PSPM 1 Exam Papers Collection

QM016'2
Mathematics

Paper 2

ISemester

Session 2006/2007
2 hours

4L QM016/2 CHOW CHOON WOOI

4 Matematik

Kertas 2

Semester I

Sesi 2006/2007

2 jam

'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

51

QM01612 CHOW CHOON WOOI

INSTRUCTIONS 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

52

QM016/2

LIST OF MATIIEMATICAL FORMULAE

Differentiation CHOW CHOON WOOI

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

d(dv\

dzy _A\A)

d-'--@-

dt
Integration

[uau=uv- lvdv

ra

5

53

QM016/2

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

x-+4 .rlx -2

(b) lim atJ:4x +.x

-6.x-++co *2

Giventhat *= | *d =;y 1-tz where / is a non-zero parameter.
t*7 '

Show that

dy zt (lt+tr'))''

d-=

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

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

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

[6 marks]

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

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

[7 marks]

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

lr*'-1, x>2.

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

continuous at x=2. [6 marks]

7

54

QM016'2

(b) Given a function f on a closed interval l-2,47 as follows: [5 marks] CHOW CHOON WOOI
/-
[6 marks]
I s,f (x)=jrIt(*-x-l-)z(t1x+3)'- -2<x<4 [6 marks]
x=4.
1
/Find the intervals on [-2,4] where is continuous
[4 marks]
6. Let x2y2+Zxy+4y={. [4 marks]
[4 marks]
(a) Findthevalues of A,.Band C if +d=x x(4x/yQ+B:9)+C'.

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

7. A function / is defined by

rI f @) = lx+ rl-2'

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

(c) Evaluate Il, Xr> *.

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

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

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

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

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

I

55

QM016/2

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

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

x CHOW CHOON WOOI

(a) Show that

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

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

hence state the largest interval where F is continuous.

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

r'@)=.8 r@).

END OF QUESTION BOOKLET

11

56

2007/2008

57

CHOW CHOON WOOI

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 WOOI

BAHAGIAN MATRIKULASI
KEMENTERIAN 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

58

QM016rl CHOW CHOON WOOI

INSTRUCTIONS 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

59

QMo16/1

LIST OF MATHEMATICAL FORMULAE

Arithmetic Series: CHOW CHOON WOOI

Tn=a+(n-t)d

s,=lDo+(n-fia]

Geometric Series:

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

60

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 CHOON WOOI
3. If z, : 4- i and Zz : | -2i, find rr-1z2. Expressthe answer inpolarform.

[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

61

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

(b) l--l CHOW CHOON WOOI

l-:-l<2.
lx+41

8. 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]

62

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 CHOW CHOON WOOI

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

of g(.x).

[7 marks]

(b) If /rt.r1 = l'(.r) * g(..r), show tliat the coefficient of xn for h(r) is

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

1613

QMo16/2 QMo16/2
Mathematics
Paper 2 Matematik

ISemester Kertas 2

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

'mffi.,

BAHAGIAN MATRIKULASI
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

64

QM016/2 CHOW CHOON WOOI

INSTRUCTIONS 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

65

QM016/2

LIST OF MATHEMATICAL FORMULAE

Differentiation

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

dx dt dx

d(dv.]

d2y AIA)

dx2 dx

dt

v
[udv =uv - lvdu

\t

5

66

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

3. Let 1 = .r ( In x )r. r > 0. Show that CHOW 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 /

rlr'-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

67

QM016/2 [6 marks] CHOW CHOON WOOI
[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]
[4 raarks]
8. Given "f (*): 2x -3 Find

(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

68

QMo'16/2
10.

y = 3- r, +1t-_ CHOW CHOON WOOI
by the line t' -' -----
"Y 1+x
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

