Cambridge
International
Examinations
Cambridge IGCSE
Mathematics
Study Guide
Answer key and markscheme
CONTENTS
Answer Key
1 Numbers
Chapter 2 Algebra
Chapter 3 Geometry
Chapter 4 Mensuration
Chapter 5 Coordinate Geometry
Chapter 6 Trigonometry and Pythagorasʼ Theorem
Chapter 7 Vectors and Matrices
Chapter 8 Probability
Chapter 9 Statistics
Chapter
1
9
21
27
28
30
33
38
41
Practice test markscheme
Practice Test (Paper 2)
47
Practice Test (Paper 4)
51
Answer Key
Chapter 1 Numbers
10 (a) H.C.F = 6
Exercise 1.1
1 (a) 7, −2, 111, 25, 0
3
1
(b) 7, −2, , 111, 2.4, 25, 0, 5
4
2
L.C.M = 1560
(b) H.C.F = 45
L.C.M = 1350
(c) H.C.F = 15
L.C.M = 4725
(d) H.C.F = 12
L.C.M = 11880
(e) H.C.F = 36
L.C.M = 2592
(f) H.C.F = 90
L.C.M = 1080
(c) π , 3
(d) 25, 0
Exercise 1.2A
2 (a) 71, 81
(b)
−
{
}
1 A = 11, 13, 17, 19
4, −3
8
(c) 6.2, − 4, , −3, 71, 81, 3.1
9
2 (a) P
P=
=
Q
=
(b) Q =
(d) 81
3 (a) 3, 7, 13
{
(c) 23, 29, 31
(d) 5, 37, 43
(e) 41, 43, 47
(f) 53, 61, 79
5
(b)
}
3 B = 1, 3, 9, 13
(b) 2, 17
4 (a) 1, 3, 25
{{4,4, 8,8, 12,
12, 16
16}}
{{3,3, 6,6, 9,9, 12,
12, 15,
15, 18
18}}
4 (a) All elements in A are in B
(b) Yes, it is because 26 is not in B
5 (a)
4
 , 25
(c) −2, 2.1, 1, 3, 5, 4.62,
3
{3, 7, 9}
6 (a) 4
(d) 3, 25
(c) 0
5 7, 14, 56, 63, 70
6 (a) 1, 2, 3, 6
(c) 1, 5, 25
(b) 4
(d)
{2} , {2, 4 } , {4 }
2
2
{12}
(b) 1, 2, 3, 4, 6, 12
(b)
(d) 1, 13
(c) 11
8 (a)
(f) 1, 2, 4, 8, 16, 32, 64
(b) 8
8 (a) 24
(b) 60
9 (a) 32 × 5
(b) 32 × 23
(c) True, since 6.5 is not a member of set A.
{−7, −6, −5, −4, −2, −1, 2, 3, 4, 6}
{−3, 5}
{−8, −5, −3, 0, 1, 2, 5, 7, 8}
9 (a) A′ =
3
(d) 2 × 3 × 5
3
(e) 2 × 7 × 9
(f) 5 × 3
(g) 2 × 3 × 5 × 7
(h) 32 × 5 × 7
© 2017 NTK Publishing Limited
{2, 3, 5, 7}
(b) 4
7 (a) 4
(c) 7 × 3 × 2
{2, 3, 5, 7, 8, 9, 11}
7 (a) 9 is a multiple of 3
(e) 1, 2, 3, 5, 6, 10, 15, 30
2
(b)
(b)
(c)
1
Cambridge IGCSE Mathematics Study Guide
10 (a)
(b)
(c)
{−8, −7, −6, −5, −2, −1, 0}
{−4, 2}
{−8, −7, −6, −5, −3, −2, −1, 0, 1, 3}
(b)
7
A∩ B ∩C = φ
( A ∪ B ) ∩ C = {10}
ℰ
A
15
B
Exercise 1.2B
1 (a)
C
33 9
4.2
{5, 10, 15}
2
10
7
5
(b) φ
(c)
2 (a)
{5, 7, 8, 9, 10, 15}
8 (a)
{5, 8, 10, 13, 15, 17, 30, 33}
(c)
{30, 33}
(b)
(b)
(b)
(c)
5 (a)
B
A
B
A
B
ℰ
{5, 9}
{22, 24}
(c) φ
4 (a)
A
{1, 2, 13, 15, 17}
(b)
3 (a)
ℰ
(c)
ℰ
{4, 5, 6}
{4, 5}
{3, 8, 20, 22}
(d)
ℰ
A
ℰ
A
B
B
a
b
C
e
c
f
d
(b)
6 (a)
{ c}
ℰ
A
9
3
B
C
2
4
8
10
(b) 8
10 (a) 0
(b) 2
(c) 5
(d) 7
(e) 18
(f) 1
5
1
2
9 (a) 7
7
11 (a) 1
(b) 0
(c) 13
© 2017 NTK Publishing Limited
Answer Key
12
ℰ
A
B
1
2
ℰ
7 (a)
Sandwich
15
3
7−x
5
20
x
10
8−x
Chocolate
Exercise 1.2C
8 (a) 15
(b) 10
ℰ
C
50
T
50
Exercise 1.3
100
1 (a) 121
(b) 50
3 (a) 28
(b) 5
(d)
3
10
27
10
(f)
1
6
(b)
−64
(e)
2 (a) 125
ℰ
M
5
5
(b) 6.25
(c) 49
(c) 5
4 (a)
Coffee
(b) 4
1 14
2 (a)
9
S
(c)
10
(e) 1.5
−1
(d) 0.2
(f)
2
5
5
(b) 15
(c) 5
(d) 5
5 (a)
ℰ
G
5
3 13
4
−4.913
5
−62.9
S
5
6
15
4 + 9 = 2+3= 5
4 + 9 = 13 ≠ 5
5
(b) 5
6 (a)
(c) 15
ℰ
B
7
Exercise 1.4
F
8
15
1 (a) 17°C
20
(b) 8
7 41 or −41
(c) 20
(b) 30°C
(c) Moscow
2 8084
(d) 7
3 (a)
© 2017 NTK Publishing Limited
−4°C
(b) 0.5°C
3
Cambridge IGCSE Mathematics Study Guide
Exercise 1.5
23
30
5
(c)
28
29
(e)
24
19
(g)
2
1 (a)
1
3
1
(c)
2
2 (a) 1
3 (a) 2.4
8
23
28
5
(d)
9
81
(f)
35
121
(h)
52
(b)
(b)
3
4
(b) 2400 mL
5 (a) Terminating decimals: 0.35, 0.6, 4.7
 2.87

Recurring decimals: 2.6,
(b) Terminating decimals: 1.3, 3.56, 2.43, 4.7
7 (a)
(c)
753
999
(d)
(e)
(f)
40%
1.23
123%
0.24
24%
0.2
22.2%
10.34
1034%
2.2
220%
Exercise 1.6
(b)
−1
(c) 0.23 < 0.32
(d)
−0.1
1
(e) 3 > 3
2
(f)
2 (a)
(b)
−1,
<0
> −1
9 < 16
0
−8, −7, −6, −5, −4, −3
27
50
76
(d)
25
(b)
45
(f)
99
1
(h) 2
18
3
2
3 (a) 0.2 < 0.2 < 0.2 < 3 0.2 <
1
0.2
(b) 0.2% < 0.2 < 0.2
1
1
<
< 0.011 < 1.2%
101 100
1
1
% < 0.025 < < 3 2
(d)
4
4
(c) 0.001 <
1
250
(b)
14
25
Exercise 1.7
23
10
(d)
131
5000
1 (a) 27
(c) 360
2 (a) 24 × 34 × 7
(c) 23 × 33 × 132
4
0.4
(c) 3, 2, 1
Recurring decimals: 3.8
(g) 2
(c)
Percentages
1 (a) 7 > 2
(b) 1.4, 7.85, 7.94, 9.3, 10.7
4
5
33
(c)
25
2
(e)
3
(b)
Decimals
(d) 3
4 (a) 0.34, 1.5, 2.4, 3.7, 6.45
6 (a)
(a)
Fractions
2
5
123
100
6
25
2
9
517
50
11
5
(b) 32
(d) 51597
(b) 52 × 112
(d) 23 × 7 × 11 × 13
© 2017 NTK Publishing Limited
Answer Key
25
4
1
(c)
3
3 (a)
(b)
1
9
3
2
3
2 = 12
(c)
5 5
8
3 (a)
(d) 4
(e)
1
32
(f)
−8
(g)
1
16
(h)
5
8
−9
Exercise 1.9
10
4 (a) 4.98 × 10
(b) 9.999 × 10
5 (a) 124 000 000
(b) 0.000 000 62
6 (a) 1.512
(b) 1.6 × 104
(d) 9.72 × 102
(e) 3.33 × 1010
(f) 2.30 × 106
(g) 2.32 × 10
(h) 3.06 × 10
7 7−23, by 40 353 607 times.
