1. SULIT
PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUA
SEKOLAH MENENGAH MALAYSIA (PKPSM) CAWANGAN MELAKA
PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 2010
MATEMATIK TAMBAHAN
Kertas 1
Dua Jam
JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU
Kertas soalan ini mengandungi 18 halaman bercetak
[ Lihat sebelah
3472/1 SULIT
Nama : ………………..………………..
Tingkatan: ………………………..……
3472/1
Matematik Tambahan
Kertas 1
Sept 2010
2 jam
1. This question paper consists of 25 questions
Kertas soalan ini mengandungi 25 soalan.
2. Answer all questions.
Jawab semua soalan.
3. Give only one answer for each question
Bagi setiap soalan berikan SATU jawapan sahaja.
4. Write the answers clearly in the space provided in the question paper.
Jawapan hendaklah ditulis pada ruang yang disediakan dalam kertas soalan.
5. Show your working. It may help you to get marks.
Tunjukkan langkah-langkah penting dalam kerja mengira anda. Ini boleh
membantu anda untuk mendapatkan markah.
6. If you wish to change your answer, cross out the work that
you have done. Then write down the new answer.
Sekiranya anda hendak menukar jawapan, batalkan kerja mengira yang telah
dibuat. Kemudian tulis jawapan yang baru.
7 The diagram in the questions provided are not drawn to scale unless
stated.
Rajah yang mengiringi soalan ini tidak dilukiskan mengikut skala kecuali dinyatakan.
8. The marks allocated for each question and sub-part of a question are
shown in brackets.
Markah yang diperuntukkan bagi setiap soalan atau ceraian soalan ditunjukkan
dalam kurungan.
9. A list of formulae is provided on page 2 to 3
Satu senarai rumus disediakan di halaman 23 hingga 3
10. You may use a non-programmable scientific calculator.
Buku sifir matematik empat angka boleh digunakan.
11 This question paper must be handed in at the end of the examination.
Kertas soalan ini hendaklah diserahkan pada akhirpeperiksaan .
Kod
Pemeriksa
Soalan
Markah
Penuh
Markah
Diperoleh
1 2
2 2
3 4
4 3
5 3
6 3
7 3
8 4
9 3
10 4
11 4
12 4
13 3
14 3
15 4
16 3
17 4
18 3
19 3
20 3
21 3
22 3
23 3
24 3
25 3
Jumlah
80
2. SULIT
The following formulae may be helpful in answering the questions. The symbols given are the ones
commonly used.
Rumus-rumus berikut boleh digunakan untuk membantu anda menjawab soalan. . Simbol-simbol yang diberi adalah yang biasa
digunakan.
ALGEBRA
1
2
4
2
b b ac
x
a
− ± −
=
2 am
× an
= a m + n
3 am
÷ an
= a m - n
4 (am
) n
= a nm
5 loga mn = log am + loga n
6 loga
n
m
= log am - loga n
7 log a mn
= n log a m
8 logab =
a
b
c
c
log
log
9 Tn = a + (n -1)d
10 Sn = ])1(2[
2
dna
n
−+
11 Tn = ar n-1
12 Sn =
r
ra
r
ra nn
−
−
=
−
−
1
)1(
1
)1(
, (r ≠ 1)
13
r
a
S
−
=∞
1
, r <1
CALCULUS( KALKULUS)
1 y = uv ,
dx
du
v
dx
dv
u
dx
dy
+=
2
v
u
y = ,
2
v
dx
dv
u
dx
du
v
dy
dx
−
= ,
3
dx
du
du
dy
dx
dy
×=
