Transcription of UNIVERSITY OF CAMBRIDGE INTERNATIONAL …
1 *0157926880* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSG eneral Certificate of Education Advanced LevelMATHEMATICS9709/32 Paper 3 Pure Mathematics 3(P3)May/June 20131 hour 45 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 [ ] at the end of each questionor 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 06_9709_32/FP UCLES 2013[Turn the equation x 2 = 13x.]
2 [3]2 The sequence of values given by the iterative formulaxn+1=xn x3n+100 2 x3n+25 ,with initial valuex1= , converges to!.(i)Use this formula to calculate!correct to 4 decimal places, showing the result of each iterationto 6 decimal places.[3](ii)State an equation satisfied by!and hence find the exact value of!.[2]3x2lnyO( , )( , )The variablesxandysatisfy the equationy=Ae kx2, whereAandkare constants. The graph of lnyagainstx2is a straight line passing through the points , and , , as shown in thediagram. Find the values ofAandkcorrect to 2 decimal places.[5]4 The polynomialax3 20x2+x+3, whereais a constant, is denoted by p x . It is given that 3x+1 is a factor of p x .(i)Find the value ofa.[3](ii)Whenahas this value, factorise p x completely.[3]5xy a3aMThe diagram shows the curve with equationx3+xy2+ay2 3ax2=0,whereais a positive constant. The maximum point on the curve isM. Find thex-coordinate ofMinterms ofa.[6] UCLES 20139709/32/M/J/1336(i)By differentiating1cosx, show that the derivative of secxis secxtanx.
3 Hence show that ify=ln secx+tanx thendydx=secx.[4](ii)Using the substitutionx= 3 tan1, find the exact value of 311 3+x2 dx,expressing your answer as a single logarithm.[4]7(i)By first expanding cos x+45 , express cos x+45 2 sinxin the formRcos x+! ,whereR>0 and 0 <!<90 . Give the value ofRcorrect to 4 significant figures and the valueof!correct to 2 decimal places.[5](ii)Hence solve the equationcos x+45 2 sinx=2,for 0 <x<360 .[4]8(i)Express1x2 2x+1 in the formAx2+Bx+C2x+1.[4](ii)The variablesxandysatisfy the differential equationy=x2 2x+1 dydx,andy=1 whenx=1. Solve the differential equation and find the exact value ofywhenx= your value ofyin a form not involving logarithms.[7]9 (a)The complex numberwis such that Rew>0 andw+3w*=iw2, wherew*denotes the complexconjugate ofw. Findw, giving your answer in the formx+iy, wherexandyare real.[5](b)On a sketch of an Argand diagram, shade the region whose points represent complex numbers which satisfy both the inequalities 2i 2 and 0 arg +2 140.
4 Calculate the greatestvalue of for points in this region, giving your answer correct to 2 decimal places.[6]10 The pointsAandBhave position vectors 2i 3j+2kand 5i 2j+krespectively. The planephasequationx+y=5.(i)Find the position vector of the point of intersection of the line throughAandBand the planep.[4](ii)A second planeqhas an equation of the formx+by+c =d, whereb,canddare planeqcontains the lineAB, and the acute angle between the planespandqis 60 . Findthe equation ofq.[7] UCLES 20139709/32/M/J/134 BLANK PAGEP ermission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort 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 UNIVERSITY ofCambridge Local Examinations Syndicate (UCLES), which is itself a department of the UNIVERSITY of
