:t/.t1!_t./tr?.i1'{,it

,r/,,1,'{

,1;,1 .t

,1"1.

tr/.i1,{

t.{.a.t.rt.

j.t.1,'

CONFIDENTIAL* (SULIT*)

;M,,1..,1

\ 1){"{'{/l ,r.1.1

t'l lf- flf' fl,7I1(Jt

I i \ l; ll i.t.\r I i; l/,',i-,,'; l,l liji\ l ;\ l,ri ; \ lj it 1) "/ t,l l/itt,,t I l'i ll/t,t I l\ t,i1.'1 },t 1. 1. 1 1.,'1 ?'.9' 1.,1 t1 f .'1 / 1 1,\ 1

) / /) 1 11 1 X.E..,1,"1 .\1.4, 11..1 \':1,'/,1 1,1 t' r 1 .,1 4,/,l l / t t' l l l. P /' k f L' \1,1,\),1.1.4ffi1,'1\:t1,1.'1

.1. 1 .\ . t t,,1 !,,\75lll,tr 1, k,1 f

:,,1 1 :. 1 1_ .t | :.,1 1l : 3i /, 1.: 1 Y ? I : 1t" f A',r.,,1,.1 h:,t/

IMATE MATI K (t)t *|, |t;ii1 .,4 .\' l. l, i 1.,.1 ?'3' f .,,i ] 1.,1 / I 1.\7 l,: 1) / : lt / k',t :1..1,\.1 1.l, \ rl I/ lr\r- : \/ lli /\^t,l /' ,q1!.,r ; i \ !,,ERTAS l) ii irir.,'1.i,';,,. i i', ;,'i r \ r,'ir ir.r,{,.i,'lili.\!"t irlr:'1.,t\ { i\:l

3:: "#;'"Tii,;.i',,iiit;1,|'/;'lj|''itt

,1/ 4:.t7..1.,11,.1?'.t/,.,1

.4t..,1Yt"/,.:,11 .11 t{,.1.,4 { .,lY.5f ,1

1L,,1' 5/.1.tr 1.,,'1,

"t:1..1.,11..'1Y31.,1

11.,1"'t,411,..,11'.*.,11/..,1Y.q..,1aJ ,.1j.w,,1,1 1,..1 Y3 1,,1'11 .11.:r/,.1.,1/.,.i1.1/..1,1f .,1?3'1.,.1

1t_,41:\7,.1.11.,.17',t/,.1

'f ?rI\i:fi ,r1f,'{?ikl,'i',{Ji!iil;',14i}X{t'1.1 .

"\,'t' /, 1 1'1 "),, 1,'1,,1 l. U 1. I 5' / 1: { I }l / A'.r..1, 1,tr', t

MAJLIS PEPERIKSAAN MALAYSIA(ueravsieN EXAMTNATToNS couNCrL)

SIJIL TINGGI PERSEKOLAHAN MALAYSIA(MALAysrA HTGHER scHoor. cnRrrncarn)

Instructions to candidates:

DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO.

Answer all questions in Section A and only one question in Section B. Answers may be writtenin either English or Baltasa Melayu.

All necessary working should be shown clearly.

Non-exact numerical answers may be given correct to three significant figures, or onedecimal place in the case of angles in degrees, unless a dffirent level of accuracy is specified inthe question.

Scientific calculqtors may be used. Programmable and graphic display calculators areprohibited.

A list of mathematical formulae is provided on pages 6 and 7 of this question paper

Arahan kepada calon:

JANGAN BUKA KERTAS SOALAN INI SEHINGGA ANDA DIBENARKANBERBUAT DEMIKIAN.

Jowab semua soalqn dalom Bahagian A dqn satu soalan sahaja dalam Bahagian B. Jawapanboleh ditulis dalam bahasq Inggeris cttau Bahqsq Melayu.

Semua kerja yang perlu hendaklah ditunjukkan dengan ielas.Jawapan berangka tak tepat boleh diberikan betul hingga tiga angka bererti, atau satu

tempat perpuluhan dalam kes sudut dalam darjah, kecuqli aras kejituan yang lain ditentukandalam soalan.

Kalkulator saintifik boleh digunakan. Kalkulator boleh atur carq dan kalkulator paparangrafi.k tidak dib enorkan.

