Mathematics (Matriculation)

QS026/2

LIST OF MATHEMATICAL FORMULAE

Trapezium Rule

J"lc.\l)h'lt.la. = ; 1(ro + t,) + 2 (y + y2 + ... t ))n r)|,u'here ft = b-a

Nervton-Raphson Method

.,ntt ,,il / (*,) n = 7.2,3,... CHOW

l,?r).

Statistics

t,
For ungrouped data, the ftth percentile,

P"l,.:1[I|-,t((tt,r)]+) x2(s+t) . :rct.t..is an integer

, if s is a non-integer

:rvhere , =",1! and [^r ] the least integer greater than ft.
100

a) n-For grouped data, the kth percentiles, Pp = Lp -rf rfi,r- 1
1oo,
|

v

5

251

QS026/2

dv1
dxSoive the differentiai equation rvitir initial conditicn 1,(0) .= 4.
,l*y

Express 1 in terins cf ..;. i5 marlul

Given trvc events A and B nj1h

P(B): +J , P(l u B): Ja and n(a j;)= 1
4
+^
CHOW
Find

(a) p(,4) {4 marlx)

.y (b) ptEi;t 12 mtirks)

(a) Holr,'many one-. t'wo-, three-, and four-digit numbers can be formed using the

digits 4.5.6. and7, rvhen each digit can be used only once?

13 marksl

(b) Hon mar-iv of rhe nrimbel'-. lormed in part (a) are odd anci greaier than 500?
i4 ntarksf

y

+ The follor.l,ing tabie represents the pictrabiiity distribution of' a cliscrete random

variable Il

Y _') -1 i 5

P(I',: "v) 0.1 0.3 0.4 0.1 0.1

Find l2 mnrksl

(u) e(ir.l > r).

(b) E(j'-3)2 ancl Var()'"-31. 15 marksl

7

252

QS026/2

5 (a) Given g(x) : (r - l;fi a 2 . By using the Newton-Raphson method starling

at ..rn:1.1. findtherootof g(r). 16 marks)

l

(b) ofB-v using trapezoidal method, obtain the approximate value lxe",:k

0

based on four subintervals, correct to four decimal places.

CHOW 14 marksl

6 The w,aiting time for 50 customers to have their food served at a restalrrant on a

particular da1'- is shou-n in tire follou,ing table.

v

Time (minutes) Number of Customer

1-5 4
9
6-i0 15
11
11- 15 6
16-20 1J
21-25 2
26-34
31-35

v (a) Calcr-r1ate the mean. median and mode of the waiting time.

18 marksl

(b) Plot an ogive. Hence. determine the percentage of customers rvho har,'e to uait

bel,ond 23 minutes. 15 mark's)

I

253

QS0?612

7 A model for the concentration 01' glucose solution in the bloodstreani, C = C(I,},

ris given b-v the differenriai equation 'i!-: =,'- 4- C. u'here is the constant rate at

rviricli a.lucose soiution enters the bloodstream and k is a positir.e constant. If

( tUi- ( ,. shori tlrat tlre coticentratiLin at al.I\ timc I is

C(ri=l(\.f, " --k,t"e))"--A.t. t' l\tnrtrktl

A:

After a .,,er1,' long peri",d of tirne. the concentration of gli.rcose is found to be 1 unit.CHOW

It {:,) = 9. rvhat is the concenti'ation of giucose at t :?-? ll marksl
k

v

8 L-r a class of 15 students o1'rvhich 7 are males.5 students n'ear spectacles.'fhere
Iare nrale students u,ho rvear speciacles. Four stuCents are chosen at random.

Find the ploi:abilitr that

(a) all lemales are chosen. 12 marksl

(b) equal irumber of rnales and iemales who lvear spectacies are chosen.

12 ntarksl

\}
(c) all males are chosen if it is known that they all do not w'ear spectacles.

12 mark;)

(d) more students u'ho do not \\-ear spectacles are chosen if it is knou,n that the1,'

are lemales i3 ntarksl

(e) all females or students -who do not n'ear spectacles are chosen.

l-3 marksl

11

254

QS026/2

I XThe probability density function of a continuous random variable is given b1'

.li/-.r\_.\ r _ lI'k--I-n-r-. l<.Y<c

I|. 0) .-\_ othcru ise.

Show that k:2. 13 marks)

Hence- CHOW 13 marks'l

(a) obtain the cumulative distribution function, tr(x)'

3 (b) determine the 81" percentile fbr the distrrbution of }- 13 marks)

(c) calcr:late E(X). 14 marks'l

10 in any large shipment of w'atermelons from a particular orchard, it is known that 2o/o

are unripe. Upon arnva| of a shipment at areceiving depot, random samplings with

replacement are conducted.

