Cambridge IGCSE Mathematics - International 0607 — 2011 May/June Paper 5 · Variant 1

This document consists of 5 printed pages and 3 blank pages. IB11 06_0607_05/4RP © UCLES 2011 [Turn over *9892833728* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/05 Paper 5 (Core) May/June 2011 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, highlighters, glue or correction fluid. You may use a 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. www.XtremePapers.com

2 © UCLES 2011 0607/05/M/J/11 For Examiner's Use Answer all the questions. INVESTIGATION PICK’S EQUATION In 1899 the Austrian mathematician Georg Pick found a method to work out the area of any polygon that has its vertices (corners) on a square grid. His method used the number of dots (p) on the perimeter of the polygon and the number of dots (i) inside the polygon. In the polygon shown, p = 7 and i = 4. 1 The diagram below shows 9 polygons, labelled Q to Y. For the rectangle R, p = 10 and i = 2. Its area is A = length × width = 3 × 2 = 6 squares. For the triangle X, p = 8 and i = 1. Its area is A = 1 2 × base × height = 1 2 × 2 × 4 = 4 squares. Q R S T U V W X Y

3 © UCLES 2011 0607/05/M/J/11 [Turn over For Examiner's Use Complete the table below. Polygon Dots on perimeter p Dots inside i Area A p + 2i – 2 Q 0 2 R 10 2 6 12 S 14 20 T 2 5 U 8 16 V 16 12 W 18 2 X 8 1 4 8 Y 4 1 2 9 2 Use the table to write down an equation connecting p + 2i – 2 with A. This is Pick’s Equation. 3 Show that Pick’s Equation gives the correct value for the area of this polygon.

4 © UCLES 2011 0607/05/M/J/11 For Examiner's Use 4 Use Pick’s Equation to find the area of this polygon. Area = squares 5 A polygon has an area A equal to 4 squares. (a) Using Pick’s Equation, a possible pair of values for p and i is p = 6 and i = 2. Find another possible pair of values for p and i. p = and i =

5 © UCLES 2011 0607/05/M/J/11 For Examiner's Use (b) The diagram below shows a quadrilateral with A = 4, p = 6 and i = 2. Draw, on the square grid below, a quadrilateral with A = 4 and the pair of values for p and i that you found in part (a). 6 For any polygon, explain why the value of p is greater than 2. 7 Use Pick’s Equation to show all the possible pairs of values for p and i when a polygon has an area A = 6.

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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2011 question paper for the guidance of teachers 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/05 Paper 5 (Core), maximum raw mark 24 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 must be read in conjunction with the question papers and the report on the examination. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the May/June 2011 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.XtremePapers.com

Page 2 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 05 © University of Cambridge International Examinations 2011 Question Answer Mark Notes Comments 1 Figure p i A p + 2i – 2 Q 4 0 1 2 R 10 2 6 12 S 14 4 10 20 T 8 2 5 10 U 8 5 8 16 V 16 5 12 24 W 18 2 10 20 X 8 1 4 8 Y 9 1 4 1 2 9 10 B10 Deduct one for each wrong or omitted entry up to the maximum of 10 2 p + 2i – 2 = 2A oe 1 B1 Condone bad form 3 p = 18 i = 15 18 + 2 × 15 – 2 (= 46) A = 23 4 A1 soi M1ft substitution into p + 2i –2 A1 cao C1 Evidence of using areas 23 SC1 (if C1 not given) e.g. counting squares must be for the pentagon 4 7 + 2 × 4 – 2 s.o.i. A = 6 1 2 2 M1 A1 OR B2 13 implies M1 Communication for three terms seen

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 05 © University of Cambridge International Examinations 2011 5 (a) (b) One from p = 10 i = 0 p = 8 i = 1 p = 4 i = 3 etc. 1 1 B1 isw B1 Communication for evidence of using (maybe correctly) p + 2i – 2 = 8 or p + 2i = 10 Other quadrilaterals are possible Corresponding to their correct p and i If (a) wrong or omitted: accept a different quadrilateral from that in the question with p = 6 and i = 2 6 p = 2 gives a line oe 1 R1 p = 3 is the smallest value to give an area Reference must be made to dots or p 7 (p) 4 6 8 10 12 14 (i) 5 4 3 2 1 0 B3 + 1 2 for each correct pair. – 1 2 for each wrong pair. Round down Communication for reasoning using Pick’s equation 1 C1 for one communication mark in questions 4, 5(a) or 7