Mathematics (Matriculation)

QS025/2

10 The registration record of a private college indicates that 40Yo of its new intakes are

international students and the remaining are local students.

(a) If 20 new students are randomly selected and the number of local students are
noted, find the probability that there are

(i) equal number of local and international students.

CHOW 12 marl<sl

(iD not less than 9 local students.

14 marl<sl

!

(b) Exactly 100 new students are randomly selected. By using a suitable

approxim ate distribution,

(r) fiod the probability that between 38 and 46 arc international students.

15 marksl

(ii) determine the value rz such that the probability that the number of

intemational students is at most m is 0.993.
[4 marks)

(u

END OF QUESTION PAPER

19 201

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS025
2011/2012

202

QSo25/1 QSo2s/1
Makematja
Paprl Matematik
IISernester
Session 2011'2012 Kertas 1
2 hours
Semester II

Sesi 2011/2012

2 jam

I _a_ CHOW
ar,a6*ri_+t

il BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA
It IATRICWIITON DIVNION
MINNLRY OF EDUCATTON M4LAYSU

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
It IATRICUI/ITION PROGMMME EXAIvIINATTON

MATEMATIK

Kertas I

2 jam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.

iDC !"9T CF-,\ S CTESJ]ON PAPER U\ITILYOIJ METOfTO DO SO,

Kertas soalan ini mengandungi 15 halaman bercetak,
This question papermnsbts of 15 pinted pagx.

@ Bahagian Matrikulasi

203

QS025/1 CHOW

I\STRI CTIONS TO CANDIDATE:
This question paper consists ofl0 questions.

-\rrsu'er all questions.
-{11 ansrvers must be written in the answer booklet provided. Use a new page for each

question.

The full marks for each question or section are shon'n in the bracket at the end of the question

r or section.
All steps must be shown clearly.

Only non-programmable scientific calcularors ;an :e used

Numerical answers may be given in the ton: oi ;. ;. surd. fracrions or up to three significant

figures, where appropriate. unless stated o:leri-:se :i: ::e cuesrlon.

Y

3
204

QS02s/1

LIST OF MATHEMATICAL FORMULAE

Trigonometn

I *sin (,eX A) = sin cos ,B cos A sin B

Icos (exB)= cos cos B + sin A sin B

tan (l+ a)= tanA + tanB

1 + tan AtanB
CHOW
sinl + sin B: 2 B B

"inA*)) "oro-

r> sinl - sin B: z ro"A*2B2 ,*A- B
cosl *cos B :2 *ro*))u *rA- B

cosl - cos B : -z "irA* B "*A- B

snZA= 2 sinA cos I

cos2A = cos2 A-sin2 A

= 2 cosz A-l

Y =l-2sin2 A

-tan 2A 2tan 4
1-tan'z A

s.ln-),4. l-cos2A

2

) . l+cos2A

cos- -4
2

5
205

QS025/1

LIST OF MATHEMATICAL FORMULAE

Differentiation and Integration

d,
fi("orx)= -cosec'x

fd,(t.. r) = ... x tan x

d ("ot..r)= -"ot.. xcotx
CHOW
f

t^, dx = ,'r(x) *"

) "f'1*1rt")

t f' (,*(-)! du_r =- rrnr l|r 7\1^*)1 l+,
J
f
v

iuctr'=ur,- lt,clu

Sphere I-=!-" S='lnr2
RightCircularCone S=/rr2+rrh
Right circular cylinder 3
S =2rr2 +Znrh
,'=!nr|h

3

V : nr2h

.J

7
206

QSo25/1

LIST OF MATHEMATICAL FORMULAE

Numerical Methods
Iteration Method:

xn+t= s(*r), n=1,2,3,... where ls'(rr)l.r

Newton-Raphson Method : /t=1,2,3,....CHOW

xn+t = .r-H,

J Conics
Circle:

(* - h)' + (y - k)' : r'
,' + y'+2gx+2fy*c=0

:lxr + Wr+ g(x+ x)+ f(y+n)+c =0

,=rlf'+g'-c

7 Parabola:
(*- h)' = 4p(y - k)
(y-k)'=4p(x-h)

F(h+p,k) or F(h,k+p)

Ellipse:

(*- h)' f. (y - k)'

'-

o' b' =l

F(h+c,k) or F(h,O+c)

I 207

QS025/1

1 Show that 4x2 -l6x+ y' +2y-8 = 0 is an equation of an ellipse with vertical major

axis. Hence, find its centre and foci.