(a) Calculate the probability of getting at most one unripe watermelon in a sample

of size 20. l1 ntarksl
v

(b) Approximate the probability of getting one to three unripe watermelons in a

sample of size 50. 15 rnarks)

(c) If the sample size is 1000, approximate the probability of getting not more

than eight unripe watermelons. 16 marks)

END OF BOOKLET

13

255

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS026
2008/2009

256

QS026/1 QS026/1
Mathematics
Paper 1 Matematik

Seruester II Kertas 1

2008t2009 Semester II
2 hours
2008t2009
I
2 jam

I

BAHAGIAN MATRIKULASI CHOW
KEMENTERIAN PELAJARAN MALAYSIA

MATRI C U LATI O N D IVI S I O N

MINISTRY OF EDUCATION MALAYSIA

PEPERIKSAAN SEMESTER PROGRAM MATRIKU LASI
MATRI C U LATI O N P ROG RAMME EXAMIN AT I O N

\{ATEN{ATIK

Kertas 1
2 jarn

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.
DO /VOI OPEIV IHiS BOOKLET UNTIL YOU ARE TOLD IO DO SO

\,

Kertas soalan ini mengandungi 13 halaman bercetak.

This booklet conslsfs of 13 printed page-s.

O Bahagian Matrikulasi

257

QS026/1 CHOW

INSTRUCTIONS TO CANDIDATE:
This question booklet consisis of 10 qLrestions.
Ansu'er all questions.
The tirll tlarks for each question or section are shou,n in the bracket at the end of the question

or sectioit.
A11 steps nrust be shorvn clearly.
Only non-prosrarnmable scientiflc calculators can 'oe useci.

Numerical ans\\'ers rnay be given in the forrn of x, € , sr"rrd. ft'actrons or up to three

signiflcant figures, rvhere appropriate, unless stated othenlise in the question.
Y

Y

3
258

QS026/1

LIST OF MATHEMATICAL FORMULAE

Trigonometn

sin (a t B): sin A cos B * cos ,4 sin B
cos (A t B)= cosAcosB T sinAsinB

tan(At B)= tanA+ tanB
1 T tan Atan B

sinA +sin B:2rinA+2B2 "n.A-B CHOW
A+B
sin,4-sin B :2cos .inA B
22

cosA t cosB :2 B .or'-u

"orA*22
cosA-cos B:-2sin A+B rinA-B

Limit

h..ms_in=ftl

h-->0 h

hIr-m+l0-cohs ft -0

Hyperbolic

:sinh (x + y) sinh x cosh y * cosh x sinh y
:cosh (x + y) cosh x cosh y + sinh x sinh y

cosh2x - sinh2:r: 1

- x:1 tanh2 sech2x

coth2x-1:cosech2x

sinh 2x:2 sinh x cosh x
cosh 2x: cosh2 x * sinh2 x

5
259

LIST OF MATHEMATICAL FORMULAE

Differentiation and Integration

/^lzt,\l / luJ
- cosecl ,r
cot .r
sec ,{ sec .r tan ,r
cosec .r
- cosec .y cot ,r
CHOW
coth ,r cosechl.r

v -r -sech sech .r tanh -r

xcosech - cosech,r coth x

,l.',//"'(l,,rr')) c[r - lrr I ft.t | -,

ir rr,A = /1r'- | r,1rr

I

Sphere V:!n'3 S='lnr'l

Right Circular Cone V = ! n,':1, 5 = 7I t' s
J
v Rightcircularcylinder V=rr2lt S:2nrlt

7
260

QS026/1

1. Use the first principle of derivative to show that

;d (cos ir) - - srn x.

dx

[5 marks]

2. If lu+ol:5 and lu-rl =1, find u.v byusing theproperty a.a=lult.

[6 marks]
CHOW
3. The end points of the diameter of a circle arc A(2,0) and B(10, 4).

Determine:

(a) the equation of the circle.

[4 marks]

(b) the equation of the tangent line to the circle at the point B.

[3 marks]

4. Shou. that 1 = cosh 2x-t stnh2x .

cosh 2-r - sinh 2-r

Hence, evaiuate

lI dx
{ ."tl-, 2, i,rrh 2,r
and leave your answer in term of e .

[7 marksl

5. Water is leaking from the bottom of a conical tank with radius 1.5 meter and height

2 meter at a rate of 0.25 cubic meter per minute. The tank was initialiy full. If the
height of water is 1 meter then find the rate of change of

(a) the water level, [7 rnarks]
(b) the radius of the \\'ater surf-ace. [2 marks]

I
261

QS026/1

6. The equation 4x2 - J't - 24x - 4y +16 = 0 represents a hyperbola.

(a) Determine the coordinates of its centre and vertices.

[7 marks]

(b) Write the equations of the as\ lrprores.
CHOW
[2 marks]

(c) Sketch the hyperbola and label its centl'e. r eriices and asrurptotes.

[3 marks]

7. (a) Bvrvriting tan,L sin'r, that 41trrr) = sect,i-.

= CoS.\' shou ,1t'

Hence, trnd 4(tan (cos2-r)).
,1i

[6 marks]

(b) if sin (2.t.r') = .r+cos (.t-, t ), eraluate r1.r rihen r'= 0. [7 marks]

dx

8. (a) Find A and B if

sin2x cos 3x = Asin 5x + B sin x.

Hence. el,aiuate [7 marks]
[6 marks]
a

J sin 2,r cos3x r/x.

0

(b) Find j sin:l.r 1Lr.

