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Answers to Extended revision exercises: Number Worksheet 1: Reviewing number concepts 1 a 9, 18, 27, 36, 45 b 17, 34, 51, 68, 85 2 a 36 b 4 3 a 60 b 36 4 a 23, 29, 31, 37, 41, 43, 47 5 a 2³ × 5² b 3³ × 7 6 a F b T c T d F e F f T g T b 18.25 c 3.716 d 5.68 e 17.74 f 3.92 d 0.5 e 0.2 f g 15.54 h 0.28 b −12°C, −4°C, 0°C, 7°C, 19°C, 29°C 9 4°C b 1112.6m c L ion’s Head would effectively be at a lower altitude because altitude here is measured as ‘height above sea level’; it would now measure 667.15 m above sea level. −26.94 −115 12 a 1442 b 14.5 c 2.452 d 6.543 e 76.983 f g 13.608 h −1999 i 1728.694 −8.057 13 a 33 and 61 b 45 and 26 14 6 metres 15 22 members 16 a 772.695 8 a −2°C, 0°C, 2°C, 4°C, 8°C, 10 a −28m b 1 c 42.75 i b 59, 61, 67, 71, 73, 79, 83, 89, 97 7 a 16.047 11 a 5.84 b 772.695 c 3.85 d 20.07 e 2 007 000 f 1.925 17 a prime (e.g. 2 → 47, 5 → 71, 10 → 151) b still prime; 40 → 1681, which is not prime. 18 a 8 b 16 c 2n © Cambridge University Press 2018 1 Answers to Core revision exercises: Algebra Worksheet 2: Making sense of algebra 1 a x −6 b c x + 2. 5 e 2x 2 a 2a + 6b 2x 2 9 e 9yz c g k 2x8 1 x 2 b 32xy o 64x12 p 2 a q x4 r 1 4x2 t 1 x y 4z2 d 2x2y - 4xy 6x 5 5x - 9y h x 2 y2 3 a 21 b 20 c -1 d 50 e 56 f g 3 h 70 i j 2 64x9 4 n 3xy m 24x 1 d x 3 f l s 2 x2 y 8 a 2.08 2 b 4.42 c 10.1 d 6 e 16 6 13 9 a x b x 15 5 −7 3 c x 2 k 0.45 l m 2 n 160 d x3 5 y2 4 10 a 4(x + 3) = 4x + 12 o 2 b 4(x - 2) + 2(x + 4) = 4x - 8 + 2x + 8 = 6x + 0 = 6x 4 a 4x + 12 c 3(x - 2) + 5(x + 1) = 3x - 6 + 5x + 5 = 8x - 1 b 5x - 10 d 2(x + 3) + 3(x + 4) = 2x + 6 + 3x + 12 = 5x + 18 c 2x2 + 8x d 3x2 - 6x e 7x2 + 7xy f g 6x - 8 h 2x - 4y i j 4x - 8x + 2y + 3y l 7x3 - 8x2 + 15x 6x + 2 k 3x - 36 5 a 13 × 22 3x + 8 2 d 24 × 32 g 210 h 2 × 52 × 72 2 2 b 2 × 13 c 2 ×3 e 25 2 11 a Total bottles in a day f 34 6 The only way that we can write the prime number 41 as a product is 41 × 1. But 1 is not a prime number. Therefore we can’t write 41 as a product of only primes. In fact this is true for any prime integer. 7 a x 6 b 1 c x 6 d x15 e 2 f g x-2 h 2x8 i j 24x3y2 2 2x3y4 bHow many more water bottles were sold than cool drink bottles cHow many more cool drink bottle were sold than water bottles d Total number of water bottles in d days e Total number of bottles in d days f8 less water bottles than the total number of bottles in d days 12 a 2 3x c 4 3 5x 2 2 1 3 c x = 27 13 a x= b −1 5 4x 2 d 2 y2 3x 2 b x = -4 d x= 1 4 © Cambridge University Press 2018 2 Worksheet 2: Making sense of algebra 14 a b x= 15 a x = 3y 1 2 x =± 1 1 − ≈ −0.290 or −0.710 4 2 2 4 b x=-y c 3x = 7 - 2y d 2 y3 = e 3x + 1 = 9y f x 3 x = ±y © Cambridge University Press 2018 3 Answers to Extended revision exercises: Shape, space and measures Worksheet 3: Lines, angles and shapes 1 Name of polygon Number Sum of of sides angles Size of one angle when the polygon is regular Triangle 3 180° 60° Quadrilateral 4 360° 90° Pentagon 5 540° 108° Hexagon 6 720° 120° Octagon 8 1080° 135° Decagon 10 1440° 144° 2 160° b NOT TO SCALE c NOT TO SCALE 3 NOT TO SCALE Z 70 X mm 40 mm 50 mm Y d NOT TO SCALE 4 a NOT TO SCALE © Cambridge University Press 2018 4 Worksheet 3: Lines, angles and shapes 5 a 15° b 150° c 51° d 64° e x = 45° and y = 100° f g 30° h 70° 100° 6 Let angle MQN = x Then angle PMQ = x (isosceles triangle) So angle MPQ = 180 - 2x (angles in a triangle add up to 180 degrees) Therefore angle MPN = 180 - (180 - 2x) = 2x (angles on a straight line) So angle PMN = 2x (isosceles triangle) And angle NMQ = x + 2x) = 3x 7 a 77° c 282° b 143° d 262° 8 1440° © Cambridge University Press 2018 5 Answers to Extended revision exercises: Data handling Worksheet 4: Collecting, organising and displaying data 1 a Discrete b Number of fizzy drinks consumed 0–4 5–9 10–14 15–19 20–24 25–29 30–34 c Frequency of students 16 8 5 6 6 5 4 Number of cans of fizzy drink consumed by students in one week 2 a Black b White c 28.9% (3sf) d 25 (to the nearest whole number) © Cambridge University Press 2018 6 Worksheet 4: Collecting, organising and displaying data 3 a Stem 5 6 7 8 9 Leaf 0 0 0 0 0 1 1 1 9 3 1 1 1 2 2 3 3 3 5 5 6 8 2 2 3 3 4 5 5 5 7 7 2 5 5 5 9 Key 5 | 0 = 50 years 9 9 9 b Age of oldest grandparent of 40 sixteen-year-old girls 4 a Pulse rate before exercise Stem Pulse rate after exercise 5 5 0 5 9 9 7 4 6 4 3 7 0 8 4 Key 9 5 7 8 10 3 11 3 5 5 12 0 1 Before exercise After exercise 0 | 5 = 50 beats per minute 8 | 4 = 84 beats per minute bIn every person, the pulse rate increased after exercise. © Cambridge University Press 2018 7 Worksheet 4: Collecting, organising