Matriculation Mathematics PSPM 1

QM016/2

1. Evaluate each of the following limits, if it exists. [3 marks]
[3 marks]
(a) lim +

x-+4 .rlx -2

(b) lim atJ:4x +.x

-6.x-++co *2
CHOW
CHOON Giventhat *= | *d =;y 1-tz where / is a non-zero parameter.
WOOI t*7 '

Show that

dy zt (lt+tr'))''

d-=

Hence find its value at the point (|, 0). [6 marks]

3. If y=e-I'lnx, showthat .t

o(#.*J*,-'(t+x)=s !

[6 marks]

4. lx , x(l
g(x) = ),*+t , l<x<4
\-.
[-r* , x)4.

Find the values of a and 6 so that g is continuous on the interval (-co, oo).

[7 marks]

ls*'+*, x<Z
(a) Given f (x\=l r, x=2

lr*'-1, x>2.

Findthevalue of msuchthat fif(x) exists. Hencefind thevalueof ksuch that / is

continuous at x=2. [6 marks]

7

QM016'2

(b) Given a function f on a closed interval l-2,47 as follows:
/-

I s,f (x)=jItr(*-x-l-)z(t1x+3)'- -2<x<4
x=4.

/Find the intervals on [-2,4] where is continuous

6. Let x2y2+Zxy+4y={.

(a) Findthevalues of A,.Band C if +d=x x(4x/yQ+B:9)+C'.

(b) Determine the value ,f *dx' at the point (Z,Z).

7. A function / is defined by

rI f @) = lx+ rl-2'

(a) Sketch the graph of f Hence, determine its domain and range.
(b) Is/differentiable in its domain? Justi$ your answer.

(c) Evaluate Il, Xr> *.
CHOW [5 marks]
CHOON
WOOI [6 marks]
[6 marks]

1

[4 marks]
[4 marks]
[4 marks]

8. Let R be aregion bounded by y=,'.f,'tn, , !:0, x:l and x:4. Find

(a) the area of R, [5 marks]

(b) the volume of revolution when R is rotated through 360o about the.r- axis.
[7 marks]

g. Express #""4+4x2+a1s .. fractions. [6 marks]
[7 marks]
partial

Hence, evaluate +4x' +l *.
Jl'Zl xoxt +x

I

QM016/2

10. The functions -f, B andh are defined by

f(x)=x' -1, g@)=Ji, *20 and h(g=L, x*0.

x
CHOW
(a) Show thatCHOON
WOOI
Jil'F(x) = @" S.,fXr) = *=L. t2 marksl
[2 marks]
ft) State the domain and range of F. [2 marks]
F.(c) Find the vertical and horizontal asymptotes of [4 marks]

(d) Sketch the graph of F. Determine its points of discontinuity and [5 marks]

hence state the largest interval where F is continuous.

(e) For x) l, find F-t(x) and hence determine real p such that

r'@)=.8 r@).

END OF QUESTION BOOKLET

11

PSPM
MATRICULATION MATHEMATICS

QM016
2005/2006

QMo1611 QM016/1

Mathematics Matematik
Faper 1
Kertas 1
ISemester
Semester I
Session 2005/2006
2 hours Sesi 2005/2006

2 jam

4LCHOW
CHOON
4- _____WOOI
-=

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATNCUIATION DIVISION
MINISTRY OF EDUCATION MALAYSIA

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI

MATNCULATION PROGMMME EXAMINATION 1

MATEMATIK

Kertas 1
2 jam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DO NOI OPEN IHIS BOOKL ET UNTIL YOU ARE TOLD TO DO SO,

Kertas soalan ini mengandungi 11 halaman bercetak.
This booklet consrsfs of11 pinted pages.

@ Bahagian Matrikulasi

CHOW QMo1611
CHOON
WOOIINSTRUCTIONS TO CANDII}ATE:
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 z, e, surd, fractions or up to three significant
figures, where appropriate, unless stated otherwise in the question.

I

r\---

3

QM016/1

LIST OF MATHEMATICAL FORMULAE

For the quadratic equation ax' + bx + c = 0:

* --bt"[t' -q*
2a
CHOW
CHOONFor air arithmetic series:
WOOI
T, = ct+(n-l)d

s, =llzo+(n_\dl

For a geometric series:

Tn = ar'-l

, s, =#,r*r t!

