MATHEMATICS 9709/03 Paper 3 Pure Mathematics 3 (P3) For Examination from 2017

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1 Cambridge International Examinations Cambridge International Advanced Level CANDIDATE NAME * * CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/03 Paper 3 Pure Mathematics 3 (P3) For Examination from 2017 SPECIMEN PAPER Candidates answer on the Question Paper. Additional Materials: List of Formulae(MF9) 1 hour 45 minutes 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. UCLES2016 [Turn over

2 2 1 Solvetheinequality 2x 5 > 3 2x +1. [4]

3 3 2 Usingthesubstitutionu=3 x,solvetheequation3 x +3 2x =3 3x givingyouranswercorrectto3significant figures. [5] [Turn over

4 4 3 Theangles and liebetween 0 and 180,andare such that tan = 3 and tan +tan = 1. Findthepossiblevalues of and. [6]

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6 6 4 Theequationx 3 x 2 6 = 0has onerealroot, denoted by. (i) Findby calculation thepairofconsecutiveintegers between which lies. [2] (ii) Show that, if a sequence of values given by the iterative formula x n+1 = x n + 6xn converges,then itconverges to. [2]

7 7 (iii) Use this iterative formula to determine correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3] [Turn over

8 5 Theequationofacurveisy=e 2x tanx,for0 x< (i) Obtainanexpressionfor dy dx andshowthatitcanbewrittenintheforme 2x a +btanx 2,where a and b are constants. [5]

9 9 (ii) Explain why the gradient of the curve is never negative. [1] (iii) Findthevalueofxforwhich thegradient is least. [1] [Turn over

10 10 6 The polynomial 8x 3 +ax 2 +bx 1, where a and b are constants, is denoted by p x. It is given that x +1 is afactorof p x and thatwhenp x isdividedby 2x +1 theremainderis1. (i) Findthevaluesofaandb. [5]

11 11 (ii) Whenaandbhavethesevalues,factorisep x completely. [3] [Turn over

12 7 ThepointsA,BandChaveposition vectors,relativeto theorigino,givenby 1 3 OA = 2, OB = 0 and 1 OC = Theplanemis perpendicular toabandcontains thepointc. (i) Findavectorequation forthelinepassing throughaandb. [2] (ii) Obtaintheequationof theplanem,givingyour answer in theformax +by +cz =d. [2]

13 13 (iii) ThelinethroughAandBintersectstheplanematthepointN. FindthepositionvectorofN and showthatcn = 13. [5] [Turn over

14 14 8 Thevariablesxand satisfy thedifferential equation dx d = x +2 sin2 2, anditisgiventhatx=0when =0. Solvethedifferentialequationandcalculatethevalueofxwhen = 1,giving youranswercorrect to 3significant figures. [9]

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16 9 Thecomplexnumber3 i is denoted byu. Its complex conjugate isdenotedbyu*. 16 (i) On an Argand diagram with origin O, show the points A, B and C representing the complex numbersu,u*andu* u respectively. What typeof quadrilateral isoabc? [4] (ii) Showing your working and without using a calculator, express u * in the form x +iy, where x u andyare real. [3]

17 17 (iii) By considering theargument of u * u,provethat tan = 2tan [3] [Turn over

18 18 10 y M O R 1 p x x2 The diagram shows the curve y = for x 0, and its maximum point M. The shaded region R 3 1 +x is enclosed by thecurve,thex-axis andthe linesx = 1 andx=p. (i) Find the exact value of the x-coordinate of M. [4]

19 19 (ii) Calculate the value of p for which the area of R is equal to 1. Give your answer correct to 3 significant figures. [6]

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. 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.

Cambridge International Examinations Cambridge International Advanced Subsidiary Level

Cambridge International Examinations Cambridge International Advanced Subsidiary Level CANDIDATE NAME *8968664976* CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/21 Paper 2 Pure Mathematics 2 (P2) May/June