(b) 0.01
(c) 132.13
(d) 0.54
−1
(c) 6000
(d) 93 000
(b) 0.2
(c) 10
(d) 0.7
(c) 737 000 000
2
−
1
−4
−2
3
,
4.2
×
10
,
2
×
10
,
2
22 × 103
(b) 0
3 (a) 70 000
4 (a) 1030
8 500 s
9
1 (a) 232.12
2 (a) 3 720 000
(c) 4.2 × 108
10
5
6
−10
20
3
(d) − =
3
9
2
(b)
5 (a) 8009.21
(c) 8000
(b) 1 500 000
(d) 93.8
(b) 8010
(d) 8009.2
(e) 8009.206
Exercise 1.8
1 (a)
6 (a)
(b)
−6
(c) 0.6
(d)
−0.2
(e)
(f) 21
−1
−16
(g) 64
(h)
(i)
(j) 12
−8
(k) 4
2 (a)
−1
(l)
(b)
−43.47
16
2
−5
(c) 20
(d) 33
(e)
(f) 3
−6
(b) 14
7 14.02
8 (a) 4 × 2 = 4
6
3
(c)
4
3
8
(b)
( 2 + 2)
2
2
=8
=1
Exercise 1.10
1 upper bound: 9.8 +
9.8 −
© 2017 NTK Publishing Limited
7×8
4
0.1
= 9.75
2
0.1
= 9.85; lower bound
2
5
Cambridge IGCSE Mathematics Study Guide
1
2
2 (a) lower bound: 5
8
1
= 33.75
2
1 (a) 1:3
1
2
17
=
(b) upper bound:
9
1
5
2
8+
0.01
= 27.26
2
0.01
(b) lower bound: 4 6.81
= 27.22
2
(b) upper bound: 5 26 +
1
= 25.5 kg
2
1
= 132.5 kg
2
1
6 (a) upper bound: 65 +
2
(b) lower bound: 65
1
58
= 8 kg
2
1
2
58 +
1
= 6 kg
2
0.01
2
= 3.77
0.01
9.81+
2
3.56
(b) a:b:c = 5:4:8
(c) a:b:c = 40:15:50 = 8:3:10
(d) a:b:c = 9:12:8
Exercise 1.11
1 (a) P =
1 3
w
4
0.01
2
= 3.79
0.01
9.81
2
3.56 +
(b)
729
4
(c) 36 m/s
2 (a) d = 3 × 10−5 L
(b) 125 000 mm
3 35 km
4 6
5 78 kg
7 (a) t =
80
n
(b) 639.6
(b) 16
(c) decrease by 1 hour
8 (a) x =
(b) upper bound:
0.01
2 3.14 +
2
2 (a) a:b:c = 1:2:6
6 (a) 743.04
7 (a) lower bound:
0.01
2 3.14
2
(d) 1:2:8
(e) 15:48:20
10
2
22
= 262
3 number of small tanks:
69
0.1
3.5
2
5 (a) lowest possible: 26 −
(b) 3:5
(c) 8:11
900 +
4 (a) upper bound: 4 6.81+
Warm up 1.11
3.6 × 105
f
(b) f = 1440
(c) 240
9 40 minutes
10 1 day
8 upper bound: 6 7.4 +
lower bound: 6 7.4
6
0.1
2
2
0.1
2
2
= 333.015
11 7 days
= 324.135
12 4
© 2017 NTK Publishing Limited
Answer Key
13 (a) (i)
(b) (i)
C = 4500N
C = 4500
C=
(ii) t =
192
N
192
t
Decrease, it is because the salary budget is
inversely proportional to the time required to
finish the project.
864000
t
t =8
(ii) 108000 =
4 months
14 (a) 8:12
(c) 115:92
(b) 36:9
(d) 144:192
(e) 4104:912
15 (a) 8:12
(c) 115:92
(d) 56.55
(e) 62.328
(f) 8.75
(b) 10% decrease
(c) 1.18% increase
(d) 1.83% decrease
(e) 7.24% decrease
(f) 13% decrease
Exercise 1.12
1 15.9%
2 (a) 85%
(b) 3:17
3 60
4 750
5 4.62%
(b) 36:9
(d) 144:192
(e) 4104:912
16 (a) 280
(b) $2240
17 (a) 5.5 kg
(b) £22
(c) £12.8
18 (a) 3 km
(b) 173.4
(c) 595.92
4 (a) 10% increase
864000
t
(ii) Inverse proportion
(c) (i)
3 (a) 11.76
6 £96
7 11%
8 5.6%
9 £403.2
10 1% decrease
(b) 3.6 km/h
11 150,000
12 20
Warm up 1.12
1 (a) $2
13 £1416
(b) £13.176
(c) 4.2 kg
(d) 6.4 g
(e) 9
(f) 464.4
2 (a) 10.2
(b) 105.176
(c) 64.5
(d) 4185.44
(e) 12.84
(f) 773.85
© 2017 NTK Publishing Limited
14 (a) £2940
(b) £1230
15 1 783 000 m3
7
Cambridge IGCSE Mathematics Study Guide
4 (a) £1416
Exercise 1.13
1 (a)
−5.12
(c) 102400
(d) 0.26
(e) 0.80
(f)
2 (a) 0.75
3
(b) 3.50
(b) 1
(d)
(e) 0.125
(f) 1.20
−5.20
(cos 45°)3 , (cos 45°)2 , cos 45°, 3 cos 45°
4 (a) >
(b) >
(c) <
(d) <
2 (a) $9.8
(c) 3.49
6 £29166.8
7 6.37%
Exercise 1.17
1 (a) 20
(b) 30, 418.5
2 (a) 1000
(b)
0
1000
1
874.1
3 (a) 13
Exercise 1.15
(c) €765.31
5 P = £4754.8
−11.57
(c) 0.231
1 (a) €6.12
(b) £1429.2
2
764
3
667.8
(b) 3.25 grams
(c) 2 years
(b) €10.51
(d) €0.13
(b) $98
(d) $58 800
4 (a) B = 100 (2)
(b) 6000
5 (a) 3 114 918
(b) 13 788 382
t
(c) Population of species A keeps decreasing.
(d) 4:1
3 4.7% increase
Chapter 2 Algebra
4 £2415
5 (a) 17 800
(b) £600
(c) £588
6 (a) €40
(c) £3405.6
Exercise 1.16
1 6%
2 3200
3 £13 665.27
8
Exercise 2.1
1 (a) 2p + 2q + 2r
(b) £34.4
(c) ab
−2x
(d)
−9
aq
(b) r =
2 (a) y = c + mx
(c)
(b)
+ 10y
p2
a+b
1 2 1
= +
R x 2y
3 (a) d =
1
u+v t
2
(
4 (a) S = 28
)
(b) d = xt
(b) V = 810
© 2017 NTK Publishing Limited
Answer Key
1
4
(c) r =
T − 2x
3
5 (a) y =
8
3
(d) r =
(c) y =
2 x − 5 Rx
3R + 1
(e) s =
A + 2t
5
6 (a) 1. x =
q − 2p
p−q
(k) x3 + x2 − 3x + 9
(b) fu + fv = uv
fv
u=
v− f
1
12 x 2 − 7 x − 10
2
(
3
−3x
)
1 (a) x2 − 4y2
(c) 4x4 −
(b) 9 − 25x2y2
1
4
(d) x2 −
(e) 5x2 + 10xy + y2
2. x = −3
(g) 4x2 + 4x +
1
x2
(h)
1
2
1
−
+ 2
2
xy x
y
2. s = 2
R2 a2
(c) 1. b =
1 − R2
2. b = 9
(j) x6 − 2x4 + 6x3 + x2 − 6x + 9
(k) x6 − 4x4 + 6x2 +
(i)
3 − y2b
4 y2 − a
2. x = − 3
14
(e) 1. x =
5 y + 4 Py
2P − 8
2. x =
(l)
−17
4
x4 + 2x3 + 3x2 + 2x + 1
1
−4
x2
x2 + y2 + z2 + 2xy + 2xz + 2yz
2 (a) 5x2 + 20x − 146
3
1 (a) 6xy − 6y
(b) 3x − 2xy + 5
2 (a) 9p2 + 2q
(b) 3p2q + p2
3 (a) 0
(b) 2abc
4 (a)
−5s
2
t − 4st
5 (a) 3xyz
2
1 (a) 2by2
(c)
(b)
−5xyz
2
+ 7xy z
(c)
(c)
Exercise 2.2B
(e)
(b) 4b − ab
(b) 2b + 3ac
(c) 2a2 − ab + 4b
(e) x2 − x − 6
(f) x2 − 3x − 10
(g) 6x2 − 19x − 11
(h)
−5x
(j)
−y
(i)
20y2 + 3y − 2
© 2017 NTK Publishing Limited
2
− 18x + 8
− 3y + 18
(d) 6z4 + z3 − 16z2 − 7z + 4
(b) ax
( a − 2)
(d) ay
(
)
7 x ( 7 x + 3)
2 (a) ax x + 2 a
(b) 2s2 − 6s2t
(b) 4xy
Exercise 2.2D
3 (a)
1 (a) a2 + 2a
2
(c) 2y − 3y + 1
Exercise 2.2A
9
x2
(f) 49 − 14xy + x2y2
bk
(b) 1. s =
rb − a
(d) 1. x =
+ 5x2 + 4
Exercise 2.2C
2 − 2 y2
y2 + 1
(d) x =
2
(l)
(f)
4 (a)
(c)
(e)
(g)
(
(b) 4by 2 2 y + b
(
2
2
)
(d) x 2 ax + a x + 4
)
( p − 2q )( r − 1) (b) ( a − 5)( x + y )
b ( 3x − 1)( a + c )
(d) ( x − 2 )( x − 2 − a )
5( x − y ) ( x − y − 2 )
2 x ( 2b − c )
2
2
( x + 1)( y + 2)
( p + q )( m + n)
( a − 2)( b − a )
( y − 1)( x − y )
(b)
(d)
(f)
(f)
(5 + r )(t + s)
(7 − a )( y − z )
( y − 1)( y + 5)
( 2 x + 1)( x − 1)( x + 1)
9
Cambridge IGCSE Mathematics Study Guide
(i)
(y
2
)(
+ y −1 h − k
)
(j)
( x − 1) ( a − b)
2
Exercise 2.2G
( 2 x + 1)( x + 3) (b)
(3 y − 2)( y + 3) (d)
5( 2 a − 1)( a − 1)
(f)
2 ( 6s + 5)( s + 1)
