Cambridge IGCSE Mathematics (with coursework) 0581 — 2010 May/June Paper 2 · Variant 2

This document consists of 12 printed pages. IB10 06_0581_22/FP © UCLES 2010 [Turn over *3836469639* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/22 Paper 2 (Extended) May/June 2010 1 hour 30 minutes Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments Mathematical tables (optional) Tracing paper (optional) READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all questions. If working is needed for any question it must be shown below that question. Electronic calculators should be used. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π , use either your calculator value or 3.142. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 70. www.XtremePapers.com

2 © UCLES 2010 0581/22/M/J/10 For Examiner's Use 1 For the diagram, write down (a) the order of rotational symmetry, Answer(a) [1] (b) the number of lines of symmetry. Answer(b) [1] 2 Calculate 3sin120° − 4(sin120°)3. Answer [2] 3 Write the following in order of size, smallest first. 2 3 2 3 − 3 3 2 2 − Answer < < < [2]

3 © UCLES 2010 0581/22/M/J/10 [Turn over For Examiner's Use 4 Write as a single fraction 8 3a + 5 4 . Answer [2] 5 Write 28 × 82 × 4-2 in the form 2n. Answer [2] 6 Change 64 square metres into square millimetres. Give your answer in standard form. Answer mm2 [2] 7 A B C The shaded area in the diagram shows the set (A ∩C ) ∩B'. Write down the set shown by the shaded area in each diagram below. A B C A B C [2]

4 © UCLES 2010 0581/22/M/J/10 For Examiner's Use 8 0.5 1 0 30° 60° 90° 120° 150° 180° –1 –0.5 y x y = sinx y = cosx The diagram shows accurate graphs of y = sinx and y = cosx for 0° Y x Y 180°. Use the graph to solve the equations (a) sinx – cosx = 0, Answer(a) x = [1] (b) sinx – cosx = 0.5. Answer(b) x = [2] 9 A fence is made from 32 identical pieces of wood, each of length 2 metres correct to the nearest centimetre. Calculate the lower bound for the total length of the wood used to make this fence. Write down your full calculator display. Answer m [3]

5 © UCLES 2010 0581/22/M/J/10 [Turn over For Examiner's Use 10 Make x the subject of the formula. 3 x P x + = Answer x = [4] 11 O Q C P T 6 cm 4 cm NOT TO SCALE Two circles, centres O and C, of radius 6 cm and 4 cm respectively, touch at Q. PT is a tangent to both circles. (a) Write down the distance OC. Answer(a) OC = cm [1] (b) Calculate the distance PT. Answer(b) PT = cm [3]

6 © UCLES 2010 0581/22/M/J/10 For Examiner's Use 12 The diagram represents the ski lift in Queenstown New Zealand. 37.1° B T h 730 m NOT TO SCALE (a) The length of the cable from the bottom, B, to the top, T, is 730 metres. The angle of elevation of T from B is 37.1°. Calculate the change in altitude, h metres, from the bottom to the top. Answer(a) m [2] (b) The lift travels along the cable at 3.65 metres per second. Calculate how long it takes to travel from B to T. Give your answer in minutes and seconds. Answer(b) min s [2]

7 © UCLES 2010 0581/22/M/J/10 [Turn over For Examiner's Use 13 M = 6 3 4 5 1 x −          . (a) Find the matrix M. Answer(a) M = [2] (b) Simplify ( x 1 ) M. Answer(b) [2] 14 y x –2 –4 4 6 8 10 12 14 16 18 2 0 10 8 6 4 2 By shading the unwanted regions of the grid above, find and label the region R which satisfies the following four inequalities. y [ 2 x + y [ 6 y Y x + 4 x + 2y Y 18 [4]

8 © UCLES 2010 0581/22/M/J/10 For Examiner's Use 15 0 y = 4 x y 4 A B 2x + y = 8 3x + y = 18 NOT TO SCALE (a) The line y = 4 meets the line 2x + y = 8 at the point A. Find the co-ordinates of A. Answer(a) A ( , ) [1] (b) The line 3x + y = 18 meets the x axis at the point B. Find the co-ordinates of B. Answer(b) B ( , ) [1] (c) (i) Find the co-ordinates of the mid-point M of the line joining A to B. Answer(c)(i) M ( , ) [1] (ii) Find the equation of the line through M parallel to 3x + y = 18. Answer(c)(ii) [2]

9 © UCLES 2010 0581/22/M/J/10 [Turn over For Examiner's Use 16 The graphs show the speeds of two cyclists, Alonso and Boris. Alonso accelerated to 10 m/s, travelled at a steady speed and then slowed to a stop. 0 2 4 6 8 10 12 14 16 10 Speed (m / s) Time (seconds) Alonso Boris accelerated to his maximum speed, v m/s, and then slowed to a stop. 0 16 v Speed (m / s) Time (seconds) Boris NOT TO SCALE Both cyclists travelled the same distance in the 16 seconds. Calculate the maximum speed for Boris. Show all your working. Answer m/s [5]

