Matriculation Mathematics PSPM 1

CHOW QSo15/1
CHOON
WOOIINSTRUCTIONS TO CAI\DIDATE :
This question paper consists of L0 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, e, stJrd, fractions or up to three significant

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

Y

CHOW CHOON WOOI

151

QS015l1

LIST OF MATHEMATICAL FORMULAE

Quadratic equation ax2 +bx+c=02

x= -t+.[6'-a*

CHOW Arithmetic series:
CHOON
WOOI To = a*(n-l)d

s, =llzo+(n_\dl

U

Geometric series:

Tn = ar*l

Sum to infinity:

s- =fr, l'l<t

t Binomial expansion:

*(a + b). = an *(i)"" t .(i)o- + + (i)"'u' + ...+ b' ,

*.where neN =@+ya

[;)

*:W(t + m)n = r + n(ax).$r*y * -2(*)' *...

laxf<twhere neZ- orn eQ

CHOW CHOON WOOI

152

QS015/1

1 Find the value of x which satisfies the equation

log, (5 - x) - log, (* - z) : 3 -lo1z(t + r).

16 marksl

CHOW2 Determine the solution set of the inequality
CHOON 2x1-1l <_x+.2
WOOI
16 marlcsl

3 Given k+2,k-4,k-7 arc the firstthree terms of a geometric series. Determine the

value of k Hence, find the sum to infinity of the series.

f6 marksl

1 Givenacomplexntrmber z:l-$i. Deteminethevalue of ,tif 7 =UvL.

[7 marksl

Ir5 -11
(a) MatrixMisgiven* tr"*that M2:7M-8.I, where ^I isthe
L_O O-l

2x2 identitymatrix. Deduce that M-t =818r-lr.

15 marksl

lr*t -r(b) Given matrix l: | 3 I24 I ana lZl =27. Findthe value

-,L |
o p*2)

ofp, wherep is an integer.

15 marlcs)

CHOW CHOON WOOI

153

QSo15/1

6 =:::,The tunctions ,f and g are defined as f (x) x +2 and 8(x) = 3-x.

(a) Find /-t(x) and s-'(x). 15 marksl
13 marks)
o) Evaluate (/.s-,)t:). 14 marksl

(c) If (g' f')(k):1, *O the value of fr.

s
CHOW
7 (a) Solve lx' -x-rl::.CHOON
WOOI
2x2 +9x - 4 15 narlcsl
x+2 l7 morl<sl
(b) .Find
the solution set of the inequality o.

CHOW CHOON WOOI

154

QSo15/1

8 The first four terms of a binomial expansion (1+ axl is
l+ *-L*'+ oxj + ...

2

Find

(a) the values of a and nwharc n*A.
CHOW
CHOON 16 narksl
WOOI
O) *=!, "" \Ethe value of p. Hence, by substitutingshow is approximately

equal to 9r2.8

[7 marks]

9 Given -f(r):h(2t+3) md g(r.),= +
2

(a) 5trourfint "f (r)it a one-to-one fuuction algebraically.

[3 marlu]

O) Find (,f "sxr) and (s'l)(r). Hence, state the conclusion about the results,

15 narksl

(c) /(x)Sketch the graphs of and g(x) on the same axes. Hence, state the

&main and range of /(x).

15 marksl

CHOW CHOON WOOI

'11
155

QS015l1

10 Given

lz23l

.n=tLtlr3st44)1.

(a) Find the determinant of matrixl.
(b) l.Find the minm, cofactor and adjoint of matrix
CHOW 12 mark*l
CHOON 15 marksl
WOOI
O G) Given /(a{ioint( A))=lAllwhere.(is 3 x 3 identity matrix, showthat

f :fr"ai"iot(,1). Hence, find l-r.

15 narlxl

(O Byusing A-rinpart (c), solve the follswing simultaneous equations.

