Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level

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1 Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level CANDIDATE NAME * * CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/11 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_11/RP UCLES2017 [Turn over

2 2 1 Thecoefficients ofx 2 andx 3 in the expansion of 3 2x 6 areaandbrespectively. Find thevalueof a b. [4]

3 2 Relativeto anorigino,theposition vectors ofpointsaandbare given by OA = and OB =, andangleaob = p 3 (i) Find the value of p. [2] ThepointC issuch that OC = 2 OA. 3 (ii) Findtheunit vectorin thedirection of BC. [4] [Turn over

4 3 (i) Provetheidentity 1 +cos sin + 4 sin 1 +cos 2 sin. [3]

5 (ii) Hencesolvethe equation 1 +cos sin + 5 sin 1 +cos = 3 cos for [3] [Turn over

6 6 4 (a) Anarithmeticprogressionhasafirsttermof32,a5thtermof22andalasttermof 28. Findthe sum of all the terms in the progression. [4]

7 7 (b) Each year a school allocates a sum of money for the library. The amount allocated each year increases by 2.5% of the amount allocated the previous year. In 2005 the school allocated $2000. Find the total amount allocated in the years 2005 to 2014 inclusive. [3] [Turn over

8 8 5 Theequationofacurveisy=2cosx. (i) Sketchthegraphofy=2cosxfor x,statingthecoordinatesofthepointofintersection with the y-axis. [2] PointsPandQlieon thecurveandhavex-coordinates of 3 1 and respectively. (ii) Find the length of PQ correct to 1 decimal place. [2]

9 ThelinethroughPandQmeets thex-axisath h, 0 andthey-axis atk 0, k. 9 (iii) Show thath= 5 andfind thevalueofk. [3] 9 [Turn over

10 10 6 Thehorizontal baseof asolidprism is anequilateral triangleofsidexcm. The sidesoftheprism are vertical. Theheight of theprism ishcm and thevolumeofthe prismis2000cm 3. (i) Expresshin terms ofxand showthat thetotal surfaceareaoftheprism,acm 2,is given by A = 3 2 x x 1. [3]

11 11 (ii) Given thatxcan vary, find thevalueofxforwhichahasastationary value. [3] (iii) Determine, showing all necessary working, the nature of this stationary value. [2] [Turn over

12 7 Acurveforwhich dy dx = 7 x2 6x passes through thepoint 3, (i) Find the equation of the curve. [3]

13 13 (ii) Express 7 x 2 6xin theforma x +b 2,whereaandbare constants. [2] (iii) Findtheset of valuesofxforwhich thegradient ofthecurveis positive. [3] [Turn over

14 14 8 X A M 12cm B 10cm O Inthediagram,OAXBis asectorofacirclewith centreoand radius 10cm. Thelengthofthechord AB is 12cm. The line OX passes through M, the mid-point of AB, and OX is perpendicular to AB. TheshadedregionisboundedbythechordABandbythearcofacirclewithcentreX andradiusxa. (i) Show that angle AXB is radians, correct to 3 decimal places. [3] (ii) Find the perimeter of the shaded region. [3]

15 15 (iii) Find the area of the shaded region. [3] [Turn over

16 16 9 Thefunctionfis definedby f : x 2 3 2x forx, x 3 2. (i) Findan expressionforf 1 x. [3]

17 17 Thefunctiongis defined byg:x 4x +a forx,whereais aconstant. (ii) Findthevalueofaforwhich gf 1 = 3. [3] (iii) Findthepossiblevalues ofagiventhat theequation f 1 x = g 1 x has twoequal roots. [4] [Turn over

18 18 10 y y = 4 5 3x Thediagram shows part ofthecurvey= O x. x (i) Find the equation of the normal to the curve at the point where x = 1 in the form y = mx +c, where m and c are constants. [5]

19 Theshaded region is boundedby thecurve,thecoordinateaxesand thelinex=1. 19 (ii) Find, showing all necessary working, the volume obtained when this shaded region is rotated through 360 about thex-axis. [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 and Advanced Level

Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level CANDIDATE NAME *9022343494* CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/12 Paper 1 Pure Mathematics