.t

Binomial expansion:

(a+b), = a, +(:).-,r.(;)o-u'+ + (i)"-'u' + +b' ,

dengan neNd*gf')J:, nt',

fu-r\'rl

5

QM016/1

1. Find the values of x satisfuing the equation [5 marks]
1og, (xo + +)= I + logo (x' + +).

2. polynomial P(r):2x3 +ax'-x+6 has (r+1) asafactorandleavesaremainder 12

when divided by (, - 3). Oetermine the values of a and b-

[6 marks]
CHOW
CHOON,/1.aThe quadratic equation .r'+2(k+1)x+(2k+5)=0 has roots arrd1.If the roots
WOOI:are equal, show that k 2. Hence, find the quadratic equation with roots
a'11r'\_
p'-----=.

-OIId

[7 marks]

4. *" I216 +.*dThe third and the sixth terms of a geometric series Determine the

values of the first term and the common ratio. Hence, find the sum of the first nine

terms of the series.

j[7 marks]

5; Solve the following inequalities:

(a) 7x2 + x-6 < x' -4 . [4 marks]
[6 marks]
i.:--

(b) x+3 >3.

x-1

7

QM016/1

6. ' __L-Expand (3-x)' up to the term x3 and determine the interval of x for which the
a+=expansion is valid. Hence, approximat" (2.9)' correct to fots decimal places'

t12 marksl

CHOW7t'. LGrilvveenn ^.t==l[l-2:r, o -o2l1, ",=-l[l-tj;, -11 ''=-Ll'r -o1 o-l
CHOON
WOOI 1; ; ]' o'l *o -tl

-,1

(a) Find matrix D = ,n-(nC)' ' [5 marks]
[7 marks]
(b) show that lADl=lnel.

8. Two factors of the polynomiat P(x)= xt +ffi'+bx-6 arc (x+l) and (x-2)'

Determine the values of a and b, and find the third factor of the polynomial' Hence'

express

r 2x2 -5x -13 1

P( x)

as a sum of partial fractions. t13 marks]

g. (a) Giventwocomplexnumbers zr=1 +3f arrdz'-2-i'express 2!32 in

ztzz

the form of a + bi, where a and' b are real numbers' Hence, determine

lrr+rr l f6 marksl
| ,,r, I

(b) Solvethe equation 5(2x+r; - 4' =16' [7 marks]

9

QM016/1

10. Consider the system of linear equations

x, -2x, *3x. -1 = 0

xt + mx2 *2x, = )

-Zxr+m'xr-4xr+4=3m

where misaconstant.

(a) VWrite the above system of linear equations in augmented matrix, I f1

[2 marks]
CHOW
CHOON
WOOI
(b) By using row operations, show that the above augmented matrix can be
reduced to

-23
:l[1

l0

Lo
m+2 -l i [6 marks]
0m

(i) m:Solve the above system of linear equations for 1.

[ 5 marks]

(iD State the condition of m for which the system of linear equations has
an infinite number of solutions and has no solution.

[2 marks]

END OF QUESTION BOOKLET

11

QM016/2 QM01612
Mathematics
Matematik
Paper 2
Kertas 2
ISemester
Semester I
Session 2005/2006
2 hours Sesi 2005/2006

2 jam

dLCHOW
CHOON
=r=p:A. a.-oWOOI
r. =

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATNCULATION DIVISION
MINISTRY OF EDUCATION MAI-AYSIA

PEPERI KSAAN SEMESTER PROGRAM MATRIKULASI
A/TATRICULATION P ROGRAMME EXAMINATION

MATEMATIK

Kertas 2
2 jam

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

Kertas soalan ini mengandungi 13 halaman bercetak.
This booklet consisfs of 13 printed pages.

@ Bahagian Matrikulasi

CHOW QM016/2
CHOON
WOOIINSTRUCTIONS TO CANDIDATE:
This question booklet consists of l0 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 zr, e, surd, fractions or up to three significant
figures, where appropriate, unless stated otherwise in'the question.

t

3

QMo16/2

LIST OF MATHEMATICAL FORMULAE

Differentiation

tf y -sk) d* *: fQ), maka *=*-*
CHOW
IntegrationCHOON
WOOI
ludr=uv- [vdu

I rl

!

Er,

QM()16/2

1. Functions "f wrdg aredefinedas

.f(r)= "" S(r) =l- x, x e R.

