1. 7
n'l:6tauo-orfionrinGsulr,lrsmfrsilurnrnrnfiarfiorz?rdnsrtuemnrivrEJ'1fr'EJrfislolnd
ufi riil sr fiuz ooigua rra Rrr,t il nr:6ntn zssz
t' a r o7 orfisrorragrf z?to6tou't51 07 Fl6un9r16{9t: 2
uasAv
aoufuarfiorifi zt frurrerru fl.c[.25s1 rxd'l 12.00 - 14.00 ll.
j jd'
fO - }['.lil6fn4........ ....1n1?111.:fl0uq
OA
nlafiu'tEI
u - !a&
1. rioaauo:iufiiivitraro 32 ila:?il 7 n#r uriroonriju 2 eou neuilr.udru 100 nsuur.{
-&91.1U
ool,rfi I fi + ila d'.:usirio 1 - 4 zianc 4 orcur.ru
norrfi 2 fiz}ria d'.ousirio 1 -28 fraac 3 ncurru
2. fraetouna$i{ r rio t - + rilurionouuuutri'fioairurnr tr{rirriroroufinnsio.r Torst#.u
nrtfisruaaocflrurario"leirJosr'eras '[:Jsroutun:cortr{i'lsrrur"t fi t n"odlri'uirr7.a"u{
4 aA { r v , o ct A- u A
gfnr*f ifinl:oou [u1i0.]gr?0u't.lGrlaoufloyi 25 uiln:39t'u*l']00u14il1?1 1
tt/ )
3. rTaaau oar,lfi zila I - zg rilurioflauuuurdonnau usinvrioocfirirqoutvirfion +
riroau 6a t. z. g. uns 4. tri'fior:nn16onrirEroufirfiu'ir nnsioofiaora3ornmsar,r(uq
i a o d yqea o a o r q i.
Yrdo ril EJir o'ln a u [o s? un? lTo1t6tosir ze uS0 air n'ir rsu't u [u? { n a iln a o{ n1Ttilq
oivu
nTeo,lr*st'loauoa$Yi 2 0{0? 0 u'r.l
4. nr:yr o rfl 1r tvivr otusirrio sou uns#r rut#ro4a o 6 o dr ur r*1o 1 To u r6ozr o
s. #rmirria aou uaen:eo1r*dr n au o a no1 n#oo aauTo s r6os1o
ffia chutiltlodu#z I#ao fr ayitilal{au
vt
a?o El'10 nlT9rOU
(0) 2 + 3 = ?
1. 2
2.3
3.4
4.5
s,' X o i A
1J0U nlnoUytnnno 5qt
6..:1:-t q ou o1'r ud on + tunrsen :grir a ou d".rfi
(0)
o@@o
tila s#o6[ouTo cr o1o'l:elo'lrn1 rnrfi'o
la l8 s N 6[ a %'[#in'[r,ra ?1 ilr i
2. eJ A A v
iuarfi nHfr 21 or,trtnil v{ .G7.2551
289
or orrfisrcf1cfn{ 2 ra#T {d 1
1.
aor,ufi 1 (z?o 1 - 4 ilaae 4 a&ur,tr,t)
g,/OOAIA'l
n 1 fl 'l 14 }[q a1 u? l,bfl 1l'l Tn ? 0..1 tflr 9l a 1 { t] 9l ..1
9l'l T'l..l
ts9t U A B C ANB Anc BNC AJBJC
6A
orurl{flillfrn 60 17 20 24 8 7 I 40
'iruruarur8nturryn (AJB) nc tvirn-u tvirto
2. 1?i'a" tlYtullorfd n totiiroiutntnrf,q ri'r au = 31 rtns an = 5a, ttd'l a',o rviT ri'utvirtgl
5q ,u A i
rnuilatililnunu.trruot#lrj t fl ogfi:rnrtvirriu c(t) u't?l le'ilid C(t) - O.7k(0.9)t-1
A e, , s u X q ' q - X g'..1a----.-: v
rfio r riJuritnloi'r d'r6o:n-lnrirurlu:rnr soo,000 u'tm unel6trsurnfliluil:'lgt'ltmlnu 405,000
, - Xqu v Aa,
ulYr o{14',1?1Tnnilil [T3J1 lln? nu
j i ua o ct -X
gn:'t{ uo n uolnm rufr t oI tn:q taa ilt 0Iun Hn :*ro'l1*?1a 40 o1.[ tljuo.]t,t
!
