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Cambridge IGCSECambridge IGCSECambridge IGCSECambridgeCambridgeCambridge IGCSE IGCSE IGCSECambridge IGCSE
MathematicsMathematicsCambridgeCambridgeCambridgeCambridgeCambridgeCambridgeCambridgeCambridgeCambridgeCambridge IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE
MathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsStudy GuideStudy GuideStudy GuideStudy GuideStudy GuideStudy GuideStudy GuideStudy GuideStudy Guide
CambridgeInternationalExaminations
Answer key and markscheme
CONTENTSAnswer Key
Chapter 1 Numbers 1
Chapter 2 Algebra 9
Chapter 3 Geometry 21
Chapter 4 Mensuration 27
Chapter 5 Coordinate Geometry 28
Chapter 6 Trigonometry and Pythagorasʼ Theorem 30
Chapter 7 Vectors and Matrices 33
Chapter 8 Probability 38
Chapter 9 Statistics 41
Practice test markscheme
Practice Test (Paper 2) 47
Practice Test (Paper 4) 51
1
Answer Key
© 2017 NTK Publishing Limited
Chapter 1 NumbersExercise 1.1
1 (a) 7, −2, 111, 25, 0
(b) 7, −2, 34
, 111, 2.4, 25, 0, 512
(c) π , 3
(d) 25, 0
2 (a) 71, 81
(b) − 4, −3
(c) 6.2, − 4, 89
, −3, 71, 81, 3.1
(d) 81
3 (a) 3, 7, 13 (b) 2, 17
(c) 23, 29, 31 (d) 5, 37, 43
(e) 41, 43, 47 (f) 53, 61, 79
4 (a) 1, 3, 25 (b) 5
(c) −2, 2.1, 1, 3, 5, 4.62, 43
, 25
(d) 3, 25
5 7, 14, 56, 63, 70
6 (a) 1, 2, 3, 6 (b) 1, 2, 3, 4, 6, 12
(c) 1, 5, 25 (d) 1, 13
(e) 1, 2, 3, 5, 6, 10, 15, 30
(f) 1, 2, 4, 8, 16, 32, 64
7 (a) 4 (b) 8
8 (a) 24 (b) 60
9 (a) 32 × 5 (b) 32 × 23
(c) 7 × 3 × 22 (d) 23 × 3 × 5
(e) 2 × 7 × 9 (f) 5 × 33
(g) 2 × 3 × 5 × 7 (h) 32 × 5 × 7
10 (a) H.C.F = 6 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
Exercise 1.2A
1 A 11,13,17,19{ }=
2 (a) P
Q
4, 8,12,16
3, 6, 9,12,15,18
{ }{ }
=
= (b)
P
Q
4, 8,12,16
3, 6, 9,12,15,18
{ }{ }
=
=
3 B 1, 3, 9,13{ }=
4 (a) All elements in A are in B
(b) Yes, it is because 26 is not in B
5 (a) 3, 7, 9{ } (b) 2, 3, 5, 7, 8, 9,11{ }
6 (a) 4 (b) 4
(c) 0 (d) 2 , 2, 4 , 42 2{ } { }{ }
7 (a) 9 is a multiple of 3
(b) 12{ } (c) 11
8 (a) 2, 3, 5, 7{ } (b) 4
(c) True, since 6.5 is not a member of set A.
9 (a) A 7, 6, 5, 4, 2, 1, 2, 3, 4, 6{ }′ = − − − − − −
(b) 3, 5{ }−
(c) 8, 5, 3, 0,1, 2, 5, 7, 8{ }− − −
2
Cambridge IGCSE Mathematics Study Guide
© 2017 NTK Publishing Limited
10 (a) 8, 7, 6, 5, 2, 1, 0{ }− − − − − −
(b) 4, 2{ }−
(c) 8, 7, 6, 5, 3, 2, 1, 0,1, 3{ }− − − − − − −
Exercise 1.2B
1 (a) 5,10,15{ } (b) φ
(c) 5, 7, 8, 9,10,15{ }
2 (a) 1, 2,13,15,17{ } (b) 5, 8,10,13,15,17, 30, 33{ } (c) 30, 33{ }
3 (a) 5, 9{ } (b) 22, 24{ } (c) φ
4 (a) 4, 5, 6{ } (b) 4, 5{ } (c) 3, 8, 20, 22{ }
5 (a) ℰ
A
ab
ef
d
c
B
(b) c{ }
6 (a) ℰ
A
B C
3
248
9
10 5
1 7
(b) A B CA B C 10
φ
( ) { }∩ ∩ =∪ ∩ =
7 ℰ
A
B C
2
4.2
933
15
5
107
8 (a) ℰ
A B
(b) A Bℰ
(c) A Bℰ
(d) A
B C
ℰ
9 (a) 7 (b) 8
10 (a) 0 (b) 2
(c) 5 (d) 7
(e) 18 (f) 1
11 (a) 1 (b) 0
(c) 13
3
Answer Key
© 2017 NTK Publishing Limited
12 A
1 2 3
5
B
ℰ
Exercise 1.2C
1 14
2 (a) ℰ
C
50 50 100
T
(b) 50
3 (a) 28 (b) 5
(c) 5
4 (a) ℰ
M
5 5
5
10
S
(b) 15 (c) 5
(d) 5
5 (a) ℰ
G
5 5
5
15
S
(b) 5 (c) 15
6 (a) ℰ
B
7 8
20
15
F
(b) 8 (c) 20
(d) 7
7 (a)
x
Sandwich
Chocolate Coffee
20
15
8 − x
7 − x
9
10
ℰ
(b) 4
8 (a) 15 (b) 10
Exercise 1.3
1 (a) 121 (b) 6.25
(c) 49 (d) 3
10
(e) 2710
(f) 16
2 (a) 125 (b) −64
(c) −1 (d) 0.2
(e) 1.5 (f) 25
3 13
4 −4.913
5 −62.9
6 4 9 2 3 5
4 9 13 5
+ = + =
+ = ≠
7 41 or −41
Exercise 1.4
1 (a) 17°C (b) 30°C
(c) Moscow
2 8084
3 (a) −4°C (b) 0.5°C
4
Cambridge IGCSE Mathematics Study Guide
© 2017 NTK Publishing Limited
Exercise 1.5
1 (a) 2330
(b) 2328
(c) 528
(d) 59
(e) 2924
(f) 8135
(g) 192
(h) 12152
2 (a) 113
(b) 34
(c) 12
(d) 3
3 (a) 2.4 (b) 2400 mL
4 (a) 0.34, 1.5, 2.4, 3.7, 6.45
(b) 1.4, 7.85, 7.94, 9.3, 10.7
5 (a) Terminating decimals: 0.35, 0.6, 4.7
Recurring decimals: 2.6, 2.87
(b) Terminating decimals: 1.3, 3.56, 2.43, 4.7
Recurring decimals: 3.8
6 (a) 45
(b) 2750
(c) 3325
(d) 7625
(e) 23
(f) 4599
(g) 2 753999
(h) 2 118
7 (a) 1
250 (b)
1425
(c) 2310
(d) 131
5000
8 Fractions Decimals Percentages
(a)25
0.4 40%
(b)123100
1.23 123%
(c)625
0.24 24%
(d)29
0.2 22.2%
(e)51750
10.34 1034%
(f)115
2.2 220%
Exercise 1.6
1 (a) 7 > 2 (b) −1 < 0
(c) 0.23 < 0.32 (d) −0.1 > −1
(e) >312
3 (f) <9 16
2 (a) −1, 0
(b) −8, −7, −6, −5, −4, −3
(c) 3, 2, 1
3 (a) < < < <0.2 0.2 0.2 0.2 10.2
3 2 3
(b) < <0.2% 0.2 0.2
(c) < < < <0.001 1101
1100
0.011 1.2%
(d) < < <14
% 0.025 14
23
Exercise 1.7
1 (a) 27 (b) 32
(c) 360 (d) 51597
2 (a) 24 × 34 × 7 (b) 52 × 112
(c) 23 × 33 × 132 (d) 23 × 7 × 11 × 13
5
Answer Key
© 2017 NTK Publishing Limited
3 (a) 254
(b) 19
(c) 13 (d) 4
(e) 1
32 (f) −8
(g) 1
16 (h)
58
4 (a) 4.98 × 10−9 (b) 9.999 × 1010
5 (a) 124 000 000 (b) 0.000 000 62
6 (a) 1.512 (b) 1.6 × 104
(c) 4.2 × 108 (d) 9.72 × 102
(e) 3.33 × 1010 (f) 2.30 × 106
(g) 2.32 × 1010 (h) 3.06 × 10−1
7 7−23, by 40 353 607 times.