(h)
−4 ( 2 m − 1)(5m + 2 ) (j)
− ( 2 p − 1)( 2 p − 3) (l)
( 2 x + 1)( x − 3)
(3 y + 1)(3 y − 4)
3( 4 a − 1)( a + 5)
3( 4s + 1)( 3s − 2 )
−3( 2 m − 1)( m − 1)
−2 ( 3 p + 7 )( p − 2 )
( x + 11y )( x − 12 y )
( 2 p + 5q )( p − 5q )
(d)
( 4m + n)( 2m − 3n)
(8s − 3t )(3s − 8t )
3 (a)
(12 x + 7 )( 2 x − 3)
(b)
(12 x − 29)( 2 x − 9)
4 (a)
1
x − y x − 4y
4
1 (a)
Exercise 2.2E
1 (a)
(c)
(e)
(f)
2 (a)
(c)
(e)
3 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(c)
( x + 3)
( 2 x − 5)
3( 2 a − b )
2
(b)
2
(d)
2
(f)
( x − 7)
− ( y − 11)
−5 ( 3a + 2b)
(c)
(e)
(g)
2 (a)
(c)
(e)
(g)
(i)
(k)
10
(g)
(i)
(k)
2 (a)
2
(c)
2
2
(3 p + 7q )(11 p + 7q )
( 4 x − yz )
2 ( p − 1)( 2 − 2 p − q )
( 2a + b − 6)( 2a + b + 6)
20s (5t − 3s )
( 2a − 7b − 6c )( 2a − 7b + 6c )
(3m + n − x + 1)(3m + n + x + 1)
(3x − 1)(3x + 1)(9 x + 1)
−4
(
2
2
Exercise 2.2F
1 (a)
(e)
(b) ( 4 y − 3)( 4 y + 3)
( 2 − x )( 2 + x )
5(1 − 2 z )(1 + 2 z ) (d) 7 x ( y − 5)( y + 5)
( 2a − b − 6)( 2a − b + 6)
33( p − 3)( p + 1)
)(
2
( p − 2)( p − 5)
2 ( m − 3)( m − 1)
6 ( n + 5)( n − 3)
−2 ( q − 7 )
7 ( x − 3)( x + 2 )
−3( z − 10 )( z + 9 )
−
2
(b)
(d)
(f)
(h)
(b)
(d)
(f)
(h)
(j)
(l)
( x + 6)( x − 1)
( x − 3)( x + 2)
( x − 5)( x + 2)
( x + 7 )( x − 4)
( a + 6)( a − 3)
4 ( y − 4 )( y − 7 )
4 ( 6 + 8)( 6 − 1)
−5 ( s + 7 )( s − 3)
11( y − 11)( y − 1)
−2 ( t − 11)( t − 3)
−
)
1
x + y x − y x − 2y x + 2y
4
(
(b)
)(
)(
)(
)
Exercise 2.2H
(
1 11a 6 + 2
)
2 xy y − x
(
)
3 a 6a + b
(
)
(
)
4 xy y − x
( x + 6)( x + 1)
( x + 2)( x + 3)
( x − 8)( x + 7 )
− ( x + 9)
(b)
5
( n + 1)( m − 1)
6
( m − x )( m − 1)
(
)(
)
7 2 6s + 5 s + 1
8
( p − qx )( p − q )
(
)(
9 s 1− t 1+ t
(
)
)(
10 −6 x + 2 y x − 2 y
(
)(
)
)
11 y − 12 y − 13
© 2017 NTK Publishing Limited
Answer Key
(
)
12 2 n + 3
2
(g)
(
13 5a − 11b
(
)
2
)(
14 3c − 5 c + 7
(
15 2 x x + 2
(
)
(
)
)(
17 m − n + 6 m − n + 6
(
)(
)
)(
)
18 q − 2 2 p + 1 2 p − 1
(
)(
(
)
(
)( )
19 x + 2 x − 8
20 3x + 2
(
)(
)(
)(
)
22 p − 1 p − 4 p + 1 p + 1
24
)
)(
(c)
)(
)
2 (a)
(b)
12
5
−
10
3
3 (a)
a
3
1
(c)
3c
(e)
−
6
7
x
4y
x
z
(g)
2y − 5
5y
(h)
2− x
3x
(i)
x−2
2x + 1
(j)
2x + 3
x−5
(l)
1
2− x
(n)
x
x−3
(p)
x +1
2
(b)
−
5 (a)
(d) 1
(b)
−
x
x+2
(b)
−
© 2017 NTK Publishing Limited
−
a
2
1
4
1
(b) − x 2
3
y
(b)
7x
b
(b)
2− x
(b)
x
2b ( y + 1)
1 (a) LCM: 12x2
−
HCF: 2x
(b) LCM: 36x2y2
1
4a
b
(d)
5c
(f)
x2
Exercise 2.3C
3
2
3
(d)
2
(c) 0
(f)
4
5a
bc
2 (a)
6 ad
x −1
3 (a)
3x
x +1
4 (a)
x
Exercise 2.3A
1 (a) 6
2x
y
1 (a)
1
2a − 1 2a + 1 4a 2 + 1
16
(
2
−
(d)
Exercise 2.3B
4
1
23
2x − 1
16
(
−3z
(e)
(o)
21 4 p − q q
x2
(b) 3a + b
a
c
1
(m)
x +1
2
2
z
8y
(k)
)
−3z
(c)
2
)(
(h)
4 (a) 2a + 1
)
16 a 5x + 6 x + 4
z
8y
−5
4y
HCF: 3xy
(
)( ) HCF: 1
HCF: ( x − 1)
LCM: 2 x ( x − 1)
LCM: 60 a ( a + b)( a − b)
(c) LCM: 2 x + 2 x − 2
(d)
(e)
2
2
HCF: 5
(
)
3
(f) LCM: 18 1 − m n
(
HCF: 3 1 − m
)
11
Cambridge IGCSE Mathematics Study Guide
( )(
HCF: ( 2 x − 3)
)(
)
(g) LCM: x + 1 x − 1 2 x − 3
2
21
10 a
(b)
4 + 6a
3ab
(c)
2x
x −1
(d)
y − 2x
2 x− y
(e)
7m + 4
m m+2
(f)
9
2 n−3
(g)
4x + 2
4x2 − 1
2 (a)
(
)
(h)
(
(
(b)
)
1− x
x−3
3x
(e)
x−3
(c)
(d)
( )
y ( y − 2x )
x ( y − x)
( x − 3)( x − 2)( 2 x + 1)
(b)
5x − 4
x − 3 x − 2 2x + 1
)(
−4 ab
a+b
2x − 3
2x + 3
1
9
1
(c)
3
12
(f)
(g) a14
(h) 8a2b11
3 4
−25a
b
1
ab
3 6
(j) 1
1
33
(b)
1
62
(c) 2
(d)
x2
y3
(e) x4
(f)
3 (a) 3
(c)
−2
1
1
x2 y 3
x
(h)
2y
(b)
−2
(d) 7
(e) 3
4 (a)
(
(b) 25 x − y
) (x
2
2
+ xy + y 2
)
1
16
(c) 2
(e)
5
21
(g)
−4
(b) 5
(d)
3
4
(f)
3
2
(b) 4
(d) 3
(e) 1
(f) 1
(g) 0
(h) 1
(i)
(j) 4
125
(e) a
9
)
Warm up 2.4
1 (a)
(d) a9b7
(g) x 2
3 3x + 5
4 (a) a3 − b3
(b) 78
(c) 37a5
2 (a)
x−3
(h)
x+3
2 (a)
)(
1
625
1
9
1
(n)
2
(l)
1 (a) 59
(f) 1 − x
x +1
(g)
2x − 1
(
(m)
(j)
4
9a
b b− a
1
8
Exercise 2.4
)
Exercise 2.3D
1 (a)
(k)
Exercise 2.5A
1 (a) 6
(b) 4
© 2017 NTK Publishing Limited
Answer Key
(c) 4
10 (a) 12
2 (a) 15
(b) 1
Exercise 2.5C
(c) 1
3 (a) 2
(b) 7
(c) 2
(d) 2
4 (a) 6
(b)
7
4
7
(c) −
6
5 (a)
1
18
−
6
5
(b)
69
88
(c) x =
5
9
or x = −
3
4
2 (a) x = 0 or x = 4
(b)
62
3
7
5
3 (a) x = 0 or x = 3
(c) x =
(c) 39
7 (a) 0
1 (a) x = 0 or x = 3
(c) x = 0 or x =
(c) 29
6 (a)
(b) 98
7
or x = −5
2
4 (a) x = −2 or x = −1
(c) x = 8 or x = −3
(b) 8
36
(c)
7
5 (a) x =
1
3
or x =
3
2
2
6
(c) x = − or x =
3
5
Exercise 2.5B
6 (a) x = 3 or x = 5
3
5
(c) x = − or x =
2
6
1 5
5
2
(b) x = 4 or x = −
(d) x = 3 or x = 4
(b) x = 0 or x = 5
(d) x = 0 or x = −4
(b) x = −
(d) x =
5
or x = 5
2
3
or x = 6
2
(b) x = 3 or x = −4
(d) x = 12 or x = −3
(b) x = −5 or x =
(d) x =
7
4
2
3
or x =
3
7
(b) x = −1 or x =
5
2
7
4
(d) x = − or x =
4
5
2 19
3 14
Exercise 2.5D
4 13
1 (a) x = 0 or x = 3
5 (a) 7
(b) 12, 16, 20
(b) 96
(c) x = −
7 7
(b) x = 0 or x =
32
9
19
19
or x =
7
7
3 (a) x = −4 or x = −12
8 90
9 (a) 36
(c) x = 2 2 or x = −2 2
2 (a) x = 7 or x = −7
6 7
(b) x = 13 or x = −13
(b) x =
(b) 204
© 2017 NTK Publishing Limited
−2
3+7
2 3+7
or x =
4
4
13
Cambridge IGCSE Mathematics Study Guide
(c) x = −5
(b) x = 3 − 7 or x = 3 + 7
8 3 2
3 2 8
or x =
(d) x = − −
−
7
7
7
7
(c) x = −
4 (a)
( x − 1)
2
(c) 2 x +
(b)
+3
x
2
3
2
+ 9.5
2
5
2
3
2
(d) 2 x
5 (a) x = 1 or x = 7
5
13
13 5
(b) x = − −
or x =
−
2
2
2
2
(
)
(c) x = 2 1 + 3 or x = 2
(
(f) x = −
)
3 −1
33
33
or x =
−2
2
2
3
4
or x =
2
5
Exercise 2.5F
1 26
2
1
8
2
or x = 4
3
1
3
or x = 1 +
width = 12
5 6
1
3
9 + 21
9 − 21
(c) x =
or x =
6
6
4
(d) x = −
3
5
(e) x = −
2
6 12
7 30
8 12 km/h
9 (a) 60 km/h
43
43
or x = 4 +
3
3
145
5
145
5
(h) x = −
or x = +
12
12
12
12
1 (a) x = 3, y = 3
(c) x = −3, y = 4
2 (a) x = 2, y = 3
13
13 3
3
2 (a) x = − −
or x =
−
2
2
2
2
14
(b) 9:30 p.m.