A
rnTorilaEl
I
A
a'l'liln
1.5 - 1.9 4
2.0 - 2.4 6
2.5 - 2.9 14
3.0 - 3.4 12
3.5 - 3.9 4
u A v v 6 & Aa i q u I 4-'-.--'-q-
Soungto.:nltruflgsnruerultm:ig0.tgt*yiiln?1ilnfl{flq tuql:'].]fl't.lul{u iln']tYl'l[9t.ilq
3.
4.
U
'[n 16 s il 6[ o r,l'[#in'[un ?1 EJ i
3. a,/ A dd ar
??tolytag?l 21 fi%?1ail y{ .F1.2551
290
or orrfisrctlrfm{ 2 urirfi 2
A.vtt
oouri 2 (ra 1 -28 ?raa& 3 acuilr,t)
. u A y A oo e a a a i,, t y A
1. untTUl{fi0.1141t.1il0'tu?u 50 nu asqo{[:ul.lR1:*10u14T0nl191ruilu0u'l{1t8u141*.]R1:*'l
e v a i a nrr*rrfi:Juluncirudf iru?u 30 nu 6uunr:*r6uorirrrdutata6sunr:srfidun'runr:uuYrrTUu s r q 0'11*?u 30 nu t:u1*R1t*'101*0u't.ttou?fi:a s r
, a ao y o v a ia a aA
0 u't.J [9] u ? tjo't u? u 42 Ftu un? 01 u? uun t: u l{Yt tTU uR1 t*'10 uil n n u
1. 12 2. 18 3. 20 4. 28
2. ritalbqril LLU ufl o.r flru n1T lil ud'.,: fi'(u
3
1 =M4
13 +23
f +23 +33
= (r'Xr')
= (r'Xo')
4
4
4
YqYqgtutatte,6l
n1 lTnl: t14 tu a il n ttuu 01J1,[u un? t{ nfl il AB ilfi1 tYt''l nu rYrl toqq{
1. n'+2n+1 2. n'+2n 3. n'+n 4. n'+1
3. rirnuo
- <3 v a cr u d
[]tst 1) fl0?[nu.]flno?ultililnT1,[9ttiluflq?una'{q
- <X y a dt v d
2) flA?[n U.]nno?uuilu't.tt1{9ttljufla? 2 11
nr n n:dturiotq nil ril q sil r{ n.lq
udctud
1. fl91? 2 lj1flntril$ltuuflq?Un
v6:adrv6:
2. flgt? 2 1J1U't.t5U9tt:jl.lflq?Un
u6'acru6'
3. fl9?Unnn5u9t[UlAfl9? 2 1'.1
l,l 'au f j c, u
4. lililfl9? 2 lj','lYtrillrflAiun
4. tri," , b lflud,ru?uoi.rto .l riotqfisr
1. d'r a > b rrd'r a" bu
3. d'ro<a< l rrd'r a2<a
'[n rfr E il 6t al,t'l#intua ?1 il ;
9,/ 9,/
2. nla<bUn?-a<-b
sl cr,/ a+b
4. n1a>bLbn? a>
2
b
4. ilrarfi nEft zt #urrnil il .Fr. zss1
29t
or nrflsrsrasr{ 2
tvi
14?$'114 3
2. Curro{nnrrqruro.r 2+^.6 6o 2-,.6$
t.l lt*l . o) rfJurryq'jl..:
s. riotq fio
1. 2rr rijudru?t*aarrnuv
/ ^2
3. *2-3x+1-[*-gl
." 2)
v
LLn? zunnil x$1
_ 13
4
9,t Cr o
n1 x LLnf x LUl.tFl'lA0U1J0.lfltln1T
12
d r r v 9,/ 6l
tJgt'l L?]'lnul,0 L9t
1. -1 2. 1
4.
