Mathematics1HLPaper1spa.docx

1、Mathematics1HLPaper1spaIB DIPLOMA PROGRAMME PROGRAMME DU DIPLME DU BI PROGRAMA DEL DIPLOMA DEL BIMATHEMATICS HIGHER LEVEL PAPER 1Wednesday 3 November 2004 (afternoon)2 hoursN04/5/MATHL/HP1/ENG/TZ0/XX88047401School codeCandidate codeINSTRUCTIONS TO CANDIDATESWrite your school code and candidate code

2、in the boxesabove.Do not open this examination paper until instructed to doso.Answer all the questions in the spacesprovided.Unless otherwise stated in the question, all numerical answers must be given exactly or to three significantfigures.Indicate the make and model of your calculator in the appro

3、priate box on your coversheet.Maximum marks will be given for correct answers. Where an answer is wrong, some marks may be given for correct method, provided this is shown by written working. Working may be continued below the box, if necessary. Solutions found from a graphic display calculator shou

4、ld be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer.1.Considerf (x) =x3 - 2x2 - 5x +k . Find the value of k if (x + 2) is a factor off (x) .Working:Answer:1 -1 22.Given that thematrixA =2 p3is singular, find the value of p.1

5、 -2 5Working:Answer:3.The sum of the first n terms of a series is givenbynS =2n2 -n , where n+.(a)Find the first three terms of theseries.(b)Find an expression for the nth term of the series, giving your answer in terms ofn.Working:Answers:(a) (b) 4.Giventhat(a+i)(2-bi)=7-i, find the value ofa and o

6、fb,wherea,b.Working:Answer:5.Ify=ln(2x-1),findd2y.dx2Working:Answer:6.Afairsix-sideddie,withsidesnumbered1,1,2,3,4,5isthrown. Find the mean and variance of thescore.Working:Answer:7. (a) Find the largest set S of values of x such that the function values.(b) Find the range of the function f defined

7、on the domainS.f(x)=1takes realWorking:Answers:(a) (b) 8.(a) Find the expansion of (2 +x)5 , giving your answer in ascending powers ofx.(b) By letting x = 0.01 or otherwise, find the exact value of 2.015.Working:Answers:(a) (b) 9.BFind the area of the shaded region.Working:Answer:10.Considerthe equa

8、tione-x=cos2x,for0x2.(a)How many solutions are there to thisequation (b)Find the solution closest to 2, giving your answer to four decimalplaces.Working:Answers:(a) (b) 11.Consider thefourpointsA(1,4,-1),B(2,5,-2),C(5,6,3)andD(8,8,4).Findthepointofintersection of the lines (AB) and (CD).Working:Answ

9、er:12.A continuous random variable X has probability density function givenbyf (x) =k (2x-x2), for 0 x 2f (x)=0, elsewhere.(a)Find the value ofk.(b) Find P(0.25 x 0.5).Working:Answers:(a) (b) 13.Giventhatz % , solve theequationz3 - 8i = 0 , giving your answers in the formz =r (cos+ isin) .Working:An

10、swer:14.Findthe total area ofthe two regions enclosed bythe curvey=x3-3x2-9x+27and the liney =x + 3 .Working:Answer:15.Find the range of values of m such that for allxm (x +1) x2 .Working:Answer:16.Find the equation of the normal to thecurvex3 +y3 - 9xy = 0at the point (2, 4).17.Usingthe substitutio

11、n 2x=sin, or otherwise, find(1-4x2)dx.18.A closed cylindrical can has a volume of 500 cm3 . The height of the can is h cm and the radius of the base is rcm.(a)Find an expression for the total surface area A of the can, in terms ofr.(b)Given that there is a minimum value of A for r 0 , find this valu

12、e ofr.Working:Answers:(a) (b) 19.(a) FindthecartesianequationoftheplanethatcontainstheoriginOandthetwopointsA (1, 1, 1) and B(2, -1, 3) .(b) FindthedistancefromthepointC(10,5,1)totheplaneOAB.Working:Answers:(a) (b) 20.The following diagram shows thelinesx-2y-4=0,x+y=5and the pointP (1, 1) . A lineis drawn from P to intersect with midpoint of QR.x-2y-4=0y10at Q, and withx +y =5at R, so that P is the862 (1,1 0x2468Find the exact coordinates of Q and of R.Working:Answer: