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3472/1 [Lihat sebelahSULIT
PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUASEKOLAH MENENGAH
NEGERI KEDAH DARUL AMAN
PEPERIKSAAN PERCUBAAN SPM 2009
Kertas soalan ini mengandungi 17 halaman bercetak
For Examiner’s use only
Question Total Marks MarksObtained
1 22 43 34 35 36 47 48 39 310 311 312 413 314 315 216 317 418 419 320 221 422 323 324 325 4TOTAL 80
MATEMATIK TAMBAHANKertas 1Dua jam
JANGAN BUKA KERTAS SOALAN INISEHINGGA 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 inthe question paper.
5. Show your working. It may help you to get marks.
6. If you wish to change your answer, cross out the workthat you have done. Then write down the newanswer.
7. The diagrams in the questions provided are notdrawn to scale unless stated.
8. The marks allocated for each question and sub-partof a question are shown in brackets.
9. A list of formulae is provided on pages 2 to 3.
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 ofthe examination .
Name : ………………..……………
Form : ………………………..……
3472/1Additional MathematicsPaper 1Sept 20092 Hours
ADDITIONAL MATHEMATICS Paper 1
Two hours
3 to 4.
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SULIT 3472/1
3472/1 [ Lihat sebelah SULIT
3
The following formulae may be helpful in answering the questions. The symbols given are the onescommonly used.
ALGEBRA
12 4
2b b acx
a- ± -
=
2 am ´ an = a m + n
3 am ¸ an = a m - n
4 (am) n = a nm
5 log a mn = log a m + log a n
6 log a nm
= log a m − log a n
7 log a mn = n log a m
8 logab =ab
c
c
loglog
9 Tn = a + (n−1)d
10 Sn = ])1(2[2
dnan-+
11 Tn = ar n-1
12 Sn =rra
rra nn
--
=--
1)1(
1)1(
, (r ¹ 1)
13r
aS-
=¥ 1 , r <1
CALCULUS
1 y = uv ,dxduv
dxdvu
dxdy
+=
2vuy = , 2v
dxdvu
dxduv
dydx -
= ,
3dxdu
dudy
dxdy
´=
4 Area under a curve
= òb
a
y dx or
= òb
a
x dy
5 Volume generated
= òb
a
y 2p dx or
= òb
a
x 2p dy
5 A point dividing a segment of a line
( x,y) = ,21çèæ
++
nmmxnx
÷øö
++
nmmyny 21
6 Area of triangle
= )()(21
312312133221 1yxyxyxyxyxyx ++-++
1 Distance = 221
221 )()( yyxx -+-
2 Midpoint
(x , y) = çèæ +
221 xx
, ÷øö+
221 yy
3 22 yxr +=
42 2
ˆ xi yjrx y+
=+
GEOMETRY
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SULIT 3472/1
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4
STATISTIC
1 Arc length, s = rq
2 Area of sector , A = 212
r q
3 sin 2A + cos 2A = 1
4 sec2A = 1 + tan2A
5 cosec2 A = 1 + cot2 A
6 sin 2A = 2 sinA cosA
7 cos 2A = cos2A – sin2 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) =BABA
tantan1tantan
m
±
12C
cB
bA
asinsinsin
==
13 a2 = b2 + c2 − 2bc cosA
14 Area of triangle = Cabsin21
1 x =N
xå
2 x =åå
ffx
3 s =N
xxå - 2)( = 2
2
xN
x-å
4 s =å
å -
fxxf 2)(
= 22
xfxf
-åå
5 m = Cf
FNL
múúúú
û
ù
êêêê
ë
é -+ 2
1
6 1
0
100QIQ
= ´
71
11
wIwI
åå
=
8)!(
!rn
nPrn
-=
9!)!(
!rrn
nCrn
-=
10 P(AÈB) = P(A)+P(B) − P(AÇB)
11 P (X = r) = rnrr
n qpC - , p + q = 1
12 Mean µ = np
13 npq=s
14 z =sm-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 xxf -®- , find the value of
Diberi5
3:1 xxf -®- , cari nilai bagi
(a) )3(-f ,
(b) p if 7)( -=pf . [ 4 marks ][4 markah]
Answer/ Jawapan : (a) ……………………..