15 marlcs)

2 Solve the differential equation ,' *ox *lc'.I = l. given thaty: 2 when x : 0.

16 marksl
CHOW
3 l.By using the substitution r = .. :-:riJ :re exacr value of the integral

?

E*J[", :-l

[7 marks]

I

1 Shm rt* ru cqudim (S - fr1' : x has a root in the interval (l , z).By using the

I{efficRryh$n method with the first approximation rr = 1, find an approximate

root of the equation correct to three decimal places.

17 marla)

?

5 A radioacti'e substance of mass N gram decays at the rate of d! = -ilv,

dt

where ft is a constant. Initially the amount of the substance was 80 gram. After

100 years it decal'ed to 20 gram.

(a) Express N in terms of the elapsed time r. [5 marks)
15 marl<sl
(b) Calculate the amount of the substance remains after r20 years.

11 208

QS025/1

6 ifFind the values of A, B, C and D

x2+2x+l:4*B _ xC-l_.rD+1

x'1x'-11 x'x''

l,(m)"Hence,evaruateCHOW

ll2 marksl

7 Find the area of the region bounded by y :sinJi, x = O, x = n2 and the x_axis.

0 If the region is rotated 360' about the x-axis, find the volume of the solid generated.

ll2 marksl

8 Given the point P(4,2,-3), rhe straight line I,+ 4-3= 45 = 4 *rathe plane

fl:?s+ y+22:9. Find

(a) ran acute angle between the straight line and the plane rI.

13 marl<sl

(b) an intersection point between the straight line Z and the plane IL

!7 15 marksl

(c) a Cartesian equation of the plane containing the point P and the straight

line L.

15 marlul

13 209

QS025/1

9 oP, Q andR are three points in space where FA = and FR=b. Given

a =2i_+2i-L
b= i+2 j +zft

(a) Find the area of the triangle PeR.

CHOW 14 marl<sl

(b) Find the parametric equations of the line z passing through the point

R (2,0,3) and parallel to vector a.

14 marks)
i7

(c) rf u = (laln*lala) *a r'= ( q b.- b q) . er aluare zz ,v. Hence, interpret the

geometrical relationship benreen 4 and r,,

15 marl<sl

1{l A liae sepent joining (-1,0} and (3,4} is a diasreter of a circle.

(a) Find an equation of the circle.

[3 marks]

v

(b) Find an equation of the tangent to the circle at the point (3,4).

14 marlal

(c) Find the points of intersection of the circle with its chord of which the

midpoint is the origin.

l8 marlcs)

END OF QUESTION PAPER

15 210

QSo252 QS025/2

Mahematkx Matematik

Papr2 Kertas 2

Semester II Semester II

Session 201 1/2012 Sesi 2011/2012
2 hours
2 jam

t :r_h: CHOW

il_ 3q-iEil"tr.-!;' E

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

AIATRICUTATTONDIWION
MINNTW OF EDUCATTON AIAIAYSU

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI

IIIATRICU-ATION PRrcMII ,tE EXAM IAUON

MATEMATIK

Kertas 2
7 jam

JANGAN BUM KERTAS SOALAN INI SEHINGGA DIBERITAHU,

rc,A'OTOPEN IIJIS QUESI/ON PMERUNNLYOU METOLDTODO SO.

Y

Kertas soalan inimengandungi 19 halaman bercetak.
This question paperconsisfs of 1g pinted pages.

@ Bahagian Matrikulasi

211

QS025/2 CHOW

I\STRUCTIONS TO CANDIDATE:
Tlus question paper consists of 10 questions.
.\nsu'er all questions.
A11 answers must be written in the ans\yer booklet provided. Use a new page for each

question.

The full marks for each question or section are shown in the bracket at the end of the question

or section.

All steps must be shown clearl1,.

Only non-programmable scientit-rc calculators can be used.

Numerical answers may be sir en in ihe lbrm of t, e, svrd, fractions or up to three significant

figures, where appropriate. unless stated othenvise in the question.

3

3
212

I

QS025/2

LIST OF MATHEMATICAL FORMULAE

Statistics

For ungrouped data, the hh percentile,

(

l'(r)+x(s+r) lI 't ls an lnteger
'Ptt'-- ) ' ifsisanon-integer
,
1 .
CHOW
L'(trl)

:where
t = ry-:! and I s ] the ieast inteeer sreater than k.