Find f '(*) and hence obtain (, . f -') (r). [5 marks]
CHOW [6 marks]
2. Using integration by parts, evaluate !' *e-"dx.CHOON
WOOI [6 marks]
- 3.:. By taking logarithm on both sides of the equation [3marks]
[4 marks]
v =(*')r; ,

show that ,' = J]!'l't x +Z)(x, )*

4. Find the following limits.

t-

(a) t*#

@) -fri*--+[-0]\-x1{1]*x x)

7

QM016/2

5. (a) /State the conditions of continuity of a function defined in the closed interval

lo, bT- [2 marks]

(b) If function / is defined by

h**', 3<x<o
f(*)=I12-*, 0<x<2

,I

LG-zf z<x<5
CHOW
CHOONsketch its graph.
WOOI
Using the conditions of continuity in (a), determine the value of x where the

function is not continuous. [5 marks]

(c) If function g is defined as

Ax+2, -3<x<1

'.r={x2 + Bx+ A, l<x<2
1-1, 2<x<5
x

'i

find the values of ,4 and B such that g is continuous in the interval [-3, 5].

[5 marks]

6. Given ! = Ax * !x,w' here A atdB are constants and x * 0.

4(a) dx*d t+.
Find
dx'

Hence, show that xzd*x.2' *9d-x2y =0. [5 marks]

(b) Find the values of A ard B if y:3 and !' = -6 when x: \. [2 marks]

(c) If y= "f(*),frnd lim f(*), ii* /(r) andlim /(i) wherethevaluesofl

ofl.and B as found in (b). Hence, sketch the graph [5 marks]

I

QM016/2

7. Express #5x3x-'(*x2'++lr)-1 in the form of partial fractions.

t;!Jl!a.Hence, evaruate y

Give the answer correct to tlree significant figures.

8. (a) Evaluate f * JFl: a.

'- (b) Given f (*)={:,'.,', ; ::, rind [',r 6)*.
g. Consider the parametric equations,

x=tz arrd y=t3-3t.

(a) fiLvulrrut" ,tthe point (3,0).

O) *Find the point (x,y) where '"not defined.
!.- (c) *Determine the interval of r such that dx' .O
CHOW [12 marks]
CHOON [5 marks]
WOOI t7 marks]

: [s marks]

t2 marksl
[6 marks]

11

I

QMo16/2

10. Let /(r)=l1xr, l.-x<e.

(a) -axis.Find the area of the region bounded aV fG) and the x [5 marks]

(b) Hence, find the volume of the solid generated by revolving the region 2n

radian
CHOW
CHOON aboutthe x-axis. Give the answer interms of e and r . [10 marks]
WOOI
END OF QUESTION BOOKLBT

t

€.

13

PSPM
MATRICULATION MATHEMATICS

QM016
2004/2005

['I ' -.- 4 QMMatoh1e6m/1atics QMo16/1
7Paper
Matematik
ISemester
Session 2004/2005 Kertas I
2 hours
Semester I

Sesi 2004/2005

2jam

4LCHOW
CHOON
:i-7-=WOOI

"ffiffid

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATNCULATION DIVISION
MINISTRY OF EDUCATION MALAYSIA

PEPERTKSMN SEMESTER PROGMM MATRIKULAST
MATMCULATION PROGMMME EXAMINATION'

MATEMATIK
Kertas 1
2 jam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DTBERITAHU.
DO NOI OPEN IHIS BOOKLET UNTIL YOU ARE TOLD IO DO SO

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

D- fin@ Bahagian Matrikulasi

,<-er'(?,1^-J^ 8+

QMol6/1
INSTRUCTIONS TO CANDII}ATE:
This question booklet consists of 10 questions.
Answer all questions.
The full marks are shown in the brackets at the end of each question or section.
All work must be clearly shown.
The usage of electronic calculator is allowed.

