FD 7.7 cm

48° 55°

b 6.5 cm

6

7.2 cm

5.8 cm 6.3 cm

R T

S

b 57°

7 a

5 cm BA

46° 71°

C

b i 4 cm ii 20 m

8

105°

Y

X

5 cm Z

7cm

b 44°

9

5.8 cm

A C

8.3 cm

D

B

4.6 cm

b 13.3 cmc 30.59 cm2


10

B7.2 cm

A

11

X Y

P

6 cm

2 cm

4 cm

12

110˚

13

8 cm

9 cm

6 cm

point P

b The angle bisectors all meet at one point.

14

A

3.6 cm


15

X

8 cm

Y

2 cm

16

5 cm

A

C

C

10 cm

8 cm

B

17

3 cm

4 cm

4 cm

10 cm BA

OC D


18

11 cm

7 cm

A

C

B

9 cm

19

7 cm

A B

C D10 cm

6 cm

Exercise B 1 FrontPlan Side

FrontPlan Side

Front SidePlan

FrontPlan

Side

5Plan Font side

6 sideFontPlan

2

3

4


Chapter 39

Exercise A 1 a 35 mph

b 62 mphc 50 mphd 64 mphe 54 mphf 69 mphg 115 mphh 45 mphi 45 mphj 1440 mph

2 a 48 km/hb 66 km/hc 62 km/hd 36 km/he 85 1

3 km/h

f 112.8 km/hg 60 km/hh 95 km/hi 2.4 km/hj 0.0864 km/h

3 48 mph

4 112 mph

5 1 hour 40 minutes

6 105 km

7 50 km/h

8Average speed Time taken Length of journey

a) 45 mph 30 minutes 22.5 miles

b) 74 mph 114

hours 92.5 miles

c) 50 km/h 3 hours 36 minutes 180 km

d) 60 km/h 54 minutes 54 km

e) 100 km/h 45 minutes 75 km

f) 90 mph 8 minutes 12 miles

g) 36.4 km/h 6 minutes 3.64 km

h) 48 km/h 1 12

minutes 1200 m

9 56 mph

10 42 mph

11 0800

12 a 60 km/hb 64.3 km/h

13 138.6 km

14 6 minutes 15 seconds

15 a 10 m/sb 18.3

. m/s

c 0.015 m/sd 2.3 m/se 0.25 m/s

16 a 72 km/hb 172.8 km/hc 0.72 km/hd 0.432 km/he 1.2 km/h

Exercise B 1 2 g/cm³

2 a 0.003 kg/cm³b 3 g/cm³

3 1250 g

4 a 3780 gb 3.78 kg

5 80 cm³

6 400 cm³

7 a 5.71 g/cm³b 19.29 g/cm³c 1.07 g/cm³d 1.74 g/cm³

8 0.78125 g/cm³

9 0.06 kg/cm³

10 a 1226.88 gb 59.86008 kg

11 6 cm

12 Density of cone = 1.55 g/cm³ Therefore it will not float.

13 7.33 cm


Chapter 40 1

Measurement Accuracy Lower Bound

Upper Bound

1 15 cm nearest cm 14.5 cm 15.5 cm

2 5.9 cm nearest mm 5.85 cm 5.95 cm

3 200 cm nearest 10 cm 195 cm 205 cm

4 23.80 s nearest hundredth of asecond

23.795 s 23.805 s

5 464 ml nearest ml 463.5 ml 464.5 ml

6 5.5 m2 1 decimal place 5.45 m2 5.55 m2

7 75.0 cm 1 decimal place 74.95 cm 75.05 cm

8 6000 g nearest 100 g 5950 g 6050 g

2 a 43.25 cm + 81.75 cm = 125 cmb 10.315 seconds + 19.175 seconds = 29.49

seconds

3 a 124.8 cm

b 29.47 seconds

4 a 38 m

b 0.299 kg

5 a 36 m b 0.297 kg

6 greatest 5 hours 1 minute, least 4 hours 59 minutes

7 greatest = 155.5 mm, least = 144.5 mm

8 4.2 m

9 Yes upper bound of sum of weights = 500 g

10 a 27.1875 m2

b 26.697375 m2

11 a 26.1375 m2

b 26.592275 m2

12 a 64.29 km/hb 7.733 cm/second

13 a 61.37 km/hb 7.731 cm/second

14 a minimum width = 25.77 cm, maximum width = 27.62 cm

b minimum width = 4.252 cm, maximum width = 4.345 cm

15 a maximum height = 9.368 cm, minimum height = 6.918 cm

b maximum height = 3.677 cm, minimum height = 3.471 cm

16 a 28.5 cm b 330.75 cm2

17 100 seconds

18 a i 15.8 ii 60.0075 iii 1.512 b i 3 ii 87.4225 iii 2.5

19 a i 1275 ii 435 iii 2.04819277 b i 562500 ii 1165 iii 20.37154879

20 a 3.265 b 3.230623819

Chapter 41

Exercise A 1

1

–1–1 1 2 3 4 x–2–3

–2–3

–4

2D A

C

B

E

3

4

y = 1

x = 3

5–4–5

y

2

1

–1–1 1 2 3 x

x = 1/2

–2–3

–2–3

2B

D

A

C

3

y

y = x

3 a reflect in line y = 12

