Cambridge IGCSE Mathematics - International 0607 — 2016 Oct/Nov Paper 4 · Variant 2

This document consists of 16 printed pages. DC (LEG/JG) 117020/3 © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 9 3 3 5 2 6 4 2 1 2 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/42 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. Answer 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.

2 0607/42/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 0607/42/O/N/16 © UCLES 2016 [Turn over Answer all the questions. 1 The number of matches in each of 140 matchboxes are counted. The table shows the results. Number of matches 168 169 170 171 172 173 174 175 176 177 178 Number of matchboxes 7 13 16 23 22 21 14 11 8 3 2 (a) Write down the modal number of matches. … [1] (b) Write down the range. … [1] (c) Find the median. … [1] (d) Find the inter-quartile range. … [2] (e) Calculate the mean. Give your answer correct to one decimal place. … [2]

4 0607/42/O/N/16 © UCLES 2016 2 Roberta starts from a point A and walks 1 km North to a point B. She then walks 2 km East to a point C, then walks 3 km South to a point D and finally walks 4 km West to a point E. North A (a) Find the distance AE. …km [3] (b) Find the bearing of E from A. … [2] (c) Find the area ABCDE. …km2 [2]

5 0607/42/O/N/16 © UCLES 2016 [Turn over 3 Ten students at a school each recorded the number of hours they spent revising before an examination. The school compared the number of hours spent revising and the examination mark. Number of hours spent revising (x) 3 4 8 9 10 12 13.5 17 21 24 Examination mark (y) 45 36 68 55 62 66 73 81 80 94 (a) What type of correlation is there between the number of hours spent revising and the examination mark? … [1] (b) Find (i) the mean number of hours spent revising, … [1] (ii) the mean examination mark. … [1] (c) (i) Find the equation of the regression line for y in terms of x. y = … [2] (ii) Estimate the examination mark for a student who spent 19 hours revising. … [1]

6 0607/42/O/N/16 © UCLES 2016 4 42° NOT TO SCALE B P C A Q O A, B and C lie on a circle, centre O. The line QBP is a tangent to the circle at B. AC = BC = BP and angle QBA = 42°. Find the value of (a) angle OAB, Angle OAB = … [1] (b) angle AOB, Angle AOB = … [2] (c) angle BCA, Angle BCA = … [1] (d) angle CBP, Angle CBP = … [2] (e) angle CPB. Angle CPB = … [2]

7 0607/42/O/N/16 © UCLES 2016 [Turn over 5 The age, h, of each of 120 passengers travelling on a train are shown in the table. Age (years) Frequency 0 < h  15 12 15 < h  20 18 20 < h  25 13 25 < h  35 27 35 < h  50 22 50 < h  90 28 (a) Calculate an estimate of the mean age of a passenger. …years [2] (b) Complete the frequency density column in this table. Age (years) Frequency Frequency density 0 < h  15 12 15 < h  20 18 20 < h  25 13 25 < h  35 27 35 < h  50 22 50 < h  90 28 [3]

8 0607/42/O/N/16 © UCLES 2016 6 Describe fully the single transformation that is the inverse of (a) a reflection in the line y = x, … … [2] (b) a rotation of 90° clockwise, centre (2, 3), … … [2] (c) a translation with vector 4 3 - J L KK N P OO, … … [2] (d) an enlargement scale factor 3, centre (0, 0). … … [2] 7 Solve the simultaneous equations. You must show all your working. x y 3 4 8 + =- x y 5 6 7 - =- x = … y = … [4]

9 0607/42/O/N/16 © UCLES 2016 [Turn over 8 (a) cosx 3 1 = for 0° < x < 90°. Find the exact value of sinx. Give your answer as a surd. sinx = … [3] (b) NOT TO SCALE 10 cm 9 cm 11 cm A C B (i) Show that cosB 3 1 = . [2] (ii) Using your answer to part (a), show that the exact value of the area of triangle ABC is 30 2 cm2. [3]

10 0607/42/O/N/16 © UCLES 2016 9 A circle of radius 5 cm is inscribed inside a square. The square has one side on the base of an equilateral triangle, ABC. The other two vertices of the square touch the triangle as shown. NOT TO SCALE A C B (a) Work out the shaded area. …cm2 [2] (b) (i) NOT TO SCALE 10 cm x cm 60° Find the value of x. x = … [2]

11 0607/42/O/N/16 © UCLES 2016 [Turn over (ii) Work out the length of a side of the equilateral triangle ABC. … cm [2] (iii) Calculate the area outside the square but inside triangle ABC. …cm2 [4]

12 0607/42/O/N/16 © UCLES 2016 10 y x –6 6 –10 10 0 f x x 3 2 6 = - - ^ ^ h h (a) On the diagram, sketch the graph of f y x = ^ h for values of x between –6 and 6. [3] (b) Write down the equations of the asymptotes of the graph of f y x = ^ h. … and …[2] (c) Solve the equation f x x =- ^ h . … [2] (d) Solve the inequality f x x 0 1 + ^ h . … [3] (e) Describe fully the single transformation that maps (i) y x 3 6 = - onto y x 3 2 6 = - - ^ h, … … [2] (ii) y x 2 6 =- - ^ h onto y x 3 2 6 = - - ^ h. … … [2]