11

262

QS026/1

9. f ffi,The tunction is defined by f (*) = u2 - u-1

(a) fFind the y-intercept and determine the horizontal asymptote of .

[3 marks]

(b) Find the critical points of f and determine the intervals where ,f is
fincreasing and is decreasing.

[7 marks]
CHOW
(c) Sketch the graph of f .

[3 marks]

10. Let P(1, 3, 2), Q(3, -1, 6), and R(5, 2, 0) be points in three dimensional space.

Determine:

(a) the direction cosines for the vector PQ.

[4 marks]

(b) whether PQ and PR are perpendicular vectors.

[4 marks]

(c) an equation of the plane containing P, Q, and R.

[4 marks]

(d) the parametric equations of the line passing through the point B(0, l, 2)

and perpendicular to the plane in part (c).
[3 marks]

END OF BOOKLET

13

263

QS026/2 QS026/2
Mathematics
Matematik
Paper 2
Kertas 2
Semester II
Semester II
2008/2009
2 hours 2048t2009

2 jam

T CHOW I

BAIIAGIAI{ MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATNC UI-ATION DIVISION
MINISTRY OF EDUCATION MAI-AYSIA

PEPERIKSMN SEMESTER PROGMM MATRIKULASI
MAT RIC UI.4TION P ROG RA MME EXAMINATION

MATEMATIK

Kertas 2
2 iam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DO NOT OPEN IHIS BOOKLET UNTIL YOA ARE TALD IO DO SO.

Kertas soalan ini mengandungi 19 halaman bercetak.

Thisbooklet consrbfs of 19 pinted pages.

@ Bahagian Matrikulasi

264

I QS026/2

iIr INSTRUCTIONS TO CANDIDATE:
TLus questron booklet consists of 10 questions.
Y ,{nsu er all qr-restions.
The firll marks for each cluestion or sectior-r are shos'n in the bracket at the end of the question
or section.

A11 steps rnust be sholvn clearly.
Only non-programmable scientitlc calcularors can be used.

;.Numerical answers may be given in the fbrnt of e . surd. fractions or up to three

si-onificant figures, r,there appropriate, unless statecl others ise in the qurestiol.
CHOW

3
265

QS026/2

LIST OF N,IATHEA,IATICAL FORMULAE

Trapeziurn Rule

.|,, /,..)dr' - Ir {(ro +.}",,) * 2 (-r'r -r .r': + ... +,r',,-r)},r,vhere /r b-o

; n

Neryton-Raphson Method

-- 191xn-rl = xn tt = r,r.i,CHOW
/'(.;, )

Statistics

For ungroupeci

II) /t\. -

rr'here

fa) i.- iFor grouped data, the ftth percentiies, P1, L1, [100,n-F,A. -1
fr
v
l

5
266

QS026/2

1. -\ btrokstore recorded the number oi books, X, sold daily u,ith the probability

Pr l' - ..) = ( .X = U.1.2. j"4. Calculare E( X ) . Hence. llncl

'^r''

20

p\t'lx- Etx) -:2l)

[5 marks]

2. (a) in hou, lnany ways can 6 u omen and 3 men be arranged in a rou.' if-
CHOW
(i) the ror,r,begins rvith a man and ends u,ith a u.ornan'.) [2 marks]
(ii) the men must be separated fiotr each other? [2 marks]

)1b ln hor."' lxany ways can a fbur-member committee be formed from 6 \\'omen

rnd :l nten if'the committee has equal number of both sexes?

[2 nrarks]

3. la) ,\Pfirr)\linlti i -,.,.., t .tr. br using the Trapezoiclal RLrle and n = 4

Gir e vou ans\\.er correct to ibLrr decimal places.
[4 marks]

(b) Using the substitution rr : l--rr , shou,that i tt ,rt .. .L. = 1.

J 3

[2 marks]
Give a reason fbr the difTerence of the values obtained in (a) and (b).

[1 mark]

7
267

QS026i2

1. The selection committee of a cornpetition uill determine the w'inners forthe first to

the t-rfth p1ace. If tr,velve females and eight males participate in the competition, in

hou'many ways can one select

(a) three females for the first to tl.re thild place winners, and two males for the

lourth and fitih place n'innels?
[2 marks]

(b) five u,inners ntich consist of tl-iree terlales and tr.r'o males?CHOW[2 marks]
(c) at least fbur f-emales ll in in the competrtion? [3 marks]

5. ofA total 30 rats are randomly captured fronr a plantation and kept to breed in an

experimental laboratory. After a month under observation, the number of rats has

increased by 10. The rate of increase per month of the population is giver, by

"rl'n-: kn(50 - n\

tlt

nhere p is ihe current population and k is a constant.

(a) Solr e tl-re dilferential equation. Give your answer lor p in terms of r.
ll0 marksl

(b) Compr"rte the number of rats after a period olone year.

