Cambridge IGCSE Mathematics (with coursework) 0581 — 2011 May/June Paper 2 · Variant 1

This document consists of 12 printed pages. IB11 06_0581_21/FP © UCLES 2011 [Turn over *2011* *9148474896* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/21 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 0581/21/M/J/11 For Examiner's Use For Examiner's Use 1 A concert hall has 1540 seats. Calculate the number of people in the hall when 55% of the seats are occupied. Answer [1] 2 Shade the required region on each Venn diagram. A B A B A ∪ B' (A ∩ B)' [2] 3 Calculate 810.25 ÷ 4−2 . Answer [2] 4 (a) Find m when 4m × 42 = 412 . Answer(a) m = [1] (b) Find p when 6 p ÷ 6 5 = 6 . Answer(b) p = [1]

3 © UCLES 2011 0581/21/M/J/11 [Turn over For Examiner's Use For Examiner's Use 5 A hummingbird beats its wings 24 times per second. (a) Calculate the number of times the hummingbird beats its wings in one hour. Answer(a) [1] (b) Write your answer to part (a) in standard form. Answer(b) [1] 6 6 cm 2 cm NOT TO SCALE A company makes solid chocolate eggs and their shapes are mathematically similar. The diagram shows eggs of height 2 cm and 6 cm. The mass of the small egg is 4 g. Calculate the mass of the large egg. Answer g [2] 7 Find the length of the straight line from Q (−8, 1) to R (4 , 6). Answer QR = [3]

4 © UCLES 2011 0581/21/M/J/11 For Examiner's Use For Examiner's Use 8 Calculate the radius of a sphere with volume 1260 cm3 . [The volume, V, of a sphere with radius r is V = 3 4 πr3.] Answer cm [3] 9 A B C D 2x° 5x° x° NOT TO SCALE AB is parallel to CD. Calculate the value of x. Answer x = [3] 10 Solve the simultaneous equations. 3x + y = 30 2x – 3y = 53 Answer x = y = [3]

5 © UCLES 2011 0581/21/M/J/11 [Turn over For Examiner's Use For Examiner's Use 11 A rectangular photograph measures 23.3 cm by 19.7 cm, each correct to 1 decimal place. Calculate the lower bound for (a) the perimeter, Answer(a) cm [2] (b) the area. Answer(b) cm2 [1] 12 A train leaves Barcelona at 21 28 and takes 10 hours and 33 minutes to reach Paris. (a) Calculate the time the next day when the train arrives in Paris. Answer(a) [1] (b) The distance from Barcelona to Paris is 827 km. Calculate the average speed of the train in kilometres per hour. Answer(b) km/h [3]

6 © UCLES 2011 0581/21/M/J/11 For Examiner's Use For Examiner's Use 13 The scale on a map is 1: 20 000. (a) Calculate the actual distance between two points which are 2.7 cm apart on the map. Give your answer in kilometres. Answer(a) km [2] (b) A field has an area of 64 400 m2 . Calculate the area of the field on the map in cm2 . Answer(b) cm2 [2] 14 Solve the equation 2x2 + 3x – 6 = 0. Show all your working and give your answers correct to 2 decimal places. Answer x = or x = [4]

7 © UCLES 2011 0581/21/M/J/11 [Turn over For Examiner's Use For Examiner's Use 15 A teacher asks 36 students which musical instruments they play. P = {students who play the piano} G = {students who play the guitar} D = {students who play the drums} The Venn diagram shows the results. G D 5 x P 2 7 4 5 1 8 (a) Find the value of x. Answer(a) x = [1] (b) A student is chosen at random. Find the probability that this student (i) plays the drums but not the guitar, Answer(b)(i) [1] (ii) plays only 2 different instruments. Answer(b)(ii) [1] (c) A student is chosen at random from those who play the guitar. Find the probability that this student plays no other instrument. Answer(c) [1]

8 © UCLES 2011 0581/21/M/J/11 For Examiner's Use For Examiner's Use 16 k cm The diagram shows a square of side k cm. The circle inside the square touches all four sides of the square. (a) The shaded area is A cm2. Show that 4A = 4k2 – πk2. Answer (a) [2] (b) Make k the subject of the formula 4A = 4k2 – πk2. Answer(b) k = [3]

9 © UCLES 2011 0581/21/M/J/11 [Turn over For Examiner's Use For Examiner's Use 17 C A B O T 24° NOT TO SCALE A, B and C are points on a circle, centre O. TA is a tangent to the circle at A and OBT is a straight line. AC is a diameter and angle OTA = 24°. Calculate (a) angle AOT, Answer(a) Angle AOT = [2] (b) angle ACB, Answer(b) Angle ACB = [1] (c) angle ABT. Answer(c) Angle ABT = [2]

