Cambridge IGCSE Mathematics - International 0607 — 2017 Oct/Nov Paper 6 · Variant 3

This document consists of 12 printed pages. DC (KN/SG) 137085/3 © UCLES 2017 [Turn over * 8 7 2 9 4 9 0 6 8 9 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 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 to 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 0607/63/O/N/17 © UCLES 2017 Answer both parts A and B. A INVESTIGATION CHEQUERED FLAGS (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation is about the number of coloured squares on a chequered flag. A chequered flag has two or more colours making a pattern of squares. There is always a black square in the top left corner. The size of a flag is m by n, where m is the number of rows and n is the number of columns. 1 (a) The flag above is a 4 by 5, black and white chequered flag. Write down the number of black squares and the number of white squares. Black … White … (b) (i) Complete both tables. Size of flag 2 by 1 2 by 2 2 by 3 2 by 4 2 by 5 2 by n Black 1 2 White 1 2 Size of flag 4 by 1 4 by 2 4 by 3 4 by 4 4 by 5 4 by n Black 2 White 2

3 0607/63/O/N/17 © UCLES 2017 [Turn over (ii) Find two expressions, in terms of n, for the number of black squares and the number of white squares on a 6 by n flag. Black … White … (c) How many different sizes of black and white chequered flags are there with 12 black squares and 12 white squares? … (d) m is an even number. For the flags in this question find an expression, in terms of m and n, for the number of black squares on an m by n chequered flag. …

4 0607/63/O/N/17 © UCLES 2017 2 This is a 3 by 5, black and white chequered flag. (a) Write down the number of black squares and the number of white squares. Black … White … (b) Is your expression in question 1(d) correct for this flag? Write down the restrictions on m and n for your expression in question 1(d). … (c) Complete both tables. Size of flag 3 by 1 3 by 2 3 by 3 3 by 4 3 by 5 Black White Size of flag 5 by 1 5 by 2 5 by 3 5 by 4 5 by 5 Black White (d) m and n are both odd numbers. Using your answer to question 1(d), find an expression, in terms of m and n, for the number of black squares and the number of white squares. Black … White …

5 0607/63/O/N/17 © UCLES 2017 [Turn over 3 This is a 4 by 6, black, white and grey three-coloured chequered flag. (a) For this flag, write down the number of squares of each colour. Black … White … Grey … (b) (i) Complete both tables. Size of flag 3 by 1 3 by 2 3 by 3 3 by 4 3 by 5 3 by n Black White Grey Size of flag 6 by 1 6 by 2 6 by 3 6 by 4 6 by 5 6 by n Black White Grey (ii) For the flags in this question, find an expression, in terms of m and n, for the number of black squares on an m by n three-coloured chequered flag. … (iii) Is your expression in part (ii) correct for a flag with 2 rows? Write down the restriction on n for your expression in part (ii). …

6 0607/63/O/N/17 © UCLES 2017 4 The number of black squares on a flag with six colours is mn 6 . (a) Show that this is not true for a 16 by 14 flag with six colours. (b) A six-coloured chequered flag has 3 black squares. Find all the possible sizes of the flag. 5 The expression p mn gives the number of each coloured square on an m by n flag with p colours. What is true about m, n and p? …

7 0607/63/O/N/17 © UCLES 2017 [Turn over B MODELLING AREAS OF POLYGONS (20 marks) You are advised to spend no more than 45 minutes on this part. This task looks at the relationship between the number of sides and the area of an enclosure. A farmer has a 24 metre length of fencing to make an enclosure. 1 (a) He makes a rectangular enclosure as shown in this diagram. 11 m 1 m Area = 11 m2 NOT TO SCALE Complete this table to show all the possible rectangular enclosures he can make with 24 m of fencing. The sides of the enclosure are always a whole number of metres. 1 by 11 and 11 by 1 are the same rectangle. Width (m) Length (m) Area (m2) 1 11 11 2 10 20 (b) What is the mathematical name of the rectangle that gives the maximum area of the enclosure? …

8 0607/63/O/N/17 © UCLES 2017 2 The farmer makes an enclosure that is an equilateral triangle with a perimeter of 24 m. The perpendicular distance from a corner to the opposite side of the enclosure is h metres. h NOT TO SCALE Show that the area of this triangle is 27.7 m2, correct to 1 decimal place. 3 (a) The farmer makes an enclosure that is a regular pentagon with a perimeter of 24 m. a° NOT TO SCALE (i) Show that a = 72.

9 0607/63/O/N/17 © UCLES 2017 [Turn over (ii) The regular pentagon can be divided into 5 isosceles triangles. The perpendicular distance from the centre of the pentagon to a side is h metres. Use trigonometry to find the value of h. a° h 24 5 m NOT TO SCALE … (iii) Use your value of h to find the area of the isosceles triangle. … (iv) Show that the area of the pentagon is 40 m2, correct to the nearest square metre.

10 0607/63/O/N/17 © UCLES 2017 (b) The farmer makes an enclosure in the shape of a regular hexagon with a perimeter of 24 m. Use the method in part (a) to find the area of this hexagon. … 4 The farmer makes an enclosure that is an n-sided regular polygon with a perimeter of 24 m. Here is an isosceles triangle from this polygon. a° h 24 n m NOT TO SCALE (a) (i) Use this triangle and the method from question 3, to show that a model for calculating the area (A m2) of a regular n-sided polygon with perimeter 24 m is ° tan A n n 180 144 = J L KK N P OO . Show all your working.