(b) ……………………... 4
2
2
1
Forexaminer’s
use only
Set A Set B
499
x 9 7
3
Diagram 1Rajah 1
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3. Given that function axxg +®2: and 9:2 -®bxxg .
Diberi fungsi axxg +®2: dan 9:2 -®bxxg .
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 += xy is a tangent to the curve kxy += 2 . Find the value of k .
Diberi garis lurus 14 += xy ialah tangen kepada lengkung kxy += 2 . Cari nilai k .
[ 3 marks][3 markah]
Answer/Jawapan : k =......…………………
Forexaminer’s
use only
3
4
3
3
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5.4)2(3 2 --= xy
Diagram above shows the graph of the function 4)2(3 2 --= xy . 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 --= xy . Q ialah titik minimum bagilengkung 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 =+- xx .Find the value ofDiberi a danb adalah punca bagi persamaan kuadratik 0792 =+- xx . Cari nilaibagi
(a) ba +(b) ab
(c) 22 ba + [ 4 marks ][4 markah]
Answer/Jawapan : (a).............................
(b)............................
(c).............................
3
5
4
6
Forexaminer’s
use only
x
y
P ·
· Q
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7. Solve the equation:Selesaikan persamaan:
xxx 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
Forexaminer’s
use only
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9. Given that 2loglog 93 =+ yx . Express y in terms of x.[ 3 marks ]
Diberi 2loglog 93 =+ yx . 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
Forexaminer’s
use only
3
9
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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 =+ ra 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 =+ ra dan hasil tambah hingga sebutan ketakterhinggaan ialah 32,cari nilai a dan r .
[4 markah]
Answer/Jawapan:a=.….………..….......
r=...............................4
12
Forexaminer’s
use only
3
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, carikan 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 -= xy Find the value of h and of k. Pembolehubah x dan y dihubungkait dengan persamaan 2310 -= xy . Cari nilai h dan k.
[3 marks][3 markah]
Answer/Jawapan : h=………………
k=….………..….
3
13
3
14
Forexaminer’s
use only
),0( k·
)2,(h·y10log
x
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15.
In the diagram above, jniOA += 8 and jiAB 23 -= . Given
unitsOA 10= , find
Dalam rajah di atas, jniOA += 8 dan jiAB 23 -= . Diberi
unitOA 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 jipa 2+= and jib --= 2 , find the value of p if ba - is parallel
to j . [ 3 marks ]
Diberi jipa 2+= dan jib --= 2 , cari nilai p jika ba - selari dengan j .
[3 markah]
Answer/Jawapan :………………………..
2
15
Forexaminer’s
use only
3
16
y
xO
A
B
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17. Solve the equation 0cotcos2 =+ xx 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 p32
radian and the area of the shaded
region is 212 cmp . Find the length of the minor arc PQ.
Diberi sudut minor POQ ialah p32
radian dan luas sektor berlorek
ialah 212 cmp . Cari panjang lengkok minor PQ.[4 marks]
[4 markah]
Answer/Jawapan: …………………
_
p32_1O
P
4
17
4
18
Forexaminer’s
use only
Q
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19. Find the gradient of the curve25
234 2 -+=x
xy at the point )3,1( .
[ 3 marks ]
Cari kecerunan kepada lengkung25
234 2 -+=x
xy pada titik )3,1( .
[3 markah]
Answer/Jawapan:………………………
20. Differentiate1214 2
+-
xx
with respect to x .
[2 marks]
Bezakan1214 2
+-
xx
terhadap x.
[2 markah]
Answer/Jawapan: …...…………..….......