6. (*+ 2)' +(* -1)' - o
1
4.
2
1
3.
2
4tt3. (-m , -3] oo)
T . riotqlilurrqrriraoura.l
1. [-a , s] 2.
B. riotouijurrynrirqo
1 t"lt(x <rl
3 t"l- >2 Hto
U? 0.1
x' -2x( ts
[-s , 3]
lx-sl <2
2 t-l -T<x(-s)
4. {-l->r Hto *<3}
9.
x( -2
3
cr,t 6l A I
rio LqF ooirr o.: (16) o
1. 1 2.
1 o. riato riluoirr o.l
1. atb*
9
3. aob-3
3.7
atb -o
I
a'ba
to.rdrfiu turioter
2. 1,3,
4, 1,3,
114.
t-
{"tb'
g
(a-1b-')'
2.
4.
11. an _
1. 1
3. 1
2n*1 + (-1)" I
dt dA
ruu il01{?1 n
3
,3, 5,7 ,9
, 3 ,7 ,17 ,
7 , 15 , 31
5, 11 ,21
ln rfi s er aau'[#in'[r,unrT e,r i
5. iuarfi n$fi zt #r,rerail T{ .Fr. zssl
292
or orrfisrrrran{ 2
tvi
u}[1m 4
12. ri'r sn u?ruzunu?nro.l
d
n 1^lOl.tLLTnT 0.1 0l,bnTilq
1+1a'+1a
248
1
'l-t-
2n
m'lt- ,^')
1
' ,n+'
1
2n
S
urd'r n
Sn+r
i_
t Lt 9,/ 6l
t?l'tnulj0 [o
1
1-
2n
2.1.
4.3.
,l- 1
z^"
t 1l2t 1- |
L 2")
t+. 'lunrmoonnrcl.r z nnnforunTu"doo#o nmrudrovrilusorruornr:nffirunuanrorusi'rururnn'ir,uruq
A t v A 9q
14T0 [yt']nu 10 n010to
tg. d'rfinuuolnn:.rtyrvl.r6.:arrgr 2g1u ornousurfi.:un:nr::ri'1 fl"ru ttarolnilnrar::d'qqq
A A A r v o u A I u & a d q tA eq
n.ttTU.t[14il 3 fl',tut[n'] 0'11*'tulfluYt'l]iluqnq1{nuil{l43.J9r0']nn:.ltY'lT{9n{[5u.1lililna?J0Lo.l
1. 3 2.4 3.5 4.6
2.
15. rita
1.
1
tl-
I .. n+1
z
Fr ?'r e.r #ru rnTufluriot q r [J ufi .: ri'd'u
1. 11- t(*,v)lx - o LLnv y < o)
3. 13 - t(*,v)lx < o Ltnvy < o)
17. ri'rHt,ton:lulto.t y = f(x1 eYtgrJ
T
6
1
1.
3
5
4.
36
1
3.
12
Tq uilt,tfl or r rvirri'urioter
* -3,3) 4. t.l. =
R)
(x-3Xx+3)
3t'
2. 12- t(*,v)lx < 0 LLnv y - 0)
4, 14- t(*,v)lx - o LLnv y - o)
rletFl?'ta.t#ufi'uf r - {,*,v)lv -
L
t.l.*3I 2 t.l.*-3)
x'-g
lx
16.