Exercise 2.5G
(f) x = 6
(g) x = 4 −
1 13, 15
4 length = 15
1
or x = 3
2
(b) x = 1 −
Warm up 2.5F
3 x=
Exercise 2.5E
1 (a) x =
135
4
2 6, 12
(d) x = 3 − 19 or x = 3 + 19
(e) x = −2 −
(d) x = 2 + 23 or x = 2 − 23
1
4
2
793
1
793 1
or x =
−
−
12
12
12
12
(c) x = 3, y = −4
(b) x = 2, y = 3
5
5
(d) x = , y = −
2
3
(b) x = 5, y = −2
(d) x = 2, y = −5
© 2017 NTK Publishing Limited
Answer Key
2
3
3 (a) x = , y = −
3
2
(b) x = 15, y = 12
(b)
3
2
(d) x = , y =
4
7
(c) x = 6, y = −5
4
4 Adrian: £12 Billy: £20
5 Chocolate: £0.5
y
x
8
Crisps: £1.6
6 Solution A: 12.5 litres Solution B: 1.6 litres
(c)
4
Exercise 2.6A
1 (a) x < −2
(b) 4x < −2
3
(c) x ≤
4
3
(e) x <
2
3
5
(d) 4x < −
2
x
(f) 4x ≥ 2
2 (a) x < 26
(b) x <
9
7
25
(e) x <
69
(c) x ≥
3 (a) x < −
y
(d)
14
3
y
(d) x ≤ 8
3
(f) x ≥
26
3
28
(b) x >
3
101
46
5
2 (a)
Exercise 2.6B
x
y
3
1 (a)
y
3
x
4
4
© 2017 NTK Publishing Limited
x
6
15
Cambridge IGCSE Mathematics Study Guide
(b)
(b)
y
y
5
4
7
x
4
x
5
4
(c)
(c)
y
y
3
1
2
1
0
1
2
4
x
2
x
6
2
3
(d)
y
(d)
y
2
14
4
3
9
x
5
6
8
3 (a)
x
y
4 (a) (i)
4
30x + 60y ≥ 1800
x + y ≤ 50
(ii) x ≥ 0
y≥0
5
x
(b)
y
50
30
50 60
x
(c) 40
16
© 2017 NTK Publishing Limited
Answer Key
5 5
3
6 90
4 n=−
7 (a) 10
5
(b) 6p stamp: 28 8p stamp: 4
2
(b) 2, 1,
3
0, 5
(b)
−4
+ 34
−2
x
4 (a) Tn = 6n − 17
(
(
n n + 1− n + 2
(b) T20 = 103
))
(ii) 3.33 m/s
50 m
(ii) 200 m
(c) 2.5 m/s
(c) 33 minutes
5
10
10
3 (a) 2.25 ms−2
4 (a) (i)
5
(b) 100 m/h
(d) No
0.6 ms−2
(b)
−1.2
ms−2
(d) 12.1 ms−1
(ii) 0.4 ms−2
(b) 512 m
1
(c) 992 m
(b) n + 1
(d) 12.4 ms−1
n
(c) 2
(c) 792 m
7 (a) Pattern 4: 20; Pattern 5: 29; Pattern 6: 40
2
(b) n + 4
8 (a) 4
(b) (i)
(c) 423 m
= −n
6 (a) 1
8
1
16
(e) 75 km/h
n−2
(c)
2.5 m/s
2 (a) 70 km
3 (a) 4n − 3
2
1
4
(iii) 5 m/s
(c) T1 = 1, T2 = 7, T3 = 79
−3,
1
1
2
1 (a) (i)
(b) T1 = 1, T2 = 10, T3 = 37
(c)
a
Exercise 2.9A
1 (a) T1 = 1, T2 = 3, T3 = −7
2 (a) 5, 17, 29
1
2
b
Exercise 2.7
5
p=8 5
5 (a) (i)
0.48 ms−2
(ii) Yes, the gradients of the two lines are equal.
n−1
(b) No
(b) (i)
91.5 seconds
(ii) 588 m
(c) (i)
Exercise 2.8
(ii) 495 m
9
1
32
2 (a) y =
83.75 seconds
3
1
z +1
4
(
)
(b)
© 2017 NTK Publishing Limited
(4 y)
1
3
−1
17
Cambridge IGCSE Mathematics Study Guide
Exercise 2.9B
1 (a)
10
54 °C
(c) (i)
T = −0.6t + 54 (ii) 45 minutes
°C / minute
K = 0.0006
4
t0 = 500
2
(b) (i)
0
5
10
15
20
(c) t > 1000
(b) 1.6 cm
(c) 0.24, it means the spring will extend 0.24 cm
when an extra 1 g of load is added.
(ii) 20 GBP
420 HKD
CA = 2 + 0.004t
(ii) CB = 0.006t
25
W (g)
(b) (i)
−0.6
M = 0.0004
6
2 (a) (i)
(ii)
5 (a) F = 4
8
L (cm)
(b) (i)
12, it is the exchange rate of GBP to HKD.
(ii) 200 m
6 (a) (i)
(iii) 70 NZD
(b) The exchange rate of EUR or NZD to USD.
(c) 1.52
7 (a)
Time t
5
10
15
20
25
2
25
100
225
400
625
(c) 0
t
3 (a)
(ii) 60 NZD
84 USD
Height H
above the 4375 4000 3375 2500 1375
ground (m)
°F
100
80
60
(b) H = −5t2 + 4500
40
(c) (i)
(ii) t = 30 seconds
4500 m
20
20
10
0
°C
10
20
30
40
50
20
(b) (i)
1.8
(ii) 32 m
(c) (i)
m = 1.8, c = 32
(ii) 70
4 (a)
Exercise 2.10
1 (a)
x
0.1
0.5
1
1.5
2
0.5
3
y
−29.9
−5.5
−2
−0.5
0.5
1.3
2
(b)
y
10
Temperature (°C)
5
60
50
0
40
5
30
10
20
15
10
0
1
1.5
2
2.5
3
20
10
20
30
40
Time (minutes)
18
x
0.5
50
25
30
© 2017 NTK Publishing Limited
Answer Key
(c) 1.7 (accept 1.8)
2 (a)
6 (a) (i)
x
−3
−2
−1
0
y
1
0
−1
−1
(b)
1
2
(ii) 1, −0.8, 4.8
(accept 0.9 – 1.1, −0.9 – −0.7, 4.7 – 4.9)
3
0.5 3.75 8.875
y = 2x + 4
(b) (i)
y
y = 3x − 10
(ii)
−1.7,
2.5, 4.2
(accept −1.8 – −1.6, 2.4 – 2.6, 4.1 – 4.3)
10
8
7 (a)
6
4
2
−3
−2
−1
0
1
2
3
4
5
y
−35
0
15 16
9
0
−5
0
21
(b)
2
3
x
y
x
1
1
2
2
30
3
20
4
10
6
3
3 (a) (i)
2
1
y=x+1
10
x
1
2
3
4
5
20
(ii) 1.5, 7.5 (accept 1.4 – 1.6, 7.4 – 7.6)
(b) (i)
0
30
y=3
40
(ii) 1.4, 6.6 (accept 1.3 – 1.5, 6.5 – 6.7)
(c) y = 0
4 (a) y = −2x + 2
(b)
−2,
(d)
3
−2,
2, 4
8 (a) 0.5, 3 (accept 0.4 – 0.6, 2.9 – 3.1)
5 (a)
x
−3
−2
−1
0
1
2
3
y
1
8
1
4
1
2
1
2
4
8
(b)
Exercise 2.11
y
10
1 (a)
8
3
2
(b)
−1.5
(accept −1.7 – −1.3)
6
(b) 1.5 (accept 1.3 – 1.7)
4
(c)
−2
(accept −2.2 – −1.8)
2
(d)
−9
(accept −9.5 – −8.5)
x
1
2
1
2
3
(e) 5.6 (accept 5 – 6)
(f) 1.3 (accept 1.2 – 1.4)
4
6
(c) x = 1
© 2017 NTK Publishing Limited
2 (a)
x
−2
−1
0
1
2
3
y
−11
−1
1
1
5
19
19
Cambridge IGCSE Mathematics Study Guide
(b)
3 (a)
y
20
15
5
x
1
5
1
2
()
f ( 4) = 0
f ( 6 ) = −3
f 2 =4
5
fg 2 x = 4 x − 1
6
f
15
20
(ii) 1
5
(c)
f : x  5x + 7
(b) f : x  4 x 2 − 9
f : x  2x + 1
(d) f : x 
2 (a) 0
(b) 2
(c) 27
3
4
()
fg x =
1
4 x 2 + 15
8 (a) 8.04
(c)
(b)
1
16
(b)
−4.5
9
151
26
33
(d) 52
()
gf ( x ) = 10 x
9 (a) a = 3
3
(c)
3
2x − 4
6
gf x = − 4
x
()
fg x =
10 (a)
()
16
3
(ii)
16
3
(iv) −1.4
(b) denominator = 0
2 (a) x = 4
(b)
(3x + 1)
3
4
x+3
x+2
x
(b) denominator = 0
()
3
3
3
= , gg 2 = 4, gg
=5
2
2
2
()
( )
( )
( )
(b) gf s = 4, fg 3 = 0, gh 3 = 3, gf
1.4
(iii)
−
11 (a) ff 2 = 2, ff
Exercise 2.12A
20
1
4x − 3
(c)
2
12
= 2
x
x
(d) 2
fg x = 250 x 3
1 (a) (i)
( )
7 (a)
Warm up 2.12A
1 (a)
− 14 x + 12
4
3
10
(c) (i)
2
(b) 2 − 29 , 2 + 29
5
5 5
5
10
2
()
g ( x ) = 5x
f x = −10 x + 17
(c) hgf
( 1) = 0, hgf (0) = 6, hgf
9
=0
2
3
3
=
2
2
(d) 2, 5
12 (a) a = 7
(b) b = −1
c = −1
(b) x = 0
© 2017 NTK Publishing Limited
Answer Key
(b)
Exercise 2.12B
1 (a)
(c)
()
f −1 x = x − 3
()
f −1 x =
4x + 7
3
5 − 6x
x =
8
()
f −1
()
15x − 9
10
−1
(b) f x =
(c)
()
3 (a)
f −1 x =
()
1
−2
x
(c)
f −1 x =
()
5
+3
3x
5x + 3
x =
1− x
7 − 5x
x =
2x + 3
4 (a)
f −1
()
(c)
f −1
()
5 (a) (i)
x+7
2
x+3
5
2 (a) f −1 x =
(c)
()
−1
(b) f x =
()
(b) f −1 x =
2 3
−
x 2
(d)
(b) f −1
2x + 5
x =
4 − 3x
()
()
ff x = x
(ii) f is a self-inverse function
(c) m = 2
5
n=
6
Chapter 3 Geometry
Exercise 3.1
1 (a) trapezium
(b) isosceles triangle
(c) parallelogram
(d) hexagon
(e) pentagon
(f) equilateral trangle
3 (a) similar
(b) neither
(c) congruent
(d) neither
(e) congruent
(f) neither
Exercise 3.2
1 (a)
O
A
2 (a)
B
© 2017 NTK Publishing Limited
21
Cambridge IGCSE Mathematics Study Guide
4 (a)
(b)
A
60°
A
B
C
O
2 (a)
B
M
∠AMC = 120°
(b)
M
A
A
60°
N
28°
112°
B
O
(b)
C
N
M
B
∠ANM = 90°
A
5 (a)
B
70°
A
112°
O
3 (a)
A
B
(b)
B
B
C
(b) They meet at the in centre (Centre of the circle
inscribed in ΔABC)
22
A
© 2017 NTK Publishing Limited
Answer Key
6 (a)
10 (a)
A
A
30°
30°
B
(b) 4.6 cm
B
C
(b) The lines (medians) meet at the centroid.
