spm trial 2009 addmath q&a (kedah)
TRANSCRIPT
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PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUASEKOLAH MENENGAH
NEGERI KEDAH DARUL AMAN
PEPERIKSAAN PERCUBAAN SPM 2009
Kertas soalan ini mengandungi 17 halaman bercetak
For Examiners use only
Question Total MarksMarks
Obtained
1 22 43 34 35 36 47 48 39 3
10 311 312 413 314 315 216 317 4
18 419 320 221 422 323 324 325 4
TOTAL 80
M AT E M AT I K TA M B A H A N
Kertas 1Dua jam
J A N G A N B U K A K E RTA S S O A L A N IN ISEHINGGA DIBERITAHU
1 This question paper consists of 25 questions.
2. Answer all questions.
3. Give onlyone answer for each question.
4. Write your answers clearly in the spaces provided inthe question paper.
5. Show yo ur working. It may help you to get marks.
6. If you wish to change you r answer, cross out the work that you have done. Then write down the newanswer.
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 pag es 2 to 3.
10. A boo klet of four-figure ma thematical tables isprovided.
.11 You m ay use a non-programm able scientific
calculator.
12 This question paper must be handed in at the end of the examination .
Name : ..
Form : ..
3472/1 Additional Mathematics
Paper 1Sept 20092 Hours
ADDITIONAL MATHEMATICSPaper 1
Two hours
3 to 4.
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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 onescommonly used.
ALGEBRA
12 4
2
b b ac x
a
- -=
2 a m a n = a m + n
3 a m a n = a m - n
4 (a m) n = a nm
5 log a mn = log a m + log a n
6 log anm
= log a m log a n
7 log a mn
= n log a m
8 log ab =a
b
c
c
log
log
9 T n = a + (n 1)d
10 S n = ])1(2[2
d nan -+
11 T n = arn-1
12 S n =r r a
r r a nn
--=
--
1)1(
1)1(
, (r 1)
13r
aS
-=
1, r
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STATISTIC
1 Arc length, s = r q
2 Area of sector , A = 212
r q
3 sin 2 A + cos 2 A = 1
4 sec 2 A = 1 + tan 2 A
5 cosec 2 A = 1 + cot 2 A
6 sin 2 A = 2 sin A cos A
7 cos 2 A = cos 2 A sin 2 A= 2 cos 2 A 1= 1 2 sin 2 A
8 tan 2 A = A
A2tan1
tan2-
TRIGONOMETRY
9 sin ( A B) = sin A cos B cos A sin B
10 cos ( A B) = cos A cos B m sin A sin B
11 tan ( A B) = B A B A
tantan1tantan
m
12C
c B
b A
asinsinsin
==
13 a 2 = b2 + c2 2bc cos A
14 Area of triangle = C ab sin21
1 x = N
x
2 x =
f
fx
3 s = N
x x - 2)(= 2
2
x N
x -
4 s = -
f
x x f 2)(= 2
2
x f
x f -
5 m = C f
F N L
m
-+ 2
1
6 10
100Q
I Q
=
71
11
w I w
I
=
8)!(
!r n
nPr n
-=
9!)!(
!r r n
nC r
n
-=
10 P( A B) = P( A)+P( B) P( A B)
11 P ( X = r ) = r nr r n q pC - , p + q = 1
12 Mean = np
13 npq=s
14 z =s
m - x
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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.
a) State the image of 9.Nyatakan imej bagi 9.
b) Find the value of x.Cari nilai x. [ 2 marks]
[2 markah]
Answer/Jawapan : (a) ..
(b) ...
2. Given5
3:1
x x f
-- , find the value of
Diberi5
3:1
x x f
-- , cari nilai bagi
(a) )3(- f ,
(b) p if 7)( -= p f .[ 4 marks ]
[4 markah]
Answer/ Jawapan : (a) ..
(b) ... 4
2
2
1
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use only
Set A Set B
49
9
x 9
7
3
Diagram 1Rajah 1
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3. Given that function a x xg + 2: and 9:2 - bx xg . Diberi fungsi a x xg + 2: dan 9:2 - bx xg .
Find the value of a and of bCari nilai bagi a dan b .
[3 marks][3 markah]
Answer/Jawapan : a =.........................b =.........................
4. Given that the straight line 14 += x y is a tangent to the curve k x y += 2 .Find the value of k .
Diberi garis lurus 14 += x y ialah tangen kepada lengkung k x y += 2 .Cari nilai k .
[ 3 marks][3 markah]
Answer/Jawapan : k =......
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3
4
3
3
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5.
4)2(3 2 --= x y
Diagram above shows the graph of the function 4)2(3 2 --= x y . Q is the minimum pointof the curve and the curve intersects the y-axis at point P. Find the equation of the straightline PQ.
[ 3 marks ]
Rajah di atas menunjukkan graf bagi fungsi 4)2(3 2 --= x y . Q ialah titik minimum bagi
lengkung itu dan lengkung tersebut bersilang dengan paksi-y di titik P. Cari persamaan garislurus PQ.[3 markah]
Answer /Jawapan : ........................
___________________________________________________________________________
6. Given that a and b are the roots of the quadratic equation 0792 =+- x x .Find the value of
Diberi a dan b adalah punca bagi persamaan kuadratik 0792 =+- x x . Cari nilaibagi
(a) b a +(b) ab
(c) 22 b a + [ 4 marks ][4 markah]
Answer/Jawapan : (a).............................
(b)............................
(c).............................
3
5
4
6
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x
y
P
Q
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7. Solve the equation:Selesaikan persamaan :
x x x 51 4)8(2 =- .
[4 marks][4 markah]
Answer/Jawapan : x =................................
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]
Answer/Jawapan : ...................................
4
7
3
8
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9. Given that 2loglog 93 =+ y x . Express y in terms of x.[ 3 marks ]
Diberi 2loglog 93 =+ y x . Ungkapkan y dalam sebutan x.