9,/ 51 9,/
lJ 0 [qnnq 0.](u
dojdraao
1. 6[ilnlT f(x) - 0 tlgl'lA0UyltlJ1A011A?%AT.] 3 A11,[?1.[
2. Tqrilr,lto.r f 6orir.: [-r,+.o)
dAo^
3. LT1,[410.1 f 9l0LS9|1J0.1A1}[?1,{QT.1
oJA
4. n1n1fr9t1J0.t f O0 f(0)q
1 n rfi s il a o ?r'[#in'Lr,u or er e.r i
6. iua'rfi nl.ffr zl fiinraal y,t .Gr. zss1
293
or orfisrcrran{ 2
tvi
14%',11'l 5
18 td'
1.
3.
1 g. nirlqr u fl'rrrnd.,: 6 qor n nirlq u nr:riuqi'u +o u'r?l
6;t a't X
LlJ1,[q.llt
af
1O N LN Lil9ITLLTN
af
n Ln ril aTnv 6
20 fiInLelqrn-oulil fiInLilerTnr 8
a'/'TAAiA
T? 3.J n U n I L O U fl'l TS.1 9t O Q'l n Tr U y ?1'l .l ?1 Tn 1 .:
u']?1
u'l?1
r. = ft*,y) lt = l*l+ r) ti"'Irrur*ro.r r 6orryorot.iruruoSrnuud'r r:ud'ro.r r 6ariotel
rrqro.rriruruof.: 2. tonrotdruruo3tlrn
ooia,ratv
rsqro:aru?uor.t?urrnurRR?lnrarillnu 1 4. rrqro{,iruruottfifiriT rurnn'ir r
9'/
a/ a.r
Hnt0lnuil
aT
n Ln Lilqrnv 10 u.,t?l
^&auanrutr6fru:nornnurrfiunr::runfin#rfirriruuilt*reuvyr.r{ +s fiTnriler: srum6osd'orsiruir
Ti
Lq U fllTNU1YI
1. 370 4. 490
CE rt,t'ruriu BA
I ril9l:
2. 410 3. 450
ABD rijulnlilln $nvfiI
A
20. a1n3il ABD rfJunreJ Lufi uruilililln.ur
gt?'t3.JU't?lJA.: AD lilufi tS.tgtT
1. 16
2. B{3
3. 20
4. 166
zt. rinviattfiurnuradrfiuorjuuHrirzur (oqn) dr*orln:rr-urilnr,, 300 rilqr e.:rnqrrfiuGo oql tq / ct
--
6rv^vaqAuyuAru6a
tuilrlilnil 60" RUttlr?Tsqu lutil;toiurnuttilru{rffla.:ilTsR'lct']: (oqe) fl.ttnntfrut5o oq 4 , -'--"'
cte^vu:Y
riiurlruflru 6o' n:..llrul:sct-u ri'1il:rololtdrria.:orn:roi'uri'.tnrr, 195 rila: :susdr.::yn.jr.:q uq
:J:vnr qrr:fiuH #r zur ltuuu?ryo-u ) ufJ ufi ril er
A
656
1656
3006
4ssJi 195 rsJA:
'[ n ufi u aJ 6[ o t,t'L#in'Lua ?1 EJ ;
!
AA
?1uiluq
1.
2.
3.
4.
