General Certificate of Education Ordinary Level ADDITIONAL MATHEMATICS 4037/12

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1 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level * * ADDITIONAL MATHEMATICS 4037/12 Paper 1 May/June hours Candidates answer on the Question Paper. No Additional Materials are required. 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 a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, 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 17 printed pages and 3 blank pages. DC (NF/SW) UCLES 2013 [Turn over

2 2 Mathematical mulae 1. ALGEBRA Quadratic Equation theequationax 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 mulae for ABC a sina = b sinb = c sinc a 2 =b 2 +c 2 2bccosA = 1 2 bcsina

3 3 1 A B TheVenndiagramshowstheuniversalset,thesetAandthesetB.Giventhatn(B )=5,n(A )=10 andn( )=26,find (i) n( A+ B), [1] (ii) n(a), [1] (iii) n( Bl + A). [1] [Turn over

4 4 2 A4-digitnumberistobeformedfromthedigits1,2,5,7,8and9.Eachdigitmayonlybeused once.findthenumberofdifferent4-digitnumbersthatcanbeformedif (i) therearenorestrictions, [1] (ii) the4-digitnumbersaredivisibleby5, [2] (iii) the4-digitnumbersaredivisibleby5andaregreaterthan7000. [2]

5 5 3 Showthat(1 cos θ sinθ ) 2 2(1 sinθ )(1 cosθ )=0. [3] [Turn over

6 6 4 Findthesetofvaluesofkforwhichthecurvey=2x 2 +kx+2k 6liesabovethex-axisfor allvaluesofx. [4]

7 7 5 Theline3x+4y=15cutsthecurve2xy=9atthepointsAandB.Findthelengthofthe lineab. [6] [Turn over

8 8 6 Thenormaltothecurvey+2=3tanx,atthepointonthecurvewherex= 4 3r,cutsthe y-axisatthepointp.findthecoordinatesofp. [6]

9 9 7 Itisgiventhatf(x)=6x 3 5x 2 +ax+bhasafactorofx+2andleavesaremainderof27 whendividedbyx 1. (i) Showthatb=40andfindthevalueofa. [4] (ii) Showthatf(x)=(x+2)(px 2 +qx+r),wherep,qandrareintegerstobefound. [2] (iii) Hencesolvef(x)=0. [2] [Turn over

10 10 8 (a) GiventhatthematrixA= 4 2 c m,find 3 5 (i) A 2, [2] (ii) 3A+4I,whereIistheidentitymatrix. [2]

11 (b) (i) 11 6 Findtheinversematrixof c m. [2] (ii) Hencesolvetheequations 6x+y=5, 9x+3y= 2 3. [3] [Turn over

12 12 9 (i) Given that n is a positive integer, find the first 3 terms in the expansion ofc1+ xm in 2 ascendingpowersofx. [2] 1 n (ii) Giventhatthecoefficientofx 2 25 intheexpansionof(1 x) c1+ xm is,findthevalue 2 4 ofn. [5] 1 n

13 13 10 (a) (i) Findy 2x-5dx. [2] 15 (ii) Henceevaluatey 2x-5dx. [2] 3 [Turn over

14 (b) (i) 14 d Find ^ ln d x x 3 x h. [2] 2 (ii) Hencefindy x ln xdx. [3]

15 15 11 (a) Solvecos2x+2sec2x+3=0for0cG xg360c. [5] 2 r (b) Solve2sin `y- j=1for0g ygr. [4] 6 [Turn over

16 16 12 AparticlePmovesinastraightlinesuchthat,tsafterleavingapointO,itsvelocityvms 1 is givenbyv=36t 3t 2 fort H0. (i) FindthevalueoftwhenthevelocityofPstopsincreasing. [2] (ii) FindthevalueoftwhenPcomestoinstantaneousrest. [2] (iii) FindthedistanceofPfromOwhenPisatinstantaneousrest. [3]

17 17 (iv) FindthespeedofPwhenP isagainato. [4]

18 18 BLANK PAGE

19 19 BLANK PAGE

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. University of 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.

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