Cambridge IGCSE Mathematics - International 0607 — 2016 Oct/Nov Paper 4 · Variant 1

This document consists of 19 printed pages and 1 blank page. DC (NF/SG) 117023/2 © UCLES 2016 [Turn over * 6 6 1 6 4 7 4 6 3 4 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/41 Paper 4 (Extended) October/November 2016 2 hours 15 minutes Candidates answer on the Question Paper. Additional Materials: Geometrical Instruments 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. Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate. Answers in degrees should be given to one decimal place. For r, use your calculator value. You must show all the relevant working to gain full marks and you will be given marks for correct methods, including sketches, even if your answer is incorrect. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 120. Cambridge International Examinations Cambridge International General Certificate of Secondary Education

2 0607/41/O/N/16 © UCLES 2016 Formula List For the equation ax bx c 0 2 + + = x a b b ac 2 4 2 ! = - - Curved surface area, A, of cylinder of radius r, height h. r A rh 2 = Curved surface area, A, of cone of radius r, sloping edge l. r A rl = Curved surface area, A, of sphere of radius r. r A r 4 2 = Volume, V, of pyramid, base area A, height h. V Ah 3 1 = Volume, V, of cylinder of radius r, height h. r V r h 2 = Volume, V, of cone of radius r, height h. r V r h 3 1 2 = Volume, V, of sphere of radius r. r V r 3 4 3 = sin sin sin A a B b C c = = cos a b c bc A 2 2 2 2 = + - sin bc A 2 1 Area = A C B c b a

3 0607/41/O/N/16 © UCLES 2016 [Turn over Answer all the questions. 1 Toby takes a journey from Johannesburg to Zurich. (a) He changes 2500 rand into Swiss francs (CHF). 1 Swiss franc = 12.43 rand. Calculate the amount Toby receives in Swiss francs. Give your answer correct to the nearest Swiss franc. … CHF [2] (b) Toby leaves Johannesburg at 19 30 and arrives in Zurich at 06 10 the next morning. Local time in Zurich is the same as local time in Johannesburg. The distance from Johannesburg to Zurich is 8350 km. (i) Calculate the average speed of the journey. … km/h [3] (ii) After arriving at 06 10, Toby takes a further 1 hour 55 minutes to reach his office. Work out the time he arrives at his office. … [1] (iii) Later, Toby takes a taxi from his office to a hotel. The taxi fare is made up of a fixed charge of 20 CHF plus 2.40 CHF per kilometre. Toby paid 36.80 CHF altogether. Work out the distance of Toby’s taxi journey. … km [3]

4 0607/41/O/N/16 © UCLES 2016 2 P T R Q 8 7 9 10 y x 6 5 4 3 2 1 –3 –4 –2 –1 –5 –6 –7 –8 –9 –10 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 1 0 2 3 4 5 6 7 8 9 10 (a) u 3 2 = - - J L KK N P OO and v 5 3 = - - J L KK N P OO (i) Find u + v. J L K KK N P O OO [1] (ii) Draw the image of triangle T under the translation by the vector u + v. [2] (iii) Calculate |u + v|. … [2]

5 0607/41/O/N/16 © UCLES 2016 [Turn over (b) Describe fully the single transformation that maps (i) triangle T onto triangle P, … … [2] (ii) triangle T onto triangle Q, … … [3] (iii) triangle T onto triangle R. … … [3]

6 0607/41/O/N/16 © UCLES 2016 3 x 2 f sinx = ^ h (a) On the diagram, sketch the graph of y = f(x) for ° ° x 360 360 G G - . 0 3 –0.5 –360° 360° y x [3] (b) Find the range of f(x). … [2] (c) Find the value of f(x) when (i) x = 3780°, … [1] (ii) x = 4050°. … [1]

7 0607/41/O/N/16 © UCLES 2016 [Turn over (d) (i) Find the four values of x from −360° to 1080° for which f(x) = 0.5 . … , … , … , … [2] (ii) The values in the answer to part (d)(i) form the first four terms of a sequence. Find the nth term of this sequence. … [2] (e) x x x 16200 360 g = - ^ ^ h h (i) On the diagram, sketch the graph of y = g(x) for ° ° x 0 360 G G . [2] (ii) Solve the equation f(x) = g(x). x = … or x = … [2]

