Cambridge IGCSE Mathematics - International 0607 — 2015 Oct/Nov Paper 4 · Variant 2

This document consists of 20 printed pages. DC (LEG/SW) 100760/1 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 7 5 1 1 2 3 6 7 2 1 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/42 Paper 4 (Extended) October/November 2015 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. Answer in degrees should be given to one decimal place. For π, 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.

2 0607/42/O/N/15 © UCLES 2015 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 0607/42/O/N/15 © UCLES 2015 [Turn over Answer all the questions. 1 (a) By writing each number correct to 1 significant figure, find an estimate for . . . . . 5 13 987 9 12 16 3 1 91 3 2 + + ^ h You must show your working. Answer(a) … [2] (b) Explain why your answer to part (a) is greater than the actual answer. Answer(b) … … [2] (c) Work out. . . . . . 5 13 987 9 12 16 3 1 91 3 2 + + ^ h Answer(c) … [1]

4 0607/42/O/N/15 © UCLES 2015 2 (a) Solve the equations. (i) log log log log x 4 3 3 4 5 2 - = - Answer(a)(i) x = … [3] (ii) sinx 4 3 1 + = for x 0 360 c c G G Answer(a)(ii) … [3] (b) Make x the subject of the formula. a x x 1 = - Answer(b) x = … [3]

5 0607/42/O/N/15 © UCLES 2015 [Turn over 3 The table gives the marks of 10 students in a geography exam and a history exam. Geography mark (x) 12 23 36 41 57 62 78 81 89 93 History mark (y) 32 43 41 51 52 60 68 65 76 80 (a) Find (i) the mean geography mark, Answer(a)(i) … [1] (ii) the mean history mark. Answer(a)(ii) … [1] (b) (i) Find the equation of the regression line for y in terms of x. Answer(b)(i) y = … [2] (ii) Estimate the history mark when the geography mark is 51. Answer(b)(ii) … [1]

6 0607/42/O/N/15 © UCLES 2015 4 The transformation P is a reflection in the x-axis. The transformation Q is a rotation of 90° clockwise about the origin. (a) Write down the transformation that is (i) the inverse of P, Answer(a)(i) … … [1] (ii) the inverse of Q. Answer(a)(ii) … … [2] (b) Describe fully the single transformation equivalent to P followed by Q. Answer(b) … … [2]

7 0607/42/O/N/15 © UCLES 2015 [Turn over 5 Find the next term and the nth term in each of the following sequences. (a) 27, 20, 13, 6, – 1, ... Answer(a) next term = … nth term = … [3] (b) 1024, 512, 256, 128, 64, ... Answer(b) next term = … nth term = … [3]

8 0607/42/O/N/15 © UCLES 2015 6 The marks, x, of 800 students in a mathematics exam are given in the table. Mark (x) Frequency 0 < x  20 62 20 < x  30 84 30 < x  40 140 40 < x  50 160 50 < x  60 142 60 < x  80 112 80 < x  100 100 (a) Calculate an estimate of the mean mark. Answer(a) … [2] (b) Complete the cumulative frequency table. Mark (x) Cumulative frequency 0 < x  20 62 0 < x  30 0 < x  40 0 < x  50 0 < x  60 0 < x  80 0 < x  100 800 [1]

9 0607/42/O/N/15 © UCLES 2015 [Turn over (c) On the grid below, draw a cumulative frequency curve. 0 10 20 30 40 Cumulative frequency Mark x 50 60 70 80 90 100 100 200 300 400 500 600 700 800 [3] (d) Use your graph in part (c) to find estimates for (i) the median mark, Answer(d)(i) … [1] (ii) the interquartile range, Answer(d)(ii) … [2] (iii) the minimum mark for a candidate to obtain a grade A, given that 15% of students gain a grade A. Answer(d)(iii) … [3]