and displaying data 5 a Favourite subject of a group of students in Dhaka b34 c 48 d English e Biology 6 a Pictogram b Each stick man represents 1 billion people 1 c billion = 500 million 2 d 200 years e 2012 f g 9 full stick men and 1 of a stick man. 5 World population (in billions) over time © Cambridge University Press 2018 8 Worksheet 4: Collecting, organising and displaying data 7 a Africa (60°) b Students’ own answers. Example: Asia (215°) A bar chart with ‘Continent’ on the horizontal axis, and ‘Percentage’ on the vertical axis. Europe (36°) North America (17°) South America (31°) Oceania (2°) © Cambridge University Press 2018 9 Answers to Extended revision exercises: Number Worksheet 5: Fractions and standard form 1 a 15.6 (1dp) b 383 000 000 (3sf) c 0.000 035 (2sf) 3 4 0.75 75 17 25 0.68 68% 33 100 0.33 33% d 1.000 (3dp) e 32 450 (nearest 10) f 0.123 (nearest thousandth) g 2 a 5 300 000 130 (nearest 10) b 9 560 000 000 000 c 108 000 000 d 8 750 000 000 e 0.0053 f g 0.000 000 91 h 0.000 000 021 45 0.000 002 08 3 a 6.5 × 107 b 3.48 × 1010 c 1.9 × 10-4 d 1.27 × 10-7 e 1.2 × 104 f g 2.75 × 107 h 3.45 × 10-13 24 35 3 c 8 40 9 e 40 b 2.8 × 10-3 g 9 4 a 8 × 107 5 5.6 × 10-3 c 1.035 × 103 d 3.312 × 104 e 4.465 × 1028 f Fraction Decimal 3.04 × 10-9 Percentage 1 3 0.333… 77 100 0.77 7 20 0.35 35 7 5 1.4 140 1 8 0.125 12.5 3 8 0.375 37.5 19 20 0.95 95 Decimal Percentage Fraction 0.583… 4 1 5 1.8 9 20 0.45 0.6 77 58.3… 180 45 60 37 70 11 8 28 d 5 45 1 f 12 b h 3 2 35 5 12 43 5 7 j k 10 5 9 l 105 i m 7 12 3 5 33.3… 6 a 7 10 28 45 q 16 5 n 2 3 16 2 9 34 243 r 315 o p 3 s 32 t u 23.5 v 7.68 w 150 x 2 7 a 25% 2040 b 66.7% (3sf) c 400% d 100% e 40.4% (3sf) f 8 a 170 233% (3sf) b 28 c 11.385 d 88.22 e $36 f 9 a 37.5 3.03 kg b 776 c 71.04 d 33.88 e $45 f 10 a 3 60.03 seconds b 45 c 2 d 400 e 40000 f 2 g 35 h 10 5 = 12 6 © Cambridge University Press 2018 10 Worksheet 5: Fractions and standard form 11 $500 12 a 600 b 90 13 a $4400 (2sf) b $5000 (2sf) c x V 1 − 100 14 On January 1st 2034 the painting is due to be worth $2520. So the value of the painting will first be worth more than $2500 during 2033. n © Cambridge University Press 2018 11 Answers to Extended revision exercises: Algebra Worksheet 6: Equations and transforming formulae 1 a 6 (x − y ) c 3x (2a + b − 3c ) e 3x ( x + 5 y ) g (2a − 3b) ( x + 2 y ) i 1 (a + 3b) 4 k 8 (x − 4) 2 a -3x + 6y b 3 (4 x − 3 y + c ) d xy ( x + y ) f ( x 2 y x 2 y 2 + 7 − 3xy 2 ( ) h 2x 6x 2 − 4 x + 1 j l 1 2 x (7 x + 6 ) 8 3 ( x − 1) (7 − 6 x ) 2 b -6x + 4x 2 c -4x + 12x d -10x2 + 10xy e -6x2 + 14xy f g -2x + 10xy h -7.2x + 12 i j -x2 - 7x l -26x + 20y 2 3 2x - 3 k -3x + xy - 2y - 4 3 a 27 c 14 e -2 4 15 -7 -8y + 56 b 0.04 8 d 3 2 f − 3 g − h 4.5 i j 4 k 2 l 7 m 24 n 40 1 5 q 21 o 5 s 168 4 a -1 c − e 0 4 5 10.5 g 1 i 1 6 p -45 r 12 t 16 b 8 9 ) 5 x = 2, Area = 275 cm2 v −u 6 a t= a g c f = h e x = yz b y =s−x−z d −c b x = y+3 d a= f nx y g D = ST h m= i b = c (a − x ) j x = y2 l a y= x k y= z2 x m x = (m − y ) 2 m o x= y n y = x −b 2 1 z−x p y= qx p b b= 1 (c − a)2 d y= 1 (x + 1)2 y= z z +1− x 7 a y= c x= e x= z y +1 f g q= p +1 p −1 h z= i yz + z x= y − 1 1 y 2 −1 y x− x y +1 2 5 9 f 13 1 h 1 6 d − © Cambridge University Press 2018 12 Answers to Extended revision exercises: Shape, space and measures Worksheet 7: Perimeter, area and volume b 286 mm c 3.51 m2 d 450 m2 c 10.7 cm d 12.8 cm e 738 cm2 f e 5.65 m f g 0.137 m2 h 2513 mm2 g 44.7 m h 276 cm i 31.4 m j k πx cm l 1 a 66.0 m 7.85 cm 51.8 cm π x + 3π cm 12 a 96 cm3 1357.1 cm b 14 235.3 m3 c 38 880 cm3 d 1200 mm3 e 374.052 mm3 f 4926.02 cm3 2 a 2.23 mm b 12.9 cm g 3.6 cm3 h 19.683 cm3 c 32.8 mm d 5.01 cm x f r= 2π i j 95.4 m2 l 1767.1 cm2 e 35.0 cm 3 a 2.62 cm (2dp) b 17.17 cm (2dp) 4 a 1.54 cm2 b 1.54m2 c 6.61 m2 1697.4 m2 k 1206.4 cm2 13 a NOT TO SCALE d 452 cm2 e π(x + 4)2 mm2 5 a 0.349 m2 6 a i b 0.131 m2 20π mm b 100π mm ii 24π mm 2 1 7 Triangle = bh 2 2 Square = a b Rectangle = ab Parallelogram = ah (a + b ) h 1 Trapezium = (a + b ) h or 2 2 Rhombus = ah 1 Kite = a (b + c ) If you split the kite into two 2 triangles, one with height b and one with height c: 1 1 ab + ac 2 2 ab ac = + 2 2 a = (b + c ) 2 1 = a (b + c ) 2 8 a 158.76 cm2 b 103.7 m2 c 121.8 cm2 9 24.4 m 10 a 23 cm b 92 cm 11 a 178.75 cm2 b 32 956.5 mm2 NOT TO SCALE 14 a 136 cm2 b 3259.4 m2 c 8280 cm2 d 1120 mm2 e 468.8 cm2 f g 15.6 cm2 h 43.74 cm2 i j 160 cm2 l 706.9 cm2 688.1 m2 k 696.1 cm2 1715.3 cm2 15 153 000 cm2 (3sf) © Cambridge University Press 2018 13 Worksheet 7: Perimeter, area and volume 16 2940 cuboids 17 a x = c 180° π 1 2π 18 a Surface area of B = 1 2 b Area = r 