Matriculation Mathematics PSPM 1

CHOW QS015l2
CHOON
WOOIARAHAN KEPADA CALON:
Kertas soalan ini mengandungi 10 soalan.

Jawab semua soalan.
Semua jawapan hendaklah ditulis pada buku jawapan yang disediakan. Gunakan muka surat
baru bagi nombor soalan yang berbeza.
Markah penuh yang diperuntukkan bagi setiap soalan atau bahagian soalan ditunjukkan
dalam kurungan pada penglrujung soalan atau bahagian soalan.
Semua langkah kerja hendaklah ditunjukkan dengan jelas.
Kalkulator saintifik yang tidak boleh diprogramkan sahaja yang boleh digunakan.
Jawapan berangka boleh diberi dalam bentuk ?E, e, st)rd, pecahan atau sehingga tiga angka

bererti, di mana-mana yang sesuai, kecuali jika dinyatakan dalam soalan.

2
101

CHOW QSo15/2
CHOON
WOOIINSTRUCTIONS TO CANDIDATE:
This question paper consists of I0 questions.
Answer all questions.
All answers must be written in the answer booklet provided. Use a new page for each

question.
The fuII 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 tT, e, surd, fractions or up to three significant

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

3
102

QSo15/2

SENARAI RUMUS MATEMATIK

Trigonometri

I I+Sln (Z* A)= sin cos .B cos sin B

CHOW (ex a) = cos A cos B + sinl sin B
CHOON
WOOItan(A*B) = tanA + tanB
1 + tanAtanB

sinA+sin,B: 2rioA+ B B

"orA*

sinA -sin.B : 2 "r"A* B *inA- B

Icos * cos B:2 "orA* u "orA- B

lcos - cos.B : -2 rioA* B "inA- I

sin2A=2sinAcosA

cos 2A = cos2l -sinz A
= 2 coszA-l
= l-Zsin2 A

tan2A = 2 tanA

l-lrrfi A

sinz A = l-cos2A

2

cos'A = l+cos2A

2

4
103

QS01sl2

LIST OF MATHEMATICAL FORN{ULAE

Trigonometry

IB +sin (l +,8) = sin ,{ cos
cos sin .B

CHOW cos (,atB)= cosAcosB + sinl sinB
CHOON
WOOItan(A* B)= tanA + tanB
1 + tanAtanB

sinA +sin B : z B B

"inA+22 "orA-

sinl - sinB: z B B

"orA*22 "inA-

cosl * cos.B :2 "orA*2.F2 "orA-,8
cosl -cos B = -2 rioA+ B rroA- B

sin2A=2sinAcosA

cos 2A = cos2 A-sin2 A

= 2 cos2 A-l

=l-2sin2 A

tan 2A = 2tanA

-l---t-a--n---'-A--

s.tn1'A. 1-cos2A
2

CO1S-I= l+cos2A
2
-

5
104

QS015/2

SENARAI RUMUS MATEMATIK

Pembezaan

f(.) f'(*)

kotx -kosek2 x
sekx sekxtanx

kosek.:r - kosek x kot x
CHOW
CHOON Jika x = f {t) dat y= s(/) maka *= #"*
WOOI
d(dv\

dd'xyz-AIdAx )

dt

Sfera V =14"n2r3 S = 4nr'
J
Kon membulat tegak 't = 'fi rs
V =! nr2h
Silinder membulat tegak J S = 2nrh

V = nr2h

a -r'.\ ''...

6
105

QSo1sl2

LIST OF MATHEMATICAL FORMULAE

Differentiation

f(.) f'(*)

cotx -cosec2x
sec .r sec x tan x
CHOW
CHOON cosecx -cosecxcotx
WOOI
If x= f(t) ana t = s(t), then !dx= 1dt"*dx

d(dv\

adzy ir\d- )
dx
dxz=

Sphere V=!n'3 S=4nr2

Right circular cone V =: *r2h =$ TE rs

Righteircularcylinder V=nr2h S=2*rh

7
106

QSo15/2

I :!:1LUngkapkan (rx5':-'4)(x+2) dalam benrukpecahan separa.

[6 markah)

Nilaikan yang berikut (ika wujud):

(a) h.xa-d+.-x-2-'--+l-*-4_i-x-Z---l1--2i-.

n.xa+.ld11--.-XJ;
CHOW [4 markah]
CHOON 13 markahl
WOOI
Cari terbitan untuk fungsi berikut: 13 markahl
14 morkahl
(a) .f (x) =sshE7;T.

