Cambridge IGCSE Mathematics (with coursework) 0581 — 2012 May/June Paper 2 · Variant 3

This document consists of 12 printed pages. IB12 06_0581_23/RP © UCLES 2012 [Turn over *7561576524* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/23 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 0581/23/M/J/12 For Examiner's Use 1 B C D E A 120° 82° 73° x° NOT TO SCALE The diagram shows a quadrilateral ABCD. CDE is a straight line. Calculate the value of x. Answer x = [2] 2 Hans invests $750 for 8 years at a rate of 2% per year simple interest. Calculate the interest Hans receives. Answer $ [2] 3 (a) Calculate 3 0.9 1.5 22 7 + and write down your full calculator display. Answer(a) [1] (b) Write your answer to part (a) correct to 4 significant figures. Answer(b) [1]

3 © UCLES 2012 0581/23/M/J/12 [Turn over For Examiner's Use 4 Solve the inequality. 3y + 7 Y 2 – y Answer [2] 5 9 cm 12 cm 5 cm 5 cm NOT TO SCALE The diagram shows a quadrilateral. The lengths of the sides are given to the nearest centimetre. Calculate the upper bound of the perimeter of the quadrilateral. Answer cm [2] 6 C A B 28° 9 cm 15 cm NOT TO SCALE Calculate the area of triangle ABC. Answer cm2 [2]

4 © UCLES 2012 0581/23/M/J/12 For Examiner's Use 7 Height (h cm) 0 < h Y 10 10 < h Y 15 15 < h Y 30 Frequency 25 u 9 Frequency density 2.5 4.8 v The table shows information about the heights of some flowers. Calculate the values of u and v. Answer u = v = [2] 8 During her holiday, Hannah rents a bike. She pays a fixed cost of $8 and then a cost of $4.50 per day. Hannah pays with a $50 note and receives $10.50 change. Calculate for how many days Hannah rents the bike. Answer days [3] 9 Make w the subject of the formula. t = 2 – a w 3 Answer w = [3]

5 © UCLES 2012 0581/23/M/J/12 [Turn over For Examiner's Use 10 The periodic time, T, of a pendulum varies directly as the square root of its length, l. T = 6 when l = 9. Find T when l = 25. Answer T = [3] 11 Boris invests $280 for 2 years at a rate of 3% per year compound interest. Calculate the interest Boris receives at the end of the 2 years. Give your answer correct to 2 decimal places. Answer $ [4]

6 © UCLES 2012 0581/23/M/J/12 For Examiner's Use 12 Without using your calculator, work out the following. Show all the steps of your working and give each answer as a fraction in its simplest form. (a) 12 11 − 3 1 Answer(a) [2] (b) 4 1 ÷ 13 11 Answer(b) [2] 13 (a) Find the value of 7p – 3q when p = 8 and q = O5 . Answer(a) [2] (b) Factorise completely. 3uv + 9vw Answer(b) [2]

7 © UCLES 2012 0581/23/M/J/12 [Turn over For Examiner's Use 14 Simplify the following. (a) ( )3 2 4pq Answer(a) [2] (b) ( ) 4 1 8 16 − x Answer(b) [2] 15 Solve the equation 2x2 + 6x – 3 = 0 . Show your working and give your answers correct to 2 decimal places. Answer x = or x = [4]

8 © UCLES 2012 0581/23/M/J/12 For Examiner's Use 16 4 cm 15 cm NOT TO SCALE The diagram shows a solid prism of length 15 cm. The cross-section of the prism is a semi-circle of radius 4 cm. Calculate the total surface area of the prism. Answer cm2 [4] 17 A =       3 1 4 2 B = ( ) 2 1 (a) Calculate BA. Answer(a) [2] (b) Find A– 1 , the inverse of A. Answer(b) [2]

9 © UCLES 2012 0581/23/M/J/12 [Turn over For Examiner's Use 18 Q R O P M X q p NOT TO SCALE O is the origin and OPRQ is a parallelogram. The position vectors of P and Q are p and q. X is on PR so that PX = 2XR. Find, in terms of p and q, in their simplest forms (a) , Answer(a) = [2] (b) the position vector of M, the midpoint of QX. Answer(b) [2]

10 © UCLES 2012 0581/23/M/J/12 For Examiner's Use 19 120 100 80 60 40 20 0 0.5 1 1.5 2 2.5 3 3.5 Time (minutes) Speed (km / h) The diagram shows the speed-time graph for part of a car journey. The speed of the car is shown in kilometres / hour. Calculate the distance travelled by the car during the 3.5 minutes shown in the diagram. Give your answer in kilometres. Answer km [4]