Exercise 3.3
1 (a)
7 (a)
A
B
B
3 cm
4 cm
A
C
4.5 cm
(b) PQ = 1.5 m, QR = 2 m, PR = 2 m
C
(b) They meet at the circum-centre. (Centre of the
circle circumscribing ΔABC)
(c)
2 (a) 1 : 6000000
(b) 565 km
3
A
B
N
45°
4.4 cm
A
1 cm represents 5 km
C
The circle circumscribes ΔABC.
Exercise 3.4A
8
1 (a) True
A
(b) True
(c) False
B
2 (a) x = 9; y = 12
(b) x = 45; y = 4; θ = 65°
3 (a) y = 5; α = 53; β = 65°
(b) x = 8.75; y = 2; α = 20°; β = 70°
C
© 2017 NTK Publishing Limited
23
Cambridge IGCSE Mathematics Study Guide
4 (a) x = 24; y = 16; g = 3; k = 2
11 (b) (i)
14.4 cm
(ii) 360 cm2
(b) x : g = 8 : 1; y : k = 8 : 1
(c) s1 : s2 : g = n : 1 :
n −1
2
Exercise 3.4B
5 (a)
(b)
1 (a) x = 13.5
(b) 5
2 (a) x = 1215
(b) x = 3
3 (a) x = 200
(b) x = 18
4 (a) 24π cm2
(b) 12π cm3
216π cm2
(c) (i)
2 cm
(b) 3888π cm3
5 (a) 6 cm
1 cm
(ii) 324π cm3
6 (a) (ii) 33.3%
4 cm
4 cm
8 cm
8 cm
(b) (i)
5.77 cm
(ii) 11.6 cm2
7 (a) 4 3 cm
1 cm
(b) (i)
2 cm
6 (a) ΔAMN ∼ ΔABC
(b) ΔAOB ∼ ΔPOQ
7 (a) ΔAXC ∼ ΔDXB
(b) ΔBED ∼ ΔBCA
(c) ΔDAE ∼ ΔBAC
(d) ΔABD ∼ ΔCAB
8 (a) α = 50°; x = 9
(c) x = 8; α = 60°
9 (a) x = 3.98
(b) x = 6; y = 12
(d) x = 12; α = 40°
(b) x = 1.61
8 (a) (i)
5 3 cm
(ii)
(15 − 5 3) cm
14.3 cm; 3.71 cm
(ii) 121 cm
(b) (i)
XY = 18.3 cm; YZ = 13.7 cm
(ii) 2.60 cm
9 (a) 144π cm2
(c) (i)
16.0 cm
10 (a) 4.17%
(b)
8
27
(ii) 2.65 cm
(b) 8.17%
(c) r = 8; t = 7.5; α = 80°
(d) x =
10 (a) x =
20
3
36 7
27 7
;y=
7
7
(b) x = 15
Exercise 3.5
1 (a) yes; 4
(b) yes; 2
(c) no
(d) no
(e) no
(d) yes; 6
(c) x = 5.25; y = 8.75;
(d) x = 16; y = 15; z = 9; t = 25
24
© 2017 NTK Publishing Limited
Answer Key
2 (a)
(b)
5 x = 26°; y = 32°
6 x = 22°; y = 46°
7 x = 55°
(c)
(d)
8 a = 111°; b = 69°
9 x = 78°; y = 102°
10 x = 65°; y = 55°; z = 65°
(e)
(f)
11 x = 20°; y = 45°
12 x = 9°
3 (a)
(c)
(b)
(d)
Exercise 3.6B
1 (a) x = 60°
(b) x = 111°
(c) x = 50°; y = 10°
(d) x = 67.5°
(e) x = 80°
(f) x = 25°; y = 50°; z = 80°
2 36°
3 25°
4 60°
4 (a) 2
(c) 2
(b) 3
(d) Infinite
5 4
5 90°
6 (a) x = 108°; y = 72°
(c) x = 27°; y = 30°
(b) x = 18°
(d) x = 40°; y = 55°
(e) x = 25°
Exercise 3.6A
7 x = 112.5°
1 a = 102°; b = 78°; c = 102°
8 n=6
2 a = 150°; b = 60°
9 180°
3 x = 20°
10 x = 60°
4 y = 18°
© 2017 NTK Publishing Limited
25
Cambridge IGCSE Mathematics Study Guide
Exercise 3.6C
Exercise 3.6E
1 (a) x = 30°
(b) y = 45°
(c) x = 30°
(d) y = 50°
(c) x = 36°; y = 72°
(d) x = 20°
(e) x = 35°; y = 65°
(f) x = 30°; y = 60°
(e) x = 100°
(f) x = 86°; y = 22°
2 (a) x = 9°; y = 18°
(b) x = 112°; y = 112°
1 (a) x = 82°
(b) x = 60°; y = 102°
2 (a) x = 100°; y = 80°
(b) x = 90°; y = 60°
(c) x = 100°; y = 30°
(d) no solution
(c) x = 60°
(d) x = 49°; y = 30°
(e) x = 118°; y = 59°
(f) x = 70°
(e) x = 12°; y = 15°
(f) x = 44°; y = 68°
3 (a)
A
C
(b) 8.5 cm
3 (a) No.
(b) Yes. Sum of any two opposite angles is 180°.
O
(c) No.
(d) Yes. ∠A + ∠B = 180°, ∠B = ∠C,
∠A + ∠C = 180°, same as ∠B + ∠D
B
BC is the diameter of the circle.
(e) Yes. ∠A + ∠C = 180° and ∠B + ∠D = 180°
4 (b) a = 120°
(f) Yes. Converse of angles in the same segment.
5 Base angles in an isosceles triangle are equal.
Exercise 3.6D
1 (a) x = 60°
(c) x = 90°; y = 60°
4 ∠CAB = 70°
Exercise 3.7
(b) x = 90°; y = 40°
1
A
(d) x = 35°
B
2 (a) x = 67°
(b) x = 80°; y = 80°
(c) x = 40°; y = 70°; z = 30°
(d) x = 30°
(e) x = 25°; y = 90°
D
C
2 (a)
A
(f) x = 118°
3 x = 100°; y = 40°
C
B
26
© 2017 NTK Publishing Limited
Answer Key
(b)
Chapter 4 Mensuration
A
Exercise 4.1
C
B
(c)
A
1 (a) 0.23
(c) 0.045
(d) 0.15
(e) 250
(f) 6 520 000
2 (a) 3750
(c) 0.00112
C
(b) 120 000
3 (a) 3.5 × 106
(b) 0.012
(d) 0.04
(b) 600
−5
(c) 6.5
(d) 5 × 10
(e) 23,000
(f) 5.1 × 106
B
3
A
4 (a) 30
B
D
C
4 (a)
(b) 1 × 107
(c) 300
(d) 100 000
(e) 0.0031
(d) 6500
Exercise 4.2
1 110 cm
O
2 100 cm
1.5 cm
3 a=5
4 32.8
5 23
5
C
B
6 179
7 (b) x = −4 or x = 1
A
(c) x = 1
8 40 cm2
D
9 3 23 + 9
2
© 2017 NTK Publishing Limited
27
Cambridge IGCSE Mathematics Study Guide
Exercise 4.3
Exercise 4.5
1 (a) perimeter = 25.1 m, area = 50.3 m2
1 (a) 113 m3
(b) perimeter = 18.8 m, area = 28.3 m2
2 (a) arc length = 27.9 m, area = 69.8 m2
(b) arc length = 15.7 cm, area = 78.5 cm2
(c) 48 m3
2 (a) 615.0 cm3
(c) 1282.9 cm3
(b) 500 m3
(d) 136 cubic units
(b) 41.9 m3
(d) 173.7 cm3
3 471 cm3
3 r = 7.14 cm
4 (a) 4.71 m
4 1367.1 cm3
(b) 16.7 m
(c) 14.1 m
5 (a) 44.6 m2
2
(c) 12.6 m
6 (a) 37.7 cm
(c) 25.1 m
(b) 18.8 m2
2
(d) 2.46 cm
(b) 22.8 m
(d) 46.8 cm
5 (a) 864 cm2
(b) 1832.6 cm3
6 (a) 3π x3
(b) x = 3
(c) 210 cm2
7 (a) 12.15, 80
(b) 530.93
7 perimeter = 15.7 cm, area = 4.03 cm2
Chapter 5 Coordinate Geometry
8 perimeter = 30.9 cm, area = 61.4 cm2
Exercise 5.1
Exercise 4.4
1 (a) 1550 cm2
(c) 193 cm2
2
(b) 294 cm2
(d) 84.3 cm2
2
2 (a) 225 cm
(b) 424 cm
(c) 384 cm2
(d) 85 cm2
3 (a) surface area = 22 cm2, volume = 157 m3
(b) surface area = 438 m2, volume = 335 m3
(c) surface area = 462 cm2, volume = 718 cm3
(d) surface area = 113 cm2, volume = 113 cm3
4 h=
A − 2π r
2π r
( ) ( ) ( )
D ( −3,5) ; E (1,5) ; F ( 7, −1)
1 (a) A 4,2 ; B −3, −1 ; C 0, −4 ;
(b) E, A, F
2 (b) A straight line
Exercise 5.2
1 (a) 3
(c) 0
(b)
−1
2
(d) undefined
2
5 (a) 1256.64 cm2
2 (a)
(b) 4720 cm3
−3
(c) undefined
(b)
−1
(d) undefined
6 surface area = 63.2 cm2, volume = 56.5 m3
28
© 2017 NTK Publishing Limited
Answer Key
3 (a) no
(b) no
(c) yes
(c) y = −2
(d) yes
4 (a) c = 1
(b) q = −15
5 (a) mAB = 2; mAD
1
1
= ; mDC = 2; mBC = −
3
3
(b)
5
3
15
2
−3
117
x+
11
11
(c) a =
89
60
;b=
2
11
(b) gradient = undefined
y-intercept = does not exist
3
(b) −
8
(d)
−
11
11
, y-intercept = −
8
4
3
(d) gradient = , y-intercept = −1
4
(c) gradient = −
5
2
2 (a) AB = 5; AC = 5; BC = 2 5
(c)
(e)
1
3
x−
2
5
2
(b) y = x − 2
7
5
(c) y = − x −
3
9
(d) y = − x + 3
11
6 (a) y =
(b) 5 + 3 5
4 (a)
(b) No
1
5
5 (a) gradient = , y-intercept =
4
8
3
4
3 (c) Area =
(d) y = −bx + ab
(b) L1: 2x − 3y − 8 = 0; L2: y − 4 = 0.