1619

2008/2009

70

CHOW CHOON WOOI

QMo1611 QM()16/1

Mathematics Matematik

Paper 1 Kertas 1

ISemester Semester I

2008/2009 2008t2009

2 hours 2 jam

& CHOW CHOON WOOI

-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

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

O Bahagian Matrikulasi

71

QM(}16/1 CHOW CHOON WOOI

INSTRUCTIONS 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

72

QMo16/1

LIST OF MATHEMATICAL FORMULAE

Arithmetic Series: CHOW CHOON WOOI

T,=o+(n-t)a

s,=|lzr+(n_lal

Geometric Series:

T, = ar'-l

'.=+! ror r<l

Binomial Expansions:

(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

573

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 CHOON WOOI

(a) Find the value of cornmon ratio, r of the series.

[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

74

QMo16/1 CHOW CHOON WOOI

,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]

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

75

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 CHOON WOOI

(b) The following table shows the quantities (kg) and the amount paid (RM) for

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

76

QM016/1

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

(a) Find
(i) the determinant of l,
(it) the minor of ,4 and
(iir) the adjoint of A.

[9 marls]

(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

77

QMo16/2 QMo16/2
Mathematics
Paper 2 Matematik

ISemester Kertas 2

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

2 jam

+ CHOW CHOON WOOI

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,

Kertas soalan inimengandungi 11 halaman bercetak.

This booklet consrsfs of 11 printed pages.

78O Bahagian Matrikulasi

QM016/2 CHOW CHOON WOOI

INSTRUCTIONS 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

79

QMo16/2

LIST OF MATHEMATICAL FORryIULAE

Differentiation

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

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

l|u'e dx

Integration

y [rd, = uv - tvclu

!7

E

80

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

[5 marks]

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

[6 marks]

3. Given that f {x}: 10 - 2.r Find the value of ,t so that
,
[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

81

QMo16/2

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

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

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

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

JiZ1 -+

[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

82

QM01612 [7 marks] CHOW CHOON WOOI
[5 marks]
9. Er aluate

(a)I1a,

'l+e-'

(b) Ji" 1r'; a, .

10. Civen f(x):oxl_x -f1fl i

Y

(a) Show that f is equivalent to

g(x) = Il.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

83

2009/2010

84

CHOW CHOON WOOI

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.

85@ Bahagian Matrikulasi

r QMo1611 CHOW CHOON WOOI

I INSTRUCTI.\S To CANDIDATE:

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

86

QM016/1

LIST OF MATHEMATICAL FOR\{LLAE

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

, = *!:!b'_1!,
2a

For an arithmetic series:

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

87

QMo16/1

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

16 marlrsl

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

l7 narksl

3 '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

88

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 WOOI

(b) the cofactor and the adjoint matrix of ,{. i{ence, detemrine the interse of, l.

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

89

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.

\ Fruit Mango Durian Rambutan Amount paid CHOW CHOON WOOI
(ke) (ke) (ke) (RM)
S,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

9101

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

2009t2010

2 jam

s CHOW CHOON WOOI

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

MATEMATIK

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.

91@ 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 CHOW CHOON WOOI
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 fr , € , surd, fractions or up to three

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

Y/

3

92

QMo16/2

LIST OF MATIIEMATICAL FORMULAE

Difrerentiation CHOW CHOON WOOI

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

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

dt

Integration

\7

Iutu:uv-lvdu

-y

5

93

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 CHOON WOOI

2 /A function is given as

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

94

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 CHOON WOOI

tinft(c) Evaluate -lx 2r

[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

95

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 CHOW CHOON WOOI

(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

(iii) through the points t U.U +!) and ( 0 1)

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

9116

2010/2011

97

CHOW CHOON WOOI

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 WOOI

a'-eI::.{ffIf:,:fs"

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

98

QS()16/1 CHOW CHOON WOOI

INSTRUCTIONS 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

99

QSo1611

LIST OF MATHEMATICAL FORMULAE

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

.--bi'lT;*4ac
2a

For an arithmetic series:

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

100