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 9709/12 Paper 1 Pure Mathematics 1 (P1) 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_12/RP UCLES2017 [Turn over

2 1 Findtheterm independent ofxin theexpansion of 2 2x 1 9 4x 2. [4]

3 2 Afunction fis defined byf:x 4 5x forx. 3 (i) Find an expression for f 1 x and find the point of intersection of the graphs of y = f x and y = f 1 x. [3] (ii) Sketch,onthesamediagram,thegraphsofy=f x andy=f 1 x,makingcleartherelationship between the graphs. [3] [Turn over

4 4 3 (a) Each year, the value of a certain rare stamp increases by 5% of its value at the beginning of the year. A collector bought the stamp for $10000 at the beginning of Find its value at the beginning of 2015 correct to the nearest $100. [2]

5 5 (b) Thesumofthefirstntermsofanarithmeticprogressionis 1 n 3n +7. Findthe1sttermandthe 2 common difference of the progression. [4] [Turn over

6 6 4 C P Q A D O 6cm B The diagram shows a semicircle with centre O and radius 6cm. The radius OC is perpendicular to thediameterab. ThepointDlies onab,anddc is anarc ofacirclewith centreb. (i) Calculate the length of the arc DC. [3]

7 (ii) Find the value of 7 areaof regionp areaof regionq, giving your answer correct to 3 significant figures. [4] [Turn over

8 5 (i) Show thattheequation cos2x tan 2 2x = 0can beexpressedas 8 2cos 2 2x +3cos2x +1 = 0. [3]

9 9 (ii) Hencesolvethe equation cos2x tan 2 2x = 0 for0 x 180. [4] [Turn over

10 6 (a) Thefunctionf,defined by f :x a +bsinx forx,is such that f 1 6 = 4 and f 1 2 = (i) Findthevaluesoftheconstantsaandb. [3] (ii) Evaluateff 0. [2]

11 11 (b) Thefunctiongisdefinedbyg:x c +dsinxforx. Therangeofgisgivenby 4 g x 10. Findthevaluesoftheconstantscandd. [3] [Turn over

12 12 7 PointsAandBlieonthecurvey=x 2 4x +7. PointAhas coordinates 4, 7 andbisthestationary point of thecurve. Theequation ofalinelisy=mx 2, wheremisaconstant. (i) Inthe casewherelpasses through themid-point ofab,findthe valueofm. [4]

13 13 (ii) Findtheset of valuesofmforwhichldoes not meet thecurve. [4] [Turn over

14 14 8 Acurveissuch that dy dx = x2 +5x 4. (i) Find the x-coordinate of each of the stationary points of the curve. [2] (ii) Obtain an expression for d2 y and hence or otherwise find the nature of each of the stationary 2 dx points. [3]

15 15 (iii) Given that thecurvepasses through thepoint 6, 2,findtheequation ofthecurve. [4] [Turn over

16 9 16 O A B C ThediagramshowsatrapeziumOABC in whichoaisparalleltocb. Theposition vectors ofaand Brelative totheoriginoaregiven by 2 OA = 2 and 6 OB = (i) Show thatangleoabis 90. [3] The magnitude of CB is threetimes themagnitudeof OA. (ii) Find the position vector of C. [3]

17 17 (iii) FindtheexactareaofthetrapeziumOABC,givingyouranswerintheforma b,whereaandb are integers. [3] [Turn over

18 18 10 y P 2, 3 y = 5x 1 O Q x The diagram shows part of the curve y = 5x 1 and the normal to the curve at the point P 2, 3. This normal meets thex-axis atq. (i) Find the equation of the normal at P. [4]

19 19 (ii) Find, showing all necessary working, the area of the shaded region. [7]

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