2x+2y+32=49
x +5y+42=74
3x + y +42=49

[3 marks]

END OF QUESTION PAPER

CHOW CHOON WOOI

13
156

QSo15/2 QS015/2
Mathematkx
PaprZ Matematik

/Semester Kertas 2
Session 2012/2013
2 hours Semester I

Sesi 201212013

2jam

GL

==-=
,,

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATRICU./TTTON DII/BION
MINNIPJ OF EDUCATION MAI-4YSU

U PEPERIKSMN SEMESTER PROGMM MATRIKULASI

AATRIC U-/TNON PROGRAMME FXIUNUruON

MATEMATIK

Kertas 2

,' i'^

JAt{CrAl{ BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.

IfiTffE'DCI I}dS Q,fiESTffi,' FAPER UNTL YW ME TW IO DO SO.
CHOW
CHOON\,
WOOI

Kertas soalan ini mengandungi 17 halaman bercetak,

Thls quNion paper mnsrisfs of 17 pintd pagx.

@ Bahagian Matrikulasi

157

CHOW QS015/2
CHOON
WOOIINSTRUCTIONS TO CAI\iDIDATE :
This question paper consists of 10 questions.
Answer all questions.
All answers must be written in the answer booklet provided. Use a newpage 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.

1. Numerical answers may be given in the form of T, e, sltrd, fractions or up to three significant
- figures, where appropriate, unless stated otherwise in the question.

v

158

QS()15/2

LIST OF MATHEMATICAL FORMULAE

Trigonometry

I ftsin (,,4 A) = sin cos .B cos A sin B

CHOW cos (r4tB)=*t AcosB + sin,4 sinB
CHOON
WOOItanI\/A.r.B-\'l'= tanA + tanB
l+@LAtanB

T sinl +sinB: 2"ioA*2B2 .o* l-B

sinl - sinB : 2 rn*A*2B2 r*A- B

cosl +cos6 :z*n*A* B ,rro- u

22

ml-G(E B:1*' l+B rio1-B

sn 2A:2smA cos,{

w2A: cG2 A-sinz A
I =2cos2 A-l

=l-Zsinz A

:tafl 2A 2*'! A

l-tarf

. 1 . l-cos2A

Sfn'.4 =

2

). -l+cosZA

COS'A = 2

-

159

QS015/2

LIST OF MATIIEMATICAL TORMULAE

Limit

hli+- 0thhfr = I
h,.-+lA-cohs ft _O
CHOW
DifferentiationCHOON
WOOI
f(.) f'(*)

cotx - cosec2x
rsec
sec x tan x

cosecr -cosecx cotx

tf y:st) *a *=fk\uenfi=*"*

d(dv)l

d&'y2&-AIA)

&

Sphere Y =! nr} S = 4nr2
3 Sl = firS
Right circular cone
Right circular cylinder , S:Zxrh
V =! nrrh

3

V = rcr2h

160

QS015/2

1 Giventhat f(*)=j ll+nt,., x<1

x=l

12-x, x>1.

Find ,li3_,f (x) anA Um /(r). Does the trq/(x) exist? State your reason.

15 marlcsl
CHOW
2 Prove that 1+tan20la$0 =sec20.CHOON
W
OOI 16 marlrsl

3 Find the following limits:

(a) 2x2 + x-- 4

r'-')o 1- x2 '

o) 3-rf,+7 13 marks)
rH+m2 t' -4 ' 14 marlcsl

4 Express 2x3 -27x2'-7!xl7+x6- 19 intheformofpartialfractions.

l7 marksl

g
161

QS015/2

s (a) ll*' - r-zl x * 0,2
x=2.
Given that f (x) = 1L#,0,

Find the yy f (-). Is /(r) continuous at x=Z?

CHOW [6 marks]
CHOON
W
OOI
lax+6, x <4
(b) A tunction/(r) t defined fAV (*)=) x' +2, 4< x <6

lr-B*, x>6.

J
a fDetermine the values of the constants and B ir (x) is continuous.

15 marlcsl

6 The polynomial P(x) =2x3 + mz +bx-24 has a factor (*-Z) and a remainder 15

nfren divided by (x+3).

(a) Find tre values of a and D.

