Cambridge IGCSE Mathematics - International 0607 — 2016 Oct/Nov Paper 6 · Variant 2

This document consists of 11 printed pages and 1 blank page. DC (NH/SG) 117170/2 © UCLES 2016 [Turn over * 5 8 2 8 2 6 2 9 9 0 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 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 0607/62/O/N/16 © UCLES 2016 Answer both parts A and B. A INVESTIGATION RECTANGLES WITHIN RECTANGLES (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks for a method to find the number of rectangles when you draw horizontal and vertical lines inside a rectangle. One horizontal line, AB, is drawn inside a rectangle PQRS. The total number of rectangles is 3. They are PQBA, PQRS and ABRS. P A S Q B R 1 (a) Another line CD is drawn inside the rectangle PQRS. The total number of rectangles is now 6. P A C S Q B D R Three of the 6 rectangles are PQBA, PQRS and ABRS. Write down the other three rectangles. …

3 0607/62/O/N/16 © UCLES 2016 [Turn over (b) Three horizontal lines, AB, CD and EF are drawn inside the rectangle PQRS. Find the total number of rectangles. P A C E S Q B D F R … (c) Four horizontal lines are drawn inside the rectangle. Find the total number of rectangles. …

4 0607/62/O/N/16 © UCLES 2016 (d) Complete the table. Number of horizontal lines inside the rectangle 0 1 2 3 4 5 6 7 Total number of rectangles 3 6 36 (e) The numbers in the bottom row of the table in part (d) form a sequence. Write down the mathematical name of these numbers. … (f) Ten horizontal lines are drawn inside the rectangle. Find the total number of rectangles. …

5 0607/62/O/N/16 © UCLES 2016 [Turn over 2 One vertical line, AB, is drawn inside rectangle PQRS. The total number of rectangles is 3. They are PABS, PQRS and AQRB. P A Q S B R Complete the table. Number of vertical lines inside a rectangle 0 1 2 3 4 5 6 7 Total number of rectangles 3 3 (a) n vertical lines are drawn inside a rectangle. The total number of rectangles is an2 + bn + c. Find the values of a, b and c. a = … b = … c = …

6 0607/62/O/N/16 © UCLES 2016 (b) Using your values for a, b and c found in part (a), factorise an2 + bn + c. Write it in the form a(n + p)(n + q). … 4 Two horizontal lines are drawn inside rectangle PQRS to form a total of 6 rectangles. For each of these 6 rectangles, 3 rectangles are formed when one vertical line is also drawn inside rectangle PQRS. P Q S R P Q S R So the total number of rectangles is 18. (a) One horizontal line and one vertical line are drawn inside rectangle PQRS. Find the total number of rectangles. …

7 0607/62/O/N/16 © UCLES 2016 [Turn over (b) Three horizontal lines and two vertical lines are drawn inside a rectangle. Find the total number of rectangles. … (c) n horizontal lines and m vertical lines are drawn inside a rectangle. Find an expression, in terms of n and m, for the total number of rectangles. Do not simplify this expression. … 5 ABCD is a square. An equal number of horizontal lines and vertical lines are drawn inside the square. Show that the number of rectangles inside ABCD could not be 762 but it could be 782.

8 0607/62/O/N/16 © UCLES 2016 B MODELLING BIRTHDAY MONEY (20 marks) You are advised to spend no more than 45 minutes on this part. 1 Rosie’s grandmother gave her $10 on her 1st birthday. On each birthday after that her grandmother gave her $5 more than she gave Rosie on her last birthday. (a) How much did Rosie receive from her grandmother on her 5th birthday? … (b) Find a model, in terms of n, for the amount in dollars, A, that Rosie received from her grandmother on her nth birthday. … (c) Use your model to show that Rosie received $105 from her grandmother on her 20th birthday. 2 (a) Find the total amount that Rosie had received from her grandmother, up to and including her 5th birthday. …

9 0607/62/O/N/16 © UCLES 2016 [Turn over (b) A model for the total amount of money, $T, that Rosie received up to and including her nth birthday, is T = kn(n + 3). (i) Use your answer to part (a) to find the value of k. … (ii) Show that Rosie received a total of $1150 from her grandmother, up to and including her 20th birthday. (c) Work out Rosie’s age when she first received a total of more than $4 000. … 3 Zahari’s grandfather gave him $10 on his 1st birthday. On each birthday after that his grandfather gave him 10% more than he gave him on his last birthday. How much did Zahari receive from his grandfather on his 5th birthday? …

10 0607/62/O/N/16 © UCLES 2016 4 The model A = 10 × 1.1n–1 can be used to calculate the amount in dollars, A, that Zahari received from his grandfather on his nth birthday. (a) Explain why the numbers 10 and 1.1 are used in the model. … … … (b) Use the model to calculate how much Zahari received from his grandfather on his 20th birthday. Give your answer in dollars correct to the nearest cent. … 5 (a) 0 1 40 n A (i) On the axes, sketch the graph of the model for A that you found in question 1(b). (ii) On the same axes, sketch the model A = 10 × 1.1n–1.

