RI(JC) H2 Maths Promo 2008

RAFFLES JUNIOR COLLEGE JC1 Promotion Examination 2008 MATHEMATICS 9740 Higher 2 September 2008 3 hours Additional mate.ials : Answer Paper List of Formulae ([,4F15) READ IHESE INSTRUCTIONS FIRST Write your name, exam number and CT group on all the work you hand in. Write in dark blue or black pen on boih sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use slaples, paper clips, highlighters, g{ue or correction fluid. Answer all the questions. Give non exact numerical answers correct to 3 significant figltes, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question You are expected to use a oraphic calculator. Unslrpported answers from a graphic calculator are allowed unless a question specifically Where unsupported answers from a graphic calcu{ator are not allowed in a question. you are requked to present the mathematical steps using mathematical notations and not calculator You arc reminded of the need for clear presentation in your answers At the end of the examinaiion, fasten all your work securely iogether- The number of marks is qiven in brackets [] at the end of each question or part queslion Can do the 11 hole question. Can do part ofqucstio[ only. /. o RJC tES 2008

,y'' Find {ir gll )- ct J3'dr. Hence find Jr3'.r d.,c. The functiors f and g are defined by [::r-+:] : , x€R, r+ 2, g.,"+-x' , r€R,.r<d. (i) Sketch the graph of f. (ii) Fiod l-1 , the inve6e tunction of f. (iii) Fiod the largest value ofd so that the composite function lg exists. Two concentric circles have radii R and ,", where R > r. R increases at a constant rate of I cm per minutc and the arca between the lwo circl€s remdirs consLlnt at 25cm?- At the instad when R = 5 cm. find the exact rate of increase of (i) the area of the srnall€rcircle, (ii) r. State precis€ly a sequence of 2 8€om€tric transformations on the graph would .esult io the graph of), = g(4 r.). Ill tU t3l of y = g(x) that I2l The graph of-y = g(Jr) has asFptotes t =2at:dt=4 n. It aLso has a maximum potnl whose Jc coordinate is 3. Given that g(jr) : g(4 - r) for all real values ofr in the dornain of g' sketch one possibl€ graph ofy = g(ir)- fYour sketch shoutd demonstrote .karu all the appropriate features of the grcph tlldt lhov that the given conditions are satisfed.l t3l (N) Find J', ller dr. (b) Find Jj.cos(") dr. tlt Ill t3t I2l 12t t3t t2t (ct U.erhesubstrurio" " '^il ro rrnd lhe exacl *rr.ot Jt-f.a, . t{2 MA 9740/5/2008 RJC tC I Pomotion Exad I4l

l A sequence ofreal numbers Jrr,J.r,Jrr,... satisfies the recurrence relation Prove algebraically that, if th€ sequence converges, then it converges to eith€r Derenrine rhe rdnge of rhe tuncrion ,,r,- rF-" .;^ l. r . r 2. (u (i0 I or 2. [21 IU (iii) 'l hc diagram belolv shows a porrion of thegmpn.,., =,/{l]l r I, for,.:0.-\t ) When rr = 1.001, by considering x,,,, -.r, ;r.., > x. fbr all integen a ) l. The above diagram shows the points l(0, coordinates (r, 0), where x > 0. t,et f(Jc) b€ the Show that and usrrrg the graph grcn rbo\e. sho$ that I2l It value of -r that minimizes Ist l) and a(5,4). P is a variable poinl widl length ofthe path IPB (t.e. AP + PB). qa= J7*o* By diff€rentiation and without using a graphic calculator, find the f(r)- Hence find the minimum leogth ofthe path,{PB. 1",r rl ^ -t (5 i)']+lb. H2 MA 97,14/s'?008 RJC JCI P6molio. Axm ITurn Ovcr

4 In arr experiment, glucose is infused into the bloodsfeam at a constant rate of B mglmin. Glucose is also conveated and removed from the bloodstream at a rat€ proporiional to the amount of glucose, r mg, present in the bloodstrean al time I minutes aficr thc start of thc experiment. Ifr = 150, the amount ofglucose in the bloodstream remains constant. Show ihat 9{= l,rrn,.,'ril I 50 Find.n in tefins of-B and , given that the bloodstream contaifls 70mg of glucose wheo I = 0. t4l I2l The diagram shows the graphs of yr = 6 - x and Ju = -n. (i) State the coodinates oftbeir points ofirtersection. (ii) Find the area ofthe finite region bounded by the 2 graphs. tU t3t (iii) The region R is bound€d bythe line,y=-Jr,apa(of lhe curve -yr =6 -x and the t-axis, as shown io the diagram. Find the volume ofthc soiid fomcd when R is rotated through 2z radians about the I axis. giving your answer co rrect to two dcc imal places- 13t tl2 MA 9740/512008 RtC JC I Pmmotio. Eram