8 500 s
9 ×
× ×− − −122 10
, 4.2 10 , 2 10 , 234 2
23
Exercise 1.8
1 (a) −1 (b) −6
(c) 0.6 (d) −0.2
(e) −16 (f) 21
(g) 64 (h) −43.47
(i) −8 (j) 12
(k) 4 (l) 16
2 (a) −1 (b) 25−
(c) 20 (d) 33
(e) −6 (f) 3
3 (a) 32
(b) 56
(c)
3258
125
= (d)
10332
209
=
−
−
Exercise 1.9
1 (a) 232.12 (b) 0.01
(c) 132.13 (d) 0.54
2 (a) 3 720 000 (b) 0
(c) 6000 (d) 93 000
3 (a) 70 000 (b) 0.2
(c) 10 (d) 0.7
4 (a) 1030 (b) 1 500 000
(c) 737 000 000 (d) 93.8
5 (a) 8009.21 (b) 8010
(c) 8000 (d) 8009.2
(e) 8009.206
6 (a) 7 8
4×
(b) 14
7 14.02
8 (a) 4 26
43
×= (b) 2 2
28
2( )+=
(c) 48
13
=
Exercise 1.10
1 upper bound: 9.8 0.12
9.85+ = ; lower bound
9.8 0.12
9.75− =
6
Cambridge IGCSE Mathematics Study Guide
© 2017 NTK Publishing Limited
2 (a) lower bound: 5 12
8 12
= 33.75
(b) upper bound: 8+ 1
2
5 12
= 179
3 number of small tanks: 900+ 10
2
3.5 0.12
= 262 2269
4 (a) upper bound: 4 6.81+ 0.012
= 27.26
(b) lower bound: 4 6.81 0.012
= 27.22
5 (a) lowest possible: 26 12
25.5 kg− =
(b) upper bound: 5 26+ 12
= 132.5 kg
6 (a) upper bound: 65+ 12
58 12
= 8 kg
(b) lower bound: 65 12
58+ 12
= 6 kg
7 (a) lower bound:
2 3.14 0.012
3.56 0.012
9.81+ 0.012
= 3.77
(b) upper bound:
2 3.14+ 0.012
3.56+ 0.012
9.81 0.012
= 3.79
8 upper bound: 6 7.4+ 0.12
2
= 333.015
lower bound: 6 7.4 0.12
2
= 324.135
Warm up 1.11
1 (a) 1:3 (b) 3:5
(c) 8:11 (d) 1:2:8
(e) 15:48:20
2 (a) a:b:c = 1:2:6
(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 w14
3 (b) 7294
(c) 36 m/s
2 (a) = × −d L3 10 5 (b) 125 000 mm
3 35 km
4 6
5 78 kg
6 (a) 743.04 (b) 639.6
7 (a) =tn
80 (b) 16
(c) decrease by 1 hour
8 (a) =×xf
3.6 105
(b) f = 1440
(c) 240
9 40 minutes
10 1 day
11 7 days
12 4
7
Answer Key
© 2017 NTK Publishing Limited
13 (a) (i) C = 4500N (ii) =tN
192
(b) (i) C = 4500 192t
C = 864000t
(ii) Inverse proportion
(c) (i) Decrease, it is because the salary budget is inversely proportional to the time required to finish the project.
(ii) =
=t
t
108000 864000
8 4 months
14 (a) 8:12 (b) 36:9
(c) 115:92 (d) 144:192
(e) 4104:912
15 (a) 8:12 (b) 36:9
(c) 115:92 (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) 3.6 km/h
Warm up 1.12
1 (a) $2 (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
3 (a) 11.76 (b) 173.4
(c) 595.92 (d) 56.55
(e) 62.328 (f) 8.75
4 (a) 10% increase (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%
6 £96
7 11%
8 5.6%
9 £403.2
10 1% decrease
11 150,000
12 20
13 £1416
14 (a) £2940 (b) £1230
15 1 783 000 m3
8
Cambridge IGCSE Mathematics Study Guide
© 2017 NTK Publishing Limited
Exercise 1.13
1 (a) −5.12 (b) 3.50
(c) 102400 (d) 0.26
(e) 0.80 (f) −11.57
2 (a) 0.75 (b) 1
(c) 0.231 (d) −5.20
(e) 0.125 (f) 1.20
3 ( ) ( )° ° ° °cos45 , cos45 , cos45 , cos453 2 3
4 (a) > (b) >
(c) < (d) <
Exercise 1.15
1 (a) €6.12 (b) €10.51
(c) €765.31 (d) €0.13
2 (a) $9.8 (b) $98
(c) 3.49 (d) $58 800
3 4.7% increase
4 £2415
5 (a) 17 800 (b) £600
(c) £588
6 (a) €40 (b) £34.4
(c) £3405.6
Exercise 1.16
1 6%
2 3200
3 £13 665.27
4 (a) £1416 (b) £1429.2
5 P = £4754.8
6 £29166.8
7 6.37%
Exercise 1.17
1 (a) 20 (b) 30, 418.5
2 (a) 1000
(b) 0 1 2 31000 874.1 764 667.8
3 (a) 13 (b) 3.25 grams
(c) 2 years
4 (a) ( )=B 100 2t (b) 6000
5 (a) 3 114 918 (b) 13 788 382
(c) Population of species A keeps decreasing.
(d) 4:1
Chapter 2 AlgebraExercise 2.1
1 (a) 2p + 2q + 2r (b) −2x + 10y
(c) ab (d) −
aq9
2 (a) y = c + mx (b) =+
r pa b
2
(c) = +R x y1 2 1
2
3 (a) ( )= +d u v t12
(b) d = xt
4 (a) S = 28 (b) V = 810
9
Answer Key
© 2017 NTK Publishing Limited
(c) =r 14
(d) =r 83
5 (a) =−y T x23
(b) + =
=−
fu fv uv
u fvv f
(c) =−+
y x RxR
2 53 1
(d) =−+
x yy
2 21
2
2
(e) =+s A t25
6 (a) 1. =−−
x q pp q
2 2. x = −3
(b) 1. =−
s bkrb a
2. s = 2
(c) 1. =−
b R aR1
2 2
2 2. b = 9
(d) 1. =−
−x y b
y a34
2
2 2. = −x 314
(e) 1. =+−
x y PyP
5 42 8
2. =−
x 174
Exercise 2.2A
1 (a) 6xy − 6y (b) 3x − 2xy + 5
2 (a) 9p2 + 2q (b) 3p2q + p2
3 (a) 0 (b) 2abc
4 (a) −5s2t − 4st (b) 2s2 − 6s2t
5 (a) 3xyz2 (b) −5xyz + 7xy2z
Exercise 2.2B
1 (a) a2 + 2a (b) 4b − ab
(b) 2b + 3ac (c) 2a2 − ab + 4b
(e) x2 − x − 6 (f) x2 − 3x − 10
(g) 6x2 − 19x − 11 (h) −5x2 − 18x + 8
(i) 20y2 + 3y − 2 (j) −y − 3y + 18
(k) x3 + x2 − 3x + 9 (l) −3x3 + 5x2 + 4
2 ( )− −x x12
12 7 102
Exercise 2.2C
1 (a) x2 − 4y2 (b) 9 − 25x2y2
(c) 4x4 − 14
(d) x2 − x9
2
(e) 5x2 + 10xy + y2 (f) 49 − 14xy + x2y2
(g) 4x2 + 4x + x1
2 (h) − +
y xy x1 2 1
2 2
(i) x4 + 2x3 + 3x2 + 2x + 1
(j) x6 − 2x4 + 6x3 + x2 − 6x + 9
(k) x6 − 4x4 + 6x2 + x1
2 − 4
(l) x2 + y2 + z2 + 2xy + 2xz + 2yz
2 (a) 5x2 + 20x − 146 (b) 4xy
(c) 2y3 − 3y2 + 1 (d) 6z4 + z3 − 16z2 − 7z + 4
Exercise 2.2D
1 (a) 2by2 (b) ax
(c) ( )−a 2 (d) ay
2 (a) ( )+ax x a2 (b) ( )+by y b4 22
(c) ( )+x x7 7 3 (d) ( )+ +x ax a x2 42 2
3 (a) ( )( )− −p q r2 1 (b) ( )( )− +a x y5
(c) ( )( )− +b x a c3 1 (d) ( )( )− − −x x a2 2
(e) ( ) ( )− − −x y x y5 22
(f) ( )−x b c2 22
4 (a) ( )( )+ +x y1 2 (b) ( )( )+ +r t s5
(c) ( )( )+ +p q m n (d) ( )( )− −a y z7
(e) ( )( )− −a b a2 (f) ( )( )− +y y1 5
(g) ( )( )− −y x y1 (f) ( )( )( )+ − +x x x2 1 1 1
10
Cambridge IGCSE Mathematics Study Guide
© 2017 NTK Publishing Limited
(i) ( )( )+ − −y y h k12 (j) ( ) ( )− −x a b12
Exercise 2.2E
1 (a) ( )( )− +x x2 2 (b) ( )( )− +y y4 3 4 3
(c) ( )( )− +z z5 1 2 1 2 (d) ( )( )− +x y y7 5 5
(e) ( )( )− − − +a b a b2 6 2 6
(f) ( )( )− +p p33 3 1
2 (a) ( )+x 32 (b) ( )−x 7
2
(c) ( )−x2 52 (d) ( )−− y 11
2
(e) ( )−a b3 22 (f) ( )+− a b5 3 2
2
