Cambridge IGCSE Mathematics - International 0607 — 2016 Oct/Nov Paper 6 · Variant 1

This document consists of 12 printed pages. DC (NH/SG) 117169/4 © UCLES 2016 [Turn over * 9 4 0 1 4 6 2 2 5 8 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) October/November 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 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. Cambridge International Examinations Cambridge International General Certificate of Secondary Education

2 0607/61/O/N/16 © UCLES 2016 Answer both parts A and B. A INVESTIGATION SQUARES ON GRIDS (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at the number of squares drawn on a grid. All squares are drawn using gridlines. 1 (a) Here is a 2 by 2 grid. Explain why there are 5 squares on a 2 by 2 grid. … … (b) Here is a 3 by 3 grid. Complete these statements about the numbers of different sized squares on a 3 by 3 grid. The number of 1 by 1 squares on a 3 by 3 grid is … The number of 2 by 2 squares on a 3 by 3 grid is 4 The number of 3 by 3 squares on a 3 by 3 grid is … So the total number of squares on a 3 by 3 grid is …

3 0607/61/O/N/16 © UCLES 2016 [Turn over (c) Complete these statements about the numbers of different sized squares on a 4 by 4 grid. You may use the grids below to help you. The number of 1 by 1 squares on a 4 by 4 grid is … The number of 2 by 2 squares on a 4 by 4 grid is … The number of 3 by 3 squares on a 4 by 4 grid is … The number of 4 by 4 squares on a 4 by 4 grid is … So the total number of squares on a 4 by 4 grid is 30

4 0607/61/O/N/16 © UCLES 2016 2 (a) Use your results from question 1 to help you complete this table. Size of grid Number of … Total number of squares 1 by 1 squares 2 by 2 squares 3 by 3 squares 4 by 4 squares 5 by 5 squares 6 by 6 squares 1 by 1 1 2 by 2 5 3 by 3 4 4 by 4 30 5 by 5 6 by 6 (b) What is the mathematical name for the numbers in the 1 by 1 squares column? … (c) Work out the total number of squares on an 8 by 8 grid. … (d) Write down an expression, in terms of n, for the number of 2 by 2 squares on an n by n grid. …

5 0607/61/O/N/16 © UCLES 2016 [Turn over 3 Here is a formula for the total number of squares, T, on an n by n grid. 3 2 T n n cn d 3 2 = + + + (a) Find the values of c and d. (b) Show that your formula gives a total of 385 squares on a 10 by 10 grid. (c) The total number of squares on an n by n grid is 1240. Find the value of n. …

6 0607/61/O/N/16 © UCLES 2016 4 Here is a 1 by 2 grid. There is a total of 2 squares on a 1 by 2 grid. Write an expression, in terms of n, for the total number of squares on a 1 by n grid. … 5 Here is a 2 by 3 grid. There is a total of 8 squares on a 2 by 3 grid. (a) Find the total number of squares on a 2 by 4 grid. … (b) Complete this table. Size of grid Number of … Total number of squares (T ) 1 by 1 squares 2 by 2 squares 2 by 1 2 0 2 2 by 2 2 by 3 8 2 by 4 2 by 5 2 by n

7 0607/61/O/N/16 © UCLES 2016 [Turn over 6 Complete this table for 3 by n grids. You may use the grid below to help you. Size of grid Number of … Total number of squares (T ) 1 by 1 squares 2 by 2 squares 3 by 3 squares 3 by 1 3 0 0 3 3 by 2 6 2 0 8 3 by 3 9 1 3 by 4 12 3 by 5 15 3 by n 3n 7 The expression for T in question 6 does not work when n = 1. The expression for T for a 4 by n grid is 10n – 10. For what values of n will the expression for T for a 4 by n grid not give the correct total? …

8 0607/61/O/N/16 © UCLES 2016 B MODELLING MEASURING ROD (20 marks) You are advised to spend no more than 45 minutes on this part. You may find some of these formulas useful. Volume, V, of pyramid, base area A, height h. 3 1 V Ah = Volume, V, of cylinder of radius r, height h. V r h 2 r = Volume, V, of cone of radius r, height h. 3 1 V r h 2 r = Volume, V, of sphere of radius r. 3 4 V r3 r = A a C b B c Area = 2 1 sin ab C Jim has a tank for oil, shown in the picture. He uses a measuring rod to put in the top of the tank to find the amount of oil. This investigation is about how Jim marks the measuring rod. Here is some information about the tank. Measuring rod 1770 mm 1030 mm Length 1770 mm Width 1030 mm Capacity 1235 litres

9 0607/61/O/N/16 © UCLES 2016 [Turn over Jim decides to model the volume of oil in his tank. 1 (a) Select the shape that Jim should use for the model. cuboid cylinder sphere cone pyramid circle … (b) Jim uses a capacity of 1200 litres and a width of 100 cm. For his model, show that the length of the tank is 153 cm, correct to the nearest centimetre. 2 (a) Give a practical reason why the length of his measuring rod should be more than 100 cm. … … (b) This is a cross-section of the tank showing the measuring rod and some oil. NOT TO SCALE Oil (i) The tank contains 600 litres of oil. Find how many centimetres from the bottom of the measuring rod Jim should mark “600 litres”. … (ii) Jim now wants to mark “300 litres” on the measuring rod. Explain why he should not mark this point halfway between the bottom of the measuring rod and the “600 litres” mark. … …

