Matriculation Mathematics PSPM 1

t QM016tl QMo16t1
Mathematics
Matematik
Paper 1
Kertas 1
ISemester
Semester I
2009/201a
2009t20fi
2 hours 2 jam

CHOW &
CHOON
WOOI :E:=J-:-:--

BAIIAGIAIY MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATNCULATION DIVISION
MINISTRY OF EDUCATION MAIAYSIA

PEPERIKSMN SEMESTER PROGMM MATRIKULASI
MATNC ULATION P ROGRA MME EXA MINATIO N

MATEMATIK

Kertas 1
2 jam

JANGAN BUKA KERTAS SOALAN lNISEHINGGA DIBERITAHU.
D0 NOIOPE,V IHIS BOOKLET UNTIL YOU ARE TALD IO DO S0.

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

@ Bahagian Matrikulasi

r QMo1611 CHOW
CHOON
I INSTRUCTI.\S To CANDIDATE: WOOI

This question booklet consists of l0 questions.
Ansu-er all questions.
The fuil marks lbr each question or section are shown in the bracket at the end of the question

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

Numericai answers may be given in the form of a. e . surd, fractions or up to three

significant figures, where appropriate, unless stated otherw.ise in the question.

=

3

QM016/1

LIST OF MATHEMATICAL FOR\{LLAE

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

, = *!:!b'_1!,
2a
CHOW
CHOONFor an arithmetic series:
WOOI
T,=a+in-l)ri

Sn=nfT.a+(n_l)dl
y

For a geometrie series:
Tn = Gr"-l

s, = -{t!i !., *i

Binomial expansion:

.y where neN una l[',jJ=- nl. rl

(n- r)t

'fu:!:#1:!-,'(\r!(r + x)' =t + nx
r' + .. . {- x' +... for ixi < r

5

QMo16/1

I Solr.e the equation 32, - l0 ( 3,-,; + 1 = 0^

16 marlrsl

2 Determine the solution set for 2, * 1 < S.
x

CHOW l7 narksl
CHOON
WOOI3 'Express -(x-- 2+["=' ++ -2-x.+r2) in r'part'-ia-'l- f--ractions.

[6 marksf

v 4 The first term and common difference of an arithmetic progression are a and -2.

respectively. The sumof the first n terms is equaltothe sumof the first 3r terms.

ifExpress a in terms of ru. Hence, shovl, that n = 7 a = 27 "

16 marksl

5 (a) Solve 25+r>.r.

14 narksl

(b) If ct and p are the roots of the quadratic equation 2x2 + x * 4 = 0, form an
17 equation whose roots are a + 2p and 2a + B.

l7 marksl

6 Given a complex number z = a +6i which satisfy the equation z2 = B+ 6i.
(a) Find all the possible values of z.

16 marl<s)

(b) Hence, express z in polar form.

[6 marla]

7

8M016/1

[: x ir.l
07 l'4atrix .4 is given as j O x 4 | and .11: -75. Find

L0 .r-10.1
(a) the value cf x.

[4 rnctrks]
CHOW
CHOON(b) the cofactor and the adjoint matrix of ,{. i{ence, detemrine the interse of, l.
WOOI
l8 mark)

8 Given a poly'nttmial P(x)= 1.'ir' * ,,.-r * o.r - -r0 has iactors (x + 2) and (x-5).

\r 16 morksl
l3 marksj
(a) Find the value of the consrants a and b. 13 marksl

(b) Factorize P(x) completeil'.

(c) Obrtain the solution sei for P(x) < 0.

!, g (a.r i xExpand 1+ - ,)1 and (1 + 3x)- i,, ur.rncling powers of up ro

the term .tr.
15 marksl

(b) Find the expansion cl (a - i1 I up to the term .rt and determine

I)r (1 + 3x)-

the range ol- .r such that this expansion is valid. Hence. by substituting

j

x = ;iJ . approximate the vaiue of J51 .oo..t to four significant tigures.

l8 marksl

I

QM016/1

10 The following table shows the quantities in kilogram (kg) and the amount paid (RM)

for three types of fruits bought from three stalls at a night market.

CHOW \ Fruit Mango Durian Rambutan Amount paid
CHOON (ke) (ke) (ke) (RM)
WOOIS,N 34.00
5 J 2 37.00
P J 4 4 29.00
2 3 aA
a

R

.Y -" yThe price in RM per kilogram (kg) for mango, dwian and rambutan are x, and z

respectively.

