Cambridge IGCSE Mathematics - International 0607 — 2015 May/June Paper 6 · Variant 3

This document consists of 12 printed pages. DC (LEG/SG) 100442/4 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 8 6 2 2 6 0 3 6 9 3 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2015 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 the 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 0607/63/M/J/15 © UCLES 2015 Answer both parts A and B. A INVESTIGATION T-VALUES (20 marks) You are advised to spend no more than 45 minutes on this part. A grid of any length and width 10 is numbered 1, 2, 3, ... . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 The grid has a letter T placed on it, as shown. The T has a horizontal bar of length 3 and a vertical bar of length 2. The T shown is shape 1 because the number in the top left square of the T is 1. T-values are found using this method. Method Calculation of T-value for shape 1 Square the number at the bottom of the T. 222 = 484 Multiply together the numbers at each end of the horizontal bar. 1 × 3 = 3 Take the second answer from the first answer to find the T-value. 484 – 3 = 481 The T-value for shape 1 is T1= 481. This investigation is about finding T-values. 1 (a) Complete this table. Shape number n Working T-value Tn 1 222 – 1 × 3 = 484 – 3 T1 = 481 2 232 – 2 × 4 = 529 – 8 T2 = 521 3 T3 = 4 T4 = 5 T5 =

3 0607/63/M/J/15 © UCLES 2015 [Turn over (b) When a T is placed at the end of a line, it still has a T-value. The T “wraps round” like this. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Work out T9. T9 = … (c) T1, T2, T3, T4, T5, ... form a sequence. Find a formula, in terms of n, for Tn. Tn = … (d) When Tn = 2641, find the value of n. n = … (e) Explain why 843 cannot be a T-value. … …

4 0607/63/M/J/15 © UCLES 2015 2 The T is now placed on a new grid that is 11 squares wide. 1 2 3 4 5 6 7 8 9 10 11 22 44 33 55 12 13 14 15 16 17 18 19 20 21 23 24 25 26 27 28 29 30 31 32 34 35 36 37 38 39 40 41 42 43 45 46 47 48 49 50 51 52 53 54 (a) Complete this statement for the numbers in the grid. In each row the numbers increase by 1 and in each column the numbers increase by … . (b) Complete the squares in this T using expressions in terms of n. n n + 1 (c) Complete this working to show that Tn = 44n + 529. The first line of working is started for you. Tn = (n + …) 2 – n(n + …)

5 0607/63/M/J/15 © UCLES 2015 [Turn over 3 The T is now placed on a grid that is 12 squares wide. Find a formula, in its simplest form, for Tn. Tn = … 4 The T is now placed on a grid that is w squares wide. (a) (i) Show that Tn = 4 w 2 + 4(n + 1)w + 1. (ii) Find the width of the grid when T9 = 1501. … (b) Use part (a)(i) to explain why Tn must always be odd. … …

6 0607/63/M/J/15 © UCLES 2015 B MOdELLING FILLING A BATH (20 marks) You are advised to spend no more than 45 minutes on this part. Karen wants to work out how much it would cost to fill a bath every day for 1 year. 1 She assumes that a bath is a cuboid that is 150 cm long, 60 cm wide and 30 cm deep. 150 cm 20 cm 60 cm NOT TO SCALE (a) Find the number of litres of water needed to fill this bath to a depth of 20 cm. …litres (b) Water costs 20 cents per 100 litres. (i) Find the cost, in dollars, of filling a bath with water to a depth of 20 cm each day for 1 year (365 days). $ …

7 0607/63/M/J/15 © UCLES 2015 [Turn over (ii) Show that a model for the cost, $C, of filling a bath to a depth, d cm, every day for 1 year is C = 6.57d. (iii) In 1 year all baths are filled to the same depth, d cm. The total cost of the water is $157.68 . Use the model in part (ii) to find the value of d. …

8 0607/63/M/J/15 © UCLES 2015 You may use this information in the remainder of the task. The area, A, of a trapezium is h a b 2 + ^ h. a h b 2 Karen notices that one end of her bath slopes. From the side, her bath looks like this. NOT TO SCALE She estimates that the angle of slope is 60°. The depth of water in the bath is 20 cm. 10 cm 20 cm 60° 150 cm A B E D C NOT TO SCALE

9 0607/63/M/J/15 © UCLES 2015 [Turn over (a) Show that AB = 5.77 cm, correct to 3 significant figures. (b) BC = 144 cm, correct to 3 significant figures. Find the volume of water in the bath, giving your answer in litres. … litres

10 0607/63/M/J/15 © UCLES 2015 (c) Show that a model for the volume of water in the bath, V litres, with angle of slope 60° and depth d cm is . . tan tan V d d 0 03 300 60 30 60 30 = - - - c c ^ c h m (d) On these axes, sketch the graph of V for d 0 30 G G . Volume (litres) Depth (cm) 30 V 0 d (e) Find the depth of water in the bath when the volume is 150 litres. …cm

11 0607/63/M/J/15 © UCLES 2015 [Turn over 3 The depth of water in her bath is d cm. Water costs w cents per 100 litres. (a) Change the model in question 2(c) to find the cost, $C, of filling her bath every day for 1 year. C = … (b) (i) Change your model in part (a) to find C when the angle of slope is θ °. θ° d cm C = … Parts (b)(ii), (b)(iii) and (c) are printed on the next page.

