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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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 3. 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 6. Lower bound
Upper bound
13.1 (1 d.p.) 13.5 (1 d.p.)
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 1
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.)
Student assessment 4
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.)
Student assessment 2 1.
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 2
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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%
Student assessment 4
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
Student assessment 1
b) 25%
Student assessment 2
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 1
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
Student assessment 2
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°
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 3
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.)
Student assessment 5
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
Student assessment 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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 6
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 1
6.
2 years
1. 09 03
7.
5 years
2.
8.
3 years
3. 49 km
Student assessment 1 1.
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
Student assessment 2
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
Student assessment 2 1.
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}
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 1
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}
Student assessment 2
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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}
Student assessment 3
{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
Student assessment 4
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 3
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
Student assessment 1
2.
Student assessment 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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
21
Solutions
Student assessment 4 1.
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.)
Student assessment 2
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
Student assessment 1
Student assessment 5
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 4 1.
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
x5
23456
1.
4.
40
page 148 180
c) 92°
62°
52°
360°
4
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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 11
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
q2
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 11
1 2 3
4
5 x 0
1
2
3
4
5
6
7
8
9 1 0 11 12
p
2 3
c) Any integer point in the unshaded region,
e.g. (5, 2) meaning 5 loaves and 2 cakes
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
29
Solutions
a) m 2
3.
n 2m
n 11
m
Student assessment 1
b) n
12
1.
a) x 5
2.
a) 2
3.
a) b)
11
y4 r
page 153
b) y 3
10 3
s
b) 4
p8
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
c) Any integer point in the unshaded region,
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
Student assessment 2 1.
a) x 7
2.
a) 1 p
page 154
b) y 9
4
b) 4
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
x7
A
Solutions
4 y
a) x b)
3.
2 x
y8
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
e5
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 1
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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) $515.46
Exercise 16.1
3
a) b) c) d) e)
0
1.
1
iii) un = 25( 5 )n−1
ii) −243, 729 iii) un = (−3)n
1.
–2
Student assessment 2
Exercise 15.4 1.
–4
Term
n
k g
Solutions
Exercise 16.3
page 171
1.
a) h = kt2 d) 6 s
2.
a)
3.
a) l = km 3
4.
a) P = kI2
b) 5 amps
v
=k e 1
17
b) 5
c) 45 m
b) 3
c) 49 J
Exercise 17.1
b) 3
c) l = 6 cm
1.
page 172
1.
a) 1.5
b) 15
c) 3
d) 12
2.
a) 10
b) 2.5
c) 1
d) 20
1 3 1 2
d) 12
3.
a)
4.
a) 5
5.
a) 3
b) 72
c)
5
b) 4
1
c)
4
Student assessment 2 1.
) m (k e c n ta s i D
80 60 40 20
d) 1
c) ±2
b) 3
page 175
100
Student assessment 1
1 3
Graphs in practical situations
d) ±1
0
10
20
30
40
50
60
70
Distance (miles)
page 173
a) 50 km 31 miles b) 40 miles 64 km, therefore
a) x
1
2
4
8
16
32
y
32
16
8
4
2
1
80 miles
128 km
c) 100 km/h 62 mph d) 40 mph 64 km/h 2.
b) 1.6
140
2.
a)
x
1
2
4
5
10
y
5
10
20
25
50
120 ) ˚F ( 100 re tu a r 80 e p m
b) x
1
2
4
5
10
y
20
10
5
4
2
x
4
16
25
36
64
y
4
8
10
12
16
e T
c)
60 40
3.
a) 0.8
b) 0.8
4.
a) 1.6 (1 d.p.)
b) 2
20
0
10
20
30
40
50
Temperature (˚C)
a) 25 °C
80 °F
b) 100 °F 35 °C c) 0 °C 30 °F d) 100 °F 35 °C, therefore 200 °F
70 °C
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
33
Solutions
3.
Approx. True
120
Exercise 17.2 a) 6 m/s d) 20 km/h
b) 4 m/s e) 160 km/h
c) 39 km/h f) 50 km/h
2.
a) 400 m d) 255 km
b) 182 m e) 10 km
c) 210 km f) 79.2 km
3.
a) 5 s b) 50 s d) 1 min 11.4 s e) 5 s f ) 4 min
100
) F (˚ re 80 u t ra e 60 p m e T
40 20
Exercise 17.3
0
10
20
30
40
1.
