1. 3472/1
Additional Mathematics Name : ………………..……………
Paper 1
Sept 2009 Form : ………………………..……
2 Hours
PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUA
SEKOLAH MENENGAH
NEGERI KEDAH DARUL AMAN
PEPERIKSAAN PERCUBAAN SPM 2009
MATEMATIK TAMBAHAN
ADDITIONAL MATHEMATICS For Examiner’s use only
Paper 1 1
Kertas Marks
Dua jam
Two hours Question Total Marks
Obtained
JANGAN BUKA KERTASSOALAN INI 1 2
SEHINGGA DIBERITAHU 2 4
3 3
1 This question paper consists of 25 questions.
4 3
2. Answer all questions. 5 3
6 4
3. Give only one answer for each question.
7 4
4. Write your answers clearly in the spaces provided in 8 3
the question paper.
9 3
5. Show your working. It may help you to get marks. 10 3
11 3
6. If you wish to change your answer, cross out the work 12 4
that you have done. Then write down the new
answer. 13 3
14 3
7. The diagrams in the questions provided are not 15 2
drawn to scale unless stated.
16 3
8. The marks allocated for each question and sub-part 17 4
of a question are shown in brackets. 18 4
9. A list of formulae is provided on pages 23 to3.
to 4. 19 3
20 2
10. A booklet of four-figure mathematical tables is 21 4
provided.
. 22 3
11 You may use a non-programmable scientific 23 3
calculator. 24 3
12 This question paper must be handed in at the end of
25 4
the examination . TOTAL 80
Kertas soalan ini mengandungi 17 halaman bercetak
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2. SULIT 2 3472/1
BLANK PAGE
HALAMAN KOSONG
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3. SULIT 3 3472/1
The following formulae may be helpful in answering the questions. The symbols given are the ones
commonly used.
ALGEBRA
b 2
b 4ac log c b
1 x 8 logab =
2a log c a
2 am an = a m + n 9 Tn = a + (n−1)d
3 am an = a m - n
n
10 Sn = [2a (n 1)d ]
2
4 (am) n = a nm
11 Tn = ar n-1
5 log a mn = log a m + log a n a(r n 1) a(1 r n )
12 Sn = , (r 1)
m r 1 1 r
6 log a = log a m − log a n
n a
13 S , r <1
7 log a mn = n log a m 1 r
CALCULUS
dy dv du 4 Area under a curve
1 y = uv , u v b
dx dx dx
= y dx or
a
du dv b
v u
u dx dx dx , = x dy
2 y , 2
v dy v a
5 Volume generated
b
dy dy du
3 = y 2 dx or
dx du dx a
b
= x 2 dy
a
GEOMETRY
1 Distance = ( x1 x2 ) 2 ( y1 y2 ) 2 5 A point dividing a segment of a line
nx1 mx2 ny1 my 2
( x,y) = ,
2 Midpoint m n m n
x1 x2 y1 y2
(x , y) = ,
2 2 6 Area of triangle
1
3 r x2 y2 = ( x1 y 2 x 2 y 3 x3 y11 ) ( x 2 y1 x3 y 2 x1 y 3 )
2
xi yj
4 ˆ
r
x2 y2
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4. SULIT 4 3472/1
STATISTIC
x w1 I 1
1 x = 7 I
N w1
n n!
fx 8 Pr
2 x = (n r )!
f
n n!
9 Cr
(x x) 2
x 2 (n r )!r!
3 = = x2
N N 10 P(A B) = P(A)+P(B) − P(A B)
f (x x)2 f x2 11 P (X = r) = nCr p r q n r , p + q = 1
4 = = x2
f f
1
12 Mean µ = np
N F
5 m = L 2 C
fm 13 npq
x
14 z=
Q1
6 I 100
Q0
TRIGONOMETRY
1 Arc length, s = r 9 sin (A B) = sinA cosB cosA sinB
1 2 10 cos (A B) = cosA cosB sinA sinB
2 Area of sector , A = r
2
3 sin 2A + cos 2A = 1 tan A tan B
11 tan (A B) =
1 tan A tan B
4 sec2A = 1 + tan2A
a b c
12
5 cosec2 A = 1 + cot2 A sin A sin B sin C
6 sin 2A = 2 sinA cosA
2 2
13 a2 = b2 + c2 − 2bc cosA
7 cos 2A = cos A – sin A
= 2 cos2A − 1 1
= 1 − 2 sin2A 14 Area of triangle = absin C
2
2 tan A
8 tan 2A =
1 tan 2 A
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5. SULIT 5 3472/1 For
examiner’s
use only
Answer all questions.
Jawab semua soalan.
1. Diagram 1 shows the relation between set A and set B.
Rajah 1 menunjukkan hubungan antara set A dan set B.
Set A Set B
9 3
49 7
x 9
Diagram 1
Rajah 1
a) State the image of 9.
Nyatakan imej bagi 9.
b) Find the value of x.
Cari nilai x. [ 2 marks]
[2 markah]
1
Answer/Jawapan : (a) ……………………..
(b) ……………………... 2
3 x
2. Given f 1 : x , find the value of
5
3 x
Diberi f 1 : x , cari nilai bagi
5
(a) f ( 3) ,
(b) p if f ( p) 7.