b reflect in the line y = −xc reflect in the line y = 11

2

4 a Reflect A in the line y = 2 to get Bb Reflect A in the line y = x to get Cc Reflect B in the line x = 31

2 to get D


5

1

–1–1 1 2 3 x–2–3

–2–3

2C

B

3

4

y

A

D

6 a

1

–1–1 1 2 3 4 x–2–3

–2–3

–4

2B

T

A3

4

5

6

7

5 6 7 8–4–5–6

y

b Rotation 90° clockwise about (5, 5)d Rotation 180° about (0, 3)

7 a Rotation 90° clockwise about the origin (0, 0) b Rotation 90° anticlockwise about (4, 5)c Rotation 180° about (0, 3) d Rotation 90° anticlockwise about the origin (0, 0)

8 a Rotation 180° about originb Rotation 90° clockwise about (2, 1) c Rotation 90° clockwise about (1, 0)

9

1

–1–1 1 2 3 4 x–2–3

–2–3–4

2

DC

A

EB

345678

–5–6

5 6 7 8 9 10–4–5–6

y

10

1

–1–1 1 2 3 4 x–2–3

–2–3

2

C

B

A

D

3

4

5

5

y

11 a Translation 5

0

b Translation 3

3−

c Translation −−

1

2

d Translation −−

6

2

Exercise B 1 a

1

10 2 3 4 x

2

B

A3

4

5

6

7

5 6 7

y

c Enlarge B by a scale factor 13 centre of

enlargement (1, 2)

2 a

1

10 2 3 4 x

2

A

B

3

4

5

6

5 6

y

c Enlarge B by a scale factor of 3, centre of enlargement (0, 0)


3

1

10

2 3 4 x

2 A

B

3

4

5

6

7

8

5 6 7 8

y

c Enlarge B by a scale factor 25

centre of enlargement (0, 0)

4 a Enlargement scale factor 2 centre of enlargement (0, 0)

b Enlargement scale factor 12 centre of

enlargement (0, 0)c Enlargement scale factor 3 centre of

enlargement (0, 3)d Enlargement scale factor 1

3 centre of enlargement (0, 3)

5

1

–1–1 1 2 3 4 x–2–3

–2–3

–4

2

3

4

5

6

–5

–6

5 6–4–5–6

y

6

1

–1–1 1 2

A B

C

3 4 x–2–3

–2–3–4

23456

–5–6–7–8–9

5 6–4–5–6–7–8–9–10

y

B’

A’C’

7 a enlargement scale factor –2 centre (0, 0) b enlargement scale factor –3 centre (0, 2) c enlargement scale factor –4 centre (4, 2)

d enlargement scale factor – 13 centre (0, –1)

e enlargement scale factor –12 centre (0, –21

2 )

f enlargement scale factor – 13 centre (0, 2)

Exercise C 1 a Translation 1

2−

b Reflection in the line x = –1c Rotation 90° anticlockwise about (0, 0)

d Translation 23−

e Rotation 180°about (1, 2)

f Translation 03

g Rotation 180°about (3, 1)

h Translation −

30

2 a Rotation 90° anticlockwise about (0, 0) b Reflection in the line x = 2 c Rotation 180° about (1, 0) d Rotation 90° clockwise about (1, –1) e Reflection in the line y = –1

3

1

–11 2 3 4 x

–2–3

2

3

4

5

5 6 7 8 9 10C

BA

y

c Rotation 180° about (5, 1)

4

1

–11 2

D

E

F

3 4 x

–2–3

2

3

4

5 6 7 8 9 10

y

c Rotation 90° anticlockwise about (7, 3)


5

1

–1–1 1 2

A

C

B

3 4 x–2–3

2

3

4

5

5–4–5

y

c Translation 12

6

1

F

D

E

–1–1 1 2 3 4 x–2–3

–2–3

–4

2

3

4

5

6

–5

–6

5 6–4–5–6

y

c Enlargement scale factor –2, centre of enlargement (0, 0)

7

1

–1–1 1 2 3 4 x

A

B

–2–3

–2–3

–4

2

3

4

5

6y = –x

–5

–6

5 6–4–5–6

y

8 Translation −

52 , reflection in the line x = 4

9 a Translation 47−

followed by a rotation 90°

anticlockwise about (1,–5) b Enlargement scale factor 2

7 about (–2, 3)

c Reflection in the line y = x followed by rotation 90º clockwise about (–3, 0)

d Translation −−

45

e Translation 74−

followed by reflection in y = 3

Chapter 42 1 What do you eat for breakfast?

