Cambridge International Examinations Cambridge Ordinary Level MATHEMATICS (SYLLABUS D) 4024/21 Paper 2 May/June 2016

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1 * * Cambridge International Examinations Cambridge Ordinary Level MATHEMATICS (SYLLABUS D) 4024/21 Paper 2 May/June 2016 Candidates answer on the Question Paper. Additional Materials: Geometrical instruments Electronic 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. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Section A Answer all questions. Section B Answer any four questions. 2 hours 30 minutes If working is needed for any question it must be shown in the space below that question. Omission of essential working will result in loss of marks. You are expected to use an electronic calculator to evaluate explicit numerical expressions. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 100. This document consists of 23 printed pages and 1 blank page. DC (N/SG) /3 [Turn over

2 2 Section A[52marks] Answerallquestionsinthissection. 1 Ashopkeeperbuyssomeplatesfromamanufacturerfor$12each. (a) Themanufacturermakesaprofitof60%. Calculatethecostofmanufacturingeachplate. (b) Theshopkeepersellseachplatefor$ Calculatethepercentageprofitmadebytheshopkeeper. (c) Inasale,eachplateisreducedfrom$17.40to$ Calculatethepercentagediscountgiven. Answer $... [2] Answer...%[2] Answer...%[2]

3 3 (d) Theshopkeeperbuys100platesat$12each. Hesells60platesat$17.40eachandxplatesat$11.31each. Theshopkeepermakesaprofitofatleast10%. Findtheleastpossiblevalueofx. Answer... [3] [Turn over

4 4 p 2 (a) Solvetheequation - 1 = p 1 J 6 9ab N 2 (b) Simplify a 3 b 2O. L P Answer... [2] Answer... [2]

5 5 2 3 q - q (c) Simplify. 3-3q (d) (i) Factorise4t t - 9. (ii) Hencesolvetheequation4t t - 9 = 0. Answer... [2] Answer... [2] Answer... [1] [Turn over

6 6 3 Thetablebelowisfor y = x + x x y (a) Usingascaleof2cmto1unitonthex-axisfor-3 G x G 2 andascaleof1cmto1unitonthey-axisfor- 4 G y G 4, plotthepointsfromthetableandjointhemwithasmoothcurve. [2] (b) (i) Useyourgraphtoestimatethesolutionsoftheequation x 2 + x - 3 = 0. y Answer x=...or... [1] (ii) Useyourgraphtoestimatethesolutionsoftheequation x 2 + x - 5 = 0. Answer x=...or... [2] x

7 7 (c) Bydrawingatangent,estimatethegradientofthecurveat ^1, 1h. Answer... [2] (d) Theequation x 2 - x - 1 = 0canbesolvedbydrawingastraightlineonthegraphof 2 y = x + x - 3. (i) Findtheequationofthisstraightline. (ii) Drawthisstraightlineandhencesolve x 2 - x - 1 = 0. Answer... [2] Answer x=...or... [2] [Turn over

8 8 4 B N A ANB,BLCandCMAarestraightlines.NMisparalleltoBCand LNisparalleltoCA. (a) ProvethattriangleANMissimilartotriangleNBL. Giveareasonforeachstatementyoumake [3] L M C

9 9 (b) AN:NB=2:3 (i) FindNM:BC. (ii) FindareaANM:areaNBL. (iii) FindareaANM:areaNMCL. Answer... :...[2] Answer... :...[1] Answer... :...[2] [Turn over

10 10 5 (a) ABisverticalandCBishorizontal. AB=31mandCB=115m. C 115 CalculatetheangleofdepressionofCfromA. A 31 B Answer... [3]

11 11 (b) 354 L Jandaretwopositionsatsea. ThebaseofalighthouseisatL. JisdueEastofLandisdueSouthofL. L=354mandJ=1100m. (i) Calculate LJ t. (ii) HencefindthebearingoffromJ Answer... [2] Answer... [1] J [Turn over

