Stpm Trial 2009 Kl
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;:(lNFIDENTIAL * J ~6ATAN
PELAJARAN W!lJI.YAH PERSEKUTUN' ." '., • . . "m.
Jr'- " - _. . .. _ . ... .... '''\HPERSEKUIUA
~:
954/2
~ ,. ....
-AN PELA.Jt,R"'" W IU\Yt\H PERSEKUTUAN KU,\LA LUMPUR
. _ ''- :- ' " .
ANPELA ... . . . " ' ':.
::::~~~~~~~~ #~~, ~~~~~
-- - -- . ._ . .. . . , ... ' .. MPUR
STPM TRIAL 2009
~~~~
AN PElAJAt(;\i\I ',VIV, rAH I--'tK~t;rl.U I UI\N rliU,i LA tIJMPUR
.lAMA I AN I"l:.UWI\KAN V':ILA Y.4.H PERSEKUTUA
JABA rAN PElAJARAN W1LAYAH PERSEKUI UA AN PHCo JAnAN W 'LAYAH PER!->EKU'TUAN " TAN PElAJ,'\RAN W ,LAYAH PERSEKUTUAN KUAli. LUMPUR JABATAN PELAJARAN W'lLAYAH PERSEKUTUAJ> PAPER 2 TAN PELAJARAN \VILAYAH PERSEKUTUAN KUALA LU f·/,PUR JABATAN PELAJARAN WtLAYAH PERSEKUTUM iAN PEL,\ J"'V\ N 'NltAYAH PERSEKUTUAN KUALA LUMPUR JABATAN PELAJARAN WtLAYAH PERSEKUTUAJI. " V .... L" u ... ,."- ,,,". . " " ' ,, rAN PEU~JARAN V>lIU,YAH PERSEKUrUAN KUALA LuMPUR .lA6ATAN PELAJARAN WILAY,\H PERSEKUT' ' " ., .•• , < . . .. . ,., • • ~ • • ~ ' '''' ' 1 PElA.JAR.i\N WILA YAH PERSEKUTUAN KUALA LUMPUR
MATHEMATICS T
~~~~~~~ ~~~~~~~~~:~;~~~ ~~~~~~~~
Three hours
.J/,BATAN PELAJARAN W1LAYf\H PERSEKUi U I "~
f' ••.• f>l >\ LUMt"t.JK .),,:),' I
: ~~~:~~~:,~ ~~:~~:~ ~~=~~~~~:~~~ ~~~~ ~~;:~~~=
,,,-I PEL AJAI1}\ 'I 'NIL-'; y,'I,H PE KS f.KlJTUAN KUALA LUMPUR
JABATAN PELAJARAN WILAYAH PERSEKUTUAN KUALA LUMPUR SIJIL TINGGI PERSEKOLAHAN MALAYSIA
•
Instructions to cand id ates:
DO NOT OPEN THIS QUESTION PAPER UNTI L YOU ARE TOLD TO DO SO. Answer a ll questions. All necessary working should be shown clearly. Non-exact numerical answers may be given correct to three Significant fig ures, or one decimal place in the case ofangles in degrees, unless a diffe rent level 0) accuracy is specified in rhe question. Mathematical tables, a
/iSI
of mathematical formlllae and graph paper are pro vided.
This question pape r consists of3 printed pa ges and I blank pa ge.
STPM 95412 *Th is qu estio n paper is CONF IDENT IAL until the exam inati on is ove r.
I Turn oyer CONF IIH~ NT I AL*
2
CONFlDENTIAL· I
Show that the homogenous equation x(y - x) dy = y(x + y) can be transfonned into dx . t lle sub' . Y"" vx. x -dv = 2. by usmg Stltut lOD [4 marks1 dx v - I
2
Forces (4i + 3j )N, (3 i + 8j)N and ( -6; - 5j )N act at a po int. Calculate the mal:oitude o f the
resultant force and the angle between the resul tant force-and the unit vector j .
}5 marks ]
. ,t.,L-k -51 . , G ( ,lL , t ) 1\
\..1.
