Cambridge IGCSE Mathematics - International 0607 — 2015 May/June Paper 6 · Variant 1

This document consists of 12 printed pages. DC (AC/FD) 101335/3 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 6 7 1 4 1 0 0 3 2 9 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) May/June 2015 1 hour 30 minutes Candidates answer on the Question Paper. Additional Materials: 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, glue or correction fluid. You may use an HB pencil for any diagrams or graphs. DO NOT WRITE IN ANY BARCODES. Answer both parts A and B. You must show all the relevant working to gain full marks for correct methods, including sketches. 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 marks for this paper is 40.

2 0607/61/M/J/15 © UCLES 2015 Answer both parts A and B. A INVESTIGATION STAIRCASES (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at the number of cubes that make different types of staircase. 1 This is an UP staircase of height 3 made using 6 cubes. It is a 3-step UP staircase because it has a height of 3 cubes. (a) Write down the number of cubes that make an UP staircase of height 2. … (b) On the grid below draw an UP staircase of height 4.

3 0607/61/M/J/15 © UCLES 2015 [Turn over (c) Complete the table for the number of cubes that make these UP staircases. Height 1 2 3 4 5 6 Number of cubes 1 6 (d) Find an expression, in terms of n, for the number of cubes that make an UP staircase of height n. … (e) Find how many cubes make an UP staircase of height 10. …

4 0607/61/M/J/15 © UCLES 2015 2 This is an UP AND DOWN staircase of height 3 made using 9 cubes. It is a 3-step UP AND DOWN staircase because it has a height of 3 cubes. (a) Find how many cubes make an UP AND DOWN staircase of height 4. … (b) Complete the table for the number of cubes that make these UP AND DOWN staircases. Height 1 2 3 4 5 6 Number of cubes 1 9 (c) Find an expression, in terms of n, for the number of cubes that make an UP AND DOWN staircase of height n. … (d) Find how many cubes make an UP AND DOWN staircase of height 10. …

5 0607/61/M/J/15 © UCLES 2015 [Turn over 3 This is a DOUBLE staircase of height 3 made using 12 cubes. It is a 3-step DOUBLE staircase because it has a height of 3 cubes. (a) Complete the table for the number of cubes that make these DOUBLE staircases. Height 1 2 3 4 5 6 Number of cubes 2 12 (b) Find an expression, in terms of n, for the number of cubes that make a DOUBLE staircase of height n. … (c) Find how many cubes make a DOUBLE staircase of height 10. … (d) Find the height of a DOUBLE staircase made from 240 cubes. …

6 0607/61/M/J/15 © UCLES 2015 4 This is a sequence of MULTIPLE staircases of heights 1, 2 and 3. These are MULTIPLE staircases because, for each staircase, the width, the height and the depth are the same. (a) Complete the table for the number of cubes that make these MULTIPLE staircases. Height 1 2 3 4 5 6 Number of cubes 1 6 18 (b) Find an expression, in terms of n, for the number of cubes that make a MULTIPLE staircase of height n. …

7 0607/61/M/J/15 © UCLES 2015 [Turn over 5 There are 1800 cubes available. Use your expressions in questions 1(d), 3(b) and 4(b) to complete the table. Type of staircase Maximum height using 1800 cubes Number of cubes left over UP UP AND DOWN 42 36 DOUBLE MULTIPLE

8 0607/61/M/J/15 © UCLES 2015 B MODELLING BOAT TRIPS (20 marks) You are advised to spend no more than 45 minutes on this part. A boat travels up and down a river. The time taken for a journey depends on the speed of the boat and the speed of the water current. 1 In this model the water is still and so the speed of the water current is zero km / h. The speed of the boat in still water is 15 km / h. (a) Find how many minutes it will take for the boat to travel 10 km. … min (b) The boat travels for 24 minutes. Find the distance that it travels. … km

9 0607/61/M/J/15 © UCLES 2015 [Turn over When a boat travels against the current it goes in exactly the opposite direction to the current. boat current When a boat travels with the current it goes in exactly the same direction as the current. boat current 2 In this model the water is not still. The speed of the water current is 2 km / h. The speed of the boat in still water is 15 km / h. (a) Show that it will now take the boat approximately 46 minutes to travel 10 km against the current. (b) The boat travels for 20 minutes against the current. Find the distance it travels. … km (c) How far will the boat travel in 46 minutes with the current? … km

10 0607/61/M/J/15 © UCLES 2015 3 The boat travels 20 km up the river before returning to where it started. The speed of the water current is 2 km / h. The speed of the boat in still water is v km / h. (a) (i) Find a model for the total travelling time, T hours, for this whole journey. T = … (ii) Show that your model simplifies to T v v 4 40 2 = - . (iii) Sketch the graph of T v v v 4 40 0 20 for 2 G G = - . T v 20 0 (iv) The model is only appropriate for v k 2 . Find the value of k and give a practical reason why k must have this value. k = … because … …

11 0607/61/M/J/15 © UCLES 2015 [Turn over (b) When the speed of the boat in still water is 18 km / h, find the time taken for the whole journey. … hours (c) The return journey takes the boat 3 hours. Find the speed of the boat in still water. … km / h 4 The boat travels 20 km up the river before returning to where it started. The speed of the boat in still water is v km / h. A whole journey takes the boat 3 hours. (a) (i) Adjust the model in question 3(a)(ii) for a water current of 3 km / h. T = … (ii) Find the speed of the boat in still water. … km / h (b) The speed of the boat in still water is now 15 km / h. Adjust the model in question 3(a)(ii) and find the speed of the water current. … km / h Question 5 is printed on the next page.