12 12 J4-1N J2 0N 6 A = OB = O L 1 3 P L 7-5 P (a) Evaluate2A B. (b) FindA 2. Answer Answer J L J L N O O [2] P N O O [2] P

13 13 (c) FindB 1. (d) A+Z=A FindZ. (e) M+2I=B,whereIisthe2 # 2identitymatrix. FindM. Answer Answer Answer J L J L J L N O O [2] P N O O [1] P N O O [2] P [Turn over

14 14 Section B[48marks] Answerfourquestionsinthissection. Eachquestioninthissectioncarries12marks. 7 (a) ACisadiameterofthecircle,centreO,radius5cm. ACB t =64. Calculatethelengthoftheminorarc BC. (b) rim Abakingtrayisanopencylinderofradius15.5cmwitharim. Theouteredgeoftherimisacircleofradius16.5cm. Answer... cm[4] A 5 O B 64 C

15 15 (i) Calculatetheareaofthetopsurfaceoftherim. (ii) 44identicalcircularholesarecutoutofthebottomofthebakingtray. Theareaofthebottomthatremainsis650cm 2. (iii) Calculatetheradiusofeachcircularhole cm Answer...cm 2 [2] Answer... cm[3] d mm Tomakeapizza,thebakingtrayiscompletelyfilledwithdoughtoadepthofdmm. Theopencylinderholds500cm 3 ofdough. Calculatethedepthofthedough,dmm,givingyouranswercorrecttothenearestmillimetre. Answer... mm[3] [Turn over

16 16 8 5q 8 (a) p = - q (b) (i) Findpwhenq=2.6. (ii) Expressqintermsofp. H x 2 Trapezium A x h Answer... [1] Answer... [2] x x + 3 Trapezium B ThelengthsoftheparallelsidesoftrapeziumAarexcmand^x - 2hcm. ThelengthsoftheparallelsidesoftrapeziumBarexcmand^x + 3hcm. TheheightoftrapeziumAisHcmandtheheightoftrapeziumBishcm. Theareaofeachtrapeziumis15cm (i) ShowthatH = andh = x - 1 2x + 3. [2]

17 17 (ii) Findanexpressionintermsofxforthedifferenceinheight,H h,betweentrapeziumaand 75 trapeziumb,andshowthatitsimplifiesto ^ x - 1 h^2 x + 3h. [3] (iii) Thedifferenceinheightis1.5cm. (a) Showthat2x 2 + x - 53 = 0. [2] (b) Findx,givingyouranswercorrectto2decimalplaces. Answer x=... [2] [Turn over

18 18 9 (a) A 2 F D ABCDrepresentstherectangularslopingsurfaceofatriangularprism. ABEFisahorizontalrectangle.CEandDFarevertical. CBE t =15,DC=5mandAD=2m. (i) CalculateAC. (ii) CalculateCE. 5 B 15 Answer... m[2] Answer... m[2] C E

19 19 (iii) Calculate FAE t. (b) (i) 9 10 θ Answer... [4] Atrianglehassidesof10cm,9cmand6cm,andanangleofθ,asshowninthediagram. Calculateθ. (ii) ThetriangleGHhassidesofacm,bcmandccm asshowninthediagram. Itisgiventhat GH t isanobtuseangle. Completethestatementbelowusingoneofthesymbols1 G = H Answer... [3] c 2 ^a + b h [1] a c G b H [Turn over

20 electriclightbulbsofBrandAweretestedtofindhowlongeachbulblasted. Theresultsaresummarisedinthetablebelow. Time (thours) Number ofbulbs tg tg tG tG tG tG tG (a) Completethecumulativefrequencytable. Time (thours) Cumulative frequency tg50 tg100 tg150 tg200 tg250 tg300 tg [1] (b) Onthegrid,drawasmoothcumulativefrequencycurvetorepresentthisinformation. LabelthiscurveBrandA. 100 Cumulative frequency Time (t hours) [2]