O) "f (x)= e2.ln(3x+4).

Diberi cosec'itr-cot x = 3, tunjukkan bahawa cotzx-cotx-2 = 0.

Seterusnya, selesaikan persamaan cosec'x-cot,r = 3 untuk 0 -< x < n.

16 markahl

8
107

QSo15l2CHOW I
CHOON 16 marlcsl
I - f.Expr*r, (,xt"I'-41),0(x,'+* 2) in the form of partial fractions.WOOI
14 marlcsl
2 Evaluate the following (if exist):
[3 marla]
\a) XH-x2 +4x-12
13 marl<sl
ft) ti*1-G. [4 markah]
:-+l 1- X
16 marksl
3 Find the derivative of the following functions:

(a) .f(*)=qs1''E;.

(b) .f (x) = e2'ln{3x+4).

4 Given cosec'x-cotx=3, showthat cot2r- catx-2=0.

Henca, solve the equation cosec'x-cotx:3 for 0 1x1tt.

9
108

QS015/2

Polinomial P(x) = 2xa +ax3 +bx'-l7x+c dengan a, b dan c adalahpemalar,

mempunyai faktor (x + 2) dan (r - 1) . Apabila P(x) dibahagikan dengan (x + 1),

bakinya adalah 8. Cari nilai bagi a, b dan c. Seterusny4 faktorkan P(x)

selengkapnya dan nyatakan pensifarnya.

19 markahl
CHOW
CHOON*uffi.6 (a) Nilaikan r-+-6 5x + I
WOOI
14 markah)

5- p*, -2 < x<-l
-l<x<2
(b) Diberi x'+ px+q,
x>2.
^rr={ x'-4
x-2'

(i) Cari nilai p dan q jika tungsi /(x) adalah selanjar untuk semua nilai

nyatabagi x.
f6 markohT

(iD Lakarkan graf f (x) menggunakan nilai p dan q ymgdiperoleh

dalam bahagian (i).
t3 rnarkahl

10
109

QS015/2

5 Apolynomial P(x) =2x4 +ax3 +bxz -17x+c where a, b and c areconstants,

has factors (x + 2) and (x - 1). When P(x) is divided by (x + 1), the remainder is 8.
Find &e values of a, b and c. Hence, factorize P(x) completely and state its zeroes.

19 marl<sl

CHOWG (a) u^WEvaluate .t-+-6 5X + I .
CHOON
WOOI 14 marlrs)

5-p*, -2<x<-l
(b) Given f(x)= x'+ px+q, -l<x32

x'-4 x>2.
x-2'

(i) Find the values of p and q lt f (x) is continuous for all real values

of x.

16 marl<sl

(ii) Sketch the graph of f (x) using the values p Md q obtained

inpart (i).
13 marksl

11
110

QSo15/2

Suatu lengkung diberi oleh persamaan berparameter

x=t--tl,1t !=t+-.

4(a) Cur, o* dalam sebutan /.
#dx
CHOW
CHOON 17 markahl
WOOI
(b) Dapatkan koordinat titik pegun bagi lengkung tersebut dan tentukan sifat titik

tersebut.
[6 markah]

(a) 1=-L8 Iika y2 -zyJfu*1+x2 =0, tu-nrrjJuBkukLaBnrbvaqhra*wvras &c- JQ+ rt.

16 markahl

(b) Air mengalir dengan kadar tetap 36trcm3s-' ke dalam suatu kon membulat
tegak yang terbalik dengan sudut sopara menegak 45o.

(i) Cankadar peningkatan kedalaman air apabila ketinggian air

adalah 3 cm.

14 marlahl

(ii) Cari masayang diambil apabilaketinggian air adalah 18 cm.

13 markahl

12
111

QSo15l2

7 A curve is given by the parametric equations

x=t--1, 1 l=t+-'

t

(a) Find dy *rd d'4 in terms of ,.

dx dx'
CHOW
CHOON 17 ntarksl
WOOI
(b) Obtain the coordinates of the stationary points of the curve and determine the
nature of the points.
16 marksl

8 (a) If y2 -2y (l+x2) +x2 = 0, show that dy_ *
dx- JG+rt'

f6 marl<sl

(b) Water is running at a steady rate of 36rrcm3s-r into a right inverted circular
cone with a semi-vertical angle of 45o.