10 0607/42/O/N/15 © UCLES 2015 7 –4 15 y x –15 6 0 f x x x 2 3 6 11 = - + ^ ^ ^ h h h (a) (i) On the diagram, sketch the graph of f y x = ^ h, for values of x between x 4 =- and x 6 = . [2] (ii) Write down the equations of the asymptotes. Answer(a)(ii) … , … [2] (iii) Write down the co-ordinates of the points where the graph crosses the axes. Answer(a)(iii) ( … , … ), ( … , … ) [2] (b) Solve the inequality. x x x 2 3 6 11 1 - + ^ ^ h h Answer(b) … [4]

11 0607/42/O/N/15 © UCLES 2015 [Turn over 8 Freddo lives in Manchester. He drives to Cambridge for a meeting. The distance from Manchester to Cambridge is 300 km. (a) Freddo leaves Manchester at 07 05 and arrives in Cambridge at 10 50. Calculate his average speed. Answer(a) …km/h [3] (b) After the meeting Freddo drives back to Manchester. His average speed for this journey is 5% more than his average speed driving to Cambridge. He leaves Cambridge at 17 45. Find the time Freddo arrives in Manchester. Answer(b) … [3] (c) Freddo’s car uses fuel at the rate of 8.1 km per litre. Fuel costs £1.45 per litre. Find the total cost of fuel for Freddo’s journey from Manchester to Cambridge and back to Manchester. Answer(c) £ … [2]

12 0607/42/O/N/15 © UCLES 2015 9 (a) A coat costs $100. The price is increased by 10% and then decreased by 10%. Find the new price of the coat. Answer(a) $ … [2] (b) A chair costs $1000. The price is increased by 20% and then decreased by 20%. Find the new price of the chair. Answer(b) $ … [2] (c) A car costs $10 000. The price is increased by x% and then decreased by x%. Find an expression, in terms of x, for the new price of the car. Give your answer in its simplest form. Answer(c) $ … [3]

13 0607/42/O/N/15 © UCLES 2015 [Turn over 10 A bag contains 3 red balls and 5 blue balls. In an experiment, three balls are chosen at random without replacement. (a) Find the probability that the three balls chosen are (i) all red, Answer(a)(i) … [2] (ii) two red and one blue, Answer(a)(ii) … [3] (iii) at least one of each colour. Answer(a)(iii) … [3] (b) This experiment is to be carried out 1680 times. Find the expected frequency of 3 red balls being chosen. Answer(b) … [2]

14 0607/42/O/N/15 © UCLES 2015 11 A is the point (2, 6) and C is the point (5, 4). The equation of the line AB is y x 4 14 + = . The equation of the line BC is y x 1 = - . (a) B is the point where the lines AB and BC intersect. Find the co-ordinates of the point B. Answer(a) ( … , … ) [3] (b) M is the midpoint of AC. Find the co-ordinates of M. Answer(b) ( … , … ) [2]

15 0607/42/O/N/15 © UCLES 2015 [Turn over (c) Find the equation of the line BM. Answer(c) … [3] (d) The point D lies on the line BM. The co-ordinates of D are (k, k + 9). Find the value of k. Answer(d) k = … [2]

16 0607/42/O/N/15 © UCLES 2015 12 In the diagram, ABC is a straight line and BFED is a rectangle. 120° 12 cm 15 cm 20 cm NOT TO SCALE B C D E F A (a) Find BC. Answer(a) … cm [3] (b) Show that angle DBC = 34.7°, correct to 3 significant figures. [3]

17 0607/42/O/N/15 © UCLES 2015 [Turn over (c) Find the perimeter of the quadrilateral ACDE. Answer(c) … cm [4] (d) Find the area of the quadrilateral ACDE. Answer(d) …cm2 [3]

18 0607/42/O/N/15 © UCLES 2015 13 120 y x –40 5 0 f x 2 100 10 x = - ^ h (a) (i) On the diagram, sketch the graph of f y x = ^ h, for x 0 5 G G . [2] (ii) Write down the x co-ordinate of the point where the graph crosses the x-axis. Answer(a)(ii) … [1] (iii) Write down the range of f(x). Answer(a)(iii) … [1] (b) Solve the equation. 2 100 10 20 x - = Answer(b) x = … [1] (c) Describe fully the single transformation that maps the graph of y 2 100 x = onto the graph of y 2 100 10 x = - . Answer(c) … [2]

