Cambridge IGCSE Mathematics - International 0607 — 2018 Oct/Nov Paper 6 · Variant 1

This document consists of 15 printed pages and 1 blank page. DC (NH/CT) 153618/3 © UCLES 2018 [Turn over * 6 9 4 9 6 4 8 1 5 8 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) October/November 2018 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 (questions 1 to 4) and part B (questions 5 to 7). 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/18 © UCLES 2018 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 to 4) DOTS IN RECTANGLES (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at the number of dots inside rectangles drawn on square dotty paper. 1 Rectangles are drawn at an angle to the horizontal. They are called diagonal rectangles. The rectangles below are drawn at an angle of 45° to the horizontal. Two sides of each rectangle have a gradient of 1. These are diagonal rectangles with gradient 1. Length 3 Length 2 Length 1 Width 1 Width 1 Width 1 Width 2 Length 1

3 0607/61/O/N/18 © UCLES 2018 [Turn over (a) Complete the tables below. Width (W ) Length (L) Number of dots inside a diagonal rectangle with gradient 1 (d) 1 1 1 1 2 2 1 3 1 4 1 5 Width (W ) Length (L) Number of dots inside a diagonal rectangle with gradient 1 (d ) 2 1 2 2 2 2 3 2 4 2 5 Width (W ) Length (L) Number of dots inside a diagonal rectangle with gradient 1 (d ) 3 1 3 3 2 3 3 3 4 3 5

4 0607/61/O/N/18 © UCLES 2018 (b) Use your results from part (a) and any patterns you notice to complete the table. Width (W ) Number of dots inside a diagonal rectangle with gradient 1 (d ) 1 2 3L − 1 3 4 5 9L − 4 (c) A formula, in terms of W and L, for the number of dots, d, inside a diagonal rectangle with gradient 1, is d = (aW + b)L - (W + c) . Find the values of a, b and c. a = … b = … c = … 2 The diagram below shows three diagonal rectangles, each of width 1 and gradient 2 1 .

5 0607/61/O/N/18 © UCLES 2018 [Turn over (a) Complete the tables below. You may use the square dotty paper to help you. Width (W ) Length (L) Number of dots inside a diagonal rectangle with gradient 2 1 (d ) 1 1 4 1 2 8 1 3 12 1 4 1 5 Width (W ) Length (L) Number of dots inside a diagonal rectangle with gradient 2 1 (d ) 2 1 8 2 2 17 2 3 2 4 2 5

6 0607/61/O/N/18 © UCLES 2018 Width (W ) Length (L) Number of dots inside a diagonal rectangle with gradient 2 1 (d ) 3 1 12 3 2 3 3 40 3 4 3 5

7 0607/61/O/N/18 © UCLES 2018 [Turn over (b) Use your results from part (a) and any patterns you notice to complete the table. Width (W) Number of dots inside a diagonal rectangle with gradient 2 1 (d ) 1 2 3 4 19L − 3 5 24L − 4 (c) Find a formula, in terms of W and L, for the number of dots, d, inside a diagonal rectangle with gradient 2 1 . …

8 0607/61/O/N/18 © UCLES 2018 3 The diagram below shows three diagonal rectangles, each of width 1 and gradient 1 3 .

9 0607/61/O/N/18 © UCLES 2018 [Turn over (a) Complete the table. You may use the square dotty paper below to help you. Width (W ) Number of dots inside a diagonal rectangle with gradient 1 3 (d ) 1 2 19L − 1 3 4 5 49L − 4 (b) Find a formula, in terms of W and L, for the number of dots, d, inside a diagonal rectangle with gradient 1 3 . …

10 0607/61/O/N/18 © UCLES 2018 4 (a) Complete the following table, using your answers to question 1(c), question 2(c) and question 3(b) and any patterns you notice. Gradient Number of dots inside a diagonal rectangle (d ) 1 1 2 1 3 1 4 (17W – 1)L – (W – 1) 1 5 (b) Use your answers to part (a) to find a formula, in terms of W, L and n, for the number of dots, d, inside a diagonal rectangle with a gradient of n 1 . … (c) There are 4833 dots inside a 4 by 12 diagonal rectangle. Find the gradient of this rectangle. …

