Mathematics (Matriculation)

QS026/2 [2 marks]
[5 marks]
1. The mean sample

**, w , 7, 9, 17, 72. 311,, 18

is 10.25 with w constant.

(a) Find w.

(b) Calculate the coefficient of variation and interpret your answer.

2. Given two events A and B with the following probabilities:CHOW

P(A) =53, pfelB) =15, P(AnB) ::

15

Find

(a) P(B). [4 marks]
[3 marks]
(b) e1a e;
I

3. Determrne the general solution of the differential e{ruation dv -l . Hence.

#:3rr.,lx'

).find the equation of the curve that passes through the point (t, - I [7 marks]

y

4. Ali plans to buy a computer and the probabiliq that he gets a loan is 0.6. The

probabilit"v that he will buy the computer if he gets the loan is 0.9. and the probability
that he u,iil stili bu1, the computer er en thouqh ri rthout getting the loan is 0.7.

(a) What is the probabilit.v that Ali .,vill buy a computer? [3 marks]

(b) If it is knorvn that Ali did not buy a computer, what is the probability

that he failed to get the loan? [3 marks]

7
301

QS026/2

: I *.5. (a) Use the trapezoidal rule r..,,ith n 4 to approximate Using definite

J ,*,integration, find the value oi I t{.\ , Compare the trvo answers and give a

reason for the difierence. [7 marks]

(b) Approximate r 7 b-'. usine \euion-Raphson nrethod and initiai value 2, up to

the second iteration. CHOW [3 marksl

6. The foilowing table shou's the ciistrlbLnroi.i tor iiie n,.imber ol rneciical leaves (in Cays)

taken within a ceilain period by'65 ernplovees of a companr'.

v

Number of Medieal Leaves (davs) Number of Employees

i -3 4
4-5
7 -9 6
ra-12
I
13 15
12
i6 - 18 18

19 -21 11

6

(a) Calculate

{i) the mean, mode and median. Hence, describe the distribution of the

v data. [7 marks]

(ii) the variance. [2 marks]

(b) \\-ithout using graphs. fincl rz such that 100r, of the number of medical leaves

exceeds lil. 14 marksl

I

302

QS026i2

A total of 13 candidates comprising of 5 mathematicians and 8 physicists will be

selected to form a committee. In how many ways can

(a) a committee of 5 members can be formed if it consists of at least 3

mathematicians? [4 marks]

(b) all the candidates be placed in a row such that they always sit in a group of the

same expertise? [2 marks]

i (c) any 3 members may be selected from the candidates for the positions ofCHOW
U
president, secretary and treasurer? [2 marks]

(d) a committee of 5 members with a mathematician as president and a physicist

as secretary can be formed?
12 marksl

8. Consider a simple electric circuit with the resistance of 3 () and inductance of 2 H.If

a battery gives a constant voltage of 24 V ard. the switch is closed when / : 0, the

current. I(t). after I seconds is given b-v

ff*), =rz, r(o): o.

(a) Obtain {r). [7 marks]

(b) Determine the difference in the amount of current flowing through the circuit

liorn the fourth to the eighth seconds. Give y'our answer correct to 3 decimal

places. [3 marks]

(c) If current is allorved to flow' through the circuit for a very iong period of time,

estimate (r). [2 marks]

11
303

8S026/2

9. Il a delir erl of microchips, it is known tliat the nnmber of defective is 2 out of i 0.

(a) If i 5 microchips are delivered, calculate the probability that

(i) at least 5 microchips are defective. 13 marksl

(ii) exaetly 11 microchips are good. 12 rnarksl

{b} If 500 microchips a,re Ce lii.ered. iind ii si-rch that the pro"t,abi1iti,. of obtainingCHOW

the number cf defective microchips exceeding n rs 0.147 . [-5 marks]

Y (c) Suppose in another shipment of microchips, the probability of defective is

0.01. If a sample of 300 microchips is taken from the shipment, estimate the

probabiiitr of getting 1 to 3 clelective microchips. i3 mar'k-sl

1S. The probability density function of a continuous random variable X is given by

llo', -l<x<2
f(x)=lI k, 2<x<4

I 0, otherwise

[

where fr is a positive constant. [3 marks]

(a) Find k

(b) Determine ,f(X),the cumulative distribution function forX [4 marks]

(c) Calculate [4 marks]
(i) Yar(X). [1 mark]

(ii) Yar(3X - 2).

