Cambridge IGCSE Mathematics - International 0607 — 2015 Oct/Nov Paper 6 · Variant 3

This document consists of 12 printed pages. DC (LEG/FD) 104389/2 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 4 5 8 9 9 6 5 0 7 9 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 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 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 0607/63/O/N/15 © UCLES 2015 Answer both parts A and B. A INVESTIGATION POSITION OF SECURITY CAMERAS (20 marks) You are advised to spend no more than 45 minutes on this part. Houses are built around squares. Security cameras give a clear view for a distance of one side of a square in any direction. On the diagrams a cross represents a security camera. One square needs a minimum of 2 cameras to view all four sides. Two squares, in a row, need a minimum of 3 cameras as shown. This investigation looks at the minimum number of security cameras for squares in different arrangements. 1 (a) (i) Three squares in one row need a minimum of 4 cameras. Draw 4 crosses on the diagram to show the positions of the cameras. (ii) Four squares in one row need a minimum of 5 cameras. Draw 5 crosses on the diagram to show the positions of the cameras. (iii) Draw crosses on the diagram to show the positions of the minimum number of cameras for five squares in one row. (b) Find an expression, in terms of n, for the minimum number of cameras for n squares in one row. …

3 0607/63/O/N/15 © UCLES 2015 [Turn over 2 There are now three rows of squares. (a) (i) What is the minimum number of cameras needed when there is 1 square in each of three rows? Draw crosses on the diagram to show the positions of these cameras. Minimum = … (ii) Two squares in each of three rows need a minimum of 6 cameras. Draw crosses on the diagram to show the positions of these cameras. Minimum = 6 (iii) Draw crosses on the diagram to show the positions of the minimum number of cameras for 3 squares in each of three rows. Minimum = … (b) Find an expression, in terms of n, for the minimum number of cameras for n squares in each of three rows. …

4 0607/63/O/N/15 © UCLES 2015 3 There are now five rows of squares. Find the minimum number of cameras for 2 and 3 squares in each of five rows. Minimum = … Minimum = …

5 0607/63/O/N/15 © UCLES 2015 [Turn over 4 (a) Complete the table to show the minimum number of cameras for an odd number of rows. Number of squares in each row 1 square 2 squares 3 squares 4 squares 5 squares n squares One row 2 3 4 5 Three rows 6 Five rows 6 Seven rows 8 (b) Find an expression for the minimum number of cameras for n squares in each of r rows, when r is an odd number. … (c) For an odd number of rows, the minimum number of cameras is 16. Find all the possible numbers of squares in each row. …

6 0607/63/O/N/15 © UCLES 2015 5 Now consider even numbers of rows with an even number of squares in each row. Two rows, each with two squares, need a minimum of 4 cameras. Two rows, each with four squares, need a minimum of 7 cameras. (a) Find the minimum number of cameras for 6 and 8 squares in each of two rows. Minimum = … Minimum = … (b) Find an expression for the minimum number of cameras for two rows each with n squares, when n is even. …

7 0607/63/O/N/15 © UCLES 2015 [Turn over 6 (a) Complete the table to show the minimum number of cameras for even numbers of rows each with an even number of squares. Number of squares in each row 2 squares 4 squares 6 squares 8 squares n squares Two rows 4 7 Four rows 7 12 Six rows 10 24 Eight rows 13 40 (b) Find an expression for the minimum number of cameras when the number of rows, r, and the number of squares in each row, n, are both even numbers. …

8 0607/63/O/N/15 © UCLES 2015 B MODELLING BACTERIA (20 marks) You are advised to spend no more than 45 minutes on this part. In an experiment a biologist recorded the number of bacteria in a dish at the end of each day for 5 days. The table shows the results. Time in days (x) 1 2 3 4 5 Number of bacteria (n) 120 170 250 370 530 1 (a) On the grid below, plot the five points and join them to form a smooth curve. 0 100 200 300 400 500 600 1 2 3 4 5 x n (b) Write down an estimate for the number of bacteria at the start of the experiment. …

9 0607/63/O/N/15 © UCLES 2015 [Turn over 2 (a) Which of the following models best fits the relationship between x and n? n = pqx n = px2 + q n = px + q … (b) Use the number of bacteria for day 3 and day 4 with your model to find a value for q. … (c) Find the value of p that corresponds to the value for q in part (b). … (d) (i) Rewrite your model substituting your values for p and q. Use your model to estimate the number of bacteria at the end of the seventh day. … (ii) Use your model to estimate the number of bacteria at the start of the experiment. … (iii) Compare your answer in part (ii) with your estimate in question 1(b). … …

