SULIT 3472t1
JABATAN PELAJARAN NEGERI JOHOR
PEPERIKSAAN PERCTIBAAN SPM 24flADDITIONAT MATHEMATICSKertas 1September2Jam
NAMA:
trGLAS:
34721r
Duaiam
2.
1
JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU
l. Tulis nama dan kelas anda pada petakyang disediakan.
Kertas soalan ini adalah dalam dwibahasa.
Soalan dalam bahasa inggeris mendahuluisoalan yang sepadan dalam bahasa Melayu.
Calon dibenarkan menjawab keseluruhanatau sebahagian soalan sama ada dalambahasa Inggeris atau bahasa Melayu,
Calon dikehendaki membaca maklumat dihalaman belakang kertas soalan .
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Soalan Markah Penuh Markah Diperolehi1 z7 23 44 )
a
6 5
7 38 49 210 J
11 412 J
13 2l4 4l5 A
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16 4t7 4l8 J
t9 J
20 22l J
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23 a
24 425 4
Jumlah 80
Kertas soalan ini mengandungi 19 halaman bercetak
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ST'LIT
/-l3F-
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4
Answer all questions.Jawab semuo soalai.
In Diagram l, set p shows the images of certain elements of set p.
Dalam Rajah 1, set Q menunjukkan imej bagi unsur-ansur tertentu dari set p.
DiagramRajah I
3472t1
[2 marks]
[2 markah)
[2 marks]
12 markahl
f(x)
Set P Set p
(a) State the type of relation between set p and set p.
Nyatakanjenis hubungan antara set p dengan set e.(b) using the function notation, write a relation between set p and set e.
Dengan rnenggmakan tatatandafungsi, tulis satu hubungan antara set p dengan set e.
Answer/ Jawapan.
(a)
(b)
2 Given the function f (x) =llx-tl , nnd the values of s such that f(x)=5.
Diberifungsi .f (x) = l+"r-l l, cari nilai-nilai x dengan keadaan f (x)=5.
Answer/ JawaDan:
2
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STILIT 3472t1
3 Given the function s f (x7=4 and g(x)= pr* l, findJJ
Diberifungsi f(i=# dan g(x)= p**I, cari
(a) "f-t(s)(b) the value of p if the function f (3 - x) = g(x) .
nilai p jikafungsi f(3-x) =g(x).
Answer / Jawapan :
(a)
(b)
[4 marks]
[4 marlmh]
[3 marks]
[3 mokah
4 A quadratic equation 4x2 + p :3(2x- l) has no real roots. Find the range of values ofp.
Suatu persamaot kuadratik 4f + p - 3(2x - l) tidak mempunyai punca-punca nyatq.
Cari julat bagi nilai p.
Answer / Jawapan :
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SI]LIT 3472t1
5 Diagram 5 showsthegraphofaquadraticfunction f(*)=-6-b)2 +cwiththemaximum
point at B(3,5). The straight line AC is parallel to the x-axis.
Rajah 5 menunjukkan graf bagi suatu fungsi lcuadratik f (x) = - (, - b)2 + c dengan titik
malaimum B(3, 5). Garis lurus AC selari dengan paksi-x.
Diagram 5Rajah 5
-4)
6
[to
a) Find the value of fr.
Cari nilai bagi k.
b) Find the equation of the curve in the form y : * (x - b)2 + c.
Cari persamaan bagi lengkung dalan bentuk y = - (x - b)2 + c.
Answeil Jawapon:
a)
b)
Find the range ofvalues ofx for x(x -6)<27 .
Cari julat bagi nilai-nilai x unnk x(x - 6) < 27 .
Answer/ Jawapan ;
[3 marks]
[3 narkah]
[3 marks]
13 markahl
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STlLIT
7 Solve the equation
Selesaikan persamaan,)
gX = ___-_l6r-r
Answer / Jawapan :
3472t1
[3 marks]
13 markahl
[4 marks
[4 markahj
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Given that log2 m = p, express logro, 8m2 interms ofp.
Diberi bahawa log2 m = p , ungkapkan log16^8mz datam sebutan p.
Answer/ Janapan:
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8
The first term of a geometric progression isp antl the sixth term is 32 p6 ,Find the common ratio in terms ofp.
sehutan pertama satu.ianjang geometri ialah p dan sebutan keenamnya iatah 32 p6Carikan nishah sepunyanya dalam sebutan p.
Answer/ Jawapan:
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[2 marks]12 markahl
Given the second term and fifth term of an arithmetic progression is 2q + 3 and l\q - |
respectively and the common difference is 2. Find the value of q.
Diheri sebutan kedua dan sebutan kelima bagi.janjang aritmetik adalah 2q * 3 dan l2q - |
masing-masing dan nilai sepunya udalah 2. Curikcn nilai bagi q. [3 marks]
[3 markah]Answer/ .Jawapnn:
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I I Diagram I I (a) shows the curve y : 4x2 - 2.' Diagram I l(b) shows the straight line graph obtained wheny :4i *2 is expressed in the
l inearform Y= -2X+4.
Rajah t I (a) menunjukkan suatu graf lengkungan y = 4x2 - 2.
Rajah I 1(b) menuniukkan graf garis lurus yang diperoleh apabila y: 4xz - 2 diungkapkan
dalam bentuk linear Y : -2X + 4.
-Diagram l l(a)Ruiah ll(a)
'(a) Express X and Y in tefihs'bf x a*i y.
Ungkapkan nilai X dan Y dalan sehitqn x dan y.
(b) Find the value of /r.
Cari nilai hagi k.
Answer/ Jawapan:
(a)
[4 marks
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(b)
Dihgram I l(b)Rajah tl(b)
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STJLIT 3472tr
12 Diagram l2 shows a straight line PQ which is perpendicular to the straight line eR atpoint Q. The equation of the straight line pR is 2_y: r - 4.
Ra.iah l2 menuniukkan garis lurus PQ yang berserenJang dengan garis lurus QR parta titik ePersamaan garis htrus QR ialah 2y : , * 4.
l0For
examiner'suse onl-v-
iI
I
Find the coordinates of P.