(iii) e(lxl > r) [3 marks]

END OF QUESTION PAPER

13
304

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS026
2005/2006

305

QS026/1
Mathematics
Paper 1

IISemester

Session 20A5/2006
2 hours

4L QS026/1

:YZ: Matematik

:l- Kertas 1

'mffi"" Sernester II
CHOW
Sesi 2005/2006

2 iam

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATRIC ULATION DIYIS ION
MINISTRY OF EDUCATION MALAYSIA

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
MATRICULATION P ROGMMME EXAMINATION

MATEMATIK !

Kertas I

2 jam

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

Kertas soalan ini mengandungi9 halaman bercetak.
This booklet conststs of 9 printed pages.

@ Bahagian Matrikulasi

306

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 each of the

question or section.
steps must be shown clearly.

Only non-progrcmmable scientific calculators can be used.
Numerical answers can be given in the fomr of z, e, surd, fractions or up to three significant
figures, where appropriate, unless stated otherwise in the question.

I

I

3
307

QS026/1

LIST OF MATHEMATICAL FORMULAE

Trigonometry

! t Isin (,4 B): sin ,4 cos B cos sin B
cos (l X B): cos A cos.B T sinl sin B

WLU * B): tanA+tanB

I + tan Atan B

Differentiation and Integration CHOW

f(*) f'(*)

cot x - cosec2 x

sec x sec x tan x

cosec x - cosec x cot x

ffidx=tnlf(*\*"

Coordinate Geometry

$erpendicular distance from the point (x,, y, ) to ttre line ax+bx*c=0 1S
t

. l*, +by, + cl

dv:-

+bz

Trapezium RuIe

b" b- a

: :I r G) a* n
X{A, + y,) +'2(1t r * ! z+ ... + y,-r}\,where I

Newton-Raphson Method

x*n.+.t =- **n - f(:'\. n:t,2,3,...
f,(*r),

Sphere V:!,rr' S=4ttr2
J

Right Circular Cone v =Lnr'h S = fils
Right Circular Cylinder J S =2nrh

Y = rr2h

5
308

QS026/1

1. If u, vandwarethreenonzero vectors suchthat u+v+w =0, showthat

U.V _ l*rl'-l"l'-l"l' [5 marks]
2

2. Use the trapezoidal rule with four subintervals to approximate

x

I o"o.o xdx.

J,

Give your answer correct to three decimal places.
CHOW [6 marks]

3. Find an equation of the circle passing tluough the origin and its centre is the focus of

the parabola x2 = 8Y -16. [6 marks]

,

4. Use Newton's method to find an approximate value of '16' Give your answer
[7 marks]
correct to three decimal Places.

! {

5. The equation of a hyperbola is given by l

y' -4x'-8x- 4Y-4=o- [7 marks]
[3 marks]
Determine: [2 marks]

(a) the coordinates of its cente, foci and vertices.
(b) the equations of its asymptotes.

Sketch the graph of the hYPerbola-

6. (a) Show that tan(A + B) = tan A+tafiB [5 marks]
A t*, B
| i*

4',O) If B is an acute angle arrdP = sin-r ,ho* that tar,p = 1 Hence, by using

2

"/S

the expansion of tan(A.+ B), show that

""'(i).,*-'(+)=i [7 marks]

7
309

QS026/1

7. The motion of an object is governed by the equation

d,v =8-b,

ctt

where v is the velocity at time t, g is the gravity and k is a constant.

(a) Find the velocity u by assuming that the object starts from rest.

[10 marks]

(b) Deduce that after a long period of time, the object will move with a constant
CHOW
velociw'kE. [2 marks]

8. ff anL= f . exDress sinx andcosx in terms of /. [2 marks]
2
[4 marks]
Hence, [6 marks]

:(a) find all values of r which satisfu 3cosx - 4sinx 5 .

7t

(b) evaluate JFo L. d.

1+sinx

9. The position vectors of the points P, Q and R are given respectively as

I p=4i+3j+1lk, g=-2i+8k, r=i+2j. i

:

(a) Show that PQR is a right angle triangle and calculate its area. [6 marks]
(b) Find an equation of the plane containing P, Q and R. [3 marks]

(c) Find parametric equations of the straight line passing through the point

(3, -5, 2) afi perpendiculgr to the plane containing P, Q and R. [4 marks]

10. Given that f(x1: Jl+tx2^ '.

(a) Determine the intervals on which / is increasing and decreasing. Hence

find the local extremum. [6 marks]

/(b) Use the second derivative test to determine the intervals on which is concave

upward and concave downward. Hence, find the inflection points.

