sah@mozac2008 - · PDF fileSULIT 2 3472/1 3472/1 SULIT INFORMATION FOR CANDIDATES 1. This question paper consists of 25 questions. 2. Answer all questions. 3. Give only one answer

SULIT 3472/1

[Lihat sebelah 3472/1 © 2008 Hak Cipta JPWPKL SULIT

…

…

________________________________________________________________________________________ Kertas soalan ini mengandungi 17 halaman bercetak

Nama : ………………………………. Tingkatan :………….………………..

JABATAN PELAJARAN WILAYAH PERSEKUTUAN KUALA LUMPUR

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1. Tuliskan nama dan tingkatan anda pada

ruang yang disediakan. 2. Kertas soalan ini adalah dalam

dwibahasa. 3. Soalan dalam bahasa Inggeris

mendahului soalan yang sepadan dalam bahasa Melayu.

4. Calon dibenarkan menjawab

keseluruhan atau sebahagian soalan sama ada dalam bahasa Melayu atau bahasa Inggeris.

5. Calon dikehendaki membaca maklumat

di halaman 2 kertas soalan ini.

Guru Pemeriksa

Soalan Markah Penuh

Markah Diperoleh

1 2 2 3 3 3 4 3 5 3 6 4 7 3 8 4 9 4

10 4 11 3 12 4 13 2 14 4 15 4 16 3 17 3 18 3 19 3 20 3 21 4 22 3 23 3 24 3 25 2

Jumlah 80

PEPERIKSAAN PERCUBAAN SPM 2008 3472/1 ADDITIONAL MATHEMATICS Kertas 1 September 2 jam Dua jam

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SULIT 2 3472/1

3472/1 SULIT

INFORMATION FOR CANDIDATES 1. This question paper consists of 25 questions. 2. Answer all questions. 3. Give only one answer for each question. 4. Write your answers clearly in the spaces provided in this question paper. 5. Show your working. It may help you to get marks. 6. If you wish to change your answer, cross out the work that you have done. Then write down the new answer. 7. The diagrams in the questions provided are not drawn to scale unless stated. 8. The marks allocated for each question are shown in brackets. 9. A list of formulae is provided on pages 3 to 5. 10. A booklet of four-figure mathematical tables is provided. 11. You may use a non-programmable scientific calculator. 12. Hand in this question paper to the invigilator at the end of the examination. MAKLUMAT UNTUK CALON 1. Kertas soalan ini mengandungi 25 soalan. 2. Jawab semua soalan. 3. Bagi setiap soalan berikan satu jawapan sahaja. 4. Jawapan anda hendaklah ditulis dengan jelas dalam ruangan yang disediakan dalam kertas

soalan ini. 5. Tunjukkan langkah-langkah penting dalam kerja mengira anda. Ini boleh membantu anda untuk mendapatkan markah. 6. Sekiranya anda hendak menukar jawapan, batalkan dengan kemas jawapan yang telah

dibuat. Kemudian tulis jawapan yang baru. 7. Rajah yang mengiringi soalan tidak dilukis mengikut skala kecuali dinyatakan. 8. Markah yang diperuntukkan bagi setiap soalan ditunjukkan dalam kurungan. 9. Satu senarai rumus disediakan di halaman 3 hingga 5. 10. Sebuah buku sifir matematik empat angka disediakan. 11. Anda dibenarkan menggunakan kalkulator saintifik yang tidak boleh diprogram. 12. Serahkan kertas soalan ini kepada pengawas peperiksaan pada akhir peperiksaan.

2

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SULIT 3 3472/1

[Lihat sebelah 3472/1 SULIT

The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. Rumus-rumus berikut boleh membantu anda menjawab soalan. Simbol-simbol yang diberi adalah yang biasa digunakan.