10 0607/63/O/N/15 © UCLES 2015 3 In this question log n represents log10 n. (a) Complete the table of values, giving log n correct to 3 significant figures. Time in days (x) 1 2 3 4 5 Number of bacteria (n) 120 170 250 370 530 log n (y) 2.08 (b) Find the mean value of x and the mean value of y. Mean value of x … Mean value of y … (c) On the grid below, plot y against x and draw a line of best fit. 0 0.5 1 1.5 2 2.5 3 1 2 3 4 5 x y

11 0607/63/O/N/15 © UCLES 2015 [Turn over (d) The equation of the line of best fit is y = mx + c. (i) Estimate the value of c from your graph. … (ii) Find the value of m. … (e) Another model for the number of bacteria, n, is log n = mx + c. Rewrite this model substituting your values for m and c. Use this model to estimate the number of bacteria at the end of the seventh day. … (f) Use this model to estimate the number of bacteria at the start of the experiment. … Question 4 is printed on the next page.

12 0607/63/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. 4 Compare the models in question 2(d)(i) and question 3(e). … …

® 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 0607/63 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 63 © 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 SECURITY CAMERAS Question Answer Mark Part Marks 1 (a) (i) oe 1 (ii) oe 1 (iii) oe 1 (b) n + 1 1 2 (a) (i) oe [Minimum =] 4 1 B1 for diagram and 4 (ii) oe 1 (iii) oe [Minimum =8] 1 (b) 2n + 2 oe 1 C opportunity χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 63 © Cambridge International Examinations 2015 A INVESTIGATION SECURITY CAMERAS Question Answer Mark Part Marks 3 9 12 1 1 C opportunity 4 (a) Number of squares in each row 1 square 2 squares 3 squares 4 squares 5 squares n squares One row 6 Three rows 8 10 12 Five rows 15 18 Seven rows 12 16 20 24 4n + 4 oe 2 B1 for 8, 9 or 10 number cells correct B1 for 4 4 + n oe (b) ( ) ( ) 1 1 1 1 2 2 r n r + + + oe 1 (c) 1, 3, 7, 15 1 C opportunity 5 (a) 10 13 1 C opportunity (b) 1 2 3 + n 1 C opportunity 6 (a) Number of squares in each row 2 squares 4 squares 6 squares 8 squares n squares Two rows Four rows 17 22 Six rows 17 31 Eight rows 22 31 4 2 9 + n 2 B1 for 4, 5 or 6 number cells correct B1 for 4 2 9 + n oe (b) ( ) 1 1 1 2 2 r n r + + oe 1 Communication seen in two of 2(b), 3, 4(c), 5(a), 5(b) 1

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 63 © Cambridge International Examinations 2015 B MODELLING BACTERIA Question Answer Mark Part Marks 1 (a) Correct curve between x = 1 and x = 5 2 B1 for 5 points correctly plotted (within 1 mm) B1 for curve through plotted points (within 1 mm) (b) Answer in range 80 to 100 1 2 (a) [n =] pqx 1 (b) [q =] 1.48 1FT FT n = px2 + q in their (a) C opportunity (c) [p =] 77.1[...] 1FT FT their q in n = pqx Or their q in n = px2 + q C opportunity (d) (i) Answer in range 1099 to 1200 1FT FT their p and their q in non- linear models C opportunity (ii) 77[.1...] 1FT (iii) Correct statement about similarity of answers 1FT FT their 1(b) and their 2(d)(ii)

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 63 © Cambridge International Examinations 2015 B MODELLING BACTERIA Question Answer Mark Part Marks 3 (a) 2.23 2.4[0] 2.57 2.72 2 B1 for accuracy to 3 s.f . and B1 for all correct if rounded (b) 3[.0] 2.4[…] 1 Correct to 1d.p. (c) 2FT B1FT for 5 correctly plotted points B1FT for correct ruled line between x = 1 and x = 5 through (3, their 2.4) (d) (i) 1.9 to 1.95 1 FT their correct line of best fit if outside range (ii) 0.15 to 0.17 1 C opportunity (e) 890 to 1390 1 C opportunity (f) 79 to 90 1 4 Correct statement comparing the models 1FT FT their 3(e) and their 2(d)(i) Communication seen in two of 2(b), 2(c), 2(d)(i), 3(d)(ii), 3(e) 1