Cambridge International Examinations Cambridge International General Certificate of Secondary Education

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1 Cambridge International Examinations Cambridge International General Certificate of Secondary Education * * ADDITIONAL MATHEMATICS 0606/22 Paper 2 October/November hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. 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. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. This document consists of 16 printed pages. DC (SLM) UCLES 2015 [Turn over

2 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation Fortheequationax 2 +bx+c=0, b b ac x = 4 2 a 2 Binomial Theorem (a+b) n =a n +( n 1 ) an 1 b+( n 2 ) an 2 b 2 + +( n r ) an r b r + +b n, wherenisapositiveintegerand( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A+cos 2 A=1 sec 2 A=1+tan 2 A cosec 2 A=1+cot 2 A Formulae for ABC a sina = b sinb = c sinc a 2 =b 2 +c 2 2bccosA = 1 2 bcsina

3 Itisgiventhat f( x) = 4x - 4x - 15x (i) Showthat x + 2isafactorof f ( x). [1] (ii) Hencefactorise f( x) completelyandsolvetheequation f ( x) = 0. [4] [Turn over

4 4 2 (i) Find,inthesimplestform,thefirst3termsoftheexpansionof ( 2-3x) 6,inascendingpowers ofx. [3] (ii) Findthecoefficientof x 2 intheexpansionof( 1 + 2x)( 2-3x) 6. [2]

5 5 3 Relativetoanorigin O,pointsA, BandChavepositionvectors c 5 4 m, c mandc mrespectively.all distancesaremeasuredinkilometres.amandrivesataconstantspeeddirectlyfrom AtoB in20minutes. (i) Calculatethespeedinkmh 1 atwhichthemandrivesfromatob. [3] Henowdrivesdirectlyfrom BtoCatthesamespeed. (ii) Findhowlongittakeshimtodrivefrom BtoC. [3] [Turn over

6 6 2 4 (a) GiventhatA = f p and B = c m,calculate2ba. [3] (b) ThematricesCandDaregivenbyC = c m and D = c m (i) FindC 1. [2] (ii) Hencefindthematrix XsuchthatCX + D = I,whereIistheidentitymatrix. [3]

7 5 (a) Solvethefollowingequationstofindpandq. 7 q - 1 2p # 2 = 4 p # 3 q = 81 [4] (b) Solvetheequation lg( 3x - 2) + lg( x + 1) = 2 - lg 2. [5] [Turn over

8 8 6 y π π 3π 2π 2 2 x Thefigureshowspartofthegraphof y = a + b sin cx. (i) Findthevalueofeachoftheintegersa,bandc. [3] Usingyourvaluesofa,bandc find d (ii) y, [2] dx

9 9 r (iii) theequationofthenormaltothecurveat `, 2 3 j. [3] [Turn over

10 10 7 r cm 8 cm h cm 6 cm Acone,ofheight8cmandbaseradius6cm,isplacedoveracylinderofradiusrcmandheighthcm andisincontactwiththecylinderalongthecylinder supperrim.thearrangementissymmetricaland thediagramshowsaverticalcross-sectionthroughthevertexofthecone. (i) Usesimilartrianglestoexpresshintermsof r. [2] (ii) Henceshowthatthevolume,Vcm 3,ofthecylinderisgivenbyV = rr - rr. [1]

11 11 (iii) Giventhatrcanvary,findthevalueofr whichgivesastationaryvalueofv.findthisstationary valueofvintermsofπanddetermineitsnature. [6] [Turn over

12 12 8 Solutions to this question by accurate drawing will not be accepted. TwopointsAandBhavecoordinates( 3,2)and(9,8)respectively. (i) FindthecoordinatesofC, thepointwherethelineabcutsthey-axis. [3] (ii) FindthecoordinatesofD,themid-pointofAB. [1]

13 13 (iii) FindtheequationoftheperpendicularbisectorofAB. [2] TheperpendicularbisectorofABcutsthey-axisatthepointE. (iv) FindthecoordinatesofE. [1] (v) ShowthattheareaoftriangleABEisfourtimestheareaoftriangle ECD. [3] [Turn over

14 14 9 Solvethefollowingequations. (i) 4 sin 2x + 5 cos 2x = 0 for0c G x G 180c [3] 2 (ii) cot y + 3 cosec y = 3 for0c G y G 360c [5]

15 15 (iii) r 1 cos `z + j =- for0 G z G 2r radians,givingeachanswerasamultipleofπ [4] 4 2 Question 10 is printed on the next page. [Turn over

16 16 10 A particle is moving in a straight line such that its velocity, vms 1, t seconds after passing a 2t -2t fixedpointoisv = e - 6e - 1. (i) Findanexpressionforthedisplacement,sm,fromOoftheparticle aftert seconds. [3] (ii) Usingthesubstitutionu = e 2t, orotherwise,findthetimewhentheparticleisatrest. [3] (iii) Findtheaccelerationatthistime. [2] Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonableefforthasbeen madebythepublisher(ucles)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwillbepleasedtomakeamendsat theearliestpossibleopportunity. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedonlineintheCambridgeInternationalExaminations CopyrightAcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadatwww.cie.org.ukaftertheliveexaminationseries. CambridgeInternationalExaminationsispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminations Syndicate(UCLES),whichisitselfadepartmentoftheUniversityofCambridge.

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *0835058084* ADDITIONAL MATHEMATICS 0606/11 Paper 1 October/November 2012 2 hours Candidates