Cambridge IGCSE Mathematics - International 0607 — 2014 Oct/Nov Paper 5 · Variant 3

This document consists of 6 printed pages and 2 blank pages. IB14 11_0607_53/FP © UCLES 2014 [Turn over *9254095192* Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core) October/November 2014 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, glue or correction fluid. You may use an HB 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.

2 © UCLES 2014 0607/53/O/N/14 Answer all the questions. INVESTIGATION CUBES 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 0607/53/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 (a) On the dotty grid below, draw a 4 by 4 by 4 cube. Mark a cross on the outside face of each small cube. (b) The 4 by 4 by 4 cube is made from 64 small cubes. (i) How many of these small cubes have 3 crosses on them? … (ii) How many of these small cubes have 2 crosses on them? …

4 © UCLES 2014 0607/53/O/N/14 4 Complete this table. You may use the dotty grid on page 6 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 5 (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. …

5 © UCLES 2014 0607/53/O/N/14 [Turn over 6 (a) The number of small cubes with 0 crosses forms a sequence of cube numbers. 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. (b) The number of small cubes with 1 cross forms a sequence. Find the nth term of this sequence. … (c) The number of small cubes with 2 crosses forms a sequence. Find the nth term of this sequence. …

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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 0607/53 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 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 53 © Cambridge International Examinations 2014 1 (a) (b) (c) 8 Response implying some faces hidden within the large cube 24 1 1 1FT bod for ‘can’t see’ FT 3 × their (a) 2 (a) (b) (c) 27 8 6 1 1 1 3 (a) (b) (i) (ii) 4 by 4 by 4 cube drawn 8 24 2 1 1 If 0 scored, B1 for one correct face C opportunity 4 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 4 B1 for 0 in row 1 column 5 B1 for 8 in row 1 column 6 B1 for 125 in row 4 column 2 B1 for 36 in row 4 column 5

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2014 0607 53 © Cambridge International Examinations 2014 5 (a) (b) (c) 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 2 1 1FT 1 B1 for either 24 FT their 9 × 6 C opportunity 6 (a) (b) (c) (n – 2)3 oe isw 6(n – 1)2 oe isw 12(n – 1) oe isw 2 1 1 B1 for [kn] – 2 Or B1 for n3 soi C opportunity Accept 6(n – 2)2 from cubes C opportunity 12(n – 2) from cubes C opportunity Communication in two of 3(a), 5(c), 6(a), 6(b) or 6(c) 1