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) February/March 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 03_9709_12/2R UCLES2018 [Turn over

2 2 1 A curve passes through the point 4, 6 and has an equation for which dy dx = x Find the equation of the curve. [4] UCLES /12/F/M/18

3 3 2 (i) Findthecoefficients ofx 2 andx 3 in theexpansionof 1 2x 7. [3] (ii) Hencefindthecoefficientofx 3 in theexpansion of 2 +5x 1 2x 7. [2] UCLES /12/F/M/18 [Turn over

4 4 3 Onacertainday,theheightofayoungbambooplantwasfoundtobe40cm. Afterexactlyonedayits height was found to be 41.2cm. Two different models are used to predict its height exactly 60 days after it was first measured. ModelAassumesthatthedailyamount ofgrowthcontinuestobeconstantattheamountfound forthefirst day. Model B assumes that the daily percentage rate of growth continues to be constant at the percentage rate of growth found for the first day. (i) Using modela,find thepredicted height in cm of thebamboo plant exactly 60 days after it was first measured. [2] (ii) Using modelb,find thepredicted height in cm of thebamboo plant exactly 60 days after it was first measured. [3] UCLES /12/F/M/18

5 5 4 Astraightlinecutsthepositivex-axisatAandthepositivey-axisatB 0, 2. AngleBAO = 1 6 radians, whereois theorigin. (i) Find the exact value of the x-coordinate of A. [2] (ii) FindtheequationoftheperpendicularbisectorofAB,givingyouranswerintheformy=mx +c, wheremis given exactly andcis an integer. [4] UCLES /12/F/M/18 [Turn over

6 6 5 (a) Express the equation 5 +2tanx = 1 +tanx as a quadratic equation in tanx and hence solve the 3 +2tanx equationfor0 x. [4] UCLES /12/F/M/18

7 7 (b) y 2 O y = ksin The diagram shows part of the graph of y = ksin +, where k and are constants and 0 < < 180. Findthevalueof and thevalueofk. [2] UCLES /12/F/M/18 [Turn over

8 8 6 P 10cm O R 2.2 rad Q ThediagramshowsasectorPOQofacircleofradius10cmandcentreO. AnglePOQis2.2radians. QR is an arcof acircle withcentrepandporisastraight line. (i) Show thatthelengthofpqis 17.8cm,correct to 3significant figures. [2] UCLES /12/F/M/18

9 9 (ii) Find the perimeter of the shaded region. [4] UCLES /12/F/M/18 [Turn over

10 10 7 E 8 4 C 3 B 1 D 2 O 7 3 A D j O k i C B A Fig.1 Fig.2 Fig.1 shows a rectangle with sides of 7 units and 3 units from which a triangular corner has been removed,leavinga5-sidedpolygonoabcd. ThesidesOA,AB,BC anddohavelengthsof7units, 3 units, 3 units and 2 units respectively. Fig.2 shows the polygon OABCD forming the horizontal baseofapyramidinwhichthepointeis8unitsverticallyaboved. Unitvectorsi,jandkareparallel tooa,odandde respectively. (i) Find CE andthe length ofce. [3] UCLES /12/F/M/18

11 11 m n (ii) Use a scalar product to find angle ECA, giving your answer in the form cos, 1 where m and n are integers. [5] UCLES /12/F/M/18 [Turn over

12 12 8 Acurvehas equationy= 1 2 x2 4x x. (i) Find the x-coordinates of the stationary points. [5] UCLES /12/F/M/18

13 13 (ii) Find d2 y dx2. [1] (iii) Find, showing all necessary working, the nature of each stationary point. [2] UCLES /12/F/M/18 [Turn over

14 14 9 Acurvehas equationy= 1 x +c and alinehas equationy=cx 3,wherecis aconstant. (i) Findtheset of valuesofcfor whichthe curve and thelinemeet. [4] UCLES /12/F/M/18

15 15 (ii) Thelineisatangenttothecurvefortwoparticularvaluesofc. Foreachofthesevaluesfindthe x-coordinate of the point at which the tangent touches the curve. [4] UCLES /12/F/M/18 [Turn over

16 16 10 Functionsfand garedefined by f x = 8 +2 forx>2, x 2 g x = 8 +2 for2<x<4. x 2 (i) (a) State the range of the function f. [1] (b) Statetherangeofthefunction g. [1] (c) Statetherangeofthefunction fg. [1] (ii) Explain why the function gf cannot be formed. [1] UCLES /12/F/M/18

17 17 (iii) Findtheset of valuesofxsatisfying theinequality 6f x +2f 1 x 5 < 0. [6] UCLES /12/F/M/18 [Turn over

18 18 11 y B y = 1 2x O A 1 2,0 x y = 1 2x 1 2x 3 Thediagramshowspartofthecurvey=1 2x 1 2x 3 intersecting thex-axis at theoriginoand ata 1 2, 0. ThelineABintersects they-axis atband has equationy=1 2x. (i) Show thatabis the tangent to thecurveata. [4] UCLES /12/F/M/18

19 19 (ii) Show thattheareaoftheshadedregioncan beexpressedas x 3 dx. [2] (iii) Hence, showing all necessary working, find the area of the shaded region. [3] UCLES /12/F/M/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/F/M/18

MATHEMATICS 9709/33 Paper 3 Pure Mathematics 3 (P3) October/November 2017

Cambridge International Examinations Cambridge International Advanced Level CANDIDATE NAME *9388870911* CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/33 Paper 3 Pure Mathematics 3 (P3) October/November