HCI 2024 H2 Math Prelims Paper 2

HCI2024H2MathematicsPreliminaryExaminationsPaper2SectionA(PureMaths) Question1 2 3 4 5 6 Marks6575710 Question1[6] Part(a)[3] Byconsidering,showthat𝑡𝑎𝑛 (𝐴−𝐵) .𝑡𝑎𝑛−1(1 𝑥)−𝑡𝑎𝑛−1(1 1+𝑥)=𝑡𝑎𝑛−1(1 𝑥2+𝑥+1) Part(b)[3] Henceshowthat ,𝑟=1 1 ∑𝑡𝑎𝑛−1(1 𝑟2+𝑟+1)=𝑘−𝑓(𝑛) Whereisaninversetrigonometricfunctionandisanexactconstanttobefound.𝑓(𝑛) 𝑘

Question2[5] Asequenceisdefinedbytherecurrencerelation ,for.𝑢𝑛+1=1 + 𝑢𝑛 1 − 𝑢𝑛 𝑛≥1 Part(a)[1] Statewhathappenstothesequencewhen.𝑢1=0 Itisnowgiventhat.𝑢1=2 Part(b)[2] Find,,,,and.𝑢2𝑢3𝑢4𝑢5 𝑢6 Part(c)[2] Byobservingthepatterninpart(b),findintermsof.𝑟=1 4𝑛 ∑𝑢𝑟 𝑛

Question3[7] Part(a)[5] Acurvehasequation .𝐶 2𝑦3−𝑦2=𝑥𝑒𝑥 Findtheequationsofthetangentswhichareparalleltothey-axis. Part(b)[2] Itisgiventhatthetangentsfoundinpart(a)makeanacuteangleofradianswiththelineπ 6 .Findthevaluesof.𝑦=𝑚𝑥+1 𝑚

Question4[5] Part(a)[2] Foranynon-parallelandnon-zerovectorsmandn,explainclearlyandshowthat (m·n)2+|m×n|2=|m|2|n|2 andaretwodistinctpoints,where=pand=q.Itisalsoknownthatpandqare𝑃 𝑄 𝑂𝑃→ 𝑂𝑄→ non-zerovectors. Twoparallellinesandhavevectorequations𝑙1 𝑙2 r=p+suandr=q+tvrespectively,wheres,t∈R. Part(b)[3] Ifv×(p−q)=0,whatcanbesaidabouttherelationshipbetweenthetwolines? Justifyyouranswer.

Question5[7] Part(a)[4] UsingstandardseriesfromtheListofFormulae(MF26),findtheMaclaurinexpansionof inascendingpowersofuptoandincludingthetermin.1 (1+𝑐𝑜𝑠𝑥)2 𝑥 𝑥4 Part(b)[3] Findthesetofvaluesofforwhichiswithin±0.5ofthepolynomialfoundinpart𝑥 1 (1+𝑐𝑜𝑠𝑥)2 (a),where.0≤𝑥≤π

Question6[10] Part(a)[1] Giventhat ,whereandarerealnumbers,showthat*.𝑢=𝑥+𝑖𝑦 𝑥 𝑦 |𝑢|2=𝑢 𝑢 Twocomplexnumbersandwithnon-zerorealandimaginarypartssatisfy𝑧 𝑤 ,where.|𝑧+𝑤|=|𝑧−𝑤| 𝑧≠𝑤 Part(b)[3] Byconsideringpart(a),showthat**.𝑧𝑤+𝑧𝑤=0

Part(c)[2] Henceshowthat*ispurelyimaginary.𝑧𝑤 Itisnowgiventhat−,andtheargumentofis,where− .𝑤=1+𝑖3 𝑧θ π<θ≤π Part(d)[4] Usingtheresultinpart(c),findthepossibleexactvaluesof.θ

HCI2024H2MathematicsPreliminaryExaminationsPaper2SectionB(Statistics) Question7 8 9 10 11 12 Marks6 7 9 12 12 14 Question7[6] Afactoryproducesalargenumberofmonitorscreens.Itisknownthat,onaverage,100𝑝% ofthemonitorscreensarefaulty.Thenumberoffaultymonitorscreensproducedeachdayis independentofthatonotherdays.Eachday,thequalitycontrolmanagerwillproduceacheck onrandomlychosenmonitorscreensproducedonthatday.𝑛 Letbethenumberoffaultymonitorscreensfound.Youmayassumethatcanbe𝑀 𝑀 modelledbyabinomialdistribution. Part(a)[2] Statetheprobabilitythatonaparticularday,thereareatleast2butnomorethan3faulty monitorscreensfound,givingyouranswerintermsofand.𝑛𝑝 Part(b)[4] Eachday,thequalitycontrolmanagerwillperformacheckon10randomlychosenmonitor screensproduced.Findthepossiblevaluesofsuchthatthereisa25%chancethatona𝑝 randomlychosenweekwith5workingdays,thereareexactly3dayswithatleast2butno morethan3faultymonitorscreensfound.

Question8[7] Aconferencehallhasfivedoors,labelledA,B,C,DandE,whicharelocatedsidebysideas shownbelow.Thedoorsaretobepaintedusingfourdistinctcolours,andeachdoorwillbe paintedwithasinglecolour. A B C D E Part(a)[1] Byconsideringthenumberofcoloursavailableforeachdoor,findthenumberofwaysto paintthefivedoorssuchthatthereisnorestrictiontothecolourofeachdoor. Part(b)[3] Findthenumberofwaystopaintthedoorssuchthattherearenoconsecutivedoorswhich areofthesamecolour. Part(c)[3] Findthenumberofwaystopaintthedoorsifallfourcoloursaretobeused.

Question9[9] Inthisquestion,youshouldstatetheparametersofanydistributionyouuse. Aceramicshopsellshandmadeceramiccups.Themouthsofthecupsareassumedtobe circularinshape.Thediameteroftheoutercircumferencesofthetoprimofthecups,,are𝑆 assumedtofollowanormaldistributionwithmeanmmandstandarddeviationmm.µ σ Part(a)[3] Itisgiventhat andthattheprobabilityofthediameterofthe𝑃(𝑆<80.5)=𝑃(𝑆>84.5) outercircumferenceofthetoprimofarandomlychosencupbeingmorethan85mmis 1.15%.Findthevalueof,andshowthat ,whencorrectedto3significantfigures.µ σ=1.10