1 (a) undefined
−
5
8
x+
18
9
4 (a) L1: 5x − 7y + 25 = 0; L2: x + 5 = 0.
Exercise 5.3
(c)
3 (a) y =
(ii) undefined
0
(iii)
(c) x = 5
(b) y =
−
(b) same gradient
6 (a) (i)
2 (a) y = x − 3
2
4
(d) y = − x −
3
3
1
2
( 20 )( 20 ) = 10 sq.units
(6,2)
1 1
,
2 2
5a b2
,
4 2
(b)
( 2,1)
(d)
( −2,3)
(f)
(0, a + 2b)
5 c = −4; d = −3
6 (a) height = 9 units
(b)
(3,6)
Exercise 5.4
1 (a) y = −x + 3
(b) y =
© 2017 NTK Publishing Limited
4
x
5
Exercise 5.5
1 (a) L2 and L6
2 (a)
−4
3
y=
2
11
x+
3
3
4
y=
1
x+3
2
(b)
−
7
2
29
Cambridge IGCSE Mathematics Study Guide
5 (a) k = −
4
3
(b) x-intercept =
9
3
y-intercept =
2
2
(c) 6y + 4x − 9 = 0
Chapter 6 Trigonometry and
Pythagorasʼ Theorem
Exercise 6.1
(d) 3y + x − 9 = 0
1 (a)
N
Exercise 5.6
1 L3 and L5
L1 and L4
070°
19
(b)
7
2 (a) 6
(c)
−6
or 1
3 (a) 2
(b)
(d) 1 or −1
N
(b) 3
(c) 25
A
5
4 (a) −
3
230°
(b) b = 1
8
8
(c) y-intercept = , x-intercept = −
3
5
(c)
N
5 (a) L1: y = −x + 7; L2: y = x + 1
(b) L1: x-intercept = 7; L2: x-intercept = −1
(c) L1
3
6 (a) L1: y = − x + 3
4
3
15
L2: y = − x −
4
4
5
(b) L1: y = − x + 15
2
2
18
L2: y = x −
5
5
A
2 (a) 050°
300°
(b) 280°
(c) 270°
3
R
7 (b) k = 3 or k = −1
5 km
(c) For k = 3, L1: 3x + 3y − 4 = 0
L2: −x + y − 3 = 0
N
N
For k = −1, L1: −x + 3y − 4 = 0
L2: 3x + y − 3 = 0
Q
080°
4 km
310°
P
30
© 2017 NTK Publishing Limited
Answer Key
4 200°
8 39.1 km
5 090°
9 8.66 cm
10 (a) x = 8.94; φ = 63.4°; θ = 26.6°
Warm up 6.2A
(b) x = 11.5; φ = 27.5°; θ = 62.5°
1 (a) x = 12.8
(b) x = 13.7
(c) x = 6; φ = 53.1°; θ = 36.7°
(c) x = 7.21
(d) x = 5.83
(d) x = 7.62; φ = 66.8°; θ = 23.2°
2 (a) sin θ =
p
q
p
; cos θ = ; tan θ =
r
r
q
a
b
a
(b) sin θ = ; cos θ = ; tan θ =
c
c
b
4
3
4
(c) sin θ = ; cos θ = ; tan θ =
5
5
3
1
2 16
6
; tan θ =
(d) sin θ = ; cos θ =
5
5
12
3 (a) x = 4.11
(b) x = 13.7
(c) x = 6.53
(d) x = 5.64
(e) x = 40
(f) x = 50
4 (a) θ = 53.1°
(b) θ = 31°
(c) θ = 63.9°
(d) θ = 49.5°
11 θ = 52.0°
12 (a) 8.94 m
(b) 35.3°
13 2.25 m
14 20.5 cm
15 270 m
16 53.1°
17 12.0 m
18 (a) 10
(b) 21.8°
19 (a) 41.5°
(b) 51.3°
20 248°
Exercise 6.2A
21 112°
1 OD = 5
22 (a) 175 km
2 EC = 9.90 cm
(b) 323°
23 1.73 km
3 AE = 17.3
4 (a) 3x2 − 24x = 40
5 r=
(b) x = 8
8
3
6 26 km
7 1.3 km
Exercise 6.2B
1 0.866 m
2 40 m
3 109 m
4 12.7 m
© 2017 NTK Publishing Limited
31
Cambridge IGCSE Mathematics Study Guide
6 78.6 cm2
5 83.9 m
6 4.1 m
Exercise 6.3B
7 55.6 m
1 (a) a = 3.54
8 10.8 m
2 (a) x = 5.87; y = 13.50
9 710 m
(b) a = 12.62; b = 4.68
3 (a) θ = 42.55°
Exercise 6.2C
1 (a)
−cos 60°
(b)
−cos 150°
(c)
−cos 90°
(d)
−cos 30°
(f)
−cos 13°
(e) cos 157°
2 (a) sin 150°
(b) sin 135°
(c) sin 60°
(d)
(e) sin 155°
(f) sin 4°
3 (a) (i)
(b) x = 17.91
1
−sin 0°
Exercise 6.3A
(b) 50.3 cm2
(c) 36.8 m2
(b) x = 14.1
(c) x = 9.67
3 (a) θ = 71.8°
(b) α = 48.59°
(b) 26.8 cm2
(c) 65.0 cm2
5 (a) 12.0 cm2
5 (a) 12.42 cm
(b) AD = 6.62 cm; DC = 9.38 cm
6 (a) PQ = 8.91 cm; PS = 3.03 cm
(b) 28.96 cm2
1 (a) x = 7.93
(b) x = 12.26
2 (a) x = 16
(b) x = 18.60
3 (a) α = 60.61°; β = 40.80°
(b) α = 95.74°; β = 33.56°
4 (a) x = 5.18; α = 34.40°; β = 98.60°
(b) x = 20.84; α = 22.58°; β = 15.42°
5 28.59 km2
6 (a) PR = 11.34 cm; QS = 13.63 cm
(c) β = 48°
4 (a) 70.9 cm2
(b) θ = 30.95°; β = 113.05°
Exercise 6.3C
(b) sin2θ + cos2θ = 1, for any angle θ .
2 (a) x = 12.0
4 (a) α = 56.76°; x = 17.72
(ii) 1
(iii) 1
1 (a) 25.8 cm2
(b) θ = 36.23°
(b) ∠ SQR = 14.11°; ∠ SPR = 30.50°
(c) RS = 9.76 cm
(b) 2.93 cm2
(c) 61.4 cm2
32
© 2017 NTK Publishing Limited
Answer Key
5 (a) 33.69°
Exercise 6.3D
(c) 30.81°
1 BD = 10.58 cm; AC = 17.44 cm
6 8.94 cm
2 (a) QS = 7.13 cm
(b) ∠ QRP = 23.33°; ∠ QRS = 46.67°
(c) 35.93 cm2
3 (a) BM = 11.28 cm
(b) ∠ AMB = 117.1°
(c) MC = 14.91 cm
N
130°
N
(b) 35.26°
8 (a) 500 m
(b) 23.75°
9 (a) 14.42 cm
(b) 29.02°
10 (a) 4.29 cm
(b) 16.08°
11 (a) 14 cm
(b) 39.87°
B
1200 m
12 37.76°
860 m
068°
A
C
(b) 1774.43 m
13 (a) 61.93°
(b) 51.24°
14 55.47°
(c) 273.34°
1857.54 m
7 (a) 54.74°
(c) 25.09°
5 (a)
6 (a) (i)
(b) 53.13°
(ii) 1717.26 m
(b) VA:VB = 1.08:1
15 (a) 207.85 km/h
(b) 32.21°
7 (a) 996.11 km
(b) 7.88°
Chapter 7 Vectors and Matrices
8 (a) (i)
(ii) 39.16 cm
Exercise 7.1
(b) (i)
12.59 cm
AP = 12.63 cm; PB = 26.53 cm
(ii) ∠ ABP = 24.87°; ∠ APB = 93.13°
(b) 51.34°
(b)
1
3
(c)
0
4
(d)
1
3
(e)
5
1
(c) 45°
2 (a) 45°
(b) 45°
(b) CD
(d) HK
3
1
2 (a)
Exercise 6.4
1 (a) 59.04°
1 (a) OA
(c) BA
(c) 35.26°
3 (a) 45°
(b) 51.34°
4 67.01°
© 2017 NTK Publishing Limited
33
Cambridge IGCSE Mathematics Study Guide
3
8 (a) b + g
y
A
4
(c)
3
C
E
4
3
2
+b
(d)
(e) g + f
B
2
−f
(b) g + f
−f
+b
(f) g + b
1
1
D
0
1
1
2
3
1
a
2
1
a+b
(c)
2
x
4
9 (a)
2
3
1
b
2
(b)
1
(d) a + b
2
4
7
4
(b)
(c)
0
4
(d)
(e)
5
4
(f)
4 (a)
5 (a)
7
1
3
8
(c)
14
36
1 (a)
a+b
a+b
(c)
(c)
(d)
b
a
(9,2)
( 2,4)
39
28
(d)
14
3
23
6
(d)
2 (a)
a
( −17,15)
( a + 4,4a − 2)
(b)
y
5
a+b
4
b
3
a+b
2
A
6 (a)
(b)
Exercise 7.2
b
a
a
25
19
8
4
(b)
b
10 (a)
1
(b)
a
a
b
5
4
b
a−b
3
2
1
B
a−b
0
1
1
(d)
b
a
a−b
2
3
2
3
x
2
3
(c)
D
(b)
C
y
5
a−b
b
4
A1
a
3
2
7 (a) (i)
a+b
(iii) −a + c + b
(b) (i)
−a
+b
(iii) a = b − c
34
(ii)
−c
+a+b
(iv) a + b + c
(ii) c
1
B1
5
4
3
2
1
C1
0
1
D1
1
x
2
3
© 2017 NTK Publishing Limited
Answer Key
( −5, −2)
( 2,4)
3 (a)
(c)
(b)
(d)
(3,4)
(3,2)
4 (a)
(c)
(b)
(d)
5
( −17,15)
( −5,4)
9 (a)
y
( −3, −4)
( −7, −4)
y
4
k=3
3
2
1
4
3
2
C
1 0
1
x
C
A
B
1
2
3
4
5
6
x
(b)
y
2
3
4
( −2, −4)
( 4, −2)
6 (a)
(c)
(b)
C
( −2,3)
( −3, −2)
7 (a)
(c)
( 2,4)
(b)
8
( 2, −3)
k= 2
(c)
y
x
y
5
4
3
P
2
P2
5
4
3
2
1
1 0
1
1
2
2
3
4
3
4
5
P1
x
C
k=
1
2
x
5
© 2017 NTK Publishing Limited
35
Cambridge IGCSE Mathematics Study Guide
(d)
y
k=
(c)
1
3
−
2
5
a− c
7
7
4 (a) (i)
C
(iii)
x
(c) (i)
g−d
2
g−d
3
(
(d)
2
5
c+ a
7
7
(ii)
2
2
−d + g
3
3
(ii)
−b
( )
)
2
k g−d
3
(
)
(d) 2:1
5 (a) (i)
10
(iii) −b
y
C
4
3
A
2
B
6 (a) (i)
1
3
2
1 0
1
1
2
3
4
x
5
2
(b) (i)
7 (a) (i)
3
(v)
11 (a) reflection about x = 0
(b) enlargement with centre at
( )
(
), scale factor 2
(c) (i)
−6,1
8 (a)
Exercise 7.3
(c)
2
2 (a) q + r
3
3
1
r− q
(c)
4
4
3 (a) c + a
36
1
c−b
2
1
b+c
(d)
2
(
)
(
)
3
q+r
4
1
q+r
(d)
4
(
)
(
)
(b)
)
3p
(ii)
(b)
−2p
−p
+u
(iv) −4p + 2u
+ 4u
(vi) 2u + p
(ii) 6
6
(iv) 12
(v) 18
(d) reflection about y = −1
(
(ii) s + t
(ii) XS = t
(iii) 6
(c) rotation with centre 0,0 clockwise 90°
1
b−c
(c)
2
s + 2t
QX = s + t
(iii) 2u
4
1 (a) c − b
(iv) 2b
(b) 4
(c) BE , EF , CF , FC , AD, DA.