11 © UCLES 2012 0581/23/M/J/12 [Turn over For Examiner's Use 20 Simplify fully. x x x x x 25 10 20 2 3 2 + − − − Answer [5] 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 2012 0581/23/M/J/12 For Examiner's Use 21 M B A D C P 5 cm 8 cm 8 cm NOT TO SCALE The diagram shows a pyramid on a square base ABCD. The diagonals of the base, AC and BD, intersect at M. The sides of the square are 8 cm and the vertical height of the pyramid, PM, is 5 cm. Calculate (a) the length of the edge PB, Answer(a) PB = cm [3] (b) the angle between PB and the base ABCD. Answer(b) [3]

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 0581/23 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 23 © 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 95 2 B1 for 85 seen or M1 x = 180 – their angle ADC, if it is clearly seen 2 120 2 M1 for 100 8 2 750 × × oe seen or SC1 870 as final answer 3 (a) (b) 3.26077… 3.261 1 1ft seen their (a) to 4 significant figures 4 y ø –1.25 2 M1 inequality with y’s and constants correctly collected 5 33 cao www 2 M1 any two of 5.5, 9.5, 12.5 seen 6 31.7 2 M1 0.5 × 9 × 15 × sin 28 7 u = 24(.0), v = 0.6 2 B1 each 8 7 cao 3 B1 for 39.5(0) or 31.5(0) or 42 M1 for (their 39.5 – 8) ÷ 4.5 or (their 42 – 10.5) ÷ 4.5 9 ( ) 3 2 t a − cao oe 3 M1 correct re-arrangement to isolate the term in w M1 correct multiplication by a M1 correct division by their 3 An incorrect answer scores a maximum of M2 10 10 3 M1 T = k√l A1 for k = 2 11 17.05 cao www 4 M1 for 280 × (1 + 100 3 )2 oe M1 subtracting 280 from 280(1 + 100 k )2 any k A1 for 17.052 or SC2 297.05 on answer line

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 23 © University of Cambridge International Examinations 2012 12 (a) (b) 12 4 12 11 − oe 12 7 cao ww 0 11 13 4 1 × oe 44 13 cao ww 0 2 2 M1 correct use of a common denominator A1 M1 inversion and operation change A1 13 (a) (b) 71 3v (u + 3w) final answer 2 2 M1 for 7×8 – 3×–5 or B1 56 and –15 B1 for 3(uv + 3vw) or v (3u + 9w) As final answer 14 (a) (b) 64p3q6 0.5x–2 or 2 2 1 x oe 2 2 B1 64puqv or kp3q6 B1 u x 2 1 oe or 2 1 kx oe 15 –3.44, 0.44 correct working must be shown 4 B1 for ) 3 )( 2 ( 4 ) 6 ( 2 − − or better seen B1 if in form r q or p ) ( − + , for p = –6 and r = 2×2 oe B1, B1 (SC1 –3.4 or –3.436… and 0.4 or 0.436...) 16 359 www 4 M1 π × 42 or 2 2 1 4 × π M1 for 0.5 × π × 8 × 15 oe M1 for 8 × 15 + their 2 ends + their curved surface area 17 (a) (b) (4 10)       − − 2 1 4 3 2 1 oe 2 2 B1 each element or correct without brackets B1 for       d b c a 2 1 or       − − 2 1 4 3 k seen 18 (a) (b) p – 3 1 q oe 2 1 p + 6 5 q oe 2 2 ft M1 RX QR + oe or –q + p + ( 3 2 )q oe ft q + 2 1 their (a) but must be vectors or M1 for QM OQ + oe 19 6(.00) www 4 M1 use of area = distance M1 complete, correct set of area statements, ignoring units M1 changing min to hours or km/h to km/min

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 23 © University of Cambridge International Examinations 2012 20 ) 5 ( 4 − + x x x oe cao 5 B2 (x – 5)(x + 4) seen or SC1 (x + a)(x + b) where ab = –20 or a + b = –1 B2 x(x – 5)(x – 5) or B1 one of x(x2 –10x + 25), (x – 5)( x – 5), (x – 5)(x2 – 5x) seen 21 (a) (b) 7.55 www 41.5 www 3 3 M2 ( 2 1 √(82 + 82))2 + 52 or 42 + 52 + 42 seen or M1 82 + 82 or 52 + 42 or 42 + 42 or 52 + (their MB)2 seen M2 sin(B) = ) a( 5 or tan(B) = MB their 5 or cos(B) = (a) their MB or M1 recognition of angle PBM