13 0607/42/O/N/16 © UCLES 2016 [Turn over 11 Find the next term and the nth term in each of these sequences. (a) 1, 8, 27, 64, 125, ... Next term = … nth term = … [2] (b) 3, 7, 13, 21, 31, ... Next term = … nth term = … [4] (c) –2, 1, 14, 43, 94, ... Next term = … nth term = … [4]

14 0607/42/O/N/16 © UCLES 2016 12 A solid hemisphere has radius 6 cm. (a) Find, in terms of r, (i) the volume of the hemisphere, …cm3 [2] (ii) the total surface area of the hemisphere. …cm2 [2] (b) Sixteen of these hemispheres, all with radius 6 cm, are made into one solid sphere. (i) Find the radius of the sphere. … cm [3] (ii) Find the ratio surface area of the sphere : total surface area of the 16 hemispheres. Give your answer in its simplest form. … : …[3]

15 0607/42/O/N/16 © UCLES 2016 [Turn over 13 (a) log log log log p q x 3 2 6 + - = Find x in terms of p and q. x = … [3] (b) Solve the equations. (i) 4 6 x = x = … [3] (ii) x x 3 2 2 3 1 + - = ^ ^h h You must show all your working. x = … or x = … [5] Question 14 is printed on the next page.

16 0607/42/O/N/16 © UCLES 2016 14 y x –2 8 0 –12 10 f x x 2 3 1 x 3 = - ^ h (a) On the diagram, sketch the graph of f y x = ^ h, for values of x between −2 and 8. [4] (b) Write down the y co-ordinates of the local minimum points. y = … and y = … [2] (c) Write down the co-ordinates of the local maximum point. ( … , …) [2] (d) Solve the equation x x 2 3 1 2 1 x 3 - = - ^ h, for all real values of x. x = … or x = … or x = … or x = … [4] 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 7 printed pages. © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/42 Paper 4 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 42 © 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 Question Answer Mark Part Marks 1 (a) 171 1 (b) 10 1 (c) 172 1 (d) 4 2 B1 for 170 or 174 seen (e) 172.1 2 M1 for attempt at ∑fx soi by 24099 or 172 or 172.1... 2 (a) 2.83 or 2.828... 3 B2 for 8 or 2 2 final answer or M2 for 2 2 2 2 + or M1 for correct sketch (b) 225 cao 2 B1 for 45 soi by e.g. 135 If 0 scored SC1 for 224.9 to 225.1 (c) 8 cao 2 M1 for 2 × 3 + 0.5 × 2 × 2 oe 3 (a) Positive 1 (b) (i) 12.15 1 (ii) 66 1 (c) (i) y = 37.2+ 2.37x 2 Range 37.20 to 37.21 and 2.369 to 2.370 B1 for 37.2 + kx, or a + 2.37x, If 0 scored, SC1 for 37 + 2.4x (ii) 82 or 82.2... 1 FT their (i)

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 42 © UCLES 2016 Question Answer Mark Part Marks 4 (a) 48 1 (b) 84 2 FT (180 – 2 × their (a)), their (a) ≠ 45 M1 for (180 – 2 × their (a)) their (a) ≠ 45 (c) 42 1 FT their (b) ÷ 2 (d) 69 cao 2 B1 for angle OBC or OAC = 21 or angle ABC = 69 (e) 55.5 2 FT their (d) M1 for (180 – their (d)) ÷ 2 5 (a) 36.7 or 36.68 to 36.69 2 B1 for at least 3 of (7.5, 17.5, 30, 42.5, 70) soi by 4402.5 Accept 37.2 or 37.18 to 37.19 for full marks and 3 of (8, 18, 30.5, 43, 70.5) soi for B1 (b) 0.8, 3.6, 2.6, 2.7, 1.47 or 1.466 to 1.467, 0.7 3 B2 for 4 or 5 correct or B1 for 2 or 3 correct 6 (a) Reflection y = x 2 B1 for each (b) Rotation, centre (2, 3) 90 [anticlockwise] or 270 clockwise 2 B1 for each (c) Translation 4 3 −       2 B1 for each (d) Enlargement, centre (0, 0) [SF] 1 3 oe 2 Allow reduction B1 for each 7 Correctly equating one set of coefficients M1 Equation ݔ= or ݕ= from one equation Note – a correct sketch showing intersection in third quadrant scores M2 (other sketches may score the M1 for y = … seen) Correct method to eliminate one variable M1 Correct substitution into other equation x = –2 1 2 y = − B2 B1 for each If zero scored SC1 for correct substitution into one of original equations and evaluation to find other variable