5
4
−a
(v) 2b + 2a
6
5
b−a
−
1
1
a+ b
2
2
1
1
a+ b
2
2
9 (a) (i)
(iii)
(b) (i)
b+v
1
u−v
2
(
3
4
1
1
a+ b
2
2
(b)
−
(d)
−a
1
− b
2
(ii)
−b
+u
)
(ii) 30
(b) a − c
© 2017 NTK Publishing Limited
Answer Key
Exercise 7.4A
1 (a) 2 × 1
(b) 1 × 3
(c) 2 × 4
(d) 3 × 3
(c)
2 (a) a = 0, b = 4
3 (a)
(c)
2
1
3 (a)
(
)
3
13
(b)
1 15
11 9
(d) does not exist
−12
−2
6
4
1
(e)
3
4
3
7 9
5 6
(g)
3
8
(h)
16 8
14 7
5 k = −4 or k = −1
(j)
14 42
2 16
10 10
6 (a)
18 2
17 7
(c)
1 4
1 12
(b)
12 16
5 1
(d)
11 1
2 6
(b) (i)
4
8
4
5
(f)
3
29
10
3
7
2
(ii)
403
178
89
42
4
3
3
2
2
1
(c) (i)
−3
(ii)
(e) (i)
1
(ii) 1
7 a=
−3
1
1
or −
3
3
8 (b) (ii)
5 a = 5, b = 0
7
4
A5 = 89A + 136I
(f)
8 2
7 9
1
19
A4 = 17A + 72I
3 9 0
6 12 9
4 (a)
(d)
5
8
4 (a) A3 = 9A + 8I
(e)
does not exist
3
5
(b)
8
2
10
(i)
6 4
10 6
2 6
1
2
9
4
2
1
(c) x = 2, y = 1
Exercise 7.4B
1 (a) 1
(c)
−21
(e) 2
(g)
2 (a)
−
5
12
1
2
1
(c)
56
(b) 2
Exercise 7.5
(d)
−1
1 (a) Reflection about the x-axis.
(f)
−
1
12
(h) 0
(b) Enlargement with centre (0,0) and scale factor 2.
(c) Reflection about the x-axis, enlargement with
centre (0,0) and scale factor 2.
(d) Reflection about the line y = −x.
1
2
6
8
3
4
7
0
(b) does not exist
(e) Rotation about the origin through 90° clockwise.
(f) Reflection about the line
(d) does not exist
© 2017 NTK Publishing Limited
= x, enlargement with
centre (0,0) and scale factor 3.
37
Cambridge IGCSE Mathematics Study Guide
(g) Rotation about the origin through 90°.
anti-clockwise, enlargement with centre (0,0) and
scale factor 2.
(d)
y
6
5
4
(h) Rotation about the origin through 90°
anti-clockwise.
2
3
C2 2
6
y
5
4
3
B2
2
1
1 0
A2 1
1
2
3
4
5
6
x
2
6
3
5
4
4
5
3
6
2
1
4
3
2
0
1
1
1
2
3
4
5
6
x
Chapter 8
2
3
Exercise 8.1
4
1
4
2
(c)
3
1 (a)
3 (a)
1
1
(b)
0
1
4 (a)
(−1, −3)
(b)
(3, −1)
(3,1)
(d)
(−3, −1)
(c)
Probability
2 (a) (i)
(b)
2
11
5
12
(ii) 0
(b) I
0 2
2 0
5 (a)
3 (a)
(b)
y
1
2
(b)
1
2
(ii)
1
4
(b)
4
30
(c) 1
6
5
4 (a) 10
4
3
1
6
5
4
3
2
1 0
1
1
2
2
3
5
(c)
38
1
4
0
2
3
4
5
B1
6
x
7
30
1
(c)
3
5 (a)
4
6
1
2
3
(iii)
10
(b) (i)
A1
2
C1
2
0
© 2017 NTK Publishing Limited
Answer Key
(d) Experimental probability will be close to
theoretical probability.
Exercise 8.2
1 (a) certain
(c) impossible
(b) certain
(d) equally likely
2 (a) P(rainy day) = 0.64
Exercise 8.5
1 (a)
4
(b) P(red ball) =
9
(c) P(boy) = 50%
1
2
H
1
2
T
1
2
H
1
2
T
H
3 2, 4, 1, 3
1
2
Exercise 8.3
1 (a) Not getting a 3 when a fair die is rolled.
1
2
(b) There is at least one boy in the class.
T
(c) There are no girls in the room.
(d) Get at least one tail when a fair coin is flipped
twice.
2 (a)
3
5
6
(b) Yes
4
5
(b) (i)
(ii)
3
2 (a)
4 0.52
1 (a)
2
3
(b) 104
1
3 (a)
5
(b) 10
9
20
1
5 (a)
6
1
2
1
2
T
1
2
1
2
T
1
2
1
2
T
1
2
T
H
H
H
3
8
6
11
Red
5
11
Green
7
11
Red
4
11
Green
(b) 80
4
2 (a)
11
4
H
Red
7
12
Exercise 8.4
1
2
5
12
(b) (i)
9
(b)
100
(iii)
Green
28
33
(ii)
15
22
7
22
(iv)
35
66
(c) It is normal to have variation between theoretical
and experimental probability.
© 2017 NTK Publishing Limited
39
Cambridge IGCSE Mathematics Study Guide
3 (a)
1
3
6 (a) p = 16, q = 15, r = 7, s = 10
(b) (i)
(b) (i)
2
3
Hit
2
3
(iii) 10
(c) (i)
Hit
1
3
Miss
(iii)
7 (a)
2
3
1
3
Hit
4
(ii)
9
(iii)
(c) (i)
4 (a)
2
27
(ii) 2
8 (a) (i)
1
3
(iii)
n
(b) (i)
ℰ
M
1
(iii)
S
8
(b) (i)
1
30
(ii)
43
110
83
110
(iv)
4
11
(b)
1
4
(ii)
7
80
(ii)
1
8
9
64
(ii)
1
16
11
16
(iv)
1
8
49
400
12
25
5
8
3
8
9 (a)
7
14
(ii)
3
10
0.6
8
(c)
9
(b) (i)
no
rain
0.14
(iii) 0.584
ℰ
F
12
0.35
Rain
0.65
no
rain
0.54
Rain
0.46
no
rain
Rain
0.4
7
(iii)
15
5 (a)
Miss
8
(iii)
9
8
55
3
4
(c) (i)
Miss
1
3
(ii) 56
56
(ii) 0.464
(iv) 0.724
B
9
15
4
(b) (i)
36
(ii) 12
(iii) 9
(c) (i)
(iii)
40
9
40
(ii)
3
10
3
8
© 2017 NTK Publishing Limited
Answer Key
10
(c) 53.3%
Dice 2
1
Dice 1
11 (a)
(c)
3
4
5
6
1
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
2
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
140 – 150
1
3
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
150 – 160
3
4
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
160 – 170
5
5
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
170 – 180
3
6
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
180 – 190
3
(b)
Height
1
12
(c)
1
9
Exercise 9.2
1
3
(b)
19
36
1
Tally
Frequency
5
36
Chapter 9
Statistics
Exercise 9.1
1 (a) qualitative
(c) qualitative
2 (a) discrete
(b) quantitative
(d) qualitative
(b) continuous
(c) continuous
0
(d) discrete
2 (a)
Number of movies
Tally
Frequency
0
1
1
3
2
3
3
3
4
3
5
2
© 2017 NTK Publishing Limited
5
10
15
20
25
30
Length of telephone call (minutes)
3 (a) discrete
(b)
4 (a) continuous
Frequency density
(b)
2
Height (h mm)
Frequency
100 < h ≤ 105
5
105 < h ≤ 110
7
110 < h ≤ 120
10
120 < h ≤ 135
42
135 < h ≤ 140
20
(b) 84
3 (a) 14
(b) 84
41
Cambridge IGCSE Mathematics Study Guide
4 (a) 12
(b) 16
Exercise 9.3
5 (a)
1 (a) mean = 5.7, median = 5.5, mode = 5.5
Frequency density
90
(b) mean = 42.4, median = 45, mode = 45
75
(c) mean = 18.375, median = 19.5, mode = 20
60
45
2 mean = 27.75, median = 25, mode = 25
30
3 (a) 42
15
0
1
2
3
4
5
6
7
Distance (km)
(b)
Distance (d km)
Frequency
0<d≤1
15
1<d≤2
60
2<d≤3
90
3<d≤4
75
4<d≤6
60
4 90
5 13
6 17.2
7 3, 5, 12, 15, 15
Exercise 9.4
1 mean = 6, mode = 7
6 (a)
Frequency density
2 mean = 23, mode = 24
3 (a) 4 < x ≤ 6
(b) 5.39
4 (a) 3030
0
120
130
140
150
160
170
180
Height (cm)
(b)
(b) 40.3
(b) We do not know the exact income of each staff
member.