Cari koordinat P.
Answer/ .Iaw,upan:
[3 marks]
[3 markuh]
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Diagram l2ktjah 12
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13 Diagram 13showstwovectors,dand E.Giu"n that l2Al : l*1.Raiah 13 menuniukktnduavektor, Of donE.Dibert bahawal*l: ldl.
Express 7E inthe form xi + yi.
Ungkapkan AB dalam bennk xi + yi.
Answer / Jawapan :
12
34721r
[2 markah
[4 marks]
14 markah
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Given O]=3i +4j , Oe :4i+5j and E6=Qk-3) i+3j .
Diber i oA:3/+4j , oc=4i+ 5j dan Bn:(2k_3) i+3j .
FindCari
(a) AC
(b) the value of fr if 7e and EEur" parallel.
nilai bagi kjikaft dan Bdadalah selari.
Answer/ Jawapan:
(a)
(b)
Diagram l3
Rajah 13
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ST]LIT t2 3472t1
[4 marks]
14 markahl
[4 marksJ
14 markahl
Solve the equation 2sin2 x=tan4S" -l for 0" Sx < 360".2
Selesaikan persamaan 2sin2 x=tan aS" -] for 0" <x < 360'.
Answer lJa,vapan:
15
Eo
16 Diagram l6 shows a semicircle ABC withcentre O andsector DBE withcentre D.
Rajah 16 menunjukkan semibulatan ABC berpusat o dan sehor DBE berpusat D.
Rajah 16
Rajah 16
Given OD:4 cm and the length of arc BE= 5.2 cm, AD = DO= OE: EC.
Diberi OD = 4 cm dan panjang lengkok BE = 5.2 cm, AD: DO = OE = EC.
CalculateHitung(a) the value of Q inradian.
nilai Q dalam radian.
(b) the area, in cm2, of the shaded region.
Iuas, dalam cm', kawasan berlorek,
luse / gunan:3.1421
Answer/ Jawapan:(a)
(b)
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STTI,IT t3 3472t1
l7 The gradient to the curve / = 2x' + x -l at point A is perpendicular to the straight line
5y: - x.
Kecerunan lengkung ! = 2x2 + x - | pada titik A adalah bersergnjang dengan garis 5y : - x.
Find/ Cari
a) the coordinate of Akoardinat A
b) the equation of the tangent to the curve at pointlpersamaan tangent kepada lengkung pada titik A.
Answeil Jawapan:
(a)
(b)
For
use only
18 Given that y : (4r+1)2and x:2r.Find *in
term ofx.
Diberi bahawa y=(4r+l)2dan x - 2r. Cari Q in rcrm ffx.dx
[3 marks][3 markahl
Answer / Jawapan :
18
r:lI lg lo
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ST]LIT 3472t1
R19 Given y =
x+1, express the approximate change'iny in terms of z, when x changes
from 3 to ( 3 -z ), where m is asmall value.
Diberi / =+, ungkapkan perubahan kecil bagi y dalam sebutan m apabilax
berubah daripada 3 to ( 3 - m ), dengan l<eadaan m ialah satu nilai yang lcecil.
l4For
examirer'suse only
Answer/ Jawapan :
Diagram 20 shows the curvey :-f(x).
Rajah20 menunjukkan garis lengkungy =f(x).
Diagram 20Rajah20
ydx=9, find the area of the shaded region.
J
I y dx=9 , cari luas kqwasan berlorek.0
Answet/ Janapan:
[3 marks][3 markah]
[2 marks]
[2 markah]
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STJLIT
I fJ0
Jika
20
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ST]LIT
2l . Given I f(x)dx=3 and I f@)dx =2. Findthevblue of frwhere &is aconstant
4
I(f t*> +zlw)dx = 7, where ft is a constant.I
34Diberi I f @) dx =3 and I f @) dx =2. Hitungkan nilai k di mana k adalah pemalar
l5 3472t1
[3 marks
[3 markah]
[3 marks
13 markah)
J
II
4iJ
if
I
Answer / Jawapan .
22 A set of positive integers consists of 6,7 , m, \, 8,3,3.
Suatu set nombor positif terdiri daripada 6, 7, m, l, 8, 3,3.
(a) Find the value of lrl ifthe mean of the data is 5.
Cari nilai bagi m jika min bagi data itu ialah 5.
(b) State the range of the values of z if the median of the datais m.
Nyatakan julat bagi nilai m jika median bagi data itu ialah m'
Answer/ Jawapan:
(a)
jika lU@) +2tac)dx = 7, jika & ialah pemalar.
(b)
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ST]LIT 3472t1
A Kenari car can accommodate I driver and 3 adults. Find the number of different waysthe selections can be made from 3 men and 4 women ifsebuah kereta Kenari boleh menempatkan seorang pemandu dan 3 orang dewasa. caribilangan cara berlainan pemilihan boleh dilakukan daripada3 orang lelaki dan 4 orangwanita jika,
(a) there is no restriction for the seating
tiada syarat dikenakan bagi tempat duduk itu(b) the driver must be a man.
pemandu adalah seorang lelaki
[Assume that all of them can drive]
[Andailran kesemua mereka boleh memanduf
Answer/ Jawapan:
(a)
(b)
l6
[4 marks]
14 markahl
24 The probability Bahari being chosen as a monitor is I
chosen is {. finO the probabiliry that5
Kebarangkalian bahawa Bahari dipitih sebagai ketua ialah
ADewi dipilih tahh a.Cari kebarangkalian bahawa
(a) neither of them is chosen as a monitor,
tidak seorang daripada mereka akan diplilih sebagai ketua,
(b) only one of them is chosen as a monitor.
hanya seorang daripada mereka dipilih sebagai ketua.
Answer/ Jav'apan:
(a)
(b)
while the probability Dewi being
f, man t"t, kebarangkalian
[4 marks][4 markah]
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25
347211
X is the mass of the nails produced in a factory and it is a continuous random variable of a
normal distribution with a mean of 52 gand standaf,d deviation of 10 g.