[2 marks]

9
268

QS026/2

6. The following table shows the frequencies of daily income of 42 fruit sellers.

Income, x (RM) Number of

ti'uit sellers

100Sx<120 4

120 <,r < 1;{0 8

140<-r<160 14

160<.r,<lltt) 12
I CHOW

I 180<.i<l(,)(.) 4

(a) Calcr-r1ate the nrean and the standard der iation of-the daily income.

v t5 marksl

(b) What is the daily income earned by the most tiuit sellers?

[2 marks]

(c) State the skeuness of the dailr'ir-rcoir-re distriLrr-rtion using the Pearson's

coeftlcient ol skeu ness.
[3 marks]

(d) If the ar erage ir.rcome of-fishmongers pe1'day is RM180 u,ith standard

der iation of RNlt2O. determine rvhose income is more stable betn'een the

tishrlongers and fiuit sellers?
[2 marks]

11

269

QS026/2

7. Corllpact discs produced by a f-actor,v are packed in boxes. Each box contains 100

co11-lPact discs. [t is knorvn tl'tal" 49A of the colrpact discs producecl are defective.

(a) Shorv that the probabilitl tirat a l.rtrr chosen at ranclom i.r,ii1 contain at most

3 def'ecti'',,e compact drscs is aLrDt-r)\i1latei1,- 0.43.

CHOW [3 marks]

(b) Find the probabilitl that among 1l Lrores chosen at random, there 11.ill be
4 boxes *hich con,.ai, et l,ost i def-ecti'e compact discs.

[3 marks]

(c) Seventy boxes are chosen at randcin'r. Find the probability that betvveen 20

boxes and 40 boxes, inclusively. u,hich contain at most 3 def-ective compact

discs.

[6 marks'l

v

13
270

QS026/2

8. A total of 2100 nerv students at a college u,ere inteLvieu.ed to f-ind oLrt if they either

receive a scholarship, loan or no financial aid. There are 150 male stuclents. ol
which 50 receive loan and 70 do not receive any financial aid. One hundred f-emale
students receive scholarship. There are 140 students n,ho do not receir.,e any

financial aid.

If a neu'student is selected at randoln. calculate tlie probability that the student is a

(a) female or a scholarship recipient .CHOW

[3 marks]

t}

(b,) loan recipient if it is know'n rhat the student is a ferrale.

[3 marks]

(c) male r,r'ho is a scholarship recipient or a f-erraie u,ho receir,es a loan.

[3 marks]

(d) fernale or non-scl'iolarship recipient.

[3 marks]

!I

15
271

QS026/2

9. The relative fieqr"rency distribution of the marks lbr a Statistics test obtaineo by a

group of 100 students in the iast semester is shorvn in the fbllorving table.

Marks Relatir. e frequencv

0 -19CHOW0.05

20-39 0.i5
4A-59
60-79 0.3 8
80-99
0.32
0.10

(a) Determine the mean and rredian ibr the distribution. [7 rnarks]

Y

(b) A student taking the statistics test is chosen at random. Using the above data,

estin,ate the probabiiity the student has at least 40 marks.
[2 marks]

(c) Trl'o students taking the Statistics test are chosen at random. Using the above

data, estimate the probability both have at least 40 marks.
[3 marks]

Y

17
272

QS026/2

10. A continuous random variable X has the cumulative function siven brr

0 .r'< I
(,.Y-r/).]
: i<.r<7
tr(-r): 12
.1 )-
l4r_- 1_- - ./\

l)

i:
CHOW
where a and b are constants. [4 marks]
[2 marks]
v (a) :Show that rz 1 dan b = 21 . [3 marks]
[3 rnarks]
(b) FindP(2<X<s). [3 marks]

(c) Calculate the median.

(dt Detelnrine the densitv function.

V

(e) Sketch the graph tf J(x)and hence f-lnd the tnode.

END OF BOOKLET

19
273

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS026
2007/2008

274

q,tlfilEi i

!{larhsmeiics

fdU€l t

Sernestar lJ
2ili)7,/2008
2 hoztr.t

@CHOW fqtvq.Ja4iv?i 4:l,ta

f=fY=ir fffieie*latik

{,**ag'i

Ser*ester []

2{.t*7?fl*g

I i4zri

E.AE{AGtrAN MATRSKUI..&SE

KEMENTERIAH FEL.&JAR.AN MALAYSEA

[I,{ATRI C IJLATI ON D T VTSI CN
MJNISTRY OF EDUCATION ALALAYSM

PEPERIKSAAru SEMESTEH FRO€RAM trATRIKULASI
A,MTRIC ULATTON P ROGRA MME LYAMINATION

F.EATEMATIK.