(, Find the rate of increasing in water depth when the water level is 3 cm.

14 marl<sl

(ii) Find the time taken when the depth of the water is 18 cm.

[3 marlts)

13
112

QSo15/2

9 (a) TentukannilaibagiRdan a, dengan ^R>0 dan 0o<a<90" supaya

3sin d - 4cosg = ft sin(d - a).

14 marlmhl

(b) Seterusnya, selesaikan persamaan 3sin0-4cos0 =2 bagi 0o< I < 360o.

14 markahl
CHOW
CHOON(c) Berdasarkan jawapan daripada bahagian (b), cari rulai 0 bagi 0o < e <36A"
WOOI
supaya f (0)=*nA j|;s,+t5 adalah minima.

Seterusnya, cari nilai minima bagi f .

13 markah)

10 (a) tCari nilai jika kecerunan bagi lengkun gan xu + fuz y *2y2 = 0 pada

titik (-11) adalah -3.

15 markahl

(b) ti"

Diberi Y=.l+cosx

(i) Cari L dun d'!, dalam sebutan x.
dx
dx'

15 markahl

(ii) Seterusnya, tunjukkan *dx-- ytd4x--(\d4x\)' = o

15 markah)

KERTAS SOALAN TAMAT

14
113

QS015/2

9 (a) Determine the values of R and a, where .R > 0 and 0" < a <90o so that

3sind - 4cosd = Rsin(d - a).

14 marksl

(b) Hence, solvetheequation 3sis0-4cosd=2 for 0o <0<360".

CHOW 14 marks)
CHOON
WOOI(c) From the answer obtained in part (b), find the value of 0 for 0 < 0 < 360o

so that f (g\= 1 is minimum.
' 3sin6-4cosd+15

/.Hence, find the minimum value of

13 marksl

10 (a) Findthevalue of &if the slope of the curve x1 +!ac'y-2y' =0 at

thepoint (-t,t) is -3.

l5 marlal

(b) th'

Given "y= rl+cosx

(i) * #Find ""u in terms of x.

(ii) "*Hence, show #-r#-(*)'=0. 15 marlal
15 markahf

END OF QUESTION PAPER

15
114

PSPM
MATRICULATION MATHEMATICS

QS015
2014/2015

115

QS015/1 QSo15/1
Mahematix
Paperl Matematik

ISemester Keftas 1

Session 2014/2015 Semester I
2 hours
Sesi 2014/2015

2 iam

KF.ME,NTE,RtAN

PENDIDIKAN
MATAYSIA
CHOW
BAHAGIAN MATRIKULASICHOON
WOOI
M4TRICU- TTON DIVBION

PEPERIKSMN SEMESTER PROGMM MATRIKULASI
IT UTNCULAfiON PROGRAIUIME EXAMINATION

MATEMATIK

Kertas L
2 jam

JANGAN BUIG KERTAS SOALAN INISEHINGGA DIBERITAHU.

N NOTOPEN IHISQUESI/ON PAPERUNNLYOU ARETOLD IODOSO.

Kertas soalan ini mengandungi 13 halaman bercetak.
This question papermrsfs of13 pnhfedp4ges.

@ Bahagian Matrikulasi

116

CHOW QSo15/1
CHOON
WOOIINSTRUCTIONS TO CANDIDATE:
This question paper consists of 10 questions.
Answer all questions.
All answers must be written in the answer booklet provided. Use a new page for each

question.

The full marks for each question or section are shown in the bracket at the end of the question
or section.
All steps must be shown clearly.
Only non-programmable scientific calculators can be used.
Numerical answers may be given in the form of fi, e, swd, fractions or up to three significant
figures, where appropriate, unless stated otherwise in the question.

3
117

QS01s/1

LIST OF MATHEMATICAL FORMULAE

Quadratic equation ax' + bx + c = 0'.

*--ott[o'-+*
2a
Arithmetie series:

T,=o+(n-a)d
CHOW
CHOON s,=;Ba+@-a)dl
WOOI
Geometric series:
Tn = arn-l

s,=ff,r*r

Sum to infinity:

S''-=14-r .rl <l

'

Binomial expansion:

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

where neN *d [f'r)/ - (n n''
- r)tr
r.

