Cambridge IGCSE™ International Mathematics answers
1
1 Number
All answers were written by the authors.
Exercise 1.1 page 25
1
The prime numbers are:
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
Exercise 1.2 page 25
1
The square numbers are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Exercise 1.3 page 25
1
a
b
92
122
c
82
d
73
e
43
f
3 2 × 23
g
5 3 × 22
h
42 × 32 × 22
Exercise 1.4 page 26
1
a
b
1, 2, 3, 6
1, 3, 9
c
1, 7
d
1, 3, 5, 15
e
1, 2, 3, 4, 6, 8, 12, 24
f
1, 2, 3, 4, 6, 9, 12, 18, 36
g
1, 5, 7, 35
h
1, 5, 25
i
1, 2, 3, 6, 7, 14, 21, 42
j
1, 2, 4, 5, 10, 20, 25, 50, 100
Cambridge IGCSE International Mathematics Third edition
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Cambridge IGCSE™ International Mathematics answers
1
Exercise 1.5 page 26
1
a
b
3, 5
2, 3
c
2, 3
d
2
e
2, 5
f
13
g
3, 11
h
5, 7
i
2, 5, 7
j
2, 7
Exercise 1.6 page 27
1
a
b
22 × 3
25
c
22 × 32
d
23 × 5
e
22 × 11
f
23 × 7
g
32 × 5
h
3 × 13
i
3 × 7 × 11
j
32 × 7
Exercise 1.7 page 27
1
a
b
4
5
c
6
d
3
e
9
f
22
g
8
h
13
i
17
j
12
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Cambridge IGCSE™ International Mathematics answers
1
Exercise 1.8 page 28
1
2
a
15
b
12
c
14
d
28
e
8
f
30
g
12
h
12
i
60
j
60
a
42
b
60
c
70
d
90
e
120
f
105
g
20
h
231
i
240
j
200
Exercise 1.9 page 28
1
a
5
b
3
c
7
d
10
e
11
f
13
g
0.1
h
0.2
i
0.3
j
0.5
2
Students check their answers
3
a
1
3
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Cambridge IGCSE™ International Mathematics answers
b
1
4
c
1
5
d
1
7
e
1
10
f
2
3
g
3
10
h
7
9
i
5
3
j
5
2
1
Exercise 1.10 page 29
1
x
0
1
4
9
16
25
36
49
64
81
100
y
0
1
2
3
4
5
6
7
8
9
10
The following answers are correct to 1 d.p.:
a
5.9
b
6.7
c
7.4
d
7.7
e
1.4
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Cambridge IGCSE™ International Mathematics answers
2
1
Students check their answers
Exercise 1.11 page 29
1
a
2
b
5
c
3
d
0.1
e
0.3
f
6
g
10
h
100
i
–2
j
–3
k
–10
l
–1
Exercise 1.12 page 30
1
a
1296
b
259
c
6561
d 100
e
7
f
2
g
27
h 36
i
64
j
64
k 5
l
9
Exercise 1.13 page 32
1
a
26
b
10
c
42
d
16
e
8
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Cambridge IGCSE™ International Mathematics answers
2
3
f
–6
a
20
b
34
c
32
d
31
e
−36
f
23
a
27
b
64
c
−36
d
3
e
144
f
1.6
1
Exercise 1.14 page 32
1
2
a
6 × (2 + 1) = 18
b
1 + 3 × 5 = 16
c
(8 + 6) ÷ 2 = 7
d
(9 + 2) × 4 = 44
e
9 ÷ 3 × 4 + 1 = 13
f
(3 + 2) × (4 – 1) = 15
a
12 ÷ 4 – 2 + 6 = 7
b
12 ÷ (4 – 2) + 6 = 12
c
12 ÷ 4 – (2 + 6) = –5
d
12 ÷ (4 – 2 + 6) = 1.5
e
4 + 5 × 6 – 1 = 33
f
4 + 5 × (6 – 1) = 29
g
(4 + 5) × 6 – 1 = 53
h
(4 + 5) × (6 – 1) = 45
Exercise 1.15 page 33
1
a
2
b
3
c
7
d
4
e
23
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Cambridge IGCSE™ International Mathematics answers
2
f
0
a
1
b
5
c
2
d
50
e
7
f
–1.5
1
Exercise 1.16 page 35
1
2
a
8
b
24
c
4
d
20
e
9
f
63
g
15
h
25
i
2
j
6
a
9
b
16
c
20
d
40
e
18
f
72
g
30
h
48
i
210
j
52
Exercise 1.17 page 36
1
a
b
14
3
18
5
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Cambridge IGCSE™ International Mathematics answers
c
47
8
d
17
6
e
17
2
f
68
7
g
58
9
h
17
4
i
59
11
j
55
7
k
43
10
l
146
13
1
Exercise 1.18 page 36
1
a
b
1
4
3
6
5
7
c
6
5
6
d
6
5
8
e
5
4
9
f
5
1
12
g
9
3
7
h
3
3
10
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Cambridge IGCSE™ International Mathematics answers
i
9
1
2
j
6
1
12
1
Exercise 1.19 page 37
1
Hundreds
Units
a
6
b
5
1
𝟏𝟏
𝟏𝟏𝟏𝟏
0
9
𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏
𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏
2
4
3
8
3
d
0
0
7
1
e
2
0
0
1
f
3
5
6
c
2
Tens
a
19.14
b
83.812
c
6.6
d
11.16
e
35.81
f
5.32
g
67.14
h
6.06
i
1.4
j
0.175
Exercise 1.20 page 38
1
a
58
= 58%
100
b
17 68
=
= 68%
25 100
c
11 55
=
= 55%
20 100
d
3 30
=
= 30%
10 100
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Cambridge IGCSE™ International Mathematics answers
e
23 92
=
= 92%
25 100
f
19 38
=
= 38%
50 100
g
3 75
=
= 75%
4 100
h
2 40
=
= 40%
5 100
1
2
Fraction
Decimal
Percentage
1
10
0.1
10%
1
5
0.2
20%
3
10
0.3
30%
4 2
=
10 5
0.4
40%
1
2
0.5
50%
3
5
0.6
60%
7
10
0.7
70%
4
5
0.8
80%
9
10
0.9
90%
1
4
0.25
25%
3
4
0.75
75%
Exercise 1.21 page 39
1
a
2 4 8 20 16
= = = =
5 10 20 50 40
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Cambridge IGCSE™ International Mathematics answers
2
b
3 6 9 15 27
= = = =
8 16 24 40 72
c
4 8 12 32 36
= = = =
7 14 21 56 63
d
5 15 20 50 55
= = = =
9 27 36 90 99
a
b
1
1
2
1
3
c
2
3
d
4
9
e
3
4
f
9
10
Exercise 1.22 page 40
1
a
b
2
1
7
1
10
c
5
9
d
1
12
e
9
40
f
1
a
b
1
20
17
60
45
88
c
17
20
d
44
195
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Cambridge IGCSE™ International Mathematics answers
3
e
9
20
f
1
18
a
b
4
3
4
3
5
10
5
c
3
1
10
d
6
7
24
e
1
f
25
36
a
b
1
1
8
1
8
9
7
40
5
3
8
c
−
d
7
20
e
−2
f
1
39
140
1
4
Exercise 1.23 page 44
1
a
b
4
3
9
5
c
1
7
d
9
e
4
11
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Cambridge IGCSE™ International Mathematics answers
f
2
a
b
3
4
5
1
8
37
1
6
3
5
c
4
21
d
2
3
e
1
4
f
7
20
a
5
6
b
2
c
1
d
4
e
1
5
f
2
3
a
3
5
b
3
c
8
15
d
12
a
13
28
b
1
2
1
7
1
6
107
120
1
4
1
2
1
−
24
c
−
d
7
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Cambridge IGCSE™ International Mathematics answers
1
Exercise 1.24 page 45
1
2
a
0.75
b
0.8
c
0.45
d
0.34
e
0.3
f
0.375
g
0.4375
h
0.2
a
2.75
b
3.6
c
4.35
d
6.22
e
5.6
f
6.875
g
5.5625
h
4.2
Exercise 1.25 page 46
1
a
b
1
2
7
10
c
3
5
d
3
4
e
33
40
f
1
20
g
1
20
h
201
500
i
1
5000
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Cambridge IGCSE™ International Mathematics answers
2
1
2
5
1
6
2
a
2
b
c
8
1
5
d
3
3
4
e
10
f
9
g
15
91
200
h
30
1
1000
i
1
11
20
51
250
41
2000
Exercise 1.26 page 48
1
White = 47%, Blue = 23%, Red = 30%
2
70%
3
a
60%
b
40%
4
a
b
5
73
100
28
100
c
10
100
d
25
100
a
27%
b
30%
c
14%
d
25%
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Cambridge IGCSE™ International Mathematics answers
6
7
a
0.39
b
0.47
c
0.83
d
0.07
e
0.02
f
0.2
a
31%
b
67%
c
9%
d
5%
e
20%
f
75%
1
Exercise 1.27 page 49
1
2
3
a
20
b
100
c
50
d
36
e
4.5
f
7.5
a
8.5
b
8.5
c
52
d
52
e
17.5
f
17.5
a
Black 6
b
Blonde 3
c
Brown 21
4
Beef 66, Chicken 24, Lamb 18, None 12
5
English 143, Pakistani 44, Greek 11, Other 22
6
Newspapers 69, Pens 36, Books 18, Other 27
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Cambridge IGCSE™ International Mathematics answers
1
Exercise 1.28 page 50
1
a
48%
b
36.8%
c
35%
d
50%
e
45%
f
40%
g
1
33 %
3
h
57% (2 s.f.)
2
1
2
Win 50%, Lose 33 %, Draw 16 %
3
3
3
A = 34.5% (1 d.p.), B = 25.6% (1 d.p.), C = 23.0% (1 d.p.), D = 16.9% (1 d.p.)
4
Red 35.5%, Blue 31.0%, White 17.7%, Silver 6.58%, Green 6%, Black 3.23%
Exercise 1.29 page 51
1
2
3
a
187.5
b
322
c
7140
d
245
e
90
f
121.5
a
90
b
38
c
9
d
900
e
50
f
43.5
a
20%
b
80%
c
110%
d
5%
e
85%
f
225%
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Cambridge IGCSE™ International Mathematics answers
4
a
50%
b
30%
c
5%
d
100%
e
36%
f
5%
5
7475 tonnes
6
6825 BRL
7
a
£75
b
£93.75
a
43
b
17.2%
8
9
1
1100
Exercise 1.30 page 53
1
2
3
a
NZ$72
b
£420
c
¥102
d
252 baht
e
HK$369.60
a
5 years
b
0.41 years (or 151 days)
c
5 years
d
8 years
e
6 years
f
7 years
a
70%
b
4%
c
3.5%
d
7.5%
e
8%
f
11%
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Cambridge IGCSE™ International Mathematics answers
4
a
400 Ft
b
NZ$200
c
€850
d
1200 baht
e
€4000
f
US$1200
5
4%
6
2 years
7
4.5%
8
9.5%
9
AU$315
1
10 6%
Exercise 1.31 page 57
1
$11 033 750
2
$52 087.50
3
$10 368
4
1331 students
5
3 276 800 tonnes
6
2 years
7
5 years
8
3 years
Exercise 1.32 page 58
1
2
a
600
b
350
c
900
d
250
e
125
f
1.5
a
56
b
65
c
90
d
20
e
0.25
f
–38
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Cambridge IGCSE™ International Mathematics answers
3
280 pages
4
12 500 families
5
22
6
12 200 000 m3
1
Exercise 1.33 page 59
1
48 books
2
16 h 40 min
3
11 units
4
a
7500 bricks
b
53 h
a
6250 litres
b
128 km
5
6
1110 km (3 s.f.)
7
a
450
b
75
c
120
a
480 km
b
96 km/h
8
Exercise 1.34 page 61
1
2
a
450 kg
b
1250 kg
a
3 : 10 : 1 : 2
b
Butter 600 g, Flour 2 kg, Sugar 200 g, Currants 400 g
c
120 cakes
3
775 m
4
800
16
= ≈ 2.3 (1 d.p)
350
7
4.95
11
= = 2.2 (1 d.p)
2.25
5
Therefore the 800 g packet is better value for money
5
a
b
24.1 km (1 d.p.)
47
= 2.1 (1 d.p)
22
972
= 1.8 (1 d.p)
530
Therefore the smaller car has the most efficient fuel consumption.
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Cambridge IGCSE™ International Mathematics answers
1
Exercise 1.35 page 62
1
a
60 : 90
b
16 : 24 : 32
c
3.25 : 1.75
d
18 : 27
e
10 : 50
2
7:1
3
Orange 556 ml (3 s.f.), Water 444 ml (3 s.f.)
4
a
11 : 9
b
440 boys, 360 girls
a
3
5
b
32 cm
5
6
4 km and 3 km
7
40°, 80°, 120°, 120°
8
45°, 75°, 60°
9
24-year old $400 000, 28-year old $466 667, 32-year old $533 333
10 Alex $2000, Maria $3500, Ahmet $2500
11 a
b
12 a
16.8 litres
Red 1.2 litres, White 14.3 litres
125
b
Red 216, Yellow 135
c
20
13 a
b
42 litres
Orange juice 495 litres, Mango juice 110 litres
Exercise 1.36 page 63
1
4
2
3
a
Speed (km/h)
60
40
30
120
Time (h)
2
3
4
1
i
90
1
1
3
50
10
2
5
12
2
12 h
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Cambridge IGCSE™ International Mathematics answers
1
4h
ii
iii 48 h
b
i
16
ii
3
iii 48
4
a
30 rows
b
42 chairs
5
6 h 40 min
6
4
7
12 h
Exercise 1.37 page 65
1
2
3
a
69 000
b
74 000
c
89 000
d
4000
e
100 000
f
1 000 000
a
78 500
b
6900
c
14 100
d
8100
e
1000
f
3000
a
490
b
690
c
8850
d
80
e
0
f
1000
Exercise 1.38 page 66
1
a
5.6
b
0.7
c
11.9
d
157.4
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Cambridge IGCSE™ International Mathematics answers
2
e
4.0
f
15.0
g
3.0
h
1.0
i
12.0
a
6.47
b
9.59
c
16.48
d
0.09
e
0.01
f
9.30
g
100.00
h
0.00
i
3.00
1
Exercise 1.39 page 66
1
2
a
50 000
b
48 600
c
7000
d
7500
e
500
f
2.57
g
1000
h
2000
i
15.0
a
0.09
b
0.6
c
0.94
d
1
e
0.95
f
0.003
g
0.0031
h
0.0097
i
0.01
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Cambridge IGCSE™ International Mathematics answers
1
Exercise 1.40 page 67
1
2
3
4
a
420
b
5.05
c
166
d
23.8
e
57.8
f
4430
g
1.94
h
4.11
i
0.575
a
130
b
80
c
9
a
1200
b
3000
c
3000
d
150 000
e
0.8
f
100
a
200
b
200
c
30
d
550
e
500
f
3000
Answers to Q.5 & 6 may vary from those given below.
5
6
a
100 m2
b
44 m2
c
410 cm2
a
200 cm3
b
4000 cm3
c
2000 cm3
Exercise 1.41 page 71
1
a
33
b
25
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Cambridge IGCSE™ International Mathematics answers
2
3
4
c
42
d
64
e
86
f
51
a
2 3 × 32
b
45 × 52
c
32 × 43 × 5 2
d
2 × 74
e
62
f
3 3 × 42 × 6 5
a
4×4
b
5×5×5×5×5×5×5
c
3×3×3×3×3
d
4×4×4×6×6×6
e
7×7×2×2×2×2×2×2×2
f
3×3×4×4×4×2×2×2×2
a
32
b
81
c
64
d
216
e
1 000 000
f
256
g
72
h
125 000
1
Exercise 1.42 page 71
1
2
a
36
b
87
c
59
d
410
e
24
f
3 5 × 66
g
4 8 × 59 × 6 2
h
24 × 510 × 68
a
44
b
53
c
2
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Cambridge IGCSE™ International Mathematics answers
3
4
d
63
e
63
f
8
g
43
h
37
a
54
b
412
c
1010
d
315
e
68
f
86
a
23
b
3
c
53
d
45
e
28
f
6 4 × 85
g
4 3 × 52
h
4 × 67
1
Exercise 1.43 page 72
1
2
a
8
b
25
c
1
d
1
e
1
f
0.25
a
1
4
b
1
9
c
3
50
d
1
200
e
1
f
1
1000
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Cambridge IGCSE™ International Mathematics answers
3
4
a
1
b
2
c
4
d
1
2
e
1
6
f
10
a
12
b
32
c
225
d
80
e
7
f
64
1
Exercise 1.44 page 73
1
2
3
4
a
4
b
5
c
10
d
3
e
9
f
10
a
2
b
3
c
2
d
2
e
6
f
4
a
8
b
32
c
27
d
64
e
1
f
9
a
25
b
8
c
32
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Cambridge IGCSE™ International Mathematics answers
d
100
e
32
f
27
1
Exercise 1.45 page 73
1
2
3
a
1
b
7
c
2
d
1
e
81
f
6
a
25
b
2
c
2
d
27
e
4
f
2
a
4
b
2
c
64
d
9
e
3
f
27
Exercise 1.46 page 76
1
d and e
2
a
6 × 105
b
4.8 × 107
c
7.84 × 1011
d
5.34 × 105
e
7 × 106
f
8.5 × 106
a
6.8 × 106
b
7.2 × 108
c
8 × 105
d
7.5 × 107
3
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Cambridge IGCSE™ International Mathematics answers
4
e
4 × 109
f
5 × 107
a
6 × 105
b
2.4 × 107
c
1.4 × 108
d
3 × 109
e
1.2 × 1013
f
1.8 × 107
5
1.44 × 1011 m
6
a
8.8 × 108
b
2.04 × 1011
c
3.32 × 1011
d
4.2 × 1022
e
5.1 × 1022
f
2.5 × 1025
a
2 × 102
b
3 × 105
c
4 × 106
d
2 × 104
e
2.5 × 106
f
4 × 104
a
4.26 × 105
b
8.48 × 109
c
6.388 × 107
d
3.157 × 109
e
4.5 × 108
f
6.01 × 107
g
8.15 × 1010
h
3.56 × 107
7
8
9
1
Mercury 5.8 × 107 km
Venus 1.08 × 108 km
Earth 1.5 × 108 km
Mars 2.28 × 108 km
Jupiter 7.78 × 108 km
Saturn 1.43 × 109 km
Uranus 2.87 × 109 km
Neptune 4.5 × 109 km
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1
Exercise 1.47 page 77
1
2
3
4
a
6 × 10–4
b
5.3 × 10–5
c
8.64 × 10–4
d
8.8 × 10–8
e
7 × 10–7
f
4.145 × 10–4
a
6.8 × 10–4
b
7.5 × 10–7
c
4.2 × 10–10
d
8 × 10–9
e
5.7 × 10–11
f
4 × 10–11
a
–4
b
–3
c
–8
d
–5
e
–7
f
3
6.8 × 105, 6.2 × 103, 8.414 × 102, 6.741 × 10–4, 3.2 × 10–4, 5.8 × 10–7, 5.57 × 10–9
Exercise 1.48 page 80
1
2
3
a
2 6
b
4 3
c
5 3
d
10 7
e
9 2
a
No simplification possible
b
7 2
c
2 2
d
No simplification possible
e
12 3
a
3
b
5
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Cambridge IGCSE™ International Mathematics answers
4
5
c
2 3
d
2 2
e
2 5
f
7 7
a
3 2
b
4 7
c
5 5
d
−5 2
e
3 10
f
7
a
5+4 2
b
4− 2
c
10 − 2 5
d
−14 + 5 3
e
3+9 2
f
42 − 17 5
1
Exercise 1.49 page 81
1
a
5
5
b
2 7
7
c
2
d
3
e
4 7
7
f
2
g
5
h
2 3
i
15
3
j
7 3
3
k
6 2
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Cambridge IGCSE™ International Mathematics answers
2
1
1+ 3 2
2
a
3+ 6
3
b
9 5
10
c
Exercise 1.50 page 81
1
2 −1
2
3 +1
2
3
4
5
6
7
8
9
(
2(
)
2 + 1) or 2 2 + 2
−5 ( 2 − 5 ) or 5 5 − 10
3 2 + 3 or 6 + 3 3
−7(1−√7)
6
or
7 7 −7
6
3+ 3
6
2�2 + √3� or 4 + 2 3
6+ 5
10 2 2
Exercise 1.51 page 82
1
2
3
a
9.20 a.m., 09 20
b
11.55 a.m., 11 55
a
4.25 p.m., 16 25
b
7.50 p.m., 19 50
a
14 30
b
21 00
c
08 45
d
06 00
e
12 00
f
22 55
g
07 30
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Cambridge IGCSE™ International Mathematics answers
4
5
6
19 30
i
01 00
j
00 00
a
07 15
b
20 00
c
09 10
d
08 45
e
14 45
f
19 40
a
7.20 a.m.
b
9.00 a.m.
c
2.30 p.m.
d
6.25 p.m.
e
11.40 p.m.
f
1.15 a.m.
g
12.05 a.m.
h
11.35 a.m.
i
5.50 p.m.
j
11.59 p.m.
k
4.10 a.m.
l
5.45 a.m.
a
i
08 05
ii
08 40
iii
09 05
iv
09 30
i
18 20
ii
18 45
iii
19 00
iv
19 15
b
7
1
h
a
08 25
b
09 35
c
09 50
d
10 05
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Cambridge IGCSE™ International Mathematics answers
8
1
a
Depart
Arrive
06 15
07 55
06 30
08 10
09 25
11 05
10 20
12 00
13 18
14 58
14 48
16 28
18 54
20 34
19 25
21 05
Depart
Arrive
06 00
08 05
06 45
08 50
08 55
11 00
09 09
11 14
13 48
15 53
14 17
16 22
21 25
23 30
22 05
00 10
Cambridge
04 00
08 35
12 50
19 45
21 10
Stansted
05 15
09 50
14 05
21 00
22 25
Gatwick
06 50
11 25
15 40
22 35
00 00
Heathrow
07 35
12 10
16 25
23 20
00 45
Heathrow
06 25
09 40
14 35
18 10
22 15
Gatwick
08 12
11 27
16 22
19 57
00 02
Stansted
10 03
13 18
18 13
21 48
01 53
Cambridge
11 00
14 15
19 10
22 45
02 50
b
9
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Cambridge IGCSE™ International Mathematics answers
1
10
London
Jo’burg
London
Jo’burg
Sunday
06 15
17 35
14 20
01 40
Monday
07 25
18 45
18 05
05 25
Tuesday
07 20
18 40
15 13
02 33
Wednesday
07 52
19 12
20 10
07 30
Thursday
06 10
17 30
16 27
03 47
Friday
06 05
17 25
20 55
08 15
Saturday
09 55
21 15
18 50
06 10
11
Amsterdam Kuala
Lumpur
Amsterdam Kuala
Lumpur
Amsterdam Kuala
Lumpur
Sunday
08 28
22 13
14 00
03 45
18 30
08 15
Monday
08 15
22 00
13 30
03 15
20 05
09 50
Tuesday
09 15
23 00
15 25
05 10
17 55
07 40
Wednesday
07 50
21 35
14 15
04 00
18 37
08 22
Thursday
07 00
20 45
13 45
03 30
18 40
08 25
Friday
10 25
00 10
15 00
04 45
17 53
07 38
Saturday
10 12
23 57
14 20
04 05
19 08
08 53
12
City
Date
Rome
9 January 2017
Moscow
13 March 2017
Cairo
29 May 2017
Mumbai
30 July 2017
Singapore
20 September 2017
Tokyo
8 December 2017
Los Angeles
23 January 2018
Brasilia
17 February 2018
Paris
16 May 2018
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1
Exercise 1.52 page 87
1
2
3
4
5
6
a
0 h 15 min 0 s
b
3 h 45 min 54 s
c
0 h 13 min 12 s
Arrival at stop 1
06 45
Arrival at stop 2
08 03
Arrival at stop 3
09 42
Arrival at stop 4
12 36 and 36 seconds
a
6 m/s
b
4 m/s
c
39 km/h
d
20 km/h
e
160 km/h
f
50 km/h
a
400 m
b
182 m
c
210 km
d
255 km
e
10 km
f
79.2 km
a
5s
b
50 s
c
4 min
d
71.4 s (3 s.f.)
e
5s
f
4 min
a
105 km/h
b
120 km/h
7
800 km/h
8
a
13.71 h
b
13 h 43 min
9
Thursday 10a.m.
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1
Exercise 1.53 page 89
1
2
3
4
a
5s
b
17 m
a
Speed A = 40 m/s
b
Distance apart = 453
a
2
m/s
3
b
6 m/s,
c
1 m/s
d
1
m
2
e
7
Speed B = 13
1
m/s
3
1
m
3
2
m/s
3
1
m
3
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1
Exercise 1.54 page 90
1
a
45 km/h
b
20 km/h
c
Paul has arrived at the restaurant and is waiting for Helena.
2
3
Exercise 1.55 page 91
1
2
a
A$37.50
b
3750 rupees
c
ZIM$8 256 000
d
4500 rand
e
L520
f
5200 yen
g
140 dinar
h
US$172.50
a
€333.33
b
€2.67
c
€0.02
d
€33.33
e
€187.50
f
€9.23
g
€142.86
h
€130.43
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1
Exercise 1.56 page 92
1
$154.82
2
$182
3
$131
4
$290.50
5
a
$195.05
b
$132.63
6
$137.50
7
€525
8
a
298 rand
b 253.30 rand
Exercise 1.57 page 94
1
$1.80 loss
2
$2.88 profit
3
$54.65 profit per seat
4
$240 extra
5
$250 loss
Exercise 1.58 page 94
1
2
3
a
11%
b
25%
a
30%
b
20%
Type A = 30%
Type B = 15.4%
Type C = 33.3%
Type C makes most profit.
4
80%
Exercise 1.59 page 95
1
a
b
i
Continents of the world
ii
Student’s own answers
i
Even numbers
ii
Student’s own answers
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Cambridge IGCSE™ International Mathematics answers
c
d
e
f
g
h
i
j
k
2
i
Days of the week
ii
Student’s own answers
i
Triangle numbers
ii
Student’s own answers
i
Months with 31 days
ii
Student’s own answers
i
Boy’s names beginning with the letter M
ii
Student’s own answers
i
Prime numbers greater than 7
ii
Student’s own answers
i
Vowels
ii
o, u
i
Planets of the solar system
ii
Student’s own answers
i
Numbers between 3 and 12 inclusive
ii
Student’s own answers
i
Numbers between –5 and 5 inclusive
ii
Student’s own answers
a
7
c
7
e
12
f
Unquantifiably finite, though theoretically infinite
h
5 or 6
i
8
1
Exercise 1.60 page 97
1
2
3
a
Q = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28}
b
R = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}
c
S = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
d
T = {1, 4, 9, 16, 25}
e
U = {1, 3, 6, 10, 15, 21, 28}
a
B = {55, 60, 65}
b
C = {51, 54, 57, 60, 63, 66, 69}
c
D = {64}
a
True
b
True
c
True
d
False
e
True
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Cambridge IGCSE™ International Mathematics answers
1
Exercise 1.61 page 99
1
2
3
a
True
b
True
c
False
d
False
e
False
f
True
a
A ∩ B = {4, 6}
b
A ∩ B = {4, 9}
c
A ∩ B = {Yellow, Green}
a
A ⋃ B = {2, 3, 4, 6, 8, 9, 10, 13, 18}
b
c
4
5
A ⋃ B = {Red, Orange, Blue, Indigo, Violet, Yellow, Green, Purple, Pink}
a
U = {a, b, p, q, r, s, t}
b
A' = {a, b}
a
U = {1, 2, 3, 4, 5, 6, 7, 8}
b
A' = {1, 4, 6, 8}
c
A ∩ B = {2, 3}
d
A ⋃ B = {1, 2, 3, 4, 5, 7, 8}
e
f
6
A ⋃ B = {1, 4, 5, 6, 7, 8, 9, 16}
a
b
(A ∩ B)' = {1, 4, 5, 6, 7, 8}
A ∩ B' = {5, 7}
i
A = {even numbers from 2 to 14}
ii
B = {multiples of 3 from 3 to 15}
iii
C = {multiples of 4 from 4 to 20}
i
A ∩ B = {6, 12}
ii
A ∩ C = {4, 8, 12}
iii
B ∩ C = {12}
iv
A ∩ B ∩ C = {12}
v
A ⋃ B = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15}
vi
7
a
i
A = {1, 2, 4, 5, 6, 7}
ii
B = {3, 4, 5, 8, 9}
iii
C' = {1, 2, 3, 4, 5, 8, 9}
iv
A ∩ B = {4, 5}
v
A ⋃ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}
vi
b
C ⋃ B = {3, 4, 6, 8, 9, 12, 15, 16, 20}
C⊆A
(A ∩ B)' = {1, 2, 3, 6, 7, 8, 9}
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Cambridge IGCSE™ International Mathematics answers
1
Exercise 1.62 page 100
1
a
b
2
a
b
3
i
ii
i
ii
A ∩ B = {Egypt}
A ⋃ B = {Libya, Morocco, Chad, Egypt, Iran, Iraq, Turkey}
P ∩ Q = {11, 13, 17}
P ⋃ Q = {2, 3, 5, 7, 11, 13, 15, 17, 19}
4
5
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Cambridge IGCSE™ International Mathematics answers
1
Exercise 1.63 page 102
1
a
5
b
14
c
13
2
45
3
a
10
b
50
4
a
b
100
Primes and squares page 105
1
22 + 32 = 13
2
42 + 52 = 41
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
The following prime numbers can be expressed as the sum of two different square numbers:
5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97
4
If a number generated by the rule 4n + 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, add 1 to the
prime number and divide the result by 2. If the answer is even then the prime number cannot be
expressed as the sum of two squares.
5
The rule works for the numbers shown, but this does not prove that it always works.
Football league page 106
1
(17 + 16 + 15 + … + 1) × 2 = 306
2
n = t(t – 1)
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Cambridge IGCSE™ International Mathematics answers
1
The school trip page 106
2
3
4
a
$970
b
$1240
c
$5510
a
$970.00
b
$124.00
c
$73.47
57 students
ICT Activity 1 page 107
3
e.g. in cell C4 enter =B4/B$3*100
Student assessment 1 page 108
1
a
b
c
2
3
4
5
6
a
1
8
1
1
5
1
4
7
=5
=
7
4
18 = 3 2
b
35
c
24 = 2 6
a
4 3
b
5 6
c
2 2
a
1− 2 2
b
49 − 12 5
a
3 5
5
b
10
2
c
4 2+4
d
7−4 3
a
Irrational as the total length is 3 3
b
Rational as the area is 3 5
(
)
2
= 45 cm
2
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Cambridge IGCSE™ International Mathematics answers
7
a
b
8
a
1
(2 + 3)2 × 4 = 100
(20 + 4) ÷ 22 = 6
10
b 11
9
a
b
c
d
e
f
10 a
b
c
d
4 × 4 × 4 = 64
15 × 15 = 225
10 × 10 × 10 = 1000
14
5
1
1
5
1
23
1
8
3
2
(2 )
=
1
= 82 = 64
e
812 = √81 = 9
f
2
1
8−3 =
1
1
83
=
1
3
√8
=
1
2
Student assessment 2 page 109
1
Fraction
Decimal
Percentage
1
4
0.25
25%
3
5
0.6
60%
5
8
0.625
62.5%
2 14
2.25
225%
2
750 m
3
€525
4
£97 200
5
a
29.2% (3 s.f.)
b
21.7% (3 s.f.)
c
125%
d
8.33% (3 s.f.)
e
20%
f
10%
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Cambridge IGCSE™ International Mathematics answers
6
8.3% (1 d.p.)
7
a
¥6500
b
61.8% (3 s.f.)
8
8 days
9
€955 080
1
10 5 years
Student assessment 3 page 110
1
a
5
1
7
b
7
2
9
c
14
1
5
2
$200, $25, $524, $10
3
$462, $4000, $4500, $5500
4
15 marks
5
35 000 television sets
6
25 000 units
7
470 tonnes
8
200 bars
9
a
HCF of 60 and 24 is 12.
60 ÷ 12 = 5 and 24 ÷ 12 = 2
Therefore ratio in simplest form is 5 : 2
b
5
2
= 2.5
130 × 2.5 = 325
Total volume is 130 + 325 = 455 ml
Student assessment 4 page 110
1
2
a
6470
b
88 500
c
65 000
d
10
a
6.8
b
4.44
c
8.0
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Cambridge IGCSE™ International Mathematics answers
3
d
63.08
e
0.057
f
3.95
a
40
b
5.4
c
0.06
d
49 000
e
700 000
f
687 000
1
Answers to Q.4–6 may vary from those given below.
4
34 000 yards
5
150 cm2
6
a
20
b
1
c
3
7
30 m3 (2 s.f.)
8
1.41%
9
a
168.02 cm2
b
10.7% (answers may vary)
10 a
875 7
=
1000 8
b
3 25 = 3 × 5 = 15 =
c
44 11
=
100 25
15
1
Student assessment 5 page 112
1
a
6 × 106
b
4.5 × 10–3
c
3.8 × 109
d
3.61 × 10–7
e
4.6 × 108
f
3 × 100
2
4.05 × 108, 3.6 × 102, 9 × 101, 1.5 × 10–2, 7.2 × 10–3, 2.1 × 10–3
3
a
1.5 × 107, 4.3 × 105, 4.35 × 10–4, 4.8 × 100, 8.5 × 10–3
b
4.35 × 10–4, 8.5 × 10–3, 4.8 × 100, 4.3 × 105, 1.5 × 107
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Cambridge IGCSE™ International Mathematics answers
4
a
1.2 × 108
b
1.48 × 1011
c
6.73 × 107
d
3.88 × 106
5
43.2 minutes (3 s.f.)
6
2.84 × 1015 km (3 s.f.)
7
$114.80
8
a
b
c
d
1
34
35
14
15
13
8
17
5
Student assessment 6 page 112
1
2
a
2 h 24 min 0 s
b
0 h 48 min 0 s
c
3 h 33 min 36 s
a
6.4 h
b
0.2575 h
1.201̇ h
c
1
km
2
3
5
4
a
30 km/h
b
6.4 h or 6 h 24 m
5
a
b
180 km
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Cambridge IGCSE™ International Mathematics answers
6
7
8
9
1
a
b
Distance from M ≈ 77 km
c
Time ≈ 1 h 13 min after start
a
b
25 min
a
B, C
b
B because it illustrates going back in time, C because it illustrates infinite speed
1.5 km2 and 3.75 km2
10 880 students
Student assessment 7 page 113
1
2
3
a
{even numbers from 2 to 8}
b
{even numbers}
c
{square numbers}
d
{oceans}
a
7
b
2
c
6
d
366
a
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Cambridge IGCSE™ International Mathematics answers
1
b
4
{a, b}, {a}, {b}, ∅
5
A' = {m, t, h}
Student assessment 8 page 113
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
3
4
b
{Ankara, Cairo}
c
{Nairobi, Pretoria}
a
b
{2, 3, 5, 7, 11, 13, 17, 19}
c
{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 23, 29}
a
b
X = {multiples of 10}
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5
1
a
b
6
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Cambridge IGCSE™ International Mathematics answers
2
2 Algebra
All answers were written by the authors.
Exercise 2.1 page 116
1
2
3
4
a
4x – 12
b
10p – 20
c
–42x + 24y
d
6a – 9b – 12c
e
–14m + 21n
f
–16x + 6y
a
3x2 – 9xy
b
a2 + ab + ac
c
8m2 – 4mn
d
–15a2 + 20ab
e
4x2 – 4xy
f
24p2 – 8pq
a
–2x2 + 3y2
b
a–b
c
7p – 2q
d
3x – 4y + 2z
e
3x –
f
2x2 – 3xy
a
12r3 – 15rs + 6rt
b
a3 + a2b + a2c
c
6a3 – 9a2b
d
p2q + pq2 – p2q2
e
m3 – m2n + m3n
f
a 6 + a 5b
3
y
2
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Cambridge IGCSE™ International Mathematics answers
2
Exercise 2.2 page 116
1
2
3
a
–a – 8
b
4x – 20
c
3p – 16
d
21m – 6n
e
3
f
–p – 3p2
a
8m2 + 28m + 2
b
x–4
c
2p + 22
d
m – 12
e
a2 + 6a + 2
f
7ab – 16ac + 3c
a
4x + 4
b
5x –
c
9
5
x– y
4
2
d
9
1
x+ y
2
2
e
7x – 4y
f
0
3
y
2
Exercise 2.3 page 117
1
2
a
2(2x – 3)
b
6(3 – 2p)
c
3(2y – 1)
d
2(2a + 3b)
e
3(p – q)
f
4(2m + 3n + 4r)
a
a(3b + 4c – 5d)
b
2p(4q + 3r – 2s)
c
a(a – b)
d
2x(2x – 3y)
e
ab(c + d + f)
f
3m(m + 3)
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3
4
5
6
a
3pq(r – 3s)
b
5m(m – 2n)
c
4xy(2x – y)
d
b2(2a2 – 3c2)
e
12(p – 3)
f
6(7x – 9)
a
6(3 + 2y)
b
7(2a – 3b)
c
11x(1 + y)
d
4(s – 4t + 5r)
e
5q(p – 2r + 3s)
f
4y(x + 2y)
a
m(m + n)
b
3p(p – 2q)
c
qr(p + s)
d
ab(1 + a + b)
e
p3(3 – 4p)
f
b2c(7b + c)
a
m(m2 – mn + n2)
b
2r2(2r – 3 + 4s)
c
28xy(2x – y)
d
18mn(4m + 2n – mn)
2
Exercise 2.4 page 119
1
2
a
0
b
30
c
14
d
20
e
–13
f
–4
a
–3
b
–30
c
20
d
–16
e
–40
f
42
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Cambridge IGCSE™ International Mathematics answers
3
4
5
a
–160
b
–23
c
42
d
–17
e
–189
f
113
a
48
b
–8
c
15
d
16
e
–5
f
9
a
12
b
–5
c
–5
d
7
e
7
f
36
2
Exercise 2.5 page 120
1
2
a
n=r–m
b
m=p–n
c
n = 3p – 2m
d
q = 3x – 2p
e
a=
cd
b
f
d=
ab
c
a
x=
4m
3y
b
r=
7 pq
5
c
x=
c
3
d
x=
y−7
3
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Cambridge IGCSE™ International Mathematics answers
3
4
5
2
e
y=
3r + 9
5
f
x=
5y − 9
3
a
b=
b
a=
6b + 5
2
c
z=
3x − 7 y
4
d
𝑥𝑥 =
2a − 5
6
4𝑧𝑧+7𝑦𝑦
3
e
y=
3x − 4 z
7
f
p=
8+q
2r
a
p = 4r
b
p=
4
3r
c
p=
n
10
d
n = 10p
e
p=
2t
q+r
f
q=
2t
−r
p
a
r=
3m − n
t ( p + q)
b
𝑡𝑡 =
3𝑚𝑚−𝑛𝑛
𝑟𝑟(𝑝𝑝+𝑞𝑞)
rt ( p + q ) + n
c
m=
d
n = 3m – rt(p + q)
e
p=
3m − n
−q
rt
f
q=
3m − n
−p
rt
3
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Cambridge IGCSE™ International Mathematics answers
6
2
a
d=
ab
ce
b
a=
cde
b
c
c=
ab
de
d
a = cd – b
e
b= d−
f
c=
a
c
a
d −b
Exercise 2.6 page 121
1
2
3
4
a
x2 + 5x + 6
b
x2 + 7x + 12
c
x2 + 7x + 10
d
x2 + 7x + 6
e
x2 + x – 6
f
x2 + 5x – 24
a
x2 + 2x – 24
b
x2 – 3x – 28
c
x2 – 2x – 35
d
x2 – 2x – 15
e
x2 – 2x – 3
f
x2 + 2x – 63
a
x2 – 5x + 6
b
x2 – 7x + 10
c
x2 – 12x + 32
d
x2 + 6x + 9
e
x2 – 6x + 9
f
x2 – 12x + 35
a
x2 – 9
b
x2 – 49
c
x2 – 64
d
x2 – y2
e
a2 – b2
f
p2 – q2
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Cambridge IGCSE™ International Mathematics answers
5
a
b
c
d
e
f
6
7
8
9
a
2
2𝑦𝑦 2 + 3𝑦𝑦 + 2𝑥𝑥𝑥𝑥 + 3𝑥𝑥
2𝑦𝑦 2 − 𝑥𝑥𝑥𝑥 − 𝑥𝑥 2
2𝑥𝑥𝑥𝑥 + 2𝑦𝑦 2 + 𝑥𝑥 + 𝑦𝑦
4𝑝𝑝2 − 𝑞𝑞 2
6𝑚𝑚2 + 15𝑚𝑚 + 8𝑚𝑚𝑚𝑚 + 20𝑛𝑛
12𝑎𝑎2 − 3𝑏𝑏 2
2p2 + 13p – 24
b
4p2 + 23p – 35
c
6p2 + p – 12
d
12p2 + 13p – 35
e
18p2 – 2
f
28p2 + 44p – 24
a
4x2 – 4x + 1
b
9x2 + 6x + 1
c
16x2 – 16x + 4
d
25x2 – 40x + 16
e
4x2 + 24x + 36
f
4x2 – 9
a
–4x2 + 9
b
16x2 – 9
c
–16x2 + 9
d
–25y2 + 49
e
8y2 – 18
f
25y2 – 70y + 49
a
3x3 + 16x2 + 23x + 6
b
4x3 + 2x2 – 16x – 8
c
–12x3 + 7x2 + 8x – 3
d
–2x3 + 9x2 + 8x – 15
e
–2x3 + 11x2 – 13x + 4
f
–12x3 – 9x2 – 5x + 2
Exercise 2.7 page 123
1
a
(x + y)(a + b)
b
(x – y)(a + b)
c
(3 + x)(m + n)
d
(m + n)(4 + x)
e
(m – n)(3 + x)
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Cambridge IGCSE™ International Mathematics answers
2
3
4
f
(x + z)(6 + y)
a
(p + q)(r – s)
b
(p + 3)(q – 4)
c
(q – 4)(p + 3)
d
(r + 2t)(s + t)
e
(s + t)(r – 2t)
f
(b + c)(a – 4c)
a
(y + x)(x + 4)
b
(x – 2)(x – y)
c
(a – 7)(b + 3)
d
(b – 1)(a – 1)
e
(p – 4)(q – 4)
f
(m – 5)(n – 5)
a
(m – 3)(n – 2)
b
(m – 3r)(n – 2r)
c
(p – 4q)(r – 4)
d
(a – c)(b – 1)
e
(x – 2y)(x – 2z)
f
(2a + b)(a + b)
2
Exercise 2.8 page 124
1
2
3
a
(a – b)(a + b)
b
(m – n)(m + n)
c
(x – 5)(x + 5)
d
(m – 7)(m + 7)
e
(9 – x)(9 + x)
f
(10 – y)(10 + y)
a
(12 – y)(12 + y)
b
(q – 13)(q + 13)
c
(m – 1)(m + 1)
d
(1 – t)(1 + t)
e
(2x – y)(2x + y)
f
(5p – 8q)(5p + 8q)
a
(3x – 2y)(3x + 2y)
b
(4p – 6q)(4p + 6q)
c
(8x – y)(8x + y)
d
(x – 10y)(x + 10y)
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Cambridge IGCSE™ International Mathematics answers
4
e
(pq – 2p)(pq + 2p)
f
(ab – cd)(ab + cd)
a
(mn – 3y)(mn + 3y)
b
1  1
1 
1
 2 x − 3 y  2 x + 3 y 