Nunerical answers can be gtven in the form ofq e, surd fractions or upto three significant

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

*

I
CHOW
CHOON
WOOI

QMo16/1 5

LIST OF MATIIEMATICAL FOR]VIULAE

For the quadratic equation or2 + bx + c: O'.

t-

-b+^lYb'-4ac
'2a
CHOW
X'or an arithmetic series:CHOON
WOOI
T = a +(n-L)d
t

t,=tlza+(n-\dj

For a geome$ric series

ln: ff n-l

Sn= a'(l- rn'\-r+l
l-r

, Binomial Expansion:

- " [zJ"(a+r)' =an + (:\n^r. fl on162+ . . + (("'J\"n- -rbr + .. +b'.
tr./

where ne-,"t/ a*n-d ft'r).:J (,"-n,t\,r,r'

QM01611

l. Solve *5 "-3tnx i 4x :2!. [5 marks]

2. Express 5.555... in the form of a geometric series. Hence find the [4 marks]
(a) sum of the ftstn terms [2 marks]
(b) infinite sum

of the series.
CHOW
CHOON
WOOI
i \_ 3. Using mathematical induction method, show that [6marks]

(lx3)+ (z)(4)+(3)(s)+...+ n(n+D=1@+t)(Zn+7). t3 **tsl
6
[3 marks]
4. Given 2.1: | - 3i and z2:2 * 5i.

(a) + bi.l-
+.Express zt zz in the form of a +

(b) Find the argument of Zrinradian.

5. (a) -2;.--Given (x +3) is one factor ofP(x) = 9 1?sc 1l* Factorise
t'liltcompletely P(-r), and express
Pl*) as a sum of partial fractions.

[8 marks]

(b) Expand ,-+ term.in increasing power up to the fourrh [3 marks]

\x+3)'

7

QM016/1 [3 marks]
[4 marks]
6. Solve the following inequalities: [7 marks]

(a) +, +2> n
x

O) 1+ logz x-6log,2 >o

(c) l,*11=,
l_r*_ ol
CHOW
CHOON7. Show that [3 marks]
WOOI
r

(a) \(o,*, - ai) = a,+t - at.

i=l

(b) ai+1 - ai =2i3i +3i*l if ai : r in terms

i3i. Hence, using (a) evaluate lZitt
i=l
of r.

[10 marks]

A polfromial has the form P(x) = 2x3 -3x2 - px+ q, withx real md p,{ constants.

t'l
(a) -when P(x) is divided by (x - 1) the remainder is e afl. Find the values ofp
roots.and Q, and factorize P(x) completery if 2 is one of the
[7 marks]

(b) Hence, form a quadratic equation oy' * Ay * c :0 if the sum of roots is the

sum of all the roots of P(x) and the product of roots is the product of all the

'roots of P(x). Also, find the roots of the quadratic equation that is formed.
[5 marks]

g. (a) Given z:x*yi, wherexand yarerealnumbers. If lIz'+*'l+i|l=r-.'
0.show that 3x2 +3y2 + 8x + 6y + 7 =
[5 marks]

(b) Givenx:3 *4i is onerootof p(.r):2x3 -g* +lU*75, find alltheroots
[7 marks]
of P(x).

I

QMo16/1

/z x l)

Show that the determinant of the matrix A = llItvx ', I
l)
ttl_,
v I10.
(a)

is (y - x)(z - x)(y - z) for rcal x, y and z. [3 marks]

[r 3l(b) Bysubstitutingx: l, y:2andz:3,thematrix,<U".o-., [l : j.lCHOW
CHOONFind the adjoint and inverse of the matrtx A.
WOOI [6 marks]

(c) The graph of a quadratic equation y = ax' + bx + c passes through the points

whose coordinates are (1, 2), (2,3) and (3, 6).

(il Obtain a system of linear equations to represent the given information.

[2 marks]

(ii) Write down the system in (i) as a matrix equation in the form of

= *:Dwhere':[:r,,1) [1 mark]

, I
i

Use the inverse of the matrix to solve the system of linear equations
in (ii). Hence, find the quadratic equation of the graph.

[3 marks]

END OF QUESTION BOOKLET

11

QM016/2 QM016/2
Mathematics
Matematik
Paper 2
Kertas 2
ISemester
Semester I
Session 2004/2045
2 hours Sesi 2004/2005

2 jam

4L

"fffi--\7-: t:

q-^\ I /-6

BAHAGIAN MATRTKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATNCULATION DIVISION
MINISTRY OF EDUCATION MALAYSIA

PEPERIKSMN SEMESTER PROGMM MATRIKULASI
MATMCULATION PROGMMME EXAMINATION

MATEMATIK

Kertas 2
2 jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.
DO NOI OPEN THIS BOOKLET UNTILYOU ARETOLD TO DO SO.

i \-.
CHOW
CHOON
WOOI

Kertas soalan ini mengandungi 11 halaman bercetak.
This booklet consists of 1 1 pnhfed pages.

