Kedah Add 1 2008

3472/1 Additional Mathematics Paper 1 Sept 2008 2 Hours

Name : ………………..…………… Form : ………………………..……

PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUA SEKOLAH MENENGAH NEGERI KEDAH DARUL AMAN PEPERIKSAAN PERCUBAAN SPM 2008

MATEMATIK TAMBAHAN ADDITIONAL MATHEMATICS

Kertas Paper 1 1 Two hours Dua jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU 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 the 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 and sub-part of a question are shown in brackets. 9. A list of formulae is provided on pages 2 3toto3.4. 10. A booklet of four-figure mathematical tables is provided. . 11 You may use a non-programmable scientific calculator. 12 This question paper must be handed in at the end of the examination .

For Examiner’s use only Question

Total Marks

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

3 4 3 3 3 3 3 3 3 4 4 3 4 3 2 3 4 4 3 3 4 2 2 3 4

TOTAL

80

Kertas soalan ini mengandungi 17 halaman bercetak 3472/1

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Marks Obtained

SULIT

2

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BLANK PAGE HALAMAN KOSONG

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The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. ALGEBRA

x=

1

−b ± b − 4ac 2a 2

2

am × an = a m + n

3

am ÷ an = a m -

4

(am) n = a nm

5

log a mn = log a m + log a n

6

log a

7

log a mn = n log a m

logab =

9

Tn = a + (n−1)d

10

Sn =

11

Tn = ar n-1

n

m = log a m − log a n n

log c b log c a

8

n [ 2a + ( n − 1) d ] 2

a (r n − 1) a (1 − r n ) = 12 Sn = , (r ≠ 1) r −1 1− r a , r <1 13 S ∞ = 1− r CALCULUS

1

2

3

dy dv du =u +v dx dx dx

y = uv ,

u dx = y= , v dy

v

du dv −u dx dx , 2 v

4 Area under a curve b

∫ y dx

=

or

a

b

=

∫ x dy a

5 Volume generated b

dy dy du = × dx du dx

∫

2 = πy dx or a

b

=

∫ πx

2

dy

a

GEOMETRY 1 Distance =

( x1 − x 2 ) 2 + ( y1 − y 2 ) 2

2 Midpoint

y + y2 ⎞ ⎛ x1 + x 2 , 1 ⎟ 2 ⎠ ⎝ 2

(x , y) = ⎜

3

r = x2 + y2

4

rˆ =

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5 A point dividing a segment of a line ⎛ nx + mx 2 ny1 + my 2 ⎞ ( x,y) = ⎜ 1 , ⎟ m+n ⎠ ⎝ m+n 6 Area of triangle 1 = ( x1 y 2 + x 2 y 3 + x3 y11 ) − ( x 2 y1 + x3 y 2 + x1 y 3 ) 2

xi + yj x2 + y2 [ Lihat sebelah SULIT

4

SULIT

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STATISTIC

1

x =

2

x =

∑x N

7

∑ fx ∑f

8

∑ (x − x )

3 σ =

N

σ=

∑ f (x − x) ∑f

5 m =

⎡1 ⎤ ⎢2 N −F⎥ L+⎢ ⎥C ⎢ fm ⎥ ⎣⎢ ⎦⎥

4

6

I=

∑x

2

=

N

2

=

9 2

∑ w1 I 1 ∑ w1 n! n Pr = (n − r )! n! n Cr = (n − r )!r!

I=

− x2

∑fx ∑f

2

− x2

10

P(A ∪ B) = P(A)+P(B) − P(A ∩ B)

11

P ( X = r) = n C r p r q n − r , p + q = 1

12

Mean µ = np

13

σ = npq x−µ z= σ

14

Q1 ×100 Q0 TRIGONOMETRY

1 Arc length, s = r θ

9 sin (A ± B) = sinA cosB ± cosA sinB

1 2 rθ 2

2 Area of sector , A = 3 sin 2A + cos 2A = 1

10 cos (A ± B) = cosA cosB m sinA sinB 11 tan (A ± B) =

4 sec2A = 1 + tan2A 2

2

5 cosec A = 1 + cot A

12

tan A ± tan B 1 m tan A tan B

c a b = = sin A sin B sin C

6 sin 2A = 2 sinA cosA 2

2

7 cos 2A = cos A – sin A = 2 cos2A − 1 = 1 − 2 sin2A 8 tan 2A =

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13 a2 = b2 + c2 − 2bc cosA 14 Area of triangle =