ALGEBRA

1 a

acbbx2

42 8

abb

c

ca log

loglog

9 Tn = a + (n – 1)d

10 dnanSn )1(22

2 am x an = a m + n

3 am an = a m – n

4 ( am )n = a m n

5 nmmn aaa logloglog 11 Tn = ar n – 1

6 nmnm

aaa logloglog 12 1,

11

11

rrra

rraS

nn

n

7 log a mn = n log a m 13 1,1

rr

aS

CALCULUS (KALKULUS)

1 y = uv , dxduv

dxdvu

dxdy

4 Area under the curve (Luas di bawah lengkung)

or (atau)ba y dx

2 2,

vdxdvu

dxduv

dxdy

vuy

b

adyx

3 dxdu

dudy

dxdy

5 Volume generated (Isipadu janaan)

2

2

π or (atau)

π

ba

ba

y dx

x dy

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SULIT 4 3472/1

3472/1 SULIT

STATISTICS (STATISTIK)

1 Nxx

7 i

ii

WIWI

2 ffxx

8 !!rn

nPrn

3 2

22

xNx

Nxx

9 !!!

rrnnC r

n

4 2

22

xf

fxf

xxf

5 Cf

FNLm

m

2

1

10 BAPBPAPBAP 11 1, qpqpCrXP rnr

rn

12 Mean (min), = np 13 npq

6 1

0

100QIQ

14

XZ

GEOMETRY (GEOMETRI)

5 22 yxr 1 Distance (Jarak)

= 221

221 yyxx

2 Midpoint (Titik tengah)

2

,2

, 2121 yyxxyx

6 22

ˆyx

yxr

ji

3 A point dividing a segment of a line (Titik yang membahagi suatu tembereng garis)

nm

mynynm

mxnxyx 2121 ,,

4 Area of triangle (Luas segitiga)

1 2 2 3 3 1 2 1 3 2 1 312

x y x y x y x y x y x y

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SULIT 5 3472/1

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TRIGONOMETRY (TRIGONOMETRI)

1 Arc length, s = r Panjang lengkok, s = j

8 BABABA sincoscossin)sin( sin( ) sin kos kos sinA B A B A B

9 BABABA sinsincoscos)cos( ( ) sin sinkos A B kosAkosB A B m 2 Area of sector, 2

21 rA

Luas sektor, 212

A j

3 sin2 A + cos2 A = 1 sin2 A + kos2 A = 1

10 BA

BABAtantan1

tantan)tan(

4 AA 22 tan1sec

2 21 tansek A A

11 A

AA2tan1

tan22tan

5 AA 22 cot1cosec 2 2kosek 1A kot A

12 C

cB

bA

asinsinsin

6 AAA cossin22sin sin 2 2sinA AkosA

13 a2 = b2 + c2 – 2bc cosA a2 = b2 + c2 – 2bc kosA

7 2 2co s 2 co s sinA A A

2

2

2cos 11 2sin

AA

2 2k o s 2 s inA k o s A A

2

2

2kos 11 2sin

AA

14 Area of triangle = Cba sin21

Luas segitiga = Cba sin21

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SULIT 6 3472/1

3472/1 SULIT

For Examiner’s

Use

Answer all questions. Jawab semua soalan.

1

Based on the above information, the relation between R and S is defined by the set of ordered pairs {( 2, 6) , (3 , 8) , (3, 10) , (4, 12) }. Berdasarkan maklumat di atas, hubungan R dan S ditakrifkan oleh set pasangan tertib {( 2, 6) , (3 , 8) , (3, 10) , (4, 12)} . State Nyatakan a) the images of 3, imej-imej bagi 3, b) the object of 12. objek bagi 12.

[2 marks] [2 markah]

Answer / Jawapan : a) ………………………

b) ……………………….

2 Function f and g are defined as 23: xxf and x

xxg

1: , 0x .

Find )(xgf .

Fungsi f dan g ditakrifkan oleh 23: xxf dan x

xxg

1: , 0x .

Cari )(xgf . [3 marks]

[3 markah]

Answer / Jawapan : …………………………………

R = { 2, 3 , 4 } S = { 6, 8, 10, 12 }

3

2

2

1

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SULIT 7 3472/1

3472/1 SULIT

For Examiner’s

Use 3 Given the function 3: hxxf and kxxf 2:-1 where h and k are constants

find the value of h and of k . [3 marks]

Diberi fungsi 3: hxxf dan kxxf 2:-1 dengan keadaan h dan k adalah pemalar, carikan nilai h dan k. [3 markah]

Answer / Jawapan : h = …………………

k = ………………….

