Cambridge IGCSE Mathematics - International 0607 — 2009 May/June Paper 6 · Variant 1

This document consists of 4 printed pages. IB09 06_0607_06/5RP © UCLES 2009 [Turn over *7384524122* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/06 Paper 6 (Extended) May/June 2009 1 hour 30 minutes Additional Materials: Answer Booklet/Paper Graphics Calculator 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. Do not use staples, paper clips, highlighters, glue or correction fluid. You may use a pencil for any diagrams or graphs. Answer both parts A and B. You must show all relevant working to gain full marks for correct methods, including sketches, even if your answer is incorrect. In this paper you will also be assessed on your ability to provide full reasons and communicate your mathematics clearly and precisely. At the end of the examination, fasten all your work securely together. The total number of the marks for this paper is 40. www.XtremePapers.com

2 © UCLES 2009 0607/06/M/J/09 Answer both parts A and B. A. INVESTIGATION REMOVING DISCS 24 marks 6 5 2 10 7 1 4 3 8 9 Ten discs, numbered 1 to 10, form a circle. You remove the disc numbered 1, and, going clockwise, leave the next one, remove the one after that, leave the next one, and so on until only one disc remains. The discs which you remove are, in order, numbered 1, 3, 5, 7, 9, 2, 6, 10, 8. The remaining disc is numbered 4. 1 Copy and complete the table showing the number on the remaining disc when you have 2 discs, 3 discs, 4 discs, …., 20 discs in the circle. Number of discs in the circle Number on the remaining disc 2 2 3 2 4 4 5 6 7 8 8 9 10 4 11 6 12 8 13 10 14 15 14 16 16 17 18 19 6 20 8

3 © UCLES 2009 0607/06/M/J/09 [Turn over 2 When you have 2, 4, 8 or 16 discs in the circle, the number on the remaining disc is always the same as the total number of discs in the circle. Assume this pattern continues. Write down the next three numbers when this happens. 3 Use question 2, and any patterns you see in your table, to find the number on the remaining disc when the circle contains (a) 65 discs, (b) 125 discs, (c) 200 discs, (d) 100 000 discs. 4 Find an algebraic expression for the number of discs in the circle, when the remaining disc is numbered 10. 5 Consider the original ten discs again. Remove the disc numbered 10, and going anticlockwise, leave the next one, remove the one after that, leave the next one, and so on until only one disc remains. You should finish with the disc numbered 7. (a) (i) Write down the numbers on the discs in the order in which you remove them. (ii) When you started with the disc numbered 1 and worked clockwise, the order was 1 3 5 7 9 2 6 10 8 Compare this order with the order you have written down in part (a) (i). Explain how the two orders are related. (b) (i) When you have n discs in the circle and work clockwise from the disc numbered 1, the number on the remaining disc is x. When you work anticlockwise from the disc numbered n, the number on the remaining disc is y. Find a rule connecting x, y and n. (ii) Find the number on the remaining disc when there are 100 discs and you work anticlockwise by first removing the disc numbered 100. Part B is on the next page.

4 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. 0607/06/M/J/09 B. MODELLING HANGING CHAIN 16 marks P Q y 0 x A B NOT TO SCALE The diagram shows two vertical poles (AP and BQ) that support the ends (A and B) of a hanging chain. The chain is symmetrical about the y-axis. The poles are each 7.52 metres high and are 4 metres apart. The lowest point of the chain is 2 metres above the horizontal ground (PQ). The units are in metres. 1 Write down the co-ordinates of A, B and the lowest point of the chain. 2 The scientist Galileo chose one of the following models for the shape of a hanging chain. Which model best fits the hanging chain in the diagram? y = ax + b y = | ax + b | y = ax2 + b y = a sin x + b 3 Use the co-ordinates of the lowest point of the chain to find the value of b in your chosen model. 4 Find the value of a in your chosen model. 5 According to your model, how high is the chain above a point on the ground that is 50 cm from one of the poles? 6 Today, engineers use the model y = k       + x w x w 1 , w > 1, for a hanging chain. (a) Use the co-ordinates of the lowest point of the chain to find the value of k. (b) Find the value of w for this model. 7 The two models give heights for the hanging chain. The difference in these heights is h. (a) Sketch the graph of h against x. (b) Find the greatest difference in height between the two models.

