Cambridge IGCSE Mathematics - International 0607 — 2016 May/June Paper 6 · Variant 2

This document consists of 11 printed pages and 1 blank page. DC (LEG/FD) 115846/3 © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 6 4 9 3 1 2 5 3 8 8 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) May/June 2016 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.

2 0607/62/M/J/16 © UCLES 2016 Answer both parts A and B. A INVESTIGATION SUMS OF CONSECUTIVE INTEGERS (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at the results when the terms of a sequence of consecutive positive integers are added together. 1 The mean of 6 positive integers is 4.5 . Calculate the sum of the 6 integers. … 2 (a) Complete the table for sequences of two or more consecutive positive integers. Sequence Number of terms Mean Sum of all the terms 5, 6, 7, 8, 9, 10 6 10, 11, 12, … , 40 31 25 2, 3, 4, 5, 6, 7, 8 35 4 42 49 (b) Describe how to calculate the mean using only the first term and the last term of a sequence of consecutive integers. … …

3 0607/62/M/J/16 © UCLES 2016 [Turn over 3 k, k + 1, k + 2, …, k + 99 is a sequence of consecutive integers. (a) Write down the number of terms in this sequence. … (b) Use the first term and the last term to find an expression for the mean in terms of k. … (c) Use your answers to part (a) and part (b) to write down an expression for the sum of all the terms of the sequence. … 4 Use the method of question 3 to show that the sum of the integers k, k + 1, k + 2, …, k + (n – 1) is n # k n 2 2 1 + - .

4 0607/62/M/J/16 © UCLES 2016 5 (a) If n is odd, explain why the value of the expression k n 2 2 1 + - must be an integer. … … (b) If n is even, explain why the value of the expression k n 2 2 1 + - must end in .5 . … … 6 The sum of a sequence of consecutive positive integers is 84. (a) Using question 4 and question 5, find all the possible values of n and the corresponding values for the mean.

5 0607/62/M/J/16 © UCLES 2016 [Turn over (b) Write down all the possible sequences of consecutive positive integers whose sum is 84. 7 Find an even number, bigger than 20, which cannot be written as the sum of consecutive integers. …

6 0607/62/M/J/16 © UCLES 2016 B MODELLING TRAFFIC FLOW (20 marks) You are advised to spend no more than 45 minutes on this part. This task looks at maximising the number of cars that can safely pass a point on a road in an hour. 1 It takes one second to react to an emergency when driving. (a) The speed of a car is 54 km/h. Calculate the number of metres that it travels in 1 second. … (b) The speed of a car is x km/h. Show that the number of metres, a, travelled in 1 second is approximately 0.278x. 2 The speed of a car is x km/h. When the driver brakes, the number of metres, b, that the car travels before stopping is kx2. When x = 50, b = 20. Find an expression for b in terms of x. …

7 0607/62/M/J/16 © UCLES 2016 [Turn over 3 For safety, the distance between cars travelling at x km/h must be a + b. a b 4 P The average length of a car is 4 metres. So the number of metres between corresponding points on a road is a + b + 4. (a) At a speed of x km/h, how many metres does a car travel in one hour? … (b) Explain why a model for the number of cars, N, safely passing point P in one hour is . N x kx x 0 278 4 1000 2 = + + where x km/h is the speed of the cars and k has the value you found in question 2. … … …

8 0607/62/M/J/16 © UCLES 2016 (c) Using your value for k from question 2, sketch the graph of N for 0 G x G 60. 0 60 x N Speed (km / h) Number of cars per hour (d) Find the maximum possible number of cars which can safely pass point P in one hour. … (e) (i) Find, correct to one decimal place, the speed that gives this maximum. … (ii) Comment on the size of this answer. … (f) When you increase the average length of a car, what is the effect on (i) the maximum number of cars that can pass point P in one hour, … (ii) the speed at which this maximum is possible? …

9 0607/62/M/J/16 © UCLES 2016 [Turn over 4 A revised model for traffic flow does not include the braking distance, b. This is because the car in front also travels the same braking distance. So the revised model uses k = 0. The model also allows 2 seconds, instead of 1 second, for the driver to react to the car in front stopping quickly. Assume the average length of a car is 4 metres. (a) Revise the model in question 3(b). N = … (b) Sketch the graph of N for 0 G x G 60. 0 60 x N Speed (km / h) Number of cars per hour

10 0607/62/M/J/16 © UCLES 2016 (c) Can 1800 cars safely pass point P in one hour? Use algebra to explain your answer.