21 21 (c) (i) Useyourgraphtoestimatethemedian. (ii) Useyourgraphtoestimatetheinterquartilerange. (d) 100BrandBbulbsgavethefollowingresults. 4bulbslasted50hoursorless. Thelongesttimeanybulblastedwas300hours. Themedianis250hours. Theupperquartileis275hours. Theinterquartilerangeis75hours. Answer... hours[1] Answer... hours[2] Onthegrid,drawandlabelthecumulativefrequencycurvefortheBrandBbulbs. [4] (e) Usingyourgraph,estimatethenumberofBrandAbulbsthatlasted275hoursorless. (f) Completethestatementbelow. Answer... [1] Brand...had...morebulbsthatlastedlongerthan275hoursthanBrand... [1] [Turn over

22 22 11 (a) TriangleABChasverticesA(2,2),B(3,5)andC(4,1). Triangle AlBlC lhasvertices Al( 4,4), Bl( 3,7)andCl( 2,3). WritedownthecolumnvectorofthetranslationthatmapstriangleABContotriangle AlBlC l. J N O Answer O [1] O L P (b) PQRSisaparallelogram. J - 4N ThepositionvectorofPrelativetoOisgivenbyOP = O. L 2 P J ThepositionvectorofQrelativetoOisgivenbyOQ= 4 N O. L 6 P Q (i) Express PQasacolumnvector. (ii) Find RS. (iii) Find RS. Answer Answer J L J L P S O N O O O P N O O O P [2] [1] Answer...units[2] R

23 23 10 (c) 5 y D ThediagramshowstriangleD. (i) Anenlargementwithcentre(5,4),scalefactor2,mapstriangleDontotriangleE. DrawandlabeltriangleE. [2] (ii) Anenlargementwithcentre(5,4),scalefactor0.5,mapstriangleDontotriangleF. DrawandlabeltriangleF. [1] (iii) TriangleGhasvertices(5,4),(4,3)and(3,5). TriangleFcanbemappedontotriangleGusingasingleenlargement. TriangleFcanalsobemappedontotriangleGusingadifferent singletransformationt. DescribefullythesingletransformationT. Answer [3] x

24 24 BLAN 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 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.

25 IGCSE is the registered trademark of Cambridge International Examinations. Cambridge International Examinations Cambridge Ordinary Level MATHEMATICS (SYLLABUS D) 4024/21 Paper 2 May/June 2016 MAR SCHEME Maximum Mark: 100 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 May/June 2016 series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. This document consists of 6 printed pages. [Turn over

26 Page 2 Mark Scheme Syllabus Paper Cambridge O Level May/June Question Answers Mark Part Marks 1 (a) 7.5(0) 2 M1 for x + 60 x = 12 soi or 100 B1 for by 160 (b) 45 2 M1 for (c) 35 2 M1 for (d) 25 3 M1 for x 11.31( 1320) or B1 276 A1 for 24.4(03...) 2 (a) 6 2 M1 for p 1 = 5(7 p) soi (b) (c) 3b a 2 q M1 for a b ab 4 9b 2 a oe oe or B1 for 3b² as numerator or k a 2 B1 for q 2 (1 q) or 3(1 q) (d) (i) (4t 1)( t+ 9) 2 B1 for ( at + c)( bt + d) with ab = 4 or cd = 9 (ii) 1 9 or ft 1ft 4 3 (a) Correct graph 2 B1 for correct scales and 4 points or wrong scales and all points. (b) (i) 2.3 ± ± (ii) 2.8 ± ± M1 for x 2 + x 3= 2soi (c) 2.4 to M1 for tangent at x =1 (d) (i) y = 2x 2 2 B1 for 2x or 2 (ii) Dependent on line drawn B1 for their line having FT gradient or FT intercept Cambridge International Examinations 2016