Kertas I

2 iar*

JANGAFI EUKA- KERTA* SSALAru IruISEHIruGGA DIBERITAHU.
DO NOT OPEru IHIS EOOKTEI UNIIL YOUARE TALD TO OO SC,

Kena* -lcel;i; ii,r, tiiei,Grir,JLrngi '!3 t-ra!t're:: Ler:ei3; 275

?,t l;, 1;uirr3,1*: g1;.:i: 3i1, i._.. r,ri .i F i-rn if i.' CAit$.S.
[6;.,-r

3q131 r i..,jAli.ii.tLr,:c;

QSSz6rt CHOW

TNST'RUCTTONS TO CA.NBtrDATE:
This question booklet consists of L0 questions.
Answer all questions.
The futi marks for each question or section are shown in the bracket at the end of the question
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 fr , € ,surd, fractions or up to three

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

2
276

QS$zcit

LES? GF M}"?FSEMATICAT, F*RMUX,Atr

Trigancxne€ry

sin (,a + B\ = sin I cos B * cos Asin B

"os(l tB)= cosAcosB 4 sin AsinB
tanA + tanB
arr(,a* B)=
i T tan AtanB
CHOW
sinA +sin B : ZrirA*2B2 ,orA-B

sinl-sin B :2ror4*B rirrl-B
22

BcosA+ cos : z B *orA- B

"orA*22

cosA-cos B: -Zrinl*B ,inA-B

Limit

I..1msi_n/-zl

h-+0 h

.. _I --cLo)s ft

hm f
h--+u 11

Ftryperbolie

sinh (x + y*) = sinh x coshy + ccsh r sinhy

:cosh (x + y) cosh x coshy + sinh x sirJ-ry

ccsh2x-sinh2x: i

i - tanh2x: sech2x

coth2x-1:cosech2.r

:sinli 2-r Zsinh x cosh x

cosh 2x: cosh2 x + sinh2,x

277

8$S26/t

LIST' OF MATX{EM.ATECAL FORMULAE

Differentiation and Integration

f {.} f'(*}

cot Jc - cos"c2,
sec r
sec x tan x

cosecx -cosecxcotx CHOW

cothx - cosech2x
sechx sechxtanhx

cosechx - cosechx cothr

I#dx:rnlr(,)l*"

Judv=Ln,- tvdu

Sphere Lr1 S = 4rrr2
l/ Gv" S= 7r r-"
S =Znrh
--

3

Right Circular Cone 1L 1-

Right eiraular cyli.nder V =-a Ttt'-h
J

V = nr2h

7
278

GS*Zei'!

;." The total cost of man,;iacmrin-q A bcxes oi ch+cciates (a iuncti'on of lime. ; ) i.t given by

C{,k|:2t2 +k+9{}'.t-

:',,i'here ir{t'| t2 + YJA: .
:Comf,ute the rate of change of the total cost with respect tc time rvhen r i -

[5 rirarks]

2. ,By using the identitl" sin 2 x + cos' .tr = 1 sho-r"' thatCHOW

ccsil\./stn-r-.ti\:\l ;!2-x
(. -,rl)\Hence.
cor^npuie coii sin t\-a/.l/i rri.t,ln6: r;sing calculaior.
\

[6 naarks]

3. Lei /(x) = sinh .t . Prove tha't f-' (.-', -- ;,,[ 't + .'[' t )

[7 marks]

4. The position vectors F, Q, F and s are given such that
(s-p)'(q-r)=0 and (s-q)'(r-p)=0'

(a) Show'that (s - r)'(P - Q): 0'

[4 marks]

(b) If F:48+5j, q:3i+2i, r:*4i+i and s=xi+1';.

finci the va.iues of r and;'.

[3 rnarks]

s. *l ' :

A straight ltne 2x* j,= 4 intersecls a hyperbola )-t+ i= at A anc, E.

(ai Find tire coordinates of ,4 and 8.

[4 marks]

{E}} Iience, find the eqr"raiicir ci a palabola that passes ti-r;ough iire points

/ " 1] and (8. (i)

[6 ntark-.1

279

8S0zSJ'l

6. Show that cos :6-x "o, 2, ( 4 cos2 2x 4) .

Hence, evaluate

G"l *'-.-o-r*z--*\{ A-co-s-z zx -z\Iax
J

0

[1] marks]

7. Find the values of p, q and r which make the ellipseCHOW

4x2 +y2 +px+Qit+r=A

touches the x-axis at the origrn and passes through the point (t,Z) -

Express the equation obtained in the standard form and hence find its foci.
[13 marlsl

8. Given that f (x) = x3 -3x2 - 9.r + 11.

(a) If /intersectsthex-axis at x=1, x= p arrdx=q,fttd pedq.

t3 marks]

(b) / /Detennine the intervals where is increasing and is decreasing.

t4 marks]

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

extremum.

[4 marksl

(d) ISketch the graph of

12 m*rks!

fi1
280

A6AAa t2

Lf fuLidEJ.i

9" Given thal tt=3i+3i-r*, anC v=6;+2's-.I{ E}xi:: (,r*2}+iZ.F-. fieterminelhe

values af a anc b.

[4 mari<si

I{ence. deierrrine

{.a's the direction angles of u. [4 m*rks]
(b) the area cf paralieiogram r.i,ith sides u and v.
CHOW [2 *aarks]

{c} the angle beti.veen u aird v.

[3 marksj

- iS. ia) Lei P{x-_r.,i f,r a poin', or a unii circle '*ith ceaire {} at tne oi'igin" such that

OP mal<es an angle acuie A rEith the posiiive x-axis. Prove that

sin'6+cos'0=1,

and hence, shoi.v that

sec' 0 =7 +tan2 o . [5 marks]

(b) Show that the equation

cos x(sin .r + cGS x) - i = 0

can be reduced to

tarr.rr(1 - tan .r) = $.