10 0607/61/O/N/16 © UCLES 2016 3 Jim works out the distance, d cm, where “300 litres” should be marked on the measuring rod. O x° B A NOT TO SCALE d cm O is the centre of the cross section and OA and OB are radii. x° is the angle between OA and OB. He uses this method. Shaded area = Area of sector OAB – area of triangle OAB (a) Show that an expression for the area of triangle OAB is 1250 sinx° cm2. (b) Show that an expression for the area of sector OAB is approximately 21.8x cm2. (c) Write down an expression, in terms of x, for the area of the shaded segment. …

11 0607/61/O/N/16 © UCLES 2016 [Turn over (d) Using your result from question 1(b) rounded to the nearest centimetre and your result from question 3(c), show that a model for the volume of oil, V cm3, is approximately V = 3340x – 191 000 sinx°. (e) On the axes, sketch the graph of this model. 420 000 80 000 Volume (cm3) V 80 150 x (f) When the tank contains exactly 300 litres of oil, use the model to find (i) the value of x, … (ii) the value of d, the distance from the bottom of the measuring rod. … (g) Write down the distance from the bottom of the measuring rod to the “900 litres” mark. … Question 4 is printed on the next page.

12 0607/61/O/N/16 © UCLES 2016 4 Jim buys oil when the tank contains 100 litres of oil. Work out the distance from the bottom of the measuring rod to where “100 litres” should be marked. … 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.

® 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 0607/61 Paper 6 (Extended) October/November 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 October/November 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 – October/November 2016 0607 61 © UCLES 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 SQUARES ON GRIDS Question Answer Mark Part Marks 1 (a) 4 small and 1 large oe 1 (b) 9 4 1 14 1 (c) 16 9 4 1 30 1 If 0 scored in parts (b) and (c), SC1 for 1, 4, 9, 16 (i.e. reverse order) 2 (a) Size Total 1 by 1 1 1 2 by 2 4 1 5 3 by 3 9 4 1 14 4 by 4 16 9 4 1 30 5 by 5 25 16 9 4 1 55 6 by 6 36 25 16 9 4 1 91 2 B1 for first 4 rows correct B1 for rows 5 and 6 correct If 0 scored in parts 1(b) and 1(c) or SC in 1(c), SC1 for first 4 rows correct, in reverse order AND SC1 for rows 5 and 6 correct, in reverse order (b) Square [numbers] 1 (c) 204 1 C opportunity (d) (n – 1) 2 oe 1 3 (a) d = 0 c = 1 6 1 1 C opportunity

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 61 © UCLES 2016 Question Answer Mark Part Marks (b) T = 1 3 103 + 1 2 102 + 1 6 10 leading to 385 1 (c) 15 1 C opportunity 4 n 1 5 (a) 11 1 (b) 2 by 1 2 0 2 2 by 2 4 1 5 2 by 3 6 2 8 2 by 4 8 3 11 2 by 5 10 4 14 2 by n 2n n – 1 3n – 1 oe 1 1 6 3 by 1 3 0 0 3 3 by 2 6 2 0 8 3 by 3 9 4 1 14 3 by 4 12 6 2 20 3 by 5 15 8 3 26 3 by n 3n 2n – 2 n – 2 6n – 4 oe 2 B1 for rows 4 or 5 correct B1 FT for their linear expressions in columns 3, 4 and 5 7 [n] < 3 oe 1 C opportunity Communication: Seen in two of the following questions 1 2 (c) For showing 91 + 49 + 64 or 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 or in tabular form 3 (a) For showing working of a correct method 3 (c) For showing working or sketch 7 For < 2 in 2 by something and < 3 in 3 by something oe

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 61 © UCLES 2016 B MODELLING MEASURING ROD Question Answer Mark Part Marks 1 (a) Cylinder 1 (b) 152.7…cm oe 2 M1 for 2 1200 π 0.5 × oe 2 (a) Must be able to hold it oe 1 (b) (i) 50 1 (ii) Cross-section narrows oe 1 3 (a) 1 2 × 50 × 50 × sinx 1 (b) 2 π 50 360 x × × 21.81x to 21.82x 1 1 (c) 21.8x – 1250sinx isw 1 (d) their 3(c) × 153 1 FT their 3(c) (e) Correct curve 2 B1 for correct shape B1 for passing through approximately (80, 79 000) and approximately (150, 406 000) (f) (i) 132 to 132.2 1 C opportunity (ii) 29.6 to 29.75 2 FT their f(i) in f(i) cos 2       FT M1 for 132 50 cos 2 their   ×     oe C opportunity (g) 70.2 to 70.3 1 FT 100 – their (f(ii)) 4 13.7 or 13.74 to 13.75 2 M1 for 87.05 cos 50 2 their  ×     implied by 36.2 to 36.3 C opportunity

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 61 © UCLES 2016 Question Answer Mark Part Marks Communication: Seen in one of the following questions 1 3 (f) (i) seen in 3(e) For line on graph (sketch) at V = 300000 3 (f) (ii) For working shown i.e. extra stage like division by 2 or cos their angle 4 seen in 3(e) For line on graph (sketch) at V = 100000 or x = 87.0[5]