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Cambridge IGCSE Mathematics Study Guide Answer Key PDF

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Cambridge IGCSE Cambridge IGCSE Cambridge IGCSE Cambridge Cambridge Cambridge IGCSE IGCSE IGCSE Cambridge IGCSE Mathematics Mathematics Cambridge Cambridge Cambridge Cambridge Cambridge Cambridge Cambridge Cambridge Cambridge Cambridge IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE IGCSE Mathematics Mathematics Mathematics Mathematics Mathematics Mathematics Mathematics Mathematics Mathematics Mathematics Study Guide Study Guide Study Guide Study Guide Study Guide Study Guide Study Guide Study Guide Study Guide Cambridge International Examinations Answer key and markscheme CONTENTS Answer 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 Numbers Exercise 1.1 1 (a) 7, −2, 111, 25, 0 (b) 7, −2, 3 4, 111, 2.4, 25, 0, 51 2 (c) π, 3 (d) 25, 0 2 (a) 71, 81 (b) − 4, −3 (c) 6.2, − 4, 8 9, −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, 4 3, 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) 3 2 × 5 (b) 3 2 × 2 3 (c) 7 × 3 × 2 2 (d) 2 3 × 3 × 5 (e) 2 × 7 × 9 (f) 5 × 3 3 (g) 2 × 3 × 5 × 7 (h) 3 2 × 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 , 4 2 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 a b e f d c B (b) c{ } 6 (a) ℰ A B C 3 2 4 8 9 10 5 1 7 (b) A B C A B C 10 φ ( ) { } ∩ ∩ = ∪ ∩ = 7 ℰ A B C 2 4.2 9 33 15 5 10 7 8 (a) ℰ A B (b) A B ℰ (c) A B ℰ (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) 27 10 (f) 1 6 2 (a) 125 (b) −64 (c) −1 (d) 0.2 (e) 1.5 (f) 2 5 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) 23 30 (b) 23 28 (c) 5 28 (d) 5 9 (e) 29 24 (f) 81 35 (g) 19 2 (h) 121 52 2 (a) 11 3 (b) 3 4 (c) 1 2 (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) 4 5 (b) 27 50 (c) 33 25 (d) 76 25 (e) 2 3 (f) 45 99 (g) 2 753 999 (h) 2 1 18 7 (a) 1 250 (b) 14 25 (c) 23 10 (d) 131 5000 8 Fractions Decimals Percentages (a) 2 5 0.4 40% (b) 123 100 1.23 123% (c) 6 25 0.24 24% (d) 2 9  0.2 22.2% (e) 517 50 10.34 1034% (f) 11 5 2.2 220% Exercise 1.6 1 (a) 7 > 2 (b) −1 < 0 (c) 0.23 < 0.32 (d) −0.1 > −1 (e) > 31 2 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 1 0.2 3 2 3 (b)  < < 0.2% 0.2 0.2 (c) < < < < 0.001 1 101 1 100 0.011 1.2% (d) < < < 1 