@ Bahagian Matrikulasi

Re6'c/,--^t^ YA fi'rl

QMo16/2

INSTRUCTIONS TO CANDIDATE:
This question booklet consists of 10 questions-
Answer all questions.
The fuIl marks are shown in the brackets at the end of each question or section.
All work must be clearly shown.
The usage of electronic calculator is allowed.

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

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

i\,
CHOW
CHOON
WOOI

QMo16/2 5

LIST OF MATIIEMATICAL FORMULAE

Differentiation

CHOW +-+If y = g (t) fand x= (r ), then +d=x dt -
CHOON
WOOI dt

Integration

' J[u'-d- v=uv-lJvdu

i

I

QMo1612 [3 marks]
[3 marks]
1. Given s$)=i/i ana h(i=*.
[3 marks]
(a) Find /(x) such that(/ . s. h\*)= * [3 marks]
(b) Determine the domain of(,f . I " h\*).
CHOW [3 marks]
CHOON2. Find each of the following iimits, if they exist.
WOOI '1
(\/a) / 1* ,) l,
[4 marks]
xri+m^0|[9"*=--l1)
[7 marks]
f\-t/) xJliim* t1!'. -?

3x +6

3. If xy =212s - y)' , find the following values at the point {1,2).

(a) dv

+d_x'

I

+(b)
ctx

'\ 4. By substitut ing u2 = x * l, determine Jt -x:Jx+1' dx .

5. rna J x'e'* dx . [7 marks]

Hence, find the volume of the solid generated by the area bounded by the curve

! = xex and thelines x = 0, x =1, ! = 0 which is rotated at2nradian about thex-axis.

[4 marks]

7

QM016/2 [6 marks]
[5 marla]
6. Ex-pressat'c;'w-2xa-9as parrial fractions.

Hence, evaluate

)1,26,;'--26x-,9*tr*'

giving the answer correct to three significant figures.
CHOW
CHOON7. (a) Given !=2x', find,! [4 marks]
WOOI ax
[8 marks]
(b) If y = e''ln{l+ x), show that
:
G+'D\'d({x4' -+dxl) = xex
t6 marksl
8- Given *=2t -Ltt a11d y=1q 1, where , is a non-zero parameter. [2 marlcs]
(a) Showthat [4 marks]

Qdx=2L\(r-2-Jt'+-)l) .

(b) Hence deduce ,h" +d<x2| for att r.

(c) 9]t2 when , : 1.

Pra6
dx'

I

QM016/2

9. (a) State the conditions for the function / to be continuous at x = c. I mark]
(b) Given that

q(x) :1Il1**'_--'_4+, x * 2

II l, x=2.
CHOW
CHOON(i) Sketch the graph of q. [2 marks]
WOOI(i, Discuss the continuity of q at x:2. [4 marks]

(c) Determine the values of ,4 and.B such that [6 marks]

I x. x<l

I

"f(x)=\Ax+8, 7<x<4

l-z*, x>4

is continuous on the rnterval (-co, co).

/10. A function is defined by i

I

f(x)=ffi

(a) IState the domain of I mark]
: L O) Find the verticirl asymptotes.
[1 mark]

(c) Determine lim ,f(x) und ,la -f @). Hence, statethehorizontal
asymptotes
[4 marks]

(d) fFind /-r and determine the range of [5 marks]

(e) ISketch the graph of [4 marks]

END OF QUESTION BOOKLET

11

PSPM
MATRICULATION MATHEMATICS

QM016
2003/2004

QMol6/l Qlvro16/1

4L:YZ: Matematik
Kertas 1
,,qHffi"U
Semester I

Sesi 2003/2004

2 iam

Mathematics

Papr 1
ISemester

Session 2003/2004

2 hours
CHOW
CHOON BAHAGIAN MATRIKULASI
WOOI KEMENTERIAN PENDIDIKAN MALAYSIA

NdATNCUUITION DIWSION

,\- MINTSTRY OF EDUCATION MAUrySA

PEPERIKSAAIT SEMI,STER PROGRAM MATRIKULASI I

SFAIESTER FXAMINATION FOR MATNC T]I-ATION PROGMMME

SEMESTER I

sEsr 2003naa4

SEMESTERI

SESSION 2ABI2OO4
I

MATEMATIK
Kertas 1
2 jam

MATHEMATICS

Paper I

2 hours

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.