2 4 π(kr )2 = 4 πk 2r 2 = k 2 (4 πr 2 ) 4 4 4 b Volume of B = π(kr )3 = πk 3r 3 = k 3 πr 3 3 3 3 = k3 × volume of A c 1 16 d 1 ( p) 3 2 = k 2 × surface area for A © Cambridge University Press 2018 14 Answers to Extended revision exercises: Data handling Worksheet 8: Introduction to probability 3 10 1 c 10 e 0 1 2 a 2 2 5 1 d 5 1 a b b 1 c 4 1 2 3 d 4 1 13 5 g 13 8 i 13 d g 0 h 35 142 65 c 71 8 a 7 a 12 13 5 h 26 e 1 2 1 c 2 1 e 2 6 a f b f b d 1 2 1 6 5 6 5 6 29 142 12 71 3 a Diagrams will vary but must show six segments of the same size, and an indication that five segments are red and one is blue. Diagrams will vary but must show five segments of the same size, each one labelled with a letter from A, B, C, D, E. b i 4 a 1 11 1 5 ii 0 iii 4 5 iv b 2 11 9 c 11 e 0 4 d 11 1 16 1 c 16 7 e 16 15 16 9 d 16 5 a 2 5 v b 3 5 Diagrams will vary but must show three segments of the same size, and an indication that one is blue, one is white and one is black. b © Cambridge University Press 2018 15 Worksheet 8: Introduction to probability 9 a c Black dice White dice 1 2 3 4 5 6 Diagrams will vary but must show eight segments of the same size, and an indication that only one segment is not coloured blue. d b i iii 2 3 4 5 6 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 1, 3 2, 3 3, 3 4, 3 5, 3 6, 3 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4 1, 5 2, 5 3, 5 4, 5 5, 5 6, 5 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6 1 18 ii 11 36 1 36 2 5 b 4 25 c 9 25 d 6 25 e 12 25 f 16 25 11 a 1 21 b 1 21 12 a 30 b 5 10 a Diagrams will vary but must show five segments of the same size, and an indication that only one segment is coloured blue, the other four are coloured black. 1 13 4 45 © Cambridge University Press 2018 16 Answers to Core revision exercises: Number Worksheet 9: Sequences and sets 1 a 17 b 28 c 80 d 11 e 1, 4, 9, 16, 25 f 5 a 5, 8, 11, 14, 17 b i b 0, -3, -6 c 25, 36, 49 1 1 1 e , , 32 64 128 3 a 2n + 4 d 24, 30, 36 f 6 a r ational because can be expressed as a fraction in its simplest terms b irrational because it’s a decimal that does not terminate and does not recur b 5n - 3 d -2n + 21 e 0.75n - 0.5 f i iii 15 002 3, 3.5, 4 c 2n . 4 a i ii 602 9, 18, 27, 36, 45 2 a -10, -12, -14 -1.2n + 10.2 n 1 2 3 4 5 6 d 1 4 7 10 13 16 d = 3n - 2 c rational because it’s an integer (and because it can 3 be expressed as a fraction ) 1 d irrational because 3 is irrational and you get an irrational number when you subtract (or add) an irrational number from another number e rational because the answer is zero iii 748 b i f n 1 2 3 4 5 6 d 5 10 15 20 25 30 rational because it is a recurring decimal g rational because it is a recurring decimal ii d = 5n h rational because the answer is the integer 2 (when you divide an irrational number by an irrational number, the answer can be rational or irrational) iii 1250 i irrational because it is not a terminating or recurring decimal (when you multiply an irrational number by an irrational number, the answer can be rational or irrational) j rational because when you square the square root of a number, you get the number and 17 is a rational number (it’s an integer) c i n 1 2 3 4 5 6 d 3 5 7 9 11 13 ii d = 2n + 1 iii 501 d i n 1 2 3 4 5 6 d 5 9 13 17 21 25 n 1 2 3 4 5 6 d 1 6 11 16 21 26 n 1 2 3 4 5 6 d 4 8 12 16 20 24 ii d = 4n + 1 iii 1001 e i ii d = 5n - 4 iii 1246 f 77 i ii d = 4n iii 1000 k irrational because this would mean multiplying a rational number (17) by an irrational number ( 17) l irrational because when you divide a rational number (1) by an irrational number (π) you get an irrational number m rational because it can be expressed as a fraction in 1 its simplest terms 3 n irrational because it cannot be expressed as a fraction in its simplest terms (and the decimal does not terminate or recur) o rational because the answer has a terminating decimal and can be expressed as a fraction in its 5 simplest terms 2 © Cambridge University Press 2018 17 Worksheet 9: Sequences and sets 7 a Integers from 9 to 16 inclusive b b 8 c {9, 11, 13, 15} d {9, 12, 15} e {9, 15} 8 a {x: x is an integer, 0 x 6} b {x: x is a letter in the word ‘Mathematics’} c {x: x is a prime, 2 x 19} d {x: x is a square, 1 x 64} c 9 a {1, 3, 6, 7, 9} b {1, 4, 7, 8, 12, 16, 20} c {1, 7} d {1, 3, 4, 6, 7, 8, 9, 12, 16, 20} 10 a 1, 1, 2, 3, 5, 8, 13, 21 b Fibonacci numbers c u13 11 a 1, 3, 6, 10, 15 b triangular numbers c n = 20 1 d un−1 = (n − 1)n 2 8 9 b 1 2 c 49 99 d 5 90 e 172 333 f 34 333 g 1 1 12 a 13 a F d T d e b T c T e f F F 14 a {2, 4, 5, 8, 11, 12, 16, 20} b {2, 3, 5, 6, 9, 11} c {2, 3, 4, 5, 6, 8, 