[3 markah]
Answer/Jawapan : ......................................
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]
Answer/Jawapan : ......
3
10
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11. The first three terms of a geometric progression are 36, 36 p and q. If the
common ratio is31- , find the value of
Tiga sebutan pertama suatu janjang geometri ialah 36, 36 p dan q . Jika nisbah
sepunya ialah31- , cari nilai bagi
(a) p ,
(b) q.
[ 3 marks ][3 markah]
Answer/Jawapan: a) p = ...........
b) q =...............................
12. The first term of a geometric progression is a and the common ratio is r . Given that
096 =+ r a 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 096 =+ r a dan hasil tambah hingga sebutan ketakterhinggaan ialah 32,cari nilai a dan r .
[4 markah]
Answer/Jawapan:a= ...........
r =...............................4
12
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11
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13. Given that A, B )4,4(- , C )7,2( are collinear and 3AB=BC, find the coordinates of A.[ 3 marks ]
Diberi A, B )4,4(- ,C )7,2( adalah segaris dan 3AB=BC, c arikan koordinat titik A.[3 markah]
Answer/Jawapan : ...
14. Diagram below shows the graph of y10log against x . Rajah di bawah menunjukkan graf y10log lawan x.
The variables x and y are related by the equation 2310 -= x yFind the value of h and of k .
Pembolehubah x dan y dihubungkait dengan persamaan 2310 -= x y .Cari nilai h dan k.
[3 marks][3 markah]
Answer/Jawapan : h=
k =....
3
13
3
14
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),0( k
)2,(h y10log
x
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15.
In the diagram above, jniOA += 8 and ji AB 23 -= . GivenunitsOA 10= , find
Dalam rajah di atas, jniOA += 8 dan ji AB 23 -= . Diberi
unit OA 10= , cari
(a) the value of n.nilai n.
(b) coordinates of B.koordinat B.
[2 marks][2 markah]
Answer/Jawapan : (a) n =
(b)..
16 Given that ji pa 2+= and jib --= 2 , find the value of p if ba - is parallelto j .
[ 3 marks ]
Diberi ji pa 2+= dan jib --= 2 , cari nilai p jika ba - selari dengan j .
[3 markah]
Answer/Jawapan :..
2
15
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3
16
y
xO
A
B
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17. Solve the equation 0cotcos2 =+ x x for 00 3600 x .[ 4 marks ]
Selesaikan persamaan 02 =+ kotxkosx bagi 00 3600 x .[4 markah]
Answer/Jawapan: ............
18. Diagram below shows a circle with centre O. Rajah di bawah menunjukkan satu bulatan dengan pusat O.
Given that the minor angle POQ is p 32
radian and the area of the shaded
region is 212 cmp . Find the length of the minor arc PQ.
Diberi sudut minor POQ ialah p 32
radian dan luas sektor berlorek
ialah 212 cmp . Cari panjang lengkok minor PQ.
[4 marks][4 markah]
Answer/Jawapan :
_
p 32 _ 1O
P
4
17
4
18
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Q
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19 . Find the gradient of the curve25
23
4 2 -+= x
x y at the point )3,1( .
[ 3 marks ]
Cari kecerunan kepada lengkung2
5
2
34 2 -+=
x x y pada titik )3,1( .
[3 markah]
Answer/Jawapan:
20 . Differentiate12
14 2
+
-
x
xwith respect to x .
[2 marks]
Bezakan1214 2
+-
x x
terhadap x.
[2 markah]
Answer/Jawapan: ............
3
19
2
20
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21 . Given that the gradient function of a curve passing through the point (1, 2) is
2)12(
3- x
+ 2 x , determine the equation of the curve.
Fungsi kecerunan bagi suatu lengkung yang melalui titik ( 1, 2) ialah
2)12(3- x
+ 2x, tentukan persamaan bagi lengkung ini.
[4 marks][4 markah]
Answer/Jawapan : ..
22 . Given that10
)32( 5-= x y and x is increasing at the rate of 2 units per second, find the
rate of change of y when21= x . [ 3 marks ]
Diberi10
)32( 5-= x y dan x bertambah dengan kadar 2 unit sesaat, cari kadar
perubahan bagi y apabila21= x . [3 markah]
Answer/Jawapan : .
4
21
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3
22
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23. 90 percent of the students of Form 5 Euler passed the April mathematics test. Amongthose who passed, 20 percent score with distinction.90 peratus pelajar Tingkatan 5 Euler lulus ujian matematik bulan April. Antaramereka yang lulus, 20 peratus skor dengan cemerlang.
(a) If a student of Form 5 Euler was selected at random, find the probability that hepassed the April mathematics test with distinction.
Jika seorang pelajar dari tingkatan 5 Euler dipilih secara rawak, carikebarangkalian dia lulus ujian matematik bulan April dengan cemerlang.
(b) If 5 students of Form 5 Euler were selected at random, find the probability thatonly one of the five students selected passed the April mathematics test withdistinction.
Jika 5 orang pelajar dari tingkatan 5 Euler dipilih secara rawak, carikebarangkalian hanya seorang daripada lima pelajar terpilih lulus ujianmatematik bulan April dengan cemerlang.
[3 marks][3 markah]
Answer/Jawapan : (a) ..
(b) ..
24. Five cards are numbered 1, 2, 3, 4 and 5 respectively. How many different odd numberscan 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]
Answer/Jawapan : 3
24
3
23
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25. A random variable X is normally distributed with mean 370 and standard deviation10. Find the value of Satu pembolehubah rawak X bertaburan normal dengan min 370 dan sisihan piawai10. Cari nilai bagi
(a) the z-score if X = 355.skor z jika X = 355
(b) )367( < X P .