[6 marks]

-o (b) Factorise P(r) completely and find all zeroes of p(x).

16 marksl

11
162

QS015/2

7 Given f (0) = 3sind -2cos0.

(a) Express f (e) inthe form of Xsin(d-a), where R >0,0< "=;.
fHence, find the maximum and minimum values of (e\.

l8 marksl
CHOW
CHOON(b) Solve f (o):E for oo <o<3600.
W
OOI 14 marks)

8 (a) =#Given that y

(i) ff.By using the first principle of derivativ e, fina 14 marksl
12 marlul
(ii) ,*#
12 marksl
(b) +Ftud of the following: 14 marlcsl
ax
(i) y=e2* tarrx.
(ii) !=xs"'.

13
163

CHOW 8S015t2
CHOON
W 9 (a) A conical tank is of height 12 m and surface diameter I m. water is pumped
OOI
into the tank at the rate of 50 m3/min. How fast is the water level increasing
when the depth of the water is 6 m?

16 morksl

(b) A cylindrical container of radius r and height h}p,s a constant volume v. The

cost of the maerials for the surface of both of its ends is twice the oost of its

,sides. state in t€ms of r and I/. Hence, find & and r in tenns of zsuch that

the cost is minimrmr.
l7 marlal

15
164

QS015/2.

10 (a) Given 3y2 -ry+x2 =3.8y using implicit differentiation,

(i) findthevalue .f * at x=1.

CHOW(ii) 4*!.show that (ay - r(*)' - r*.2 = o 16 marksl
CHOON 12 marksl
WO) Consider the parametric equations
OOI [3 marlal
x =3t -ut',-, =3t *? where t * o.
14 marl<s)
t

O show tn* fl=;fi;.

(ii) + t:NFid_ when l.

t,

END OF QUESTION PAPER

17
165

PSPM
MATRICULATION MATHEMATICS

QS015
2011/2012

166

QS015/1 QS{t15/1
thMb
PWl Matematik
SS2eersmsteioosntue2r rI0slI/2012
Kertas f

Semester I

Sesi 2011/2012

2jam

4L
qef1:Y---

== 5--

BAHAGIAN MATRIKT]LASI
KEMENTERIAN PELAJARAN MALAYSIA

n IATRIC\I-,IflON DIVNON
MIMtrRY OF EDUCATION MAI-AYSIA

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
I,IATRICU-4UON PROGRAMME EX,4MINATION

MATEMATIK

Kertas I

2 jam
I 'vCHOW
CHOON
WOOI
JANGAN BUKA KERTAS SOAI.AN INISEHINGGA DIBERITAHU.
DO NOT @EN IHIS QUESNON PA,PER UI,NL YOU A,RE lCI.D IO DO SO,

Kertas soalan ini mengandungi 15 halaman bercetak.
This quesiriat paperconslsts of 15 pnfied pages.

@ Bahagian Matrikulasi

167

CHOW QS015/1
CHOON
WOOIINSTRUCTIONS TO CANIDIDATE:
This question paper consists of 10 questions.
Answer all questions.
All answers must be written in fte mswer booklet provided. Use a new page for each

question.

The full ma*s for each question or section ae 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 fi, e, strd, fractions or up to three significant

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

-

3
168

QSo1s/1

LIST OF MATHEMATICAL FORMULAE

Quadratic equation ax2 +bx*c=0l
--:--

-b=r b- -|ac

Arithmetic series:CHOW
CHOON
Tn = ct*(n-l)dWOOI

s, =lrlz"+(n_ldl

ry

Geometric series:
T, = arn-l

s,=ff,r*t

Sum to infinity:

s*=*,l.l.r

ry Binomial expansion:

(a+b)^ = an +(i)"".(;)"-u'+ + (:)"'', + +bn ,

*.where neN [;j =@lW

(t+ax)n =t+n(ax).9tax12 *n(n-)!n-z) @13 +...

lo*1.1 where neZ- or n ee

5
169

QSo15/1

I Solve the equation 3zx+t - 28 (3') * 9 : 0.