c
(p2 + q2)(p + q)(p – q)
d
4(m2 – 3y2)(m2 + 3y2)
e
(4x2 + 9y2)(2x + 3y)(2x – 3y)
f
(2x – 9y2)(2x + 9y2)
2
Exercise 2.9 page 125
1
2
3
4
a
60
b
240
c
2400
d
280
e
7600
f
9200
a
2000
b
9800
c
200
d
3200
e
998 000
f
161
a
68
b
86
c
1780
d
70
e
55
f
5
a
72.4
b
0.8
c
65
d
15
e
1 222 000
f
231
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Cambridge IGCSE™ International Mathematics answers
2
Exercise 2.10 page 126
1
2
3
4
5
6
a
(x + 4)(x + 3)
b
(x + 6)(x + 2)
c
(x + 12)(x + 1)
d
(x – 3)(x – 4)
e
(x – 6)(x – 2)
f
(x – 12)(x – 1)
a
(x + 5)(x + 1)
b
(x + 4)(x + 2)
c
(x + 3)2
d
(x + 5)2
e
(x + 11)2
f
(x – 6)(x – 7)
a
(x + 12)(x + 2)
b
(x + 8)(x + 3)
c
(x – 6)(x – 4)
d
(x + 12)(x + 3)
e
(x + 18)(x + 2)
f
(x – 6)2
a
(x + 5)(x – 3)
b
(x – 5)(x + 3)
c
(x + 4)(x – 3)
d
(x – 4)(x + 3)
e
(x + 6)(x – 2)
f
(x – 12)(x – 3)
a
(x – 4)(x + 2)
b
(x – 5)(x + 4)
c
(x + 6)(x – 5)
d
(x + 6)(x – 7)
e
(x + 7)(x – 9)
f
(x + 9)(x – 6)
a
(2x + 2)(x + 1)
b
(2x + 3)(x +2)
c
(2x – 3)(x + 2)
d
(2x – 3)(x – 2)
e
(3x + 2)(x + 2)
f
(3x – 1)(x + 4)
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Cambridge IGCSE™ International Mathematics answers
7
g
(2x + 3)2
h
(3x – 1)2
i
(3x + 1)(2x – 1)
a
𝑥𝑥(𝑥𝑥 − 2)(𝑥𝑥 + 1)
b
c
d
e
f
g
h
2
𝑥𝑥(𝑥𝑥 + 1)(𝑥𝑥 − 1)
𝑥𝑥(𝑥𝑥 + 2)(𝑥𝑥 + 1)
𝑥𝑥(𝑥𝑥 + 3)(𝑥𝑥 − 1)
2𝑥𝑥(𝑥𝑥 − 1)(𝑥𝑥 − 4)
𝑥𝑥(𝑥𝑥 + 5)2
𝑥𝑥(3𝑥𝑥 + 1)(𝑥𝑥 − 1)
2𝑥𝑥(3𝑥𝑥 − 2)(2𝑥𝑥 + 3)
Exercise 2.11 page 127
1
2
3
P
2m
a
x=
b
x= ±
T
3
c
x= ±
y2
m
d
x=
± p2 − q2 − y2
e
x=
± y 2 − n2 − m2
f
x= ±
a
x=
b
x= ±
P
Qr
c
x= ±
Pr
Q
d
x= ±
n
m
e
x= ±
wst
r
f
x= ±
wr
p+q
a
x = (rp)2
p2 − q2 + y2
4
P
Qr
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62
Cambridge IGCSE™ International Mathematics answers
4
b
 mn 
x=

