Cambridge IGCSE Mathematics - International 0607 — 2014 Oct/Nov Paper 6 · Variant 1

This document consists of 11 printed pages and 1 blank page. IB14 11_0607_61/2RP © UCLES 2014 [Turn over *3861217397* Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) October/November 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 0607/61/O/N/14 Answer both parts A and B. A INVESTIGATION CUBES (20 marks) You are advised to spend no more than 45 minutes on this part. Identical small cubes fit together to make larger cubes. There are no gaps between these small cubes. For each cube that is made, a cross is marked on each outside face of each small cube. The diagram shows the first three cubes that can be made. Diagram 1 1 by 1 by 1 Diagram 2 2 by 2 by 2 Diagram 3 3 by 3 by 3 This investigation is about the number of crosses that can be marked on cubes. Look at the 1 by 1 by 1 cube. It is made from 1 small cube. It has 6 crosses on it (3 crosses are on the faces not seen on the diagram). 1 Look at the 2 by 2 by 2 cube. (a) How many small cubes is this cube made from? … (b) Explain why there are only 3 crosses on each small cube. … (c) Find the total number of crosses on the 2 by 2 by 2 cube. …

3 © UCLES 2014 0607/61/O/N/14 [Turn over 2 Look at the 3 by 3 by 3 cube. (a) How many small cubes is this cube made from? … (b) How many of these small cubes have 3 crosses on them? … (c) There are 12 small cubes with 2 crosses on them. There is 1 small cube with no crosses on it. How many small cubes have only 1 cross on them? … 3 Complete this table. You may use the dotty grid on page 7 to help you. Size of cube Total number of small cubes Number of small cubes with 0 crosses 1 cross 2 crosses 3 crosses 2 by 2 by 2 0 0 3 by 3 by 3 1 12 4 by 4 by 4 64 8 24 5 by 5 by 5 27 54 8

4 © UCLES 2014 0607/61/O/N/14 4 (a) To work out the number of crosses on the 3 by 3 by 3 cube, complete the following. 1 small cube with 0 crosses gives 0 crosses … small cubes with 1 cross gives 6 crosses 12 small cubes with 2 crosses gives … crosses … small cubes with 3 crosses gives … crosses Total = … crosses (b) The total number of crosses can also be worked out by the following method. Complete the following. The number of crosses on one face of the 3 by 3 by 3 cube is … So the total number of crosses on all the 6 faces is … (c) Find the total number of crosses on a 4 by 4 by 4 cube. … (d) Find an expression, in terms of n, for the total number of crosses on an n by n by n cube. …

5 © UCLES 2014 0607/61/O/N/14 [Turn over 5 The number of small cubes with 0 crosses forms a sequence. Size of cube 2 by 2 by 2 3 by 3 by 3 4 by 4 by 4 5 by 5 by 5 n by n by n Number of small cubes with 0 crosses 0 1 8 27 For an n by n by n cube, find an expression, in terms of n, for the number of small cubes with 0 crosses. Write your answer in the table above. 6 For an n by n by n cube, an expression for the number of small cubes with one cross is 6(n – 2)2. In an n by n by n cube, can the number of small cubes with 0 crosses equal the number of small cubes with one cross? Show your working.

6 © UCLES 2014 0607/61/O/N/14 7 For an n by n by n cube, find an expression, in terms of n, for the number of small cubes with 2 crosses. … 8 (a) In an n by n by n cube there are 64 small cubes that have 0 crosses on them. How many small cubes does it have altogether? … (b) In another n by n by n cube there are 60 small cubes that have 2 crosses on them. In this cube, how many small cubes have only 1 cross on them? …

7 © UCLES 2014 0607/61/O/N/14 [Turn over

8 © UCLES 2014 0607/61/O/N/14 B MODELLING FISH PONDS (20 marks) You are advised to spend no more than 45 minutes on this part. Volume of a cylinder of radius r, height h, is πr2h Volume of a sphere, radius r, is 3 3 4 r π Fish ponds can be either hemispherical or cylindrical. d r d cylindrical pond hemispherical pond 1 (a) A hemispherical pond has radius r metres. When r = 3, show that the volume of the pond is 18π m3. (b) A cylindrical pond has a radius of d metres and a depth of d metres. Show that the volume of this pond is πd 3 cubic metres. (c) The radius and the depth of the cylindrical pond in part (b) is the same as the radius of the hemispherical pond in part (a). Show that the cylindrical pond holds more water than the hemispherical pond. (d) A hemispherical pond of radius r metres has the same volume as a cylindrical pond of radius and depth d metres. Show that r d 3 3 2 = .