8 0607/41/O/N/16 © UCLES 2016 4 9 cm 9 cm NOT TO SCALE The diagrams show a solid hemisphere and a solid cone. Both the hemisphere and the base of the cone have radius 9 cm. The volumes of the two shapes are equal. (a) Show that the perpendicular height of the cone is 18 cm. [2] (b) (i) Calculate the total surface area of the hemisphere. … cm2 [2]

9 0607/41/O/N/16 © UCLES 2016 [Turn over (ii) Calculate the curved surface area of the cone. … cm2 [3] (c) The hemisphere is made from metal. The metal is melted down and made into spheres of radius 2 cm. Calculate the number of spheres that are made. … [3]

10 0607/41/O/N/16 © UCLES 2016 5 U P Q n(U) = 25 n(P) = 18 n(Q) = 12 P Q 3 n , = l ^ h . (a) Show that P Q 8 n + = ^ h . [2] (b) An element is chosen at random from U. Find the probability that the element is a member of (i) P Q , , … [1] (ii) P Q , l. … [1] (c) An element is chosen at random from P. Find the probability that this element is also a member of Q. … [1] (d) The probability of a single event is 3 2 . Describe this event in terms of P and Q. … [1]

11 0607/41/O/N/16 © UCLES 2016 [Turn over 6 x y A B 0 NOT TO SCALE A is the point (0, 6) and B is the point (4, 0). (a) Find the equation of the perpendicular bisector of AB. … [5] (b) x y A C B 0 NOT TO SCALE The line y = 2x + 3 cuts the y-axis at C. The perpendicular bisector of AB cuts the y-axis at D. Find the length CD. CD = … [2]

12 0607/41/O/N/16 © UCLES 2016 7 P Q C B X D A 20 cm 10 cm 9 cm NOT TO SCALE The diagram shows a triangular prism with a horizontal base ABCD. X is a point on the line AQ. AB = 20 cm, BC = 10 cm, CQ = 9 cm and angle BCQ = 90°. (a) Calculate angle QBC. Angle QBC = … [2] (b) Calculate angle BAQ and show that it rounds to 33.9°, correct to 1 decimal place. [3]

13 0607/41/O/N/16 © UCLES 2016 [Turn over (c) AX = 22 cm. Calculate the length of BX. BX = … cm [3]

14 0607/41/O/N/16 © UCLES 2016 8 y x 0 –2 –4 4 8 x x x 1 3 f = + + ^ h (a) On the diagram, sketch the graph of y = f(x) for values of x between −4 and 4. [2] (b) Find the zeros of f(x). … [2] (c) Solve the inequality f(x) < 0. … [3] (d) The asymptotes of the graph are x = a and y = x + b, where a and b are integers. Find the value of a and the value of b. a = … b = … [2] (e) x x x 1 g = + ^ h Describe fully the single transformation that maps the graph of y = f(x) onto the graph of y = g(x). … … [2]

15 0607/41/O/N/16 © UCLES 2016 [Turn over 9 In one day a delivery company delivers 93 parcels. The histogram shows information about the masses, m kg, of these parcels. 20 10 0 0 1 2 3 5 Mass (kilograms) Frequency density 7 9 4 6 8 10 m (a) Complete the frequency table. Mass (m kg) m 0 1 1 G m 1 2 1 G m 2 3 1 G m 3 5 1 G m 5 10 1 G Frequency [3] (b) Calculate an estimate of the mean mass. … kg [2] (c) Two parcels are chosen at random. Find the probability that they both have a mass greater than 1 kg. Give your answer as a decimal, correct to 3 significant figures. … [2]

16 0607/41/O/N/16 © UCLES 2016 10 (a) Solve. 7x + 2 = 11 x = … [2] (b) Write as a single fraction, in its simplest form. x x 2 1 3 1 + + - … [2] (c) Simplify the following. (i) x y x y 4 8 3 4 4 2 … [2] (ii) x x x 2 3 9 2 2 - - - … [4]

17 0607/41/O/N/16 © UCLES 2016 [Turn over 11 f(x) = 3x + 1 g(x) = logx (a) Find the value of g(f(33)). … [2] (b) Find the value of x when g(x) = f(−1). x = … [2] (c) Find (i) f −1 (x), f –1 (x) = … [2] (ii) g −1 (x). g −1 (x) = … [2]

