ANGLO-CHINESE JUNIOR COLLEGEMAITIEMATICS DEPARTMENT

MATHEMATICSHigher 2Paper I

JC I PROMOTIONAL EXAMINATION

Additiooal Materials: List ofFormulae (MFl5)

9740

3 october 2008

Time allowed: 3 hours

Can do the rvholc question.

Can do part ofquestion only. /

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Pose t afl

expant l- v4 r r rn a..erding t6scrs of r up to and rncludrng lhe tenn rn2{\Iriod the rangc ofr for which the expansion is valid.

Fxpress lhe sum ol (hefirsr ', lennsoflhe"crics

2 +l +4+6 +8 +9 + 16 +12 + 32 + l5 r-& + l8 +128+21+256+....

irrrheturm )ra, r 6' ). whirc.r and /, arc constanLs to Lre lound

Hence, or othenvise, find the sum ofthe lirst 2, tenlls ofthe given series in tems ofn.131

Find Fin(ln-tdr

6rr ,t2 +4r I

(,'+rl:.'+:) rvhere l, B, C and 1) arc

l5l

I2i

I3t

Irlttt

lTufl, over

Isl

Express

constants to be dctcmrincd.

llcnce, or othcnvisc, find

Ax+D Ct+Drn rh. h,rm +-

r'+l Ir'+2

ft! :i: !a,J(,'*t)(:" *:)

Ill

tlt

There are 6 discs, of which 2 are red and the other 4 are of differcnt colours. T hescdiscs are olthe sam€ size and the 2 red discs arc idcntical. Find the number ofways t{)

anangc :rll thc 6 discs in a circle il(i) the 2 .ed discs arc adjacent ta each olher; and l2l(ii) the 2 rcd discs arc not adjacent to each olher l2l

Deduce the total number ofr,^ys that all thc 6 discs can be arranged in a circle. tll

Thc region R is bourded by the curves y = (., J)2 and Jz2 = tl t-l between the;r

points of irtersection. Show on a sketch the region R. Find the volume of the solidfomred when R is rotatcd complctely about the r-axis- I4l

Functions falrd g are dcfincd by

- Lrr 5 ^ ^ |I \F- , t 0, r/ ' cll{.\,,r " )r-l 5

(i) Find thc cxact coordinates of the stationary point on the graph of f, ard sketch the

graph of f.(ii) Deduce the range of a

(iii) Dctermine ifthc composite fuoction gfelisls

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without using a catculator, solve the inequdlity xtl ,\ t z

.

Hence, deduce the solutron of the inequality 4+ r 4 . .'r'+) "-l

The curve ^S

has equation u = 4:1 where I and B are constants. There is a verticalRxasFnptote at r=20 and a horizontal aslmptote at J,,=2.

t3t

t3l

(0(i0

Find the valu€s ofl and B. t3lWith the values for I and B found above, sketch the graph of S, showing clearly theequations of the asymp0otes and the coordinates of the points of inteNection of .t with

13j

I1

the axes.

Prore bv rnduction. thar i / -l I for all oosirive rnLepen ,' 21(r . t)l (/l rt),

Hence. or orherwise. fi'd tl ' ' I in r..-s ofz.=\ '! '/

- / ,\51619 drg

'.1'6 66 f l l :i Ia\rtl

The diagram shows a sketch of part of the gaph of y=2ta9.shaded rectangle and the area of the region between the graph

) < r <.1 show ,n^, 13 ( z"*9)a" , tJz \ xl

show also that f '1r,,*9)*.s.Jz \ x)

14l

I2t

I1t

By considering the

and the x-axis for

trt

tlt

to be

t4l

IIHence deduce that _ lnl.5< , where p and q are posirive inregerspqdetermined-

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12 A curve C is give[ pammetrically by the equations

x=at2. y=zat.

Show that the normal at the point with pa-ram€ter I has equation

y +a=2it +at3.

The normal at the point P(9a,6a) cuts C again at the point g.ot Q.

t4t

Find the coordinates

I4t

t/ t, A curve C is given by th€ parametric equations

x = tl +i, y = t(t1 +iJ

(i) Find the Caftesian equation of C and hence show, ushg an algebraic method, that thccurvc is strnmetrical about the J.-axis. I2l

(ii) Shoq using an algebraic method, thar rhere is no part o f the curve for which -r<3_ [tl(iii) Sketch the curve and label the axial intercepts. I2l

(iv) Find the exact area ofthe rlrgion bounded bythe curve Cand the line x=4r2 +3. [41

civenrr,r, ) tnlcos,).sho$ rhar lr-10'1' ,, o

(i) By repcated differe(tiation qf this result, hnd the series cxpansion ofypowers ofj up to and includiog the term in x4.

(ii) Bypunrng ., t,.ho* thal ln 2 rs aDnroximdlcly L* o

4 ' 'tb t5tlo(iii) Use the expansion ofy in (i) to deduce rhe Maclaurin's seties lor tan2r..

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ir ascending

Ist

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[Turn Over

15(a)

The diagam above shows the graph of f, with tfuee as)4nptotes at ; = 2, r = -l and

y=2.'fhe graph of f crosses the -r-axis at ,r=4 and r=-2.6, and ther-axis at/ o\

r . L lr has a minimum pornl at i l.- l. Sketch. on separare r),er. the graphs of\ 4./

/ r\{i) r = rllx l+-]; ana

... I

l(r)

13l

I4I

showing the aslmptotes, the coordinates ofany poiots of intersection with the axes andany slationary points.

(b) The diagrams below show thc graphs of y'z=g(r) and ],=g'(,r), the derivativcfunction ofg, for a certain function g. Sketch the graph ofg, showing the statioDarypoints of g, the as'.rnptotes and the coordinates of the points of ilte.section of g withthc axes. I4l

!v

- End of Plper,lnqL Chi^ee Junio. Coles.

Il2 Mathqnatics 9740: u003 JC I Ptuftotional Eud.ario. Pape.

Pase t of7

(*.1.)

r' =g(r)

(u'.-J:')v = c'Qe)

I

2 a=3,b=2,2(2" ll +1n(n +l)

3

f't""tt't cos(hx)]+&, where I I

2

4 Solving using G-C. or ma

l"1,'-,1 j[,F)"'.nualLy 2, R:0,C:O, D= I

* r"1,,+! *",,*"."I

5 24,60

6 By CC, points ofinte.sectiotr a( r=1, r=6. Reqlired volume- l87.5z =589 clr units

7

naneeorris {re

m ,r .1|, era*" *,..i",.

8 I

-s-f<02

or -t>l -r< lor;r>l

et Il=20,A=-2

l0 l1nl

ll p=3,q=2

l2 ( tztel;,, +4

l3 128, ^^,5

x'1.,. l1

l4v,2-L.,Ll6 t5-t6

tnn 2" = 2r+ 9rl +..-3

lt I '!2t2

Anglo-Chinese Junior CollegeII2 MatAeftatits 9740

2008.tC I PROMOIIONAL EXAMINTION Solutlons

,anql Ch 4e Jnnr CaltEEtH2 Mademarics 97.10: 2008lC I Pmmorio.al €nmindion PtF

Prq.6 ofr

, = r(1,1*|)

(r+,;)

(ii)

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