305847293-IGCSE-Mathematics-Textbook-Answers.pdf

Solutions

1

Number and language

Exercise 1.1

Prime numbers are: 1. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

1.

a) b) c) d) e) f) g) h) i) j)

page 6

1, 2, 3, 6 1, 3, 9 1, 7 1, 3, 5, 15 1, 2, 3, 4, 6, 8, 12, 24 1, 2, 3, 4, 6, 9, 12, 18, 36 1, 5, 7, 35 1, 5, 25 1, 2, 3, 6, 7, 14, 21, 42 1, 2, 4, 5, 10, 20, 25, 50, 100

Exercise 1.3 1.

a) 3, 5 e) 2, 5 i) 2, 5, 7

5

b) 2 e) 22 h) 3

Exercise 1.5

d) 2 h) 5, 7

c) Rational

page 9

1.

a) 5 f ) 13

2.

Students check their answers

3.

a) f)

b) 3 g) 0.1

b) g)

c) 7 h) 0.2

c) h)

d) 10 i) 0.3

e) 11 j) 0.5

d) i)

e) j)

page 9

1.

The following answers are correct to 1 d.p.: a) 5.9 b ) 6.7 c) 7.4 d) 7.7 e) 1.4

2.

Students check their answers

a) 2 f) 6 k) –10

page 10

b) 5 g) 10 l) –1

c) 3 h) 100

Exercise 1.10 2.

2

11 13

2

c) 2 3 f ) 23 7 i) 3 7 11

d) 0.1 i) –2

e) 0.3 j) –3

page 10

c) 6 h) 13

d) 3 i) 17

e) 9 j) 12

2.

a) 42

b) 60

c) 70

d) 90

e) 120

f ) 105

g) 20

h) 231

i) 240

j) 200

page 8 b) Rational e) Rational h) Rational

a) d)

$35 $160

b) $318 e) $90

c)

$88

3. 165 m 4. 695 m

Student assessment 1

page 7

b) 5 g) 8

a) Rational d) Rational g) Irrational

b) Irrational

1.

a) 4 f ) 22

1.

a) Rational d) Rational

Exercise 1.9

1.

Exercise 1.6

3.

1. 146 °C

page 7

3 5 5 7

c) Rational f ) Rational

Exercise 1.8

c) 2, 3 g) 3, 11

2

a) 2 d) 23 g) 32 j) 32

b) Irrational e) Rational

page 6

b) 2, 3 f ) 13 j) 2, 7

Exercise 1.4 1.

a) Irrational d) Rational

Exercise 1.7

page 5

Exercise 1.2

2.

page 11

1.

a) Rational d) Rational

b) Irrational e) Rational

c) Rational f ) Irrational

2.

a)

b) 3

c)

3.

a) 81 d) 0.49

b) 225

c) 0.04

4.

a) 12.25

b) 16.81

c) 0.0225

c) Irrational f ) Rational i) Rational

Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013

1

Solutions

5.

a) 15 d) 35

b) 0.1 e) 73

c) 0.9 f ) 117

6.

a) 64

b) 0.001

c)

8 27 4 5

7.

a) 3

b) 100

c)

8.

a) $84 d) $74

b) $91 e) $43

c) $45 f ) $15

4.

a) 130 d) 4

5.

c), e) and f) are incorrect.

Accuracy

Exercise 2.1

6.

a) 69 000 d) 4000

page 12 b) 74 000 e) 100 000

c) 89 000 f) 1 000 000

2.

a) 78 500 d) 8100

b) 6900 e) 1000

c) 14 100 f) 3000

3.

a) 490 d) 80

b) 690 e) 0

c) 8850 f) 1000

Exercise 2.2

a) 120 m2

b) 40 m2 3

a) 200 cm

b) 4000 cm

Exercise 2.5

1.

a) 5.6 d) 157.4 g) 3.0

b) 0.7 e) 4.0 h) 1.0

c) 11.9 f) 15.0 i) 12.0

2.

a) 6.47 d) 0.09 g) 100.00

b) 9.59 e) 0.01 h) 0.00

c) 16.48 f) 9.30 i) 3.00

Exercise 2.3

page 13

c) 400 cm2 3

page 17

1.

5.5 Upper bound 6.5 ii) 5.5 x 6.5 b) i) Lower bound 82.5 Upper bound 83.5 ii) 82.5 x 83.5 c) i) Lower bound 151.5 Upper bound 152.5 ii) 151.5 x 152.5 d) i) Lower bound 999.5 Upper bound 1000.5 ii) 999.5 x 1000.5 e) i) Lower bound 99.5 Upper bound 100.5 ii) 99.5 x 100.5

2.

a) i) Lower bound

page 13

1.

c) 9 f) 250

Answers to Q.6 and 7 may vary slightly from those given below: 7.

2

b) 80 e) 200

a) i) Lower bound

Upper bound

3.75 3.85



1.

a) 50 000 d) 7500 g) 1000

b) 48 600 e) 500 h) 2000

c) 7000 f) 2.57 i) 15.0

2.

a) 0.09 d) 1 g) 0.0031

b) 0.6 e) 0.95 h) 0.0097

c) 0.94 f) 0.003 i) 0.01

Exercise 2.4 a) 419.6 d) 23.8 g) 1.9

1.

page 14 b) 5.0 e) 57.8 h) 4.1

c) 166.3 f) 4427.1 i) 0.6

Answers to Q.2–4 may vary slightly from those given below: 2.

a) 1200

b) 3000

c) 3000

3.

d) 150 000 a) 200 d) 550

e) 0.8 b) 200 e) 500

f) 100 c) 30 f) 3000

2

x 3.85 15.55 ii) b) 3.75 i) Lower bound Upper bound 15.65 ii) 15.55 x 15.65 c) i) Lower bound 0.95 Upper bound 1.05 ii) 0.95  x 1.05 d) i) Lower bound 9.95 Upper bound 10.05 ii) 9.95 x 10.05 e) i) Lower bound 0.25 Upper bound 0.35 ii) 0.25 x 0.35

3.

a) i) Lower bound

4.15 Upper bound 4.25 ii) 4.15 4.25 x b) i) Lower bound 0.835 Upper bound 0.845 ii) 0.835 x 0.845 c ) i) Lower bound 415 Upper bound 425 ii) 415 x 425


c) 2000 cm3

Solutions

d) i) Lower bound

4950 Upper bound 5050 ii) 4950 x 5050 e) i) Lower bound 0.0445 Upper bound 0.0455 ii) 0.0445  x 0.0455 f ) i) Lower bound 24 500 Upper bound 25 500 ii) 24 500  x 25 500 4.

a)

5.3

5.35

b) 5.35  M 5.

a)

11.7

7.

5.5

Upper bound

g) Lower bound 11.8

11.85

Upper bound

11.9

h) Lower bound

Upper bound

11.85

615 m3 Upper bound 625 m3 b) 615  x 625

i) Lower bound

a) Lower bound

a) Lower bound

625 m Upper bound 635 m b) 395  W 405

Exercise 2.6 1.

5.45

a) Lower bound

Upper bound b) Lower bound Upper bound c) Lower bound Upper bound d) Lower bound Upper bound e) Lower bound Upper bound f ) Lower bound

5.45

11.75

b) 11.75  T 6.

5.4

2.

page 18

a) Lower bound

Upper bound b) Lower bound

Upper bound

c) Lower bound

Upper bound d) Lower bound Upper bound e) Lower bound Upper bound f ) Lower bound Upper bound g) Lower bound Upper bound h) Lower bound Upper bound i) Lower bound Upper bound j) Lower bound Upper bound k) Lower bound Upper bound l) Lower bound Upper bound

263.25 297.25 3295.25 3455.25 4925.25 5075.25 3.76 (2 d.p.) 4.26 (2 d.p.) 2.83 (2 d.p.) 3.19 (2 d.p.) 8.03 (2 d.p.) 8.66 (2 d.p.) 44.95 (2 d.p.) 52.82 (2 d.p.) 39.77 (2 d.p.) 42.23 (2 d.p.) 16.14 (2 d.p.) 18.88 (2 d.p.) 3.55 (2 d.p.) 7.12 1.47 (2 d.p.) 1.63 (2 d.p.) 18.51 (2 d.p.) 28.59 (2 d.p.)

Upper bound 3.

a) Lower bound b) c) d) e)

Upper Lower Upper Lower Upper Lower Upper Lower Upper

bound bound bound bound bound bound bound bound bound

f ) Lower bound

Upper bound

g) Lower bound

Upper bound h) Lower bound

Upper bound i) Lower bound

Upper bound

6.7 6.9 29.69 (2 d.p.) 30.80 (2 d.p.) 147.76 (2 d.p.) 150.25 (2 d.p.) 13.3 13.5 1.75 (2 d.p.) 1.81 (2 d.p.) 0.39 (2 d.p.) 0.46 (2 d.p.) 34.10 (2 d.p.) 40.03 (2 d.p.) 0.98 (2 d.p.) 1.02 (2 d.p.) 0 0.04 20 002.5 20 962.5 1.06 (2 d.p.) 1.15 (2 d.p.) 1 116 250 1 188 250 88.43 (2 d.p.) 91.60 (2 d.p.) 131.75 (2 d.p.) 139.34 (2 d.p.) 18.10 20.39 (2 (2 d.p.) d.p.) 2.24 (2 d.p.) 2.53 (2 d.p.) 3.45 (2 d.p.) 3.97 (2 d.p.) 60.34 (2 d.p.) 68.52 (2 d.p.)

Exercise 2.7

page 19

1. Lower bound

3.5 kg 4.5 kg

Upper bound 2. Lower bound


Upper bound

21.8 cm 22.2 cm 13.7 m 13.74 m 2

4. Lower Upperbound bound

74.13 cm cm2 (2 75.88 (2 d.p.) d.p.)

5. Lower bound

68 425 m2 75 625 m2


Upper bound

13.1 (1 d.p.) 13.5 (1 d.p.)


3

Solutions

7. Lower bound

9.0 cm (1 d.p.) 11.1 cm (1 d.p.)

Upper bound 8. a) Lower bound

Upper bound b) Lower bound Upper bound 9. a) Lower bound

Upper bound b) Lower bound Upper bound

53.1 cm (1 d.p.) 53.7 cm (1 d.p.) 224.3 cm2 (1 d.p.) 229.7 cm2 (1 d.p.) 11.2 cm (1 d.p.) 11.4 cm (1 d.p.) 70.5 cm (1 d.p.) 71.3 cm (1 d.p.)

5.

0.0035  x

6.

a) b) c) d)

Student assessment 3 1.

a) Lower bound

Upper bound b) Lower bound

10. Lower bound = 11.5 g/cm 3 (1 d.p.)

Upper bound

c) Lower bound

13.6 g/cm3 (1 d.p.)

Upper bound

0.0045

4.825  x 4.835 5.045  y 5.055 9.95  z 10.05 99.995  p 100.005

Upper bound

11. Least 93.3 h (1 d.p.) Greatest 138.5 h (1 d.p.)

d) Lower bound


e) Lower bound

a) 2800 d) 1000

1. 2. 3.

b) 7290

Upper bound page 21

Upper bound f ) Lower bound

c) 49 000

Upper bound

a) 3.8 d) 1.58

b) 6.8 e) 10.0

c) 0.85 f) 0.008

2. Lower bound

a) 4 d) 10

b) 6.8 e) 830

c) 0.8 f) 0.005

3. Lower bound

Answers to Q.4–6 may vary from those given below: 5.

40 m2

6.

a) 25

7. 92.3 cm

c) 4

(1 d.p.)


3.

a) 124.5  V 125.5 cm3 b) 4.99  L 5.01 cm (2 d.p.)

7.5

4.

a) 25.10  C b) 50.14  A

40.5

5. 22 cups

page 22

a) 6.5

7

39.5

40

b) c)

299.5

300

300.5

2000

2000.5

d) 1999.5

a) 205  x c) 2.95  x

2.

3. Length: 349.5

215 3.05 

L

Width: 199.5  W

4.

58.85

4

page 23

1. 255 kg 2. 31 575 cm 3 (5 s.f.)


0.46 kg (2 d.p.) 0.48 kg (2 d.p.)

Upper bound b) 4

3

1

5. Lower bound

4. 18 000 yards

6381.75 6.2 (1 d.p.) 6.3 (1 d.p.) 47.7 (1 d.p.) 54.7 (1 d.p.) 0.8 (1 d.p.) 1.2 (1 d.p.) 0.5 (1 d.p.) 0.9 (1 d.p.)

10.5 cm 13.5 cm

Upper bound 0.8  x

965.25 1035.25 6218.75

118.7 cm2 (1 d.p.) 121.1 cm2 (1 d.p.)

Upper bound

4.

page 22

b) 63.5  x 64.5 d) 0.875  x 0.885

6.

25.16 (2 d.p.) 50.39 cm2 (2 d.p.)

a) Lower bound

4.5 cm Upper bound 5.5 cm b) Upper and lower bounds of 100 matches ÷ 100 Lower bound 5.43 cm Upper bound 5.44 cm

350.5 200.5 58.9

58.95


Solutions

4

Integers, fractions, decimals and percentages

Exercise 4.1 a) 9 f ) 72

1.

page 30

b) 16 g) 30

Exercise 4.2

c) 20 h) 48

d) 40 i) 210

e) 18 j) 52

page 31

1.

a) f) k)

b) g) l)

c) h)

d) i)

e) j)

2.

a) 7 f) 1

b) 6 g) 9

c) 6 h) 3

d) 6 i) 9

e) 5 j) 6

2.

a) 4.5 e) 9.27 i) 4.356

b) 6.3 f ) 11.36 j) 9.204

c) 17.8 g) 4.006

a) 19.14 e) 35.81 i) 1.4

b) 83.812 f ) 5.32 j) 0.175

c) 6.6 g) 67.14

Exercise 4.4 1.

page 31 d) 3.07 h) 5.027 d) 11.16 h) 6.06

page 32

a) 9

b) 3

2.

a) 6 (4 6) ÷ 3 20 b ) (9 3) (7 2) 54

3.

a) 29 830

b) 41 492

4.

a) 2

b) 7

5.

a) 224

b ) 28

6.

a) 1127.4

b) 526.1

Exercise 4.6

page 35

1.

c)

a)

b)

d)

e)

f)

page 35

1.

a) 1

b)

c)

d) 1

e) 1

f) 1

2.

a) 1

b) 1

c) 1

d)

e) 1

f)

3.

a)

b)

c)

d)

e) 

f) 1

4.

a)

b)

c)

d)

e)

f)

5.

a) 5

b) 5

c) 3

d) 6

e) 1

f)

6.

a) 5

b) 7

c)

d)

e)

Exercise 4.8

2

f) 1

page 37

1.

a) 8

b) 

c)

d)

e)

f)

a) 

58%

b)

68%

2.

a)

b)

c)

d)

e)

f)

c)

55%

d)

30%

3.

a)

b) 2

c) 1

d) 4

e)

f)

e)

92%

f)

38%

4.

a)

b)

c)

d)

e)

f) 1

g)

75%

h)

40%

5.

a)

b) 3

c)

d) 12

Exercise 4.9

2. Frac t i o n

De c i ma l

Pe rc e n t a ge

1 10

–

0.1

10

–15

0.2

20

3 10

–

0.3

30

–4 = –25 10

0.4

40

–12

0.5

50

–35

0.6

60

10 –45

0.7 0.8

70 80

9 10

–

0.9

90

–14

0.25

25

–34

0.75

75

–7

6

page 34

1.

Exercise 4.7

Exercise 4.3 1.

Exercise 4.5

page 37

1.

a) 0.75 e) 0.3˙ i) 0.6˙3˙

b) 0.8 f ) 0.375

c) 0.45 g) 0.4375

d) 0.34 h) 0.2˙

2.

a) 2.75 b) 3.6 e) 5.6˙ f ) 6.875 i) 5.4˙ 28571˙

c) 4.35 g) 5.5625

d) 6.22 h) 4.2˙

Exercise 4.10

page 38

1.

a) f)

b) g)

c) h)

d) i)

e)

2.

a) 2

b) 6

c) 8

d) 3

e) 10

f) 9

g) 15

h) 30

i) 1


Solutions

Exercise 4.11 1.

a)

1 3

2.

a) c)

92 90

3. 4.

b)

page 40

7 9

55 990

= 181 = 1451

5

c)

42 99

= 14 33

b) d)

39 638 9900

62 990

=

d)

65 99

31 495 19 = 4 4950

1 5 1 6

Student assessment 1 1. a) 12 2. a) 30% e) 80% i) 31%

b) 33 b) 29% f ) 219% j) 7%

3. a) 23

b) 18

c) c) g) k)

d) 90 d) 70% h) 75% l) 200%

5. 360.2 6. a) 2

b) 9

7. a) 0.4

b) 1.75

c) 0.8˙1˙

d) 1.6˙

8. a) 4

b)

c) 1

d) 2

37 99

b)

10.

650 900

8 99

c) 1 19 90

= 13 18


page 42

1. a) 21

b) 27

c) 22

d) 39

2. a) 60% e) 150% i) 77% 3. a) 0

b) 49% f ) 327% j) 3% b) 19

c) 25% g) 5% k) 290%

d) 90% h) 35% l) 400%

a) 60%

b) 40%

2.

a) 87.5% c) 29.16%

b) 73.3% d) 14.29% (2 d.p.)

3.

a) 0.39 e) 0.02

b) 0.47 f) 0.2

c) 0.83

d) 0.07

4.

a) 31% e) 20%

b) 67% f) 75%

c) 9%

d) 5%

a) 25% d) 180%

b) 66.6˙% e) 490%

c) 62.5% f) 387.5%

2.

a) 0.75 d) 0.07

b) 0.8 e) 1.875

c) 0.2 f) 0.16˙

3.

a) 20 d) 36

b) 100 e) 4.5

c) 50 f) 7.5

4.

a) 8.5 d) 52

b) 8.5 e) 17.5

c) 52 f) 17.5

5.

a) Black 6

b) Blonde 3

c) Brown 21

6. Lamb 66 Chicken 24 Turkey 12 Duck 18 7. Australian 143 Pakistani 44 Greek 11 Other 22 8.

7. a) 0.875

b) 1.4

c) 0.8˙

d) 3.2˙85714˙

8. a) 6

b)

c) 3

d) 3

7 99

b)

10.

11 50

a) 48% d) 50% g) 33 %

2. Win 50% b) 2

9. a)

Newspapers 69 Pens 36 Books 18 Other 27

Exercise 5.3

4. 18 032

6. a) 1

1 1000

3.

c) 3 992

page 44

1.

1.

5. 340.7

page 44

1.

Exercise 5.2

4. 22 977

9. a)

Exercise 5.1

page 41

6 50% 6% 340%

Further percentages

A C

page 45 b) 36.8% c) 35% e) 45% f) 40% h) 57% (2 s.f.)

Lose 33 % Draw 16 %

34.5% (1 d.p.) B 23.0% (1 d.p.) D

25.6% (1 d.p.) 16.9% (1 d.p.)

4. Red 35.5%

Blue 31.0% White 17.7% Silver 6.6% Green 6.0% Black 3.2%

Exercise 5.4

page 47

1.

a) 187.5 d) 245

b) 322 e) 90

c) 7140 f) 121.5

2.

a) 90 d) 900

b) 38 e) 50

c) 9 f) 43.5


7

Solutions

a) 20% d) 5%

3.

b) 80% e) 85%

a) 50% d) 100%

4.

c) 110% f) 225%

b) 30% e) 36%

c) 5% f) 5%

Student assessment 3 1. $3500

$12 000

2.$500

$250

3.

5. 7475 tonnes

$1

$56

$20 000

4. $15 000

$6825

7.

a) $75

b) $93.75

5. 40 000 tonnes

8.

a) 43

b) 17.2%


9.

1100

Exercise 5.5

1. $200 2. $462

page 48

a) 600 d) 250

b) 350 e) 125

c) 900 f) 1.5

3. 15 marks

2.

a) 56 d) 20

b) 65 e) 0.25

c) 90 f) 38

5. 25 000 units

3. 280 pages 4. 12 500 families

6.

12 200 000 m 3 page 49

1. 640 m 2. $345.60 3. $10 125 4. a) 20% c) 22.5% e) 7%

6. 46 500 units

Ratio and proportion

Exercise 6.1 1.

48

2.

16 h 40 min

page 52

3. 11 units 4. b) 41.7% (3 s.f.) d) 85.7% (3 s.f.) f) 30%

5. 16% profit 6. a) $36

4. 35 000

6


b) 25%


page 49

1. 750 m 2. $525

a) 7500 bricks b) 53 h

5.

a) 6250 litres

6.

1110 km (3 s.f.)

7.

a) 450

b) 75

8.

a) 480 km

b) 96 km/h

b) 128 km

Exercise 6.2

c) 120

page 54

1.

a) 450 kg

2.

a) Butter 600 g

a) 16.8 litres b) Red 1.2 litres

3. $97 200

b) 1250 kg

Flour 2 kg Sugar 200 g Currants 400 g b) 120 cakes

4. a) 29.2% (3 s.f.) c) 125%

b) 21.7% (3 s.f.) d) 8.33% (3 s.f.)

3.

e) 20% 5. 8.3%

f ) 10%

4.

a) 125 b) Red 216 c) 20

6. a) $650

b) 61.8% (3 s.f.)

5.

a) 42 litres b) Orange juice 495 litres

White 14.3 litres

Yellow 135

Mango juice 110 litres 8

page 51

$25 $524 $10 $4000 $4500 $5500

1.

22

$137 500

15

6.

5.

page 50


Solutions

6.

32

7.

210

2. 16 : 24 : 32

8.

90

3. 3.25 : 1.75

Exercise 6.6

Exercise 6.3

page 55

1. 60 : 90

page 58

4. 18 : 27

1. 22 cm by 16.5 cm

5. 10 : 50

2. 28 cm by 21 cm

6. 7 : 1

3. 5 : 2

7. Orange 556 ml (3 s.f.) 8. a) 11 : 9 b) 440 boys

Water 444 ml (3 s.f.) 4. Min. 32 : 1 Max. 35 : 1 5. a) i) 28 cm2 ii) 112 cm 2

360 girls

6. a) i) 9 cm

9.

2

2

ii) 81 cm

b) 9 : 1

10. 32 cm

7. a) i) 30 cm3 ii) 240 cm

11. 4 km and 3 km

8. a) i) 64 cm3 ii) 1728 cm

12. 40°, 80°, 120°, 120°

9. a) 16 cm

13. 45°, 75°, 60°

28 yr old $466 667

a)i) 12 h b) i) 16 a) 30 rows

5.

6 h 40 min

7.

40

23

30

120

90

50

10

4

1

1 –3

1

2 –5

2

12

iii) 48 h iii) 48

5. a) 2 km b) 48 cm 6. a) 26 litres of petrol and 4 litres of oil b) 3250 ml 7. a) 1 : 40 8. Girl $1040

b) 42 chairs

b) 13.75 cm

Boy $960

9. 24°, 60°, 96°

4

10. a) 15 s

18 h

11. 6 h

b) 8 copiers

12. 2

page 57

a) 160 d) 110

b) 250 e) 225

c) 175 f) 128

a) 93 d) 157

b) 116 e) 154

3.

a) 40 d) 36

4.

a) 22 d) 5

5.

50

2.

page 59

b) 6 km/h

3. 1200 g

ii) 4 h ii) 3

Exercise 6.5 1.

c) 1 : 4

4. 200 g Speed (km/h) 60

4. 6.

b) 4 cm

b) 27 : 1

2. 16 cm and 14 cm

page 56

4

Time (h)

3.

2

1. a) 15 km

Maria $3500 Ahmet $2500

Exercise 6.4 2.

b) 8 : 1 3


32 yr old $533 333

1.

3

10. Student’s own answer.

14. 24 yr old $400 000 15. Alex $2000

2

b) 4 : 1


page 60

1. a) 30 km

b) 30 km/h

c) 80 f) 85

2. a)

b) 45 cm

3. a) 375 g

b) 625 g

b) 50 e) 15

c) 35 f ) 52

4. a) 450 m 5. a) 1 : 25

b) 80 cm b) 1.75 m

b) 6 e) 18

c) 17 f ) 13

6. 300 : 750 : 1950 7. 60°, 90°, 90°, 120°


9

Solutions

8. 150° 9. a) 13.5 h

Exercise 7.3

b) 12 pumps

10. 53˚

a) 8 d) 1

b) 25 e) 1

c) 1 f)

2.

a) d)

b) e) 1

c) f)

3.

a) 1 d)

b) 2 e)

c) 4 f ) 10

4.

a) 12 d) 80

b) 32 e) 7

c) 225 f) 64

1

8 3 cm

53˚ 37 ˚ 2

4 cm

6 3 cm

11. a) 4 min 48 s

b) 1.6 litres/min

Exercise 7.4

12. 21.6 cm by 9 cm 7

Indices and standard form

Exercise 7.1 1.

a) 33 d) 64

2.

a) 23 d) 2

3.

4.

a) b) c) d) e) f)

4 5 3 4 7 3

32 74 4 5 3 4 7 3

5 3 4 2 4

a) 32 d) 216 g) 72

a) 36 d) 410 g) 48

2.

a) 44 d) 63 g) 43

3.

a) 5315 d)

59

10

5 3 6 2 4

5

5 3 6 2 4

c) 32 f) 33

62

43 42

b) 4 e) 4

c) 3 f) 0

2.

a) 4 d) 1

b) 1 e) 2

c) f) 0

3.

a) d)

4.

a) 3 d) 2

52 65

3.