19 0607/42/O/N/15 © UCLES 2015 [Turn over 14 A fraction P has denominator x. The numerator of the fraction is 3 less than the denominator. (a) Write down fraction P in terms of x. Answer(a) … [1] (b) The numerator and the denominator of fraction P are each increased by 3 to give fraction Q. Write down fraction Q in terms of x. Answer(b) … [1] (c) Q P 40 9 - = (i) Write down an equation in x and show that it simplifies to x x 3 40 0 2 + - = . [3] (ii) Solve the equation x x 3 40 0 2 + - = . Answer(c)(ii) x = … or x = … [2] (iii) Write down the original fraction, P. Answer(c)(iii) … [1] Question 15 is printed on the next page.

20 0607/42/O/N/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. 15 Solve the inequalities. (a) x 2 1 5 3 1 - Answer(a) … [3] (b) log 2 10 x 2 ^ h Answer(b) … [2]

® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the October/November 2015 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/42 Paper 4 (Extended), maximum raw mark 120 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 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 – October/November 2015 0607 42 © 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 Question Answer Mark Part Marks 1 (a) 10 2 B1 for 3 correct terms of 9 2 20 5 1000 2 3 + + or B1 for either of 2 or 8 soi (b) Numerator over-estimates, oe and denominator under-estimates oe 2 B1 for each (c) 8.55 or 8.546… 1 2 (a) (i) 40.5 oe 3 M1 for correct use of b alog M1 for correct use of b a log log ± (ii) 210, 330 with no extras in range 3 B2 for 210 or 330 ignoring any extras from using 30. or M2 for appropriate sketch or M1 for 5.0 sin − = x A1 for 30 or – 30 soi (b) [ ] 2 1 1 x a = − oe 3 M1 Correct squaring M1 Correct multiplication M1 Collection of terms M1 Correct factorisation and division by their( ) 2 1 a − If answer incorrect, maximum possible is M2 3 (a) (i) 57.2 1 (ii) 56.8 1 (b) (i) x y ] 0 [ 54 .0 9. 25 + = or 25.92 to 25.93, 0.5397… 2 B1 for 25.9 + mx, or B1 for c + 0.54x, If 0 scored, SC1 for 26 + 0.5x or better (ii) 53 or 53.4 to 53.5 1FT FT their (b)(i)

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 42 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 4 (a) (i) Reflection in x-axis 1 (ii) Rotation 90° [anticlockwise] [about] origin oe 2 B1 for rotation (b) Reflection y = –x 1 1 5 (a) –8 34 – 7n oe 1 2 M1 for k n + −7 or kn + 34 oe 0 ≠ k (b) 32 n       × 2 1 2048 oe e.g. 1 2 1 1024 −       × n or n − 11 2 1 2 M1 for k n+       2 1 oe soi, where k is an integer 6 (a) 49.3 or 49.33 to 49.34 2 M1 for mid-points soi, at least 3 of (10, 25, 35, 45, 55, 70, 90) implied by 39 470 (b) 146, 286, 446, 588, 700, 800 1 (c) Correct graph 3 All marks in (c) and (d) are dependent on increasing curve. B1 for plotting points at upper group limit B1FT for correct vertical plots (d) (i) 46 to 49 1 (ii) 26 to 30 2 B1 for 33 to 35, or 61 to 63 soi (iii) 74 to 77 3 M1 for 800 15 .0 × or 800 85 .0 × oe M1 for correct use of their 680.