16 marksl

2 The functions "f and g are defined as:

f(x)=JA, x>1

g(x)=x', x>0.
CHOW
CHOONFind the inverse function, f-t (*) and determine its range. Then, evaluate
WOOI
("F " s)el. 16 marksl

v

3 The ninth term and the sum of the first fifteen terms of an arithmetic progression are
24 arrd 330 respectively. Find the first term, a and the common difference, d.

Hence, find the least possible value n, such that the sum of the first n terms is

-qreater than 500.

[6 marlrs]

, -,f

[r4 Matrix .{ is given as lZ 3 -3 l.

lz 2 -t)

[: x+y -21
(a) Giventhecofactormatrixof e is | 0 | 2 | *n... x>0.
[-: x'
-t_]

rDetermine the values of and y.

{3 marl<sl

(b) Given A2-4A+1=0,showthat A3=l5A-41 where 1 isthe3x3

identity matrix. Hence, find, A3.

14 marks)

7
170

QS015/1 CHOW lI markl
CHOON
5 Giventwocomplexnumbers zr:Ja3i and zr=2-i. WOOI13 marl<sl
(a) State z, ""d 4. 16 marl<sl

(b) ifDetermine the value of k | = k1. 12 morksl
zr 12 marksl

(c) Find zrzr.Hence,showthat 21 zz=2F2. 15 marl<sl
14 marks)
v
6 (a) Given -f (x)= e ' and g(x)= x'.

(i) fFind the domain and range of and g.

(ii) Show that (g " -f)(*) = e-" .

(b) Given

I

I (i) Find ft-'(.r).

(ii) Sketchthegraph for h(x) and h-t(x).

I

171

QS015/1 15 marlal
17 marksl
7 (a) Solve the equation log(r-4)+ 2log3:r.*(;)

(b) Find the solution set of the inequality

141.,

lx+ll
CHOW
CHOON
WOOI
y I (a) -(i)"Given that the sum of the first n terms, S, of a series as E = ,

Find an expression for the zth term. Show that the series is a geometric series

and find the sum to infinity, ^S-.

16 marlal

I

(b) Expand (r4)' intheasce,ndingpowersiof r uptothetermin x3.

,EHence, by substituting r = 3, evaluat" correct to three decimal places.

y [6 marks)

11
172

QSo15/1

9 (a) A tunction /(x) is defined by f(i=4x-o for x * 6.

Show that f (x) is a one-to-one function.
Find the values of x such that (f . "f)(x) = 0.

CHOW(b) Given -f (r)=.'/l= x l7 marks)
CHOON 16 marksl
WOOI and Su\(-r,)= -1.

2

Find /[s'(j))

v

13
173

QS015/1

l0 The following table shows the quantities (unit) and the amount paid (RM) for pens

bought from three shops.

Pen Pilot Kilometrico Papermate Amount paid
Shop (unit) (uni0 (unit) (RM)

S I p 2p 18.00
I 3q 31.00
T q 4r 37.00
U 1
r
CHOW
CHOONGiven the price in RM per unit of pilot, kilometrico and papermate pens be x, y Md
WOOI
z respectively.

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

ll mark)

(b) BWrite the system in the form of a matrix equation AX = where
/x\

* =l ,1.

l,)

I mark)

(c) Giventheminor 4r, ozt arrtd azz ofmatrix A is 9, 12 and 8

v respectively. Find the values of p, q and r.

14 marksl

(d) Find the determinant, cofactor, adjoint and A-t of matrix l. Hence, find the
values of x, y and z.

19 marksl

END OF QUESTION PAPER

15
174

QS(}15I2 QS015/2

ftffitsndis Matematik

Pw2 Kertas 2

ISemester Semester I

Session 201l/2012 Sesi 2011/2012
2 ja,m
2 hours

A
ffid

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

IqI4TRICUI-AflON DIWSION
MIMSIRY OF EDUCANON M4I-AYSIA

PEPERIKSMN SEMESTER PROGRAM MATRIKU LASI
NATRICUI.,ITION PROGRAMME EXAMINATTON

I MATEMATIK
I Kertas 2

2 jam

E JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.

