Cambridge IGCSE Mathematics - International 0607 — 2018 May/June Paper 6 · Variant 2

* 4 3 6 6 8 9 7 4 9 9 * This document consists of 14 printed pages and 2 blank pages. DC (NF/CT) 153612/2 © UCLES 2018 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) May/June 2018 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 (Questions 1 to 3) and B (Questions 4 to 6). You must show all 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 0607/62/M/J/18 © UCLES 2018 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 to 3) TILE PATTERNS (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at the number of grey tiles and the number of white tiles in a sequence of square patterns. All the tiles are the same size and each is a square. The grey tiles and white tiles make borders around a single grey tile. The single grey tile is the first grey border. The first white border is 8 white tiles surrounding the grey tile. The second grey border is 16 grey tiles surrounding the 8 white tiles. 1 (a) On this grid, complete the diagram to show the second white border.

3 0607/62/M/J/18 © UCLES 2018 [Turn over (b) On this grid, complete the diagram to show the third grey border. 2 This question looks at the patterns that finish with a white border. Pattern finishes with Number of white tiles in border Total number of white tiles in pattern 1st white border 8 8 2nd white border 32 3rd white border 40 4th white border 128 5th white border 72 200 (a) Complete the table.

4 0607/62/M/J/18 © UCLES 2018 (b) The values in the last column, for the total number of white tiles in these patterns, form a sequence. Find an expression, in terms of n, for the total number of white tiles in the pattern that finishes with the nth white border. … (c) There are 3 white tiles along one side of the pattern that finishes with the first white border. (i) Complete the table. Pattern finishes with Number of white tiles along one side 1st white border 3 2nd white border 3rd white border nth white border (ii) Use your answer to part (c)(i) to write down an expression, in terms of n, for the total number of all tiles in the pattern that finishes with the nth white border. …

5 0607/62/M/J/18 © UCLES 2018 [Turn over (d) Use your answers in part (b) and part (c) to show that an expression for the total number of grey tiles in the pattern that finishes with the nth white border is n n 8 8 1 2 - + . 3 (a) Michel uses the tiling method of question 2 to cover a square floor. The first tile is grey and the final border is white. The square floor measures 5.7 m by 5.7 m. Each tile measures 30 cm by 30 cm. Work out the number of white tiles and the number of grey tiles Michel needs to cover the floor completely. number of white tiles = … number of grey tiles = …

6 0607/62/M/J/18 © UCLES 2018 (b) Jasmine has 500 white tiles and 500 grey tiles. Each tile measures 30 cm by 30 cm. She uses the same method of tiling, but she can finish with a grey border or a white border. Find the side length of the biggest square floor that she can cover completely. Find how many tiles of each colour will not be used in this case. side length of biggest square floor = … number of white tiles not used = … number of grey tiles not used = …

7 0607/62/M/J/18 © UCLES 2018 [Turn over B MODELLING (QUESTIONS 4 to 6) GOING WITH THE FLOW (20 marks) You are advised to spend no more than 45 minutes on this part. In this task you will look at models for how the speed of water in a river changes with its height above the river bed. The river is 2 metres deep. An engineer measures the speed of water at different heights above the river bed. The water flows parallel to the river bed. The speed of the water at height x metres above the river bed is v cm/s. 2 m x m v cm/s surface water river bed NOT TO SCALE The table shows the data that the engineer collects. Height above river bed (x metres) Speed of water (v cm/s) 0.2 15.1 0.4 17.5 0.6 18.9 0.8 20.0 1.0 21.0 1.2 21.7 1.4 22.2 1.6 23.0 1.8 23.5 2.0 24.0 The data is plotted on the grid on the next page.