3
19
2
20
Forexaminer’s
use only
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SULIT 15 3472/1
3472/1 [ Lihat sebelah SULIT
21. Given that the gradient function of a curve passing through the point (1, 2) is
2)12(3-x
+ 2x , 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-=
xy 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-=
xy dan x bertambah dengan kadar 2 unit sesaat, cari kadar
perubahan bagi y apabila21
=x . [3 markah]
Answer/Jawapan: …………………….
4
21
Forexaminer’s
use only
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, 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 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 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]
Answer/Jawapan: ………………………3
24
3
23
Forexaminer’s
use only
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25. A random variable X is normally distributed with mean 370 and standard deviation10. Find the value ofSatu 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( <XP .
[4 marks][4 markah]
Answer/Jawapan: (a)………………………
(b)………………………
END OF QUESTION PAPERKERTAS SOALAN TAMAT
4
25
j2kk
SULIT3472/2Additional MathematicsKertas 2September, 20092 jam 30 minit
PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUASEKOLAH MENENGAH
NEGERI KEDAH DARUL AMAN
PEPERIKSAAN PERCUBAAN SPM 2009
ADDITIONAL MATHEMATICSKertas 2
Dua jam tiga puluh minit
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 questionsfrom 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
SULIT
j2kk
SULIT
3472/2 [Lihat sebelahSULIT
2
The following formulae may be helpful in answering the questions. The symbols given are the onescommonly used.
ALGEBRA
1. x =a
acbb2
42 -±- 8.abb
c
ca log
loglog =
2. aaa nmnm +=´ 9. dnaT n )1( -+=
3. aaa nmnm -=¸ 10. ])1(2[2
dnanS n -+=
4. aa mnnm =)( 11. 1-= nn arT
5. nmmn aaa logloglog +=12.
rra
rraS
nnn -
-=
--
=1
)1(1
)1( , r ≠ 1
6. log log loga a am m nn= - 13.
raS-
=¥ 1 , r < 1
7. mnm an
a loglog =
CALCULUS
1. y = uv,dxduv
dxdvu
dxdy
+=4 Area under a curve = ò
ba dxy or
= òba dyx
2. y =vu ,
2vdxdvu
dxduv
dxdy -
=
5. Volume of revolution = ò
ba dxy 2p or
= òba dyx2p
3.dxdu
dudy
dxdy
´=
GEOMETRY
1. Distance = 212
212 )()( yyxx -+- 4. Area of triangle
= 1 2 2 3 3 1 2 1 3 2 1 3
1( ) ( )
2x y x y x y x y x y x y+ + - + +
2. Mid point
( x , y ) = ÷øö
çèæ ++
2,
22121 yyxx
5. 22 yxr +=
3. Division of line segment by a point
( x , y ) = ÷÷ø
öççè
æ++
++
nmmyny
nmmxnx 2121 ,
6.2 2
ˆxi yj
rx y
+=
+% %
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SULIT
3472/2 [Lihat sebelahSULIT
3
STATISTICS
1.N
xx å= 7åå=
i
ii
WIWI
2.åå=
ffxx 8
)!(!rn
nPrn
-=
3.N
xxå -=
2)(s = 2
2x
Nx
-å 9!)!(
!rrn
nCrn
-=
4.å
å -=
fxxf 2)(
s = 22
xf
fx-
åå 10 P(AÈB) = P(A) + P(B) – P(AÇB)
11 P ( X = r ) = rnrr
n qpC - , p + q = 1
5. m = L + Cf
FN
m ÷÷÷
ø
ö
ççç
è
æ -21 12 Mean , m = np
13 npq=s
6. 1000
1 ´=QQI 14 Z =
sm-X
TRIGONOMETRY
1. Arc length, s = rq 8. sin ( A ± B ) = sin A cos B ± cos A sin B
2. Area of sector, A = q221 r 9. cos ( A ± B ) = cos A cos Bm sin A sin B
3. sin ² A + cos² A = 110 tan ( A ± B ) =
BABA
tantan1tantan
m
±
4. sec ² A = 1 + tan ² A11 tan 2A =
AA2tan1
tan2-
5. cosec ² A = 1 + cot ² A12
Cc
Bb
Aa
sinsinsin==
6. sin 2A = 2sin A cos A 13 a² = b² + c² – 2bc cos A7. cos 2A = cos ² A – sin ² A = 2 cos ² A – 1 = 1 – 2 sin ² A
14 Area of triangle = 1 sin2
ab C
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SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
4
Section ABahagian A[ 40 marks ]
[ 40 markah ]
Answer all questions.Jawab semua soalan.