DO NOIOPEN 7HlS QUESTTO,\/ PAPER UNNL YOU ARE TOI-D IO DO SO.
CHOW
CHOON
WOOI

Kertas soalan ini mengandungi 15 halaman bercetak.
ThisquMion paperconslsts of 15 printed pages.

@ Bahagian Matrikulasi

175

CHOW QS015r2
CHOON
WOOIINSTRUCTIONS TO CAITDIDATE :
This question paper consists of 10 questions.
Answer all questions.
All answers must be written in the ailiwer booklet provided. Use a new page for each

question.
The fuIl 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 scie,ntific calculators can be used.

'v Numerical answers may be given in the form of r, e, srttd, firactions or up to three significant

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

3

3
176

QS015/2

LIST OF MATHEMATICAL FORMULAE

Trigonometry

Isin (,{ t B)= sin,4 cos B + cos sin B

cos (.1t B) = cos .-1 cos -B + siri,-1 sin B
CHOW
CHOON \ '/ I +tan ( -4+ .B ) = L'ul '
WOOI
tan.{ tan B

'v sin,4 +sin,B :2rinA+ B B

22 "orA-

sin A -sin B :2 B ,inA- B

"rrA*22

cos -4 t cos B: 2 cos A+ B B

22 "oro-

cos,{ - cos :.B -z rinA*2B2 ,inA- B

sin 2J = 2 sin.-{ cos --1

cos 2.{ = cos: .l -sinr ,l

v = I cos: -l -1
= L-2sinr I

2tan A

1-tan'A

sin' A = l-cos2A

2

cos' A = 1+cos2A

5
177

QS015/2 LIST OF MATHEMATICAL FORMULAE

Limit

li*'inft =1

h">o h
CHOW
CHOON lim l-cos ft =0
WOOIh--+0 h

Differentiation

=y f(*) f'(*)

cot r - cosec2x
xsec
sec x tan x

cosecx -cosec xcotx

rf y=g(r) and *=f(t),tnen f=*"*

d(dv\
dd'xv2-Alddx- )

dt

.,

Sphere v =! nr3 s = 4xr2
3
-^S 7c rs
Right circular cone V =L nrzh
3 S =2nrh

Rightcircularcylinder V = nrzh

7
178

QS015'2 15 marksl

I -Expr"*. (,t!xr:-1+?)u' in the form of partial fractions. [3 marks]
13 marl<sl
2 Evaluate the following limits:
(a) lrm xo -16

x-+2 X-2

-.

..EI*G

(b) rl+lm@- J,
CHOW
CHOON3 Fkd +dx forthefollowingequations: 13 marlal
WOOI 13 marksl
(a) ! =32'*1.

O G) ery+y-5a.

4 The surface area of a balloon in the shape of a sphere is decreasing at the rate of

2 cmzf min Find the rate at which the volume is decreasing when the radius of the

balloon is 5 cm.

l7 marks)

9
179

QS()15l2 -3The function f (x) = x' - 6x' +9x is defined on the interval [0, 5].

s (a) Find the critical points of f (x) on this interval and determine whether the

critical points are local minimum or maximum.

[6 marks]

CHOW(b) ffi.Find the horizontal and vertical asymptotes for /(x) =
CHOON
WOOI [7 marks)

a6 The polynomial p(*) = x' -Zx' + ax +b, where ba1|J are constants, has a
factor of (x-2) dand leaves aremainder of when it is divided by (x-a).

(a) aFind the values of and b.

f6 marlcsl

(b) Factorize f (.r) completely by using the values of a and b obtained from
bpart 6(a). Hence, find the real roots of p(x)= 0, where a and are not

equal to zero.
16 marksl

Giyen that ., =^,-lLt+t' ,:ry, where / is a non zero parameter.

(a) Show that dy -l-+---t:2- 16 marksl
16 marksl
dx tt

(b) ,rno t4 when t =1.