(a) Form a system of linear equations which represent the total expenditure per

stall calculated based on the weight bought and price per kilograrn. Hence,
write the system in the form of a matrix equation AX = B.

[3 marks]

(b) Find the determinant, minor and adjoint of matrix l.

16 marl<s)

y (c) Based on part (b) above, find l-1. Hence, solve the rnatrix equation.

14 marl<s)

(d) Suppose the price per kilogram for mango, durian and rambutan has increased

by RM2. RM2 and LVI1. respectiveiy. Obtainanewmatrixrepresenting

the amount spent on each tvpe of fruit to be bought.

12 marlcs]

END OF QUESTION BOOKLET

11

QM016/2 QM(}16/2 tr
Mathematics
Matematik
I IFaoer2
Semester Kertas 2
2009/2010
2 hours Semester I

2009t2010

2 jam

s

Em- JF: '

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATRIC UL,ATI ON D IVISI O N

MINISTRY OF EDUCATION MALAYSU

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
MATNC ULATION P ROGRAMME EXAMINATION
CHOW
CHOON MATEMATIK
WOOI
Kertas 2
2 jam

JANGAN BUKA KERTAS SOALAN TNISEHINGGA DIBERITAHU.
DO NOT OPEN IHIS BAOKLET UNTIL YOU ARE TOLD IO DO SO,

Kertas soalan ini mengandungi 11 halaman bercetak,

This booklet consrsfs of 11 printed pages.

@ Bahagian Matrikulasi

0M016/2

I I\STRL-CTIO\S To CANDIDATE: I
I This question booklet consists of l0 questions.

Answer all questions.

T'he 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.
CHOW
CHOONOnly non-programmable scientific calculators can be used.
WOOI
Numerical answers may be given in the form of fr , € , surd, fractions or up to three

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

Y/

3

QMo16/2

LIST OF MATIIEMATICAL FORMULAE

Difrerentiation

tf y=s(/) *d *=fft],*"n !&=c4d"t 4dx

d(dv\
d'y _Ald. )
dx2 dx

dt
CHOW
CHOONIntegration
WOOI
\7

Iutu:uv-lvdu

-y

5

QMo16/2

1 A function g is defined by

g(x)=-L x>1.

../x - 1
Find g-1(x) and state its domain and range.

15 marl<sl
CHOW
CHOON2 /A function is given as
WOOI
Ilx+t]. x<0

f(x)=I j 2" x=0

|. .'.. x>0.

Find lim /ix;. xl-i-m+0/*(x) and lim /(x).

!7 x-+0- :-+0

/Hence, deterrnine whether is continuous at x = 0. Give a reason to your answer.

16 marksl

3 I;.f.If :.]. .y + e'. sho\\- rhar , =,

4 Eraluare r: r-l ar. 16 marks]
[7 marl<s]
.|, .rr_,
14 marl<sl
l7
18 marlal
5 A parametric curve is givenby *=, -:, r- =t+!, t +0.

(a) Find * in terms of r and evaluate it at r = -2.

(bl +Find the r alue of ut r = 1. and evaluate r1-ia^451[Y4) .

dr-

7

QMo16/2

6 (a) Show that y - ^[7i .O for all real values of y.

[2 marl<s)

(b) eJ -X

Let _f be a function defined by -f f x ) - Find .f-t (*).

16 marksil
CHOW
CHOONtinft(c) Evaluate-lx 2r
WOOI
[3 morks]

7 /A function is defined by

34, x=-4

= 0. x=2

"f (x) = 17, x=4

xJ'f)--+3xx-'6--4 , x*-4,x+-3,x+2,x+4

(a) Evaluate !y; f (x).

14 marlcsl

(b) Find the interval(s) where / is continuous on the interval l-4,41.

[8 marl<s]

8 (a) Given a function g defined by

_ | *," , x(1

t,s(x) = j lrnr), , x>1.

e3

Er aluate J , g\x) dx.

[6 marks)

O) Use inte$ation b1'parts to showthat

'i\-eig-' *-l dr = (.r- thp, * i - i-:]- a,
+l

"le" 17 marlal

I

QMo16/2

9 (a) Let f and g beiunctionssuchthat f(x)=xtg(x') with g(1)=Z and

8'(1) = l. Find f 'tt1.

