Tampitres Judior College2008 JC 1 Promotional Examination

H2 Mathematics 9?40

Answer l/, questions (100 marks).

Sketch the graph of y = /(r) for the given graph below-

v

t3ll.

2.

3.

I. t5lUsiog intcgration by partr, fiM the exact value of

(D

(it(iii)

Find rh. exnansion of 3' Jt+ z'in ascerdirg powers of.I, up to and including the term in rz-State the set ofvalues ofx for which the expansion is valid-Deduce the equation ofthe tangent to the curye

I +r'' Jt +2,

at the point wherer = 0.

t3ltrl

t2)

4.

, "l /8,,\L'=l'-il' l.," 'J.

Hcnce.how rhar , '. "lln'.u. *h* I

,,1_ i)

5. Ci\enlhat t-c srr\.provetlrar n#-r. .o. , . ana nence trnd fl

Obrain Maclaurin's series for . sln r up lo and includrng rhe lerm in \'By using the Maclauin's series for e" sifl n and the staodard series expansion

show that .:'sin2.r- 2-t+ 2x? -i .3

A metailic cuboid has a squa'e base of width 2t m and height l' m A cylindrical hole ofdiameter ir m and height ), m is bore through the cuboid as shown in the diagram. The

vollrme ofthe remaining solid, V, is kept coastant at I m3.4

Lt

Show that the total exposcd surface area of the .emaining solid (includinS the curyed

surface area created by the hoilow). is given by

t7l

t3l

t3lforcos -r,

I2l

6. The diagram shows the srapl$ of y = f(r)

'/.

Sketch, on sepamle clearly lab€lled diagrams. the graphs of

t it y =1rtt -., r-l

t';l v = -1t( ')(iii) yr = f(r)

A sketch ofthe curvc

al + bx +16

i) Write down the vatue ofc.ii) Find the valu€ of.r and show that b = 8.

t2l

tll

t3l

n+cwhere d, D and c are constanti, is shown, not to scale, h thc diagram. 'Itre cqualions olthe as)4rptotes are -r = 3 ard y =r + I l.

tr lpl

iii) find the mrmerical value of the 1wo slalionary points, giving your aoswers comect toone decimal place.

iv) Copy th€ above sketch and, by drawing a sketch ofaoother curve ol the formt - L\ . I Z . in lhe same diatmm. show thar thc equ.rrron

-r' 4,l. 8n 16=0has exactly one aeal root.

t2l

tll

8a) The region R is bounded by the curves , y = -f and the y-aris from r - 0I

Show that the exact value ofthe area ofregion R is

l=t*"

1 tr h2, where f is a constant to

Isl

b) -the diagram below shows the graph of}, = sin r and y = 1,6, i6.6a-rao.

,y = slil )l:

(iD

(iii)

Show tbat fsnf

xdr = 11 1"inZrnC.

I'2

'[ he two curves meet at A-(i) Using g.aphic calculator, find dle coordinates of point A, giving your answer

coFect to one decimal place. tll

121

'fhp regron enclosed bv theFaphsof l_srn)rand v Ir rs rourrc.l rllough

2z radians about tl1e r-axis to form a solid ofrevolution o"f rrclume I/. Find thenumerical value of the volume ta giving your answer correct to two signilicant

4

figures. t2I

9a) Determine the de.ivative of each ofthe foLlowing with respcct to x:2(i)

b) Find the gradient of the curve given by the equation sin(t)+3.},?

(t t\{ i . ; J.

rv;ne vow ans$er Ln exacr rom

(ii)

t3t

t3I

point

t4l

=I at the4

l0a) A circular disc of radius ," is cut into 15 sectors. The areas of the sectors form anarithmetic progression. The area of the largest sector is ttree times the area of thesmallesl sector. Find, in tcfms of[, the angle ofthe smallest sector. t5]

A geometric progression has non-zero first term and common ration r, where r > 0 and

rr l. The sum of the first 20 terms is three times the sumolth€ fi.st l0 te.ms. Showthat a2 3u+ 2 - 0. whcre u - rro.Hence find the value ofr. Docs the sum to ifla rity exist? Give your reason. t6l

b)

ll- a) Use the substitution lt = 2r- I io find the e).act laLue ot ['::-l- .lr. t5]a (1, t).

b) i) cL\en rhar ,qt' 6t ) can be wnrren rn' (r.\ - l)(j! -l)

and -B a-re integers, f1nd the \.?lues of,.1 and B

ii.) Hrnce, lmd

,n" a,* ,*1*{. *l.." a

t3l

I3tI#ffi".

