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950/1.95.1r ' I\ lathcnrat ic 's S,Nlathernat ics TPaper I3 hclrrrs

PERSIDANGAN KEBA\GSAA]Y PEN{GETTIASEKOLAH N,IENE,I \GAH ([ A\ryANGAN MELAKA)

PEPERI KSAAN PERCTJBAANSIJIL TI | {GGI PERSEKOL.\HAI\ MAL,\YSIA 2010

MATH E MATICS S, 1\ I \THEMATICS T

PAPE R I

3 hout ' i

Instruct ions to candidates :

Ansv.er all queslion,v. '4nsv,crs mo.1t 17e v,ritteru in ,-'

All neces.\'er)' v'ot'king,should he shown clearly.

Non-exact numerical anstvers mey be given correcidecimul plctce in the c($e. o,{angles in degrees, unlc,in the cluestion,

.'i' English or klalu.v.

'ltree ,rignificant Jigres, ot" one' tli,fibre.nt level af oct'nruc,l; is ,rpec:ilied

lr'lothemutical tables, s list of mathematiculformulae .:':cl graph paper ore prctviderl.

Arahan kepada calon:

Jawcth semua soalen. ,km'upcrn boleh dinlis clnlsm i.. ,',;sct Inggeri,s' atuu hahasa X,felatu.

Sent"ua kerja yang perltt hendcrklah ditrgtf ukkan clengr,'' . !q,s.

"lrnuopan herangkn tak tepctt boleh cliheriktn belul hilr:::-t tiga angkd hererti, utcru satutempat perpuluhan clalam kes sudut dalam da(ah, keo. ': aras kejituan )]ang lctin ditentukandulant soalan.

Sifir mutemutik, senarai rtoTtt{.\' rnatentatik, dan kertss p'.' ,rl'clibekalkan.

This question paper consists ol -l printed pages(Kertas soalan ini terdiri daripada 4 halaman bercetak,)

s'rPM TRTAL (MELAKA) 95011 ,95411* 'l'his question paper is CONFIDENTIAL Llntil the exarnination is over.* Kertas soalan ini S{JLIT sehingga peperiksaan kertas ini tamat.

[Turn over (Lihat sebelah)CONFIDEIVTIAL*

SU[,IT*

CONFIDENTIAL* 1

shorv that ,

pandq.,

Using the laws of the algebra of sets.

An(A-B) ' :A-R' '

Using a suitable sr"rbstitution, t'ind

The first three tertns in thc e rp31,t1'-'l l tri ( 2 + ar

32 - 40x + bx2 . Fincl thc r a lucs i ' i - l . a and b

[4 marks]

[4 r irarksl

in ascendrng110\\ ers of r. i lre

[5 marks]

f r. shorv that

marks]

, . i . : r t1:e l i t te

i - i n iarks]

. :ne \ -axis. r -er is arrd the l ine x : n is

lire irapezinrn rulc vn'ith seven ordinates,

[5 rnarks]

i J-{l - ; chr{q'++x

I f y : p cos( ln x) * c l s in( l : r '

. d2 t. ' th,.t ' - '; i +.Y+ --) = t) .

d\- QX

I rL-

Find the equat iotrs r l f tu , ' s i l ' - : - - i l i ines, eacl :

4x + 3v - 2 l - - 0 ant l i rc lSS,: . - : : .1 ' ' - ighthe p,r

- ; : - -' l ' l ie region enclosed b: ' t i re c i . r" : l : : e t

rotated contpletelr atror- t t l i rc . - . : ' . .^>. By usl l ' - i

est imate the volume of the scl , . l ienerated.

The functions f and g

f :x-+. . / i , x>0

g:x-+3lnx.x>0

follor.. s :

(a)

(b)

(c)

Sketch the graph of g and cir e a reasott ri hr

Find g -l arrd state its domain.

Find the composite function of fg-r and state

tite inverse tunction exists. 13 marks]

[3 marks]

its range. [3 marks]

954/l l(athematics TIiSI 2010 Trial ('\lclukul

CONFIDENTIAL*

(a) The polynomial xa -2x3 - pxz + q is denoreS by f(x).

by (x - 2)'.

(i) Find the values of p and q.

(ii) Hence, show that f(x) is never negative,

(b) Show that for all real values of x,

x '+x+1

x+l

does not lie between -3 and 1.

ThepointP lies on acurve whichhasparametric equations x: ( + t andy :2t+ l.