___________________________________________________________________________

4 A quadratic equation (p – 1)x2 – 8x = 4 has two different roots. Find the range of values of p. [3 marks]

Suatu persamaan kuadratik (p – 1)x2 – 8x = 4 mempunyai dua punca berbeza. Cari julat nilai p. [3 markah]

Answer / Jawapan:……………………………. ___________________________________________________________________________

5 Find the range of the values of x for (x – 5)(x + 1) 16. [3 marks] Cari julat nilai x bagi (x – 5)(x + 1) 16. [3 markah] Answer / Jawapan: ……………………………….

3

5

3

4

3

3

[Lihat sebelah

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SULIT 8 3472/1

3472/1 SULIT

For Examiner’s

Use 6 Diagram 1 shows the graph of khxxf 2)(21)( , where c, h and k are constants.

Rajah 1 menunjukkan graf bagi khxxf 2)(21)( dengan keadaan hc , dan

k adalah pemalar.

a) Given that (2, 6) is the maximum point of the graph, state the value of h and of k . Diberi (2, 6) adalah titik maksimum bagi graf tersebut, nyatakan nilai h dan nilai k .

b) The graph intersects the f(x)-axis at (0, c), find the value of c. Graf tersebut menyilang paksi-f(x) pada (0, c), cari nilai c.

[4 marks] [ 4 markah]

Answer / Jawapan: (a) h =………………

k = ………………

(b) c = ……………….

_________________________________________________________________________ 7 Solve the equation 5 12 x = x8 . [3 marks] Selesaikan persamaan 5 12 x = x8 . [3 markah]

Answer /Jawapan: …………………….

3

7

f(x)

(0, c)

O

(2, 6)

x

4

6

Diagram 1 Rajah 1

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SULIT 9 3472/1

3472/1 SULIT

For Examiner’s

Use 8 Solve the equation 033log4log3 xx . [4 marks]

Selesaikan persamaan 033log4log3 xx . [4 markah]

Answer /Jawapan: x = …………………………

9 The third and the fifth terms of a geometric progression are 108 and 12 respectively.

Sebutan ketiga dan kelima dalam suatu janjang geometri ialah 108 dan 12 masing-

masing.

Given that all the terms are positive, find the

Diberi semua sebutan adalah positif, cari

(a) common ratio,

beza sepunya,

(b) first term.

sebutan pertama. [4 marks] [4 markah]

Answer / Jawapan: (a) .. ………………………… (b) ………………………….. ___________________________________________________________________________ 10 Given that x + 1 , 12, 4x + 4 are three consecutive terms of a geometric progression,

find the possible values of x. [4 marks] Diberi x + 1 , 12, 4x + 4 adalah tiga sebutan berturut-turut bagi suatu janjang geometri, cari nilai x yang mungkin . [4 markah]

Answer / Jawapan: ……………………………

4

9

4

8

4

10

[Lihat sebelah

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SULIT 10 3472/1

3472/1 SULIT

For Examiner’s

Use 11 The sum for the first six terms of an arithmetic progression is 39. If the sixth term is 8, find the first term. [3 marks]

Hasil tambah enam sebutan pertama suatu janjang aritmetik ialah 39. Jika sebutan keenam ialah 8, cari sebutan pertama. [3 markah]

Answer /Jawapan: …….……………………….. ___________________________________________________________________________ 12 Diagram 2 shows the straight line obtained by plotting log4 y against x. The variables x and y are related by the equation y = Abx + 1 where A and b are constants. Rajah 2 menunjukkan garis lurus yang diperoleh dengan memplot log4 y melawan x. Pembolehubah x dan y dihubungkan oleh persamaan y = Abx + 1 dengan keadaan A dan b adalah pemalar.