(c) Sketchthe graph of/. [6 marks]
[3 marks]

END OF QUESTION PAPER

310

QS02612
Mathematics

Paper 2

Semester II

Session 2005/2046
2 hours

6L QS026/2

==-YE, Matematik

: l: Kertas 2

'ffiffir" Semester II
CHOW
Sesi 2005/2006

2 jaru

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATKIC UI-/ITION DIT4SION

MINISTRY OF EDUCATION MAI-,|YSIA

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
MATNC ULATION P ROGRAMME EXAMINATION

MATEMATIK

Kertas 2
2 iam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU,

DO NOTOPEA/ THIS BOOKLET UNTILYOU ARETOLD TO DO SO.

NURULHUDA BT. IAIE
.l,ensyarah Matematik
ncxej ltatrtkubsi Kedah

Kertas soalan ini mengandungi 11 halaman bercetak.
This booklet consr.sfs of 11 printed pages.

@ Bahagian Matrikulasi

311

QS026/2 CHOW

INSTRUCTIONS TO CAFIDIDATE:
This question booklet consists of 10 questions.
Answer all questions.
The fulImarks for each question or section are shown in the bracket at the end of each of the
question or section.
All steps must be shown clearly.

. Only non-programmable scientific calculators can be used.

Numerical answers can be given in the form of n, e, surd, fractions or up to three significant
figures, where appropriate, unless stated otherwise in the question.

a

!

3
312

QS026/2

1. The cumulative distribution for the ages of 50 patients at a clinic on a particular day is
shown in the table below.

Age (Years) Number of Patients

<2 0
5
<12
<22 t2
<32
<42 29
<52 38
<62 45
50
CHOW
Calculate the [2 marks]
[4 marks]
(a) median.
(b) mode.

=2.- ifHow many ways can all the alphabets from the word PUTRAJAYA be arranged

(a) the first letter is P and the last letter is A? [2 marks]

(b) - all the letters A must be next to each other? [2 marks]

(c) t i[2 marks]

they begin with a consonant and end with a vowel?

r' 3s. l h a cooking competition, 6 out of 10 contestants are females. In how many ways can

4. onechoose

(a) the champion, first and second runners-up? [2 marks]

(b) four winners consisting of at least two females? [2 marks]

(c) winners for first, second and third place consisting of trvo females and a male?

[2 marks]

-'

',

Given A and B are two events with the following probabilities:

P{il=l, P(A lB')=; and P(AnB)=*.

(a) Find P(AuB). tfunhrtsl
[2 marks]
(b) Determine whether A and B are independent events.

313

QS026/2 [6 marks]

5. Consider a sample consisting of the following observations:

24 t2 12 t220 32 4 t4

(a) Calculate the mean, median and mode.

(b) A new sample is formed using all the above observations with the three values
in part (a) included. For this new sample, calculate the

(i) first quartile. CHOW

(ii) standard deviation.

[6 marks]

'6. A game involves rolling a fair die followed by drawing a marble from either an urn A

or B. If the outcome is less than 3, then a marble is drawn at random from urn A.

Otherwise, a marble is drawn at random from um B. Urn A contains 3 red'marbles, 4

blue marbles, and 3 green marbles. Urn B contains 3 red marbles and 1 blue marble.

(a) Find the probability that '!

(i)t

a red martle is chosen.

(ii) the outcome of the die is less than 3 if it is known that the marble

drawn is red.

(iii) it is a red or a green marble.

[9 marks]

(b) The similar game is repeated but two marbles are drawn instead from either
um A or B. Find the probability that both marbles are red if the first martle is

taken

(i) with replacement.

(ii) without replacement

[4 rnarks]

7
314

QS02612

.7 The probability distribution of a discrete random variable Xis given as follows:

X1 01 2 J
3p 2p
P(X: x) p p p

Showthatp:l. [2marks]
8 t2 marksl

(a) Find the cumulative distribution frrnction ofX.
(b) Find

(i) the mean.
CHOW
€r) the median.

(iii) r((:x - 0').

[8 marks]

8. The continuous random variableX has the probability density function of

( r, ,-1<xs1

f(x)=]la0{lxl+x , otherwise.

showtht a :* 4: , [5 marks]

-\(.(a)
r\. form.

Find
Pl 0 < X <;z)I arrd give your answer in the surd [2 marks]

(b) X.Find the median and the expected value of [6 marks]

\'-' g. The number of short messages (SMS) received by a teenager in half an hour has a

Poisson disribution with mean p.

(a) If the probabiliry of receiving no SMS within half an hour is 0.0025, show that

p:6 (to the nearest integer).

[2 marks]

(b) Using the value of p-- 6, find the probability that

(i) he receives less than six SMS in half an hour.