Senarai rumus matematik dibekalkan pada halaman 6 dan 7 dalam kertas soalan ini.

This question paper consists of 7 printed pages and 1 blank page.(Kertas soalan ini terdiri daripada 7 halaman bercetak dan t halaman kosong.)

@ Majlis Peperiksaan Malaysia 2014

srPM 954i2 [Turn over (Lihat sebelah)*This question paper is CONFIDENTIAL until the examination is over. CONFIDENTIAL*(*Kertas soalan ini SULIT sehingga peperiksaan kertas ini tamat.) (SULIT*)

I The function f is defined by

[,-.'. x(0,(x):{ I, x:0,

|..'- t, x>0.

(a) Determine the existence of the limit of f(x) as x approaches 0.

CONF'IDENTIAL* 2

Section A 145 marl<s)

Answer all questions in this section.

14 marksl

(6) State, with a reason, whether f is continuous atx:0. Hence, give the interval(s) on which

13 marksl

13 marl<s)

[7 marl<sl

13 marksl

15 marksl

[7 marl<s]

l8 marl<sl

f is continuous.

(c) Sketch the graph of f.

2 The equation of a curve is y : *t "s-x.

(a) Find the stationary points on the curve, and determine it's nature.

(b) Sketch the curve.

3 Find the value or f '(r + 2x) ln(l + x)dx.

Jo

4 Using the substitution z : cos y, flnd the general solution.of the differential equation

dv1 1:+-coty= icosec),.

5 If Y: acos-rr, show that

,2" d-v dv(l -.r-)ff-r:_ -y:O,fbr-l <x< l.

Hence, find the Maclaurin series for "cos-'*

,, to and including the term in xa.

fr_

J,^lz - " a*'

6 Use the trapezium rule with five ordinates to estimate, to three decimal places, the value of

95412*This question paper is CONFIDENTIAL until the examination is over.

[5.marks]

CONFIDENTIAL*

I Fungsi f ditakrifkan oleh

Il-.'. x<0,If(x):{ l, x:0,[.'- t, x> o.

(a) Tentukan kewujudan had bagi f(,r) semasa x menghampiri 0.

0-t)Dengan yang demikian, cari

dalamra .

6 Gunakan petua trapezirtm denganperpuluhan, nilai bagi

d'y

Lfsiri

SULIT* aJ

Bahagian Al45 markahl

Jawab semua soalan dalqm bahagian ini.

14 markahl

(b) Nyatakan, dengan satu sebab, sama ada f selanjar di x : 0. Dengan yang demikian, berikanselang yang mana f adalah selanjar.

(c) Lakar graf f.

2 Persamaan satu lengkung ialah y : x' e'-u.

(a) Cari titik pegun pada lengkung itu, dan tentukan jenisnya.

(b) Lakar lengkung itu.

ft3 Cari nilai | ( +2x) ln(l +x)dx.

Jn'

4 Dengan menggunakan gantian z -kos y, cari penyelesaian am bagi

dv 1- l--=- + -kot )'= - kosek y.dxxx'

5 Jika y: eoon'', tunjukkan bahawa

13 markahl

13 markah)

l7 markahl

13 markah)

15 markahl

persamaan pembezaan

l7 markah)

l8 markahl

untuk menganggar, hingga tiga tempat

15 markahl

[Lihat sebelahSULIT*

i'

dv- x-r - u:0- basi -l <x < l.

d-x

Maclaurin bagi eko'-' sehingga dan termasuk sebutan

lima ordinat

|e-*

95412*Kertas soalan ini SULIT sehingga peperiksaan kertas ini tamat.

CONF'IDENTIAL* .4

Section B ll5 marl<sl

Answer only one question in this section.

7 The curve/: ln(4x) is shown in the diagram below.

the curve. 15 marl<sl

(b) Calculate the area of the shaded region bounded by the curve, the tangent and the -r-axrs.

15 marksl

(c) Calculate the volume of the solid formed when the shaded region is revolved completelyabout the y-axis.

8 The function g is defined by

15 marks)

1

--xE(x)=e2

The tangent to the curve at the point P passes through the origin O.