[4 marks ][4 markah]
Answer/Jawapan : (a)
(b)
END OF QUESTION PAPER KERTAS SOALAN TAMAT
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SULIT3472/2Additional MathematicsKertas 2September, 20092 jam 30 minit
PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUASEKOLAH MENENGAH
NEGERI KEDAH DARUL AMAN
PEPERIKSAAN PERCUBAAN SPM 2009
ADDITIONAL MATHEMATICS
Kertas 2
Dua jam tiga puluh mini t
JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU
1. This question paper consists of three sections : Section A, Section B and Section C.
2. Answer all questions in Section A, four questions from Section B and two questions from Section C .
3. Give only one answer/solution to each question.
4. Show your working. It may help you to get your marks.
5. The diagrams provided are not drawn according to scale unless stated.
6. The marks allocated for each question and sub - part of a question are shown inbrackets.
7. You may use a non-programmable scientific calculator.
8. A list of formulae is provided in page 2 and 3.
This question paper consists of 19 printed pages and 1 blank page.
3472/2 [Lihat sebelah
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The following formulae may be helpful in answering the questions. The symbols given are the onescommonly used.
ALGEBRA
1. x =a
acbb
2
42 -- 8.a
bb
c
ca log
loglog =
2. aaa nmnm += 9. d naT n )1( -+=
3. aaa nmnm-= 10. ])1(2[
2d na
nS n -+=
4. aamnnm =)( 11.
1-= nn ar T
5. nmmn aaa logloglog +=12.
r r a
r r a
Snn
n --=
--=
1)1(
1)1(
, r 1
6. log log loga a am
m nn
= - 13.r
aS -
=1
, r < 1
7. mnm ana loglog =
CALCULUS
1. y = uv ,dxdu
vdxdv
udxdy += 4 Area under a curve
= ba dx y or
= ba dy x
2. y = vu
, 2vdxdv
udxdu
v
dxdy
-
=
5. Volume of revolution
= ba
dx y 2p or
= ba dy x2p
3.dxdu
dudy
dxdy =
GEOMETRY
1. Distance = 2122
12 )()( y y x x -+-4. Area of triangle
= 1 2 2 3 3 1 2 1 3 2 1 31
( ) ( )2
x y x y x y x y x y x y+ + - + +
2. Mid point
( x , y ) =
++
2,
22121 y y x x
5. 22 y xr +=
3. Division of line segment by a point
( x , y ) =
+
++
+nm
mynynmmxnx 2121 ,
6.2 2
xi yj
r x y
+=
+% %
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STATISTICS
1. N
x x
= 7 =
i
ii
W
I W I
2.=
f fx
x 8)!(
!r n
nPr
n-
=
3. N
x x -=2)(
s = 22
x N x - 9 !)!(
!r r n
nC r
n-
=
4. -=
f x x f 2)(
s = 22
x f
fx -
10 P(A B) = P(A) + P(B) P(A B)
11 P ( X = r ) = r nr r n q pC - , p + q = 1
5. m = L + C f
F N
m
-21 12 Mean , m = np
13 npq=s
6. 10001 = Q
Q I 14 Z = s
m - X
TRIGONOMETRY
1. Arc length, s = r q 8. sin ( A B ) = sin A cos B cos A sin B
2. Area of sector, A = q 221
r 9. cos ( A B ) = cos A cos B m sin A sin B
3. sin A + cos A = 110 tan ( A B ) =
B A B A
tantan1tantan
m
4. sec A = 1 + tan A11 tan 2 A =
A A2tan1
tan2-5. cosec A = 1 + cot A
12C
c B
b A
asinsinsin
==
6. sin 2 A = 2sin A cos A 13 a = b + c 2 bc cos A7. cos 2 A = cos A sin A
= 2 cos A 1= 1 2 sin A
14 Area of triangle =1
sin2
ab C
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3472/2 Additional Mathematics Paper 2 [Lihat sebelah
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Section A Bahagian A[ 40 marks ]
[ 40 markah ]
Answer all questions. Jawab semua soalan.
1. Solve the simultaneous equations 3 2 0 x y- - = and ( )2 1 3 x y - = .Give your answers correct to three decimal places.
[5 marks ]
Selesaikan persamaan serentak 3 2 0 x y- - = dan ( )2 1 3 x y - = . Beri jawapan anda betul kepada tiga tempat perpuluhan.
[5 markah ]
2.
In the diagram, the gradient and y-intercept of the straight line PQare 2 and 3 respectively. R is a point on the x-axis.
(a) Find the value of h and k .
(b) Given that PQ is perpendicular to QR , find the x-intercept of QR .
(c) Calculate the area of PQR .
[3 marks ]
[3 marks ]
[2 marks ]
Dalam rajah, kecerunan dan pintasan-y bagi garis lurus PQ masing-masing ialah 2 dan 3 . R ialah titik pada paksi-x.
(a) Cari nilai h dan nilai k .
(b) Diberi bahawa PQ berserenjang dengan QR, cari pintasan-xbagi QR.
(c) Hitungkan luas PQR.
[3 markah ]
[3 markah ]
[2 markah ]
x
y
0
Q(4, k )
P (h, 3) R
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3. (a) Sketch the graph of 2 sin 2 y x= - for 0 2 x p .
(b) Hence, using the same axes, sketch a suitable straight line to find
the number of solutions for the equation sin22
x x
p = for
0 2 x p . State the number of solutions.
(a) Lakar graf bagi 2 sin 2 y x= - untuk 0 2 x p .
(b) Seterusnya, dengan menggunakan paksi yang sama, lakar satugaris lurus yang sesuai untuk mencari bilangan penyelesaian
bagi persamaan sin22 x
xp
= untuk 0 2 x p .
Nyatakan bilangan penyelesaian itu.
[4 marks ]
[3 marks ]
[4 markah ]
[3 markah ]
4. Given that ( )2 1 x x - is the gradient function of a curve which passesthrough the point )1,1( -P . Find
(a) the gradient of the tangent to the curve at P,
(b) the equation of the curve,
(c) the coordinates of the turning point at x = 1 . Hence determinewhether the turning point is a maximum or a minimum point.