X merupakan jisim bagi paku yang dihasilkan di sebuah kilang dan jisim merupakan
pembolehubah rawak selanjar bagi tabutan normal dengan min 52 g dan sisihan piawai
1o g.
(a) Find the range of z - score if the mass of each nail must be between 53.259 and
54.25s..
Cari julat skor - z jika jisim bagi setiap paku mesti antara 53.259 dan 54.259.
Diagram 25 shows standard normal distribution graph of the mass of the nails.
The probabil i tyof k<z < 0 is 0.3485.
Rajah25 menunjukkan graf taburan normal bagi iisim paku. Kebarangkalian
bagi k4z <0 ialah 0.3485. f(f)
0
Diagram 12Rajah 12
Find the mass of a nail which has P ( z < k ).
Cari jisim bagi sebatang paku yang mempunyai P (z < k )
[4 marks]
[4 markah]
Answer/ Jawapctn:
(a)
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l7
(b)
(b)
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ST]LIT l8
END OF QUESTION PAPERTHE UPPER TAIL PROB4PILITY e(z) FORTHE NORMAL DrsrRrBUTroN N(0, l)KEBARANGKALIAN HT]TANG ATAS OhI BAGI T.ABIIRAN NNPM{I N/N I\
"f (z)
3472t1
Example lContoh:
If x - N(0, 1), then
JikaX -N(0, l ) ,
P(X> k)= 8&)
P(X>2.17=QQ.\)
makn
= 0.0179
3472t1
QQ) =| - , ,| | 1z) clz
k
,
[Lihat halaman sebelahSULIT
L
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SULIT
JABATAN PELAJARAN NECSRI JOHOR
PEPARIKSAAN PERCUBAAN SPHI 2OI!ADDITIONAL IVIATHEMATICS
3477t2
PAPAR 2Sept" 20ll
^1.z-t'nm2'
3412t2
Dua jam tiga puluh rninit
JANGAN BUKA KERTAS SOALAN INI S$TIINGGA DIBHRTTAHI.]
l. Kertcs soalan ini adalah tlalam dwibahasa.
2. Saalan dalam bahasa hggeris mendahalui soalan yang sepadan dalam hahasd tlBlayu.
3. Cslon dikehendaki membacu *oi,klo*o, cli halaman belakang kertas soalsn ini.
4. Calon dikehendaki meneerailutn halamanTl dan iknt sebagai muks hadapanhersuma-sama dengan jawapan anda-
Kertas soalan ini mengandungi 2l halaman bercetak dan I halaman tidak bercetak
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Thefollowingfarmulae uay be helpful in answering the qaestions.the oies eommonly used.
3{72t2
The symbols given are
2a
{ x ao= onin
{ * {a 6n'n
(fro - **n
logumn - l0g gn * log n
Iog 3 * logum - lopnn
log6mn = nlogum
Iog,# = $t"glog. (r
Tn=0+(n*l ld
s^: llza+(n*t:)d\
Tr: Qr n-l
g = a(r ' - l ) =o(!-r ' ) , r + lr - l l * r
s- =* , lrlot
*b+ r|bz -4ac
ALGEBRA
II
)
3
4
5
6
7
I
l0
il
t7
I3
CALCIJLUS
dv dv duV:l IV.3rU-+V-
dx dt d\
du dvr v- -u--*dY- dx dx
/ - - r T--*-* i - :vdxv'
dv dv du-'.,,1-- =--:-X-
dx du dn
5 Volurnebfr= ltw"t -
4
btt= llDC-Ja
generated
dx ar
dy
1
)
Distance=W
Midpoint(x.+x. y,+/r ' l(x,y)=l-*- ,
2 )l r l -$. f
^ xt+vI4 --E t t1
V;- + Y-
347212 g Hak CiptaJahatan Pelojaran Negeri.lohor
5 A point dividing a segment of a line
(+y)= {Y*m,\ l t+mtz\\ m+n m+n )
6 Area <lf triangle =
I r .1lt*rl,
* xtlt * xsh,) * (xry, * xtlz + xr!)l
ONOMATRY
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$ULTT t
STATISTICS
3472n
T.rxz tut
,ltI
; : IFZ,T
; -YI,w,' -T*oF- =
f,l'
' (n * 71t.
nn nl
' ' ' = (o*r)fr
P (A v B) *P (A) + F (B)-P (A n B)
P(X-r) = "C,P'q ' - ' , P*q =l
IVIean, F * np
o = J;pq
x- l tz=-6
3 c-
4o*
5m
6 I *&xW}Qo
f 1r-ol= L+12 lc
l f , , )
q
l0
11
t2
t3
14
Xr.' -zE7-'
TRIGONOMETRY
I Arc length, s * r0
2 hreaofsector, A= Lr'e2
3 sin 2A + coszA * l
4 set?A: | +imnzA
5 cosecz A= | + cs{ A
6 sin 2l * 2 siM cosl
7 cos 24: *a*A-sinz A :2 cos2l - |* l- 2 sinzA
t tan2g* ztu!-l-tanr A
347212 @ Hak Cipta Jabetan Pelajaran Negeri JohorSULIT
sin (l tB)= sinl cos.B + cos.4 sin"B
cos (l * B) = aa$ cos3 F siM sin3
I l tan(A*ts) .=tanA*tanB
lTtan AtarNI]
abci - t = -'k s in l s inB sinC
13 a2* b?+c2 -2btccsA
14 Area of triangle : labsinC2
I
IO
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SULIT4
Section'dBahagian A,
[40 narks]
140 markaltl
Answer all questions in this section
Jawab semua soalan.
Solve the following simultaneous equations :
Selesaikan pers amaan serentak b erikut :
y- x=3
y' + x-2ry =7
Giventhe quadratic function ! = 3 - 8x- 2xz = -2(x + b)' + c.
Diberi fungsi kuadratik ! = 3- 8r- 2)cz = -2{x + bl + c.
a) Find the values of b and c.
Tentukan nilai bagi b dan c.