11
180

QS01s/2 15 marl<sl
[6 marks]
8 (a) lf y=sin(x'+l), showthat
x!t2(-?*,4x3y=g. 14 marksl
13 marksl
dx" dx
16 morlal
(b) Fild the gradient of a curr e x eu = e2' - e3v at (0, 0).
CHOW
CHOONv9(a)Givenf I *'-64 ' x+4
WOOI
(*)=l ,-o
L 40, x=4.

(D Find li]l t (*).

(ii) Is / continuous at x=4? Giveyourreason.

(b) Determine the values of A arrd B such that the function

v *- a, x<-l
+3Ax+ B, -l < x<l
h(x)= |
x>1.
l2x: 4,

[

is continuous for all values of x.

13
181

QS015'2

10 (a) Given t*13=4,1-l and tanL=\.
Express t^# intheformof a+Ji where a and b areintegers.

.Hence, show that r^g\6A\/= +Jt.
CHOW
CHOON 16 marl*l
WOOI
(b) aFind R and suchthattheexpression gsing +l2cos? canbeexpressed
in the form of Rsn(O + a), ofrer" R > 0, 0o < a < 90".

Hence,if 9sind+l2cos0:5, solve for 0 intheinterval O" <0 <360".

Y 19 marksl

END OF QUESTION PAPER

v

15
182

PSPM
MATRICULATION MATHEMATICS

QS016
2010/2011

183

0s016/1 QS()16/1 I

Mathematics Matematik

Paper 1 Kertas 1

ISemester Semester I
Session 2010/201 I
2iamSesi 2010/2011
Ir 2 hours

4L:!,:CHOW
CHOON
a'-eI::.{ffIf:,:fs"WOOI

BAIIAGIAI{ MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATNCULATION DIVISION
MINISTRY OF EDUCATION MAI- YSIA

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI

MATRIC UL,ATION P ROGMMME EXAMINATION

MATEMATIK

Kertas 1
2 jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU.
DO NOI OPEN IHIS BOOKLET UNTIL YOU ARE TOLD IO DO SO,

Kertas soalan ini mengandungi 15 halaman bercetak.

This booklet conslsfs of 15 printed pages.

@ Bahagian Matrikulasi

184

CHOW QS()16/1
CHOON
WOOIINSTRUCTIONS TO CANDIDATE:
This question booklet consists of 10 questions.
Answer all questions.
The full marks for each question or section are shown in the bracket at the end of the question

or section.
Al1 steps must be shown ciearly.
Only non-programmable scientific calculators can be used.

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

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

Y

3
185

QSo1611

LIST OF MATHEMATICAL FORMULAE

For the quadratic equation ax) + bx * c = 0 :

.--bi'lT;*4ac
2a
CHOW
CHOONFor an arithmetic series:
WOOI
Tn=o+(n-l)d

S, =!f2.a+(n-t)dl

v
For a geometric series:

T' = ar'-l

t.=ff 'r*1

Binomial erpansion:

(n\ +(l2n\y'-'^b'^+ ...+/rl,lr\f' 'u' + ...+ b' ,

ta - b)" = a" -1,
)o'-'u

- where neN *df')=-n!,
\'J (n - r)r. v1

-+.z(t + x)" =r+ nx *... *n(n -t)":(n - r +l) x' +... for lxl < r

5
186

QSo16/1

1 Dividing M(x)=r'* ax+b by (r+1) and G-t) givearemainderof -12 and
-16 respectively. Determine the vaiues of a and b.

16 marl<s)

2 Solve the equation

CHOW lnx- 3 = -2.
CHOON
WOOI lnx

[6 marlrs]

3 Thequadraticequation x2+3mx+2=0 hasroots aand pwhere m isa
v constant. Form a quadratic equation with roots (" * F)' and (a - p)' interms

of m.

l7 marl<sl

4 The sum S, ofthe first rz terms ofan arithmetic progression is givenby

5,,= pr1*cytt. Thesumof thefirstfiveandtentermsare 40 and 155 respectively.

(.a) pFrnd the values ol and q.