(t + ax)' = ! + n(ax). @?b-y * n(n - D@ - 2)-(*I * . .

laxl<t where neZ-orneQ

5
118

QS01il1

I Solve the equation 3' +3G-*) =12.

16 marl<s)

2 Soive the inequality +6-x . -x]--l
CHOW
CHOON 16 marks)
WOOI
[r o ol [r o ol

l, r-l v3 Givenmatrices a=l + I 0 I and B=l z I 0 I where B istheinverseof A.
b L, rj

Find x, "y and e in terms of a and b.

16 marlrsl

4 Using algebraic method. find the least value of n for which the sum of the first n terms

of a geometric series

0.88 + (0.88)'] - (0.88)3 + (0.88)a +...

is greater than half of its sum to infinity.
[7 marlu]

5 (a) State the interval for r such that the expansion for (4 + 3r)i 1, valid.

[2 marksl

(b) *lt1iExpand {+ in ascending power ofx up to the term in 13.

14 marlul

(c) Hence, by substifuting an appropriate value ofx, evaluat. 1S;i correct to three

decimal places.

14 marlcsl

7
119

QSo15/1

6 (a) Given f(*)=2x+l and g(r) =x'+2x-1.

(i) Find (/ - g)(x). [2 marksl
14 marksl
(ii) Evaluate (3s -zf)(t).
CHOW
CHOON(b) EGiven .f (*) =state the domain and range of /(r).
WOOI
Hence, on the same ixes, sketch the graph of /(x) and f-t(x).

16 marlal

7 lr;t z = a+bi be a nonzero complex number.

ft(a) !show that =

14 marks)

O) Show that if i = -2, then z is a complex number with only an imaginary part.

13 marlal

(c) Find the value of a and b if z(2-i) =(i+1)(t+;).

15 marksl

I

120

QSo15/1

8 (a) Solve the following equation le*, +x_l tl= +.

(b) Find the solution set for the inequality 16 marlu)
l7 marlesl
( x+2\ _

2-l
CHOW\x-t-<4)).
CHOON
WOOI
9 Two companies P and Q decided to award prizes to their employees for three work

ethical values, namely punctuality (x), creativity (y)and efficiency (e). Company p

decided to award a total of RM3850 for the three values to 6, 2 and3 employees

respectively, while compary Q decided to award RM3200 for the three values to

4' I and 5 employees respectively. The total amount for all the three prizes is

RMl000.

(a) Construct a system of linear equations to represent the above situation.

13 marksl

(b) By forming a matrix equation, solve this equation system using the elimination

method.

[7 marks\

(c) with the same total amount of money spent by company p and e, is it possible

for company P to award 15 employees for their creativity instead of 2
employees? Give your reason.

13 marla)

11
121

QSo15/1

l0 (a) Determinewhether f(x)=* *O g(x)= T areinversefrurctionof

each other by computing their composite functions.

[5 marks)

(b) Given .{(x)=ln(l-3r).
CHOW
CHOON(1) Determine the domain and range of f (x). Then sketch the graph of
WOOI
f (x).

[6 marks]

(ii) Find /-r(x), if it exists. Hence, state the domain and range of f-t(x).

[4 marks]

END OF QUESTION PAPER

13
122

QS015/2 QS015/2

Mathematia Matematik

Papr2 Kertas 2

ISemester Semester I

Session 2014/2015 Sesi 2014/2015
2 hours
2 jam

KEN,IENTERIAN

PENDIDiKAN
MALAYSIA
CHOW
BAHAGIAN MATRIKULASICHOON
WOOI
M4TRICUATION DIVNION

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
II,IATNC UIATTON PRrcMMME EXAMINATION

MATEMATIK

Kertas 2
2 jam

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DONOTOPEN THIS QUESflON PAPERUNNLYOU ARE IOLD IODOSO.

Kertas soalan ini mengandungi 15 halaman bercetak.
This quxlbn paperconsisfs of 15 pinted pages.

@ Bahagian Matrikulasi

123

CHOW QS015/2
CHOON
WOOIINSTRUCTIONS TO CANDIDATE:
This question paper consists of 10 questions.
Answer all questions.
All answers must be written in the answer booklet provided. Use a new page for each

question.

