PSPM 1 Exam Papers Collection

Kedah Matriculation College

Mathematics

(Matriculation)

From
2003/2004

To
2020/2021

Exam Papers Collection

Table of Contents Page
3
Year 16
2003/2004 29
2004/2005 43
2005/2006 57
2006/2007 70
2007/2008 84
2008/2009 97
2009/2010 114
2010/2011 131
2011/2012 148
2012/2013 166
2013/2014 182
2014/2015 211
2015/2016 234
2016/2017 257
2017/2018 276
2018/2019 295
2019/2020
2020/2021

2

2003/2004

3

CHOW CHOON WOOI

QMol6/l Qlvro16/1

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

Sesi 2003/2004

2 iam

Mathematics

Papr 1
ISemester

Session 2003/2004

2 hours

BAHAGIAN MATRIKULASI
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

4

QMOl6/1 CHOW CHOON WOOI
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

\--

5

QMor6/1 5

LIST OF MATHEMATICAL FOR]VIULAE

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

x=

2a

For an arithmetic series:

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

6

QMol6 / r 7 CHOW CHOON WOOI

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.

3. 5x7 +l7x+17 as a sum of partial fractions. t7l

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

7

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]

7. Solve the following inequalities. tsl CHOW CHOON WOOI
(a) x'+x-12.>0. t8l

(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

8

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 CHOON WOOI
l

FindAB 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

9

QMor6/2 QMol6/2
Matematik
4L:iaZ- CHOW CHOON WOOI
Kertas 2
:-=-=5==
Semester I

Sesi 2003/2004

2 jam

Mathematics

Papr 2

ISemester

lkssion 2003/2004

2 hours

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

10

QMol6/2 3

INSTRUCTIONS TO CANDIDATE: CHOW CHOON WOOI
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.

t

11

QM016/2 5

LIST OF MATHEMATICAL FORMULAE -'

Differentiation CHOW CHOON WOOI

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

Integration

Jlu- dv=uv-Jfvdu

t .{t
!

12

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 : t3I CHOW CHOON WOOI
t3l
(a) -hr-.m+1_rX'.--l1
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.

13

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.

is continuous. (a), CHOW CHOON WOOI

=;^Hence, by using the definition of .f '(a) f (x) - f determine

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

14

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

15

2004/2005

16

CHOW CHOON WOOI

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

Sesi 2004/2005

2jam

4L CHOW CHOON WOOI

:i-7-=

"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- fin8+'?,1^-J^17@ Bahagian Matrikulasi

,<-er(

QMol6/1 CHOW CHOON WOOI
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

18

QMo16/1 5

LIST OF MATIIEMATICAL FOR]VIULAE

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

t-

-b+^lYb'-4ac
'2a

X'or an arithmetic series:

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'

19

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] CHOW CHOON WOOI
(a) sum of the ftstn terms [2 marks]
(b) infinite sum

of the series.

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

20

QM016/1 [3 marks] CHOW CHOON WOOI
[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

7. Show that [3 marks]

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).

2I 1

QMo16/1

/z x l)

Show that the determinant of the matrix A = lIltvx ', 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.l CHOW CHOON WOOI
Find the adjoint and inverse of the matrtx A.
[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 22

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

2 jam

4L CHOW CHOON WOOI

"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 \-.

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

23@ Bahagian Matrikulasi

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

QMo16/2 CHOW CHOON WOOI
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\,

24

QMo16/2 5

LIST OF MATIIEMATICAL FORMULAE

Differentiation

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

dt

Integration

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

i

I

25

QMo1612 [3 marks] CHOW CHOON WOOI
[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\*).
[3 marks]
2. Find each of the following iimits, if they exist.
'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]

267

QM016/2 [6 marks] CHOW CHOON WOOI
[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.