13 marksl

(b) Hence, find the ruth term of the arithmetic progression and the values of the
.-l7 first term, a and the common difference, d.

14 marlc;l

7
187

QS016/1 14 marl<sl
[8 marks]
5 Solve the following inequalities.
(a) _ai-x2 -. +,_r >- 40.
2x- -3x-2

(b) ll*^-'rl<I 2.
l.r + 31
CHOW
CHOON6 (a) Given two complex numbers Zr :2 + i and zz =I-ZL
WOOI
v (i) Express ,,t *l in the form x+ yi, where x and y are real
z)
numbers and iz is the conjugate of zr.
14 marksl

(ii) Hence, find the modulus of z,'+-l-. 12 marksl
[6 marlrs]
z2

(b) Find the square roots of -3 + |i.

..,v

I

188

QSo16/1

7 The following table shou,s the price (RM) per type of 0.5 kg cakes sold at the shops

P, Q and R together u'ith the total expenditure if a customer buys a number of each
type of cake tiom the listed shops.

CakeCHOW Banana Chocolate Vanilla Total
CHOON Expenditure
Jl'pes WOOI 5 8 5
4 6 aM)
Shops 5 9 6
1 36
P 30

a 40

R

Let the number of banana, chocolate and vanilla cakes bought from each shop be x,

.v and z respectively.

] (a) :Write the matrix equation AX B using the above information.

11 marlc)

(b) l.Obtain the adjoint matrix of Hence, find the inverse of matrix ,4.

{8 marksl

(c) Determine the values of .r. -r, and z using the inverse matrix of ,,4 obtained

in (br.

l2 marksl

8 A polynomiat /G) = px3 *(p* q)*' +(p +2q) x +l has a factor (x+l).

3

(a) Express q in terms of p.

13 marks)

(b) Write /(x) in terms of p and x. Determine the quotient when f(r) t

divided by (x + 1).

l3 marlal

(c) p ifHence, find the value of x = 3 is one of the roots for "f(*)=0. Using

the value of p, factonze f (x) completely.

[5 marks]

11

189

QSo16/1

(a) Giventhat 1=0.015151515... = p+q+s+..., where p,q and s arethe
u
Iffirst three terms of geometric progression. p = 0.015, state the value of q
and s in decimal form. Hence, find the value of u.

14 marlcs)

CHOW I
CHOON
WOOIFind the expansion(t-*)t* to the term x2. State the range of x

".

for which the expansion is valid. Show trhrra*rt 11//su- "z : z'\'(t-iric)i). '
"Hence, by substituting x :2, approximat t,lT correct to four siguificant

figures.

- 19 marl<s)

Y

13
190

QS016/1

10 Thegraphof aquadratic function !=o)cz +bx+c, where a, b and c areconstants

passes through the points (- 2, -10), (1, 8) and (2,6).

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

12 marks)

O) Write the system of linear equations in the form of a matrix equation AX: B,

where
CHOW
CHOON r:lu[,.ll 12 marks)
WOOI [2 marks]
Lc_l

Y

(c) Find the determinant of the matrix A.

(d) By using the Cramer's Rule, solve the matrix equation.

17 marks)

(e) Hence. urite the quadratic function of the graph and determine whether the

graph has a masimum or minimum value.
[2 marks]

-

END OF QUESTION BOOKLET

15
191

QS016/2 QS016/2
Mathemattcs
Matematik
Paper 2
Kertas 2
ISemester
Semester I
Session 2010/2011
2 hours Sesi 2010/2011

2 jam

CHOW
CHOON
WOOI
I dL I
=Y=a"%:: XlX-:-F-,"S'

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATRIC ULATION DIVISION
MINISTRY OF EDUCATION MALAYSIA

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
MATRIC ULATION P ROGRAMME EXAMINATION

MATEMATIK

Kertas 2
2 iam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DO NOT OPEN IHIS BOOKLET UNTILYAU ARE TOLD IO DO SO.

Kertas soalan ini mengandungi 15 halaman bercetak.

Thisbooklet conslsfs of 15 printedpages.