14 markl

(b) Givenacurve y=r+1.
Y

(i) Determine the gradient ofthe cun'e ] =, * I at ..r = b in terms of b.
,r

(ii) Find the value of b rf a straight line rijth the gradient in (i) passes
CHOW
CHOON(iii)through the points t U.U +!) and ( 0 1)
WOOI
t)

Hence. find the equation of a line perpendicular to the line in (ii) at

(0,4)

19 marl<sl

10 Aregion R isboundedbythecurve y:x(x-2) andline !=x. 12 markl
(a) Sketch the graphs and shade the region R.

(b) Find the area of R.

13 marks)

(c) Find the volume of the solid obtained when the part of R above the x-axis is

rotated through 360o about the x-axis.

15 mqrksl

v (d) Let R forms the surface of water in a pond where the depth of the water at

any point (x, y) in R is given by x + 5. Find the volume of the water in the

pond.

15 marl<sl

END OF QUESTION BOOKLET

11

PSPM
MATRICULATION MATHEMATICS

QM016
2008/2009

QMo1611 QM()16/1

Mathematics Matematik

Paper 1 Kertas 1

ISemester Semester I

2008/2009 2008t2009

2 hours 2 jam

&

-Y:

-r=-

BAHAGIAN MATRIKULASI
KEMENTERIAN PELAJARAN MALAYSIA

MATRIC ULATION DII/ISION
MINISTRY OF EDUCATION MALAYSIA

PEPERIKSAAN SEMESTER PROGMM MATRIKULASI
IATRICULATION P ROGRAMME EXAMINATION

MATEMATIK

Kertas 1
2 jam

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

t
CHOW
CHOON
WOOI

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

O Bahagian Matrikulasi

CHOW QM(}16/1
CHOON
WOOIINSTRUCTIONS TO CANDIDATE:
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 n, e, surd, fractions or correct to three
significant figures, where appropriate, unless stated otherwise in the question.

3

QMo16/1

LIST OF MATHEMATICAL FORMULAE

Arithmetic Series:

T,=o+(n-t)a

s,=|lzr+(n_lal

Geometric Series:

T, = ar'-l

'.=+! ror r<l
CHOW
CHOONBinomial Expansions:
WOOI
(a + b)' = o' +(i)"".(;)"-u' +-. -.(:)"' u' +. . + b', where,e e N and

(I[",\Jt- -;Gvlt-4

(t+x)' =r+nx*"fu2:!') *, +...*n(n-l)":(n-r+l) x, +... for lxl<1

rt

5

QMo16/1

1. (-2 r2* rQ

Ex'press /(!t_-x;f'f-l{+x) in partial fractions.

[5 marks]

2. The fifth term and the tenth term of a geometric series are 3125 and,243respectively.

CHOW(a) Find the value of cornmon ratio, r of the series.
CHOON
WOOI [3 marks]

(b) Determine the smallest value of n such that S-; S' < 0.02, where S, is',
s_
the sum of the first r term and S* is the sum to infinity of the geometric

series.

[3 marks]

3. +Solve the equation 3log, 3 + 1og, V; =
J

[7 marks]

4. Determine the interval of x satis$ing the inequality lx+Zl>tO-x2 . [7 marks]
, - 5. The roots of the quadratic equation 2x2 =4x-1 are a and p.

(a) Find the values of az + p2 and. a3 p + aB3 .

[5 marks]

(b) Form a new quadratic equation whose roots are (o -Z) ana (p -Z).

[5 marks]

7

QMo16/1

,2
(a)

zl-zz

a l-1-lbwhere and arereal numbers. Hence, determine

' lzr-zzl

[5 marks]

(b) Giventhat z=x + ry, where x and y aretherealnumbers and Z isthe

complex conjugate of z. Find the positive values of x and y so that

12

-Z+z -=5-t.

[6 marks]
CHOW
CHOON
WOOI
of1 (a) The rth term of an arithmetic progression is (1+ 6r). Find in terms n, the

sum of the first n terms of the progression.

[4 marks]

(b) (D Showthat I"lg-x=3!(\r-I9))-'

[3 marksl

I

I(ii) Find the first three terms in the binomial expansion of (t- +e)i -' t,

ascending powers of x and state the range of values of x for which

this expansion is valid.

[3 marks]

'(iii) Find the first three terms in the expansion o, :( + x) in ascending

"lg-x

powers of x.