50
Temperature (˚C)
c) 4 min
page 176
60 50
i) a) 25 °C 77 °F b) 100 °F 38 °C c) 0 °C 32 °F d) 100 °F 38 °C, therefore 200 °F 76 °C
) 40 m ( e c 30 n ta is 20 D
ii) The rough conversion is most useful at lower temperatures (i.e. between 0 and 20 °C). 4.
page 176
1.
10
a)
10
Peak
0
8 ) ($ 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%
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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.
Acceleration is 0.2 m/s 2
3.
Acceleration is 7 m/s 2
4.
Acceleration is 1.75 m/s 2
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
35
Solutions
Exercise 17.7 1.
Student assessment 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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 2 1.
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 4 1.
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
47
Solutions
Student assessment 1 1.
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
1 2 3
4 x
2 x
Solutions
Student assessment 2
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
49
Solutions
Student assessment 3 1.
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
Student assessment 4
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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)
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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)
Student assessment 1 4
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
Student assessment 2
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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.
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
No
No
Yes
leg na ir t
Solutions
Student assessment 1 1.
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.
Student’s own construction
3. 4.
C
c) Approx. 23 m 4.
a) Student’s own construction b) Approx. 24 m2
Student’s own construction
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.
Student’s own construction
2.
Student’s own construction
Exercise 21.3
Y
50 m X
20
40
m
m
page 226
1.
Student’s own construction
2.
Student’s own construction
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
Student assessment 1 1.
Student’s own construction
2. 3.
Student’s own construction a) Student’s own construction b) Approx. 53 m2
page 229
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
55
Solutions
Student assessment 2
Exercise 22.3
page 230
1.
Student’s own construction
2.
a) 22 cm
3.
a) Student’s own construction b) Approx. 23 cm2
a) i) 8 m ii) 4.8 m c) 252 cm2
b) 22 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
Student assessment 1
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
c) 35.2 cm2
Solutions
Student assessment 2 1.
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
Student assessment 2
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.
a)b)c) Student’s own diagrams
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)
a)b)c) Student’s own diagrams
3 41 5
page 243
1.
3.
3
Exercise 23.1 1.
3
H
Student assessment 1
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°
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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°
Student assessment 1
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°
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 4 1.
2.
6m
page 264
CDA OCB
CBA OAB
AOB
COB
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
Solutions
2.
4. Compound
5.
S
Building
45°
X
Y
3.
Student assessment 1
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.
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 2 1.
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:
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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.
Student assessment 1
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
Student assessment 2 1.
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 4
(3 s.f.)
3
5.
2
Student assessment 3
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
Student assessment 2
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.)
Student assessment 1
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
c) 3270 cm3
Solutions
Student assessment 5
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.)
Student assessment 6 3
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
) 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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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 10 1
d)
1
1
2
4 3 2 10 1
3 x
1
2
3 x
4 3 2 10 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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
73
Solutions
c)
y 9
Student assessment 2
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.
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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 11
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 11
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
Student assessment 1
5
2
1. Student’s own diagram
6
4
2. Student’s own diagram
7
6
3. Student’s own diagram
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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°
Student assessment 1
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
Student assessment 2 1.
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.)
Student assessment 3
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
77
Solutions
Student assessment 4
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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.
Student assessment 1
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.)
Student assessment 2
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!
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 1
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)
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 5
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
Student assessment 6
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
Student assessment 3
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.
Student assessment 4
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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.
Student’s own matrices
6.
Student’s own matrices
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.
Student assessment 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)
Student assessment 2 1.
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.
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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'
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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 10 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
a) Student’s own diagram b) Student’s diagram should show a rotation
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
a) Student’s own diagram b) Student’s diagram should show a rotation
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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 20 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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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)
Student assessment 3 1.
1.5 0 0 1.5
d)
– 0 0 –
page 433
Student assessment 4
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)
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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)
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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)
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
Solutions
Exercise 36.3 1.
Student assessment 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)
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
99
Solutions
Student assessment 2 1.
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Median = 3 Range = 5
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
Median = 1 Range = 5
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
101
Solutions
Student assessment 1 1.
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Student assessment 1
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
F re q . d e n s i t y
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.
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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)
Student assessment 2 1.
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
F re q . d e n s i t y
0. 4
0. 6
1. 6
Oceania (1.8°) 35
750 Asia (218°) 4140
106
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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.
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
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
Cambridge IGCSE Mathematics 3rd edition © Hodder & Stoughton Ltd 2013
111