7. (a) Given !=2x', find,! [4 marks]
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'

2I7

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 WOOI

(i) Sketch the graph of q. [2 marks]
(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

28

2005/2006

29

CHOW CHOON WOOI

QMo1611 QM016/1

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

2 jam

4L CHOW CHOON WOOI

4- _____
-=

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

30

QMo1611 CHOW CHOON WOOI

INSTRUCTIONS 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

31

QM016/1

LIST OF MATHEMATICAL FORMULAE

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

* --bt"[t' -q*
2a

For air arithmetic series:

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

32

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 CHOW CHOON WOOI

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

[6 marks]

,/1. aThe quadratic equation .r'+2(k+1)x+(2k+5)=0 has roots arrd1.If the roots
: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

33

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

7t'. LGrilvveenn ^.t==l[l-2:r, o -o2l1, ",=-l[l-tj;, -11 ''-=Ll'r -o1 o-l CHOW 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

34

QM016/1

10. Consider the system of linear equations CHOW CHOON WOOI

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]

(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

35

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

2 jam

dL CHOW CHOON WOOI

=r=p:A. a.-o
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

36

QM016/2 CHOW CHOON WOOI

INSTRUCTIONS 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

373

QMo16/2

LIST OF MATHEMATICAL FORMULAE

Differentiation CHOW CHOON WOOI

tf y -sk) d* *: fQ), maka *=*-*

Integration

ludr=uv- [vdu

I rl

!

38rE,

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 CHOON WOOI
[6 marks]
2. Using integration by parts, evaluate !' *e-"dx.
[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)

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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 CHOW CHOON WOOI
f(*)=I12-*, 0<x<2

,I

LG-zf z<x<5

sketch its graph.

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]

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QM016/2 [12 marks] CHOW CHOON WOOI
[5 marks]
7. Express #5x3x-'(*x2'++lr)-1 in the form of partial fractions. t7 marks]

t;!Jl!a.Hence, evaruate y : [s marks]

Give the answer correct to tlree significant figures. t2 marksl
[6 marks]
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

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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 CHOW CHOON WOOI

radian

aboutthe x-axis. Give the answer interms of e and r . [10 marks]

END OF QUESTION BOOKLBT

t

€.

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2006/2007

43

CHOW CHOON WOOI

QM016'1
Mathematics

Paper 1

ISemester

2006/2007
2 hours

& QMo16/1 CHOW CHOON WOOI

'ffiJ Matematik

Kertas 1

Semester I

2006D0a7

2 jam

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAY SIA

WTRICULATION DIYISION
MINISTRY OF EDUCATION MALAYSU

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
MATNCULATION P ROGRAMME EXAMINATION

MATEMATIK
Kertas 1
7 jam

JANGAN BUKA KERTAS SOALAN INISEH]NGGA DIBERITAHU.
DO NOT OPEN THIS BOOI<LET UNTIL YOU ARE TOLD TO DO SO.

Kertas soalan ini mengandungi 13 halaman bercetak.
This booklet oonsisfs of 13 prtnbd pages.

@ Bahagian Matrikulasi

44

QM016/1 CHOW CHOON WOOI

INSTRUCTIONS TO CAF{DIDATE:
This question booklet consists of 10 questions.
Answer all questions.
The full marks allocated for each question or section is shown in the bracket at the end of
each question or section.
All steps must be shown clearly.
Only non-programmable scientific calculator can be used.
Numerical answers can be given in the form of rc, e, surd fractions or up to three significant

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

3

45

QM016/1

LIST OF MATHEMATICAL FORMULAE

Arithmetic Series: CHOW CHOON WOOI

T,=e+(n_l)d

,s,=tV"+fu_lal

Geometric Series: for r<l

T' = arn-l

',=#

Binomial.Expansion:

I

(a + b)' = a' +(i)"",.(i)".* . .(:).-, b, +... + b., where n e N and

("\= rt(n"-, r)t

[rJ

fu)+"!x,(r +x)" =t+ra.fuP., +...+ +... ror lxl< r

465

QMo16/1

lr -21

IfP =

[o -r]
l; ;l1. ando:[l o -ll. rrna matrix R such that
-l 0l'

[o z 21
rR+ 2(Po) =l-z 4 31.
L-4 5 3l
CHOW CHOON WOOI
[5 marks]

2. By substituting a = 2', solve the equation [6 marks]
[7 marks]
4' +3 =2'*2 .

3. Obtain the solution set for

lzx+{ > -x' +4 .

