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8/3/2019 KEDAH ' 09--STPM- P2 + answers
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1 The le"9th of the sides PR. P.O and PO of triangle PQR are, (x y) and (x 2y)respechvely Show th ::!t cos _ PR:::t '" [2 ma r_.s l
Hence, find the range of va lues of ; for which _ PRO hes between 120 and 180"
SO lurion:p
6 R.. ,Cos L PRO = (H -y)1+X' _ (x+2y)'2x{x t y)
",' - 2.1)' - 3 /:-,(-" y )
(x t y)l.. .. - 3)')2 x ( x ~ y )
Since 120" < L PRQ CJS L PRQ " COS 180 0
1 x - 3y- - > -- > -,2 2x
[3 ma rks)
M'
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< , 151.2. . Two vanables t and v. are connected by the differential equaticn '% f " 2.v(1 v). where
12 0. O ~ v < I . l lndv = Vowhenl "O1 '-10,,"Showlhal v "
Solution:
= 2. I + c
n ( 1 + IflV " 21+c
In(6 }=21 +c
Subsll ue . t" O. " " Vo
Ifl ( .J&.... ) " cI - V"c " In K
n ( I ~ } : 2t + In K
In(K(IV_ v}) " 21
[6 marks]
4 . C ' ' " ~Ml ~ " T I ~ ( j h
"' ' " : \ .-I... .."p ""II'
r ,I.'> U ' .I->e."""5
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" e"
3. Prove tholt cot 0 tan 0 : 2 cot 20HQnc., or OIh p l"W!S.@. finr1lhft V:llllf!S of fI that sa lis fies the equation
Solution :colO-lane I I 2,{:. , for O s 0 :; 360'
LHS _ cotO _ unOcosO si nO,. sinO - C{j;Ocoslo"'si:t1e
sillOcosO~ < K ? n
'" ~ 2C".2co120
0018 - lImO = 2.J3'.2 0012" " 2J300120",'/3
IUUI 20 "'J j20 .. 30' , 2 1 0 ~ , 390' ,570'(} : IS' , IOS",lYS' , 285'
4
M'
161
[3 marilsJ
[4 marks I
M'
MAl
""'A'Al 17 1
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Tile dIag ram above shows a cncle APQC intersects with the circle ABOO al points A andQ . G iveolhal APB arld BOC are two straight L " and 0 is the centre of the circle APQC.Prove U,a\(a) a PBa is an isosceles tr ianglo,(b) CO", APSolution:
(a) LeI L BPQ '" uLAce", LBPQ "" (Extel10r aogle cychc qu.odrila\eraLequal to
iolCfioJ oppos ite angLe)
LAOO" UACB "2( (Mllte at tile cerme IS
[5 marks)[J marts]
" twice angle al the dJC\Jmfereoce) 91
L AOO LPBQ " IBet (OppoSIte angles cr-lic quadrilateralare supPlemefllary)
/ PBO" ISO' 2(>.
B'
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.....POO. _ BPO POB" 1!!if (Sum angles 01 a lnangle)
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The pOs ition l ! o r s of the pomts A, S and C relatrve to ill1 ong 'f' 0 are i i, b and a + 2brespectively AS and OC al 0 .... hefe '" and = qExp rC33 00 In lerms of(a) a, b and P.(b) a, b and q .Hence, evaluate p and q
[2 mark s1[1 mark]
[3 marks]Given a " 41 + kJ. and b " 7i J. find Ihe positive value of k rt L ODA = 90 "
Solution:
Ibl
AD DS=p . l-p.-. OD = pb +(1-p j a
A
[4 marKs1
_ c
"'B1
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pb+ (1 -p ) a " Q( a
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6 In an ecologn:al researd l II was found tl1al Ihe rate of decreaso of Ihe populat.on of acertain $peaes of frogs . Y " represented by the equal lOn = - kyo where I '5 lhe lime
dImeasured ,n weeks and k os Ihe reproductive COOSlanl(a) Given that the ,nitiat population was 500. sc;Nve the differertial equal on I 3 mns J(b) Two weeks later . the population had dropped to 300. lind k and show Ihal
(c) If the population was reduced to jllst one qLarter of the " , ~ i a l population ,n Tweeks. lind the v a l u ~ of T [2 marks I
When the poflUlalion had reduced to hall 01 M .mli,,1 populatio1 . water pol lu tion flltherdecreased the popu i(lllOn at a constant rate . W . Show h ~ 1
ky + W = Ae - kl , where A is iI constant.By anumlng Inat the reI=HNu ;Iiv" ,;u",.lanl. k. never chllnged. c alculll!e W if the frog"became extinCI after ;;motne! 4 weeks.Solution:
(. )
.:Q:" - /WIyf-hl
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