9 © UCLES 2014 0607/61/O/N/14 [Turn over 2 Theo wants to work out how many fish he can put in a pond. A fish needs 5 litres of water for every 0.1 cm of its length. (a) The average length of 15 fish is 18 cm. Find the number of cubic metres of water needed for these fish. … (b) There are F fish in the pond, with an average length of L cm. Find a model for the number of cubic metres of water, W, needed. … (c) Theo’s pond has a volume of 20 m3. (i) Use your model to find the maximum number of fish, of average length 24 cm, that can be put in this pond. … (ii) If Theo’s pond is hemispherical find its radius. … (iii) If Theo’s pond is cylindrical, and the radius and depth are the same, find its radius. …

10 © UCLES 2014 0607/61/O/N/14 3 Theo decides to have a cylindrical pond. The radius, r metres, does not have to be the same as the depth, d metres. (a) For a pond with volume 20 m3 find a model for the depth, d, in terms of r. … (b) Sketch the graph of d against r for 1 Y r Y 5. 0 5 r d 7 (c) What practical problem is there when the radius is less than one metre? … (d) Find the radius of a pond when the depth is 1 m. …

11 © UCLES 2014 0607/61/O/N/14 4 The water level must be 30 cm below the top of the sides of the pond. (a) Modify your model in question 3 (a). … (b) Explain how this affects your graph in question 3 (b).

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® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the October/November 2014 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 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 October/November 2014 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 – October/November 2014 0607 61 © Cambridge International Examinations 2014 A INVESTIGATION CUBES 1 (a) (b) (c) 8 Response implying some faces hidden within the large cube 24 1 1 1FT bod ‘can’t see’ 3 × their (a) 2 (a) (b) (c) 27 8 6 1 1 1 3 Size of cube Total number of small cubes Number of small cubes with 0 crosses 1 cross 2 crosses 3 crosses 2 by 2 by 2 8 0 0 0 8 3 by 3 by 3 27 1 6 12 8 4 by 4 by 4 64 8 24 24 8 5 by 5 by 5 125 27 54 36 8 2 B1 for 125 and 36 or B1 for first 3 rows correct 4 (a) (b) (c) (d) 1 small cube with 0 crosses gives 0 crosses 6 small cubes with 1 cross gives 6 crosses 12 small cubes with 2 crosses gives 24 crosses 8 small cubes with 3 crosses gives 24 crosses Total = 54 crosses 9 54 96 6n2 oe 1 1 1 1 C opportunity C opportunity 5 ( )3 2 − n oe isw 2 B1 for [kn] – 2 for n3 soi C opportunity 6 Yes oe and n = 8 oe or 216 seen 1 SC1 for n = 2 and cubes = 8 with working shown e.g. sketch

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2014 0607 61 © Cambridge International Examinations 2014 7 ( ) 2 12 − n oe 1 C opportunity 8 (a) (b) 216 150 1 2 C opportunity B1 for n = 7 soi If 0 scored SC1 FT their 7 = 60 followed by their n in ( )2 2 6 − n n must be integer C opportunity Communication seen in at least two of 4(c), 4(d), 5, 7, 8(a) or 8(b) 1

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2014 0607 61 © Cambridge International Examinations 2014 B MODELLING FISH PONDS 1 (a) (b) (c) (d) 3 3 3 4 2 1 × × × π oe d d × × 2 π [cylinder =] 27π [and] [hemisphere =] 18π oe 3 3 d r 3 2 π π = 1 1 1 1 seen through working accept [ ] 3 2 3 H r π = and [ ] 3 C r π = 2 (a) (b) (c) (i) (ii) (iii) 13.5 [m3] W = 0.05FL oe 16 [fish] 2.1… to 2.19 1.85[…] [m] or 1.86[m] 3 1 2FT 1 1 M2 for 1.0 5 18 15 × × oe or M1 1.0 18 15× or better soi by 2700 or 1.0 5 20× or better C opportunity B1 for 16.6[…] or FT their 16.6[…] C opportunity C opportunity Accept cube root of 20 π If 0 scored in (i) and in (ii) SC1 for same converting error in both C opportunity

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2014 0607 61 © Cambridge International Examinations 2014 3 (a) (b) (c) (d) 2 20 d r π = oe Too deep oe 2.52[m] 2.522 to 2.523… 1 2 1 1 1 for shape 1 for not reaching either axis between y = 7 and x = 5 C opportunity 4 (a) (b) 2 20 0.3 d r π = + Translates [up by] 0.3 oe 1FT 1FT FT their 3(a) + 0.3 FT their + 0.3 Communication seen in two or more of 2(a), 2(c)(i), 2(c)(ii), 2(c)(iii) or 3(d) 1