Mathematics (Matriculation)

8S025/2

5 Data berikut dikumpulkan dari beberapa pesakit X di sebuah klinik dan ditunjukkan

oleh gambar rajah dahan-dan-daun seperti di bawah:

2 889
J 12366
4 0157 9
5 2336668
6 0235
7 24

80
CHOW
Berdasarkan gambar rajah diberi,

(a) cari mod, median, kuartil pertama dan kuartil ketiga.

14 marknhl

(b) >lcari min dan sisihan piawai diberi Xx: 1335 dan :71783.

14 markahl

(c) hitung pekali kepencongan Pearson dan nyatakan kepencongan taburan data

tersebut.
13 markahl

10 151

QS025/2

5 The following data are collected from a number of patients X inaclinic and is

represented by the stem-and-leaf diagram as below:

2 889
aJ 12366
4 01579
5 2336 6 68
6 0235
7 24

80
CHOW
Based on the given diagram,

(a) find the mode, median, first and third quartiles.

14 marksl

(b) find the mean and standard deviation given that I.x: 1335 andZf :71783.

14 marks)

(c) calculate Pearson's coefficient of skewness and state the skewness of the data

distribution.

[3 marksf

11 152

QS025/2

6 Tujuh kotak yang sama dilabel dengan nombor 1,2,3,4,5,6 dan7. Jika lima kotak

dipilih secara rawak,

(a) cari bilangao cara berlainan untuk menyusun kotak tersebut dalam satu barisan

supaya

(i) terdapat dua kotak bemombor ganjil dan tiga kotak bernombor genap.

13 marknhl
CHOW
(ii) terdapat hanya satu kotak bernombor genap.

13 markah)

(b) cari kebarangkalian terdapat hanya dua kotak bernombor ganjil yang

bersebelahan.
15 markahl

12 153

QS025/2

6 Seven identical boxes are labeled with numbers 1, 2,3,4,5, 6 and 7. If five boxes are

chosen at random,

(a) find the number of different ways to uurange the boxes in a row such that

(i) there are two odd and three even numbered boxes.

CHOW 13 marlcsl

(ii) there are only one even numbered box.

13 marks)

(b) find the probability that there are only two odd numbered boxes next to each

other.
15 marksl

13 154

QS025/2

7 Di sebuah kolej terdapat 150 pelajar yang mengambil kursus Kimia, Fizik dan

Biologi. Dalam kalangan pelajar tersebut, 92 ialahperempuan. Terdapat 48 pelajar
mengambil kursus Kimia yang mana 28 daripadanya ialah perempuan. Separuh
daripada 68 pelajar yang mengambil kursus Fizik adalah perempuan.

(a) Bina jadual kontingensi bagi data diberi.

CHOW [3 markah]

(b) Seorang pelajar dipilih secara rawak. Cari kebarangkalian pelajar tersebut

(i) mengambil kursus Biologi. ll markah)
(ii) lelaki dan diketahui ia mengambil kursus Biologi.
(iii) mengambil kursus Biologi atau perempuan. 12 markahl
12 marknhl

(c) Dua pelajar dipilih secara rawak, cari kebarangkalian sekurang-kurangnya

seorang pelajar ialah perempuan dan mengambil kursus Biologi.
14 marknhl

14 155

QSo2512

7 In a college there are 150 students taking courses in Chemistry, Physics and Biology.

. Among the students,92 arc females. There are 48 students taking Chemistry which 28
are females. Half of the 68 students taking Physics are females.

(a) Construct the contingency table for the given data.

[3 marlrs)

(b) A student is chosen at random. Find the probability that the studentCHOW

(i) takes Biology.

ll marlcl

(iD is a male, given that he takes Biology.

12 marlal

(iii) takes Biology or afemale.

12 marl<sl

(c) Two students are chosen at random, find the probability at least one student is

a female and takes Biology.
14 marksl

15 156

QS025l2 CHOW

8 Sebiji telur dikelaskan sebagai gred A jika beratnya adalah sekurang-kurangnya

100 gram. Katakan kebarangkalian telur yang dihasilkan di suatu ladang dikelaskan
sebagai gred A ialah 0.4.