(a) Show that the coordinates of the point "

tr fo, ),

urra find the equation of the tangent to

,rr(fr,for all values of x.

(a) Show that g"(x) + g'(x) + g(x):0.

Hence, show that the Maclaurin series for g(x) is

+.-*,,*fir-.(b) Use the Maclaurin series obtained in (a) to

(i) find the expansion of rgl? in ascending powers ofx up to the term inx3,' l+2x

(ii) find the value of 6* 8(')..r_o X

15 marksl

15 marl<sl

13 marksl

12 marlcsl

1

J

954t2*This question paper is CONFIDENTIAL until the examination is over. CONFIDENTIAL*

SULIT*

Bahagian B 115 markahl

tawob satu soqlan sahaja dalom bahagian ini.

Lengkungy: 1n(4x) ditunjukkan dalam gambar rajah di bawah.

Tangen kepada lengkung itu di titikP melalui asalan O.

(a) Tunjukkan bahawa koordinat titik P iahh (!4, ), uu, cari persamaan bagi tangen kepada

lengkung itu. 15 markahl

(b) Hitung luas rantau berlorek yang dibatasi oleh lengkung, tangen, dan paksi x. [5 markah]

(c) Hitung isi padu bongkah yang terbentuk apabila rantau berlorek dikisar sepenuhnyadi sekitar paksi y.

15 markahl

8 Fungsi g ditakrifkan oleh

1_

g(x)=e 2

bagi semua nilai "r.

(a) Tunjukkan bahawa g'(x) + g'(x) + g(x):0.Dengan yang demikian, tunjukkan bahawa siri Maclaurin bagi g(x) ialah

15 markahl

+.-*;*fit- 15 markahl

(6) Gunakan siri Maclaurin yang diperoleh di (a) untuk

(i) mencari kembangan +9 dalam kuasa x yang menaik hingga sebutan dalarl x3,- I+Zx - "J----o l3markah]

(ii) mencari rrllul 1ru6 8@.

""(fr,

95412*Kertas soalan ini SULIT sehingga peperiksaan kertas ini tamat.

12 markahl

[Lihat sebelahSULIT*

CONF'IDENTIAL* 6

MATHEMATICAL FORMULAE

Dffirentiation

d_, I6, (sln x):G

d -ldx

(cos-' *): G

d_, 1

dx (tant' x): | + i

$ rtrrl e(r)l: f'(x) g(.x) + f(x) g'(x)

d I rt*) I r'(r) e(x) * f(x) g'(x)

e ts(,) l- -- rgt*)i-

Integration

if'(')JT(, d': ln lflx)l + c

rdv tdulu-dx:uv - lv=-dxJ dx JdJr

Maclourin series

e':l+r**+"'+*.

ln(l+x):x -+.+ .+(-1)*' +. .., -r<x(1

, xt xt v2r+lsinx:x- t+n -. . . +(-l)' er+ U +...

cosx : r - * * ^d

-"' +(-1)' &.

Numerical methods

Newton-Raphson method

x,+t: xn- #, n: o, 1,2,3, . . .

Trapezium rule

l), * = |nlto+ 2(y,+ !z* " '* y,-,) + y,), h : +

9s4t2*This question paper is CONFIDENTIALuntil the examination is over. CONFIDENTIAL*

Pembezaan

S{'''-' il:#d-16, (kos'*): G

d,l6r(tan'x): l+x,

* tn e(x)l: f'(-r) g(x) + (x) g'(x)

a f rlry I r'(r) e(x) - (x) e'(x)eLs(,)l:-G(,x-

Pengamiran

lf'G)J6*: ln lrlx;l + c

rdv tduJ, a*M:r,tv- Jra, tu

Siri Maclaurin

e':l+-*fi+'.'+i.ln(1 + x):x-+.+ .+(-1)*'+..., -1 <.r(1

-3 -5 -2r I

sinx:x- i+o -. .' +(-l)" 12fo ,y.

+' . .

) a )v

kosx: | - h * 4,u -. . . + (-l)' OO.

+. . .

Kaedah berangka

raeaahlNewton-Raphson

x,+t: x,- #, n : 0,1,2,3,. . .