8 0607/62/M/J/18 © UCLES 2018 Speed of water (cm/s) Height above river bed (m) 14 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 15 16 17 18 19 20 21 22 23 24 v x

9 0607/62/M/J/18 © UCLES 2018 [Turn over 4 (a) The engineer models the data using v ax bx 12 2 = + + where a and b are constants. (i) Use the values of v when x = 1 and when x = 2 to write down two equations, each in terms of a and b. … … (ii) Solve your equations to find the value of a and the value of b. Write your values in the model below. v = … x2 + … x + 12 (b) The table shows the values that this model gives. x 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 v 14.3 16.3 18.1 19.7 21.0 22.1 22.9 23.5 23.9 24.0 (i) Plot these values and draw the graph of the model on the grid on page 8. (ii) Give the range of values of x for which this model overestimates the speed of the water. …

10 0607/62/M/J/18 © UCLES 2018 5 In this question, all logarithms are base 10. The engineer decides to test another model for how v changes with x. This model is v k x 2 m = b l where k and m are constants. (a) Using x = 2.0 and v = 24.0 find the value of k for this model. k = … (b) (i) Take logarithms of both sides of this model to complete the equation below. log v = … log x 2 b l + …

11 0607/62/M/J/18 © UCLES 2018 [Turn over (ii) The graph shows the data values of log v plotted against log x 2 b l and a line of best fit. –1 –0.9 –0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2 –0.1 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34 1.36 1.38 0 log v log x 2 b l Use the graph to calculate the value of m for the model, and write down the model. m = … v = … x 2 b l … (c) Use this model to find the height above the river bed where the water is moving at 12 cm/s. …

12 0607/62/M/J/18 © UCLES 2018 6 A factory releases some dirt into the river. The dirt moves parallel to the river bed at the same speed as the water. At the same instant that the factory releases the dirt, the engineer lowers a metal grid into the river. The metal grid is 20 m from the factory. The metal grid does not affect the speed of the water but will stop the dirt. 2 m x m 20 m dirt v cm/s u cm/s surface metal grid river bed NOT TO SCALE (a) The metal grid moves vertically downwards at speed u centimetres per second. Show that the number of seconds it takes the bottom of the metal grid to move from the surface to a height x metres above the river bed is u x 100 2- ` j . (b) The dirt moves along the river towards the grid at the speed given by the model in question 5(b)(ii). Find an expression, in terms of x, for the time it takes the dirt to move 20 m along the river. …

13 0607/62/M/J/18 © UCLES 2018 [Turn over (c) Use part (a) and your answer to part (b) to show that the metal grid stops the dirt at height x if . u x x 1 2 2 2 .0 2 H - ` bj l . Question 6(d) is printed on the next page.

14 0607/62/M/J/18 © UCLES 2018 (d) (i) On the axes below, sketch the graph of . u x x 1 2 2 2 .0 2 = - ` bj l for x 0 2 G G . 0 2 u x (ii) Find the smallest value of u that stops all the dirt. …

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IGCSE™ is a registered trademark. This document consists of 7 printed pages. © UCLES 2018 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) May/June 2018 MARK SCHEME Maximum Mark: 40 Published 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 International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the May/June 2018 series for most Cambridge IGCSE™, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 2 of 7 Generic Marking Principles These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE 1: Marks must be awarded in line with: • the specific content of the mark scheme or the generic level descriptors for the question • the specific skills defined in the mark scheme or in the generic level descriptors for the question • the standard of response required by a candidate as exemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE 2: Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positively: • marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate • marks are awarded when candidates clearly demonstrate what they know and can do • marks are not deducted for errors • marks are not deducted for omissions • answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE 4: Rules must be applied consistently e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 3 of 7 MARK SCHEME NOTES The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M Method marks, awarded for a valid method applied to the problem. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be earned or implied. B Mark for a correct result or statement independent of Method marks. When a part of a question has two or more ‘method’ steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. The notation ‘dep’ is used to indicate that a particular M or B mark is dependent on an earlier mark in the scheme. Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working nfww not from wrong working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 4 of 7 Question Answer Marks Partial Marks A INVESTIGATION TILE PATTERNS 1(a) Correct border 1 1(b) Correct border 1 2(a) Pattern finishes with Number of white tiles in border Total number of white tiles 1st white border 8 8 2nd white border 24 32 3rd white border 40 72 4th white border 56 128 5th white border 72 200 2 B1 for any 2 correct 2(b) 8n2 oe 2 M1 for their three correct second differences or B1 for recognising factor of 8 in the total number of white tiles. 2(c)(i) 7 11 4n –1 oe 2 B1 for 7 and 11 or for 4n–1 C opportunity 2(c)(ii) (4n –1)2 oe isw 1 FT their(4n –1) 2(d) their (4n –1)2 – their 8n2 M1 16n2 – 8n + 1 may be seen in 4(c)(ii) 16n2 – 8n + 1 – 8n2 A1 answer given 3(a) [number of white tiles =] 200 [number of grey tiles =] 161 4 B1 for 19 or 361 B1 for n = 5 seen or substituted in their 8n2 or 8n2 – 8n + 1 or B1 for at least 4 terms of sequence 1 17 49 97 161 … B1 for each correct answer C opportunity

