UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level

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1 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level * * MATHEMATICS (SYLLABUS D) 4024/21 Paper 2 October/November 2013 Candidates answer on the Question Paper. Additional Materials: Geometrical instruments Electronic calculator READ THESE INSTRUCTIONS FIRST 2 hours 30 minutes 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, highlighters, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Section A Answer all questions. Section B Answer any four questions. 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. 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 24 printed pages. DC (LEG/CGW) 84304/2 R UCLES 2013 [Turn over

2 2 Section A [52marks] Answerallquestionsinthissection. 1 (a) Therateofexchangebetweendollars($)andpounds( )is$1.56= 1. Therateofexchangebetweeneuros( )andpoundsis 1.10= 1. (i) Amychanges 300intodollars. CalculatehowmanydollarsAmyreceives. (ii) Benchanges 770intopounds. CalculatehowmanypoundsBenreceives. Answer $...[1] (iii) Chrischanges$780intoeuros. CalculatehowmanyeurosChrisreceives. Answer...[1] Answer...[2]

3 3 (b) DebbiechangedsomedollarsintoJapaneseyen. Therateofexchangewas81dollars=1yen. Emmachangedthesamenumberofdollarsintoyen. TherateofexchangeforEmmawas82dollars=1yen. Emmareceived3feweryenthanDebbie. Giventhatthenumberofdollarschangedeachtimeisx,findx. Answer...[3] [Turn over

4 4 2 (a) ConstructthetriangleABCinwhich BAC t =40 andac=8cm. CisabovethelineAB,whichisdrawnforyou. A B [2] (b) Constructthelocusofallthepointsoutsidethetrianglethatare2cmfromtheperimeterof thetriangle. [2] (c) FindandlabelthepointP, insidethetriangle,thatis6.5cmfromaand equidistantfrombandc. [2]

5 5 3 ThelineAB joinsthepointa( 2,1)tothepointB(6,5). (a) FindthecoordinatesofthemidpointofAB. Answer (...,...) [1] (b) FindthegradientofAB. Answer...[1] (c) AB intersectsthey-axisatthepoint(0,c). Findc. Answer...[2] (d) Express ABasacolumnvector. (e) Cisthepoint(5,2)andDisthepoint(h,k). ThelinesAB andcd areequalinlengthandparallel. FindthecoordinatesofeachofthepossiblepointsD. Answer [1] Answer (...,...)and(...,...)[3] [Turn over

6 6 4 Thetableshowsthedistributionofthemassesof100babiesatbirth. Mass (xkg) Number ofbabies 1.5<xG2 2<xG <xG3 3<xG <xG4 4<xG <xG (a) Writedownthemodalclass. Answer...[1] (b) thispartofthequestionusethegridbelow. Usingascaleof4cmtorepresent1kg,drawahorizontalx-axisfor1 G x G 5. Usingascaleof2cmtorepresent5babies,drawaverticalaxisforfrequencyfrom0to30. Usingyouraxes,drawafrequencypolygontorepresenttheseresults. [2]

7 7 (c) (i) Completethecumulativefrequencytablebelow. Mass(xkg) x G 2 x G 2. 5 x G 3 x G 3. 5 x G 4 x G 4. 5 x G 5 Cumulativefrequency [1] (ii) Onthegridbelowdrawasmoothcumulativefrequencycurvetorepresenttheseresults Cumulative frequency x Mass (kg) [2] (d) yourcurvetoestimate (i) themedianmass, Answer...kg[1] (ii) the10thpercentile. Answer...kg[1] [Turn over

8 8 2 5 (a) Solve x =. Answer...[1] (b) Factorise (i) 5x + 5y, Answer...[1] (ii) 9x Answer...[1] (c) (i) Factorise2x 2 + 5x Answer...[1] (ii) youranswertopart (c)(i)tosolvetheequation2x 2 + 5x - 12 = 0. Answer x=...or...[1]

9 9 (d) Asourceoflightisobservedfromadistanceofdmetres. Theamountoflightreceived, Lunits,isinverselyproportionaltothesquareofthedistance. GiventhatL=9whend=2,findthevalueofLwhend=3. Answer...[2] [Turn over

10 10 6 (a) A D C B InthetriangleABC, ABC t = 90c, ACB t = 50candBC = 31m. DisthepointonAC suchthatbda t = 90c. (i) ShowthatCD=19.93m,correctto2decimalplaces. [2] (ii) CalculateAD. Answer...m[3]

11 11 (b) S P Q R TwoboatsareatthepointsPandQ. RSisaverticalcliffofheight52m. PSQ t = 10c and Q S t R = 55c. (i) StatetheangleofdepressionofP froms. (ii) Calculatethedistance,PQ,betweentheboats. Answer...[1] Answer...m[3] [Turn over

12 12 7 (a) A E D C B IntriangleABC,Disthepoint onbc suchthatad bisectsbac t and E isthepointonabsuchthatae = AC. (i) ShowthattrianglesAEDandACD arecongruent. [3] (ii) Giventhat ABD t = xc, EDB t = y cand ACB t = zc, findxintermsofyandz. Answer x=...[2]

13 13 (b) S P R Q IntrianglePQR,QSbisects PQR t andrsbisectsprq t. PQR t = 42cand PRQ t = 54c. FindreflexangleQSR. Answer...[2] [Turn over