3 (a) ( )( )+ +− p q p q4 3 7 11 7
(b) ( )−x yz42
(c) ( )( )− − −p p q2 1 2 2
(d) ( )( )+ − + +a b a b2 6 2 6
(e) ( )−s t s20 5 3
(f) ( )( )− − − +a b c a b c2 7 6 2 7 6
(g) ( )( )+ − + + + +m n x m n x3 1 3 1
(h) ( )( )( )− + +x x x3 1 3 1 9 12
Exercise 2.2F
1 (a) ( )( )+ +x x6 1 (b) ( )( )+ −x x6 1
(c) ( )( )+ +x x2 3 (d) ( )( )− +x x3 2
(e) ( )( )− +x x8 7 (f) ( )( )− +x x5 2
(g) ( )+− x 92 (h) ( )( )+ −x x7 4
2 (a) ( )( )− −− p p2 5 (b) ( )( )+ −− a a6 3
(c) ( )( )− −m m2 3 1 (d) ( )( )− −y y4 4 7
(e) ( )( )+ −n n6 5 3 (f) ( )( )+ −4 6 8 6 1
(g) ( )−− q2 72 (h) ( )( )+ −− s s5 7 3
(i) ( )( )− +x x7 3 2 (j) ( )( )− −y y11 11 1
(k) ( )( )− +− z z3 10 9 (l) ( )( )− −− t t2 11 3
Exercise 2.2G
1 (a) ( )( )+ +x x2 1 3 (b) ( )( )+ −x x2 1 3
(c) ( )( )− +y y3 2 3 (d) ( )( )+ −y y3 1 3 4
(e) ( )( )− −a a5 2 1 1 (f) ( )( )− +a a3 4 1 5
(g) ( )( )+ +s s2 6 5 1 (h) ( )( )+ −s s3 4 1 3 2
(i) ( )( )− +− m m4 2 1 5 2 (j) ( )( )− −− m m3 2 1 1
(k) ( )( )− −− p p2 1 2 3 (l) ( )( )+ −− p p2 3 7 2
2 (a) ( )( )+ −x y x y11 12 (b) ( )( )+ −m n m n4 2 3
(c) ( )( )+ −p q p q2 5 5 (d) ( )( )− −s t s t8 3 3 8
3 (a) ( )( )+ −x x12 7 2 3 (b) ( )( )− −x x12 29 2 9
4 (a) ( )( )− −x y x y14
4
(b) ( )( )( )( )+ − − +x y x y x y x y14
2 2
Exercise 2.2H
1 ( )+a11 6 2
2 ( )−xy y x
3 ( )+a a b6
4 ( )−xy y x
5 ( )( )+ −n m1 1
6 ( )( )− −m x m 1
7 ( )( )+ +s s2 6 5 1
8 ( )( )− −p qx p q
9 ( )( )− +s t t1 1
10 ( )( )+ −− x y x y6 2 2
11 ( )( )− −y y12 13
11
Answer Key
© 2017 NTK Publishing Limited
12 ( )+n2 32
13 ( )−a b5 112
14 ( )( )− +c c3 5 7
15 ( )+x x2 22
16 ( )( )+ +a x x5 6 4
17 ( )( )− + − +m n m n6 6
18 ( )( )( )− + −q p p2 2 1 2 1
19 ( )( )+ −x x2 8
20 ( )+x3 22
21 ( )( )−p q q4
22 ( )( )( )( )− − + +p p p p1 4 1 1
23 ( )−x116
2 14
24 ( )( )( )− + +a a a116
2 1 2 1 4 12
Exercise 2.3A
1 (a) 6 (b) 67
(c) 125
(d) 1
2 (a) −103
(b) 32
(c) 0 (d) 32
3 (a) a3
(b) −a
14
(c) c13
(d) bc5
(e) −xy4
(f) −
y5
4
(g) zy8
(h) − zx3 2
2
4 (a) 2a + 1 (b) 3a + b
(c) zy8
(d) − zx3 2
2
(e) −xy
2 (f) xz
(g) −yy
2 55
(h) − xx
23
(i) −+
xx
22 1
(j) +−
xx
2 35
(k) −ac
(l) − x1
2
(m) +x1
1 (n)
−x
x 3
(o) +x
x 2 (p) +x 1
2
Exercise 2.3B
1 (a) a4
5 (b) −
a2
2 (a) bcad6
(b) −14
3 (a) −xx1
3 (b) − x1
32
4 (a) +xx
1 (b)
yx7
5 (a) ( )+−
xb y2 1
(b) −bx2
Exercise 2.3C
1 (a) LCM: 12x2 HCF: 2x
(b) LCM: 36x2y2 HCF: 3xy
(c) LCM: ( )( )+ −x x2 2 2 HCF: 1
(d) LCM: ( )−x x2 12 HCF: ( )−x 1
(e) LCM: ( )( )+ −a a b a b60 2
HCF: 5
(f) LCM: ( )− m n18 13
HCF: ( )− m3 1
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Cambridge IGCSE Mathematics Study Guide
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(g) LCM: ( )( )( )+ − −x x x1 1 2 32
HCF: ( )−x2 3
2 (a) a
2110
(b) + aab
4 63
(c) −x
x2
1 (d)
( )−−
y xx y
22
(e) ( )
++
mm m7 4
2 (f)
( )−n9
2 3
(g) +−
xx
4 24 12
(h) ( )( )−
−
y y xx y x
2
Exercise 2.3D
1 (a) ( )−a
b b a9 4
(b) +
− aba b4
(c) −−
xx1
3 (d) −
+xx
2 32 3
(e) −x
x3
3 (f) 1 − x
(g) +−
xx
12 1
(h) −+
xx
33
2 (a) ( )( )( )− − +x x x3 2 2 1
(b) ( )( )( )−
− − +x
x x x5 4
3 2 2 1
3 3x + 5
4 (a) a3 − b3 (b) ( )( )− + +x y x xy y252 2 2
Warm up 2.4
1 (a) 19
(b) 4
(c) 13
(d) 3
(e) 1 (f) 1
(g) 0 (h) 1
(i) 125 (j) 4
(k) 18
(l) 19
(m) 1625
(n) 12
Exercise 2.4
1 (a) 59 (b) 78
(c) 37a5 (d) a9b7
(e) a (f) a b
13 6
(g) a14 (h) 8a2b11
(j) −25a3b4 (j) 1
2 (a) 133 (b)
162
(c) 2 (d) xy
2
3
(e) x4 (f) x y
1
213
(g) x92 (h) x
y2
3 (a) 3 (b) −2
(c) −2 (d) 7
(e) 3
4 (a) 116
(b) 5
(c) 2 (d) 34
(e) 521
(f) 32
(g) −4
Exercise 2.5A
1 (a) 6 (b) 4
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Answer Key
© 2017 NTK Publishing Limited
(c) 4
2 (a) 15 (b) 1
(c) 1
3 (a) 2 (b) 7
(c) 2 (d) 2
4 (a) 6 (b) 74
(c) −76
5 (a) 1
18 (b)
6988
(c) 29
6 (a) −65
(b) 623
(c) 39
7 (a) 0 (b) 8
(c) 367
Exercise 2.5B
1 5
2 19
3 14
4 13
5 (a) 7 (b) 12, 16, 20
(b) 96
6 7
7 7
8 90
9 (a) 36 (b) 204
10 (a) 12 (b) 98
Exercise 2.5C
1 (a) x = 0 or x = 3 (b) x = 4 or x = −52
(c) x = 53 or x = −
94
(d) x = 3 or x = 4
2 (a) x = 0 or x = 4 (b) x = 0 or x = 5
(c) x = 0 or x = 75
(d) x = 0 or x = −4
3 (a) x = 0 or x = 3 (b) x = −52
or x = 5
(c) x = 72
or x = −5 (d) x = 32
or x = 6
4 (a) x = −2 or x = −1 (b) x = 3 or x = −4
(c) x = 8 or x = −3 (d) x = 12 or x = −3
5 (a) x = 13 or x =
32 (b) x = −5 or x =
74
(c) x = −65
or x = 23 (d) x =
23 or x =
37
6 (a) x = 3 or x = 5 (b) x = −1 or x = 52
(c) x = −56
or x = 32 (d) x = −
45
or x = 74
Exercise 2.5D
1 (a) x = 0 or x = 3 (b) x = 13 or x = −13
(c) x = 2 2 or x = −2 2
2 (a) x = 7 or x = −7 (b) x = 0 or x = 329
(c) x = −197
or x = 197
3 (a) x = −4 or x = −12
(b) x = +−2 3 74
or x = +2 3 74
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(c) x = −5
(d) x = −−87
3 27
or x = −3 2
787
4 (a) ( )− +x 1 32
(b) x 52
214
(c) 2 x + 32
2
+ 9.5 (d) 2 x 32
2135
4
5 (a) x = 1 or x = 7
(b) x = −−52
132
or x = −132
52
(c) ( )= +x 2 1 3 or ( )= −x 2 3 1
(d) = −x 3 19 or = +x 3 19
(e) x = −−2 332
or x = −332
2
(f) x = −32
or x = 45
Exercise 2.5E
1 (a) x = 12
or x = 3
(b) x = −1 13
or x = +1 13
(c) x = +9 216
or x = −9 216
(d) = −x 43
(e) = −x 52
(f) x = 6
(g) = −x 4 433
or = +x 4 433
(h) = −x 512
14512
or = +x 512
14512
2 (a) = −−x 32
132
or = −x 132
32
(b) = −x 3 7 or = +x 3 7
(c) = −−x 112
79312
or = −x 79312
112
(d) = +x 2 23 or = −x 2 23
Warm up 2.5F
1 13, 15
2 6, 12
Exercise 2.5F
1 26
2 18
3 =x 23
or x = 4
4 length = 15 width = 12
5 6
6 12
7 30
8 12 km/h
9 (a) 60 km/h (b) 9:30 p.m.