11 0607/61/O/N/18 © UCLES 2018 [Turn over B MODELLING (QUESTIONS 5 to 7) LADDERS (20 marks) You are advised to spend no more than 45 minutes on this part. This task looks at the safe positions for placing a ladder against a wall. A ladder is x metres long. It leans against a vertical wall. The bottom of the ladder is 1.5 m from the base of the wall. The ladder touches the wall y metres above the ground. NOT TO SCALE x y 1.5 5 (a) Show that . y x 2 25 2 = - . (b) (i) Sketch the graph of . y x 2 25 2 = - on the axes below. –6 0 6 6 y x (ii) Only one part of the graph fits this practical situation. Give a reason why the other part does not. … …

12 0607/61/O/N/18 © UCLES 2018 6 Safety rules for ladders say that the angle between the bottom of the ladder and the ground must be more than 76°. NOT TO SCALE x z (a) To use a ladder safely, show that a model for its position is . z x 0 242 1 . (b) To use a ladder safely the angle between the bottom of the ladder and the ground must be less than 82°. Find a second inequality connecting z and x. …

13 0607/61/O/N/18 © UCLES 2018 [Turn over (c) On the axes, shade the region defined by the inequalities in part (a) and part (b). z x 0 0 1 2 3 4 5 6 1 2 (d) A ladder is 3 m long. To use this ladder safely, a 1 z 1 b. Use your graph in part (c) to find the value of a and the value of b. … 1 z 1 …

14 0607/61/O/N/18 © UCLES 2018 7 Ladders can be extended to increase their original length. A ladder is extended by 0.9 times its original length. The bottom of this ladder is 1.5 m from the base of the wall. 1.5 y x + 0.9x (a) Find a formula for y in terms of x. … (b) (i) When the ladder is extended it reaches higher up the wall. This increase in height, y, is C metres. Using the model in question 5(a), show that a model for C is . . . C x x 3 61 2 25 2 25 2 2 = - - - .

15 0607/61/O/N/18 © UCLES 2018 (ii) On the axes below, sketch the graph of C for 0 1 x 1 6. C 6 x 0 (c) This part is about the smallest increase in height that the ladder reaches up the wall. (i) Find this smallest increase in height. … (ii) Write down the original length of the ladder. … (iii) Show how you can decide whether the extended ladder is safe.

16 0607/61/O/N/18 © UCLES 2018 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. BLANK PAGE

This document consists of 7 printed pages. © UCLES 2018 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) October/November 2018 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 International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the October/November 2018 series for most Cambridge IGCSE™, Cambridge International A and AS Level components and some Cambridge O Level components.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 2 of 7 +Generic Marking Principles These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE 1: Marks must be awarded in line with: • the specific content of the mark scheme or the generic level descriptors for the question • the specific skills defined in the mark scheme or in the generic level descriptors for the question • the standard of response required by a candidate as exemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE 2: Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positively: • marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate • marks are awarded when candidates clearly demonstrate what they know and can do • marks are not deducted for errors • marks are not deducted for omissions • answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE 4: Rules must be applied consistently e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 3 of 7 MARK SCHEME NOTES The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M Method marks, awarded for a valid method applied to the problem. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be earned or implied. B Mark for a correct result or statement independent of Method marks. When a part of a question has two or more ‘method’ steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. The notation ‘dep’ is used to indicate that a particular M or B mark is dependent on an earlier mark in the scheme. Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working nfww not from wrong working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 4 of 7 Question Answer Marks Partial Marks A INVESTIGATION DOTS IN RECTANGLES 1(a) 1 2 3 4 5 2 5 8 11 14 3 8 13 18 23 3 B1 for each correct table 1(b) L oe 3L − 1 5L – 2 oe 7L – 3 oe 9L − 4 2 B1 for two correct cells C opportunity 1(c) 2 −1 −1 2 B1 for two correct values C opportunity 2(a) 4 8 12 16 20 8 17 26 35 44 12 26 40 54 68 2 B1 for two correct tables 2(b) 4L 9L − 1 14L – 2 oe 19L − 3 24L – 4 2 B1 for two cells correct C opportunity 2(c) d = (5W– 1)L– (W– 1) oe 1 C opportunity