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 42 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 7 (a) (i) Correct graph 2 M1 for graph in 2 sections, with each section approximately correct. (ii) x = 1.5 oe y = 3 1 1 (iii) (0, –3.67) or (0, –3.667 to –3.666) or 11 0, 3   −     (–1.83, 0) or (–1.833…, 0) or 11, 0 6   −     1 1 (b) 5.5 5.1 < < x oe and x < –1 3 1 B2 for 1.5 ⩽ x ⩽ 5.5 oe or B1 for 1.5 and 5.5 seen or for x ⩽ 5.5 or 1.5 ⩽ x Condone ⩽ Ignore inclusion of –4 or 6 throughout 8 (a) 80 3 B1 for 3h 45 min oe or better M1 for 300 time in hours their oe (b) 21 19 to 21 20 3 M2 for ( ) 300 1.05 their × a oe or M1 for 1.05 × their(a) oe or for 300 new speed their if > their(a) (c) 107 or 107.4... 2 M1 for 45 .1 1.8 600 × or SC1 for 7. 53 45 .1 1.8 300 = × or 53.70…

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 42 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 9 (a) 99 2 M1 for use of 1.1 × 0.9 oe (b) 960 2 M1 for use of 1.2 × 0.8 oe (c) 2 10000 x − oe 3 M2 for use of      −      + 100 1 100 1 x x oe or B1 for      ± 100 1 x oe soi 10 (a) (i) 336 6 oe 2 M1 for 6 1 7 2 8 3 × × (ii) 336 90 oe 3 M2 for 6 5 7 2 8 3 3 × × × or M1 for 6 5 7 2 8 3 × × If M0 scored, then B1 for RRB, RBR, BRR (iii) 336 270 56 45 oe 3 M2 for 6 4 7 5 8 3 3 × × × + their (a)(ii) or for 1 – their (a)(i) 6 3 7 4 8 5 × × − or M1 for 5 4 3 8 7 6 their × × + (a)(i) or for 6 5 7 2 8 3 6 4 7 5 8 3 × × + × × (b) 30 2FT M1 for 1680 × their (a)(i) 11 (a) Correctly eliminate 1 variable 3 = x 2 = y M1 B1 B1 or appropriate sketch If B0 scored, M1 for correct substitution to find 2nd variable. (b) (3.5, 5) 2 B1 for each (c) 16 6 − = x y oe 3 M1 for gradient 5.0 3 = oe soi M1 for substitution B or M into y = mx + c oe (d) 5 2 M1 for (k, k + 9) substituted into their (c) if linear

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 42 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 12 (a) 30.4 or 30.41… 3 M1 for 120 cos 20 15 2 20 15 2 2 2 × × × − + = x A1 for 925 (b) 4. 30 120 sin 20 sin their B = 34.71 to 34.73… M2 A1 M1 for 120 sin 4. 30 sin 20 their B = becomes M2 if 34.71 to 34.73… seen (c) 116 or 115.8…. 4 B1 for angle A = 34.7 or 34.71 to 34.73… or angle B = 55.3 or 55.26... to 55.29 M1 for 12 ( 21.1) sin 34.7 AB their = = oe M1 for 12 ( 17.3) tan 34.7 AF their = = oe (d) 414 or 413.7 to 413.9 3 M2 for 120 sin 20 15 5.0 3. 17 12 5.0 15 12 × × × + × × + × their oe or M1 for any correct area. 13 (a) (i) Correct graph 2 M1 for graph with correct shape. (ii) 3.32 or 3.321 to 3.322 1 (iii) ( ) [ ] 10 f − > x 1 Ignore ⩽ 90 (b) 1.74 or 1.736 to 1.737 1 (c) Translate       −10 0 1 1

Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 42 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 14 (a) x x 3 − 1 (b) 3 + x x 1FT (c) (i) All correct with no errors 40 9 3 3 = − − + x x x x 2 ( 3)( 3) 9 ( 3) 40 x x x x x − − +   =   +   oe or better x x 27 9 360 2 + = oe 0 40 3 2 = − + x x M1 M1 A1 their Q – their P i.e. at least one more correct line and no errors or omissions (ii) −8 5 1 1 (iii) 5 2 1 Allow final answer 8 11 − − but not 8 11 15 (a) x < 0.5 and 3 4 > x 3 M1 for sketch fit for purpose B1 for x > 3 4 or for x < 0.5 or for 0.5 and 3 4 soi

Page 8 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 42 © Cambridge International Examinations 2015 Question Answer Mark Part Marks (b) x > 33.2 or 33.21 to 33.22 2 M1 for appropriate sketch or M1 for 10 2 log > x