Exercise 2.5G
1 (a) x = 3, y = 3 (b) x = 2, y = 3
(c) x = −3, y = 4 (d) x = 53, y = −
52
2 (a) x = 2, y = 3 (b) x = 5, y = −2
(c) x = 3, y = −4 (d) x = 2, y = −5
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Answer Key
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3 (a) x = 23
, y = −32
(b) x = 15, y = 12
(c) x = 6, y = −5 (d) x = 34
, y = 27
4 Adrian: £12 Billy: £20
5 Chocolate: £0.5 Crisps: £1.6
6 Solution A: 12.5 litres Solution B: 1.6 litres
Exercise 2.6A
1 (a) x < −2 (b) 4x < −2
(c) x ≤ 34
(d) 4x < −52
(e) x < 32
(f) 4x ≥ 2
2 (a) x < 26 (b) x < 143
(c) x ≥ 97
(d) x ≤ 8
(e) x < 2569
(f) x ≥ 326
3 (a) x < −3
28 (b) x > 101
46
Exercise 2.6B
1 (a)
x
y
4
4
(b)
x
y
4
8
(c)
x
y
3
4
(d)
x
y
3
5
2 (a)
x
y
3
3
6
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(b)
x
y
7
4
(c)
x
y
0
1
3
42
2
1
(d)
x
y
2
4 93
6
2
43
6
3 (a)
x
y
4
5
4
(b)
x
y
4
5
4 5
55
4 54
(c)
x
y
12
2 6
3
(d)
5
8
14
x
y
4 (a) (i) 30x + 60y ≥ 1800 x + y ≤ 50
(ii) x ≥ 0 y ≥ 0
(b)
x
y
50
30
50 60
(c) 40
17
Answer Key
© 2017 NTK Publishing Limited
5 5
6 90
7 (a) 10
(b) 6p stamp: 28 8p stamp: 4
Exercise 2.7
1 (a) T1 = 1, T2 = 3, T3 = −7
(b) T1 = 1, T2 = 10, T3 = 37
(c) T1 = 1, T2 = 7, T3 = 79
2 (a) 5, 17, 29 (b) 2, 1, 23
(c) −3, 0, 5
3 (a) 4n − 3 (b) −4 + 34
(c) −2n−2x
4 (a) Tn = 6n − 17 (b) T20 = 103
5 ( )( )+ − +
= −
n n n
n
1 2
6 (a) 1 5 10 10 5 1
(b) n + 1
(c) 2n
7 (a) Pattern 4: 20; Pattern 5: 29; Pattern 6: 40
(b) n2 + 4
8 (a) 4n−1 (b) No
Exercise 2.8
1 932
2 (a) ( )= +y z14
13 (b) ( ) −y4 1
13
3 =p 8 5
4 = −n 12
5 a 1 2 8
b12
14
116
Exercise 2.9A
1 (a) (i) 2.5 m/s (ii) 3.33 m/s
(iii) 5 m/s
(b) (i) 50 m (ii) 200 m
(c) 2.5 m/s
2 (a) 70 km (b) 100 m/h
(c) 33 minutes (d) No
(e) 75 km/h
3 (a) 2.25 ms−2 (b) −1.2 ms−2
(c) 423 m (d) 12.1 ms−1
4 (a) (i) 0.6 ms−2 (ii) 0.4 ms−2
(b) 512 m
(c) 992 m
(d) 12.4 ms−1
(c) 792 m
5 (a) (i) 0.48 ms−2
(ii) Yes, the gradients of the two lines are equal.
(b) (i) 91.5 seconds
(ii) 588 m
(c) (i) 83.75 seconds
(ii) 495 m
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Exercise 2.9B
1 (a)
5 10 15 20 250
2
4
6
8
10
L (c
m)
W (g)
(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.
2 (a) (i) 420 HKD (ii) 20 GBP
(b) (i) 12, it is the exchange rate of GBP to HKD.
(ii) 200 m
(c) 0
3 (a)
201020 30 40 500
20
20
40
60
80
100
10
°F
°C
(b) (i) 1.8 (ii) 32 m
(c) (i) m = 1.8, c = 32
(ii) 70
4 (a)
10 20 30 40 500
10
20
30
40
50
60
Temperature (°C)
Time (minutes)
(b) (i) 54 °C (ii) −0.6 °C / minute
(c) (i) T = −0.6t + 54 (ii) 45 minutes
5 (a) F = 4
M = 0.0004
K = 0.0006
t0 = 500
(b) (i) CA = 2 + 0.004t
(ii) CB = 0.006t
(c) t > 1000
6 (a) (i) 84 USD (ii) 60 NZD
(iii) 70 NZD
(b) The exchange rate of EUR or NZD to USD.
(c) 1.52
7 (a) Time t 5 10 15 20 25t2 25 100 225 400 625
Height H above the
ground (m)4375 4000 3375 2500 1375
(b) H = −5t2 + 4500
(c) (i) 4500 m (ii) t = 30 seconds
Exercise 2.10
1 (a) x 0.1 0.5 1 1.5 2 0.5 3y −29.9 −5.5 −2 −0.5 0.5 1.3 2
(b)
x
y
10
5
00.5 1 1.5 2 2.5 3
5
10
15
20
25
30
19
Answer Key
© 2017 NTK Publishing Limited
(c) 1.7 (accept 1.8)
2 (a) x −3 −2 −1 0 1 2 3y 1 0 −1 −1 0.5 3.75 8.875
(b) y
x1
2
4
8
6
10
2 312
4
6
23
3 (a) (i) y = x + 1
(ii) 1.5, 7.5 (accept 1.4 – 1.6, 7.4 – 7.6)
(b) (i) y = 3
(ii) 1.4, 6.6 (accept 1.3 – 1.5, 6.5 – 6.7)
4 (a) y = −2x + 2
(b) −2, 3
5 (a) x −3 −2 −1 0 1 2 3
y18
14
12
1 2 4 8
(b) y
x1
2
4
8
6
10
2 312
4
6
23
(c) x = 1
6 (a) (i) y = 2x + 4
(ii) 1, −0.8, 4.8 (accept 0.9 – 1.1, −0.9 – −0.7, 4.7 – 4.9)
(b) (i) y = 3x − 10
(ii) −1.7, 2.5, 4.2 (accept −1.8 – −1.6, 2.4 – 2.6, 4.1 – 4.3)
7 (a) x −3 −2 −1 0 1 2 3 4 5y −35 0 15 16 9 0 −5 0 21
(b) y
x1
02 3 54123
10
20
30
40
10
20
30
(c) y = 0
(d) −2, 2, 4
8 (a) 0.5, 3 (accept 0.4 – 0.6, 2.9 – 3.1)
(b)
Exercise 2.11
1 (a) −1.5 (accept −1.7 – −1.3)
(b) 1.5 (accept 1.3 – 1.7)
(c) −2 (accept −2.2 – −1.8)
(d) −9 (accept −9.5 – −8.5)
(e) 5.6 (accept 5 – 6)
(f) 1.3 (accept 1.2 – 1.4)
2 (a) x −2 −1 0 1 2 3y −11 −1 1 1 5 19
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(b)
x
y
1 2 315
10
2
5
10
15
20
15
20
(c) (i) 5 (ii) 1
Warm up 2.12A
1 (a) +f x x: 5 7 (b) −f x x: 4 92
(c) +f x x: 2 1 (d)
−f x
x: 1
4 3
2 (a) 0 (b) 2
(c) 27 (d) 2
3 ( )( )
=
=
fg x x
gf x x
250
10
3
3
4 ( )
( )
=−
= −
fg xx
gf xx
32 46 4
Exercise 2.12A
1 (a) (i) 1.4 (ii) 163
(iii) 163
(iv) −1.4
(b) denominator = 0
2 (a) x = 4 (b) x = 0
3 (a)
( )( )
= +
= − +
−f x x
g x x x
10 17
5 14 122
(b) − +25
295
, 25
295
4 ( )( )( )
=
=
= −
f
f
f
2 4
4 0
6 3
5 ( ) = −fg x x2 4 1
6 f 2x
= 12x2
7 (a) ( ) =+
fg xx
14 152 (b)
116
(c) 9
151
8 (a) 8.04 (b) −4.5
(c) 2633
(d) 52
9 (a) a = 3 (b) 34
(c) ( )++
− xx3 1
3
10 (a) +xx
2 (b) denominator = 0
11 (a) ff 2( ) = 2, ff 32
= 32
, gg 2( ) = 4, gg 32
= 5
(b) gf s( ) = 4, fg 3( ) = 0, gh 3( ) = 3, gf 92
= 0
(c) hgf 1( ) = 0, hgf 0( ) = 6, hgf 32
= 32
(d) 2, 5
12 (a) a = 7 (b) b = −1 c = −1
21
Answer Key
© 2017 NTK Publishing Limited
Exercise 2.12B
1 (a) ( ) = −−f x x 31 (b) ( ) = +−f x x 72
1
(c) ( ) = +−f x x 35
1
2 (a) ( ) = +−f x x4 73
1 (b) ( ) = −−f x x15 910
1
(c) ( ) = −−f x x5 68
1
3 (a) ( ) = −−f xx1 21 (b) ( ) = −−f x
x2 3
21
(c) ( ) = +−f xx53
31
4 (a) ( ) = +−
−f x xx
5 31
1 (b) ( ) = +−
−f x xx
2 54 3
1
(c) ( ) = −+
−f x xx
7 52 3
1
5 (a) (i) ( ) =ff x x
(ii) f is a self-inverse function
(c) =
=
m
n
256
Chapter 3 GeometryExercise 3.1
1 (a) trapezium (b) isosceles triangle
(c) parallelogram (d) hexagon
(e) pentagon (f) equilateral trangle
2 (a)
(b)
(c)
(d)
3 (a) similar (b) neither
(c) congruent (d) neither
(e) congruent (f) neither
Exercise 3.2
1 (a) A
B
O
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(b)
A
B
O
2 (a) A
M
B
O
112°28°
(b) A
N
B
O
112°70°
3 (a) A
B
C
(b) They meet at the in centre (Centre of the circle inscribed in ΔABC)
4 (a) A
M BC
60°
∠AMC = 120°
(b) A
M BC
60°
N
∠ANM = 90°
5 (a) A
B
(b)
A
B
23
Answer Key
© 2017 NTK Publishing Limited
6 (a) A
BC
(b) The lines (medians) meet at the centroid.