[1 mark ]
[3 marks ]
[3 marks ]
Diberi ( )2 1 x x - ialah fungsi kecerunan bagi suatu lengkung yangmelalui titik )1,1( -P . Cari
(a) kecerunan tangen kepada lengkung itu di P,
(b) persamaan lengkung itu,
(c) koordinat bagi titik pusingan pada x = 1 . Seterusnya tentukansama ada titik pusingan itu adalah titik maksimum atau titik minimum.
[1 markah ]
[3 markah ]
[3 markah ]
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5. A set of fifty numbers, 1 2 3 50, , ,...... x x x x , has a mean of 11 and astandard deviation of 8.
(a) Find
(i) x ,(ii)
2 x
(b) If each of the numbers is multiplied by 1.8 and thenincreased by 5, find the new value for the
(i) mean,(ii) variance.
[3 marks ]
[3 marks ]
Suatu set yang terdiri daripada lima puluh nombor,1 2 3 50, , ,...... x x x x , mempunyai min 11 dan sisihan piawai 8.
(a) Cari
(i) x ,(ii)
2 x
(b) Jika setiap nombor didarab dengan 1.8 dan ditambahdengan 5, cari nilai yang baru untuk
(i) min ,(ii) varians .
[3 markah ]
[3 markah ]
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6.
A strip of metal is cut and bent to form some semicircles. Thediagram shows the first four semicircles formed. The radius of thesmallest semicircle is 5 cm. The radius of each subsequentsemicircle is increased by 3 cm.
(a) If the radius of the largest semicircle is 104 cm, find the numberof semicircles formed.
(b) Calculate the total cost needed to form all the semicircles in (a) if the cost of the metal strip is RM4 per meter.[ Round off your answer to the nearest RM ]
Satu jalur logam dipotong dan dibengkok untuk membentuk beberapa semi bulatan. Rajah menunjukkan empat semi bulatan
pertama yang telah dibentukkan. Jejari semi bulatan yang terkecilialah 5 cm . Jejari semi bulatan yang berikutnya bertambahsebanyak 3cm setiap satu.
(a) Jika jejari bagi semi bulatan yang terbesar ialah 104 cm , caribilangan semi bulatan yang telah dibentuk.
(b) Hitungkan jumlah kos yang diperlukan untuk membentuk semuasemi bulatan dalam (a) jika kos jalur logam ialah RM4 semeter.[ Bundarkan jawapan anda kepada RM yang terdekat ]
[3 marks ]
[4 marks ]
[3 markah ]
[4 markah ]
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Section B Bahagian B
[ 40 marks ][ 40 markah ]
Answer four questions from this section. Jawab empat soalan daripada bahagian ini.
7.
The table shows the values of two variables, x and y, obtained froman experiment. Variables x and y are related by the equation
1=+ y xl m , where m and l are constants.
(a) Plot y1
against x1
, using a scale of 2 cm to 0.1 unit on both
axes. Hence draw the line of best fit.
(b) Use your graph in 7(a) to find the value of
(i) m ,
(ii) l .
[5 marks ]
[5 marks ]
Jadual menunjukkan nilai-nilai bagi dua pembolehubah, x dan y, yang diperoleh daripada satu eksperimen. Pembolehubah x dan y
dihubungkan oleh persamaan 1=+ y xl m
, dengan keadaan m dan l
adalah pemalar
(a) Plot y1
melawan x1
, dengan menggunakan skala 2 cm
kepada 0.1 unit pada kedua-dua paksi. Seterusnya, lukis
garis lurus penyuaian terbaik.(b) Gunakan graf di 7(a) untuk mencari nilai
(i) m ,
(ii) l .
[5 markah ]
[5 markah ]
x 1.5 2.0 2.5 4.0 5.0 10.0 y 0.96 1.2 1.4 2.0 2.2 3.0
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8.
The diagram shows straight line x + y = 4 that intersects with thecurve y = ( x 2 ) 2 at points P and Q.
Find
(a) the coordinates of Q,
(b) the area of the shaded region A,
(c) the volume generated, in terms of , when the shaded region B is revolved through 360 o about the x-axis.
[2 marks ]
[4 marks ]
[4 marks ]
Rajah menunjukkan garis lurus x + y = 4 bersilang denganlengkung y = ( x 2 ) 2 pada titik P dan Q .
Cari
(a) koordinat Q ,
(b) luas rantau berlorek A ,
(c) isipadu janaan, dalam sebutan , apabila rantau berlorek B
dikisarkan melalui 360 o pada paksi-x.
[2 markah ]
[4 markah ]
[4 markah ]
O x
y = ( x 2 ) 2P
Q A B x + y = 4
y
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9.
In the diagram, PQRS is a quadrilateral. The diagonals PR and QS
intersect at point T . It is given PQ = 2~
x , PS = 3~
y and
SR =~
x ~
y .
(a) Express in terms of ~
x and~
y :
(i) QS
(ii) PR .
(b) Given that QT = m QS , PT = n PR , where m and n areconstants, express
(i) QT in terms of m,~
x and~
y ,
(ii) PT in terms of n,~
x and~
y .
(c) Using PQ = PT + TQ , find the value of m and of n.
[10 marks ] Dalam rajah, PQRS ialah sebuah sisiempat. Pepenjuru-pepenjuru
PR dan QS bersilang di titik T . Diberi PQ = 2~
x , PS = 3~
y
dan SR =~
x ~
y .
(a) Ungkapkan dalam sebutan~
x dan~
y :
(i) QS
(ii) PR .
(b) Diberi QT = m QS , PT = n PR , dengan keadaan m dan nialah pemalar, ungkapkan
(i) QT dalam sebutan m,~
x dan~
y ,
(ii) PT dalam sebutan n,~
x dan~
y .