(a) Jika 15 biji telur dipilih secara rawak daripada ladang tersebut, hitung

kebarangkalian bahawa lebih 20% daipadanya adalah bukan telur gred A.
14 markahf

(b) Seorang peniaga membeli 500 biji telur dari ladang tersebut.
(i) Anggar peratusan bahawa peniaga ini telah membeli daripada

220hingga23O telur gred A.
15 markahl

(iD Jika kebarangkalian tidak lebih daripada m btJi telur gred A yang

dibeli ialah 0.9956, tentukan rulal m.
14 markahl

16 157

QS()25/2

8 An egg is classified as grade A if it weighs at least 100 grams. Suppose eggs lay at a

particular farm has the probability of 0.4 being classified as grade A eggs.

(a) If 15 eggs are selected at random from the farm, calculate the probability that

more than 20Yo of them are not grade A eggs.

14 marks)

(b) A retailer bought 500 eggs from the farm.CHOW

(i) Approximate the percentage that the retailer would have bought from

220 to230 grade A eggs.

15 marl<sl

(ii) If the probability not more than m of the eggs bought are of grade A is

0.9956, determine thevalue m.
14 marlal

17 158

QS025/2

9 Katakan pembolehubah rawak X mewakili nombor yang diperoleh apabila sebiji

dadu tak saksama dilambung. Kebarangkalian dadu tak saksama ini mendapat nombor
ganjil adalah tiga kali ganda kebarangkalian mendapat nombor genap dalam setiap

lambungan.

(a) Jika dadu dilambung sekali,

CHOW(i) binajadual taburan kebarangkalianbagi X.

14 markahl

(ii) cari kebarangkalian mendapat nombor kurang daripada 3.

12 markahl

(iii) cari min dan varians bagi X.

15 markahl

(b) Jika dadu dilambung 100 kali, cari nilai jangkaan mendapat nombor 6.

12 markahl

18 159

QS02s/2

9 XLet be the random variable representing the number obtained when a biased dice is

rolled. The probability of the biased dice to give odd numbers is three times higher
than even numbers when it is rolled.

(a) If the dice is rolled once,

CHOW(i) construct a probability distribution table for X. 14 marl<sl
(iD find the probability of getting a number less than 3.
(iii) find the mean and variance of X. f2 marksl
f5 marlal

(b) If the dice is rolled 100 times, find the expected value of getting the number 6.

12 marksl

19 160

QS025/2

10 XFungsi taburan longgokan suatu pembolehubah rawak selanjar, diberikan seperti

berikut:

[0, x<o

r(,)=]|,{,*01, o<x<4
L| "-l, x> 4

(a) Hitung r(lx-tl< t). CHOW 13 markahl
[4 markah)
(b) Cari median.
l8 marknhl
(c) Tentukan fungsi ketumpatan kebarangkalian bagi X.

Seterusnya, nilaikan e(ZX'? -t).

KERTAS SOALAN TAMAT

20 161

QS025/2

10 The cumulative distribution function of a continuous random variable, X is given as

follows:

[0, x<o
r(r)=l]r(r*+), o<x<4

t5z

1I 1, x> 4

(a) Calculate r(lx-tl<t). CHOW [3 marks]
14 marlaf
(b) Find the median.
18 marksl
(c) Determine the probability density function of X .

Hence, evaluate n(lX'z -f).

END OF QUESTIONS PAPER

21 162

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS025
2013/2014

163

QS025/1 QS025/1
Matlrcmalix
Matematik
Paperl
Kertas 1
Semester II
Semester II
Session 2013/2014
2 hours Sesi 2013/2014

2 jam

KEMENTE,RIAN

PtrNDIDIKAN
MAIAYSIA

BAHAGIAN MATRIKULASI CHOW

MATruCUI.4TION DIVISION

PEPERIKSMN SEMESTER PROGMM MATRIKULASI
MATRIC U-/ITION PROGRAMME FXAMINATION

MATEMATIK

Kertas I

2 jam

I NJANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
NOT OPEN IHIS QUESI/ON PAPERUMILYOU ARETAD IODO SO,

Kertas soalan ini mengandungi 17 halaman bercetak.
This quedion paperconsrsts of 17 pinted pqes.