Diagram 2 Rajah 2 Find the value of

Cari nilai bagi (a) A, (b) b. [4 marks] [4 markah]

Answer /Jawapan: (a) A = …………………..

(b) b = …………………..

3

11

4

12

log4 y

x

(0, 2)

(–2, 1)

O

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SULIT 11 3472/1

3472/1 SULIT

For Examiner’s

Use

13 Two straight lines 16

yax and 32 xy are perpendicular to each other , find the

value of a. [2 marks]

Dua garis lurus 16

yax dan 32 xy adalah berserenjang, cari nilai a .

[2 markah] Answer / Jawapan: a = ……...……………… ___________________________________________________________________________ 14 Given that points P(-6, -8), Q(-4, -4) and R(k, -14) are the vertices of a triangle. Diberi bahawa P(-6, -8), Q(-4, -4) dan R(k, -14) adalah bucu-bucu sebuah segitiga.

(a) Express the area of PQR in terms of k. Ungkapkan luas PQR dalam sebutan k.

(b) Find the value of k when the points P, Q and R are collinear. Carikan nilai k apabila P, Q and R adalah segaris.

[4 marks] [4 markah]

Answer / Jawapan : (a).…………………………….

(b)……………………………..

2

13

4

14

[Lihat sebelah

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SULIT 12 3472/1

3472/1 SULIT

For Examiner’s

Use 15 Diagram 3 shows a parallelogram PQRS where QTS is a straight line. Rajah 3 menunjukkan sebuah segiempat selari PQRS dengan keadaan QTS ialah garis lurus.

Given that ~

15 pPQ , ~

10 qPS and TQST 23 , express, in terms of ~p and

~q :

Diberi ~

15 pPQ , ~

10 qPS dan TQST 23 , ungkapkan, dalam sebutan ~p dan

~q :

(a) QS ,

(b) TR .

[4 marks]

[4 markah]

Answer / Jawapan: (a) QS =………..…………..

(b) TR = …………………...

4

15

S R

P Q

T

Diagram 3 Rajah 3

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SULIT 13 3472/1

3472/1 SULIT

For Examiner’s

Use 16 Diagram 4 shows vector OP .

Rajah 4 menunjukkan vektor OP .

(a) Express OP in the form

yx

,

Ungkapkan OP dalam bentuk

yx

,

(b) Find the unit vector in the direction of OP . Cari vector unit dalam arah OP .

[3 marks]

[3 markah]

Answer / Jawapan: (a) ……………………………..

(b) ……………………………. ___________________________________________________________________________ 17 Given that tx cos and 0 ≤ x ≤ 90, express, in terms of t : Diberi bahawa kos x t dan 0 ≤ x ≤ 90, ungkapkan, dalam sebutan t : (a) sin x (b) tan ( x090 ) [3 marks] [3 markah]

Answer / Jawapan:(a) ……………………………..

(b) …………………………….

y

3

16

x

P(−3 , −4) Diagram 4 Rajah 4

O

3

17

[Lihat sebelah

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SULIT 14 3472/1

3472/1 SULIT

For Examiner’s

Use 18 Diagram 5 shows a sector POQ of a circle with centre O and POR is a right-angle

triangle. Rajah 5 menunjukkan sektor POQ bagi sebuah bulatan berpusat O dan POR adalah segitiga bersudut tegak.

Given that ORP = 90, PR = OR = 4 cm and ORQ is a straight line. Diberi bahawa ORP = 90, PR = OR = 4 cm dan ORQ adalah garis lurus.

Find the perimeter of the shaded region. Cari perimeter bagi kawasan yang berlorek.

[3 marks]

[3 markah]

Answer / Jawapan:…….………………………..

19 The curve )(xfy is such that 121

xdxdy . The gradient of the normal to the curve at

kx is 31 , where k is a constant.

Find the value of k.

Suatu lengkung )(xfy adalah dengan keadaan 121

xdxdy . Kecerunan normal

lengkung itu di kx ialah 31 , di mana k ialah pemalar .

Cari nilai k. [3 marks] [3 markah]

Answer /Jawapan: k = ………..………….

