itzpipey_amath_november_2024_P1_QP

CANDIDATENAME: ……………………………………………………………… CLASS: ……………………… INDEXNUMBER: ……......... ADDITIONALMATHEMATICS 4049/01 Paper1 November2024 Secondary4Express 2hours15minutesSetter:itzpipey:))) Candidatesanswer onthequestionpaper.NoAdditional Materialsarerequired. READTHESEINSTRUCTIONSFIRST Writeyour name, class, andindexnumber inthespacesat thetopof thispage.Writeindarkblueor blackpen.YoumayuseanHBpencil for anydiagramsor graphs.Donot usestaples, paper clips, glueor correctionfluid. Answer all thequestions.Givenon-exact numerical answerscorrect to3significant figures, or 1decimal placeinthecaseof anglesindegrees, unlessadifferent level of accuracyisspecifiedinthequestion.Theuseof anapprovedscientificcalculator isexpected, whereappropriate.Youareremindedof theneedfor clear presentationinyour answers. Thenumber of marksisgiveninbrackets[ ] at theendof eachquestionor part question.Thetotal number of marksfor thispaper is90. Thisdocumentconsistsof19printedpagesand1blankpage.

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3 1 Solvefor and if . [4]𝑎 𝑏 24 − 𝑎 68 − 3 = 2 + 𝑏 32

4 2 Thecurve has stationarypoints. Bysketchingsuitablecurveson𝑦 =−𝑥 3 + 5𝑥 − 3𝑥𝑙𝑛𝑥 𝑘theaxesbelow, indthevalueof . Youaretolabelallaxialinterceptsintheirexactform.𝑘Pleaseshowyourworkings. [5]

5 3 (a) Statetherangeofprincipalvaluesfor (i) [1]𝑠𝑖𝑛 −1 𝑥, (ii) [1]𝑐𝑜𝑠 −1 𝑥. (b) Youaregiventhat , where isafunctionof . Showthat 𝑠𝑖𝑛 4 𝑥 − 𝑐𝑜𝑠 4 𝑥𝑇(𝑥) = 𝑠𝑖𝑛 4𝑥 𝑇(𝑥) 𝑥 . [3]𝑇(𝑥) =− 12 𝑐𝑜𝑠𝑒𝑐2𝑥

6 4 (a) Byrepresenting7999999intheform , explainwhy7999999isnotaprime𝑎 3 −𝑏 3 number. [3] (b) Represent asasumofpartialfractions. [5] 48𝑥 2 + 3𝑥 − 214𝑥 2 −7𝑥 3

7 5 Deducewhetherthecurve isincreasingordecreasingfor . [4]𝑦 = 𝑒 −𝑠𝑖𝑛 2 𝑥 0 < 𝑥 < π4

8 6 Thediagramshowsatriangle inscribedinacircle, with asthediameter. and are𝐵𝐶𝐷 𝐵𝐷 𝐴 𝐸pointsonatangentofthecircleat suchthat isparallelto andBCisparallelto𝐶 𝐴𝐵 𝐶𝐷 𝐷𝐸. (a) Showthattriangles and aresimilar. [3]𝐴𝐵𝐶, 𝐵𝐶𝐷 𝐶𝐷𝐸 (b) Explainwhetheracirclecouldbedrawnintersectingpoints and , andstate𝐴,𝐵 𝐶whichlinewouldbethediameter. [2]

9 (c) Showthat [3]𝐴𝐵 × 𝐶𝐷 + 𝐵𝐶 × 𝐷𝐸 = 𝐵𝐷 2 .

10 7 Thecurve intersectstheline atthepoint𝑦 = 12𝑥 3 + 56𝑥 2 + 57𝑥 − 26 𝑦 = 2𝑥 − 1. Findthecoordinatesoftheotherintersectionpoint(s)betweentheandline. [6]( 13 , 13 )