:

4. The sumlof the frst t terms of an arithmetic series is 777 - The first term is -3 and the

ft-th term is 77 . Obtain the value of k and the eleventh term of the series.

[7 marks]

- 5. (a) Findthevalues of A,B,CandDfortheexpression 4x3 -3xz +6x-27

xa + 9x2

+-#-Hin the form of partial fraction, where A, B, Cand D

are constants' t5 marks]

o(b) -t00
l-z ol _I_1155 . Show that AB = kI

Givenl=l-+ G -zlanda= )?
L6 -4 -2)

55

where k is a constant and ,I is an identity matrix. Find the value of ft and hence

obtain l-1.

[5 marks]

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47

QM016ll

6. (a) Given the complex number z and its conjugate 7 satisfu the equation
zZ +22 :12+61. Find the possible values of z.

[6 marks]

(b) An equation in a complex number system is given by [3 marks] CHOW CHOON WOOI
[3 marks]
11

= =-(-"--
where zr =l+2i ")and "Zz =2- i. Find

(D the value of z inthe Cartesian form a + ib

(ii) the modulus and argument ofz.

(a)7. Find the soiution set of the inequality

11 [5 marts]
3-Zx x+4

I 't

(b) Solve the following inequalrty equation for all x is real numbers. Wriie your

answer in set form.

4-Pll+-2x*lI> |

[7 marks]

8. (a) Show that (x-3) is a factor ofthe polynomial

P(x):S-2*-5x+6'

Hence, factarize P(x) completely.

[4 marks]

(b) It f(x) = ax2 + bx +c leaves remainder 1, 25 and I ondivision by (x- 1),
(x + 1) and (x - 2) respectively, find the values of a, b and c. Hence, show

that l(x) has two equal real roots.

[9 marks]

9 .{

48

QM016/1

9. (a) Find the first four terms in the binomial expansion of the following functions:

(i) -,lt+2. [2 marks]

(ii) 1 [2 marks]

(r - *)' CHOW CHOON WOOI

.(b) Hence, expand -1lf(-t1+-2r)." in ascending power of x up to the term containing
l, Hfx'. By putting , - show that .h2000 is approxim""fv

[9 marks]

10. A doctor prescribed to a patient 13 units of vitamin A,zzunits of vitamin D and 31
units of vitamin E each day. The patient can choose from the combination of three

brands of capsules; L, M and N. Each capsule of brand L contains I unit each of

vitamins A, D and E- Each capsule of brand M contains 1 unit of vitamins A, 2 units
of vitamin D, and 3 units of vitamin E. Each capsule of brand N contains 4 units of
vitamins A, 7 units of vitamin D and 10 units of vitamin E. The above information is

summarized in the following table:

_1

Type of Vitamins Brand of Capsules Total Unit of Vitamins

A LMN t3

D I 14 22
E 31
1 27

I J l0

By usingx as the number of capsules of brand L,ythe number of capsules of brand M

and z the number of capsules of brand N,

(a) form a system of linear equations from the above information.

[2 marks]

(b) write the above system of linear equations in the form of matrix equation:
(c)
AX: B, where,4 is the coefficient matrix, Xis the variable matrix, and B is the
constant matrix. Solve the system of equations by using the Gauss-Jordan

elimination method.
[8 marks]

determine the possible combinations of the number of capsules of brand L, M
and N to be taken each day.

[3 marks]

11

49

QM016/1

(d) Ifbrand L costs 10 cents per capsule, brand M costs 30 cents per capsule and

brand N costs 60 cents per capsule. Determine the combination that will CHOW CHOON WOOI

minimize the patient's daily cost.
[2 marks]

END OF QUESTION BOOKLET

'-J

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