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- L,f* '/ L.jm\( Grp,'Vuw'\Conf ident ia lx
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