[3 marks]

9

QMo16/1

' '1 I z 2 -:l

n=| o l. ttnu Pe and
(a)t, , , -r,l8.
, ['Given the matrice, = and 3.1
Ll
2 2) L-3 0

hence, determine P-t . r

[4 marks]
CHOW
(b) The following table shows the quantities (kg) and the amount paid (RM) forCHOON
WOOI
the three types of items bought by three housewives in a supermarket.

Housewives Sugar (kg) Flour (kg) Rice (kg) Amount Paid (RM)

Aminah 3 6 a 16.s0
J

Malini 6 3 6 21.30

Swee Lan J 6 6 2l.00

The prices in RM per kilogram (kg) of sugar, flour and rice are x, y and z

respectively.

(i) F'orm a system of iinear equations from the above information and

write the system of linear equations in the form of matrix equation

AX=8.

[3 rnarks]

(ii) Rewrite AX = B above in the form kPX: B, where A: kP

( P is the matrix in (a) ) and, k is a constant. Determine the value of
k and hence find the values of x, y and z.

[6 marks]

9. Polynomial PG) = mxt -8xz +nx+6 can be divided exactly by *' -2x-3. Find

the values of m and n. Using these values of m and n, factorize the polynomial

completely. Hence, solve the equation

3x4 -14x3 +llx2 + l6x-lz = a
" using the polynomial P(.r).

[13 marks]

11

QM016/1

I [:10. ,Matrix is given by = I ]'l
lz -3 -rl

(a) Find
(i) the determinant of l,
(it) the minor of ,4 and
(iir) the adjoint of A.
CHOW
CHOON [9 marls]
WOOI
(b) l-tBased on part (a) above, f,ind . Hence, solve the simultaneous equations

Y+ z=1
2

5x+y - z =9

2x -3y -1, =i.
2

[6 marks]

END OF BOOKLET

13

QMo16/2 QMo16/2
Mathematics
Paper 2 Matematik

ISemester Kertas 2

Session 2008/2009 Semester I
2 hours
Sesi 2008/2009

2 jam

+

a€. I -_-s
----

BAHAGIAN MATRIKULASI
KEMBNTERIAN PELAJARAN MALAYSIA

MATRIC ULATION DIVISION

I MINISTRY OF EDUCATION MALAYSIA
PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI
MATNC ULATION P ROGRAMME EXAMINATION

}IATEN{ATIK

Kertas 2
2 jam

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

Kertas soalan inimengandungi 11 halaman bercetak.
This booklet consrsfs of 11 printed pages.

O Bahagian Matrikulasi

CHOW QM016/2
CHOON
WOOIINSTRUCTIONS TO CANDIDATE:
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 ru, e, surd, fractions or up to three significant
figures, where appropriate, unless stated otherwise in the question.

Y/

3

QMo16/2

LIST OF MATHEMATICAL FORryIULAE

Differentiation

lf y =g(r) and * = f(t).tt',rn lL=*"*

i CHOW,,c,-i'.!2= 4dt\(dqx))
CHOON
|lu'e WOOI dx

Integration

y [rd, = uv - tvclu

!7

E

QMo16/2 dv
dx
1. Given Iny = sx! ,

[5 marks]

) lf y=Jzx\Sxl , determine the domain of dy and find the respective intervals
dx
in which Q, o und Q.o.
I dx dx

[6 marks]
CHOW
CHOON3.Giventhatf{x}:10- 2.r Find the value of ,t so that
WOOI ,
[7 marks]
,[;)t7
f-, (*z)= Hence, find

4. Let f (x) =l+x -11 and g(x) = x +2.

(a) Find the interval of x for which "f (x) < s@). [4 marks]
[3 marks]
If(b) h(x)= f (x)+29(x), express h(x) as a piecewise function.

v

Let J Gx).lr=a a' x- + a- x + -la g'here a is non-zero.

(a) Find a if l-(0) : 6 [2 marks]
[3 marks]
(b) Determine f (x) .
[5 marks]
(c) Determine the domain and range of f (x). Hence, state the
interval in which f is one to one.

7

QMo16/2

6. (a) By using the partiai fraction method, shor,v that

l---:--- rr 1 r)

*2-4 4\.r-2 x+?-)

find n:-:l'; +]^
CHOW I{ence. dr .
CHOON JiZ1 -+
WOOI
[6 marks]

(b) : ''Sketch the region boundeci br the cuil-es l' .re J' = .vr . .l ) 0
and the line r: 2. Iind its area.