@ Bahagian Matrikulasi

164

QS025/1 CHOW

INSTRUCTIONS TO CANDIDATE:
This question paper consists of 10 questions.
Answer all questions.
All answers must be written in the answer 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 clearly.
Only non-programmable scientific calculators can be used.

Numerical answers may be given in the form of r, €, srtrd, fractions or up to three significant

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

Y.

3 165

QS025/1

LIST OF MATHEMATICAL FORMULAE
Trigonometry

sln (eXA)= sinl cos B * cos A sin B

(etB)=coSl cos B + sin A sin B

tanA 1- tanB
1 T tan AtanB
tan (.ttB) = CHOW

sinl + sin,B: 2 B u

rinA+22 "rrn-

3 sinl - sin B: 2 B ,inA- B

"o"A*22

cos I * cos B :2 "orO2*2' "orn- U

cosl -cos B : -z "ioA* B "ioA- B

sin2A: 2 sinA cos ,4

cos 2A = cos2 A-sin? A

= 2 cos2 A-l

= 1-2sin2 A

-

2tanA

l-tan'A

l-cos2A

sin' A =

2

cos' A = l+cos2A

5 166

I

QS02s/1 CHOW

LIST OF MATHEMATICAL F'ORMULAE
Differentiation and Integration

d,

fr("orr)= -cosec'x

d,
f (secr) = secrtanx
d,
f ("ot".*) = -"ot". xcotx

I f'1*1,"') dx = er(x) +c

Iffirf'(r\ dx = rn l,t-(z.)r l*,

Iudv = tw - !vdu

sphere Y:! n" S = 4nr2
3
Right circular cone S = ttrz t nrh
Right circular cylinder V =: nr2h
3 S =2nr2 +2rrh

V : nr2h

7 167

QS025/1

LIST OF MATHEMATICAL FORMULAE

Numerical Methods
Iteration Method:

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

Newton-Raphson Method : CHOWlt=1,2,3,....

xn+t = -r-#,

!; Conics

Circle:

(x h)2 +(y-k)'=r'
*' + y'+2gx+2fy+c=0
nr t Wr + g@+x,)+ .f(y + y1)+ c = 0

,=Jr+{-c

v Parabola:

(*-h)'=4p(y-k)

(v-k)'=4p(x-h)

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

Ellipse:

(x-h)' _._ (y-k)' _ r

or- ' b-) ' -l

F(h+s,l(1 or F(h,k+c)

I 168

QS025/1

1 Determine [cot'zesint 2e d0.

[5 marks]

2 Two circles of radius 5 units pass through the origin with their centres lie on the line

x * y = 1. Show that the equations of the circles are

x' + y' +6x-8y = 0 and x' + y' -8x+ 6y =0.CHOW16 marks)

J 3 pGiven nonzero vectors and q are perpendicular. Prove that

(a) ly* gl' =lzl' *lsl' . 13 marl<sl
(b) le* gl=le.- gl. 14 marlcsl

aO 4 Given the points A(1,2,-2), 8(2,4,6) and C(-4,3,-1). Find the area of the triangle

ABC.
17 marksl

11
169

QS025/1

5 According to the Newton's law of cooling, the rate of change of temperature of an

object is proportional to the difference in temperature between the object and the

surrounding temperature, M. The law is given by the following differential equation

4L=-r(r-u)

dt\

where f (r) is the temperature of the object at time / and ft is a constant.CHOW

(a) Express Z in terms of time r.

15 marlcsl

(b) A bowl of soup is removed from an oven with temperature at 60". What is the

temperature after 5 minutes if k = 0.04 assuming that the temperature of the

surrounding is 26.9".

16 marlrsl

6 (a) *.*'r'Use trapezoidal rule with four subintervals to estimat "'1 correct to

0

tl, four decimal places.

