Cambridge IGCSE Mathematics (with coursework) 0581 — 2012 May/June Paper 2 · Variant 2

This document consists of 12 printed pages. IB12 06_0581_22/RP © UCLES 2012 [Turn over *9871983067* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/22 Paper 2 (Extended) May/June 2012 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 2012 0581/22/M/J/12 For Examiner's Use 1 The ferry from Helsinki to Travemunde leaves Helsinki at 17 30 on a Tuesday. The journey takes 28 hours 45 minutes. Work out the day and time that the ferry arrives in Travemunde. Answer Day Time [2] 2 T R I G O N O M E T R Y From the above word, write down the letters which have (a) exactly two lines of symmetry, Answer(a) [1] (b) rotational symmetry of order 2. Answer(b) [1] 3 For this question, 1 < x < 2 . Write the following in order of size, smallest first. x 5 5x 5 x x O 5 Answer < < < [2] 4 1 2 1 + 3 1 + 4 1 = 12 p Work out the value of p. Show all your working. Answer p = [2]

3 © UCLES 2012 0581/22/M/J/12 [Turn over For Examiner's Use 5 A lake has an area of 63 800 000 000 square metres. Write this area in square kilometres, correct to 2 significant figures. Answer km2 [2] 6 x is a positive integer and 15x – 43 < 5x + 2 . Work out the possible values of x. Answer [3] 7 8 cm 6 cm r NOT TO SCALE The perimeter of the rectangle is the same length as the circumference of the circle. Calculate the radius, r, of the circle. Answer r = cm [3]

4 © UCLES 2012 0581/22/M/J/12 For Examiner's Use 8 A car company sells a scale model 10 1 of the size of one of its cars. Complete the following table. Scale Model Real Car Area of windscreen (cm2) 135 Volume of storage space (cm3) 408 000 [3] 9 P B A 170 m 78.3° 58.4 m NOT TO SCALE The line AB represents the glass walkway between the Petronas Towers in Kuala Lumpur. The walkway is 58.4 metres long and is 170 metres above the ground. The angle of elevation of the point P from A is 78.3°. Calculate the height of P above the ground. Answer m [3]

5 © UCLES 2012 0581/22/M/J/12 [Turn over For Examiner's Use 10 500 400 300 200 100 0 10 20 30 40 50 60 Time (min) Speed (m / min) The diagram shows the speed-time graph for a boat journey. (a) Work out the acceleration of the boat in metres / minute2. Answer(a) m / min 2 [1] (b) Calculate the total distance travelled by the boat. Give your answer in kilometres. Answer(b) km [2]

6 © UCLES 2012 0581/22/M/J/12 For Examiner's Use 11 y varies directly as the square of (x – 3). y = 16 when x = 1. Find y when x = 10. Answer y = [3] 12 North 5 km 8 km 150° C B A NOT TO SCALE A helicopter flies 8 km due north from A to B. It then flies 5 km from B to C and returns to A. Angle ABC = 150°. (a) Calculate the area of triangle ABC. Answer(a) km2 [2] (b) Find the bearing of B from C. Answer(b) [2]

7 © UCLES 2012 0581/22/M/J/12 [Turn over For Examiner's Use 13 The taxi fare in a city is $3 and then $0.40 for every kilometre travelled. (a) A taxi fare is $9. How far has the taxi travelled? Answer(a) km [2] (b) Taxi fares cost 30 % more at night. How much does a $9 daytime journey cost at night? Answer(b) $ [2] 14 A B 3 2 1 0 1 2 3 4 5 6 7 8 x y (a) Describe fully the single transformation that maps triangle A onto triangle B. Answer(a) [3] (b) Find the 2 × 2 matrix which represents this transformation. Answer(b)           [2]

8 © UCLES 2012 0581/22/M/J/12 For Examiner's Use 15 60 50 40 30 20 10 0 10 20 30 40 50 Height (cm) Cumulative frequency The cumulative frequency diagram shows information about the heights of 60 tomato plants. Use the diagram to find (a) the median, Answer(a) cm [1] (b) the lower quartile, Answer(b) cm [1] (c) the interquartile range, Answer(c) cm [1] (d) the probability that the height of a tomato plant, chosen at random, will be more than 15 cm. Answer(d) [2]

9 © UCLES 2012 0581/22/M/J/12 [Turn over For Examiner's Use 16 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 x y The diagram shows the graph of y = 2 x + x 2 , for 0 < x Y 8. (a) Use the graph to solve the equation 2 x + x 2 = 3. Answer (a) x = or x = [2] (b) By drawing a suitable tangent, work out an estimate of the gradient of the graph where x = 1. Answer(b) [3]