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2009 question paper for the guidance of teachers 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/06 Paper 6 (Extended), maximum raw mark 40 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 2009 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 2009 0607 06 © UCLES 2009 M marks are given for a correct method. A marks are given for an accurate answer following a correct method. B marks are given for a correct statement or step. D marks are given for a clear and appropriately accurate drawing. P marks are given for accurate plotting of points. E marks are given for correctly explaining or establishing a given result. C marks are given for clear communication. Abbreviations cao correct answer only cso correct solution only ft follow through oe or equivalent soi seen or implied ww without working www without wrong working

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0607 06 © UCLES 2009 Section A Question Answer Mark Notes Comments 1 Number of discs Last disc Number of discs Last disc Number of discs Last disc 2 2 9 2 17 2 3 2 10 4 18 4 4 4 11 6 19 6 5 2 12 8 20 8 6 4 13 10 7 6 14 12 8 8 15 14 16 16 7 B7 1 for each shaded square [7] 2 32, 64, 128 2 B1 for 32 B1 for 64 and 128 [2] 3 (a) (b) 2 122 1 2 B1 B2 M1 for 2(125 – their 64) or their 128 – 2(their 128–125) oe A1 for correct evaluation Dependent on 3 values in Question 2 ft from their 128

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0607 06 © UCLES 2009 (c) (d) 144 68928 2 3 B2 OR M1 for 2(200 – their 128) or 256 – 2(256 – 200) oe A1 for correct evaluation B3 OR M1 evidence of 16 or 17 or 65536 or 131072 seen M1 for 2(100 000 – 216) or 217– 2(217 – 100000) oe A1 for correct evaluation If 0 scored, SC2 34464 ft with 2×their 128 for 256 6. 16 2 log 100000 log = [8] 4 2n + 5 for n > 2 3 B1 for 2n B1 for n > 2 oe B1 for + 5 or 5 more than (2, 4,) 8, 16, 32… oe OR SC3 for 2n+2 + 5 Ignore “n=” and subsequent “working” [3]

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0607 06 © UCLES 2009 5 (a) (b) (i) (10) 8, 6, 4, 2, 9, 5, 1, 3 (7) (ii) corresponding terms add to 11 (i) x + y = n + 1 oe (ii) 29 1 1 1 2 B1 B1 ft with consistent pattern B1 ft with their consistent pattern B2 ft with their consistent pattern OR M1 72 identified OR anticlockwise table Accept one omission or error. Accept a diagram indicating this [5] 2 C1 for communication by one of: C2 for communication by at least two of: Showing strategy Comparing Checking. Award marks for: Strategy shown in question 2 question 3(b) 3(c) 3(d) Variables defined in question 4 Indicating comparison of corresponding terms in question 5(a)(ii) Strategy shown in question 5(b)(ii) Checking of a result [2] [Total: 27 scaled to 24]

Page 6 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0607 06 © UCLES 2009 Section B Question Answer Mark Notes Comments 1 (–2, 7.52) (2, 7.52) (0, 2) B2 SC1 for ( ± 4, 7.52) B1 One for each point, seen anywhere Accept inclusion of units [3] 2 (y =) ax2+b 1 B1 Accept mention of quadratic [1] 3 2 1 B1ft from (0, their 2) [1] 4 1.38 2 B2 OR M1 for substituting ( ± 2, 7.52) or their A or B into their function soi, A1 correct evaluation Allow follow-through Condone –2² appearing [2] 5 5.1(1m) or 5.1(05m) 2 B2 OR M1 for substituting x = ± 1.5 A1 correct evaluation OR SC1 correct answer from substituting x = ± 0.5 Allow follow-through Implied by 2.345 or 2.09 or better [2]

Page 7 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0607 06 © UCLES 2009 6 (a) (b) 1 www 2.7(17..) www 2 3 M1 Substituting (0, their 2) to get 2 = k (w0 + 0 1 w ) A1 k = 1 or 2 2 their M1 Substitute their A or B M1ft A1ft w = 2.7(17….) Allow follow-through (or k not yet found) ( ± 2, 7.52) gives 7.52 = w2 + 2 1 w w2 = 2 7.52 7.52 4 2 ± − Allow extra decimal places [5] 7 (a) (b) 0.4(m) to 0.45(m) 2 1 G1 approximate shape G1 through (0,0) dependent B1ft Accept reflection in x-axis Allow domain beyond –2 to 2. Follow-through only if quadratic in question 2. [3] 2 C1 for communication by one example of: C2 for communication by at least three examples of: Showing strategy Checking Scale. Working shown in Question 4 Question 5 Question 6(a) Question 6(b) Scale in Question 7(a) [2] [Total: 19 scaled to 16]