2 2

a) 6.8 d) 7.5

4.

c) 64 f) 256

a) 3800 d) 101 000

5.

a) 6 d) 3

105 109

6.

1.44

1011 m = 1.44

2 2

c) 59 f ) 35

510

b) 3 e) 43 h) 4

66

68 c) 2 f) 8

c) f)

106 107

c) 3 f) 5

b) 4.8 107 e) 7 106

c) 7.84 1011 f ) 8.5 106

b) 7.2 108 e) 4 109

c) 8 f) 5

b) 4 250 000

c) 90 030 000

107 1013

b) 2.4 e) 1.2

8

10 1022

c) 1.4 f ) 1.8

105 107

108 107

108 km

b) 2.04 1011 c) 3.32 1011 e) 5.1 1022 f ) 2.5 1025

7.

a) 8.8 d) 4.2

8.

a) 2 d) 2

9.

a) 4.26 105 b) 8.48 109 d) 3.157 109 e) 4.5 108 g) 8.15 1010 h) 3.56 107

102 104

5

page 67

a) 6 105 d) 5.34 105

2 2

4 2

b) 7 e) 8

2.

page 63 b) 87 e) 24 h) 24

b) e)

Exercise 7.5

5

6 2 2

3

d and e

12

52

a) 2 d) 3

1.

b) 3 105 e) 2.5 106

c) 4 f) 4

106 104

c) 6.388 107 f ) 6.01 107

10

b) 648 e)

3

a) 2 d) 45 g) 43

52

b) 53 e) 63 h) 37

4

4.

b) 45 e) 62

b) 81 e) 1 000 000 h) 125 000

Exercise 7.2 1.

c) 42 f ) 51

page 65

1.

page 62 b) 25 e) 86

page 64

1.

Exercise 7.6

c) 810 6 f)

1.

3

c) 5 2 (= 2 ) f) 64 2

67

8

8

5

2.

a) 6 10 4 d) 8.8 10 a) 6.8 10 d) 8 10 9

8 4

page 68 b) 5.3 10 e) 7 10 7 b) 7.5 e) 5.7


10 10

5

c) 8.64 10 4 f) 4.145 10 4

7

c) 4.2 10 10 f) 4 10 11

11

Solutions

3.

a) 0.008 d) 0.000 010 1

b) 0.000 42

4.

a) d)

4 5

b) e)

5.

6.8 105 6.741 10 4 5.57 10 9

6.2 3.2

Exercise 7.7 1.

a) 4

b) 5

3 7 103 10 4

c) 0.0903


c) 8 f) 3

1.

a) 8 d) 4

8.414 102 5.8 10 7

2.

a) 20 700

3.

7.41 4.21

10 9 107

3.6 6.2

10 5 107

5.5 4.9

10 3 108

4.

a) 6

106 10 1

8.2 5.2

105 104

4.4

10

106 10 1

8.2 4.4

105 10 3

5.2

104

c) f)

4 5

page 70 c) 10

8 d) 3

e) 9

2.

a) 2

b) 3

c) 2

d) 2

e) 6

f) 4

3.

a) 8

b) 32

c) 27

d) 64

e) 1

f) 9

4.

a) 25

b) 8

c) 32

d) 100 e) 32

f ) 27

5.

a)

b)

c)

d)

e)

f)

6.

a)

b)

c)

d)

e)

f)

d) 1

e) 81

f) 6

Exercise 7.8 1.

a) 1

b) 7

c) 2

a) 25 b) 2

c) 2

d) 27

e) 4

f) 2

3.

a) 4

c) 64

d) 9

e) 3

f) 27

Student assessment 1 3

1. 2.

a) 2 a) 4

3.

a) 800

2

5 4

page 71 2

3 6

a) 3 d) 6

5.

a) 4

b) 9

6.

a) 7

b)

b) 6 e) 32

9

3 42

c) 2 f) 1

2.

a) 6 b)

6

2

1. 2.

4.

3

5

6

b) 2

6

6

d)

page 71

14

6

2

2

2

3. 4.

a) 27 000 a) 27 d) 33

5.

a) 5

b) 16

6.

a) 2.5

b)

c) 0.0523

108 105

108

b) 5.6

b) 125 b) 77 312 c) 26 1 e) 4 f) 22 c) 49

d) 48

c) 0

d)

1010 10 7

3

105

c) 2

4.73 1015 km correct to three significant figures (3 s.f.)

6

a) 6 10 d) 3.61 10

7

a) 8 112 000

b) 4.5 e) 4.6

10 108

page 73 3

b) 440 000

8

4.05 10 1.5 10 2

3.6 7.2

10 10

a) 1.5 107 4.8 100 b) 4.35 10

4.3 8.5 8.5 1.5

105 10 3 10 3 107

105

4

5.

a) 3 d) 6

b) 9 e) 1

6.

a) 1.2 108 d) 3.88 106

b) 1.48

7.

43.2 minutes (3 s.f.)

8.

2.84

c) 3.8 109 f ) 3 100 c) 0.000 305

2

9 101 2.1 10

3

4.35

10

4.8

100

3 4

c) 3 f) 8

1011 c) 6.73

107

1015 km (3 s.f.)


2

0.5

b) 0.001 45

b) 8 e) 5

a) 1.2 d) 2.5

4.3

d) 1

1

c)

c) 7.5 f ) 6.4


3.

1 2

8.

7

c) 5

Student assessment 2 a) 2

6

8 a) 2 d) 5

page 72

b) 7.2 10 4 e) 4.75 109

7. 6 minutes

b) 27 5

4.

1.

6.

5

b) 2 b) 6

4

7

2

5.

page 70

2.

b) 2

b) 6

f) 10

106 10 4

page 74

1.

a) 9 e) 49

b) 3 f ) 0.5

c) 3 g) 5

d) 125 h) 16

2.

a) 1 e) 2

b) 9 f) 8

c) 4 g) 1

d) 25 h) 45


11

Solutions

3.

a)

y

8

10

Money and finance

9 8

Exercise 8.1

7

1.

a) c) e) g)

2.

a) 400 c) 2.13 e) 326.09 g) 125

6 5 4 3 2 1 O

3 2 1

1

2

b) Approx. 2.9


A$31.25 ZIM$9400 L299 160 dinar

page 74

1.

$154.82

2.

$182

3.

$131

a) 2 e) 3

b) 81 f) 2

c) g) 6

d) 64 h) 32

4.

$290.50

5.

a) $195.05

2.

a) 15 e) 4

b) 4 f) 1

c) 3 g) 27

d) 3125 h) 10

6.

$137.50

3.

a)

1.

7. 8.

y

b) 3500 rupees d) 3300 rand f) 4120 yen h) US$195 b) 2.86 d) 45.45 f) 11.65 h) 115.38

Exercise 8.2

3 x

page 75

page 76

b) $132.63

525 a) 298 rand

b) 253.30 rand

20 18

Exercise 8.3

16

1. $1.80 loss

page 77

14

2. $2.88 profit

12

3.

10 8

4. $240 extra

6

5. $250 loss

4

Exercise 8.4

2

3 2 1

b) Approx.

$54.65 profit per seat

O

2.6

1

2

3 x

a) 11%

b) 25%

2.

a) 30%

b) 20%

3. Type A

30% Type B 15.4% Type C 33.3% Type C makes most profit.

4.

80%

Exercise 8.5 1. a) $72 2. a) t 12

page 78

1.

page 80

b) $420

5

b) t

7


Solutions

3. a) r

7

b) r

4

4. a) P

200

b) P

850

Time

9

5. r

4

Exercise 9.1

6. t

2

1. 08 40

7. r

4.5

2. 18 45

8. r

9.5

3. 08 25

9. $315

a) 2 h 18 min b) 1 h 24 min

4.

r

6

10.

Exercise 8.6

page 85

5. 1st: 30 min

page 82

2nd: 32 min 44 s 4th: 35 min 7s

3rd: 34 min 17 s 5th: 36 min

1.

$11 033 750

2.

$52 087.50

6.

3.

$10 368

7. 21 45

4.

1331 students

8. 11 15

5.

3 276 800 tonnes


6.

2 years

1. 09 03

7.

5 years

2.

8.

3 years

3. 49 km


a) 2880 rupees

2.

a)

3.

$122.40

b) HK$83.33

€625

4. $26.16 5.

a) $72 d) 7.5%

6.

$6000

7.

30%

b)

($12.96

€192.31

$13.20)

b) 8% e) $1250

a) $82.50

2.

3.5%

3.

5 years

4.

a) $24.36

5.

a) i) $10 625

ii) $9031.25 b) 16 years

c) 4 years

4.

b) $4800

page 84 c) $2187.50

page 86

5 h 54 min

11 h 5 min


page 86

1. 08 18 2.

2 h 15 min

3.

154 km

4.

a) 3 h 30 min or 3.5 h b) 90 km/h

10


page 83

2.10 a.m. on Wednesday

Set notation and Venn diagrams

Exercise 10.1

page 88

1. a) i) Continents of the world

ii) Student’s own answers b) $2969.24

c) $9953.45

b) i) Even numbers

ii) Student’s own answers c) i) Days of the week

ii) Student’s own answers d) i) Months with 31 days

ii) Student’s own answers Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013

13

Solutions

e) i) Triangle numbers

4.

a) {a, b, p, q, r, s, t} b) A' {a, b}

5.

a) b) c) d) e) f)

ii) Student’s own answers f) i) Boy’s names beginning with the letter m

ii) Student’s own answers g) i) Prime numbers greater than 7

ii) Student’s own answers h) i) Vowels

ii) o, u 6.

i) i) Planets of the solar system

a) i)

iii) C

ii) Student’s own answers

b) i)

ii) Student’s own answers

ii) iii) iv) v) vi)

k) i) Numbers between

5 and 5 ii) Student’s own answers

a) 7 c) 7 d) 7 f) Unquantifiably finite, though theoretically 7.

infinite h) 5

1.

c) S d) T e) U

{1, 4, 9, 16, 25} {1, 3, 6, 10, 15, 21, 28}

a) B b) C c) D

{55, 60, 65} {51, 54, 57, 60, 63, 66, 69} {64}

b) R

2.

page 89

{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28} {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29} {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

a) Q

a) {p, q, r}, {p, q}, {p, r}, {q, r}, {p}, {q}, {r}, { } b) {p, q}, {p, r}, {q, r}, {p}, {q}, {r}

4.

a) True e) False

Exercise 10.3

c) True g) True

8.

{multiples of 4 from 4 to 20} B {6, 12} C {4, 8, 12} C {12} B C {12} B {2, 3, 4, 6, 8, 9, 10, 12, 14, 15} B {3, 4, 6, 8, 9, 12, 15, 16, 20} {1, 2, 4, 5, 6, 7} {3, 4, 5, 8, 9} {1, 2, 3, 4, 5, 8, 9} B {4, 5} B {1, 2, 3, 4, 5, 6, 7, 8, 9} B)' {1, 2, 3, 6, 7, 8, 9}

a) i)

W {1, 2, 4, 5, 6, 7, 9, 10} ii) X {2, 3, 6, 7, 8, 9} iii) Z' {1, 4, 5, 6, 7, 8, 10} iv) W Z {2, 9} v) W X {2, 6, 7, 9} vi) Y Z { } or

Exercise 10.4 1.

a)

a) True d) False

2.

a) A b) A c) A

B B B

{4, 6} {4, 9} {yellow, green}

3.

a) A b) A c) A

B B B

{2, 3, 4, 6, 8, 9, 10, 13, 18} {1, 4, 5, 6, 7, 8, 9, 16} {red, orange, blue, indigo, violet, yellow, green, purple, pink}

93

A

B

Libya Morocco

1.

14

A B C' A A (A b) C A

d) False h) False

page 91

b) True e) False

a) i)

{even numbers from 2 to 14} {multiples of 3 from 3 to 15}

b) Z

3.

b) True f) True

A A B A A C

ii) iii) iv) v) vi)

i) 9

Exercise 10.2

A

ii) B

j) i) Numbers between 3 and 12

2.

{1, 2, 3, 4, 5, 6, 7, 8} A' {1, 4, 6, 8} A B {2, 3} A B {1, 2, 3, 4, 5, 7, 8} (A B)' {1, 4, 5, 6, 7, 8} A B' {5, 7}

c) False f) True

Iran Egypt

Turkey

Chad

b) i) A ii) A

B B

Iraq

{Egypt} {Libya, Morocco, Chad, Egypt, Iran, Iraq, Turkey}


Solutions

2.

P

a)

Q

4.

a)

P

C

2 11

15

3

10 13

5

20

17

25

19 5

7 2

b) i) P ii) P 3.

{11, 13, 17} {2, 3, 5, 7, 11, 13, 15, 17, 19}

Q Q

A

30

B

1

2

B

6

b)

8 4

3

X

1.

a) {even numbers from 2 to 8} b) {even numbers} c) {square numbers} d) {oceans}

2.

a) 7

3.

a)

Y

a

b

d l e

g

h k

c

100


10

4.

8

b) 2

page 94

c) 6

d) 366

A

B

A

B

i m

f

j

b) Z

5.

P

Q 11 7

5 15 1

4

10

9

4.

{a, b}, {a}, {b}, { }

5.

A'

1.

Exercise 10.5 1.

a) 5

2.

45

3.

a) 10

{m, t, h}


R

page 94

b) 14

c) 13 2.

b) 50

page 95

a) {odd numbers from 1 to 7} b) {odd numbers} c) {triangle numbers} d) {countries in South America} a) 12 b) 3 c) 7 d) Student’s own answer


15

Solutions

3.

a)

4.

B

A

a) M

N 2

4

9

b)

5 10

B

A

6 7 8 1 3

4. 5.

P'

5.

{1, 3, 5, 7}


{multiples of 10}

b) X

{o, r, k}, {w, r, k}, {w, o, k}, {w, o, r}, {w, o, r, k}

a)

G

C

page 95

1.

{2, 4}, {2, 6}, {2, 8}, {4, 6}, {4, 8}, {6, 8}, {2, 4, 6}, {2, 4, 8}, {2, 6, 8}, {4, 6, 8}

2.

a)

19

7

6

J

K

2

London Paris Rome

Ankara Cairo

Washington

Nairobi

Pretoria

a)

c) {Nairobi, Pretoria}

M

N

5

3

a) 32 b) {a, e, i, o, u}, {a, e, i, o}, {a, e, i, u}, {a, e, o,

2.

a)

u}, {a, i, o, u}, {e, i, o, u}

23

7

9

11 14

13

16

page 96

1.

2 6

8

10


12

4 15

S

b) 6

b) {Ankara, Cairo}

22

13

Canberra

3.

8

X

Y

cat

29 17

lion

tiger

elephant

19 18

20

cheetah

leopard

zebra

puma

b) {2, 3, 5, 7, 11, 13, 17, 19} c) {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 23, 29}

gazelle

jaguar

anaconda tarantula mosquito Z

b) {lion, cheetah} 16


c)

d)

Solutions

3.

a) Let the number liking only cricket be x.

11

U C

F

x

3x

2x + 6

Exercise 11.1

4

b) 15 4.

Algebraic representation and manipulation

c) 5

d) 16

a) 5

b) 35

c) 40

d) 50

f ) 12

g) 10

h) 78

i) 78

a) 4x 12 c) 42x 24y e) 14m 21n

b) 10p 20 d) 6a 9b 12c f) 16x 6y

2.

a) 3x2 c) 8m2 e) 4x2

b) a2 ab ac d) 15a2 20ab f) 24p2 8pq

3.

a) 2x2 3y2 c) 7p 2q e) 3x y

4.

a) b) c) d) e)

e) 15

Topic 1 Mathematical investigations and ICT Primes and squares page 100 1.

22 + 32 = 13 42 + 52 = 41

2.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

3.

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97

4.

If a number generated by the rule 4 n + 1 is a prime, then it can be expressed as the sum of two square numbers. (This was proved by Fermat in the 17th century.) Alternatively,

page 104

1.

9xy 4mn 4xy

12r3

15rs 6rt a3 a2b a2c 6a3 9a2b p2q pq2 p2q2 m3 m2n m3n f) a6

Exercise 11.2 1.

b) a b d) 3x 4y 2z f ) 2x2 3xy

a) a c) 3p

a5 b

page 104

8 16

b) 4x 20 d) 21m 6n 2

5.

add 1 to number the result bythe 2. Ifprime the answer is and evendivide then the prime number cannot be expressed as the sum of two squares.

2.

The rule works for the numbers shown, but this does not prove that it always works.

3.

Football leagues 1. 2.

page 100

(17 + 16 + 15 + … + 1) × 2 = 306 n = t(t – 1)

ICT activity 1

ICT activity 2

y

2.

a) x2 c) x2 e) x2

2x 2x 2x

3.

a) x2 c) x2 e) x2

4.

a) x2 c) x2 e) a2

2

f) p 3p b) x 4 d) m 12 f ) 7ab 16ac b) 5x d) x f) 0

4y

5x 7x

page 101

Student’s graphs and responses will vary except: 6. The gradient of the tangent at a point represents the speed of the runner at that point.

4

a) x2 c) x2 e) x2

4. Student’s spreadsheet 5. Student’s report

a) 4x c) x e) 7x

1.

page 101

e.g. in cell C4 enter =B4/B$3*100

3 8m2 28m 2p 22 a2 6a 2

Exercise 11.3

2. Student’s spreadsheet 3.

e) a) c) e)

page 105

6 10

b) x2 d) x2 f) x2

7x 7x 5x

12 6 24

24 35 3

b) x2 d) x2 f) x2

3x 2x 2x

28 15 63

5x 6 12x 32 6x 9

b) x2 d) x2 f) x2

7x 10 6x 9 12x 35

9 64

b) x2 d) x2 f) p2

49

6

x

2

b

3c

y y

y2 q2


17

Solutions

Exercise 11.4 1.

2.

3.

4.

5.

6.

a) 2(2x 3) c) 3(2y 1) e) 3(p q)

b) 6(3 2p) d) 2(2a 3b) f ) 4(2m 3n

a) a(3b 4c c) a(a b) e) ab(c d

5d) b) 2p(4q d) 2x(2x f) f) 3m(m

a) 3pq(r 3s) c) 4xy(2x y) e) a) c) e)

a) b) c) d)

f) b) d) 3s) f)

n) s) 4p)

1.

4r)

6(7x 9) 7(2a 3b) 4(s 4t 5r) 4y(x 2y)

c) 20 f) 42

3.

a) d)

160 17

b) e)

23 189

c) 42 f) 113

4.

a) 48 d) 16

b) e)

8 5

c) 15 f) 9

5.

a) 12 d) 7

b) 5 e) 7

Exercise 11.6 1.

a) n c) n e) a

r

m

3p

2m

cd

b) m d) q

4m 2.

a) x c) x e) y

18

2p

c

d) x

9



f) x

y

5

5y

3 3

4z 

f) p

3x 7

e)

a) r

t(p  q) rt (p q)

a) d

e) b

9



b) t

3 8

7y  q

2r



n

c) p

10 2t

f) q



r

3m

n

r(p

 q)

p

n





q



p

rt 3m n rt

4z

q)

n

ab

b) a

ce ab

d) a

de a

f) c

c

dec b cd

b a

d



b

page 108 7y 6 b) 3y2 25y 28 17y 8 d) 4y2 6y 2 23y 20 f) 18y2 15y 3

1.

a) 2y c) 2y2 e) 6y2

2.

a) 2p2 13p 24 c) 6p2 p 12 e) 18p2 2

b) 4p2 23p 35 d) 12p2 13p 35 f) 28p2 44p 24

3.

a) 4x2 4x 1 c) 16x2 16x 4 e) 4x2 24x 36

b) 9x2 6x 1 d) 25x2 40x 16 f) 4x2 9

4.

a) 4x2 9 c) 16x2 9 e) 8y2 18

b) 16x2 9 d) 25y2 49 f ) 25y2 70y

Exercise 11.8 7

q

n

3m 3m

d

3r 2t

5



2

4

p

3 rt( p

6b

Exercise 11.7 2

ab

b) r

3y

5

n

7pq

c

3 3r

p

3x

d) x

4

10p

e) p

page 107

f) d

b

c) 5 f ) 36

7y 

3x

d) n

d) n

c) 14 f) 4

30 40

b) a

b) p

c) c

b) e)



4r

c) m

page 106

3 16

5

6

3m 5.

6.

b) 30 e) 13

2a

a) p

f) q

a) d)

2.

4.

2n) 3c2)

m(m2 mn n2) 2r2(2r 3 4s) 28xy(2x y) 18mn(4m 2n mn)

a) 0 d) 20

e) y

b) 3p( p 2q) d) ab(1 a b) f) b2c(7b c)

Exercise 11.5

a) b c) z

3r 2s ) 3y) 3)

b) 5m(m b) b2(2a2

12( p 3) 6(3 2y) 11x(1 y) 5q( p 2r

a) m(m c) qr( p e) p3(3

3.

page 106

49

page 109

1.

a) (x c) (3 e) (m

y)(a b) x)(m n) n)(3 x)

b) (x d) (m f) (x

y)(a b) n)(4 x) z)(6 y)

2.

a) (p c) (q e) (s

q)(r s) 4)(p 3) t)(r 2t)

b) (p d) (r f) (b

3)(q 2t)(s c)(a


4) t) 4c)

r

Solutions

3.

a) (y c) (a e) (p

x)(x 7)(b 4)(q

4.

a) (m c) (p e) (x

3)(n 4q)(r 2y)(x

4) 3) 4)

2.

3.

4.

a) (a c) (x

b)(a 5)(x

2)(x y) 1)(a 1) 5)(n 5)

2) b) (m 3r)(n 2r) 4) d) (a c)(b 1) 2z) f) (2a b)(a b)

Exercise 11.9 1.

b) (x d) (b f) (m

page 109

b) 5)

b) (m d) (m

n)(m 7)(m

e) a) c) e)

(9 x)(9 x) f) (10 y)(10 y) (12 y)(12 y) b) (q 13)(q 13) (m 1)(m 1) d) (1 t)(1 t) (2x y)(2x y) f) (5p 8q)(5p 8q)

a) b) c) d) e) f)

(3x 2y)(3x 2y) (4p 6q)(4p 6q) (8x y)(8x y) (x 10y)(x 10y) ( qr 2p)( qr 2p) (ab cd)(ab cd)

a) b) c) d) e) f)

(mn 3y)(mn 3y) y)( x y) ( x (2x 9y2)(2x 9y2) (p2 q2)(p2 q2) (p 4(m2 3y2)(m2 3y2) (4x2 9y2)(4x2 9y2) (2x

b) 9800 e) 998 000

5)(x 3) 4)(x 3) 12)(x 3)

5.

a) (x c) (x e) (x

4)(x 6)(x 7)(x

2) 5) 9)

b) (x d) (x f) (x

5)(x 6)(x 9)(x

6.

a) (2x c) e) g) i)

1)(x

(2x (3x (2x (3x

a) x

1) b) (2x

3)(x

2)

3)(x 1)(x 1)2

2) 4)

P

page 112 

p2

q2

y2

 y2

m2

e) x



n2



 p

c) x





c) x



3



2

 T

b) x

2m

d) x

a) x

4) 7) 6)

3)(x 2) d) (2x 2)(x 2) f) (3x 3)2 h) (3x 1)(2x 1)

P

y2

y2

q

b) x

Qr

 m

4

d) x



c) 2400 f) 9200

f) x



a) x

(rp)2

c) 200 f) 161

b) 86 e) 55

c) 1780 f) 5

4.

a) 72.4 d) 15

b) 0.8 e) 1 222 000

c) 65 f) 231

page 111

3.

d) x f) x

P



Qr

4)(x 3) b) (x 12)(x 1) d) (x 6)(x 2) f) (x

6)(x 2) 3)(x 4) 12)(x 1)

a) (x c) (x e) (x

5)(x 3)2 11)2

4)(x 5)2 6)(x

2) 7)

n e) x m



wr  p q

r2g

4π 2 4m2r

b) x e) x

 Pr Q

a) t

u

v

a 2

c) s e) a



wst r



mn

2

k

c) x

p 4m2r

g2

p2

p2

Exercise 11.13 1.

a) (x c) (x e) (x

b) (x d) (x f ) (x

b) (x d) (x f) (x

9y )

a) 68 d) 70

1)

3) 3) 2)

2

3.

2.

5)(x 4)(x 6)(x

2.

page 110

1.

a) (x c) (x e) (x

q)(p2 + q2)

q)(p

b) 240 e) 7600

Exercise 11.11

4.

f) x

Exercise 11.10

a) 2000 d) 3200

8)(x 3) 12)(x 3) 6)2

2

3y) (4x

2.

12)(x 2) b) (x 6)(x 4) d) (x 18)(x 2) f) (x

1.

3y)(2x

a) 60 d) 280

a) (x c) (x e) (x

Exercise 11.12

2

1.

n) 7)

3.

u2

v

2a 2(s ut) 2

t

page 113 b) u d) u





2

2as 2

s

at

 t

f) t



2(s

ut)

a


19

Solutions

2.