(c) Dengan mengguna PQ = PT + TQ , cari nilai m dannilai n. [10 markah ]
P
T
S
R
Q
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10.
The diagram shows the cross-section of a cylindrical roller withcentre O and radius 20 cm resting on a horizontal ground PQ .OAB is a straight line that represents the handle of the roller and
OA : AB = 1 : 3.Calculate(a) POA in radian,(b) the perimeter, in cm, of the shaded region,(c) the area, in cm 2 , of the shaded region.
[3 marks ]
[3 marks ]
[4 marks ]
Rajah menunjukkan keratan rentas sebuah penggelek berbentuk silinder dengan pusat O dan jejari 20 cm yang terletak di atas
lantai mengufuk PQ. OAB ialah garis lurus yang mewakili pemegang penggelek itu dan OA : AB = 1 : 3. Hitungkan
(a) POA dalam radian ,(b) perimeter , dalam cm, kawasan berlorek,(c) luas , dalam cm2 , kawasan berlorek .
[3 markah ]
[3 markah ]
[4 markah ]
QP
O
A
B
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11 (a) In a survey carried out in a school, it is found that 40% of itsstudents are participating actively in co-curricular activities.
(i) If 6 students from that school are chosen at random,calculate the probability that at least 4 students areparticipating actively in co-curricular activities.
(ii) If the variance of the students who are active in co-curricularactivities is 288, calculate the student population of theschool .
[5 marks ](b) The masses of chicken eggs from a farm has a normal distribution
with a mean of 62 g and a standard deviation of 8 g. Any egg thathas a mass exceeding 68 g is categorised as grade double-A.
(i) Find the probability that an egg chosen randomly from thefarm has a mass between 60 g and 68 g.
(ii) If the farm produces 3000 eggs daily, calculate the number of eggs with grade double-A.
[5 marks ](a) Dalam satu tinjauan yang dijalankan ke atas murid-murid di
sebuah sekolah, didapati 40% daripada murid-murid sekolah itumengambil bahagian secara aktif dalam aktiviti kokurikulum .
(i) Jika 6 orang murid daripada sekolah itu dipilih secara rawak,hitungkan kebarangkalian bahawa sekurang-kurangnya 4orang murid adalah aktif dalam aktiviti kokurikulum .
(ii) Jika varians murid-murid yang mengambil bahagian secaraaktif dalam aktiviti kokurikulum ialah 288 , hitungkan bilanganmurid dalam sekolah itu .
[5 markah ](b) Jisim telur ayam dari sebuah ladang adalah mengikut satu taburan
normal dengan min 62 g dan sisihan piawai 8 g . Sebarang telur dengan jisim melebihi 68 g dikategorikan sebagai telur gred double-A
(i) Cari kebarangkalian bahawa sebiji telur yang dipilih secararawak dari ladang itu mempunyai jisim di antara 60 g dan68 g.
(ii) Jika ladang itu menghasilkan 3000 biji telur setiap hari,hitungkan bilangan telur yang mempunyai gred double-A.
[5 markah ]
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Section C Bahagian C
[ 20 marks ][ 20 markah ]
Answer two questions from this section . Jawab dua soalan daripada bahagian ini.
12. A particle P starts from a fixed point O and moves in a straightline so that its velocity, v ms -1 , is given by v = 8 + 2t t 2, wheret is the time, in seconds, after leaving O.[Assume motion to the right is positive.]
Find
(a) the initial velocity, in ms -1 , of the particle,
(b) the value of t at the instant when the acceleration is 1 ms -2,
(c) the distance of P from O when P comes to instantaneousrest,
(d) the total distance, in m, travelled by the particle P in the first5 seconds.
Suatu zarah P mula dari suatu titik tetap O dan bergerak disepanjang garis lurus. Halajunya v ms -1 , diberi olehv = 8 + 2t t 2 , dengan keadaan t ialah masa, dalam saat, selepasmelalui O.[Anggapkan gerakan ke arah kanan sebagai positif]
Cari
(a) halaju awal, dalam ms -1 , bagi zarah itu,
(b) nilai bagi t apabila pecutannya ialah 1 ms -2 ,
(c) jarak P dari O apabila P berada dalam keadaan rehat seketika,
(d) jumlah jarak yang di lalui, dalam m , oleh zarah P dalam5 saat yang pertama.
[1 mark ]
[2 marks ]
[3 marks ]
[4 marks ]
[1 markah ]
[2 markah ]
[3 markah ]
[4 markah ]
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13. The pie chart shows five items, A, B, C, D and E used in makingcakes. The table shows the prices and the price indices of theseitems.
E
D
C
B
A
30
75
120
Items
Bahan
Price (RM)per kg for the
year 2003
Harga (RM) per kg padatahun 2003
Price (RM)per kg for the
year 2006
Harga (RM) per kg padatahun 2006
Price index for theyear 2006 based on
the year 2003
Indeks harga padatahun 2006
berasaskan tahun2003
A 0.40 x 150
B 1.50 1.65 110C 4.00 4.80 y D 3.00 4.50 150 E z 2.40 120
(a) Find the value of (i) x,(ii) y,(iii) z.
(b) Calculate the composite index for the cost of making these
cakes in the year 2006 based on the year 2003.
(c) The total expenditure on the items in the year 2006 isRM 5000. Calculate the corresponding total expenditurein the year 2003.
(d) The price of each item increases by 20 % from the year 2006to the year 2008. Find the composite index for totalexpenditure on the items in the year 2008 based on the year2003.
[3 marks ]
[3 marks ]
[2 marks ]
[2 marks ]
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Carta pai menunjukkan lima bahan, A, B, C, D dan E yangdigunakan untuk membuat sejenis kek. Jadual menunjukkan hargabahan dan nombor indeks bagi kelima-lima bahan tersebut.
(a) Carikan nilai(i) x,(ii) y,(iii) z.