Sketchthe graph ofthe quadratic functiony =3-8x-2xz .
Lakarkan graffungsi kuadratik y =3-8x-2xz .
write the function of the saph if the graph in b) is reflected on the x-ru<is.
Tulis persamaan bagi grafjilca graf dalam b) dipanfi:lkan pada paksi-r.
b)
c)
3472t2
[5 marks]
[5 markahl
[3 marks]
p marlcahl
[3 marks]
p markahl
[l mark]
} markahl
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SULIT 3472/2
Diagram 3 shows fourthe semicircles AH, BG, CF andDEwith center O
Rajah 3 menunjuklcan empat semibulatan AE, BF, CG and DH dengan pusat O.
Diagram 3Rajah3
(a) Given OE :3 cm. Show that the length of the circumference of the semicircles form a
arithmetic progression. Hence state the corlmon difference in term of a
[3 marks]
Diberi OE --3 cm. Tunjukkan bahawa panjang lilitan bagi semibulatan membentuk
janjang aritmetik Seterusnya nyatakan bua sepunyanya dalam sebutan r .
&) (i) Find the sum of the length of the circumference of the semicircles
Carikan jumlah semua panjang lilitan semibulatan tersebut
(ii) Calculate in which term that the circumference of the semicircle is 2ln
Kira pada semibulatan yang ke berapakah panjang lilitannya ialah 2ln
347212 O Hak Cipta Jabatan Pelajaran Negeri Johor
13 markahl
[2 marks]
[2 markah]
[2 marks]
12 markahl
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4 (a) Sketchthegraphof y=2cos2x*2 for 0(x(n
Lakarkan graf ! =2cos2x*2 untuk 0 < xS n
(b) Hence, usingthe same axes? draw a suitable straight line to find the number of
solutions to the equation cos2r = L -l for 0 1 x 3 n.IE
Seterusnya dengan menggunakan paksi-paksi yang sama, lukiskan garis lurus yang
sesuai untuk mencari bilangan penyelesaian bagi persamaan cos2x = L -lTE
bagini la i 0(x(n.
SULIT
State the number of solutions.
Ny atakan b i lan gan p enyele s ai an i tu.
347212 O Hak Cipta Jabatan Pelajaran Negeri Johor
3472t2
[ 4 marks ]
14 markahl
[3 marks ]
[3 markah]
5. Table 5 shows the frequency distribution of the marks of group ofpupils in an Additional
Mathematics test.
Jadual 5 menunjukkan taburan kekerapan markah bagi selatmpulan murid dalam satu
uj i an )rI at emati k T amb ah an.
Marks
Mwka.h
,Number of pupils
Bitqr,gan mufid.
40-49 9
50-59 24
60-59 m
70 -79 9
80-89 5
90 -99
Table 5Jadual 5
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SULIT
(a)
In Diagram 6, ABC is a triangle.
Dalam Rajah 6, ABC ialah sebuah segitiga.
(b)
7
It is given that the median mark of the distribution is 58,25.Calculate thevalue of m.
Diberi bahawa markah median bagi taburan itu ialah 58.25.Hitung nilai bagi m
Calculate the variance of the marks
Hitung varian bagi markah itu
3472/2
[3 marks]
[3 markah)
[3 marks]
[3 markah]
It is given that
Diberi bahawa
dE:b,
OB:b,
OA: a,
OA: a,
1'
Diagram 6 Rajah 6
I - a-
ox =:oB and AC=aAB .23
1- . -
OX =-OB and AC=aAB23
a) Express in term of a and b
(i) oc
GDfr [3 marks]
[3 markah]
b) (i) Given thatE =k&.Express Nintermofk, aandb [2marks]
Diberi bahawa fr = kfr .Nyatakan d doh* sebutan k, a dan b. 12 markahl
(ii) If O, Y and C are collinear, find the value of k [3 marks]
Jika O, Y dan C adalah segaris, tentukan nilai bagi k. f3 markahl
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lt
SULIT 3472t2
Section BBahagianB
[40 narks]
[4A markah]
Answer four questions from this section.Jawab empat soalan daripada bahagian ini.
Used the gaph paper to answer this question.
Gunakan leertas graf untuk menjawab saalan ini.
Table 7 shows the value of two variables, x and y,obtained from an experiment.
Variables x and y arc related bythe equation 2y - a- !, *hu" a andb are constants.x
Jadual T menunjukkan nilai-nilai bagi daa pembolehubah x dan y yang diperoleh daripadasatu eksperimen. Pembo,lehubah x dan y dihubanghbn aleh persamean 2y - o = ! , d"rgon
keadaan a dan b ialah pemalar.
7 cmto 2 units onthe
Ptot xy melawan x, dengan menggunakan skala 2 cm hepada 1 unit di paksi-x dan Z cmkepada 2 unit di paksi-xy. seterusnya lukis garis lurus peaywaion terb;ih
(b) use your graph from 7(a) to find the value of
t5 markahl
Gunakan graf daripadaT(a) bagi menghitung nilai
(Da
(ii) b
( r i i ) ywhenx:4.5
y apabila x: 4.5[5 marks]
[5 markah)
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....'....-'--
il
Tahle 7JadualT
(a) Plot xy againstx, using a scale of 2 cm to 1 unit on the x-axis andry,"axis. Hence draw the line of best fit.
i l it l
=
:-
=
=
=
;5
E,F*dh"*
)c t 2 3 4 5 6
v 5 3.5 3.1 2.7 2.6 2.5
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SULIT s472n
[3 marks ]
13 marlcahl
[ 4marks ]
14 narkahl
I
I
Diagram 8 shows the point 8 which is the point of intersection between curve! = -x2 +8 and straight line AB. Given,4B is a tangent to the curve. The gradient of
the normal to the curve at point B is +
Raj,ah 8 menunjukkan titik B yang merupalcan titik persilangan antara y = -x2 +8dengan garis lurus AB. Diberi AB adalah tangen kepada lenglang . Kecerunan normal
tenglwng pada ritikB adalah 14
Diagrarn 8Rajah 8
FindCari
the equationof AB
persamaanAB
the area of shaded regionP
luas kawasan rdntaa berlorek P
a)
b)
iilirirti
li
l i i
c) the volume generated when the region which are bounded by the,curve, y-axis and y : 4i,q revolved 360o about the y-axis. [3 marks ]
tsipadu yang terjana apabila kawasan ltang dilenglcungi oleh lengkung,paksi y dangaris 1t : 4 diputarkan 360 o di paksi-y. [3 marlmhl
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SULIT 3412t2
9 Solution by scale drawing is not accepted.
Penyelesaian secara lukisan berskala tidak diterima.