1 absin C 2

2 tan A 1 − tan 2 A

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Answer all questions. Jawab semua soalan. 1.

x

f

For examiner’s use only

px - 3

-1

1

2

q

Refer to arrow diagram above for the function f : x → px − 3 . Dengan merujuk kepada gambarajah anak panah di atas bagi fungsi f : x → px − 3 . Find the value of Cari nilai bagi (a) p, (b) q. [ 3 marks ] [3 markah] Answer/Jawapan : (a) ……………………..

(b) ……………………...

1 3

2. Given g : x → 3 x − 2 and gf : x → 3x 2 + 4 . Diberi g : x → 3 x − 2 dan gf : x → 3 x 2 + 4 .

Find Cari (a) g −1 ( 2) , (b) f (x ) .

[ 4 marks ] [4 markah]

2

Answer/ Jawapan : (a) ……………………..

(b) ……………………... 3472/1

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For SULIT examiner’s use only

3.

6

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x−7 , n x−7 Diberi fungsi songsang bagi f : x → 3 x + m ialah f −1 : x → , n find the value of cari nilai bagi

Given that the inverse function of f : x → 3 x + m is f

−1

:x→

(a) m, (b) n, (c) f −1 f (5) . [3 marks] [3 markah]

Answer/Jawapan : (a) m=......................... (b) n =......................... (c).................................

3 3

4

One of the roots of the equation x 2 + 4 x =

2k + 1 is three times the other root. Find 2

the value of k . [ 3 marks ] k + 2 1 Salah satu punca bagi persamaan x 2 + 4 x = adalah tiga kali ganda punca 2 yang satu lagi. Cari nilai k. [3 markah]

4 3

Answer/Jawapan : .........………………… 3472/1

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For examiner’s use only

y

5.

•

(2, 10+k)

10 • x

Diagram above show the graph of the function y = p − ( x − q) 2 . The curve intersects the y-axis at 10 and has the maximum point (2, 10+k). Find the value of (a) q, (b) p, (c) k. [ 3 marks ] Gambar rajah di atas menunjukkan graf bagi fungsi y = p − ( x − q) . Lengkung tersebut bersilang dengan paksi-y pada 10 dan mempunyai titik maksimum (2, 10+k). Cari nilai bagi 2

(a) q, (b) p, (c) k. [3 markah] Answer /Jawapan: (a) ……........................ (b) ……........................

5

(c).................................. 3

___________________________________________________________________________ 6. The curve of the quadratic function f ( x) = px 2 − 5 x + p cuts the x-axis at two distinct points. Find the range of values of p. [ 3 marks ]

Lengkung bagi fungsi kuadratik f ( x) = px 2 − 5 x + p memotong paksi-x pada dua titik yang berbeza. Cari julat bagi nilai p. [3 markah]

6

Answer/Jawapan : .................................. 3472/1

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3

8

SULIT For examiner’s use only

7.

Solve the equation

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3x + 3 = 3x + 234

Selesaikan persamaan

.

[ 3 marks ]

3 x +3 = 3 x + 234 .

[3 markah]

7

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

3

8.

During a given period, the number of people in each car passing a certain location on a road was recorded and the results are shown in the following table. Write down the inequality which must be satisfied by x if (a) the mode of the number of people in a car is 2, (b) the median of the number of people in a car is 2.

[3 marks]

Pada satu masa tertentu, bilangan orang di dalam kereta yang melalui sesuatu lokasi di sebatang jalan tertentu itu dicatat dan keputusan yang diperolehi seperti jadual di bawah. Tuliskan ketaksamaan yang mesti dipenuhi oleh x jika (a) mod untuk bilangan orang di dalam kereta ialah 2, (b) median untuk bilangan orang di dalam kereta ialah 2.