Toast Cereal Porridge Fruit Other Nothing

2 a It is a leading question.b The response section only has Yes and No. It

needs to have Unsure.

3 a It needs to include a time e.g. How often do you eat fruit and vegetables in a day? or How often do you eat fruit and vegetables in a week?

b 4 and 8 are included twice (1–4 and 4–8, 4–8 and 8–12) There is nowhere to record 0 times.

4 a It is a personal question.b What age group do you belong to?A 20−29B 30−39C 40−49D 50−59E 60−69

5 a It is too general, it is not related to the caravan park.

b It does not include an option for other. It does not include an option for none.

c Which of these activities would you like to do while staying at the caravan park? Tennis ? Swimming ? Crazy Golf ? Football ? Other ? None ?

If other, please specify…………………….

6 Do you go to the cinema? ? Yes ? No If yes, how often do you go in a month?

? 1-2 ? 3-4 ? 5-6 ? more than 7 times ? less than once a month


7 a What type of programme on the radio do you like most?

b When do you listen to the radio? ? morning ? afternoon ? evening ? never

c Do you like competitions and phone-ins on the radio?

? Yes ? Nod Question is too personal and not needed for

this questionnaire

8 a Do you live in the countryside or in the town? ? countryside ? town

How often do your children go to the park |in a week? ? 1-2 times ? 3-4 times ? 5 or more times ? 0 times

b How many texts do you send a day? ? 0 ? 1-10 ? 11-20 ? 21-30 ? more than 30 What were your most recent exam results?c Do you wear glasses or contact lenses? ? Yes ? No How often do you eat carrots in a week? ? 0 times ? 1-2 times ? 3-4 times ? 5 or more timesd Would you describe yourself as musical? Yes ? No ?

Would you say you are good at Maths? Yes ? No ?

Chapter 43

Exercise A 1 a Mark Frequency

0–9 210–19 520–29 1030–39 940–49 4

b

1

2

Freq

uenc

y

3

4

5

7

8

9

10

Mark in test0−9 10−19 20−29 30−39 40−49

6

2 a Height (cm) frequency155 h 160 2160 h 165 6165 h 170 8170 h 175 10175 h 180 11180 h 185 7185 h 190 4190 h 195 2

b

12

Freq

uenc

y

345

789

1011

155 160 165 170 175Height (cm)

180 185 190 195

6

3 a 8 t 9b 40c 30%d 7.7 minutes

4

1

2

Freq

uenc

y

3

4

5

7

8

9

10

11

12

3 4 5 6 7

Height (cm)

8 9

6


5

5

10

Freq

uenc

y

15

20

25

35

40

50 10 15 20 25Time minutes

30 x

y

30

6 a, b

5

10

Freq

uenc

y

15

20

25

3.63.4 3.8 4.0 4.2Vital lung capacity (litres)

4.4 4.6 x

30

c 124 pupilsd There are more boys than girls with larger

vital lung capacities.

7 a

1 4 7 8

2 1 2 5 8 9 9

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

4 0 1 2 3 5 6 7 7

5 0

Key 3|2 = 32 yearsb i 36 ii 39 iii 36

8 a

Women Men

3 2 1 7

7 3 3 7 9

9 4 2 1 0 4 7 7 7 8

4 3 2 2 2 1 5 1 2 7 8

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

7 1 7 3 4 6 7

4 2 8 1 2

Key 8|1 = 81 yearsb 8 years

9 a 3.8 cmb 5.2 cmc 5.168̇1̇ cmd 10

10 a 27%b 34%c 32d 41%e 39%f 26%


Exercise B 1 a negative correlation

b no correlationc positive correlation

2 a

10

Rai

nfal

l (m

m)

20

40

50

60

70

80

90

0 10 20Temperature (˚c)

30

30

b no correlationc nothing can be deduced

3 a b

10

Wei

ght

of T

omat

oes

(kg)

20

40

50

60

70

80

90

100

0 20 3010 40 50Amount of fertiliser (g/m2)

60 70 80

30

b positived reading from pupil's line of best fit

4 a c

10

0

Mile

age

(Tho

usan

ds o

f m

iles)

