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) May/June 2018 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 in the space provided. If additional space is required, you should use the lined page at the end of this booklet. The question number(s) must be clearly shown. 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. This document consists of 20 printed pages. JC18 06_9709_12/2RP R UCLES2018 [Turn over
2 1 Thecoefficient ofx 2 intheexpansion of x a +x 5 is330. Findthe valueoftheconstanta [5] UCLES /12/M/J/18
3 3 2 Theequationofacurveisy=x 2 6x +k,wherekis aconstant. (i) Findtheset of valuesofkforwhich thewholeofthecurvelies abovethex-axis. [2] (ii) Findthevalueofkforwhich theliney +2x = 7 is atangent to thecurve. [3] UCLES /12/M/J/18 [Turn over
4 4 3 A company producing salt from sea water changed to a new process. The amount of salt obtained eachweek increased by2% ofthe amount obtained in theprecedingweek. It is giventhat in thefirst weekafterthechangethecompany obtained 8000kg ofsalt. (i) Find the amount of salt obtained in the 12th week after the change. [3] (ii) Findthetotal amountofsaltobtained in thefirst 12 weeks afterthechange. [2] UCLES /12/M/J/18
5 5 4 Thefunctionfis such thatf x = a +bcosx for 0 x 2. It is given thatf 1 3 = 5and f = 11. (i) Findthevaluesoftheconstantsaandb. [3] (ii) Findtheset of valuesofkforwhich theequation f x = k has nosolution. [3] UCLES /12/M/J/18 [Turn over
6 6 5 C D B O A The diagram shows a three-dimensional shape. The base OAB is a horizontal triangle in which angle AOB is 90. The side OBCD is a rectangle and the side OAD lies in a vertical plane. Unit vectors i and j are parallel to OA and OB respectively and the unit vector k is vertical. The position vectorsofa,banddaregiven by OA = 8i, OB = 5jandOD = 2i +4k. (i) Express each of the vectors DA and CAin terms ofi,jand k. [2] UCLES /12/M/J/18
7 7 (ii) Use a scalar product to find angle CAD. [4] UCLES /12/M/J/18 [Turn over
8 6 O r 8 2 rad B A T The diagram shows points A and B on a circle with centre O and radius r. The tangents to the circle at A and B meet at T. The shaded region is bounded by the minor arc AB and the lines AT and BT. AngleAOBis 2 radians. (i) In the case where the area of the sectoraob is the same as the area of the shaded region, show that tan = 2. [3] UCLES /12/M/J/18
9 9 (ii) In the case where r = 8cm and the length of the minor arc AB is 19.2cm, find the area of the shaded region. [3] UCLES /12/M/J/18 [Turn over
10 10 7 Thefunctionfis definedby f : x 7 2x 2 12xforx. (i) Express 7 2x 2 12xin theforma 2 x +b 2,whereaandbareconstants. [2] (ii) Statethecoordinates ofthe stationary point on thecurvey=f x. [1] UCLES /12/M/J/18
11 11 Thefunctiongis defined byg:x 7 2x 2 12xforx k. (iii) Statethesmallest valueofkforwhich g has aninverse. [1] (iv) Forthis valueofk,findg 1 x. [3] UCLES /12/M/J/18 [Turn over
12 12 8 PointsAandBhavecoordinates h, h and 4h +6, 5h respectively. Theequationoftheperpendicular bisectorofabis 3x +2y = k. Find thevalues oftheconstantshandk. [7] UCLES /12/M/J/18
13 UCLES /12/M/J/18 [Turn over
14 9 Acurveissuch that dy dx = 4x +1 and 2, 5 isapoint onthecurve. 14 (i) Find the equation of the curve. [4] UCLES /12/M/J/18
15 15 (ii) A point P moves along the curve in such a way that the y-coordinate is increasing at a constant rateof0.06unitspersecond. Findtherateofchangeofthex-coordinatewhenPpassesthrough 2, 5. [2] (iii) Show that d2 y dx 2 dy isconstant. [2] dx UCLES /12/M/J/18 [Turn over
16 16 10 (i) Solvetheequation 2cosx +3sinx = 0,for0 x 360. [3] UCLES /12/M/J/18
17 17 (ii) Sketch,on thesamediagram,thegraphsofy=2cosxandy= 3sinxfor0 x 360. [3] (iii) Use your answers to parts (i) and (ii) to find the set of values of x for 0 x 360 for which 2cosx +3sinx > 0. [2] UCLES /12/M/J/18 [Turn over
18 18 11 y y = x x P Q y = 4 O x The diagram shows part of the curve y = x The line y = 4 intersects the curve at the points P x andq. (i) Show thatthetangents to thecurveatpandqmeet at apointon theliney=x. [6] UCLES /12/M/J/18
19 19 (ii) Find, showing all necessary working, the volume obtained when the shaded region is rotated through 360 about thex-axis. Giveyouranswerin terms of. [6] UCLES /12/M/J/18 [Turn over
20 20 Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown 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. UCLES /12/M/J/18
Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level
Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level CANDIDATE NAME *8124433764* CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/13 Paper 1 Pure Mathematics
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