Number of People in a car Number of cars

1 50

2 x

3 59

4 64 [3 markah]

8

Answer/Jawapan : (a)...................................

(b)……………………...

3

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9

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9.

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For examiner’s use only

Given log8 2 = p and log8 5 = q. Express log8 4 5 in terms of p and q. [ 3 marks ] Diberi log8 2 = p dan log8 5 = q. Ungkapkan log8 4 5 dalam sebutan p dan q. [3 markah]

9

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

10.

3

Given that log 2 ( y + 6), log 2 ( y + 2) and log 2 y are the first three terms of an arithmetic progression, find (a) the value of y. (b) the common difference of the arithmetic progression. [ 4 marks] Diberi log 2 ( y + 6), log 2 ( y + 2) dan log 2 y adalah tiga sebutan pertama bagi suatu janjang arimetik, cari (a) nilai y. (b) beza sepunya janjang arimetik. [4 markah]

10

Answer/Jawapan : .(a)...……………...………..

(b)........................................ 3472/1

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4

10

SULIT For examiner’s use only

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11. The sum of the first n terms of an arithmetic progression is given by S n = 2n(n + 3) . Find

(a) the first term. (b) the common difference of the progression. [ 4 marks ] Hasil tambah n sebutan yang pertama bagi suatu janjang arimetik diberi oleh S n = 2n(n + 3) . Cari (a) sebutan pertama (b) beza sepunya janjang itu. [4 markah]

Answer/Jawapan: a)…...…………..….......

11

b) .................................. 4

12. The sum of the first n terms of the geometric progression 64, 32, 16, … is 126.

Find the value of n. Hasil tambah n sebutan pertama bagi janjang geometri 64, 32, 16, …. ialah 126. Cari nilai n.

[ 3 marks ] [3 markah]

12

Answer/Jawapan:…...….………..…....... 3

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SULIT 13.

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The straight line x + py = q passes through the point (1, 2) and is perpendicular to the line 2 x − y + 7 = 0 . Find the value of p and of q . [ 4 marks ]

For examiner’s use only

Garis lurus x + py = q melalui titik (1, 2) dan berserenjang dengan garis 2 x − y + 7 = 0 . Cari nilai bagi p dan q. [4 markah]

Answer/Jawapan : p = ………………..…….

13 4

q = ……………….....…...

14. Given that the points A(3, 6), P(5, t) and B(8, 1) are collinear, find

(i) (ii)

AP : PB the value of t . [ 3 marks ]

Diberi titik-titik A(3, 6), P(5, t) dan B(8, 1) adalah segaris, cari (i) (ii)

AP : PB nilai t . [3 markah]

Answer/Jawapan : (i)……………… (ii)…………..….

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For examiner’s use only

12

SULIT 15

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Given that a = 3i + j, b = −2i + j. If ma + nb = 8i + j. Find the value of m and of n. [ 2 marks ] Diberi a = 3i + j, b = −2i + j. Jika ma + nb = 8i + j. Cari nilai m dan n. [2 markah]

15

Answer/Jawapan : m =.…………………. n =.………………….

2

16

Diagram below shows a triangle ABC. It is given that 2CD = 3DB, E is the midpoint of AB, AC = 3x and AB = 4y . Express ED , in terms of x and y. Rajah di bawah menunjukkan sebuah segitiga ABC. Diberi bahawa 2CD = 3DB, E ialah titik tengah AB, AC = 3x dan AB = 4y. Ungkapkan ED dalam sebutan x dan y. C

D A

E

B

[ 3 marks ] [3 markah]

16

Answer/Jawapan :………………………..

3

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17.

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For examiner’s use only

Solve the equation 3 tan 2 x + 5 tan x = 4 sec 2 x for 0 0 ≤ x ≤ 360 0 .

[ 4 marks ] Selesaikan persamaan 3 tan 2 x + 5 tan x = 4 sec 2 x bagi 0 0 ≤ x ≤ 360 0 . [4 markah]

17

Answer/Jawapan: …...…………..…....... 18.