7 (a)
A B
C
(b) They meet at the circum-centre. (Centre of the circle circumscribing ΔABC)
(c)
A B
C
The circle circumscribes ΔABC.
8
A
B
C
10 (a)
A30° 30°
B
(b) 4.6 cm
Exercise 3.3
1 (a) B
A C
3 cm 4 cm
4.5 cm
(b) PQ = 1.5 m, QR = 2 m, PR = 2 m
2 (a) 1 : 6000000
(b) 565 km
3
A
1 cm represents 5 km
4.4 cm45°
N
Exercise 3.4A
1 (a) True (b) True
(c) False
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°
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4 (a) x = 24; y = 16; g = 3; k = 2
(b) x : g = 8 : 1; y : k = 8 : 1
(c) s1 : s2 : g = n : 1 : n 1
2−
5 (a)
(b)
4 cm 4 cm 8 cm
2 cm
8 cm
1 cm
1 cm
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 (b) x = 6; y = 12
(c) x = 8; α = 60° (d) x = 12; α = 40°
9 (a) x = 3.98 (b) x = 1.61
(c) r = 8; t = 7.5; α = 80°
(d) x = 203
10 (a) x y36 77
; 27 77
= =
(b) x = 15
(c) x = 5.25; y = 8.75;
(d) x = 16; y = 15; z = 9; t = 25
11 (b) (i) 14.4 cm (ii) 360 cm2
Exercise 3.4B
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
(c) (i) 216π cm2 (ii) 324π cm3
5 (a) 6 cm (b) 3888π cm3
6 (a) (ii) 33.3%
(b) (i) 5.77 cm (ii) 11.6 cm2
7 (a) 4 3 cm
(b) (i) 5 3 cm (ii) 15 5 3 cm( )−
8 (a) (i) 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 (b) 827
(c) (i) 16.0 cm (ii) 2.65 cm
10 (a) 4.17% (b) 8.17%
Exercise 3.5
1 (a) yes; 4 (b) yes; 2
(c) no (d) no
(e) no (d) yes; 6
25
Answer Key
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2 (a) (b)
(c) (d)
(e) (f)
3 (a) (b)
(c) (d)
4 (a) 2 (b) 3
(c) 2 (d) Infinite
5 4
Exercise 3.6A
1 a = 102°; b = 78°; c = 102°
2 a = 150°; b = 60°
3 x = 20°
4 y = 18°
5 x = 26°; y = 32°
6 x = 22°; y = 46°
7 x = 55°
8 a = 111°; b = 69°
9 x = 78°; y = 102°
10 x = 65°; y = 55°; z = 65°
11 x = 20°; y = 45°
12 x = 9°
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°
5 90°
6 (a) x = 108°; y = 72° (b) x = 18°
(c) x = 27°; y = 30° (d) x = 40°; y = 55°
(e) x = 25°
7 x = 112.5°
8 n = 6
9 180°
10 x = 60°
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Exercise 3.6C
1 (a) x = 30° (b) y = 45°
(c) x = 30° (d) y = 50°
(e) x = 35°; y = 65° (f) x = 30°; y = 60°
2 (a) x = 9°; y = 18° (b) x = 112°; y = 112°
(c) x = 100°; y = 30° (d) no solution
(e) x = 118°; y = 59° (f) x = 70°
3 (a) C
B
A
O
(b) 8.5 cm
BC is the diameter of the circle.
4 (b) a = 120°
5 Base angles in an isosceles triangle are equal.
Exercise 3.6D
1 (a) x = 60° (b) x = 90°; y = 40°
(c) x = 90°; y = 60° (d) x = 35°
2 (a) x = 67°
(b) x = 80°; y = 80°
(c) x = 40°; y = 70°; z = 30°
(d) x = 30°
(e) x = 25°; y = 90°
(f) x = 118°
3 x = 100°; y = 40°
Exercise 3.6E
1 (a) x = 82° (b) x = 60°; y = 102°
(c) x = 36°; y = 72° (d) x = 20°
(e) x = 100° (f) x = 86°; y = 22°
2 (a) x = 100°; y = 80° (b) x = 90°; y = 60°
(c) x = 60° (d) x = 49°; y = 30°
(e) x = 12°; y = 15° (f) x = 44°; y = 68°
3 (a) No.
(b) Yes. Sum of any two opposite angles is 180°.
(c) No.
(d) Yes. ∠A + ∠B = 180°, ∠B = ∠C, ∠A + ∠C = 180°, same as ∠B + ∠D
(e) Yes. ∠A + ∠C = 180° and ∠B + ∠D = 180°
(f) Yes. Converse of angles in the same segment.
4 ∠CAB = 70°
Exercise 3.7
1 A
D
C
B
2 (a) A
B
C
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Answer Key
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(b) A
B
C
(c) A
B
C
3 A B
CD
4 (a)
O 1.5 cm
5 C B
A
D
Chapter 4 MensurationExercise 4.1
1 (a) 0.23 (b) 120 000
(c) 0.045 (d) 0.15
(e) 250 (f) 6 520 000
2 (a) 3750 (b) 0.012
(c) 0.00112 (d) 0.04
3 (a) 3.5 × 106 (b) 600
(c) 6.5 (d) 5 × 10−5
(e) 23,000 (f) 5.1 × 106
4 (a) 30 (b) 1 × 107
(c) 300 (d) 100 000
(e) 0.0031 (d) 6500
Exercise 4.2
1 110 cm
2 100 cm
3 a = 5
4 32.8
5 23
6 179
7 (b) x = −4 or x = 1 (c) x = 1
8 40 cm2
9 +3 23 92
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Cambridge IGCSE Mathematics Study Guide
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Exercise 4.3
1 (a) perimeter = 25.1 m, area = 50.3 m2
(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
3 r = 7.14 cm
4 (a) 4.71 m (b) 16.7 m
(c) 14.1 m
5 (a) 44.6 m2 (b) 18.8 m2
(c) 12.6 m2 (d) 2.46 cm2
6 (a) 37.7 cm (b) 22.8 m
(c) 25.1 m (d) 46.8 cm
7 perimeter = 15.7 cm, area = 4.03 cm2
8 perimeter = 30.9 cm, area = 61.4 cm2
Exercise 4.4
1 (a) 1550 cm2 (b) 294 cm2
(c) 193 cm2 (d) 84.3 cm2
2 (a) 225 cm2 (b) 424 cm2
(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 ππ
=−h A rr
22
2
5 (a) 1256.64 cm2 (b) 4720 cm3
6 surface area = 63.2 cm2, volume = 56.5 m3
Exercise 4.5
1 (a) 113 m3 (b) 500 m3
(c) 48 m3 (d) 136 cubic units
2 (a) 615.0 cm3 (b) 41.9 m3
(c) 1282.9 cm3 (d) 173.7 cm3
3 471 cm3
4 1367.1 cm3
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
Chapter 5 Coordinate Geometry
Exercise 5.1
1 (a)
( ) ( ) ( )( ) ( ) ( )
− − −
− −
A B C
D E F
4,2 ; 3, 1 ; 0, 4 ;
3,5 ; 1,5 ; 7, 1
(b) E, A, F
2 (b) A straight line
Exercise 5.2
1 (a) 3 (b) −12
(c) 0 (d) undefined
2 (a) −3 (b) −1
(c) undefined (d) undefined
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Answer Key
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3 (a) no (b) no
(c) yes (d) yes
4 (a) c = 1 (b) q = −15
5 (a) = = = =− −m m m m2; 13
; 2; 13AB AD DC BC
(b) same gradient
6 (a) (i) 0 (ii) undefined
(iii) 53
(b) 152
Exercise 5.3
1 (a) undefined (b) −38
(c) −34 (d) −
52
2 (a) = = =AB AC BC5; 5; 2 5
(b) +5 3 5
3 (c) Area = ( )( ) =12
20 20 10 sq.units
4 (a) ( )6,2 (b) ( )2,1
(c) 12
, 12
(d) ( )−2,3
(e) 5a4
, b2
2 (f) ( )+a b0, 2
5 c = −4; d = −3
6 (a) height = 9 units (b) ( )3,6
Exercise 5.4
1 (a) y = −x + 3 (b) =y x45
(c) y = −2 (d) = −−y x23
43
2 (a) y = x − 3 (b) = +y x518
89
(c) x = 5 (d) y = −bx + ab
3 (a) = +−
y x311
11711
(b) No
(c) = =a b6011
; 892
4 (a) L1: 5x − 7y + 25 = 0; L2: x + 5 = 0.