[6 marks]

(b) Given f (x)= x' -5x+3. Show that f (x):0 can be written as

x = g(x) = -:x-.'-5 With the initial value xr = 0.5, find the roots of f (x) by

using iteration method. Hence, calculate the root of f (x) accurate to three

decimal places.
[6 marksl

13
170

QS()25/1 13 marks)
16 marltsl
7 Given 9x' -72x +16y2 +32y = -16 is the equation of an ellipse. [3 marks]

(a) Write the equation in standard form.

(b) Find the foci, vertices, lengths of the major and minor axes.

(c) Sketch the graph.
CHOW
Express 7x2 +3x+2 as partial fractions. Hence, evaluate 4 (,7+xzr+)^'3(dx*x-+z2)*"'

(x+l)'(*-z) II

|2 marl<sl

b

15
171

QS025/1

9 Giventwoplanes 7\:x+2y+ z=1, frz:2x-y+42=1 andthe straightline
,. x-2 y+3 z-l
245

(a) Find an acute angle between the planes n, and nr.

[5 marksl

(b) Write the equation of Z in parametric form. Hence, find the intersection pointCHOW

betweenthe straight line Z andthe plane tr,.

[5 marks)

I(c Find a Cartesian equation of the plane which is orthogonal to the straight line
l. and passes through the point (1.2.-3).

[3 marks]

10 Given -l(.r) = 1n.r.

(a) /.Sketch the graph of Shade the region R which is bounded by -f (x), x-axis,
x:l and x=2.

[3 marksl

(b) Find the area of R.

[5 marl<sl

(c) Find the volume of the solid generated when the region R is rotated 360"

about the x-axis.

[7 morksl

END OF QUESTION PAPER 172

17

QS025/2 QS025/2

Mathemahcs Matematik

Papr2 Kertas 2

Sernester II Semester II

Session 2013/2014 Sesi 2013/2014

2 hours 2 jam

KEME,NTE,RIAi{
PE,I.{DIDIKAN
MALAYSIA

BAHAGIAN MATRIKULASI CHOW

]vIAn?IC U-ATION DIWSION

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
]IIATRICUU TION PROGMMME EXAMINAUON

MATEMATIK

Kertas 2

2 jam

JANGAN BUKA KERTAS SOALAN INISEH]NGGA DIBERITAHU.
DONOTOPEN IHISQUESI/ON PAPERUNTILYOU ARE IOLD IODOSO,

:,

Kertas soalan ini mengandungi 19 halaman bercetak.

his question paperconsbfs of 19 pinted pqes.

@ Bahagian Matrikulasi

173

QS02sl2 CHOW

INSTRUCTIONS TO CANDIDATE:
This question paper consists of 10 questions.
Answer all questions.
All answers must be written in the answer 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 clearly.
Only non-programmable scientific calculators can be used.

U Numerical answers may be given in the form of E, e, st)rd,fractions or up to three significant

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

-

3
174

QS025l2

LIST OF MATHEMATICAL FORMULAE

Statistics

For ungrouped data, the hh percentile,

-PK,-= I|lx",(') +-r('* r) , if s is an integer
if s is a non-integer
2

I , CHOW

L '(t" ])

=:::where t :and Is ] the least integer greater than ft.
100

y l(L\"-F,-,1, *l /For grouped data, the kth percentrilretsrs, rPkk = Lkr _ \100

^'
I

l___r ),
-]
[, 4 .
For grouped data, the mode, M = Lu*
Lq+ d2 )
".

Variance

-, ->f,*,'-nI(-zl f,*,)'

- Binomial Distribution

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

Poisson Distribution

x-1@)

4=+,P(X = x =0,1,2,3,...

5
175

QS025/2

1 The weights (in kg) to the nearest integer of ten lecturers in a college are given as

follows:

51, 74, 59, 59, w, 51, 60, 70, w + 8, 56

where w is an integer. If the mean weight is 65 kg, determine the value of w. Hence,
find the mode and the 80th percentile.

16 morlcsl
CHOW
2 The events A andB are independent withP(,4) = x, P(B)= x+ 0.3 and

P (A aB) = 0.0+. Determine the value of x. Hence , find P (A., B)' .