4

Diagram below shows two sectors, OAB and OCD with centre O. Rajah di bawah menunjukkan dua sektor, OAB dan OCD dengan pusat O. 2 cm

A

D

O

30 0 5 cm

C

B

Find Cari (a) ∠ AOB , in radian. (a) ∠ AOB , dalam radian. (b) the area of the shaded region ABCD. (b) luas bagi rantau berlorek ABCD. [4 marks] [4 markah]

18

Answer/Jawapan: (a)………………… (b)………………… 3472/1

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4

For examiner’s use only

14

SULIT 19.

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Given that the volume of a spherical balloon is increasing at the rate of 50π cm 3 s −1 , find the rate of change of the radius of the balloon when its radius is 5 cm. 4 [ V = π r 3 ]. [3 marks] 3 Diberi bahawa isipadu bagi suatu belon berbentuk sfera bertambah dengan kadar 50π cm 3 s −1 , cari kadar perubahan jejari bagi belon tersebut apabila jejarinya ialah 5 cm. 4 [ V = π r 3 ]. 3 [ 3markah]

19

Answer/Jawapan:………………………

3

20.

Given that

Diberi

∫

4 2

∫

4 2

k 6 dx = , find the value of k . 2 7 (3x − 5)

k 6 dx = , cari nilai k. 2 7 (3x − 5)

[3 marks]

[3 markah]

20

Answer: …...…………..….......

3

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SULIT 21.

15

(a)

If

(b) If

2

∫ ∫

4

∫

Given that

g ( x) dx =10 .

2

For examiner’s use only

k g ( x) dx = 45 , find the value of k .

4 4

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[ g ( x) + p] dx = 80 , find the value of p .

2

[ 4 marks ] Diberi

∫

4 2

g ( x) dx =10 .

Jika

∫

(b) Jika

∫

(a)

2 4 4 2

k g ( x) dx = 45 , cari nilai k . [ g ( x) + p] dx = 80 , cari nilai p . [4 markah]

Answer/Jawapan: (a) k = ……………………..

21

(b) p =.……………..………

22.

Given y =

4

dy 1 in terms of x . , where u = 2 − 3x 2 . Find 3 dx u [ 3 marks ]

Diberi y =

dy 1 , dan u = 2 − 3x 2 . Cari dalam sebutan x. 3 dx u [3 markah]

22

Answer/Jawapan: ……………………. 3472/1

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3

SULIT For examiner’s use only 23.

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In a multiple choice test consisting of 10 questions, each question has five answers to choose from. One mark will be given for each correct answer and no marks will be deducted for wrong answers. If a candidate chooses an answer randomly for each question, what is the expected score of the candidate? [ 2 marks ] Dalam satu ujian pelbagai pilihan yang mengandungi 10 soalan, setiap soalan diberikan 5 pilihan jawapan. Satu markah akan diberikan untuk setiap jawapan yang betul. Tiada markah dipotong untuk jawapan yang salah. Jika seorang calon menjawab secara rawak, apakah skor jangkaan bagi calon itu ? [2 markah]

23 2

Answer/Jawapan: …...…………..…....... 24. A student committee consisting of six members is to be formed from 7 boys and 5 girls. Find the possible number of committees that can be formed if

(a) there are no restrictions. (b) at least one girl must be chosen. [ 3 marks ] Satu jawatankuasa pelajar yang terdiri daripada enam ahli akan dibentuk daripada 7 pelajar lelaki dan 5 pelajar perempuan. Cari bilangan jawatankuasa berlainan yang boleh dibentuk jika (a) tiada syarat dikenakan (b) sekurang-kurangnya seorang pelajar perempuan mesti dipilih. [3 markah]

24

Answer/Jawapan: (a)………………………

3

(b)……………………… 3472/1

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25. A random variable X is normally distributed with mean 45 and standard deviation 6. Find the value of Satu pembolehubah rawak X bertaburan normal dengan min 45 dan sisihan piawai 6. Cari nilai bagi

(a) P( X > 48) (b) k if P( X > k ) = 0.6915 . (b) k jika P( X > k ) = 0.6915 . [4 marks] [4 markah]

25

Answer/Jawapan: (a)……………………… (b)………………………

END OF QUESTION PAPER KERTAS SOALAN TAMAT

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