Diagram 9 shows a triangle ABC where O is the origin. Point D lies on the straight hne AB.
Rajah 9 menunjukkan segitiga ABC di mana O ialah asalan . Titik D terletak pada garisIurus AB.
10
Diagrarn 9Rajah9
Given thatAD: DB --3 :2,fnd coordinate of.B.
Diberi bahawa AD : DB : 3 : Z cari koordinat bagi B.
Calculate the area, in unit2, oftrianglel-BC.
Hitung luas, dalam unif , segi tiga ABC.
GivenAP:PC:L::2.
Diberi AP : PC: I :2.
i) Find the equation of the locus ofP.
Cari persamadn bagi lolats P.
ii) Determine whether locus P intercepts the x-axis.
Tentukan sama ada lolan P memintas paksi-x atau tidak.
3412n O Hak Cipta Jabatan PelajaranNegeri Johor
a)
b)
c)
[2 marks]
12 mtarlcahl
{2 marksl
[2 markah]
[6 marks]
15 markahl
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,rSULIT 347212
10 Diagram 10 shows arectangle ABCD.
OAED is a sector of a circle with centre O and radius 4 cm. It is given that O is the midpoint
of AC and ZAOD: L.222 radian.
Rajah l0 menuniuklcan sebuah segi empat tepat ABCD'
OAED ialah se6or bagi sebuah bulatan berpusat O dan jejari 4 cm. Diberi bahawa O ialah
titik tengah AC dan IAOD : 1.222 radian'
Diagram 10Rajah L0
t1
I
I
Calculate
Hitung
a) the area of sector AODE
luas sector AODE
b) the perimeter ofthe shaded region
peri met er kaw a s an berlor ek
c) the area of the shaded region
luas kawasan berlorek
[2 marks]
[2, markahl
[4 marks]
14 markahl
[4 marks]
14 markahl
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: l
SULIT t2
11
3472/2
a) I0o/o of starfruits from an orchard are rotten. 8 starfruits are chosen randomlv. Findthe probability,
l0o/o daripada buah belimbing dari sebuah kebun adalah busuk.dipi li h secara raw ak. T entukan keb arangkali an,
(D none of rotten starfruits is chosen.
tiada buah belimbing busuk dipilih.
8 biji buah beltmbing
(ii) at least two rotten starfruits are chosen.
selatrang-kurangnya 2 biji buah belimbing busuk dipitih.
[2 marks]
[2 narkah]
[3 marks]
[3 markah]
b) A school with 1000 students take partin a cross-country event. The Cross-countryevent started at 0800. Time taken for the students to finish the event is normallydistributed with a mean of 45 minutes ancl a variance of 100 minutesz.
Sebuah sekolah yang mempunyai 1000 orang pelajar mengambil bahagian dalamAcara merentas desa. Acara merentas desa bermula pada 08A0. Tempoh masa untukmenamatkan acara adalah tertabur secara normal dengan min 45 minit dan varians100 minit2.
(D Find the probability of students who finished the event after t hour.
[2 marks]
Cari kebarangkalian pelajar yang menamatkan acara merentas desa selepasI jam.
12 narkahl
(ii) rf 242 students finished the event in less than t minutes, find the value of /.
[3 marks]Jika 242 orang pelajar tamat acara kurang dari t minutes, cari nilai t.
[3 narkah]
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SULIT 13
Section CBahagian C
120 marks)
[20 markah]
Answer two questions from this section.Jawab dua soalan daripada bahagian ini.
3472t2
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12 Table 12 shows the price of four ingredients , A, B, C andD used in making akind ofbiscuit.
Jadual 12 menunjukkan harga bagi empat bahan, A, B,C dan D yang digunakan untukmembuat sejeni s bi skut.
IngredientBahan
Price:per kg (RM)Harea sekilosram RbI)
Year 1998Tahun 7998
Year 1999Tahun 1999
A 3.00 3.45
B 1)
C 1.80 2.15
D 6.00 z
Table 12Jadual 12
(a) The index number of ingredient D in the year 1999 based on the year I 998 is 90.Calculate the value of z. [2 marks]
Nombor indeks bagi bahan D dalam tahun 1999 berasaskan tahun 1998 ialah 90.Hitungkan nilai bagi z 12 marhahl
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SULIT
(b)
3472t2
The index number of ingredient B inthe year 1999 based on the year 1998 is 130. The
price per kg of ingredient B in the year L999 is RM 3 more than its comesponding price
in the year 1998. Calculate the values of r and y. [3 marks]
Nomber indeks bagi bahan B dalam tahun 1999 berasaskan tahun 1998 iatah I30.
Harga sekilogram bahan B dalam tahun 7999 adalah RM 3 lebih mahal daripada
harganya yang sepaclan dalam tahun 1998. Hitungkan nilai bagi x dan y.
[3 markah]
(c) The composite index for the cost of making the biscuit in the year 1999 based on theyear 1998 is 114.2. Calculate,
Indeks gubahan kos membuat biskut itu dalam tahun 1999 berasaskan tahun l99B ialah1I4.2. Hitungkan,
i) the price of the biscuit in the year 1998 if its corresponding price in the year 1999 isRM1.75.
harga sebungkus biskut itu pada tahun I998jika harganya yang sepadan dalamtahun 1999 ialah RWl.75.
ii) the value of p ifthe amount of ingredientA, B, C andD used are in the ratio of3 :5 : p:6.
nilai p iiha huantiti bahan A, B, C and D yang digunakan adalah mengikut nisbah3:5:p:6.