16 marksl

3 A discrete random variable Xhas a Poisson distribution with parameter ).. By using

fi,(X:its probability distribution tunction, show that P 1*= y) where y
y + 1) =

is an integer. Given that E(X): 1.5, find P(X =Z).

[6 marks]

4 The probability distribution function of a discrete random variable Xis given as

follows:

X I 2J 456

i i I iP(X = x) J
1 1 J?

k kk

where t is a constant. Determine the value of ft. Hence, calculate Yar (2X -l).

l7 marlcsl

7
176

QS02s/2

5 The following table shows the frequency distribution of the number of children at a

childcare center according to their ages (in years).

Age CHOWFrequency

t.0 -2.0 t2

2.0 - 3.0 8
3.0 - 4.0 2
4.0 - 5.0 4
5.0 - 6.0 10

(a) Calculate the mean, mode and median. Hence, state the shape of the

distribution of the data and give your reason.
[8 marlcsl

(b) Calculate and interpret the value of the third quartile.

13 marl<sl

v

9
177

(a) In the final of a science quiz competition, teams A and B sit in rows facing

each other. Each team consists of two females and two males. Find the number

ifof different seating arrangements of all the contestants

(i) in each team, contestants of the same gender request to sit next to each

other.
13 marksl
CHOW
(ii) contestants of the same gender do not sit next to each other in team A,

or in team B, contestants of the same gender sit at both ends of the

row. 13 marlrsl

J

(b) A test consists of ten true-false questions. How many possible arrangements of

ifanswers can be obtained

(i) all questions must be answered?

12 marlcsl

(ii) only six randomly chosen questions must be answered?

14 marksl

J

i

11 178

QS025/2

7 The number of vehicles owned by residents in a housing estate, X is a discrete

random variable with probability distribution function

=)f @)+,t r=o12.

' x=1' 2' 3' 4'

IG*,;.

(a) Verify thatf (x) is a probability distribution function.CHOW

12 marlcsl

O (b) Find the probability that a resident has more than two vehicles.
[2 marl<sl

(c) Find the cumulative distribution function forXand hence determine the

median.
15 marksl

(d) :Let Y 30X + l0 be the monthly fee (in RM) imposed by a security company.

Find the expected amount of monthly fee to be paid.
13 marlrs)

o

ili

uII

IIiI

[.\

I

I

I

13 179

QSo25/2

8 The probability that a person is a carrier of Thalassemia is 0.03. If a person is actually

a carrier, the probability a medical diagnostic test will give a positive result,
indicating that he is a carrier, is 0.92. If the person is actually not a carrier, the
probability of a positive result is 0.03. Draw a tree diagram to represent the given

information.
12 marlal

(a) CHOWA medical diagnostic test is said to be efficient if 5% of the time it gives a
correct positive result. Determine if the test is efficient.

13 marksl

r? O) Find the probability that the test gives a negative result.

12 marksl

(c) What is the probability that the diagnostic test gives a negative result and the
person tested is not a carrier?

12 marl<sl

(d) Two persons who took the test are randomly chosen. What is the probability

that both give positive results?
13 marlcsi

15 180

QS025/2

9 The time for patients to experience complications in a week after a heart surgery, Xis

a continuous random variable with probability density function given by

f(i=l I aQ-x), 0<x<l

I o, otherwise.

*'here a is a constant. Show that a:2.CHOW

12 marlcsl

g Hence, find the

ra) cumulative distribution function ofXand estimate its median.

14 marksl

(b) mean and variance ofX. Calculate Var (3 -2X).

17 marksl

-

17 181

QS(}25/2 CHOW

10 The lifetime of D sized batteries produced by a local factory is normally distributed

with mean 1 1.5 months and standard deviation 0.8 months.

(a) Suppose a battery is selected at random from the factory's production line.
(i) Calculate the probability that the battery's lifetime is between 9.5 and

1 1.5 months, correct to one decimal place.
13 marksl

(iD If the probability that the battery's lifetime is less thanh months is

0.975, determine the value of /r.
[4 marks]

(b) Suppose ten batteries are selected at random from the factory's production

line, calculate the probability that at most three batteries have lifetime between
9.5 and 11.5 months.

