catiiolic junior collegt' 2oo8 jci promotional examinations

8/14/2019 Catiiolic Junior Collegt' 2oo8 Jci Promotional Examinations

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CATIIOLIC JUNIOR COLLEGT'2OO8 JCI PROMOTIONAL EXAMINATIONSHIGHER 2 MATIIBMATICS

7 Octot er 2008ADDITIONAL MATERIALS : List of Formulae (MFl5)

0800 hr - ll00 hr

INSTRUCTIONS TO CANDIDATESWrite your name and class on all the u,ork you hand in.wrile in dark blue or black pen or both sides offhe papcr-You may usc a soft pencil for any diagrams or graphs.Do not usc staples, pape. clips, highlighters, gluc o. concction fluid.Ansrver AII lhe questions.Give non-exacf nume.ical answers corrct 10 I significant figures, or I decimal place in the caseof anglcs in degres, unless a diflerent levcl ofaccuracy is spccificd in the questiooYou a.e expected to usc a graphic calculatorUnsuppoded arswers lrom a graphic calculator arc aLlowed unless a question specifically statosotherwise. Wherc unsuppoded answeN fro,r a gmphic calculator are nol allowed ir a qur\ti,rn,you are required to prescnt thc mathcmatical steps usiog orathematical notations ard notcalculalor coftmandsYou arc rerninded olthe need for clear presentation in your a swers.At the end of the exard[ation, arrange your answrs iu NUMERICAI, ORDER. Attachthis covcr shcet to the fro t ofyour atrswer scrhf.

1.

Can do the 11hole question. ,/Can do part of ques tion only. ./.

This quertrun papcr consi"ts of 5 pnnted pages


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1 CilcnZrl=:( +I)l2n+I), evaluate I(r-3rxr+l) lvhere,' is a constart, leaving-()your answer in tenlrs of n.Th.rumol fust ,r remrr of x \erie: is S, 4'r -n t ln-1.22Find an expression, in thc simplest tbrm, fbr the ,?th term and hence show that the serLes rs

t3l

tslfhe range ofvalues ofr for

t4lt2)

t3t

the sum ofan arithmtic progression and a geometric progression- I4l3- A certain thrce digit numberMexcceds fourteel times th sum ofits digits by 6. The digitin the tens placc is two more than thc sum ofthe other hvo digits. If its digits are rcvcrscd,the resulting numbcr exceeds Mby 297. Find M.

I

4 til Frpand ,.r' _^ un ru anJ in'luJinf llre lcnn rn \ . srrrin!which this expansion is valid-(ii) Hence, by subsiituting r = 1, shorv that '5 = #5. (D

Using a graphical mcthod, solvc thc ancquality l2-r1lncquatity lz16 --rrl > 4 - 3-tr, r + 0.

0r)

Sketch the curvc defincd parametrically by the equationsx=tt+t, y=2t I, l


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I

9

In this question, learc yolc answen in thcir simplestlotm(a) Diffcrertiatc each ofthe following u,ith respect 10 r:(i) ;r'rr(sc'cz:r)(ii) tan ''f- l, r>o1b) Frnd d v ,n term. ol , and, if ry ',.{(r)ir.dr

'r 4r +RThe curvr a has equatLon I I - - lt is given that Chas a verlica{ as}mptote atx+qr=2.(i) Dte.mine the value of4.(ii) Find thc equation ofthe other zrslrDptol ofC.

in tcnns of a and Ir,2l*o'ft*"")rvhere a is a positive constant alrd .; + 1.Dcducc trrxr In-. ilrrf . icivcnrhar v c 5,n, ih.\'*'H-{: : ')

l2lt2l13t

(iii) Draw a sketch ofC,labelling clearly any axial intercepts, aslmptotes and stationarypoints. tll(iv) Dcduce the nuube.ofreal roots ofthe equation r.r -4Jrl8=2x) -7r+(t. t2l10. By considcring &,, r/, r , where ,,,, - I . 6r,1

ttltrl

t4lpl

By diflrentiating fhis rcsult further, find the Maclaurin's series for y in ascending powersof r up to and including the tenn in rr-Verify the coffectness ofth series found by using the standard scrics cxpansion fb. s' andsin.r. t71

1 2. thc r ' lcrm ofc scqucnce rs tsir en bl r, . ^ -- 2 lor r 1. 2, 1. ...' (2/ IX]" + l)(i) Write down the first thrce tenns ofthe sequence, and helce shte fhe values ofLu, tot n = l, : and J.

{ir) Vat e n conte. tur< lbr a l"nnuld lbr ll, inrhefornroll-fr/r).\thcrc /tnl rs afunction in terms ofr. Prove the cory''elnre by induction.

t2l

t6l


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t5.

10 cm

A closed container is constructed using a sheet of metalwith area 100 z cm2. The containercomprises 2 shapes, a cone and a cylinder. The slant height, 1, ofthe cone is l0 cm_ Giventhat the cone and the cylindcr share the same height r, and radius r.100 /'? r0/{a) \ho$ lhar /r - tr ltl(b) find the exact radil8 when the volumc of the conlainea is maximum. t6]lVolumeofr-one-lrr'lr r urve.lSurtn.cAr.eol ( one ?]"/l'l(c) The diagram below shows the graph of I = 1(r) with stationary pornt (0, ll) and threeasJirnptotes andf=r,wherea>0andb>0.

Sketch the graph ofy = /'(r). tabelting ctearly all as],rnptote(s) and axial inrercept(s)t3;

*- End of Paper *-


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Qn/No foDic Set Answers1 Sigma Notation ,'-i$+r)(,+t)2 AP&GP r = +"lfl*;, r' t4i3 System of Linear:quations r=1, y='t, z-44 3inomial Series r t( ^ 3r') 5rrllri :=-l l+- +- +- |' ' J2 -.,, J2 | 4 r2 128./l,l.z5 rarametric Equations (') (0, -l), (0, -l). (3/1.0),(2. t)

iio (;,,6 nequalities Hence, Jr < -1 ori> I7 Functions (i) a':rrJ2 c', .'e(--,-)iio r=1. o','-!'[, l). "1B lifferentiation GXJ)

(aXu)

(bi

2-r(rta 2ri+ln{sedr))IJ"' tdt =- t9 3.aphing Techniques i) q='2 (;t ,=\ 2iv) 2 real rcots

10 \4eihod of Difference ii-d;-=;, I +dN 211 Vlaclaurin's Series e'sirrr=r+r?+1-rt312 \,4athematical lnduction

" +'1 ,','f'1i'if' ;ur i, =t- I= )n +l14 lntegration 1u11;1 l*-L1',yI L 4+t-o, {1,' * s}- *a-']

',1' "1,;,""(;).'.15 Application oflifferentiation :b) r = l!"*

catiiolic junior collegt' 2oo8 jci promotional examinations

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