18 0607/41/O/N/16 © UCLES 2016 12 (a) In 2015, Ahmed had a monthly salary of $1375. In 2016, his monthly salary is $1540. (i) Calculate the percentage increase in Ahmed’s monthly salary. … % [3] (ii) Work out $1375 as a percentage of $1540. … % [1] (iii) In 2015, Ahmed’s monthly salary of $1375 was 10% more than his monthly salary in 2014. Calculate Ahmed’s monthly salary in 2014. $ … [3]

19 0607/41/O/N/16 © UCLES 2016 (b) Samia invested $500 in each of two Schemes. Scheme A 3% per year simple interest. Scheme B 2.5% per year compound interest. (i) Calculate the difference between the value of Scheme A and the value of Scheme B after 5 years. Show all your working. $ … [5] (ii) Find the number of complete years it will take for the value of Scheme B to be greater than the value of Scheme A. … [4]

20 0607/41/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 6 printed pages. © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/41 Paper 4 (Extended) October/November 2016 MARK SCHEME Maximum Mark: 120 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 41 © 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 Qu. Answer Mark Part Marks 1 (a) 201 2 M1 for 2500 ÷ 12.43 (implied by 201.1…) (b) (i) 783 or 782.5 to 783.3…. 3 B1 for 10h 40min oe 10.66..., 10.67, 10 2 3 , 640 M1 for 8350 ÷ their journey time (ii) [0]8 05 oe 1 (iii) 7 3 M2 for (36.8 – 20) ÷ 2.4 oe or M1 for 20 + 2.4 × distance = 36.8 oe 2 (a) (i) 8 5 −     −   1 (ii) Image at (–4, –1), (2, –1), (2, 3) 2FT SC1FT for translation 8 k −       or 5 k     −   (iii) 9.43 or 9.433 to 9.434 2 M1 for 2 2 ( ( 8)) ( ( 5)) − + − their their oe (b) (i) Reflection y-axis oe 1 1 (ii) Enlargement 0.5 oe (10, –10) 1 1 1 (iii) Stretch [factor] 0.25 oe x-axis oe invariant 1 1 1 3 (a) Correct sketch 3 B1 for shape including 2 minimum points and 2 maximum points B1 for all above x-axis (b) 0.5 ⩽ f(x) ⩽ 2 2 Allow written separately or in words B1 for each SC1 for 0.5 ⩽ x ⩽ 2 200 0 -200 3 2.5 2 1.5 1 0.5 0 -0.5 200 0 -200 3 2.5 2 1.5 1 0.5 0 -0.5 200 0 -200 3 2.5 2 1.5 1 0.5 0 -0.5 200 0 -200 3 2.5 2 1.5 1 0.5 0 -0.5

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 41 © UCLES 2016 Qu. Answer Mark Part Marks (c) (i) 1 1 (ii) 2 1 (d) (i) –90, 270, 630, 990 2 B1 for –90 and 270 with no others from –360 to 360 (ii) 360n – 450 oe 2FT FT only if clear linear sequence B1FT for 360n + k or kn – 450 (e) (i) Correct sketch 2 B1 for parabola vertex upwards (ii) 122.4 or 122 or 122.4… 326.2 or 326 or 326.2… 1 1 4 (a) 3 2 2 π 9 3 1 π 9 3 × × or equation with parts clearly cancelled leaving 2 and 9 M2 M1 for 2 3 1 2 π 9 π 9 3 3 h × × = × oe (b) (i) 763 or 764 or 763.4 to 763.5… 2 M1 for 2 2 π 9 2π 9 × + × or SC1 for 509 or 508.9 to 509.0… or 162π (ii) 569 or 569.0 to 569.1 3 M2 for 2 2 π 9 9 18 × × + or M1 for 2 2 9 18 + (c) 45 3 M2 for 3 3 2 π 9 3 4 π 2 3 × × or equation with parts clearly cancelled (implied by 45.56 to 46) or M1 for 3 3 4 2 π 2 π 9 3 3 n × × = × 5 (a) 18 – x + x + 12 – x + 3 = 25 oe Completion to x = 8 with at least one step M1 A1 B1 for Venn diagram completed with the 10, 8, 4 and 3 (b) (i) 22 25 oe 1 0.88 (ii) 21 25 oe 1 0.84 200 0 -200 3 2.5 2 1.5 1 0.5 0 -0.5 200 0 -200 3 2.5 2 1.5 1 0.5 0 -0.5 200 0 -200 3 2.5 2 1.5 1 0.5 0 -0.5 200 0 -200 3 2.5 2 1.5 1 0.5 0 -0.5