3

18

3

19

Diagram 5 Rajah 5

P

O

Q

4 cm R

4 cm

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SULIT 15 3472/1

3472/1 SULIT

For Examiner’s

Use

20 Given that ,)12(

3)( 3

xxh evaluate ).1(h [3 marks]

Diberi ,)12(

3)( 3

xxh hitung nilai ).1(h [3 markah]

Answer / Jawapan: ……………………….….

___________________________________________________________________________

21 Given that 24642

dxxk

, where k 0, find the values of k. [4 marks]

Diberi bahawa 24642

dxxk

, di mana k0 , cari nilai-nilai k. [4 markah]

Answer /Jawapan: k = ………..…………

4

21

3

20

[Lihat sebelah

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SULIT 16 3472/1

3472/1 SULIT

For Examiner’s

Use 22 The numbers 54321 ,,,, xxxxx has interquartile range of 8 and variance of 25.

Nombor–nombor 54321 ,,,, xxxxx mempunyai julat antara kuartil 8 dan varians 25. Determine Tentukan (a) the new interquartile range, and julat antara kuartil yang baru, dan (b) the new standard deviation, of the numbers : sisihan piawai yang baru, untuk nombor–nombor :

1 2 3 4 51 1 1 1 12 , 2, 2, 2, 24 4 4 4 4

x x x x x .

[3 marks] [3 markah] Answer / Jawapan : (a) ………..………………..

(b) ………………………….

23 The probability that Siti qualifies for a college education is 72 while the probability

that Alex qualifies is 31 .

Kebarangkalian Siti layak memasuki sebuah kolej ialah 72 manakala kebarangkalian

Alex layak ialah 31 .

Find the probability that Cari kebarangkalian bahawa

(a) both of them qualify for the college education,

kedua-duanya layak memasuki kolej,

(b) only one of them qualifies for the college education.

hanya seorang daripada mereka layak memasuki kolej. [3 marks]

[3 markah]

Answer / Jawapan:(a) ………..………………..

(b)………………………….

3

22

3

23

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SULIT 17 3472/1

3472/1 SULIT

For Examiner’s

Use 24 In a survey, it is found that 30% of the population in a housing area are teenagers. If

a sample of 10 people is selected at random, find the probability that 2 of them are teenagers. [3 marks]

Dalam suatu tinjauan, didapati 30% daripada penduduk di suatu kawasan perumahan terdiri daripada remaja. Jika satu sample 10 orang dipilih secara rawak, cari kebarangkalian bahawa 2 orang daripadanya ialah remaja.

[3 markah] Answer / Jawapan: ….…………………………….

___________________________________________________________________________ 25 Diagram 6 shows a standard normal curve. Given that )( aZP = 0.6103. Find the value of a. Rajah 6 menunjukkan graf taburan normal. Diberi )( aZP = 0.6103. Cari nilai a. [ 2 marks] [ 2 markah]

Answer/ Jawapan : a =……………………………. __________________________________________________________________________ END OF QUESTION PAPER

KERTAS SOALAN TAMAT

2

25

3

24

a 0

f(z)

z

Diagram 6 Rajah 6

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SULIT 3472/1(SP)

[Lihat sebelah 3472/1(SP) SULIT

3472/1 Matematik Tambahan Kertas 1 Ogos 2008 2 jam

JABATAN PELAJARAN WILAYAH PERSEKUTUAN KUALA LUMPUR

PEPERIKSAAN PERCUBAAN

SIJIL PELAJARAN MALAYSIA 2008

SKEMA PEMARKAHAN

ADDITIONAL MATHEMATICS

Paper 1

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SULIT 3472/1(SP)

