for Cambridge O Level Answer Book

87 Pages·2016·1.38 MB·English

Mathematics
for Cambridge O Level
Answer Book
1
&2
Gareth Rowlands
Published by
PEAK PUBLISHING LIMITED
London
UK
© Peak Publishing Limited 2016
Text © Gareth Rowlands 2016
First published 2016
ISBN: 1 845222 15 6
All rights reserved.
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Printed in Pakistan
Contents
Math Bk-1 Selected Answers: Math Bk-2 Selected Answers:
1 Number ................................................ 4 1 Series ................................................. 46
2 Fractions ............................................... 8 2 Graphs in Practical Situations ............. 48
3 Decimal fractions and percentages ..... 13 3 Graphs of functions ............................ 51
4 Squares, Cubes and indices ................ 17 4 Functions ........................................... 53
5 Ratio and Proportion .......................... 19 5 Symmetry ........................................... 56
6 Sets ..................................................... 23 6 The Circle ........................................... 59
7 Mensuration: Additions ..................... 26 7 Trigonometry ...................................... 62
8 Commercial Mathematics .................. 30 8 Statistics ............................................. 67
9 Algebra 1 ............................................ 32 9 Matrices ............................................. 71
10 Algebra 2 ............................................ 35 10 Probability .......................................... 74
11 Lines and Angles ................................. 38 11 Vectors ............................................... 79
12 Polygons ............................................. 41 12 Geometric Transformation ................ 83
Math Bk-1 Answers
O Level Math Bk-1 Selected Answers
Unit 1: Number
Exercise 1.1 a
1 Set D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 2 Set E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
3 Set F = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} 4 Set G = {–5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5}
5 Set G = {+1, +2, +3, +4, +5} 6 9 and –2
7 all of them 8 a no b yes c yes d yes
9 √�2� and π
Exercise 1.1 b
1 a 9 > 8 9 is greater than 8.
b 27 > 16 27 is greater than 16.
c 0.5 < 1 0.5 is less than 1.
d –3 < 5 –3 is less than 5.
e 0.503 > 0.062 0.503 is greater than 0.062.
f 0.724 > –0.87 0.724 is greater than –0.87.
g 1 < 6.5 1 is less than 6.5.
h 9 = 0.9 3 = 0.75 9 > 3 9 is greater than 3.
10 4 10 4 10 4
i 0 < 0.1 0 is less than 0.1.
2 a k ⩽ 7 means that the values of k can be 7 or less than 7.
b x ⩾ 0.8 means that the values of x can be 0.8 or more than 0.8.
c 0 < d < 4 means that the values of d are greater than 0 but less than 4.
Exercise 1.1 c
1 a 9 < 32 b 386 > 96 c 2,546 < 2,654 d 54,876 < 55,786
e 0.985 > 0.895 f 0.75 < 0.76 g 3 < 4.3 h 0.6 > 2
5
i 2 > 3 j 0.02 > 0 k 0.004 > 0.003
3 5
2 a 1087 > 654 > 187 > 98 > 12 > 1
b –3.1 < –2.5 < –1.1 < 0 < 5.5 < 6.4 < 7.6
c 7 < 0.4 < 1 < 7 < 1.5
20 2 10
3 a y is less than or equal to 0.6.
b t is greater than or equal to 2.5.
c x is less than 10 but greater than 0.