 p 
c
x=
d
x=
r2g
4π 2
e
x=
4m 2 r
p2
f
x=
4m 2 r
p2
a
𝑥𝑥 = −
b
c
𝑥𝑥 =
𝑥𝑥 =
2
2
k
g2
8𝑦𝑦
3
−3𝑦𝑦
4−𝑎𝑎
2
𝑦𝑦+3
Exercise 2.12 page 127
1
t=
b
u=
± v 2 − 2as
c
s=
d
1
s − at 2
2
u=
t
e
a=
f
2
v−u
a
a
v2 − u 2
2a
2 ( s − ut )
t2
t= ±
2( s − ut )
a
A
a
r=
b
A
± ( )2 − r 2
h=
πr
c
𝑢𝑢 =
d
v=
π s2 + t 2
𝑣𝑣𝑣𝑣
𝑣𝑣−𝑓𝑓
fu
u− f
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Cambridge IGCSE™ International Mathematics answers
2
2
e
 t 
l =
 g
 2π 
f
 2π 
g = l

 t 
2
Exercise 2.13 page 128
1
2
3
4
a
xp
yq
b
q
y
c
p
r
d
d
c
e
bd
c2
f
p
a
m2
b
r5
c
x6
d
xy2
e
abc3
f
r3
pq
a
2x
y
b
4q2
c
5n
d
3x3y
e
3p
f
2
3mn
a
2a
3b
b
2y
x
c
2
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Cambridge IGCSE™ International Mathematics answers
5
d
3xy
e
3
f
6xy
a
8y
3
b
5p
2
c
p3
s
d
ef
c2
e
4qm
9p
f
x3
yz
2
Exercise 2.14 page 129
1
2
a
4
7
b
a+b
7
c
11
13
d
c+d
13
e
x+ y+z
3
f
p2 + q2
5
a
3
11
b
c−d
11
c
4
a
d
2a − 5b
3
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Cambridge IGCSE™ International Mathematics answers
3
4
5
e
2x − 3y
7
f
−
a
1
2
b
3
2a
c
5
3c
d
7
2x
e
3
2p
f
−
a
3p − q
12
b
x − 2y
4
c
3m − n
9
d
x − 2y
12
e
5r + m
10
f
5s − t
15
a
7x
12
b
9x − 2 y
15
c
m
2
d
m
p
e
x
2y
f
4r
7s
2
1
2x
1
2w
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Cambridge IGCSE™ International Mathematics answers
2
Exercise 2.15 page 130
1
2
3
4
a
5
6
b
8
15
c
11
28
d
11
15
e
29
36
f
24
35
a
3a + 2b
6
b
5a + 3b
15
c
7 p + 4q
28
d
6a + 5b
15
e
9 x + 20 y
36
f
10 x + 14 y
35
a
a
6
b
2a
15
c
11 p
28
d
11a
15
e
29 x
36
f
24 x
35
a
m
10
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Cambridge IGCSE™ International Mathematics answers
5
6
7
b
r
10
c
−
d
29 x
28
e
23 x
6
f
p
6
a
p
2
b
2c
3
c
4x
5
d
m
3
e
q
5
f
w
4
a
3m
2
2
x
4
b
7m
3
c
−
d
5m
2
e
p
3
f
36q
7
a
pr − p
r
b
x + xy
y
c
mn + m
n
m
2
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Cambridge IGCSE™ International Mathematics answers
d
a + ab
b
e
2xy − x
y
f
2 pq − 3 p
q
2
Exercise 2.16 page 131
1
2
3
a
3x + 4
2x2
b
10 − x 2
2 x4
c
7x + 4
x( x + 1)
a
(3 x + 4)
( x + 1)( x + 2)
b
m−7
( m + 2 )( m − 1)
c
3p − 7
( p − 3)( p − 2 )
d
w + 11
( w − 1) ( w + 3)
e
3y
( y + 4 )( y + 1)
f
12 − m
( m − 2 )( m + 3)
a
x
x+2
b
y
y+4
c
m+2
m−3
d
p
p −5
e
m
m+4
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Cambridge IGCSE™ International Mathematics answers
4
5
f
m +1
m+2
a
x
x+3
b
x
x+4
c
y
y −1
d
x
x+3
e
x
x+2
f
x
x +1
a
x
x +1
b
x
x+3
c
x
x−3
d
x
x+2
e
x
x−3
f
x
x+7
2
Exercise 2.17 page 132
1
2
a
x = –4
b
y=5
c
y = –5
d
p = –4
e
y=8
f
x = –5.5
a
x = 41
b
x=5
c
x=6
d
y = –8
3
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70
Cambridge IGCSE™ International Mathematics answers
3
4
5
6
7
e
y=4
f
m = 10
a
m=1
b
p=3
c
k = –1
d
x = –21
e
x=2
f
y=3
a
x=6
b
y = 14
c
x=4
d
m = 12
e
x = 35
f
p=8
a
x = 15
b
x = –5
c
x = 7.5
d
x=8
e
x = 2.5
f
x = 10
a
x=5
b
x = 14
c
x = 22
d
x=5
e
x=8
f
x=2
a
y = 10
b
x = 17
c
x = 13
d
y = –9
e
x=4
f
x = 6.5
2
Exercise 2.18 page 134
1
a
i
3x + 60 = 180
ii
x = 40
iii 40°, 60°, 80°
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b
i
3x = 180
ii
x = 60
2
iii 20°, 80°, 80°
c
i
18x = 180
ii
x = 10
iii 20°, 50°, 110°
d
i
6x = 180
ii
x = 30
iii 30°, 60°, 90°
e
i
7x – 30 = 180
ii
x = 30
iii 10°, 40°, 130°
f
i
9x – 45 = 180
ii
x = 25
iii 25°, 55°, 100°
2
a
i
12x = 360
ii
x = 30
iii 90°, 120°, 150°
b
i
11x + 30 = 360
ii
x = 30
iii 90°, 135°, 135°
c
i
12x + 60 = 360
ii
x = 25
iii 35°, 80°, 90°, 155°
d
i
10x + 30 = 360
ii
x = 33
iii 33°, 94°, 114°, 119°
3
a
i
11x – 80 = 360
ii
x = 40
iii 40°, 80°, 80°, 160°
b
i
10x + 60 = 360
ii
x = 30
iii 45°, 90°, 95°, 130°
c
i
16x + 8 = 360
ii
x = 22
iii 44°, 96°, 100°, 120°
d
i
9x + 45 = 360
ii
x = 35
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2
iii 35°, 55°, 120°, 150°
e
i
6x + 18 = 180
ii
x = 27
iii 50°, 50°, 130°, 130°
4
5
6
a
20
b
25
c
14
d
25
e
31
f
40
a
50
b
13
c
40
d
40
a
5
b
2
c
7
d
1.1
e
25
f
15
Exercise 2.19 page 139
1
2
3
4
a
132.7 cm2 (1 d.p.)
b
r=
c
12.6 cm (1 d.p.)
a
1206 cm3 (4 s.f.)
b
5.4 cm (1 d.p.)
a
37.8 °C (1 d.p.)
b
F=
c
50 °F
a
125 m
b
t=
c
25.8 s (1 d.p.)
A
π
9
C + 32
5
2s
a
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5
6
7
8
9
2
1
(a + b) h
2
a
A=
b
20 cm2
c
5 cm
a
C = 2.5n + 5
b
n=
c
30 km
a
$9.40
b
C = 10 – 0.2x
c
C = 10 −
a
T = xe
b
T = xe + yl
c
C = 20 – (xe + yl + zc)
a
C = 15.5n + 20
b
i
n=
ii
23.5 metres
C −5
2 12
xy
100
C − 20
15.5
10 a
L = 50 – 2x
b
W = 30 – 2x
c
H=x
d
V = x(50 – 2x)(30 – 2x)
e
1872 cm2
Exercise 2.20 page 148
1
a
x = 4, y = 2
b
x = 6, y = 5
c
x = 6, y = –1
d
x = 5, y = 2
e
x = 5, y = 2
f
x = 4, y = 9
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2
3
4
5
6
7
2
a
x = 3, y = 2
b
x = 7, y = 4
c
x = 1, y = 1
d
x = 1, y = 5
e
x = 1, y = 10
f
x = 8, y = 2
a
x = 5, y = 4
b
x = 4, y = 3
c
x = 10, y = 5
d
x = 6, y = 4
e
x = 4, y = 4
f
x = 10, y = –2
a
x = 5, y = 4
b
x = 4, y = 2
c
x = 5, y = 3
d
x = 5, y = –2
e
x = 1, y = 5
f
x = –3, y = –3
a
x = –5, y = –2
b
x = –3, y = –4
c
x = 4, y = 3
d
x = 2, y = 7
e
x = 1, y = 1
f
x = 2, y = 9
a
x = 2, y = 3
b
x = 5, y = 10
c
x = 4, y = 6
d
x = 4, y = 4
a
x = 1, y = –1
b
2
x = 11 , y = 8
3
c
x = 4, y = 0
d
x = 3, y = 4
e
x = 2, y = 8
f
x = 1, y = 1
2
3
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2
Exercise 2.21 page 146
1
2
3
4
a
x = 2, y = 3
b
x = 1, y = 4
c
x = 5, y = 2
d
x = 3, y = 3
e
x = 4, y = 2
f
x = 6, y = 1
a
x = 1, y = 4
b
x = 5, y = 2
c
x = 3, y = 3
d
x = 6, y = 1
e
x = 2, y = 3
f
x = 2, y = 3
a
x = 0, y = 3
b
x = 5, y = 2
c
x = 1, y = 7
d
x = 6, y = 4
e
x = 2, y = 5
f
x = 3, y = 0
a
x = 1, y = 0.5
b
x = 2.5, y = 4
c
x=
1
,y=4
5
d
x=
3
1
,y=
4
2
e
x = 5, y =
f
x=
1
3
1
,y = 1
2
Exercise 2.22 page 148
1
10 and 7
2
16 and 9
3
x = 1, y = 4
4
x = 5, y = 2
5
60 and 20 years old
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6
2
60 and 6 years old
Exercise 2.23 page 148
1
a
7
b
3
c
6
d
7
e
10
f
2
3
4
16
2
3
a
7
b
6
c
6
d
10
e
3
a
8
b
5
c
8
d
6
e
6
a
6, 18, 26
b
160, 214, 246, 246
c
50°, 80°, 100°, 140°, 170°
d
80°, 80°, 80°, 160°, 160°, 160°
e
150°, 150°, 150°, 150°, 120°, 120°, 120°, 120°
Exercise 2.24 page 151
1
2
a
–4 and –3
b
–2 and –6
c
–1 and –12
d
2 and 5
e
2 and 3
f
2 and 4
a
–5 and 2
b
–2 and 5
c
–7 and 2
d
–2 and 7
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Cambridge IGCSE™ International Mathematics answers
3
4
e
–5 and 3
f
–3 and 5
a
–3 and –2
b
–3
c
–8 and –3
d
4 and 6
e
–4 and 3
f
–2 and 6
a
–2 and 4
b
–4 and 5
c
–6 and 5
d
–6 and 7
e
–7 and 9
f
–9 and 6
2
Exercise 2.25 page 151
1
2
3
4
a
–3 and 3
b
–4 and 4
c
–5 and 5
d
–11 and 11
e
–12 and 12
f
–15 and 15
a
–2.5 and 2.5
b
–2 and 2
c
–1.6 and 1.6
d
−
1
and 1
2
2
e
−
1
and 1
3
3
f
−
1
and 1
20
20
a
–4 and –1
b
–5 and –2
c
–4 and –2
d
2 and 4
e
2 and 5
f
–4 and 2
a
–2 and 5
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Cambridge IGCSE™ International Mathematics answers
5
6
7
8
b
–5 and 2
c
–3 and 6
d
–6 and 3
e
–4 and 6
f
–6 and 8
a
–4 and 3
b
–6 and –2
c
–9 and 4
d
–1
e
–2
f
–9 and –8
a
0 and 8
b
0 and 7
c
–3 and 0
d
–4 and 0
e
0 and 9
f
0 and 4
a
–1.5 and –1
b
–1 and 2.5
c
–1 and
d
–5 and − 1
2
e
1.5 and 5
f
1
1
−1 and
2
3
a
–12 and 0
b
–9 and –3
c
–8 and 4
d
–7 and 2
e
–6 and 6
f
–10 and 10
2
1
3
Exercise 2.26 page 151
1
–4 and 3
2
–6 and 7
3
2
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2
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
x=4
c
11 m × 6 m
Exercise 2.27 page 154
1
2
3
4
a
–3.14 and 4.14
b
–5.87 and 1.87
c
–6.14 and 1.14
d
–4.73 and –1.27
e
–6.89 and 1.89
f
3.38 and 5.62
a
–5.30 and –1.70
b
–5.92 and 5.92
c
–3.79 and 0.79
d
–1.14 and 6.14
e
–4.77 and 3.77
f
–2.83 and 2.83
a
–0.73 and 2.73
b
–1.87 and 5.87
c
–1.79 and 2.79
d
–3.83 and 1.83
e
0.38 and 2.62
f
0.39 and 7.61
a
–0.85 and 2.35
b
–1.40 and 0.90
c
0.14 and 1.46
d
–0.5 and –2
e
–0.39 and 1.72
f
–1.54 and 1.39
Exercise 2.28 page 158
1
a
(3, 2)
b
(5, 2)
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Cambridge IGCSE™ International Mathematics answers
2
3
c
(2, 1)
d
(2, 1)
e
(–4, 1)
f
(4, −2)
a
(3, −2)
b
(−l, −1)
c
(–2, 3)
d
(−3, −3)
e
Infinite solutions
f
No solution
2
The lines in part (e) are the same line. The lines in part (f) are parallel.
Exercise 2.29 page 161
1
a
i
ii
b
(1, 0) and (2, 0)
i
ii
(–6, 0) and (2, 0)
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c
ii
d
No zeros
i
ii
f
(3, 0) and (5, 0)
i
ii
e
2
i
No zeros
i
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ii
2
a
i
ii
b
(4, 0) and (6, 0)
i
ii
d
(–2, 0) and (3, 0)
i
ii
c
2
(3, 0)
(–2, 0) and (8, 0)
i
ii
(–5, 0)
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2
Exercise 2.30 page 163
1
2
3
a
x<4
b
x>1
c
x≤3
d
x≥7
e
x<2
f
x>2
a
x<7
b
x ≥ –2
c
x > –3
d
x ≥ –12
e
x > –24
f
x ≥ –3
a
x<2
b
x ≤ 12
c
x≥2
d
x ≥ –2
e
x > –0.5
f
x≥2
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2
Exercise 2.31 page 163
1
2
a
2<x≤4
b
1≤x<5
c
3.5 ≤ x < 5
d
2 ≤ x < 4.2
a
1<x≤5
b
–1 ≤ x < 1
c
2<x<3
d
No solution
Exercise 2.32 page 166
1
a
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Cambridge IGCSE™ International Mathematics answers
2
b
c
d
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Cambridge IGCSE™ International Mathematics answers
2
e
f
2
a
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Cambridge IGCSE™ International Mathematics answers
2
b
c
d
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Cambridge IGCSE™ International Mathematics answers
2
e
f
Exercise 2.33 page 168
1
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Cambridge IGCSE™ International Mathematics answers
2
2
3
4
Exercise 2.34 page 173
1
a
b
c
−7 < 𝑥𝑥 < 3
𝑥𝑥 < −8 and 𝑥𝑥 > 8
𝑥𝑥 ≤ −4 and 𝑥𝑥 ≥ −2
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2
3
2
− ≤ 𝑥𝑥 ≤ 6
d
Exercise 2.35 page 174
1
2
3
a
c8
b
m2
c
b9
d
m3n6
e
2a4b
f
3x3y2
g
uv3
2
h
x2 y3 z 2
3
a
12a5
b
8a5b3
c
8p6
d
16m4n6
e
200p13
f
32m5n11
g
24xy3
h
a(d + e)b(d + e) or (ab)(d + e)
a
𝑎𝑎
b
c
d
𝑥𝑥 4
2
𝑚𝑚5
𝑚𝑚2 𝑛𝑛
Exercise 2.36 page 175
1
2
a
2
b
4
c
3
d
3
e
4
f
0
a
4
b
1
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Cambridge IGCSE™ International Mathematics answers
3
4
5
6
2
c
3
2
d
–1
e
2
f
3
a
–3
b
–4
c
–5
d
−
e
–2
f
−
a
–3
b
–7
c
–3
d
2
e
8
f
5
a
1
2
b
1
3
c
1
2
d
1
3
e
1
2
f
1
6
a
1
3
b
1
2
c
1
4
d
1
3
2
3
1
2
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e
1
6
f
1
3
2
Exercise 2.37 page 176
1
a
7 cm
b
0 cm
c
5 hours
d
2
3
4
1
hours
2
e
Approx. 5
a
Approx. 2.5
b
−
a
Approx. 4.3
b
Approx. 3.3
c
Approx. 2.3
a
Approx. 0.3
b
Approx. 2.4
1
2
Exercise 2.38 page 179
1
a
b
i
3n + 2
ii
32
i
4n − 4
ii
36
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Cambridge IGCSE™ International Mathematics answers
c
d
e
f
2
i
n − 0.5
ii
9.5
i
−3n + 9
ii
−21
i
3n – 10
ii
20
i
−4n − 5
ii
−45
2
a
Position
1
2
5
12
50
n
Term
1
5
17
45
197
4n – 3
Position
1
2
5
10
75
n
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
Position
2
5
7
10
50
n
Term
1
10
16
25
145
3n – 5
b
c
d
e
f
3
a
Position
1
2
5
20
50
n
Term
–5.5
–7
–11.5
–34
–79
–1.5n – 4
i
+4
ii
4n + 1
iii 201
b
i
+1
ii
n−1
iii 49
c
i
+3
ii
3n – 13
iii 137
d
i
+0.5
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Cambridge IGCSE™ International Mathematics answers
ii
2
0.5n + 5.5
iii 30.5
e
i
+4
ii
4n − 62
iii 138
f
i
−3
ii
−3n + 75
iii −75
Exercise 2.39 page 184
1
Tn = n2 + 1
2
Tn = n2 – 1
3
Tn = n2 + 5
4
Tn = n2 + 8
5
Tn = n2 – 3
6
Tn = 2n2 + 2
7
Tn = 2n2 – 2
8
Tn = 3n2 + 2
9
Tn = 4n2 – 4
10 Tn = 5n2 – 4
Exercise 2.40 page 184
1
Tn = n3 + 10
2
Tn = n3 – n
3
Tn = n3 – 5
4
Tn = n3 + n2
5
Tn = n3 + n2 + 5n
6
Tn = n3 + 3n2 + 5n – 2
7
Tn = n3 + n2 – n
8
Tn = n3 + 5n + 7
Exercise 2.41 page 187
1
a
Exponential
b
Exponential
c
Not exponential
d
Exponential
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Cambridge IGCSE™ International Mathematics answers
2
e
Not exponential
f
Not exponential
a
i
3
ii
162, 486
2
iii Tn = 2(3)n−1
b
i
1
5
ii
1
1
,
25 125
1
iii Tn = 25  
5
d
i
−3
ii
−243, 729
n−1
iii Tn = (−3)n
3
4
a
−6, −12, −24
b
8
a
−4
b
1
4
c
−65 536
5
€38 203.20
6
a
Because 0.85 ≠ 0
b
$3276.80
c
Because 0.8n ≠ 0 where n is the number of years
Exercise 2.42 page 188
1
2
3
4
5
i
Tn = n2 + 1
ii
26, 37
i
Tn = n3 + 2
ii
127, 218
i
Tn =
ii
12
i
Tn =
ii
16, 32
i
Tn = n(n + 1)
n2
2
1
, 18
2
1
× 2𝑛𝑛 or Tn = 2(n – 1)
2
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Cambridge IGCSE™ International Mathematics answers
6
7
8
2
ii
30, 42
i
Tn = n2 + n3
ii
150, 252
i
Tn = n3 – n2
ii
100, 180
i
Tn = 3 × 2𝑛𝑛
ii
96, 192
Exercise 2.43 page 192
1
2
3
4
a
3
b
21
c
27
d
3
e
10
a
0.5
b
8
c
24.5
d
8
e
16
a
24
b
3
8
c
1
9
d
1
a
0.25
b
25
c
4
d
1
16
Exercise 2.44 page 192
1
a
b
i
y ∝ x3
ii
y = kx3
i
y∝
ii
y=
1
x3
k
x3
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Cambridge IGCSE™ International Mathematics answers
c
d
e
f
i
ii
t = kP
i
s∝
1
t
k
t
ii
s=
i
A ∝ r2
ii
A = kr2
i
T∝
ii
T=
2
10.5
3
a
1
2
b
2
4
32
5
a
1
8
b
0.4
6
2
t∝P
1
g
k
g
75
Exercise 2.45 page 193
1
2
3
4
a
h = kt2
b
k=5
c
45 m
d
6s
a
v=k e
b
3
c
49 J
1
km 3
a
l=
b
3
c
l = 6 cm
a
P = kI2
b
5 amps
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Cambridge IGCSE™ International Mathematics answers
5
2
a
b
i
Note
A quadratic relationship 𝑦𝑦 α 𝑥𝑥 2 .
Students may choose a different relationship, but then the checking will reveal it is
incorrect.
ii
c
d
6
a
b
𝑘𝑘 = −1.5
Check 𝑦𝑦 = −1.5𝑥𝑥 2
i
ii
c
d
𝑦𝑦 = 𝑘𝑘𝑥𝑥 2
𝑘𝑘 = 2
𝑦𝑦𝑦𝑦 √𝑥𝑥
𝑦𝑦 = 𝑘𝑘 √𝑥𝑥
Check 𝑦𝑦 = 2√𝑥𝑥
House of cards page 195
1
155
2
5612
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3
2
The formula can be proved using the method of differences explained in Topic 2 as follows:
Height of house
1
2
3
4
5
Number of cards
2
7
15
26
40
1st difference
5
8
2nd difference
11
14
3
3
3
2
3
4
Comparing with the algebraic table below:
2nd difference
2a
2a
9a + b
25a + 5b + c
5
16a + 4b + c
7a + b
3a + b
1st difference
5a + b
a+b+c
Term
9a + 3b + c
1
4a + 2b + c
Position
2a
It can be deduced that:
3
2
1
therefore b =
3a + b = 5
2
therefore c = 0.
a+b+c=2
3
1
1
This produces the rule c = n2 + n which factorises to c = n(3n + 1).
2
2
2
2a = 3
therefore a =
Chequered boards page 195
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
mn
. Where
2
both m and n are odd, the number of black and white squares differ by one. The number of
mn – 1
mn + 1
and the number of white squares is
, assuming that the bottom
black squares is
2
2
right-hand corner is white.
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2
ICT Activity 1 page 197
1
2
3
ICT Activity 2 page 197
1
x ≈ 2.7
2
x ≈ 2.6
3
x ≈ 2.1
4
x ≈ 0.6
Student assessment 1 page 197
1
a
6x – 9y + 15z
b
8pm – 28p
c
–8m2n + 4mn2
d
20p3q – 8p2q2 – 8p3
e
–2x – 2
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2
3
4
f
22x2 – 14x
g
2
h
5 2
x −x
2
a
8(2p – q)
b
p(p – 6q)
c
5pq(p – 2q)
d
3pq(3 – 2p + 4q)
a
0
b
–7
c
29
d
7
e
7
f
35
a
n = p – 4m
b
y=
4x − 5z
3
c
y=
10 px
3
y
d =
2
3w
−x
m
pqt
4mn
e
r=
f
q = r(m – n) – p
Student assessment 2 page 198
1
2
3
a
(q + r)(p – 3r)
b
(1 – t2)(1 + t2)
c
750 000
d
50
a
x2 – 2x – 8
b
x2 – 16x + 64
c
x2 + 2xy + y2
d
x2 – 121
e
6x2 – 13x + 6
f
9x2 – 30x + 25
a
(x – 11)(x + 7)
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4
2
b
(x – 3)2
c
(x – 12)(x + 12)
d
3(x – 2)(x + 3)
e
(2x – 3)(x + 4)
f
(2x – 5)2
a
f = ±
b
t= ±
p
m
m
5
2
 A
c=
p 
 −q
πr 
5
6
7
8
ty
y −t
d
x=
a
x4
b
nq
c
y3
d
4
q
a
2m
11
b
−
c
3x
4y
d
6m + 13n
30 p
a
9p
20
b
m
10
c
−p
12
a
7 x − 23
( x − 5)( x − 2 )
b
a −b
a+b
c
1
x+3
3p
16
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2
Student assessment 3 page 199
1
2
3
4
5
a
0.53 m3 (2 d.p.)
b
r=
c
5.05 cm
a
65.6 °C (1 d.p.)
b
–11.1 °C (1 d.p.)
c
F=
d
320 °F
a
15 hours
b
H = 1200(T – k)
c
5000 m
a
523.6 cm3 (1 d.p.)
b
r=
c
r = 8.4 m (1 d.p.)
a
$251.50
b
𝑛𝑛 =
c
V
πh
9
C + 32
5
3
3V
4π
𝑥𝑥 − 1.50
0.05
n = 470
Student assessment 4 page 200
1
2
3
4
a
–6