4 % 0.025 1 4 2 3 Exercise 1.7 1 (a) 27 (b) 32 (c) 360 (d) 51597 2 (a) 2 4 × 3 4 × 7 (b) 5 2 × 11 2 (c) 2 3 × 3 3 × 13 2 (d) 2 3 × 7 × 11 × 13 5 Answer Key © 2017 NTK Publishing Limited 3 (a) 25 4 (b) 1 9 (c) 1 3 (d) 4 (e) 1 32 (f) −8 (g) 1 16 (h) 5 8 4 (a) 4.98 × 10−9 (b) 9.999 × 10 10 5 (a) 124 000 000 (b) 0.000 000 62 6 (a) 1.512 (b) 1.6 × 10 4 (c) 4.2 × 10 8 (d) 9.72 × 10 2 (e) 3.33 × 10 10 (f) 2.30 × 10 6 (g) 2.32 × 10 10 (h) 3.06 × 10−1 7 7−23, by 40 353 607 times. 8 500 s 9 × × × − − − 1 22 10 , 4.2 10 , 2 10 , 2 3 4 2 2 3 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) 2 5 − (c) 20 (d) 33 (e) −6 (f) 3 3 (a) 3 2 (b) 5 6 (c) 3 2 5 8 12 5 = (d) 10 3 3 2 20 9 = − − 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 2 6 4 3 × = (b) 2 2 2 8 2 ( ) + = (c) 4 8 1 3 = Exercise 1.10 1 upper bound: 9.8 0.1 2 9.85 + = ; lower bound 9.8 0.1 2 9.75 − = 6 Cambridge IGCSE Mathematics Study Guide © 2017 NTK Publishing Limited 2 (a) lower bound: 5 1 2 8 1 2 = 33.75 (b) upper bound: 8+ 1 2 5 1 2 = 17 9 3 number of small tanks: 900 + 10 2 3.5 0.1 2 = 262 22 69 4 (a) upper bound: 4 6.81+ 0.01 2 = 27.26 (b) lower bound: 4 6.81 0.01 2 = 27.22 5 (a) lowest possible: 26 1 2 25.5 kg − = (b) upper bound: 5 26 + 1 2 = 132.5 kg 6 (a) upper bound: 65+ 1 2 58 1 2 = 8 kg (b) lower bound: 65 1 2 58+ 1 2 = 6 kg 7 (a) lower bound: 2 3.14 0.01 2 3.56 0.01 2 9.81+ 0.01 2 = 3.77 (b) upper bound: 2 3.14 + 0.01 2 3.56 + 0.01 2 9.81 0.01 2 = 3.79 8 upper bound: 6 7.4 + 0.1 2 2 = 333.015 lower bound: 6 7.4 0.1 2 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 w 1 4 3 (b) 729 4 (c) 36 m/s 2 (a) = × − d L 3 10 5 (b) 125 000 mm 3 35 km 4 6 5 78 kg 6 (a) 743.04 (b) 639.6 7 (a) = t n 80 (b) 16 (c) decrease by 1 hour 8 (a) = × x f 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) = t N 192 (b) (i) C = 4500 192 t C = 864000 t (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 m 3 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 , cos45 3 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 3 1000 874.1 764 667.8 3 (a) 13 (b) 3.25 grams (c) 2 years 4 (a) ( ) = B 100 2 t (b) 6000 5 (a) 3 114 918 (b) 13 788 382 (c) Population of species A keeps decreasing. (d) 4:1 Chapter 2 Algebra Exercise 2.1 1 (a) 2p + 2q + 2r (b) −2x + 10y (c) ab (d) − aq 9 2 (a) y = c + mx (b) = + r p a b 2 (c) = + R x y 1 2 1 2 3 (a) ( ) = + d u v t 1 2 (b) d = xt 4 (a) S = 28 (b) V = 810 9 Answer Key © 2017 NTK Publishing Limited (c) = r 1 4 (d) = r 8 3 5 (a) = − y T x 2 3 (b) + = = − fu fv uv u fv v f (c) = − + y x Rx R 2 5 3 1 (d) = − + x y y 2 2 1 2 2 (e) = + s A t2 5 6 (a) 1. = − − x q p p q 2 2. x = −3 (b) 1. = − s bk rb a 2. s = 2 (c) 1. = − b R a R 1 2 2 2 2. b = 9 (d) 1. = − − x y b y a 3 4 2 2 2. = − x 3 14 (e) 1. = + − x y Py P 5 4 2 8 2. = − x 17 4 Exercise 2.2A 1 (a) 6xy − 6y (b) 3x − 2xy + 5 2 (a) 9p 2 + 2q (b) 3p 2q + p 2 3 (a) 0 (b) 2abc 4 (a) −5s 2t − 4st (b) 2s 2 − 6s 2t 5 (a) 3xyz 2 (b) −5xyz + 7xy 2z Exercise 2.2B 1 (a) a 2 + 2a (b) 4b − ab (b) 2b + 3ac (c) 2a 2 − ab + 4b (e) x 2 − x − 6 (f) x 2 − 3x − 10 (g) 6x 2 − 19x − 11 (h) −5x 2 − 18x + 8 (i) 20y 2 + 3y − 2 (j) −y − 3y + 18 (k) x 3 + x 2 − 3x + 9 (l) −3x 3 + 5x 2 + 4 2 ( ) − − x x 1 2 12 7 10 2 Exercise 2.2C 1 (a) x 2 − 4y 2 (b) 9 − 25x 2y 2 (c) 4x 4 − 1 4 (d) x 2 − x 9 2 (e) 5x 2 + 10xy + y 2 (f) 49 − 14xy + x 2y 2 (g) 4x 2 + 4x + x 1 2 (h) − + y xy x 1 2 1 2 2 (i) x 4 + 2x 3 + 3x 2 + 2x + 1 (j) x 6 − 2x 4 + 6x 3 + x 2 − 6x + 9 (k) x 6 − 4x 4 + 6x 2 + x 1 2 − 4 (l) x 2 + y 2 + z 2 + 2xy + 2xz + 2yz 2 (a) 5x 2 + 20x − 146 (b) 4xy (c) 2y 3 − 3y 2 + 1 (d) 6z 4 + z 3 − 16z 2 − 7z + 4 Exercise 2.2D 1 (a) 2by 2 (b) ax (c) ( ) − a 2 (d) ay 2 (a) ( ) + ax x a 2 (b) ( ) + by y b 4 2 2 (c) ( ) + x x 7 7 3 (d) ( ) + + x ax a x 2 4 2 2 3 (a) ( )( ) − − p q r 2 1 (b) ( )( ) − + a x y 5 (c) ( )( ) − + b x a c 3 1 (d) ( )( ) − − − x x a 2 2 (e) ( ) ( ) − − − x y x y 5 2 2 (f) ( ) − x b c 2 2 2 4 (a) ( )( ) + + x y 1 2 (b) ( )( ) + + r t s 5 (c) ( )( ) + + p q m n (d) ( )( ) − − a y z 7 (e) ( )( ) − − a b a 2 (f) ( )( ) − + y y 1 5 (g) ( )( ) − − y x y 1 (f) ( )( )( ) + − + x x x 2 1 1 1 10 Cambridge IGCSE Mathematics Study Guide © 2017 NTK Publishing Limited (i) ( )( ) + − − y y h k 1 2 (j) ( ) ( ) − − x a b 1 2 Exercise 2.2E 1 (a) ( )( ) − + x x 2 2 (b) ( )( ) − + y y 4 3 4 3 (c) ( )( ) − + z z 5 1 2 1 2 (d) ( )( ) − + x y y 7 5 5 (e) ( )( ) − − − + a b a b 2 6 2 6 (f) ( )( ) − + p p 33 3 1 2 (a) ( ) + x 3 2 (b) ( ) − x 7 2 (c) ( ) − x 2 5 2 (d) ( ) − − y 11 2 (e) ( ) − a b 3 2 2 (f) ( ) + − a b 5 3 2 2 3 (a) ( )( ) + + − p q p q 4 3 7 11 7 (b) ( ) − x yz 4 2 (c) ( )( ) − − − p p q 2 1 2 2 (d) ( )( ) + − + + a b a b 2 6 2 6 (e) ( ) − s t s 20 5 3 (f) ( )( ) − − − + a b c a b c 2 7 6 2 7 6 (g) ( )( ) + − + + + + m n x m n x 3 1 3 1 (h) ( ) ( )( ) − + + x x x 3 1 3 1 9 1 2 Exercise 2.2F 1 (a) ( )( ) + + x x 6 1 (b) ( )( ) + − x x 6 1 (c) ( )( ) + + x x 2 3 (d) ( )( ) − + x x 3 2 (e) ( )( ) − + x x 8 7 (f) ( )( ) − + x x 5 2 (g) ( ) + − x 9 2 (h) ( )( ) + − x x 7 4 2 (a) ( )( ) − − − p p 2 5 (b) ( )( ) + − − a a 6 3 (c) ( )( ) − − m m 2 3 1 (d) ( )( ) − − y y 4 4 7 (e) ( )( ) + − n n 6 5 3 (f) ( )( ) + − 4 6 8 6 1 (g) ( ) − − q 2 7 2 (h) ( )( ) + − − s s 5 7 3 (i) ( )( ) − + x x 7 3 2 (j) ( )( ) − − y y 11 11 1 (k) ( )( ) − + − z z 3 10 9 (l) ( )( ) − − − t t 2 11 3 Exercise 2.2G 1 (a) ( )( ) + + x x 2 1 3 (b) ( )( ) + − x x 2 1 3 (c) ( )( ) − + y y 3 2 3 (d) ( )( ) + − y y 3 1 3 4 (e) ( )( ) − − a a 5 2 1 1 (f) ( )( ) − + a a 3 4 1 5 (g) ( )( ) + + s s 2 6 5 1 (h) ( )( ) + − s s 3 4 1 3 2 (i) ( )( ) − + − m m 4 2 1 5 2 (j) ( )( ) − − − m m 3 2 1 1 (k) ( )( ) − − − p p 2 1 2 3 (l) ( )( ) + − − p p 2 3 7 2 2 (a) ( )( ) + − x y x y 11 12 (b) ( )( ) + − m n m n 4 2 3 (c) ( )( ) + − p q p q 2 5 5 (d) ( )( ) − − s t s t 8 3 3 8 3 (a) ( )( ) + − x x 12 7 2 3 (b) ( )( ) − − x x 12 29 2 9 4 (a) ( )( ) − − x y x y 1 4 4 (b) ( )( )( )( ) + − − + x y x y x y x y 1 4 2 2 Exercise 2.2H 1 ( ) + a 11 6 2 2 ( ) − xy y x 3 ( ) + a a b 6 4 ( ) − xy y x 5 ( )( ) + − n m 1 1 6 ( )( ) − − m x m 1 7 ( )( ) + + s s 2 6 5 1 8 ( )( ) − − p qx p q 9 ( )( ) − + s t t 1 1 10 ( )( ) + − − x y x y 6 2 2 11 ( )( ) − − y y 12 13 11 Answer Key © 2017 NTK Publishing Limited 12 ( ) + n 2 3 2 13 ( ) − a b 5 11 2 14 ( )( ) − + c c 3 5 7 15 ( ) + x x 2 2 2 16 ( )( ) + + a x x 5 6 4 17 ( )( ) − + − + m n m n 6 6 18 ( )( )( ) − + − q p p 2 2 1 2 1 19 ( )( ) + − x x 2 8 20 ( ) + x 3 2 2 21 ( )( ) − p q q 4 22 ( )( )( )( ) − − + + p p p p 1 4 1 1 23 ( ) − x 1 16 2 1 4 24 ( ) ( )( ) − + + a a a 1 16 2 1 2 1 4 1 2 Exercise 2.3A 1 (a) 6 (b) 6 7 (c) 12 5 (d) 1 2 (a) −10 3 (b) 3 2 (c) 0 (d) 3 2 3 (a) a 3 (b) − a 1 4 (c) c 1 3 (d) b c 5 (e) − x y 4 (f) − y 5 4 (g) z y 8 (h) − z x 3 2 2 4 (a) 2a + 1 (b) 3a + b (c) z y 8 (d) − z x 3 2 2 (e) − x y 2 (f) x z (g) − y y 2 5 5 (h) − x x 2 3 (i) − + x x 2 2 1 (j) + − x x 2 3 5 (k) − a c (l) − x 1 2 (m) + x 