c4 SrSl.(a) Find thebinomial expansion of1+ + sr;i, 1"1 a 15in ascendingpowers of_r, upto and including the termin,r2.Give eachcoefficientin itssimplestform.(b) Find rhe exacrralueol (4 - 5jr) *h.n ,.-- |l0Giveyour answer in the formtrD, whereft is a constantto be determined.(1)(c) Substitute ,, = I ,n,o your binomialexpansionfrom part (a) and hencefind anapproximare valuc lor rEGireyouranswerin rhef norm ! wherep andq are inlcgers.(2)2.ThecurveChasequationx2-3xy-4y2+64=0(a) Find !l in ,..r, of x and y.dx (s)(b) Find the coordinatesof the points on C where I = 0'dx(Solutionsbasedentirely ongraphical or numericalmethodsare not acceptable.)LyL+64 =ox-L-l' -\6rl.t-=Q-z\3.Figure 1Figure1shows a sketch of partof the curvewithequation ! = 4x-,"1',,r- OThecurvemeetsthex-axis at the origin O and cutsthe x-axisat thepoint,4.(a) Find,in termsof 1lr.2,the x coordinateof the point l.(b) Find rrJ xe7'dx(3)The finiteregionR, shown shaded in Figure1,is boundedby thex-axisandthecurve withequationl!=4x-ys1',x)0(c) Find,by integration, the exactvalue forthe areaof R.Give youranswerin termsof ln2(3)(2\a.)r ltd* fuv' = uv --fu'v 7 (A:)L y = 2e'l, ^ ,-. r (;;=t v,""t^|zr-&\ - I ze**a- vr -Jr,z*il* - +;t^ *-( zt*z*^i. )^4lrr2-c) (= j;;-*.*"dr- = [i*: -r*]n*4"r^"J:'ol;il[- = ( :zt "..z)' - 8 \n z Jn n 4J^k 1- (o-o= 32-(tnz)"- 32\.'.2 + lg-t:iII+4)lil= Sz(tnz)" - 32\y,.2 + \z---2_4. Wth respect to a fixed origin e the lines l, and l. aregiven by the equations,,,, lril .,1 i) 1.:r._l il .,1 ;)I il (_,/ :' l_il -l_;)where ) and p are scalar parameters andp is a constant.The lines /, and /, intersec t at the point A.(a) Find the coordinates ofl.(b) Find the value ofthe constantp.tt' i,T.li* acute angle between t, and t,, g'in(3)g your answer in degrees to 2 decimalThe point .Blies on /, where trz : , ('),0, l:1-,,1: :n.onesr disrance tiom rhe poinr B to the lir J slgnificanl figures. -"' Lrr! PU'IIro to me lrne /,. giving your answer toQ)(3)a) Fr=rr. " frt*)= (fj-ti) o s,B+3A -, r^tr'.: A( e-i, i-*r-z*s) -'. *(srtre)b) @ -3r\=5t9(-r) a ii+tr=l :-}.=k---2G g-lt*) = -z-s(-r) :) P-\z = -2+S .'- pls-,_9.3!Ee"'z-d) e(iiil s(r )ts(rr,t,-t) o"' 2,_ tq,li6 {soo or=/[i).(i) | =, (o,g =rerit-ut \er ;[= (+] (i)- L?").. tfi t=F'-r-lo''\-zoo-'- shorte.ototrrfanq -- J6o r Srn3t-Ez' ?.$ortt25. A cr.rrve C hasparametricequationsx=4r+3, y=4t+a+1, t+02t'(a) Find thevalueol !I ut the point onC wheret :2, givngyou.ransweras a fractionoirin itssimplestform.(3)(b) Show thatthe cartesian equationof the curve C can be written in the formx2 +ax+b, x+3x-3where a and b are integersto bedetermined.(3)6-Diagramnot to scaleFigure2 shows a sketchof the curue with equation f, = ]@:iX, + gThefinite regionR,shownshadedin Figure2, is bounded by the curve,and the 7-axis.(a) Use the substitution:r: 1+ 2sin6 to showthatIt:= klcos'?deI-t6Figure2wheref is a constantto be determined.(b) Hence find, by integration, the exact area ofR., o(x(the x-axis,(s)(3 - x)(x + 1)d"r(3), a.-=l @r2(orolg 3-L = 2-?-Srr9 ,2(r-aE t cr-r\ = 2+ZSrnO --Z(lln3It-__c^) lJtg-2cr6ar lcs t+23'aQJo abc- - 2 Cosgdx=L+ ec'n1GI^L= 4 1 rl l-Srn29 x (ob JO?L=3 t t2srng --3-qZS,ng -2"s-' +J'icosze'*('0 ,tg-[b2SrnO = -\,h\'r/ Costg = ZG>20 -\ --) t(ozo *LL = Corg'tt.-> flL'orc = z$-'o+\ a0 '2 fts'^*-,u]:EEfrTr= Lxlx f S,nzO *rrJ_'u^ , IS,^2O . *]*6T=(o*tt) (-g-g) ,*E7, (a) Express =- in partialfractions.P(P _ 2)(3)A team ofbiologists is studying apopulation ofa particular species of animal.The population is modelled by the differential equationd,Pli=rptp-2)cos2t, r)0Xrl,iTI_:J* poputation in thousands, andr is the time measurecl in years sincethe starrGiventhatp = 3 whent:0,(b) solve thisdifferential equation to showthatD_ 6I^J - ";stnzr(c) find the time taken forthepopuration to reach 4000for the first time.Give your answer in y"urc to1 significant n*"r. " ^-a.:. ( = -ln333lSm2tlr?5'nzt =9 e4L-- or-/3\'-e-'-' L j,n2t=Z\,^(a)t,O.4+3-)8.Diagramnot to scaleF'igure 3Figure 3 shows a sketch of partof th6 curveC withequationv:3',The point P lies onC andhas coordinates(2,9).Theline/ is a tangent to C at P. Theline/ cutsther-axisat thepoint Q.(a) Findthe exactvalue of the * coordinateof Q.(4)Thefinite regionR,shownshaded in Figure3, is bounded by the curveC,the.r-axis,the y-axisand the linei. This regionR is rotated through 360oabout the x-axis.(b) Use integration to find the exact valueof the volume ofthesolidgenerated.Give youranswerin the form -L wherep andq areexact constants.q'lYou may assumethe formula V = ! w2h for thevolumeoJ a cone.l3 (6)S-q --ql^".(:, -L) arH S ls oB trT2ln3 JO--Ln.j J63 hA3