4 Area under a curve ( Luas dibawah lengkung)
= ∫
b
a
y dx or
= ∫
b
a
x dy
5 Volume generated ( Isipadu Janaan)
= ∫
b
a
y2
π dx or
= ∫
b
a
x2
π dy
3472/2 SULIT
2
5 A point dividing a segment of a line
Titik yang membahagi suatu tembereng garis
( x,y) = ,21
+
+
nm
mxnx
+
+
nm
myny 21
6 Area of triangle ( Luas Segitiga ) =
)()(
2
1
312312133221 1
yxyxyxyxyxyx ++−++
1 Distance (Jarak) = 2
21
2
21 )()( yyxx −+−
2 Midpoint ( Titik Tengah )
(x , y) =
+
2
21 xx
,
+
2
21 yy
3 22
yxr +=
4 2 2
ˆ
xi yj
r
x y
+
=
+
GEOMETRY
3. STATISTICS
1 Arc length, s = rθ
( Panjang lengkok) s = j θ
2 Area of sector , L =
21
2
r θ
( Luas sektor L = θ2
2
1
j )
3 sin 2
A + cos 2
A = 1
4 sec2
A = 1 + tan2
A
5 cosec2
A = 1 + cot2
A
6 sin 2A = 2 sinA cosA
7 cos 2A = cos2
A – sin2
A
= 2 cos2
A - 1
= 1 - 2 sin2
A
8 tan 2A =
A
A
2
tan1
tan2
−
TRIGONOMETRY
9 sin (A ±B) = sinA cosB ± cosA sinB
10 cos (A ±B) = cosA cosB sinA sinB
11 tan (A ±B) =
BtanAtan
BtanAtan
1
±
12
C
c
B
b
A
a
sinsinsin
==
13 a2
= b2
+ c2
- 2bc cosA
14 Area of triangle = Cabsin
2
1
( Luas Segitiga )
1 x =
N
x∑
2 x =
∑
∑
f
fx
3 σ =
N
xx∑ − 2
)( =
2_2
x
N
x
−
∑
4 σ =
∑
∑ −
f
xxf 2
)(
=
2
2
x
f
fx
−
∑
∑
5 m = C
f
FN
L
m
−
+ 2
1
6
1
0
100
Q
I
Q
= ×
7
1
11
w
Iw
I
∑
∑
=
8 )!(
!
rn
n
Pr
n
−
=
9 !)!(
!
rrn
n
Cr
n
−
=
10 P(A∪ B) = P(A)+P(B)- P(A∩ B)
11 P (X = r) = rnr
r
n
qpC −
, p + q = 1
12 Mean µ = np
13 npq=σ
14 z =
σ
µ−x
3
4. 3
Answer all questions.
Jawab semua soalan
1 Diagram 1 shows the relation between two sets of number .
Rajah 1 menunjukkan satu hubungan diantara dua set nombor.
Diagram 1 / Rajah 1
Based on the above information, the relation between P and Q is defined by the set
of ordered pairs { (-2, 1 ), (-1, 0 ), ( 0, 1 ), ( 1, 2 ), (2, 3 )}.
Berdasarkan maklumat diatas hubungan antara P dan Q ditarifkan sebagai set pasangan tertib
{ (-2, 1 ), (-1, 0 ), ( 0, 1 ), ( 1, 2 ), (2, 3 )}.
State,
Nyatakan,
(a) the image of 2.
Imej bagi 2
(b) the object of 0.
Imej bagi 0
[2 marks]
[ 2 markah ]
Answer/Jawapan: (a) ……………………..
(b) ……………………...
2 Given the function g : x → x2
+1 , find the values of g -1
(10)
Di beri g : x → x2
+1 . Cari nilai –nilai bagi g -1
(10)
[ 2 marks ]
[ 2 markah ]
Answer/Jawapan: …………………....…..
[ Lihat sebelah
3472/1 SULIT
2
4
2
For examiner’s
use only
.P ={ -3, -2, -1, 0, 1, 2 }
Q ={ -1, 0, 1, 2, 3 }
2
4
1
5. 3. The function f is defined by f: x → kx2
+ p and the function g is defined by g: x →1 + 2x.
Given the composite function fg : x → x2
+ x + 6, find the values of k and p.
Fungsi f ditakrifkan sebagai f: x → kx2
+ p dan fungsi g ditarifkan sebagai g: x →1 + 2x
Diberi fungsi gubahan fg : x → x2
+ x + 6 , Cari nilai k dan nilai p
[4 marks]
[ 4 markah]
Answer/ Jawapan : k = ....…………p = …………
4 Given that
p
1
is one of the roots of the quadratic equation px2
+ 7x − 2p = 0, find the values of p.
