Cambridge IGCSE Mathematics - International 0607 — 2014 May/June Paper 6 · Variant 3

This document consists of 12 printed pages. IB14 06_0607_63/2RP © UCLES 2014 [Turn over *6238789273* Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2014 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 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 © UCLES 2014 0607/63/M/J/14 Answer both parts A and B. A INVESTIGATION TOTALS (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at possible totals when you make addition sums using only two different positive integers. The integers have no common factor unless otherwise stated. Example Using only 3 and 4 you can make a total of 23. 23 = 3 + 3 + 3 + 3 + 3 + 4 + 4. This is written as 23 = 5 × 3 + 2 × 4. In the table below, when a total is possible, the calculation is shown. When a total is not possible, the cell is crossed out. Totals using only 3 and 4 Total 1 2 3 4 5 6 7 8 Calculation 1 × 3 1 × 4 2 × 3 1 × 3 + 1 × 4 2 × 4 You can make all the totals bigger than 8 by adding 3 to the possible totals 6, 7, 8, … to get 9, 10, 11, … . The largest total that is not possible is 5. 1 (a) Complete the tables below. One integer is always 2. Totals using only 2 and 3 Total 1 2 3 4 5 6 7 8 Calculation 1 × 2 1 × 3 2 × 2 2 × 3 2 × 2 + 1 × 3 The largest total that is not possible is 1.

3 © UCLES 2014 0607/63/M/J/14 [Turn over Totals using only 2 and 5 Total 1 2 3 4 5 6 7 8 Calculation 1 × 2 2 × 2 3 × 2 1 × 2 + 1 × 5 4 × 2 The largest total that is not possible is ……. Totals using only 2 and 7 Total 3 4 5 6 7 8 9 10 Calculation 3 × 2 1 × 7 4 × 2 5 × 2 The largest total that is not possible is ……. (b) From your answers in part (a) complete the following statement. The largest total, using only 2 and y, that is not possible is …………

4 © UCLES 2014 0607/63/M/J/14 2 (a) Complete the tables below. One integer is always 3. Totals using only 3 and 5 Total 3 4 5 6 7 8 9 10 Calculation 1 × 3 1 × 5 2 × 3 3 × 3 The largest total that is not possible is 7. Totals using only 3 and 7 Total 8 9 10 11 12 13 14 15 Calculation 1 × 3 + 1 × 7 4 × 3 2 × 7 5 × 3 The largest total that is not possible is 11. Totals using only 3 and 8 Total 10 11 12 13 14 15 16 Calculation 4 × 3 2 × 3 + 1 × 8 5 × 3 2 × 8 The largest total that is not possible is 13. (b) 3 and 6 have a common factor of 3. Explain why, using only 3 and 6, you cannot find the largest total that is not possible.

5 © UCLES 2014 0607/63/M/J/14 [Turn over 3 (a) Some answers for 5 and y are shown in the table. y Largest total that is not possible 5 6 19 5 7 23 5 8 27 5 9 31 Find an expression, in terms of y, for the largest total that is not possible using only 5 and y. (b) Some answers for 7 and y are shown in the table. y Largest total that is not possible 7 2 5 7 3 11 7 4 17 7 5 23 7 6 29 Find an expression, in terms of y, for the largest total that is not possible using only 7 and y.

6 © UCLES 2014 0607/63/M/J/14 4 Some answers for x and y are shown in the table. x The expression for the largest total that is not possible 2 3 2y – 3 5 7 11 10y – 11 (a) Complete the table by copying the results from questions 1(b) and 3. Write down an expression, in terms of y, for the largest total that is not possible when x = 13. (b) Write down an expression, in terms of x and y, for the largest total that is not possible. 5 (a) Using only 24 and 25, calculate the largest total that is not possible. (b) Using only 24 and 25, show how it is possible to get a total of 320.

7 © UCLES 2014 0607/63/M/J/14 [Turn over 6 The Example on page 2 used only 3 and 4. Totals using 3 and 4 Total 1 2 3 4 5 6 7 8 Calculation 1 × 3 1 × 4 2 × 3 1 × 3 + 1 × 4 2 × 4 The largest total that is not possible is 5. The smallest possible total, after which all totals are possible, is 6. So 6, 7, 8, 9, … are all possible. (a) Use your answer to question 4(b) to write down an expression, in terms of x and y, for the smallest possible total after which all totals are possible. (b) Show that your expression in part (a) is equal to (x – 1)(y – 1). (c) Find all the pairs of integers so that the smallest possible total, after which all totals are possible, is 24.