, In the fi gure above, AS is the diameter of circle O. DE is a tangent at 0 which is perpendicular to BE. AD and BE produced meet at C. Prove that AS = Be. [7 marks1
::--:-. ,~c , I .1 ... - ') "\....
4
tan ) x - 3 tan x
Show that tan ] x = -c:--,"--.,-3tan 2 x- l Hence, by taking
( =
.....
.;. A. t\ ..,
Y.
tan x, find all the roots of the equation
I ' - JI 'Ji - ] 1+Ji = O.
5
6
The rate of increase in temperature of a thennometer is proportional to the difference between the room temperature, Xo and the temperature of the thcmometer, x. Write a diITerential equation to express thi s informati on. A thennometer with a reading of lODe is brought into a room with room temperature 3 iDe . The thennometer reads 24 DC after f minutes. Find the time taken for the reading to be 30DC. . 't. - \b'~L'I'L ' [10 marks1
'-~ BCD is a paralle logram . E and F are points on the diagonal AC such that AE - rc. ~ove that BFDE is a parallelogram.
15 marks]
(b) A ship P sails at a spced 40 km h' l along the directi on N45 D E. Anothe r ship Q sa ils at a speed of 65 kmlf' along the direction S60oE. Find the re lative vc!ocity of P with respect to Q. Initiall y Q is exactly 100 km north of P. Find the shortest distance between P and Q.
f8 marksJ 7
A discrete random variable X takes integer values probabilities gi ven by P( X = x) :o 2 J 2 5 6
betwcc~ 0~j ~\nelusive with
~ . Find the ex pectation of X.
(4 marks)
95412 ~Th is
question paper is CONFIDENTIAL lmtil the examination is over,
CONFIDENTIAL·
3
/"ctlf'/FIDENTlAL'
8
The letters in the word 'ADA!1; are randomly arranged in a lim!. Find the probabi lity that the arrangements (a) begin and end with 'A'
1
~ .,~
( b ) have separated consonants " .'to ~
f5 marksJ
A box containSl fcri\.mangoes.~f the mangoes are of vari ety A and are all good. The remai ning ma;~ are of varicly-B and of these, th ree are good and three are not good. Events X and Yare defin~~s fo llows: X: A random selection ~angoes which containSI9fvatiety A a~variety B.
t'
It
Y: A random selection of 4 mangoes which contains exactly 3 good mangoes. Find P( X), P( Y) and P( X n Y). State, with a rcason, whether the events X and Yare independent . "f.
10
IS·,
'{
The times spent in visiting a particular bookstore
by(~
and ~are
~
arks] to be
<\SS U
independent and are nonna lly di stributed with means'ii{ minu~nd 25 minutes and standard dev iat ions 2.2 minutes and 0.8 minute respective ly. Find the probabihty that
-
c
(a) the time spent by Ali is longer than that spent by Ravi in a particular visi t,
[5 marks]
'/<.-1 (b) the tota l time spem by Al i in 3 separate visits exceeds twice the ti me spent by Rav i in a si nglevi~ it. 't\1"-"1."t'lj ~ "2'( [4 marks]
,
II
Souvenir c ups man!ifactured by a certai n company are pac ked in boxes of(lOO. On the average o ne cup irvSO cups is defective. A box which contain s,) or more defec~ps will be rejected. UsingVsuitable approximation, find the probab ility that a given box wi ll be rejected. A de li very van carries ~uch boxes. The number o f boxes. out ol 20, that wi ll be rejected is denoted bV~nn i ne P( Y:5 ~) and fin d the expected number of rejected boxes. lJI marks1
12
The we ight, to the nearest kg, of 90 new born babies are summarized in the table below.
W~4'!' (~) 1.0 - 1.4 1.5 - 1.9
N lImber of Babies 8
2.0 - 2.4 2.5 - 2.9
12 18 24
3.0 -
20
]A
3.5 - 3.9
-
8,
(a) Draw the frequen cy cu rve for the data above. State the mode for the weight o f the (4 marks] babies. ( b) Plot the cumulative frequency graph for the data above. Esti mate the median and [7 marks ) the se nti.i.t:.'.:~r.tilc range of weight for the babies. (c) Find the probabi lity that a baby chosen at random has a weight excet!ding 3.5 kg.