[5 marks]
[5 markah]
I4
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SULIT
13 Diagram l3 shows quadrilateralABCD.
Rajah'13 menunjukkan sebuah sisi empat ABCD.
Diagram 13
Rajah 13
The area of triangle ABD is 16 cm2 and /BAD is acute.
Luas segi tiga ABD iatah 16 cmz dan ZBAD ialah tirus.
Calculate
Hitung
(a) ZBAD
(b) the length, in cm, of BD.
panjang, dalam cm, bagi BD.
(c) lCBD
(d) the are4incm2, of quadrilaIeralABCD.
luas, dalam cm2, sisi empatABCD.
15 347212
f2 marksl
12 markahl
12 marksl
12 markahl
13 marksl
13 markahl
13 marlesl
13 markahl
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3472t2
A particle moves in a straiglrt line passes through a fixed point O. It velocity, u ms'l, isgrven by u: f - lt * 12, where f is the timg in second, after passingthrough o.
Satu zarah .bergerak di sepanjang suaru garis lurus dan melalui satu titik tetap O.Halajunya,vms- ' ,d iber io lehv:?-7t*12,dengankeadaant ia lahmasa,dalamsaat,
selepas rnelalui O.
[Assume motion to the right is positive]
lrAnggapkan gerakan lce arahkanan sebagai positifl
(a) Find the tinie when the partiCle is momentaril5r at rest.
Cari masa apabila zarah berehat seketika.
The time interval during which the acceleration ofthe particle is negative
Julat masa apabila pecutan zarah itu adalah negatif,
(ii)
[4 marks]
14 markahl
(b) sketch the velocity-time saph of motion of the particle for 0 < r < 4
Lakar graf halaju melawan masa bagi pergerakkan zarah itu untuk 0 < t < 4.
[ 2 marks ]
[2 markah]
(c) Calculate the total distance travelled during the first 4 seconds after leaving 0,
Hitungiumlah iarak yang dilalui dalam 4 sat yang pertama selspas melalui 0.
[ 4 marks ]
14 markahl
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SULIT t6
l4
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SULIT
15
s472t2
A factory produce two types of circuit boards, P and Q. The production of each tlpe of
circuit board involves two process, assembling and welding. Table 15 shows the time
taken to assemble and weld both types of circuit boards.
Sebuah bilang mengeluarkan dua jenis papan litar, P dan Q. Pengeluaran setiap ienispapan litar melibatkan dua proses iaitu peffiasangan dankimpalan. Jadual 15
menunjukkan masa yang diambil untuk meffiasang dan mengimpal kedua-duaienis
papan litar tersebut.
Cifcuit Board
',',"P.xpqnlitar':
Tinre Taken:, (minUte$;.
Masa yang diambil
Assembling'
P.emasaigarc
Weldme ,
'PW'Wililn
P 50 40
v 30 60
Table 15Jadual 15
The factory produces r circuit boards of type P and y circuit boards of type Qpoday is based on the following constrains.
Kilang itu menghasilkan x papan litar jenis P dan y papan litarienis Q setiap hari
b er d a s arkan kekangan-kekan gan b erikut :
D The maximum total time taken for assembling both types of circuitboard
is 600 minutes.
Jumlah maksimum masa yang diambil untuk menggabung kedua-dua jenis
papan litar tersebut ialah 6A0 minit.
11) The total time taken for welding both types of circuit boards is at least 300
minutes.
Jumlah ffiasa yang diambil untuk mengimpal kedua-duaienis papan litar
adalah s elurang-kur angnya 304 minit.
T7
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SULIT 3472/2
IID The ratio of the number of circuit boards oftype P to the number of circuitboards of tlpe Q is at most 9: 10.
Nisbah bilangan papan litar jenis P kepada papan litar jenis Q adalahselebth-lebihnva9 : 10.
(a) write three inequalities, other than x ) 0 and y >- O,which satisfy theconstraints above . t3 marks]
Tulislcan tiga lcetaksamasn selain daripada x) 0 dan y > a yang memuaskankekangan-lcekangan di atas. 13 marlcahl
(b) Using a scale of 2 cm to 2 circuit boards on the both axes, construct and shade theregion R in which every point stratifies all the constrains above. [3 marks]
Menggunakan skala 2 cm kepada 2 papan lttar di paksi-x dan 2 cm kepada 5papan litar di paksi-y, bina dan lorelckan rantau R yang dapat memuaskan
13 markahlkekangan-kekangan yang dib eri.
(c) Using your graph from (b), find
Menggunakan graf yang diberi, cari
i. the maximum number of the circuit board of type p produced per day if 3circuit boards oftype P are produced per day.
Bilangan maksimum papan litar dari jenis Q yang dapat dihasilkan setiaphari jika 3 papan litar jenis P dihasilkan setiap hari.
ii. The maximum total profit obtained per day if the profit from one circuit boardof type P is RM 40 and one circuit board of tlpe Q is RM20.
Jumlah keuntungan maksimum setiap hari jika keuntungan satu papan litarjenis P ialah RM 40 dan satu papan litar jenis Q ialah RM 20.