3472/1(SP) SULIT

2

Question Solution Sub Mark

Full Mark

1

(a) 8,10 (b) 4

1

1 2

2

)23()( xgxgf

= 23

)23(1

xx

= 23

33x

x , 32

x

1

1,1

3

3

)(1 xfh

xh

31

21

h

6k

1

1 1

3

4

(p - 1)x2 - 8x - 4 = 0 (-8)2 – 4(p – 1)(– 4) > 0 p > –3

1 1

1

3

5

x2 – 4x – 21 0 (x – 7)(x + 3) 0 or

x -3 , x 7

1

1

1

3

6 (a) h = –2 k = 6

(b) c = 6)20(21 2

c = 4

1 1

1 1

4

7

log xx 8log5 1012

10

(2x+1)log 8log5 1010 x

0.699 (2 x +1) = 0.9031x

x = 4949.0

699.0

= 1.412

1

1

1

3

7 -3


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SULIT 3472/1(SP)

3472/1(SP) SULIT

3

8 log 03

log4

33

xx

04log3log 32

3 xx Let y = log x3

2 3 4 0( 4)( 1) 0y yy y

or (log3 x + 4) (log3 x - 1) = 0 y = - 4 or y = 1 log 43 x or log 13 x

x811 , 3x (Both)

1

1

1

1

4

9 a) ar 2 = 108 or ar 4 = 12

r = 31

b ) a (31 ) 2 = 108

a = 972

1

1

1

1

4

10 r =

112x

or r = 12

44 x

1

12x

= 12

44 x

4 x 2 + 8 x – 140 = 0 ( x – 5 ) ( x + 7) = 0 x = 5, x = −7 (Both)

1

1

1 1

4

11

6 6[ 8] 39 OR [2 5 ] 39 and a + 5d =82 2

a = 5

a a d

1, 1

1

3

12 (a) log4 y = (blog4 A) x + log4 A or log4 A = 2 A = 16 (b) blog4A = = b =

1

1

1

1

4

[Lihat sebelah

2 – 1 . 0 – (–2)


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SULIT 3472/1(SP)

3472/1(SP) SULIT

4

13

14

6 (2) 1a

12a

(a) Area = 86

1444

86

21

k

= 844328562421

kk

= k43621

= 18 2k

(b) 18 2 0k k = 9

1

1

1

1

1 1

4

15

(a) QS QP PS

uuur uuur uuur

= ~~

1015 qp

(b) TR TQ QR

uur uuur uuur

3 5

SQ QR uuur uuur

~ ~

3 (15 10 ) 1059 4

p q q

p q

%

% %

1

1 1

1

4

16

(a)

43

OP

(b) Magnitude of OP = 5

3145

or

5453

1 1

1

3

2


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SULIT 3472/1(SP)

3472/1(SP) SULIT

5

17

(a) 21 t (b) xx cot)90tan( 0

= 21 t

t

1

1

1

3

18

POR = 0.7854 rad PO = 6569.53244 22 QR = 5.6569 4 = 1.6569 Perimeter = ( 0.7854 32 ) + 4 + 1.6569 = 10.1 or 10.0998

1 1 1

3

19

Gradient of tangent = -3

8

3121

k

k

1 1

1

3

20

3

4

'4

( ) 3(2 1)( ) 9(2 1) (2)

18(1)[2(1) 1]

(1) 18

h x xh x x

h

h

1 1

1

3

21

2

2

4 6 242

kx x

( 2k2 + 6k ) – ( 8 + 12 ) = –24 k 2 +3k +2 = 0

( k +1) k + 2 ) = 0 k = –1 , k = –2 (Both)

1 1

1 1

4

22 (a) New interquartile range = 28

41

1(b) New standard deviation 254

5 = 1.254

or

1

1

1

3

or

[Lihat sebelah


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SULIT 3472/1(SP)

3472/1(SP) SULIT

6

23 (a)

2 1 7 3221

(b)

2 2 1 57 3 3 7

921

1 1 1

3

24 n = 10, p = 0.3, q = 0.7, r = 2 P(X = 2) = 2102

210 )7.0()3.0( C

= 0.2335

1 1 1

3

25

3897.06103.01

From the table, 28.0a

1 1

2

END OF MARK SCHEME


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Documents

sah@mozac2008 - · PDF fileSULIT 2 3472/1 3472/1 SULIT INFORMATION FOR CANDIDATES 1. This question paper consists of 25 questions. 2. Answer all questions. 3. Give only one answer