@ Bahagian Matrikulasi

192

CHOW QS016/2
CHOON
WOOIINSTRUCTIONS TO CANDIDATE:
This question booklet consists of 10 questions.
Answer all questions.
The firll 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.

Numerical answers may be given in the form of fi, e, svrd, fractions or up to three

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

!7

3
193

QSo16/2

LIST OF MATHEMATICAL FORMULAE

Differentiation

If y=g(r) *a x=f|\then !d=x4d"t4dx

d(dv)l

dd'xv2-AldAx )

dt
CHOW
CHOONIntegration
WOOI
v
ludv=uv- lvdu

rt

E

194

QS()16/2 12 marl<sl
14 marl<s)
1 !Find foreach of the following:
aIx [6 marl<s]
(a) 1 = (ln x)' . 16 marksl

(b) xy' - ye* =3. l7 marl<sl
CHOW
CHOON2 fofFind the exact value f ,t,' -t a,
WOOI
.y

3 If f is a tunction with .f '(l) =2, find xlir+nl"f GL- f Q)

riX _1

4 ff{fExpr.r, as partial fractions.

Hence, evaluate J[ 2!'x]'+*31xt ,r.

v

7
195

QSo16/2

5 Given the functions f and g as follows:
f(x)=2-*',

8Q) = x +2'

(a) Find /.g and g.f.

(b) fState the domain and range of " g.

(c) Find (g " f)'.

v

(d) rDetermine the value of such that f .g(x) = g" f (x).
CHOW 14 marlcs]
CHOON
WOOI 13 marla)
12 marksl

13 marl<sl

6 (a) State the conditions of continuity of a function at a point x = c.

12 marksl

(b) A tunction / defined by

I x-21 - -5<.r<2
x'+3x-10
fG)=1 A
2<x<3
v Ax+B , x=3

is continuous at x :2 and x : 3. [6 marlrs)
[5 marl<s)
(i) Find lrry_ /(x).

(ii) Determine the values of the constants A and B.

I

196

QSo16/2

7 (a) Evaiuate.

(i) rli+m€!+lt -{Ix++1za x-L

. 2- VrxT'-5 13 marl<s)
14 morl<sl
til-il..-i -1 .r + 3 13 marks)
CHOW (ili
CHOON
WOOI -.

(b) If li* /(')- 5 = 1, find lri-m4"/(x).

x+{ X-2

v

8 Consider the curve given by the equation .f (x) = 2 - x' .

(a) Sketch the region bounded by the curves -f (x) , g(x) = x2, the lines
x : 0 and x:2. Hence, find the area of the region.

l7 marks)

(b) Find the volume of solid generated when the region bounded by the curve
/(x), lines x=1 and x=2 isrotatedcompletelyaboutthe x-axis.

15 marl<s)

Y

11

197

QS016/2

9 Consider the parametric equations

x=2t-t-' , !=2tlt'' , />1.

(a) Show that

CHOW dy 2r2 -1
CHOON dx 2r: -l
WOOI
(b) +Er aluate at the point (1, 3). 13 marl<sl
(L\ 14 marks)

Y 16 marksl

(c) 4!pin6 in term of r. Hence, show that

dx'

ddrxy2= g

Y3'

-

13
198

QS016/2

10 Atunction / isdefinedby f(x)="5'Y2x-:r'-+"R-x. -r'- 4

'

(a) Find the vertical and horizontal asymptotes of /.

13 marl<s)

(b) /Find the coordinates of,the point where the curve cuts the horizontal

asymptote.

[2 marlcs]
CHOW
CHOON(c) Determine the coordinates of the point where /'(x) = Q.
WOOI
13 marksl

(d) :fBy writing y (x), show that

(y-5)r'+(f-8)x-4=0

Hence, for real x, show that f (x) < -4 or f (x)>4. 14 marl<s)
(e) Sketchthe graph of f . 13 marl<s)

END OF QUESTION BOOKLET

15
199

PSPM
MATRICULATION MATHEMATICS

QM016
2009/2010