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 41 © UCLES 2016 Qu. Answer Mark Part Marks (c) 8 18 oe 1 4 9 , 0.4444… (d) element chosen from Q is also in P oe 1 6 (a) 2 5 3 3 = + y x oe 5 B1 for (2, 3) seen B1 for gradient of AB = 3 2 − B1FT for gradient = 2 3 M1 for correct method in finding c. (b) 1 13 oe 2 FT 3 – their 5 3 in (a) (but not if 0) M1 for 3 – their 5 3 in (a) 7 (a) 42.[0] or 41.98 to 41.99 2 M1 for tan = 9 10 oe (b) tan = 2 2 9 10 20 + oe 33.91 to 33.93 M2 A1 or M1 for 2 2 9 10 + or 2 2 2 9 10 20 + + (c) 12.4 or 12.39 to 12.40... nfww 3 M1 for 2 2 20 22 2 20 22cos33.9 + −× × A1 for 153 to 154 8 (a) Correct sketch 2 B1 for one correct branch (b) –2.62 or –2.618… –0.382 or –0.3820 to –0.3819 1 1 If 0 scored, M1 for correct use of quadratic formula oe (c) x < –2.62 –0.382 < x < 0 1FT 2FT FT only if 2 negative roots in (b) FT only if 2 negative roots in (b) B1 each (d) [a= ] 0 [b =] 3 1 1 (e) Translation 0 3     −   oe 1 1 4 2 0 -2 4 6 4 2 0 -2 4 2 0 -2 4 6 4 2 0 -2 4 2 0 -2 4 6 4 2 0 -2 4 2 0 -2 4 6 4 2 0 -2

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 41 © UCLES 2016 Qu. Answer Mark Part Marks 9 (a) 18, 20, 15, 20, 20 3 B2 for 4 correct B1 for 3 correct (b) 3.3[0] or 3.295 to 3.296 2FT M1 for at least 3 mid-values seen, 0.5, 1.5, 2.5, 4, 7.5 If 0 scored, SC1 for 2.26 or 2.258… or for 4.33 or 4.333… or 4.3 (c) 0.649 cao 2 M1 for 75 74 93 92 × their their their their (implied by 5550 8556 or 0.6486 to 0.6487 oe) 10 (a) 9 7 oe 2 M1 for 7x = 11 – 2 oe (b) 5 1 6 + x final answer 2 M1 for 3(x + 1) + 2(x – 1) seen (c) (i) 2 2x y final answer 2 B1 for 2 terms correct (ii) 3 1 + + x x final answer 4 B1 for (x – 3)(x + 3) B2 for (x – 3)(x + 1) or or SC1 for (x + a)(x + b) where ab = –3 or a + b = –2 11 (a) 2 2 B1 for [f(33) =] 100 or M1 for log(3 1) + x (b) 1 100 or [0].01 2 M1 for g(x) = 3(–1) + 1 oe (c) (i) 1 3 − x oe 2 M1 for 3 1 = + x y or 1 3 −= y x (ii) 10x 2 M1 for log = x y or 10 = y x 12 (a) (i) 12 3 M2 for 1540 1375 100 1375 − × oe or M1 for 1540 100 1375 × or for 1540 1375 1375 − (ii) 89.3 or 89.28 to 89.29 1 (iii) 1250 3 M2 for 1375 ÷ 1.1 oe or M1 for associating 1375 with 110%

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 41 © UCLES 2016 Qu. Answer Mark Part Marks (b) (i) 500 + 500 3 5 100 × × oe 500 × 1.0255 500 × 1.0255 – 500 500 3 5 100 × × amount – amount or interest – interest 9.3[0] or 9.295 to 9.296 M2 and M1 or M2 and M1 M1 A1 or M1 for 500 3 5 100 × × oe (575, 565.704…) or M1 for 500 × 1.0255 (65.704…, 75) (ii) 16 4 B3 for final answer of 15 or 15.28 to 15.29 seen or 15 reached by trial and improvement or M2 for sketch leading to answer or trial and improvement with at least two steps beyond 5 years or M1 for 500 + 500 3 100 × × x = 500 × 1.025x oe , implied by one trial