Exercise 1.2
1 a
−6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6
4
Math Bk-1 Answers
b i
−6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6
ii
−6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6
iii
−6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6
iv
−6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6
v
−6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6
2 a +5 b +4 c +7 d −8 e −12
f −7 cm g −3.5 m h +80 kg i +5 years j +30 days
k +9x² l −7y³ m −13pq² n +½q²r² o +0.25r²s³
Exercise 1.3 a
1 +10 2 +16 3 +25 4 +5 5 +7
6 +40 7 +7 8 +7 9 −1 10 −14
11 +12 12 +8 13 +9 14 +11 15 +10
16 +25 17 −10 18 −10 19 +2 20 −2
21 +7 22 +6 23 0 24 +12 25 +14
26 +13 27 +21 28 +6 29 +1 30 +31
Exercise 1.3 b
1 +8 2 +3 3 +15 4 +16 5 +12
6 +11 7 +20 8 +31 9 +44 10 +71
11 −9 12 −17 13 +1 14 +21 15 +39
16 +3 17 −6 18 0 19 +8 20 −1
Exercise 1.3 c
1 −9°C, −4°C, −1°C, 0°C, 5°C, 6°C 2 −4°C 3 Rs 450 4 16 m
5 4 points
Exercise 1.3 d
1 +12 2 +12 3 +15 4 +49 5 −36
6 −56 7 −45 8 −48 9 −108 10 −55
11 −36 12 +48 13 +35 14 +36 15 −64
16 +30 17 +30 18 +18 19 −36 20 +60
Exercise 1.3 e
1 +3 2 +4 3 +2 4 −5 5 +3
6 −2 7 −2 8 −3 9 +6 10 +2
11 +7 12 +4 13 −4 14 +7 15 −5
16 +3
5
Math Bk-1 Answers
Exercise 1.4
1 +4 2 −24 3 +2 4 +1 5 +9
6 +72 7 +34 8 +31 9 −27 10 +13
11 +41 12 −18 13 +50 14 +141 15 +2
Exercise 1.5
1 a 1, 2, 4, 8 b 1, 2, 5, 10 c 1, 13
d 1, 3, 5, 15 e 1, 2, 3, 4, 6, 8, 12, 24 f 1, 2, 4, 8, 16, 32
g 1, 2, 4, 6, 9, 12, 18, 36 h 1, 43 i 1, 2, 4, 8, 16, 32, 64
j 1, 3, 5, 9, 15, 45
2 a 8 .... 1, 2, 4, 8 b 9 .... 1, 3, 9 c 12 .... 1, 2, 3, 4, 6,12
10 .... 1, 2, 5, 10 15 .... 1, 3, 5, 15 18 .... 1, 2, 3, 6, 9, 18
HCF = 2 HCF = 3 HCF = 6
d 14 .... 1, 2, 7, 14 e 16 .... 1, 2, 4, 8, 16 f 15 .... 1, 3, 5, 15
35 .... 1, 5, 7, 35 24 .... 1, 2, 3, 4, 6, 8, 12, 24 45 .... 1, 3, 5, 9, 15, 45
HCF = 7 HCF = 8 HCF = 15
g 26 .... 1, 2, 13, 26 h 36 .... 1, 2, 4, 9, 12, 18, 36
39 .... 1, 3, 13, 39 48 .... 1, 2, 3, 4, 6, 8, 12, 24, 48
HCF = 13 HCF = 12
3 a 4, 8, 12 ... b 6, 12, 18 ... c 7, 14, 21 ... d 3, 6, 9 ...
e 8, 16, 24 ... f 12, 24, 36 ... g 10, 20, 30 ... h 25, 50, 75 ...
i 15, 30, 45 ... j 9, 18, 27 ...