The full marks for each question or section are shown in the bracket at the end of the question
or section.
All steps must be shown clearly.
Only non-programmable scientific calculators can be used.
Numerical answers may be given in the form of tr, e, stJrd, fractions or up to three significant
figures, where appropriate, unless stated otherwise in the question.

3
124

QSo15/2

LIST OF MATHEMATICAL FORMULAE

Trigonometry

Isln (e+ A)= sin,4 cos .B + cos sin.B
CHOW
CHOON (exA)=cos AcosB + sinl sinB
WOOI
tan (A+ B) = tanA * tan B

1 + tanAtanB

sin I + sin B: 2 ,inA+2B2 "orA- B

sinl - sinB: 2 B rinA- B
"orA*22

cosA* cos B:2 B B

"o"A*22 "rrA-

cosA- cos B : -2 rinA+ B ,inA- B

sin2A=2sinAcosA

cos 2A= cos2 A-sin2 A

= 2 cos2 A-l
= l-Zsin2 I

tan 2A= 2tan 4

l-tar," A

sinz A = l-cosZA

2

cos' A = l+cos2A

5
125

QS015/2

LIST OF MATHEMATICAL FORMULAE

Limit

li* si'ft = I

h-+0 h

hlm+0l-cohs fr =0
CHOW
CHOONDifferentiation
WOOI
f(*) "f'(*)

cot x - cosec2,
xsec
sec x tan x

cosecr -cosec xcotx

*"*rf y =g(r) and * = fk), tnen ff=

d(dv\

dd'xv2-AldAx )

dr

Sphere V A nr3 S = 4nr2

=1 =g rE rs
J -,S 2 nrh

Right circular cone Y =! nr2h
3

Right circular cylinder V = nrzh

7
126

QSo15/2

Given that (x-2) is afactot of thepolynomial "f(*)= axt -l}x' +bx-2 where
a and b arc real numbers. lt f (x)is divided by (x + 1) the remainder is -24,
find the values of aand D. Hence, find the remainder when /(x) is divided

by(2x+1).

16 marksl

CHOW2 Solve the equation 2cos2 x -l =sinx for0 I x 12n. Give your answer in terms of a.
CHOON
WOOI 16 marksl

3 Find the relative extremum of the curve ! = x3 -4x2 +4x.

16 marksl

Car X is travelling east at a speed of 80 km/h and car Y is travelling north at 100 km/h
as shown in the diagram below. Obtain an equation that describes the rate of change
of the distance betlr,een the two cars.
Hence, evaluate the rate of change of the distance between the two cars when
car X is 0.15 km and car Y is 0.08 km from P.

Car X

[7 marks)

9
127

QSo15/2

Expand (x+a)(x+ b)' , o and b are real numbers with b > 0. Hence, find the

values of a and b if (x+ a)(x+b)'=*t -3x-2.

-Express #xo -4x'+5x-l in the form of partial fractions.
x'-3x-2

ll2 marksl
CHOW
CHOON(a) Express sin 6x - sin 2x in a product form. Hence, show that[6 marlcsl
WOOI l7 marksl
sin 6x - sin2x + sin 4x = 4 cos 3r sin 2x cos x .

(b) Use the result in (a) to solve

sin 6x - sin 2x + sin 4x = sin 2x cos x
for 0<x<180".

7 Find the limit of the following, if it exists.

-.(a) lx[--f>l-.3-_vrx+q)3J

(b) l.l.m--Z--:x.-l 13 marksl
x+-* *z -g 13 marks)
r ^f 14 marksl

l,

(c) l,xl.m+4x',-l/3X- x_2-4 .

11

128

QSo15t2 x<0
0<x<4
lt+r-,
x>4
8 lr-,Given rhat /(,r)= ]-
lc,
u'here C isaconstant.

CHOW(a) Determine whether /(x) is continuous at x = 0.
CHOON
WOOI 15 marksl

(b) fGiven that (x) is discontinuous at x = 4, determine the values of c.

13 marksl

(c) fFind the vertical asymptote of (x).

14 marksl

13
129

I

QS015/2

9 Consider the parametric equations of the curve

.tr=cos30 and y=sin30, 0<0<2r.

