Cambridge IGCSE Mathematics - International 0607 — 2013 May/June Paper 6 · Variant 1

This document consists of 8 printed pages. IB13 06_0607_06/2RP © UCLES 2013 [Turn over *6915781925* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/06 Paper 6 (Extended) May/June 2013 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, highlighters, glue or correction fluid. You may use a 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. www.XtremePapers.com

2 © UCLES 2013 0607/06/M/J/13 For Examiner's Use Answer both parts A and B. A INVESTIGATION DIAGONALS OF RECTANGLES (20 marks) You are advised to spend 45 minutes on part A. Rectangles are drawn on a grid. The sides of each rectangle lie on gridlines and the length is greater than or equal to the width. This investigation looks for a method for calculating the number of small squares through which a diagonal passes. 1 The diagram shows a rectangle with length 5 and width 3. The diagonal crosses 4 vertical gridlines inside the rectangle. Write down (a) the number of horizontal gridlines that the diagonal crosses inside the rectangle, (b) the total number of gridlines that the diagonal crosses inside the rectangle. 2 A rectangle has length x and width y. x and y do not have a common factor. (a) Write down an expression for (i) the number of vertical gridlines that a diagonal crosses inside the rectangle, in terms of x, (ii) the number of horizontal gridlines that a diagonal crosses inside the rectangle, in terms of y, (iii) the total number of gridlines, N, which a diagonal crosses inside the rectangle, in terms of x and y. Write your answer in its simplest form. N =

3 © UCLES 2013 0607/06/M/J/13 [Turn over For Examiner's Use (b) S is the number of squares through which the diagonal passes. For example, the diagonal in question 1 passes through 7 squares. (i) Write S in terms of N. S = (ii) Write S in terms of x and y. S = (c) Show that your formula for S in part (b)(ii) gives the correct value for an 8 by 5 rectangle. Use the grid to show clearly how many squares the diagonal passes through.

4 © UCLES 2013 0607/06/M/J/13 For Examiner's Use 3 In question 2, x and y did not have a common factor. In this question, x and y do have a common factor. (a) (i) Show clearly that your formula for S does not give the correct value for a 9 by 6 rectangle. (ii) 9 and 6 have a common factor of 3. Show how you use the value of S for a 3 by 2 rectangle to calculate S for a 9 by 6 rectangle.

5 © UCLES 2013 0607/06/M/J/13 [Turn over For Examiner's Use (b) Use your method in part (a)(ii) to find S for each of these rectangles. (i) 93 by 90 (ii) 60 by 35 4 The diagonal of a rectangle passes through 6 squares. Use question 2 and question 3 to find the length and the width of each possible rectangle.

6 © UCLES 2013 0607/06/M/J/13 For Examiner's Use B MODELLING DRILLING A TUNNEL (20 marks) You are advised to spend 45 minutes on part B. On the plan, A is south of B and C is east of B. AB = 500 metres and BC = 300 metres. Engineers want to drill a tunnel from A to C. The tunnel has one or more straight sections. 1 Calculate the length of the shortest possible tunnel from A to C. Give your answer correct to the nearest metre. m 2 Write down the length of the tunnel if the engineers drill through as little hard rock as possible. m 3 P is a point which is x metres south of B. The engineers decide to drill from A to P to C. Through normal rock, from A to P, the drill moves forward at 2 metres per hour. Through the hard rock, from P to C, the drill moves forward at 1 metre per hour. 300 m 500 m B C A Normal rock Normal rock Hard rock NOT TO SCALE North 300 m x m 500 m B C A P Normal rock Normal rock Hard rock NOT TO SCALE North

7 © UCLES 2013 0607/06/M/J/13 [Turn over For Examiner's Use (a) Explain why the time in hours, T, that it takes to drill the tunnel, can be modelled by this equation. 2 90000 2 500 x x T + + − = (b) All the measurements are accurate. Write down a practical reason why the time given by the model may be different from the actual time. (c) On the diagram, sketch the graph of T against x. 400 500 600 Time (hours) T x 0 500 Distance (metres) (d) (i) Find, to the nearest metre, the position of P which gives the minimum time to drill the tunnel. metres from B (ii) Find this minimum time correct to the nearest 10 hours. hours

8 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. University of 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. © UCLES 2013 0607/06/M/J/13 For Examiner's Use 4 To drill through normal rock costs 2 thousand dollars per hour. To drill through the hard rock costs 3 thousand dollars per hour. (a) The total cost of drilling the tunnel is n thousand dollars. Write down a model for n in terms of x. n = (b) (i) Find, to the nearest metre, the position of P which gives the minimum cost. metres from B (ii) Write, in full, this minimum cost to the nearest ten thousand dollars. $ 5 The model for the time taken to drill the tunnel is 2 90000 2 500 x x T + + − = . (a) The position of B and C are fixed. Investigate the position of P which gives the minimum time when A is more than 500 m south of B. (b) If AB = d metres explain, using part (a), why the minimum time in hours is k d T + = 2 , where k = 260 correct to 3 significant figures.