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 42 © UCLES 2016 Question Answer Mark Part Marks 8 (a) 8 2 2 or 3 3 or 8 9 3 M2 for 2 2 3 1 3 − or M1 for 32 – 12 If 0 scored, SC1 for 0.943 or 0.9428... or 0.889 (b) (i) [cosB] = 2 2 2 9 10 11 2 9 10 + − × × oe 2 M1 for 2 2 2 11 9 10 2 9 10 cosB = + −× × × (ii) 0.5 × 9 × 10 × their exact (a) M2 M1 for 0.5 × 9 × 10 × their (a) (their (a) must be < 1) Leading to 30 2 A1 Cancelling seen or 180 2 6 or 90 2 3 or 60 2 2 seen 9 (a) 21.5 or 21.45 to 21.46… 2 M1 for 100 − π × 52 oe (b) (i) 5.77 or 5.773 to 5.774 2 M1 for tan60 = 10 x oe (ii) 21.5 or 21.54 to 21.55 2 M1 for 10 + 2 × their (b)(i) oe or 10 + 10 sin60 oe (iii) 100 to 101.0… nfww 4 M3 for 0.5 × 10 × their (b)(i) + 0.5 × 10 × their (b)(i) + 0.5 × 10 × 10 sin 60 oe or M2 for any 2 of these or M1 for any 1 of these OR M3 for 0.5 × (their(b)(ii))2 × sin60 – 102 oe or M2 for 0.5 × (their(b)(ii))2 × sin60 oe or M1 for their attempt at area of triangle ABC – 100

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 42 © UCLES 2016 Question Answer Mark Part Marks 10 (a) Fully correct curve f(x)=3-6/(x-2) -6 6 -10 10 x y 3 B2 for both branches but with serious 'curl back' and/or overlap. or B1 for 1 branch (b) x = 2 y = 3 2 B1 for each (c) [x = ] −4 [x = ] 3 2 B1 for each (d) x < –4 2 < x < 3 1 2 FT their –4 from (c) FT their 2 from (b) and their 3 from (c) B1 for each (e) (i) Translation 2 0    2 B1 for each (ii) Translation 0 3    2 B1 for each 11 (a) 216 n3 oe 1 1 (b) 43 n2 + n + 1 oe 1 3 M2 for pn² + qn + c p, q, c ≠ 0 or M1 for second differences = 2 or pn² + c or pn² + qn (c) 173 n 3 – n2 – n – 1 oe 1 3 FT their (a) – their (b) FT their (a) – their (b) M2 for 3 2 pn qn rn c + + + p, q, r, c ≠ 0 or n3 – their (b) or M1 for third differences = 6 or for 3 2 pn qn c + + or 3 2 pn qn rn + + or 3 pn rn c + +

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 42 © UCLES 2016 Question Answer Mark Part Marks 12 (a) (i) 144π 2 M1 for 2 3 × π × 63 (ii) 108π 2 M1 for 2π × 62 + π × 62 If 0 scored SC1 for 72π (b) (i) 12 or 11.99 to 12.01 nfww 3 M2 for ( )( ) 16 4 π 3 their a i × oe or M1 for 4 3 × π × r3 = their(a)(i) × 16 (ii) 1 : 3 or 1 :1 3 cao nfww 3 M2 for 4 × π × (their(b)(i))2 : 16 × their(a)(ii) oe or M1 for 4 × π × (their(b)(i))2 or 16 × their(a)(ii) 13 (a) 3 2 6 p q final answer 3 M1 for correct use of alogb M1 for correct use of loga ± logb (b) (i) 1.29 or 1.292… 3 M2 for log6 log 4 or log4 6 or sketch of 4x y = and y = 6 oe or M1 for xlog4 = log6 or sketch of 4x y = (ii) 6x² – 5x – 7 = 0 B2 or B1 for 3 terms correct in expansion 2 6 9 4 6 x x x − + − 2 ( 5) ( 5) 4 6 ( 7) 2 6 x −− ± − − × × − = × M1 FT their three term quadratic or for sketch of parabola with minimum point Alternative If sketch of parabola with minimum point and y = 1 and no three term quadratic seen, allow B3 x = 1.57 or 1.574... x = –0.741[01...] B2 B1 for each

Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 42 © UCLES 2016 Question Answer Mark Part Marks 14 (a) Fully correct curve f(x)=2^x-x^3/3 -2 2 4 6 8 -10 -8 -6 -4 -2 2 4 6 8 10 12 x y 4 B1 for correct graph for x < 0, minimum point seen above x-axis B1 for correct graph for 0 < x < 2, maximum point seen higher than minimum point B1 for minimum point seen below x-axis, 2 < x < 8 If 0 or 1 scored, SC2 instead for ‘correct curve’ except stationary point of inflexion instead of LH minimum and maximum (b) 0.729 or 0.7287... –10.3 or –10.26... 2 B1 for each (c) (1.31 or 1.311 to 1.312, 1.73[0] ) 2 B1 for each co-ordinate (d) –2.82, 0.364, 4.23, 5.76 or –2.824 to –2.823 0.3643 to 0.3644 4.228 to 4.229 5.758… 4 B1 for each If 0 scored SC2 for −2.8, 0.36, 4.2, 5.8 or SC1 for three of these.