(ii) he receives less than six SMS in one hour.

(iii) two teenagers selected at random will receive at least six SMS in half
an hour'

[g marks]

o
315

QS()2612

10. In any of its shipments, a company found that the probability of bad oranges it

supplies is 0.2. At the receiving terminal, a sample is taken at random and the number
ofbad oranges is recorded

(a) A shipment will be rejected if there are more than l0o/o bad oranges in the

sample taken. Calculate the probability that a particular qhipment will be

accepted if a sample of size 20 is taken.

[4 marks]

(b) Using the normal approximation, estimate the probabiiity of obtaining 180 to

210 bad oranges if 1000 oranges are inspected at random.

[7 marks]
CHOW
(c) In another shipment, the probability of obtaining bad oranges is 0.03. The r--,'
probability of rejecting this shipment is 0.022. Using the Poisson
aapproximation, determine the maximum allowable number of bad oranges in

sample of size 300 such that the shipment is accepted.

[4 marks]

END OF QUESTION PAPER I

11

316

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS026
2004/2005

317

QSo26/1 QSo26/1
Mathematics
1Paper Matematik
IISemester
Session 2004/2A05 Kertas 1
2 hours
Semester II

Sesi 2004/2005

2!am

4L:\7: CHOW

-:l::---a-:

BAHAGIAhI MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATNCULATION DIVISION
MINISTRY OF EDUCATION MADLYSIA

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI
MATNC ULATION P ROGRAMME EXAMINATION

MATEMATIK

Kertas I

2 iam

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

Kertas soalan inimengandungi 11 halaman bercetak,
Thisbooklet conslsfs of11 pinted pages.

@ Bahagian Matrikulasi

318

QS026/1 CHOW

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

Answer aII the questions.
The fulImarks for each question or section are shown in the bracket at the end of each of the
question or section.

All steps must be shown clearly.

Only non-programmable scientific calculators can be used.
Numerical answers can be given in the form of n, e, surd, fractions or up to three significant
figures, where appropiate, unless stated otherwise in the question.
Y

3 319

QS026/1

LIST OF MATHEMATICAL FORMULAE

Trigonometry sin (l t B) = sin,4 kos B + kos Asin B

Ikos (Z + B) = kos kos B T sin Asin B

tant\lA* B-\/l= taflA+tanB
1+tanAtanB

Differentiation and Integration

f (.) f'(*) CHOW

cotx - csc'x

sec.x secrtanx

csc_r - cscxcolr

.y

#dxr flr) = rn l/(x)l + c

Coordinate Geometry

Perpendicular distance from the point (x1, y 1) to the line ax +by+c=0is

u)--

Trapezium Rule

a

v

NewtorRaphson Method

rn+r = ,""- {,9"1, n= 1,2,3, ...

J'lx")

Sphere v =! n13 S = 4nrZ
J S= 7[i's
Right circular cone
Right circular cylinder V =! nrzh S =2nrh
J

V = tr2h

5 320

QS026/1

1. Showthat s1ind+1+ 1 =2seczl. [2 marks]
[4 marks]
1- sind

Hence,evaluate .$ro (\,1-s!in3-'x* !'l*.

1+sin3x/

2. Use the trapezoidal rule to approximate lrJt--'a;. with 6 subintervals, giving
CHOW
your answer correct to three decimal places. [6 marks]

3..!7 Solve the following differential equation, [6 marks]

*= tn-"; ,Y(o) = 1'

L4, rina rt [3 marks]
dx [4 marks]
!(a) = tan3 (x3 +2) .
(b) sin(x-y)=ycosx.

5. Find the vertices, foci and the equation of asymptotes of the hyperbola

9xz -16y2 +54x+64y-127 =0. tgmarksl
Sketch the hyperbola and label the vertices, foci and its asymptotes. [3 marks]

(a) Given that /(x)=coSr, 0<x<77. State the domain and range of /-t(x).
/ f-' axes.Sketch the graphs of and
orthe same coordinate [6 marks]

(b) If t*(;)= /, find sinx and cosx in terms of r. Hence, solve

cosx+7sinx=5, for 0 1 x1 tt.

[6 marks]

7 321

QS026/1

-7. Given that f (x) = 3x4 4x3 +l .

(a) Find the intervals of x where /(x) is increasing and decreasing. [4 marks]
[3 marks]
(b) Use the first derivative test to determine the relative maximum [5 marks]

or minimum (if any).