1. Solve the simultaneous equations 3 2 0x y- - = and ( )2 1 3x y - = .Give your answers correct to three decimal places.
[5 marks]Selesaikan persamaan serentak 3 2 0x y- - = dan ( )2 1 3x 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-x bagi 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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SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
5
3. (a) Sketch the graph of 2 sin 2y x= - for 0 2x p£ £ .
(b) Hence, using the same axes, sketch a suitable straight line to find
the number of solutions for the equation sin 22x xp= for
0 2x p£ £ . State the number of solutions.
(a) Lakar graf bagi 2 sin 2y x= - untuk 0 2x p£ £ .
(b) Seterusnya, dengan menggunakan paksi yang sama, lakar satu garis lurus yang sesuai untuk mencari bilangan penyelesaian
bagi persamaan sin 22x xp= untuk 0 2x p£ £ .
Nyatakan bilangan penyelesaian itu.
[4 marks]
[3 marks]
[4 markah]
[3 markah]
4. Given that ( )2 1x 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 determine whether the turning point is a maximum or a minimum point.
[1 mark]
[3 marks]
[3 marks]
Diberi ( )2 1x 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 tentukan sama ada titik pusingan itu adalah titik maksimum atau titik minimum.
[1 markah]
[3 markah]
[3 markah]
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3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
6
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) xå ,
(ii) 2xå
(b) If each of the numbers is multiplied by 1.8 and then increased 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) xå ,
(ii) 2xå
(b) Jika setiap nombor didarab dengan 1.8 dan ditambah dengan 5, cari nilai yang baru untuk
(i) min , (ii) varians .
[3 markah]
[3 markah]
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SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
7
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 number of 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 membentukbeberapa semi bulatan. Rajah menunjukkan empat semi bulatanpertama 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]
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
8
Section BBahagian 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=+yxlm , where m and l are constants.
(a) Ploty1 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=+yxlm , dengan keadaan m dan l
adalah pemalar
(a) Ploty1 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 10.0y 0.96 1.2 1.4 2.0 2.2 3.0
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
9
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 π , when the shaded region
B is revolved through 360o 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 π , apabila rantau berlorek Bdikisarkan melalui 360o pada paksi-x.
[2 markah][4 markah]
[4 markah]
O x
y = ( x – 2 ) 2P
QAB x + y = 4
y
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
10
9.
In the diagram, PQRS is a quadrilateral. The diagonals PR and QSintersect 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-pepenjuruPR 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 n ialah 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 dan nilai n. [10 markah]
P
T
S
R
Q
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
11
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 andOA : AB = 1 : 3.Calculate(a) Ð POA in radian,(b) the perimeter, in cm, of the shaded region,(c) the area, in cm2 , of the shaded region.
[3 marks]
[3 marks]
[4 marks]
Rajah menunjukkan keratan rentas sebuah penggelek berbentuksilinder dengan pusat O dan jejari 20 cm yang terletak di ataslantai mengufuk PQ. OAB ialah garis lurus yang mewakilipemegang penggelek itu dan OA : AB = 1 : 3.Hitungkan
(a) Ð POA dalam radian,(b) perimeter , dalam cm, kawasan berlorek,(c) luas, dalam cm2 , kawasan berlorek.
[3 markah]
[3 markah]
[4 markah]
QP
OA
B
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
12
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 the farm 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 4 orang murid adalah aktif dalam aktiviti kokurikulum.
