Cambridge IGCSE Mathematics (with coursework) 0581 — 2011 May/June Paper 2 · Variant 2

This document consists of 12 printed pages. IB11 06_0581_22/FP © UCLES 2011 [Turn over *9520291400* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/22 Paper 2 (Extended) May/June 2011 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 2011 0581/22/M/J/11 For Examiner's Use 1 In the right-angled triangle ABC, cos C = 5 4 . Find angle A. A B C NOT TO SCALE Answer Angle A = [2] 2 Which of the following numbers are irrational? 3 2 36 3 + 6 π 0.75 48% 8 3 1 Answer [2] 3 Show that 1 9 5 ÷ 1 9 7 = 8 7 . Write down all the steps in your working. Answer [2]

3 © UCLES 2011 0581/22/M/J/11 [Turn over For Examiner's Use 4 5 3 < p < 3 2 Which of the following could be a value of p? 27 16 0.67 60% (0.8)2 9 4 Answer [2] 5 A meal on a boat costs 6 euros (€) or 11.5 Brunei dollars ($). In which currency does the meal cost less, on a day when the exchange rate is €1 = $1.9037? Write down all the steps in your working. Answer [2] 6 Use your calculator to find the value of 2 3 . Give your answer correct to 4 significant figures. Answer [2]

4 © UCLES 2011 0581/22/M/J/11 For Examiner's Use 7 Solve the equation 4x + 6 × 103 = 8 × 104. Give your answer in standard form. Answer x = [3] 8 p varies directly as the square root of q. p = 8 when q = 25. Find p when q = 100. Answer p = [3] 9 Ashraf takes 1500 steps to walk d metres from his home to the station. Each step is 90 centimetres correct to the nearest 10 cm. Find the lower bound and the upper bound for d. Answer Y d < [3]

5 © UCLES 2011 0581/22/M/J/11 [Turn over For Examiner's Use 10 The table shows the opening and closing times of a café. Mon Tue Wed Thu Fri Sat Sun Opening time 0600 0600 0600 0600 0600 (a) 0800 Closing time 2200 2200 2200 2200 2200 2200 1300 (a) The café is open for a total of 100 hours each week. Work out the opening time on Saturday. Answer(a) [2] (b) The owner decides to close the café at a later time on Sunday. This increases the total number of hours the café is open by 4%. Work out the new closing time on Sunday. Answer(b) [1] 11 Rearrange the formula c = b a − 4 to make a the subject. Answer a = [3]

6 © UCLES 2011 0581/22/M/J/11 For Examiner's Use 12 Solve the simultaneous equations. x – 5y = 0 15x + 10y = 17 Answer x = y = [3] 13 O y° z° x° 54° Q R P T NOT TO SCALE The points P, Q and R lie on a circle, centre O. TP and TQ are tangents to the circle. Angle TPQ = 54°. Calculate the value of (a) x, Answer(a) x = [1] (b) y, Answer(b) y = [1] (c) z. Answer(c) z = [2]

7 © UCLES 2011 0581/22/M/J/11 [Turn over For Examiner's Use 14 60 students recorded their favourite drink. The results are shown in the pie chart. 66° 120° Banana shake Apple juice Lemonade Cola NOT TO SCALE (a) Calculate the angle for the sector labelled Lemonade. Answer(a) [1] (b) Calculate the number of students who chose Banana shake. Answer(b) [1] (c) The pie chart has a radius of 3 cm. Calculate the arc length of the sector representing Cola. Answer(c) cm [2]

8 © UCLES 2011 0581/22/M/J/11 For Examiner's Use 15 Write the following as a single fraction in its simplest form. 5 1 + + x x – 1 + x x Answer [4] 16 C O B A a c P M Q NOT TO SCALE O is the origin and OABC is a parallelogram. CP = PB and AQ = QB. = a and = c . Find in terms of a and c, in their simplest form, (a) , Answer(a) = [2] (b) the position vector of M, where M is the midpoint of PQ. Answer(b) [2]

9 © UCLES 2011 0581/22/M/J/11 [Turn over For Examiner's Use 17 Simplify (a) 32x8 ÷ 8x32, Answer(a) [2] (b) 3 2 3 64      x . Answer(b) [2] 18 4 3 2 1 1 2 3 4 5 6 7 0 y x C B A The lines AB and CB intersect at B. (a) Find the co-ordinates of the midpoint of AB. Answer(a) ( , ) [1] (b) Find the equation of the line CB. Answer(b) [3]