A

a) r

π

f

v

c) u

t

e) l

g

2

A

r2

πr

u

f) g

d)

2

t

2 7(p + 2) b) a = ± 4 + (b – 3) 3t

a) x = ±

a)

f

2π

l

Exercise 11.16 1.

fu

2

2π





d)

f

v

3.

b) h

2 s t2

2.

a) d)

Exercise 11.14 1.

page 113

3

a) 0.53 m (2 d.p.)

h

a) 66 °C (2 s.f.)

9C

c) F 3.

4.

5

a) $251.50

a) d)

2.

b) n

470

yq d c

a) m

2

d) xy2 3.

a)

2x y

d) 3x3y 4.

a)

2a 3b

d) 3xy 5.

a) d)

6. 20

b) r

8.42 m (3 s.f.)

xp

8y 3

ef

b) e)

y bd c2

b) r

5

e) abc3 b) 4q2 e) 3p b)

2y x

e) 3 b)

3

5p 2

k)

5.



3V 4π

x

2.

a) d)

r3 pq

3.

2

a) d)

3mn

c) 2

4.

f ) 6xy c)

p3 s

a) d)

5.

a) d)

9b b) 8

13 3

e)

5b 3

1 2 7

x

e) b) e)

2x

7x

b) e) b)

12 m

e)

p

6

6a

5b 15

a

6 11a 15 m

10 29x 28 p

2

m

3 3m 2 5m 2

2b

b y

e) b) e) b) e) b) e) b)

z

3 c

d

11 2x

3y 7

3 2a 3

f) c)

11 13 p2

4 a

1 2x 5 3c 1

f)

2y

x

4 5r

m

10 9x

2y 15

x

c) f) c) f)

2y

q2

5

f) c)

2p

2w 3m

n

9 5s t 15 m

2 4r 7s

page 116

b) 5a

e)

c)

7

b)

11 2a

a) 3a

6

c) 5n f)

a)

d)

r

f) p

f)

d

3p q 12 x 2y d) 12

p

c) x

c

page 115

a

Exercise 11.17 1.

c)

b)

7

a)

d)

1.5 0.05

page 114 q

c2

20x a) 3

1200(T

b) H

Exercise 11.15 1.

11 °C (2 s.f.)

d) 320 °F

a) 524 cm3 (3 s.f.)

c) n

d) 4.

32

a) 15 hours c) 5000 m

c) r 5.

b)

a)

π

c) 5.05 cm 2.

3.

V

b) r

4

15

9x

20y 36

2a 15 29x 36 r

10 23x 6 2c 3

q

5 7m 3 p

3


3b

c) 7p f) c) f)

c) f) c) f)

4q

10x

14y 35

11p 28 24x 35 x

c) f)

28

4 p

6 4x 5

w

4 m

2 36q 7

Solutions

6.

a) d)

7.

pr

p

b)

r a

ab

e)

b

f)

3.

4.

1.

q

b) 8pm 28p d) 20p3q 8p2q2 f) 22x2 14x h) x2 x b) p(p 6q) d) 3pq(3 2p

3.

a) 0 d) 7

4.

a) n

14

page 118

3x 4 a) (x 2)(x 1) 3p 7 c) (p 3)(p 2) –12 e) (y 4)(y 1) a)

a)

a)

2)(m 1) 11 d) (w 1)(w 3) 12 m f) (m 2)(m 3) y m 2 b) c) y 3 m 3 m m 1 e) f) m 4 m 2

x

2

x p

5

p x

b)

3

x x

e)

3

x x

b)

1

x x x

b)

e)

2

(m

x x

f)

2

x x x

f)

3

x

3

x

b) x(x 4y) d) 8xy(3 2x

a) q c) y e) p

b) 26 e) 12

3p

x

2mt 3 xyt

2rs

b) n d) w f) x

3m 2y

page 120

a) a



rn

d) g

p3q r2

12m 5n a) 16 9y

p2

r2



4π 2 l t2

4n2

b) x4

c)

5t 2 e) 6 3m

3 f) 2s 3r 5r

b)

4y 6r

e) 7 12 5m 13 a) (m 2)(m 3) c)

6)

5w f) x = 3

7 − 2y 3

d) 7.

 m

8x

5

b) h

2s

a) b2 d)

6.

u2

v

e) x = 5.

y

p

4.

y)

n)

n)

b) (x 4)(2 x d) (x 1) f) (3x 2)2



w(m

r(m

a) (x 16)(x 2) c) (x 6)(x 3) e) (2x 1)(x 3)

8r

x

x

m

3.

3x 5

y

3w

b) x2 14x 49 d) x2 5x 14 f) 25y2 70y 49

c) s

c) 43 f ) 12

z 5

a) x2 8x 15 c) x2 10x 25 e) 6x2 13x 8

page 119

a) 4(3a b) c) 4p2(2p q)

3

2.

2

b) 15x 27x d) 15x3y 9x2y2 f) 14m2 14m h) m2 3m

4x

(m 5n)(x 5) (2x 9y)(2x 9y) 7600 (x2 y2)(x2 y2) (x – y)(x + y)(x2 + y2)

2

7

4q)

c) 29 f ) 35

a) b) c) d)

x x

f) q

8p3

1.

1 x

c)

3

x

pqt

4m n


x x

d) y

3

e) r

3

y

b) y

10px

c) y

y

c)

4

x

4m

p

w

a) 10a 30b 15c c) 15xy2 5y3 e) 12 p g) 4x 3

a) 21 d) 7

b) 7 e) 7

7

m

page 119

a) 6x 9y 15z c) 8m2n 4mn2 e) 2x 2 g) 2

d) 25d − 54

15


2.


c) 3c − 2

d)

1.

m

n 2pq 3p

a) 8(2p q) c) 5pq(p 2q)

d) 4.

x

y

mn

2.

d) 3.

c)

b) 11b − 20

6

Exercise 11.18

2.

xy

y 2xy

a) 5a + 12

4

1.

x

c)

m

12x −5x − 2

f) b)

3 y y

3 3

x x

7


21

Solutions


a) (q r)(p b) (1 t2)(1 c) 750 000

page 121

3r ) t2) = (1

t)(1 d) 50

2.

a) x2 2x 8 c) x2 2xy y2 e) 6x2 13x 6

3.

a) (x c) (x

t2)

t)(1

b) x2 16x 64 d) x2 121 f) 9x2 30x 25

Algebraic indices

Exercise 12.1

3)

page 123

1.

a) c8 d) m3n6 uv3 g)

b) m2 e) 2a4b x2y3 z2 h)

c) b9 f) 3x3y2

2.

a) 12a5

b) 8a5b3

c) 8p6

2

7) b) (x 3)2 12) d) 3(x 2)(x

11)(x 12)(x

12

3

4 6

e) (2x 4.

a) f

3)(x

4) f) (2x

 p



A

c) p

q

e ) p = 4q 5.

6.

7.

a)

1. t

r(s − 1) 2+ r

2.

b) nq

c) y

4

3 e) n 2

2 f) 21bc 4

q

2m

b)

11 6m d)

13n

x

3p

3 b)

3

4y

e ) 4y + 9

a

b

a

b

2

f) 32m n

page 124

3

a) c

b) g

c) q–2

d) m–1 or

a) a2

b)

c) t

3x

c)

16

e ) 14 − y

30p 7x 23 a) (x 5)(x 2) 1 c)

y

5 11

e) 200(dp e) h) ab

Exercise 12.2

4

a) x d)

t=

m

ty

d) x

f)

5



2

πr

d) 16m 3n g) 24xy

5)2

b) t

m

13

16

3 5

2.

a) ( b)2

7

1 3



3

b) a

d) a 7

5

b) ( b)8

3  4

3



c) a c) ( b)–2

3 20

page 122

h V

20

5

3

6

4.

a) ( b)19

b) ( b)–7

c) ( b)14

d) ( b)–11

5.

2 a) x

−

b) 2y

1 3

−

1.

a) a3b2

2.

a) m × m × m

3.

a) a7

b) p5 × q9

4.

a) r4

b)

5.

a) p–9

b) h

3.

a) 5.39 cm (3 s.f.)


b) 3.68 m (3 s.f.)

1.

a) a 8

2.

a) ( b4)

z2

d) 0.713 m (3 s.f.) 4.

a) 4.44 s (3 s.f.) c) 2.28 m (3 s.f.)

22

b) l

T2g 4π

2

3. 4.

1

b2 b) h7

or b–2

7



9

a) a

page 126

c) b3

a) 2410 cm2 c) 10.9 cm

y2

1

b) r × r × r × r

2.

2π r

9

d)

b) d2e5

π

r

5 2

2x 3

b) r

d 2

d) a

c) 4p

a) 0.204 m3 (3 s.f.) c) r 5.40 cm

c) x

– 16

a) a

1.

A

3

d) ( b)–4

3.

12

c) a

5  2



r6

page 125

b) a

5

3

or

d) m



a) a



9

Exercise 12.3 1.

r–6

p–6

1 m p6

11  6

a) ( t )3


c)

1 n–3

b) a

d) e6

or n3

page 126

– 25

3

b) ( b–2) 19



b) a 6

15

b) ( t )–19

Solutions

13

Exercise 13.1 1.

a) x d) p

2.

a) x d) y

3. 4.

d) i) 9x

45 360 ii) iii) 35°, 55°, 120°, 150° e) i) 6x 18 180 ii) iii) 50°, 50°, 130°, 130°

Equations and inequalities page 127

4 4

b) y e) y

5 8

c) y f) x

8

b) x e) y

5 4

c) x f) m

6 10

21

b) p e) x

3 2

c) k f) y

3

b) y e) x

14 35

c) x f) p

4 20

4

a) m d) x

1

a) x d) m

6 12

5 5.5

1

5.

a) x d) x

15 8

b) x e) x

5 2.5

c) x f) x

7.5 10

6.

a) x d) x

5 5

b) x e) x

14 8

c) x f) x

22 2

7.

a) y d) y

10 5

b) x e) x

17 4

c) x f) x

13 6.5

Exercise 13.2 1.

2.

3x 60 180 ii) iii) 40°, 60°, 80° b) i) 3x180 ii) iii) 20°, 80°, 80° c)i) 18 x180 ii) iii) 20°, 50°, 110° d) i) 6x180 ii) iii) 30°, 60°, 90° e) i) 7x 30 180 ii) iii) 10°, 40°, 130° f ) i) 9x 45 180 ii) iii) 25°, 55°, 100°

x

40

x

60

x

10

x

30

x

30

x

25

a)i)

x

30

x

30

x

25

x

33

11 x 80 360 ii) x 40°, 80°, 80°, 160° 10 x 60 360 ii) x 45°, 90°, 95°, 130° 16 x 8 360 ii) x 44°, 96°, 100°, 120°

40

iii) b) i) iii) c)i) iii) d) i) iii) 3.

a)i)

iii) b) i) iii) c)i) iii)

12 x360 ii) 90°, 120°, 150° 11 x 30 360 ii) 90°, 135°, 135° 12 x 60 360 ii) 35°, 80°, 90°, 155° 10 x 30 360 ii) 33°, 94°, 114°, 119°

22

x

27

a) 20 e) 31

b) 25 f) 40

c) 14

d) 25

5.

a) 50

b) 13

c) 40

d) 40

6.

a) 5 e) 25

b) 2 f) 15

c) 7

d) 1.1

Exercise 13.3

page 133

a) x c) x e) x

4 6 5

y y y

2

2.

a) x c) x e) x

3 1 1

y y y

3.

a) x c) x e) x

5 10 4

4.

a) x c) x e) x

5 5 1

a) x c) x

4

y

3

d) x

2

y

x x x x

1 2 4 5

y y y y

1 3 6 1

f) b) d) f)

x x x x

2 5 4

y y y 3 y

a) x c) x e) x

1 4 2

y y y

0 8

1.

5.

6.

7.

e) a) c) e)

f) x

6 5 4

y y y

5 2 9

2 b) x 1 d) x 10 f) x

7 1 8

y y y

4 5 2

y y y

4 5 4

b) x d) x f) x

4 6 10

y y y

3 4

y y y

4 3 5

b) x d) x f) x

4 5

y y 3 y

2

b) x

1 d) x 2

5 y

2 b) x

1 b) x

Exercise 13.4

30

35

4.

page 129

a) i)

x

d) x f) x

2 2 3

3 y

4 7 9 10 4 3

11 3 1

y y y

8 4 1

page 135

1.

a) x c) x e) x

2 5 4

y y y

3 2 2

b) x d) x f) x

1 3 6

y y y

4 3 1

2.

a) x c) x e) x

1 3 2

y y y

4 3 3

b) x d) x f) x

5 6 2

y y y

2 1 3

3.

a) x x c) e) x

10 2

y y y

73 5

b) x x d) f) x

65 3

y y y

42 0

4.

a) x c) x e) x

1

y y y

0.5 b) x 4 d) x

2.5

y y y

4

5

f) x

1


23

Solutions

Exercise 13.5

5.

page 135

1. 10 and 7

x

1 y

4

4.

x

5 y

2

5.

60 and 20 years old

6.

60 and 6 years old

Exercise 13.6

a) 0 and 8 d) 4 and 0

7.

a) 1.5 and c) 1 and e) 1.5 and 5

8.

a) d)

a) 7 d) 7

b) 3 e) 10

c) 6 f ) 15

1.

2.

a) 7 d) 10

b) 6 e) 3

c) 6

3.

3.

a) 8 d) 6

b) 5 e) 6

c) 8

4.

a) b) c) d) e)

6, 18, 26 160, 214, 246, 246 50°, 80°, 100°, 140°, 170° 80°, 80°, 80°, 160°, 160°, 160° 150°, 150°, 150°, 150°, 120°, 120°, 120°, 120°

1.

a)

4 and

c) 1 and e) 2 and 3

3

b)

12

d) 2 and 5 f ) 2 and 4

5 and 2 2 and 7

2.

a) d)

3.

a) 3 and d) 4 and 6

4.

a) d)

page 139 2 and

2 and 5 5 and 3

c) f)

7 and 2 3 and 5

2 b) e)

3 4 and 3

c) f)

8 and 3 2 and 6

b) e)

4 and 5 7 and 9

c) f)

6 and 5 9 and 6

2 and 4 6 and 7

Exercise 13.8

page 139

a) c) e)

3 and 3 5 and 5 12 and 12

b) d) f)

4 and 4 11 and 11 15 and 15

2.

a) c) e)

2.5 and 2.5 1.6 and 1.6 and

b) d) f)

2 and 2 and and

3.

a) 4 and c) 4 and e) 2 and 5

4.

a) d)

1.

24

2 and 5 6 and 3

1 2

5 and 2 4 and 6

1

c) f)

b) d) f)

12 and 0 b) 7 and 2 e)

9 and 4 9 and 8

c) 3 and 0 f) 0 and 4

1 and 2.5 5 and 1 and

9 and 3 c) 6 and 6 f)

8 and 4 10 and 10

page 140

6 and 7

4.

4

5.

Height = 3 cm, base length = 12 cm

6.

Height = 20 cm, base length = 2 cm

7.

Base length = 6 cm, height = 5 cm

8.

a) 9x

14

50 b)

Exercise 13.10

4

x

c) 11 m

6m

page 142

1.

a) c) e)

3.14 and 4.14 6.14 and 1.14 6.89 and 1.89

b) 5.87 and 1.87 d) 4.73 and 1.27 f) 3.38 and 5.62

2.

a) c) e)

5.30 and 1.70 3.79 and 0.79 4.77 and 3.77

b) d) f)

3.

a) 0.73 and 2.73 c) 1.79 and 2.79 e) 0.38 and 2.62

b) 1.87 and 5.87 d) 3.83 and 1.83 f) 0.39 and 7.61

4.

a) 0.85 and 2.35 c) 0.14 and 1.46 e) 0.39 and 1.72

b) d) f)

Exercise 13.11 1.

page 144

4

a) x

1234

1 1234

3 and 6 6 and 8

f)

2

b) x

b) 5 and 2 d) 2 and 4 f ) 4 and 2 b) e)

b) 0 and 7 e) 0 and 9

2 c)

4 and 3

2.

6

b) e)

6 and 2

b) e)

Exercise 13.9

page 137

1.

Exercise 13.7

4 and 3 1

6.

2. 16 and 9 3.

a) d)

c) x  3 0123

d) x  7 7

8

9


10

5.92 and 5.92 1.14 and 6.14 2.83 and 2.83

1.40 and 0.90 2 and 0.5 1.54 and 1.39

Solutions


a) 9x 90 360 c) 50°, 60°, 100°, 150°

2.

2

3.

135°, 85°, 95°, 95°, 130°

page 146 50

b) x

b) 0.44 and 4.56 (2 d.p.)

5.

a)

B

x

5. 2.77 and

x

4

4 and 5

4.

C

1.27 (3 s.f.)

6.

3

7.

All values other than 0

x

c) 10 cm, 8 cm, 6 cm 14

Student assessment 5 a)

x

page 147

cm

( x  3) cm

Linear programming

Exercise 14.1 1.

a) x 2 d) y  6

2.

a) 2 y  4 c) 5  m 7

1.

54 15 cm Width

b) 1  p d) 3 x

4.

a) x b) x

5.

a)

18 x y 134 y 116

70

y

6

4 3

40

360

2 1 O

x

2 1 1

x

8

page 150

y

5

0.317 and 6.32 (3 s.f.)

b)

1 2 3

2 3

x

1

c) 5 cm, 12 cm, 13 cm

Student assessment 6 a) x

2.

b) x 85 y 55 b) 5x 100 e) 92° 82° 72°

3.

26

b) c)

y

30 x

1.

120 4x 2

y

4x 2 64 0 x

5 4

7

12 cm

a) x, x 8, x 23 b) 3x 31 134; 55, 47, 32

3.

c) x  f) p 

8

Perimeter = 54 cm

b) 4x 6 c) Length

a)

page 149

b) y  2 e) t 0

Exercise 14.2

2.

A

2

x5

23456

1.

4.

40

page 148 180

c) 92°

62°

52°

360°

4


4

5 x

2 3

Solutions

b)

e)

y

y

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

2

c)

1

1

O1 2 3 1

4

2

5 x

1

2

2

3

3

4

5 x

4

5 x

4

5 x

y

y

f)

8

8 7

7 6

6

5

5

4

4

3

3

2

2 1

1

2 1 O 1 2 3 1

4

2

5 x

1

O1 2 3

1 2

2

3

3

d)

O1 2 3

1

y

2.

8

a)

y

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

O

2 1 1 2 3

1 2 3

4

5 x

2 1 O 1 1

2 3

2 3


27

Solutions

b)

e)

y

y

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

O

2 1 1

1 2 3

4

2 1 O 1 1

5 x

2

2

3

3

c)

f)

y

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

2 1 1

1 2 3

4

5 x

1 2 3

4

5 x

O

5 x

2 11

2

2

3

3

d)

4

y

8

O

2 3

Exercise 14.3

y

page 152

8

1.

7

7

5

6

4

5

3

4

2

3

1

2 1

O

1 2 3

y

8

6

2 1 2 3

4

5 x

1 O

2 1 1

1 2 3

2 3

28


4

5 x

Solutions

2.

Exercise 14.4

y

page 153

8

a) x  5

1.

7 6

8 x

y  12

y

4

12

3

11

2

10

1 O

y

b)

5

2 1 1

2

9

1 2 3

4

8

5 x

7

2

6

3

5 4

3.

y

3

8 2

7 1

6 5

0

1

2

3

4

5

6

7

8

9 1 0 11 12

x

4

c) Any integer point in the unshaded region,

3

e.g. (5, 3) meaning 5 car trips and 3 minibus trips

2 1

2.

O

2

1

1

1 2 3

4

5 x

a) p  5

q2

q  10

p

b)

2

q

3

12 11

4.

10

y

9

8

8

7

7

6

6

5 4

5

3

4 3

2

2

1

1

O

2 11

1 2 3

4

5 x 0

1

2

3

4

5

6

7

8

9 1 0 11 12

p

2 3


e.g. (5, 2) meaning 5 loaves and 2 cakes


29

Solutions

a) m  2

3.

n  2m

n  11

m


b) n

12

1.

a) x  5

2.

a) 2

3.

a) b)

11

y4 r

page 153

b) y  3

10 3

s

b) 4

p8

9 s

r

12

s

10

12

9

11

8

10 7 6

9

5

8 7

4 3

6

2

5 4

1

3 0

1

2

3

4

5

6

7

8

9 1 0 11 12

m

2


1

e.g. (2, 4) meaning 2 long curtains and 4 short ones a) 3  L

4.

9 S

6 L

0

1

2

3

4

5

6

7

8

9 1 0 11 12

r

c) Student’s own answer

S  10

4.

b)

a) 12 b)

A

E

20 A

10 E  A

3

S E

12

20

11

18

10

16

9

14

8

12

7

10

6

8

5

6

4

4

3

2

2 0

1

0

1

2

3

4

5

6

7

8

9 1 0 11 12

c) Any of the points in the unshaded region,

e.g. (3, 0) meaning 3 large oranges and no small ones.

30

2

4

6

8 10 1 2 1 4 1 6 1 8 2 0

c) Student’s own answers

L


a) x  7

2.

a) 1  p

page 154

b) y  9

4

b) 4


x7

A

Solutions

4 y

a) x b)

3.

2 x

y8

b)

y

Position

1

2

5

10

75

n

8

Term

5

11

29

59

449

6n – 1

Position

1

3

8

50

100

n

Term

2

0

–5

–47

–97

–n + 3

Position

1

2

3

10

100

n

Term

3

0

–3

–24

–294

–3n + 6

7

c)

6 5 4 3

d)

2 1

0

1

2345678

x

e)

c) Student’s own answers a) c b) e

4.

e  40

c  16

e5

c

2e

40

Position

2

5

7

10

50

n

Term

1

10

16

25

145

3n – 5

f)

36

Position

1

2

5

20

50

n

–5.5

–7

–11.5

–34

–79

–1.5n – 4

32

Term

28 24

3.

20 16 12 8 4

0

4

8 12 1 6 2 0 2 4 2 8 3 2 3 6 4 0

c) Student’s own answers 15

1.

a) i) 3n + 2 c) i) n − 0.5 e) i) 3n − 10

ii) 20 2.

ii) 0.5 n + 5.5 iii) 30.5 f) i) −3 ii) −3 n + 75 iii) −75 page 161

2. un = n2 – 1 3. un = n2 + 5

b) i) 4n − 4 ii) 36 d) i) −3 n + 9 ii) −21 f) i) −4 n − 5 ii) −45

ii) 9.5

ii) 3 − n13 iii) 137 e) i) +4 ii) 4 − n62 iii) 138

1. un = n2 + 1

page 157

ii) 32

b) i) +1 ii) n−1 iii) 49 d) i) +0.5

Exercise 15.2

Sequences

Exercise 15.1

c

a) i) +4 ii) 4 + 1n iii) 201 c) i) +3

4. un = n2 + 8 5. un = n2 – 3 6. un = 2n2 + 2 7. un = 2n2 – 2 8. un = 3n2 + 2

a)

9. un = 4n2 – 4

Position

1

2

5

12

50

n

Term

1

5

17

45

197

4n – 3

10. un = 5n2 – 4


31

Solutions

Exercise 15.3 1.

un = n3 + 10

2.

un = n3 – n

4.

page 161

5.

3.

un = n3 – 5

4.

un = n3 + n2

5.

un = n3 + n2 + 5n

6.

un = n3 + 3n2 + 5n – 2

u5 = 27, u100 = 597 u5 = 32 , u100 = –46

a) b)



a)

Position

1

2

3

10

25

n

Term

17

14

11

–10

–55

–3n + 20

2

6

b) Position

10

80

n

1

7.

un = n3 + n2 – n

8.

un = n3 + 5n + 7

page 164

a) Geometric c) Not geometric e) Not geometric a) i) 3

2

b) Geometric d) Geometric f) Not geometric b) i)

ii) 162, 486

ii)

iii) un = 2(3)n−1 d) i) −3

1 5 1 1 25, 125

a) −6, −12, −24

b) 8

4

a) −4

5.

$38 203.20

6.

a) Because 0.8 5 ≠ 0 b) $3276.80 c) Because 0.8n ≠ 0 where n is the number of

b) 14

c) −65 536

year


b) 3 years

2.

a) $253.50

b) $8.08

3.

a) – 13

b) 243

4.

un = n3 + 3n2 + 4

5.

un = n3 + n2 + 3n + 5

16



2. 3.

f) i)81, 243

a) 4n2 d) n

b) 6n + 7 e) 10n 10

a) b)

32

2 3

i) un = 4n – 3 ii) 37 i) un = –3n + 4 ii) –26

c) 6n f) n3

3 1

c) 10

page 170

a) 3

b) 21

c) 27

d) 3

e) 10

2.

a) 0.5

b) 8

c) 24.5

d) 8

e) 16

3 8

3.

a) 24

b)

4.

a) 0.25

b) 25

a) i) y  x3 1  3 x

d) i) s

e) i) A  r

2.

10.5

3.

a) 12

4.

32

5.

a) 8

6.

75

1

page 170

ii) y = kx3

ii) s = 2

f) i) T 

d) 16

k x3

ii) t = kP

1  t 1 g

k t

ii) A = kr2 ii) T =

b) 2

b) 0.4


d) 1 1

c) 4

ii) y =

c) i) t  P

1 9

c)

Exercise 16.2

i) i) i i) ii) i)

Terms increasing by 9 Terms decreasing by 6 Terms halving Terms decreasing by 6 Descending orderof cube numbers i i) Terms multiplied by 3

page 166

1.