(b) Hitungkan nombor indeks gubahan bagi kos penghasilan kek itu pada tahun 2006 berasaskan tahun 2003 .
(c) Jumlah kos bahan-bahan tersebut pada tahun 2006 ialahRM 5000 . Hitungkan jumlah kos yang sepadan pada tahun2003 .
(d) Harga bagi setiap bahan bertambah sebanyak 20% daritahun 2006 ke tahun 2008 . Cari nombor indeks gubahanbagi jumlah kos ke atas bahan-bahan tersebut pada tahun2008 berasaskan tahun 2003 .
[3 markah ]
[3 markah ]
[2 markah ]
[2 markah ]
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14. Use graph paper to answer this question.
A factory produces two components, S and T for a digitalcamera, by using machines P and Q. The table shows the timetaken to produce components S and T respectively.
In any given week, the factory produces x units of component S
and y units of component T . The production of the componentsper week is based on the following constraints:
I : Machine P operates not more than 2000 minutes.
II : Machine Q operates at least 1200 minutes.
III : The number of component T produced is not more than threetimes the number of component S produced.
(a) Write three inequalities, other than x 0 and y 0, whichsatisfy all the above constraints.
(b) Using a scale of 2 cm to 10 units on both axes, construct andshade the region R which satisfies all of the aboveconstraints.
(c) Use your graph in 14(b) to find
(i) the maximum number of component S that could beproduced, if the factory plans to produce only 30 unitsof component T,
(ii) the maximum profit per week if the profit from a unitof component S is RM20 and from a unit of component T is RM30.
Time taken (minutes) Masa diambil (minit)
Component
Komponen Machine P Mesin P
Machine Q Mesin Q
S 40 15
T 20 30
[3 marks ]
[3 marks ]
[4 marks ]
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17
Gunakan kertas graf untuk menjawab soalan ini.
Sebuah kilang menghasilkan dua komponen, S dan T bagisesuatu kamera digital dengan menggunakan mesin P dan Q.
Jadual menunjukkan masa yang diambil untuk menghasilkankomponen-komponen S dan T.
Dalam mana-mana satu minggu, kilang tersebut menghasilkan xunit bagi komponen S dan y unit bagi komponen T.Penghasilan komponen-komponen tersebut adalah berdasarkankekangan berikut:
I : Mesin P beroperasi tidak melebihi 2000 minit.
II : Mesin Q beroperasi sekurang-kurangnya 1200 minit.
III : Bilangan komponen T yang dihasilkan tidak melebihi tigakali ganda bilangan komponen S yang dihasilkan.
(a) Tuliskan tiga ketaksamaan, selain x 0 dan y 0 , yangmemenuhi semua kekangan di atas.
(b) Menggunakan skala 2 cm kepada 10 unit pada kedua-dua paksi, bina dan lorek rantau R yang memenuhi semuakekangan di atas.
(c) Gunakan graf anda di 14 (b) untuk mencari
(i) bilangan maksimum bagi komponen S yang bolehdihasilkan jika kilang tersebut bercadang untuk menghasilkan 30 unit komponen T sahaja,
(ii) keuntungan maksimum seminggu jika keuntungan yangdiperoleh dari satu unit komponen S ialah RM20 dandari satu unit komponen T ialah RM30 .
[3 markah ]
[3 markah ]
[4 markah ]
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15. The diagram shows a cyclic quadrilateral PQRS with PQ = 9 cm,PR = 11 cm and QR = 7 cm.
(a) Find PQR.
(b) Given that PS = 6 cm, find the length of RS.
(c) Calculate the area of PQRS .
S R
Q
P
11 cm
7 cm
9 cm
[3 marks ]
[4 marks ]
[3 marks ]
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Rajah menunjukkan satu sisiempat kitaran PQRS denganPQ = 9 cm , PR = 11 cm , dan QR = 7 cm .
(a) Cari PQR.
(b) Diberi PS = 6 cm, cari panjang RS.
(c) Hitungkan luas bagi PQRS.
[3 markah ]
[4 markah ]
[3 markah ]
END OF QUESTION PAPER KERTAS SOALAN TAMAT
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SULIT JPNKd/2006/ 3472/1
Additional Mathematics paper 1
Nama Pelajar : Tingkatan 5 : .3472/1
Additional Mathematics Paper1September 2009
PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUASEKOLAH MENENGAH
NEGERI KEDAH DARUL AMAN
PEPERIKSAAN PERCUBAAN SPM 2009
ADDITIONAL MATHEMATICSMARKING SCHEME
Paper 1
.
SULIT 3472/1
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Question Solution/ Marking Scheme Answer Marks
8B2: 22y x
B1: 87
119752 y x
12,10 y x
13,9 y x 3
9 B2: 2log4log 292
3 y xor y x
B1: 2log3log
log2
9loglog
log 99
9
3
33 y
xor
y x
281
x y 3
10B2: 513 d and a
B1: 3710125 d aor d a 62 3
11
(a) B1:31
3636 p
(a) 48
(b) 4
2
1
12B3: 48
21
aor r
B2: 32
961
096)1(32
aa
or r r
B1: 321 r
a
21,48 r a 4
13
B2 : 4437
44
23 yand
x
B1 : 4437
44
23 yor
x
( 6, 3) 3
14
B2 : 23230
22 hor
hk
or k
B1 : 23log 10 x y
h =34
k = 2
3
15(a) 6
(b) (11, 4)
1
1
3472/1 Additional Mathematics Paper 1 SULIT
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SULIT 3472/14
Question Solution/ Marking Scheme Answer Marks16 B2: 02 p
B1: ji p 3)2( p = 2 3
17 B3:
)42(330,210,270,90 0000 of out any
B2 : )(21
sin,0cos both x x
B1 : 0sincos
cos2 x x
x
0
0
00
330
,270
,210,90
4
18B3 : )
32(6 S
B2 : 6r
B1: 12)32
(21 2r
12.57 cm( 4 ) cm
4
19B2:
2)1(2
3)1(8
B1:22
38
x x
dxdy
2
13
3
20 B1: 12x y or
2
2
)12(
)14(2)8)(12(
x
x x xdxdy
2 2
21
B3 : c123
2
B2:2
22
)12(3 21 xand
x
B1:2
22
)12(3 21 xor
x
25
)12(23 2x
x y 4
3472/1 Additional Mathematics Paper 1 SULIT
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3472/1 Additional Mathematics Paper 1 SULIT
5
Question Solution/ Marking Scheme Answer Mark
22
B2 : 210
2)32(5 4xdt dy
B1 : 210
2)32(5 4
dt dx
or x
dxdy
32 units persecond
3
23(b) B1: 411
5 )82.0()18.0(C (a) 0.18
(b) 0.4069
1
2
24 B2: 34
13
PP
B1: 34
13 Por P
72 3
25(a) B1 :
10370355
(b) B1 :10
370367
(a) 1.5
(b) 0.3821
2
2
END OF MARKING SCHEME
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SULIT JPNKd/2006/ 3472/2
Nama Pelajar : Tingkatan 5 : . 3472/2
Additional Mathematics
September 2009
PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUASEKOLAH MENENGAH
NEGERI KEDAH DARUL AMAN
PEPERIKSAAN PERCUBAAN SPM 2009
ADDITIONAL MATHEMATICS
Paper 2
.