DO NOT OPEN THIS QUESruON BOOKLET UNTIL YOU ARE INSTRUCTED.

Kertas soalan ini mengandungi 1l halaman bercetak.
This question booklet consists of 11 printed pages.

@ Hak cida Bahaghn Matrikutasi 2fi)3

@ Matriculation Division Copyright 2Co3

QMOl6/1
INSTRUCTIONS TO CANDIDATE:
This question booklet consists of 10 questions.
Answer all questions.
The full marks are shown in the brackets at the end of each question or section.
All work must be clearly shown.
The usage of electronic calculator is allowed.

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

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

,\*.

t

\--
CHOW
CHOON
WOOI

QMor6/1 5

LIST OF MATHEMATICAL FOR]VIULAE

For the quadratic equation ox2 + bx+ c: 0.
*b+

x=

2a
CHOW
CHOONFor an arithmetic series:
WOOI
Tr= o+(n-lld

s =!lza+1n-tldl
n /'

For a geometric series

"t ln: ar n-l

S^n a(1- rn ) - r+l

l-r

Binomial Expansion:

('l(a+b)' =an + [) ",-'r-, [;) on'b2+ ..+l("\ la n-r br + ...+ bn,

where neNand[:):6h

QMol6 / r 7

l. By substituting a = 3' , solve the equation tsl
t6l
9' +3 = 28(3'-1) .

) Find the sum of even numbers between 199 and 1999.
CHOW
CHOON3. 5x7 +l7x+17 as a sum of partial fractions. t7l
WOOI
Express (x+2)(x+l)z

\-

-14" The sum of the first fourterms of a geometric series with common ratiq ir ro
2

Determine the tenth term and the infinite sum, ,S- . t7I

1

(a) r matrix A =[l6 -41 i

Let 0lI :
Lr5.
If A2-pA-qI=0 where pard qare realnumbers, lis aZx2identity
q.matrix and 0 is a 2 x 2 null matrix, findp and
t4I

(b) Given a matrix equationAX: B as

iLi ill Lil

(i) Find the determinant of matrix,4. l2l

l' -2",, [-; i q(ii) Given the cofactor matrix
= -i-l, find p nd l2l

1l

(iii) Determine the adjoint matrix of.4 and hence find the inverse ofr4. tzl

QM016 / 1 9

6. If (x - l) and (x + 2) are factors of the expression 4xo - 6.13 + ax' + bx - 72,
determine a and D. Hence, factorise the expression completely.
[ 1]

CHOW7. Solve the following inequalities. tsl
CHOON(a) x'+x-12.>0. t8l
WOOI
(b) 12'- ll = r
lx+21

8. (a) Using the principle of mathematical inductiorq prove that t6l
2+4+6+...+ 2n =n2 +n, where nisapositive integer.

(b) The sum ofthe ftst nterrns of an arithmetic sequen ce is !q+n + 20).
,t

. (i) Write down the expression for the sum of the first (z - 1) terms. 121

I Find the first term and the common difference ofthe above ,lqu.r"..

(iD t5I

(a) Solve 3ln2x =3+1n27 . l4l

2-i='(b) Given a complex number , = -. 121
(i) State z in the form of a + ib where a and D are real numbers.
(iD Find the modulus and argument ofz. t3l

(c) Given the complex numbers u, v and rry suchthat :l_ll_-.+_t_. If v=l_3i
uvw

and w =2+i, state a in the form of a + bi where a and b are real numbers.

t4l

QM016 / 1 11

10. (a) Matrices A and B are given as tsl

I t z 3l B =|l-+t -1 -41
3s l
,a=l-r o 41, -l
L o 2 2) [r r I
-r
CHOW l
CHOON
WOOIFindAB and hence find A-1

(b) .{A company produces three grades of mangoes: Y and Z. The total profit
Ifrom I kg grade x,2 kg grade and 3 kg grade Z mangoes is RM20. The

profit from 4 kg grade Z is equal to the profit from 1 kg grade X mangoes.
The total profit from 2 kg grade land Zkggrade Z mangoes is RM10.