[6 marks]

(r 7. Given

fe'+A, x<o
f(x)=Jr'-2x+3, 0<x<1

[x+8, x>1.

(a) Determine the values of A and B for f tobe continuous.

[4 marks]

(b) Find the minimum value of f

[3 marks]

(c) Is 7 differentiable? Justify your answer by using the first principle of

differentiation.

fHint: e* =71; n x2- 1...1

Y

[5 marks]

8. Given that
!=e'+e-' and x=e-'

ta) Find the point (x,l,) on the curve vrhere ! = O . [6 marks]
u_! [7 marks]

(b) Solve for r if
it/-a-t)-v1t|2 +a7'v-I=U.
\d*' ) dx

I

QM01612 [7 marks]
[5 marks]
9. Er aluate

(a)I1a,

'l+e-'

(b) Ji" 1r'; a, .
CHOW
CHOON10. Civen f(x):oxl_x -f1fl i
WOOI
Y

(a) Show that f is equivalent to

g(x) = lI.xx+>2'r

I
Ii- x+' 2'
x<l

[3 marks]

(b) Determine the asymptotes and the points of discontinuit5, of g.

[6 marks]

(c) Sketch the graph of g.

[3 ma"rks]

Y (d) Find the points of intersection of g(x) u,ith the straight line

.|, --..,\ -Tl Z. -

[3 marks]

BND OF BOOKLET

11

PSPM
MATRICULATION MATHEMATICS

QM016
2007/2008

QM016/1 4L QMo16/1
lvlathematics Matematik
Paper 1 :-
Kertas 1
ISemester :r_i:
Semester I
2A07/2008
2 hours 2007/2008

2 iam
CHOW
CHOON BAHAGIAN MATRIKULASI
WOOIKEMENTERIAN PELAJARAN MALAYSIA

IVATR]C ULATION DTVISIOI{
MINISTRY OF EDUCATION ALALAYSIA

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI
MATNC ULATION P ROGKAMME EX4MINATIO|V

MATEMATIK

Kertas L
2 jam

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

Kertas soalan ini mengandungi 11 halaman bercetak.
This baoklet consrsts of 11 printed pages.

@ Bahaoian Matrikulasi

CHOW QM016rl
CHOON
WOOIINSTRUCTIONS TO CAI\IDIDATE:
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 correct to three
significant figures, where appropriate, unless stated otherwise in the question.
Y

v

3

QMo16/1

LIST OF MATHEMATICAL FORMULAE

Arithmetic Series:

Tn=a+(n-t)d

s,=lDo+(n-fia]
CHOW
CHOONGeometric Series:
WOOI
T' = arn-l

s,=4:-Ll for r<1
I-r

Binomial Expansions:

+ b)' a' +('\n-' n(')o'- o' .. ( n)o'-' u' +'.. b', n e N
[r] 12)(a= u +. + + where and
(,J

u lfr.),J_-;Gnt:4

(t+x)' =7+nx.fuP.'+....MI*dx' +.,. for lxl <1

5

QM016/1

1. Given that g1r _ 3(zr_:), and 7t8v+6x = 64'y Find the values of x and y.

[6 marks]

) Express 2x +l in partial fractions. [6 marks]
(x+2)(x2 -2x+4)
CHOW
CHOON3. If z, : 4- i and Zz : | -2i, find rr-1z2. Expressthe answer inpolarform.
WOOI
[6 marksl

t7 4. The sum of the first n terms of an arithmetic series is !2Q' "-S). If the second and

fourth terms of the arithmetic series are the second and the third terms of a geometric
series respectively, find the sum of the first eleven terms of this geometric series.

[7 marks]

f,. The quadratic equation x' + k(x +Z) - (* +6)= 0 has roots a and p, where ft is a

constant.

I(a)
1Find a quadratic equation u,ith roots una p in terms of t.

t4

[5 marks]

(b) Find a2 + p2 rnterms of k. Hence, determine the minimum value of a2+p2.

v [4 marks]

6. (a) Find a cubic polynomiat A(*)= (x + o)(, + b[x + c) satisffing the

following conditions:

thecoefficientsof x'is 1, }eD=0,Q(2)=0, and QQ)=-5.