11 0607/63/O/N/17 © UCLES 2017 [Turn over (ii) Write down a condition for n. … (b) Use the model to show that the area of a regular octagon with a perimeter of 24 m is 43.5 m2, correct to 1 decimal place. (c) Sketch the graph of ° tan A n n 180 144 = J L KK N P OO for n 3 50 G G on the grid below. A 50 0 3 50 n (d) The farmer wants to use the 24 m of fencing to make a regular polygon with an area of 44.0 m2, correct to the nearest 0.1 m2. Find the number of sides of this polygon. … Questions 4(e), 4(f) and 4(g) are printed on the next page.

12 0607/63/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. (e) Use your graph to write down an estimate of the greatest possible area of a polygon with a perimeter of 24 m. … (f) (i) When n is very large, the shape of the polygon is approximately a … (ii) Calculate the exact value of the greatest possible area of a shape with a perimeter of 24 m. … (g) Change the model in question 4(a) to find the area of any regular polygon with a perimeter of P. …

® IGCSE is a registered trademark. This document consists of 6 printed pages. © UCLES 2017 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 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.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 2 of 6 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

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 3 of 6 Question Answer Marks Guidance A INVESTIGATION CHEQUERED FLAGS 1(a) 10 10 1 1(b)(i) Size of flag 2 by 1 2 by 2 2 by 3 2 by 4 2 by 5 2 by n Black 1 2 3 4 5 n White 1 2 3 4 5 n Size of flag 4 by 1 4 by 2 4 by 3 4 by 4 4 by 5 4 by n Black 2 4 6 8 10 2n White 2 4 6 8 10 2n 2 B1 for at least five unshaded numerical pairs correct and B1 for all four algebraic expressions correct 1(b)(ii) 3n oe 3n oe 1 1(c) 8 1 C opportunity 1(d) 2 mn oe 1 2(a) 8 7 1 2(b) [No] one of m or n must be even 1 2(c) Size of flag 3 by 1 3 by 2 3 by 3 3 by 4 3 by 5 Black 2 3 5 6 8 White 1 3 4 6 7 Size of flag 5 by 1 5 by 2 5 by 3 5 by 4 5 by 5 Black 3 5 8 10 13 White 2 5 7 10 12 1 At least four of the unshaded pairs correct 2(d) 1 2 2 + mn oe [black] 1 2 2 − mn oe [white] 2 B1 for each

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 4 of 6 Question Answer Marks Guidance 3(a) 8 8 8 1 3(b)(i) Size of flag 3 by 1 3 by 2 3 by 3 3 by 4 3 by 5 3 by n Black 1 2 3 4 5 n White 1 2 3 4 5 n Grey 1 2 3 4 5 n Size of flag 6 by 1 6 by 2 6 by 3 6 by 4 6 by 5 6 by n Black 2 4 6 8 10 2n White 2 4 6 8 10 2n Grey 2 4 6 8 10 2n 2 B1 for all numerical unshaded triples correct and B1 for all algebraic triples correct 3(b)(ii) 3 mn oe 1 3(b)(iii) No and n must be a multiple of 3 oe 1 4(a) 16 14 6 × = 37.3[…] or is not integer 1 4(b) 1 × 18 18 × 1 2 × 9 9 × 2 3 × 6 6 × 3 1 C opportunity 5 m or n or mn is a multiple of p oe 1 Communication: seen in one of the following questions 1 1(c) Showing (at least 2 different) pairs of factors of 24 4(b) 6 mn = 3 or mn = 18

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 5 of 6 Question Answer Marks Guidance B MODELLING AREAS OF POLYGONS 1(a) Width m Length m Area m2 1 11 11 2 10 20 3 9 27 4 8 32 5 7 35 6 6 36 1 1(b) Square 1 2 6.93 or better 27.71 2 B1 Accept 6.9 to 6.93 or 4 3 or 48 B1 Accept 27.68 to 27.72 3(a)(i) 360 ÷ 5 oe leading to 72 1 3(a)(ii) 3.3[0...] 1 C opportunities 3(a)(iii) 7.93 or 7.92 to 7.93 1 FT 2.4 × their 3.3 correctly evaluated to 3 s.f. C opportunity 3(a)(iv) 39.6 seen or 39.6 to 39.65 1 3(b) 41.6 or 41.4 to 41.6 1 C opportunities 4(a)(i) 0.5 × 24 n × 12 360 tan 2 n n × n isw 1 4(a)(ii) Integer or n ⩾ 3 oe 1 4(b) 144 180 8tan 8       = 43.45[…] or 43.45 to 43.50 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 6 of 6 Question Answer Marks Guidance B MODELLING AREAS OF POLYGONS 4(c) Correct shape 1 4(d) 9 1 C opportunity 4(e) 45.8 1 Accept 45 to 46 4(f)(i) circle 1 4(f)(ii) 144 π 2 M1 for [r = ] 12 π or SC1 for 45.8[…] 4(g) 2 180 4 tan P n n 1 C opportunity Communication: seen in two of the following questions 1 3(a)(ii) 2 a = 36 or 2.4 tan36 or 2.4tan54 or 2.4sin54 sin36 3(a)(iii) 0.5 × 4.8 × their 3.3 oe 3(b) 2 tan30 or 2sin60 sin30 or 2tan60 4(d) 43.9[…] for 9 or 44.3[…] for 10 4(g) 360 2n or 180 n and 0.5 P n or 2 P n seen 10 20 30 40 50 0 50 n A