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 5 of 7 Question Answer Marks Partial Marks 3(b) (largest side length possible =) 8.7 or 870 (number of white tiles not used =) 108 (number of grey tiles not used =) 51 4 M1 for [white] their 8n2 = 500 or their (4n – 1)2 = 1000 or 961 or 729 or n = 7 or 8 soi or 8 32 72 128 200 288 or 8 + 24 + 40 + 56 + 72 + 88[+…] or for [grey] 8n2 – 8n + 1 = 500 or (4n – 3)2 = 1000 or 961 or 1 17 49 97 161 241 or 1 + 16 + 32 + 48 + 64 + 80[+…] B1 for each correct answer If 0 or 1 scored, replace with SC2 for 29 soi or 392 or 449 or 841 seen. C opportunity Communication: Seen in three of the following questions 1 2(c)(i) Two differences of 4 seen or difference = 4 corner tiles shown 3(a) 570 30 or 5.7 0.3 or 570 570 30 30 × × or 324900 900 oe 3(a) further correct working to find second number of tiles 3(b) Correct units on answer line for length of side after working with 30 or 0.3 seen. 3(b) annotation of working at least twice, e.g. 29 ‘grey tiles’

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 6 of 7 Question Answer Marks Partial Marks B MODELLING GOING WITH THE FLOW 4(a)(i) a + b + 12 = 21 oe isw 4a + 2b + 12 = 24 oe isw 2 B1 for each equation If 0 scored, then SC1 for [v =]a + b + 12 and [v =]4a + 2b + 12 4(a)(ii) v = –3 x2 + 12 x + 12 2 FT their linear equations, each with a and b terms, leading to a ≠ 0. B1 for one correct. or M1 for correctly eliminating one variable. 4(b)(i) Smooth curve drawn through at least 9 correct points. 2 B1 for 9 or 10 correct points plotted 4(b)(ii) 1 < x < 2 oe 1 5(a) 24[.0] 1 C opportunity 5(b)(i) [log v = ] m log 2       x + log their24 2 Strict FT from their 5(a) B1 for each C opportunity 5(b)(ii) [m =] 0.2 or 1 5 v = their k 2       their m x 1 5(c) 0.06[25] oe or 6[.25] cm 2 M1 for 12 = their 24 0.2 2       their x or better A1FT for m ≠ 1 leading to 0 < x < 2 C opportunity 6(a) [Distance = (2 – x)m =] 100(2 – x) cm or [Speed = ] 100 u m/s B1 Time = distance speed oe B1 6(b) 0.2 20 100 24 2 ×       x or better isw 1 FT their v from 5(b)(ii)

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 7 of 7 Question Answer Marks Partial Marks 6(c) 0.2 100(2 ) 20 100 24 2 − ×       x u x - or better isw 1 answer given 6(d)(i) Correct sketch 1 C opportunity 6(d)(ii) 1.397 to 1.4 1 C opportunity Communication: Seen in two of the following questions 1 5(a) [k](1)m 5(b)(i) log v = log 2       m x + log k 5(c) Correct units (normally metres) 6(d)(i) scale for u u x 2 0