11 0607/62/M/J/16 © UCLES 2016 5 There is one speed, greater than 0 km/h, at which both models give the same number of cars per hour. Find this speed. …

12 0607/62/M/J/16 © UCLES 2016 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 the registered trademark of Cambridge International Examinations. This document consists of 5 printed pages. © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) May/June 2016 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 will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2016 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 2016 0607 62 © Cambridge International Examinations 2016 Abbreviations awrt answers which round to 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 SUMS OF CONSECUTIVE INTEGERS Question Answer Marks Part Marks 1 27 1 C opportunity 2 (a) Sequence Mean Sum 5, 6, 7, 8, 9, 10 6 7.5 45 10, 11, 12, …….. , 40 31 25 775 2, 3, 4, 5, 6, 7, 8 7 5 35 9, 10, 11, 12 4 10.5 42 4, 5, 6, 7, 8, 9, 10 OR 24, 25 7 2 7 24.5 49 5 B1 for each row C opportunity (b) add and divide by 2 oe 1 3 (a) 100 1 (b) 2 99 2 k + oe final answer 1 (c) their (a) × their (b) isw 1FT 50(2k + 99) oe 4 number of terms = n mean = 2 1 2 k n + − or [mean =] 1 2 k k n + + − 2 B1 for each statement 5 (a) [2k +] n – 1 is even and even + even = even or even / 2 is an integer 1 (b) [2k +] n – 1 is odd and odd + even = odd or odd / 2 = … .5 1

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 62 © Cambridge International Examinations 2016 Question Answer Marks Part Marks 6 (a) [1 and 84] 3 and 28 7 and 12 8 and 10.5 [12 and 7] [28 and 3] [84 and 1] [21 and 4] [4 and 21] 3 B1 for each pair, allowing reversed order (b) for any 2 correct sequences 1 27, 28, 29 9, 10, 11, 12, 13, 14, 15 7, 8, 9, 10, 11, 12, 13, 14 7 Any one of 32, 64, 128, … 1 C opportunity Communication seen in one of 1, 2(a), 2(b), 7 1

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 62 © Cambridge International Examinations 2016 B MODELLING TRAFFIC FLOW Question Answer Marks Part Marks 1 (a) 15 1 C opportunity (b) 1000 60 60 x × oe 1 2 1 125 x2 or 0.008x2 or 8×10–3 x2 oe 2 M1 20 = k 502 or better 3 (a) 1000x 1 (b) Numerator = distance in one hour Denominator = distance between cars oe 1 (c) Correct shape 2 B1 for a curve with a single max turning point, above the x–axis at x = 60 soi C opportunity (d) 1570 or 1572 to 1573 1FT FT their k, 0.002 ⩽ k ⩽ 0.8 (e) (i) 22.3 to 22.4 [km / h] 1FT FT their k, 0.002 ⩽ k ⩽ 0.8 (ii) It is a low speed oe 1 Dependent on (e)(i) < 45 (f) (i) decreases oe 1 (ii) increases oe 1

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 62 © Cambridge International Examinations 2016 Question Answer Marks Part Marks 4 (a) 1000 4 0.556 x x + oe isw 1 C opportunity (b) 1 correct shape, through (0,0) implied, and reaching x = 50 (c) 1000x = 7200 + (1800 × their 0.556)x or 1000 0.556 4 1800 x their x = + (or better) No, and their correct x given or No, and correct working leading to “x is negative” or No, and correct working leading to an impossible equation M1FT A1 FT 1000 4 0.556 x their x + only C opportunity If x found then must be correct. 5 Anything which rounds to 35 [km / h] 1FT FT their k, 0.002 ⩽ k ⩽ 0.1 and 1000 4 0.556 x their x + Communication in three of 1(a), 3(c), 4(a) and 4(c). 2 C1 if seen in two of them.