100

,fot\l,-r,l(.For grouped data. the hh perce:rr:le-. p = L, -'\ I00/ "-',-l

1 ,.

-'l l

Variance

. Ii..,-11n'Lf-'.: r.,):''
:
n-l

Binomial Distribution

X - B(n, p)

f (X = *) ='C,p'(l- p)'-' , x =0,1,2,3,...,n

Poisson Distribution

x - P(),)

p(x = x)-e-^)", x=0,1,2,3,...
x!

5
213

CHOW QS025/2

I : : :Given P(A) 0.5, P(B) 0.6 and P(Au B) 0.8. Calculate the probability that both

events A and B occur. Hence, verify that A and B are independent events.
16 marksl

2 Six yellow balls are labeled with numbers l, 2, 3,4, 5 and 6, and four red balls are

labeled with letters P, Q, R and S. All the ten balls are of similar size. In how many

different ways can one

(a) arrange all the balls in a straight line such that balls of the same colour are

next to each other?

.> 12 marl<sl

O) choose and arrange equal number of yellow and red balls in a straight line

zuch that balls of the same colour are next to each other?
[4 marks]

3 A t€am of four members qill be formed by selecting randomly from a goup

consisting of four stud€nts and six lecturcrs.

calculate the numberof differcmways to form ateam consisting of [2 marlcs)
[2 marks]
(a) no students at all. 12 marks)

(b) equal number of students and iecturers.
(c) more students than lecturers.

7
214

QSo2512

4 The frequency distribution of the age (in years) of 80 patients in a clinic is given in

the table below.

Age 10-15 15 - 20 20 -25 25 -30 30-35 35-40

Number of 5 15 24 18 10 8

Patients

Find the mean and mode. Hence. calculate and interpret Pearson's coefficient of CHOW

skewness given that the standard der iation is 6.798 years.

l7 marl<sl

5 On the average, a hospital rmeives 6 emergency calls in l5 minutes. It is assumed that

the number of emergency calls received follows the Poisson distribution.

(a) Find the prrobability that

r :.: ncre than 15 emergency calls are received in an hour.

[3 marl<s)

:: the hospital will receive the first emergency call between 9.00 am and

9.05 am.
13 marlrsl

(b) FinJ the number of emergency calls received, m, if it is known that the

probabilitv at most m emergency calls received in half an hour is 0.155.
14 marks)

9
215

QSo25/2

6 The following is the stem-and-leaf diagram for a sample of heights (in cm) of a type

of herbal plant. All observations are integers.

13 loo
t4 I 588

15 0258
16 0688
t7 l5

19 2

23 0
CHOW
.t.S (a) Calculate the mean.

12 marks)

(b) Find the vaiues of the median, flust and third quartiles.

14 marlrsl

(c) Constnrct tbe box-and-whiskers plot and comment on the data distribution.

[6 marksl

J

11
216

Qgl25f2

7 It is found that30o/o of the population of an island are overweight. Among the

overweight, the probability of those who do not have any chronic illness is 0.4 and
among those who are not overweight, the probability that they do not have any
chronic illness is 0.65.
Draw a tree diagram to represent the given information.

l2 marks)

(a) Hence, if a person is randomly chosen from that population, find the

probability that he
CHOW
(i) does not have any chronic illness. 12 marl<sl

v

(ii) is overweight knowing that he does not have any chronic illness.

12 marksl

(b) If nvo persons are randomlr chosen tiom the population, find the probability

that

v (i) both of them do not have any chronic illness. [2 marla]
(ii) only one of them has chronic illness. 12 marksl
t (iii) at least one of them does not have any chronic illness.
12 marl<s)

13
217

QS02s/2

8 The probability that a type of antibiotics can cure a certain disease is 0.95.

(a) If five patients are given the antibiotics, find the probability that

(i) exactly three patients are cured after finishing the course of antibiotics.

[3 marks]

(ii) at least one patient is cured after finishing the course of antibiotics.

12 marlal

(b) If 500 patients are given the antibiotics, find the
CHOW
(i) probability that more than 480 patients are cured.

14 marksl

(ii) largest possible vaiue n such that the probability that at least r patients

recovered alter tlnlshrns the course of antibiotics is 0.9.
14 marksl

9

15
218

QS02s/2

A nurse works five days in a week. The number of days in a week she works overtime
is a discrete random variableXwith probability function

al:.' - il. x =0,1,2
I' x =3r4
J1l
x=5
,,rr={ iL(' - 2),
J'

CHOW:L(.r - 1).