Hence, solve foi x on the intervai 1,0, 7n].

[5 r*arks]

(ci Fir-rd the aree enciosecl brt,iae clirve -f (x)=iar;:t(;, - tan.:r) ai-i.i the.r-a.*:is in

the fir.ct quadrant.

[5 xxari*]

ENT} GF'E{}OKE-ET

281

.*,.,

QS026/2
Mathematics
Paper 2

Semester II

2007/20a8
2 hours

:&v: QS$261?

r-t: 'm*t meo n F:]aten:atik
CHOW
Kertas 2

Semrester FI

2**it28*8

j i.r*

BAHAGIAN MATRIKULASI
KEMENTETTIAN PELAJAR,A,N MALAYSIA

MATRIC ULATI ON D TYISIO N
MINISTRY OF EDUCATIAN MAL,/TYSIA

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI

hfATzuC ULATI ON P ROG RA MME EXA MINATIO N

MATEMATIK

Kertas 2
2 jam

JA,NGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DO NOI OPEN THIS BOOKLET ANNL YOU ARE TALD TO DO SO

Kei't*s soaieil ini m*;igandungi 15 haiarn;n herceiak.

,rhis hcokici cc,rsisis 0f 1-q t:,.;iiis3 trt*r.

lO *ehacl tir irl;iili.uiasi

282

QS026/2 CHOW

Ttr\uNrituSvriTvrrAu Ti(-TINNS Tr Tv t CLr A NDID AT[',:
This question booklet consists of 10 questions.
Answer all questions.
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 cleariy.

Only non-programmable scientific calculators can be used.

Numerical answers may be given in the form of fr , € , surd, fractions or up to three

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

\+}
283

T !qT TTII ft,{ATI{FM A'I-ECAI TiT'}PMT IE AF

Trapeziurm Rule

)fu,ftrltr = h tt + )'n\) +2a(/vt -vz +... * !tr-1 r) ,uh7ere/ir = b-u
)|
i\(n 7

Nervtan-Raphson Method CHOW

f (r,\

v-._v_t/(

'llltl fl -t / \

\xn)

Statistics

::

For ungrouped data, the hh percentile,

*(, ) *-'(, *, ) ,

P,-: t[ 2 if s is an integer
if s is a non-integer
I '

[ '(t'])

,=!*o*'here *d :1b;
I s ] the least integer greater than fr.

[l.a), -t * t*,]:-'--': *lFor grouped data. the kth percentites . Pk to \ I

I.

L.r |

284

Q5026/2

1. Given a sample of the heights (in cm) of seedlings in an experiment as follows:

76.A, 68.1, J3.4, 8A.2, 75.4, 18.3

(q> Find the mean and standard deviation of the above data.

['{ marks]

(b) If it was later discovered that the measuring scales were readifig acm below

the correct height, state the changes (if any) on the mean and the variance.

[2 rnarks]
CHOW
2. The probability of a woman giving bi*h to ababy boy is 0.6. If the woman gave birth

to 3 children, find the probability that

(a, the number of sons exceeds the number of daughters. [3 marks]
(n all three children are of the same gender. [3 marks]

3. The probability density function of a discrete random variable X is

)=+P( X = * nhere,r =1.1.3.+.5.6

and fr isaconstant.

(a) +Shcw that k = 2l

(b,) Find Var(x). [2 marks]
[4 marks]

285

QS026/2

4. The probability that a student passes Mathematics is 0.4. If the student passes

Mathematics, the probability that the student will pass Physics is 0.7. If the student
fails Mathematics, the probability the student will pass Physics is 0.63.

(a) Calculate the probability that the student passes

(i) Physics.

[2 marks]

(ii) Mathematics if the student passes Ph,vsics.
CHOW
[2 marks]

(b) Determine whether the events of a student passing Mathematics and Phi'sics

are independent.

[2 marks]

5. A parachutist jumps offan aeroplane on a regular training session. When his

parachute opens, he travels vertically downward with a velocity v". The velocity of

the parachutist at time / minutes is v and his acceleration is given by

A,

6r=&-uv

*'here g is acceleration due to gravity and a is a constant.

(a) Showrhat v -scx-(\as -r,".)n-"'.

[9 martrs]

(b) Determine the difference in the velocities of the parachutist from the fifth to

the tenth minutes.

[2 marks]

(c) Find the velocit)'' of the parachutist after a very long period.

[2 marks]

u
286

QS026/2

6' By sketching the graph of v = x3 and ! = 4x - 2, show that the equation
x' *4x+2:0 has three real roots'

[3 marks]

(l) Show that one of the real roots of the equation lies between I and2-

[3 marks]

(b)., By using the Newton-Raphson method and initial value 2, determine the
real root that lies between I and2, correct to two decimal places.

[6 marks]
CHOW
{"7.Theconsulttaabtileonbteimloews(.inminutes)for100patientsataprivateclirricisgiveninthe \/'

Time interval ( minutes), x. Frequency, /

0-9 9
10-19 34
20-29 20
30-39 18
40-49 9
s0-59
60-69 J

Given Z* f = 262A md lx' f = 91858.