l0 (i) Without using a calculator, solve exact form. 5 422the rnequalrly / r , lea\rng your ans\rer _ir _r+ I A * B wh€re I and B are constants 2r +I 2r+3 t41 t3t(ii) Hmce. solve the inequality /'4e' -?L.1*r,;ng yo* unswer in exact form e'rl ll The n'h term of a gcometric progression is i' l(r + l)' , whe.e jr + I, r.+0. (i) Determine the range of values ofJ. for which t the sum to infrnity exists. (ii) civen Jr = l, find J (a) the exact value ofs, (b) the least value ofn for which lS, -Sl is tess ttran l0{, where 3, is the rhe fi rsr'l tcmri ol- th. progres.ron. 13l to be Izt ,/ tz c'renntat y e'f .^ 1. ,ho$ rh.rl ot' ,)11 r '!l- u o lJ {i} LxDress ' - In the lorD(2r +1)(2r +t) dete.mined. (ii) Hence find the surr ofthe serics F ind the Mac [au.in's serie s for ], up to and including the term in jrr . Hence find the expansion for " t'-G up to afil including the tenn in .xJ 12l sun1 of I2t t6l t|l t4l 12t 3.s s.] 7.9 9.ll 11.13 (2n+l).(7n+3) giving your answer in the form f (z), where I is a constant. tllriii) Henc€ find the lum of , ).7.9 2.9.I I 2.l l.lJ H2 MA 9740/3/2008 R'C JCI Pnmtion Exam lTuro Over

A(o.2) c(r,-2) Th€ diagram shows the graphof/ = f(n). The graph passes through the points ,.{(0,2), A (2,0) and C (3, -2)- The equations ofthe ast mptotes are) = 0 andr: a. (i) Sketch, on separate clearly tabeled diagrams, the graphs of (a) ./= f'('), (h, u=-L.r(a Ill 13t (iD G) r=-.f(r. Copy the above sketch ofl = f(.r) and curve on the sarne diagram, state the [f1;r;]'--,r+r = o. 13l by dmwing a sketch of anothcr suitable number of real roots ot the equation t2t 15 A sqluence ofreal numbers vr,r,,vr,._. is defined by the following equation v,. '_t( t) '. , n= Calculate the values of v, for n=1,2,3and4. Conjecture a formula for v" in terms of/l and induction END OF PAPER (0 (iD n>1. I2l prove your conjecture by mathematical tst H2 MA 9740/5/2008 RJC tC I Pomodon Exam [Turn Ovcr

Rafflcs Junior College Il2 I{athematics Ycar s (2008) 2008.IC1 H2 Mathematics Promotion Examination Solutions l(i) (i0 -L (1') + c (ii0 2(i\ Note that the.arge ofl - R\ {l} (iD IL '-1i'! "P r:, tl rcL?,r+L (iii) "= J1 3(,) Hcnce, required .ate = l0z crn'?/ ;n. C (ii) rcqdircd ratc = F "'t,",. 4 BITHER l- 'I rans late the graph 4 unils in the negat ive r.-d;r€ct ion. lhcn I Reflect the Araptt about/in they-axis OR L Reflect the grap,r abou/in the'-axis. then IL Trans late thc graph 4 uni ts in thc posilive r dir€ction. Graph of! = qG): Any grzph that l. has as).mptotes r = 2,] = 4 -r ard a maximu poi (3,10), t0 < I 2. is s),mmetric about the iine jr = 2- 3. llas asl,mptotes I = r, (with all 3 aslanptoles intelsccling at (2, 2).) 4. lns a maximum poinl (l,yo),yo < i. s(a) i.13 (to 3 s.1) (Lom GC) (b) t? ) Hl N,l^ 9740/5/2008 RrC xr I Pronolio. Llxrm lTurn Ovcr

(c) r,'1 2 6(0 l: I or2. (ii) The range ofthe linction isu,2l. 7 Hence (J.) is minimum when r : l5l?, and on substituting jr gives f(t) = .,fi4 or8.60 (to 3.s.t 8 .r = 150 -80c e(D Coordinates of the points of intersection are (-3, 3) and (2, 2). (ii) 125 un(s' 6 or 20 8 unrts2 1to 3 s 11 (ii) Voluore ofR= 10.36 umtsr (o 2 d.p.) l0(0 0-r< j or r>l 1 (ii) -r<0 ", ,',-'l:;) lr(i) r" -r(-r+l);-1.62<r<0.618, r+ 1,0. (roundcd offto 3 s.t) (ii)(a) 6 - or0 24; 25 l0 t9 (ii)rb) Leastn=ll l2 )('.]'.' j"'.$'').,.-n =(" t3(i) ,t=!t s= !2) (ii) L rr I ) 6 2l2'+1, (iiD I 28 H2 MA9740/5/2008 tuC JCI Pooolion Exam lTurn Over

1a(a)(i) (ii) (iii) H' MA 9r4O/S /OO8 tu( l.l Pm'nonon Fr.m lTuro Orer

10 (D) tf().)12 -i+1=0 - f(i) = tG-l There are 3 rcal roots to the equation. (Since there are 3 points ofirtersection be(veen gmphs of t =