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 9709/13 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_13/RP UCLES2017 [Turn over

2 2 1 An arithmetic progression has first term 12 and common difference 6. The sum of the firstnterms exceeds Calculate the least possible value of n. [4]

3 3 2 Find the set of values of a for which the curve y = 2 and the straight line y = ax +3a meet at two x distinct points. [4] [Turn over

4 (i) Findtheterm independent ofxintheexpansion of x 3x. [2] (ii) Findthevalueofaforwhich there isno termindependent ofxin theexpansionof 1 +ax 2 2 x 3x 6. [3]

5 5 4 The function f is such that f x = 2x x for 2 1 < x < k, wherekis aconstant. Find the largest valueofkfor whichfisadecreasingfunction. [5] [Turn over

6 5 (i) Show thattheequation 6 cos +4 sin +1 +5sin 5 = 0may beexpressedas5cos2 cos 4 = 0. [3]

7 (ii) Hencesolvethe equation cos +4 sin sin 5 = 0for [4] [Turn over

8 8 6 Thefunctions fand g aredefined by 2 f x = x 2 1 forx< 1, g x = x 2 +1 forx>0. (i) Findan expressionforf 1 x. [3]

9 9 (ii) Solvetheequation gf x = 5. [4] [Turn over

10 10 7 A 5 B 3 D P Q C The diagram shows a rectangle ABCD in which AB = 5 units and BC = 3 units. Point P lies on DC andapisanarcofacirclewithcentreb. PointQliesonDC andaqisanarcofacirclewithcentre D. (i) Show that angle ABP = radians, correct to 4 decimal places. [1] (ii) Calculate the areas of the sectors BAP and DAQ. [3]

11 11 (iii) Calculate the area of the shaded region. [3] [Turn over

12 12 8 y A y = 3 2x O B y = 4 3 x x Thediagram shows parts ofthegraphsofy=3 2x andy=4 3 xintersectingat pointsaandb. (i) Find by calculation the x-coordinates of A and B. [3]

13 13 (ii) Find, showing all necessary working, the area of the shaded region. [5] [Turn over

14 9 Relativeto anorigino,theposition vectors ofthepointsa,bandc aregiven by OA =, and OC = OB = A fourth point, D, is such that the magnitudes terms respectively of a geometric progression. (i) Findthemagnitudes AB, BC and CD are the first, second and third AB, BC and CD. [5]

15 15 (ii) Given that D is a point lying on theline throughbandc, find thetwo possibleposition vectors ofthepointd. [4] [Turn over

16 16 10 Acurvehasequationy=f x anditisgiventhatf x =ax 2 +bx,whereaandbarepositiveconstants. (i) Find, in terms ofaandb, the non-zero valueofxfor which the curve has astationary point and determine, showing all necessary working, the nature of the stationary point. [3]

17 17 (ii) Itisnowgiventhatthecurvehasastationarypointat 2, 3 andthatthegradientofthecurve atx = 1 is 9. Find f x. [6] [Turn over

18 18 11 y B 5, 2 y = x O A 1, 0 x Thediagram shows thecurvey= x and pointsa 1, 0 andb 5, 2 lyingon thecurve. (i) Findtheequation ofthelineab,givingyouranswer intheformy=mx +c. [2] (ii) Find, showing all necessary working, the equation of the tangent to the curve which is parallel to AB. [5]

19 19 (iii) Find the perpendicular distance between the line AB and the tangent parallel to AB. Give your answer correct to 2 decimal places. [3]

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.

MATHEMATICS 9709/31 Paper 3 Pure Mathematics 3 (P3) May/June 2018

Cambridge International Examinations Cambridge International Advanced Level CANDIDATE NAME *7084725728* CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/31 Paper 3 Pure Mathematics 3 (P3) May/June 2018