9, 11, 12, 16, 20} d {2, 5, 11} e {2, 5, 11} f {2, 3, 4, 5, 6, 8, 9, 11, 12, 16, 20} 15 a 108 d 61 b 47 c 51 e f 26 57 f g 16 16 a © Cambridge University Press 2018 18 Worksheet 9: Sequences and sets g j h k i l © Cambridge University Press 2018 19 Answers to Core revision exercises: Algebra Worksheet 10: Straight lines and quadratic equations 1 a b c x y −1 0 1 2 3 2 3 4 5 6 x y −1 0 1 2 3 3 3 3 x −1 0 1 2 3 1 2 −1 0 1 2 y d e f g h i j 2 −1 − 1 2 x y −1 0 1 2 3 −6 −4 −2 0 2 x −1 0 1 2 3 y 1 2 0 −1 −1 x y −1 0 1 2 3 3 1 −1 −3 −5 x y −1 0 1 2 3 2 −1 0 1 2 x y −1 0 1 2 3 −6 −4 −2 0 2 7 7 7 7 7 −1 0 1 2 3 −1 0 1 2 3 0 −1 −2 −3 −4 x y x y − 1 2 1 2 3 a gradient = 1, equation: y = x + 1 b gradient = -1, equation: y = -x + 5 4 4x c gradient = , equation: y = +4 3 3 d gradient = 0, equation: y = 2 1 x e gradient = − , equation: y = − − 2 5 5 4 a y = 4x + 4 b y = -3x + 13 c y = 0.5x + 0.9 3 d y = 0.5 − x ( or 4 y + 3x = 2) 4 e y = 2.5x + 3.5 (or 5x - 2y = -7) 5 a x2 + 12x + 11 b x2 + 5x - 84 c y2 - y - 12 d x2 - 16 e p2 - 17p + 72 f g h2 - 10h - 75 h p2 - 13p + 30 i x2 - 6x + 9 4x2 - 2x - 12 6 ( a) and (h), (c) and (g), (d) and (e), (i) and (j) are all pairs of perpendicular lines. ( b) and (f) are not perpendicular to any other lines, or to each other. 1 7 gradient of AB = -3, gradient of DE = 3 x 13 8 equation: y = − + 3 3 © Cambridge University Press 2018 20 Worksheet 10: Straight lines and quadratic equations 9 gradient AB = − gradient AC = 2 4 , gradient BC = , 3 7 7. 4 4 7 gradient AB × gradient AC = − × = −1 , therefore 7 4 triangle ABC is right-angled. 10 a 12x2 - 19x - 21 c 6p - 13pq - 28q 2 b 2 + 13x - 45x2 2 e x -1 3 g p + 2pq + q 2 2 2 h (x + 7)2 i j (x - 5)2 k 2(x + 3)2 l 2(x2 - 8) m (x + 1)(x + 2) n (x - 2)(x - 4) o (x + 8)(x - 3) q (x + 1)(x - 3) p (x + 5)(x - 4) r 3(x + 5)(x - 5) s (xy + 5)(xy - 5) t (x - 6)2 h p + 3p q + 3pq + q e x = 3 or x = -2 f g x = -1 or x = -6 h x = -11 i j x = 7 or x = -6 l x = -2 or x = -5 2 3 x + y - z + 2xy 2 2 2 11 a x3 + 3x2 - 6x - 8 x = 3 or x = 1 k x = 6 or x = -8 b x - 19x + 30 17 a i 3 2 2x4 - 6x3 + 6x2 - 2x c t = 64 or t = 4 d d = ± 3 or ± b 343 cm3 -1 c i 1 2 1 d i 0 e i 4 f 1 b i i 18 b x=8 1 1 −1 2 , 1 2 ii 4.24 iii ii 4.47 iii (6, 3) ii 2 2 = 2.83 iii (3, -2) ii 10 iii (0, 0) 1 ii 4.12 iii − , 6 2 ii 5.66 iii (2, -1) 1 iii , 0 2 g i ii 13.9 h i 3 4 ii 10 iii (5, 5) i − ii 6.71 iii 1 2 15 a (x + 4)(x - 4) c (x - 3)(x - 1) ( x ) +1= 2 x ⇒ ( x) −2 ⇒ ( x − 1) 1 1, −5 2 b 5(x + 2)(x - 2) d (x + 4)(x - 7) 2 2 2 12 − 7 i x +1= 0 b p = 36 or p = 25 x 3 3x 2 3x + + +1 8 4 2 12 a V = 8x3 + 36x2 + 54x + 27 g 14 a i 2 ⇒ y 2 − 7 x + 12 = 0 ii x = 9 or x = 16 e x3 + 6x2 + 12x + 8 3x 2 13 a V = x3 - 9x - 4 2 ( x) −2 ⇒ d -x4 + 8x2 - 16 x = 1 or x = -6 x − 7 x + 12 = 0 c 2x - 10x - 44x - 32 f b x=2 d x = 7 or x = -5 2 k m4 - 4m3n + 6m2n2 - 4mn3 + n4 3 2(1 + xy)(1 - xy) c x = -1 or x = -2 j 4 g (4x + 3)(4x - 3) (7 + x)(7 - x) f 6p q + 8pq - 9p - 12q 2 3 x - 2x y + y 4 f 16 a x = -7 2 2 2 i d x - 2x - 8 4 e (x + 8)(x - 8) x +1= 0 2 ⇒ x = 1 or 1(repeated solution) ⇒ x = 1is the only solution (Alternative answer: Let y = y+ x 1 =2 y y2 + 1 = 2 y (× y ) (−2 y ) y2 − 2 y + 1 = 0 ( y − 1)( y − 1) = 0 factorise y = 1, so as 1 = 1, x = 1) © Cambridge University Press 2018 21 Worksheet 10: Straight lines and quadratic equations 19 a a = 2, b = -7 20 x = 4 or x = 8 b min value = -7, when x = -2 c x = 0.65 and x = - 4.65 © Cambridge University Press 2018 22 Answers to Extended revision exercises: Shape, space and measures Worksheet 11: Pythagoras’ theorem and similar shapes d If (a, b, c) is a Pythagorean Triple then a2 + b2 = c2 1 A and K If (ka, kb, kc) is a Pythagorean Triple then (ka)2 + (kb)2 = (kc)2 B, F and I C and G ⇒ k 2a2 + k 2b2 = k 2c 2 H and J 2 a 12 cm b 14.1 cm (3sf) c 25 mm d 44 cm e 50 mm f g 1m h 1.7 m e Any two of: 3 149 m 7, 24, 25 4 3.7 km 10, 24, 26 5 a y = 18 mm 18, 24, 30 b y = 4.29 cm 24, 32, 40 c x = 1.2 cm, y = 1.3 cm 24, 70, 74 d x = 27.5 mm, y = 30 mm 24, 143, 145 b yes c no 0 no 7 a 8.25 (3sf) b 4.24 (3sf) c 18.0 (3sf) d 5.10 (3sf) e 17.7 (3sf) f g 6.40 (3sf) h 8 a ABD = ACD, SSS or RHS or ASA or SAS b MNO = QPO, SSS or SAS c no a2 g CAB = DEF, ASA h RQP = RTS, RHS no (sides are not marked as the same length) 9 64 : 27 10 a 6 m blarger tank volume = 402 m (3sf), smaller tank volume = 170 m3 (3sf) 12 a 2.87 m 2 c2 u2 − v 2 u2 + v 2 (uv ) + 2 = 2 2 u2v 2 + ⇒ 2 u 4 − 2u2v 2 + v 4 u2 + 2u2v 2 + v 2 = 4 4 4u2v 2 + u 4 − 2u2v 2 + v 4 = u 4 + 2u2v 2 + v 4 ⇒ u 4 + 2u2v 2 + v 4 = u 4 + 2u2v 2 + v 4 c Let a = prime (p), then a = p × 1 (so u = p, v = 1) If b and c differ by 1 then: ⇒ b +1= c ⇒ p2 − 12 p2 + 12 +1= 2 2 ⇒ p2 1 p2 1 − +1= + 2 2 2 2 ⇒ p2 1 p2 1 + = + 2 2 2 2 3 b 1:4 2 = b2 (If a = 17 then uv = 1 × 17 because 17 is a prime number and its only factors are 1 and itself. If you substitute the values of 1 and 17 into the formulae for b and c then you get b = 144 and c = 145.) no 11 a 1 : 2 + b 17, 144, 145 e CAB = CDE, SSS or SAS i 14 a ⇒ d WXY = YZW, SSS f 24, 45, 51 3 (a2 + b2 ) 2 a 2 + b2 = c 2 15 cm 6 a no (÷ k ) k 2 (a2 + b2 ) = k 2c 2 b 0.0298 m3 13 a 152 + 202 = 225 + 400 = 625 = 252 b 62 + 82 = 36 + 64 = 100 = 102 c (3k)2 + (4k)2 = 9k2 + 16k2 = 25k2 = (5k)2 © Cambridge University Press 2018 23 Worksheet 11: Pythagoras’ theorem and similar shapes 15 Diagram of the cuboid with sides x cm, y cm and z cm: Let x = a, y = b, z = c and the diagonal of the base face (x × y) = e The diagonal of the cuboid forms a triangle with the e = a 2 + b2 diagonal of the base face: So, e = a2 + b2 and d = e 2 + c 2 ⇒ d2 = ( a 2 + b2 ) +c 2 2 d 2 = a 2 + b2 + c 2 d = a 2 + b2 + c 2 © Cambridge University Press 2018 24 Answers to Extended revision exercises: Data handling Worksheet 12: Averages and measures of spread 1 a mean = 4.58, median = 4.5, mode = 3, range = 7 b mean = 5.27, median = 6, mode = no mode, range = 9 c mean = 7.6, median = 12, mode = 12 , range = 21 d mean = 5.14, median = 5, mode = no mode, range = 7 e mean = 7.45, median = 8, mode = 8, range = 11 4 a 126 b 23 5 a Anna = 40, Zane = 40 b no c mean, no extreme data 6 a Shoe size 4 Frequency 7 fx 28 5 6 30 2 mode = 11 mins, median = 23.5 mins, range = 53 mins 6 14 84 3 a 7 8 56 8 6 48 f mean = 49, median = 48, mode = 48, range = 18 g mean = 11.5, median = 11.5, mode = 11.8, range = 1 h mean = 74.6, median = 76.5, mode = 77, range = 16 Outcome 1 Frequency 2 fx 2 2 4 8 9 6 54 3 6 18 10 3 30 4 3 12 5 2 10 6 2 12 mean = 3.26, median = 3, mode = 3, range = 5 b c 6.6 d 6 e m ean, there is no extreme data and will be an understandable average, OR mode, owners of shoe shops will want to know the most common shoe size so they can stock enough shoes of that size Outcome 123 Frequency 8 fx 984 124 4 496 125 5 625 126 6 756 127 7 889 8 a 74 128 8 1024 129 9 1161 9 mean = $33.67, modal class = $40 – $49.99, median class = $30 – $39.99 mean = 126, median = 127, mode = 129, range = 6 c b 6 Outcome 7 Frequency 12 fx 84 8 9 72 9 16 144 10 13 130 11 20 220 12 20 240 mean = 9.89, median = 10, mode = no mode, range = 5 7 a I t is the age where there are the same number of people below and above it. b Th e median is quite low so there are a lot of children and young people. This may mean the birth rate is high or that life expectancy is low. The median is getting larger so life expectancy may be increasing. b 56 c 98 10 a median = 3 upper quartile = 4.25 lower quartile = 2 interquartile range = 2.25 0 1 2 3 4 5 6 7 b median = 30 upper quartile = 42.5 lower quartile = 20 © Cambridge University Press 2018 25 Worksheet 12: Averages and measures of spread interquartile range = 22.5 mean = = 0 10 20 30 40 50 60 70 n + (n + 1) + (n + 2) + (n + 3) + (n + 4) + (n + 5) 6 6n + 15 6 =n+2 c median = 3 1 2 mean of 3rd and 4th largest integers = mean of c + d: upper quartile = 5.25 lower quartile = 2 mean of c + d = interquartile range = 3.25 = 0 2 4 6 8 10 13 a Mixture 1 Leaf a=n c = (n + 2) e = (n + 4) = n + (n + 1) + (n + 2) + (n + 3) + (n + 4) 5 5n + 10 5 b =n+2 This is the third integer: c = n + 2 12 Let a, b, c, d, e and f be represented as follows: b = (n + 1) Mixture 1 e = (n + 4) f = (n + 5) 11 15789 01146789 268 123567 c Mixture 2 d = (n + 3) Mixture 2 Leaf Range Q1 Median Q3 IQR 26.35 30.25 32.45 6.1 Mixture 1 8 27.83 28.65 30.03 2.2 Mixture 2 4.6 a=n c = (n + 2) 25 26 27 28 29 30 31 32 33 320 85 7 9540 9753 210 d = (n + 3) 1 2 Stem 9532 532 b = (n + 1) mean = 2n + 5 2 =n+2 12 11 Let a, b, c, d and e be represented as follows: (n + 2) + (n + 3) 2 0 25 26 27 28 29 30 31 32 33 34 The data for Mixture 2 is more symmetrical than the data for Mixture 1 and the variation in Mixture 1 is higher. The data suggests that Mixture 2 has a lower viscosity than Mixture 1. © Cambridge University Press 2018 26 Answers to Extended revision exercises: Measurement Worksheet 13: Understanding measurement 1 a 4500 b 970 7 12.25 kg, 12.75 kg c 970 d 2.324 65 8 a i e 80 f g 5800 h 26 670 i 900 j 0.077 b 3851.35 k 0.0125 l 3 c 1502.79 2 a 9 km, 100 m 0.059 b 69 015 cm, 15 cm 4462 ii 3493.5 iii 10 466.1 iv 38 544 v 5440.5 vi 14 977.6 9 a 0.2 m c 285 mm, 2 mm d 4.99 m, 98 cm c 10 feet e 0.7 km, 15 m f e 1.8 m 3 a 700 5.5 km, 50 m b 1200 c 6420 d 70 000 e 0.009 543 f 4 a 56 000 Upper bound = 1.35 kg