(a) Find 4L and express your answer in terms ofd.

dx

CHOW(b) Find the value ot Ldxf4f x =O 14 marksl
CHOON 14 marksl
WOOI- 44dxz(c) Show ,,nur3 cosa 0 sin?' [5 marksl
15 marksl
*iHence, calculate " 0 =L
ll0 marksf
(a) J;.10 Use the first principle to find the derivative ofg(x) =

(b) Given that ev + xy +ln{l+Zx) =1, x > 0.

lff ffishow that (ev *
*,' (U*)' . rff- =o

#Hence, find the value "f at the point (0,0).

END OF QUESTION PAPER

15
130

PSPM
MATRICULATION MATHEMATICS

QS015
2013/2014

131

QS015/1 QS015/1
Mathemalix
Paprl Matematik
1Semester
Session 2013/2014 Kertas 1
2 hours
Semester I

Sesi 2013/2014

2iam

CHOW BAHAGIAN MATRIKULASI
CHOON KEMENTERIAN PENDIDIKAN MALAYSIA
WOOI
MATRICUATION DIVBION
MINNIRY OF EDUCATION MAIaffSA

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
MATRICU-ATTON PROGRAMME EX,4MINATION

MATEMATIK

Kertas 1
2 jam

I JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DO NOIOPEN 7H'S QUESTTON PAPER UAINLYOU ARE TAD IO DO SO.

I

Kertas soalan ini mengandungi 13 halaman bercetak.
This quxtion paperconssfs of 13 pinted pages.

@ Bahaglan Matrikulasi

132

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

question.

The full marks for each question or section are shown in the bracket at the end of the question
or section.
All steps must be shown clearly.
Only non-programmable scientific calculators can be used.
Numerical answers may be given in the form of n, e, strtd, fractions or up to three significant

\- figures, wltere appropriate, unless stated otherwise in the question.

133

QS015/1

LIST OF MATHEMATICAL FORMULAE

Quadratic equation m2 +bx+c =0:,

*--ut'[* -+*
2a

CHOWArithmetic series:
CHOON
WTn = o+(n-t)d
OOI
s, =llzo+(n-r)dl

Geometric series:

Tn = arn-\

t,=ffl*l

Sum to infinity:

s. =r_",lrl<t
\- Binomial expansion:

(a+b)'=an +(:)"",.(;)"-,uz + ..+(:)"-,. + ..+bn,

where neN and(:)=@+W

*(t + ax), = t + n(m). t*f * n(n -t)(n - z) (*)' * ...

laxf <t where neZ- or n eQ

134

QS015/1

1 [; -" *. dGiven matrice, ,n=
r,*3l , =1" l.l. the values of c, and,e
looz) [o;,)
"such that AB: 14 .I, where .I is the identity matrix. Hence, determine l-r.

16 marksl

CHOW2 Consider the function f (*)= I + ln x, x) l. Determine -f-t (x) and state its range.
CHOONHence, evaluate f'(3).
WOOI
16 marksl

3 Find the value of x which satisfies the equation

logrx=(log, x)2, x>1.

l7 marksl

4 Solve the equation 22x-2 -T*t =2' -23.

17 marksl

5 #,Given g(x) = - * lwhere fr is a constant.

(a) Find the value of /r if (g. S)(r): r.

(b) Find the value of & so that g(x) is not a one-to-one function. 15 marksl
15 marksl

135

QS015/1

6 f(rt:Given e3' + 4, x e "R.

(a) Find /-r(x).

15 m.arksf

O /(r)On the ffurre axes, skctch the graphs ofCHOWarrd 7-t (x) . State the domain of
CHOON
WOOI"f(x) and ,f-'(;),

16 marlcsl

4-2i 4+2i 2

16 narkd

(b) Given logo2=ril atrd lo&u7=z. Expressr intemsof n and n if

(l4t*txs'*)= z'

16 marksl

136

CHOW QS015/1
CHOON
WOOI8 An osteoporosis patient was advised by a doctor to take enough magnesium,

vitamin D and calcium to improve bone density. In a week, the patient has to take
8 units magnesium, I I units vitamin D and 17 units calcium. The following are three
types of capsule that contains the three essential nutrients for the bone:

Capsule of type P: 2 units magnesium, 1 unit vitamin D and I unit calcium.

Capsule of type Q: I unit magnesium, 2 units vitamin D and 3 units calcium.