1.

page 165

–5

2

Variation

b) i) y

i i)45, 54 i)30, 24 i i) 2.25,1.125 i) 12, 18 i) 27, 8 i

35

a) $515.46

Exercise 16.1

3

a) b) c) d) e)

0

1.

1

iii) un = 25( 5 )n−1

ii) −243, 729 iii) un = (−3)n

1.

–2


Exercise 15.4 1.

–4

Term

n

k g

Solutions

Exercise 16.3

page 171

1.

a) h = kt2 d) 6 s

2.

a)

3.

a) l = km 3

4.

a) P = kI2

b) 5 amps

v

=k e 1



17

b) 5

c) 45 m

b) 3

c) 49 J

Exercise 17.1

b) 3

c) l = 6 cm

1.

page 172

1.

a) 1.5

b) 15

c) 3

d) 12

2.

a) 10

b) 2.5

c) 1

d) 20

1 3 1 2

d) 12

3.

a)

4.

a) 5

5.

a) 3

b) 72

c)

5

b) 4

1

c)

4


) m (k e c n ta s i D

80 60 40 20

d) 1

c) ±2

b) 3

page 175

100


1 3

Graphs in practical situations

d) ±1

0

10

20

30

40

50

60

70

Distance (miles)

page 173

a) 50 km 31 miles b) 40 miles 64 km, therefore

a) x

1

2

4

8

16

32

y

32

16

8

4

2

1

80 miles

128 km

c) 100 km/h 62 mph d) 40 mph 64 km/h 2.

b) 1.6

140

2.

a)

x

1

2

4

5

10

y

5

10

20

25

50

120 ) ˚F ( 100 re tu a r 80 e p m

b) x

1

2

4

5

10

y

20

10

5

4

2

x

4

16

25

36

64

y

4

8

10

12

16

e T

c)

60 40

3.

a) 0.8

b) 0.8

4.

a) 1.6 (1 d.p.)

b) 2

20

0

10

20

30

40

50

Temperature (˚C)

a) 25 °C

80 °F

b) 100 °F 35 °C c) 0 °C 30 °F d) 100 °F 35 °C, therefore 200 °F

70 °C


33

Solutions

3.

Approx. True

120

Exercise 17.2 a) 6 m/s d) 20 km/h

b) 4 m/s e) 160 km/h

c) 39 km/h f) 50 km/h

2.

a) 400 m d) 255 km

b) 182 m e) 10 km

c) 210 km f) 79.2 km

3.

a) 5 s b) 50 s d) 1 min 11.4 s e) 5 s f ) 4 min

100

) F (˚ re 80 u t ra e 60 p m e T

40 20

Exercise 17.3

0

10

20

30

40

1.

50

Temperature (˚C)

c) 4 min

page 176

60 50

i) a) 25 °C 77 °F b) 100 °F 38 °C c) 0 °C 32 °F d) 100 °F 38 °C, therefore 200 °F 76 °C

) 40 m ( e c 30 n ta is 20 D

ii) The rough conversion is most useful at lower temperatures (i.e. between 0 and 20 °C). 4.

page 176

1.

10

a)

10

Peak

0

8 ) ($ t s o C

2.

4

0

6

8 1 0

50 ) 40 m ( e 30 c n ta 20 is D

2468 Time (min)

b) 8 min $6.80 c) 8 min d) Extra time 1 min 20 s

4

Time (s)

2

$9.60

10 0

5.

2

4

6

8 1 0

Time (s)

120

a) 5 s

100 0 2 1 f o t u o s k r a M

2

Off Peak

6

80 60 40

b) 17.5 m

3.

a) Speed A = 40 m/s Speed B b) Distance apart 453 m

4.

a) m/s d) m

b) 6 m/s, e) 7 m

m/s

13 m/s c) 1 m/s

20

Exercise 17.4 1. a) 45 km/h b) 0

20

40

60

80

100

Percentage

a) 80 c) 54 34

67% 45%

page 178 20 km/h c) Paul has arrived at the restaurant and is waiting for Helena.

b) 110 92% d) 72 60%


Solutions

2.

Exercise 17.5 ) 150 m k ( e c 100 n a t s i D 50

0

1

2 Time (h)

3

page 180

1.

Acceleration is 0.375 m/s 2

2.


3.

Acceleration is 7 m/s 2

4.


5.

Deceleration is 0.25 m/s 2

6.

Deceleration is 1 m/s 2

3. 7.

) 30 m (k e c 20 n a t s i D 10

) s / m ( d e e p S

0

4.

1

2 Time (h)

40 30 20 10 0

2

4

6 8 Time (s)

10

12

a) s r’ o d o y F m o fr e c n a t is D

8. 4

35

) m3 (k e s u o2 h

) /s 25 m ( d 20 e e 15 p S

Fyodor

1

10 5 5

10 15 20 Time (min)

b) After 20 min a)

30

Yin

0

5.

3

50

c)

25

Distance

30 0

2

4

6

2 km

) m k ( 150 P m o 100 fr

Exercise 17.6

e c n 50 a t is D 0 18 00 18 20 18 40 19 00 19 20 19 40 20 00 Time

8 Time (s)

10

12

page 180

1.

a) 1.5 m/s2

b) 0 m/s2

c) 0.5 m/s2

2.

a) Cheetah d) 15 m/s2

b) 7.5 m/s2

c) 5 m/s2

3.

a) b) c) d) e)

0.5 m/s2 0.25 m/s2 0.104 m/s2 (3 s.f.) Travelling at a constant speed of 30 m/s Stationary

≈

b) Distance Time 18from 57 Q 79 km c) d) The 18 10 train from station Q arrives first. ≈

6.

a) a: 133 km/h b) d:100 km/h

b: 0 km/h c: 200 km/h e: 200 km/h


35

Solutions

Exercise 17.7 1.


page 182

a)

1. 10

) /s m ( d e e p S

page 184

a) 2000

8 s e 1500 e p u R 1000

6 4 2

500

0

2

4

6

8

10

Time (s)

b) 0.5 m/s2 2.

0

c) 75 m

a) Time (s)

0

Speed (m/s)

6543210

0.5

10

20

30

40

50

60

70

80

b) 50 yuan 1250 rupees c) 1600 rupees 64 yuan 1

1.5

2

2.5

3

2.

a) 80

b) 60 d n a 40 R

6

) /s 5 (m

20

d4 e e p3 S

0

2

4

6

8 1 0

km

2 1

i) 5 km: 50 rand ii) 8.5 km: 71 rand

0

0.5

1

1.5 2.0 Time (s)

2.5

b) 80 rand: 10 km

3.0

3.

a)

c) 9 m 3.

a) 1.5 m/s 2

b) 2400 m

4.

a) 390 m

b) 240 m

c) 40 s

40

B A

) 30 ($ t s o 20 C

5. 21.45 km 6. 720 m

10

7.

a) 0.37 m/s 2 d) 204 m

b) 2.16 m/s2 e) 4 m

c) 208 m 0

50

100 150 200 250 300 350 400 Units

b) If the customer uses under 200 units/

quarter then he or she should use account type B.

36


Solutions

4.

a)

2. 200

500

50

) M (D 300 t s o C 200

) m (k 150 e c n a t 100 s i D

0

400

1

2

100

3

Time (h)

0

b) 180 km 5.

a)

1

234567 Time (h)

a) b) $295

) m200 (k

3.

M m150 o fr e c 100 n ta is D 50

0

c)

≈ $408

d) 3 h

a) 4 ) m (k e c n a t is D

1

2

3

3 2 1

Time (h) 0

b) Distance from M

≈

5

10

77 km

c) Time ≈ 1 h 13 min after start


15

20

25

Time (min)

b) 25 min

page 185

4.

K

a) B, C b) B because it illustrates going back in time,

C because it illustrates infinite speed 250

Student assessment 3 200 150

page 186

1.

a) 2 m/s 2

b) 225 m

c) 10.6 s (3 s.f.)

2.

a) 4 m/s 2

b) 3 m/s2

c) 102 m

d) 9.83 s

100 50

250 200 150 100 50

a) 40 °C b) 100 K

0

50

˚C

233 K 173 °C


37

Solutions

3. a)

18

120

) h / 100 m (k 80 d 60 e e p 40 S

Graphs of functions

Exercise 18.1 1.

x

4

y

10

20 0

3

2

4

0123

1

0

2

2

0

4

3

4 x

10

1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 Time (s) y 14

b) 189 m (3 s.f.) 4.

page 190

12

a) 1 m/s2 b) 750 m c) 1237.5 m

10 8 6


2.

3.

page 187

4 2

2

a) 2.5 m/s b) 180 m c) 3.5 s (1 d.p.)

 5 4  3  2 1 0 2 4

a) 0.278 m/s 2 (3 s.f.) b) 93.8 m (3 s.f.) c) 97.2 m (3 s.f.)

6

2.

x

3

2

y

12

50

a) 160 140 120

1

34

2

3

30

2

4 3 2  1 0 2

1

2

3

4

5

6 x

4 1

2

3

4 5 6 Time (s)

7

8

6

9

8 10

b) 162.5 m a) 3.33 m/s2 (3 s.f.) b) 240 m c) 212.5 m

12 14

3.

x y

1

01

9410149


23

4 5

4

20

38

0

6

40

4.

1

y

) h / 100 m (k 80 d e e 60 p S

0

2

1

45

5 12

Solutions

6.

y

x

2

1

0

1

2

3

1

9

10

y

9

9

1

8

3

3

y

7

10

6

8

5

6

4

4

3

2

2

3 2 1 0 2

1

 2 1 0

1

2345

1234

x

4

6 x

6

4.

x

4

3

2

y

9

4

1

1

0

0

1

1

7.

2

4

x

3

2

y

15

4

1

2

6

5

0

3

3

9

9

2

5 4 3 2 1 0 2

1

2

6

3 x

3

4 6

4 3 2 1 0 3

8

6

10 12

9 12

14

15

1

2

4

3

x

18

5. 8. y

0

y

y

x

1

9

4 9

3 0

2 7

0

1 12

15

1 16

2

34

15

12

5

6

7 0

9

x

2

y

12

1 0

0 6

1 6

2

3

0

12

y 14

y 12

12

9

10

6

8

3

6

5 4 3 2 1 0 3

4 1 2 3 4 5 6 7

x

2

6 9

3 2 1 0 2

12

4

15

6

18

8

1 2 3 4

x


39

Solutions

9.

1

x y

Exercise 18.3

0123

7

4

7

2

1.

11

page 192

1.6 and 2.6

2. No solution y 12

3. 2 and 4

10

4.

3.5 and 2.5

8

5. 0.3 and 3.7

6

6. 0 and 3.5

4

7. 8.

2

2 1 2

12 34

0.2 and 2.2 and 2

Exercise 18.4

x

1.

4

y

6

2

8

10.

page 193

x

2

1

01

y

25

9

1

2 1

9

1

3 25

4 3 2 1

O 1234

x

y

2 1

0

1

2

3

1

x

–2

2

–4 –6

2.

–8

y

–10 –12

4

–14

3

–16

2

–18

1

–20 –22

4 3 2 1 1

–24

2

–26

3

Exercise 18.2 1 and 1

2.

3 4 and 3

4. 5.

4

page 191

2 and 3

1. 3.

O 1234

2

6. 0.5 and 3 7. 8. 40

1 and 2 Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013

x

Solutions

3.

2. i)

y

3

x f(x)

2

2.5

3

ii)

1

1

0

1

2

3

3.5

4

4.5

5

5.5

2

y

6

4 3 2 1

O 1234

5

x

4

1

3 2

2

1 O

3 2 1 1

Exercise 18.5 1. i)

ii)

1

2

3 x

2

page 194

3

x

3

2

10123

f(x)

7

4

1

2

5

8

11

3. i)

4

x f(x)

y

5

3 3

2

1

1

0

1

1

3

5

2 7

12

ii)

10

5

6

4

4

3

2 O

3 2 1 2 4 6 8

y

6

8

2 1

2

3 x

1 O

4 3 2 1 1

1

2 x

2 3 4 5 6 7


41

Solutions

4. i)

3

x f(x)

2

17

1

7

ii)

0

1

6. i)

123

1

1

7

2

x f(x)

17

1

17

6

ii)

y

0

12

1

2

9

y

18 16

18

14

16

12

14

10

12

8

10

6

8

4

6

2

4 2

O

3 2 1 2

1

2

3 x

3 2 1

5. i) x

7. i) 5

f(x) 5.5

ii)

4 2

3 0.5

2

1

2

2.5

0

12

2

0.5 2

3 5.5

ii)

O

1

2

3 x

x

2

1

0

12

f(x)

16

2

0

2

16

y

y

6

20 5

16 4

12

3

8

2

4

1 O

5 4 3 2 1 1 2 3

42

O

2

3 x

2 1 4

1

8 12 16


2

x

Solutions

8. i)

3

x f(x)

4.5

ii)

2 3

1

0

12

4.5

3

1.5

3 3

10. i)

x f(x)

10.5

3

2

0.22

0.5

ii)

y

1 2

0

12

–

2

3 0.5

0.22

y

12 2

10 8 6 4

1

2 O 3 2 1 123 2

x

4 3 2 1

9. i)

ii)

x

3

2

1

0123

f(x)

1

1.5

3

–

3

1.5

1

11. i)

y

x f(x)

5 4

ii)

O 123

x

3

2

1

0

12

8.88

5.75

2

–

4

3 6.25 9.11

y

3 2

10

1

8

3 2 1 O123 1 2 3 4

x

6 4 2 O 123

3 2 1 2

x

4 6 8


43

Solutions

Exercise 18.6 1.

i)

3

2

1

0.04

0.11

0.33

x f(x)

3.

page 195

ii)

i)

f(x)

0123 1

3

x

3

9

2

3.125

3.25

ii)

27

10123 3.5

4

5

7

11

y

y

12 10

27

8

24

6

21 4 18

2

15

3 2 1 0 2

12 9

1

2

3 x

6 3

3 2 1

2.

i)

3

x f(x)

0 123

2

1

11

1

x

0 11

4.

i)

x f(x)

123 1

1

ii)

3

2

1

2.875

1.75

0.5

0123 1

3

y 12

ii)

y 10 8

2

6 4 2

1

4 3 2 1 0 2

4 3 2 1

4

3 2 1

44

0 123

x


x

6

11

Solutions

5.

i)

x f(x)

3

2

3.125

Exercise 18.7

10123

2.25

1.5

1

1

5 i)

2

page 197

1.

ii) 2

y

16

ii)

14

y

12 5

10 8

4

6 4

3

2 O

4 3 2 1 2

2

1

2

3

4 x

1

i)

yii)

2.

2

8

4 3 2 1 0

4 3 2 1

x

7 6

6.

i)

x f(x)

3

2

1

0

1

23

8.96

3.89

0.67

1

2

5

5 4

18

3

ii)

2

y 21

1

18

4

15

3

2

1 O 1

1

2

3

4 x

12 9 6

i)

3

4 3 2 1 0 3

3.

y

4 3 2 1

24 x

18

6

12

9

6

12

ii) 3

30

O 123

3 2 1 6

x

12 18 24 30


45

Solutions

i)

4.

ii) 24

y

Exercise 18.8

20 10 O

4 3 2 1 10

1. 1 2 3 4

page 198 ± 2.8

b) x

a) y

x

9

20

8

30

7

40

6

50

5

60 70

4 3

80

2

90

1

100

O

4 3 2 1 1

110

1 2 3

4 x

120

2.

i)

y ii)

5.

1.7

b) x

a) y

4

5 40 4 32 3 24 2 16 1 8 O

    4 3 2  11

1234

3

x

2

O1 2 3

x

8

24

4

32

5

6.

1

16

3

i)

2

ii) 0.7

y

3.

1.5

b) x

a) y

20 8

16

7

12

6

8

5

4

4

O

3 2 1 4

3

8 12

2 1

3 2 1

O

1 2 3

1

2

3 x

16 20

46


x

Solutions

4.

b) x

a)

0.4, 0.5, 2.4

y

9. a) 101 b) 1120 c) P

6

4500

5

4000

4

3500

3

3000

2

2500

1

2000 1500

4 3 2 1 O 1 1

2 3

4 x

1000 500

2 3

0

4

d) Approx. 4.5 hours

5

10. a)

6

0

t

P

5.

b) x

a)

t

1234567

0.7, 3

1

1000 500

b)

2

3

250

125

4567 63

31

16

8

8

9

10

4

2

1

P

y 1100

2 1

27

1000

24

900

21

800

18

700

15

600

12

500

9

400

6

300

3

200

O

100

1 2 3

4

5 x 0

6.

a) 7 cm b) 0 cm e) Approx. 5 hours

c) 5 hours

7.

a) Approx. 2.5

b)

8.

a) Approx. 4.3 c) Approx. 2.3

b) Approx. 3.3

1 2 34

5 67

8 9 10

t

c) Approx. 90 insects


47

Solutions


a)

b)

page 200

y

10

b)

8 6 4 2 O

5 4 3 2 1 2

1 2 3

x

4

2.

6 8

a) x y

7 8

6 3

5

4

0

3

1

0

2

1

3

b)

8

0

1

2

15

24

35

10

4.

a)

8

36

7

32

6

28

5

24

4

20

3

16

2

12

1

8

7 6 5 4 3 2 1

4 O

7 6 5 4 3 2 1 4

1

2

a)

x

5.

y

7 and

O

1

2

1

2

a)

y

2

3 2 1 2

O

1 2

b) x 3.

y

y

1

3 x

4 6 8

4 3 2 1

10

O

12 14 16

1

18

b) x

48

0.4 and 2.6


1 2 3

4 x

2 x

Solutions


b)

page 201

y

1

1.

O

5 4 3 2 1 1

1

x

2 3 4 5 6 7 8

2.

a)

9

x

7

6

5

y

12

6

2

4 0

3 0

2

1

0

1

2

2

6

12

20

30

4.

a)

y

15

y

b)

4 O

7 6 5 4 3 2 1 4

12 1

9

2 x

6

8

3

12

O

6 5 4 3 2 1 3

16 20

1 2 3

4 x

6

24

9

28 32

12 15

36

3.

a)

b) i) x

y

3.7 and 2.7 ii) x

1.8 and 2.8

10

5.

8

a)

y

6

2

4 2 O

3 2 1 2

1 1 2 3

4

5

6 x

4 3 2 1

4 6

O

1

2

3

4 x

1

8 10

2

12 14

b) x

2 and 1


49

Solutions


a) i) Linear ii) Quadratic b) Student’s own answer

2.

a) i)

5

x

4

3.

page 201

b) i) 5.5

a)

ii)

2.5

y

12 10

3

2

1

01

2

2

0

8

2

6

f(x)

10

4

0

ii)

4

10

4 2

y

12 10

6

8

2

10

O

1

2

ii)

4.

3

2

1

012

9.3

6.5

4

–

a)

21

3 4

18

6.5 9.3

15 12 9

8

6

6

3

4

O

2 1 3

2 O

4 x

y

y 10

3 2 1 2

3

12

x

4

f(x)

2

6

4

5 4 3 2 1 2

x

1

2 4

8

b) i)

O

4 3 2 1

1

2

1 2 3

4

5 x

6

3 x

9

4 6 8 10

0.8 and 3.3

b) x


page 202

1.

a) i) Reciprocal ii) Exponential b) Student’s own answer

2.

a) i)

x f(x)

50

1.6 and 3.1

c) x

3

2

2.9

1.8


1

0123

0.5 1

3

6

11

Solutions

ii)

b) i) 0

y

4.

12 10 8

ii) 4

a)

x

3

2

1

0.5

y

4.9

4.8

4

1

0.25 0 0.25 0.5 11

– 11

1 1

2 4

3 4.8

4.9

6

b)

4

y

2 O

3 2 1

12 1 2 3

x

10

2 4

8 6 4

b)

i)

2

x f(x)

ii)

3

2

9.0

3.9

10123 0.7

1

O

2

5

3 2 1 2

18

1

2

3 x

4

y

6

18 15

c) x

± 0.4

± 0.4 and ± 2.6

d) x

12

19

9 6

Exercise 19.1

3 O

3 2 1 3

1

2

3 x

6 9

3.

a)

Functions

y

30

page 204

1.

a) 6 e) 2

b) 10 f) 2

c) 3 g) 10

d) 5 h) 1

2.

a) 10 e) 5

b) 22 f) 18

c) 8 g) 23

d) h)

3.

a) 2 e) 1.5

b) 28 f) 12

c) 20.5 g) 34.5

d) 14 h) 13.5

4.

a) 19 e) 20

b) 26.5 f) 8

c) g)

25

7 1

4 6

d) 8.2 h) 3.5

20 15

Exercise 19.2

10

1.

a) 2 e) 0.125

b) 6.5 f) 4

c) 2.375 g) 2.5

2.

a) 4 e) 3.5

b) 9 f) 16

c) g)

3.

a) 0.5 e) 5

b) 2 f) 2.75

c) 4 g) 35

d) 0.25 h) 4.25

4.

a) 4 e) 3.5

b) 1.5 f) 0.5

c) 2.75 g) 0.375

d) 0.25 h) 0.875

5 O

5 4 3 2 1 5

10

1 2 3

x

15 20

page 204

1

d) 0.5 h) 0.7 d) h)


6

51

Solutions

Exercise 19.3 1. 2.

b) 52 f) 3.25

c) 4 g) 12

a) 70 e) 7

b) 187 f) 4

c) g)

b) 3.5 f) 2

c) 4 g) 6

3.

a) e)

14 28

4.

a) d) g)

5 6.875 6.25

5.

6.

b) 32.5 e) 7.5 h) 8.125

a) 2x

3

a) 12x2

4

24x

a) 4x c) 4x2 e)

2x 2

2.

a)

3.

52

4 3) 18

x

6

5.

a) 4.5

b) 6

6.

a) 8

b)

6x

1 1

c) 6x

2

2

1.

a) x

2.

a) 2x d) x

3.

a)

2 8x

6

6x 5x 57x

20

e)

5

x

2 4x 2 3

b) 4(x e)

5

f) 3x

2

2) 4

x

6

11

c) 0

2

c) 0.5

page 207 3 c) 2x

b) 6x

b)

1

3x

d) x

c) 8x

5

c)

2

2 7

x

3

13 20

4.

a) 1

b) 8

5.

a) 50

b)

2.9

c) 39

d)

2

c) 10

d)

1.6

page 208

1.

a) 9

b)

1

c)

16

2.

a) 5

b)

1

c)

5.5

3. 4.

a) a)

5.

a) 5

6.

4x

1

c) x

x

6

c)


1

c)

b) x

8 2 1

2x

c) c) 2

2

6 4

x

b) 0 2x b)

18

c) f)

x

6

3 5x 7 8

c) 15

3

b) 2

14


b)

a) 2(x d)

4

16

b) x

4

0.5 68

page 206

e)

d) 2x

d) h)

f) x

f) 36x2

x

b)

3

3x 2

d) x

b) 1.5

a) 4

c) 2x2

b) 16x d) 4x2

3

a) x

2 3x2

0.5

a) 3

5

3x 2 19x

Exercise 19.5 1.

b)

4.

8

4

a) 2

d)

e) 3x2

2

2. 3.

page 205

b)

b) 4

5 4

c) 5 f) 11.2

b) 4x x e)

1

a) 6

Exercise 19.7

b) 0 e) 0 h) 9

page 206

1.

d) h)

c) 0 f) 15

a) 9 d) Infinite g)

d) 27x2 f ) 12x2 3.

2 4.25

b) 24 c) 0 e) 13.5 f)  h) 21.9 (3 s.f.)

d) x 2.

d) 3 h) 5

a) 0 d) 10.5 g) 97.5

Exercise 19.4 1.

Exercise 19.6

page 204

a) 19 e) 4

a) 13

2.

a)

2

b) 1

c)

3.

a)

2

b) 0

c) 18

4.

a)

5.

a) 1

6.

32x

b)

9

x

3

2

page 208

1.

b) 4x b) 2

6

c) x

24 12 3x 10 f) 8


c) 1

2

1

Solutions

Topic 2 Mathematical investigations and ICT House of cards page 209 1.

Modelling: Stretching a spring page 210 1.

40

155

2.

8475

3.

The formula can be proved using the method of differences (see Chapter 15) as follows: He i g h to h f o u se

1

Number of cards

2

2

3

7 5

1st difference

4

15 8

5

26 11

3

2nd difference

3

1

2 c

+

Term

14 0

3

1st difference

3

4

c

c

+

+

3.

5 c

c

+

+

b

b

b

3

4

b

2

+

+

+

+

+

a

a

a

a

4

5

6 1

9

b

b

b

+

+

+

+

a

a

a

a

5

2a

7

2a

9

3a + b = 5 a + b +=c 2

therefore a = therefore b = therefore

c = 0.

This produces the rule c = 23 n2 + 12 n which factorises to c = 12 n(3n + 1).

Chequered boards

500

100

200 300 Mass (g)

400

500

40

4.

y ≈ 0.06x

5.

16.5 cm

6.

The spring is likely to snap (or exceed its elastic limit).

page 209

1.

Student’s own diagrams and results, presented in a logical table.

2.

