Cambridge IGCSE Mathematics - International 0607 — 2017 Oct/Nov Paper 6 · Variant 1

* 3 0 8 8 3 0 4 2 8 6 * This document consists of 10 printed pages and 2 blank pages. DC (ST/SW) 134794/2 © UCLES 2017 [Turn over CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) October/November 2017 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 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. Cambridge International Examinations Cambridge International General Certificate of Secondary Education

2 0607/61/O/N/17 © UCLES 2017 Answer both parts A and B. A INVESTIGATION EQUABLE SHAPES (20 marks) You are advised to spend no more than 45 minutes on this part. In this investigation lengths are given in centimetres. The area of a shape is A square centimetres and its perimeter is P centimetres. This task investigates the dimensions of equable shapes. The shape is equable if A = P. All the diagrams in this investigation are not to scale. 1 3.6 4.5 This rectangle is equable. Show by calculation that A = P. … … 2 y 10 (a) Write down an expression, in terms of y, for the area of this rectangle. … (b) Write down an expression, in terms of y, for the perimeter of this rectangle. … (c) This rectangle is equable. Using your answers to part (a) and part (b) write an equation in terms of y. Solve this equation. …

3 0607/61/O/N/17 © UCLES 2017 [Turn over 3 y x This rectangle is equable. (a) Write down the equation A = P in terms of x and y. … (b) Show that (x - 2)(y - 2) = 4 is equivalent to your equation in part (a). (c) Use part (b) to find all the equable rectangles that have integer lengths and widths. …

4 0607/61/O/N/17 © UCLES 2017 4 This isosceles triangle has base 6 and height 7.2 . Its area is 2 1 × base × height = 3 × 7.2 = 21.6 . Show that it is equable. 3 3 7.2 5 This isosceles triangle has base 2a and height h. a a h (a) Find, in terms of a and h, an expression for A and an expression for P. When A = P, show that ah a a h 2 2 2 2 - = + .

5 0607/61/O/N/17 © UCLES 2017 [Turn over (b) (i) By squaring both sides of the equation in part (a) show that a2h – 4a2 = 4h. (ii) Write a2 in terms of h. … (iii) For which values of h, the height of an equable isosceles triangle, is your formula for a2 valid? … (c) An equable isosceles triangle has a height of 4.5 . Find its perimeter. …

6 0607/61/O/N/17 © UCLES 2017 B MODELLING CARBON DIOXIDE MEASUREMENTS (20 marks) You are advised to spend no more than 45 minutes on this part. This task models data from Mauna Loa on the island of Hawaii. Since 1958, scientists at Mauna Loa have been measuring the amount of carbon dioxide in the atmosphere. The units of measurement are parts per million by volume (ppm). The scientists use a sine wave to model the graph of the amount of carbon dioxide in the atmosphere. Here is the graph of y = sin x°. Its period is 360. –1 –0.5 30 0 0.5 1 60 90 120 150 180 210 240 270 300 330 360 x y 1 (a) –1 –0.5 30 0 0.5 1 60 90 120 150 180 210 240 270 300 330 360 x y (i) On the grid, sketch the graph of y = sin(2x)° for x 0 360 G G . (ii) Write down the period of y = sin(2x)°. … (b) Write down the period of y = sin(6x)°. … (c) Find the value of k when the period of y = sin(kx)° is 40. …

7 0607/61/O/N/17 © UCLES 2017 [Turn over 2 The graph below models the change in the amount of carbon dioxide at Mauna Loa in one year. x is the number of months since the start of the year. x = 0 is the beginning of January and x = 12 is the end of December. y is the amount of carbon dioxide above a given level. –2 –4 1 0 2 4 2 3 4 5 6 7 8 9 10 11 12 x y The equation of the graph is y = a sin(bx)°. (a) Write down the period of the graph and find b. period = … b = … (b) Write down the value of a and the equation of the graph. …

8 0607/61/O/N/17 © UCLES 2017 3 The table shows the amount of carbon dioxide at the beginning of each year. Year Number of months since the beginning of 2012 (x) Amount of carbon dioxide in ppm (y) 2012 0 393 2013 12 395 2014 24 397 2015 36 399 2016 48 401 2017 60 403 (a) On the grid, plot these points. 390 0 12 24 36 48 60 x y 395 400 405 (b) Find the equation for y in terms of x that fits the data. …

