HCI 2024 H2 Math Prelims Paper 1

HCI2024H2MathematicsPreliminaryExaminationsPaper1 Question 1 2 3 4 5 6 7 8 9 10 11 Marks 4 5 5 7 7 11 9 12 14 14 12 Question1[4] Thecurve cutstheaxesat and ,where isaconstantsuch𝑦 = 𝑓(𝑥) (0, 1−𝑝𝑝 ) (1 − 𝑝,0) 𝑝that .Itisgiventhat exists.0 < 𝑝 < 1 𝑓 −1 State,ifpossibletodoso,thecoordinatesofthepointswherethefollowingcurvescuttheaxes. (a)𝑦 = 𝑓(𝑥) + 1 (b)𝑦 = 𝑓(𝑥 − 𝑝) (c)𝑦 = 𝑓(3𝑥 − 𝑝) (d)𝑦 =𝑓 −1 (𝑥)

Question2[5] Part(a)[1] Itisgiventhat and arenon-zeropolynomials.𝑓(𝑥) 𝑔(𝑥) Whensolvinganinequality ,astudentwrites . 𝑓(𝑥)𝑔(𝑥) ≥ 1 𝑓(𝑥) > 𝑔(𝑥) Commentonthestudent’sworking.[1] Part(b)[4] Findtheexactsetofvaluesof forwhich . [4]𝑥 2𝑥 2 −𝑥−9𝑥 2 −𝑥−6 ≥ 1

Question3[5] Theregionboundedbythecurvewithequation ,thelines , and𝑥 = 𝑦 2𝑦−𝑦 2 𝑦 = 1 𝑦 = 1.6 they-axisisrotatedthrough2πradiansaboutthey-axistoformasolidornament. Part(a)[4] Findtheexactvolume oftheornament,givingyouranswerintermsofπ.𝑉1 Part(b)[1] Anornamentdesignerdesignsadifferentornamentbyrotatingtheregionboundedbyanothercurvewithequation ,where ,thelines , andthe𝑥 = 𝑏𝑦 2𝑦−𝑦 2 𝑏 > 0 𝑦 = 1 𝑦 = 1.6 y-axis.Theregionisnowrotatedthrough2πradiansaboutthey-axis.Thevolumegeneratedisnow .Statetheratioof to .𝑉2 𝑉1 𝑉2

Question4[7] Part(a)[4] The11th,15thand23rdtermsofanarithmeticprogressionarethreedistinctconsecutivetermsofageometricprogression.Findthecommonratioofthegeometricprogression. Part(b)[3] Thesum, ,ofthefirst termsofasequence, , isgivenby𝑆𝑛 𝑛 𝑣1 𝑣2 𝑣3 .𝑆𝑛 = 3 𝑛+2 −(−2) 𝑛+2 −56 Findanexpressionfor intermsof ,simplifyingyouranswer.𝑣𝑛 𝑛

Question5[7] Part(a)[4] Byusingthesubstitution ,where ,showthat𝑥 = 𝑠𝑒𝑐 θ 0 ≤ θ ≤ π2 ,2 2 ∫ 1 𝑥 2 −1 𝑑𝑥=θ1 θ2 ∫𝑔(θ) 𝑑θ Where and areexactconstantstobestated,θ1 θ2 and isasingletrigonometricfunctiontobedetermined.𝑔 Part(b)[3] Hence,findtheexactvalueof .2 2 ∫ 1 𝑥 2 −1 𝑑𝑥

Question6[11] Thefunctionsand aredefinedby𝑓 𝑔 , for ,𝑓:𝑥 ↦ 𝑙𝑛 [(𝑥 + 4) 2 − 9] 𝑥 ∈ 𝑅, 𝑥 > 𝑘 , for .𝑔:𝑥 ↦ 3−2𝑥1+2𝑥 𝑥 ∈ 𝑅, 𝑥 > 12 Part(a)[2] Findtheleastvalueof forwhichthefunction exists.𝑘 𝑓 −1 Usethevalueof foundinpart(a)fortherestofthisquestion.𝑘 Part(b)[2] Withoutfinding ,findtheexactvalueof if .𝑓 −1 α 𝑔( 32 ) =𝑓 −1 (α)

Thefunctionisdefinedbyℎ ,for ,whereisaconstant.ℎ:𝑥 ↦ 1 4 𝑥(𝑎−𝑥) 0 < 𝑥 < 𝑎 𝑎 Part(c)[3] Sketchthegraphof , statingthecoordinatesofthestationarypointandthe𝑦 = ℎ(𝑥)equationsofanyasymptotes.

Part(d)[2] Giventhatthecompositefunction exists,findtherangeofvaluesof .𝑔ℎ 𝑎 Part(e)[2] Byconsidering anditsstationarypoint,orotherwise,findthevalueof forwhich𝑦 = 1ℎ(𝑥) 𝑎onlyhasonerealroot.[ℎ(𝑥)] 2 = 1

Question7[9] Itisgiventhat ,where, ,and arenon-zerorealconstants.𝑓(𝑥) = 𝑎𝑥 5 + 𝑏𝑥 3 + 𝑐𝑥 𝑎 𝑏 𝑐 Part(a)[1] Showthat − =− .𝑓( 𝑥) 𝑓(𝑥) Part(b)[3] Itisgiventhat hasonlyonerealrootandoneofthenon-realrootsis ,𝑓(𝑥) = 0 𝑝 + 𝑞𝑖where and arenon-zerorealconstants.Find,intermsof and,alltheothernon-real𝑝 𝑞 𝑝 𝑞rootsof ,justifyingyouranswers.𝑓(𝑥) = 0

Part(c)[2] Giventhat −,statethevaluesof and .0 3 ∫𝑓(𝑥) 𝑑𝑥= 5 −3 3 ∫ 𝑓(𝑥) 𝑑𝑥 −3 3 ∫ 𝑓(|𝑥|) 𝑑𝑥 Let and .𝑎 = 1 𝑏 = 3 Part(d)[3] Byconsidering , findtherangeofvaluesof such