(c) Find the intervals of x where the graph /(x) is concave up and

concave down. Hence, find the inflection points (if any).
CHOW
8. (a) Find the foci of 9x2 + 4y' = 36 and sketch its graph. [5 marks]

(b) By using implicit differentiation, find the gradient of the tangent to the curve

9x2 + 4y2 = 36. Hence, find the coordinates on the curve with madient 2 .

2
[7 marks]

r.9. The figure below shows a triangle ABC circumsuibed in aa circle of radius The

sides AB and AC are equal in length and the angle BAC is 0.

(a) {.ProvethatAB = 2rcos H.n r, ifl- isthe areaofthetriangleABC, show
2
that L = r'(1+ cos 0)sin0.
[4 marks]

#=(b) show that -r2(sing +2sin2o). [3 marks]

(c) If the value of 0 varies, find the maximum area of the triangle in terms of r.

[5 marks]

322
l/

QS02611

10. A(6, 3, 3), B(3, 5, 1) and C(-1, 3, 5) are points in a three-dimensional space. Find

ii!"

(a) the vectors BA and BC in terms of unit vectors i, j dan k. Hence, show

BIthat is perpendicular to Be , [6 marks]

(b) a unit vector that is perpendicular to the plane containing the points A, B and
C, [6 marks]

(c) a Cartesian equation of the plane described in (b).CHOW[3 marks]

END OF QUESTION PAPER

11 323

QS026/2 QS026/2
Mathematics
Matematik
Paper 2
Kertas 2
IISemester
Semester II
Session 2004/2005
2 hours Sesi 2004/2005

2 jam

4L CHOW

-t7:

",XYff'-"

BAHAGIAN MATRIKULASI
KEMENTERIA}I PELAJARAN MALAYSIA

MATRIC UL TION D IYIS ION

MINISTRY OF EDUCATION MALAYSIA

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
MATMC ULATION P ROGMMME EXAMINATION

MATEMATIK

Kertas 2
2 jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.
DA NOT OPEN IHIS BOOKLET UNNL YAU ARE TOLD IO DO SO.

Kertas soalan ini mengandungi 11 halaman bercetak,
This booklet consrsfs of 11 pinted pages.
O Bahagian Matrikulasi

324

QSo26'2 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 each of the

question or section.

All steps must be shown clearlY.

Only non-programmable scientific calculators can be used.

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

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

3 325

QS02612

l. (a) The following data set represents the number of candies that was obtained by

8 children in a game.

,3 0 2 | 2 0 2 3

Determine the median. [2 marks]

The sample variance of a set of data with r: l0 is 209 10

lr
(b) lxi90 i=1 is 65, find

the sample mean. CHOW [5 marks]

2. A gymnastic team consists of 5 men and7 women.

(a)'Y In how many ways can the team be lined up if all the men have to be together?

[2 marks]

(b) Find the number of ways a team of 5 members can be formed if the team

consists of

(i) 3 men and 2 $'omen, [2 marks]

(ii) at least 3 men [3 marks]

J. {.a } !iou'many words consisting of three alphabets can be formed from the word

TSL}{AMI if

(i) none of the alphabets can be repeated? [2 marks]

(ii)Y
every alphabet can only be used once in each word and no word

starts with M? [2 marks]

(b) How many ways can two alphabets be chosen from the word TSLIN and one

alphabet from the word AMI? [2 marks]

:4. Given that the two events E and F rvith P(4 = 0,1 and P(D 0.3 are independent.
(a) State a condition for the two events E and F to be independent. [1 mark ]

(b) Find P(En fl. Are the events E and F mutually exclusive? [3 marks]
[3 marks]
(c) Find P(r'l r1.

5 326

QS026/2

5. The workers in a factory need to attend a competency course and pass three tests. The

probability of passing the first test is 0.9 and if a worker passes a test, the probability
that the worker will pass the subsequent test is 0.7. Instead, if the worker fails, the
probability that the worker will fail the subsequent test is 0.8.

(a) Construct a tree diagram for the events. [3 marks]

(b) Find the probability that aworker will pass the first and the third test.CHOW

(c) Find the probability that a worker will pass at least two tests. [3 marks]
[3 marks]

y

The probability density function of a continuous random variable X is given by

fx, 0<x<l [4 marks]
[3 marks]
"f(x)=I 1r-*, l<x<k [6 marks]

[0, otherwise

ufiere t is a positive constant.

(a) Show that k:2.

(b) Calculate P(0.5 < X <3).