12 0607/63/M/J/15 © UCLES 2015 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. (ii) Describe the effect on the volume of water in her bath when θ decreases. … … (iii) Why is the design of this bath not suitable when θ is small? … … (c) Use your model from part (b) to find the cost of filling her bath every day for 1 year when • θ = 50° • d = 25 cm • water costs 21 cents per 100 litres. $ …

® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the May/June 2015 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 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 May/June 2015 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 – May/June 2015 0607 63 © Cambridge International Examinations 2015 Abbreviations 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 1 (a) 561 601 641 2 B1 for one from 561, 601 and 641 If 0 scored SC1 for 242 – 3×5, 252 – 4×6, 262 – 5×7 (b) [T9=] 801 1 C opportunity (c) 40n + 441 oe 2 B1 for 40n + k or jn + 441 (j > 0) or B1 for (n + 21)2 and B1 for –n(n + 2) or better (d) 55 1FT FT their (c) if answer is linear C opportunity (e) All T-results end in 1 oe [and this ends in 3 oe] or [n = ] 10.05 or 843 – 441 in not divisible by 40 oe 1 2 (a) 11 or eleven 1 (b) (top right) n + 2 oe (bottom) n + 23 oe 1 1 (c) [(n + 23)(n + 23) – n(n + 2) oe] n2 + 46n + 529 – n2 – 2n 2 B1 for n2 + 46n + 529 B1 for– n2 – 2n 3 48n + 625 2 M1 for (n + 25)2 – n(n + 2) 4 (a) (i) (n + 1 + 2w)2 – n(n + 2) n2 + n + 2w + n + 1 + 2w +2wn + 2w + 4w2 – n2 – 2n M1 A1 or better Methods based on extending sequences or justifying by substitution do not score (ii) 15 2 M1 for attempt at solving 4w2 + 40w + 1 = 1501 by factorising, formula, sketch, completing the square C opportunity (b) [even +] even + 1 = odd 1 No wrong statements Communication seen in one of 1(b), 1(d), 4(a)(ii) 1

P B 1 2 Pag B ge (a (b (a (b (c) (d (e) 3 a) b) ( (i a) b) ) d) ) (i) (ii) iii) M 18 13 1 24 ta A [D B 18 MOD 80 31. 50 4 an 6 Anyt DE BC 8[.1 DE 4[0 × 10 6 60 = thin ] E = 2 + D 1… ELL 0] × 000 60 = A ng 15 DE …] C LIN × × 0 36 AB 10 rou 50 − × E Cam NG × 100 65 B o und ta − 10 × d mb © G × 0 0 [ or ta ding 30 an 6 000 60 × brid Ca 2 ]. 0 an g to 0 60 0 0 o dge amb × 2 30 o 1 oe M e IG brid d o = 66 Mar GC dge oe 10 AB k S CSE e In 0 B o Sch E – nter oe hem – M rnat me May tion e y/J nal un Ex ne 2 xam 1 1F 1 1 1 4 1 1 1 1 20 mina 1 FT 1 1 1 4 1 1 1 1 15 ations C FT w C C 10 B B or us M [A C C s 20 op T t wron op op 0 s sin 1 f 1 f r on sing M1F Alm op op 015 ppo thei ng w ppo ppo sin n60 for for ne t g re FT mos ppo ppo 5 ortu ir ( wo ortu ortu 30 0 3 tan [Ar trap ecta for st] l ortu ortu unit (a) orki unit unit or 30 n 6 rea pez ang r th line unit unit ty × 0 ing ty ty 1 3 × 0 [ a =] zium gles heir ear ty ty S 0.2 ta × =1 ] (1 m ( s an r ar thr Syl 0 × 3 30 an6 7.3 44 (sid nd rea rou llab 060 365 0 60 3..]o + k de 1 tria × ugh bu 07 5 ÷ etc oe k)× 144 ang 100 60 h (0 s 10 c. × 2 2 4) c gles 00 0 0, 0) P 00 w 2 0 o calc s oe ) Pa 6 with oe cula pe 63 hou ated er ut d

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 63 © Cambridge International Examinations 2015 3 (a) (30 ) 30 0.001095 300 tan 60 tan 60 d dw −   − −     2 Accept 0.03 365 (30 ) 30 300 100 100 tan 60 tan 60 d w d × × − − − ×       or better M1 for 2 of the operations 100 365 w × × (b) (i) (30 ) 30 0.001095 300 tan tan d dw θ θ −   − −     1FT FT their 3(a) (ii) Decreases oe 1 (iii) No place to sit oe or Base of bath sloping oe 1 Not stable Not enough water (c) Anything truncating to 155 1FT FT their b(i) C opportunity Communication seen in two of 1(a), 1(b)(i), 1(b)(iii), 2(d), 2(e), 3(c) 1