Cambridge IGCSE Mathematics - International 0607 — 2015 Oct/Nov Paper 6 · Variant 1

This document consists of 12 printed pages. DC (RW/FD) 104122/3 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 0 3 7 6 2 2 7 2 5 5 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) October/November 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 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/O/N/15 © UCLES 2015 THE INVESTIGATION STARTS ON PAGE 3.

3 0607/61/O/N/15 © UCLES 2015 [Turn over Answer both parts A and B. A INVESTIGATION SUMS OF TWO SQUARES (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at the results when two square numbers are added together. 1 Here is a list of the first 11 prime numbers. 2 3 5 7 11 13 17 19 23 29 31 (a) In the list there are 4 numbers that are one more than a multiple of 4. These are called Pythagorean Primes. The smallest one is 5 and the largest one is 29. Write down the other two. 5, ….….… , ….….… , 29 (b) The 17th century French mathematician Albert Girard proved that every Pythagorean Prime equals the sum of two square numbers. Write your answers to part (a) as the sum of two square numbers. Two have been written down for you. 5 = 12 + 22 ….….… = ….….… + ….….… ….….… = ….….… + ….….… 29 = 22 + 52 (c) Another Pythagorean Prime is 101. Write 101 as the sum of two square numbers. 101 = ….….… + ….….…

4 0607/61/O/N/15 © UCLES 2015 2 The sum of two square numbers can equal a square number. For example, 32 + 42 = 9 + 16 = 25 = 52 We say that 3, 4, 5 is a Pythagorean Triple. (a) Show, by calculation, that 7, 24, 25 is a Pythagorean Triple. (b) Each row in this table is a Pythagorean Triple. Complete the table. Use patterns of numbers in the table to help you. 3 4 5 5 12 13 7 24 25 9 40 11 60 13 113

5 0607/61/O/N/15 © UCLES 2015 [Turn over (c) What is the connection between the square of the smallest number and the other two numbers in each Pythagorean Triple in the table? … … (d) Use your answer to part (c) and the patterns of numbers in the table to complete the following Pythagorean Triples. (i) ………. , ………. , 421 (ii) 101 , ………. , ……….

6 0607/61/O/N/15 © UCLES 2015 3 Sometimes the sum of two square numbers can equal the sum of another pair of square numbers. For example, 52 + 52 = 12 + 72 (Both sums equal 50.) (a) Show that (x + y)2 + (m – n)2 = (x – y)2 + (m + n)2 simplifies to xy = mn. (x + y)2 + (m – n)2 = (x – y)2 + (m + n)2 xy = mn

7 0607/61/O/N/15 © UCLES 2015 [Turn over (b) x, y, m and n are different positive integers with x > y and m > n. When xy = mn = 6 one solution is x = 3, y = 2 and m = 6, n = 1. The substitution of these values into (x + y)2 + (m – n)2 = (x – y)2 + (m + n)2 gives these equal sums of square numbers. 52 + 52 = 12 + 72 Find all the possible solutions when xy = mn = 12. For each solution, write the equal sums of square numbers. (c) Complete the following equal sums of square numbers. 92 + ………. = 52 + ……….

8 0607/61/O/N/15 © UCLES 2015 B MODELLING POPULATION GROWTH (20 marks) You are advised to spend no more than 45 minutes on this part. This modelling task compares three different models of population growth. Ten fish are put into a lake to breed. The maximum number of fish that can live in the lake is 100. After ten years the number of fish stays approximately the same. The table shows the number of fish, y, in the lake at the end of x years. Number of years (x) 0 1 2 3 4 5 6 7 8 9 10 Number of fish (y) 10 18 27 41 61 78 88 93 97 99 100 This information is shown below. 0 10 20 30 40 50 60 Number of fish 70 80 90 100 1 2 3 4 5 Number of years 6 7 8 9 10 x y 1 (a) Why is it correct to join the points? … … (b) Comment on the rate of increase in the number of fish when the number of fish approaches 100. … …

9 0607/61/O/N/15 © UCLES 2015 [Turn over 2 The data can be modelled using the cubic function y = ax3 + bx. (a) Find an equation in a and b so that the model gives the value of y in the table when (i) x = 1, … (ii) x = 5. … (b) Solve the simultaneous equations from part (a) and write down the model. …

10 0607/61/O/N/15 © UCLES 2015 3 The data can also be modelled by the trigonometric function y = a + cos b x 18 ° ^ h . (a) Find an equation in a and b so that the model gives the value of y in the table when (i) x = 0, … (ii) x = 10. … (b) Solve the equations from part (a) and write down your model. …

11 0607/61/O/N/15 © UCLES 2015 [Turn over 4 The data can also be modelled by the logistic function y k 2 100 2 x x # = + . (a) The model gives the value of y in the table when x = 0. Find the value of k. … (b) Comment on the accuracy of the model when x = 5. … … Question 5 is printed on the next page.

12 0607/61/O/N/15 © UCLES 2015 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. 5 (a) Sketch the graphs of your models in questions 2, 3 and 4 for x 0 15 G G . The original data has been shown again. 0 10 20 30 40 50 60 Number of fish 70 80 90 100 5 10 15 x y Number of years (b) Give two reasons why the logistic model is the best. … … …

® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the October/November 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 October/November 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 – October/November 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 SUMS OF TWO SQUARES Question Answer Mark Part Marks 1 (a) 13 17 1 (b) 13 = 22 + 32 17 = 12 + 42 1 (c) [101 =] 12 + 102 1 2 (a) 49 + 576 = 625 oe 2 B1 for two correct squares (b) 41 61 84 85 15 112 3 B1 for each column In third column FT their 84 either by pattern (+1) or by Pythagoras (correct to at least 1 dp) (c) equal sum oe 1 C opportunity (d) (i) 29, 420 1 C opportunity (ii) 5100, 5101 1 C opportunity 3 (a) Each bracket correctly squared 4xy = 4mn 1 1 (b) 132 + 42 = 112 + 82 82 + 12 = 42 + 72 132 + 12 = 112 + 72 4 B2 for one correct statement B1 for each further correct statement If 0 scored then B1 for one solution (c) [92 +] 132 [= 52 +] 152 2 M1 for x = 7, y = 2 soi C opportunity Communication seen in one of 2(c), 2(d)(i), 2(d)(ii) or 3(c) 1

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 61 © Cambridge International Examinations 2015 B MODELLING POPULATION GROWTH Question Answer Mark Part Marks 1 (a) Any correct statement implying why it is correct to do so 1 (b) Any correct statement about size or change of rate 1 2 (a) (i) a + b = 18 oe 1 (ii) 125a + 5b = 78 oe 1 (b) y = –0.1x3 + 18.1x 2FT B1FT for [a =] – 0.1 B1FT for [b = ]18.1 If 0 scored B1FT for two inaccurate answers C opportunity 3 (a) (i) a + b = 10 oe 1 (ii) a – b = 100 oe 1 (b) y = 55 – 45 cos(18x)° 2FT B1FT for [a =] 55 B1FT for [b =] – 45 C opportunity 4 (a) [k =] 9 nfww 2 M1 for k + 1 100 = 10 (b) Accurate oe dependent on k 1FT FT their k 5 (a) 4FT B1FT for each correct shape B1FT for all 3 y-intercepts correct C opportunity (b) Accurate oe Levels out after 10 years oe 2 B1 for each Communication seen in one of 2(b), 3(b) or 5(a) 1