Petua trapezium

[), * = lnUo+ 2(y,+ y,+. . . t y,_,) + y,J, h : +

SULIT* 7

RUMUS MATEMATIK

954t2*Kertas soalan ini SULIT sehingga peperiksaan kertas ini tamat. SULIT*

8

BLANK PAGE(HALAMAN KOSONG)

954t2

Srpucot+ Se.n 2 (r\4arhr qg/ 2)Section A

t) a) o"g f rxl exists onnof ??

^'$- f cx) = Jl:- r*e*

: l- eo

: l-l

: o -Yl1

sincz ;S"f{x; = JfB_ f (*r , {gf (x)

b) f(o1 =1 _M1

sincs .l,I-ftxl I fCo)X-+o'cf )t=o. - Aa

Jn+err,als , t g : |r'e

Ma - rrre*hod

AL - ar8xer

lI*f$=Jib €x-,

: eo -l: l-l

: O -tvla

&xrsf$. - A1-44(I- I

* lj[1 f txt , *hus f i5 nat crrtinur'ot/e

,geRJ -,A1 (s-l

c)- cort€Gt ards. - Al- c0FreCt CLrfye - AL

correl la}ia3) - At

3={(xt

(3ml

,) a\ -ts4 ffaflonoJS Pt = *urning Pa

b _ds_ ="dx

* (ztj(-r"9- X.x3 ( el-:x;

3=dU

-.-J- :

dx

:

:

13 r3-ar

ex* (e**)

gx" [g3-")

x, d*'(3

{u - ,- -t,ll.dx ' \'/

)e L3--^ (g-ax) = o

q:heo x--Q r _![ : A

3-2x = o

-z*'* -3.-3x-= _A^

-Mr

.'. (O,91 isq point o$

infleairo - Af

- t4a

( t,t , 3.37s ) is a

xte3-=* = o

)<t =ox=o

whl]-., x= * ,U=3.s"s(*)

oTxirhu,o po;nt -&

x<o x?o v>O

-ds-dx *veo *ye

/rh

fir."t

b) (1,s,3.3?S)

(3nn )

J' 13 e'-n

correct

coHesf

cD$@+

otis - AL

crrlve - 41,

la&) - Aa

3) J: (r+rx) tn(r+x) dx

g1: lD (l*x) dv = lr 2^

clu= fr:-ML r:,t+x'

: c,,8tr6 (3s,{) Aa (-Srn I

AA{Ml

+) z =<(:sU

*; -sins(S) _Mr

3g- - -J /dz.\clx - sing \ti.)

: - csecg &) - M1

-# * * .at y = * ""secy

dtl I lcosV \+Fr i(+fr:uj=*(st)-(rfu)(,U+$(ffi)=+(rfol Mr

: a-tntxl_t:X

_tx

(*)*# + (-*) (*). '**(*)*(*)k)= -x-3 -v,.

-Lt-?tx : J z -dx

+ . - [+J+.,

COSg - L --t .-a. ' \zx -r\,

c€sg s #+cx -A1oR

u= cos-'(=b+A) -Aa

tch€,ra, A=CX (7m)

s) ! = gcos-f X

,n g : cos-r|

uffi)= -#J,-x, (e): -!

(

U ; CoS-'xJ

cosg: x

(-s;nV)H*J=y

dur4--dx - s;oy

-_t- Jl;'"

*

lvlL

(f--r')*(S):-y - M1

(r*x')*(*1 + (*)(-r*J(,-*'i* (S) =-*(J,*)(-:*) - x(#)#r =-*(r-x,) (#)- xg5 : - (JF*,)(#)

(,-t')(-$*)-**: -(-y)(t-x')ffi) -rS -S =o (shonrn) 41

(-ex)(S)+(r=x)t$h)-S - xS -#=o _ Mj.