(ii) Jika varians murid-murid yang mengambil bahagian secara aktif dalam aktiviti kokurikulum ialah 288, hitungkan bilangan murid 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 telurdengan jisim melebihi 68 g dikategorikan sebagai telur gred‘double-A’
(i) Cari kebarangkalian bahawa sebiji telur yang dipilih secara rawak dari ladang itu mempunyai jisim di antara 60 g dan 68 g.
(ii) Jika ladang itu menghasilkan 3000 biji telur setiap hari, hitungkan bilangan telur yang mempunyai gred ‘double-A’.
[5 markah]
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
13
Section CBahagian 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 – t2, 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 – t2, 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]
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
14
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 150B 1.50 1.65 110C 4.00 4.80 yD 3.00 4.50 150E 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 thesecakes in the year 2006 based on the year 2003.
(c) The total expenditure on the items in the year 2006 is RM 5000. Calculate the corresponding total expenditure in 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]
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
15
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]
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
16
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 Sand 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 three times the number of component S produced.
(a) Write three inequalities, other than x ³ 0 and y ³ 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 ofcomponent T is RM30.
Time taken (minutes)Masa diambil (minit)
Component
Komponen Machine PMesin P
Machine QMesin Q
S 40 15
T 20 30
[3 marks]
[3 marks]
[4 marks]
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
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 tiga kali ganda bilangan komponen S yang dihasilkan.
(a) Tuliskan tiga ketaksamaan, selain x ³ 0 dan y ³ 0, yangmemenuhi semua kekangan di atas.
(b) Menggunakan skala 2 cm kepada 10 unit pada kedua-duapaksi, 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 untukmenghasilkan 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]
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
18
15. The diagram shows a cyclic quadrilateral PQRS with PQ = 9 cm,PR = 11 cm and QR = 7 cm.
(a) Find Ð 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]
j2kk
SULIT September, 2009
3472/2 Additional Mathematics Paper 2 [Lihat sebelahSULIT
19
Rajah menunjukkan satu sisiempat kitaran PQRS denganPQ = 9 cm, PR = 11 cm, dan QR = 7 cm.
(a) Cari Ð PQR.
(b) Diberi PS = 6 cm, cari panjang RS.
(c) Hitungkan luas bagi PQRS.
[3 markah]
[4 markah]
[3 markah]
END OF QUESTION PAPERKERTAS SOALAN TAMAT
j2kk
SULIT JPNKd/2006/3472/1
Additional Mathematics paper 1
Nama Pelajar : ………………………………… Tingkatan 5 : …………………….3472/1AdditionalMathematicsPaper1September 2009
PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUASEKOLAH MENENGAH
NEGERI KEDAH DARUL AMAN
PEPERIKSAAN PERCUBAAN SPM 2009
ADDITIONAL MATHEMATICSMARKING SCHEME
Paper 1
.