b
6
c
4
d
2.4
a
0.5
b
4
c
9.5
d
5
a
6
b
15
c
22
d
6
a
8.5
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5
2
1
3
b
4
c
8.5
d
12
a
x = 7, y = 4
b
p = 1, q = 2
c
x = 7, y = 1
d
m = 3, n = 5
Student assessment 5 page 200
1
a
9x – 90 = 360
b
x = 50
c
50°, 60°, 100°, 150°
2
2
3
135°, 85°, 95°, 95°, 130°
4
–4 and 5
5
2.77 and –1.27 (3 s.f.)
6
3<x≤5
7
All values other than 0
Student assessment 6 page 201
1
2
a
x – y = 30, x + y + 40 = 180
b
x = 85, y = 55
a
b
5x + 100
c
92°
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2
d
e
3
92° + 82° + 72° + 62° + 52° = 360°
a
b
120 – 4x2
c
x2 – 16 = 0 x = 4
4
b
0.44 and 4.56 (2 d.p.)
5
a
6
7
c
10 cm, 8 cm, 6 cm
a
x≤7
b
y≤9
a
1≤p<4
b
4<x≤7
Student assessment 7 page 202
1
a
2 2 × 35
b
214
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2
3
4
5
6
a
6×6×6×6×6
b
1
2× 2× 2× 2× 2
a
27 000
b
125
a
27
b
77 × 312
c
26
d
33
e
4–1
f
22
a
5
b
16
c
49
d
48
a
2.5
b
–0.5
c
0
d
2
9
2
Student assessment 8 page 202
1
2
a
2
b
81
c
1
3
d
64
e
3
f
2
g
6
h
32
a
15
b
4
c
3
d
3125
e
4
f
1
g
27
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h
3
a
b
4
2
10
Approx. –2.6
Plotting the graphs 𝑦𝑦 = 3𝑥𝑥 2 and 𝑦𝑦 = 6 − 17𝑥𝑥 on the same axes gives:
1
3
The points of intersection occur at 𝑥𝑥 = −6 and 𝑥𝑥 = .
1
3
Solution can be deduced from the graph as −6 ≤ 𝑥𝑥 ≤ .
Student assessment 9 page 203
1
a
b
2
i
Tn = 4n – 3
ii
T10 = 37
i
Tn = –3n + 4
ii
T10 = –26
a
T5 = 27, T100 = 597
b
T5 = 3 , T100 = –46
2
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3
2
a
Position
1
2
3
10
25
n
Term
17
14
11
–10
–55
–3n + 20
Position
2
6
10
80
n
2
Term
–4
–2
0
35
1
n–5
2
–4
b
4
5
6
a
$515.46
b
3 years
a
$253.50
b
$8.08
a
r= −
b
T1 = 243
c
n = 10
1
3
7
Tn = n3 + 3n2 + 4
8
Tn = n3 + n2 + 3n + 5
Student assessment 10 page 204
1
2
3
4
a
1.5
b
15
c
3
d
12
a
10
b
2.5
c
1
d
20
a
1
3
b
72
c
1
3
d
12
a
5
b
5
4
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5
c
1
2
d
1
a
1
3
b
4
3
c
±2
d
±1
2
Student assessment 11 page 204
1
a
b
2
a
8
5
x
1
2
4
8
16
32
y
32
16
8
4
2
1
= 1.6
x
1
2
4
5
10
y
5
10
20
25
50
x
1
2
4
5
10
y
20
10
5
4
2
x
4
16
25
36
64
y
4
8
10
12
16
b
c
3
4
a
0.8
b
0.8
a
1.6 (1 d.p.)
b
2
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3
3 Functions
All answers were written by the authors.
Exercise 3.1 page 210
1
a, b and c are functions
2
Domain: –1 ≤ x ≤ 3 Range: –3 ≤ f(x) ≤ 5
3
Domain: –4 ≤ x ≤ 0 Range: –10 ≤ f(x) ≤ 2
4
Domain: –4 ≤ x ≤ 4 Range: 0 ≤ f(x) ≤ 8
5
Domain: –3 ≤ x ≤ 3 Range: 2 ≤ f(x) ≤ 11
6
Domain: x ∈ ℝ
Range: f(x) ≥ 2
7
Domain: 0 ≤ x ≤ 4
Range: –14 ≤ f(x) ≤ 2
8
Domain: –3 ≤ x ≤ 1 Range: –29 ≤ f(x) ≤ –1
Exercise 3.2 page 213
1
2
3
a
Student’s five linear functions in the form f(x) = ax + b with a constant
c
Student’s sketches of their functions
d
Student’s conclusion that b represents the y-intercept
a
Student’s five linear functions in the form f(x) = ax + b with b constant
c
Student’s sketches of their functions
d
Student’s conclusion that a represents the gradient
a
Student’s graphics calculator should look similar to that shown
b
Student’s graphics calculator should look similar to that shown
Exercise 3.3 page 215
1
2
3
4
a
Student’s five quadratic functions in the form f(x) = a𝑥𝑥 2 + bx + c with a and b constant
c
Student’s sketches of their functions
d
Student’s conclusion that c represents the y-intercept
a
Student’s five quadratic functions in the form f(x) = ax2 + bx + c with b and c constant
c
Student’s sketches of their functions
d
Student’s conclusion that a affects the steepness of the curve
a
Student’s five quadratic functions in the form f(x) = a𝑥𝑥 2 + bx + c with a and c constant
c
Student’s sketches of their functions
d
Student’s conclusion that b affects the position of the curve
a
Student’s graphics calculator should look similar to that shown
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b
3
Student’s graphics calculator should look similar to that shown
Exercise 3.4 page 217
1
a
b
Student’s investigation
Student’s conclusion that a affects the steepness/tightness of the graph
2
a
b
Student’s investigation
Student’s conclusion that d affects the y-intercept
3
a
b
Student’s graphics calculator should look similar to that shown
Student’s graphics calculator should look similar to that shown
Exercise 3.5 page 220
1
a
b
2
a
3
Student’s investigation
For a > 0, as a increases the graph moves diagonally away from the axes
a < 0 produces a reflection in the x-axis
b
Graphs are above x-axis and pass through the point (0, 1)
a
i
ii
iii
4
b
Each is a reflection of the other in the y-axis
a
b
Student’s graphics calculator should look similar to that shown
Student’s graphics calculator should look similar to that shown
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3
Exercise 3.6 page 223
1
2
3
a
b
y=2
c
y=0
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4
5
3
a
b
y = 0 and x = –2
a
0
Translation �−2
�
b
Translation �40�
Exercise 3.7 page 224
1
a
b
2
a
b
Translation �30�
0
Translation �−2
�
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3
a
3
i
ii
b
i
ii
4
a
i
y = 𝑥𝑥 2 + 4
y = (𝑥𝑥 + 2)2
ii
b
i
ii
𝑦𝑦 =
𝑦𝑦 =
1
𝑥𝑥−4
1
−
𝑥𝑥
2
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c
3
i
Asymptotes at x = 4 and y = 0
ii
Asymptotes at x = 0 and y = –2
Exercise 3.8 page 232
1
2
a
b
x = –3, x = 1 and x = 2
c
y-intercept at (0, 6)
d
Maximum at (–1.53, 13.1), minimum at (1.53, –1.13) (3 s.f.)
a
b
x = – 1 and x = 6
c
y-intercept at (0, –12)
d
Maximum at (–1, 0), minimum at (3.67, –102) (3 s.f.)
e
x
–3
–2
–1
0
1
2
3
4
5
6
7
y
–72
–16
0
–12
–40
–72
–96
–100
–72
0
128
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3
3
a
b
x = –4, and x = 2
c
y-intercept at (0, –64)
d
Minima at (–4, 0), and (2, 0), maximum at (–1, 81)
e
4
x
–5
–4
–3
–2
–1
0
1
2
3
y
49
0
25
64
81
64
25
0
49
a
b
x = –4.57, and x = –3.29, x = 1.29 and 2.57 (3 s.f.)
c
y-intercept at (0, 50)
d
Minima at (–4, –14) and (2, –14), maximum at (–1, 67)
e
x
–5
–4
–3
–2
–1
0
1
2
3
y
35
–14
11
50
67
50
11
–14
35
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5
3
a
b
x = –1, and x = 2
c
y-intercept at (0, –1)
d
Maxima at (–1, 0) and (2, 0) minimum at (0.5, –1.27)
e
6
7
8
x
–4
–3
–2
–1
0
1
2
3
y
–81
–25
–4
0
–1
–1
0
–4
x = –1.21 and 1
x = –1.20 and 1.81
x = 0.14 and 1.13
Exercise 3.9 page 234
1
a
i
f(x) = (x + 2)(x + 1)
ii
b
i
f(x) = (x + 4)(x – 1)
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3
ii
c
i
f(x) = (x + 5)(x – 2)
ii
d
i
ii
f(𝑥𝑥) = (𝑥𝑥 + 7)(𝑥𝑥 + 6)
y-intercept at (0, 42)
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e
i
3
f(x) = (x + 3)(x – 3)
ii
f
i
f(x) = (x + 8)(x – 8)
ii
2
Student’s observations. The x-intercept on the graph corresponds to the x-value required to
make the function zero.
Exercise 3.10 page 236
1
a
b
b = –2 and c = 0
f(𝑥𝑥) = 𝑥𝑥 2 + 7𝑥𝑥 + 6
b = 7 and c = 6
𝑏𝑏 = − and 𝑐𝑐 = −
d
f(𝑥𝑥) = 𝑥𝑥 2 − 𝑥𝑥 − 12
a
f(𝑥𝑥) = 𝑥𝑥 2 −
a = 1, b = 6, c = 8
b
a = 1, b = –6, c = 5
c
a = –1, b = 0, c = 4
c
2
f(x) = 𝑥𝑥 2 – 2x
5
𝑥𝑥
2
−
3
2
b = –1 and c = –12
5
2
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2
10
120
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3
d
a = –1, b = –7, c = –6
e
a = 1, b = –2, c = 1
f
a = –1, b = –6, c = –9
a
f(𝑥𝑥) = 2𝑥𝑥 2 − 10𝑥𝑥 + 12
b
c
d
e
f
3
1
2
f(𝑥𝑥) = 𝑥𝑥 2 + 2𝑥𝑥 − 6
f(𝑥𝑥) = −𝑥𝑥 2 − 6𝑥𝑥 − 9
f(𝑥𝑥) = −3𝑥𝑥 2 + 12𝑥𝑥
9
2
f(𝑥𝑥) = 𝑥𝑥 2 − 𝑥𝑥 + 2
1
4
5
2
f(𝑥𝑥) = 𝑥𝑥 2 + 𝑥𝑥 + 4
Exercise 3.11 page 240
1
a
i
y-intercept (0, –5), x-intercept (5, 0) and (–1, 0)
ii
iii (2, –9)
b
i
y-intercept (0, 24), x-intercept (–4, 0) and (–6, 0)
ii
iii (–5, –1)
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c
i
3
y-intercept (0, 5), x-intercept (5, 0) and (1, 0)
ii
iii (3, –4)
d
i
ii
e
y-axis at (0, −
63
4
1
2
1
2
), x-axis at (4 , 0) and (−3 , 0)
1
2
iii ( , −16)
i
y-axis at (0, 6), x-axis at (−4 + √10, 0) and (−4 − √10, 0)
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3
ii
iii (–4, –10)
f
i
y-axis at (0, 25), x-axis at (5, 0)
ii
iii (5,0)
2
a
i
ii
b
(–3, –1)
iii f(𝑥𝑥) = (𝑥𝑥 + 3)2 − 1
i
ii
c
f(𝑥𝑥) = 𝑥𝑥 2 + 6𝑥𝑥 + 8
f(𝑥𝑥) = 𝑥𝑥 2 − 5𝑥𝑥
5
2
( ,−
25
)
4
5
2
iii f(𝑥𝑥) = (𝑥𝑥 − )2 −
i
ii
f(𝑥𝑥) = 𝑥𝑥 2 − 9
25
4
(0, –9)
iii f(𝑥𝑥) = (𝑥𝑥 − 0)2 − 9
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d
i
ii
e
ii
f
a
b
4
a
7
2
25
4
f(𝑥𝑥) = −𝑥𝑥 2 + 10𝑥𝑥 − 25
(5, 0)
iii f(𝑥𝑥) = −(𝑥𝑥 − 5)2
i
ii
3
7 25
)
2 4
( ,
iii f(𝑥𝑥) = −(𝑥𝑥 − )2 +
i
3
f(𝑥𝑥) = −𝑥𝑥 2 + 7𝑥𝑥 − 6
f(𝑥𝑥) = −𝑥𝑥 2 + 4𝑥𝑥 + 21
(2, 25)
iii f(𝑥𝑥) = −(𝑥𝑥 − 2)2 + 25
In the form
f(x) = a(x – h)2 + k
Coordinates of the vertex
Example
f(x) = (x +1)2 – 9
(–1, –9)
1a
f(x) = (x – 2)2 – 9
(2, –9)
1b
f(x) = (x + 5)2 – 1
(–5, –1)
1c
f(x) = (x – 3)2 – 4
(3, –4)
1d
f(x) = (x – )2 – 16
( , –16)
1e
f(x) = (x + 4)2 – 10
(–4, –10)
1f
f(x) = (x – 5)2
(5, 0)
2a
f(x) = (x + 3)2 – 1
(–3, –1)
2b
f(x) = (x – )2 –
2c
f(x) = (x – 0)2 – 9
1
2
5
2
25
4
2d
f(x) = –(x – )2 +
2e
f(x) = –(x – 5)2
2f
f(x) = –(x – 2)2 + 25
7
2
25
4
1
2
5
2
( ,−
25
)
4
(0, –9)
7 25
)
2 4
( ,
(5, 0)
(2, 25)
Student’s explanation similar to ‘A quadratic written in completed square form
𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − ℎ)2 + 𝑘𝑘 has a vertex at (h, k)’
(2, 4)
b
(–5, –3)
c
(6, 4)
d
(0, –3)
e
(–5, 3)
f
(4, 0)
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5
3
a
b
c
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Cambridge IGCSE™ International Mathematics answers
3
d
e
f
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6
a
i
ii
b
i
ii
c
i
ii
d
i
ii
7
a
i
ii
b
i
ii
c
i
ii
d
i
ii
3
f(𝑥𝑥) = (𝑥𝑥 + 3)2 + 6
f(𝑥𝑥) = 𝑥𝑥 2 + 6𝑥𝑥 + 15
f(𝑥𝑥) = (𝑥𝑥 − 2)2 − 4
f(𝑥𝑥) = 𝑥𝑥 2 − 4𝑥𝑥
f(𝑥𝑥) = (𝑥𝑥 + 1)2 + 6
f(𝑥𝑥) = 𝑥𝑥 2 + 2𝑥𝑥 + 7
f(𝑥𝑥) = (𝑥𝑥 + 4)2
f(𝑥𝑥) = 𝑥𝑥 2 + 8𝑥𝑥 + 16
f(𝑥𝑥) = −(𝑥𝑥 + 3)2 + 6
f(𝑥𝑥) = −𝑥𝑥 2 − 6𝑥𝑥 − 3
f(𝑥𝑥) = −(𝑥𝑥 − 2)2 − 4
f(𝑥𝑥) = −𝑥𝑥 2 + 4𝑥𝑥 − 8
f(𝑥𝑥) = −(𝑥𝑥 + 1)2 + 6
f(𝑥𝑥) = −𝑥𝑥 2 − 2𝑥𝑥 + 5
f(𝑥𝑥) = −(𝑥𝑥 + 4)2
f(𝑥𝑥) = −𝑥𝑥 2 − 8𝑥𝑥 − 16
Exercise 3.12 page 244
1
a
(1, 6)
b
(–4, –6)
c
(0, –4)
d
(–2, –6)
e
(6, 2)
f
(–1, 0)
2
a and d, b and g, c and f, e and h
3
4
d
a
ii
b
iii
c
i
d
iv
a
i
5
ii
b
i
ii
c
i
ii
f(𝑥𝑥) = 2(𝑥𝑥 + 1)2 − 5
f(𝑥𝑥) = 2𝑥𝑥 2 + 4𝑥𝑥 − 3, 𝑎𝑎 = 2, 𝑏𝑏 = 4, 𝑐𝑐 = −3
f(𝑥𝑥) = −2(𝑥𝑥 − 1)2 + 4
f(𝑥𝑥) = −2𝑥𝑥 2 + 4𝑥𝑥 + 2, 𝑎𝑎 = −2, 𝑏𝑏 = 4, 𝑐𝑐 = 2
f(𝑥𝑥) = 3(𝑥𝑥 + 4)2 − 4
f(𝑥𝑥) = 3𝑥𝑥 2 + 24𝑥𝑥 + 44, 𝑎𝑎 = 3, 𝑏𝑏 = 24, 𝑐𝑐 = 44
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d
i
ii
e
i
ii
f
6
a
b
i
e
f
+ 5𝑥𝑥 +
f(x) = –(x –3)2
21
,
2
1
2
𝑎𝑎 = , 𝑏𝑏 = 5, 𝑐𝑐 =
3
2
3 2
𝑥𝑥 +
2
f(x) = (x +5)2 – 12
51
,
2
3
2
i
(0, –3)
ii
�−1 − �2 , 0� and �−1 + �2 , 0�
i
i
15x +
a = , b = 15, c =
5
51
2
5
(0, 2)
�1 − √2, 0� and �1 + √2, 0�
(0, 44)
2
i
�−4 −
ii
(–7, 0) and (–3, 0)
i
(0, –9)
ii
(3, 0)
i
�0,
ii
21
2
f(x) = –𝑥𝑥 2 + 6x – 9, a = –1, b = 6, c = –9
f(x) =
ii
d
f(𝑥𝑥) =
3
ii
ii
c
1
2
1 2
𝑥𝑥
2
f(𝑥𝑥) = (𝑥𝑥 + 5)2 − 2
�0,
21
�
2
√3
, 0� and �−4 +
2
√3
, 0�
51
�
2
�−5 − 2√2, 0� and �−5 + 2√2, 0�
7
p=3
8
a
t=2
b
v = –2
Exercise 3.13 page 249
1
2
3
4
a
Linear
b
f(x) = x + 5
a
Reciprocal
b
f(x) =
a
Cubic
b
f(x) =𝑥𝑥 3 − 2𝑥𝑥 2 − 𝑥𝑥 + 2
a
b
5
3
𝑥𝑥
Reciprocal
4
𝑥𝑥
a
f(x) = −
b
f(x) = – x + 6
Linear
1
3
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6
7
Cubic
b
f(x) =2𝑥𝑥 3 − 3𝑥𝑥 2 − 17𝑥𝑥 − 12
a
b
8
3
a
a
b
Cubic
f(x) = –3𝑥𝑥 3 − 3𝑥𝑥 2 + 18𝑥𝑥
Cubic
f(x) =𝑥𝑥 3 − 4𝑥𝑥 2 − 4𝑥𝑥 + 22
Exercise 3.14 page 252
1
a
b
c
2
f(x) = 2𝑥𝑥
f(x) = 6𝑥𝑥
1 𝑥𝑥
f(x) = �5�
a0 = 1 for all real values of a
Exercise 3.15 page 254
1
2
3
4
a
4
b
10
c
1
a
4
b
0
c
2
a
f(g(x)) = 6x
b
f(g(x)) = 4(x – 3)
c
f(g(x)) = 9x – 1
d
f(g(x)) = 3x – 1
e
f(g(x)) = 6x
f
f(g(x)) = –2x – 6
a
𝑥𝑥 =
b
c
1
4
3
5
𝑥𝑥 = −
x=2
3
2
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5
6
f(g(x)) = 8x – 18
b
g(f(x)) = 8x – 8
c
i
–2
ii
0
i
f(g(x)) = 2(2x – 3)
ii
g(f(x)) = 4x – 3
i
–6
ii
–3
a
b
7
3
a
c
No, because 4x – 6 = 4x – 3 has no solution
a
Student’s proof
b
𝑥𝑥 = −1 ± √6
Exercise 3.16 page 255
1
a
b
c
d
e
f
f�g(𝑥𝑥)� = 2(𝑥𝑥 − 1)2
f�g(𝑥𝑥)� = 4𝑥𝑥 2 + 1
f�g(𝑥𝑥)� = 4𝑥𝑥(𝑥𝑥 + 1)
f�g(𝑥𝑥)� = 2(3𝑥𝑥 − 2)2
f�g(𝑥𝑥)� = 𝑥𝑥 2 + 4𝑥𝑥 + 4
h
f�g(𝑥𝑥)� = (3 − 𝑥𝑥)2
a
i
g
2
f�g(𝑥𝑥)� = (𝑥𝑥 + 1)2
ii
iii
b
c
3
2
i
f�g(𝑥𝑥)� = (2𝑥𝑥 − 1)3
ii
–1
iii
1
iv
–27
i
f�g(𝑥𝑥)� = 23𝑥𝑥
iv
b
3
3
𝑥𝑥+1
Undefined
iii
a
f�g(𝑥𝑥)� =
iv
ii
3
1
f�g(𝑥𝑥)� = 2 (2 − 3𝑥𝑥)2
1
8
1
8
f�g(𝑥𝑥)� = 𝑥𝑥 2 + 2𝑥𝑥 − 2
–3
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c
4
a
b
c
3
−1 ± √3
f�g(𝑥𝑥)� = 2(𝑥𝑥 − 1)2
2
4 and –2
Exercise 3.17 page 257
1
2
a
x–3
b
x–6
c
x+5
d
x
e
𝑥𝑥
2
f
3x
a
𝑥𝑥
4
𝑥𝑥−5
2
𝑥𝑥+6
3
b
c
d
a
4𝑥𝑥+2
3
5𝑥𝑥−7
8
b
4(x + 2)
c
𝑥𝑥+24
12
𝑥𝑥−18
6
𝑥𝑥+4
6
3𝑥𝑥+10
8
2
𝑥𝑥−3
1
5(𝑥𝑥+2)
e
f
3
2x – 4
d
e
f
g
h
2(x – 3)
or
1
5𝑥𝑥+10
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3
Exercise 3.18 page 258
1
2
3
4
5
6
a
6
b
4
c
–1
a
2
b
–0.5
c
–6
a
3
b
1.5
c
2
a
4
b
–2
c
–11
a
4.5
b
6
c
0
a
8
b
–2
c
0.5
Exercise 3.19 page 259
1
a
0
b
1
c
2
d
3
2
Student’s explanation
3
a
b
1
10
10−1 =
1
10
Exercise 3.20 page 260
1
a
b
c
d
log 2 8 = 3
log 7 343 = 3
log10 100 000 = 5
log 3 1 = 0
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e
f
2
a
b
c
26 = 64
34 = 81
54 = 625
1
9
a
5−3 =
1
125
b
y=6
c
a=5
d
b=0
e
x = 32
f
g
𝑥𝑥 =
h
n=3
e
f
3
log 𝑝𝑝 q = 4
3−2 =
d
3
1
4
log 2 = –2
𝑥𝑥 0 = 1
x=1
𝑚𝑚
1
27
1
=
5
Exercise 3.21 page 261
1
a
2
b
6
c
log10 2
d
e
f
log10 5
log10
3
2
g
1
h
1
4
1
3
Exercise 3.22 page 263
1
a
x = 2.82
b
a = 1.60
c
y = –1.46
d
p = –0.602
e
x = 1.86
f
y = 8.99
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2
g
x = 1.74
h
m = –1.86
i
x = 15625
j
b = 32
k
x = 1.26
l
x = 0.36
a
x > 3.86
b
y > 1.43
c
x > 4.64
d
x > –1.36
3
x = 13
4
m=7
3
Exercise 3.23 page 264
1
a
b
When 𝑡𝑡 = 0, 𝑃𝑃 = 3. Therefore, three flamingos at the start
56 (Accept 57)
c
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d
3
i
Approx 4.4 days
ii
2
4.41 days (2 d.p)
e
The exponential equation increases in value very quickly. After 10 days there are an
estimated 1071 flamingos. There comes a point where the numbers are not sustainable.
a
𝑇𝑇 = 2500 × 1.05𝑛𝑛
b
17 years (the solution to the equation is 16.21, but this must be rounded up to 17)
3
a
𝒕𝒕 (min)
𝑻𝑻 (℃)
0
1
2
3
4
5
6
97
75.3
59.8
48.7
40.8
35.1
31.1
b
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c
3
i
Approx 2.3 minutes
𝑛𝑛 = 2.4 minutes
ii
d
21℃. As 𝑡𝑡 increases, the value of 76 × 1.4−𝑡𝑡 gets closer to zero.
Paving blocks page 266
1
For a 2 × 4 design there are: 8 equilateral shaped blocks, 4 isosceles shaped blocks, 10 rhombus
shaped blocks
2–4
5
6
7
Student’s own responses
a
Number of isosceles blocks is twice the width
b
i = 2w
a
Number of equilateral blocks is twice the height
b
e = 2h
b
r = w(h – 1) + h(w – 1)
Regions and intersections page 267
1
Dimension of
curve stitch
Number of points of
intersection
Number of enclosed
regions
1×1
0
1
2×2
1
3
3×3
3
6
4×4
6
10
etc.
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2
The pattern is the sequence of triangle numbers.
3
𝑝𝑝 =
(𝑚𝑚−1)𝑚𝑚
𝑟𝑟 =
𝑚𝑚(𝑚𝑚+1)
2
3
2
The pattern for the number of enclosed regions is also the sequence of triangle numbers. But
the sequence is one on from that obtained for the sequence of points of intersection.
Modelling: Parking error page 268
1
2
Quadratic
3
𝑦𝑦 =
1
20
𝑥𝑥 2
4
11.25 m
5
49 seconds
6
Due to opposing forces such as friction, the car may not continue to accelerate as it rolls down
the hill.
ICT Activity page 269
2
3
a
c
𝑦𝑦 = ±√𝑥𝑥 2 − 𝑎𝑎2
d
a affects the position and curvature of the curve
Student's graph and equations of asymptotes for 𝑥𝑥 2 − 𝑦𝑦 2 = 1; asymptotes are y = x and y = –x.
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3
Student assessment 1 page 270
1
a
b
Range –18 ≤ f(x) ≤ 18
Range f(x) ≥ –4
2
a is the gradient of the line and b is the y-intercept
3
a
4
b
y=x+3
c
y = 2x + 1
d
y=x–2
Several solutions possible such as
a
b
c
5
1
2
𝑦𝑦 = − 𝑥𝑥 − 4
a
b
c
1 2
𝑥𝑥 +
2
1 2
𝑥𝑥 −
4
2
2
2
−𝑥𝑥 + 2
𝑦𝑦 = 0
i
ii
𝑦𝑦 = −3
𝑦𝑦 = 0
Student assessment 2 page 271
1
a
b
1
2
A'(–2, –2) B'�0, −3 � C'(4, –6)
1
2
A′(1, 1) B′ �3, − � C′(7, −3)
