Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level CANDIDATE NAME *9022343494* CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/12 Paper 1 Pure Mathematics 1 (P1) May/June 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 06_9709_12/RP UCLES2017 [Turn over
1 (i) Findthecoefficient ofxinthe expansion of 2 2x 1 x 5. [2] (ii) Hencefindthecoefficientofxin theexpansion of 1 +3x 2 2x 1 5. [4] x
3 2 The point A has coordinates 2, 6. The equation of the perpendicular bisector of the line AB is 2y = 3x +5. (i) Find the equation of AB. [3] (ii) Find the coordinates of B. [3] [Turn over
3 (i) Prove the identity 1 cos tan 2 4 1 sin 1 +sin. [3]
(ii) Hence solve the equation 5 1 2 cos tan = 1,for0 360. [3] 2 [Turn over
6 4 O 2 rad rcm A B rcm D C ThediagramshowsacirclewithradiusrcmandcentreO. PointsAandBlieonthecircleandABCD is arectangle. AngleAOB = 2 radians andad = rcm. (i) Express theperimeter oftheshadedregioninterms ofr and. [3]
7 (ii) Inthe casewherer = 5 and = 1,find theareaof theshaded region. [4] 6 [Turn over
8 5 Acurvehas equationy=3 + 12 2 x. (i) Findtheequationofthetangenttothecurveatthepointwherethecurvecrossesthex-axis. [5]
9 (ii) Apointmovesalongthecurveinsuchawaythatthex-coordinateisincreasingataconstantrate of0.04 units persecond. Find therateofchangeofthey-coordinatewhenx=4. [2] [Turn over
10 6 y A 1, 4 x+y = 5 y = 4 x B 4, 1 O x The diagram shows the straight line x +y = 5 intersecting the curve y = 4 at the points A 1, 4 and x B 4, 1. Find,showingallnecessaryworking,thevolumeobtainedwhentheshadedregionisrotated through 360 about thex-axis. [7]...................................................
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12 7 (a) Thefirsttwotermsofanarithmeticprogressionare16and24. Findtheleastnumberoftermsof theprogressionwhich must betakenfortheirsum toexceed 20000. [4]
13 (b) A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric progressionisformedbysquaringeachofthetermsoftheoriginalprogression. Findthesumto infinity of the new progression. [4] [Turn over
8 Relativeto anorigino,theposition vectors ofthreepointsa,bandc aregivenby OA = 3i +pj 2pk, wherepandqareconstants. 14 OB = 6i + p +4 j +3k and OC = p 1 i +2j +qk, (i) Inthe casewherep=2,useascalarproducttofindangleaob. [4]
15 (ii) Inthe casewhere ABis parallelto OC,findthevaluesofpandq. [4] [Turn over
16 9 Theequationofacurveisy=8 x 2x. (i) Find the coordinates of the stationary point of the curve. [3] (ii) Findanexpressionfor d2 y dx 2 andhence,orotherwise,determinethenatureofthestationarypoint. [2]
17 (iii) Findthevaluesofxatwhich theliney=6meets thecurve. [3] (iv) Statetheset ofvalues ofkforwhichtheliney=kdoes not meet thecurve. [1] [Turn over
18 10 Thefunctionfis definedby f x = 3tan 1 2 x 2, for 1 2 x 1 2. (i) Solvetheequation f x +4 = 0,giving youranswercorrect to 1decimal place. [3] (ii) Findan expressionforf 1 x andfind thedomain off 1. [5]
19 (iii) Sketch,on thesamediagram,thegraphsofy=f x andy=f 1 x. [3]
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 www.cie.org.uk 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.