10 © UCLES 2011 0581/21/M/J/11 For Examiner's Use For Examiner's Use 18 Q P M X S R NOT TO SCALE In the diagram, PQS, PMR, MXS and QXR are straight lines. PQ = 2 QS. M is the midpoint of PR. QX : XR = 1 : 3. = q and = r. (a) Find, in terms of q and r, (i) , Answer(a)(i) = [1] (ii) . Answer(a)(ii) = [1] (b) By finding , show that X is the midpoint of MS. Answer (b) [3]

11 © UCLES 2011 0581/21/M/J/11 [Turn over For Examiner's Use For Examiner's Use 19 0 15 10 5 2 4 6 8 10 12 14 Time (minutes) Speed (metres per second) The diagram shows the speed-time graph of a train journey between two stations. The train accelerates for two minutes, travels at a constant maximum speed, then slows to a stop. (a) Write down the number of seconds that the train travels at its constant maximum speed. Answer(a) s [1] (b) Calculate the distance between the two stations in metres. Answer(b) m [3] (c) Find the acceleration of the train in the first two minutes. Give your answer in m/s2. Answer(c) m/s2 [2] Question 20 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 0581/21/M/J/11 For Examiner's Use 20 f(x) = x3 g(x) = 2x − 3 (a) Find (i) g(6), Answer(a)(i) [1] (ii) f(2x). Answer(a)(ii) [1] (b) Solve fg(x) = 125. Answer(b) x = [3] (c) Find the inverse function g−1(x) . Answer(c) g –1(x) = [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 0581/21 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 21 © 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 1 847 1 2 correct regions shaded 1, 1 3 48 2 B1 for 3 and 16 seen 4 (a) 10 (b) 5.5 oe 1 1 5 (a) 86400 (b) 8.64 × 104 1 1ft 6 108 2 M1 for 33 or 27 or 3 3 1       or 27 1 seen 7 13 3 B1 for 12, 5 seen M1 for (their 12)2 + (their 5)2 or M2 √[(–8 – 4)2 + (1 – 6)2] oe or M1 if √ missing 8 6.70 3 M1 for ( r3 = ) 1260 × π 4 3 oe seen M1 for 3 of their r3 seen or implied 9 22.5 oe 3 B2 180 = 5x +2x + x oe or better B1 for 2x or 6x marked in the correct place on the diagram. 10 x = 13 y = –9 3 M1 for consistent multiplication and addition/subtraction A1 for x = 13 or A1 for y = -9 11 (a) 85.8 (b) 456.8625 cao 2 1 M1 for 23.25 and 19.65 seen 12 (a) (0)8(.)01 (am) (b) 78.4 or 78.38 to 78.39 1 3 Not 8.01pm M2 for 827 ÷ 10.55 or M1 for figs 827 ÷ their time 13 (a) 0.54 (b) 1.61 2 2 M1 for 0000 10 000 20 7.2 × oe or SC1 for figs 54 in answer SC1 for figs 161 or M1 2002 or 20 0002 seen

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0581 21 © University of Cambridge International Examinations 2011 14 –2.64, 1.14 cao with working 4 B1 for ( )( ) 6 2 4 32 − − or better seen anywhere B1 for p = –3 and r = 2 × 2 or better as long as in the form r q p + or r q p − After B0B0, SC1 for –2.6 or –2.637(45…) and 1.1 or 1.137(45…) 15 (a) 4 (b) (i) 36 12 oe 0.333 (ii) 36 11 , 0.306 or 0.3055 to 0.3056 (c) 15 8 oe 0.533(3…) 1 1 1 1 16 (a) Answer given (b) k = (±) ( ) π 4 4 − A or ( ) π 4 2 − A 2 3 M1 (A =)k2 – π 2 2      k E1 A = k2 – 4 2 k π correctly completed to 4A = 4k2 – πk2 M1 factorising (must contain a π) M1 division (by coefficient of k2) M1 square root 17 (a) 66° (b) 33° (c) 123° 2 1 2 M1 for 90° clearly identified as A B1 for OBA or OAB = 57° 18 (a) (i) -r + q or q – r (ii) ½(3q – r) oe (b) correct working 1 1 3 Must be simplified M1 for MX = ½ r + ¾ their (–r + q) M1 using a different route for XS or ½ MS E1 dep correct simplification and conclusion 19 (a) 480 (b) 9900 (c) 0.125 or 8 1 1 3 2 M1 for attempt at area under graph M1 for 0.5 × 15 × (their (a) + 14 × 60) oe or 0.5 × 15 × (8 + 14) oe M1 for numerical vertical/horizontal or numerical use of v = u + at but t ≤ 120 or t ≤ 2 20 (a) (i) 9 (ii) 8x3 cao (b) 4 www (c) 2 3 + x 1 1 3 2 M1 for (2x – 3)3 = 125 M1 2x – 3 = 5 M1 for x ± 3 = 2y or x = 2 3 ± y