(b) L1: 2x − 3y − 8 = 0; L2: y − 4 = 0.
5 (a) gradient = 14
, y-intercept = 58
(b) gradient = undefined
y-intercept = does not exist
(c) gradient = −118
, y-intercept = −114
(d) gradient = 34
, y-intercept = −1
6 (a) = −y x12
35
(b) = −y x27
2
(c) = −−y x 53
(d) = +−y x911
3
Exercise 5.5
1 (a) L2 and L6
2 (a) −4 (b) −72
3 = +y x23
113
4 = +y x12
3
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5 (a) = −k 43
(b) x-intercept = 92 y-intercept =
32
(c) 6y + 4x − 9 = 0
(d) 3y + x − 9 = 0
Exercise 5.6
1 L3 and L5 L1 and L4
2 (a) 6 (b) 197
(c) −6 or 1 (d) 1 or −1
3 (a) 2 (b) 3
(c) 25
4 (a) −53 (b) b = 1
(c) y-intercept = 85
, x-intercept = −83
5 (a) L1: y = −x + 7; L2: y = x + 1
(b) L1: x-intercept = 7; L2: x-intercept = −1
(c) L1
6 (a) L1: = +−y x34
3
L2: = −−y x34
154
(b) L1: = +−y x52
15
L2: = −y x25
185
7 (b) k = 3 or k = −1
(c) For k = 3, L1: 3x + 3y − 4 = 0L2: −x + y − 3 = 0
For k = −1, L1: −x + 3y − 4 = 0L2: 3x + y − 3 = 0
Chapter 6 Trigonometry and Pythagorasʼ Theorem
Exercise 6.1
1 (a) N
070°
(b) N
230°A
(c) N
300°A
2 (a) 050° (b) 280°
(c) 270°
3
N
N
R
P
Q
080°4 km
5 km
310°
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Answer Key
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4 200°
5 090°
Warm up 6.2A
1 (a) x = 12.8 (b) x = 13.7
(c) x = 7.21 (d) x = 5.83
2 (a) θ θ θ= = =pr
qr
pq
sin ; cos ; tan
(b) θ θ θ= = =ac
bc
ab
sin ; cos ; tan
(c) θ θ θ= = =sin 45
; cos 35
; tan 43
(d) θ θ θ= = =sin 15
; cos 2 165
; tan 612
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°
Exercise 6.2A
1 OD = 5
2 EC = 9.90 cm
3 AE = 17.3
4 (a) 3x2 − 24x = 40 (b) x = 8
5 =r 83
6 26 km
7 1.3 km
8 39.1 km
9 8.66 cm
10 (a) x = 8.94; φ = 63.4°; θ = 26.6°
(b) x = 11.5; φ = 27.5°; θ = 62.5°
(c) x = 6; φ = 53.1°; θ = 36.7°
(d) x = 7.62; φ = 66.8°; θ = 23.2°
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°
21 112°
22 (a) 175 km (b) 323°
23 1.73 km
Exercise 6.2B
1 0.866 m
2 40 m
3 109 m
4 12.7 m
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5 83.9 m
6 4.1 m
7 55.6 m
8 10.8 m
9 710 m
Exercise 6.2C
1 (a) −cos 60° (b) −cos 150°
(c) −cos 90° (d) −cos 30°
(e) cos 157° (f) −cos 13°
2 (a) sin 150° (b) sin 135°
(c) sin 60° (d) −sin 0°
(e) sin 155° (f) sin 4°
3 (a) (i) 1 (ii) 1
(iii) 1
(b) sin2 θ + cos2 θ = 1, for any angle θ .
Exercise 6.3A
1 (a) 25.8 cm2 (b) 50.3 cm2
(c) 36.8 m2
2 (a) x = 12.0 (b) x = 14.1
(c) x = 9.67
3 (a) θ = 71.8° (b) α = 48.59°
(c) β = 48°
4 (a) 70.9 cm2 (b) 26.8 cm2
(c) 65.0 cm2
5 (a) 12.0 cm2 (b) 2.93 cm2
(c) 61.4 cm2
6 78.6 cm2
Exercise 6.3B
1 (a) a = 3.54 (b) x = 17.91
2 (a) x = 5.87; y = 13.50
(b) a = 12.62; b = 4.68
3 (a) θ = 42.55° (b) θ = 36.23°
4 (a) α = 56.76°; x = 17.72
(b) θ = 30.95°; β = 113.05°
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
Exercise 6.3C
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
(b) ∠ SQR = 14.11°; ∠ SPR = 30.50°
(c) RS = 9.76 cm
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Answer Key
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Exercise 6.3D
1 BD = 10.58 cm; AC = 17.44 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
5 (a)
1200 m860 m
068°
130°
N
N
CA
B
(b) 1774.43 m
(c) 273.34°
6 (a) (i) 1857.54 m (ii) 1717.26 m
(b) VA:VB = 1.08:1
7 (a) 996.11 km (b) 7.88°
8 (a) (i) 12.59 cm (ii) 39.16 cm
(b) (i) AP = 12.63 cm; PB = 26.53 cm
(ii) ∠ABP = 24.87°; ∠APB = 93.13°
Exercise 6.4
1 (a) 59.04° (b) 51.34°
(c) 45°
2 (a) 45° (b) 45°
(c) 35.26°
3 (a) 45° (b) 51.34°
4 67.01°
5 (a) 33.69° (b) 53.13°
(c) 30.81°
6 8.94 cm
7 (a) 54.74° (b) 35.26°
8 (a) 500 m (b) 23.75°
9 (a) 14.42 cm (b) 29.02°
(c) 25.09°
10 (a) 4.29 cm (b) 16.08°
11 (a) 14 cm (b) 39.87°
12 37.76°
13 (a) 61.93° (b) 51.24°
14 55.47°
15 (a) 207.85 km/h (b) 32.21°
Chapter 7 Vectors and MatricesExercise 7.1
1 (a) � ���OA (b)
� ���CD
(c) � ���BA (d)
� ���HK
2 (a) 31
(b) 13
(c) 04
(d) 13
(e) 51
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3
x
y
0 2
2
4
4
1
1
3
3
A
C
E
D
B
11
2
3
4
234
4 (a) 74
(b) 84
(c) 04
(d) 71
(e) 54
(f) 38
5 (a)
a a + b
b (b) a
b
a + b
(c)
a b
a + b
(d) a
b
a + b
6 (a) a
a − b
b
(b) a
a − b
b
(c)
a
a − b
b
(d) a
a − b
b
7 (a) (i) a + b (ii) −c + a + b
(iii) −a + c + b (iv) a + b + c
(b) (i) −a + b (ii) c
(iii) a = b − c
8 (a) b + g (b) g + f
(c) −f + b (d) −f + b
(e) g + f (f) g + b
9 (a) a12
(b) b12
(c) +a b12
(d) +a b12
10 (a) 2519
(b) 3928
(c) 1436
(d)
143236
Exercise 7.2
1 (a) ( )9,2 (b) ( )−17,15
(c) ( )2,4 (d) ( )+ −a a4,4 2
2 (a)
0 2 31
1
2
3
4
5
2 11
2
3
45 3x
y
A
B
C
D
(b)
0 2 31
1
2
3
4
5
2 11
2
3
45 3x
y
A1
B1
C1
D1
35
Answer Key
© 2017 NTK Publishing Limited
3 (a) ( )− −5, 2 (b) ( )−17,15
(c) ( )2,4 (d) ( )−5,4
4 (a) ( )3,4 (b) ( )− −3, 4
(c) ( )3,2 (d) ( )− −7, 4
5
x
A
BC
y
0 2
2
1
3
4
4 51 3 6234
2
11
3
4
6 (a) ( )− −2, 4 (b) ( )2,4
(c) ( )−4, 2
7 (a) ( )−2,3 (b) ( )−2, 3
(c) ( )− −3, 2
8
x
P
P2
P1
y
0 21
2
4
5
1
3
43 524 135
2
4
5
1
3
9 (a)
xk = 3
C
y
(b)
x
C
y
k = 2
(c)
x
C
y
k = 12
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(d)
x
C
yk = 1
3
10
x
y
0 21
2
4
5
6
1
3
43 524 135
2
4
1
3
C
BA
11 (a) reflection about x = 0
(b) enlargement with centre at ( )−6,1 , scale factor 2
(c) rotation with centre ( )0,0 clockwise 90°
(d) reflection about y = −1
Exercise 7.3
1 (a) c − b (b) ( )−c b12
(c) ( )−b c12
(d) ( )+b c12
2 (a) +q r23
(b) ( )+q r34
(c) −r q34
14
(d) ( )+q r14
3 (a) c + a (b) a − c
(c) −− a c27
57
(d) +c a27
57
4 (a) (i) g − d (ii) ( ) +−d g23
23
(iii) ( )−g d23
(c) (i) ( )−k g d23
(d) 2:1
5 (a) (i) b − a (ii) −b − a
(iii) −b (iv) 2b
(v) 2b + 2a
(b) 4
(c) � ��� � ��� � ��� � ��� � ��� � ���BE EF CF FC AD DA, , , , , .