7. q/AdAqr
?.1ta11i nun 21 fi%?1Ftil ?rt .F1.2551
294
or orrf,not'16fa{ 2 uriT fd 6
22. ri'r sr r LLt,lt*6ro ur arfin 6u u 1 6 eru 6u{ a1 ru dr oi'uld'd'.:fi
56 60 65 69 73 73 75 78 83 85 87 87 89 93
a./4{9,rVl 9,/lVltA4{a'/,={
t+n LT u ua I { 610 u Ld'sr y rLtAurvirlH i 6.r e v il t,rn tr u uil :v tl'r tu 20o/o
1. 66.6 2. 67.0 3. 91.0
'[#rre{%R1nsir,r-'tu {dritnt,to'[# aourirn'rerz7a 23 un v 24
^ *u-tu rfler.:ri'narin#runr:; tfiInniru) to.t{Tou6r'rr 2tL9,llAR']1{A
nsirunr 1o Fru rijuoY.,rfi'q
A
?t CU.'0
at
rA
nniln0q
94 97
Yue./
? s .1 T-lAt gt'9t g LtUl*
$.1
n'j'l
4. 91 .4
IA
nnilT'luLtn3HRJ.,lqo,
4. 60
g1U
87
463
2023
5
0
1
2
3
q(U
9
578
0054
75
zs. folqfio
, f,
^
3 u v
' rrl a At I v af v
1. n1tilnulfl1nila18.]u'1fi1.lRflilR',]:rljA{nnilHtotufl1T14tu{ iln',]lYtlnu 22 nlnnTil.l 'u
av Y - - t gill t 9,l a A I I u
2. T.tflU10itll',tfiilnflilR'rTsl0innilt't loufl1:t1uun;nnilil [ouR]Tfiru{iln1tytlnilqflat
I
v6sv'9,ia.r^-A'tv
3. uI, ud'r1.[1a{il']14uRflilR1:r1J0{nRilil [9lufl1Tnru.]tlflrndilil toufl1:5't uiln1[yi''lnu49{otru
qa7--rf&artv
4. ilr'ud'tuttntd't uuu lt'u 0{ 11'114 1.[R fl lJ Rl:e1 0.1 H to u fi"'lT?l.1 20 fiil il nl tyt1 nilsql
e e u A 4 yu a e^ i
^
aY u vl r a af u
24. n1[0'r14u't?]outu't9t[14fltJR1TstouYt't{1a{ru [oufl'rTto:o.]uil illi']l{un tNti'lu zo ?l [nn:rulq 'u
a e , a A y o e'f , XAu , a ^J
j ! - - a
filn[nUOA{A1U[.]U[y{3Jttfl? O'lU?1*9.1[9lUfi"lTnnilUYl9l0.:0'lU[.:Uly{eJtilO.]01nillnilnflilR'ITJ[RU{q
a ct j t d 6d d
nqrilunLuOTrsuet
1. 35 2. 40
2s . rio ru n ro fd, d.: rJ :v n o u d'r u
$q
.)'/ i t 9v
lJ O TJ NTq?1 fl O.] :JTUN O U O? EJ
ruq
,ioAA,,
K?rlruu{ruurl'rqr16'rura.:Toilnroyr14u{ [[nrn?rruu:J::J:':utorriorunrofiaar6o@9tq{q
9', 6f o Qrt
lJ 0 [9tA1ilnl0U
1. 2,4 2. 2,7 3. 4,7
3. 55
20 19 21 18 22 23 17
25 24 26 23 27 28 22
'[ a rfi g il 6[ a ?bl#in'[ua ?1 il i
8. 7
iuarfi n fift zt #lmrail r{ .Gt.2ss1
295
or nrflsrgT asr{ 2
jvi
141,[1tt 7
o/d A dA
26, un tSuugt t,lr4 il.: n.: yr v Lil u urT u t* 4 itr Isr u fi dr t,r? t*H r.ir u fi q un r Lnro al il enyt.: qi otrj fi
AA
?T'IYI 1 2 3 4
IA
HU? UNA 3 2 1 2
tnTo C B A
27.
e u A &u "
I e r.lll y r v y a I 6t ctit r t v e q
n1untTUilnuila0{nlTYt1tn:otilnulfi [9tm1nu 2.75 t1'19t0.1flo]l?{17t 4 tlt totn:otm1nulja[91
q er o u cr - !