(i) Obtain a system of linear equations to represent the given information.

t3l

(ii) Write down the system in (i) as a matrix equation- tll

(iii). Use the Cramer's rule to solve the system of linear equation. Hence,
I grade. ,state the pro{it per kg for each
[U]

i

END OF QUESTION PAPER

QMor6/2 QMol6/2
Matematik

Kertas 2

Semester I

Sesi 2003/2004

2 jam

Mathematics

Papr 2

ISemester

lkssion 2003/2004

2 hours
CHOW 4L:iaZ-
CHOON
WOOI:-=-=5==

BAHAGIAN MATRII(ULASI
KEMENTERIAN PENI}IDIKAN MALAYSIA

MATNCULATION DIWSION
MIMSTRY OF EDUCATION MALATSU

PEPERIKSAA}I SEMESTER PROGRAM MATRIKTILASI 1
SEMESTER FXn ILNITION FOR MATNCUI-ATION P RAGMMME

SEMESTER I

sEsr 200312004

SEMESTERI

' sasslolr 2003/2004

MATEMATIK
Kertas 2
2 jam

MATHEMATICS
Paper 2
2 hours

JANGAN BT]KA I(ERTAS SOALAN INI SEIIINGGA DIBERITAHU.

DO NOT OPEN TTIIS OUESTION BOOKLET UNTILYOU ARE INSTRUCTED.

Kertas soalan ini mengandungi 1l halaman bercetak
This questionbooklet consists of 11 printed pages.

@ Hak Cipta Bahagian Matrikulasi 2003
@ Ma*iculation Diyision Copyngft 2OO3

QMol6/2 3

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

Answer all questions.
The full marks are shown in the brackets at the end of each question or section.

All work must be clearlY shown.
The usage of electronic calculator is allowed.
Numerical answers can be given in the form of zq e, surd, fractions or up to three significant

r:\- figures, where appropriatg unless otherwise stated in the question.
CHOW
CHOONt
WOOI

QM016/2 5

LIST OF MATHEMATICAL FORMULAE -'

Differentiation

If y =g(t) and, x= 7(l), then *=*-*.

Integration

Jlu- dv=uv-Jfvdu
CHOW
CHOON
WOOI
t .{t
!

QMor6/2

h(x)-1. Giventhat/(x)= 2x+l and &(r) =2x? +4x+1, findafunction g suchthat
U"sXr) = tsl

Write g(x) in the form of a(r+ b)z + c, where a, b and c are constants. tl]

2. Find the following limits, if they exist :CHOW t3I
CHOON t3l
(a) -hr-.m+1_rX'.--l1WOOI
t t4I
..(b) l1m Ji -z
x--)a X -9 ;
-
tzl
J. If y - vs-',
l4l
(a) lf"in'*d 4- *"'6* d'! t3l
6
&2' t4l
l8l
(b) show that {&t "&*zL*y = o.

4. Find the values of A B and C which satisfy

x+2 A '-L- Br+C
xz +2
l-x

Hence, nnd [Cart+rq*

5. Given that x+t=xt and Zty-yz =3, find & .dv-

-datnd dt

E&Hencq find the values of when x:2.

QM016/2 9

6. Find a value of t so that the function t4l
t8l
fk\=1lk', if x<2.
lZx+k,
if x>2.

CHOWis continuous. (a),
CHOON
WOOI=;^Hence, by using the definition of .f '(a)f(x) - fdetermine

x-+a x-a

whether f'(2) exists or not.

7. Parametric equations of a curve is given by
X=t-'2-+-t1:- and 'V=3t-t"=2++-ll

Find

(a) . dY in terms of /. t6l

& 1
t6I
O) # t:when r.

[r'+1, for r<0. t4l

(a) Find lim /(r). 121
x--+0
l2l
(b) /Is the function continuous at x:0? Give your reasons.
(c) Sketchthe graph of/fromx : -Ztor = 1. t4]
(d) Find the area of the region bounded by the cunre/ x-a:ris and

: 1.the lines x: - 2, x

QM016/2 11

I l{,r !9. Shade the region bounded by the curves = = x andy = 2. 141

Find, in terms of n, the volume of the solid generated when the region is rotated

through 360" about the x-axis, l8l

10. gGven that f , and h as follows :CHOW
CHOON
WOOI'f(') = l'l'

g(r) = x2 -1,

I , x*0.

tdx)='

x

(a) Find F(x): (f ogoh\x). State its domain and rarge. [4]

(b) Find all the asymptotes of F and determine the interval where F is continuous.

12)

(c) Find the values ofx when F(x) = 5. I tsl
(d) Sketch the graph ofF.
!

t4l

END OF QUESTION PAPER