(a) Find

A. the mode and median [4 marks]

(ii' the rnean standard deviation and Pearson skewness coefficient.

[5 marks]

(b) Hence, state with reason, which of the above measures of central tendency

better describes the distribution of the data.

[2 rnarks]

1f

287

QSo26/2

&. (a) There are 10 men, 15 women and 12 children participating in a family day

event.

(i) In how many ways can a group of 7 men, 13 women and 10 children

be formed if a particular lady and a particular child must be in that

grouP?

[3 marks]

t$l Thirly participants are required in an event. L.r how many wa,vs can

this group be fcrmed if each group must consist of at least 8 men?

[3marks]

(b) A fir,e-ciigit number may be formed from the ciigits 7,2,3,4,5.6,7 and 8

w'ith no repetition. Hovn'many
CHOW
{4 five-digit numbers having values betrveen 10000 and 50000 can be

: formed?

[3 marks]

(u) iffi!;8;, even numbers having values more than 60000 can be

[3 rnarks]

9. The time taken by a student (in hours) to stud-v is given by a continuous variable X,

rvith a cumulative density function of

I o if xSo

r(r):]Ir-r(ro-r)' if o<x<io
It' r if x>10

where k isaconstant. [2 marks]
[2 marks]
(a) Determine the value of k. [2 marks]
[3 marks]
&) Find P(:<x.e). [4 marksj

{t4 Determine the probabilit-v densit.v function of X for 0 S x < 10 .
fdi' Fincl the median off .
te) Obiain the variance oi-tr.

IJ
288

QS026/2

10" Assume that the number of e-mails received by a student daily has a Poisson

distribution with a mean of 5.

(a) (i) Determine the probabiliti' that the student receives betr*,'een 5 and 13
.. e-mails daily.

[2 marks]

&{S If the probability of a student receiving not more than m e-mails in a

day is A.616, determine the value of rz.

[4 marks]
CHOW
(b) if i 5 days are randornll, chosen. find the probability that the student receives

betu,een 5 ar-rd 13 e-mails daily for a period of 9 da,vs.

[3 marks]

(c) If 150 days are randomly chosen. use the normal approrimation to find the

probability that the student receives between 5 and 13 e-mails daily for less
than 70 days.

[6 marks]

END OF BOOKLET

t5

289

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS026
2006/2007

290

QS026/1

Mathematics

Paper 1

IISemester

2006/2007

2 hours

,4ffLiJ.TgY::= QS026/1
CHOW
Matematik

Kertas 1

Semester II

2406D007

2 iam

BAIIAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATRICULATION DIYISION
MINISTRY OF EDUCATION MALAYSIA

I PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI
MATNC ULAT I ON P ROG RA MME EXAMINATI ON

MATEMATIK

Kertas 1
2 jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.
DA NOT OPEAJ IHIS BOOKLET UNTIL YOU ARE TOLD TO DO SO

Kertas soalan ini mengandungi 13 halaman bercetak.
This booklet consrsfs of13 pinted pages.

@ Bahagian Matrikulasi

291

QS026/1 CHOW

INSTRUCTIONS TO CANDIDATE:
This question booklet consists of 10 questions.
Answer all questions.
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 clearly.

Only non-programmable scientific calculators can be used.

Numerical answers may be given in the form of fr , e , surd, fractions oI up to three

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

v

3
292

QS026/1

LIST OF MATHEMATICAL FORMULAE

Trigonometry

sin (,a t B)= sinlcosB * coslsinB

(l t B)= Icos cos B T sin --1 sin B
B)=
"or tanA + tan B

tan(At 1 T tan Atan B

sin I + sin B :2 ,in'1+2B2 .o, '' - B CHOW
sinl-sin B:2cosA+2B2ri, l-B
A+ B
lcos * cos B: 2 cos .or'-'
22
cosl -cos B : -2 rinA+ B ri, I - B

Limit

h..m_si-nlft

h-+0 h

.. 1-cos /r

lrm

h--+t) nI

--[)

Hyperbolic

v :sinh (.r + y) sinh x cosh y * cosh x sinh y

:cosh (x + y) cosh x cosh y + sinh x sinh y

cosh2x - sinh2x: 1
I - tanh2x: sech2x

cothzx-1:cosech2x

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

5
293

LIST OF MATHEMATICAL FORMULAE

Differentiation and Integration

f (.) t'(,)

cot x - cosecl.,
xsec
sec .r tan.r
COSCC -I
- COSEC -I COt .T
CHOW
coth x - cosech2x

!7 sechx -sech xtznhx

cosechx - cosechx coth;r

'l4"f 4(*)dx = rn lr(41*,

!udv=tty- [vdu

Sphere V=!n'3 S=4nr2

3 S=firs

RightCircularCone V=!nr2h S=2nrh
Rightcircularcylinder J

V=nr2h

7
294

QS026/1

1. Given that tan-r (-Z)= a and tan-'(:) = p,frndthe value of cot(a + p). [5 marks]

2. Thevectors a, b and c aresuchthat bxc=3i and cxa=2i+k,wherei, j and kare

unit vectors. Express (a +b)x(a-b *-lc) in terms of i, j and k.