(c) Find the mean and variance ofX.

v

7. A discrete random variable Xhas a probability distribution function given by

iP(X = = *E12-r|I ,, : o, l, 2,3,4, 5
where misaconstant.
<.zX

(a) Determine the value of rn. [2 marks]
[2 marks]
(b) Calculate P(X = x), x : 0, 1,2,3,4,5, [8 marks]

(c) Find the mean and variance for l' = 5 - 2X .

7 327

QSo26/2

8. The following table shows the distribution for the weight of 50 parcels in a post

office.

Weieht fte) Number of Parcels

0.5 - s.5 4
-5.5 10.5
-10.5 15.5 6
i 5.5 - 20.5
t2
20.5 -25.5 t6

2s.5 - 30.s 10
2
CHOW
(a) Calculate the mean and the first quartile of the above sample. [5 marks]

(b) Construct a cumulative frequency table "less than" and subsequently draw an

ogive "less than" on a graph paper. [3 marks]

(c) Based on the graph, answer the following questions.

(i) It is found that 25o/o of the parcels exceed the maximum allowabie
weight.weight. Approximate the allowable maximum
[2 marks]

(ii) Approximate the percentage of parcels whose weight exceeds 23 kg.

[2 marks]

9. The distribution of the number of car breakdorvns on a highway in any one day is
Poisson *ith mean -?,5.

(a) Find the probability that

rii exactly 2 cars break down on a particular day. [2 marks]
rii, at most 5 cars break down on a particular days. [2 marks]

(iii) between 100 to 1 I I cars break donn on rhe davs for the month
of April,
[6 marks]

(b) An auto repair company places 3 of its trucks to provide assistance in car

breakdowns along the above highway everyday. Find the probability that on a
particular day the company could not provide any such assistance. 12 marks]

9 328

QS026/2

10. It is known that i0% of the patients with high fever are confirmed to be suffering

from dengue fever.

(a) If l5 patients with high fever are randomly chosen, find the probability that

(i) less than 6 are confirmed to be suffering from dengue fever.

[3 marks]

(ii) exactly 10 patients with high fever are confirmed to be free of
dengue fever.
CHOW t3 marks]

(b) If 100 patients with high fever are randomly chosen,

(i) approximate the probability that 9 to 14 patients are confirmed to be
suffering from dengue fever.
[5 marks]

(ii) find the value of n such that the probability of more than m patients

that are confirmed to be suffering from dengue fever is 0.025.
[4 marks]

END OF QUESTION PAPER

i

I

11 329

PSPM 2 CHOW
MATRICULATION MATHEMATICS

QS026
2003/2004

330

QS026/1 QS026/1

+ Matematik

'mF Kertas 1

Semester II

Sesi 2003/2004

2 jam

Mathematics

Paper 1

IlSemester

Session 2003/2004

2 hours

CHOW BAHAGIAN MATRIKULASI
KEMENTERIAN PENDIDIKAN MALAYSIA

MAT RI C U LATI ON D IVI S I ON

MINISTRY OF EDUCATION MALAYSIA

PEPERIKSAAN SEMESTER PHOGRAM MATRIKULASI
SEM ESTER EXAMINATION FOR MATRICIJLATION PROGRAMME

SEMESTER II

sEsr 2003/2004

SEMESTER II

sEss/oN 2003/2004

MATEMATIK

Kertas 1
2 jam

MATHEMATICS

Paper I

2 hottrs

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DO NOT OPEN THIS QUESTION BOOKLET UNTIL YOU ARE TOLD TO DO SO,

Kertas soalan ini mengandungi 16 halaman bercetak. 331
This question booklet consists of 16 printed pages.

@ Hak Cipta: Bahagian Matrikulasi
@ Copyright: M atriculation Division.

QS026/1 CHOW

INSTRUCTIONS TO CANDIDATE :
This question booklet consists of 10 questions.
Answer all the questions.
The full marks for each question or section is shown in the bracket at the end of the question

or section.

All steps must be clearly shown.
Only non-programmable scientific calculators can be used.
Numerical answers can be given in the form of z, e, surd, fractions or up to three significant
figures, where appropiate, unless stated otherwise in the question.

I
1

3 332

as026/1

LIST OF MATHEMATICAL FORMULAE

Trigonometry ttsin (a B)-- sin A cos B cos Asin B

cos(at B)= cos Acos B T sin Asin B

/\ /\ tanAttan8
lTtanAtanB

Differentiation and Intergration CHOW f'(*)

f (.) - csc2,

cotx secx tau

secr - csc.r cot x
csc.r

l#-dx=rnl/(")l*"

Coordinate GeometrY

Perpendicular distance from the point (x1, y r) to the line ac + bx + c = 0 is

u^--l*rJT+b;vbr'+cl

Trapezium Rule

n=+iru, d.=rL0o* y,)+z(vr*vr+ ... +vn-r)), where

a

Newton-Raphson Method /k") , o =1,2,3, ...

an+l - ^n f'(*")

Sphere v =! nr3 S = 4nr2
J
Right Circular Cone S=zrs
Right Circular Cylinder v =! nr2h
3 S:Zrrh

v = nr2h

5 333

QS02611

t. A straight line x-2y = 0 intersects a circle x' + y' -8x+6y-15=0 at the points P

and Q. Find the coordinate of P and Q. Hence, find the equation of the circle

1)passing through P, Q and the point (1, [6 marks]

2. Find the shortest distance between the lines 4x+3y+12= 0 and 4x+Jy -15= 0.

Determine the equation of the line equidistant between the two lines above. [5 marks]
CHOW
3. Using the Newton-Raphson method and by taking rh = 0.6, estimate the root of the

\* equation x' -3x' * 1 = 0 correct to three decimal place. [6 marks] ,

4. Determine the general solution of the differential equation 169 -3r- *3.
dx

Hence, 0nd the particular solution of the equation if it has a stationary pciint

corresponding to.r = 1. [J marks]

5. Given an equation 4x2 -3y2 -8x* 18y-59= 0.

a) Show that the equation represents a hyperbola. [4 marks]

b) Determine the coordinates at the center and vertices of the hyperbola.

[3 marks]

c) Find the slopes of the asymptotes. [1 marksJ

d) Sketch the hyperbola and label its asymptotes. [3 marks]

334
7

QS026/1

6. a) 9Find for each of the following cases:
dx

i. y=cos(2 - 3r). [2 marks]
[2 marks]
ii. !=x2 sin 2x. [2 marks]

iii. y+cos y= .t+cos.r.CHOW

b) Given the function ,f(x) =2x3 -6xz -L4x+2,

i. Determine the points on the graph of /(x) where the slope is 4.

[4 marks]

11. Find the normal line equation at the point obtained in (i) for x > 0

[2 marks]

a

t

i

7. Given two vectors a=2i+j+qk and b=ei-Zj+2qk.

a) Determine the values of q such that a and b has the same magnitude.[4 marks]

b) ;["-If q -4, find the angle between the vectors b and ;t) [3 marks]

c) i. Find the value of q if a x b = 8i - 4j+ 2k. [3 marks]

ii. Determine the Cartesian equation of the plane passing through point

( l, 0, 2 ) and perpendicular to a and b. [3 marks]

9 335

Q502611 CHOW '!

8. Evaluate each of the following integrals: [4 marks]
[4 marks]
plu [4 marks]

a) JJ sin Sxcosxdx.

b) It *

I

c) [ 3' ,in' o cosz o do.
Jo

9. Given that f (0) = 4cos0 + 3sin 0 .

I

a) Express /(e) in the form of rcos(0-o) for r>0and Ooco,<90o.

[4 marks]

b) Sketch the curve for f(0). [2 marks]

c) Find all the values of 0 which satisfy f (0) =2. [3 marks]

d) IFind all the values of which satisfy lf O>l<2. [4 marks]

11 336

0s026/1

I0. a) Given that f (x) = xo -Zxz +1.

i. Determine the intervals of x-values where/(x) is increasing and

decreasing. i5 marksl

ii. Use the frrst derivative test to determine the local extremum of

f (x). CHOW [4 marks]

Based on the information obtained above, sketch the graph of f (x).

b) A ladder 5 meter long is leaning against a vertical wall and is sliding down the

wall. When the foot of the ladder is 3 meters away from the wall, the top of

the ladder slides down at the rate of 0.5 m/s. At what rate is the foot of the '

ladder moving away from the wall at this instant. [6 marks]

.l

:

END OF QI]ESTTON PAPER

13 337

QS026/2

Matematik

Kertas 2

Semester II

Sesi 2003/2004

2 jam

Mathematics

Paper 2

Semester II

Session 2003/2004

2hours

-ffin':^\7G- L

z-\ t:o
CHOW
BAHAGIAN MATRIKULASI
KEMENTE,RIAN PENDIDIKAN MALAYSIA

M AT RI C U I-ATI ON D IVI S I O N

Y OF EDTJCATION MALAYSIA

PEPERIKSAAN SEHESTER PROGRAM MATRIKULASI

lffiifs EM ESr E ExA M tNAr ro Ns[fl u t-Ar r o N p Ro G RA M M E

:

sEsl 2003/2004

SEMESTER II

sEssloff 2M3n004

MATEMATIK

Kertas 2
2 jam

MATHEMATICS
Paper 2
2 hours

-

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DO NOT OPENTHIS QUESTION BOOKLET UNTILYOU ARETOLDTO DO SO,

Kertas soalan ini mengandungi 12 halaman bercetak. 338
This question booklet consists of 12 printed pages.

o Hak CISa Baha$an Matrikulasi

@ Cnpyndt Maticulation Diviskn.