9 0607/61/O/N/17 © UCLES 2017 [Turn over 4 The expressions for y in question 2 and question 3 combine together to give the complete model. Use one of the operations, addition, subtraction, multiplication or division, to write down the complete model. … 5 The graph shows the complete model for the amount of carbon dioxide between the beginning of 2012 and the beginning of 2017. Label the points marked on the axes. x Number of months since the beginning of 2012 Amount of carbon dioxide (ppm) y

10 0607/61/O/N/17 © UCLES 2017 6 Answer this question using your model from question 4. (a) Find the highest amount of carbon dioxide in the atmosphere between the beginning of January 2012 and the beginning of January 2017. Give your answer correct to 1 decimal place. … (b) Find the amount of carbon dioxide at the end of May 2015. Give your answer correct to 1 decimal place. … 7 Assume that the graph in question 5 continues in the same way. Find when the amount of carbon dioxide first reaches 410 ppm. Write down the year and the month. Year … Month … 8 Explain why this model may not be appropriate for finding the amount of carbon dioxide in the atmosphere in the year 2050. … …

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12 0607/61/O/N/17 © UCLES 2017 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. BLANK PAGE

® IGCSE is a registered trademark. This document consists of 4 printed pages. © UCLES 2017 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) October/November 2017 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 October/November 2017 series for most Cambridge IGCSE®, Cambridge International A and AS Level components and some Cambridge O Level components.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 2 of 4 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

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 3 of 4 Question Answer Marks Partial Marks A INVESTIGATION EQUABLE SHAPES 1 3.6 × 4.5 = 16.2 2 × (3.6 + 4.5) oe = 16.2 2 B1 for each 2(a) 10y isw 1 2(b) 2y + 20 oe isw 1 2(c) 2.5 1 C opportunity 3(a) xy = 2x + 2y oe 1 3(b) xy – 2x – 2y + 4 = 4 isw 1 3(c) 3 by 6 4 by 4 2 B1 for each Deduct 1 for any extras If 0 scored B1 for 1 × 4 and 2 × 2 soi 4 2 2 3 7.2 + oe B1 7.8 + 7.8 + 6 = 21.6 oe B1 5(a) A = ah P = 2a + 2 2 2 + a h 2 B1 for each If 0 scored SC1 for both correct expressions 5(b)(i) a2h2 – 4a2h + 4a2 = 4a2 + 4h2 leading to the final answer with at least one correct step. 2 B1 for either side of the equation correct 5(b)(ii) 2 4 [ ] 4 = − h a h oe 1 C opportunity 5(b)(iii) h > 4 1 5(c) 27 2 B1 for [a2 =] 36 or better C opportunities Communication: Seen in one of the following questions. 1 2(c) 10y = 2y + 20 5(b)(ii) a2(h – 4) = 4h 5(c) Correct substitution of h shown 5(c) 6 × 4.5 or 7.5 seen with 12 + 2×7.5 oe

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 4 of 4 Question Answer Marks Partial Marks B MODELLING CARBON DIOXIDE MEASUREMENTS 1(a)(i) 2 B1 for correct maxima and minima B1 for correct period 1(a)(ii) 180 1 1(b) 60 1 1(c) 9 1 C opportunity 2(a) [period = ] 12 [b = ] 30 2 B1 for each C opportunity 2(b) 4 soi 1 3(a) Correct 6 points 2 B1 for 5 correct points 3(b) 1 393 6 = + y x oe 2 B1 for + 393 or 1 6 x C opportunity 4 y = 4 sin30x[°] + 1 393 6 + x 1 FT y = their functions added together 5 12 24 36 48 60 on the x-axis 395 400 405 on the y-axis 1 6(a) 405.5 1 6(b) 401.8 1 FT their model as above C opportunity 7 2019 February 2 B1 for 85[. ……] 8 Valid comment about extrapolation 1 Communication: Seen in one of the following questions. 1 1(c) 360 ÷ 40 or 360 ÷ 9 = 40 2(a) 360 ÷ 12 soi 3(b) numerical or graphical indication of the gradient fraction giving 1 6 6(b) 41