Diberi bahawa
p
1
ialah salah satu punca bagi persamaan kuadratik px2
+ 7x − 2p = 0, Cari nilai-
nilai p
[ 3 marks ]
[ 3 markah ]
Answer/Jawapan: ..…………………………………..
5 Diagram 5 shows the graph of a quadratic function f(x) = 3(x + p)2
+ 2, where p
is a constant. The curve y = f(x) has the minimum point (4, q), where q is a constant.
For examiner’s
use only
For examiner’s
use only
4
3
3
4
6. ( 4, q )
5
Rajah 5 menunjukkan graf fungsi kuadratik f(x) = 3(x + p)2
+ 2 , dimana p ialah pemalar
Lengkung y = f(x) mempunyai titik minimum (4, q), dimana q is satu pemalar.
State,
Nyatakan,
(a) the value of p,
nilai p
(b) the value of q,
nilai q
(c) the equation of the axis of symmetry.
persamaan paksi semetri
[ 3 marks ]
[ 3 markah]
Answer : (a) ……........................
(b) ……........................
(c)..................................
6 Find the range of values of x for which 27)6( ≤−xx .
[ Lihat sebelah
3472/1 SULIT
For examiner’s
use only
3
5
DIAGRAM 5 / RAJAH 5
7. Cari julat nilai- nilai x dimana 27)6( ≤−xx
[3 marks]
[ 3 markah]
Answer/ Jawapan: ........……........................
7 Solve the equation 02781 321
=− −+ xx
.
Selesaikan persamaan 02781 321
=− −+ xx
[ 3 marks ]
[ 3 markah]
Answer /Jawapan: x = ………………………...
8. Given log7 2 = h and log7 5 = k. Express log7 2.8 in terms of h and k.
Diberi log7 2 = h dan log7 5 = k. Ungkapkan log7 2.8 in terms of h and k.
[ 4 marks ]
[ 4 markah ]
Answer /Jawapan : = .................................
9 Solve 1)1(log)4(log 33 =+− xx
Selesaikan 1)1(log)4(log 33 =+− xx
[3 marks]
4
8
3
2
6
3
7
For examiner’s
use only
8. 7
[ 3
markah]
Answer/Jawapan: ……..……...……….....
10 The first three terms of an arithmetic progression are 6, t− 2, 14,...….
Tiga sebutan pertama satu janjang arithmatik ialah 6, t − 2, 14,...….
find,
cari,
(a) the value of t,
nilai t
(b) the sum of the first ten term .
hasil tambah sepuluh sebutan pertama
[ 4 marks ]
[ 4 markah ]
Answer /Jawapan: (a)…………….(b) ..………....
11 The sum of the first n terms of the geometric progression, 64, 32, 16, ….. is 126.
Hasil tambah n sebutan pertama suatu janjang geometri, 64, 32, 16, ….. ialah 126
Find,
Cari,
(a) the value of n,
nilai n
(b) the sum to infinity of the geometric progression.
Hasiltambah ketakterhingaan janjang geometri ini,
[ 4 marks ]
[ 4 markah]
Answer/Jawapan: (a) n = ………………(b)……………
12 x and y are related by the equation
m
x ny
x
+ = , where m and n are constants.
A straight line is obtained by plotting xy against x2
, as shown in Diagram 12 .
[ Lihat sebelah
3472/1 SULIT
4
4
10
For examiner’s
use only
3
9
4
4
11
9. x dan y dihubungkan oleh persamaan
m
x ny
x
+ = , dimana m dan n ialah pemalar.
Satu garisurus diperolehi dengan memplotkan xy melawan x2
, sebagaimana ditunjukan
dalam Rajah 12
Calculate the value of m and of n.
Kira nilai m dan nilai n
[4 marks]
[ 4 markah]
.
Answer/Jawapan: m =………………………
n=……………………………..
13 Given that the point P ( )3,2− divides the line segment that joining ),4( tA − and
)8,(rB in
the ratio AP : PB = 1 : 4. Find the value of r and of t.