27 Page 3 Mark Scheme Syllabus Paper Cambridge O Level May/June Question Answers Mark Part Marks 4 (a) Complete proof 3 B2 for 2 pairs of equal angles 1 pair with reason. B1 for 1 pair of equal angles. (b) (i) 2 : 5 2 B1 for NM : BL = 2 : 3 oe or NM = LC (ii) 4 : 9 1 (iii) 1 : 3 2 B1 for such as ANM = 4 ABC 25 or NBL 9 = ABC 25 5 (a) 15.1 or 15.08(.. 3 M1 for 6 (a) tanθ = or tanθ = A1forθ = 15.1or θ = (b) (i) 18.8 or M1 for sinθ = 1100 (b) (c) (d) (e) (ii) 251 or 251.2(. 1ft 270 their LJ final ans oe isw 2 B1 for at least 2 elements correct in a 2 x 2 matrix 2 B1 for at least 2 elements correct or M1 for soi B1 for det B = 10 soi or B1 for soi 0 1 Cambridge International Examinations 2016

28 Page 4 Mark Scheme Syllabus Paper Cambridge O Level May/June Question Answers Mark Part Marks 7 (a) 4.53 to B2 for BOC = 52 or after B0 B1 for ABC ˆ = 90 or triangle OBC isosceles or BAC = 26 M1 for 52 2π 5ft 360 (b) (i) 101 or 32π or 100 to M1 for π(16.52 ) or15.5 (ii) 0.87 to B1 for π or 44πr 2 and 2 π M1 for r = 44π (iii) 7 3 M1 for π d = (a) (i) 1.92 (3 1 (ii) 8 p + 5 A1 for 0.66 to M1 for 8 p 5 q = + or pq= 8 5q or 8 p = 5 q (b) (i) H and h correctly derived 2 M1 for correct substitution in the formula for the area of a trapezium. (ii) 75 correctly ( x 1)(2x+ 3) derived (iii) (a) Equation correctly derived. 2 B1 for (b) B1 for 15(2x+ 3) 30( x 1) 3 M1 for soi ( x 1)(2x+ 3) B1 for 30x x+ 30 soi B1 for 75 = 1.5 ( x 1)(2x+ 3) ( 53) soi or 1± their soi Cambridge International Examinations 2016

29 Page 5 Mark Scheme Syllabus Paper Cambridge O Level May/June Question Answers Mark Part Marks 9 (a) (i) 5.38 to 5.39 or 29 2 M1 for ( AC ) = CE (ii) to M1 for = sin15 oe 2 AF (iii) 68.8 to M1 for = cos15 oe or BC²=BE²+ (their CE)² or 2 any complete alternative method A1 for and M1 for tan FAE ˆ = 5 2cos15 oe or 5 their( AF) (b) (i) 80.9(4. Or 81 3 B1 for 10 2 = cosθ or (ii) > 1 10 (a) (2) (4) (100) B2 for cosθ = (b) Correct curve 2 P1 for at least 5 correct plots (c) (i) 195 ft 190 and < (ii) B1 for one quartile correct in ranges 225 to 235 or 160 to 175 (d) Correct curve 4 P3 for at least 4 correct plots or (e) 92 ft 1 (f) B 15 ft A 1ft Their B1 + B1 for any two correct points soi. Cambridge International Examinations 2016

30 Page 6 Mark Scheme Syllabus Paper Cambridge O Level May/June (a) Question Answers Mark Part Marks (b) (i) (ii) ft 1 4 (iii) 8.94 or 8.94 to 8.95 or 80 oe (c) (i) Triangle vertices (5,4), (13,0), (9,8) (ii) Triangle F (5,4), (7,3), (6,5) 1 (iii) Rotation 180 Centre (5,4) 1 8 k 2 B1 for or k 4 2 M1 for 2 B1 for 2 correct 2 2 ( 8) + ( 4) oe ft 3 B2 for Rotation with either centre or angle. B1 for Rotation. Cambridge International Examinations 2016

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