Where either m or n is even, the number of black and white squares is given by Where both

400

Linear

0

3 2 1 2

200 300 Mass (g)

10

2a

It can be deduced that: 2a = 3

100

) 30 m (c n o i 20 s n e t x E

5 2

b

3

2nd difference

2.

b

a

10

40

Comparing with the algebraic table below: Po s i t i o n

) 30 m (c n o i s 20 n e t x E

mn . 2

m and n are odd, the number of

black and white squares differ by one. The mn – 1 number of black squares is and the

2 mn + 1 number of white squares is , assuming 2

that the bottom right-hand corner is white.


53

Solutions

ICT activity 1 1.

page 211

20

y

Geometrical vocabulary

4

Exercise 20.1 3

1.

2

1

2

2.

a) and f) – hypotenuse right-angle b) and c) – ASA d) and e) – SAS g) and h) – SSS

Exercise 20.2

O 4

x

el gn at ce R

4

2

O

2

3.

4

x

y

4 3 2 1

4

2

O

2

4

x

1.

x ≈ 2.7

2.

x ≈ 2.6

3.

x ≈ 2.1

4.

x ≈ 0.6

54

m ar go le ll raa P

er a uq S

s ub m o h R

et i K

la re ta ilu q E

Opposite sides equal in length

Yes

Yes

Yes

No

Yes

n/a

All sides equal in length

No

Yes

No

No

Yes

Yes

All angles right angles

Yes

Yes

No

No

No

No

Both pairs of opposite sides parallel

Yes

Yes

Yes

No

Yes

n/a

Diagonals equal in length

Yes

Yes

No

No

No

n/a

Diagonals intersect at right angles

No

Yes

No

Yes

Yes

n/a

Allanglesequal

ICT activity 2

page 218

1.

y

2

page 216

Yes

Yes

No

page 211

Exercise 20.3

page 219

Student’s own diagrams


No

No

Yes

leg na ir t

Solutions


a) Acute b) Obtuse d) Right angle

page 220

a) Student’s own construction b)

3.

c) Reflex

2. Student’s own diagram 3.

a) Obtuse scalene triangle b) Right-angled scalene

A

36 m

B

4. Student’s own diagram 5.

All sides equal in length Both pairs of opposite sides are parallel Diagonals intersect at right angles

20 m

6. Student’s own diagram 21

Geometrical constructions and scale drawings

Exercise 21.1

D

page 222

1.

Student’s own construction

2.


3. 4.

C

c) Approx. 23 m 4.

a) Student’s own construction b) Approx. 24 m2


5.

a) Student’s own construction b) Approx. 12 km

a) Student’s own construction attempt b) It is not possible as AC BC < AB

6.

a) Student’s own construction

Exercise 21.2 1.


2.


Exercise 21.3

Y

50 m X

20

40

m

m

page 226

1.


2.


Exercise 21.4

b)

page 223

W

30 m Z

page 228

c) Approx. 16 m

1.

a) 300 m d) 416 m

b) 250 m

c) 300 m

2.

a) 10 cm

b) 8 cm c) 6 cm d) 6.8 cm



2. 3.

Student’s own construction a) Student’s own construction b) Approx. 53 m2

page 229


55

Solutions


Exercise 22.3

page 230

1.


2.

a) 22 cm

3.

a) Student’s own construction b) Approx. 23 cm2

a) i) 8 m ii) 4.8 m c) 252 cm2

b) 22 m

2.

a) 220 cm2 d) 3125 cm3

b) 1375 cm2

c) 200 cm3

3.

a) 54 cm 2

b) 3

c) 729 cm3

b) 24.4 cm

Similarity

22

4.

a) 1 : n

5. 112 cm

Exercise 22.1

page 231

3.75 cm

b) 9 cm

4. p

4.8 cm

5. e

10 cm f 2

6. a) 10 cm

4.5 cm r

q

b) 1 : n

3

7.5 cm

2 cm 2

2.

a) 1 : 8

3. 16 cm

b) 1 : 7

3

4.

a) Not similar. Student’s own reasons b) 1 : 3

5.

a) 16 cm 2

b) 64 cm 2 b) 6 cm2

b) 1.6

c) 25.6 cm

6.

a) 30 km 2

7. a) 10 cm

b) 2.5

c) 150 cm 2

7.

a) 10 cm

8. a) 33 cm2

b) 6 cm

10. No, as the corresponding angles may not be

the same. 11. No, as, despite the corresponding angles being

the same, the slanting side lengths may not be in the same ratio as the horizontal sides.

1. 50 cm

2

2. 10 cm

2

3.

page 234

a) i) 456 cm (3 s.f.) ii) 90 cm b) Triangle I

4. 43.56 cm

2

5. 56.25 cm

2

b) 100 g

page 237

1. A and C H 2 2. 1 : h 3.

a) Yes. Student’s own explanation b) 5 cm c) 8 cm d) 6 cm

4.

15 m

5.

a)

2

(3 s.f.)

2

iii) 40 cm

6 cm

2

10.8 cm

b) 233.28 cm3 c) 250.56 cm2 6.

a) 12.8 cm3

7. 1250 cm

b) 880 cm2

2

8. 27 000 cm

56

30 cm

c) 144 cm2

3.6 cm 2

6. 18.1 cm

20 cm


9. 9.6 cm

Exercise 22.2

page 236

1. 20 cm

2. A, C and F are similar. B and D are similar. 3. a) 6 cm

3

Exercise 22.4

and 90°. 6.25 cm y

2

6. 0.64 litres

1. a) Interior angles are the same, i.e. 60°, 30° b) c) x

page 235

1.

3


c) 35.2 cm2

Solutions


a) 5

2.

a) 1 :

3.

x

H

2

a) 1000 cm

Proof

5.

11.8 mm (3 s.f.)

cm z

6.4 cm


b) 600 cm2

3 2

b) 12 m

3

ii) ii) ii) ii) ii) ii)

3 2 4 4 Infinite Infinite

b) 2 e) Infinite h) 4

Exercise 23.2

page 242

1.

a) Isosceles c) 50° e) 3.58 cm (3 s.f.)

2.

a) True

b) True

3.

a) False

b) True

Exercise 23.3 1.

a) 70°

2.

a) 10.9 cm (3 s.f.) b) 9.33 cm (3 s.f.) c) 3.48 cm (3 s.f.)

b) Perpendicular bisector d) 50° f) 9.33 cm (3 s.f.) c) False

page 243

b) 72°

a)b)c) Student’s own diagrams

2.


3.

Proof

4.

a) 24°

5.

88.0 cm (3 s.f.)

c) 21°

b) 9.14 cm (3 s.f.)

Angle properties

Exercise 24.1

c) 3 f) Infinite

d) True

page 244

1.

24

page 241

Student’s planes Student’s planes Student’s planes Student’s planes Student’s planes Student’s planes

a) 2 d) 4 g) Infinite

a)b) Student’s own diagrams

50°

g) h) i) i) Student’s Student’s planes planes ii) ii) Infinite 9 2.

2.

4.

Symmetry a) i) b) i) c) i) d) i) e) i) f) i)


3 41 5

page 243

1.

3.

3

Exercise 23.1 1.

3

H


h

a) 4 m by 3 m

8. 3200 cm 23

y

7.5

(3 s.f.)

6. 18.75 cm 7.

b) 1 :

41 cm

4. 156 cm 5.

2

h

page 239

4.5 y

b) x

page 247

1.

p

54°

q

63°

2.

a

55°

b

80°

c

100°

3.

v

w

60°

x

120°

y

z

120° 60°

4.

a

50°

b

130°

c

45°

d

135°

5.

p t

45° 135°

q

135°

r

45°

s

45°

6.

d

70°

e

30°

7.

a

37°

8.

a

36°

Exercise 24.2 1.

a) 70° e) 45°

2.

a) b) c) d) e) f)

a x p d a d p s

page 248

b) 55° f ) 110°

30° 50° 130° 35° 27.5° 27.5° 45° 112.5°

60°

b y q e b e q

c) 60°

45° 80° 15° 55° 27.5° 97.5° 45°

d) 73°

z r f c

70° 60° 55° 55°

r

67.5°


57

Solutions

Exercise 24.3 1.

a

115°

2.

x

40°

3. 4.

75°

m

n

65°

s

1.

55°

2.

80°

75°

3.

90°

115°

4.

100°

5.

80°

6.

20°

7.

x

54°, y

18°

8.

x

50°, y

25°

140°

y

t

Exercise 24.8

page 250

z

u

140°

115°

5.

h

120°

i

60°

j

120°

k

60°

6.

a e

80° 140°

b

20°

c

20°

d

20°

7.

p

40°

q

130°

r

50°

8.

p t

75° 70°

q u

30° 70°

r

50° 40°

Exercise 24.4

v

s

page 253

1.

a) 720°

b) 1260°

c) 900°

2.

a) 135°

b) 90°

c) 144°

3.

a) 72°

b) 30°

c) 51.4° (1 d.p.)

4.

a) 18 e) 20

b) 10 f ) 120

c) 36

d) 8

5.

a) 5 e) 40

b) 12 f ) 360

c) 20

d) 15

6.

12

Exercise 24.5

Exercise 24.9

80°

d) 150°

page 257

page 258

1.

a

72°

2.

b

33°, c

3.

d

48°, e

32°

4.

f

30°, g

120°, h

120°, i

5.

k

55°, l

55°, m

55°, n

6.

p

65°, q

40°

66°

Exercise 24.10

page 254

80°, b

30°, j

page 259

65°

1.

a

2.

c

110°, d

98°, e

3.

f

115°, g

75°

4.

i

98°, j

60°

2.

135°

5.

l

95°, m

55°, n

85°, p

95°, q

3.

20°

4.

32°

6.

r

70°, s

120°, t

60°, u

110°

5.

110°

6.

22.5°


Exercise 24.6 1.

35°

page 255 2.

3.

40°

4.

45°

5.

24°

6.

26°

8.

8 cm

7. 13 cm 9.

a) x

54°

64°

4.

a

58

135°, b

a) m 50° b) w 55° c) a 70° e 30°

n x b

60°

90° p 70° y 110° c

40° q 55° z 110° d

140° 55° 70°

6. 72° 7. a) 90° b) 6.5 cm

125°, c

130°, d

110°, e

85°

page 259

45° 60° d

5. 360°

b) 54°, 108°, 162°, 54°, 162°

55°

135° r 60° c

4. 1260°

page 256

2. 125°, 145° 3.

2.

90°

3. 162°

17.7 cm (1 d.p.)

Exercise 24.7 1.

135° q 120° b

1.

60°

90°, k

70°

1.

a) p b) a

30°

55°

8. 58°


Solutions

9. 30°

c) iii)

TPS TQS TRS PTR PSR PTQ PSQ QTR QSR d) ii) OAX, OCY iii) DCB BAD CDA CBA

10. 25° 11. 152°

Student assessment 2 1. 2.

130° q 50° b

a) p b) a

page 261

50° 50° d

2. 3.

ABC

50°

BCD ABC

40° 65°

n x

70° p 50° y

70° q 50° z

50° 45°

130° b 40° b 70°

25° c 100° c

25° d 40° d

50° 70°

5. 6.

50° 85°

36°

50°

4.

a) m b) w c) a d) a e

130° r 50° c

3. 165°

25

4. 1800° 5. 30°

OAB

CAO

40°

28° and 56° respectively 67.5°

Loci

Exercise 25.1

6. a) 90° b) 13 cm

25°

BAD ADC

1.

page 266

8m 1m

7. 28° 8. 125° 9. 45°

6m m 1

10. 42°

m 1

11. 38°

1m

Student assessment 3 ii) i) iii) ii)

1.

a) b) c) d)

2.

a) 42°

page 262 OBA OBC DAB DCB ADC CBA DAC DBC ADB ACB CAB iii) ACB ABC

8m 2m

6m

b) 21°

3.

DAB

117°

4.

BDC

25°

5.

OQR RPQ OPR

15° ORQ PRQ 75° ORP 15°

ABC DAB

92° 115° OPQ ROP

6.

35°, 54°, 91° respectively

7.

95°, 85°, 85°, 92° respectively

a) i) DAB b) i) AOC ii) OAB, iii) ABO

DCB CBA OCB CBO

90°

8m

3.

150° 3m


2.

6m

page 264

CDA OCB

CBA OAB

AOB

COB


59

Solutions

4.

8. C is on the

8m

circumference of a circle with AB as its diameter. 6m A

B

3m

5. a)

b)

c)

d) 9. Student’s own construction 10.

a

P

c

Q

b L1

L2 G

6.

L3

Y SHIP IN

Exercise 25.2 1.

a)

page 267 b)

THIS AREA Sea

Ayshe

P

Ayshe

P

X Land

7. a) Student’s own diagrams. L, M and N will

all lie on the circumference of a circle, the centre of the circle being the point equidistant from L, M and N. b) There would be no point equidistant from all three (except in the infinite!).

60

Q

Q

Belinda

Belinda

c) Ayshe

d) P

Ayshe

P

Q

Q

Belinda

Belinda


Solutions

2.

4. Compound

5.

S

Building

45°

X

Y

3.


R

1.

page 269

P

Building

Exercise 25.3

page 268 S

1. 2. O

a) b)

O 1m

2. 1m

O

1m 2m

1m

O

3.

3.

8m

5m

2m

4.

Ship’s path is perpendicular bisector of AB.


61

Solutions

5.

2.

20 m

Scale 1 cm = 2 m 12 m

7m

5m

3.

6.

P

R

Q

P

4. O

R

Q

P


R

Q

page 270 5.

J

K

A

B

A

D

A

B

D

C

B

A

D

C

Topic 3 Mathematical investigations and ICT Fountain borders

L

None the friends can see each other, as shownofabove.

62

1.

Student’s results

2. 3.

T = 2(m + n + 2)

page 271

There are many ways to prove the algebraic rule, for example:


Solutions

The srcinal pool considered:

ICT activity 1

page 272

2.

The ratios are the same.

3.

a) The ratios remain the same. b) Ratios are still equal to each other (but

probably of a different value from Q.1 (d)). 4.

has the same number of tiles as a rectangular pool of dimensions 11 × 6 units:

The ratios change as ED no longer parallel to AB.

ICT activity 2

page 273 Student’s own demonstrations 26

Measures

Exercise 26.1

In the diagram below it can be seen that the number of tiles along the length and width of the pool is twice the length and width. This leaves the four tiles needed for the corners.

page 276

1.

a) 100 b) 1000 c) e) Millilitre

2.

a) m, cm e) cm, m i) tonne

d)

b) cm c) g d) ml f ) tonne g) litres h) km j) litres

Exercise 26.2

page 277

1.

a) mm

b) m

c) mm

d) m

e) m

2.

a) 85

b) 230

c) 830

d ) 50

e) 4

3.

a) 5.6 e) 0.012

b) 6400 c) 0.96 d) 4

4.

a) 1.15 e) 0.008

b) 250

c ) 0.5

d) 0.07

6 units

Exercise 26.3 1.

11 units

Hence T = 2m + 2n + 4 which factorises to T = 2(m + n + 2).

Tiled walls

page 272

a) 3800 e) 500

Exercise 26.4

page 278

1.

a) 4500 ml d) 1000 ml

b) 1530 ml

2.

a) 1.2 litres b) 1.34 litres d) 1.4 litres

2.

Student’s diagrams Student’s ordered table of results

Exercise 26.5

3.

c = (l – 1)(w – 1)

1.

4.

t = 2(l – 1) + 2( w – 1)

1.

page 278

b) 28.5 c) 4280 d) 0.32

a) 100 000 cm 2 c) 5 000 000 m2 e) 830 mm 2

c) 7050 ml c) 1.4 litres

page 279 b) 2 000 000 mm2 d) 3 200 000 m2


63

Solutions

2.

3.

4.

a) 0.05 m2 c) 0.001 km2 e) 0.000 25 km 2

b) 150 cm2 d) 0.04 m2

a) 2 500 000 cm3 c) 2 000 000 000 m 3 e) 30 000 000 mm 3

b) 3400 mm 3 d) 200 000 cm 3

a) 0.15 m3 c) 0.000 85 km 3 e) 0.000 015 m 3

b) 24 cm 3 d) 0.3 cm 3

Exercise 27.3 1.

4

2.

3

3. 23.5 m 4.


page 285

2

a) 16 m2, 24 m2

b) 100 m2

Exercise 27.4

c) 15

page 287

1.

a) 25.1 cm

b) 22.0 cm

c) 28.9 cm

d) 1.57 m 2 a) 50.3 cm d) 0.196 m2

b) 38.5 cm2

c) 66.5 cm2

1.

a) 26

b) 88

page 280 d) 875

2.

2.

a) 4200

b) 3.94 c) 4100 d) 720

3.

3.

a) 1.8

b) 3200 c) 83

a) 2.39 cm (3 s.f.) c) 0.637 m (3 s.f.)

b) 0.5 cm d) 1.27 mm (3 s.f.)

4.

a) 5600 mm2 b) 2 050 000 cm3

4.

a) 4.51 cm (3 s.f.) c) 3.23 m (3 s.f.)

b) 6 cm d) 4.31 mm (3 s.f.)

5.

a) 0.008 67 m 3 b) 0.444 km 3

Exercise 27.5

c) 6.8


a) 310

b) 6400

d) 250

page 280

c) 4

2.

a) 0.0036

b) 0.55

c) 6.5

3.

a) 3.4

b) 6700

c) 730

2

4. 5.

a) 30 000 mm a) 0.1004 m3

1.

a) 1.57 m (2 d.p.)

2.

188 m (3 s.f.)

3. 264 mm

d) 1.51

4.

a) 144 cm2 c) 30.9 cm2 (3 s.f.)

b) 28.3 cm2 (3 s.f.)

5.

a) 57.1 m (3 s.f.)

b) 178.3 m2 (3 s.f.)

d) 300 2

Perimeter, area and volume page 282

a) 6 cm2 d) 60 cm 2

b) 32.5 cm2 e) 108 cm2

2.

a) 64 cm 2 b) 1168 mm2 d ) 937.5 mm2

Exercise 27.2 1. 58.5 cm 2. 84 cm

1.

a) 460 cm2 d) 33.52 cm2

2.

a) 2 cm

3.

a) 101 cm2 (3 s.f.) c) 279 cm2 (3 s.f.)

4.

a) 1.2 cm d) 7.0 cm

c) 300 cm2

page 289

b) 208 cm2

b) 4 cm

c) 6 cm

a) 24 cm

2.

a) 216 cm2

d) 5 cm

c) 1.7 cm

page 289

2

1.

c) 147.78 cm2

b) 276 cm2 (3 s.f.) d) 25.6 cm2 (3 s.f.)

b) 0.5 cm

Exercise 27.7

3.

2

b) 2 cm b) 15.2 cm (3 s.f.)

2

a) 94.2 cm (3 s.f.)

b) 14 cm

4. 4.4 cm

2

3. 118.7 cm

2

4. 157.5 cm

2

64

page 284

c) 20 cm 2 f ) 55 cm 2

(3 s.f.)

Exercise 27.6

27

1.

b) 637 times (3 s.f.)

d) 46

b) 5000 m b) 0.000 005 005 km 3

Exercise 27.1

2

page 287

Exercise 27.8 1.

3

a) 24 cm d) 8.82 cm3

page 290

b) 18 cm3


c) 27.6 cm3

Solutions

2.

a) 452 cm3 (3 s.f.) c) 196 cm3 (3 s.f.)

3.

a) 108 cm3 d) 6.2 cm3

b) 140 cm3

c) 42 cm 3

4.

a) 70 cm 3 d) 137.5 cm3

b) 96 cm3

c) 380 cm3

Exercise 27.9

b) 277 cm3 (3 s.f.) d) 0.481 cm3 (3 s.f.)

a) 16 cm

2.

d) 21.5% (3 s.f.) a) 42 cm 2 b) 840 cm3

c) 3217 cm3

3. 6.3 cm 4. 2.90 m

a) 6.28 cm d) 23.6 cm

b) 2.09 cm

c) 11.5 cm

a) 32.7° (1 d.p.) c) 57.3° (1 d.p.)

b) 229.2° (1 d.p.) d) 114.6° (1 d.p.)

3.

a) 12.2 cm (3 s.f.) c) 18.6 cm (3 s.f.)

b) 4.58 cm (3 s.f.) d) 4.01 cm (3 s.f.)

page 293

1.

a) 48.8 cm (3 s.f.)

b) 105 cm (3 s.f.)

2.

a) 3.67 cm (3 s.f.) c) 68.8° (1 d.p.)

b) 49.7 cm (3 s.f.)

a) 12 cm

b) 54 cm

Exercise 27.12

page 294 b) 205 cm2 (3 s.f.) d) 44.7 cm2 (3 s.f.)

2.

a) 18.5 cm (3 s.f.) c) 1.75 cm (3 s.f.)

b) 20.0 cm (3 s.f.) d) 12.4 cm (3 s.f.)

3.

a) c)

48° 20°

b) d)

Exercise 27.13 1. 79.2 m

2

b) 0.64 cm (2 s.f.) 13.7 cm (3 s.f.) Width 11.3 cm (3 s.f.) d) 5.43 cm2 (3 s.f.)

Exercise 27.14

34° 127°

a) 905 cm3 (3 s.f.) c) 2310 cm3 (3 s.f.)

b) 3590 cm 3 (3 s.f.) d) 1.44 cm3 (3 s.f.)

2.

a) 3.1 cm c) 36.3 cm

b) 5.6 cm d) 0.6 cm

2

3

(3 s.f.)

3.

11.9 cm (1 d.p.)

4.

a) 4190 cm3 (3 s.f.)

6.

A

7.

3:2

4.1 cm, B

3.6 cm, C

Exercise 27.16

b) 39.3 cm (3 s.f.)

3.

a) 4.19 cm (3 s.f.) c) 62.8 cm3 (3 s.f.)

b) 114 cm2 (3 s.f.)

4.

a) 9.40 cm (3 s.f.)

b) 0.60 cm (2 s.f.)

3.1 cm

page 297

a) 452 cm2 (3 s.f.) c) 1890 cm2 (3 s.f.)

b) 254 cm2 (3 s.f.) d) 4 cm2

2.

a) 1.99 cm (3 s.f.) c) 3.09 mm (3 s.f.)

b) 1.15 cm (3 s.f.) d) 0.5 cm

3.

1:4

4.

a) 157 cm2 (3 s.f.) c) 707 cm2 (3 s.f.)

b) 15 cm

5.

a) 804.2 cm2

b) 5.9 cm (1 d.p.)

page 299

3

2. 133 cm

a) 118 cm (3 s.f.) c) 8.66 cm (3 s.f.)

c) 48%

1.

2

2.

b) 8000 cm3

5. 10.0 cm

1. 40 cm

(3 s.f.)

page 297

1. 6.3 cm

Exercise 27.17

page 295

page 296

1.

c) 47.7° (1 d.p.)

a) 33.5 cm2 (3 s.f.) c) 5.65 cm2 (3 s.f.)

1.

a) 20° c) Length

2. 86.7 cm

Exercise 27.11

10.6 cm (3 s.f.)

Exercise 27.15

page 293

2.

3.

5.

(3 s.f.)

Exercise 27.10 1.

Width

e) 41.7 cm2 (3 s.f.)

page 291

b) 4096 cm3

1.

3

c) 34.9 cm2 (3 s.f.) d) Length 17.1 cm (3 s.f.)

3

(3 s.f.)

3

3. 64 cm 4. 70 cm

3


65

Solutions

Exercise 27.18 1.

7 cm

2.

5 cm

3.

a) 8 cm

4.

a) 3.6 cm c) 88.7 cm

Exercise 27.23

page 299

1. c) 378 cm3

(3 s.f.)

1. 90 cm 2.

page 300

1. 6.93 cm 2 (3 s.f.) 2. 189 cm (3 s.f.) 2

4. 1120 cm

3. 9.86 cm

2

a) 693 cm (3 s.f.) c) 23.6 cm (3 s.f.)

Exercise 27.20

2

b) 137 cm (3 s.f.)

2.

a) 6.91 cm (3 s.f.) b) 10.9 cm (3 s.f.) c) 0.818 cm (3 s.f.) d) 51.3 cm (3 s.f.)

3.

a) i)7.96 cm b) i)15.9 cm c) i)6.37 cm

b) 264 cm3 (3 s.f.) d) 166 cm3 (3 s.f.)

i i)4.63 cm

Exercise 27.21

ii i) 843 cm 3 ii i) 2230 cm3 ii i) 168 cm 3 ii i) 70.7 cm

3.88 cm (3 s.f.)

2.

a) 33.0 cm (3 s.f.) c) 7.31 cm (3 s.f.)

3.

a) 2304 cm3 b) 603 cm3 (3 s.f.) c) 1700 cm3 (3 s.f.)

Exercise 27.22 2. 771 cm

3

3. 3170 cm

b) 5.25 cm (3 s.f.) d) 211 cm3 (3 s.f.)

b) 275 cm3 (3 s.f.)

page 304

a) 10.2 cm

a) 339 mm

3

b) 34 cm2

c) 101.3 cm2 2

b) 283 cm

c) 633 cm2

3

b) 9.82 cm

page 307

27.0 cm Area 58.1 cm2 b) Circumference 47.1 mm Area 177 mm2

3. 326 cm 4.

2

a) 56.5 cm2

5. 418 cm

b) 108 cm2

(3 s.f.)

c) 254.5 cm2

3

a) 1012 cm2

b) 523 cm2

page 308

1.

a) 11.8 cm (3 s.f.)

b) 35.3 cm (3 s.f.)

2.

a) 272.2° (1 d.p.)

b) 5.7° (1 d.p.)