SULIT 3472/1
j2kk
SULIT 3472/1 2
SPM Trial Examination 2009 Kedah Darul Aman Marking Scheme
Additional Mathematics Paper 1
Question Solution/ Marking Scheme Answer Marks
1
(a) 3 (b) 81
1 1
2
(a) B1: 35
3
x or xxf 53)(
(b) B1: 753 p or p
5
)7(3
(a) 18 (b) 2
2
2
3 B2: 934 aorb
B1: aax )2(2
a = −3 b = 4
3
4
B2: 410)1)(1(4)4( 2 kork
B1 : or 0142 kxx 142 xkx
5
3
5
B2: 620
)4(8
orm
B1: )4,2()8,0( QorP
86 xy
3
6
(c) B1: )7(292
(a) 9 (b) 7
(c) 67
1 1 2
7 B3: xxx 1033
B2 : xx or 1033 22
B1: xxx 5213 )2()2(2
x = 2
1
4
3472/1 Additional Mathematics Paper 1 SULIT
j2kk
SULIT 3472/1 3
Question Solution/ Marking Scheme Answer Marks
8 B2: 22 yx
B1: 87
119752
yx
12,10 yx
13,9 yx
3
9 B2: 2log4log 29
23 yxoryx
B1: 2log3log
log2
9log
loglog 9
9
9
3
33 y
xor
yx
2
81
xy
3
10 B2: 513 danda
B1: 3710125 daorda
62
3
11
(a) B1: 3
1
36
36
p
(a) 48
(b) 4
2 1
12 B3: 48
2
1 aorr
B2: 32
961
096)1(32
aa
orrr
B1: 321
r
a
2
1,48 ra
4
13
B2 : 44
374
4
23
yand
x
B1 : 44
374
4
23
yor
x
( −6, 3)
3
14
B2 : 23230
22
horh
kork
B1 : 23log10 xy
h = 3
4
k = −2
3
15
(a) 6
(b) (11, 4)
1 1
3472/1 Additional Mathematics Paper 1 SULIT
j2kk
SULIT 3472/1 4
Question Solution/ Marking Scheme Answer Marks
16 B2: 02 p
B1: jip 3)2(
p = −2
3
17 B3:
)42(330,210,270,90 0000 ofoutany
B2 : )(2
1sin,0cos bothxx
B1 : 0sin
coscos2
x
xx
0
0
00
330
,270
,210,90
4
18 B3 : )
3
2(6 S
B2 : 6r
B1: 12)3
2(
2
1 2 r
12.57 cm ( 4 ) cm
4
19 B2:
2)1(2
3)1(8
B1: 22
38
xx
dx
dy
2
13
3
20 B1: 12 xy
or
2
2
)12(
)14(2)8)(12(
x
xxx
dx
dy
2
2
21
B3 : c
12
32
B2: 2
2
2
)12(3 21 xand
x
B1:2
2
2
)12(3 21 xor
x
2
5
)12(2
3 2
x
xy
4
3472/1 Additional Mathematics Paper 1 SULIT
j2kk
SULIT 3472/1
3472/1 Additional Mathematics Paper 1 SULIT
5
Question Solution/ Marking Scheme Answer Mark
22
B2 : 210
2)32(5 4
x
dt
dy
B1 : 210
2)32(5 4
dt
dxor
x
dx
dy
32 units per
second
3
23 (b) B1: 41
15 )82.0()18.0(C (a) 0.18
(b) 0.4069
1 2
24 B2: 34
13 PP
B1: 34
13 PorP
72
3
25 (a) B1 :
10
370355
(b) B1 : 10
370367
(a) −1.5
(b) 0.3821
2 2
END OF MARKING SCHEME
j2kk
SULIT JPNKd/2006/3472/2
Nama Pelajar : ………………………………… Tingkatan 5 : ……………………. 3472/2 Additional Mathematics
September 2009
PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUA SEKOLAH MENENGAH
NEGERI KEDAH DARUL AMAN
PEPERIKSAAN PERCUBAAN SPM 2009 ADDITIONAL MATHEMATICS
Paper 2
.
MARKING SCHEME SULIT 3472/2
j2kk
2
MARKING SCHEMEADDITIONAL MATHEMATICS PAPER 2
SPM TRIAL EXAMINATION 2009
N0. SOLUTION MARKS1
2
2
2
3 23 2 (3 2 1)
0 6 6 32 2 1 0
2 ( 2) 4(2)( 1)2(2)
1.366 0.3663(1.366) 2 3( 0.366) 2
2.098 3.098
y xx x
x xx x
x
x or xy 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 xP h hhQ k kk
= +- Þ - = +
= -
Þ = +
=
OR
3 3 2 30
3 2 114 0
hh
k k
+= Þ = -
--
= Þ =-
1 2122
11 14 2
1: 132
0 26
m m
yx
QR y x
y x
= Þ = -
-= -
-
= - +
= Þ =
OR
11 0 14 2
111 22
26
x
x
x
-= -
-
= - +
=
x-intercept = 26
Area3 4 26 313 11 0 32
1 33 78 12 28621 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
j2kk
3
N0. SOLUTION MARKS3
(a)
(b) 2 sin 2 22xxp
- = -
or
22xyp
= -
Draw the straight line 22xyp
= -
Number of solutions = 4 .