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2
a and b
3
a and b
i
ii
4
a
3
𝑦𝑦 = 2 and 𝑥𝑥 = 0
𝑦𝑦 = 0 and 𝑥𝑥 = 4
Local maximum at (–5, 110)
Local minimum at (1, 2)
b
Root at x = –8.025
c
y-intercept at (0, 10)
Student assessment 3 page 272
1
f(𝑥𝑥) = 𝑥𝑥 2 + 𝑥𝑥 − 6. Therefore b = 1 and c = –6.
3
f(𝑥𝑥) = 3𝑥𝑥 2 + 6𝑥𝑥 − 24
4
b = 20, c = –56 d = 48
5
a
2
a
b
b
(–1, –3)
(5, 1)
The graph of f(𝑥𝑥) = 2𝑥𝑥 intersects the y-axis at (0, 1).
5
g(𝑥𝑥) is a translation of f(𝑥𝑥) by � �; therefore g(𝑥𝑥) = 2𝑥𝑥−5
0
Substitute 𝑥𝑥 = 0 into the equation 𝑦𝑦 = 2𝑥𝑥−5:
1
1
𝑦𝑦 = 2−5 = 5 =
2
32
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3
Student assessment 4 page 273
1
2
3
a
f(g(x)) = 12x + 8
b
g(f(x)) = 12x + 2
a
–3
b
–8
c
2x – 3 ≠ 2x – 4
a
f(g(𝑥𝑥)) = 4𝑥𝑥 2 + 4𝑥𝑥 − 3
b
4
a
b
5
a
b
1
2
𝑥𝑥 = and −
f −1 (𝑥𝑥) =
h
−1 (𝑥𝑥)
x = 8.3
3
2
(4𝑥𝑥−2)
5
= 2(𝑥𝑥 − 6)
x=9
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4
4 Coordinate geometry
All answers were written by the authors.
Exercise 4.1 page 277
1
2
Rectangle
3
Isosceles triangle
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4
4
Parallelogram
Exercise 4.2 page 277
1
b
c
(1, 1)
72 units2
2
a
b
Parallelogram
72 units2
c
The base length and perpendicular height of MNRS are the same as the base length and
perpendicular height of PQRS, so their areas are the same.
b
2
3
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4
Exercise 4.3 page 277
1
a
b
2
B(7, 4), C(7, –4), D(0, –8), E(–7, –4), F(–7, 4)
a
b
S(–7, 3), T(–6, 6), U(2, 6), V(3, 3), W(3, –5)
c
(–2, –1)
Exercise 4.4 page 278
1
B at 1.5, C at 2.4, D at 4.8
2
F at 0.9, G at 1.5, H at 1.75
3
I at 4.4, J at 5.2, K at 5.9, L at 6.3, M at 6.8
4
Q at 2.4, R at 4.6, S at 5.8, T at 6.4, U at 7.8, V at 8.8
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4
Exercise 4.5 page 278
1
A(1, 1.5), B(1.2, –1.5), C(–0.9, –1.6), D(–1.8, 0.7)
2
E(1, 1.8), F(3, –2.4), G(–3.6, –1.6), H(–1.6, 3.6)
3
J(1, 0.5), K(0.75, –0.5), L(–0.85, –0.25), M(–0.55, 0.25)
4
P(37.5, 25), Q(25, –37.5), R(–37.5, –7.5), S(–42.5, 45)
Exercise 4.6 page 281
1
a
ii
5.66 units
iii (3, 4)
b
ii
4.24 units
iii (4.5, 2.5)
c
ii
5.66 units
iii (3, 6)
d
ii
8.94 units
iii (2, 4)
e
ii
6.32 units
iii (3, 4)
f
ii
6.71 units
iii (−1.5, 4)
g
ii
8.25 units
iii (−2, 1)
h
ii
8.94 units
iii (0, 0)
i
ii
7 units
iii (0.5, 5)
j
ii
6 units
iii (2, 3)
k
ii
8.25 units
iii (0, 4)
l
2
a
b
ii
10.8 units
ii
(0, 1.5)
i
4.24 units
ii
(2.5, 2.5)
i
5.66 units
ii
(5, 4)
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c
d
e
f
g
h
i
j
k
l
i
8.94 units
ii
(4, 2)
i
8.94 units
ii
(5, 0)
i
4.24 units
ii
(−1.5, 4.5)
i
4.47 units
ii
(−4, −3)
i
7.21 units
ii
(0, 3)
i
7.21 units
ii
(5, −1)
i
12.4 units
ii
(0, 2.5)
i
8.49 units
ii
(1, −1)
i
11 units
ii
(0.5, −3)
i
8.25 units
ii
(4, 2)
4
Exercise 4.7 page 284
1
a
b
c
d
e
f
g
i
1
ii
−1
i
1
ii
−1
i
1
ii
−1
i
2
ii
i
−
3
1
2
ii
i
−
2
1
3
ii
i
−
4
1
2
ii
−
1
4
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h
i
j
k
ii
i
0
ii
Infinite
i
Infinite
ii
0
i
1
4
ii
l
2
a
b
c
d
e
f
g
h
i
j
k
l
4
1
2
i
–2
–4
ii
3
2
i
−
−1
ii
1
i
−1
ii
1
i
−2
ii
1
2
i
2
3
1
2
i
−
ii
2
i
−1
ii
1
i
−2
ii
1
2
i
−
ii
2
3
2
3
i
3
2
3
2
1
4
ii
−
i
−
ii
4
i
–1
ii
1
i
0
ii
Infinite
i
–4
ii
1
4
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4
Exercise 4.8 page 286
1
y=7
2
y=2
3
x=7
4
x=3
5
y=x
6
y = 𝑥𝑥
1
2
7
y = −x
8
y = −2x
Exercise 4.9 page 287
1
a
b
c
d
e
f
2
a
b
c
d
e
f
3
a
𝑦𝑦 = x + 1
𝑦𝑦 = x + 3
𝑦𝑦 = x − 2
𝑦𝑦 = 2x + 2
1
2
1
𝑥𝑥
2
𝑦𝑦 = 𝑥𝑥 + 5
𝑦𝑦 =
−1
𝑦𝑦 = −x + 4
𝑦𝑦 = −x − 2
𝑦𝑦 = −2x – 2
1
2
3
− 𝑥𝑥
2
𝑦𝑦 = − 𝑥𝑥 + 3
𝑦𝑦 =
+2
𝑦𝑦 = −4𝑥𝑥 + 1
1
2
1
−
2
Q.1: a 1, b 1, c 1, d 2, e , f
Q.2: a −1, b −1, c −2, d
1
2
3
2
, e − , f –4
b
The gradient is equal to the coefficient of x
c
The constant being added/subtracted indicates where the line intersects the y-axis
4
b
Only the intercept c is different
5
The lines are parallel
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Exercise 4.10 page 291
1
2
3
4
a
m = 2, c = l
b
m = 3, c = 5
c
m = l, c = − 2
d
m= ,c=4
e
m = −3, c = 6
f
m = – , c =1
g
m = −1, c = 0
h
m = −1, c = −2
i
m = −2, c = 2
a
m = 3, c = l
b
m=− ,c=2
c
m = −2, c = −3
d
m = −2, c = −4
e
m= ,c=6
f
m = 3, c = 2
g
m = 1, c = −2
h
m = −8, c = 6
i
m = 3, c = 1
a
m = 2, c = −3
b
m= ,c=4
c
m = 2, c = −4
d
m = −8, c = 12
e
m = 2, c = 0
f
m = −3, c = 3
g
m = 2, c = 1
h
m=– ,c=2
i
m = 2, c = –
a
m = 2, c = −4
b
m = 1, c = 6
c
m = −3, c = −1
d
m = −1, c = 4
e
m = 10, c = −2
f
m = –3, c =
1
2
2
3
1
2
1
4
1
2
1
2
1
2
3
2
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5
6
g
m = −9, c = 2
h
m = 6, c = −14
i
m = 2, c = –
a
m = 2, c = −2
b
m = 2, c = 3
c
m = 1, c = 0
d
m= ,c=6
e
m = –1, c =
f
m = −4, c = 2
g
m = 3, c = −12
h
m = 0, c = 0
i
m = −3, c = 0
a
m = 1, c = 0
b
m = – , c = –2
c
m = −3, c = 0
d
m = 1, c = 0
e
m=− ,c=−
3
2
4
3
2
2
3
1
2
2
3
9
2
f
m = , c = –4
g
m=– ,c=0
h
m=
i
m=
1
4
2
5
1
7
– ,c=–
3
6
2
8
− ,c=−
3
3
Exercise 4.11 page 293
1
a
b
c
d
e
f
i
y = 2x − 1
ii
2x − y = 1
i
y = 3x + 1
ii
3x − y = –1
i
y = 2x + 3
ii
2x − y = –3
i
y=x−4
ii
x−y=4
i
y = 4x + 2
ii
4x − y = –2
i
y = −x + 4
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g
h
i
2
a
b
c
d
e
f
g
h
i
4
ii
x+y=4
i
y = −2x + 2
ii
2x + y = 2
i
y = −3x – 1
ii
3x + y = –1
i
y= x
ii
x − 2y = 0
i
y= x+
ii
x – 7y = –26
i
y= x+
ii
6x – 7y = –4
i
y= x+
ii
3x – 2y = –15
i
y = 9x – 13
ii
9x – y = 13
i
y=– x+
ii
x + 2y = 5
i
y=–
ii
3x + 13y = 70
i
y=2
ii
y=2
i
y = −3x
ii
3x + y = 0
i
x=6
ii
x=6
1
2
1
7
26
7
6
7
4
7
3
2
15
2
1
2
3
x
13
5
2
+
70
13
Exercise 4.12 page 293
1
a
b
c
2
1
2
𝑦𝑦 = − 𝑥𝑥 +
𝑦𝑦 = 2𝑥𝑥 + 2
19
2
𝑦𝑦 = 2𝑥𝑥 − 13
d
They have the same gradient of 2. Therefore they are parallel.
a
i
ii
b
i
ii
𝑦𝑦 = 4𝑥𝑥
1
4
𝑦𝑦 = − 𝑥𝑥
𝑦𝑦 = 2𝑥𝑥 − 6
1
2
𝑦𝑦 = − 𝑥𝑥 − 1
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c
1
3
𝑦𝑦 = 𝑥𝑥 +
ii
d
i
ii
3
a
b
c
d
𝑦𝑦 =
4
𝑦𝑦 = −3𝑥𝑥 − 9
i
2𝑥𝑥 − 𝑦𝑦 = 1
𝑥𝑥 + 2𝑦𝑦 = 3
1
− 𝑥𝑥
3
𝑦𝑦 = 3𝑥𝑥
6 18
�
5 5
� ,
13
3
+4
12
units2
5
108
is
units2
5
Area 𝐴𝐴 is
Area 𝐵𝐵
Ratio 𝐴𝐴 ∶ 𝐵𝐵 is 1 : 9
Exercise 4.13 page 295
1
a
b
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c
d
e
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4
f
g
h
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4
i
2
a
b
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4
c
d
e
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155
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4
f
g
h
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156
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4
i
3
a
b
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4
c
d
e
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4
f
g
h
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4
i
Plane trails page 296
1
Student’s investigation
2
Student’s ordered table similar to the one shown
Number of planes (p)
Maximum number of
crossing points (n)
1
0
2
1
3
3
4
6
5
10
…
3
1
2
The sequence of crossing points are the triangle numbers. 𝑛𝑛 = 𝑝𝑝(𝑝𝑝 − 1)
Hidden treasure page 297
2
The results up to 20 contestants are given in the table below:
Number of
contestants (n)
1
2
3
4
5
6
7
8
9
10
Winning chest (x)
1
2
2
4
2
4
6
8
2
4
Number of
contestants (n)
11
12
13
14
15
16
17
18
19
20
Winning chest (x)
6
8
10
12
14
16
2
4
6
8
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4
3
Student’s observed pattern: key pattern is that n = x 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.
x = 2(n – T), where x = the winning chest, n = number of contestants and T = the nearest power
of 2 below n.
5
Modelling: Stretching a spring page 299
1
2
Linear
3
4
y ≈ 0.06x
5
y = 0.06 × 275 = 16.5 cm
6
The spring is likely to snap (or exceed its elastic limit).
7
0 – 500 g
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4
Student assessment 1 page 301
1
2
a
i
m = 2, c = 3
ii
b
i
m = –1, c = 4
ii
c
i
m = 2, c = –3
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4
ii
d
i
3
2
m= ,c=
ii
3
4
5
5
2
a
y = 2x – 5
b
y = –4x + 3
a
5 units
b
26 units
a
Gradient of PQ =
12−2
1−5
=
To find equation of PQ:
5
2
𝑦𝑦 = − 𝑥𝑥 + 𝑐𝑐
5
2
12 = − + 𝑐𝑐
b
10
−4
=−
Therefore 𝑐𝑐 =
5
2
Equation of PQ is 𝑦𝑦 = − 𝑥𝑥 +
29
2
29
2
5
2
2
5
Gradient of perpendicular line to PQ is passing through midpoint with coordinates (3,7)
2
7 = 5 × 3 + 𝑐𝑐 Therefore 𝑐𝑐 =
29
5
2
5
Equation of perpendicular line passing through midpoint is 𝑦𝑦 = 𝑥𝑥 +
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5
23
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5
5 Geometry
All answers were written by the authors.
Exercise 5.1 page 308
Rectangle
Square
Parallelogram
Kite
Rhombus
Equilateral
triangle
1
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
All angles equal
Yes
Yes
No
No
No
Yes
Exercise 5.2 page 309
Student’s own diagrams
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Exercise 5.3 page 310
1
a
b
c
d
e
f
g
h
i
j
2
a
d
h
It is not possible to show all the lines of symmetry as there is an infinite number.
b
c
e
f
g
It is not possible to show all the lines of symmetry as there is an infinite number.
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3
4
a
b
c
d
e
f
5
a
b
c
d
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Exercise 5.4 page 312
1
a
b
c
d
e
f
g
2
a
h
i
b
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c
5
d
e
3
Regular polygon
Number of lines of symmetry Order of rotational symmetry
Equilateral triangle
3
3
Square
4
4
Pentagon
5
5
Hexagon
6
6
Heptagon
7
7
Octagon
8
8
Nonagon
9
9
Decagon
10
10
Exercise 5.5 page 313
1
a
b
c
i
Student’s planes
ii
3
i
Student’s planes
ii
2
i
Student’s planes
ii
4
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d
e
f
g
h
2
i
Student’s planes
ii
4
i
Student’s planes
ii
Infinite
i
Student’s planes
ii
Infinite
i
Student’s planes
ii
Infinite
i
Student’s planes
ii
9
a
2
b
2
c
3
d
4
e
Infinite
f
Infinite
g
Infinite
h
4
5
Exercise 5.6 page 317
1
2
3
a
b
47°
24°
c
100°
d
150°
e
110°
f
158°
a
b
a = 90°, b = 166°, c = 104°
d = 24°, e = 55°, f = 125°, g = 156°
c
h = 44°, i = 316°
d
j = 35°, k = 85°, l = 240°
e
m = 30°, n = 140°, o = 60°, p = 330°
f
q = 112°, r = 339°, s = 337°, t = 45°
Student’s own angles
Exercise 5.7 page 319
1
Student’s own diagram leading to:
Distance ≈ 11 km
Bearing ≈ 035°
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2
b
3
5
Student’s own diagrams leading to:
a 343°
034°
Student’s own diagrams leading to:
a 120°
b
102°
Exercise 5.8 page 321
1
2
3
a
b
75°
96°
c
96°
d
146°
a
b
111°
37°
c
22°
d
62°
a
b
40°
45°
c
42°
d
30°
Exercise 5.9 page 322
1
2
3
4
Student’s own diagram and measurements
Student’s own diagram and measurements
Student’s own diagram and measurements
Student’s own observations
Exercise 5.10 page 523
1
2
3
4
Student’s own diagram and measurements
Student’s own diagram and measurements
Student’s own diagram and measurements
Student’s own observations
Exercise 5.11 page 325
1
2
p = 54°, q = 63°
a = 55°, b = 80°, c = 100°
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3
4
5
6
7
8
5
v = 120°, w = 60°, x = 120°, y = 60°, z = 60°
a = 50°, b = 130°, c = 45°, d = 135°
p = 45°, q = 135°, r = 45°, s = 45°, t = 135°
d = 70°, e = 30°
a = 37°
a = 36°
Exercise 5.12 page 327
1
2
Rhombus, scalene triangle, isosceles triangle, square, kite, rectangle, parallelogram, trapezium,
hexagon, octagon
a Kite
b Parallelogram
c
Rhombus
d
Square
e
Trapezium
f
Trapezium
g
Rectangle
h
Parallelogram
Exercise 5.13 page 328
1
2
a
b
70°
55°
c
60°
d
73°
e
45°
f
110°
a
b
a = 30°, b = 45°
x = 50°, y = 80°, z = 70°
c
p = 130°, q = 15°, r = 60°
d
d = 35°, e = 55°, f = 55°
e
a = 27.5°, b = 27.5°, c = 55°, d = 27.5°, e = 97.5°
f
p = 45°, q = 45°, r = 67.5°, s = 112.5°
Exercise 5.14 page 320
1
a = 115°
2
x = 40°, y = 140°, z = 140°
3
m = 75°, n = 75°
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4
s = 65°, t = 115°, u = 115°
5
h = 120°, i = 60°, j = 120°, k = 60°
6
a = 80°, b = 20°, c = 20°, d = 20°, e = 140°
7
p = 40°, q = 130°, r = 50°
8
p = 75°, q = 30°, r = 50°, s = 80°, t = 70°, u = 70°, v = 40°
5
Exercise 5.15 page 332
1
2
3
4
5
6
a
720°
b
1260°
c
900°
a
135°
b
90°
c
144°
d
150°
a
72°
b
30°
c
51.4° (1 d.p.)
a
18
b
10
c
36
d
8
e
20
f
120
a
5
b
12
c
20
d
15
e
40
f
360
12
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7
8
9
5
See table below:
Number of
sides
Name
Sum of
exterior
angles
Size of an
exterior
angle
Sum of
interior
angles
Size of an
interior
angle
3
Equilateral triangle
360°
120°
180°
60°
4
Square
360°
90°
360°
90°
5
Pentagon
360°
72°
540°
108°
6
Hexagon
360°
60°
720°
120°
7
Heptagon
360°
51.4°
900°
128.6°
8
Octagon
360°
45°
1080°
135°
9
Nonagon
360°
40°
1260°
140°
10
Decagon
360°
36°
1440°
144°
12
Dodecagon
360°
30°
1800°
150°
a
42°, 84°, 126°, 126°, 168° (and 174°)
b
138°, 96°, 54°, 54°, 12° and 6° respectively
CAB = 15°
Exercise 5.16 page 334
1
a
Interior angles are the same, i.e. 60°, 30° and 90°.
b
5
8
c
x = 6.25 cm, y = 3.75 cm
2
A, C and F are similar. B and D are similar.
3
a
6 cm
b
9 cm
4
p = 4.8 cm, q = 4.5 cm, r = 7.5 cm
5
a
10 cm2
b
1.6
c
25.6 cm2
a
33 cm2
b
6 cm
6
2
3
1
3
7
9.6 cm
8
No, as the corresponding angles may not be the same.
9
No, as, despite the corresponding angles being the same, the slanting side lengths may not be in
the same ratio as the horizontal sides.
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Exercise 5.17 page 337
1
50 cm2
2
10 cm2
3
a
i
456 cm2 (3 s.f.)
ii
90 cm2
iii 40 cm2
b
Triangle I
4
43.56 cm2
5
56.25 cm2
6
18.1 cm2 (3 s.f.)
Exercise 5.18 page 338
1
2
3
a
220 cm2
b
1375 cm2
c
200 cm3
d
3125 cm3
a
54 cm2
b
3
c
729 cm3
a
1 : n2
b
1 : n3
4
112 cm3
5
0.64 litres
Exercise 5.19 page 339
1
20 cm
2
a
1:8
b
1:7
3
16 cm3
4
a
Not similar. Student’s own reasons
b
1:3
a
16 cm2
b
64 cm2
c
144 cm2
a
30 km2
5
6
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7
b
6 cm2
a
10 cm × 20 cm × 30 cm
b
100 g
5
Exercise 5.20 page 341
Students should give a valid reason for each answer.
1
60°
2
135°
3
20°
4
32°
5
110°
6
22.5°
Exercise 5.21 page 343
1
35°
2
60°
3
40°
4
45°
5
24°
6
26°
7
13 cm
8
8 cm
9
17.7 cm (3 s.f.)
Exercise 5.22 page 345
1
55°
2
80°
3
90°
4
100°
5
x = 54°, y = 18°
6
x = 50°, y = 25°
Exercise 5.23 page 346
1
a = 72°
2
b = 33°, c = 66°
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3
d = 48°, e = 32°
4
f = 30°, g = 120°, h = 120°, i = 30°, j = 30°
5
k = 55°, l = 55°, m = 55°, n = 55°
6
p = 65°, q = 40°
5
Exercise 5.24 page 348
1
a = 80°, b = 65°
2
c = 110°, d = 98°, e = 70°
3
f = 115°, g = 75°
4
i = 98°, j = 90°, k = 90°
5
l = 95°, m = 55°, n = 85°, p = 95°, q = 55°
6
r = 70°, s = 120°, t = 60°, u = 110°
7
v = 80°, w = 45°, x = 55°, y = 100°
8
𝛼𝛼 = 63°, 𝛽𝛽 = 63°, 𝜙𝜙 = 54°
Exercise 5.25 page 354
1
2
3
a
Isosceles
b
Perpendicular bisector
c
50°
d
50°
a
True
b
True
c
False
d
True
a
False
b
True
Exercise 5.26 page 351
1
a
70°
b
72°
c
21°
Towers of Hanoi page 352
1
3
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5
2
15
3
Student’s investigation
4
The results up to 8 discs are given below:
Number of
discs
Smallest
number of
moves
1
1
2
3
3
7
4
15
5
31
6
63
7
127
8
255
5
The numbers of moves are 1 less than the powers of 2.
6
1023
7
Number of moves = 2n – 1, where n = number of discs.
8
Time taken to move 64 discs is 264 – 1 seconds
This equates to 5.85 × 1011 years, i.e. 585 billion years.
Therefore according to the legend we needn’t be too worried!
Fountain borders page 353
1
Student’s results
2
T = 2(m + n + 2)
3
There are many ways to prove the algebraic rule, for example:
The original pool considered on p33 (Fig.1) has the same number of tiles as a rectangular pool
of dimensions 11 × 6 units (Fig.2).
Fig. 1
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5
Fig. 2
Fig. 3
In Fig.3 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.
Hence T = 2m + 2n + 4 which factorises to T = 2(m + n + 2).
Tiled walls page 353