Ja\/

k=!where ,t is a constant, Shou'that
5

13 morks)

(a) Find the probabilln' she ri orks o\ errime everyday in a week.

12 marl<sl

tb r Canculate the rrlbalil::. that she ur11 u-ork overtime for at least three days in

a n'Eg['

12 marksl

(c) Determine the most likely number of day's in rveek she will work overtime.

12 marl<s)

!

(d) Find the expected number of day in week she will work overtime. Hence,

evaluate E(3X+1).

14 marks)

17 219

QS025/2

10 The continuous random variable Xhas the cumulative distribution function

[o'- 'r<o

.E(x)=laV.r, - 3.r:. 0 <x <1

lr, .r>l

t

where a is constant. Shou'that a = -1.CHOW 12 marl<sl

Hence, 16 marlul

C (a) calculate the mean ad vaime ofX. 12 marlcsl

o) r"a rlx - E(x).*] 15 marlal

(c) if I: 4X-3, find the E(f and Var(y).

:

END OF QUESTIONS PAPER

19
220

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS026
2010/2011

221

QS026/1 QS0zSlt
Mathematics Maternatik
1Paper
IISemester K*rtas '{
1Session 2010/201 Semester {E
2 hours Sesi 2{}1*/2fiXtr

2 iarn

I ffflL ffi CHOW

--
--._ilG-;-=

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATNCUfuITTON DIVISION
MINISTRY OF EDUCATION MAI-AYSU

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI
MATRIC ULATION P ROGRAMME EXAMINATION

MATEMATIK

Kertas 1
2 iam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.
DO NOT OPEN IHIS BOOKLET UNTIL YOU ARE TOLD IO DO SO

Kertas soalan ini menganciungi X5 halaman bercetak.
This baoklet consjsfs af trS printed pages.

@ Bahagian l,lairikulasi

222

QS026i1 CHOW

INSTR.UCTIOF{S TO CANDIDATE :
This question booklet consists of XS quesiions"
Ans'ur,er all questions in the ansr"^/er bookiet provicied.
Use a new-page for each question.
'Ihe full marks for each question or section are shou,n in the bracket at the end of the qirestion

or section.

All steps must be shown clearly.

Only non-programmable scientific calculators can be used.

Numerical answers may be given in the form of 7r, e, surrd, fiactions or up to three significant

figures, where appropriate, unless stated otherwise in the question.
Y

Y,

{
223

QS02611

LIST OF M.AT}IEMATICAL FORIVIULAE

Trigoncmetry

sin(l 1 B):sinl cosB*cos,4 sinB

cos(,atB) = cos I cos B+sin A sin B

tan\l.-/4*Bl= tanA* tan B
1+tan AlanB
CHOW
sinl + sinB : 2 rinA*2B2 "oro-'

sin,4 - sinB : 2 B rrrA- B

"orA*22

cos,4 * cos : 2 B B

-B "orA*22 "orA-

cosl -cos B: -2 rinA+ B rinA- B

Limit

li- thi:l

h-->o h

h*-+l0-cohsft_O

Hyperbolic

U sinh (x + y): sinh x cosh y * cosh x sinhy
cosh (x + y): cosh r coshy + sinh x sinhy

cosh2x-sinh2r: i

I - tanh2x sechlx

coth2r-1:cosech2-r

:sinh 2"r 2sinh;r cosh r

cosn z.r : cl]s,nt ; + stru,1l -}.'

q
224

QS026/1

LIST OF' MATHEMATICAT, FOR.MULAE

Differentiation and Integration

f(.) f'(,)

cotx - "os""',
sec tc
cosec .r secxtanx

- cosec x eat x
CHOW
cothx - cosech2x
sechx - sechxtanhx
y cosechx - cosechx cothx

It f(-*f)Jldr : rn lr(,) | *.

ff l,A = lir'- | r,/zr

I

Sphere I-=1^'i S=4xr2

RightCircularCone L'=!n'2h S=7Err
Rightcircularcylinder 3
S:2nrh
V=nr2h

7
225

QS026/1

1 if x:sec# arrl] y=Ztafil, find 4 interms c{ 0.
dx

15 marksl

2 Find an equation of the circle that passes through the points (1,4'), (2,2) and (-1,3).

Hence, find the radius of the circle.

