8/12/2019 4724 Winter 06 Question Paper

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RECOGN SIrlG ACHIEVEMENT

OXFORD CAMBRIDGE AND RSA EXAMINATIONSAdvanced SUbsidiary General Certificate of EducationAdvanced General Certificate of EducationMATHEMATICS 47 4Core Mathematics 4Monday 23 JANUARY 2006 Afternoon 1hour 30 minutesAdditionalmaterials: 8pageanswerbooklet Graphpaper ListofFormulae(MF1)

TIME 1 hour30minutesINSTRUCTIONS TO CANDIDATES Write your name, centre numberandcandidate number in the spaces provided on the answerbooklet. Answerall thequestions. Give non-exact numerical answers correct to 3 significant figures unless a different degree ofaccuracyisspecifiedinthequestionorisclearlyappropriate. Youarepermittedtouseagraphicalcalculator inthispaper.INFORMATION FOR CANDIDATES Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion. Thetotalnumberofmarksforthispaperis72. Questionscarryingsmallernumbersofmarksareprintedearlierinthepaper,andquestionscarryinglargernumbersofmarkslaterinthepaper. You are reminded of the need for clear presentation in your answers.

S ur ~ - 1 Imp I y 2 . [3]x 92 Given that sin y = y+ find : in terms of x and y [5]3 i) Find the quotient and the remainder when ~ - ~ +x +7 is divided by - x+3 : [4]

ii) Hence, or otherwise, determine the values of the constants and b such that, when~ - +ax +b is divided by - x+5, there is no remainder. [2]

8/12/2019 4724 Winter 06 Question Paper

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24 (i) Useintegration by partsto find fx sec x dx. [ ]2(ii) Hencefind f x tan x dx. [3]

5 A curve is given parametricallyby the equationsx::: r y =2t.i) Find: in termsoft, givingyouranswerin its simplestform. [2]

(ii) Showthatthe equationofthe tangentto the curveat p2, 2p) ispy=x+p . [2]

(iii) Findthecoordinatesofthepointwherethe tangent at (9, 6) meetsthetangentat (25, -10). [4]

6 i) Showthat the substitutionx = sin2 e transforms fV x dx to f2 sin2 ede [ ](ii) Hencefind dx. [5]i V xI xo7 Theexpression 11 + 8x 2 is denotedby f x).2 - x) 1 + x)

(i) Express f x) in the form + ---. .- + C 2 whereA, B andC areconstants. [5]2 - x l+x l+x)(ii) Given that [x] < 1, find thefirst 3 terms in the expansionof f x) in ascendingpowersofx. [5]

8 (i) Solve thedifferential equation dy 2 - xdx = y -3 '

giving the particularsolutionthatsatisfies the conditiony = 4 whenx = 5. [5](ii) Showthat this particularsolution canbe expressedin the form

x - a)2 + y - b) 2 =k,

wherethe valuesoftheconstantsa, b andk are to be stated. [3](iii) Hencesketchthegraphofthe particularsolution, indicatingclearlyitsmainfeatures. [3]

9 Two lines have vectorequations

andr U) +t n r D scn,wherea is a constant.

(i) Calculate the acuteanglebetweenthe lines. [5](ii) Given thatthesetwolinesintersect, find a andthepointofintersection. [8]

47241Jan06