6 (a) (i) s + 2t (ii) s + t
(b) (i) = +� ���QX s t (ii) =
� ���XS t
7 (a) (i) 3p (ii) −p + u
(iii) 2u (iv) −4p + 2u
(v) −2p + 4u (vi) 2u + p
(c) (i) 6 (ii) 6
(iii) 6 (iv) 12
(v) 18
8 (a) +− a b12
12
(b) +− a b12
12
(c) +a b12
12
(d) −−a b12
9 (a) (i) b + v (ii) −b + u
(iii) ( )−u v12
(b) (i) 34
(ii) 30
37
Answer Key
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Exercise 7.4A
1 (a) 2 × 1 (b) 1 × 3
(c) 2 × 4 (d) 3 × 3
2 (a) a = 0, b = 4
3 (a) 313
(b) ( )− −12 2
(c) 1 1511 9
(d) does not exist
(e) 3 9 06 12 9
(f) 8 6 42 10 6
10 2 6
(g) 38
(h) 16 814 7
(i) does not exist (j) 14 422 1610 10
4 (a) 8 27 9
(b) 1 41 12
(c) 18 217 7
(d) 12 165 1
5 a = 5, b = 0
Exercise 7.4B
1 (a) 1 (b) 2
(c) −21 (d) −1
(e) 2 (f) −1
12
(g) −5
12 (h) 0
2 (a) 12
1 32 4
(b) does not exist
(c) 1
566 78 0
(d) does not exist
3 (a) 2 91 4
(b) 3 55 8
(c) 12
6 44 3
(d) 1
197 44 5
(e) 13
7 95 6
(f) 3
29
103
43
72
32
4 (a) A3 = 9A + 8I
A4 = 17A + 72I
A5 = 89A + 136I
(b) (i) 11 12 6
(ii) 403 89178 42
5 k = −4 or k = −1
6 (a) 4 28 1
(c) (i) −3 (ii) −3
(e) (i) 1 (ii) 1
7 a 13
= or 13
−
8 (b) (ii) 21
(c) x = 2, y = 1
Exercise 7.5
1 (a) Reflection about the x-axis.
(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.
(e) Rotation about the origin through 90° clockwise.
(f) Reflection about the line = x, enlargement with centre 0,0( ) and scale factor 3.
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Cambridge IGCSE Mathematics Study Guide
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(g) Rotation about the origin through 90°.anti-clockwise, enlargement with centre 0,0( ) and scale factor 2.
(h) Rotation about the origin through 90° anti-clockwise.
2
x
y
0 2 31
2
1
3
4
5
6
4 5 623 141
2
3
4
3 (a) 11
(b) 01
4 (a) ( )− −1, 3 (b) 3, 1( )−
(c) 3,1( ) (d) 3, 1( )− −
5 (a) 0 22 0
(b)
x
y
0 2 31
2
1
3
4
5
6
4 5 62
2
3
5
1
4
6
4 3 16 5
A1
B1
C1
(c) 14
0 22 0
(d)
x
y
0 2 31
2
1
3
4
5
6
4 5 62
2
3
5
1
4
6
4 3 16 5A2
B2
C2
Chapter 8 ProbabilityExercise 8.1
1 (a) 14
(b) 512
(c) 23
2 (a) (i) 211
(ii) 0
(b) I
3 (a) 12
(b) 12
(c) 1
4 (a) 10
(b) (i) 12
(ii) 14
(iii) 310
5 (a) 730
(b) 430
(c) 13
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Answer Key
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Exercise 8.2
1 (a) certain (b) certain
(c) impossible (d) equally likely
2 (a) P(rainy day) = 0.64
(b) P(red ball) = 49
(c) P(boy) = 50%
3 2, 4, 1, 3
Exercise 8.3
1 (a) Not getting a 3 when a fair die is rolled.
(b) There is at least one boy in the class.
(c) There are no girls in the room.
(d) Get at least one tail when a fair coin is flipped twice.
2 (a) 56
(b) Yes
3 45
4 0.52
Exercise 8.4
1 (a) 23
(b) 80
2 (a) 411
(b) 104
3 (a) 15
(b) 10
4 920
5 (a) 16 (b)
9100
(c) It is normal to have variation between theoretical and experimental probability.
(d) Experimental probability will be close to theoretical probability.
Exercise 8.5
1 (a)
H
T
HT
H
T
121
2
12
12
12
H
T
12
12
12
HT
12
12
HT
12
12
HT
12
12
(b) (i) 3 (ii) 38
2 (a)
Red
Green
Red
Green
Red
Green
712
611
512
511
711
411
(b) (i) 2833
(ii) 1522
(iii) 722
(iv) 3566
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3 (a) 13
(b) (i)
Hit
Miss
Hit
Miss
Hit
Miss
23
23
13
13
23
13
(ii) 49
(iii) 89
(c) (i) 227
(ii) 2 13
n
4 (a) ℰ
M
1 7
14
8
S
(b) (i) 130
(ii) 310
(iii) 715
(c) 89
5 (a) ℰ
F
12 15
4
9
B
(b) (i) 36 (ii) 12
(iii) 9
(c) (i) 940
(ii) 310
(iii) 38
6 (a) p = 16, q = 15, r = 7, s = 10
(b) (i) 56 (ii) 56
(iii) 10
(c) (i) 855
(ii) 43110
(iii) 83110
(iv) 411
7 (a) 34
(b) 14
(c) (i) 49400
(ii) 780
(iii) 1225
8 (a) (i) 58
(ii) 18
(iii) 38
(b) (i) 964
(ii) 116
(iii) 1116
(iv) 18
9 (a)
Rain0.4
0.35
0.65
0.54
0.46
0.6 norain
Rain
norain
Rain
norain
(b) (i) 0.14 (ii) 0.464
(iii) 0.584 (iv) 0.724
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Answer Key
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10 Dice 2
Dice 1
1 2 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( )
3 3,1( ) 3,2( ) 3,3( ) 3,4( ) 3,5( ) 3,6( )
4 4,1( ) 4,2( ) 4,3( ) 4,4( ) 4,5( ) 4,6( )
5 5,1( ) 5,2( ) 5,3( ) 5,4( ) 5,5( ) 5,6( )
6 6,1( ) 6,2( ) 6,3( ) 6,4( ) 6,5( ) 6,6( )
(b) 1
12 (c) 1
9
11 (a) 13
(b) 1936
(c) 536
Chapter 9 StatisticsExercise 9.1
1 (a) qualitative (b) quantitative
(c) qualitative (d) qualitative
2 (a) discrete (b) continuous
(c) continuous (d) discrete
3 (a) discrete
(b) Number of movies Tally Frequency
0 1
1 3
2 3
3 3
4 3
5 2
(c) 53.3%
4 (a) continuous
(b) Height Tally Frequency
140 – 150 1
150 – 160 3
160 – 170 5
170 – 180 3
180 – 190 3
Exercise 9.2
1
50 10 2015 3025
Freq
uenc
y de
nsity
Length of telephone call (minutes)
2 (a) 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
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4 (a) 12 (b) 16
5 (a)
1 2 43 65
Freq
uenc
y de
nsity
Distance (km)
0
15
30
45
60
75
90
7
(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
6 (a)
120 130 150140 170160
Freq
uenc
y de
nsity
Height (cm)
0 180
(b) Height (h cm) Frequency
120 < h ≤ 130 5
130 < h ≤ 140 10
140 < h ≤ 150 20
150 < h ≤ 160 30
160 < h ≤ 170 5
(c) 147.9 cm
7 33
Exercise 9.3
1 (a) mean = 5.7, median = 5.5, mode = 5.5
(b) mean = 42.4, median = 45, mode = 45
(c) mean = 18.375, median = 19.5, mode = 20
2 mean = 27.75, median = 25, mode = 25
3 (a) 42 (b) 40.3
4 90
5 13
6 17.2
7 3, 5, 12, 15, 15
Exercise 9.4
1 mean = 6, mode = 7
2 mean = 23, mode = 24
3 (a) 4 < x ≤ 6 (b) 5.39
4 (a) 3030
(b) We do not know the exact income of each staff member.
