meridian junior college

8/14/2019 Meridian Junior College

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MERIDIAN JUNIOR COLLEGE2OO8 JCl PROMOTIONAL EXAMINATION

MATTIEMATICSHigher 2 9740

Friday

Additional materials:

26 Septernber 20082 hours 30 rnins

WritinS paperList ofFomrulae (MF l5)

READ THESE INS'I'RUCTIONS FIRSTWritc you. name and civics group on all the work you hand in.Write ;n dark bluc or black per on both sidcs oI the paperYou rnay usc a sofl pencil for any cliagrams or graphs-Do not use stapler, paper clips. highlightcN, glue or correclio fluid-Ans$ cr all thc qucstiors.(livc noD c{ac{ nume.ical answers oorrcct to 3 sigrrificant figurcs, or I dcci al placc inthc casc of angles io degrees, unlcss a dillcrcnt level of accuracy is spccificd io thcYou are expcctcd (o usc a Sraphic calculator-Unsuppo(cd ans$,crs from a graphic calculator are allowed unless 3 qucstiospecifically stales olhcflvisc.Whcre unsuppo.led answers from a graphic calculator ar not allowed in a question,you arc rcquired to present the mrthcmatical stePs using mathematical ootations andnot calclllator commands-\'ou ar.. rL' nded ^l th, nc'rd for . Icdr pfesenrrri,)n in your air' wer"The number of marks is given in brackcts [ ] at the end of each questioo or pailquestionAt thc cnd ofthe examinalion. listcn all your tvork securly together

Can do the rvholc question.Can do parl ofquestion only. /.


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indrcatina ctearl\ rhe equarioos olc,),lnplotes and the arialketch the curve of I =ilrtercepts in exact form. t5t

prove bv rnducr;on rha, -L" -_L '-- I' (2)(s) (s)(8) (8)(rr) (rn-r)(rt,,)) 2Qn+2)Hence, ororherwr.e. euuturr" tirl f ] I" _Lz_(lr_t)(v I 2) lExpand

-L .,. 16.! a 1 7,l-3-tin ascending powers of n , up to ard including the teml in nz .Dctcrmine the range ofvalues of r for which the expansion is valid.

By e\pressrng +", I rn panrat rracrion,. rrnd }t:L:-U,,.-'

t51

t5ltll

t6l

J' (i) Express lr':+-r+1 in the form ,4(r+B)'?+C, wherc A, B and C are constants to bedetennined. tll(ii) Solve ".#using a non-calculalor method.-"ttll) Hence, solve x'!-.-l-r'+ I

l3ll2l


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is 2 andz(a)3

The sum to inftnity of a geometric progressionprogression is 2 . Find the common mtio.

the scodd term of

t4l

t4lIr" arc respectivcly t/ = 7rl lI

tll

thet3l

(b) An arithmetic progrcssion has /r terms and a common difference d, where / > 0 - P.ovethat the difference between the sum ofthe last f terns and the sum ofthe first k terms is(n r)u.

(, (a) A cuwe is given parametrically by the equationsx=3t'1-6t, y=4t3 5.

Find the coordinatcs of the point on thc curve where the normal is parallel t l the x,axis.ttl

instant when its radius is l0 cm, itsFind the rate of increase, at the same

(b) A spherical balloon is being infialed, and at thcsurface area is increasing at a rate of6-4 cm'?tr.rnstant, of(i) the radius,(ii) the volume[The volume and the surface area of a sphere with rudiusa d A:47tr' .l

The funclion h is defined bvh:-rr-+ 5-(-t-2)', .t


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The sketches below show the graphs of I = If (41 and r''z : f(r) .

On separate diagrams, sketch the graphs of:fi) y = i(x).(ii) y=r(2x)+3,(iii) ),: f("1).

12)trlt3t

showing clearly tbe axial intercepts, asymptotes and coordinatcs ofthe tuming points ifpossible.

A curve has equatron -y' + fe' = +.,c . find f in tenns o fr. and ]'- ill./'o o'(b) A curve has paranetric equationsu Y-ln(sinr)t+l

Find 4 in terms of t.dr t3l

v'= r(")/=lfG)l

(c) t,u"n y .-Jt-'*srr.,showrhal o,* +? tll


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fl (a) The plane 11 passes through the point-2i+j+k and i 3j + 2k.(0 Find the equation of JI in scalarA(6, 5,1) also lies on l'I.

l(2,q0) and is parallel to rhe vctorsproduct forD, and \rify that the

(ii) Find the position vector of the foor of the perpeodicular from theC(-2,1, 3) to the plane 17.

pointt3l

pointt4l

Cisl2l

Hence, Iind the equation of ..1u fhe plane parallel to IZ such that thecquidistaot ftom the plales I1 and Z, .

(b) The line 1 and the plane P have equations/: r=2(u+v),deR,p: r.(ux v) = 0.

where u and v ale non-zero and non-parallel vcctors(i) Show that 1 is contained in P. t|lf 'r l'riit Forrhecase u -l : l""O ' | : {, n"a ," (qudron,,frhe tmc cnnrJined rn p['J l,Jthat is perpendicular to.I and passes through rhc origin. l2l

point


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II2 MATITS (9740) JCI PRONTOTI0NAL EXAMINATION 2OO8SoLUTIoNSCurve SketchitrsCoordinares of y-inrerc+t *. [0, -])coordinatcs oF' inrcrc"o, "* lY ,- obli,:tue aslanptote is l, =-t I- ve(ical a-s],rnptote is Jr: 2

,] [+.)2 Mathematical Inductiolr

"l ' l=r"--l .*a I oBinomial It

Ihcretbrc.,_',, rloJ4.2'l 14.14\ I/r , .(\tJn"ioni"\dl'dl"r J.:q I.4 Summation Notatiou / Method of frifferetce2n'!2n I 2+ I In'+n n+l n , i'"':'" '=r"* ' ,= n'+n N+l5 lneoualitv(D - t- 72r'+-t + I= 2lr + ) +

(iD I3

(ii0 tlJa< : or Ja >:J: J-r


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6(a) C,omctric Progrssioa/ = I1{n-a as ld.t t o. r = --i3

7 ADDlication of Differentiation ReInarks(a) . point on curve rs ( 3. -l) Notegradient of nonnal - 0a.r.. n.rt,-ntu 9=O

b(D 0.0255 cms-'(to 3 s.e)(bxit

8 Functions(i) k:2(iD

y=h 'h(x)

r =h(')

(iiD h.G)=2-.6=9 Transformation of graDhs(0


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(iD

r= i(2r)+3

(iiD

10 Techoiques of Differentiation Remarks(a) dv = 4-v'&i 2ry +3el(b) dvl1=-ll+r) corI Preferable to differentiate r. usingquotiept rule.(.) dy- "(' v). I -.r' (shown) The derivativcs of the inverse trigofunctions (sin I :r, cos-' -'r, tan I -r)can be found in the formula list-

Note thatcos '.r+(cosr) '.

Henced,*(cos 'r)* {cosx} '(-srn r).


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lt(aXi) uo,",,",., ,., . [l]=,(aXii) r0lReq.d position ve4tor i, *=ll,,,JOXD

rhererore, equatiooor " ". [i]=,.^""'

l,n,..,".o"arionfor hetineisr-rl li] r.-

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