Show that the point P is equidistant frorn the y-axis and the point A(2,1) [4marks]

The tangent to the curve at P meets the line y: 1 at B. Show that AABP is an isosceles

triangle. [6 marks]

Solve the simultaneous equationsxt + 2xY :3

y'-xY=4

Find the values of x that satist'the €Quat1-'n

3(4*)-10(2)+3=0,

giving your answers correct to 2 dec.ral places

[6 marks]

[4 marusJ

(a)

(b)

10 It is given that f(x) is divisible

[5 marks]

[4 marks]

[4 marks]

954/l Mathematics TI/SI 2010 Trial (Melaka)

( r'.

I .1' '| , ,

11

l

(ii) Write down

(n)t lx= lb It t\c/

Hence, determine

CONFIDENTIAL*

(a) Show that the determinant of the matrix A =

real x, 7, and z,

the s1'stenr n ii r lS r tnatrix

is (y - x)(.2 - xXy - z) for

[3 marks]

(b) By substituting x:

inverse of the matrix A.

1, y:2 and z= 3 into the matr i r A, f ind the adjoint and the

f 6 nrrrksl

(c) The graph of a quadratic equatioii ) = ax2* bx *i :.rS:e S tirrough the points u'hosc

coordinates are (1,2 ) , (2.3) anC t l . ( - ) .

( i) Obtain a systenr of i inear e ql. . : i i rrns to reprr-sr:: l ' , ; re Siven infonnation.

[2 marks]

ir the lbrnr ri.\-\ : B rvhere

l3 nralk]

. t

-,/z

tl; )

12 For the curve y : i2:r- i ' , . s ' r3r i the equat iots i , : : : . '4x( 1- ' ; t

coordinates of the stationar\ pr.)inls,

Sketch this curve.

The line v : x and the cllrve \i l - r+1)-= ' : : t :e:se. ' l* r (1-x)

. ' .s\ lt:tI.,:- 's 3nd find the

[5 marks]

[4 marks]

A. r,vhose x-coorciinate is

o . Shorv that o is aroot oftirr equation -1r'' - -{r * i : l,-l and

at the point

.< ()

[4 marks]

ruse '.:ic \3\\1on-Raphson method once toBy taking - l o, a first approrinration to cl+

1__1,s1

+

pq

find a second approximation to u, Give )'our ans\\'er in the form of

integers.

o r.vhere p and q are

[3 marks]

----------END-------

954/l I,[athematics TllSl 2010 Triol (I{elakn)

\-r^^^'l f'.- lt-tul'l @r) vr-rmA { Fr+MARKING scHEME -- TRrAL EXAMTNATTo* tmdtELAKA 2010 t '| -MATHEMATICS T/S: PAPER I

Question AnswersI LHS=Arr(A-B) '

:An(AnB') '=A.r(A'uB):(AnA')u(AnB)_ Qv (AnB)= Av [An(B') 'J I: (A_B, l j

B1MIMI

A1t4l

DifferenceDe MorganDistributive

/ r i

eo;h

2 Let U : X" =+ du:2x dx

I#dx-it*: I r - , ( r ) l- - l - tan' l - l l+0-2L2 \2 ))

I [ - , ( " ' ) l_ - l tan- ' l - l l+c

4L \ .2 / j

B1

M1

M1

A1

t4l

In terms of u

Integrate hisfunction

3 (2+ax)n(n\ (n\

_ 2, +l ' " lZ,- t ax+l ' " 12,- , a, x, +. . .

\ .1 ' [2/ r

:2n +2n-, nax+ry2n-2 o2 x, +. . .

2n :32

zn-r na: -40^n(n- l )

zn-t 2 at :b

" ' [=5, a:

I

2b=20

M1

A1

M1

MI

A1tsl

Forming 3equations

Solving 3 eqn

For 3 correctans

4 y = p cos(ln x) + q sin (ln x)dv -D:-=.-:.-srn(tn x) + 9cos(tn x)axxJdv

x| : -p sin( ln x) + q cos ( lnx)dx

*+ * q---P cos( lnx)- n s in( lnx)dx-dtxx

* 'd, ' l * ** cos ( tnx) - g s in ( ln x)dx- dx

-, d2v dv. ' .x ' +x ' *V=U

dx' dx

M1A1

M1A1

t5l

A1

MathematicsTI/SI 2010 Trial - Melaka -- \V{ARKING SCHEME

Question Answers n

) bem

ht lines are:

y-3-7(x-1)

