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[ Lihat sebelah

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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

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

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2

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 acx

a

− ± −=

2 a

m × a

n = a

m + n

3 am ÷ a

n = 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

dnan

−+

11 Tn = ar n-1

12 Sn = r

ra

r

rann

−

−=

−

−

1

)1(

1

)1( , (r ≠ 1)

13 r

aS

−=∞

1 , r <1

CALCULUS( KALKULUS)

1 y = uv , dx

duv

dx

dvu

dx

dy+=

2 v

uy = ,

2v

dx

dvu

dx

duv

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

y2π dx or

= ∫b

a

x2π dy

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

1312312133221 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

ˆxi yj

rx y

+=

+

GEOMETRY

SULIT

2STATISTICS

1 Arc length, s = rθ

( Panjang lengkok) s = j θ

2 Area of sector , L = 21

2r θ

( Luas sektor L = θ2

2

1j )

3 sin 2A + cos

2A = 1

4 sec2A = 1 + tan

2A

5 cosec2 A = 1 + cot

2 A

6 sin 2A = 2 sinA cosA

7 cos 2A = cos2A – sin

2 A

= 2 cos2A - 1

= 1 - 2 sin2A

8 tan 2A = A

A2tan1

tan2

−

TRIGONOMETRY

9 sin (A ± B) = sinA cosB ± cosA sinB

10 cos (A ± B) = cosA cosB m sinA sinB

11 tan (A ± B) = BtanAtan

BtanAtan

m1

±

12 C

c

B

b

A

a

sinsinsin==

13 a2 = b

2 + c

2 - 2bc cosA

14 Area of triangle = Cabsin2

1

( Luas Segitiga )

1 x = N

x∑

2 x = ∑∑

f

fx

3 σ = N

xx∑ − 2)( =

2_2

xN

x−

∑

4 σ = ∑

∑ −

f

xxf2)(

= 2

2

xf

fx−

∑∑

5 m = Cf

FN

Lm

−

+ 2

1

6 1

0

100Q

IQ

= ×

7 1

11

w

IwI

∑

∑=

8 )!(

!

rn

nPr

n

−=

9 !)!(

!

rrn

nCr

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

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4

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: …………………....…..

2

2

For

examiner’s

use only

.

P ={ -3, -2, -1, 0, 1, 2 }

Q ={ -1, 0, 1, 2, 3 }

2

1

5

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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 px

2 + 7x − 2p = 0, find the values of p.

Diberi bahawa p

1ialah salah satu punca bagi persamaan kuadratik px

2 + 7x − 2p = 0, Cari nilai-nilai

p

[ 3 marks ]

[ 3 markah ]

Answer/Jawapan: ..…………………………………..

For

examiner’s

use only

4

3

3

4

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6

( 4, q )

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. 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)..................................

For

examiner’s

use only

3

5

DIAGRAM 5 / RAJAH 5

7

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6 Find the range of values of x for which 27)6( ≤−xx .

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 : = .................................

For

examiner’s

use only

4

8

3

6

3

7

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8

9 Solve 1)1(log)4(log 33 =+− xx

Selesaikan 1)1(log)4(log 33 =+− xx

[3 marks] [ 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)……………

4

10

For

examiner’s

use only

3

9

4

11

9

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12 x and y are related by the equation m

x nyx

+ = , where m and n are constants.

A straight line is obtained by plotting xy against x2, as shown in Diagram 12 .

x dan y dihubungkan oleh persamaan m

x nyx

+ = , 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=……………………………..

4

12

For

examiner’s

use only

xy

• (12, 2 )

• ( 6, 0) x

2

DIAGRAM 12/ Rajah 12

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10

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( 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)… ……………………………

For

examiner’s

use only

3

13

3

14

11

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15 Diagram 15 shows a triangle POQ

Rajah 3 menunjukkan segitiga POQ

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) ..............................................

O P X

C

Y

Q

4

15

For

examiner’s

use only

SULIT

12

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

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: …...…………..…......

4

17

For

examiner’s

use only

3

16

13

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A

C

R

S

O B 1.6 π

rad

8 cm

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.

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 10.5cms− . 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]

For

examiner’s

use only

3

19

3

18

DIAGRAM 18/RAJAH 18

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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 ]

Answer / Jawapan :................................................

21. A set of numbers 1 2 3 4, , , ,...,n

x 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 1n

x x x x+ + + + .

Satu set nombor 1 2 3 4, , , ,...,n

x 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 1n

x x x x+ + + + .

[ 3 marks ]

[ 3 markah]

Answer::/Jawapan : median = ……………………..

For

examiner’s

use only

3

20

3

21

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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.

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) ..................................

For

examiner’s

use only

3

22

3

21

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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

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) .............................................

3

24

For

examiner’s

use only

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25 Diagram 25 shows a standard normal distribution graph.

Rajah 25 menunjukkan graf taburan normal piawai

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

For

examiner’s

use only

-k k z

f(z))

0.803

DIAGRAM 25/ RAJAH 25

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