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2013 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/06 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 May/June 2013 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components. www.XtremePapers.com

Page 2 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0607 06 © Cambridge International Examinations 2013 A – INVESTIGATION: DIAGONALS OF RECTANGLES 1 (a) (b) 2 6 B1 B1 FT 2 (a) (i) (ii) (iii) x – 1 y – 1 x + y – 2 B1 B1 B1 Accept (i) and (ii) swapped correct answer only (b) (i) N + 1 o.e. B1 (ii) x + y – 1 o.e. B1 (c) diagonal drawn with the 12 squares marked on the diagonal. 8 +5 – 1 = 12 B1 B1 8 + 5 = 13 and 13 – 1 =12. Must use (b)(ii) If 0 then SC1 for 12 seen. 3 (a) (i) (ii) 12 squares clearly marked on the diagram. 9 + 6 –1 = 14 o.e. 3 × 4 = 12 seen or implied C1 C1 R1 2 simultaneous intersections marked. FT their 2(b)(ii) ≠12 Accept 3 ×(3 + 2 – 1) but not 9 + 6 – 3 (b) (i) [3 × 60 =] 180 B2 B1 for 60 seen Accept using x + y – 3 or x + y – common factor o.e. Without wrong working. Communication mark for numerical method used in either (i) or (ii) e.g. Common factor = 3 soi and 3 × (31 + 30 – 1) o.e. Common factor = 3 soi and 93 + 90 – 3 (ii) [5 × 18 =] 90 B2 B1 for 18 seen Without wrong working.

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0607 06 © Cambridge International Examinations 2013 4 6 × 1, 5 × 2, 4 × 3 6 × 2 6 × 3 6 × 6 B3 soi by diagrams in the answer space 5 + 2 – 1 etc. is insufficient B1 for any one row B2 for two or three rows Communication mark for showing correct numerical method or statement once. e.g. x + y – 1 = S, S = 6 OR x + y = 7 OR 5 + 2 – 1 = 6 But not 5 + 2 = 7 – 1 = 6 Deduct 1 for each incorrect extra rectangle, but ignore repeated rectangles. 1 Communication seen in questions 3(b) or 4 TOTAL 20

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0607 06 © Cambridge International Examinations 2013 B – MODELLING: DRILLING A TUNNEL 1 583[.09…] 2 M1 for 5002 + 3002 Communication mark for correct method, use of square root and no bad form seen. 2 800 1 3 (a) 500 – x seen as a distance [PC2 = ] 3002 + x time
their distance speed soi R1 R1 R1 2 500 x − on its own does not gain credit. Explanation with numerator = distance and denominator = speed (b) time added to change the drill’s direction R1 Any reason for extra time related to the context, not ‘rounding’, ‘incorrect’ or ‘different’ numbers. (c) 100 200 300 400 500 500 600 x T G1 G1 Shape continuous curve required. Minimum in the left half. and not more than 1 cm from T = 500 Reaches T axis between 500 and 600 Dependent on shape (but not the restrictions) Not less than ½ cm from 500 or 600. (d) (i) 173 If not B3 then B1 173 B1 510 SC2 for 173.2 … and 509.8 … OR 173 … and 509.8 … OR 173.2 … and 510 (ii) 510 3

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0607 06 © Cambridge International Examinations 2013 4 (a) ( ) 2 90000 3 500 x x + + − o.e. B1 B1 SC1 Thousands included in the expressions. Ignore subsequent working. (b) (i) 106 B1 FT FT answers between 0 and 500 from their (a) of similar form producing a graph with a minimum. Accept 105.5 to 106.499 … (ii) 1 350 000 B1 FT FT (b)(i) from their (a) of similar form. Accept 1350 thousand. Accept range from 1 348 000 to 1 349 000 5 (a) x stays the same o.e. B1 x = 173 on its own is insufficient. Communication: method description or substitution of a d > 500 soi (b) Substitute x = 173[.2…] in 2 90000 2 x x d T + + − = o.e. [...] 4. 346 5. 86 2 + − = d 260 2 + = d OR One substitution of a minimum to check that 260 2 + = d T Two more substitutions of minima. M1 A1 [B1] [B1] Accept 346.3[…] 259.8 to 259.9 instead of –86.5 + 346.4[…] e.g. substituting 500 and/or their answer to 3(d)(iii) 1 Communication seen in 1 or 5(a). TOTAL 20