14 14 Section B [48marks] Answerfourquestionsinthissection. Eachquestioninthissectioncarries12marks. 8 (a) B 14 A O 10 Inthediagram,thecircleseachhavecentreO. ABisachordofthelargercircleandalsoatangenttothesmallercircle. AB =14cmandtheradiusofthelargercircleis10cm. Findtheradiusofthesmallercircle. Answer... cm[3] (b) S Q T R P Inthediagram,PQ andrsarechordsofacirclethatintersectatt. (i) ShowthattrianglesPST andrqtaresimilar. [3]

15 15 (ii) S 5 T x 11 Q R P ST =5 cm,tr=11cmandtq=xcm. GiventhatPQ = 18 cm,showthatxsatisfiestheequation x 2-18x + 55 = 0. [2] (iii) Solvetheequation x 2-18x + 55 = 0. Giveeachsolutioncorrectto1decimalplace. (iv) FindthedifferencebetweenthelengthsofPTandTQ. Answer x=...or...[3] Answer... cm[1] [Turn over

16 16 9 Thenumberofbacteriainacolonytrebles everyhour. Thecolonystartswith50bacteria. Thetablebelowshowsthenumberofbacteria(y)inthecolonyaftert hours. Time (t hours) Numberof bacteria(y) (a) Completethetable. [1] (b) Onthegridontheoppositepageplotthepointsinthetable,andjointhemwitha smoothcurve. [3] (c) yourgraphtofindthenumberofbacteriainthecolonywhent=3.2. Answer...[1] (d) (i) Bydrawingatangent,estimatethegradientofthecurvewhent=2.5. (ii) Whatdoesthisgradientrepresent? Answer...[2] Answer......[1] (e) Giventhattheequationofthegraphis y = ka t, findkanda. (f) Thenumberofbacteriainanothercolonyisgivenbytheequation y Answer k=...a=...[1] = t. (i) Onthesameaxes,drawagraphtorepresentthenumberofbacteriainthiscolony. [2] (ii) Statethevalueoftwhenthenumberofbacteriaineachcolonyisthesame. Answer...[1]

17 17 y Number of bacteria t Time (t hours) [Turn over

18 18 10 Afueltankerdeliversfuelinacylindricalcontaineroflength9.5mandradius0.8m. (a) Afterseveraldeliveries,thefuelremaininginthecontainerisshowninthediagram. 9.5 A O 0.8 B AB ishorizontal,oisthecentreofthecircularcross-sectionand AOB t = 90c. (i) Calculatethecurvedsurfaceareaofthecontainerthatisincontactwiththefuel. (ii) Calculatethevolumeoffuelremaininginthecontainer. Answer... m 2 [2] Answer... m 3 [4] (iii) Calculatethisvolumeremainingasapercentageofthevolumeofthewholecontainer. Answer...%[2]

19 19 (b) Thefuelispumpedthroughacylindricalpipeofradius4.5cmatarateof300cm/s. (i) Calculatethevolumepumpedin1second. Answer... cm 3 [1] (ii) Calculatethetimetaken,inminutes,topump25000litresoffuel. Giveyouranswercorrecttothenearestminute. Answer... minutes[3] [Turn over

20 20 11 ThediagramshowstrianglesAandB. y A B x (a) (i) DescribefullythesingletransformationthatmapstriangleA ontotriangleb. Answer......[2] (ii) Findthematrixthatrepresentsthistransformation. Answer f p [2] (b) TriangleB ismappedontotrianglecbythetransformationrepresentedbythe matrix 2 0 c m. 0 1 (i) Onthegridabove,drawandlabeltriangleC. [2] (ii) Givethenameofthistransformation. Answer...[1]

21 21 (iii) Find the matrix that represents the inverse transformation that maps triangle C onto triangleb. Answer f p [2] (iv) FindtheratioareaoftriangleC :areaoftriangleb. Answer...:...[1] (c) FindthematrixthatrepresentsthesingletransformationthatmapstriangleA ontotrianglec. Answer f p [2] [Turn over

22 22 12 (a) A 65 C B 45 IntriangleABC, ABC t = 45cand BAC t = 65c. AC is5cmshorterthanbc. 5 sin 65 (i) ShowthatBC =. sin 65 - sin 45 [3] (ii) FindthelengthofBC. Answer... cm[1]

23 23 (b) P Q 6 R S IntrianglePQR,PQ=13cm,QR=6cmandRP=10cm. QRisproducedtoS. (i) Findthevalueofcos PRQ t,givingyouranswerasafractioninitslowestterms. (ii) Hencewritedownthevalueofcos PRS t. Answer...[3] Answer...[1] Turn over for The rest of ThiS question [Turn over

24 (c) 24 F E D TriangleDEGhasthesameareaastriangleDEF,butisnotcongruenttotriangleDEF. ThepointG islowerthandeandge=ef. DrawthetriangleDEGinthediagramabove. [1] (d) IntriangleLMN, LMN t = 30candML=2MN. WhentheareaoftriangleLMNis18cm 2,calculateMN. Answer... cm[3] Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonableefforthasbeen madebythepublisher(ucles)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwillbepleasedtomakeamendsat theearliestpossibleopportunity. UniversityofCambridgeInternationalExaminationsispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocal ExaminationsSyndicate(UCLES),whichisitselfadepartmentoftheUniversityofCambridge.

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