(r-x')(*)*3x** -2.$=o

( r-x")(*) + (-zx) ({s) -3 S -srffi -.2$ "o

Lgl x:o )

rJ--tt-\-J

-d[ - -e\cJX

- 141

dadxr

-d1,-dxr

.L. g"

- ^.rL- -Je'

85 *ar,J t4aalau;ria *heor€r6,

3 = q",oS-' \

g*-'x: q+ + q3, - (::ir'*J=fs ,'-t::1'***l.Ia

, e* * g* x + $e* x' * *er ^'*&e*:t*+ . - _ Al

(8rn)

6) JlJ5=dx

Le+ f(x) = J2;]-M1

r - l-on--rr 4

: o,2S -M1( s ordino*es , ar irrreaals)

fo *,nS {taPerur^ rr<l€ r - M1

X f(xlo.o | ,+14+

o.2g I ,IN87

0.513 t.36q30 '?5 1..25('2

r ,00 I 'OOOo

* ( ro q€3)

l,3ro (3 d.P) - 41

(srn )

-Sec*icn B

l)q) y,rn(+x) j(x,,u)Cn5,o (A, o,;

du +J_--6-;- 6(

- J- i x*., x*X-M1 + = *

u :l \

cuhen5 =l r

I -* ln kxi

e'.+v P (*'l) (sho"r:n)\Nf.

\,- o -41,_E

,rmcli€rr+ c{ fJ- fclrlgent t *l

=+-MLrk

J*r = (+) (,-q) :;re':n,(x-x';tl -1. 4,,- I.,' e^ I

g:{x -Aa (sm-t

b) Area = J: *- dx f rn 4x J, -t.,q.

P-Mr: [:tl*I e Jo

. *e -(o'4)= *e **

: Q. OSqS un*€z

g= €x

x'*a,r'. +lf _ Ma

_ vl.*l- -3- [ xrn4x-*] j

-l rn 4x Jx

u: ln rtx

,,r= *

xrn+x - fx ln*x - x

dv=

va

lJr

I

x

c)

{, -41

u= ln 4X x'=J

eu= 4xatv - z+

(..Irn.1

P5grb - V1g

t/olurne : C, t e* dy "J , # u, dy _ rn

.*I; en* 5i^,f: Y"t:*"((+*-t)*r: ff+l : Sat+1. : o,r$36 units3,

i o, ;rpp oits I (sJ,p)

: ftl+.-+]_ ft="(*_o,)_m. -Aa

- et- -rr a)- */l - \ C3)'[ s) TEn (Srnjo-J': ftn-*

Bq) g(x) : e-*'. sin (€-")

g,(x) = -*e-t, s;n(€-x) * +"*(+xX. *,J

*r_

3' tx I = - *S (x ) + -$-<"s (E') (e-*'J

g" (,r ) = -*e' (y) . +(€)(-snBr)(d") -(+**r)(-fe+1'Wa

g,,(x) +*s'(r): + L- su) l' + "", $r (en) (-*) _ vta

S",*1 +{r'(x) " -*s(x) t (-j) (Ot*l , jgfrl )

3', (*f + jS'(x) = -*3(x) -*g'(x)- {9tlt

S" (xJ * *g'(x) +i3'(x) + $gtxt + 4g (x) : o - Vla

$"(x) +g'(x)+g(x)=o (shu'^rn) -41

g"'(,x) + g" (x) + 3'(x,) =o - vL

g''txI *g"'(x;'+ g"(Y) =o - VL

Lef X*€ , -P1-t

g(x) =o g"(xJ=-5 g''(x;=Eg'tx] =S J"'(xJ: o

tsrn )

Bg t,(s,tj Mqdcrr,n *he,orerno , - V1,!.

g(, ) . d' ",n (5-;;

e*'"in(8x,. F) , + ,-tl*.- +x*_ -

(shown]?

(Srn )

=€* -+x.+$r*

b) [i ) 9(x)I * 2x L GX binorniaj eypansion

: g(x) [tr+r*l-'l -M1

( i' 1 linr E(x)x-0 X

( 1+er )-' . |+ +f (ax) , C+f leD' * (-r) (::)r:lr (:x)3+.

: l*2x ++xL 8x3+-' -M1

: (+- - $*".**xq*. ) ( l *rx *+x. -sxir )

, g* * JE x, *- 2 rJ:;.3 - *". f Tr, + ..

'+* -Sp*' * Ig x3 - - Ar (3rn;

' tinr' X*+t>

Ex+ -Jlx - Jr]-xr.1-:JA'78

* tirh X- "x+G

(+ - +, + S xu--- ) -^4a

_i;- -5-x -Aa(:rn

)