(c) 3000 < x ≤ 4000
Height (h cm)
Frequency
120 < h ≤ 130
5
130 < h ≤ 140
10
140 < h ≤ 150
20
150 < h ≤ 160
30
160 < h ≤ 170
5
5 (a) 52 mm
(b) 0.071
(c) 147.9 cm
7 33
42
© 2017 NTK Publishing Limited
Answer Key
(c)
Exercise 9.5
30
(b)
Length (x cm)
Cumulative frequency
x ≤ 0.5
3
x≤1
8
x ≤ 1.5
18
x≤2
33
x ≤ 2.5
45
x≤3
50
25
Cumulative frequency
1 (a) 1.5 < x ≤ 2
20
15
10
5
0
30
35
40
3 (a) 6.8
Time (hour)
Cumulative frequency
t≤2
1
40
t≤4
3
35
t≤6
11
30
t≤8
21
25
t ≤ 10
27
t ≤ 12
30
45
Cumulative frequency
25
(d) 32
50
20
15
(c)
10
0
0.5
1
1.5
2
2.5
3
Length (cm)
(d) 1.8
Cumulative frequency
30
5
25
20
15
10
2 (a) 30 < x ≤ 35
(b)
20
Age
(b)
(c)
15
5
Age
Cumulative frequency
x ≤ 20
1
x ≤ 25
4
x ≤ 30
12
x ≤ 35
24
x ≤ 40
30
© 2017 NTK Publishing Limited
0
2
4
6
8
10
12
Time (hour)
(d) 6.8
4 (a) 43.1
43
Cambridge IGCSE Mathematics Study Guide
(b)
Speed (v miles/
hour)
Cumulative
Frequency
v ≤ 30
3
v ≤ 40
11
v ≤ 50
21
v ≤ 60
28
v ≤ 70
30
Exercise 9.6
1 (a)
x
Moderate positive correlation
(b)
Cumulative frequency
(c)
y
y
25
20
15
x
10
Strong positive correlation
5
0
(c)
20
30 40 50 60
Speed (miles/hour)
y
70
(d) 48 miles/hour
5 (a) 40
(c) 5
6 (a) 15%
(b) 20
x
Zero correlation
(d) 1.7 kg
(b) 10%
(d)
y
(c) 3 hours
7 (a) 95 cm
(c) 92 cm
8 (a) 48 cm
(b) 98 cm
x
(d) 6 cm
(b) 5 mins
Strong negative correlation
(e)
y
(c) 10
9 (a) 100
(c) 18 minutes
(b) 96 minutes
(d) 88
x
Weak negative correlation
44
© 2017 NTK Publishing Limited
Answer Key
(f)
4 (a)
y
50
Number of days (y)
45
x
Strong negative correlation
2 (a) positive
(b) positive
(c) positive
(d) negative
40
35
30
0
50
55
60
65
70
Age (x)
(e) negative
(b) Moderate positive correlation
3 (a)
(c)
32
50
45
Number of days (y)
Temperature (C°)
30
28
26
24
40
35
22
30
20
0
30
35
40
45
50
55
60
65
0
70
50
55
60
(c)
5 (a)
32
30
100
90
28
80
26
70
24
60
Score
Temperature (C°)
70
(d) 41
(b) Positive correlation
22
50
40
20
0
65
Age (x)
Number of cold drinks sold
30
30
35
40
45
50
55
60
Number of cold drinks sold
(d) 43
65
70
20
10
0
50
60
70
80
90 100 110 120 130 140 150
Weight (kg)
(b) No correlation
© 2017 NTK Publishing Limited
45
MARKSCHEME
CAMBRIDGE INTERNATIONAL
MATHEMATICS
Practice Test
Paper 2 (Extended)
Cambridge IGCSE Mathematics Study Guide
1
2
ℰ
A
2
(a)
9
(b)
1
2
(a)
B
let y = − x + c
put ( 3,1) into the equation,
1
1 = −3 + c
c=4
y = −x + 4
1
let y = 0,
−x + 4 = 0
x=4
( 4,0)
10
1
x x −1
−
=1
2
3
3x − 2 ( x − 1) = 6
3x − 2 x + 2 = 6
1
1
1
x=4
(b)
3
2
1
8.44
S − π r = 2π rh
1
S − πr
=h
2π r
1
180 ( n − 2 )
= 150
n
2
2
5
6
8
1
3000 × 1.52 = 4560
2

let x = 0.12

100 x = 12.12
99 x = 12
12 4
x=
=
99 33
7
(a)
2 ( x 2 − 3x + 2 )
= 2 ( x − 2 )( x − 1)
(b)
x =9−4
x2 = 5
x= 5
x = 2.2 (correct to 2 sig. fig)
y = 2x + 3
y
6
2y x 2 = 0
4
3
2
R
1
2
1 0
1
1
2
3
4
2
1
1
1
1
x=2
13
1
1
5
6
7
x
5
3x − 6 y = 2 ..........(1)
−2 x
+ 5 y = 13 .......( 2 )
(1) × 3:
6 x − 12 y = −4 ......( 3)
( 2 ) × 3:
+ 15 y = 39 ......( 4 )
( 4 ) × ( 3) :
4x − 1
= ( 2 x − 1)( 2 x + 1)
1
7
−6 x
= ( 2 x ) − 12
1
5
2
2
48
2
12
180 n − 360 = 150 n
30 n = 360
n = 12
1
x 2 + 22 = 32
1
2
4
11
3 y = 35
35
y=
3
1
1
© 2017 NTK Publishing Limited
Practice Test Paper 2 (Extended) Markscheme
35
= 2
3
3x = 2 + 70
68
x=
3
∴ 3x 6
14
(a)
(b)
15
(a)
(b)
3
4
18
1
17
V
120
405
V=
8
1
V = π r2h
3
405 1 ( )2
= π 3 h
8
3
3( 405)
h=
= 5.37 cm
8π ( 32 )
angle BAD = angle OAD + angle OAB
= 20° + 30°
= 50°
angle BCD + angle BAD = 180°
(Supplementary angle of cyclic quad)
angle BCD = 130°
(b)
2
Total distance = area under the graph
1
10 × 2
5× 2
+ 15 × 2 +
2
2
= 10 + 30 + 5
= 45 km
1
The train is travelling at constant
speed 2 km/min for 15 minutes.
1
15 × 60 = 900 seconds
1
From the graph, the speed of the train
at 8 minutes is 1.6 km/min.
1
1
= 1.6 × 60 km/h
= 96 km/h
1
1
 3
OP = a
4
1
(c)
1
20
(a)
(b)
1
(c)
2
 1
position vector of Q = OQ = c
3
  
BQ = BC + CQ
2
= −a − c
3
2
1
1
21
AC 2 = 7
1
AC = 7
1
k
1
s
k
0 1
1 0
2
=
1
1
3=
(a)
1
AC 2 = 22 + 32 − 2 ( 2 )( 3) cos60°
p=
19
3
=
Reflection about the line y = x.
(b)
If BC = CD,
angle OCB = angle OCD
= angle ODC
130°
= 65°
=
2
16
(a)
A
B
C
5
4
k=6
∴ p=
6
s
1
when s = 9,
p=
6
9
=
6
3
=2
© 2017 NTK Publishing Limited
1
49
MARKSCHEME
CAMBRIDGE INTERNATIONAL
MATHEMATICS
Practice Test
Paper 4 (Extended)
Cambridge IGCSE Mathematics Study Guide
1
(a)
(i) sand 375 kg
stones 125 kg
2
3
0.7
1
(ii) 50 Litres
2
(iii) $100
2
3
0.8
0.2
0.4
0.3
0.6
(b)
10.7 cm
3
(e)
0.14
1
(a)
38.5 m
1
(f)
0.68
2
(b)
26.3 m
3
(g)
(c)
18.6 m
3
(d)
18 packets
4
(a)
a
b− a
2
(b)
4 + 3x2
2
(i) n > −5
2
(c)
9
4
3
(ii)
3
(b)
−4, −3, −2, −1
x=
(c)
9
z2
or x =
(d)
(a)
y
3
z
5
(e)
6
2
y
ℰ
P
0
2
0
16
1
(i)
4
2
(ii) 5
2
1
4− x
3
4
3
2
3
12, 17, 35
1
2
(ii) 36
1
4
(iii) 48
2
7
3
1
(c)
56.8
3
(a)
−2.5, 2.5, 3.33
2
y
(b)
6
5
3
4
S
(b)
(i) 6
3
(i) 56
4
T
1
(a)
(b)
−0.732, 2.73
2
(a)
(d)
3
(i)
(ii)
4
(d)
1
3
2
1
(ii) 2
1
(iii) 8
1
1
2
1
0 0.5
1
1
2
3
x
4
2
3
4
(c)
52
2
[ or 0.4]
5
2
5
6
4
© 2017 NTK Publishing Limited
Practice Test Paper 4 (Extended) Markscheme
(c)
(d)
x = 0.5, 2 (must be exact)
4
y
(i)
2
6
9
(a)
(b)
5
14
2
1
6
2
2
4
3
2
or
1
2
1
0 0.5
1
1
2
x
4
3
2
3
2
1
4
1
2
1
3
1
3
2
1
6
1
3
(c)
det C = 0 [accept matrix is singular]
1
(d)
m=4
2
n=3
2
−17, 17
4
AB, AC, AD, BC, BD, CD
2
4
5
6
(ii) slope = 3
8
(a)
accept 4 to 2
y
(i)
6
(ii)
5
R′
4
3
(e)
2
10 (a)
2
(b)
BD, BE, CD, CE, DE
2
(c)
1
1
(d)
(i) 1
3
(ii) 45
1
[ or 0.0152]
(iii)
66
1
3
2
1
5
4
3
2
S′
1 0
1
1
2
3
4
5
x
2
3
4
4
5
6
(b)
1
Enlargement, scale factor = − ,
2
centre ( 0,0 )
3
(c)
2c − 2a − 2b
2
(d)
a+b
2
(e)
(f)
3
parallelogram
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