4 a 21 b 18 c 28 d 24 e 30 f 60
g 72 h 39
Exercise 1.6
1 a 18 b 12 c 27
2 a k = 22 b 18
3 a 22 × 33 b 23 × 33 × 5 c 75
4 a 1, 2, 3, 6, 9, 18 b 23 × 72
Exercise 1.7
1 a 9 b 6 c −16 d −24 e 30 kg
f 30 years g −23pd2 h x3 i −2y3x2 j −10xy
2 a 5 b −16 c −18 d 49
3 a 3 b −5 c −2 d −10
4 a 21 b −18 c 54 d −15
5 a 3 b −9 c −100 d −8
6 a 6 b 47 c −40 d −36
7 1, 2, 4, 5, 10, 20, 25, 50, 100
8 9, 18, 27, 36, 45 …
6
Math Bk-1 Answers
Revision Exercise
1 a Set C = {-4, -3, -2, -1, 0, +1, +2, +3} b Set D = {-7, -6, -5, -4, -3, -2, -1, 0, +1, +2}
c Set E = {+2, +3, +4, +5, +6, +7, +8} d Set F = {-2, -1, 0, +1, +2}
e Set G = {-3, -2, -1, 0, +1, +2, +3}
2 a Set C = {4, 5, 6, 7, 8} b Set D = {8, 9, 10, 11, 12}
c Set E = {1, 2, 3, 4, 5} d Set F = {5, 6, 7}
e Set G = {4, 5, 6, 7, 8}
3 a Set C = {4, 5, 6, 7, 8} b Set D = {0, 1, 2, 3, 4, 5, 6}
c Set E = {4, 5, 6, 7, 8} d Set F = {1, 2, 3, 4}
e Set G = {0, 1, 2, 3, 4, 5}
4 a Set C = {21, 22, 23, 24, 25, 26, 27, 28} b Set D = {0, 1, 2, 3, 4}
c Set E = {-3, -2, -1, 0, +1, +2} d Set F = {11, 12, 13, 14, 15, 16, 17, 18}
5 a +7 b −3 c −17 d +9 e +11
f +9 kg g −4.8 m h −50 cm i +6½ years j +7 days
k +5x3 l −9y2 m +1 q²r² n −4p²q
4
6 a +6 b +16 c +5 d +13 e −1
f +6 g 0 h 10 i −8
7 a +6 b +15 c +8 d +23 e +13 f −11
g −8 h −1 i +42
8 a +15 b −24 c −32 d +30 e −27 f −45
g +60 h +56 i −90
9 a +2 b −4 c +4 d −7 e +6 f −9
g +8 h −3 i −5
10 a +6 b −24 c +5 d +7 e +2 f +73
g +5 h +24 i −48 j +3 k +31 l −30
m +58 n +451 o −7
11 a 1, 2, 3, 6 b 1, 2 c 1, 3, 9
d 1, 7 e 1, 2, 4, 7, 14, 28 f 1, 2, 4, 7, 8, 14, 28, 56
g 1, 2, 7, 14 h 1, 37
12 a 4 b 3 c 1 d 2 e 4 f 1
g 1 h 3
13 a {5, 10, 15, 20 ...} b {8, 16, 24, 32 ...} c {11, 22, 33, 44 ...}
d {13, 26, 39, 52 ...} e {15, 30, 45, 60 ...} f {16, 32, 48, 64 ...}
g {9, 18, 27, 36 ...} h {18, 36, 54, 72 ...}
14 a 10 b 21 c 66 d 54 e 60 f 21
g 130 h 72
Past Paper Questions
1 Background knowledge and thinking process:
a To express as a product of prime numbers think through the different ways prime numbers may be
multiplied to give 154.
b To find the lowest common multiple list the prime factors of 154 and 49.
7
Math Bk-1 Answers
Solution:
a 154 = 2 × 7 × 11
b 154 = 2 × 7 × 11 49 = 7 × 7
The lowest common multiple (LCM) = 2 × 7 × 7 × 11
= 1 078
2 Background knowledge and thinking process:
a Always work with multiplication and division first.
b Work out the bracket first and observe the BODMAS rules.
Solution:
a 24 ÷ 6 + 2 × 9 b 3 + 2 (4 − 5)
= (24 ÷ 6) + (2 × 9) = 3 + 2 (−1)
= 4 + 18 = 22
= 3 − 2
= 1
3 Background knowledge and thinking process:
a Perform prime factorization of 99.
b Find the LCM of 24 and 99.
Solution:
a 99 = 3 × 3 11 b n = 8
= 32 × 11
4 Background knowledge and thinking process:
Insert brackets.