[3 marksl

(c) If 100 batteries are selected at random from the factory's production line,

approximate the probability that from 48 to 51 batteries have lifetime between
9.5 and 11.5 months.

15 marl<sl

END OF QUESTION PAPER

19 182

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS025
2012/2013

183

QS025rl QS02ry1
thMitx
PWI Matematik
IISemester
Session 2012/2013 Kertas I
2 hours
Semester II

Sesi2012/2013

2jam

=+t CHOW
,,

BAHAGIAIY MATRIKULASI
KEMENTBRIAIY PELAJARAN MALAYSIA

MATRICWINONDII/NION
MINNTRY OF EDUCATION M4IAWU

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
IIATRICUIATION PRrcRAMME FXAMINAflON

MATEMATIK

Kertas 1
2 jam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.

N IfrT AtrN THIS QUESI/OI'J PAPER UI{NLYOU ARE IOLD IO DO SO,

Y

T

I

Kertas soalan ini mengandungi 15 halaman bercetak. 184

This question paper consisfs of 15 pintd pagx.

@ Bahagian Makikulasi

QSo2511 CHOW

INSTRUCTIONS TO CAITIDIDATE :
This question paper consists of 10 questions.
Answer all questions.
All answers must be written in the answer booklet provided. Use a new page for each

question.

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

A1l steps must be shown clearly.
Only non-programmable scientific calculators can be used.

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

figures, ufrere appropriate, unless stated otherwise in the question.

v

3 185

QS025/1

LIST OF MATHEMATICAL FORMULAE

TrigonometrT

tsin (,a t B) = sin,4 cos ,B cos r4 sin.B

cos (l+B) = ror A cos B T sinr4 sin B

t*n (\At g/1=lTtantaAnALttaannBB
CHOW
sinl + sin.B : 2 B B

rinA+22 "rr'-

3 sinl-sinB: 2"orA*2B2 " A-B

cos.,4 * cos B :2 "orA* B "orn-'

cosl - cos B : -2 "ioA* B ,inA- B

srn2A= 2 sinA cosl

cos2A = cos2 A-sin2 A

:2 cosz A-l

= l-2sinz A

tan 2A - 2tana

l-tan A

l-cosZA

sinz A =

2

cos' A = l+cos2A

2

5 186

QS02511

LIST OF' MATTIEMATICAL I'ORMULAE
Differentiation and Integration

(1"ot*1: -cosec2'r

dxt

d.

f (secx) = secrtanx
fd,("ot..*) = -"ot.r rcotx
CHOW
I f' 1*1"'u) dx = "r(x) * "

!#dr = rn lt(.)l*"

Iudv=tw- [vdu

Sphere V=!n'3 S=4nr2
Right circular cone 3 S = fir' * rrh

Right circular cylinder V =: n r2h S =2nr2 +2ttrh
J

V = rr2h

7 187

QS025/1

LIST OF MATHEMATICAL FORMULAE
Numerical Methods
Iteration Method:

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

Newton-Raphson Method :

xn+t = *r-;f6(,*") n=1,2,3,....
CHOW
v Conies

Circle:

(x - h)' + (y - k)' = r'
x' + y'+2gx+2fy+c=0
nr * yyr+ g(x+xr)+ f (y+ yr)+c =0

,=Jf'+g'-c

Parabola:

V

(*-h)'=4p(y-k)

(y-k)'=AP(x-h)

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

Ellipse:

t(*--jh;)-'=, (vl -k)'

F(h!c,k) or F(h,k!c)

I 188

QS025/1

1 IFind the values of and B ., 2x2_+ 4x +3 _ A(x2 + x +l) + B(2x +l)

x'+x+! x'+x+l

Hence, find J72xx2'+-+x4+x1+3 o*.

[7 marl<s)

2 Solve the differential equation .**y = xsinx, !(n) =1.CHOW

17 marksl

3 Find the center and radius of the circle x' + y' +2x =4. Obtain the equation of the

tangent to the circle at the point (0,2).

