itzpipey_amath_november_2024_P2_MS

1 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 (a) Findtheequationofthenormaltothecurve attheorigin. [3]𝑦 = 𝑡𝑎𝑛( 𝑥2 − π) (b) Aparticlemovesalongthecurve. When , isdecreasingatarateof .𝑥 = π3 𝑦 π 𝑢𝑛𝑖𝑡𝑠/𝑠Findtherateofchangeof atthisinstant. [3]𝑥

4 2 Showthatthecurve andtheline willnever(𝑚− 1)𝑥 2 − 2𝑦 2 = 𝑚𝑥𝑦 + 1 𝑦 = 𝑚𝑥 + 1intersectforallvaluesof . [5]𝑚

5 3 (a) Giventhat and , express intermsof inthesimplest𝑙𝑜𝑔3 𝑝 = 𝑞 𝑙𝑜𝑔3 𝑞 = 𝑟 𝑙𝑜𝑔27 (𝑝𝑞 6 ) 𝑟form. [4] (b) Solve . [5]𝑙𝑔(𝑥 2 + 1) + 𝑙𝑛(𝑥 2 + 1) = 𝑒

6 4 (a) Explainwhyallthepowersof inthebinomialexpansionof areodd. [3]𝑥 (2𝑥 − 1𝑎𝑥 ) 9 (b) Findtherangeofvaluesof suchthatthetermindependentof intheexpansionof𝑎 𝑥 + hasapositivecoeficient. [5]( 𝑎𝑥 7 1𝑥 5 )(2𝑥 − 1𝑎𝑥 ) 9

7 5 Forthefunction , where isaconstant, theremainder𝑓(𝑥) = 𝑝𝑥 3 +𝑝 2 𝑥 2 + 18𝑥 − 54 𝑝when isdividedby isthesameaswhenitisdividedby .𝑓(𝑥) 𝑥 − 1 𝑥 + 2 (a) Findthepossiblevaluesof [3]𝑝.

8 (b) Givenfurtherthat isdivisibleby , ind . [3]𝑓(𝑥) 𝑥 + 6 𝑓(𝑥)

9 6 (a) Thevariables and areconnectedbytheequation𝑥 𝑦 2 3𝑥 − 𝑚 𝑥 = (𝑛𝑥− 𝑦) 3 where and areconstants. Thediagramshowsthestraightlinegraph𝑚 𝑛 obtainedbyplotting against , ofwhich and are 𝑦𝑥 2 𝑥 𝑥 − 23 (2, 6.5) (64,− 9) pointson. Findthevaluesof and andhenceexpress intermsof . [6]𝑚 𝑛 𝑦 𝑥

10 (b) and areconnectedbytheequation Explainhowastraightlinegraph𝑉 𝑡 𝑉 = 𝐴𝑒 𝑘𝑡+1 .maybedrawnusinggivenvaluesof and andhowthevaluesof and maybe𝑉 𝑡 𝐴 𝑘foundusingthestraightlinegraph. [4]