[2 marks ) (d) If 10% o f the babies with the lowest weight are transferred to the intensive care unit, estimate the nHlximum weight orlhe babies transferred there. [2 marksJ 95412 ' This question paper is CON FIDEN riAL until thc
c~aminali on
is O\cr.
CONFIDENT IAL '
$'\ 0.61
~~. '"
.s
'(? 'I ~ \ O~() 1. .
PELAJARAN W!lJI.YAH PERSEKUTUN' ." '., • . . "m.
Jr'- " - _. . .. _ . ... .... '''\HPERSEKUIUA
~:
954/2
~ ,. ....
-AN PELA.Jt,R"'" W IU\Yt\H PERSEKUTUAN KU,\LA LUMPUR
. _ ''- :- ' " .
ANPELA ... . . . " ' ':.
::::~~~~~~~~ #~~, ~~~~~
-- - -- . ._ . .. . . , ... ' .. MPUR
STPM TRIAL 2009
~~~~
AN PElAJAt(;\i\I ',VIV, rAH I--'tK~t;rl.U I UI\N rliU,i LA tIJMPUR
.lAMA I AN I"l:.UWI\KAN V':ILA Y.4.H PERSEKUTUA
JABA rAN PElAJARAN W1LAYAH PERSEKUI UA AN PHCo JAnAN W 'LAYAH PER!->EKU'TUAN "
MATHEMATICS T
~~~~~~~ ~~~~~~~~~:~;~~~ ~~~~~~~~
Three hours
.J/,BATAN PELAJARAN W1LAYf\H PERSEKUi U I "~
f' ••.• f>l >\ LUMt"t.JK .),,:),' I
: ~~~:~~~:,~ ~~:~~:~ ~~=~~~~~:~~~ ~~~~ ~~;:~~~=
,,,-I PEL AJAI1}\ 'I 'NIL-'; y,'I,H PE KS f.KlJTUAN KUALA LUMPUR
JABATAN PELAJARAN WILAYAH PERSEKUTUAN KUALA LUMPUR SIJIL TINGGI PERSEKOLAHAN MALAYSIA
•
Instructions to cand id ates:
DO NOT OPEN THIS QUESTION PAPER UNTI L YOU ARE TOLD TO DO SO. Answer a ll questions. All necessary working should be shown clearly. Non-exact numerical answers may be given correct to three Significant fig ures, or one decimal place in the case ofangles in degrees, unless a diffe rent level 0) accuracy is specified in rhe question. Mathematical tables, a
/iSI
of mathematical formlllae and graph paper are pro vided.
This question pape r consists of3 printed pa ges and I blank pa ge.
STPM 95412 *Th is qu estio n paper is CONF IDENT IAL until the exam inati on is ove r.
I Turn oyer CONF IIH~ NT I AL*
2
CONFlDENTIAL· I
Show that the homogenous equation x(y - x) dy = y(x + y) can be transfonned into dx . t lle sub' . Y"" vx. x -dv = 2. by usmg Stltut lOD [4 marks1 dx v - I
2
Forces (4i + 3j )N, (3 i + 8j)N and ( -6; - 5j )N act at a po int. Calculate the mal:oitude o f the
resultant force and the angle between the resul tant force-and the unit vector j .
}5 marks ]
. ,t.,L-k -51 . , G ( ,lL , t ) 1\
\..1.
, In the fi gure above, AS is the diameter of circle O. DE is a tangent at 0 which is perpendicular to BE. AD and BE produced meet at C. Prove that AS = Be. [7 marks1
::--:-. ,~c , I .1 ... - ') "\....
4
tan ) x - 3 tan x
Show that tan ] x = -c:--,"--.,-3tan 2 x- l Hence, by taking
( =
.....
.;. A. t\ ..,
Y.
tan x, find all the roots of the equation
I ' - JI 'Ji - ] 1+Ji = O.