[4 marl<s]
[4 markah]
END OF QUESTION PAPERKERTAS SOALAN TAMAT
18
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THA UPPERTATLKENARAN6KALTA
l9
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347212 2au Hak
| | (zt dzk
Az)
Example / Contah:
lf X * N(0, l), then
,I ikaX *N(0, l ) ,
p{x> k)- 8{k,
P{X> 2.1)= 812.t'tz
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SULIT
947211AdditionalMatfiematicsKertas 1'September 20112 Jam
3472t1
JABATAN PELAJARAN NEGERI JOHORPEPERIKSAAN PERCUBAAN SPM 2011
ADDITIONAL MATHEMATICSKertas I
MARKING SCHEME
Kertas soalan ini mengandungi\ nrl"m"n bercetakG
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ADDITIONAL MA'THEMATICSMARKING SCTIEME - PAPER 1
NO SOLUTION SUBMARKS
FULLMARKS
I (a)
(b)
many to one
f :x+fBI
B1
)
2 aJ
I : _
2
3x=-2
and .r: -1
or r= - l
2
B1
2
aJ (a)
(b)
t7
x-2_-t
3
_1J
Q-x)-23
2
BI
2
B1
4
4 aJp>--'4
(4)'z -4(4)(p +3) < 0
4x2-6x+p+3=0
J
82
B1
3
5 (a)
(b)
6
0+k .2
.f (x) = -(x -3)2 + 5
2
B1
1
J
6
-3<x39
(x + 3)(x -9)
2
82
B1
J
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7 3.3x-.-3+4x/-" : z' "' or equivalent
23* or 2a!-x)
B2
B1
3
8 3+2p4+p
logrS+2logrm use two laws of logarithmlogrl6+logrm
use one law of logarithm
change to any base
4
B3
B2
BI
4
9 r:2p
prs :32p62
BI2
10 I
l$q-4:6
3d =10q - 4
aJ
B2
B1
3
1l (a)
(b)
v- !I - - ' ;
x-
IA=-
/.
4-0 ')- - -L
0-k
I
I
2
B1
4
t2 prn i \
. ' r I- l
r l 1
' . , . |- t -L
1
3
82
B1
-J
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13 -8i-4j .
-5 i
(r)l , l or i+ j
AC: AO + OC
J
,(l= (T ;') Ior
I2k -3
J
2
B1 2
t4 (a)
(b)
2
B1
2
BI
4
t5 JU", 150" 210"1 3300
30o, l50o or 210o,330"
. lsmx=tf or 30" is seen
, in ' " =14
4
B3
B2
B1
4
16 (a)
(b)
0.65
80 :5.2
79.74
l . r; (8'){t.r+21 - *tglto.oslzl
A(1,2)
dvL:4x+ldx
1.r : ' \Y - {
- . ' )- - - \, r -1
is seen
2
B1
2
BI
4
I7 (a)
(b)
2
BI
I
B1
1
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23 (a)
(b)
35
'co
60
tc1* 6c,
2
B1
2
BI
4
24 (a)
(b)
1851-x-85
23403154-x-+*x-85 85
2
BI
2
B1
4
25 (a)
(b)
0. t25 <z<0.225
53.25 - 52 or 54.25 - 52- tL- 10
41.7 g
k: -1.03
2
BI
2
B1
4
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4(2x +I)
2(2x +I)(2) or 8(4r + l)fl)' "2 '
(2x +l)2 or 2
J
82
BI
ffi,-*,*
= -rr* +l)-2 or & =- m
t8
27 is seen
s*a'ft!'^',=t
6+7 +x+1+8+3+3
2
B1
1
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SULIT
347212AdditionalMathematicsKertas 2September 20112tz Jam
3472t2
JABATAN PELAJARAN NEGERI JOHORPEPERIKSAAN PERCUBAAN SPM 2011
ADDITIONAL MATHEMATICSKertas 2
MARKING SCHEME
Kertas soalan ini mengandungi 17 halaman bercetak
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BAHAGIAN A
3472t2
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No Solution Submarks
Totalrnarks
1 y=3+x Or x=y-3 t rEliminate x or y*(3+x) '+x-2x*(3+x)=7
Ory'+y-3-2.(y-3)y= 7 Solve the quadratic
equation by using theOror equivalent
x2 -x-2 = 0f- tV+10=0
factorization.
(x+1Xx-2)=O
(v-2XY-5)=0
x=-1,x=2,orY=2,Y=5
Y=2,Y=5or
x=-1,x=2
Note:
OW-1 if the working of solving quadratic eguation is not shown.
5
I
5
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No. Solution Submarks
Totalmarks
Zal
b)
c)
a^ay=-21(x+z) ' - ; - r ' l
-2(x+2) '*11
OR equivalent method
c = 11
(- 2.11t 1
l\ll i ] \
ffi.shape, n ElMax. point : (-2,111
try-intercept=3
El
Y=2(x+2)2_f i M
3
3
17
3a)
b)i)
i i )
3rc or 8n or %cMana-mana satu tiga sebutan berturut-turut yang betul.tr
Dapatkan sekurang-kurangny a d:ua beza,sepunya bagisebutan berturutan f
-, \
dr :6n-3n dz:9n-6n \ ^ ' . /
-
d:3n I * l
Use Sn = !V^+ (n - 1)dl3
4Sr = ,, l2(.3n) + (3)(*3zr)l
L
OR 9r+&r+9r+l2t
UseTn-a+(n-1)d21n=3n+(n_1)Jn
OR other valid method.Note:lf listing method is used all terms must be correctly listed, accept forcorrect answer.
1
2
2
2 7
3472t2
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SULIT 347212
Solut ion
Graph CosI period in 0 S x< 7
Shape graph cos
Amplitude: 2
Graph shifted up by 2 unit
Drawing of the straight line from the equation involvingx and y, either gradient OR y intercept of straightLine must be correct.
2xlT
Straight line drawn correctly andNumber of solutions : 2
Submarks
Totalmarks
I
a)
;
EEiilE
All must be conecl
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No Solution Submarks
Totalmarks
6a)i)i i)
b)
Use the triangle law)_
For : AB= AO +J
OC
orrc=XO* OC
1)OC= 1a - :b
aaJJ
XC= 1"* lu36
AY = kAX=k(Ao * oxil pr I=k(-a+:b)
z
Use the triangle lawFor OY= Ol + nY
OY= X OC
1(1 -k)a. i*o=tr l ,
=a+k(- "*LO,
=(1 - t<p+1ru,)
- :b)a'J
Compare thecoefficient of a and b.
A
1-k=11.a
J
1.)