Solution:
4 + 6 × 7 – 5 = 16
= 4 + [6 × (7 – 5 )] = 16
Unit 2: Fractions
Exercise 2.1 a
1 a 4 b 7 c 5 d 3
6 10 6 4
e 4 f 8
7 12
2 a 1 = 9 b 1 = 8 c 3 = 12 d 2 = 30
2 18 4 32 5 20 3 45
e 30 = 5 f 35 = 5 g 60 = 5 h 18 = 2
48 8 49 7 108 9 99 11
3 a 3 = 9 b 4 = 16 c 5 = 35 d 3 = 36
4 12 5 20 6 42 7 84
e 9 = 63 f 7 = 42 g 5 = 40 h 8 = 64
10 70 12 72 9 72 11 88
8
Math Bk-1 Answers
4 a 1 b 2 c 4 d 4
2 3 5 9
e 7 f 3 g 3 h 2
9 8 8 3
Exercise 2.1 b
17 23 39
1 a = 17 ÷ 8 b = 23 ÷ 7 c = 39 ÷ 16
8 7 16
= 2, remainder 1 = 3, remainder 2 = 2, remainder 7
1 2 7
= 2 = 3 = 2
8 7 16
1 1 2 5 1 6
2 a 1 b 1 c 2 d 1 e 4 f 4
2 4 3 7 6 7
3 4 7 2
g 4 h 6 i 5 j 12
11 9 12 7
2 (3 × 4) + 2 3 (5 × 8) + 3 4 (9 × 7) + 4
3 a 4 = b 5 = c 7 =
3 3 8 8 9 9
12 + 2 40 + 3 63 + 4
= = =
3 8 9
= 14 = 43 = 67
3 8 9
9 15 29 43 56 68
4 a b c d e f
4 4 6 8 9 9
43 69 101 48 73 113
g h i j k l
12 7 12 5 11 15
Exercise 2.2 a
1 3 1 5 3 2 1
1 a + b + 4 c 3 + 2 + 1
3 5 2 8 5 3 4
= (5 × 1) 1+5 (3 × 3) = 4 (4 × 1) + (1 × 5) = 6(12 × 3) + (20 × 2) + (15 × 1)
8 60
5 + 9
= 4 + 5 36 + 40 + 15
15 = 4 = 6
8 60
10 2
= =
15 3 = 49 = 4 + 1 1 = 691 = 6 + 1 31
8 8 60 60
1 31
= 5 = 7
8 60
7 7 13 1 7 1
2 a b c d 1 e 1 f 3
10 8 15 21 30 4
11 1 1 11 19 11
g 3 h 6 i 8 j 7 k 7 l 9
14 8 9 30 20 18
5 5 7
m 10 n 11 o 11
24 6 8
9
Math Bk-1 Answers
Exercise 2.2 b
7 2 7 1 3 5
1 a − b 5 − 2 c 4 − 1
8 3 9 2 5 6
(3 × 7) + (8 × 2) (2 × 7) − (9 × 1) (6 × 3) − (5 × 5)
= = 3 = 3
24 18 30
21 − 16 14 − 9 18 − 25
= = 3 = 3
24 18 30
= 5 = 3 5 = 2 + 30 + 18 − 25
24
18 30 30

30 + 18 − 25
= 2
30
23
= 2
30
1 3 1 8 13 1
2 a b c d e f 2
10 8 3 21 30 4
1 2 1 11 1 1
g 3 h 4 i 7 j 3 k 3 l 3
8 9 6 35 9 2
9 13 7 5 5
m 2 n 2 o 3 p 2 q 4
10 18 40 8 14
Exercise 2.2 c
3 1 3 2 3 1 7 2
1 a × b × c 2 × 3 d 4 × 1
5 4 8 5 7 2 8 3
= 230 = 63 = 17 × 71 = 3913 × 5
40 7 2 8 3
20 1 1
= 3 = 17 = 8 1 = 65 = 8 1
20 2 2 8 8
6 1 1 5 1 2
2 a b c d e f 3
35 2 12 12 3 3
3 3 1
g 3 h 8 i 9 j 7 k 12 l 9
5 5 3
1 13 14
m n 6 o p 14 q 11
6 14 17
Exercise 2.2 d
5 2 1 3 5 3
1 a ÷ b 1 ÷ c 2 ÷ 1
6 3 2 7 8 4
= 5 × 31 = 31 × 7 = 213 × 41
6 2 2 3 8 7
2 1 2 1
= 5 = 1 1 = 7 = 3 1 = 3 = 1 1
4 4 2 2 2 2
1 1 1 5 7
2 a 3 b 1 c 1 d e f 3
3 9 2 7 8
1 1 2 2
g 6 h 11 i 2 j 1 k 2 l 1
2 2 3 3
17 7 35 11 19
m 2 n 2 o 1 p 1 q 1
26 18 41 45 60
10