3. 232 cm

2

(3 s.f.)

4.

a) 531 cm2

b) 1150 cm3

5.

a) 1210 cm2 (3 s.f.) b) 2592 cm3


(3 s.f.)

3

5.

2


a) 81.6 cm3 (3 s.f.) c) 8 : 27

3

2

a) 39.3 cm2

6.

page 303

1.

1. 81.8 cm

50.3 mm

1. 104 cm 2 2. a) Circumference

3

4.

201 mm2


a) 56.5 cm3 (3 s.f.) c) 1.34 cm3 (3 s.f.)

i i)12.7 cm i i)8.41 cm i i)3.97 cm

34.6 cm

95.0 cm2

4. 6.

page 303

1.

d) i)3.82 cm

a) Circumference

Area

(3 s.f.)

page 306

2

b) Circumference

(3 s.f.)

2

(3 s.f.)

Area

2

3. 73.3 cm

b) 1650 cm2 (3 s.f.)


b) 21.7 cm3 (3 s.f.) 3

Exercise 27.19

5.

2

2. 1130 cm b) 384 cm3

page 305

a) 415 cm2 (3 s.f.)

page 309

1.

a) 178 cm (3 s.f.)

b) 68.3 mm (3 s.f.)

2.

a) 143.2° (1 d.p.)

b) 2.9° (1 d.p.)

(3 s.f.) 3. 95.5 cm (3 s.f.) 4. a) 603.2 cm2 b) 1072.3 cm3 2

4.

a) 654 cm3

b) 12.5 cm

c) 2950 cm3

5.

66

a) 22.5 cm

b) 126 cm3


c) 3270 cm3

Solutions


page 309

1.

a) 22.9 cm (3 s.f.) c) 985 cm2 (3 s.f.)

b) 229 cm2 (3 s.f.) d) 1830 cm3 (3 s.f.)

2.

a) 905 cm3 (3 s.f.) c) 13.4 cm (3 s.f.)

b) 12 cm d) 958.2 cm2

3.

a) 10 cm c) 71.8 cm3 (3 s.f.) e) 41.1 cm3 (3 s.f.)

Square length (cm)

b) 82.1 cm3 (3 s.f.) d) 30.8 cm3 (3 s.f.)


page 310

Tray d i m e n s i o n s ( c m )

Vo l u m e

L eng th

Wi dt h

He i g h t

1

38

28

1

1064

2

36

26

2

1872

3

34

24

3

2448

4

32

22

4

2816

5

30

20

5

3000

6 7

28 26

18 16

6 7

3024 2912

1.

a) 3620 cm (3 s.f.) c) 905 cm2 (3 s.f.)

b) 3620 cm3 (3 s.f.) d) 1920 cm2 (3 s.f.)

8

24

14

8

2688

2.

a) 43.3 cm2 (3 s.f.)

b) 173 cm2 (3 s.f.)

9

22

12

9

2376

b) 26.9 cm

10

20

10

10

2000

11

18

8

11

1584

12

16

6

12

1152

13

14

4

13

728

14

12

2

14

336

15

10

0

15

0

3.

2

a) 314 cm (3 s.f.)

Topic 4 Mathematical investigations and ICT Metal trays page 311 1.

2.

a) length = 38 cm

width = 28 cm height = 1 cm b) 1064 cm3

4.

x = 5.7 cm

5.

Maximum volume = 3032 cm3

a) length = 36 cm

Tennis balls

width = 26 cm height = 2 cm b) 1872 cm3 3.

page 312

1.

Cuboids of the following dimensions should be considered. Note each unit represents the diameter of one tennis ball and only different combinations are considered. 1 × 1 × 12 1 × 2 × 6 1 × 3 × 4 2 × 2 × 3

2.

Total surface area of a cuboid is given by the formula: A = 2(lw + lh + wh) Total surface area of the four options are as shown (to nearest whole number):

Student’s investigation and ordered table of results similar to this one.

ns o is )st ne in u im ( D

thg ) ne cm L (

thd ) i m c W |(

th ) gi m e c( H

1×1×12

6.6

6.6

79.2

2178

1×2×6

6.6

13.2

39.6

1742

1×3×4

6.6

19.8

26.4

1655

2×2×3

13.2

13.2

19.8

1394


) ec 2 m far (c u ae S r a

67

Solutions

The optimum dimensions of the box are 13.2 cm × 13.2 cm × 19.8 cm. Cross-sections of possible designs are shown below:

3.

4.

Student’s investigations

5.

Student’s conclusion based on their calculations

ICT activity 1.

Exercise 28.3

d)

3

a) y c) y e) y

3.

a) 1) a) 1 d) 2 2) a) 1

5.

1.

2.

c) –2

3.

f) infinite

A horizontal line has a zero gradient.

3.

A vertical line has an infinite gradient.

4.

Gradient of A = 2 Gradient of B = 0 Gradient of C = –3

4.

Gradient of D = – 12 1 2

5.

Gradient of F is infinite.

Exercise 28.2 1.

a) x y d) g) y

37 x

page 321

b) y y e) h) y

c) y x f)

2 x

7x

6.

2x 7.

68

3

x

c) y

2

x

x

5 f) y

b) y d) y f) y

x 2 x 3 4x 1

b) 1 e) b) 1

1

x

c) 1 f) c) 2

The lines are parallel.

Exercise 28.4

2.

Gradient of E =

x 4 2x 2 x 2

2.

page 319

b) 2 e) 0

– 14

2x

page 322

b) y 2 e) y

4. Only the intercept c is different.

Straight-line graphs a) 1

1.

1

x

indicates where the line intersects the y-axis.

page 312

Exercise 28.1

a) y d) y

e) to the coefficient f) 4 of x. b) Thed) gradient is equal c) The constant being added/subtracted

Possible formulae are given: In cell B2: =A2/360*2*PI()*10 C2: =B2 D2: =C2/(2*PI()) E2: =SQRT((100-D2^2)) F2: =1/3*PI()*D2^2*E2

28

1.

page 325

a) c) e) g) i)

m m m m m

2 c 1 1 c 2 3 c 6 1 c 0 2 c 2

b) d) f) h)

m m m m

a) c) e) g) i)

m m m m m

3 c 1 2 c 3 c 6 1 c 2 3 c 1

b) d) f) h)

m m m m

a) c) e) g) i)

m m m m m

2 2 2 2 2

b) d) f) h)

m m m m

a) c) e) g) i)

m m m m m

2 c 3 c 10 c 9 c 2 c

b) d) f) h)

m m m m

1 c 6 1 c 4 3 c 6 c 14

a) c) e) g) i)

m m m m m

2 c 2 1 c 0 1 c 3 c 12 3 c 0

b) d) f) h)

m m m m

2 c

a) m c) m e) m

1 c 0 3 c 0 2 c

b) m d) m f) m

a) m c) m

c

3 4

c c c c c

3 c

0 1 4 1 2 2

0 0


b) m d) m

3 c

5 4

c

1

c

1 c

2 2

c

2 c

4

3 c 2 8 c 6 4

c

8 c 3 c

12 3 2

c

3 6

c

4 c 2 0 c 0 2

c

1 c c

0 4

c

1 c

4

Solutions

Exercise 28.5

page 327

1.

Any line with a gradient of 1

2.

a, b and d are parallel

3.

y = 4x

4.

a) y = –3x + 4

b) y = –3x – 2

c) y = –3x – 27 5.

a) y = 12 x + 3

b) y = 12 x – 14

Exercise 28.6 1.

a)

b)

y

c)

y

y

8

8

8

7

7

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

4 3 2  1 0 1

d)

page 328

1

2

4 3 2  1 0 1

3 x

1

2

4 3 2 1 0 1

3 x

1

2

3 x

1

2

3 x

Cambridge IGCSE Mathematical 3rd edition © Hodder & Stoughton 2013

69

2

2

2

3

3

3

4

4

4

5

5

e)

y

5

f)

y

y

8

8

8

7

7

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

4 3 2  1 0 1

1

2

3 x

4 3 2  1 0 1

1

2

3 x

4 3 2 1 0 1

2

2

2

3

3

3

4

4

4

5

5

5

Solutions

y

g)

h)

y

y

8

8

7

7

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

4 3 2 1 0 1

2.

a)

2

4 3 2  1 0 1

3 x

1 1

2

3 x

4 3 2 1 0 1

2

2

3

3

3

4

4

4

5

5

5

y

b)

8

y

c)

8

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

2

4 3 2  1 0 1

3 x

1

2

3 x

4 3 2 1 0 1

2

2

2

3

3

3

4

4

4

5

5

5

y

e)

8

y

f)

8

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1 3 x

4 3 2  1 0 1

1

2

3 x

1

2

3 x

8

7

2

3 x

y

7

1

2

8

7

1

1

y

7

4 3 2 1 0 1

70

1

2

4 3 2 1 0 1

d)

i)

8

1 1

2

3 x

4 3 2 1 0 1

2

2

2

3

3

3

4

4

4

5

5

5


Solutions

y

g)

y

y

8

8

7

7

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

4 3 2 1 0

4 3 2 1 0

h)

i)

8

1

4 3 2 1 0 1

2

2

2

3

3

3

4

4

4

5

5

1

2

3 x

1

2

3 x

1

2

3 x

1

2

3 x


71

1

3.

y

a)

2

3 x

5

y

b)

c)

y

8

8

8

7

7

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

4 3 2 10 1

d)

1

1

2

4 3 2  10 1

3 x

1

2

3 x

4 3 2 10 1

2

2

2

3

3

3

4

4

4

5

5

e)

y

5

f)

y

y

8

8

8

7

7

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

4 3 2  1 0 1

1

2

3 x

4 3 2  1 0 1

1 1

2

3 x

4 3 2 1 0 1

2

2

2

3

3

3

4

4

4

5

5

5

Solutions

g)

h)

y

7

7

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

2.

2.

72

2

3 x

4 3 2 1 0 1

1

2

4 3 2 1 0 1

3 x

2

2

3

3

3

4

4

4

5

5

5

page 330

b) (5, 2) e) ( 4, 1)

c) (2, 1) f) (4, 2)

Exercise 28.9

page 333

2x 1 2x 3 4x 2 2x 2

b) d) f) h)

1.

a) (3, 2) b) ( 1, 1) c) ( 2, 3) d) ( 3, 3) e) Infinite solutions f ) No solution

Exercise 28.8 1.

1

8

2

Exercise 28.7 a) (3, 2) d) (2, 1)

y

i)8

4 3 2  1 0 1

1.

y

8

2.

page 331

a) ii) 5.66 units (3 s.f.) iii) (3, 4) b) ii) 4.24 units (3 s.f.) iii) (4.5, 2.5) c) ii) 5.66 units (3 s.f.) iii) (3, 6) d) ii) 8.94 units (3 s.f.) iii) (2, 4) e) ii) 6.32 units (3 s.f.) iii) (3, 4) f) ii) 6.71 units (3 s.f.) iii) ( 1.5, 4) g) ii) 8.25 units (3 s.f.) iii) ( 2, 1) h) ii) 8.94 units (3 s.f.) iii) (0, 0) i)ii) 7 units iii) (0.5, 5) j)ii) 6 units iii) (2, 3) k) ii) 8.25 units (3 s.f.) iii) (0, 4) l) ii) 10.8 units (3 s.f.) iii) (0, 1.5) a) i) 4.24 units (3 s.f.) b) i) 5.66 units (3 s.f.) c) i) 8.94 units (3 s.f.) d) i) 8.94 units (3 s.f.) e) i) 4.24 units (3 s.f.) f) i) 4.47 units (3 s.f.) g) i) 7.21 units (3 s.f.) h) i) 7.21 units (3 s.f.) i) i) 12.4 units (3 s.f.) j) i) 8.49 units (3 s.f.) k) i) 11 units l) i) 8.25 units (3 s.f.)

ii) ii) ii) ii) ii) ii) ii) ii) ii) ii) ii) ii)

(2.5, 2.5) (5, 4) (4, 2) (5, 0) ( 1.5, 4.5) ( 4, 3) (0, 3) (5, 1) (0, 2.5) (1, 1) (0.5, 3) (4, 2)

a) c) e) g) i)

y y y y y



x x

3 x

3x

y y y y

x

b) y d) y f) y

x

1 4 x 4 3x 1

x

9x

13 x

h) y

2 6

Exercise 28.10 1.

2

x

a) y c) y e) y g) y i) x

1

3x

page 335

i) a) −1

ii) 1

iii)

y=x–3

i) b) −1

ii) 1

iii)

y=x–5

c) i) −2

1 2

ii)

1

iii) y = 2 x – 5

1

d) i) − 2 ii) 2

iii)

i) e) −1

ii) 1

f) i) −2

ii) 3

g) i) − 2 2

h) i) 3 i) i) − 14

i)j) −1 k) i) 0 l) i) −4

ii)

y = 2x – 20

iii) 1 2 2 3

1 2

iii) y = 3 x – 3 2

ii) −

y=x+9

iii) y = 2 x +

ii) 4

3

iii) y = − 2 x + 13 y = 4x + 28

iii)

ii) 1

iii)

ii) Infinite ii)

1 4


3 2 4 3

y=x–8

iii)

x=6 1

iii) y = 4 x –

13 4

Solutions

2.

3.

1

1

5

6.

a)

y

a) 3

b) y = 3 x + 3

c) –3

d) y = –3x + 15

e) y = –3x + 35

f) (8, 11)

g) (2, 9)

h) y = 3 x + 3

5

i) 6.3 units

j) (6, 7)

4

a) y =

–

8 7

1

2 5x

9

6

25

8 5

5

3 2

9

b) y = − 2 x – 2

1

c) 15.2 units

5 4 3  2  1 0 1

d) Midpoint AB = (4, 0)

x

4

page 337

b) –3

Solution is (2, 2) b)

5 x 2

2

a) y = 2x + 4

b) y =

3.

a) m = –3 c = 4

b) m = 3 c = 6

4.

c) m = – 2 c = 2 y = – 23 x + 6

5.

a) b) c) d)

1

1234

3

Student assessment 1 a) 1

x

2

Midpoint AC = (–3, 3) Midpoint BC = (2, 5)

1

1234

+4

y 9 8

3

7 6 5 4 3

y 9

2

8 y

7



1

2x

5 4 3  2  1 0 1

6 5

2

4

3

3 y



4

3

2 y 5



–x 2



4



Solution is (1, –1)

1 3

2





10



123

4

x

1



2



x

2



3



4




73

Solutions

c)

y 9


8

1

a) 2 b) 1

2

a) y

2x

3

b) y

3 x 2

5 2

a) m

1 2

c

0

b) m

4

c

6

c

5 2

7 6 5 4

3.

3 2 1

5 4  3 2 1 0 1

1234

x

2

3 2

4.

c) m y 5x

5.

a) b) c) d)

3

y

4

3

x

8

3x

y

page 339

7

Solution is (–2, 4)

6

d)

5

y 9

4

8

y

7

x 4

3 4 2

6 5

4

3

4

2

1

O

1

2

x

1

3

y

2

2 3

2 1

5 4  3 2 1 0 1

1234

6.

x

a)

y

8

2

7

3

6

4

5

Solution is (–2, 0) 7. a)i) 13 units

4

ii) (0, 1.5)

b) i) 10 units

3 2

ii) (4, 6)

1

8. a) y

3x

9. a) y

4

2 x 7

2x

b) y 31 7

7 x 2

b) y

7 12

2

10. a) i) (5, 6)

ii) y iii) y

O

 4 3  2 1 1

3x

3

9 1 x 3

6

Solution is (3, 3)

b) The diagonals are perpendicular as the

product of their gradients is

74

1.


1

2

3 x

Solutions

b)

7. a)i) 5 units b) i) 26 units

y

8 7

ii) (0, 4.5) ii) (–5, 2)

2x 5 9. a) y = – 27 x + 55 7

8. a) y

6 5

4x 3 b) y = – 27 x – 19 7 b) y

10. a) (1, 6)

4

b) y = 43 x + 21 4

3

c) Substitute y = 10 12 into the equation

2

y = 43 x + 21 and rearrange to find x = 7. 4

1 O

4 3 2  11

1

2

3 x

Topic 5 Mathematical investigations and ICT Plane trails page 341

2 3

Solution is (2, 3) c)

d) 7.5 units

y

8

1.

Student’s investigation

2.

Student’s ordered table similar to the one shown

7 6

Number of planes (p)

5 4

Maximum number of crossing pointsn) (

3

1

0

2

2

1

1

3

3

O

4 3 2  11

1

2

3 x

2

4

6

5

10

…

…

3

3.

Solution is (–3, –1) d)

The sequence of the number of crossing points is the sequence of triangular numbers. n = 12 p(p – 1)

y

8 7 6 5 4 3 2 1 O

4 3 2  1 1

1

2

3 x

2 3

Solution is (–1, –1) Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013

75

Solutions

Hidden treasure

page 342

29

1–2. The results for up to 20 contestants are given

in the table below:

Bearings

Exercise 29.1

1. Student’s own diagrams

Number of contestants n) (

Winning chest x()

1

1

2

2

3 4

2 4


5

2

1. Student’s own diagram

6

4


7

6


8

8

2.

Student’s own diagrams leading to:

3.

Student’s own diagrams leading to:

a) 343° a) 120°

3.

page 347

30

b) 034° b) 102°

Trigonometry

9

2

10

4

11

6

12

8

13

10

14

12

15

14

16

16

17

2

Exercise 30.2

18

4

1.

19

6

a) 2.44 cm d) 2.44 cm

b) 18.5 cm e) 43.8 cm

c) 6.19 cm f) 31.8 cm

20

8

2.

a) 38.7° d) 49.8°

b) 48.6° e) 32.6°

c) 38.1° f) 14.5°

Student’s observed pattern: key pattern is that x = n when n is a power of 2.

4.

31 contestants, winning chest is 30. 32 contestants, winning chest is 32. 33 contestants, winning chest is 2.

5.

x = 2(n – T), where x = the winning chest, n = number of contestants and T = the nearest power of 2 below n.

ICT activity

page 343

Student’s own investigation

Exercise 30.1

page 350

1.

a) 1.82 cm d) 4.87 cm

b) 4.04 cm e) 37.3 cm

c) 19.2 cm f) 13.9 cm

2.

a) 14.3 cm d) 4.10 cm

b) 8.96 cm e) 13.9 cm

c) 9.33 cm f) 6.21 cm

3.

a) 49.4° d) 63.4°

b) 51.1° e) 50.4°

c) 51.3° f) 71.6°

Exercise 30.3 1.

a) 36.0 cm d) 81.1° g) 70.5°

2.

a) 5 cm c) a) c) e)

page 353

b) 15.1 cm e) 6.7 cm h) 2.1 cm

Exercise 30.4 1.

page 352

c) 48.2° f ) 16.8 cm

page 354

b) 11.4 mm (3 s.f.)

d) 13.2 cm (3 s.f.) 12 cm 11.0 cm (3 s.f.) b) 14.8 cm (3 s.f.) 7.86 cm (3 s.f.) d) 7.35 cm (3 s.f.) 3 cm f) 13.9 cm (3 s.f.)

3. 71.6 km 76

page 348


Solutions

4. 66.9 km 5.

a) 225°

4.

135°

90°

b) 73.8 km

7.

a) 8.5 km

b) 15.5 km (3 s.f.)

8.

a) 13.3 m (3 s.f.)

b) 15.0 m (3 s.f.)

Exercise 30.5 c) 16.7 cm

b) 19.5 cm d) 42.5° b) 215.2° (1 d.p.)

a) 228 km (3 s.f.) c) 103 km (3 s.f.) e) 415 km (3 s.f.)

b) 102 km (3 s.f.) d) 147 km (3 s.f.) f) 217° (3 s.f.)

4.

a) 6.71 m (3 s.f.) c) 15.3 m (3 s.f.)

b) 19.6 m (3 s.f.)

5.

a) 48.2° (3 s.f.) c) 8 cm e) 76.0 cm2 (3 s.f.)

b) 41.8° (3 s.f.) d) 8.94 cm (3 s.f.)

Exercise 30.6

page 357

1. a) 12.2 km (3 s.f.)

b) 9.5° (1 d.p.)

2. a) 10.1 km (3 s.f.)

b) 1.23 km (3 s.f.)

3. a) 22.6° (1 d.p.)

b) 130 m

4. a) 0.342 km (3 s.f.)

b) 0.940 km (3 s.f.)

5. a) 64.0 m (3 s.f.)

b) 30.2 m (3 s.f.)

c) 53°, 127° f) 19°, 161°

page 361

1.

a) –cos 160° d) –cos 85°

b) –cos 95° e) –cos 33°

c) –cos 148° f) –cos 74°

2.

a) –cos 82° d) –cos 37°

b) –cos 36° e) –cos 9°

c) –cos 20° f) –cos 57°

3.

a) cos 80° d) cos 135°

b) –cos 90° e) cos 58°

c) cos 70° f) cos 155°

4.

a) cos 55° d) cos 82°

b) cos 73° e) cos 88°

c) cos 60° f) cos 70°

page 356

2. a) 20.8 km (3 s.f.) 3.

b) 9°, 171° e) 16°, 164°

Exercise 30.8

6. 57 009 m

1. a) 43.6°

a) 70°, 110° d) 34°, 146°


page 362

1.

a) 4 cm d) 3.91 cm

b) 43.9 cm

c) 20.8 cm

2.

a) 36.9° d) 33.8°

b) 56.3°

c) 31.0°

3.

a) 5 cm d) 28.5 cm

b) 6.63 cm

c) 9.29 cm


a) 160.8 km

2.

a) tan c)

5 x

page 363

b) 177.5 km

5

7.5

b) tan

x

7.5 (x 16)

(x

16)

d) 32 m

e) 8.9° (1 d.p.)

6. 6.93 km (3 s.f.) 3.

a) 285 m (3 s.f.) c) 297° (3 s.f.)

b) 117° (3 s.f.)

4.

a) 1.96 km (3 s.f.) c) 3.57 km (3 s.f.)

b) 3.42 km (3 s.f.)

7. a) 7.46 km (3 s.f.)

b) 3.18 km (3 s.f.)

8. a) 2.9 km (1 d.p.) c) 11.4° (1 d.p.)

b) 6.9 km (1 d.p.) d) 20.4 km (1 d.p.)

9. a) 2.68 km (3 s.f.) b) 3.5° (1 d.p.)

b) 1.02 km (3 s.f.) d) 16.83 km (2 d.p.)


b) 48.4° (1 d.p.)

1.

10. a) 225 m

Exercise 30.7

a) 4003 m

page 364

b) 2.35° (3 s.f.)

2. Student’s graph

page 360

1.

a) sin 120° d) sin 40°

b) sin 100° e) sin 52°

c) sin 65° f) sin 13°

2.

a) sin 145° d) sin 132°

b) sin 130° e) sin 76°

c) sin 150° f) sin 53°

3.

a) 19°, 161° d) 72°, 108°

b) 82°, 98° e) 13°, 167°

c) 5°, 175° f) 28°, 152°

3.

a) sin 130° d) –cos 60°

b) sin 30°

4. 5.

a) 38° a) 24.6° (1 d.p.) c) 5.94 km (3 s.f.)

c) –cos 135°

b) 106° b) 32.6° (1 d.p.) d) 16 41


77

Solutions


Exercise 31.2

page 364

1. Student’s graph

cos 52°

a) 4.71 m d) 3.06 cm

b) 12.1 cm e) 10.7 cm

c) 9.15 cm

2.

a) 125.1° d) 37.0°

b) 108.2° e) 122.9°

c) 33.6°

b) cos 100°

2.

a)

3. 4.

a) 678 m (3 s.f.) c) 718 m (3 s.f.) a) 21.8° (1 d.p.) c) 2.2 km e) 15.3° (1 d.p.)

5.

a)

b) 11.6° (1 d.p.) b) 8.5° (1 d.p.) d) 10.5° (1 d.p.) f ) 1.76 km (3 s.f.)

Exercise 31.3 1.

3. 90˚

180˚

Further trigonometry

b) 70.9 mm 2

c) 122 cm2

2.

a) 24.6° d) 63.2°

b) 13.0 cm

c) 23.1 cm

b) 8.93 cm

c) 5.96 mm

2.

d) 8.64 cm a) 33.2° d) 44.0°

b) 52.7°

c) 77.0°

a) 25°, 155° (nearest degree) b) A 10 cm

a) 3.90 m2 (3 s.f.)

b) 6.93 cm (3 s.f.)

2.

a) 5.83 cm (3 s.f.) c) 18.9° (1 d.p.)

b) 6.16 cm (3 s.f.)

3.

a) 6.40 cm (3 s.f.) c) 61.9° (1 d.p.)

b) 13.6 cm (3 s.f.)

4.

a) 75.3° (1 d.p.)

b) 56.3° (1 d.p.)

5.

a) i) 7.21 cm (3 s.f.) i i) 21.1° (1 d.p.) b) i) 33.7° (1 d.p.)

6.

20˚

b) i) 20.6° (1 d.p.)

i i) 61.7° (1 d.p.)

a) 6.5 cm c) 70.7 cm (1 d.p.)

b) 11.3 cm (3 s.f.)

8.

a) 11.7 cm

b) 7.55 cm (3 s.f.)

9.

a) TU

QU 6 cm 4 cm

i i) 68.9° (1 d.p.)

a) i) 8.54 cm (3 s.f.) i i) 28.3° (1 d.p.)