MARKING SCHEME
SULIT 3472/2
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2
MARKI NG SCHEM EADDITIONAL M ATHEMATICS PAPER 2
SPM TRIAL EXAMINATI ON 2009
N0. SOLUTION M ARK S
1
2
2
2
3 23 2 (3 2 1)
0 6 6 3
2 2 1 0
2 ( 2) 4(2)( 1)2(2)
1.366 0.366
3(1.366) 2 3( 0.366) 2
2.098 3.098
y x x x
x x
x x
x
x or x
y y
= -= - -=> = - -
- - =
- - -=
= = -= - = - -
= = -
P1K1 Eliminate y
K1 Solve quadraticequation
N1
N1
52
(a)
(b)
(c)
( )
2 3
( , 3) 3 2 3
3
(4, ) 2 4 3
11
y x
P h h
h
Q k k
k
= +- - = +
= - = +
=
OR
3 32 3
03
2 114 0
hh
k k
+ = = --- = =-
1 21
22
11 14 2
1: 13
20 26
m m
y x
QR y x
y x
= = -
- = --
= - +
= =
OR
11 0 14 2
111 2
226
x
x
x
- = --
= - +
=
x-intercept = 26
Area
3 4 26 313 11 0 32
133 78 12 286
21
3852192.5
- -=
- -
= - - + -
= -
=
K1 Use equation orgradient
N1
N1
P1 For 212
m = -
K1 Use equation orgradient
N1
K1 Use formula andfind the areatriangle
N1
8
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3
N0. SOLUTION M ARK S3
(a)
(b) 2 sin 2 22 x
xp
- = -
or
22 x
yp
= -
Draw the straight line 22 x
yp
= -
Number of solutions = 4 .
P1 Negative sineshape correct.
P1 Amplitude = 1[ M aximum = 3and Minimum =1 ]
P1 Two full cycle in0 x 2p
P1 Shift up the graph
N1 For equation
K1 Sketch thestr aight l ine
N1
7
3
1 2 2 x
yp
= -
y
2p 2p xO p
2 sin 2 y x= -
2
23p
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4
N0. SOLUTION M ARK S4
(a)
(b)
(c)
( ) ( )21 1 1 2dydx
= - - - = -
3 2
4 3
4 3
( )
4 31 1
14 35
12
54 3 12
y x x dx
x x y c
c
c
x x y
= -
= - +
= + +
=
= - +
( )2
22
2
2
2
1 00@1
3 2
1 1 0
1 1 5 1
4 3 12 3
dy x xdx
x
d y x x
dx
d y x
dx
y
= - ==
= -
= = >
= - + =
11,
3
Minimum point.
N1
K1 Integrate gradientfunction
K1 Substi tute ( 1,1)into equation y
N1
K1 Find2
2d y
dx
N1 N1
7
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5
N0. SOLUTION M ARK S5
(a)(i)
(ii)
(b)(i)
(ii)
11 55050
x x= => =
22
2
11 850
50(64 121)
9250
x
x
- =
=> = +=
New mean = 11 1.8 5 24.8 + =
New variance = 2(8 1.8) 207.36 =
K1 Use formula ofmean and/orstandard deviation
N1
N1
K1 Find new meanand/or new
varianceN1
N1
66
(a)
(b)
( )
5
3
5 1 3 104
1 33
34
l r
a
d
n
n
n
p
p
p
p p p
===+ - =
- ==
( ) ( )3434
2 5 33 32
S p p = +
1853
5821
58.21
cm
m
p ===
Total cost
4 58.21
232.8
233
RM
RM
RM
= ==
P1 Value of a and/or d
K1 Use T n = 104 p
N1
K1 Find S 34
N1
K1 RM4 S34
N1
7
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6
N0. SOLUTION M ARKS7
(a)
(b)(i)
(ii)
l l m 111 +-=
x y
y1 1.04 0.83 0.71 0.50 0.45 0.33
x1
0.67 0.5 0.4 0.25 0.2 0.1
l 1
= y-intercept
l = 5
l m - = gradient
m = 6.25
P1
N1
N1
K1 for correct axesand scale
N1 for all pointsplotted corr ectly
N1 for l ine ofbest-fit
K1 for y-intercept
N1
K1 for gradient
N1
10
y1
x10.2
0
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7
N0. SOLUTION M ARKS
8(a)
(b)
(c)
( x 2 ) 2 = 4 xQ(3, 1 )
A = [ ]dx x x ---3
0
2)2()4(
= dx x x -3
0
2 )3(
=3
0
32
323
- x x
=29
Note : If use area of trapezium and dx y , give marks accordingly .