Could also be written as: b 11 000 d 485 000 e 0.149 f 1.05 kg combined mass < 1.35 kg 11 Lower bound = 5.88 m 0.000 019 6 Upper bound = 5.92 m 5 a Day Time in Time out Lunch Monday 8.15 a.m. 5.25 p.m. 45 mins Tuesday 8.17 a.m. 5.30 p.m. 30 mins Wednesday 8.23 a.m. 5.50 p.m. 45 mins Thursday 8.22 a.m. 6.00 p.m. 60 mins Friday 7.58 a.m. 7.00 p.m. 45 mins Hours worked 8 hours 25 mins 8 hours 43 mins 8 hours 42 mins 8 hours 38 mins 10 hours 17 mins b 44 hours 45 mins c $221.51 6 a 33.5, 34.5 d 4.5 m 10 Lower bound = 1.05 kg 4 423 000 c 34 400 b 1 foot d $194.93 b 12 877.5, 12 878.5 c 550, 650 (continuous data) or 649 (discrete data) Could also be written as: 5.88 m width < 5.92 m 12 a 4106.363 abc < 5182.427 b - 2.205 (a - b) < - 2.095 c 3452.095 c - (a - b) < 3551.205 1 d 0.685 < 0.840 ab a e 0.169 < 0.208 b c < 1694.033 f 1564.626 b − a (a + b) < 1.212 g 1.166 b 13 4.6 cm x < 6.4 cm d 15.335, 15.345 e 12.685, 12.695 f 4.25, 4.75 g 665, 675 (continuous data) or 674 (discrete data) h 3.1415, 3.1425 © Cambridge University Press 2018 27 Answers to Extended revision exercises: Algebra Worksheet 14: Further solving of equations and inequalities 1 a x = 4, y = 0 b x = 12, y = 4 c x = 1, y = 4 d x = 5.43, y = -3.86 e x = 8.57, y = 3.43 f g x = 4, y = 3 h x = 2.5, y = 2 2 a x = 2, y = 5 c x = 0, y = 3 c x = -3, y = 9 b x = 5, y = 2 d x = -2, y = 1 3 a x = 0.5, y = 1.5 d x = 2, y = 5 b x = 1, y = − 4 a x4 1 2 b x6 c x6 x = 4, y = 2 © Cambridge University Press 2018 28 Worksheet 14: Further solving of equations and inequalities d x<8 c e x –6 f x 18 1 3 d Also acceptable is: x g 55 3 x>− 5 8 e 5 a f b g © Cambridge University Press 2018 29 Worksheet 14: Further solving of equations and inequalities Profit = 5c + 4t and this has its maximum value in the shaded region when t = 13 and c = 27. He should therefore order 13 T-shirts and 27 caps. h 8 a x = 2, y = 6 b x = 1.67, y = 2.67 9 a (x − 3) (2x − 3) b 2(x + 1) (5x − 7) c (3x + 2) (4x + 5) d 6(x + 2) (x − 1) 10 a 0.65, −4.65 6 b −0.84, −7.16 c 1.54, − 0.87 d 0.82, −1.82 11 a 1.84, −0.18 b 1.18, − 0.85 c 0.92, −3.25 d x = 1.70, x = − 4.70 e 1.45, −3.45 f g 1 h −0.62, 1.62 12 a xy t 10 and c 25 c 2t d 2x2 + 3x e x+3 f g x−1 h 2x – 1 15c + 8t 360 b x (2 x + 1) 6(x + 1)(4 x − 5) c x+2 2 d 7 x − 11 (x + 3)(x − 5) e 2x − 7 (x + 4)2 f 2(x 2 + 4) x2 − 4 h 4 x 2 − 3x + 3 x − x3 i c t = 10 2 (x − 4)(x − 2)(x + 1) 14 a i 40 x+1 (2 x − 1) (x + 1) 3 2 g 2 x − 418 x − 213x + 117 x − 13x + 36 t + c 40 32 ii − 3 iii 5 b when x = –7 answer is zero c − 3 x −7 30 2 1 15 x = , y = 3 4 c = 25 t + c = 40 20 10 15t + 8c = 360 c = 2t 0 b y c x − 2xy 13 a 7 Inequalities 4.44, 0.56 10 20 t 16 −b + b2 − 4ac −b − b2 − 4ac − 2a 2a 2 −b + b − 4ac + b + b2 − 4ac = 2a 2 2 b − 4ac = 2a 2 b − 4ac = a © Cambridge University Press 2018 30 Worksheet 14: Further solving of equations and inequalities 17 If x = 1 + 1 denominator is 1 + 1 1+ 1+… then x − 1 = 1 + so x − 1 = 1 1+ 1 1+… =x 1 1+… 1 so x − 1 = x 1− 5 1+ 5 x= , 2 2 −1 1 1+ 1 1+… 1 1+ 18 2 © Cambridge University Press 2018 31 Answers to Extended revision exercises: Shape, space and measures Worksheet 15: Scale drawings, bearings and trigonometry 1 a 5 a 10.1 b 14.3 b c 8.09 d 11.9 e 8.46 f 6 185 m c d 1.74 7 a 64.2° NOT TO SCALE b 4.36 m 8 50.3° 9 a x = 30° or x = 150° b x = 120° or x = 240° e c x = 44.4° or x = 135.6° d x = 60° or x = 240° f e x = 210° or x = 330° NOT TO SCALE 2 f x = 30° or x = 60° or x = 210° or x = 240° g x = 30° or x = 210° h x = 135° or x = 225° i x = 45° or x = 225° j x = 40° or x = 80° or x = 160° or x = 200° or x = 280° or x = 320° 10 a x = 190° or x = 310° b x = 56.3° or x = 236.3° c x = 72.2° or x = 117.8°, 297.8°, 252.2° 11 AB = 9.90 cm, AC = 5.43 cm 12 E = 22.2°, F = 34.8°, DE = 89.2 mm 13 a 992 mm2 b 585 mm2 14 54 m 15 10.2 cm 3 a 150° b 160° 4 a 36.9° b 23.2° c 45.6° d 66.0° e 68.0° f 9.6° 16 a 4.83 m c 5.97 m 17 a 9.28 km (3sf) b 58.3° d 6.77 m b 268.0° (1dp) 18 a A = 150° B = 190° b A = 134.730 km, B = 153.209 km © Cambridge University Press 2018 32 Answers to Extended revision exercises: Data handling Worksheet 16: Scatter diagrams and correlation 1 a iA negative correlation. The more hours of watching TV, the less the test score. iv iiA positive correlation. The longer the length of arm, the higher the bowling speed. iiiNo correlation. The month of birth has no effect on mass. ivA negative correlation. the more cigarettes smoked daily, the less the length of life. vA positive correlation. The taller one is, the bigger the shoe size b i ii v 2 Mid-year mark x coordinate iii 78 57 30 74 74 88 94 83 70 61 64 49 End-ofyear test mark y coordinate 73 51 39 80 74 73 88 69 63 67 68 54 End-ofMid-year year test mark x mark y coordinate coordinate 92 86 41 50 75 64 84 77 55 58 90 80 89 87 95 96 67 70 45 50 70 64 29 34 © Cambridge University Press 2018 33 Worksheet 16: Scatter diagrams and correlation a, c b Strongly correlated d 66 or 67 e Quite accurate as there is such a strong correlation. 