Diberi bahawa titik P ( )3,2− membahagi segmen garis yang menghubungkan
4
12
For examiner’s
use only
xy
• (12, 2 )
• ( 6, 0)
x 2
DIAGRAM 12/ Rajah 12
10. 9
),4( tA − dan )8,(rB dalam nisbah AP : PB = 1 : 4. Cari nilai r dan nilai t.
[3 marks]
[3 markah]
Answer/Jawapan: ………………………..
14 Given that 2 2a i j=− +
% % %
, 2 3b i j= −
% % %
and 2c a b= −
% % %
.
Diberi bahawa 2 2a i j=− +
% % %
, 2 3b i j= −
% % %
dan 2c a b= −
% % %
.
Find,
Cari,
(a) c
%
(b) unit vector in the direction of c
%
.
Vektor unit dalam arah c
%
[3 marks]
[3 markah]
Answer : (a)…………………………………
(b)… ……………………………
15 Diagram 15 shows a triangle POQ
Rajah 3 menunjukkan segitiga POQ
[ Lihat sebelah
3472/1 SULIT
3
13
3
14
For examiner’s
use only
11. DIAGRAM 3
RAJAH 3.
Given that OP
→
= p and OQ
→
= q .Point X is lies on OP where OX : XP = 2 : 1 and
point Y is lies on OQ where OY : YQ = 3 : 1 . Straight line OX and line PY intersect at point
C.
Diberi OP
→
= p dan OQ
→
= q . Titik X terletak pada OP di mana OX : XP = 2 : 1 dan
titik Y adalah titik pada OQ di mana OY : YQ = 3 : 1 . Garis lurus QX dan garis lurus PY
bersilang pada titik C.
Express in terms of p and q
(a) PY
→
(b) QX
→
[ 4 marks]
[4 markah]
Answer/Jawapan: (a) ..........................................
(b) ..............................................
16 Given that θ is an acute angle and q
p
=θsin , Find in terms of p and /or q
Diberi bahawa θ ialah sudut tirus dan
q
p
=θsin , Cari dalam sebutan p dan/atau q
O P
X
C
Y
Q
4
15
For examiner’s
use only
12. 11
a) cos θ
kosθ
b) tan ( 180 - θ)
[3 marks]
[ 3 markah ]
Answer /Jawapan (a) .......................................
(b) .......................................
17 Solve 3cos 2θ + 4 cos θ +1 = 0 for 00
3600 ≤≤θ
Selesaikan 3cos 2θ + 4 cos θ +1 = 0 untuk 00
3600 ≤≤θ
[4 marks]
[4 markah]
Answer/Jawapan: …...…………..…......
18 Diagram 18 shows a circle ABC with centre O, of radius 8 cm. SR is an arc of
a circle with center O. The reflex angle AOC is 1.6π radian.
Rajah 18 menunjukkan satu bulatan ABC dangan pusat O dan berjejari 8 cm, SR ialah
lengkuk sebuah bulatan berpusat di O . Sudut reflek AOC ialah 1.6π radian.
[ Lihat sebelah
3472/1 SULIT
For examiner’s
use only
4
4
17
3
4
16
13. A
C
R
S
OB 1.6 π rad
8 cm
Given that A and C are midpoints of OS and OR respectively, find the area of shaded
region, in terms of π .
Diberi bahawa A dan C ialah titik tengah kepada OS dan OR , Cari luas kawasan
berlorek dalam sebutan π
[ 3 marks]
[ 3 markah ]
Answer / Jawapan : ………………………. cm2
19. The radius of circle decreases at the rate of 1
0.5cms−
. Find the rate of change of the
area of a circle when the radius is 4 cm. .[ Given the area of a circle is A = πr2
]
Jejari sebuah bulatan berkurang dengan kadar 0.5 cms-1
.Cari kadar perubahan luas
bulatan apabila jejarinya ialah 4 cm, [ Diberi luas bulatan A = πr2
]
[ 3 marks]
[ 3 markah]
Answer/Jawapan: …………………………
20 Given that
5
1
( ) 5g x dx =∫ , find the value of m if
5
1
[ 2 ( )] 3mx g x dx m− = −∫
Diberi bahawa
5
1
( ) 5g x dx =∫ , cari nilai m jika
5
1
[ 2 ( )] 3mx g x dx m− = −∫
[ 3 marks ]
[ 3 markah ]
For examiner’s
use only
3
19
3
18
DIAGRAM 18/RAJAH 18
14. 13
Answer / Jawapan :................................................