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 5 of 7 Question Answer Marks Partial Marks 3(a) 9L 19L − 1 29L − 2 39L – 3 oe 49L − 4 2 B1 for two correct cells 3(b) d = (10W – 1)L– (W – 1) oe 1 C opportunity 4(a) (2W– 1)L– (W– 1) (5W– 1)L– (W– 1) (10W – 1)L– (W – 1) (17W – 1)L– (W – 1) (26W – 1)L– (W – 1) oe 1 For last row 4(b) d = ((n2 + 1)W – 1)L – (W – 1) oe 1 C opportunity 4(c) 1 10 2 M1 for 4833, 4 and 12 substituted into their 4(b) C opportunity Communication: seen in three of the following questions with at least one not a differences 1 1(b) At least 3 differences of 5 seen 1(c) At least 3 differences of 2 seen or substitute and solve equations leading to at least one correct value 2(b) At least 3 differences of 9 or 14 seen 2(c) At least 3 differences of 5 seen 3(b) At least 3 differences of 10 seen 4(b) First differences of 3, 5, 7 and at least 2 differences of 2 seen or 1, 4, 9. 16 seen 4(c) further working after substitution to solve quadratic 2(c), 3(b), 4(b) d = written in all three answers

0 © Qu 5 5( 5(b 6 6 6 6 7 7( 7(b 0607/61 © UCLES 2018 estion B M 5(a) x2 (b)(i) Co b)(ii) Le 6(a) co 6(b) > z 6(c) Co 6(d) 0.3 7(a) y (b)(i) ( b)(ii) Co ODELLING = y2 + 1.52 o orrect sketch ength [of ladd os76 > z x oe 0.139 > x oe orrect region 38… to 0.44 ( ) 2 1.9 = − x ( ) 2 1.9 1.5 − x orrect sketch C Answer G LADDERS oe h der] cannot b n between lin … < z < 0.70 2 1.5 − oe 2 2 1.5 − − x h Cambridge P r S be negative es 0… to 0.75… 2 5 oe IGCSE – M PUBLISHED Page 6 of 7 M oe … Mark Schem D Marks 1 C op 2 B1 f B1 f C op 1 1 C op 1 C op 3 B1 f B1 d depe orig 2 FT orig B1 f If 0 SC1 C op 1 C op 1 1 C op e P pportunity for correct sh for two branc pportunity pportunity pportunity for each corr dep for area endent on tw gin their (c) if bo gin for each in co scored, 1 if correct bu pportunity pportunity pportunity Octobe Partial Mar hape ches within t rect ruled bro between line wo straight lin oth lines go orrect order but positions er/Novembe 201 ks tolerance oken line es shaded, nes from the through the reversed er 18

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 7 of 7 Question Answer Marks Partial Marks 7(c)(i) 2.06… 1 7(c)(ii) 1.695 to 1.70 1 7(c)(iii) 3.22 to 3.23 1 Values to show that ladder is not safe 1 C opportunity Communication: seen in five of the following questions 2 1 mark for three opportunities seen 5(a) writing the word Pythagoras 5(b)(i) Intercept at 1.5 6(a) [cos76 =] 0.2419… 6(b) [cos82 =] 0.1391… 6(d) vertical line drawn at x = 3 cutting both lines 7(a) y2 + 1.52 = (x + 0.9x)2 or similar 7(b)(ii) for appropriate scale on C axis 7(c)(iii) working to find values for ladder not being safe