7.

B

a) 75°, 105° (nearest degree) b) P

page 372

a) 5.66 cm (3 s.f.) c) 54.7° (1 d.p.)

8 cm

B

b) 222 m3 (3 s.f.)

1.

8 cm

4 cm

2

Exercise 31.5

page 367

page 370

a) 70.0 cm2 d) 17.0 cm 2

4.

a) 8.91 cm

C

73.9 m (3 s.f.)

3. 16 800 m

1.

4.

b) 116.9° (1 d.p.) d) 33.4° (1 d.p.)

1.

45°

Exercise 31.1

3.

a) 42.9 m (3 s.f.) c) 24.6° (1 d.p.)

Exercise 31.4

1

31

page 369

e) 35.0 m (3 s.f.) 2. 370 m

1

b)

page 368

1.

TQ 10 cm 8.49 (3 s.f.)

b) 90°, 36.9°, 53.1° c) 24 cm2

40˚

Q

78

R

R


Solutions

Exercise 31.6

page 375

1.

a) RW d) WU

b) TQ e) QV

c) SQ f) SV

2.

a) JM d) HO

b) KN e) JO

c) HM f) MO

3.

a) d)

TPS RTV

b) e)

UPQ SUR

c) f)

VSW VPW

Topic 6 Mathematical investigations and ICT Numbered balls page 378 1.

If a ball is odd ( n), the next ball is n + 1. n If the ball is even (n), the next ball is 2 .

2.

65, 66, 33, 34, 17, 18, 9, 10, 5, 6, 3, 4, 2, 1 The first ball must be odd. Start at 1 and work backwards. 513, 514, 257, 258, 129, 130, 65, 66, 33, 34, 17, 18, 9, 10, 5, 6, 3, 4, 2, 1

4.

a) 5.83 cm (3 s.f.)

b) 31.0° (1 d.p.)

3.

5.

a) 10.2 cm (3 s.f.) c) 51.3° (1 d.p.)

b) 29.2° (1 d.p.)

4.

6.

a) 6.71 cm (3 s.f.)

b) 61.4° (1 d.p.)

7.

a) 7.81 cm (3 s.f.) c) 12.4° (1 d.p.)

b) 11.3 cm (3 s.f.)

8.

a) 14.1 cm (3 s.f.) c) 7.48 cm (3 s.f.)

9.

a) 17.0 cm (3 s.f.) c) 7 cm

Towers of Hanoi 1.

3

b) 8.49 cm (3 s.f.) d) 69.3° (1 d.p.)

2.

15

b) 5.66 cm (3 s.f.) d) 51.1° (1 d.p.)

4.


3. Student’s investigation

The results up to 8 discs are given below: N u m be r o f d is cs

page 377

1. 134° (3 s.f.) 2.

3. 4.

a) 11.7 cm (3 s.f.) b) 12.3 cm (3 s.f.) c) 29.1° (1 d.p.) a) 18.0 m (3 s.f.) b) 26.5° (1 d.p.) c) 28.8 m (3 s.f.) d) 278 m2 (3 s.f.) a) 12.7 cm (3 s.f.) c) 93.5 cm2 (3 s.f.)

b) 66.6° (1 d.p.) d) 14.7 cm (1 d.p.)


page 377

a) 10.8 cm (3 s.f.) c) 30.2° (1 d.p.)

b) 11.9 cm (3 s.f.) d) 41.0° (1 d.p.)

2.

a) 9.81 cm (3 s.f.) c) 19.6 cm (3 s.f.)

b) 30°

3.

a) 5.83 cm (3 s.f.) c) 7.81 cm (3 s.f.) e) 19.0 cm2 (3 s.f.)

b) 6.71 cm (3 s.f.) d) 46.6° (1 d.p.) f) 36.7° (1 d.p.)

1.

page 378

S ma ll e st n u mbe r o f moves

1

1

2

3

3

7

4

15

5

31

6

63

7

127

8

255

5.

The number of moves are 1 less than the powers of 2.

6.

1023

Number of moves = 2 n – 1, where n = number of discs. 8. Time taken to move 64 discs is 2 64 – 1 seconds This equates to 5.85 × 10 11 years, i.e. 585 billion years. 7.

Therefore according to the legend we needn’t be too worried!


79

Solutions

ICT activity 1.

page 379

ii) 0.819 (3 d.p.) iii) –0.866 (3 d.p.) or – c)

3 2

The graph of y = sin x intersects the line y = 0.7 in two places as shown. d) 30° and 150° 2.

Vectors

32

b) i) 0.940 (3 d.p.)

Exercise 32.1 1.

2 4

a)

page 383

b) 3

a)

3 2

e)

6 1

f)

6 1

g)

3

h)

5

i)

1

5

a) 4

b) 0

4

4 2 6

d)

g) 3

3.

3 c)

3 0

e)

4 2

f)

0 4

h)

2 6

i)

4 4

0 b) Two solutions c) 225°

2 4

d)

1 2.

c)

1

3.

a)

f

d

a c b

e

Solutions are 0°, 180° and 360°

g

i

h

b)

Exercise 32.2 Solutions are 38.2° and 141.8° (1 d.p.)

1.

page 384

a) b) c) d) e) f) b

a



d

a

b

b

a



a

a a

b



d

b c d



a b



c

c



b

a c

d b

80


Solutions

2.

a

3.

a) b) c) d) e) f)

b

b

a, a

d

d

a, b

c

c

b

a

b

a

d

c

b) b e) a

c) b f) a

a) 2a d) b a g) 2b

b) a e) 2(b a) h) b 2a

c) b f) 2b i) a

c c

4.

d

c

c

b

b

d

b

a) 2

b)

4 d) 1 0

e)

Exercise 32.3 d i

c e b j

a) 4

0 6

a f b k

2a g

b)

12 3 2 5

2 2

e)

g)

105

h)

3 2

c

h

2.

a) |AB| 4.0 units c) |CD| 7.2 units e) |2AB| 8.0 units

b

5.0 units | b| 7.0 units | e|

D G

b 2a b (8b – 5a)

page 389

a) i) 2a ii) 2a – b b) Proof

2.

a)b) Proofs

3.

a) i) 4a

iii) (2a + 3b)

ii) 2a iii) 2(a + b) iv) a

c) f)

c) b f) 2c

2 4 8 9

4.

a) i) (2q – p)

ii) (p – q)

b) Proof


10 6 c a

1. 2.

23

a) a

b) 7 b

4 2

6 2

c)

2

2 4

page 390

4 0

0 5

c

2 1

e

3.

4.1 units | c| 7.3 units | f |

4.5 units 6.4 units

b d

b) |BC| 5.4 units d) |DE| 13.0 units f) | CD| 7.2 units

a a

e

b

d c

e

c

e

b

c) 15.5 units f) 19.6 units

2e

2e

b

e

page 387

B 3 0 1 1

a b

b) b d) (8b 15a) f) 5a 2b h) (10a 8b)

15a)

1.

4.

3 3

a

2b

b) Proof

i)

a) 4.1 units b) 18.4 units d) 17.7 units e) 31.8 units

A

5a (8b

Exercise 32.7

page 386

|a| |d|

1.

a) c) e) g) i)

d

1.

Exercise 32.5

5 2

a

b) b e) 2c

Exercise 32.4

f)

page 385

d)

a) 2a d) a b

1 1

c)

9 0

6

3.

a) 2a d) b a

b

b

d

a

3.

2. 3.

c

2.

c) b f ) 2a i) b 2a

c

b

1.

b) a e) 2b h) b a

c

a

b

a

a) a d) b g) a b

a d c

page 388

1. d

c

4.

Exercise 32.6

E H

1 2

C 4 2 0 3

F

3 2 2 4

a)

1 5

b)

1 5

c)

1 11

d)


0 14

81

Solutions

Student assessment 2 2 7

1.

a)

2.

a) 0

5 4

c)

1 3

1.

a) b)

b)

6 0

c)

1 5

2. 3.

3 d) 3


page 390

b)

4 1

e)

3 3.

page 392

i) b Proof

ii) (4a – b)

a)

Proof

b) Proof

a) b)

i) b – a ii) a iii) a + b i) 4 : 25 i i) 20 : 25


iii) (b – a)

page 393

e

e a

c a

b

e c

d

c

2a

c

e

2a

1.

a) a

b) –b

c) (1 + 2)b

2.

a) 5b

b) 5b – a

c)

3.

a) i) a

ii) b – a

(a + 3b)

b) 2 : 3

d

33 b

4.

7 11

a)

b)

3 9

c)

6 6


d)

18 28

page 391

a) |AB | = 7.21 units b) | a| = 9.22 units | b| = 8.06 units | c| = 13 units

2.

a) 17.5 units

3.

2 A= 4

4.

b) 2.69 units

–1 B= 2 0 3

D=

Exercise 33.1 1. a) P = 2 × 3 d) S = 4 × 5

E=

C=

–3 –1

1 4

6500 900 7200 1100 7300 1040

4.

3 4 2 1 0 6 2 0 1 3 0 2

DF = BC

c)

CF = –DE

→

7.


page 392

→

2. 3.

b) Q = 2 × 4 e) T = 5 × 1

5. (8

→

b)

→

3.

6.

a) Student’s own vector →

1.

page 394

2. Student’s own matrices

→

1.

Matrices

a) | FG | = 5 units b) | a| = 6.1 units | b| = 12.4 units | c| = 14.1 units a) 29.7 units a) 2a

b) 11.4 units b) –b

c) b – a

6 9 3) 37 49 74 58 76 62 89 56 20 35 15 45 25 40 30 30 10 0 0 25 5 10 10

8.

8000 3000 5000 8000 6000 10 000 5000 11 000 9000 9000 13 000 6000

9.

6 12 43 18 6 9 6 9 15 28 18 12 12 6 12 19 30 12 9 9 9

10. Student’s own matrix

82


c) R = 4 × 2 f) F = 1 × 5

Solutions

Exercise 33.2 1.

c)

9 3 24 b) 19 35 5

24 3 18

d)

e) (–3

Exercise 33.4

page 398

14 8 a) 7 16

1.

18 78 a) 14 54

–10 13 17 2 –13 18 5 10 11

2.

a)

14 –9

3.

a)

f) 15 –5

1 –1)

–5 10 2.

3.

a) 5 4

b) 2 2 2

1 0

c)

–5 0 4

e)

–3 18 –1 –12

9 11 5 9 8 10 3 d) –3 14 3 4 5 9 g) 6 0 a) 15

a) 3 1 4

1 7 c) 4 2 5.

4 –14 –1 2 0 4 –9 –1 –5 –3 –4 f) –21 13 –3 2 9 11 –1 5 b) 15 5 c) 6 6 9 8 7 –5 10 6 11 –2 e) 9 –1 f) –9 8 2 13 –4 9 3 –1 13 –7 h) –12 12 i) –6 10 d)

–5 13 2 3 3 4.

4 4 5 7

3 4 6 6

–5 0 5 1 1 1 4 2 1 3 3

a)

d)

8 12 14 6 10 5 0 20

12 72 0 24 18 18 8 8 b) 6 6 b)

52 12 18 44 22 12

28 4 –8 –31 19 40 6 4 –2 –3 –2 1 a) –12 –8 4 18 12 –6

Exercise 33.5 0 7 24 8

1.

VW =

2.

VW =

3.

VW = (–11)

–10 –20 11 WV = –25 39

76 108 = 19 64

4 –10 0

21 6 3 0

c)

28 20 e) 16 32

9 3 0 6 21 42 7

f) 14

a)

4 1 0 2

b)

4 3 2 1

c)

3 6 0.75 1.5

d)

6 4 0.8 2

e)

2.5 7.5 7.5 10

f)

12 3 3 6

b)

22 38 13 2 7 17 19 50

page 402 –12 4 –18 20

18 4 0 0

–6 –30 15 –27 –6 12 54 12 4.

VW =

7 7 9 –6 –4 –14

5.

VW =

–33 41 –10 17 –11 –8

1.

2 1 AI = 3 2

2.

AI =

–2 –4 3 6

3.

AI =

4 8 –2 4

4.

AI =

5.

AI = (–5

page 400

b)

30 20 9 24

–9 –18 14 –1 –30 5 46 –9

WV = 0

0

b) (27 –17)

WV =

Exercise 33.6

Exercise 33.3

2.

9

b) 15 races

a) 444

265 312 189 204 b) – 140 132 121 68 c) 267

1.

4.

2 2 2

page 402

WV is not possible WV is not possible

page 403

IA =

2 1 3 2

IA = IA =

–2 –4 3 6 4 8 –2 4

3 2 1 6 –2 5 –6)

IA is not possible IA is not possible


83

Solutions

6.

7. 8.

4 –3 5 –6 3 2 1 4

IA is not possible

When AI exists, it is equal to A. For a 2 × 2 matrix AI = IA = A.

Exercise 33.7 b) 3

c) 4

d) 2

2. 3.

a) –4 a) 18

b) –10 b) 14

c) –10 c) –54

d) –1 d) 4

4.

Student’s own matrices

5.


6.


7.

a) 16 e) –104 i) 147

c) 18 g) 254

5 –7 –7 10

d) –1

e)

–1.8 –0.8 2 1

3 2

d) 936 h) –576

1 –1 –0.8 1

c)

2.

See answers for Q.1.

3.

It has no inverse as the determinant = 0.

4.

A, B, D have no inverse. – – –9 4 a) b) 7 –3 2 1

5.

d)

–

0

e)



–

f)

–

1.

a)

3 –1 –8 2

c)

–10 –7 –4 1

d) –4

e)

3 6

f)

3 –15 10 –36 26

3.

a) 1 a)

b) 66

–9 –6 –1 1

d) 1

b)

e)

6 16 –8 2 –12 14

f)

d)

d) 18

–3 – 4 67.5 –59 –75.5 66

a)

page 409

y 6 4 2

2

0

2

4

2

1 5 1 –2 0 –1.5

4 6

b) y = 2

84

–32)

Transformations

–5

5 –4 5

–2 –4

c) 66

–10 9 c) Not possible

11

c)

1 0

b) (–19

9 –8

Exercise 34.1

b) –1 –12

10

1 1 6 –3

12 –3 –12 15

page 406 6

3

– 2 –1

page 407

11 7 6 –12 b) 8 –5

34

4 –1 –7 8

a)

–

a) 42

1.


–2

6 – –11 5

c)

d) –4

8 –7 b) –9 8 – d) –

–

2.

4.

f) Not possible

c) –6

–

page 406

b)

b) 6

1

a)

b) (–83 –14)


4 –5 a) –7 9 –1

a) 1

c)

a) 3

1.

3. 4.

1.

Exercise 33.8

a) 45

page 404

b) 131 f) –576

18 –17 0 30 15

2.


6

8 1 0

x

Solutions

2.

a)

5.

y

6 4 2

6

4

4

2

2

0

2

4

6 4 2

6x

0

2

2

4

4

6

6

2

4

6x

b) y = – x – 2

a)

6.

y

2

a)

y

6

6

4

4

2

2

0

2

4

6

8 6 4 2

8 1 0

x

2

0

4

4

6

6

2

7.

2

x

y

a)

6

6

4

4

2

2

0 2

b) y = x – 2

x

b) y = 0, x = –3 y

a)

2

2

b) y = –1 4.

y

6

b) x = –1 3.

a)

2

4

6

8 1 0

x

8 6 4 2

0

2

4

4

6

6

b) y = 1, x = –3, y = x + 4, y = –x – 2


85

Solutions

8.

a)

d)

y

8 6 4 2

6

4

4

2

2 2

0

1.

a)

6 4 2

x

0

2

2

4

4

6

6

b) y = 1, x = –3, y = x + 4, y = –x – 2

Exercise 34.2

y

6

2.

a)

2

4

6x

2

4

6x

2

4

6x

2

4

6x

y

page 410

6

y

4

6 2 4

6 4 2

2

6 4 2

0

2

4

6x

4

2

6

4 6

b)

b)

6

6

4

4

2

0

6 4 2 2

4

0

2

6x

2

4

4

6

6

c)

y

y

2

6 4 2

0

2

c)

y 6

y 6

4

4

2

2

6 4 2

0

2 4

2

4

6x

6 4 2

0

2 4 6

6 86


Solutions

d)

2.

y 6 4 2

6 4 2

2

0

4

6x

2 4 6

3.

3. y 8 6 4 2

10 8 6 4 2

0

2

2

4

6

8

x

4.

y 5

4

4

6

3

8

2 1

10

0

12

1

2

3

4

5

1

6 x

14

5.

y 4

Exercise 34.3 1.

page 411

3 2 1 –4 –3 –2 –1 0 –1

1

2

3

4

x

–2 –3 –4


87

Solutions

6.

4.

y

y

a)

4

4

3

3

2

2

1

1 –2 –1 0 –1

1

2

3

4

5

–4 –3 –2 –1 0 –1

x

1

2

3

x

4

–2

–2

–3

–3

–4

Exercise 34.4 1.

b) 90° anti-clockwise about (0, 0)

page 411 5.

a)

a)

y 4 3 2 1 –1 0 –1

1

2

3

4

5

6

x

–2 –3

b) 180° clockwise/anti-clockwise 2.

–4

a)

b) 90° anti-clockwise about (2.5, 0) 6.

y 4

a)

3 2 1 –4 –3 –2 –1 0 –1

b) 90° anti-clockwise 3.

1

2

3

–2 –3

a)

–4

b) 90° clockwise about (0,

Exercise 34.5

page 413

Student’s own diagrams.

b) 180° clockwise/anti-clockwise

88


1)

4

x

Solutions

Exercise 34.6

4.

page 414

1.

A→B=

–6 0

A→C=

3 6

2.

A→B=

0 –7

A→C=

–6 1

3.

A→B=

0 6

4.

A→B=

A→C=

5

A→C=

0

Image

6 –3 –3 –6

Exercise 34.7

Object

5.

page 415

1. Image

Image

Object

Object

6. Object

2. Object

Image Image

Exercise 34.8

page 418

1. A'

3. A

Image

C

Object

C'

B B'

Scale factor is 2


89

Solutions

2.

Exercise 34.9

page 419

1. A

O

D A'

D'

B

C C'

B'

Scale factor is 1.5

3.

'

'

2.

'

'

4. C'

A'

3.

A

C

B

Scale factor is 3

5.

B'

4. D

A 1 Scale factor is – 4

O

C

90

D'

A'

C'

B' B


Solutions

Exercise 34.10

4.

page 420

A

B

1. C' D'

C'

A

B B'

D

A'

C C Scale factor is

B'

1 – 3

A' Scale factor is

3

5. B'

2.

A

B C' O

C'

C A

D

Scale factor is

B B'

A'

D'

C

2.5

A'

6. A

3.

D C' A

C'

B

B'

D'

D'

O B D

A'

C

C Scale factor is

B'

0.5

A'


91

Solutions

Exercise 34.11

3.

page 422

y

4

1.

y

10

8 6 4 2

8

D'

C'

4

A'

B'

2

A

B

A"

4.

0

2 4 6 8 D 2 C"

R'

S'

y

4

C

P'

8 6 4 2

10

C'

6

D'

2 4 6 8 A"

0 D"

2

5. x

0

6

6.

y

Q' 4

S

2 4 6 8

10 x

10 8 6 4 2 S'

P'

0

2 4

y

R' Q

R

P

S

Q'

6

2

Q'

0 P' 2

R'

S'

2 4 6 8

10 x

4

92

Q

R

P

S

2

6

8 6 4 2

10 x

2

4

4

S' S

2 4 6 8

2

2.

R P'

4

R

P 0

Q

R'



R'

Q

2 P'

8 6 4 2

10 x

Q'

P

S'

4

S

2

page 424 y

P

y

8 6 4 2

1.

R

2 4 6 8

0

4

Exercise 34.12

Q

2

6 A'

B'

2

8 6 4 2

R'

4

B"

C 4

D

10x

2

8 B

S

S'

y

A

P 2 4 6 8 P' Q'

6

x

Q'

2.

R

4

D"

B"

0

2

6

8 6 4 2

Q

2

6


2 4 6 8

10 x

Solutions

7. Q.1 represents a reflection in the line y = x. Q.2 represents a reflection in the line y = –x.

Exercise 34.13 1.

Q.3 represents a clockwise rotation of 90° about the srcin. Q.4 represents an anti-clockwise rotation of 90° about the srcin. Q.5 represents an enlargement of scale factor 2 with its centre at the srcin. Q.6 represents an enlargement of scale factor –2 with its centre at the srcin.

c) 2.

c)

X 2 Y

3.

Z Z'

Y'

4 2 0 2

2

4

6

8

10 1 2 1 4

x

6

4.

10

b) 4.5 units

c) 40.5 units

9. a)

2

d) 9

2 Q

1

4 3 2 1 0 1 P

P'

e) 9 c)

2

5.

R' R

12345

c) 13.5 units2

c) 6.

d) 2.25

A

y 3

C

−

c)

1 4

3 4

1

8 7 6 5 4 3 2 10 1

 0 0 –

about the srcin 60° anti-clockwise followed by an enlargement from srcin ( 2)

e) 2.25

2 B

1

2.5 followed

a) Student’s own diagram b) Student’s diagram should show a rotation

10. a) C'

–2 0 0 2

enlargment from the srcin by reflection in the x-axis

S S'

0.5 followed

a) Student’s own diagram b) Student’s diagram should show an

x

3

b) 6 units 2

a) Student’s own diagram b) Student’s diagram should show an

enlargment from the srcin by reflection in the y-axis

y

Q'

180° about the srcin –1 0 0 –1


X' 2

2

90° clockwise about the srcin followed by enlargement from the srcin 1.5 0 – c)  0

4

8

the srcin 0 –0.5 –0.5 0


8. a) y 4

page 426

a) Student’s own diagram b) Student’s diagram should show a reflection in y x followed by enlargement from

−

−

3 4 1 4

2x

2 A'

b) 2 units 2

B'

c) 12.5 units2

3

d) 6.25

e) 6.25


93

Solutions

Exercise 34.14 page 429

Student assessment 1 page 431

1.

1.

a) c)

y X' Y'' X''

y

a)

6

4 Y Y' 2

X

4 2

Z'

4 2 0 Z''  2

Z

2

4x

2

O

4

2

6

4

2

4

6

8

10 x

6

b) Reflection in the y-axis y

d) Rotation 45° anti-clockwise about the

srcin

e)

2.

b)

2 − 2 2 − 2

2 − 2 2 2



a) c)

2 − 2 2 − 2

f)

2 − 2 2 2



x

y

= –x + 4

2.

a) (3, 2)

3.

a)

6 0

2 is also a mirror line



90° clockwise –3 –5

b)



b)

4.

y K'

4 2

K

L'

L

M' M

6

4 20 2

K'' L''

x

J2 4 6 J' J''

O

4 6

M''

b) Enlargement scale factor 2, with centre at

the srcin d) Reflection in the line y

3. 4.

  –1 0 0 –1



x

  0 –1 –1 0 

Student assessment 2 page 432

0 – f) – 0

– 0

0 b) – 0

4

b)

2

0 a)  a)

c) d)

0 1

 Rotation 180° about the srcin

f) Reflection in the line y

1.

y

6

2

1 0

e) Reflection in the line y

94



0 –2 e) –2 0







x

O

24

6

2 4 6

x


81 0 x

Solutions

2.

a) Student’s own construction b) 180° clockwise/anti-clockwise

3.

a)

0 –3

5.

a) b)

y

4

–8 2

b)

P'

3

4.

P

2 S' S

R'

1

R Q' 1 Q 2 3 4

Q''

4 3 2 1 0 1 R''

x

S''

2 3

P''

4

c)


1.5 0 0 1.5

d)

– 0 0 –

page 433


a) C

D

1.

page 435

a) Z' W

B'

X

A' A

Y

Y'

B

D'

C'

b) The scale factor of enlargement is –0.5. 2.

Z

An enlargement of scale factor 2. Centre of enlargement (3, 3) b) A reflection about the line y = –x – 1 a)

3.

a) A reflection in the line x = 0 b) An enlargement by scale factor –0.5.

4.

a) b)

X' W'

Centre of enlargement (0, –1)

b) The scale factor of enlargement is –2.

y

J"

2.

J

6 4

K"

L"

6 4 2

2

L

O

L'

6 0 b) An enlargement of scale factor 2. Centre of enlargement (6, 8) a) A translation of vector

K 2

4

6x

2 J'

K'

4 6

c)

–1 0 0 1 Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013

95

Solutions

3.

y

a)

5.

6

Student’s investigation and ordered table

P

Number of horizontal lines

4 2 Q' R' 2

6 4

O

2

Q 2

4

6x

P'

4 6

b)

4 25

Number of triangles

0

6

1

12

2

18

3

24

…

…

R

0.16

c)

6.

4 or 0.16 25

n = 6(h + 1)

ICT activity 1

page 438

Student’s help sheet

Topic 7 Mathematical investigations and ICT A painted cube page 436 1.

a) 8

b) 12

c) 6

2.

A: 8, B: 24, C: 24, D: 8

3.

A: 8, B: 96, C: 384, D: 512

4.

When (n – 2) ≥ 0, A: 8, B: 12( n – 2), C: 6(n – 2)2, D: (n – 2)3

ICT activity 2 3. Reflection in y = x d) 1

4.

a) Reflection in y = –x b) Clockwise rotation about the srcin of 90° c) Anti-clockwise rotation about the srcin

of 90° d) Enlargement of scale factor 2, centred at

the srcin. e) Englargement of scale factor – 2, centred

at the srcin

When (l – 2) ≥ 0, (w – 2) ≥ 0 and (h – 2) ≥ 0,

5.