10 © UCLES 2010 0581/22/M/J/10 For Examiner's Use 17 O Q R S T U V W P NOT TO SCALE 8 m The diagram shows the junction of four paths. In the junction there is a circular area covered in grass. This circle has centre O and radius 8 m. (a) Calculate the area of grass. Answer(a) m2 [2] (b) O Q P 12 m 45° NOT TO SCALE The arc PQ and the other three identical arcs, RS, TU and VW are each part of a circle, centre O, radius 12m. The angle POQ is 45°. The arcs PQ, RS, TU, VW and the circumference of the circle in part(a) are painted white. Calculate the total length painted white. Answer(b) m [4]

11 © UCLES 2010 0581/22/M/J/10 [Turn over For Examiner's Use 18 (a) f(x) = 1 – 2x. (i) Find f(-5). Answer(a)(i) [1] (ii) g(x) = 3x – 2. Find gf(x). Simplify your answer. Answer(a)(ii) [2] (b) h(x) = x2 – 5x – 11. Solve h(x) = 0. Show all your working and give your answer correct to 2 decimal places. Answer(b) x = or x = [4] Question 19 is printed on the next page.

12 Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. © UCLES 2010 0581/22/M/J/10 For Examiner's Use 19 The braking distance, d metres, for Alex’s car travelling at v km/h is given by the formula 200d = v(v + 40). (a) Calculate the missing values in the table. v (km/h) 0 20 40 60 80 100 120 d (metres) 0 16 48 96 [2] (b) On the grid below, draw the graph of 200d = v(v + 40) for 0 Y v Y 120. 120 110 100 90 80 70 60 50 40 30 20 10 100 90 80 70 60 50 40 30 20 10 0 Distance (metres) d v Speed (km / h) [3] (c) Find the braking distance when the car is travelling at 110 km/h. Answer(c) m [1] (d) Find the speed of the car when the braking distance is 80 m. Answer(d) km/h [1]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2010 question paper for the guidance of teachers 0581 MATHEMATICS 0581/22 Paper 22 (Extended), maximum raw mark 70 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes must be read in conjunction with the question papers and the report on the examination. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.XtremePapers.com

Page 2 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0581 22 © UCLES 2010 Qu. Answers Mark Part Marks 1 (a) 1 (b) 1 1 1 Allow none 2 0 2 M1 4sin3120 evaluated and rounding to 2.6 or better (2.598…) or 2 3 3 3 2 – 3 , 2 – 2 3 , 3 2 , 3 2 M1 correct decimals seen 4 40 32 15 + a oe 2 B1 15a + 32 seen or SC1 40 32 40 15 + a on answer line 5 210 2 M1 26 or 2–4 seen 6 6.4 × 107 2 M1 64 × 1002 × 102 or 64 000 000 oe 7 (A ∪ B ∪ C)' (A ∪ C)' ∩ B 1 1 or A' ∩ B' ∩ C' or A' ∩ (B ∪ C)' or A' ∩ C' ∩ B 8 (a) 43 to 47 (b) 64 to 68 1 2 SC1 23 to 27 9 63.84 cao 3 M1 figs 1995 M1 32 × their lower bound 10 1 3 − = P x 4 M1 for each of the four moves completed correctly 11 (a) 10(.0..) (b) 9.80 1 3 M2 √((a)2 – 22) or M1 PT2 + 22 = (a)2 12 (a) 440 (b) 3 min 20 sec 2 2 M1 sin 37.1 or cos 52.9 = 730 h oe M1 65 .3 730 13 (a)     + − 5 4 3 6 x x but not     + − 5 ) ( 4 3 6 x x 2 B1 6x – 3 or B1 4x + 5 in a (2 × 1) matrix on answer line (b) (6x2 + x + 5) cao 2 M1 any 1 × 1 matrix in answer space 14 4 Mark the position of the letter R (or the worst unshaded region if R is missing) as follows

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0581 22 © UCLES 2010 15 (a) (2, 4) (b) (6, 0) (c) (i) (4, 2) ft (ii) y = – 3x + 14 oe 1 1 1ft 2 From (a) and (b) M1 sub their (c)(i) into y = –3x + c oe 16 16 4 1 or 16.3 5 M1 finding the area under graph A1 130 M1 2 1 × 16 × v M1 equating and solving 17 (a) 201 2 M1 π × 82 (b) 87.9 or 88.0 4 M1 360 45 × 2 × π × 12 ….. d M1 2 × π × 8 ……………..e M1 ft for their (4d + e) which must come from multiples of π SC2 43.9 or 44.0 18 (a) (i) 11 (ii) 1 – 6x 1 2 M1 3(1 – 2x) – 2 (b) –1.65, 6.65 4 M1 2 5 k ± M1 √[(–5)2 – 4 × 1 × (–11)] or better A1 A1 19 (a) 6, 30, 70 (b) graph (c) 82.5 or ft ±1 (d) 108 or ft ±1 2 3 1ft 1ft B1 for 2 correct P2 7 plots correct from table P1 5 or 6 plots correct from table C1 smooth curve through the points in the given range within one small square of the plots or the correct position