10 © UCLES 2011 0581/22/M/J/11 For Examiner's Use 19 f(x) = x2 g(x) = 2x h(x) = 2x – 3 (a) Find g(3). Answer(a) [1] (b) Find hh(x) in its simplest form. Answer(b) [2] (c) Find fg(x + 1) in its simplest form. Answer(c) [2]

11 © UCLES 2011 0581/22/M/J/11 [Turn over For Examiner's Use 20 A B C (a) On the diagram above, using a straight edge and compasses only, construct (i) the bisector of angle ABC, [2] (ii) the locus of points which are equidistant from A and from B. [2] (b) Shade the region inside the triangle which is nearer to A than to B and nearer to AB than to BC. [1] Question 21 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 2011 0581/22/M/J/11 For Examiner's Use 21 (a) A = ( ) 3 2 B =       −4 6 (i) Work out AB. Answer(a)(i) [2] (ii) Work out BA. Answer(a)(ii) [2] (b) C =       1 1 1 3 Find C–1, the inverse of C. Answer(b) [2]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2011 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 2011 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 2011 0581 22 © University of Cambridge International Examinations 2011 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 Qu. Answers Mark Part Mark 1 53.1 2 B1 C = 36.9 seen, must have C stated or marked on the diagram or M1 sinA = 5 4 or tanA = 3 4 but must have A stated 2 6 3 + , π 2 –1 for each error or omission 3 Working must be shown 2 M1 9 14 and 9 16 M1 8 7 16 14 = oe or visible cancelling 4 0.82 2 M1 conversion of 27 16 (= 0.5(9...)) and 0.82 (= 0.64) to decimals seen 5 (6)€ or euros (with correct working) 2 M1 one of 6 × 1.9037 or 11.5 ÷ 1.9037 or 11.5 ÷ 6 seen 6 3.322 cao 2 B1 3.3219(…) or 3.32(20) seen 7 1.85 × 104 3 B2 18500 oe seen or M1 4x = 74000 or x = 2 × 104 – 1.5 × 103 8 16 3 M1 p = q k A1 k = 1.6 or 8/5 9 1275, 1425 3 B1 85 or 95 or 0.85 or 0.95 M1 their LB or UB × 1500 where 85 Y LB < 90 90 < UB Y 95 10 (a) (0)700 or 7 am (b) 1700 or 5 pm 2 1 M1 100 – (5 × their(22 – 6) + their(13 – 8)) or better soi 11 c bc + 4 or b c + 4 cao 3 M1 correct move completed M1 second correct move completed M1 third correct move completed 12 x = 1 y = 0.2 or 5 1 only 3 M1 consistent mult and add/subtraction A1 one value correct after M awarded 13 (a) 72 (b) 36 (c) 54 1 1 2ft ft 90 – (b) M1 POQ = 108

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0581 22 © University of Cambridge International Examinations 2011 14 (a) 84 (b) 15 (c) 6.28 1 1 2 M1 3 π 2 360 120 × × × oe 15 ) 5 )( 1 ( 3 1 + + − x x x www 4 M1 (x + 1)2 – x(x + 5) oe B1 x2 + x + x + 1 B1 denominator(s) (x + 1)(x + 5) or x2 + 6x + 5 16 (a) 2 1 a – 2 1 c oe (b) 4 3 a + 4 3 c oe 2 2 M1 correct but unsimplified e.g. 2 1 a + – 2 1 c M1 correct but unsimplified 17 (a) 4x–24 or 24 4 x (b) 16 2 x 2 2 B1 4xn B1 24 x k or kx–24 for any numerical k, n B1 k x2 or B1 16 n x SC1 ( 4 x )2 18 (a) (6, 1½) (b) y = – 5 1 x + 4 oe 1 3 B1 correct numerical format B1 correct m B1 correct c 19 (a) 8 (b) 4x – 9 (c) 22(x + 1) or 22x + 2 or 4x + 1 1 2 2 M1 2(2x – 3) – 3 seen M1 (2x + 1)2 seen 20 (a) (i) (ii) R (b) 2 2 1 B1 correct line B1 2 sets of correct arcs B1 correct line B1 two sets of correct arcs correct region, shaded or shown by the letter R 21 (a) (i) ( ) 0 brackets essential (ii)       − − 12 8 18 12 (b)       − − 3 1 1 1 2 1 2 2 2 M1 6 × 2 + 3 × –4 or 12 + –12 M1 any 2 × 2 matrix with 2 elements correct B1       d b c a 2 1 seen or B1       − − 3 1 1 1 k seen