MATHEMATICS 9709/33 Paper 3 Pure Mathematics 3 (P3) October/November 2017

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1 Cambridge International Examinations Cambridge International Advanced Level CANDIDATE NAME * * CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/33 Paper 3 Pure Mathematics 3 (P3) October/November 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 11_9709_33/RP UCLES2017 [Turn over

2 2 1 Findthequotient andremainderwhenx 4 is divided byx 2 +2x 1. [3]

3 3 2 Twovariablequantitiesxandyarebelievedtosatisfyanequationoftheformy=C a x,wherecand a are constants. An experiment produced four pairs of values of x and y. The table below gives the corresponding values ofxandlny. x ln y By plotting lny against x for these four pairs of values and drawing a suitable straight line, estimate thevalues ofc anda. Give youranswers correct to 2significant figures. [5] lny x [Turn over

4 4 3 Theequationx 3 = 3x +7 has onereal root,denoted by. (i) Show bycalculation that lies between 2 and 3. [2]

5 Two iterative formulae, A and B, derived from this equation are as follows: 5 x n+1 = 3x n , (A) Each formula is used withinitialvaluex 1 = 2.5. x n+1 = x3 n 7. (B) 3 (ii) Show that one of these formulae produces a sequence which fails to converge, and use the other formulato calculate correct to 2 decimal places. Give theresult of each iteration to 4 decimal places. [4] [Turn over

6 6 4 (i) Provetheidentity tan 45 +x +tan 45 x 2sec2x. [4]

7 7 (ii) Hencesketchthe graphofy=tan 45 +x +tan 45 x for 0 x 90. [3] [Turn over

8 8 5 Theequationofacurveis 2x 4 +xy 3 +y 4 = 10. (i) Show that dy dx = 8x3 +y 3 3xy 2 +4y3. [4]

9 9 (ii) Hence show that there are two points on the curve at which the tangent is parallel to the x-axis and find the coordinates of these points. [4] [Turn over

10 10 6 The variables x and y satisfy the differential equation dy dx = 4cos2 ytanx, for 0 x < 1 2, and x = 0 when y = 1. Solve this differential equation and find the value of x when 4 y = 1. [8]

11 [Turn over

12 12 7 (a) The complex number u is given by u = 8 15i. Showing all necessary working, find the two squarerootsofu. Giveanswersintheforma+ib,wherethenumbersaandbarerealandexact. [5]

13 13 (b) On an Argand diagram, shade the region whose points represent complex numbers satisfying both theinequalities z 2 i 2 and 0 arg z i 1. [4] 4 [Turn over

14 14 8 Let f x = 4x2 +9x 8 x +2 2x 1. (i) Express f x intheforma + B x +2 + C 2x 1. [4]

15 4 (ii) Henceshow that 1 15 f x dx = ln [5] [Turn over

16 16 9 y R O 2 x Thediagramshowsthecurvey= 1 +x 2 e 1 2 x forx 0. TheshadedregionRisenclosedbythecurve, thex-axis and thelinesx=0andx=2. (i) Find the exact values of the x-coordinates of the stationary points of the curve. [4]

17 17 (ii) Show thattheexact valueofthe areaofris18 42 e. [5] [Turn over

18 18 10 Theequationsoftwolineslandmarer=3i j 2k + i +j+4k andr=4i +4j 3k + 2i +j 2k respectively. (i) Show that the lines do not intersect. [3] (ii) Calculate the acute angle between the directions of the lines. [3]

19 19 (iii) Find the equation of the plane which passes through the point 3, 2, 1 and which is parallel to bothlandm. Give youranswerin theformax +by +cz = d. [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.

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