16 mark"sl
CHOW
3 Given three vectors a: 2t+ Pi + 4k, b :i- 3k and c : 5i + 61+ 2\. Find the value

v of B such that

(a) 4 is perpendicular to b.

13 marksl

(b) axb=c.

14 marksl

4 Prove that cosh2 x - sinh2 r = 1. Hence, find the value of tanh -x if sinhx = 1.
4

l7 marksl

v

c
226

QS026/1

5 (a) Use the f,rrst principle of derivatil'e to show that

t

(.1

;(sm-r) = cos x.

15 marks)

(b) Given y:sin(x2), *u q* ,"0 # intermsof x. Hence, or
otherwise, show that ,# -*. :4x3y g.

15 marlwl
CHOW
v

6 Given /'(x) -gxx|'--%g .

(a) /.Determine the vertical and horizontal asymptotes of

13 marlrsl

(b) Determine &e interval of x on which / is increasing a:rd / is decreasing.

15 marksl

(c) Sketchthegraph of -f . [3 marksl
r

4!t1

227

QS025l1 13 marl<sl
15 marl<sl
7 Given the points A(1,3.7'), B(4, -1,2), C{12,0. 1) and D$,2,0). 15 marks)

Find

(a) a vector equation of the itne AB.

(b) an equation of the plane ABC inthe Cartesian form"

(c) the acute angle between the plane ABC and the plane ABD.

Y
CHOW
8 .\ ;ircie C rasses thror-rgh the origin and has its centre at the point (3,-3).

(a) (-)'i:ai: ii-ie e cil:iicl ,:i :ire ctrcle C.

13 marksl

(b) If the line l.' = .\r - 6 meets the circle C at the points P and Q. determine

the coordinates of P and Q.

15 marhsl

(c)- Find the coordinates of the points onthe circle C where the tangents are

parallel to the line PQ.

15 marksf

13
228

QS02611

9 The curve ! : x4 + cx' + bx' tras a point ol infiection at (-2.0).

(a) Find the values of a and b.

15 marksl

I(b) l, i tr)Show that another point of inflection of the curve [- z',ioJ

CHOW 14 marl*l

(c) Use the second derivative iest to find the coordinates of the local extremum of

the curve.
14 marksl

l0 (a) Prove that cos3x : 4cos3 x -3cosx.

14 marlul

'a i s: -he .:'1.r,' 3 ijentitr i.r

(i) find all the solutions in the interv'al -180" < r < i 80o of the equation
:2cos3x + cos 2x + 1 0.

l7 marksf
v

(ii) showthat cos'2x:,i(.o* 6x+3cos2x). Hence, evaluate

T

Io .ort 2.rd.r.

Jo

14 mcrk,s]

E,NB OF QUESTION EOGKLET

,t(

229

QSo26/2 QSo26/2
Mathematics
ZPaper lVlatematik

Semest<r II Kertas 2

.Sessloir 2A10'2011 Sernester I{
2 hours
Sesi 2010/2011

2 iam

$= CHOW I

:_q>_-_--

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

. MATNCULATION DIVISION
V MINISTRY OF EDUCATION MAIAYSIA

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI

MAT RI C ULAT I O ld P RO G R4 lv{ME EXA MI }{ATI O N

}IATE}IATIK

Kertas 2
2 jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.
DO NOI OPEN IHIS BOOKLET UNTIL YOU ARE TOLD IO DO SO,

v

Kertas soalan ini menEancungi 2t halarnan beicetak,

This boakler con-*rsis .t{ 21 printed pages.
O Eahaciian Mairikliasi

230

UJUIDi I CHOW

I\STRT CTIO\S TO CA\DIDATE:
This ques:i.'n bc,.lklet consists of 10 questions.
-\lst'er all questions in the answer booklet provideci.

L se a ne\\'page for each question.
The i;11 marks for each question or section are shown in the bracket at the enci of the question
or seciion.
A11 steps must be shoinn clearly.
Only non-programmable scientific calculators can be used.

N,.rmerical answers may be given in the form of ?t , e, stJrd, fractions or up to three significant

figures, where appropriate, unless stated otherw'ise in the question.

v

231
3

LIST OF MATHEMATICAL F'ORMULAE

Trapezium Rule r'ttl *2(y\rr'r +y2+...-!,, rt")) ,where4= u -
'!t n
:t-h-a
Ji,,; ,")d-t='-l\o2+!r\t')v

Newton-Raphson Method

l CHOW

j lt : r

rI xn+l= *, - [r9), , 1,2,3,...

i |x"l

Statistics

For ungrouped data, the ftth percentile,

lI(I,'--(, ) + x(s-l ) 1I s ls an lnteQ.er

Pt= 2' ' iis is a n.rn-inreser

tt[. ])

nhere ,=+ and Is]:the ]eastintegergreater thank.
100

frlf4),-r,-,1-l [1oo/, ,rFor grouped data, the kth percentiles Pk = ^-'l

L "l

-lC.