[4 marks]

(b) A polynomial P(x)=qv' -4x' +bx+18 has a factor (x + 2) and a remainder
(2x + 18) when divided by (x + 1). Find the values of a and b. Hence,

factorize P(x) completely.
[8 marks]

7

QM01611 [6 marks]
[7 marks]
7. Solve the foliowing inequalities:
x+4 2x-l

(b) l--l

l-:-l<2.
CHOW lx+41
CHOON
WOOI8. A sy'stem of linear equations is given as

ax -2y -3:: b
l.r-.y -42: 2
3 4x * 3,- -22 : 14

where a and & are constants. [9 marks]

(a) F-ind.r and z in terms of a and b using Cramer's rule.
(b) Determine the conditions of a and b for rvhich the above system

(i) has a unique solution.
(ii) has no sciution.

[4 marks]

[t a 2'i

!' g. tGiven thar. A =1,2 ,1, where aandb are constants.
ii-2 2 b)

[+,rl(a) If of u 2l
l;l = -l:. evaluate the determinant matrix 12 2 using
14 b)

determinant properties.

[4 marks]

(h) Giventhat A) -4A:51,wherelisa3 x 3 identitymatrix. Showthat a=2

andb=1.Hence.findA-1.

[9 marks]

QMo16/1

10. Given that /(x)' = .l+L, x*-1 ancl g(x) 1-, x+2 .
= -2-x

(a) Expand /(x) and g(x) as a series of ascending powers of x up to the term

containing -r'. Hence, estimate the value of (t.O)-' using the first four terms

of g(.x).

[7 marks]
CHOW
CHOON(b) If /rt.r1 = l'(.r) * g(..r), show tliat the coefficient of xn for h(r) is
WOOI
r-l r - 1 i-l.n.e. ot^tain the coefficienl of .rl flo. h(x\

.i

Y [5 marks]
[3 marks]
(c) Find the coefficient of .xr lbr l't

8('{)

END OF QUESTION BOOKLET

-7

11

QMo16/2 QMo16/2
Mathematics
Paper 2 Matematik

ISemester Kertas 2

Session 2007/2008 Semester I
2 hours
Sesi 2007/2008
4L
2 iam
+!}==

'mffi.,
CHOW
CHOON BAHAGIAN MATRIKULASI
WOOI KEMENTERIAN PELAJARAN MALAYSIA

MATRIC ULATI ON D IVIS ION
MINISTRY OF EDUCATION MALAYSIA

PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI
MATRIC ULATION P ROGRAMME EXAMINATION

MATEMATIK

Kertas 2
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 11 halaman bercetak,
This booklet conslsfs of 11 printed pages.

@ Bahagian Matrikulasi

CHOW QM016/2
CHOON
WOOIINSTRUCTIONS TO CANDIDATE:
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 n, e, surd, fractions or correct to three
significant figures, where appropriate, unless stated otherwise in the question.
V

-,

3

QM016/2

LIST OF MATHEMATICAL FORMULAE

Differentiation

If y =g(r) and x = .fQ),then !/=4"*

dx dt dx
CHOW
CHOONd(dv.]
WOOI
d2y AIA)

dx2 dx

dt

v
[udv =uv - lvdu

\t

5

QM016/2

1. A tunction/is detlned by "/(r) = x2 -2x-3 for 0 < -r < 5. State the range of .l'and
one.determine whether /is one to
[6 marks]

2. If y3 =ln(r3y2)for.x>0,y>0,then frnaffwheny:i. [6marksl

CHOW3. Let 1 = .r ( In x )r. r > 0. Show that
CHOON
WOOI r-.d-d-rlr'v-. -.r a-l'yl + t = ],v. [6 marks]

dx

1. Gir'en h(x) :+ Detining ht (.x) - (tt " ft\x\ determine the function ll: (.r) anci
)f-J

hence deduce the inverse of /u (.r). Evaluate h" (9).

[7 marksl

5. C.iven 2x'+9rr*4x-7 'g' ("r)-2:-xA:+1* B Detcrmine the frrnctiongt;.) and
2r-+9.r-4 =
.r+-l

find the r elues ot' .r and B. l-lencc.find .;l lxl - 9'i': + 4'r - 7 ,.-

r.r.- -9.r-J .

