Price per tonne of sand ($) A B C

Transcription

1 . Burkig would like to purchase cemet, iro ad sad eeded for his costructio project. He approached three suppliers A, B ad C to equire about their sellig prices for the materials. The total prices quoted are $9880, $0090 ad $060 respectively. The breakdow of uit prices for the materials is show i the table below. Supplier Price per toe of cemet ($) Price per toe of iro ($) Price per toe of sad ($) A B C Calculate the amout of each material he eeds. [3] Suppose he triples his order for cemet ad sad while keepig that for iro the same, determie the supplier he should choose to miimise cost. []. The sum of the first terms of a progressio is give by S e. (i) Fid U, the th term of the progressio. [] Prove that U, U, U3, is a geometric progressio. [] (iii) Determie if the sum S coverges. [] 3. A committee cosistig of chairperso, vice-chairperso, secretary, treasurers is to be chose from a group of 5 males ad 5 females. Fid the umber of ways to choose the committee if (i) there are o restrictios o the geder, [] the chairperso ad vice-chairperso must be of differet geder. [] The committee is to be seated aroud a roud table for oe of their discussio. Idetical chairs with labels for the differet appoitmets are arraged aroud the table. Fid the umber of ways the chairs ca be arraged. [] 4. r Prove that l r l rl( r) l( r). r [] Hece, fid i terms of, 3 4 l l 3 l 4... l 3 4 5, ( )! givig your aswer i the form l, where k is a costat to be foud. k [4] RIVER VALLEY HIGH SCHOOL 9740/0/04

2 5. Use the method of iductio to prove that 3 ( ) r. [4] 4 Hece, fid r 3 ( r ). [] r0 6. A large meetig room has oe of its walls fitted with a projectio scree of metres high. The scree is metre above the floor. Let the distace from the scree to someoe stadig directly i frot of it be x metres ad the perso s viewig agle of the scree be (see diagram). m Scree m x m Perso viewig 3 Show that ta ta. [] x x d a bx Show that where a ad b are costats to be foud. Hece fid d x ( x )( x 9) the value of x so that the viewig agle is maximum. [5] Suggest a reaso why the diagram is ot realistic. [] 7. The curve C has equatio y x. x 3 (i) State the equatios of the asymptotes of C. [] Sketch C, showig its asymptotes ad statig the coordiates of the turig poits as well as the poits of itersectio with the axes. [3] (iii) By drawig a sketch of aother suitable curve o the same diagram, show that the equatio x x 9 x 3 has o real roots. [] RIVER VALLEY HIGH SCHOOL 9740/0/04

3 3 8. A wet porous substace left i ope air loses its moisture m at a rate proportioal to the moisture cotet. If oe of such substace has moisture cotet m 0 iitially ad loses half of the moisture cotet i 3 hours, how much loger will it take for the substace to lose 80% of its origial moisture cotet? [7] 9. (i) Give that y bx for bx, fid, up to ad icludig the term i d y. Use the aswer to fid the Maclauri s series dx x. [4] cos x Fid the series expasio for six up to ad icludig the term i x, give that x is a sufficietly small agle. [4] 0. (i) Use the substitutio x ta to show that x x dx ta C. [5] (4 x ) 6 x 4 The regio R is bouded by the curve y 4, the lie y, the x-axis ad x the y-axis. Fid the exact volume of the solid formed whe R is rotated π radias about the y-axis. [4]. (a) Fid xe x dx. [3] (b) The curve C has parametric equatios ( t ), e t x y, for t. (i) Show that the equatio of the ormal to the curve at t = 0 is y x 3. [4] Fid the exact value of the area of the regio bouded by C, the lie y x 3, the x-axis ad the y-axis. [4] RIVER VALLEY HIGH SCHOOL 9740/0/04

4 4 SH. Research is beig carried out ito how the cocetratio of a drug i bloodstream varies with time, measured from the time whe the drug is give. Observatios at successive times give the data show i the followig table. Time (t miutes) Cocetratio (x micrograms per litre) (i) Sketch a scatter diagram for the data. [] Calculate the product momet correlatio coefficiet betwee x ad t. Commet o whether a liear model would be appropriate for the relatioship betwee x ad t. [] It is suggested that the relatioship betwee x ad t ca be modelled by the formula bt x ae, where a ad b are costats. (iii) For this model, show that the relatioship betwee l x ad t is liear ad calculate its product momet correlatio coefficiet. [3] (iv) Usig a suitable regressio lie, estimate, to the earest miute, the time at which the cocetratio is at 5μg/l. Give a reaso for the choice of the regressio lie ad commet o the reliability of the estimate. [4] 3. Fuctios f ad g are defied by f: x x x8, x, g: x x, x. x (i) Show that Fid f exists. [] f i similar form. [3] (iii) Show that the composite fuctio fg does ot exist. [] Fuctio h is defied as follows h: x kx, x. x (iv) By cosiderig a trasformatio of the fuctio g, or otherwise, state the rage of values of k such that the composite fuctio fh exists. [] (v) Fid the rage of fh whe k = 3. [] (vi) Fid the rage of values of x for which f(x) > g(x). [3] RIVER VALLEY HIGH SCHOOL 9740/0/04

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