1
Student’s diagrams
2
Student’s ordered table of results
3
c = (l – 1)(w – 1)
4
t = 2(l – 1) + 2(w – 1)
t = 2l + 2w – 4
Gift wrap page 354
1
1200 cm2
2
12x2 cm2
3
10x2 cm2
4
One of the following
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5
5
A = 3p(2p + 2r) = 6p(p + r)
6
Minimum area = 12p2
7
Maximum area = 30p2
a and b
8
The region between the two graphs represents the possible areas of wrapping paper used.
9
Choose a value of 𝑝𝑝 that falls in the area enclosed between the two curves and straight lines,
e.g. p = 12cm and A = 3000cm2.
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10 Student’s answers will vary depending on the value of p and A chosen, e.g. using p = 12cm and
A = 3000cm2 and rearranging the formula A = 6p(p + r) to make r the subject, gives a solution
r = 29.7 cm (1 d.p).
ICT Activity 1 page 356
2
The ratios are the same.
3
a
The ratios remain the same.
b
Ratios are still equal to each other (though probably of a different value from 1d).
4
Ratios change as ED no longer parallel to AB.
ICT Activity 2 page 354
Student’s own demonstrations
Student assessment 1 page 357
1
Student’s own diagram leading to:
a 66 km, 050°
b
2
3
75 km, 313°
Student’s own diagram:
The ship is due north of the lifeboat and is 64 km away.
Student’s own diagram:
Airport B is 329 km from the aircraft on a bearing of 234°.
Student assessment 2 page 357
1
a
H
1:  
h
b
H
1:  
h
2
3
2
x=
3
156 cm2 (3 s.f.)
4
a
1000 cm3
b
600 cm2
41 cm, y =
5
18.75 cm3
6
a
4 m by 3 m
b
12 m2
7
3 41
cm, z = 6.4 cm
5
3200 cm3
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180
5
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5
Student assessment 3 page 359
1
a
m = 50°, n = 70°, p = 70°, q = 50°
b
w = 85°, x = 50°, y = 50°, z = 45°
c
a = 130°, b = 25°, c = 25°, d = 50°
2
58°
3
30°
4
25°
5
152°
6
28°
7
125°
8
45°
9
42°
10 38°
11 a
b
67° Alternate segment theorem
226°
Obtuse ∠ AOB = 134°, angle subtended at centre by an arc is twice the angle on the
circumference subtended by same arc.
c
Reflex ∠ AOB = 360° − 134° = 226°, angle around a point is 360°.
23°
Δ AOB is isosceles, therefore ∠ AOB =
180−134
2
= 23°.
Student assessment 4 page 361
1
a
i
b
i
ii
iii
c
iii
d
ii
iii
∠DAB and ∠DCB, ∠CDA and ∠CBA
∠AOC and ∠CBA, ∠OCB and ∠OAB
∠OAB, ∠OCB
∠ABO = ∠CBO, ∠AOB = ∠COB
∠TPS = ∠TQS = ∠TRS, ∠PTR = ∠PSR, ∠PTQ = ∠PSQ, ∠QTR = ∠QSR
∠OAX, ∠OCY
∠DCB = ∠BAD, ∠CDA = ∠CBA
2
36°
3
∠ABC = 50° ∠OAB = 25° ∠CAO = 40°
4
∠BCD = 40° = ∠BAD ∠ABC = 65° = ∠ADC
5
28° and 56° respectively
6
67.5°
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7
a
82° Alternate segment theorem
b
8° ∠ AOB = 164°, angle subtended at centre by an arc is twice the angle on the
circumference subtended by same arc. ΔAOB is isosceles, therefore
∠ ABO =
180−164
2
5
= 8°.
Student assessment 5 page 362
1
Proof
2
50°
3
Proof
4
a
b
∠ ACB = 52°
∠ BAX = 90 – 38 = 52°, angle between a tangent and a radius is a right angle.
Therefore ∠ ACB = 52°, alternate segment theorem.
∠ AXB = 76°
Δ ABX is isosceles as AX = BX, as lengths of tangents are equal.
Therefore ∠ AXB = 180 − (2 × 52) = 76°.
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6
6 Mensuration
All answers were written by the authors.
Exercise 6.1 page 366
1
2
a
b
100
1000
c
1
1000
d
Millilitre
a
b
m, cm
cm
c
g
d
ml
e
cm, m
f
tonne
g
litres
h
km
i
litres
Exercise 6.2 page 367
1
2
a
b
mm
m
c
mm
d
m
e
m
a
b
85
230
c
830
d
50
e
0.4
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3
4
a
b
5.6
6400
c
0.96
d
4
e
0.012
a
b
1.15
250
c
0.5
d
0.07
e
0.008
6
Exercise 6.3 page 368
1
a
b
3800
28.5
c
4280
d
0.32
e
500
Exercise 6.4 page 368
1
2
a
b
4500 ml
1530 ml
c
7050 ml
d
1000 ml
a
b
1.2 litres
1.34 litres
c
1.4 litres
d
1.4 litres
Exercise 6.5 page 369
1
a
b
100 000 cm2
2 000 000 mm2
c
5 000 000 m2
d
3 200 000 m2
e
830 mm2
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2
3
4
a
b
0.05 m2
150 cm2
c
0.001 km2
d
0.04 m2
e
0.000 25 km2
a
b
2 500 000 cm3
3400 mm3
c
2 000 000 000 m3
d
200 litres
e
30 litres
a
b
0.15 m3
24 cm3
c
0.000 85 km3
d
0.3 cm3
e
0.000 015 m3
6
Exercise 6.6 page 371
1
2
a
b
6 cm2
32.5 cm2
c
20 cm2
d
60 cm2
e
108 cm2
f
55 cm2
a
b
64 cm2
1168 mm2
c
300 cm2
d
937.5 mm2
Exercise 6.7 page 375
1
2
a
b
25.1 cm
22.0 cm
c
28.9 cm
d
1.57 m
a
b
50.3 cm2
38.5 cm2
c
66.5 cm2
d
0.196 m2
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3
4
a
b
2.39 cm (3 s.f.)
0.5 cm
c
0.637 m (3 s.f.)
d
1.27 mm (3 s.f.)
a
b
4.51 cm (3 s.f.)
6 cm
c
3.23 m (3 s.f.)
d
4.31 mm (3 s.f.)
6
Exercise 6.8 page 376
1
a
b
2
3
4
5
188 m (3 s.f.)
84π
36(4 – π) cm2
a 8(4 + π) m = 57.1 m (to 3 s.f.)
b 16(8 + π) m2 = 178 m2 (to 3 s.f.)
1.57 m (2 d.p.)
637 times (3 s.f.)
Exercise 6.9 page 378
1
2
a
b
6.28 cm
2.09 cm
c
11.5 cm
d
23.6 cm
a
i
32.7° (1 d.p.)
ii
minor arc
i
229.2° (1 d.p.)
ii
major arc
i
57.3° (1 d.p.)
ii
minor arc
i
114.6° (1 d.p.)
ii
minor arc
b
c
d
3
a
b
12.2 cm (3 s.f.)
4.58 cm (3 s.f.)
c
18.6 cm (3 s.f.)
d
4.01 cm (3 s.f.)
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6
Exercise 6.10 page 378
1
2
3
a
11
π
4
+ 34 cm
b
18π + 48 cm
a
b
3.67 cm (3 s.f.)
49.7 cm (3 s.f.)
c
68.8° (1 d.p.)
a
b
12 cm
54 cm
c
47.7° (1 d.p.)
Exercise 6.11 page 380
1
2
3
a
b
33.5 cm2 (3 s.f.)
205 cm2 (3 s.f.)
c
5.65 cm2 (3 s.f.)
d
44.7 cm2 (3 s.f.)
a
b
18.5 cm (3 s.f.)
20.0 cm (3 s.f.)
c
1.75 cm (3 s.f.)
d
12.4 cm (3 s.f.)
a
b
48°
34°
c
20°
d
127°
4
124 cm2
5
2:1
Exercise 6.12 page 380
1
2
96 −
16
π
3
m2
a
b
118 cm2 (3 s.f.)
39.3 cm2 (3 s.f.)
c
8.66 cm (3 s.f.)
3
a
b
4.19 cm (3 s.f.)
114 cm2 (3 s.f.)
4
a
b
9.17 cm (3 s.f.)
0.83 cm (2 s.f.)
c
34.9 cm2 (3 s.f.)
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d
6
Length = 20 cm (3 s.f.)
Width = 10.8 cm (3 s.f.)
e
77.1 cm2 (3 s.f.)
Exercise 6.13 page 383
1
a
b
58.5 cm2
84 cm2
c
106 cm2 (3 s.f.)
d
157.5 cm2
Exercise 6.14 page 383
1
2
3
4
4
3
23.0 m2
a 16 m2, 24 m2
b
100 m2
c
15
Exercise 6.15 page 386
1
2
3
4
a
b
460 cm2
208 cm2
c
147.78 cm2
d
33.52 cm2
a
b
2 cm
4 cm
c
6 cm
d
5 cm
a
b
101 cm2 (3 s.f.)
276 cm2 (3 s.f.)
c
279 cm2 (3 s.f.)
d
25.6 cm2 (3 s.f.)
a
b
1.2 cm
0.5 cm
c
1.7 cm
d
7.0 cm
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6
Exercise 6.16 page 386
1
a
b
24 cm2
2 cm
2
a
b
216 cm2
i 15.2 cm (1 d.p.)
ii
3
a
b
4
4.4 cm
54
π
− 2 cm
94.2 cm2 (3 s.f.)
14 cm
Exercise 6.17 page 388
1
2
3
4
a
b
24 cm3
18 cm3
c
27.6 cm3
d
8.82 cm3
a
b
452 cm3 (3 s.f.)
277 cm3 (3 s.f.)
c
196 cm3 (3 s.f.)
d
0.481 cm3 (3 s.f.)
a
b
108 cm3
140 cm3
c
42 cm3
d
6.2 cm3
a
b
70 cm3
96 cm3
c
380 cm3
d
137.5 cm3
Exercise 6.18 page 389
1
a
b
16 cm
4096 cm3
c
3217 cm3
d
21.5% (3 s.f.)
2
a
b
42 cm2
840 cm3
3
6.3 cm
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4
5
6
2.90 m3 (3 s.f.)
a 264 cm2
b 216 cm3
Exercise 6.19 page 391
1
2
a
b
905 cm3 (3 s.f.)
3590 cm3 (3 s.f.)
c
2310 cm3 (3 s.f.)
d
1.44 cm3 (3 s.f.)
a
b
3.1 cm
5.6 cm
c
36.3 cm
d
0.6 cm
Exercise 6.20 page 391
1
6.3 cm
2
331
3
4
5
6
7
12
π cm3
11.9 cm (1 d.p.)
48%
10.0 cm
A = 4.1 cm, B = 3.6 cm, C = 3.1 cm
3:2
Exercise 6.21 page 396
1
2
3
4
5
a
b
452 cm2 (3 s.f.)
254 cm2 (3 s.f.)
c
1890 cm2 (3 s.f.)
d
4 cm2
a
b
1.99 cm (3 s.f.)
1.15 cm (3 s.f.)
c
3.09 mm (3 s.f.)
d
0.5 cm
1:4
707 cm2 (3 s.f.)
5.9 cm (1 d.p.)
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6
Exercise 6.22 page 396
1
2
3
4
40 cm3
133 cm3 (3 s.f.)
64 cm3
70 cm3
Exercise 6.23 page 396
1
2
3
4
7 cm
5 cm
a 8 cm
b 384 cm3
c
378 cm3
a
b
3.6 cm
21.7 cm3 (3 s.f.)
c
88.7 cm3 (3 s.f.)
Exercise 6.24 page 399
1
2
3
a
b
56.5 cm3 (3 s.f.)
264 cm3 (3 s.f.)
c
1.34 cm3 (3 s.f.)
d
166 cm3 (3 s.f.)
a
b
6.91 cm (3 s.f.)
10.9 cm (3 s.f.)
c
0.818 cm (3 s.f.)
d
51.3 cm (3 s.f.)
a
i
7.96 cm
ii
12.7 cm
iii 843 cm3
b
i
15.9 cm
ii
8.41 cm
iii 2230 cm3
c
i
6.37 cm
ii
3.97 cm
iii 168 cm3
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d
i
6
3.82 cm
ii
4.63 cm
iii 70.7 cm3
Exercise 6.25 page 400
1
3.88 cm (3 s.f.)
2
a
π cm
b
2
21
c
√855
d
3
4
21
4
cm
4
cm
211 cm3
1700 cm3 (3 s.f.)
8 : 27
Exercise 6.26 page 401
1
2
3
4
625
24
π cm3
771 cm3 (3 s.f.)
3170 cm3 (3 s.f.)
a 654 cm3
b 12.5 cm
c
2950 cm3
Exercise 6.27 page 403
1
a
b
2
3
1130 cm2 (3 s.f.)
a 314 cm2 (3 s.f.)
b 10.7 (3 s.f.)
415 cm2 (3 s.f.)
1650 cm2 (3 s.f.)
Tennis balls page 404
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
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2
6
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):
Dimensions
(units)
Length (cm)
Width |(cm)
Height (cm)
Surface area
(cm2)
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
The optimum dimensions of the box are 13.2 cm × 13.2 cm × 19.8 cm.
Modelling: Metal trays page 404
1
a
b
Length = 38 cm, Width = 28 cm, Height = 1 cm
1064 cm3
2
a
b
Length = 36 cm, Width = 26 cm, Height = 2 cm
1872 cm3
3
a
15 cm
𝑉𝑉 = (40 − 2𝑥𝑥)(30 − 2𝑥𝑥)𝑥𝑥 = 4𝑥𝑥 3 − 140𝑥𝑥 2 + 1200𝑥𝑥
b
4
a
b
Student’s sketch of the
graph shown.
𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 = 3032 cm3
𝑥𝑥 = 5.7 cm (1 d.p)
5
0 < 𝑥𝑥 < 15. 0 and 15 are not included in the range as these values would not produce a tray.
ICT Activity page 406
1
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
2
Student’s own answer
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6
Student assessment 1 page 407
1
2
90 cm2
a Circumference = 27.0 cm, Area = 58.1 cm2
b Circumference = 47.1 mm, Area = 177 mm2
3
48 − 9π
2
4
5
6
cm2
a
b
39.3 cm2
34 cm2
c
101.3 cm2
a
b
10.2 cm2
283 cm2
c
633 cm2
a
b
108π mm3
3.125π cm3
Student assessment 2 page 408
1
2
3
104 cm2
326 cm2
a 56.5 cm2
b 108 cm2
c
4
5
254.5 cm2
418 cm3
a 322π cm2
b
333
2
π cm2
Student assessment 3 page 410
1
a
b
11.8 cm (3 s.f.)
35.3 cm (3 s.f.)
2
a
b
272.2° (1 d.p.)
5.7° (1 d.p.)
3
729
𝜋𝜋
cm2
4
a
b
531 cm2
1150 cm3
5
a
b
1210 cm2 (3 s.f.)
2592 cm3
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6
Student assessment 4 page 411
1
a
b
178 cm (3 s.f.)
68.3 mm (3 s.f.)
2
a
b
143.2° (1 d.p.)
2.9° (1 d.p.)
3
4
95.5 cm2 (3 s.f.)
a 603.2 cm2
b 1072.3 cm3
5
3270 cm3
Student assessment 5 page 412
1
a
b
22.9 cm (3 s.f.)
229 cm2 (3 s.f.)
c
985 cm2 (3 s.f.)
d
1830 cm3 (3 s.f.)
2
a
b
288π cm3
12 cm
3
a
b
10 cm
82.1 cm3 (3 s.f.)
c
71.8 cm3 (3 s.f.)
d
30.8 cm3 (3 s.f.)
e
41.1 cm3 (3 s.f.)
a
b
3620 cm3 (3 s.f.)
3620 cm3 (3 s.f.)
c
905 cm2 (3 s.f.)
d
1920 cm2 (3 s.f.)
4
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7
7 Trigonometry
All answers were written by the authors.
Exercise 7.1 page 418
1
2
3
a
5 cm
b
11.4 cm (3 s.f.)
c
12 cm
d
13.2 cm (3 s.f.)
a
11.0 cm (3 s.f.)
b
14.8 cm (3 s.f.)
c
7.86 cm (3 s.f.)
d
7.35 cm (3 s.f.)
e
3 cm
f
13.9 cm (3 s.f.)
a
Right-angled
b
Right-angled
c
Not right-angled
d
Right-angled
4
71.6 km
5
67 km
6
a
225° – 135° = 90°
b
73.8 km
7
57 009 m
8
a
8.5 km
b
15.5 km (3 s.f.)
a
13.3 m (3 s.f.)
b
15.0 m (3 s.f.)
9
Exercise 7.2 page 421
1
a
b
c
i
8m
ii
4.8 m
2√2m
16√2 m2
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2
3
a
10 cm
b
2.5
c
150 cm2
a
b
10.9 cm (3 s.f.)
9.33 cm (3 s.f.)
7
Exercise 7.3 page 425
1
2
3
a
1.82 cm
b
4.04 cm
c
19.2 cm
d
4.87 cm
e
37.3 cm
f
13.9 cm
a
14.3 cm
b
8.96 cm
c
9.33 cm
d
4.10 cm
e
13.9 cm
f
6.21 cm
a
49.4°
b
51.1°
c
51.3°
d
63.4°
e
50.4°
f
71.6°
Exercise 7.4 page 427
1
a
2.44 cm
b
18.5 cm
c
6.19 cm
d
2.44 cm
e
43.8 cm
f
31.8 cm
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2
a
38.7°
b
48.6°
c
38.1°
d
49.8°
e
32.6°
f
14.5°
7
Exercise 7.5 page 429
1
36.0 cm
2
15.1 cm
3
48.2°
4
81.1°
5
6.7 cm
6
16.8 cm
7
70.5°
8
2.1 cm
Exercise 7.6 page 430
1
2
3
4
a
43.6°
b
19.5 cm
c
16.7 cm
d
42.5°
a
20.8 km (3 s.f.)
b
215.2° (1 d.p.)
a
228 km (3 s.f.)
b
102 km (3 s.f.)
c
103 km (3 s.f.)
d
147 km (3 s.f.)
e
415 km (3 s.f.)
f
217.0° (1 d.p.)
a
6.71 m (3 s.f.)
b
19.6 m (3 s.f.)
c
15.3 m (3 s.f.)
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5
a
48.2° (3 s.f.)
b
41.8° (1 d.p.)
c
8 cm
d
8.94 cm (3 s.f.)
e
76.0 cm2 (3 s.f.)
7
Exercise 7.7 page 438
1
2
3
4
a
sin 120°
b
sin 100°
c
sin 65°
d
sin 340°
e
sin 240°
f
sin 275°
a
sin 145°
b
sin 130°
c
sin 150°
d
sin 292°
e
sin 236°
f
sin 213°
a
19°, 161°
b
82°, 98°
c
5°, 175°
d
210°, 330°
e
240°, 300°
f
225°, 315°
a
70°, 110°
b
9°, 171°
c
53°, 127°
d
214°, 326°
e
196°, 344°
f
199°, 341°
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7
Exercise 7.8 page 438
1
2
3
4
5
a
cos 340°
b
cos 275°
c
cos 328°
d
cos 265°
e
cos 213°
f
cos 254°
a
cos 262°
b
cos 216°
c
cos 200°
d
cos 177°
e
cos 149°
f
cos 126°
a
cos 80°
b
cos 90°
c
cos 70°
d
cos 135°
e
cos 58°
f
cos 155°
a
b
cos 90°
cos 75°
c
cos 60°
d
cos 82°
e
cos 88°
f
cos 70°
The tangent of 90° is undefined because the ratio of opposite side to adjacent side becomes 1
divided by 0, and division by 0 is undefined.
Exercise 7.9 page 440
1
a
b
14.5°, 165.5°
45°, 315°
c
210°, 330°
d
120°, 300°
e
78.5°, 281.5°
f
116.6°, 296.6°
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2
a
b
3
a
b
4
sin 𝜃𝜃 =
cos 𝜃𝜃 =
cos 𝜃𝜃 =
tan 𝜃𝜃 =
7
3
5
4
5
√91
10
3
√91
opposite
adjacent
sin 𝜃𝜃 = hypotenuse, cos 𝜃𝜃 = hypotenuse
When sin 𝜃𝜃 = cos 𝜃𝜃,
opposite
hypotenuse
=
adjacent
hypotenuse
therefore opposite = adjacent.
If opposite side length = adjacent side length then θ = 45°
Exercise 7.10 page 444
1
2
3
a
8.91 cm
b
8.93 cm
c
5.96 mm
d
8.64 cm
a
33.2°
b
52.7°
c
77.0°, 103°
d
44.0°, 136°
a
25°, 155° (nearest degree)
b
4
a
75°, 105° (nearest degree)
b
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Exercise 7.11 page 446
1
2
3
a
4.71 m
b
12.1 cm
c
9.15 cm
d
3.06 cm
e
10.7 cm
a
125.1°
b
108.2°
c
33.6°
d
37.0°
e
122.9°
a
12.3 cm (1 d.p)
b
5.1 cm (1 d.p)
Exercise 7.12 page 448
1
a
42.9 m (3 s.f.)
b
116.9° (1 d.p.)
c
24.6° (1 d.p.)
d
33.4° (1 d.p.)
e
35.0 m (3 s.f.)
2
371 m (3 s.f.)
3
73.9 m (3 s.f.)
Exercise 7.13 page 449
1
2
a
70.0 cm2
b
70.9 mm2
c
122 cm2
d
17.0 cm2
a
24.6°
b
13.0 cm
c
23.1 cm
d
63.2°
3
16 800 m2
4
a
3.90 m2 (3 s.f.)
b
222 m3 (3 s.f.)
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Exercise 7.14 page 451
1
2
3
4
5
a
12.2 km (3 s.f.)
b
9.5° (1 d.p.)
a
10.1 km (3 s.f.)
b
1.23 km (3 s.f.)
a
22.6° (1 d.p.)
b
130 m
a
0.342 km (3 s.f.)
b
0.940 km (3 s.f.)
a
64.0 m (3 s.f.)
b
30.2 m (3 s.f.)
6
6.93 km (3 s.f.)
7
a
5.88 km (3 s.f.)
b
1.57 km (3 s.f.)
a
7.46 km (3 s.f.)
b
3.18 km (3 s.f.)
a
2.9 km (1 d.p.)
b
6.9 km (1 d.p.)
c
11.4° (1 d.p.)
d
20.4 km (1 d.p.)
10 a
2.68 km (3 s.f.)
b
1.02 km (3 s.f.)
c
3.5° (1 d.p.)
d
16.83 km (2 d.p.)
8
9
11 a
b
225 m
48.4° (1 d.p.)
12 1.9 km
13 a
PR = 360 m
b
PQ = 240 m
14 r = 5.41 cm (3 s.f.)
15 a
55°
b
XZ = 6011 m
c
YZ = 4316 m
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Exercise 7.15 page 457
1
2
3
4
5
a
5.66 cm (3 s.f.)
b
6.93 cm (3 s.f.)
c
54.7° (1 d.p.)
a
5.83 cm (3 s.f.)
b
6.16 cm (3 s.f.)
c
18.9° (1 d.p.)
a
6.40 cm (3 s.f.)
b
13.6 cm (3 s.f.)
c
61.9° (1 d.p.)
a
75.3° (1 d.p.)
b
56.3° (1 d.p.)
a
i
7.21 cm (3 s.f.)
ii
21.1° (1 d.p.)
i
33.7° (1 d.p.)
ii
68.9° (1 d.p.)
i
8.54 cm (3 s.f.)
ii
28.3° (1 d.p.)
i
20.6° (1 d.p.)
ii
61.7° (1 d.p.)
b
6
a
b
7
8
9
a
211 cm2
b
176 cm3
a
6.5 cm
b
11.3 cm (3 s.f.)
c
70.7 cm (3 s.f.)
a
11.7 cm (3 s.f.)
b
7.55 cm (3 s.f.)
10 a
TU = TQ = 10 cm, QU = 8.49 cm
b
90°, 36.9°, 53.1°
c
24 cm2
11 6.93 cm2 (3 s.f.)
12 189 cm2 (3 s.f.)
13 73.3 cm2 (3 s.f.)
14 1120 cm2 (3 s.f.)
15 a
693 cm2 (3 s.f.)
b
137 cm2 (3 s.f.)
c
23.6 cm (3 s.f.)
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Exercise 7.16 page 461
1
2
3
4
5
6
7
8
9
a
RW
b
TQ
c
SQ
d
WU
e
QV
f
SV
a
JM
b
KN
c
HM
d
HO
e
JO
f
MO
a
b
∠ TPS
∠ UPQ
c
∠ VSW
d
∠ RTV
e
∠ SUR
f
∠ VPW
a
5.83 cm (3 s.f.)
b
31.0° (1 d.p.)
a
10.2 cm (3 s.f.)
b
29.2° (1 d.p.)
c
51.3° (1 d.p.)
a
6.71 cm (3 s.f.)
b
61.4° (1 d.p.)
a
7.81 cm (3 s.f.)
b
11.3 cm (3 s.f.)
c
12.4° (1 d.p.)
a
14.1 cm (3 s.f.)
b
8.48 cm (3 s.f.)
c
7.48 cm (3 s.f.)
d
69.3° (1 d.p.)
a
17.0 cm (3 s.f.)
b
5.66 cm (3 s.f.)
c
7 cm
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d
7
51.1° (1 d.p.)
Exercise 7.17 page 468
1
2
0

 −3 
a
Translation 
b
Translation 
c
Translation 
 −60 

 0 
 −90   270 
 or 

 0   0 
a
b
3
a
Functions used are: y = cosx + 2, y = cosx + 1, y = cosx, y = cosx – 1 and y = cosx – 2.
b
Functions used are: y = tan(x + 30), y = tan(x + 60), y = tan(x + 90), y = tan(x + 120) and
y = tan(x + 150).
Note: other translations are possible.
Pythagoras and circles page 473
1–3
Student’s own drawings and calculations
4
Yes
5
Yes
6
Student’s own investigation
Numbered balls page 474
1
If a ball is odd (n), the next ball is n + 1. If the ball is even (n), the next ball is
2
65, 66, 33, 34, 17, 18, 9, 10, 5 ,6, 3, 4, 2, 1
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.
2
11
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3
The first ball must be odd. Start at 1 and work backwards.
4
513, 514, 257, 258, 129, 130, 65, 66, 33, 34, 17, 18, 9, 10, 5, 6, 3, 4, 2, 1
7
The ferris wheel page 474
1
2.5 cm
2
9.9 cm (1 d.p)
3
a
60°
b
Proof
4
Proof
5
a
105 m
b
26.5 m (1 d.p)
a
9.6 mins
b
Due to symmetry, the next time occurs at 30 − 9.6 = 20.4 minutes after entering the
wheel.
6
7
The plot of the graph is given below.
Intercept with y-axis occurs at (0, 15).
Coordinates of maximum point is (15, 135).
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7
ICT Activity page 476
1
b
i
0.940 (3 d.p.)
ii
0.819 (3 d.p.)
iii –0.866 (3 d.p.) or –
3
2
c
The graph of y = sin x intersects the line y = 0.7 in two places as shown.
d
2
3
30° and 150°
a
b
Two solutions
c
225°
a
Solutions are 0°, 180° and 360°.
b
Solutions are 38.2° and 141.8° (1 d.p.).
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Student assessment 1 page 477
1
a
5 cm
b
x = 4.5 cm, y = 7.5 cm
2
125 km
3
32.4 m (3 s.f.) 48.6 m (3 s.f.)
4
288 m (3 s.f.)
5
a
3.42 km
b
3.57 km
a
90°
b
6.5 cm
6
7
11.8 mm (3 s.f.)
8
13 cm
9
a
30 cm
b
188 cm (3 s.f.)
Student assessment 2 page 479
1
2
3
4
5
a
43.9 cm
b
3.92 cm
a
56°
b
34°
a
6.63 cm
b
28.5 cm
a
tan θ =
5
x
b
tan θ =
7.5
( x + 16 )
c
5
7.5
=
x ( x + 16 )
d
32 m
e
8.9° (1 d.p.)
a
1.96 km (3 s.f.)
b
3.42 km (3 s.f.)
c
3.57 km (3 s.f.)
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Student assessment 3 page 481
1
a
10.8 cm (3 s.f.)
b
11.9 cm (3 s.f.)
c
30.2° (1 d.p.)
d
49.0° (1 d.p.)
2
Student’s graph
3
a
–cos 52°
b
cos 100°
a
9.81 cm (3 s.f.)
b
30°
c
19.6 cm (3 s.f.)
a
678 m (3 s.f.)
b
11.6° (1 d.p.)
c
718 m (3 s.f.)
4
5
Student assessment 4 page 482
1
a
b
2
3
4
sin 𝜃𝜃 =
tan 𝜃𝜃 =
√11
6
√11
5
a
b
23.6°, 156.4° (1 d.p)
150°, 330°
c
95.7°, 264.3°
d
90°
3
By drawing the graph of cosine, you can see that the maximum value of cosine is 1, so cos x =
2
has no solutions.
a Angle ACB = 90° due to angle in a semi-circle = right-angle
b
3 3√3
�or
2
�2 ,
3
2
� ,−
3√3
�
2
Student assessment 5 page 483
1
a
21.8° (1 d.p.)
b
8.5° (1 d.p.)
c
2.2 km
d
10.5° (1 d.p.)
e
15.3° (1 d.p.)
f
1.76 km (3 s.f.)
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2
3
4
7
a
b
θ = 45°, 225°
a
5.83 cm (3 s.f.)
b
6.71 cm (3 s.f.)
c
7.81 cm (3 s.f.)
d
46.6° (1 d.p.)
e
19.0 cm2 (3 s.f.)
f
36.7° (1 d.p.)
a
0
Translation  
 −5 
b
 −120 
Translation 