.E,

232

QS02612

1 *, !Obtain the general solution dx -Zy = 4"" .

15 marlxl

2 The summarl'' statistics of the length (in cm) of a sampie of 50 a.Jult insects oiaCHOW

certain species is as ibllows

E-r:45,I-x2:8i.

Calcuiate the mean and variance. Hence. comment on the distribution oltite sampie

based on the coefficient of variation.
{6 marks}

3 By using the substitution u =lnx. el,aiuate i:!a-1-r, up to f-ive decinial places.
!
12 marksl

ff '{? is approximated using the trapezoidal method based on five equal

subintervals, compute the error.

15 marksl

I
233

QS026/2

4 The cumulative probability distribution function of a discrete ranciom variable X is

0 , x<-i

1

o

P(x <-r=11 2<x<4
CHOW
I 4<x<6 13 marksl
{4 marksl
6

v 1 , x>6

(a) Construct the probability distribution table of X.

(b) Calcuiate Var(,Y).

v

q
234

QS026/2

5 The time (in minutes) used by 120 students surfing the intemet to perform a certain

project is given in the following relative cumulative frequency table.

Time (r), iri minutesCHOWRelative cumulative frequency
x<0
0
x <20
Ja
x<40 4U
x<60
x<80 i9
x< 100
60

1

L

J
53

60

1

Find

(a) the median and mean.

16 marksl

(b) Pearson's skewness coefficient and comment on the value obtained.

14 marlcs)

11
235

QS026/2

6 {Neulon's law of cooling states tirat hot iiquid ai temperaiure 11 coois at urut"
dr
proporlional to the ditTerence betr,veen irs ieniperature and iemperaiure of the
surrounding environment 110.

IShow that H = Ae-o' + Ho, where k is the cooling rate constant and is an

integral constant.
13 marlrsl
CHOW
A hot tea at 76oC is left in a room of 22oC.

(a) Find the Ne*ton's cooling equation.

{2 marksl

(b) Using a container X. it is found that after 10 minutes in the room, the

temperature of tea has decreased b1' 1OoC, Determine the temperature of tea

1ianer ninutes in the rrrLrrif .

[4 marlul

(c) Using a different container Y, whose ft : 0.10. determine the time taken for

the tea to cool down to room temperature.
12 marksl

(d) h-r which of the two containers, X or Y, does the tea cools down to room

temperature faster?
l1 mark]

I

,t?
236

QS02612

Three boxes A, B anC C has identicai green and red dice as shown in the lollowing

table.

iJOX

Colour A BC

Green 4 53
Red
5 64
CHOW
(a) If all the dice in box r\ are aranged in a row. in hou, man.y different

anangements can this be done?
12 marksf

v

tbr -\ die is randomll drarvn lrom each of the boxes. If all dice drawn are of the
same colour. in hou manv different \\'avs can this be done?

13 marksl

(cJ Four dice are randomll draun rvithout replacement tlom bor B. How many

different ways can the dice be drawn such that there are equal number of red
and green dice?

13 marksl

(d) A die randoml-v drau,n from box A is put into box B. Subsequently, a die

v draun lrom box B is put into box C. Finally, a die is draum from box C

and the colour is noted. Calculate the probability that the die drawn from C

is green.
14 marks]

15
237

QS026/2

I In a chess tournament between A and B, the probabilit-v A r,vins is 0"2, B wins is
0.5 andthe probability of adraw-is 0.3. if A and B rvere io meet inthree games,

calcuiate the probability that

(a) two games are draw.

14 marksl

(b) A and B win altemately. CHOW

13 marks)

(c) either A or B wins all the games. 13 marksl
(d) B wins at least two games. 13 marlcsl

17
238

QS02612

A continuous landom variable. I has a probability densiiy ftinction

0" <-rJ'<)

oiherwise

la

Shor,v thal- c=1 and E{Y} =

1C f

CHOW 15 ntarksl

F{ence.

(a) calculate P()'> E(f)).

[3 marks]

(b) determine the mode of the distribution.