Capsule of type R: 4 units magnesium, 6 units vitamin D and l0 units calcium.

yLet x, and z represent the number of capsule oftypes P, Q andR respectively that

the patient has to take in a week.

(a) Obtain a system of linear equation to represent the given information and write
[,]

the system in the form of matrix equation AX = B, where X =ltty l.
lz)

13 marks)

(b) Find the inverse of matrix,4 from part (a) by using the adjoint method. Hence,
findthevalues of x,y and z.

[8 marks]

(c) The cost for each capsule of type P, Q andrR are RMl0, RMl5 and RMlT

respectively. How much will the expenses be for 4 weeks if the patient follows

the doctor's advice?

12 marl<s)

11

137

QS015/1

9 (a) In an arithmetic progression, the sum of the first four terms is 46 and the

seventh term exceeds twice of the second term by 5. Obtain the first term and
the common difference for the progression. Hence, calculate the sum of the
first ten even terms of the progression.

16 marl<s)

(b) A ball is dropped from a height of 2 m. Each time the ball hits the floor, it
]bounces vertically to a height that is of its previous height.

4
CHOW
CHOON(i) Find the height of the ball at the tenth bounce.
WOOI
12 marksl

(ii) Find the total distance that the ball will travel before the eleventh

bounce.
15 morksl

10 (a) Find the solution set of lZ -lxl < lx + 31. 18 marlal

(b) If x+1< 0, show that 13 marlcsl
(i) 2x-l<0. 14 marksl

(ii) #,r.

END OF QUESTION PAPER

13
138

QS015/2 QS015,2
Mahematia
Matematik
Papr2
Kertas 2
ISemester
Semester I
Session 201 3/2014
2 hours Sesi 2013/2014

2 jam

BAHAGIAN MATRIKULASICHOW
KEMENTERIAN PENDIDIKAN MALAYSIACHOON
WOOI
M4TRICWTIONDIVNION
MINNIRY OF EDUCANON MAI-4WA

PEPERIKSMN SEMESTER PROGMM MATRIKULASI
MATNC UIANON PROGMMME FX,4MINANON

MATEMATIK

Kertas 2
2 ja,m

I NJANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
NOT OtrN 7HlS 8UES77ON PAPER UAINLYOU ME TUD IO DO SO.

I

Kertas soalan ini mengandungi 19 halaman bercetak.
This question paper oonsrsfs of 19 pinted pages.

O Bahagian Matrikulasi

139

QS015l2

INSTRUCTIONS TO CANDIDATE:

This question paper consists of 10 questions.

Answer all questions.

All answers must be written in the answer booklet provided. Use a new page for each

question.

CHOWThe firll marks for each question or section are shown in the bracket at the end of the question
CHOON
WOOIor 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, surd, fractions or up to three significant

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

140

QS(lr5/2

LIST OF' MATHEMATICAL tr'ORMULAE

Trigonometry

sin (l t B) = sin,4 cos B + cos,4 sin B

CHOW I +cos (; t B) = cos cos .B sin,4 sin.B
CHOON
WOOItan(A+.8)= tanA + tan8

1 + tanAtanB

sinl + sinB : 2 rinA+ B ro"A-',

sin,{ - sinB : 2 "orA* B *inA- B
cosl * cos B =2 "orA* B "or4- B
cosr4 - cos B : -2 rinA* B "inA- B

sin 2A=2sinA cosl

cas 2A = cos2 A-stnz A

= 2 cosz A-l

= l-2sinz A

tan 2A = 2tanA
l-tanz A

l-cosZA

gin2 A =

2

l+cos2A
cos' A =

2

141

QS015/2

LIST OF MATHEMATICAL FORMULAE

Differentiation

f(.) f'(*)

cot x )

- cosec"x

CHOW xsec sec x tan x
CHOON
WOOIcosecx -cosecxcotx

If x= f (t) ana t = s(t), then !dx= 4dt*Ldx

d(dv\

dd2xyz=!,aldAx )

Sphere Y=!n'3 S=4nr2

Right circular cone V =: nr2h $ = nrs
3
S =2nrh
Rightcircularcylinder V = r,r2h

142

QS01s/2

I Expr.r,x,f+3x+2 in partial fractions form.

15 marksl

2 Statethevaluesof Rand a suchthat 3sind+6cosd=r?sin(9+a) where R>0

and 0" <a<90". Hence, solve 3sind+6cosd=.rE for 0 <0<180".