12 0607/61/M/J/15 © UCLES 2015 5 (a) There is a change in the boat’s journey. Explain how the journey has changed when the model in question 3(a)(ii) becomes T v v 4 80 2 = - . … … (b) Describe fully the single transformation that maps the graph of T v v 4 40 2 = - onto the graph of T v v 4 80 2 = - . … … 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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. 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.

® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the May/June 2015 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 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 should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2015 series for most Cambridge IGCSE®, Cambridge International A and AS Level components and some Cambridge O Level components.

Page 2 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 61 © Cambridge International Examinations 2015 Abbreviations cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent SC Special Case nfww not from wrong working soi seen or implied A INVESTIGATION STAIRCASES 1 (a) 3 1 (b) 1 (c) Height 1 2 3 4 5 6 Cubes 1 3 6 10 15 21 1 (d) 2 1 1 2 2 n n + oe 1 C opportunity (e) 55 1FT FT their (d) provided an expression in n with numerical coefficients C opportunity 2 (a) 16 1 (b) Height 1 2 3 4 5 6 Cubes 1 4 9 16 25 36 1 (c) n2 1 oe (d) 100 1

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 61 © Cambridge International Examinations 2015 3 (a) Height 1 2 3 4 5 6 Cubes 2 6 12 20 30 42 1FT FT of all values double their 1(c) (b) n2 + n oe 1 C opportunity (c) 110 1FT FT their (b) provided an expression in n of correct order with numerical coefficients C opportunity (d) 15 1 C opportunity 4 (a) Height 1 2 3 4 5 6 Cubes 1 6 18 40 75 126 2 B1FT for any two (b) 3 2 1 1 2 2 n n + oe 1 C opportunity 5 Type of staircase Max height using 1800 cubes Number of cubes left over UP 59 30 UP/DOWN 42 36 DOUBLE 41 78 MULTIPLE 15 0 3FT FT their UP ( 2 1 1 2 2 n n + ) , their DOUBLE (n2 + n), their MULTIPLE 3 2 1 1 ( ) 2 2 n n + if expression in n with numerical coefficients of correct order with at least two terms B1 for each pair (row) if B0 then allow B1 for correct column of maximum heights C opportunity Communication seen in three of 1(d), 1(e), 3(b), 3(c), 3(d), 4(b), 5 1

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 61 © Cambridge International Examinations 2015 B MODELLING BOAT TRIPS 1 (a) 40 1 C opportunity (b) 6 1 C opportunity 2 (a) 46.1[53...] or 46.2 seen or 46 60 13 10 = × oe 1 may convert to metres per min etc; condone e.g. … = 46 seconds; allow × 60 to be implied by units stated (b) 4.33[3…] or 1 3 4 isw 1 C opportunity (c) 13.0[3…] or 1 30 13 1 C opportunity 3 (a) (i) 2 20 2 20 ] [ − + + = v v T oe isw 1 (ii) 20( 2) 20( 2) [ ] ( 2)( 2) v v T v v − + + = + − oe isw or ) 2 ( 20 ) 2 ( 20 ) 2 )( 2 ( + + − = + − v v v v T oe isw 2 B1 for either correct numerator or denominator/left hand side or right hand side; must be from correct two partial fractions (iii) v T 0 2 B1 correct shape with 2 branches B1 asymptote at v = 2 only soi (iv) [k =] 2 with valid reason in context 1 not for e.g. T would be negative (b) 2.25 oe 1 (c) 13.6[2...] to 13.63 isw 1 isw halving after correct answer seen

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 61 © Cambridge International Examinations 2015 4 (a) (i) 9 40 ] [ 2 − = v v T 1 or 2 2 3 40 ] [ − = v v T isw (ii) 13.9[7…] to 14.0 1FT FT from their 9; FT correct to at least 3 sf (b) 5 1 C opportunity 5 (a) Valid explanation 1 e.g. total distance travelled is now 80 [km] or it now travels 40 [km] each way oe (b) Stretch, [scale factor =] 2, v-axis invariant 2 B1 for stretch with either factor 2 or v-axis invariant Communication seen in two of 1(a), 1(b), 2(b), 2(c), 3(c), 4(b) 1