10 © UCLES 2012 0581/22/M/J/12 For Examiner's Use 17 (a) Find the co-ordinates of the midpoint of the line joining A(–8, 3) and B(–2, –3). Answer(a) ( , ) [2] (b) The line y = 4x + c passes through (2, 6). Find the value of c. Answer(b) c = [1] (c) The lines 5x = 4y + 10 and 2y = kx – 4 are parallel. Find the value of k. Answer(c) k = [2]

11 © UCLES 2012 0581/22/M/J/12 [Turn over For Examiner's Use 18 f(x) = (x + 2)3 – 5 g(x) = 2x + 10 h(x) = x 1 , x ≠ 0 Find (a) gf (x), Answer(a) gf(x) = [2] (b) f –1(x), Answer(b) f -1(x) = [3] (c) gh(– 5 1 ). Answer(c) [2] 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 2012 0581/22/M/J/12 For Examiner's Use 19 Find the values of x for which (a)         −7 2 0 0 1 x has no inverse, Answer(a) x = [2] (b)           −8 0 0 1 2 x is the identity matrix, Answer (b) x = or x = [3] (c)         −2 0 0 1 x represents a stretch with factor 3 and the x axis invariant. Answer (c) x = [2]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2012 question paper for the guidance of teachers 0581 MATHEMATICS 0581/22 Paper 2 (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. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the May/June 2012 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 2012 0581 22 © University of Cambridge International Examinations 2012 Abbreviations cao correct answer only cso correct solution only dep dependent ft follow through after error isw ignore subsequent working oe or equivalent SC Special Case www without wrong working soi seen or implied Qu Answers Mark Part marks 1 Wednesday 22 15 or 10 15pm 2 B1 B1 2 (a) (b) I cao I N cao 1 1 3 x – 5 5 x x 5 5x 2 M1 evaluating all 4 expressions for one value in the range. (1 and 2 are out of range) 4 25 (correct working essential) 2 M1 for 18 + 4 + 3 with denominator 12 must be soi (oe is possible) 5 64000 or 6.4 × 104 2 SC1 for 63800 or 6.38 × 104 or figs 64 or 6.4 × 10k in answer space. 6 1, 2, 3, 4 3 M1 10x < 45 A1 x < 4.5 7 4.46 or 4.456 to 4.459 cao 3 B1 for 28 seen M1ft for π 2 28 their oe or better. 8 13500 408 3 M1 135 × 102 or 408000 ÷ 103 oe A1 A1 9 452 3 M1 tan 78.3 = 4. 58 x M1 “282” + 170 SC2 282 in answer space 10 (a) (b) 50 15 1 2 M1 finding area under graph SC1 15000 11 196 3 M1 y = k(x – 3)2 A1 k = 4 M1 y = k x 2) 3 ( − A1 k = 4 1

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 22 © University of Cambridge International Examinations 2012 12 (a) (b) 10(.0) 210 2 2 M1 2 1 × 8 × 5 × sin 150 M1 30° correctly placed at B or C oe 13 (a) (b) 15 11.7(0) 2 2 M1 for 4.0 )3 9 ( − oe M1 for 9 × 1.3 oe 14 (a) (b) Shear, SF2, x axis invariant       1 0 2 1 3 2ft B1 shear B1 SF2 B1 x axis invariant       1 0 1 k 2 marks if k = 2 or their SF in (a) 1 mark for any other k, k ≠ 0 15 (a) (b) (c) (d) 29 to 29.5 20 to 20.5 14 to 14.5 15 13 oe or 0.867 1 1 1 2 M1 8 seen 16 (a) (b) 0.7 to 0.8 and 5.2 to 5.4 –2 to –1 but must have a tangent at x = 1 for full marks 2 3 B1 B1 M1 drawing tangent at x = 1 M1 for using ystep/xstep on their tangent wherever it is drawn 17 (a) (b) (c) (–5, 0) –2 2 2 1 or 2 5 2 1 2 B1 (k, 0) or (–5, k) M1 4 5 = 2 k oe 18 (a) (b) (c) 2(x + 2)3 or 2x3 + 12x2 + 24x + 16 3√(x + 5) – 2 0 2 3 2 M1 v. clear evidence of f(x) × 2 then add 10 M1 correct first step M1 correct second step M1 g(–5) seen or 2 × –5 + 10 19 (a) (b) (c) 3 2 1 3 and –3 5 2 3 2 M1 2x – 7 = 0 M1 x2 – 8 = 1 A1 x = 3 A1 x = –3 M1 x – 2 = 3