8 © UCLES 2014 0607/63/M/J/14 B MODELLING DESIGNING AN OPEN BOX (20 marks) You are advised to spend no more than 45 minutes on this part. 25 cm x cm x cm Engineers use an expensive metal to make a box for a machine. The method to make a net is: • take a square piece of the metal of side 25 cm • cut a square, of side x cm, from each corner 1 Complete the inequality for x. < x < 2 Find a model for the area, A cm2, of the net by writing A in terms of x. A =

9 © UCLES 2014 0607/63/M/J/14 [Turn over 3 The engineers fold the net to make a box. The diagram shows the 8 edges of the open box that are strengthened with a seal. Find a model for the length of the seal, L cm, by writing L in terms of x. Write your answer in its simplest form. L =

10 © UCLES 2014 0607/63/M/J/14 4 (a) Show that a model for the volume of the box, V cm3, is V = 625x – 100x2 + 4x3. (b) Assume the measurements on the net are given accurately. Give a reason why the actual volume of the box may be different to V. (c) Sketch the graph of V = 625x – 100x2 + 4x3. O V x (d) Find the maximum volume of the box.

11 © UCLES 2014 0607/63/M/J/14 [Turn over 5 For the machine to work efficiently, the box must have a capacity of at least one litre. (a) Complete this new inequality for x. < x < (b) Because of cost, the engineers want the area, A cm2, to be less than 450 cm2. Use your answer to part (a) and question 2 to show that this is not possible. (c) The engineers decide to allow an area of metal which is less than 500 cm2. Complete this inequality for x. < x < Question 6 is printed on the next page.

12 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. 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 2014 0607/63/M/J/14 6 The cost of making the box is made up of three items. • $2 per square centimetre for the area of metal (A cm2) • $1 per centimetre for the length of the seal (L cm) • $500 for the labour (a) Find, in terms of x, a model for the cost, $C. C = (b) The company wants to sell the box at a profit of 20%. Write down a model for the selling price, $S. S = (c) Find the lowest selling price for a box with a capacity of one litre. $

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2014 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 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 2014 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

Page 2 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0607 63 © Cambridge International Examinations 2014 A INVESTIGATION TOTALS 1 (a) [1 ×]2 + [1 ×]3 4 × 2 or [1 ×]2 + 2 × 3 X [1 ×] 5 2 × 2 [1 ×]2 + [1 ×]7 3 B2 for four or five correct or B1 for two or three correct (b) y – 2 oe 1 2 (a) [1 ×]3 + [1 ×]5 2 × 5 3 × 3 2 × 3 + [1 ×]7 oe X [1 ×]3 + [1 ×]8 1 B2 for four or five correct or B1 for two or three correct (b) you only get multiples of 3 oe 1 Equivalent to having just one number 3 (a) 4y – 5 oe final answer 1 Condone n, x, etc C opportunity (b) 6y – 7 oe final answer 1 C opportunity 4 (a) 12y – 13 oe final answer 1 (b) (x – 1)y – x oe 1 5 (a) 551 1 C opportunity (b) 5 × 24 + 8 × 25 1 6 (a) their 4(b) + 1 1FT (b) xy – x – y + 1 1 B1 dependent (c) 2, 25 3, 13 4, 9 5, 7 3 B2 for 2 or 3 pairs or B1 for 1 pair C opportunity Communication seen in one of 3(a), 3(b), 5(a) or 6(c) 1

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0607 63 © Cambridge International Examinations 2014 B MODELLING DESIGNING AN OPEN BOX 1 0 < x < 12.5 2 B1 for each limit SC1 limits reversed 2 625 – 4x2 oe 1 3 100 – 4x or 4(25 – x) or 2(50 – 2x) 2 M1 for 4x + 4(25 – 2x) or better C opportunity 4 (a) (25 – 2x)(25 – 2x)x soi (625 – 50x – 50x + 4x2)x or (625 – 100x + 4x2)x or (25 – 2x)(25x – 2x2) M1 M1 (b) Loss of metal through cutting or thickness of the metal or the width of the seal. 1 (c) Correct shape through (0, 0) (intention), with right skew with one turning point (maximum) 2 B1 curve with one turning point (maximum) at (0, 12.5) or SC1 for correct cubic drawn beyond 12.5 C opportunity (d) 1160 [cm3] or 1157[.4 … cm3] 1 5 (a) 2.5 < x < 6.1 2 B1 for each limit SC1 if limits reversed C opportunity (b) 625 – 4 × 6.12 = 476[…] or Solving 625 – 4x2 = 450 giving x = 6.6 1FT FT their 6.1 if answer > 450 no FT for this method (c) 5.59[…] < x [< 6.1] 1 C opportunity

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0607 63 © Cambridge International Examinations 2014 6 (a) 2(625 – 4x2) + (100 – 4x) + 500 oe [=1850 – 4x – 8x2] 3 FT their 2 and 3 (b) 1.2 (their (a)) oe isw 1FT FT only if (b) is quadratic with at least two terms (c) their (b) with x = their 6.1 from 5(a) 1FT Communication seen in two from 3, 4(c), 5(a), 5(c), 6(c) 2 1 Communication seen in one question