(fl1l4ilglt14fl'l6i1OUn3[t1*Ut:JUO.:U tnTO A = 4 , B = 3 , C = 2 , D = 1 llfli F = 0)
1. A 2. B 3.C 4. D
ATAe''e)ie/
a 6r Lil L[n T il ry{ u nq { n n :gilu nl Tn Tv 0't u lJ 0 .,]
{ 0 il n yt tri n r .t T ?'l$
Aa/O.rd.tg,/51
u 9t?1 u 6fil y,l uSTgH?'l.,] Ft'l n n1 { nl illl 0 lgt
Aa/,iA
1. 0'l1,tuu3.J4:' d9
e,/riaA
2, ilr'uo1u > gtltilnutnljnilad:, d9
,iAa"A
3.9t1tflnutn1jnila d.l, oag
,iAa./A
4. 9lltilnutn1nilA = il5'U011,[ = 0'll,[Uuu
(rig ht-skewed d istri bution I rir.: u ud
o,oQ-tAA
n n uilv 6['t 9t cu ?1 6[9t ?1
arq
zg. tunr:iir:r o nm r 6 o rfi urfi m n-u rdo r "nr:t"i'u-n:[fl :6 n " Io uldttu u dr:r o
llw a a A e t A a a yq
ag ton?1ilfl I tl4umnna o:u't [Ton 0 n ofl 0 to
1.'irurur{rnrruhinr:rurn rfi ul:J
v ao A - - e
2. 9t 0.1tJFl','ln']il tn u?nunnt*ilyila0u
"
J v a d e ll rl
3. n1n1 il tR U? nUO?1 ufi9t rfi ua a{ lil rill.[Ft''ln'Iill.i''l
4. flqr ou uu u iir:r o qio r fi nm eri unsil:s nu nr:nlrfi ur fiu nr:tfu-n:ror:6 erqtql
**
-[a
tfr er il 6[ o lt'[#in'[ua ?1 ar {
9. 7
r$ne errfinet'rasr{ 2
297 eir #nirr n'r :'[ n ufr s a.r
aoufit
sfonfrz
rilnE iztr'nrfisrG[16[a I z
1)13 2)3e 3)2 4)60
1) 4 2)3 3)2 4)2 5) 3 6) 3
7)1 8)1 e)4 1o)1 11)4 12)1
13) 4 14) 2 15) 3 16) 2 17) 4 18) 4
1e)2 20) 1 21)2 22) 4 23)3 24)2
25) 1 26)2 27)3 28)4
-[a
rfr er il a a u'l#inlun tr ru i
10. rilns arfing.,rasr{ z 298 airrlnirrnr:'tn rfierru
iror:'rs#fio6fou imnrf,srcf'rc[ ni z
nar {d 1
ila 1 aou 13
ornTon ri n (A J B J C) - 40
9v
?lo 2 aou 39
6[aT a - a +(n-1)d$n1
ornTonri a = 31
8
a+7d:311
ornTonri a = 5a
92
a.+ 8d - 5(a +d)1'1 '
-4a + 3d = 0
1
Ox+ ;4a + 28d = 124
1
@.@ ; 31d - 124
.'.d:4
Lt?lu d - 4 tuO
?1+28 = 31
"'
a1 =3
01n oro = ?1 + 9d
?ro = 3 + (9)(4)
.'. dto = 39
ln rfr g e.r a a ut#intnn rr ar i
f,qr
n (A JB JC) = n(A) +n(B) + n(C) -n(A r)B) -n(A nC) -n(B -tC) +n(A nB nC)
40 - 17 +20 +24 8 7- 9+x
X-3
.'. n Itn JB) nc ]