(c) 3000 < x ≤ 4000
5 (a) 52 mm (b) 0.071
43
Answer Key
© 2017 NTK Publishing Limited
Exercise 9.5
1 (a) 1.5 < x ≤ 2
(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
(c)
0.5 1 1.5 2 2.5 30
5
10
15
20
25
30
35
40
45
50
Length (cm)
Cum
ulat
ive
freq
uenc
y
(d) 1.8
2 (a) 30 < x ≤ 35
(b) Age Cumulative frequency
x ≤ 20 1
x ≤ 25 4
x ≤ 30 12
x ≤ 35 24
x ≤ 40 30
(c)
15 20 25 30 35 400
5
10
15
20
25
30
Age
Cum
ulat
ive
freq
uenc
y
(d) 32
3 (a) 6.8
(b) Time (hour) Cumulative frequency
t ≤ 2 1
t ≤ 4 3
t ≤ 6 11
t ≤ 8 21
t ≤ 10 27
t ≤ 12 30
(c)
2 4 6 8 10 120
5
10
15
20
25
30
Time (hour)
Cum
ulat
ive
freq
uenc
y
(d) 6.8
4 (a) 43.1
44
Cambridge IGCSE Mathematics Study Guide
© 2017 NTK Publishing Limited
(b) Speed (v miles/hour)
Cumulative Frequency
v ≤ 30 3
v ≤ 40 11
v ≤ 50 21
v ≤ 60 28
v ≤ 70 30
(c)
20 30 40 50 60 700
5
10
15
20
25
Speed (miles/hour)
Cu
mu
lati
ve fr
equ
ency
(d) 48 miles/hour
5 (a) 40 (b) 20
(c) 5 (d) 1.7 kg
6 (a) 15% (b) 10%
(c) 3 hours
7 (a) 95 cm (b) 98 cm
(c) 92 cm (d) 6 cm
8 (a) 48 cm (b) 5 mins
(c) 10
9 (a) 100 (b) 96 minutes
(c) 18 minutes (d) 88
Exercise 9.6
1 (a) y
x
Moderate positive correlation
(b) y
x
Strong positive correlation
(c) y
x
Zero correlation
(d) y
x
Strong negative correlation
(e) y
x
Weak negative correlation
45
Answer Key
© 2017 NTK Publishing Limited
(f) y
x
Strong negative correlation
2 (a) positive (b) positive
(c) positive (d) negative
(e) negative
3 (a)
30 35 40 45 50 55 60 65 700
20
22
24
26
28
30
32
Number of cold drinks sold
Tem
pera
ture
(C°)
(b) Positive correlation
(c)
30 35 40 45 50 55 60 65 700
20
22
24
26
28
30
32
Number of cold drinks sold
Tem
pera
ture
(C°)
(d) 43
4 (a)
50 55 60 65 700
30
35
40
45
50
Age (x)
Num
ber o
f day
s (y)
(b) Moderate positive correlation
(c)
50 55 60 65 700
30
35
40
45
50
Age (x)
Num
ber o
f day
s (y)
(d) 41
5 (a)
50 60 70 80 90 100 110 120 130 140 1500
10
20
30
40
50
60
70
80
90
100
Weight (kg)
Scor
e
(b) No correlation
MARKSCHEME
CAMBRIDGE INTERNATIONAL MATHEMATICS
Practice Test
Paper 2 (Extended)
48
Cambridge IGCSE Mathematics Study Guide
© 2017 NTK Publishing Limited
1 ℰA B
2 9 (a) = +−y x clet
( )put 3,1 into the equation,
= +== +
−
−
ccy x
1 34
4
1
1
2 (a) 2 1 (b) yx
x
let 0,4 0
44,0( )
=+ =
=−
1
10
( )
−−
=
− − =− + =
=
x x
x xx x
x
21
31
3 2 1 63 2 2 6
4
111
(b) 2 1 11 xxx
xx
2 39 45
52.2 (correct to 2 sig. fig)
2 2 2
2
2
+ == −=
==
1
1
1
3 8.44 1
4 S − πr2 = 2πrh
ππ
−=
S rr
h2
2
1
1
12
x
R
y
0 2 3
2
1
1
4
5
6
7
3
4 5
x = 2
6 7 2
2
11
y = 2x + 3
2y x 2 = 0
5
5( )−
=nn
180 2 150
− ===
n nnn
180 360 15030 360
12
2
1
6 3000 × 1.52 = 4560 2
7 ===
= =
xxx
x
let 0.12100 12.12
99 121299
433
1
113 x y
x y
x y
x y
y
y
3 6 2 .......... 12 5 13 ....... 21 3:
6 12 4 ...... 32 3:6 15 39 ...... 44 3 :
3 35353
( )( )
( )( )
( )( )
( ) ( )
− =+ =
×− =×+ =×
=
=
−
−
−
1
1
8 (a) ( )( )( )
− += − −
x xx x
2 3 22 2 1
2 1
1
(b)
( )( )( )
−
= −= − +
x
xx x
4 1
2 12 1 2 1
2
2 2 1
1
49
Practice Test Paper 2 (Extended) Markscheme
© 2017 NTK Publishing Limited
∴
3x 6 353
= 2
3x = 2+ 70
x = 683
1
18 (a) Reflection about the line y = x. 2
(b) 0 11 0 2
19 (a) Total distance = area under the graph
=×
+ × +×
= + +=
10 22
15 2 5 22
10 30 545 km
1
1
14 (a) 34
3
= V120
V = 4058
1
1
(b) V r h
h
h
13
4058
13
3
3 4058 3
5.37 cm
2
2
2
π
π
π ( )
( )
( )
=
=
= =
1
1
(b) The train is travelling at constant speed 2 km/min for 15 minutes.
15 × 60 = 900 seconds
1
1
(c) From the graph, the speed of the train at 8 minutes is 1.6 km/min.
= ×=
1.6 60 km/h96 km/h
1
115 (a) = +
= ° + °= °
BAD OAD OABangle angle angle 20 3050
1
1 20 (a) OP a34
= 2
(b) angle BCD + angle BAD = 180°(Supplementary angle of cyclic quad)angle BCD = 130°
If BC = CD,angle OCB = angle OCD = angle ODC
=°
= °130
265
1(b) position vector of Q OQ c1
3
= = 2
(c) = +
= −−
BQ BC CQ
a c23
1
1
21
B
A
C 5
16 ( )( )= + − °=
=
ACAC
AC
2 3 2 2 3 cos607
7
2 2 2
2
111
17 =
=
=
p ks
k
k
34
6
∴ =ps
6
when s = 9,
= =
=
p 69
632
1
1
1
MARKSCHEME
CAMBRIDGE INTERNATIONAL MATHEMATICS
Practice Test
Paper 4 (Extended)
52 © 2017 NTK Publishing Limited
1 (a) (i) sand 375 kg
stones 125 kg
1
1
(d)
0.7
0.8
0.4
0.6
0.2
0.3
3
(ii) 50 Litres 2
(iii) $100 2
(b) 10.7 cm 3 (e) 0.14 1
2 (a) 38.5 m 1 (f) 0.68 2
(b) 26.3 m 3 (g) 16 2
(c) 18.6 m 3 5 (a) (i)14
1
(d) 18 packets 4 (ii) 5 2
3 (a)−a
b a2 (b) 4 + 3x2 2
(b) (i) > −n 5 2 (c)94
3
(ii) − − − −4, 3, 2, 1 3 (d)− x43
2
(c) x = 9z2 y
or x = 3z
2
y
3 (e)43 3
6 (a) 12, 17, 35 3
(b) (i) 56 1
(d) (i) 2 (ii) 36 1
(ii) −0.732, 2.73 4 (iii) 48 2
4 (a) ℰP
22
0
0
1 1
3
3
T
S
4 (c) 56.8 3
7 (a) −2.5, 2.5, 3.33 2
(b)
0 210.5 3
23
1
456
42 1
23
1
456
x
y
4
(b) (i) 6 1
(ii) 2 1
(iii) 8 1
(c) [ ]25
or 0.4 2
Cambridge IGCSE Mathematics Study Guide
53© 2017 NTK Publishing Limited
(c) x = 0.5, 2 (must be exact) 4 9 (a) 14 12 4
2
(d) (i)
0 210.5 3
23
1
456
42 1
23
1
456
x
y 2(b) 1
62 12 2
or
13
16
13
13
2
(c) det C = 0 [accept matrix is singular] 1
(d) m = 4
n = 3
2
2
(ii) slope = 3 accept 4 to 2 3 (e) −17,17 4
8 (a) (i)
(ii)
x
y
0 21 3
2
3
1
4
5
6
4 52 13
2
3
1
4
5
6
45
R′
S′
2
2
10 (a) AB, AC, AD, BC, BD, CD 2
(b) BD, BE, CD, CE, DE 2
(c) 1 1
(d) (i) 1 3
(ii) 45 1
(iii) [ ]166
or 0.0152 4
(b) Enlargement, scale factor = −12
,
centre ( )0,0 3
(c) 2c − 2a − 2b 2
(d) a + b 2
(e) 3
(f) parallelogram 1
Practice Test Paper 4 (Extended) Markscheme
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