1 1 (n) − x x 3 (o) + x x 2 (p) + x 1 2 Exercise 2.3B 1 (a) a 4 5 (b) − a 2 2 (a) bc ad 6 (b) − 1 4 3 (a) − x x 1 3 (b) − x 1 3 2 4 (a) + x x 1 (b) y x 7 5 (a) ( ) + − x b y 2 1 (b) − b x 2 Exercise 2.3C 1 (a) LCM: 12x 2 HCF: 2x (b) LCM: 36x 2y 2 HCF: 3xy (c) LCM: ( )( ) + − x x 2 2 2 HCF: 1 (d) LCM: ( ) − x x 2 1 2 HCF: ( ) − x 1 (e) LCM: ( )( ) + − a a b a b 60 2 HCF: 5 (f) LCM: ( ) − m n 18 1 3 HCF: ( ) − m 3 1 12 Cambridge IGCSE Mathematics Study Guide © 2017 NTK Publishing Limited (g) LCM: ( )( )( ) + − − x x x 1 1 2 3 2 HCF: ( ) − x 2 3 2 (a) a 21 10 (b) + a ab 4 6 3 (c) − x x 2 1 (d) ( ) − − y x x y 2 2 (e) ( ) + + m m m 7 4 2 (f) ( ) − n 9 2 3 (g) + − x x 4 2 4 1 2 (h) ( ) ( ) − − y y x x y x 2 Exercise 2.3D 1 (a) ( ) − a b b a 9 4 (b) + − ab a b 4 (c) − − x x 1 3 (d) − + x x 2 3 2 3 (e) − x x 3 3 (f) 1 − x (g) + − x x 1 2 1 (h) − + x x 3 3 2 (a) ( )( )( ) − − + x x x 3 2 2 1 (b) ( )( )( ) − − − + x x x x 5 4 3 2 2 1 3 3x + 5 4 (a) a 3 − b 3 (b) ( ) ( ) − + + x y x xy y 25 2 2 2 Warm up 2.4 1 (a) 1 9 (b) 4 (c) 1 3 (d) 3 (e) 1 (f) 1 (g) 0 (h) 1 (i) 125 (j) 4 (k) 1 8 (l) 1 9 (m) 1 625 (n) 1 2 Exercise 2.4 1 (a) 5 9 (b) 7 8 (c) 3 7a 5 (d) a 9b 7 (e) a (f) a b 1 3 6 (g) a 14 (h) 8a 2b 11 (j) −25a 3b 4 (j) 1 2 (a) 1 33 (b) 1 62 (c) 2 (d) x y 2 3 (e) x 4 (f) x y 1 2 1 3 (g) x 9 2 (h) x y 2 3 (a) 3 (b) −2 (c) −2 (d) 7 (e) 3 4 (a) 1 16 (b) 5 (c) 2 (d) 3 4 (e) 5 21 (f) 3 2 (g) −4 Exercise 2.5A 1 (a) 6 (b) 4 13 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) 7 4 (c) − 7 6 5 (a) 1 18 (b) 69 88 (c) 29 6 (a) − 6 5 (b) 62 3 (c) 39 7 (a) 0 (b) 8 (c) 36 7 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 = − 5 2 (c) x = 5 3 or x = − 9 4 (d) x = 3 or x = 4 2 (a) x = 0 or x = 4 (b) x = 0 or x = 5 (c) x = 0 or x = 7 5 (d) x = 0 or x = −4 3 (a) x = 0 or x = 3 (b) x = − 5 2 or x = 5 (c) x = 7 2 or x = −5 (d) x = 3 2 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 = 1 3 or x = 3 2 (b) x = −5 or x = 7 4 (c) x = − 6 5 or x = 2 3 (d) x = 2 3 or x = 3 7 6 (a) x = 3 or x = 5 (b) x = −1 or x = 5 2 (c) x = − 5 6 or x = 3 2 (d) x = − 4 5 or x = 7 4 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 = 32 9 (c) x = − 19 7 or x = 19 7 3 (a) x = −4 or x = −12 (b) x = + −2 3 7 4 or x = + 2 3 7 4

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