= (z-*) + x + (g-*)
= (t-s) + 3 + (g-s)
= 13
o
@
@
@q
11. rcr6rrJ Frrfisrflran{ 2 299 eir rln irr nr :'[ n u8 ar sr
?lo 3 srou 2
;ntrari t - o
5oo,ooo - o.zk (o.o1o-,
7
500,000 - g
n
500,000x9 - k
7
rJ rurTnrt en lyirrYu 405,000
C(t) - 405,000
o']n C(t) _ 0.7k (0.9)t-1
o
405,000- O.Tx500,000xYx(O.o)t-1
7
o.e = (o.oy,-,
1 - t-1
t= 2
tv
?to 4 aou 60
d
LNTq LilN U
1.5 - 1.9
2.0 - 2.4
2.5 - 2.9
3.0 - 3.4
3.5 - 3.9
A
Ft?1iln
4
6
14
12
4
A
Ft?'.liln6[yfr8J
4
10
24
36
40
founvto.rer?'rarfrnynru#ruvt'yrfiifin?1ilfin.:nq -
24
x 100 - 60$t 40
-[n
rfr s il a a ltt#in'Lr,ror er er i
12. 7
ranEJ arfiororasr{ 2 300 airri'nirrnr:'ta rfisru
9f or,tfi 2
9'/ 9,/
?lo 1 aau na 4 28
6uA
a.rA9,/AdAjrttct/
Un tT U 1*A 0.: tT U UR'l 1*'l O Uil TA q q ltO U'.l.l 1*0 U 1 R'l l9'l
riu6o n(nua) = so
alnrilrirru% n(AUB) = x+30
.'. x+30 = 50
x= 20
6uA
6u%R'ue16uo dr{ rd urHtoR'ue'rfi :luo dru6 ur drur u 42 nua'q
orn3rJrir.:Uu x+y - 42(u
20+Y - 42
Y = 22
.'. drururin 6uufd 6u%n1r*16u
= x + 30-y
= 20+30-22
=28
ln rfi s ar a a u'l#intuntr eu i
13. 7
rons nrf,srnrasr{ z 301 dtlnirrnr:'tnrfisEr
tv
?ro 2 grou 3 n'+ n
13
?JA
i,'i-1 [ ;
('.
') ]
(
^')(
t')
ornTovrri
9v
aou ?to
n4qt 1
q
13 +23 +33+... ..+n3
[;
(n*')
]
n2 (n + 1)2
4
n' (n + 1)2
{ n2 (n+1)2
n(n+1)
n2+n
4
-
(AB) 2
4
-
(AB)2
4
= AB2
-AB
-AB
-AB
lv
?ro 3 2 #qr'5' 2 "u1u'r.rrfiqriju#sr{un
rna 2q
at cX <rt
fln? rnuf $nFt?uurJ
tllsr 1 LLnr 2
I
e.t 6'
fln?un
frr{ 2 rj
ffn{ 2 11
A
HTO
n:rJq
vd
5[n?un
vd
6[n?un
A
I
la rfi n ar a a r,rt#in'[nn m ar #
ffn{ 2 11
14. r$nEJ srrfisrFflan{ z 302 rir rin irr nr :'[ a ufr s, &r
ty 91,/
uo 4 aou ?ra 2
9/A
lJ 0 2 9.lO
rir
9'/ 9,/
nl a<bLtn?-a<-b
o
!1 {T1r nlilUql a < b
9,/
e't 9/
-1 u'l9ltil?l.] 2 lj'l.l .'. -a > -bq
9y lr,/ .2
?ro 5 aou ?ro 3 x
nn tllT'ly$
_3x+1 :
[._:)'
_+
n rfJuonrrnuv dlr,hu zn qi'oruflluoarrnuyoiru1.
2.
3.
nn Lil{r'rv (r*,t6 ) x (r-,t6 ) -
fie L?lrry ( *
.