16 marksl
CHOW
CHOON\-3(a)Findthevalueofm if ti* T**?t =1.
W
OOI r-+0 4x -8X'

(b) E-6. 13 marl*l

Evaluate rr-ir+n0 X 14 marks)

4 (a) LFind if y =cosec{sin[rn(x+r)]].

13 marksl

tive of y = cos3x and express your answer in the
(b) deriva fObtain the second

simplest form.

[4 marks]

143

CHOW QS015/2
CHOON
WOOI5 A cubic polynomial P(x) has remainders 3 and I when divided by (, - 1) and

(x -2), respectively.

(a) Let Q@) be a linear factor such that P(x) = (x - 1)(x -2)Q$) + ax + p,

where a and B are constants. Find the remainder when P(x) is divided by
(x -t)(x -2).

15 marlcsl

(b) Use the values of a and B frompart (a) to determine Q(x) if the coefficient

of x3 for P(x) is l and P(3)=7. Hence,solvefor x if P(x)=7-3x.

16 marlesl

11
144

QS015/2

6 (a) State the definition of the continuity of a function at a point. Hence, find the

value of d such that

f(x\=[,"'*o' x<o

[3x+5, x>0

CHOW is continuous at x = 0. [5 marl<s]
CHOON
W(b) /A tunction is defined by
OOI
f (x)={;,,_;,, ;:i 13 marksl
14 marksl
/Determine the value(s) of fr if is:
(D continuous for all x e IR.

(ii) differentiable for all x e lR.

13
145

QS015/2

7 (a) *Find the derivative of /(x) = by using the first principle.

(b) Use implicit differentiation to find: [4 marks)

(i) ! x ykrx=e'-!. 13 marksl
frc 15 marksl

(ii) thevalue $+&ciyf x12- I =3 when r=1.
CHOW
I A curve is defined by parametric equationsCHOON
r=ln (l+r), | = e" for / >-1.W
OOI
(a) #Find Le *o interms of t.

16 marksl

O) Show that the curve has only one relative extremum at (0,1) and determine

the nature of the point.
16 marlal

15
146

QS01s/2

(a) A cylindrical container of volume l28n m3 is to be constructed with the same

material for the top, bottom and lateral side. Find the dimensions of the

container that will minimise the amount of the material needed.

16 morl<sl

CHOW(b)Gravelis poured onto a flat ground at the rate of * -' per minute to form a
CHOON
WOOI 20

conical-shaped pile with vertex angle 60o as shown in the diagram below.

Compute the rate of change of the height of the conical pile at the instant
/ = l0 minutes.

l7 marksl

17
147

QS015/2 14 marl<sl

ro (a) show,n"ffiffi=*r(ry)

(b) Use trigonometric identities to verify that

CHOW(D ztan9
CHOON
Wsind: 2
OOI
l+tarr'9'
2

13 marlxl

r-tarf q

(iD cosd =

l+tarr'9'
2

13 marlal

Hence, solvetheequation 3sind+cosd =2 for 0 <e <180'. Giveyour

answers correct to three decimal places.

15 marl<sl

(.*
END OF QUESTION PAPER

19
148

PSPM
MATRICULATION MATHEMATICS

QS015
2012/2013

149

QS01st1 QS015/1
Mathematia
Paprl Matematik

1Semester Kertas 1
Session 2012/2013
2 hours Semester I

Sesi 2012/2013

2 jam

q^GeLf5-J:s
CHOW ==4-::t-6 I :!
CHOON l:-.\
WOOI
BAIIAGIAN MATRIKULASI
KEMENTERIAII PELAJARAN MALAYSIA

I,IATRICU-,ITION DIVBION
MINNTRY OF EDUCATION MAL4YSU

v' PEPERIKSMN SEMESTE R PROG RAM MATRIKULASI

IT UTNC UI-4TTON PRrcRAMME EXAMI|,{ATION

MATEMATIK

Kertas L
2 ja'm

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU.
DO NOT OPEN IFI/S QUESNON PAPER UI{NL YOU ARE TAD IO DO SO.

v

CHOW CHOON WOOI

Kertas soalan inimengandungi 13 halaman bercetak.
This question paper consists of 13 pinted pagx.

@ Bahagian Makikulasi

150