UNIVERSITY OF CAMBRIDGE INTERNATIONAL …

1 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSG eneral Certificate of Education Advanced Subsidiary LevelMATHEMATICS9709/02 Paper 2 Pure Mathematics 2(P2)October/November 20081 hour 15 minutesAdditional Materials:Answer Booklet/PaperGraph PaperList of Formulae (MF9)READ THESE INSTRUCTIONS FIRSTIf you have been given an Answer Booklet, follow the instructions on the front cover of the your Centre number, candidate number and name on all the work you hand in dark blue or black may use a soft pencil for any diagrams or not use staples, paper clips, highlighters, glue or correction non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles indegrees, unless a different level of accuracy is specified in the use of an electronic calculator is expected, where are reminded of the need for clear presentation in your the end of the examination , fasten all your work securely number of marks is given in brackets [ ]

2 At the end of each question or part total number of marks for this paper is carrying smaller numbers of marks are printed earlier in the paper, and questions carrying largernumbers of marks later in the document consists of3printed pages and1blank page. UCLES 2008[Turn over*6986198037* the inequality|x 3|>|2x|.[4]2 The polynomial 2x3 x2+ax 6, whereais a constant, is denoted by p(x). It is given that(x+2)isafactorofp(x).(i)Find the value ofa.[2](ii)Whenahasthisvalue,factorisep( x)completely.[3]3xlny( , )(0, )OThe variablesxandysatisfy the equationy=A(b x),whereAandbare constants. The graph of lnyagainstxis a straight line passing through the points(0, )and( , ), as shown in the the values ofAandb, correct to 2 decimal places.[5]4(i)Show that the equationsin(x+30 )=2cos(x+60 )can be written in the form(3 3)sinx=cosx.[3](ii)Hence solve the equationsin(x+30 )=2cos(x+60 ),for 180 x 180.]

3 [3]5 Show that 21(1x 42x+1)dx=ln1825.[6]6 Find the exact coordinates of the point on the curvey=xe 12xat whichd2ydx2=0.[7] UCLES 20089709/02/O/N/0837(i)By sketching a suitable pair of graphs, show that the equationcosx=2 2x,wherexis in radians, has only one root for 0 x 12 .[2](ii)Verify by calculation that this root lies between and 1.[2](iii)Show that, if a sequence of values given by the iterative formulaxn+1=1 12cosxnconverges, then it converges to the root of the equation in part(i).[1](iv)Use this iterative formula, with initial valuex1= , to determine this root correct to 2 decimalplaces. Give the result of each iteration to 4 decimal places.[3]8(i)(a)Prove the identitysec2x+secxtanx 1+sinxcos2x.(b)Hence prove thatsec2x+secxtanx 11 sinx.[3](ii)By differentiating1cosx,showthatify=secxthe ndydx=secxtanx.[3](iii)Using the results of parts(i)and(ii),findtheexactvalueof 14 011 sinxdx.

4 [3] UCLES 20089709/02/O/N/084 BLANK PAGEP ermission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Everyreasonableeffort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher willbe pleased to make amends at the earliest possible of CAMBRIDGE INTERNATIONAL Examinations is part of the CAMBRIDGE Assessment Group. CAMBRIDGE Assessment is the brand name of UniversityofCambridge Local Examinations Syndicate (UCLES), which is itself a department of the UNIVERSITY of