5
6
The rate of increase in temperature of a thennometer is proportional to the difference between the room temperature, Xo and the temperature of the thcmometer, x. Write a diITerential equation to express thi s informati on. A thennometer with a reading of lODe is brought into a room with room temperature 3 iDe . The thennometer reads 24 DC after f minutes. Find the time taken for the reading to be 30DC. . 't. - \b'~L'I'L ' [10 marks1
'-~ BCD is a paralle logram . E and F are points on the diagonal AC such that AE - rc. ~ove that BFDE is a parallelogram.
15 marks]
(b) A ship P sails at a spced 40 km h' l along the directi on N45 D E. Anothe r ship Q sa ils at a speed of 65 kmlf' along the direction S60oE. Find the re lative vc!ocity of P with respect to Q. Initiall y Q is exactly 100 km north of P. Find the shortest distance between P and Q.
f8 marksJ 7
A discrete random variable X takes integer values probabilities gi ven by P( X = x) :o 2 J 2 5 6
betwcc~ 0~j ~\nelusive with
~ . Find the ex pectation of X.
(4 marks)
95412 ~Th is
question paper is CONFIDENTIAL lmtil the examination is over,
CONFIDENTIAL·
3
/"ctlf'/FIDENTlAL'
8
The letters in the word 'ADA!1; are randomly arranged in a lim!. Find the probabi lity that the arrangements (a) begin and end with 'A'
1
~ .,~
( b ) have separated consonants " .'to ~
f5 marksJ
A box containSl fcri\.mangoes.~f the mangoes are of vari ety A and are all good. The remai ning ma;~ are of varicly-B and of these, th ree are good and three are not good. Events X and Yare defin~~s fo llows: X: A random selection ~angoes which containSI9fvatiety A a~variety B.
t'
It
Y: A random selection of 4 mangoes which contains exactly 3 good mangoes. Find P( X), P( Y) and P( X n Y). State, with a rcason, whether the events X and Yare independent . "f.
10
IS·,
'{
The times spent in visiting a particular bookstore
by(~
and ~are
~
arks] to be
<\SS U
independent and are nonna lly di stributed with means'ii{ minu~nd 25 minutes and standard dev iat ions 2.2 minutes and 0.8 minute respective ly. Find the probabihty that
-
c
(a) the time spent by Ali is longer than that spent by Ravi in a particular visi t,
[5 marks]
'/<.-1 (b) the tota l time spem by Al i in 3 separate visits exceeds twice the ti me spent by Rav i in a si nglevi~ it. 't\1"-"1."t'lj ~ "2'( [4 marks]
,
II
Souvenir c ups man!ifactured by a certai n company are pac ked in boxes of(lOO. On the average o ne cup irvSO cups is defective. A box which contain s,) or more defec~ps will be rejected. UsingVsuitable approximation, find the probab ility that a given box wi ll be rejected. A de li very van carries ~uch boxes. The number o f boxes. out ol 20, that wi ll be rejected is denoted bV~nn i ne P( Y:5 ~) and fin d the expected number of rejected boxes. lJI marks1
12
The we ight, to the nearest kg, of 90 new born babies are summarized in the table below.
W~4'!' (~) 1.0 - 1.4 1.5 - 1.9
N lImber of Babies 8
2.0 - 2.4 2.5 - 2.9
12 18 24
3.0 -
20
]A
3.5 - 3.9
-
8,
(a) Draw the frequen cy cu rve for the data above. State the mode for the weight o f the (4 marks] babies. ( b) Plot the cumulative frequency graph for the data above. Esti mate the median and [7 marks ) the se nti.i.t:.'.:~r.tilc range of weight for the babies. (c) Find the probabi lity that a baby chosen at random has a weight excet!ding 3.5 kg.
[2 marks ) (d) If 10% o f the babies with the lowest weight are transferred to the intensive care unit, estimate the nHlximum weight orlhe babies transferred there. [2 marksJ 95412 ' This question paper is CON FIDEN riAL until thc
c~aminali on
is O\cr.
CONFIDENT IAL '
$'\ 0.61
~~. '"
.s
'(? 'I ~ \ O~() 1. .
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