7x-y- i0:0

rne

2

aig

g l i

IIIII4); lr )

str

nd

nd

e

IIIII

4;J

S1

rhr )- l

)

m

+

'o sl

anl

)an

rf tl

-4;J

+)_Vl

m4

:0-0

ntot t( -aI

t ' t

\J

T7T- l r

. l

.J )

(=lm"

\J:A7)=c

he tw

:- 1)

l0 :0

radie

- t

rt -It -\

I_l

- t

I

-1-7)

1:

the

x-

20

I*I

t-lt*

m)2

m-m-

rm

of

: (

y+

: l

III

r \

iVt l

48rtXr

-o

I

3 gri

0_

4)- ln3)-48i

lXrI-07ons

= - ;

f7r

43

- t

II

7io

+

the

150

t '[ ;

2_

+

atir

r4

2m-InJ

1ua1

+3

) .x

Let

tan

(1 -

7n(7n

m=

Eq,

y+

i.e.tsl

B1

M1

A1

M1

A1

For either 1

For bothCorrect.

6 -lln l l l ilxr=; lv , :e + | II o l " I II o I r | |l * r : t lyz:e. I I

!I

l *o=T lyr : r , | |I

I sn | -1 I Il * r=?lv,= ' . I I

---/

* ui) .']l-3

+ea

h:L6

Volume

: r r f , r - ' t "d*

f t ( " \ l ( -+ += n t" [ ; ,JLt . ' [ '

a+ea +e- '

_ 6.37 unit 3

tsl

BI

B1

B1

M1

A1

Mathematics TI/SI 20!0 Trial - il{elaka -- I|{ARKI^|C SCHEME

Question Answers7

(ii)

(iii)

(i) DI+vll , /l /t /l /t tl l l--lT- -->xt lt it /l rI

The inverse function exists because g is a one to onefunction.Let3lnx:y

v:_x:e3

t

g-l : -> n' i , x e![ l

Dr- , : E

fg ' t (*)- f (e l )r-;I_

= 1e3:

:e6

fg' l : x+ ui , * . frRange of fg'': { y: y > 0}

DI

B1

MI

A1

B1

MI

A1

B1

tel

For the shape

For theasymptote x = 0and the point(1,0)

8 P(f+1,2r+1)Distance of P from the v-axir - 1' * I

JpA: J(r t + l -2)= +(2r+t-1) 'f ;

- ^ l f

+2t2 +l

- t2+t

.'. P is equidistant from the y-axis and the point A(2,

x: t2+ l ,dx A.__l I

dtdy _2 _Ldr2tt

Equation of tangent at P:l .

y-(2t+l)- : (x- t ' - l )t

lot1) lAl

B1

M1

B1

M1

y-2t+1dY -2dr

MathematicsTl/Sl 2010 Trial - Melaka -- I'IARKING SCHEME

lQuestion Answers

When y: 1, 1 - (2t + l ) : l ( x - t ' -1)

x:1-t2 t

B( 1-f , 1)AR:2-(1 - t t )

= 1+t2AP:1+t2+AB:AP.'. AABP ia an isosceles triangle

A1

B1

M1AI

[1 0]

9 (a) x" + Zxy =3 ------ ---------( I )y2-xy-4 --(2)(1) x2 +Zxy _ 3(2) y ' - xt , 4

(4* '+ 1lxy - 3) ' t ) : 0

(ax-y)(x + 3y) -0. ' .y = 4x or x: -3v

M1

Ai

For Q.E in x2containing y

For both

Subst y - 4x into (1)x2 +2x(4x) =39x' :3 . /

,//i+- /

X:r--F-. ! : : - - - - : , /

.,/3 \ 3 ,/St ,bstx--3yinrot I ) . /9y ' r2y(-3y):3 , /1 "

\ r /

Y'= |

y:11,*=*3

M]

\{

A1

A1

Subst. to forrn 2

QE

Solve the 2 QE

9 (b) 3(2'*)-10(2-)*3:oLet u :2*

3u2-10u+3=o(3u - lXu - 3) : 0

1^u- i - oru:J

3II

2* =1 or 2^: 33

Ix los 2 : loq- or

vv1

J

,1rog -* :

-3 or

log 2

x_ - 1.58 (2dp) or x :

x log2: log3

log 3

log 2

1 58 (2 dp)

