dunman high school, senior high promotional examination

8/14/2019 Dunman High School, Senior High Promotional Examination

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Name Reg.No. Year5( )DUNMAN HIGHPROMOTIONALHigher 2

SCHOOL, SENIOR HIGHEXAMINATION 2OO8

MATHEMATICS

Additional Materialsl

974030 September 2008

3 hoursList of Formulae (MF15)

READ THESE INSTRUCTIONS FIRSTWrite your Name, Class and lndex number on all lhe work you hand inWite in dark blue or black pen on both sides ofihe paper.You may use a soft pencilfor anv disorams orgphsDo nol use paper clips, highlighte.s, glue or coffection fluid.Answer all the queslions.Give non exact numerical answers correct to 3 siqnificant figures, or 1 decimal place in ihe case of anqles indegrees, un{ess a diffe.ent levelofaccuracy is specified in lhe queslion.You are expected lo !se a araphic calculalorUnsupported answers fro.n a graphic calculaior are allowed unless a question specifically states otherwise.Where unsupporled answers from a graphic calculator are not allowed in a question, you are required lo presentthe mathematical steps using mathematical notations and not calculator cornmandsYou are renrinded of the need for clea. presentalion in your answers.At ihe end of lhe examination, a(ach the question paper to the front of your answer sc.ipt.The numbe. of ma*s is given in brackels [] al the end of e:ch queslion or part question.

Can do the ivholc question.Can do part ofquestion only. /


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Differentiate the following with respect to -r:

rn) .-'(''/i-2"),lea\ lng your ansuers rn tbe srmplesl form

pror.e br inducrion Lha, i'r'i ' ' r 1 ,"f t"-rt'l' a rrr l,: I

t21t3l

t5l

Sketch on the same diagram" the graphs of(a) (y - l)'z +;r'? +4;r+3 = 0,indicating cleady the cnhe of this geometrical shapeintercepts,r+l(b) I =- rvherek>1.indicating clearly the asj,.rnplotes aod the axial intercepts-

Hence deduce the nurnber ofreal roots to the equationl'\rr ll '., ,4.i rr -ull:r+l )

and any axialt21

t31

Without using the calculator, solve the inequaltyHence solve the following inequalilies1;,t -18 ' 1.1-2,Yl+l,_ l8(rt -> r.

18 ^-r+l

tll

t4l

t2l

12)

Find the first tbree terms inthe expansiolof $ in u.."o,lrng powers of.r. [31./l Lr 1+),2Dedu,e that rhe equarion ofthe tangenr (o lhc curve v - ;::! ar rhe poinl trhere./l-2r9y = 2.6r + 0..6 .

;r = 0, is given byT2T


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The diagram shows the graph of _t, = (jr) . It passes thmugh the origir O and rhe pohtP(3, 0), and has arymptotes r = 2 and y =+2 .

Sketch on sepa.ate clearly labelled diagrams, the graphs of, =lf (l"Dl,i)

(i0 f(r+l)

A curve is dfined by the equation 8y'+:rr - +-t'y = - 4.(i, Frna !{ mrermsofnandy. L2Jdr

I(iil The curwe meets the Line y= -r at the poinr P. Find the equation of thenormal at point P. L4l

Find the following integrals:

{a) f------l-a'.(2r+r)(r r)(br f5 E'' d,

t3ll3l

t3lt3l


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4Functions f and g ar defind by

f :rt'>ln(t+2), 0


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ll A sequence o f real numbers r! , n, , ,rr , ... satisf,es the rccurrcncc rclationI r)r,-al r.r .lwheie a i" a r.at con.tarrr and r, I\ r"/It is given that the sequence converges for one ofthesc two cases:)6Case l: o and Cd5e ) o35(i) Use your calculator to find dle values ofj:r, Jrr, -y4 and -r5. correct to 2 decmalplaces for each of two abovc valocs of d - Ilcrcc state which value of a

causcs thc sequcnce to convcrge. t3lAs r -) 6, r" + d . Taking the a value h pad (i) thal calrses the sequcncc toconvcrgc, find tbe exacl value d. tll

Rlative to a fixed ongin O, tire points.,{, B and C havc posilion vectors giverespeotively byr=2irlj k, b'.-ti-2j+lk, c=4i+j 2k.Piod the area oftrianglc lBC, corrcct to 3 signilicant figures. l3lHence deiemrine thc lcngth ot the perpondicular from ,4 to the side '1C of thct2)

respectively, where .r is a constant.Find, in tenns ofd, the cosinc ofthe acute aogle bctween the lines 1l and /1. [2]Ileduce the value ofd such that 4 is pcrpendicular to /:!. tlllior the case where a = 2

(i0

(a)2

triaoglc.

(b) The Lines /1 and 6 have cquatious,,. llt.l;l -,, . iil,,l,tl"J l,J l'.1 ' I J

(i) Show that the position \.l-lor of the foot of the7 6.point p(3. '.0) to rho linc /r is - i r :j5)(ii) Firrd the position vector ofthe irnage p'

peryendicular from thet3l

. the rctlcctrnn nt? rn the line /r.t21


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ll (a) The rth tenn ofa progression is given by 4 =2".Find :2/ in temrs ofn.

(ii) Fhd the Ieast value of tl such ihat(i)

4, = (5 r 103 x0.4).(iii) Find the nuorber of bacteria letl immcdiarely atier 240 miD(iv) Statc, wiG justification, whether the spray is et'fectivc inbacterial growth after a long period oftime.

trlf,.fr'f"...f, >50000, where /? is

t4l

t2li2l

oootrolling thet2t

(b)no|Covt*A1e{

A chcmist is experimenting on a new typc ofanii-bacteria[ spray. In a containerof 5x los bactcri4 this spray is activated evcry 40 min ftom the stad ofthcexperiment. Each spmy instantly kills 90% ofthe bacteria p.csent at that timc.The duflbe. ofbacteria multiplies by 2 times every 20 min from the siart ofthecxperiment.(You may assume tllat wheneve. th spmy is activated, the bactcria nlultipliesfi$t befbre it gcts killed.)(i) Catculale the number of bacteria leR imrnediately atler 40 rnin, i.e.immediately aftcr the first spray has been activalc.d. L2l(ii) tlisgiventhat B rcprcsents the number o f bactcria Iefl immcdiately a[te.4O/lmir. For examplc, Br =number of bacteria Icft immediately after40min, B, =Irumber of bacteria leti irnraediately attcr 8Ornin and so on_

Show thet


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the sequence to converge-

t2 " lt)ra oftriangle IBC =8.75; Required length:5.83;l3(a)(i) I r?(r + l)(ii) I Hence. n = 8, since, is even.(b)li) | A( rhe 40'nin. bdcreria multiplie. by 4 timcsNumber ofbacteria left

=5xl0Nx4x0.l= 2x 103

(i0 Since every 40min, the bacteria is 0.4 times the p.evious number,The.efore by G.P.,

= (5 x 103X0.4 ), wherea=0.4x(5x103)1...) I For 240 mrn, n =6, thereforeI 4 =(5 ro'xo.4"r

= 2048000(iv) I Ves, because as z +o,4 +0

The oumber ofbacteria will converpe lo zero afier a long penod of lime.

dunman high school, senior high promotional examination

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