3 a Th ere is no correlation. As one variable increases (x), there is no increase or decrease in the other variable. b Th ere is no correlation. As one variable increases (y), there is no increase or decrease in the other variable. © Cambridge University Press 2018 34 Answers to Extended revision exercises: Number Worksheet 17: Managing money 1 a $200 b $886.24 b 33.3% c 2 $1080 3 £1500 4 a $2700 b $3680 c $980 5 $19.25 6 $29 400 7 a $817.60 b $952 c $672 d $1036 8 $830.72 9 a $63.05 b $4656.79 c $1109.16 10 a $2000 11 a Years Simple interest Compound interest b $9000 7 6300 8280.39 8 7200 9925.63 12 a 2.04x b 3.1216x c $200 000 © Cambridge University Press 2018 35 Answers to Extended revision exercises: Algebra Worksheet 18: Curved graphs 1 a b c x -6 -5 -4 -3 -2 y -33 -22 -13 -6 x y −6 50 −5 37 −4 26 x y −6 4 −5 1 −4 0 2 a x = 1, x = 3 0 1 2 3 4 5 6 -1 -1 2 3 2 -1 -6 -13 -22 -33 −3 17 −2 10 −1 5 0 2 1 1 2 2 3 5 4 10 5 17 6 26 −3 1 −2 4 −1 9 0 16 1 25 2 36 3 49 4 64 5 81 6 100 b x = 0, x = 4 c x = 4.2, x = -0.2 3 x = 1, y = 0/ or x = 3.5, y = 1.25 © Cambridge University Press 2018 36 Worksheet 18: Curved graphs 4 x −5 −4 −3 −2 y −0.4 −0.5 −0.67 −1 −1 −2 1 2 2 3 4 5 1 0.67 0.5 0.4 x −1 −0.5 0 0.5 1 1.5 2 2.5 3 4 y 3 0.875 0 −0.875 −1 0.375 4 10.625 21 35.875 x −5 −4 −3 −2 −1 1 2 3 4 5 y 0.2 0.25 0.33 0.5 1 −1 −0.5 −0.33 −0.25 −0.2 x −1 −0.5 0 0.5 1 1.5 2 2.5 y 4 4.25 2 0.25 2 3.58 5.5 7.85 © Cambridge University Press 2018 37 Worksheet 18: Curved graphs 5 a 8 a gradient = - 4 y b gradient = 12 y 9 f(x) = x2 − 4x + 3 3 y = x2 – 2x – 7 x 0 –1 1 3 –1.8 0 3.8 x b f(x) = -1 [when x = 2] c x=2 6 –7 y y = x + 4x + 3 2 a i 4 4 ii -6 b x = 3.8, x = -1.8 (1dp) 3 10 a and c y = x+ 2 2 –1.6 1 –2.6 –4 –3 –2 x –1 –0.4 –0.6 1 –1 (x,y) = (-0.4, 1.6) and (-2.6, -0.6) 7 a y 2 f 1 x –4 –3 –2 –1 0 –1 1 2 3 4g –2 –3 g b x = 1 or x = -1 c 1.5 units © Cambridge University Press 2018 38 Worksheet 18: Curved graphs b x = 1 and x = -1.5 (answers within the range of -1.5 to - 1.6 are acceptable) d (1.7, 4.4) and (-1.2, -1.4) (1dp) e A t the points of intersection, the two equations are equal, so: 2x2 + x - 3 = 2x + 1 I f you rearrange this equation, you get 2x2 - x - 4 = 0. 11 a and b c x=1 d It is the tangent to the curve at the point (1,1). e 2 dy 13 a dx dy b dx dy c dx dy d dx dy e dx = 3x2 = -2 = 6x2 + 4 = 6x - 10 = 9 4x 14 y = 4x - 5 1 4 +2 15 a x = 1 b x=2 16 a y y = x3 – 3x2 4 3 2 1 –3 –2 (3, 0) (0, 0) –1 0 –1 1 2 3 4 x 5 –2 –3 –4 c ±1.41 12 a and b b (2, –4) y 4 3 y = x (x – 1)(x + 1) 2 (–0.58, 0.38) 1 (–1, 0) (0, 0) (1, 0) –3 –2 –1 0 1 2 3 –1 (0.58, –0.38) x 4 –2 –3 –4 © Cambridge University Press 2018 39 Answers to Extended revision exercises: Shape, space and measures Worksheet 19: Symmetry and loci 1 a C = 1, A = 1, M = 1, B = 1, R = 0, I = 2, D = 1, G = 0, E = 1 b I = 2, all other letters have roational symmetry of order 1. 2 Students’ own answers: drawing and naming of a rectangle or rhombus. 3 Answers will vary: student must copy shape and draw its mirror image according to the line of symmetry they have chosen. Student must draw in the line of symmetry. 4 a 7.75 cm b 13.9 cm c 25.4 cm 5 Working and reasoning may vary but size of angle should be as given below. Students must show their working and valid reasoning using statements that demonstrate knowledge of angle properties and relationships. a 40° b 22° c 45° d 40° e 122° f g 64° h 60° i j 33° l 90° 144° k 37° 64° 6 22 cm 7 75° 8 a angle QSP = 80° (alternate segment theorem) b angle SQP = 60° (angle sum of triangle) c angle PBQ = 60° (angle sum of triangle) d angle QRS = 140° (PQRS is cyclic quadrilateral) © Cambridge University Press 2018 40 Answers to Extended revision exercises: Data handling Worksheet 20: Histograms and frequency distribution diagrams 1 a 1.3-1.4 kg b 1 kg c 5 5 a d 85 2 a 60-80 1 time < 21 b About 23 21 time < 31 c N o. We are not given the distribution of marks in each bar. Histogram to show the distribution of the mass of 200 students 3 seconds 31 time < 41 41 time < 46 seconds 1 time < 21 21 time < 31 31 time < 41 41 time < 46 Ages 10 9 3 b 21 time < 31 c 4 a Frequency 8 Frequency 8 Frequency density 0.4 10 1 9 0.9 3 0.6 Frequency 10 age < 15 6 15 age < 20 24 20 age < 25 3 b H istogram to show the age of young