Cambridge IGCSE Mathematics - International 0607 — 2016 May/June Paper 6 · Variant 3

This document consists of 12 printed pages. DC (LK/SG) 115876/2 © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 4 3 7 2 1 8 9 1 6 3 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2016 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/M/J/16 © UCLES 2016 Answer both parts A and B. A INVESTIGATION AREAS AND PERIMETERS (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at the connection between the perimeter and area of a rectangle. All diagrams are not to scale. 7 4 The area of this 4 by 7 rectangle is 28 and its perimeter is 22. 1 (a) 10 3 Find the area and perimeter of this rectangle. Area = … Perimeter = …

3 0607/63/M/J/16 © UCLES 2016 [Turn over (b) x 3 The area and perimeter of this rectangle have the same value. (i) Find x. … (ii) Write down its perimeter. … (c) x 7 (i) Write down an expression for the area of this rectangle. … (ii) Write down an expression for the perimeter of this rectangle. … (iii) The area and the perimeter have the same value. Write down an equation and solve it to find x. …

4 0607/63/M/J/16 © UCLES 2016 2 x y (a) For this rectangle, find an expression, in terms of x and y, for (i) the area, … (ii) the perimeter. … (b) The area and the perimeter have the same value. Show that y x x 2 2 = - .

5 0607/63/M/J/16 © UCLES 2016 [Turn over 3 y x x 2 2 = - (a) Find y when x 12 = . … (b) Find y when x 1 = . … (c) Sketch the graph of y x x 2 2 = - for values of x between 0 and 15. y x 0 15 (d) For which values of x is the graph not valid for rectangles with area equal to perimeter? Give your answer as an inequality. …

6 0607/63/M/J/16 © UCLES 2016 4 x y When the area is less than the perimeter, y x x 2 2 1 - . (a) Explain this using your method in question 2(b). Assume that x 2 2 . (b) On your sketch in question 3(c), mark clearly a point where y x x 2 2 1 - and x 2 2 . Label this point Q. (c) Check that the co-ordinates of your point Q give a rectangle in which the area is less than the perimeter.

7 0607/63/M/J/16 © UCLES 2016 [Turn over 5 Investigate if it is possible to find a cube where the volume is numerically equal to its total surface area.

8 0607/63/M/J/16 © UCLES 2016 B MODELLING HOW MUCH GRASS CAN THE GOAT EAT? (20 marks) You are advised to spend no more than 45 minutes on this part. A farmer owns a goat and a large field with lots of grass in it. 10 m 1 The farmer ties the goat to a point in the field with a rope that is 10 m long. Find the area of grass that the goat can eat. … 2 (a) With the same 10 m rope, she ties the goat to an outside corner of a barn of length 25 m and width 15 m. 10 m 15 m 25 m NOT TO SCALE Find the area of grass that the goat can eat. …

9 0607/63/M/J/16 © UCLES 2016 [Turn over (b) With a 20 m rope she ties the goat to the same corner of the barn. 20 m 15 m 25 m NOT TO SCALE G Explain what happens when the goat continues past point G. 3 With the original 10 m rope, she ties the goat to an outside corner of a barn of length 15 m and width 8 m. (a) Draw a sketch to show the shape of the area of grass the goat can now eat. (b) Show that the area of grass the goat can eat is approximately 239 m2.

10 0607/63/M/J/16 © UCLES 2016 4 The farmer now uses a rope that is x metres long. The barn is of length 15 m and width 8 m. (a) The goat can eat an area of grass the same shape as that in question 2(a). (i) Complete the inequality for x. … x 1 1 … (ii) Write down a model for the area, A m2. A = … (b) The goat can eat an area of grass in a shape of the same form as that in question 3(a). (i) Complete the inequality for x. … x 1 1 … (ii) Find a model for this area, A m2. A = …

11 0607/63/M/J/16 © UCLES 2016 [Turn over (c) (i) When x 15 23 1 1 , find a model for the area of grass, A m2, that the goat can eat. A = … (ii) The goat needs to reach 700 m2 of grass. Use your model to find the length of the rope. … Question 4(d) is printed on the next page.

12 0607/63/M/J/16 © UCLES 2016 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. (d) The goat needs to reach 500 m2 of grass. Find the length of the rope. …

® IGCSE is the registered trademark of Cambridge International Examinations. This document consists of 5 printed pages. © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2016 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 will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2016 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 – May/June 2016 0607 63 © Cambridge International Examinations 2016 Abbreviations awrt answers which round to 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 AREAS AND PERIMETERS Question Answer Marks Part Marks 1 (a) 30 26 1 (b) (i) 6 1 (ii) 18 1FT FT 2 × (their 6) + 6 (c) (i) 7x oe 1 (ii) 14+2x oe isw 1 (iii) 2.8 oe FT1 FT their c(i) and c(ii) if same form C opportunity 2 (a) (i) xy oe 1 (ii) 2 2 + x y oe 1 (b) 2 2 − = xy y x ( ) 2 2 − = y x x 1 1 3 (a) 2.4 1 C opportunity (b) –2 1 C opportunity (c) 2 correct curves 2 B1 for each branch SC1 for correct curve but branches joined C opportunity (d) [ ] 0 2

x 1 5 10 15 −10 −5 5 10 15 x y

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 63 © Cambridge International Examinations 2016 Question Answer Marks Part Marks 4 (a) 2 2 2 2 ( 2) 2 < + − < − < xy x y xy y x y x x 1 (b) Point clearly between x-axis, x = 2 and curve 1 (c) Valid check using co-ordinates where Area < Perimeter 1 Not dependent on (b) 5 [Yes,] showing solution of 6 1 C opportunity Communication in 2 from 1(c)(iii), 3(a), 3(b), 3(c) or 5 1

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 63 © Cambridge International Examinations 2016 B MODELLING HOW MUCH GRASS CAN THE GOAT EAT? Question Answer Marks Part Marks 1 314 or 314.1… 1 2 (a) 236 or 235.6… 1FT FT 3 4 (their 314) C opportunity (b) Quarter circle shown on diagram or 5m radius implied 1 3 (a) 1 A 3 4 circle and a 1 4 circle of smaller radius C opportunity (b) 236 + π oe or 238.8 or 238.76 … 2FT FT their 2(a) M1 for 2 1 2 4 π × × oe C opportunity 4 (a) (i) 0 8 < < x 2 B1 for each limit (ii) 2 3 4 x π oe 1 (b) (i) 8 15 < < x 2 B1 for each limit (ii) 2 2 3 1 ( 8) 4 4 x x π + π − oe isw 2FT FT their (a)(ii) M1 for 2 1 4 k + π (c) (i) (their (b)(ii)) + 2 1 ( 15) 4 x π − 2FT FT their (b)(ii) M1 for (their (b)(ii)) + 2 1 4 k π or ( ) 2 1 15 4 x + π − C opportunity

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 63 © Cambridge International Examinations 2016 Question Answer Marks Part Marks (ii) 16.5 [m] 1FT FT any model including a term in ( ) 2 − x a C opportunity (d) 14.1 [m] 2 M1 for attempt at solving with 500 in any model including a term in ( ) 2 − x a C opportunity Communication in 3 of 2(a), 3(a), 3(b), 4(c)(i), 4(c)(ii) or 4(d) 2 C1 if seen in 2 of these