12- A sequeflce 4,r.,,!lr,... issuchthat r,=f and'2)

nln+tlln+2)

(i) Usc the method ofmathemalical Induction to prove that ,"=

-I - L4l

n(u +1,

(ii) tsy considerins u, -!,., inpart(t,Iind i- -: ^ t3l- 3 n(n +I)rn +2)

(iii) Give a reason why the senes in part (ii) is convergent and state the sum toinfinity. t2I

(iv) [Ience find the value of[r r r 'lI r r r...1. tfll2r4 145 4.s6 I

-

End of Paper ^.-*

solurion! rp. H2Math.mrtics Pmmorionrl Eyrhinaridn 200&

ffi=r,*"x,*,.r=tL+.1)I-i+;,'+ I

2

..1" -

Exb son R \ahd 'r

)rl< I =U.1

t- t

) - Jr+ rJ-

- r- I +;r +

w'."*=0. n9=-r,

Eqla(ion or uns.nr : '},-1=-(;-0)'l=l-J

- ,2,,..-"1:1,-l

=.,"f,:l=l

Thercfoft loral $rficc deqI ,l

/=.(rar.:l(rr'-' :l -:"li )L l:/] \lr

=G:1.,.,*."r.--LlI rl-ls-:1,,,fja LL\ l/ | 16 - r./,

d |16-rr.

For A tu be mLmum or nmmun q=o

(,6 ,),-f!:4J-L=o

( b-'l,r-13,2) ^

Ll6- ",'' -"

d. t o-rl.. --:

- Ir:- :-.o-

Th.Gror. A iinininum wrrn r,= 3+'

d\

--+ = ]. r-5r .r+ )e 6c I

y=0.*=' ,*i=,.{4=,. , ="-,-,lrl.'lrl-t2) t6.r

-.rl.,,..,- -. ti,-r- I

T = + {r,-x)

(-r,- i)

l1r,-t.1

tAi

iiD

rr+&+16'

'-f r-3- i: +rrr l6- (r+ )tr-r)+,JCDmparin! co.fi ciens orr,

UsihgCC, inlm!fr Coi -(t00.2R0)Maintrm poitrr - (-4 0, 0)

- rr-3r1=r:+3J+ 6

- ri{r-3) =tr +3r+ t6

rhe grsphs or '},=Jr -d '=!A+ i",.^-' "'

.qusiion rr 4i2-3r-16=0 hs .xxdly o.e mt roor.

Arsords,on R= I l-.1- --l*i tr+i l+'/

=l* ,-i^,"t,]"=,.,-, lrcr

(iil

=l 1"2,

*= f2

Cdordindes olA = (1,9,0.9)

JJl 2l

=11. - 1.,...1-.2L 2 l

=1.-1";:.*c

Volumc gdn.rded aboul J-qh

= { 1, sm,r d. -40.e,|.,r

= "ll'-l*,,1 ' laoei ,,

="1' '-1,.,,1 1,,",, ,, ,'-) r il

or $ing cc, answ.r i t.7

o. " I "

".,, -ll.l o.-*,,'r" l1 )

/sr .Ld '= rni T-:Lrncr-7i ,,-l

dy rl B Id.=ils-'--'l

(i i)

(i)

.os(.r/) r-+l l'6':=0

-lr(cr?)+6j l= -.r cor r)d, -rcGGr.)

At de poinr

9_

,-11$l

-;"' 1/l

;E-':6;rlTJ=rI - a+lad=I = {"7d

',. =lfr'"- 'u.

since sredors.crd.= 1"p

-!d

,(L-):o) jd0-y'0)

r l-r! -3(L-r ")i /)"-lr"+2=0

= r'-lr+2 =0 sh.rc!=r D

- (,-2)(!-D = 0

- ,"2 or r-1

S. docs oor.xisr sinc. r =2 e > 1.

{:= ?

L.

= '"r, -.1- l

rer 9rr -6rr5 , 1+ ,1 + '(3r-r)G-r) 3r.r r 1

9r: 6l+5 =3(rJ-1Xr-r)+,4(j-l)+,3(rr-l)Whenr = l, 3=2, =r=4wheir:0, 5 = I -,4 -,1 37 =-6

f r.' -o.. * s" J t,, - 11 **"rj -2rn 3r- r1+ 4rn J-11+c

(t

Asumclh!!Pr is ru. ro.lomc [.2 .

I. r(tr+ )

'' ' *(*r LJ(k+ 2J

=_L- Ir(*+, *(r+D(l+2)

- (*.1X* -r)

"{*-tit.r,'{-

. . Pr ru! = Phr hd

By Mrih.aatical hduclion, P. is !fte.

i ntn+htn+2t -''

Ld P, b. ih" $doNd " ,, = -7+ " r.r , € z'

LHs = !, =1 (By rhc d.rbnior oiibe sqrmc)

RHs--L=.1r(L+rl r

--,.

1

'' : (A +trll +tl

sinte | -o as

lL IL:.r4 r.r5 456

L+ r i-t* ,o'-r)1".r,-rxLl r I'aa-i-i

N J 6, d. serie s co !e.sen.

t0