QS026/2 CHOW

INSTRUCTIONS TO CANDIDATE :
This question booklet consists of 10 questions.
Answer all the questions.
The full marks for each question or section is shown in the bracket at the end of the question
or section.
All steps must be clearly shown.
Only norrprogramrnable scientific calculators can be used.
Numerical answers can be given in the form of 78, e, surd, fractions or up to three significant

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

3 339

as026/2

1. The number of accidents recorded yearly in a certain district for eight consecutive

years are 96,82,80, x ,94,82,96, and (.r+6). If the sample mean is 88, find the

value of x and the sample median. [5 marks]

2. Fifteen glasses of similar colour and size, are labeled each with a number from I to

15. In how many ways can
CHOW
(a) 8 glasses be selected such that 5 are labelled with odd numbers and the rest

with even numbers? t4 marksl

(b) 5 glasses be an'anged in a row such that the first 3 glasses are labelled with

odd numbers and the other two with even numbers? [3 marks]

3. (a) Fol any sets R and S, P(R) = P(S'n R) + P(S n R). Show that t

P(S I R) = 1- P(S'l R). [2 marks]

(b) If P(R)=0.5, P(S')=0.3 and P(S'IR) =0.4,evaluate P(RttS). [5marks]

4. The average number of students entering the main door of a library between 9:00 to

9:30 am during the school holidays is 10.

(a) Calculate the probability that 3 to 5 students will enter the library at that

particular time. [3 marks]

(b) If the probability of less than rz students entering the library at that time is

0'583, find the value of m' [3 marks]

5 340

os026/2 [4 marks]
[8 marks]
5. The probability distribution function of a random variable X is

r(x =r)=o(r[/'] for x=0, 1,2,3

where t is a constant. Determine

(a) the value of k. Hence, find P(0. X .3)

(b) E(X), Var(X) and Var(2X -3).
CHOW
6. The monthly earnings of operators in a particular factory are normally distributed with
mean RM780 and standard deviation RM8.

(a) If the factory has 900 operators, how many operators earn between RM770 to

RM800 a month? [7 marks]

(b) If 67Vo of the operators earn more than RMd monthly, what is the value of d?
' [5 marks]

7. The probability density function of a continuous random vanable X is given by

(z 0<-r<3 ,

lcx ,

"f(x) = .{ others
I
l.0,

where c is a constant. Find

(a) the value of c and f(t < x < Z) [4 marks]

(b) the median of X. [3 marks]
[5 marks]
(c) E(X) and Var(X).

7 341

os026/2

8. A total of OA students of a private college ride 3 types of motorcycles (KRISS,
SUZUKI and HONDA) to campus. From the total, 75 riders are males. Out of 50
students who ride KRISS, 30 are females. There are 30 males who ride SUZUKI and
5 females who ride HONDA. If one student who rides a motorcycle to campus is

chosen at random, find the probability that the student

(a) rides a KRISS or a SUZUKI. [5 marks]
CHOW
(b) is female or rides a SUZUKI. [2 marks]

(c) rides a SUZUKI given that the student is female. [2 marksl

(d) is male who rides a SIZUKI orfemale who rides a KRISS. [2 marks]

9. The probabiiitl, that a person is cured ftom pneumonia after being given a new type of

medrcine is 0.4.

(a) If a sample of 20 patients is randomly selected,

(i);

find the mean and standard deviation of patients that will be cured.

(ii) find the probability that 4 to 12 patients will be cured. [3 marks]
[4 marks]

(b) If a sample of 100 patients is randomly selected, find the probability that less

than 66 patients will not be cured. [6 marks]

I 342

QS026/2

10- The frequency distribution table for ages (in years) of a sample of 50 participants in a

motivation program is as follows:

Age Class Limit Number of Participants

7 -9 4

L0-t2 l0

13-15 L2
16-18 18
6
t9-2t CHOW

(a) Find the mean, median, mode and standard deviation of the above sample.
marksl

,10

(b) Draw an ogive (less than) and subsequently, approdmate the firsr and third
quartiles, and the percentage of participants whose ages exceed 16 years old.

r rnarksl
J5

END OF QUESTION PAPER

11 343