21. A set of numbers 1 2 3 4, , , ,..., nx x x x x has a median of 5 and a standard deviation of 2.
Find the median and the variance for the set of numbers
1 2 36 1,6 1,6 1,.......,6 1nx x x x+ + + + .
Satu set nombor 1 2 3 4, , , ,..., nx x x x x mempunyai median 5 dan sisihan piawai 2.
Cari median dan variance bagi set nombor 1 2 36 1,6 1,6 1,.......,6 1nx x x x+ + + + .
[ 3 marks ]
[ 3 markah]
Answer::/Jawapan : median = ……………………..
variance =.……………..………
22 A box contains 6 black balls and p white balls. If a ball is taken out randomly from the
box, the probability of getting a white ball is
7
4
. Find the value of p.
[ Lihat sebelah
3472/1 SULIT
For examiner’s
use only
3
20
3
21
15. Sebuah kotak mengandungi 6 biji bola hitam dan p biji bola putih . Jika sebiji bola
diambil secara random dari kotak itu kebarangkalian mendapat sebiji bola putih ialah
7
4
. Cari nilai p.
[ 3 marks ]
[3 markah]
Answer/Jawapan: …...…………..…….......
23 An expedition team consisting of 10 members to be chosen from a group of 4 teachers
and 12 students.
Satu kumpulan expedisi mengandungi 10 ahli yang akan dipilih daripada kumpulan 4
orang guru dan 12 orang pelajar.
(a) Calculate the number of teams that can be formed.
Kira bilangan kumpulan yang boleh dibentuk .
(b) If the team must consist of at least 2 teachers, calculate the numbers of teams that
could be formed.
Jika kumpulan expedisi itu mesti mengandungi sekurang-kurangnya 2 orang guru
kira bilangan kumpulan yang boleh dibentuk.
[3 marks]
[ 3 markah]
Answer/ Jawapan: (a) …...…………..……..
(b) ..................................
24 In a survey, the probability that a family owning one unit of computer is 0.6.
N families were selected at random. The standard deviation of the numbers of
family owning one unit of computer is
5
24
3
3
22
3
3
21
For examiner’s
use only
16. 15
Dalam satu tinjauan , kebarangkalian sebuah keluarga mempunyai satu unit computer
ialah 0.6. N keluarga dipilih secara rawak . Sisihan piawai keluarga yang mempunyai
computer ialah
5
24
Find,
Cari,
(a) the value of N
nilai N
(b) the mean of the numbers of family owning one unit of computer.
mean keluarga yang mempunyai satu unit computer
[3 marks]
[ 3 markah]
Answer/Jawapan:(a) ………………..………
(b) .............................................
25 Diagram 25 shows a standard normal distribution graph.
Rajah 25 menunjukkan graf taburan normal piawai
[ Lihat sebelah
3472/1 SULIT
3
24
For examiner’s
use only
17. The probability represented by the area of the shaded region is 0.803.
Kebarangkalian yang diwakili oleh kawasan berlorek ialah 0.803
(a) Find the value of P( Z > k )
Cari nilai P( Z > k )
(b) X is a continuous random variable which is normally distributed with a mean of
µ and a standard deviation of 2. If the value of X is 85 when the Z-score is k,
find the value of µ .
X ialah pembolehubah rawak selanjar yang bertabur secara normal piawai
dengan mean µ dan sisihan piawai 2. Jika nilai X ialah 85 bila skor- Z ialah k ,
cari nilai µ .
[3 marks]
[ 3 markah]
Answer : (a)…………………………………
(b)… ……………………………
END OF THE QUESTION PAPER
KERTAS SOALAN TAMAT
3
25
-k k z
f(z))
0.803
DIAGRAM 25/ RAJAH 25