A:B:8,4(l – 2) + 4( w – 2) + 4( h – 2), C: 2(l – 2)(w – 2) + 2(l – 2)(h – 2) + 2( w – 2) (h – 2), D: (l – 2)(w – 2)(h – 2)

Triangle count

page 437

9

2.

Student’s investigation and ordered table

3

1

6

2

9

3

12

…

…

n = 3(h + 1)

4.

12

96

Number of triangles

0

3.

35

Probability

Exercise 35.1

page 443

1. Student’s own drawing 2. Student’s own answers

1.

Number of horizontal lines

page 439

Exercise 35.2 1. a)

b)

page 444

c)

d)

e) 0

f) 1

2. a) i)

ii)

b) Total

1

3. a)

b)

c)

d) 1

4. a)

b)

5. a)

b)

c)

d)

ii)

b) i) c) g)

ii)

6. 7. a) i) 8. a) e)

b) f)

d) h)

9. a) RCA RAC CRA CAR ARC ACR b) c) d) e)


Solutions

10. a) e)

b) f)

c) g)

d) h)

Exercise 35.3 page 445 1.

a) 140 b) i)

2.

iii) 

ii) 

iii)

iv)

ii) 

iii) 

iv)

a) 32 b) i) 

3.

ii) 

a) 32 cards in pack b) i) ii) iii)

iv)

v)

3.

a)

b)

c)

d) 

e) 1

4.

a) 

b)

c) 

5.

128 red, 80 blue, 112 green

6.

a)

7.

a) 50 000 b) You have to assume that all entrants have an equal chance of winning.

2.

a) 70 b) i) 

1.

2

2.

25

3.

a)

b)

c)

d)

e)

4.

a)

b)

c)

d)

e)

5.

35 blue, 28 red, 21 yellow, 49 green, 7 white

6.

300

7.

14

8.

200

9.

2

Exercise 36.1 page 452 1.

a) Dice 1 1234 2 e c i D

Student assessment 1 page 449 1.

a)

b)

c) 

d) 0

2.

a) e)

b)

c)

d)

3.

a) e) 1

b)

c) 

d)

4.

a)

b)

c)

d)

5.

160 red, 96 blue, 64 green

6.

a) 

b)  

c)

e) 

f) 

g) 

7.

c)

Further probability

36

Exercise 35.4 page 448

b) 

d)

a) The girl’s results are likely to be more reliable as she repeated the experiment more times. b) It is likely that the dice is biased towards

1

1,1

2,1

3,1

4,1

2

1,2

2,2

3,2

4,2

3

1,3

2,3

3,3

4,3

4

1,4

2,4

3,4

4,4

b)  2.

c) 

6

1,6

2,6

3,6

4,6

5,6

6,6

5

1,5

2,5

3,5

4,5

5,5

6,5

4

1,4

2,4

3,4

4,4

5,4

6,4

3

1,3

2,3

3,3

4,3

5,3

6,3

2

1,2

2,2

3,2

4,2

5,2

6,2

1

1,1

2,1

3,1

4,1

5,1

6,1

Die 2 Dice 2

1

23456 Dice Die 11

the number 3.

Student assessment 2 page 450 1.

a)

b)



d)

c)

d) 0

a)  g)

b)  h) 

c)  i) 

d) 

e) 


f)

97

Solutions

Exercise 36.2 1.

3.

page 454

W

a) 1

2

1

3

1

2

2

3

1

2

3

3

b) i) 2.

ii)

iii)

a) M

M F F M M F F M F F

b) i)

ii)

iii)

WWW

L

WWL

111

D

WWD

2

112

W

WLW

3

113

L

WLL

1

121

D

WLD

2

122

W

WDW

3

123

L

WDL

1

131

D

WDD

2

132

W

LWW

3

133

L

LWL

1

211

2

212

D W

LWD LLW

3

213

L

LLL

1

221

D

LLD

2

222

W

LDW

3

223

L

LDL

1

231

D

LDD

2

232

W

DWW

3

233

L

DWL

1

311

D

DWD

2

312

W

DLW

3

313

L

DLL

1

321

D

DLD

2

322

W

DDW

3

323

L

DDL

1

331

D

DDD

2

332

3

333

v)

L

W

D

W

L

L

D

W

L

D

D

b) i) 4.

ii)

a) DB

M MMMM M F MMMF

M

W

1

iv)

F

98

a)

M MMFM W

iii)

iv)

DB

DB DB

W

DB W

LB

DB LB

P

DB P

DB

W DB

W

W W

LB

W LB

P

W P

F

MMFF

M

MFMM

F

MFMF

M

MFFM

F

MFFF

M

FMMM

F

FMMF

DB

P DB

M

FMFM

W

P W

F

FMFF

LB

P LB

M

FFMM

P

P P

F

FFMF

M

FFFM

F

FFFF

LB

P

b) i)

ii)

DB

LB DB

W

LB W

LB

LB LB

P

LB P

iii)

iv)


Solutions

Exercise 36.3 1.


page 456

a)

Roll1

Roll2 



Six

Six



Outcomes Six, Six

 

Not six Six, Not six



Not six Not six, Not six

Not six, Six

Six

Not six

1.

a) i) b) Infinite

2.

a)

Probability 

  

 



  

 





 

  

  







2.

a)

3.

a) 0.35

iii)

iv)

L

0.65

OT

0.35

L

OT 0.65

0.65 0.35 0.65 0.35

OT

0.65 0.35 0.65

b)i) 0.275 (3 s.f.)

iii) 0.444 (3 s.f.) 4. 5.

T

1T

23

a)

45

2H

L OT L OT

L,L,L L, L, OT L,OT,L L, OT, OT

L OT L OT

OT,L,L OT, L, OT OT, OT, L OT, OT, OT

4H

5H

6H

3T

4T

5T

6T

iii)

A

B

P

W

b) i) 4.

ii)

K

B

M

L

ii)

6.

0.027

7.

a) 0.752 (0.563) b) 0.753 (0.422) c) 0.7510 (0.056)

KB

M

KM

L

KL

K

BK

M

BM

L

BL

K

MK

B

MB

M

MM

L

ML

K

LK

B

LB

M

LM

L

LL

BB

P

BP

W

BW

B

PB

P

PP

W

PW

B

WB

P

WP

W

WW

iii) 0.5

0.5

H 0.5

0.5

H

HHH

0.5 0.5

T

HHT

H

HTH

H T 0.5 0.5

0.5

H

0.5

T

0.5 T

B

B

a)

b) 0.0129 (3 s.f.) d) 0.586 (3 s.f.)

a)

6

3H

2T

ii)

v)

ii) 0.123 (3 s.f.) iv) 0.718 (3 s.f.)

a) 0.0588 (3 s.f.) c) 0.414 (3 s.f.)

b) i)

1H

c) 0.35

L

0.65

1 H

B

b) 0.35

ii)

b) i) 3.

b) i) ii) c) Add up to 1

page 457

iii)

T

HTT

H

THH

0.5 0.5

T

THT

H

TTH

0.5

T

TTT

b) i)

ii)

iv)

5.

a)

b) 

c)

6.

a)

b)

c)

7.

0.009 95 (3 s.f.)

iii)


99

Solutions


page 458

a) Dice 1 123456 1

234567

2

345678

2 3 ec i D4

1.

456789 5

6

7

8

9

10

5

6

7

8

9

10

11

6

7

8

9

10

11

12

b) i) c) 25

ii)

2.

a) i) b) i)

ii) ii)

3.

a)

iii)

1 16

because there are 16 possible routes and only one results in the marble landing in Tray 1. 4

B

WB

P

b) i)

ii)

iii)

4.

a)

b)

c)

5.

a) 0.72

b) 0.729

6.

a) 0.6 0.6

0.4

T

0.6

H

0.4

T

b)i) 0.216 a) i) 0.8 b) i) 0.06

T

ii) 0.064 ii) 0.7 ii) 0.56

WP

0.6

H

HHH

0.4 0.6

T

HHT

H

HTH

H

H

Tray 5:

iv)

0.4 0.6 0.4

Tray 4:

W WW

W

T

HTT

H

THH

0.4 0.6

T

THT

H

TTH

0.4

T

TTT

= =

3 8 1 4

5.

Student’s investigation

6.

210 1024

=

105 512

7.

Each numbertointhe each row ofofPascal’s triangle corresponds number routes to landing in each tray of the game.

8.

The binomial expansion generates the numbers in Pascal’s triangle and therefore the number of routes to landing in each tray of the game.

Dice sum

page 461

1. Dice 1

iii) 0.648 123456

iii) 0.3 iii) 0.38

2 cei D

100

6 16 4 16 1 16

Tray 3:

W BW BP

1

4. Tray 2: 16 = 4

BB

P

RLL RRL

LLRL LRLL RLLL LRLR LRRL RLLR RLRL RRLL Tray 4: RRRL RRLR RLRR LRRR Tray 5: RRRR

iv) B

LRL RLR

Tray 2: LLLR Tray 3: LLRR

iv)

iii)

Tray 1: LLL Tray 2: LLR Tray 3: LRR Tray 4: RRR

2. Tray 1: LLLL

3.

B

7.

Topic 8 Mathematical investigations and ICT Probability drop page 460

1

234567

2

345678

3

456789

4

5

6

7

5

6

7

6

7

8

8

91

8

9

101

1

9

10

11

12


0

Solutions

2. 36

3.

Mean = 14 yrs 3 mths Median = 14 yrs 3 mths Mode = 14 yrs 3 mths Range = 8 mths

4.

Mean = 26.4 Mode = 28

3. 7 3

1

3

1

4. 36 = 12 5. 36 = 6 6. You are four times more likely to get a 5 than

a 2.

5.

7. Dice 1

Mean = 13.9 s (3 s.f.) Median = 13.9 s Mode = 13.8 s Range = 0.6 s

6. 91.1 kg 7. 103 points

1234

2 ec i D

Median = 27 Range = 5

1

2345

2

3456

Exercise 37.2

3

4567

1.

4

5678

page 468

Mean = 3.35 Mode = 1 and 4


2. Mean = 7.03 8. 16

3.

9. 5 4

Median = 7 Mode = 8 Range = 5 b) The mode, as it gives the highest number of flowers per bush

1

10. 16 = 4 11. Student’s investigation 12. a) m2

b) m + 1

c)

13. a) m × n

m 2 m

=

1 m

b) The total can take any integer value in the

range n + 1 c)

n nm

=



m + 1.

1 m

a) Mean = 6.33 (3 s.f.)

Exercise 37.3 1.

page 469

a)

Height (m)

Fre qu e n cy

Mi d-i n te rv al Frequency× value mid-interval value

ICT activity: Buffon’s needle experiment page 462

1.8–

2

1 .85

1.9–

5

1 .95

1.–9. Student’s experiment and results entered in

2.0–

a spreadsheet

2.1–

2

10. The value of p should tend to π.

2.2– 2.3–2.4

37

Mean, median, mode and range

Exercise 37.1 Mode = 1 2.

Mean = 6.2 Mode = 7


9 .75

10

2 .05

20. 5

22

2. 15

47. 3

2. 25

15. 75

2. 35

9.4

7 4

b) Mean = 2.13 m (3 s.f.) c) Modal class = 2.1–2.2 m 2.

a) Mean = 33 h (2 s.f.) b) Modal class = 30–39 h

3.

a) Mean = 6.2 cm b) Modal class = 6.0–6.5 cm

page 467

1. Mean = 1.67 (3 s.f.)

3 .7

Median = 6.5 Range = 9


101

Solutions


2.

a) b) c) d)

page 470

3. Fr a c t i o n

Mean = 16 Median = 16.5 Mode = 18 Range = 39

a) 28 b) i) Mean 7.75

$

D e g re e s

Clothes

1 –3

800

Transport

1 –5

480

72

Ents.

–14

600

90

Saved

13 –– 60

520

78

120

ii) Median = 8 iii) Mode = 8 iv) Range = 5 3.

a) Mean = 19 b) Modal class is 19

20

M

Student assessment 2 1. Mean = 86.8 m

Median = 90.5 m Range = 18 m

Mode = 93 m 2.

page 471

a) 26 b) i) Mean = 7.73 (3 s.f.)

4.

ii) Median = 7.5 iii) Mode = 10 iv) Range = 6 3.

Male Management 18% (65°)

Social 8% (29°)

a) Mean ≈ 10.0 kg (3 s.f.) b) Modal class is 10.0 M

10.1 kg

Non-skilled 12% (43°)

Clerical 22% (79°)

38

Collecting and displaying data Exercise 38.1

Skilled 24% (86°)

Professional 16% (58°)

page 476 Female

1. S lee p

M e a l s S po r t

TV

Sc ho o l

Ayse

8 h20

2 h

5h

2h

6 h 40

Ahmet

8 h40

2 h

5 h20

2 h

6h

Non-skilled 24% (86°)

Social 10% (36°) Management 4% (14°)

2.

Clerical 38% (137°)

Skilled 16% (58°)

Music

Professional 8% (29°)

a) Student’s own two statements b) Professional 8 million 8%

Management

102

8 million


4%

640 000 320 000

Solutions

Exercise 38.2 1.

page 477

Student’s own survey results and pie charts

2. Student’s report

Exercise 38.3 1.

page 481

Student’s answers may differ from those given below. a) Possible positive correlation (strength depending on topics tested) b) No correlation c) Positive correlation (likely to be quite strong) d) Negative correlation (likely to be strong). Assume that motorcycles are not rare/vintage. e) Factors such as social class, religion and income arelikely to affect results. Therefore little correlation is likely. f) Negative correlation (likely to be strong) g) 0–16 years likely to be a positive correlation h) Strong positive correlation

2.

a)

Sunshine and rainfall correlation 160 140

) m120 m ( ll 100 fa in 80 a R 60

40 20 0

2

4

6

8

10

12 1

4

Sunshine (hours)

b) Graph shows a very weak negative correlation. 3.

a)

Adult illiteracy and infant mortality correlation 0 140 0 0 1 120 r e p 100 y ti l 80 a rt o 60 m t n 40 fa In 20

0

20

40

60

80

Adult illiteracy (%)

b) Positive correlation c) Student’s answer. However, although there is a correlation, it doesn’t imply that one variable affects

the other


103

Solutions

d)

Female and male life expectancy correlation 90 80

y c 70 n a t 60 c e p 50 x e e fi 40 l le 30 a M 20

10 0

10

20

30

40

50

60

70

80

90

100

Female life expectancy

a) Moderate/strong positive correlation b) Approx. 31 tomatoes c) Approx. 60 cm

4.

Exercise 38.4

page 485

Exercise 38.5 1.

1. 9

page 487

a)

Time (min)

0–

10–

15–

Freq.

6

3

137

F re q . d e n s i t y

0. 6

0.6

20– 25– 30– 40–60

8 7 y c

6

n e u q e r F

5

3 2 1 1 2 3

5 64 7 8 Distance (km)

2. 8 7 6 y c n e u q e r F

5

0.6

4 0 .4

y it s n e d y c n e u q e r F

2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

4

5 10 15 2 0 25 3 0 35 4 0 45 5 0 55 6 0 Time (min)

3 2 1 145 150 155 160

165 170 175 180 185

Height (cm)

104

1. 4

4

b)

4

0

2.6

3


0. 2

Solutions

2.

a)

4.

Ti m e (m in )

Fre q u en c y

Fre q ue n cy d e ns it y

0

t

30

8

0.3

30

t

45

5

0.3

45

t

60

8

0.5

60

t

75

9

0.6

75

t

90

10

0.7

90

t

1

20

a)

12

7 y t i s n e d y c n e u q e r F

6 5 4 3 2 1 0

0.4

5 10 15 20 2 5 30 3 5 40 4 5 50 5 5 60 6 5 70 7 5 80 Age (years)

b) b) Student’s own answers 0.8 y t i s n e d y c n e u q e r F


0.7 0.6

1. 90

0.5

80

0.4

2

0.3 0.2 0.1 0

3.

15

30

45 60 75 90 1 05 120 Time (min)

) 70 m k 60 0 0 50 0 0 40 1 ( a 30 re A 20

10

a)

0

Age (years)

0–

1–

Freq.

35

4 8 1 4 0 180 260 280


3 5 12

5– 28

10– 20–

18

13

40–

Nigera

60–90

Republic of Congo

South Sudan

Kenya

150

14

5 2.

a)

Distance travelled and time taken correlation

b) 36

50

32 y t i s n e d y c n e u q e r F

page 489

) n i 40 m (

28

30

24 20

e m i 20 T

16

10

12

0

8

5

10 15 20 25 30

Distance (km)

4 0

10

20 30 40 50 60 Age (years)

70

80 90

b) (Strong) positive correlation c) It depends on their mode of transport.


105

Solutions

d)

2.

Distance travelled and time taken correlation

a)

s 70 e v lo 60 g f o 50 s ri a 40 p f o 30 r e b 20 m u 10

50 ) in 40 m (

30

e im 20 T

10 0

5

Gloves sold and outside temperature correlation

10 15 20 25 30

N

Distance (km)

e) Approx. 9.5 km 3.

0

5

10 15 20 25 30

Mean outside temperature ( C)

a)

°

Age Juniors

0–

Intermediates Full members Seniors

N u m be r Fre q ue n cy de n si ty

32

2

16–

80

16

21–

273

7

60–80

70

b) Negative correlation c) Student’s own answer d) Gloves sold and outside temperature correlation s 70 e v lo 60 g f o 50 s ri a 40 p f o 30 r e b 20 m u 10 N

3.5

b) 16 y ti s n e d y c n e u q e r F

14 12 10 8

0

5

10 15 20 25 30

Mean outside temperature ( C)

6

°

4 2

Approx. 30 pairs 0 10 20 30 4 0 50 60 7 0 80

3.

a)

Age (years)


America (48.4°) 920 Europe (39.5°)

Africa (52.4°) 995

page 490

Points

0–

5–

10–

15–

Number of games

2

3

8

9


0. 4

0. 6

1. 6

Oceania (1.8°) 35

750 Asia (218°) 4140

106


0. 9

25– 35–50 12 1. 2

3 0. 2

Solutions

b)

2.

a)

2.0

C l a s sA

1.8 y it s n e d y c n e u q e r F

Score

1.6 1.4 1.2

0

x

20

1

1.0

20

x

40

5 60

40

x

60

6

0.4

60

x

0.2

80

x

0.8 0.6

0

5 10 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0

39

1.

C la ss C

1

1 1

3 02 12

3 00

0 0

4 4

15 3

4 18

2

5

8

4

9

4

12

8

17

Class A 18 16

Cumulative frequency

Exercise 39.1

80 1

b)

Points

ClassB

Cum. Cum. Cum. Freq. Freq. Freq. freq. freq. freq.

y c 14 n e u 12 q rfe 10 e iv t 8 a l u 6 m u C 4

page 492

a)

2

Finishing time (h)

0– 0.5– 1.0– 1.5– 2.0– 2.5– 3.0–3.5

Freq.

0

0

6

34

16

3

1

C u mf.re q .

0

0

6

40

56

59

60

0

20

40

60

80 1 00

Test score

Class B

b) 60

18

50

y c 14 n e u 12 q e rf 10 e iv t 8 la u 6 m u C 4

16

y c n e 40 u q e fr e 30 iv t a l u 20 m u C

2

10

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

20

40

60

80 1 00

Test score

Finishing time (h)

c) Median ≈ 1.8 h d) As many runners finished before as after

the median.


107

Solutions

b)

Class C 18

28

16 y c 14 n e u 12 q e fr 10 e v 8 ti la u 6 m u C 4

26 24 22 y c 20 n e u 18 q e rf 16 e v ti 14 a l u 12 m u C 10

2 0

20

40

60

80 1 00

Test score

8

c) Class A median

≈ 50 Class B median ≈ 70 Class C median ≈ 78 d) As many students were above as below the median.

3.

2007

30

6 4 2 150 155 160 165 170 175 180

185

Height (cm)

a) 20 07

Height (cm) Freq.

20 08

2008

20 09

Cum. Cum. Cum. Freq. Freq. freq. freq. freq.

30 28

150–

6

6

2

2

2

2

26

155–

8

14

9

11

6

8

24

160–

11

25

10

21

9

17

165–

4

29

4

25

8

25

170–

1

30

3

28

2

27

175–

0

30

2

30

2

29

180–185

0

30

0

30

1

30

22 y c 20 n e u 18 q e rf 16 e v ti 14 la u 12 m u C 10

8 6 4 2 150 155 160 165 170 175 180 Height (cm)

108


185

Solutions

b)

2009

40

30

36

28 y c n e u q e fr e v it a l u m u C

26 24 22 y c 20 n e u 18 q e rf 16 e v ti 14 a l u 12 m u C 10

32 28 24 20 16 12 8 4 0

20

40

8

2 150 155 160 165 170 175 180

185

4.

a)

Height (cm)

c) Median (2007)

Type A M a ss ( g)

161 cm Median (2008) ≈ 162 cm Median (2009) ≈ 164 cm d) As many students are taller than the ≈

75–

Exercise 39.2

3.

F re q u e n c y

4

100–

median as shorter than the median.

2.

100

c) Qualifying distance ≈ 66 m d) Inter-quartile range ≈ 28 m e) Median ≈ 50 m

4

a) b) a) b) a)

80

Distance thrown (m)

6

1.

60

page 495

Class A ≈30 Class B ≈30 Class C ≈ 40 Student’s own responses 2007 ≈7 cm 2008 ≈8 cm 2009 ≈ 8 cm Student’s own responses

C u m .f re q .

4 7

11

125–

15

26

150–

32

58

175–

14

72

200–

6

225–250

78 2

80

Type B

Distance thrown (m)

0–

20–

40–

60–

80–100

Freq.

4

9

15

10

2

75–

Cum. freq.

4

13

28

38

40

100–

16

16

125–

43

59

150–

10

69

M as s ( g )

F re q u e n c y

0

0

175–

7

200–

4

225–250

C u m .f re q .

76 80 0

80


109

Solutions

b)

Student assessment 1 Type A

1.

80 y c n e u q rfe e v ti la u m u C

60 50

Freq.

2

3

5

7

6

4

2

1

C u m .f re q .

2

5

10

17

23

27

29

30

40

b)

30

32

20

0

75

y c n e u q e rf e iv t a l u m u C

100 125 150 175 200 225 250 Mass (g) Type B

80

24 20 16 12 8 4

60

0

50 40

20

40 60 Exam mark (%)

80

100

c)i)

Median ≈ 57% ii) Lower quartile ≈ 45% Upper quartile ≈ 69% iii) Inter-quartile range ≈ 24%

30 20

2. 0

75

100 125 150 175 200 225 250 Mass (g)

c) Median type A

≈ 157 g Median type B ≈ 137 g d) i) Lower quartile type A ≈ 140 g Lower quartile type B ≈ 127 g ii) Upper quartile type A ≈ 178 g Upper quartile type B ≈ 150 g iii) Inter-quartile type range type A Inter-quartile type range type B e) Student’s own report

110

28

70

10

5.

20– 30– 40– 50– 60– 70– 80– 90–100

Mark

70

10

y c n e u q e rf e v ti a l u m u C

page 497

a)

a) Student’s own explanation b) Student’s own explanation

≈ ≈

38 g 23 g

a) Ma rk ( % )

Fre q u e n c y

C u m u l a t i v e f re q u e n c y

31–40

21

41–50

55

76

51–60

125

201

61–70

74

275

71–80

52

327

81–90

45

372

91–100

28

400


21

Solutions

c) ‘A’ grade ≈ 75% d) Lower boundary

b)

≈ 66% Upper boundary ≈ 72% e) Inter-quartile range ≈ 25%

400 y 350 c n300 e u q250 rfe e 200 iv t la 150 u m100 u C

Topic 9 Mathematical investigations and ICT Heights and percentiles

page 499

50

0

20

40 60 80 Exam mark (%)

100

c) i) Median ≈ 60%

ii) Lower quartile ≈ 52% Upper quartile ≈ 73% iii) Inter-quartile range ≈ 21% 3.

a) Ma rk ( % )

1–10

Fre q u e n c y

10

11–20

1. 2.

Approx. 167 cm Approx. 168 cm (Note: This corresponds to the 25th percentile not the 75th.)

3.

Approx. 151 cm

4.

Student’s calculations could include mean, median, inter-quartile range and comparisons with printed charts.

5.

It is likely that different cultures have different charts as some races are taller on average than others.

C u m u l a t i v e f re q u e n c y

10

30

Reading ages

40

1.

21–30

40

80

31–40

50

130

41–50 51–60

70 100

200 300

61–70

240

540

71–80

160

700

page 501

Possible answers include length of sentences, number of words with 3 or more syllables, size of type, etc.

2. Student’s choices 3. Student’s calculations 4.

Students should choose articles on a similar topic. Ignore proper nouns. Choose more than one article from each paper.

81–90

70

770

ICT activity

91–100

30

800

1.–6. Student’s data, graph and comparisons

page 501

b) 800 y 700 c n600 e u q500 e fr e 400 v ti la 300 u m200 u C

100 0

20

40 60 80 Exam mark (%)

100


111

305847293-IGCSE-Mathematics-Textbook-Answers.pdf

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