'k=t l23
3
5 8l
3472t2
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SULIT
BAHAGIAN BNo Solution Sub
marksTotalmarks
7a)
tr'ilii)
i i i )
| | | | - | | .^ | . - l l - " Ilxy l5 l 7 | e.3 110.8 113 115 |
lf table is not shown award N1 mark if all the points are plotted correctly.Plot xy against x(conect Jxes and uniform scales)
I6 *points plotted conectly a")
YLineofbestfit
5'-\---l
xy=;*. tE Aor implied. / I
Use*m=l , - / I
2 ( Kt ) / \ From sraPh xY asainst x
: : \ / \€)*="\ /
a=46 d=' f,='
x=4.5xY=12"'y=1.67
e,Note:ss-1i lPart of the scale is not uniform at the x-axis and/ or the xy-axis.OR not using the given scales. OR not using graph paper.
1
4
5
10
3472t2
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SULIT 3472t2
Solution Totalmarks
i '(a)
y_4:*_4(x_ 2)Y: '4x+12
or equivalent
Find the area of trapezium , .4.1
Gantikan x:2 dalam * 4LdY
To find the value of the gradient of thetangent to the curye
dy :-4dx
unit 2
I;{12+al[z] : 16I
2.
or I-Or+llh: 160
*Ar - *Az
ro - rTr
Integrate
I"$ - Y)dYl - z l
, l t r -L IL- 2 )
Substitute limit 4 to 8the integration
I - z l8* l^ Y I
n l ty- ' It -11la ls J9
I
(c)
into
i l ih: f sehel :h
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b)
c) i)
ii)
{c)
- l )+ l r . '\ /1 a\l : l I 1 l
B (5,9)
2AD:PC E
z,[1xa'1'aray = JQi-6f +O-6f
x' +y' +4x+20y+24 = 0
Use
15
v:0. , .can be impliedx2 +4x+24 = 0*-q"" : -go
@
lil] does not intercept
347212
Submarks
10
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SULIT t t
t Jse /, f g to f ind the area of sector AODE= %g)2(.222)
9.776 il 9.78
3472t2
i - i i r : i seb: ia l ' ,SU L!T
I
| 10a)
cl <COD = 3.142 - 1.222 = 1.92 rador 180"-35"- 35" = 110"
find the area of triangle DOCY,(4)zsin11O"
Area of shaded region(9.776 + 7.5175)
Find length of DC2(4)cos35' or 2(4)sin55"
Perimeter of the shaded region"4.888+*6.553+4+4
17.2935 ll 17.294 ll 17.29
10
Use S = r 0 to find the length of arc AED
19.M1 il 19.44
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c! i l 1- l -347212
i subI marks
Totaimarks
t l
, l ic li i[---_l
11a) i )
Solut ion
I andq=1 Ei0 10 L
Use Binomial formula
P(x=o)-'cotfri'tfrl'
P(x > 2) = 1 - P(x = 0) - P(x = 1)
- 'c , r*) ' , * , ' - '
0.1869
use Z - $ - tt) to find the value of z6
p(z>qtff!") or P(Z' #,or?(2t 1.5)
p(2.#=0.2q2 E
t -45 = -0.7l0
c,tfrl'tftl
b)i)
i i )
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SULIT
Solution
x: 10, y: 13
Use I : Q' *eo
Untuk bahan A orIe : I15, Ic :
i suu I TotatI marks i marks
3472t2
IL ihat sebelahSULIT
13
BAHAGIAN C
t2a) o
Use I : = ' xeoz
6.00
z:5.40
Z x 100:130xory-x:3
b)
c)i)
menghapuskan x atau y(Eliminate x or y)
100
C120
Nota : if steps to solve the quadratic equations is not shown OW - I
u* ,Exroo=r14.20(;D)
Atau setara -l-
RMI.53M
(1 ls)3 + (130)5 + (120)p + (90)6 = rr4.23+5+ p+6 l0
p:11 M
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SULIT l1t+ 3472t2
[L ihat sebelahSUI. IT
No Solution Submarks
Totalmarks
13a)
b)
c)
d)
Use Area of ABD = 16 to f ind <BAD r/ , , , -\
% (7)(6)sin0 = 16 \:7\
<BAD = 49.63" il 49"38,1 Nl I
Use cosine rule in A ABD to find B))_____
BD2= 62 + z2 -2(6)(7)cos4e.63" ( *t )
I
\
5 se1l-NI
Use sine rule in A BCD to find <BCD5.531 6
Sin < BCD Sin70"
<BCD = 60.02'
<CBD= 180"- 60.0:
Use Tz (a)(b)sinO to find the area of triangle BDC% (6X5.531)sin49.98'= 12.6659
Area of quadrilateral ABCD= *(area BDC) + 16
2
2
3
3
10
347212 Hak Cipta JPNJ 2011
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SULIT 15 3472t2
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:!r-f
No Solution Submarks
fotalmarks
14a) i )
i i )
b)
c)
Used t2 -7t+12:o
t -3 ) ( t -4 ) :0
t :3 , t :4
used dv-a
dvdt
<0tof indt
<0
2t-7 <0
t<l2
For shape of the graph
tr y- intercept and x-intercept
t
lntergrate I t' -lt+ 12 dt
s =4 -?' *2,32
Use
272
Br+. t l
Irmrt $ or JI
' 6 '
Find the total distance27 I-+
4l
J
V
4
2
4 10
347212 Hak Cipta JPNJ 2011
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$,,'I
fi.r€8,ii;
9t&|I
lc
(a)
c) i)
(b)
i i )
MM
Mi i .
16
50;+30y<600
or 5x+3y<60
40x+60y>300
or 2x +3y >-I5
L=Lv10or l,Ox <9y
Draw correctly at least one straight line from the*inequalities which invoves x and y.
Draw correctiy aiithree "straight lines.Note : Accept dotted lines.
The correct region R shaded
ilt.
Solut ion
Whenx = 3, maximum number of circuit boards oftype Q = 15 E(fromx=3intheregion)
Maximum point at (7, 8) | t r l t ]
Use 40x + 20y for point in the*region R40(7) + 20 (8)
Note:
ss-1i f
RM44O
In (a) the symbol " = " is not used at al l or more than threeInequalit ies are given.
In (b) does not use the scale given or does not use graphpaper Cr interchange between the x-axis and the y-axis.
347212
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10
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