11 0607/62/O/N/16 © UCLES 2016 (b) After their 1st birthdays, on which birthday did Zahari first get more money than Rosie? … (c) Apart from their 1st birthdays, which of these two situations, A or B, gives Zahari more money than Rosie on an earlier birthday? His grandfather gave him A $5 on his 1st birthday, with an increase of 15% on each birthday B $7.50 on his 1st birthday, with an increase of 12% on each birthday. Give reasons for your choice. … … 6 Zahari’s grandmother gave him $d on his 1st birthday. On each birthday after that his grandmother gave him 10% more than she gave him on his last birthday. (a) Change the model A = 10 × 1.1n–1 to show this situation. … (b) Find the value of d that will give Zahari $148 on his 22nd birthday. …

12 0607/62/O/N/16 © UCLES 2016 BLANK PAGE 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 0607/62 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 62 © 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 RECTANGLES WITHIN RECTANGLES Question Answer Mark Part Marks 1 (a) PQDC ABDC CDRS 3 B1 for each (b) 10 1 C opportunity (c) 15 1 C opportunity (d) Number of lines 0 1 2 3 4 5 6 7 Number of rectangles 1 3 6 10 15 21 28 36 2 B1 for any two of 1, 21 and 28 C opportunity (e) Triangle [numbers] 1 (f) 66 1 C opportunity 2 Number of lines 0 1 2 3 4 5 6 7 Number of rectangles 1 3 6 10 15 21 28 36 1 FT their 1(d) 3 (a) [a =] 1 2 , [b =] 3 2 , [c =] 1 oe 3 B1 each value C opportunity (b) 1 2 (n + 2)(n + 1) Final answer 1 FT their a, b, c C opportunity 4 (a) 9 1 C opportunity (b) 60 1 C opportunity (c) 1 2 (n + 2)(n + 1) × 1 2 (m + 2)(m + 1) oe isw 1 FT (their 3(b) in terms of n) × (their 3(b) in terms of m)

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 62 © UCLES 2016 Question Answer Mark Part Marks 5 ( )( ) 2 1 2 1 2 n n   + +     oe seen Valid working to point where 76 gives non- integer solution and 78 gives integer solution 1 1 Formula and sketch give 11 and 10.8… Trial and Improvement gives 11 and (11 and 12) Communication: seen in three of the following questions 1 1 (b) Method of counting, 4 + 3 + 2 + 1 (implied addition), or list or drawing of 9 or 10 rectangles 1 (c) Method of counting, 5 + 4 + 3 + 2 + 1 (implied addition), or list or drawing of 14 or 15 rectangles 1 (d) Differences of 4, 5, 6, 7 shown correctly 1 (f) Working shown, e.g. sequence continued …, 45, 55, 66 or adding 11 + 10 + 9 + 8 + ... or substitution into formula 3 (a) Use of correct method (formula, difference, simultaneous equations) to find one coefficient and attempt to find another 3 (b) 1 2 (…………) or 1 2 (……)(……) 4 (a) 3 × 3 = 9 or 9 distinct rectangles drawn 4 (b) 6 × 10 = 60

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 62 © UCLES 2016 B MODELLING BIRTHDAY MONEY Question Answer Mark Part Marks 1 (a) 30 1 (b) [A =] 5n + 5 or [A=] 10 + 5(n – 1) isw 1 (c) 5 × 20 + 5 or 10 + 5(20 – 1) or 100 + 5 or 10 + 5 × 19 1 Accept 105 = 5n + 5 leading to 20 2 (a) 100 1 C opportunity (b) (i) 2.5 oe 1 FT their 2(a) C opportunity (ii) 2.5 × 20(20 + 3) or 2.5 × 20 × 23 oe leading to 1150 or 10 + 15 + … + 105 = 1150 1 (c) 39 2 M1 FT for substitution of their k and one correct step or sketch 3 14.64 1 C opportunity 4 (a) 10 is the first amount oe 1.1 is 110% or 1 + 10% or 1 + 100 10 or 100 110 1 1 (b) 61.16 1 C opportunity 5 (a) (i) Correct line 1 straight line with positive gradient, starting from above 0 (ii) Correct sketch 1 Correct shaped curve starting from above 0 If 0 scored in (i) and (ii) SC1 for both correct but starting at 0 (b) 30 1

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 62 © UCLES 2016 Question Answer Mark Part Marks (c) A, with 25 (or 24.0[4]) and 27 (or 26.7[4]) or A, with sketch showing two curves (labelled) and straight line and A crossing straight line before B 2 A may be implied by $5 or first option etc. B1 for A, and incomplete evidence e.g. A with 25 (or 24.0[4]) or 27 (or 26.7[4]) or A, with valid calculations for age above 25 for A and B or A and unclear diagram or missing line 6 (a) A = d × 1.1n − 1 1 (b) 20 1 C opportunity Communication: seen in two of the following questions 1 2 (a) 10 + 15 + 20 + 25 + 30 (implied addition) or use of appropriate formula 2 (b) (i) Substitution for T FT and one further correct step 3 Some working and 11, 12.1[0], 13.31, 14.64[1] or 10 × 1.14 oe 4 (b) Substitution e.g.10 × 1.119 6 (b) Substitution e.g. 148 = d × 1.121