- [10 marks]

6. /The iunctit-.n is det-ined as

v
\' - '--Y-t rl /

lrr'-31-l x=3

.f(x) = A, 3<x <4
2x-8,
.r > 4.
(-,

(a) Find lim f (x) and lim /(x). [5 marks]

x+3- x+3*

(b) fUse the definition of continuity to determine the values of A and B if is

continuous at x - 3. [3 marks]
(c) For what values of C is/discontinuous at x : 4? [4 marks]

7

QM016/2 [6 marks]
[3 marks]
7. Grren .l'(.r)= 2x) +1. x>0 and g(r)=x-3,tjnd [3 marks]
(a) the inverse of .f andg and verify that (g "./')-' = .f -' o g-t .
(b) the functio n h if (S. /)-' o /r(.r) = I " [1 mark]
[2 marks]
.x [2 marks]

(c) the values of x for rvhich J-o g = g. .f . [4 marks]
CHOW [4 raarks]
CHOON8.Given"f(*):2x -3 Find
WOOI
(x-1)(x+3)

(a) the domain of l,
V (b) the x-intercept and y-intercept of /,

(c) the vertical asymptote(s) ofl

(d) ,[g,f (r) "nd ,tt]]./(x) . Hence, state the horizontal

asymptote of /.

Sketch the graph of f

9. (a) :Find dy u'hen x 0 for each of the foilowing:

dx

(i) .,, = in(t * i;' * 1), [3 marksl
[4 marksl
-
[6 marks]
(ii) v - G-.1

(b) Given-r=3/-)t1it . )'=2t+"-. t+0. Show that

dy _?_1r i3_)

dx 3 :[:i'+z]

Hence trnd drl l' '
dr)

o

QMo'16/2
10.

CHOW y = 3- r, +1t-_
CHOON by the line t' -' -----
WOOI 1+x
"Y
In the figure above, R is the region bounded the curve

and the y-axis. Find

(a) the area of R. [7 marks]

(b) therolumeofsoiidobtainediihenRisrotatedthrough 3600aboutthex-axis.

Give your answer in term of n. [8 marks]

END OF QUESTTON BOOKLET

Y

11

PSPM
MATRICULATION MATHEMATICS

QM016
2006/2007

QM016'1
Mathematics

Paper 1

ISemester

2006/2007
2 hours

QMo16/1

Matematik

Kertas 1

Semester I

2006D0a7

2 jam
CHOW &
CHOON
WOOI'ffiJ

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

CHOW QM016/1
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

QM016/1

LIST OF MATHEMATICAL FORMULAE

Arithmetic Series:

T,=e+(n_l)d

,s,=tV"+fu_lal
CHOW
CHOONGeometric Series: for r<l
WOOI
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

5

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 [5 marks]
WOOI
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]

7

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

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.
CHOW [3 marks]
CHOON [3 marks]
WOOI
(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 .{

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.(b) Hence, expand -1lf(-t1+-2r)." in ascending power of x up to the term containing
WOOIl, 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

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

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

CHOW END OF QUESTION BOOKLET
CHOON
WOOI '-J

13

QM016'2
Mathematics

Paper 2

ISemester

Session 2006/2007
2 hours

QM016/2

Matematik

Kertas 2

Semester I

Sesi 2006/2007

2 jam
CHOW 4L
CHOON
WOOI4

'mFs

I.} BAHAGIAN MATRIKI]LASI
\- KEMENTERIAI\ PELAJARAN MALAYSIA

MATNCUU|TION DIWSION
MINISTRY OF EDUCATION MAL,ffSA

PEPERIKSMN SEMESTER PROGRAM MATRIKULASI
}VI,4TNCULATION P ROGMMME EXAMINATION

MATEMATIK

Kertas 2
2 jam

JANGAN BUKA KERTAS SOALAN INISEH]NGGA DIBERITAHU.
DO NOT OPEN THIS BOOKLET UNTILYOU ARETOLD TO DO SO.

Kertas soalan inimengandungi 11 halaman bercetak.
This booklet consrsfs of 11 printed pages.

@ Bahagian Matrikulasi

CHOW QM01612
CHOON
WOOIINSTRUCTIONS TO CAI{DIDATE :
This question booklet consists of l0 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 answ€rs can be given in the form of tE, e, surd, fractions or up to three significant

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

{
!

\-

3

QM016/2

LIST OF MATIIEMATICAL FORMULAE

Differentiation

*"*If y = s(t) and x = .f (t), tnen fi=

d(dv\

dzy _A\A)

d-'--@-

dt
Integration

[uau=uv- lvdv
CHOW
CHOON ra
WOOI

5