MATHEMATICS 9709/32 Paper 3 Pure Mathematics 3 (P3) February/March 2017

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1 Cambridge International Examinations Cambridge International Advanced Level CANDIDATE NAME * * CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/32 Paper 3 Pure Mathematics 3 (P3) February/March 2017 Candidates answer on the Question Paper. Additional Materials: List of Formulae(MF9) 1hour45minutes READ THESE INSTRUCTIONS FIRST WriteyourCentrenumber,candidatenumberandnameinthespacesatthetopofthispage. Writeindarkblueorblackpen. YoumayuseanHBpencilforanydiagramsorgraphs. Do not use staples, paper clips, glue or correction fluid. DONOTWRITEINANYBARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion. Thetotalnumberofmarksforthispaperis75. Thisdocumentconsistsof19printedpagesand1blankpage. JC17 03_9709_32/RP UCLES2017 [Turn over

2 2 1 Solvetheequation ln 1 +2 x = 2,givingyouranswercorrect to3decimal places. [3]

3 3 2 Solvetheinequality x 4 < 2 3x +1. [4] [Turn over

4 4 3 (i) By sketching suitablegraphs,show that theequation e 1 2 x = 4 x 2 has onepositiveroot andone negative root. [2] (ii) Verify by calculation that the negative root lies between 1 and 1.5. [2]

5 5 (iii) Usetheiterativeformulax n+1 = 4 e 1 2 x n todeterminethisrootcorrectto2decimalplaces. Give the result of each iteration to 4 decimal places. [3] [Turn over

6 6 4 (i) Express 8cos 15sin in the form Rcos +, where R > 0 and 0 < < 90, stating the exactvalueofrandgiving thevalueof correct to 2decimal places. [3]

7 7 (ii) Hence solve the equation 8cos2x 15sin2x = 4, for0 <x<180. [4] [Turn over

8 8 5 Thecurvewithequationy=e ax tanx,whereaisapositiveconstant,hasonlyonepointintheinterval 0 < x < 1 at which the tangent is parallel to thex-axis. Find the value ofaand state the exact value 2 of the x-coordinate of this point. [7]

9 [Turn over

10 10 6 Thelinelhas equationr=i +2j 3k + 2i j+k. Theplanephasequation 3x +y 5z = 20. (i) Show thatthelinellies intheplanep. [3]

11 11 (ii) A second plane is parallel to l, perpendicular to p and contains the point with position vector 3i j+2k. Findtheequationofthisplane,givingyouranswerintheformax +by +cz =d. [5] [Turn over

12 12 7 hm A water tank has vertical sides and a horizontal rectangular base, as shown in the diagram. The area of the base is 2m 2. At time t = 0 the tank is empty and water begins to flow into it at a rate of 1m 3 per hour. At the same time water begins to flow out from the base at a rate of 0.2 hm 3 per hour, wherehm is thedepthofwaterinthe tank at timethours. (i) Form a differential equation satisfied by h and t, and show that the time T hours taken for the depthofwatertoreach 4m isgivenby T = dh. [3] h

13 13 (ii) Using thesubstitutionu=5 h,find thevalueoft. [6] [Turn over

14 14 8 Throughout this question the use of a calculator is not permitted. Thepolynomialz 4 +3z 2 +6z +10 is denoted byp z. The complex number 1 +iisdenotedbyu. (i) Showingall yourworking, verify thatuisaroot oftheequation p z = 0. [3] (ii) Findtheotherthreeroots oftheequation p z = 0. [7]

15 15 [Turn over

16 16 9 Let f x = x 6 x 2 +x 4 +x 2. (i) Express f x inpartialfractions. [5]

17 17 (ii) Henceobtaintheexpansionoff x inascendingpowersofx,uptoandincludingtheterminx 2. [5] [Turn over

18 18 10 y P O e Q x Thediagramshowsthecurvey= lnx 2. Thex-coordinateofthepointPisequaltoe,andthenormal to thecurveatpmeets thex-axis atq. (i) Find the x-coordinate of Q. [4] (ii) Show that lnxdx =xlnx x+c,wherecis aconstant. [1]

19 19 (iii) Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the x-axis and the normal PQ. [5]

20 20 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher(ucles) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate(UCLES), which is itself a department of the University of Cambridge.

MATHEMATICS 9709/32 Paper 3 Pure Mathematics 3 (P3) May/June 2017

Cambridge International Examinations Cambridge International Advanced Level CANDIDATE NAME *9035132693* CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/32 Paper 3 Pure Mathematics 3 (P3) May/June 2017