20

40

50

60

70

80

90

100

0 2 31 4 5Price (Thousands of £)

6 7 8 9

30

b negatived reading from pupil's line of best fit

Exercise C 1 T = 24

2 C = 10

3 T = 107

Chapter 44

Exercise A 1 a 5

b 9c 4.16d 4

2 a 100b 6c 100.875d 101

3 a 1b 6c 1.636d 1

4 a 1b 7c 2.092d 2


5 a 0–4b 0–4c 5.065

6 a 4–7b 4–7c 7.417

7 a 120 x 130b 130 x 140c 131.273

8 a 58 x 60b 58 x 60c 59.56

9 a 0.3 x 0.6b 0.6 x 0.9c 0.875d 0.25m

Exercise B 1 3

2 7

3 27

4 7

5 151 cm

6 71.6%

7 60%

8 35.1 g

9 £1.80

10 7 marbles

11 Tim, mean = 194 g, range = 220 g Zoe, mean = 208 g, range = 180 g Zoe's potatoes bigger as she had higher mean. Less variation with her potatoes as range was smaller.

12 Aóife; Range = 15, No modal group, Mean = 7.526, Median group 8-9

Clare; Range = 9, Modal group 8-9, Mean = 8.756, Median group 8-9

13 Exposed site: Mean = 10.2, Modal group = 9 ı 10, Median group = 10 ı 11, Range = 8

Sheltered site: Mean = 11.64, Modal group = 11 ı 12, Median group = 11 ı 12, Range = 7

14 a Mean = £26625b Median = £18,095c Median

15 a mode – suitable reasonb mean/median – suitable reasonc median– suitable reasond mean/median – suitable reasone mean/median – suitable reasonf mode – suitable reasong mean/median – suitable reasonh mode – suitable reason

Chapter 45

Exercise A 1 a 190

b 80c 11.6%

2 a Mass (m grams) Cumulative frequencym 40 7m 50 19m 60 43m 70 75m 80 93m 90 98

m 100 100

b

0

20

40

60

80

100

20 40 60

Cum

ulat

ive

freq

uenc

y

Mass (g)

UQ

M

LQ

80 100

c Median = 62, interquartile range = 17

3 a

Height (cm) Frequency Cumulative frequency 10 15 15 20 23 38 30 36 74 40 42 116 50 24 140 60 10 150


100

20

40

60

80

100

120

140

160

UQ

Cum

ulat

ive

freq

uenc

y

M

Height (cm)

LQ

20 30 40 50 60

b Median = 30.5, interquartile range = 18.5c 150 – 103 = 47

Exercise B 1 a 24

b 19.5c 24 – 14 = 10d 420e 480 – 260 = 220

2 a 46b 62 – 30 = 32c 5d 29 – 10 = 19e 40 – 35 = 5

Exercise C 1

0 2 4 6 8 10 12 14

2

0 2 4 6 8 10 12 14 16 17

3

20 22 24 26 28 30 32 34

4

32 34 36 38 40 42 4430

5

0 2 4 6 8 10 12 14 16

6

2 4 6 8 10 12 14 16 18

Exercise D 1

a i £320 ii £300 iii £150b i £200 ii £200 iii £100e.g.c Men have a higher median wage than women.

Men have a bigger range than women.

2 Girls spend longer on average (girls’ median = 64 seconds, boys’ median = 36 seconds); the spread of times for the boys is very slightly less than the spread for the girls (girls’ interquartile range = 35 seconds; boys’ interquartile range = 34 seconds).

3 a

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Alice

Ronnie

30

b

c Ronnie has a bigger range than Alice. Alice has a bigger interquartile range than

Ronnie. Medians very similar but Alice’s median is

slightly higher than Ronnie’s.

4 a i 156 cm ii 15 cm iii 60 cm


b i 160 cm ii 19 cm iii 60 cmc The boys have a higher median than the girls

The boys have a larger IQ range than the girls. d i ii

120 130 140 150 160 170 180 190

Boys

Girls

5 Kevin if you want the chance of a really fast time, as his mean time is faster, so Kevin is generally faster but is not as consistent (higher IQ range). Dermot if you want to rely on his time, as he is more consistent (lower IQ range).

6 a Make B – if you are buying one battery you need it to be a reliable one.

b Make A – if you are buying a large quantity the spread is less important and the higher average life is more important.