P1 Negative sineshape correct.
P1 Amplitude = 1[ Maximum = 3and Minimum =1 ]
P1 Two full cycle in0 £ x £ 2p
P1 Shift up the graph
N1 For equation
K1 Sketch thestraight line
N1
7
3
1 22xyp
= -
y
2p 2p xO p
2 sin 2y x= -
2
23p
j2kk
4
N0. SOLUTION MARKS4
(a)
(b)
(c)
( ) ( )21 1 1 2dydx
= - - - = -
3 2
4 3
4 3
( )
4 31 114 35
125
4 3 12
y x x dx
x xy c
c
c
x xy
= -
= - +
= + +
=
= - +
ò
( )2
22
2
2
2
1 0
0@1
3 2
1 1 0
1 1 5 14 3 12 3
dy x xdxx
d y x xdx
d yxdx
y
= - =
=
= -
= Þ = >
= - + =
11,3
æ öç ÷è ø
Minimum point.
N1
K1 Integrate gradientfunction
K1 Substitute (-1,1)into equation y
N1
K1 Find2
2d ydx
N1 N1
7
j2kk
5
N0. SOLUTION MARKS5
(a)(i)
(ii)
(b)(i)
(ii)
11 55050
xx= => =å å
22
2
11 850
50(64 121)9250
x
x
- =
=> = +
=
å
å
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 newvariance
N1
N1
66
(a)
(b)
( )
53
5 1 3 1041 3334
l rad
nnn
ppp
p p p
===
+ - =
- ==
( ) ( )3434 2 5 33 32
S p pé ù= +ë û
1853582158.21
cmm
p===
Total cost4 58.21232.8233
RMRMRM
= ´==
P1 Value of a and/or d
K1 Use Tn = 104p
N1
K1 Find S34
N1
K1 RM4 ´ S34
N1
7
j2kk
6
N0. SOLUTION MARKS7
(a)
(b)(i)
(ii)
llm 111
+´-=xy
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
l1 = y-intercept
l = 5
lm
- = gradient
m = – 6.25
P1
N1
N1
K1 for correct axes and scale
N1 for all points plotted correctly
N1 for line of best-fit
K1 for y-intercept
N1
K1 for gradient
N1
10
y1
x10.2
0
j2kk
7
N0. SOLUTION MARKS
8(a)
(b)
(c)
( x – 2 )2 = 4 – xQ(3, 1 )
A = [ ]dxxxò ---3
0
2)2()4(
= dxxxò -3
0
2 )3(
=3
0
32
323
úû
ùêë
é-
xx
=29
Note : If use area of trapezium and ò dxy , give marks accordingly.
V = ò -2
0
4)2( dxxp
=2
0
5
5)2(úû
ùêë
é -xp
= p5
32
K1 Solve for xN1
K1 use
dxyyò - )( 12
K1 integrate correctly
K1 correct limit
N1
K1 integrate
ò dxy 2p
K1 integrate correctlyK1 correct limit
N1
10
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8
N0. SOLUTION MARKS9(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 )
= – 2m~x + 3m
~y
PT = n PR
= n (~x + 2
~y )
= n~x + 2n
~y
PQ = PT + TQ
2~x = n
~x + 2n
~y + 2m
~x – 3m
~y
= (n + 2m)~x + (2n – 3m)
~y
n + 2m = 2
2n – 3m = 0
m =74
n =76
K1 for using vector triangleN1
N1
N1
N1
K1 for substituting & grouping into components
K1 for equating coefficients correctly
K1 for eliminatingm or n
N1
N1
10
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9
N0. SOLUTION MARKS10(a)
(b)
(c)
1
1cos4
1cos4
POA
POA -
Ð =
Ð =
= 75.52 @75 31"o o
= 1.318 rad.