 0 
c
 90 
Translation  
0
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8
8 Vectors and transformations
All answers were written by the authors.
Exercise 8.1 page 488
1
a
b
c
d
e
f
g
h
i
2
a
b
c
d
e
f
g
h
i
3
�24�
�31�
�−2
�
−4
3
�−2
�
�−6
�
1
6
�−1
�
�−3
�
−1
�−5
�
−5
�−1
�
3
�44�
�04�
�−3
�
0
�−2
�
6
�−4
�
−2
0
�−4
�
�30�
2
�−6
�
�−4
�
−4
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8
Exercise 8.2 page 489
1
2
a + b = b + a, a + d = d + a, b + c = c + b
3
4
a
b
c
d
e
f
�24�
0
�−6
�
1
�−1
�
�10�
�09�
�52�
Exercise 8.3 page 491
1
d = –c
1
2
h= b
2
a
b
c
d
e
�46�
e = –a
1
2
i=– b
f = 2a
3
2
j= b
1
2
g= c
3
2
k=– a
�−12
�
−3
2
�−4
�
�−2
�
2
�−2
�
−5
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f
g
3
�−10
�
−5
i
�32�
a
2a
h
8
�
�−8
9
10
�−6
�
b
–b
c
b+c
d
a–b
e
2c
f
2c – a
Exercise 8.4 page 493
1
|a| = 5.0 units, |b| = 4.1 units, |c| = 4.5 units, |d| = 7.0 units, |e| = 7.3 units, |f| = 6.4 units
2
a
|AB| = 4.0 units
b
|BC| = 5.4 units
c
|CD| = 7.2 units
d
|DE| = 13.0 units
e
|2AB| = 8.0 units
f
|–CD| = 7.2 units
a
4.1 units
b
18.4 units
c
15.5 units
d
17.7 units
e
31.8 units
f
19.6 units
3
Exercise 8.5 page 494
1
a
a
b
a
c
b
d
–b
e
2b
f
–2a
g
a+b
h
b–a
i
b – 2a
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Cambridge IGCSE™ International Mathematics answers
2
3
4
8
a
2a
b
b
c
b–a
d
b–a
e
–a
f
a – 2b
a
–2a
b
–a
c
b
d
b–a
e
2(b – a)
f
2b – a
g
a – 2b
h
b – 2a
i
–a – b
a
5a
b
e
8
b
3
1
(8b – 15a)
3
1
(8b – 15a)
15
f
5a – 2b
g
8
b
5
1
(10a + 8b)
5
1
(8b – 5a)
5
c
d
h
i
b – 2a
Exercise 8.6 page 495
1
a
b
i
2a
ii
2a – b
iii
1
(2a
4
+ 3b)
Proof
2
a, b Proofs
3
a
i
4a
ii
2a
iii 2(a + b)
iv
3
a
2
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4
b
Proof
a
i
ii
b
8
1
(2q – p)
2
2
(p – q)
7
Proof
Exercise 8.7 page 497
1
2
3
4
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216
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8
5
6
7
8
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8
Exercise 8.8 page 498
1
2
3
4
5
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8
6
Exercise 8.9 page 499
1
2
3
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8
4
5
6
Exercise 8.10 page 500
1
a
b
180° clockwise/anti-clockwise
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2
b
3
180° clockwise/anti-clockwise
a
b
5
90° anti-clockwise
a
b
4
8
a
90° anti-clockwise
a
b
90° anti-clockwise
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Cambridge IGCSE™ International Mathematics answers
6
7
8
a
b
90° clockwise
1
Rotation 180°
2
Rotation 90° clockwise
3
Rotation 180°
4
Rotation 90° clockwise
5
Rotation 90° clockwise
6
Rotation 90° anti-clockwise
Exercise 8.11 page 502
1
2
3
4
A → B = �−6
�,
0
A → C = �36�
A → B = �06�,
6
A → C = �−3
�
0
A → B = �−7
�,
A → B = �50�,
A → C = �−6
�
1
A → C = �−3
�
−6
Exercise 8.12 page 502
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8
2
3
4
5
6
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7
1
2
3
4
5
6
8
�
�−3
−5
�−5
�
4
4
�−6
�
�25�
�60�
�01�
Exercise 8.13 page 505
1
2
3
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8
4
5
6
1
2
2
3
1
2
1
3
1
scale factor
2
scale factor
3
scale factor
4
scale factor
5
scale factor 4
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8
Exercise 8.14 page 507
1
2
3
4
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8
Exercise 8.15 page 508
1
2
3
4
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5
a and b
6
a and b
8
Exercise 8.16 page 511
1
a
b
2
y=2
a
b
x = –1
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3
b
4
y=x–2
a
b
6
y = –1
a
b
5
8
a
y=–x–2
a
b
y = 0, x = −3
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7
b
8
8
a
y = 1, x = −3, y = x + 4, y = –x – 2
a
b
y = 1, x =−3, y = x + 4, y = –x – 2
Exercise 8.17 page 512
1
a
b
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8
c
d
2
a
b
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c
d
3
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Exercise 8.18 page 514
1
2
A painted cube page 515
1
a
8
b
12
c
6
d
1
2
A: 8, B: 24, C: 24, D: 8
3
A: 8, B: 96, C: 384, D: 512
4
When (𝑛𝑛 − 2) ≥ 0,
5
A: 8,
B: 12(𝑛𝑛 − 2),
C: 6(𝑛𝑛 − 2)2 ,
D: (𝑛𝑛 − 2)3
When (𝑙𝑙 − 2) ≥ 0, (𝑤𝑤 − 2) ≥ 0 and (ℎ − 2) ≥ 0,
A: 8,
B: 4(𝑙𝑙 − 2) + 4(𝑤𝑤 − 2) + 4(ℎ − 2),
C: 2(𝑙𝑙 − 2)(𝑤𝑤 − 2) + 2(𝑙𝑙 − 2)(ℎ − 2) + 2(𝑤𝑤 − 2)(ℎ − 2),
D: (𝑙𝑙 − 2)(𝑤𝑤 − 2)(ℎ − 2)
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Triangle count page 515
1
9
2
Student’s investigation and ordered table
Number of
horizontal lines (h)
Number of
triangles (t)
0
3
1
6
2
9
3
12
etc.
3
n = 3(h + 1)
4
12
5
Student’s investigation and ordered table
Number of
horizontal lines (h)
Number of
triangles (t)
0
6
1
12
2
18
3
24
etc.
6
n = 6(h + 1)
ICT activity page 517
1
25 seconds
2
Proof
3
23.1 seconds
4
Using Pythagoras:
Distance 𝑃𝑃1 𝑃𝑃0 = �𝑦𝑦 2 − 402 = �𝑦𝑦 2 − 1600
Distance 𝑃𝑃1 𝐶𝐶 = √1002 − 402 = 20√21
5
𝑥𝑥 = 𝑃𝑃1 𝐶𝐶 − 𝑃𝑃1 𝑃𝑃0 = 20√21 − �𝑦𝑦 2 − 1600
6
Theoretically, 𝑦𝑦 ≥ 40 due to avoiding a negative square root. In reality, 40 ≤ 𝑦𝑦 ≤ 100 as the
athlete will not swim from A past point C.
63.3m
7
21.3 seconds
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8
Below is part of a suggested spreadsheet for modelling the data.
Student assessment 1 page 520
1
a
b
c
2
a
b
c
d
e
3
4
a
b
c
d
�
�−2
7
�−5
�
−4
�−1
�
−3
�03�
�−6
�
0
�−1
�
−5
�33�
4
�−1
�
7
�
�−11
�−3
�
9
6
�−6
�
�−18
�
28
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Student assessment 2 page 519
1
2
3
a
����⃗| = 5 units
|FG
b
│a│ = 6.1 units, │b│ = 12.4 units, │c│= 14.1 units
a
29.7 units
b
11.4 units
a
2a
b
–b
c
b–a
Student assessment 3 page 519
1
2
a
5b
b
5b – a
c
1
(𝐚𝐚
2
a
i
ii
3
+ 3𝐛𝐛)
1
𝐚𝐚
2
b–a
b
2:3
a
a
b
–b
c
�1 + √2�𝐛𝐛
Student assessment 4 page 520
1
2
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3
a
b
4
5
8
0
�
�−3
�−8
�
2
a
b
The scale factor of enlargement is –2.
c
The inverse transformation is an enlargement of scale factor − 12 , with the same centre of
enlargement.
6
a
A reflection in the line x = 0
b
An enlargement by scale factor –0.5. Centre of enlargement (0, –1)
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9
9 Probability
All answers were written by the authors.
Exercise 9.1 page 526
1
Answers (apart from (c)) may vary slightly:
d
a
e
b
0
2
c
1
Student’s own answers
Exercise 9.2 page 528
1
2
3
4
a
1
6
b
1
6
c
1
2
d
5
6
e
0
f
1
a
i
1
7
ii
6
7
b
Total = 1
a
1
250
b
1
50
c
1
d
0
a
5
8
b
3
8
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5
6
1
13
b
5
26
c
21
26
d
3
26
e
Student’s own answers
a
i
1
10
ii
1
4
i
1
19
ii
3
19
b
7
8
9
a
a
1
37
b
18
37
c
18
37
d
1
37
e
21
37
f
12
37
g
17
37
h
11
37
a
RCA RAC CRA CAR ARC ACR
b
i
1
6
ii
1
3
iii
1
2
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iv
9
a
1
4
b
1
2
c
1
13
d
1
26
e
3
13
f
1
52
g
5
13
h
4
13
9
1
24
Exercise 9.3 page 527
1
a
Dice 2
Dice 1
b
1
4
c
1
4
d
9
16
1
2
3
4
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
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9
2
a
1
36
b
1
6
c
1
18
d
1
6
e
1
4
f
3
4
g
5
18
h
1
6
i
11
18
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9
Exercise 9.4 page 530
1
a
b
i
1
27
ii
1
3
iii
1
9
iv
1
3
v
5
9
vi
1
3
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2
b
3
9
a
i
1
16
ii
3
8
iii
15
16
iv
5
16
a
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b
c
4
i
1
27
ii
10
27
iii
19
27
iv
8
27
9
Student’s own answer
a
b
i
1
16
ii
1
8
iii
9
16
Exercise 9.5 page 531
1
a
b
i
1
6
ii
11
36
iii
5
36
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2
3
iv
125
216
v
91
216
c
They add up to 1, because either (iv) or (v).
a
4
25
b
54
125
c
98
125
9
a
b
i
0.275
ii
0.123
iii 0.444
iv 0.718
4
0.027
5
a
0.752 (0.56)
b
0.753 (0.42)
c
0.7510 (0.06)
Exercise 9.6 page 533
1
a
25
81
b
16
81
c
20
81
d
40
81
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2
3
4
5
a
5
18
b
1
6
c
5
18
d
5
9
a
1
25
b
3
10
a
1
45
b
1
3
a
940
b
6
9312
9
=
115
1164
8372 2093
=
9312 2328
a
1
17
b
11
850
c
703
1700
d
997
1700
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7
b
8
9
a
i
1
15
ii
1
15
iii
3
5
a
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b
9
a
b
1
2
P(G, G) = ×
P(Y, Y) =
𝑛𝑛−1
2𝑛𝑛−1
𝑛𝑛+3
2𝑛𝑛+3
×
=
9
𝑛𝑛−1
2(2𝑛𝑛−1)
𝑛𝑛+2
2𝑛𝑛+2
=
(𝑛𝑛+3)(𝑛𝑛+2)
(2𝑛𝑛+3)(2𝑛𝑛+2)
Exercise 9.7 page 536
1
2
3
a
b
13
30
a
5 1
=
35 7
b
14 2
=
35 5
c
13
35
45 5
=
108 12
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4
9
a
b
i
5
1
=
100 20
ii
20 1
=
100 5
iii
25 1
=
100 4
Exercise 9.8 page 540
1
0.93
2
0.8
3
a
b
c
1
6
2
3
1
2
4
1
36
if each month is equally likely, or
if taken as 30 days out of 365. (Leap years
144
5329
excluded − though this could be an extension.)
5
a
0.1
b
0.9
c
0.3
a
0.675
b
0.2
c
0.875
a
60 2
=
90 3
b
54 3
=
90 5
6
7
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9
22 11
=
90 45
76 38
8 a
=
142 71
22 11
b
=
142 71
12 6
c
=
142 71
32 8
9 a i
=
100 25
36 9
ii
=
100 25
10 1
iii
=
100 10
43
b i
225
ii 0.347 (3 d.p.)
iii 0.653 (3 d.p.)
10 a–c Student’s own response
c
d
Number of people
Probability of them not having the same
birthday
2
 364 

 = 99.7%
 365 
3
 364   363 
99.2%

×
=
 365   365 
4
 364   363   362 
98.4%

×
×
=
 365   365   365 
5
 364   363   362   361 
97.3%

×
×
×
=
 365   365   365   365 
10
 364   363 
 356 
88.3%

×
 × ... × 
=
 365   365 
 365 
15
 364   363 
 351 
74.7%

×
 × ... × 
=
 365   365 
 365 
20
 364   363 
 346 
58.9%

×
 × ... × 
=
 365   365 
 365 
23
 364   363 
 343 
49.3%

×
 × ... × 
=
 365   365 
 365 
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e
9
Only 23 people are needed before the probability of two sharing the same birthday is
greater than 50%.
Exercise 9.9 page 544
1
a
The dice is likely to be fair as the frequencies are approximately the same.
b
This dice is unlikely to be fair as the frequencies are very different.
c
2
3
Number
1
2
3
4
5
6
Frequency
10
25
32
5
20
8
Relative
frequency
10 1
=
100 10
25 1
=
100 4
32 8
=
100 25
5
1
=
100 20
20 1
=
100 5
8
2
=
100 25
d
36 times
e
Approximately 460 times
f
Approximately 35 more sixes
a
Approximately 5 crows
b
Approximately 8 hours
c
He shouldn’t use the original data because the appearance of birds is likely to be seasonal.
a
i
Approximately 138 days
ii
The punctuality of buses is constant all year round.
b
Approximately 12 times
c
(5 × 23) + (10 × 11.5) + (15 × 46) + (20 × 11.5) = 1150 minutes = 19 hours 10 minutes
d
i
4 791 667 hours ≈ 547 years
ii
That the punctuality of buses is independent of time of year, area of the city and length
of journey. (In reality, all three of these are likely to be factors.)
4
Student’s own experiment, report and conclusion
5
a
Student’s drawing-pin experiment
b
The surface the pin lands on, the size of the pin’s head, the length of the pin’s point
c
Student’s experiment and conclusion
Probability drop page 546
1
Tray 1: LLL
Tray 2: LLR LRL RLL
Tray 3: LRR RLR RRL
Tray 4: RRR
2
Tray 1: LLLL
Tray 2: LLLR LLRL LRLL RLLL
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Tray 3: LLRR LRLR LRRL RLLR RLRL RRLL
Tray 4: RRRL RRLR RLRR LRRR
Tray 5: RRRR
3
1
because there are 16 possible routes and only one lands in Tray 1.
16
4
Tray 2:
5
Student’s investigation
6
210 105
=
1024 512
7
Each number in each row of Pascal’s triangle corresponds to the number of routes to landing in
each tray of the game.
4 1
=
16 4
Tray 3:
6 3
4 1
1
=
Tray 4: =
Tray 5:
16 8
16 4
16
Dice sum page 547
1
Dice 2
Dice 1
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
2
36
3
7
4
3 1
=
36 12
5
6 1
=
36 6
6
You are four times more likely to get a total score of 5 than a total score of 2.
7
Dice 2
Dice 1
1
2
3
4
1
2
3
4
5
2
3
4
5
6
3
4
5
6
7
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4
8
16
9
5
10
4 1
=
16 4
5
6
7
9
8
11 Student’s investigation
12 a
m2
b
m+1
c
m 1
=
m2 m
13 a
b
c
m×n
The most likely totals (t) are in the range 𝑛𝑛 + 1 ≤ 𝑡𝑡 ≤ 𝑚𝑚 + 1.
n
1
=
nm m
Infection screening page 548
1
2
a
b
350
c
10.9%
a
b
490; this is 72.1%
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3
Large screenings for a disease with a low incidence in the population run the risk of producing
a high percentage of false positives. In Scenario 2, a person receiving a positive test result is
nearly three times as likely to not be infected as they are to be infected.
4
99.8%
ICT Activity: Buffon’s needle experiment page 549
1–9
Student’s experiment and results entered in a spreadsheet
10 The value of
2
should tend to π.
p
Student assessment 1 page 551
1
2
3
a
1
7
b
12
365
c
1
365
d
1
1461
a
1
6
b
1
2
c
2
3
d
0
a
i
3
5
ii
1
19
b
4
Infinite
a
1
2
3
4
5
6
H
1H
2H
3H
4H
5H
6H
T
1T
2T
3T
4T
5T
6T
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254
9
Cambridge IGCSE™ International Mathematics answers
b
5
ii
1
4
iii
1
4
i
1
16
ii
1
8
iii
1
4
i
1
8
ii
3
8
iii
1
8
iv
7
8
9
a
b
7
1
12
a
b
6
i
a
12 4
=
27 9
b
8
27
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8
9
c
6 2
=
27 9
a
9
16
b
1
16
c
1
4
9
0.009 95 (3 s.f.)
Student assessment 2 page 552
1
a
b
2
i
16 4
=
52 13
ii
36 9
=
52 13
iii
16 4
=
52 13
iv
1
52
v
0
a
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b
i
66 11
=
120 20
ii
60 1
=
120 2
3
53%
4
0.88
5
0.4
6
a
9
It is likely to be biased as the frequencies are not approximately equal.
b
Number
1
2
3
4
5
6
Frequency
11
12
6
18
16
17
Relative
frequency
11
80
12
80
6
80
18
80
16
80
17
80
c
Approximately 17 times
d
Approximately 226 times
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10
10 Statistics
All answers were written by the authors.
Exercise 10.1 page 556
1
Discrete
2
Continuous
3
Discrete
4
Continuous
5
Discrete
6
Discrete
7
Continuous
8
Discrete
9
Continuous
10 Continuous
Exercise 10.2 page 558
1
Student’s own results
Exercise 10.3 page 561
1
a
Class 11X
Score
Frequency
31–40
1
41–50
5
51–60
2
61–70
5
71–80
8
81–90
5
91–100
5
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Class 11Y
b
Score
Frequency
31–40
3
41–50
8
51–60
6
61–70
3
71–80
2
81–90
4
91–100
5
Assuming group sizes of 10, the following observations can be made:
- Both classes have students in each group from 30 to 99.
- 11X tends to have more students in the higher percentage groups whilst 11Y tends to have
more students in the lower percentage groups.
2
Number of
apples
Frequency
1–20
9
21–40
6
41–60
7
61–80
11
81–100
7
101–120
4
121–140
4
141–160
2
Exercise 10.4 page 565
1
a
Category
Amount spent ($)
Angle
Clothes
110
146.7
Entertainment
65
86.7
Food
40
53.3
Transport
20
26.7
Gifts
20
26.7
Other
15
20.0
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b
2
a
Category
Amount spent ($)
Percentage
Clothes
10
4.1%
Entertainment
150
61.2%
Food
10
4.1%
Transport
20
8.2%
Gifts
5
2.0%
Other
50
20.4%
b
3
a
b
14
3
c
The boys tend to be taller than the girls.
d
Student’s own bar chart
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a
Fishing boat 2 as it caught 651 kg of fish type A, compared with boat 1’s 304 kg
b
Fishing boat 1, percentage of fish type A =
Fishing boat 2, percentage of fish type A =
10
304
= 38%
800
651
= 35%
1860
Therefore boat 1 had a higher percentage of fish type A in its catch.
c
Exercise 10.5 page 567
1
2
3
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Exercise 10.6 page 569
1
Student’s own response
2
Student's own comments
3
Student's own comments
Exercise 10.7 page 573
1
a
Mean = 1.67, Median = 1
Mode = 1, Lower quartile = 1
Upper quartile = 2, Range = 5
b
Mean = 6.2, Median = 6.5
Mode = 7, Lower quartile = 5
Upper quartile = 7.5, Range = 9
c
Mean = 26.4, Median = 27
Mode = 28, Lower quartile = 25
Upper quartile = 28, Range = 5
d
Mean = 13.95, Median = 13.9
Mode = 13.8, Lower quartile = 13.8
Upper quartile = 14.1, Range = 0.6
2
91.1 kg
3
103 points
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Exercise 10.8 page 576
1
Mean = 3.35, Median = 3, Mode = 1 and 4
2
Mean = 7.03, Median = 7, Mode = 7
3
a
Mean = 6.33, Median = 7, Mode = 8
b
The mode as it gives the highest number of flowers per bush.
Exercise 10.9 page 578
1
2
3
a
29.1
b
30−39
a
60.9
b
60−69
a
Mean ≈ 6
b
Modal group = 0 ⩽ t < 5
c
1
minutes late
3
The stationmaster can make the claim if the average used refers to the modal result.
Exercise 10.10 page 582
1
a
Finishing time (h)
Frequency
Cumulative freq.
0 < h ⩽ 0.5
0
0
0.5 < h ⩽ 1.0
0
0
1.0 < h ⩽ 1.5
6
6
1.5 < h ⩽ 2.0
34
40
2.0 < h ⩽ 2.5
16
56
2.5 < h ⩽ 3.0
3
59
3.0 < h ⩽ 3.5
1
60
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b
2
c
Median ≈ 1.8 hours
d
As many runners finished before as after the median.
a
Class A
Class B
Class C
Score
Freq.
Cum.
freq.
Freq.
Cum.
freq.
Freq.
Cum.
freq.
0 < x ⩽ 20
1
1
0
0
1
1
20 < x ⩽ 40
5
6
0
0
2
3
40 < x ⩽ 60
6
12
4
4
2
5
60 < x ⩽ 80
3
15
4
8
4
9
80 < x ⩽ 100
3
18
4
12
8
17
b
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c
Class A median ≈ 50
Class B median ≈ 70
Class C median ≈ 78
d
3
As many students were above as below the median.
a
2008
2009
2010
Height (cm)
Freq.
Cum.
freq.
Freq.
Cum.
freq.
Freq.
Cum.
freq.
150 < h ⩽ 155
6
6
2
2
2
2
8
14
9
11
6
8
160 < h ⩽ 165
11
25
10
21
9
17
165 < h ⩽ 170
4
29
4
25
8
25
170 < h ⩽ 175
1
30
3
28
2
27
175 < h ⩽ 180
0
30
2
30
2
29
180 < h ⩽ 185
0
30
0
30
1
30
155 < h ⩽ 160
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b
c
Median (2007) ≈ 161 cm
Median (2008) ≈ 162 cm
Median (2009) ≈ 164 cm
d
There is an increase in the median each year so we could conclude that students are getting
taller.
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Exercise 10.11 page 584
1
a
Class A ≈ 30
Class B ≈ 30
Class C ≈ 40
2
b
Student’s own response
a
2008 ≈ 7 cm
2009 ≈ 8 cm
2010 ≈ 8 cm
b
3
Student’s own response
a
Distance thrown (m)
Frequency
Cumulative frequency
0 ≤ d < 20
4
4
9
13
40 ≤ d < 60
15
28
10
38
80 ≤ d < 100
2
40
20 ≤ d < 40
60 ≤ d < 80
b
c
Qualifying distance ≈ 66 m
d
Interquartile range ≈ 28 m
e
Median ≈ 50 m
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10
a
Type A
Mass (g)
Frequency
Cumulative
frequency
75 ≤ m < 100
4
4
7
11
15
26
32
58
14
72
6
78
2
80
100 ≤ m < 125
125 ≤ m < 150
150 ≤ m < 175
175 ≤ m < 200
200 ≤ m < 225
225 ≤ m < 250
Type B
Mass (g)
Frequency
Cumulative
frequency
75 ≤ m < 100
0
0
16
16
43
59
10
69
7
76
4
80
0
80
100 ≤ m < 125
125 ≤ m < 150
150 ≤ m < 175
175 ≤ m < 200
200 ≤ m < 225
b
c
225 ≤ m < 250
Median type A ≈ 157 g
Median type B ≈ 137 g
d
i
Lower quartile type A ≈ 140 g
Lower quartile type B ≈ 127 g
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ii
10
Upper quartile type A ≈ 178 g
Upper quartile type B ≈ 150 g
iii
Interquartile type range type A ≈ 38 g
Interquartile type range type B ≈ 23 g
5
e
Student’s own report
a
Student should compare the median values to decide whether this claim is true.
b
What is the definition of reliable? Lower interquartile range? Smaller range? The accuracy
of the claim will depend on method used.
Exercise 10.12 page 592
1
2
Students’ 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
Strong negative correlation
e
No correlation
f
Depends on age range investigated. 0−16 years likely to be a positive correlation. Ages 16+
little correlation.
g
Strong positive correlation
a
b
Graph shows a very weak negative correlation. Student’s answer as to whether this is what
he/she expected.
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10
a
b
Student’s own response
c
Student’s own response
d
4
a and d
b
(Strong) Positive correlation
c
A student who lives nearer the school may walk and take longer to get to school compared
with another student who lives slightly further away but who cycles to school.
e
≈ 9 km or 10 km
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10
a
b
Negative correlation
c
If the department store tracks weather forecasts, it can predict when colder spells are due
and have more gloves ready for sale.
Heights and percentiles page 594
1
≈ 167 cm
2
≈ 168 cm (Note: This corresponds to the 25th percentile not 75th.)
3
≈ 151 cm
4
Student’s calculations could include mean, median, interquartile 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.
Reading ages page 596
1
Possible answers include length of sentences, number of words with three 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.
ICT Activity page 596
1–6
Student’s data, graph and comparisons
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Student assessment 1 page 597
1
a
Mean = 75.4, Median = 72, Mode = 72
b
Mean = 12, Median = 9, Mode = 6
c
Mean = 11.5, Median = 10, Mode = 5
2
61 kg
3
a
2.4
b
2
c
3
4
5
6
a
Group
Mid-interval
value
Frequency
10–19
14.5
3
20–29
24.5
7
30–39
34.5
9
40–49
44.5
2
50–59
54.5
4
60–69
64.5
3
70–79
74.5
2
b
39.2
a
Discrete
b
Continuous
c
Continuous
d
Continuous
e
Continuous
a
Time (min)
10 < t ≤ 15
15 < t ≤ 20
20 < t ≤ 25
25 < t ≤ 30
30 < t ≤ 35
35 < t ≤ 40
40 < t ≤ 45
Motorway
frequency
3
5
7
2
1
1
1
Motorway
cum. freq.
3
8
15
17
18
19
20
Country roads
frequency
0
0
9
10
1
0
0
Country roads
cum. freq.
0
0
9
19
20
20
20
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b
c
i
Motorway median ≈ 21 min
Country roads median ≈ 26 min
ii
Motorway lower quartile ≈ 17 min
Motorway upper quartile ≈ 25 min
Country roads lower quartile ≈ 23 min
Country roads upper quartile ≈ 28 min
iii Motorway IQR ≈ 8 min
Country roads IQR ≈ 5 min
d
Student’s explanations
e
Student’s explanations
7
8
Key: Before coaching
After coaching
a y = 0.07x + 3.8
b
1|5 = 5.1
5|2 = 5.2
There doesn’t appear to be a correlation between a footballer’s salary and his popularity.
The report is misleading as there is no evidence to support that salary affects popularity
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Student assessment 2 page 598
1
a
Mean = 5.4, Median = 5, Mode = 5
b
Mean = 13.8, Median = 15, Mode =18
c
Mean = 6.1, Median = 6, Mode = 3
2
78.4 kg
3
a
2.8
b
3
c
3
4
5
6
a
Group
Mid-interval
value
Frequency
0–19
9.5
0
20–39
29.5
4
40–59
49.5
9
60–79
69.5
3
80–99
89.5
6
100–119
109.5
4
120–139
129.5
4
Mark (%)
Frequency
Cumulative
frequency
31–40
21
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
b
75.5
a
Discrete
b
Discrete
c
Continuous
d
Discrete
e
Continuous
a
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b
c
i
Median ≈ 60%
ii
Lower quartile ≈ 52%
Upper quartile ≈ 73%
iii
7
IQR ≈ 21%
a
Mark (%)
Frequency
Cumulative
frequency
1–10
10
10
11–20
30
40
21–30
40
80
31–40
50
130
41–50
70
200
51–60
100
300
61–70
240
540
71–80
160
700
81–90
70
770
91–100
30
800
b
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c
Grade 9 ≥ 74%
d
Fail < 50%
e
200 students failed.
f
200 students achieved a grade 9.
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