Chapter 46

Exercise A 1

20

1

0

Freq

uenc

y D

ensi

ty

2

40 60 80 100 120 140Height (h cm)

160 180 x

y

2

0.1

0.2

0.3

Freq

uenc

y D

ensi

ty

0.4

0.5

1000 300200 400 500Amount raised (£)

600 700 800 9001000

3

0.2

0.4

0.6

Freq

uenc

y D

ensi

ty

0.8

1.0

1.2

1000 300200 400 500Amount earned (£)

600 700 800 9001000

4 a 10b 47c 72

5 a Money Raised (£) Frequency

0 < m 50 15

50 < m 100 25

100 < m 200 30

200 < m 500 45

500 < m 800 18

b £257.14

6 a 24b 262c 60.1̇7 6715̇ minutes

7 They have the same range Most people leaving on the 10.00 are older, than people leaving on the 0800

Exercise B 1 a Not everyone is represented in the telephone

book. Young people are not included. b Only people who use the train will be

included.c Only people who eat out are included.

2 a Roisin needs 3 pupils from each class. She should use random numbers to obtain 3 pupils from each class or pick names from a hat.

b Roisin needs 6 boys and 9 girls to make up her sample.

3 a To ensure each of the 4 departments are represented fairly.

b A 17, B 5, C 25, D 13 or A 18, B 5, C 25, D 12c Use random numbers or pick names from a hat.

4 a The method is not suitable, the people being sampled were all at the football match.

b The method is suitable.c The method is not suitable, nothing is

sampled from in between.


5 a Not suitable. These people will own a car (or some may be passengers).

b Not suitable. Not everyone is registered to vote.c Not suitable. People who do not use the bus

are excluded.

6 a 24 b 6

7 a 23 males 27 females b 38 adults 12 children

8 a 25b 15c 80

9 a 60 b 60

10 a 2 b 10

11 84

12 250

13 200

14 a 500 b Time period too big

Chapter 47

Exercise A 1 a score Relative frequency

1 0.152 0.173 0.184 0.165 0.176 0.18

b Yes, For a fair dice each score should have probability 1

6 = 0.166... or 0.17.

All the relative frequencies for Pete's dice are close to this value.

2 a Flavour Relative frequencyPlain 0.28Salt and vinegar 0.22Cheese and onion 0.38Other 0.13

b There is large number of trials.

3 a 0.84 b 0.16

4 a score Relative frequency1 0.072 0.173 0.314 0.305 0.15

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

5 = 0.2

5 0.15

6 27

7 0.25

8 0.21

9 0.27

10 a 0.17b 0.47c 0.32d 0.27e 0.46

11 a 0.85b 0.65

12 a 0.003b i 0.15 ii 0.01 iii 0.91 iv 0.18c i 0.0081 ii 0.0009

13 a 710

b 12

c 45

14a i 0.09 ii 0.3 iii 0.15b i 0.19 ii 0.49 iii 0.61 iv 0.45

Exercise B 1 1

12

2 0.42

3 14


4 a 0.21 b 0.09

5 a 0.42b 0.46

6 a 0.225 b 0.775 c 0.275

7 a 715

b 115

c 815

d 715

8 a 0

b 121

c 27

d −57

9 a 31105

b 1021

10 a 16169

b 1169

c 8169

Exercise C 1 a

Tea

Coffee

Muesli

Muesli

Toast

Toast

Grapefruit

Grapefruit

b 16

c 13

2 a

R

W

R

W

11

15

11

15 415

R

W

1115

415

415

b i 16225

ii 88225

3 a

cycles

bus

0.3

0.1

0.4

0.2

0.5 Shopping

car

0.5

canteen

gym

0.1

0.4

0.5 Shopping

canteen

gym

0.10.4

0.5Shopping

canteen

gym

b i 0.15 ii 0.1

4 a

Gazette

News

13

23

water

orange

cola

15 2

5

25

water

orange

cola

15 2

5

25

b i 115

ii 25

5 a

boy

girl1220

820

boy

girl

719

1219

boy

girl

819

11 19

b i 3395

ii 4895

6 a

red

black

613

red

black

612

612

red

black

512

712

713


b i 726

ii 1926

7 a

0.3

0.5

0.5

0.7

rain

rain

not rain

not rain

0.3

0.7

rain

not rain

b 0.51

8 a

0.6

0.4

more

not

0.8

0.2

more

not

0.4

0.6

more

not

b i 0.48 ii 0.28

9 0.14

10 0.315

11 a 0.196b 0.616.

Documents

ANSWERS TO HigHER gCSE MATHEMATiCS FOR CCEA PRACTiCE · PDF fileH gCSE M CCEA P B H S ANSWERS TO HigHER gCSE MATHEMATiCS FOR CCEA PRACTiCE BOOK Chapter 1 1 a 18, 24, 60 b 25, 60 c