17 marksl

4 (a) if a and y are nonzero vectors, showthat u.(uxu)=e.

[3 marl<s]

(b) Find a unit vectorperpendicular to u = -2i+3j -3Land y =2i_k.

14 marl<sl

tD

5 (a) show that xa =3x2 - t has a solution on the interval (t,z).

[2 marksl

(b) Use Newton-Raphson's method with xo = I to estimate the solution for part (a)

correct to four decimal places.

[7 marksl

11 189

QS02s/1

6 (a) + g,Solve the differential equation clx= 5y' given that y= 2 when x = 0.

[4 marl<s]

(b) Assume that P(t) represents the size of a population at any time r and the
increment in population size at time / can be modeled by the differential

ry=equation 0.005P(r) with an initial conditionP(0):1500. Determine
CHOW
the size of this population after 10 years.

16 marksl

j

7 (a) Find sin'x coso x dxby using the substitution a = cos x.
J

(b) Evaluate l'xbxdr. 16 marl<sl

16 marlu)

8 An equatiorr x'-4x-4y*8 =0 represents aparabola.
(a) Determine the vertex, focus and directrix of the parabola.

16 marl<sJ

(b) show that the tangent lines to the parabola at the points A(-2,5) and Be,l)

intersect at the right angle.

[7 marks]

13 190

QS025/1

9 (a) !Sketch and shade the region.R bounded by the curves = x2 + 2, the line
2y-x=2, x=0 and x=2. Hence,findtheareaofR.

17 marksl

(b) If the region ft in part (a) is rotated through 2r radiatabout the r-axis, find

the volume of the solid generated.
16 marl*)
CHOW
10 TheplanefllcontainsalineLwithvectorequation l=tj andapoint f (3,-1,2).

(a) Find a Cartesian equation of II,.

16 marksl

O) Given a second plane fI2 with equation x + 2y * 3z = 4, calculate the angle

between lI, and llr.

14 marks)

(c) Find a vector equation for the line of intersection of II, and IIr.

[5 marks]

Y

END OF QUESTION PAPER

15 191

QS025/2 QS02sr2
Matnnnlics
Matematik
Papr2
Kertas 2
Semester II
Semester II
Session 2012/2013
2 hours Sesi 201212013

trL 2 iam

;:|:3 CHOW

qailF4: I :-.\

BAHAGIAN MATRIKULASI
KEMENTERIAI\ PELAJARAN MALAYSIA

M4TruCUIITTION DII4SION
MINNTPJ OF EDUCATION MAIAYSU

PEPERIKSMN SEMESTER PROGRAM MATRIKULAS!

I,UTRIC WIflON PROGRAMME EX,4MINATION

MATEMATIK

Kertas 2

2 jam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DO NOT OtrN 7HlS QUESTTON PAPER UNNL YOU ARE TOLD IO DO SO.

It

I
I

Kertas soalan inimengandungi '19 halaman bercetak.

This quesliwt paper consisfs of 19 pfirtd pages.

@ Bahagian Makikulasi

192

QS02s/2 CHOW

INSTRUCTIONS TO CANDIDATE:
This question paper consists of 10 questions.
.A.ns*'er all questions.
AIl ansq'ers must be written in the answer booklet provided. Use a new page for each

quesrion.

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

rr seciion.

-{i1 steps must be shown clearly.
Cniv non-programmable scientific calculators can be used.

\umerical answers may be given in the form of fi, e, surd, fractions or up to three significant

:-rgures. u'here appropriate, unless stated otherwise in the question.

-

3 193

QS025/2

LIST OF MATHEMATICAL FORMULAE

Statistics

For ungrouped data, the hh percentile,

+-x(s+t) ,

pr= 1tx(s) 2 , ifsisaninteger
if s is a non-integer
L '(t, ]) CHOW

where t =::: :and I s ] the least integer greater thank.
100

r- [(rt I

"tf*]Forgroupeddata,thekthpercentiles , pk = 4 *l\'oo,l ^-'o-' 1,

\-ariance

,, _Zf,*,'-)(Zf,*,)'

n-l

Binomial Distribution

X - B(n,p)
? p(X : *)= "C,p'(l- p),-,, x:0,1,2,3,...,n

Poisson Distribution

X - PQ')

4=+,P(X : x =0,1,2,3,...

