Cambridge IGCSE Mathematics - International 0607 — 2017 May/June Paper 5 · Variant 3

This document consists of 12 printed pages. DC (ST/AR) 134779/2 © UCLES 2017 [Turn over * 6 0 5 7 7 6 6 9 0 4 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core) May/June 2017 1 hour 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 all the questions. 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 24. Cambridge International Examinations Cambridge International General Certificate of Secondary Education

2 0607/53/M/J/17 © UCLES 2017 Answer all the questions. INVESTIGATION REGULAR STARS This investigation is about the construction of regular stars and their properties. Here are some regular stars. 1 You can make regular stars by extending the sides of regular polygons. For example, this regular polygon makes a regular star with 10 sides and 5 points. (a) Use a straight edge to draw the regular stars made from these regular polygons.

3 0607/53/M/J/17 © UCLES 2017 [Turn over (b) Draw the starting polygon inside this regular star. (c) (i) Complete this table. Number of sides (P) of the starting polygon Number of sides (S) of the star 5 10 6 7 8 9 (ii) Write down a formula for S in terms of P. …

4 0607/53/M/J/17 © UCLES 2017 (d) This is a point angle. The sum of the 5 point angles in this regular star is 180°. (i) Complete the table. Regular star Number of points Sum of star’s point angles 5 6 7 8 9 180° 360° 540° 720° (ii) Is it possible for a regular star, made from a regular polygon, to have the sum of its point angles equal to 1450°? Explain how you decide. … …

5 0607/53/M/J/17 © UCLES 2017 [Turn over (e) (i) The regular pentagon making a regular star is shown in bold. The sum of the interior angles of a pentagon is 540°. Use this information to calculate the value of p. p° NOT TO SCALE … (ii) This diagram shows part of a different regular star. It also shows, in bold, part of the regular polygon that makes it. NOT TO SCALE a° b° Find an equation connecting a and b. Write your answer in its simplest form. …

6 0607/53/M/J/17 © UCLES 2017 2 You can also make stars by placing two congruent regular polygons on top of each other and rotating one of the polygons about their common centre. For example and and

7 0607/53/M/J/17 © UCLES 2017 [Turn over (a) Complete this table. Number of sides (P) of the starting polygon Number of points of the star Number of sides (S) of the star 3 4 5 10 20 6 (b) Write down an equation connecting P and S. …

8 0607/53/M/J/17 © UCLES 2017 3 You can also make regular stars by joining dots that are equally spaced round a circle. Here is a star made by joining every second dot round a circle with 5 equally spaced dots. This 3-point star is made by connecting every second dot round a circle with 6 equally spaced dots. Regular polygons are also regular stars and their vertices are the points of the star. (a) Draw the stars made by connecting every second dot round these circles. 7 dots 8 dots 9 dots 10 dots

9 0607/53/M/J/17 © UCLES 2017 [Turn over Complete this table. Number of equally spaced dots Number of points of the star 5 5 6 3 7 8 9 10 11 12 (b) There are 370 equally spaced dots round a circle. Every second dot is joined. Find the number of points of the star. …

10 0607/53/M/J/17 © UCLES 2017 4 In question 3 you made stars by joining every second dot round a circle. You can also make stars by joining every third dot. Starting from 1, dots are numbered clockwise. This gives a way to code the star. 1 4 7 2 5 8 3 6 1 1 2 3 4 5 6 7 8 (a) Here is the code for a star. 1 4 2 5 3 1 (i) Draw this star on the diagram below. 1 2 3 4 5 (ii) Write down a different code for the star you have drawn. …

11 0607/53/M/J/17 © UCLES 2017 [Turn over (b) Here are some more stars and their codes. 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 1 3 5 7 1 1 2 3 4 5 6 7 8 1 4 7 2 5 8 3 6 1 (i) Write down the connection between the code and the number of points of the star. … … (ii) Write down the connection between the code and the number of dots round the circle. … … Questions 4(c) and 4(d) are printed on the next page.

12 0607/53/M/J/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. (c) Make a sketch showing the numbered dots and the regular star with this code. 1 4 7 1 (d) Find three codes, each starting with 1, which make a star with 10 points. You may use these circles to help you. … … …

® IGCSE is a registered trademark. This document consists of 4 printed pages. © UCLES 2017 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core) May/June 2017 MARK SCHEME Maximum Mark: 24 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 2017 series for most Cambridge IGCSE®, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 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

0 © Q 0607/53 © UCLES 2017 Question 1(a) 1(b) 1(c)(i) 1(c)(ii) 1(d)(i) 1(d)(ii) 1(e)(i) 1(e)(ii) 2(a) S = 2P oe 900 No oe and 1450 is not 540 ÷ 5 or 1 36 2b – a = 180 C Answ a multiple of 108 or 72 see 0 oe Cambridge P er f 180 oe en IGCSE – M PUBLISHED Page 3 of 4 M Mark Schem D Marks 1 Allo 1 1 1 1 C op 1 1 1 B0 i C op 3 M2 or M1[ 180 or 1 2 1 e Pa ow one incorr pportunity f from 180 ÷ pportunity for 2(180 – b [either of the – b ( ) 1 180 2 a − M artial Mark rect extensio ÷ 5 b) + a = 180 e other angles May/June 201 s on oe s in triangle = 17 =]

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2017 © UCLES 2017 Page 4 of 4 Question Answer Marks Partial Marks 2(b) 4 S P = oe 1 3(a) 2 B1 for 4 or 5 cells correct If 0 scored B1 for 4 correct diagrams. 3(b) 185 1 C opportunity 4(a)(i) 1 4(a)(ii) Correct code 1 4(b)(i) Number of numbers [in the code] – 1 = number of points [of the star] oe 1 4(b)(ii) Number of dots round the circle is a multiple of the number of numbers[ in the code] – 1 oe 1 4(c) 1 4(d) 3 correct codes 2 B1 for two correct codes Communication: Seen in two of the following questions 1 1(d)(i) Difference shown or e.g. 720 + 180 1(e)(i) At least two of 180 – 108 = 72 180 – 2 × 72 = 36 oe or 180 – 144 = 36 108 −72 = 36 2 × 72 = 144 oe 3(b) 370 ÷ 2 or 370 is even