Transform to Q.E

B1

B1

MI

A1

For both

Mathemqtics TI/Sl 2010 Trtal - Melakn -- .\,f.lRKlitG SCHEME

I

5

Question Answers10 (aXi)

(i i)

f (x) : xo -Zxt -PX2+q(x - 2)':+ repeated factorsf '(x) - 4x3 - 6x2.- 2px

t(2)-0 +-4p+q:0f '12;*6 +32-21-4p:04p:8=+p-2*4(2) f q:0: .p-2 q-8

f(x)=*o*zxt-2x2+8:(x2 - 4x + a) ( x2 +2x +7)

l

BI

MIAI

M1

A1

BI

ForDifferentiation

Forming 2 eqn .Both correct

Solveforp&q

Forbothp&q

Get the otherfactor

: (x - 2) ' [ (x +l) ' +1]Since (*-2)' > 0 Y x e $tAnd[(x+1)t+1]t0Vxefr. ' . f (x)>0Vx e $li.e. (x) is never negative.

M1

M1

A1tel

Completing thesquare

l0 (b)- x2+r+lLetv: -" x+l

p<+y:x2*x+ 112 + ( I -y)x + (1 - y) : o

For real x, b2 .- 4ac > 0. ' . (1 -y)t - 4(txt - y) > o(1 -yXy+3)<0

B1

M1

Form Quad Eqn

Forrealx,yS-?ory21

.'. For real x. x' + x +l'

does not lie tretween -3 andx+1

M1

A1

t4l

Mathematics TI/Sl 2010 Trial - Melako --,I,tARKIxrG SCHEME

Answers

1l (a) lAl = *t(y - z) - x(y'-t' ) + l(y'z - z'y)- xz(y - z) - x(Y-z XY + z) + Yz(Y - z)= (y -z)lx2 - x(y + z) + Yzf= (y -z)l*t - *y - xz + Yz)- (y - z)[x(r -y) - z(x - y)]

- (y - zXx - l 'Xx-z):(y-xXz -xXy-z)

Cofactor of a1r: - 1 Cofactor of az::6

Cofactor of an : 5 Cofactor of a3 t= -1

Cofactor of al-r : -6 Cofactor of a32= 3

Cofactor of a21: I Cofactor of a33 : -2

Cofactor of a21= -8

.', Adj A = 5 -8

-6 6

lAl - (y-r) tz-r)(y -z)- (2- l X3 - l r t2 - 3)

a

l

-8

r \- t I

^lI

1l- l

I; r l

- , t I

1?(I

I

I

l t ' lo- lo 2 t l

[e 3 r , :

B1

ivl l

Ai

StatementcorrectAttempt tofactorize with(Y-z) as factor

Next 4 correct

For 3 €eil,all correct

B1

-1\" l, lr l- L/l r

1

--_ l

;I

(o) ( - t 2 -1)f2)

1,1= ; l s -B 3 i l , '[ , ,J

- [ -6 6 - t i [u j

r") ( t )

[: j =":[; l

B1

t6l

M1A1

tzl

M1

, '2"1-, I I-1

- Ii Ii

- l\0/

t3l

l,{athematics Tl/sl 2010 Trial - Melaka - i\URKINC SC.HElvtE

A1

AnswersThe equations of the 3 asymptotes are

dy 4x( l - x)12(2x + I)(2) - (2x + 1) ' [4 - 8"] -0dx

(2x+l)(ax-1)-aa

4x' (1 - x) '

16x

- -u

' ( t - " ) t

=+(2x + 1)(4x - I ) :0- i 124

Y:0, Y-3

.'. The coordinates of the stationarr

Iand (- .3)

'4 '

-1pointsare(

2,0)

For labeling(114,3),(-112,0) x=l andy- - l

Yl

.'1\ Ji,l rz:/\i

/ \ l:__-_l t -

ilI

- i - -

t , (

(U4,3)

MathematicsTliSl 20l0Trial - ,Velaku -- l\4.4RKL\G SCHEIvtE

(2x * i t-+! ,

4x( l - x t

4x3*- l r - . -. ' . 0. the r - : -*

4x3*.1.r- =

.-\nswers

r - : l j 0 iA, is a root oi the equat ion

Letf(x)= - i . . ' - - - , , - II

f ( - - l - - - - -4' 1.^

(o): l>o

A1

BI

MathemaficsTI/st 20t0 Triar - rferukct -- ^4ARKI,\'.

scHErvtE