[2 marks]

(c ) sholv that the median. iir ol the distribution satisfies the equation

nrt -9m+5=0.

13 marks)

19
239

l

QS02612 CHOW

10 The distribution of the weights of all sugar sachets produeed by a parlicular laciory is

assumed tc be normal with mean 25 gm and standard deviation 2 gm.
Show that the probabiiity of a randomly selecied sachet r,veighs within 1 gm of the
mean is 0.383.

14 marksl

(a) If te:r sachets are randomly selected, find the probability that betu,een four

and seven sachet weigh within 1 gm of the mean.

13 marksl

ft)\J Determine the sample size, n such that the probability that none of the sachet

ueiehs rvithin 1 gm of the mean is 0.021.

14 marksl

)i c I- ..ne lil:li:eJ sacheis are ran,.iLrinlr selected. approximate the probabiliti,

that less rhan -10 sachets ueieh uithin 1 em of the mean.

14 marks)

END OF QUESTION BOOKLET

240

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS026
2009/2010

241

QS026/'l QSo26i1

Mathematics Matematik

Paper 1 Kertas 1

Semester II Semester II

2009/20 I 0 2009t2010

2 hours 2 jam

Ir CHOW

4L

.-7:

:J

",*4:l:f --

--

BAHAGIAN MATRIKULASI
KEMEI\TERIAN PELAJARAN MALAYSIA

MATKIC ULATION D IVIS ION
MINISTRY OF EDUCATION MALAYSIA

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI
MA TKI C ULATI O N P ROGRA MME EXA MINA TI O I''I

\T.{.TENIATIK
Kertas 1
2 jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.
DO NOI OPEN IHIS BOOKLET UNTIL YOU ARE TOLD IO DO SO

Kertas soalan ini mengandungi 13 halaman bercetak.

This booklet consnfs of 13 printed pages.

@ Bahagian Matrikulasi

242

QS026/1 CHOW

I\ STRTiCTIOF{S TO CA}.{}IDATE:

This question booklet cc'nsisis of 1CI questions.
An s',l,er all c}restions.
The fuli marks for each question or section are shown in the bracket at the end of the qtiestion
or section.
Al1 steps must be sliot,,n clearli'.
Oniv irc',n-prograrnmabie scientific calculators can t-.e rised.

r.Numerical answers inay be given in the lorrn oi e , surd, lractions or uo to three

significant figures. where apprcpriate. unless stated othenvise in the qLrestion.
Y

v

3
243

QS026/1

LIST OF MATHEMATICAL FORMULAE

Trigonometry

sin (l t B): sin,4 cos B t cos Asrn B

cot (,e + B)= cos I cos B + sin I sin B

tan (,1 + Bl = tanl + tattB
I + tan.ltanB
CHOW
sinl + sin B :2rinA+ B .or''-'
22

sinl-sin B:2cos '1+B rin'l-B

)2

cosl + cos E - 2 cos -1+ B .o.''- B

22

cos;1 -cos B:*2rinA*B rinI-B

Limit

.1.rmsin_/=r l

h-->o h

hlit-n>_i0-cohs /r = t)

Hr perbolic

-v sinh (.r - -1 ) sinh x cosh 7., + cosh x sinh y
:cosh (.r - 1l) cosh r cosh y + sinir x sinh y

coshl.r-sinh2r:1

1 - tanh2x: sech2x

coth2x- i -cosech2x

:sinh 2x 2 sinh x cosh x
:cosh 2x cosh2 x + sinh2 x

5
244

QS026/1

LIST OF NTATHEN,IATICAI, FORF,{U LAE

Differentiation and Integration

f (.) _f,(,)

cot x - ror*c',
xsec
sec -x tan x

COSCCX -COSCCXCOTX CHOW

colh x -- coseclil .r

v sechx -sech,--;ranh,r
cosechx *cosechxccthr

'"li-li'1'.(.Ir.;)) .11. = ir, ..i , ) -L

lLiili.it\ ,.. _,,,. !,.r{1,
-Ltt J,

Spherc t^2 '\ = + 7r l'
I' = TTrr
- 'LC)- r-L,.. \
Itight Circular Cone t' - I _
y Right circuiar cl.linder ,\ = 2 x rh
'- 1'

J' '':

,, : h.tt t.)