Arc PA = 20 ( 1.318 ) = 26.36
PQ2 = 802 - 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 cm2
K1 Use correctlytrigonometricratio
N1
K1 Use s rq=
K1 Use PythagorasTheorem
K1
N1
K1 Use formula12
A bh=
K1 Use formula21
2A r q=
K1
N1
10
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10
N0. SOLUTION MARKS11(a)(i)
(ii)
(b)(i)
(ii)
p = 0.4 q = 0.6 n = 6
P ( )4X ³
( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )4 2 5 1 6 06 6 6
4 5 6
4 5 6
0.4 0.6 0.4 0.6 0.4 0.60.13824 0.036864 0.0040960.1792
P X P X P X
C C C
= = + = + =
= + +
= + +=
2 288npqs = =
( )( )288 1200
0.4 0.6n = =
62m = 8s =
( )
( )( ) ( )
60 68
60 62 68 628 8
0.25 0.75
1 0.25 0.751 0.4013 0.22660.3721
P X
P Z
P Z
P Z P Z
< <
- -æ ö= < <ç ÷è ø
= - < <
= - > - >
= - -=
( ) ( )68 0.75 0.22663P X P Z> = > =
0.22663 ⅹ 3000 = 679.89
= 679 @ 680
P1 Value of p and/or qAND p + q =1
K1 Use P(X = r)= n Cr prqn–r
N1
K1
N1
K1 Use Z =s
m-X
K1N1
K1
N1
10
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11
N0. SOLUTION MARKS12(a)
(b)
(c)
8 ms-1
dvadt
= =0
2 – 2t = 1
t =12
8 + 2t – t2 = 0
(t – 4 ) (t + 2) = 0
t = 4 t = –2 (not accepted)
s vdt vdt= +ò ò4 5
0 4
= t tt t t té ù é ù
+ - + + -ê ú ê úê ú ê úë û ë û
4 53 32 2
0 4
8 83 3
= ( )é ù é ùæ ö æ ö æ ö+ - - + + - - + -ç ÷ ç ÷ ç ÷ê ú ê úè ø è ø è øë û ë û
64 125 6432 16 0 40 25 32 163 3 3
= + -80 103 3
= 30 m
N1
K1
N1
K1
K1
N1(for t = 4 only)
K1
(for andò ò4 5
0 4
)
K1(for integration)
K1(for summation)
N1
10
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12
N0. SOLUTION MARKS13(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 xI + + + +=
150 30 110 90 120 75 150 120 120 45360
= 46800360
= 130
P03 = ( )100 5000130
= RM 3846.2
/ .I x=08 03 130 1 2 (or 130 + 130x0.2)
= 156
N1
N1
N1
P1
K1
N1
K1
N1
K1
N1
10
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13
N0. SOLUTION MARKS14(a)
(b)
(c)
(d)
40x + 20y £ 2000 or2x + y £ 100
15x +30y ³ 1200 orx + 2y ³ 80
y £ 3x
(20, 60)
(35, 30)
2x + y = 100y = 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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14
N0. SOLUTION MARKS15(a)
(b)
(c)
112 = 92 + 72 – 2(9)(7)cosÐPQR
cosÐPQR = 9126
ÐPQR = 85o 54’
ÐPSR = 180o – 85o 54’
= 94o 6’
'sin sinPRSÐ
=094 6
6 11
ÐPRS = 32o 57’
\ ÐRPS = 180o – 94o 6’ – 32o 57’ = 52o 57’
' 'sin sinoo
RS=
1152 57 94 6
RS = 8.802 cm
Area = ' '( )( )sin ( )( . )sinoo +
1 19 7 85 54 6 8 802 94 62 2
= 31.42 + 26.34
= 57.76
K1
K1
N1
P1
K1
K1
N1
K1, K1(for usingarea= ½absincand summation)
N1
10
END OF MARKING SCHEME
j2kk