5
194

QS025/2

The mean and median of the ordered sample data 1, 2, 4,7, x, !, ll,12,15,2y are g.7

and 8.5 respectively. Determine the values ofx andy. Hence, find the variance.

[6 marl<sl

A box consists of five grape-flavoured sweets and four strawberry-flavoured sweets.
All the sweets are of the same size. A child chooses at random four sweets from the
box. Find the probability that
CHOW
(a) all sweets are of the same flavour. [3 marksl
14 marksl
(,

(b) less than three sweets are strawberry-flavoured.

3 A fair die is throun once. A random variable represents the score on the uppermost

face of a die. If the score is two or more, then the random variable Xis the score. If

the score is one, the die is to be thrown once again and the random variable Xis the
sum of scores of the two throws. Construct the probability distribution table forX.

16 marksl

y
4 The number of motorcycles arriving at the main entrance of a university during peak
hours has a Poisson distribution with mean three per minute. Find the probability that

(a) at most one motorcycle will arrive in one minute. 13 marks)
(b) exactly five motorcycles will arrive in two minutes. [3 marl<s)

7
195

8s02sf2

5 The following table gives the cumulative frequency distribution for the weights (kg)

of fifty hampers during a festival at a supermarket.

Weight (kg)CHOWCumulative frequency
<2.5
< 5.5 0
< 8.5 5
< 11.5
< 14.5 t5

< 17.5 28
40
50

(a) Find the mean, median and standard deviation.

17 marl<sl

(b) Hence, calculate the Pearson's coefficient of skewness and interpret your

aruIwer.

13 marl*)

(c) State with reason whether mean or median is a better measure of location.
fl mark)

j

9 196

8S025r2

6 A security code is to be formed by using three alphabets and four digits chosen from

the alphabets {a, b, c, d, e} and digits {1,2,3,4,5,6}. All the digits and alphabets

can only be used once. Find the number of different ways the security code can be

formed if

(a) there is no restriction imposed.

13 marksl

CHOWO) all alphabets are next to each other and all digits are next to each other.

13 marks)

1} (c) it consists of at least two consonants.

15 marlrs)

T

11 197

QS025/2

Every year two teams, Unggul and Bestari meet each other in a debate competition.
Past results show that in years when Unggul win, the probability of them winning the
next year is 0.6 and in years when Bestari win, the probability of them winning the
next year is 0.5. It is not possible for the competition to result in a tie. Unggul won the
competition in 2011.

(a) Construct a probability tree diagram for the three years up to 2014.

CHOW 12 marksl

(b) Find the probability that Bestari will win in20l4.

[3 marksl

(c) If Bestari wins in 2014, find the probability that it will be their first win for at

least three years.
13 marksl

(d) Assuming that Bestari wins in 2014, find the smallest value of n such that the

probability of Unggul wins the debate competition for n consecutive years
after 2Al4 is less thaa 0.05.

15 marlcs)

13 198

QS025/2

8 A discrete random variableX has a probability distribution function

'r)-I r'l: ,f-x

Ip\x):1 32 x=1.2.3-4

I x:5

wherefrisaconstant.

(a) Sho* tlrat k = !. CHOW
16

s 12 marksl
[2 marl<sl
O) Find P( < X <3). 14 marlrs)

(c) Calculate the mean ofXand hence, calculate E(2X -3). 15 marl<sl

(d) Find the variance ofXand hence, calculate Var(9 -2X).

5

15
199

QS025/2

9 The continuous random variable Xhas the probability density function

9*, o<x<1,

5

,,rr={!{r-,)', r< x12,

0, otherwise.

(a) Find the cumulative distribution function ofX.CHOW 15 marksl
(b) Find

(i) P(0.5 <x < r.5).

12 marl<sl

(ii) P(x>15)

[2 marks)

(c) Calculate the median ofXcorrect to three decimal places.

[3 marl<s)

t

17 200