adults attending a youth camp 6 14.3 (3sf) © Cambridge University Press 2018 41 Worksheet 20: Histograms and frequency distribution diagrams 7 a Mass Cumulative frequency 0 < m 3 3 < m 3.5 3.5< m 4 4 < m 4.5 4.5 < m 6 8 57 92 b c i 8 99 100 c 3.4 kg ii 3.7kg iii 0.5kg iv 43 Swimming time (x minutes) Frequency density 3 0 x < 10 10 x < 15 15 x < 25 25 x < 30 30 x < 40 9 4.1 6.6 2.5 © Cambridge University Press 2018 42 Answers to Extended revision exercises: Number Worksheet 21: Ratio, rate and proportion 1 a 188 km/h b 1.25 runs/min b c $12.52 $/m 2 a 3:7 b 5:8 c 8:9 d 3:2 e 1:2 f g 9 : 17 h 10 : 9 3 a 50 m Graph showing the relationship between the price of books and the number that can be bought for $600 9 : 44 b 300 m c 750 m 4 167 seconds or 2.78 minutes 5 stone = 21 wheelbarrows, cement = 7 wheelbarrows 6 1 cm : 90 km 7 a 12 c 21 b 2.4 d 0.16 8 625 : 1.44 9 a (a) 8 mm (b) 16 mm (c) 21 mm Example diagram labelling: NOT TO SCALE 12 a 192 b 300 c 168 d 200 e 137 (3sf) 13 a 53.3 (3sf) c 60 b (a) 6 m (6000 mm acceptable) (b) 12 m (12 000 mm acceptable) (c) 15.75 m (15 750 mm acceptable) 10 a 2 hours 40 mins c 4 hours 26 mins 40 sec b 2 hours 30 mins d 1 hour 40 mins e 50 000 mins or 34 days 17 hours 17 mins 11 a Price $1 $2 $5 $7.50 $10 $12 $15 $20 $25 $50 No of 600 300 120 80 60 50 40 30 24 12 books b 20 d 46.7 (3sf) e 48 14 a 280 cm2 c 4:1 2 15 a i y ∝ x c i 1 x2 m ∝T d i A∝ b i 16 a 1.5 c 8 17 a 4k y∝ 1 M b 1120 cm2 ii y = kx 2 k ii y = 2 x ii m = kT ii A = b 15 k M 2. 5 b k 2 © Cambridge University Press 2018 43 Worksheet 21: Ratio, rate and proportion 18 a i c i 200 m/min or 12 km/h ii 5.09 km/h b 4.25 km 19 a i 0.75 km ii 0.833 km iii 7.58 km c 5 km d 3.79 km/h 0 km/h ii 3 km/h 20 r = 3 iii 8 km/h iv 2 km/h 21 8192 b 1 0 until 10:20 the speed remains constant at 8 km/h 22 a F = 0.02125v2 b 200 m/s rom 10:20 to 10:30 the speed drops uniformly to F 2 km/h © Cambridge University Press 2018 44 Answers to Extended revision exercises: Algebra Worksheet 22: More equations, formulae and functions 2 3 c 27 1 a 4 d b 36 d 124 2 26.5 cm, 33.5 cm 3 144 km 4 Pauline = 11, Alice = 22 5 4 p.m. 6 80 km 7 0.98 m 8 b2 + 25b = 2000. Using the quadratic formula, b = 339 or –589, but as this is a length, –589 is an impossible answer, so the width is 339 mm. ote that the curves are symmetrical about y = x N when 9 96 km 10 a x = y −2 3 11 a 3 T b l= g 2π b 1.5 c -1.5 d 0.75 e 9 f g 36 h 27 i j 36 k 0.375 5 12 a 3 c 1 13 a x 9 c 7 14 a 3 x + 1 + 1 c g −1 (x ) = x 2 − 1 24 27 x 0 for y = x2 - 1 2 and x - 1 for y = x + 1 . 15 a x = - 2 and x = - 6 b x > 1 and x < - 1 c -3x3 d -2<x<3 e - 4 x < 1.5 f all values can be included b -5 d -3 x +1 b x −1 b 1± 5 2 © Cambridge University Press 2018 45 Answers to Extended revision exercises: Shape and space Worksheet 23: Transformations 1 A to B: reflection in the line x = -1 6 ab A to C: rotation 90°, clockwise, about (5, 1) 1 A to D: rotation 180°, about − , 0 2 A to E: rotation 90°, clockwise, about (0, 0) 2 c 1:2 7 a 26.4 c 14.9 d 4:1 b 3.0 d 11.1 8 a −6 e −7 9 3 −14 −3 4 a 10 9 c 12 −2 b 18 4 d −15 −2 e −5 5 a i AF = -a + b ii OE = -a + b b AD = AO + OD = -2a, BC = -a, so AD = 2BC © Cambridge University Press 2018 46 Worksheet 23: Transformations b b y D′′ y 12 5 10 A′ 6 2 4 C′′ B′ 2 1 –6 –4 D′ –2 0 B′′ 2 –2 4 6 8 10 –5 –4 –4 –3 –2 –1 0 –1 1 –2 Z –6 Centre (0, 1) Y x –10 –8 C 4 A 3 A′′ 8 B ∴ Scale factor (–1) x 2 3 4 5 X C′ Centre (0, 1) –8 Scale factor -1 –10 c Rotation of 180° about centre (0, 1). 11 a Rotation of 90° anti-clockwise about (1, 1). 9 a (-1, 2) b Scale factor -2 y c A″(3, -1), B″(3, 1), C″(-1, 1), D″(-1, -1) 5 10 a 4 5 A' 2 1 –3 –2 –1 0 –1 –2 3 C 4 A 3 C' Z –5 –4 Y –3 –1 0 –1 –2 x 2 3 4 1 X B B' 1 2 x 1 –2 Z′′ Z′ –3 5 –4 Y′′ –3 Y′ 2 3 4 5 X′ X′′ y =– 3 2 x– 3 c A(3, 1) B(3, 3) C(2, 1) © Cambridge University Press 2018 47 Answers to Extended revision exercises: Data handling Worksheet 24: Probability using tree diagrams and Venn diagrams 1 a b 7 a 1 16 2 a 1 b i 27 ii 7 27 19 27 iv 1 27 iii 8 a 1 8 3 a 0.49 3 8 b 0.09 b i ii c 0.21 d 0.42 e 0.51 4 a Newspaper Online 10 15 16 9 b 9 c P(newspaper) = d P(neither) = 5 a 10 1 or 0.2 5 9 or 0.18 50 b P(maths) = 0.84 c P(maths or physics) = 0.96 5 9 6 a b 14 14 15 c 28 b n=4 9 a A 12 B 18 17 28 © Cambridge University Press 2018 48 Worksheet 24: Probability using tree diagrams and Venn diagrams b i 0.47 ii 0.24 iii 0.63 iv 0.63 11 a 14 24 c They are not. P(A) + P(B) ≠ P(A or B) 10 a Music 12 10 24 Maths 25 78 G B 13 23 G 10 23 B 14 23 G 9 23 B b 0.33 25 c 0.51 b P(music) = 0.264 c P(music given maths) = 0.243 © Cambridge University Press 2018 49