[6 marks]

3. (a) Show tt x L(cosh.t) = sinhCHOW.r, [2 marks]
dx' [5 marks]

(b) Evaluate Jl ,sinhr2,1,.- to three decrrnal places.

Milk is being poured into ahemispherical boulof radrus 4 cm atthe rate of 3ncm'/sec.

If the depth of the milk in the borvl is /z cm, its volume Zis

I (.., h3\ 1

=n)4h- --lc3m, '.

[

m :1

At the instant the ilk is cm deep. find

(a) the rate of change of ft. 14 marksl
(b) the rate of change of the radius of the milk's surface. [3 marks]

s. Showthat sinr-sin B =2."'(#J",[#)

fHence, evaluate cos4xsin 2xdx, giving your answer in a fraction form.
[0 marks]

I

295

QS026/1

6. Let Z be a line passing through the centre of the circle x' + y' - 2x - 2y = 7

and perpendicular to the line 3x + 4y = 7. Find

(a) the coordinates of the points of intersections of I and the circle. [9 marks]

(b) the equations of the tangents to the circle parallel to3x +4y =J. [3 marks]

7. Theconic sectiongivenby 9y'--1.r:.18r -16.r-.13 =0 is ahyperbola.CHOW

(a) Express the equation in the standard iorm. [3 marks]

(b) Determine the coordinates of the cenre. the r ertices and the foci of the

hyperbola. [4 marks]

(c) Find the equations of the asvmptotes t2 marks]

(d) Sketch the graph. [3 marks]

8. The points A(-2,1,2), B(5, -7, -3) and C(3, 3, 1) lie on the plane fI1.

The equation of a second plane IIz, is given as 2x - y -22 = 5.

y

(a) Find the vectors VE andii . [3 marks]

(b) Determine the Cartesian equation of II1. [5 marks]

(c) Find the acute angle between llr andfl2, giving your answer in degrees.

[5 marks]

11
296

QS026/1

g. The function/is defin edby f (x) = *'r- Ol .

x+Y

(a) Determine the asymptote(s) of I [2 marks]
(b) Find the critical number(s) of f and determine the intervals where/ is
i6 marks]
increasing and /is decreasing. [2 marks]
[3 marks]
(c) Find the coordinates of the local extrema of f.

(d) Sketch the graph of I
CHOW
10. (a) Shor,vthat 4sin20+3cos20 -3sind-3 : sind(8cos6-6sin e -r.

Y [3 marks]

(b) Express 8cosd - 6sind rn the tirrm of Rsin(6 - a). rvhere R is positive

and a is an acute angle rn racian [5 marks]
(c) Bl,using the f-acts in (a)and (b). solre

4sin2d = 3sind - 3cos 20 + 3
for 0 < 0 < tr. Give your answers in radian correct to three significant figures.

[7 marks]

END OF QUESTION PAPER
Y

13
297

QS026/2
llathematics

Paper 2

Semester II

2A05/2007
2 hours

4L QSo2612

€:7: Matematik

'mFI -n Kertas 2
-+
CHOW Semester II

2006D407

2 jam

BAHAGIAN MATRIKULASI
KEMEI{TERIAI{ PELAJARAN N{ALAYSIA

}AATNC ULATIO|V D IVIS IOI|
],{IAT ISTRY O F E D U CATIO A' I,TA LA }'S iA

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI

M4 TRI C UL,4 TI O l\r P RO G k4 l,{l,tE EX4 MIhtA TI O N

MATEMATIK

Kertas 2
2 jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.
DO NOT OPE,V IHIS BAOKLET UNTIL YOU ARE TOLD IO DO SO,

Kertas soalan ini mengandungi 13 halaman bercetak.
This booklet consr.sfs af 13 printed pages,

Gr Bahagian Matrikuiasi

298

QS026/2 CHOW

INSTRT CTIONS TO CANDIDATE:
This question booklet consists of 10 questions.
Ansu.er all questions.
The full marks fbr each question or section are shown in the bracket at the end of the question

or section.

All steps must be shon'n c1earl1.

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

Numerical answ'ers may be gir.en rn the tbnn of z. e . surd, fractions or up to three

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

Y

3
299

QS026/2

LIST OF MATHEMATICAL FORMULAE

Trapezium Rule

f ,',r)d-'r = : l(r'o *,r,,) + 2tyt * yz + ...* !n-t)),where ft : b-a

Nervton-Raphson Ilethoci

f,ntl= Xn - -l (t,, ) ]1 = i.-.-..,., CHOW

J",r\,,),

Statistics

ry

Fcr uug-rouped data. the Ath percenriie.

(.- il's is an integer

tD=k''..l * I),ix(r)-r(41r-t) , if s is a non-integer
I t([, ])

\\n,efe S nxk :and i s ] the least integer greater than k"
=.-
100

.I ( r\For uroupe; iata. the kth percentiles , Pk = !-k'l\-1: 00ln,-F,.A-1

ft

5
300