[ 4 marks ]
[4 markah]
2
Answer/ Jawapan : (a) ……………………..
4
(b) ……………………...
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6. For SULIT 6 3472/1
examiner’s
use only
3. Given that function g : x 2 x a and g 2 : x bx 9 .
Diberi fungsi g : x 2 x a dan g 2 : x bx 9 .
Find the value of a and of b
Cari nilai bagi a dan b .
[3 marks]
[3 markah]
Answer/Jawapan : a =.........................
3 b =.........................
3
4. Given that the straight line y 4 x 1 is a tangent to the curve y x2 k.
Find the value of k .
Diberi garis lurus y 4 x 1 ialah tangen kepada lengkung y x2 k.
Cari nilai k .
[ 3 marks]
[3 markah]
4
3 Answer/Jawapan : k =......…………………
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7. For
examiner’s
SULIT 7 3472/1 use only
5.
y
y 3( x 2) 2 4
P
x
Q
Diagram above shows the graph of the function y 3( x 2) 2 4 . Q is the minimum point
of the curve and the curve intersects the y-axis at point P. Find the equation of the straight
line PQ.
[ 3 marks ]
Rajah di atas menunjukkan graf bagi fungsi y 3( x 2) 2 4 . Q ialah titik minimum bagi
lengkung itu dan lengkung tersebut bersilang dengan paksi-y di titik P. Cari persamaan garis
lurus PQ.
[3 markah]
5
Answer /Jawapan: ……........................ 3
___________________________________________________________________________
6. Given that and are the roots of the quadratic equation x 2 9x 7 0.
Find the value of
Diberi dan adalah punca bagi persamaan kuadratik x 2 9x 7 0 . Cari nilai
bagi
(a)
(b)
2 2
(c) [ 4 marks ]
[4 markah]
Answer/Jawapan : (a)............................. 6
(b)............................
4
(c).............................
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8. SULIT 8 3472/1
For
examiner’s
use only 7. Solve the equation:
Selesaikan persamaan:
2 x (8 x 1 ) 45 x .
[4 marks]
[4 markah]
7
Answer/Jawapan : x =................................
4
8. The set of positive integers 2, 5, 7, 9, 11, x, y has a mean 8 and median 9. Find the
values of x and of y if y > x.
[3 marks]
Satu set integer positif 2, 5, 7, 9, 11, x, y mempunyai min 8 dan median 9. Cari
nilai-nilai bagi x dan y jika y > x.
[3 markah]
8
Answer/Jawapan : ...................................
3
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9. SULIT 9 3472/1 For
examiner’s
use only
9. Given that log 3 x log 9 y 2 . Express y in terms of x.
[ 3 marks ]
Diberi log 3 x log 9 y 2 . Ungkapkan y dalam sebutan x.
[3 markah]
9
Answer/Jawapan : ...................................... 3
10. The sixth and eleventh terms of an arithmetic progression are 12 and 37 respectively.
Find the value of the sixteenth term of this arithmetic progression.
[3 marks]
Sebutan keenam dan kesebelas bagi suatu janjang aritmetik ialah 12 dan 37 masing-
masing. Cari nilai bagi sebutan keenambelas bagi janjang aritmetik ini.
[3 markah]
10
3
Answer/Jawapan : .……………...………..
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10. SULIT 10 3472/1
For
examiner’s
11. The first three terms of a geometric progression are 36, 36 − p and q. If the
use only 1
common ratio is , find the value of
3
Tiga sebutan pertama suatu janjang geometri ialah 36, 36 − p dan q. Jika nisbah
1
sepunya ialah , cari nilai bagi
3
(a) p ,
(b) q.
[ 3 marks ]
[3 markah]
11 Answer/Jawapan: a) p = ..…………..….......
b) q =...............................
3
12. The first term of a geometric progression is a and the common ratio is r . Given that
a 96r 0 and the sum to infinity is 32, find the value of a and of r .
[ 4 marks ]
Sebutan pertama bagi suatu janjang geometri ialah a dan nisbah sepunya r .
Diberi a 96r 0 dan hasil tambah hingga sebutan ketakterhinggaan ialah 32,
cari nilai a dan r .
[4 markah]
12
Answer/Jawapan:a=.….………..….......
4 r=...............................
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11. SULIT 11 3472/1 For
examiner’s
use only
13. Given that A, B ( 4,4) , C (2,7) are collinear and 3AB=BC, find the coordinates of A.
[ 3 marks ]
Diberi A, B ( 4,4) ,C ( 2,7) adalah segaris dan 3AB=BC, carikan koordinat titik A.
[3 markah]
13
Answer/Jawapan : ………………..……. 3
14. Diagram below shows the graph of log10 y against x .
Rajah di bawah menunjukkan graf log10 y lawan x.
log10 y
(h,2)
x
(0, k )
The variables x and y are related by the equation y 103 x 2
Find the value of h and of k.
Pembolehubah x dan y dihubungkait dengan persamaan y 103 x 2
.
Cari nilai h dan k.