7
245

QS026/1

An ellipse u,'ith centre at the origin passes through the points (0,3) and (1.1)

Find the equation and the foci of the ellipse. l7 marks)

Pror.'e that 1 + tan x tan2x = sec 2-t. 15 marksl

Given u = 2\-2)*L. Find the vectors which have magnitude 6 and parallel to u.CHOW

16 marlal

v Using the definition of hyperbolic functions, prove that

1 =ooshx-sinh.r.
coshr+sinhx

Hence. find

dx. 17 marksl

(cosh.r - sinh "r)-

Given that f (x) = xj +8

x

/.:,, (a) State the asymptote of ll mark)
(b) Find the critical and inflection points of ./.
15 marksl

(c) Determine the rnterl'als rvhere./ is increasing and ./ is decreasing.

13 marksl

(d) Sketch the graph ofl 13 ntarksl

9
246

QS026/1 13 marksl
l8 nturks]
5 (a) Prove that fbr 0 + ntr, r,vhere ru is an urteger,

I g.sin

cos6(,1 * 2sinr o + sin' o) = tandseca

ibi Br rrsrng (a) an.J the substitution u :ran0, evaluate

CHOWIi _ s-"i"n"d )A
J':..,t6fi - lsin' 0 - sin' 0)

y Thepoints A(i,3,2). B(3. -1.6) and C(5.2.0) lieontheplane tI. Aline I

passes through the points P(7 ,2, 21 and Q(0. 1. -l). Flnd

(a) ,48" A,--'; ancl irence. ubtain an equation of tire piane I{ in Carlesian form.

l7 mark,sl

(.b) the parametric eqr-rations of the [ire L 13 marks)

(a) the point of intersection of l, and n. 13 ncctrksl

\' (a) If cos(r + 1) : 2;rsin,v u'here 0 < y < t, flnd I' Iand u-1l" . at -'i - U.
dx
16 marks)

(b) 5iGiven J, = sin(lnx), shor,v that r' * r**.y = 0.

l5 mctrksj

11
247

QS026/1

9 (a) The position of a particle moving along a straight line at any time / > 0 is

given by s(r) = 4t -l)(t - 2), where s is the distance of the particle from the

origin. Find the velocity of the particle at the instant when the acceleration

becomes zero. 14 marksl

(b) A closed right circular cylindrical container of radius r and height ft is to be

constructed with volume 4,000 .*3. The cost for the construction is RM 1.00
p", om' for the curved surface while RM 2.00 per cm' for the top and

bottom surfaces. State h in terms of r and hence, find the radius of the

cylinder so that the cost of the construction is minimum.
l8 marlcsl

v
CHOW
10 Given the circles 13 marl<s)

(' : .r-' -.l r - 2x_ 21+l =0
('.:.r:*-]t=1.

Find

(a) the centre and the radius of the circle C, .

_ (b) the equations of the tangents from the point (0, 3) to the circle C..

15 marksl

(c) the equation of the circle that passes through the point (-5, 0) and the points

of intersection of the circles Ci, and C,. l7 marks)

END OF BOOKLET

13
248

QS026/2 QSo26/2

Mathematics Matematik

Paper 2 Kertas 2

Semester II Semester II

2009/20 I 0 2009t20t0
2 hours 2 jam

4LCHOW

r oXHs I4-_Y:-^:. l,l:"'

BAHAGIAN MATRIKULASI
KEMENTERIAN PEI-AJARAN MALAYSI A

M TN C N"LA TT O N D I VI S I O N
li].,11]V ],\ TRY O F E D U C A1-T O I,,TA LAY S IA

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI

lU TNC: ULA TI O l\r P RO G R4. MM E EXl l,II NA'fI O i{

MATEMATIK

Kertas 2
2 jam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU,
DO NAT OPEN IHiS BOOKLET UNTILYOU ARE TOLD IO DO SO.

Kertas soalan ini mengandungi 13 halaman bercetak

This booklet consisls of 13 printed pages

O Bahagian Matnku asi

249

0s026/2 CHOW

INSTRUCTI O\S TO C--A}{D{DATE :

This qiiestron booklet consists of 10 questions.
-\nsri er all questions.
Ihe luli ntarlis for each question or section are shorvn in the bracket at the end of the question
or section.

Ali steps mlrst be shou,n ciearlr,.

Oni,v non-programmable scientific caicrilators can be rised.

Nr-rmerical answers mav tre given rn the form oi :r. e . surcl, fiactions or up to three

significant figures, rvhere appropriate. unless stnted othenr.ise in rhe questiol.
Y

3

J
250