V = -2
0
4
)2( dx xp
=2
0
5
5)2(
- xp
= p
5
32
K1 Solve for xN1
K1 use
dx y y - )( 12K1 integrate
correctly
K1 correct limit
N1
K1 integrate
dx y 2p
K1 integratecorrectly
K1 correct limit
N1
10
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8
N0. SOLUTION M ARKS9(i)
(ii)
(b)(i)
(ii)
(c)
QS = QP + PS
= 2~
x + 3~
y
PR = PS + SR
=~
x + 2~
y
QT = m QS
= m ( 2~
x + 3~
y )
= 2 m~
x + 3 m~
y
PT = n PR
= n (~
x + 2~
y )
= n~
x + 2 n~
y
PQ = PT + TQ
2~
x = n~
x + 2 n~
y + 2 m~
x 3 m~
y
= ( n + 2 m)~
x + (2 n 3m)~
y
n + 2 m = 2
2n 3m = 0
m =74
n =76
K1 for using vectortriangle
N1
N1
N1
N1
K1 for substitut ing& grouping intocomponents
K1 for equatingcoefficientscorrectly
K1 for eliminating m or n
N1
N1
10
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9
N0. SOLUTION M ARKS10(a)
(b)
(c)
1
1cos
41
cos
4
POA
POA -
=
=
= 75.52 @75 31"o o
= 1.318 rad.
Arc PA = 20 ( 1.318 ) = 26.36
PQ 2 = 80 2 202
PQ = 77.46
Perimeter
= 60 + 26.36 + 77.46
= 163.82 cm
Area OPQ = ( ) ( )1 20 77.46 774.62
=
Area sector POA = ( ) ( )21 20 1.318 263.62
=
Area of shaded region
= 774.6 263.6
= 511 cm 2
K1 Use correctlytrigonometricratio
N1
K1 Use s r q =
K1 Use Pythagoras
Theorem
K1
N1
K1 Use formula12
A bh=
K1 Use formula
212
A r q =
K1
N1
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10
N0. SOLUTION MARKS11(a)(i)
(ii)
(b)(i)
(ii)
p = 0.4 q = 0.6 n = 6
P ( )4 X
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )4 2 5 1 6 06 6 64 5 6
4 5 6
0.4 0.6 0.4 0.6 0.4 0.6
0.13824 0.036864 0.004096
0.1792
P X P X P X
C C C
= = + = + =
= + += + +=
2
288npqs = =
( )( )288
12000.4 0.6
n = =
62m = 8s =
( )
( )( ) ( )
60 68
60 62 68 628 8
0.25 0.75
1 0.25 0.75
1 0.4013 0.2266
0.3721
P X
P Z
P Z
P Z P Z
< = - -=
( ) ( )68 0.75 0.22663P X P Z > = > =
0.22663 3000 = 679.89
= 679 @ 680
P1 Value of p and/or qAND p + q =1
K1 Use P(X = r)= n C r prqnr
N1
K1
N1
K1 Use Z =s
m - X
K1N1
K1
N1
10
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11
N0. SOLUTION M ARK S12(a)
(b)
(c)
8 ms -1
dva
dt
= =0
2 2t = 1
t = 1 2
8 + 2t t 2 = 0
(t 4 ) (t + 2) = 0
t = 4 t = 2 (not accepted)
s v dt v dt + 4 5
0 4
=t t
t t t t
+ - + + -
4 5 3 3 2 2
0 4
8 8 3 3
= ( ) + - - + + - - + -
64 125 64 32 16 0 40 25 32 16
3 3 3
= + -80 10 3 3
= 30 m
N1
K1
N1
K1
K1
N1(for t = 4 only)
K1
(for and
4 5
0 4
)
K1(for integration)
K1(for summation)
N1
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12
N0. SOLUTION M ARK S13(a)
(b)
(c)
(d)
. x= 150 1000 4
(or formula finding y /z)
x = RM 0.60
y = 120
z = RM 2.00
45o
( ) ( ) ( ) ( ) ( ) x x x x x I
+ + + += 150 30 110 90 120 75 150 120 120 45 360
=46800
60
= 130
P03 = ( )100
5000130
= RM 3846.2
/ . I x08 03 130 1 2 (or 130 + 130x0.2)
= 156
N1
N1
N1
P1
K1
N1
K1
N1
K1
N1
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13
N0. SOLUTION MARKS14(a)
(b)
(c)
(d)
40x + 20y 2000 or 2x + y 100
15x +30y 1200 or
x + 2y 80
y 3x
(20, 60)
(35, 30)
2x + y = 100 y = 3x
x + 2y = 80
y = 30
100
90
80
70
60
50
40
30
20
10
10080604020 9070503010x
y
At least one straight line is drawn correctly from inequalitiesinvolving x and y.
All the three straight lines are drawn correctly
Region is correctly shaded
35
Maximum point (20, 60)
Maximum profit = 20(20) + 30(60)
= RM 2200
N1
N1
N1
K1
N1
N1
N1
N1
K1
N1
10
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N0. SOLUTION M ARK S15(a)
(b)
(c)
11 2 = 9 2 + 7 2 2(9)(7)cos PQR
cosPQR =9
126
PQR = 85o 54
PSR = 180 o 85 o 54
= 94 o 6
'
sin sinPRS =0
94 6 6 11
PRS = 32 o 57
\ RPS = 180 o 94 o 6 32 o 57= 52 o 57
' 'sinsin
oo
RS = 11 52 57
94 6
RS = 8.802 cm
Area = ' '( )( )sin ( )( . )sinoo +1 1 9 7 85 54 6 8 802 94 6
2 2
= 31.42 + 26.34
= 57.76
K1
K1
N1
P1
K1
K1
N1
K1, K1(for using
area= absincand summation)
N1
2kk