[3 marks]
[3 markah]
14
Answer/Jawapan : h=………………
k=….………..….
3
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12. SULIT 12 3472/1
For
examiner’s
use only
15. y
A
B
x
O
In the diagram above, OA 8i n j and AB 3i 2 j . Given
OA 10 units , find
Dalam rajah di atas, OA 8i n j dan AB 3i 2 j . Diberi
OA 10 unit , cari
(a) the value of n.
nilai n.
(b) coordinates of B.
koordinat B.
[2 marks]
[2 markah]
15 Answer/Jawapan : (a) n = ………………
(b).………………….
2
16 Given that a pi 2 j and b 2i j , find the value of p if a b is parallel
to j.
[ 3 marks ]
Diberi a pi 2 j dan b 2i j , cari nilai p jika a b selari dengan j .
[3 markah]
16
3
Answer/Jawapan :………………………..
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13. SULIT 13 3472/1 For
examiner’s
use only
17. Solve the equation 2 cos x cot x 0 for 0 0 x 360 0 .
[ 4 marks ]
Selesaikan persamaan 2kosx kotx 0 bagi 0 0 x 360 0 .
[4 markah]
17
Answer/Jawapan: …...…………..…....... 4
18. Diagram below shows a circle with centre O.
Rajah di bawah menunjukkan satu bulatan dengan pusat O.
P
__
1
O 2
3
Q
2
Given that the minor angle POQ is radian and the area of the shaded
3
region is 12 cm 2 . Find the length of the minor arc PQ.
2
Diberi sudut minor POQ ialah radian dan luas sektor berlorek
3
ialah 12 cm 2 . Cari panjang lengkok minor PQ.
[4 marks]
[4 markah]
18
4
Answer/Jawapan: …………………
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14. For
SULIT 14 3472/1
examiner’s
use only
3 5
19. Find the gradient of the curve y 4x2 at the point (1, 3) .
2x 2
[ 3 marks ]
3 5
Cari kecerunan kepada lengkung y 4x2 pada titik (1, 3) .
2x 2
[3 markah]
19
3 Answer/Jawapan:………………………
4x2 1
20. Differentiate with respect to x .
2x 1
[2 marks]
4x2 1
Bezakan terhadap x.
2x 1
[2 markah]
20
2
Answer/Jawapan: …...…………..….......
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15. SULIT 15 3472/1 For
examiner’s
use only
21. Given that the gradient function of a curve passing through the point (1, 2) is
3
+ 2x , determine the equation of the curve.
( 2 x 1) 2
Fungsi kecerunan bagi suatu lengkung yang melalui titik ( 1, 2) ialah
3
+ 2x, tentukan persamaan bagi lengkung ini.
( 2 x 1) 2
[4 marks]
[4 markah]
21
Answer/Jawapan: …………………….. 4
(2 x 3) 5
22. Given that y and x is increasing at the rate of 2 units per second, find the
10
1
rate of change of y when x . [ 3 marks ]
2
(2 x 3) 5
Diberi y dan x bertambah dengan kadar 2 unit sesaat, cari kadar
10
1
perubahan bagi y apabila x . [3 markah]
2
22
Answer/Jawapan: ……………………. 3
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16. For
SULIT 16 3472/1
examiner’s
use only 23. 90 percent of the students of Form 5 Euler passed the April mathematics test. Among
those who passed, 20 percent score with distinction.
90 peratus pelajar Tingkatan 5 Euler lulus ujian matematik bulan April. Antara
mereka yang lulus, 20 peratus skor dengan cemerlang.
(a) If a student of Form 5 Euler was selected at random, find the probability that he
passed the April mathematics test with distinction.
Jika seorang pelajar dari tingkatan 5 Euler dipilih secara rawak, cari
kebarangkalian dia lulus ujian matematik bulan April dengan cemerlang.
(b) If 5 students of Form 5 Euler were selected at random, find the probability that
only one of the five students selected passed the April mathematics test with
distinction.
Jika 5 orang pelajar dari tingkatan 5 Euler dipilih secara rawak, cari
kebarangkalian hanya seorang daripada lima pelajar terpilih lulus ujian
matematik bulan April dengan cemerlang.
[3 marks]
[3 markah]
23 Answer/Jawapan: (a) ……………………..
(b) ……………..………
3
24. Five cards are numbered 1, 2, 3, 4 and 5 respectively. How many different odd numbers
can be formed by using four of these five cards?
[ 3 marks ]
Lima kad masing-masing ditulis dengan nombor 1, 2, 3, 4 dan 5. Berapa nombor ganjil
boleh dibentuk dengan menggunakan empat daripada lima kad ini ?
[3 markah]
24
3
Answer/Jawapan: ………………………
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17. SULIT 17 3472/1
25. A random variable X is normally distributed with mean 370 and standard deviation
10. Find the value of
Satu pembolehubah rawak X bertaburan normal dengan min 370 dan sisihan piawai
10. Cari nilai bagi
(a) the z-score if X = 355.
skor z jika X = 355
(b) P ( X 367) .
[4 marks]
[4 markah]
25
Answer/Jawapan: (a)………………………
4
(b)……………………… 4
END OF QUESTION PAPER
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