=X
3 2 13
_t
z) 4
2 3x+
4. nn Ly,trlr I r" I
913
44
1
aou z?ag -I2
(x+z)'+("-r)'- o
^2
+ 4x + 4 + *2 2x + 1 _ o
2x2 + 2x + 5 _ 6
2x2 + 2x 1 _ 0
^2
3x
tv
sa6
-2tln
zunsil,l"l
'l'. Jo)
r-'- fr)
[ 4 ,[ 4 )
d'lgn (Conjagate) to{ff1fi'urravfi'u
e./ 3vr . ey aa o c'vl e,/
?tilt1EJ t?ta 101,[ Lila 0{ u6[9t.:rlytrfl Lc{
-
nlaau il Loaytilu6 q
2
4
A 6l e'/
[1 lT'lY Ltl S [U6[AT x -il
(t/t d 9'/ Vl 9'/
nilnunqo{ Lq 1 uY,
U
13
44
1
2
-b +
la rfi s il a a ?b'[#5ntr,rn r,r ar {
15. -
rans nrflErarasri 2
303 air r['n ? g'r n 1 :-[ n tfi s e.r
lv
sa7
ila 8 nail ila1
I
lx
-2
2+5
3
aau z?a 1 t-3 , sl
*2 2x
^2 2x - 1s
(*-s) (*+3)
Lsetrirqou6o [-3 , 5]
H
@-30s@
t.lt ( x <rI
5l
x-5
X
X
lv
so I aau ila+ 11
3
(,'0 );
3
( 24't;
)
23 (-
8+3 =
3)
11
-27
1 n rfr cr au a o r,u'[#in'[u er rr ar i
16. 7
rans nrf,srdran{ 2 304 airrinirrnr;'[n rfrsil
#a 10 aau ila 1 atbo
{a3b2:,
( a-t b-' 1z
t-v"3b2
a3 b2
a-3 b-6
rfiut'
"3
b4
9l'/ 9,/
?to 11 mou zta 4 1 , 3 , 5 , 11 ,21
2n+1 + (-1)n
3
A'ln a
22 1
3
23 + 1
3
24 1
3
25 + 1
3
26 + 1
11
21
1 n 16 s, er r a ut#in'lr,uor rr er i
17. 7
rilns nrfinorasr{ 2 30s eirrlnirrnr:'tn tfieril
z?a 12 aau
1
1-
2nlv
?to 1
1-
nu'ir 1+ I +
2
1
fl
1
+
4
[;)
m
[;.1
1
+
I
ldra
... + tulAol,[nTilLTIJ1nfUA
aq
n
f=
a1
1r
H'l
S n
Sn*t
a1 ( ,-rn*1
1r
1
11
^n*1z
9,/ ly
ga 13 aou zta 4 6
:-- -Y-. e..-^.r L' &
dr ur uL 6[lAyl'.t.1 Yl LLq n q1.] n u?l { 14 u ql
1 ,n*1
aulu't + t,tnrc[?::ri'q
1
2
(
'-'^ )
1 l.1)'*'
2)
1
1
2n
1 rn
[;)'
2
X
nr.,rtyn{ -+ ou5'u'lqq
cy
6 f flU?]'l.l
d
l,$FITfl?TTFI -+ rfru.:tHil
'[ n rfi s e.r a o ut#inlu o rr a.r i
18. radEJ Frrf,srcrran{ z 306 iirrfinirrnra-ta rfisil
9y 91,/
?ro 14 aou ?ra 2 !
6
n(S)=6X6=36
rirur.ro E 6 o rtsrr o.: rnsrnr:n[fi r.r nu?nr a{ usi'rurrnn'irnto ryir n-u 1 0
E = {(+,0) , (6,4) , (5,5) , (5,6) , (6,5) , (6,6) }
3 t-l- # -3,3)
*2g
lv 9y
?to 15 aou ?to
a1n y
"'
n(E) = 6
a./ y ,_ n(E) 6 1
o.tl,ll,t p(E) -' n(S) 36 6
.'.(*-s)(x+e)+o
x + 3 , -3
r4ilr'r EJ lil a qir cir uqiol'Lri ryir n-u 0
r?o to srou fio z r, : ftx,y;lx < 0 unry = o)
nrtru#rulnt-uf r firilufirrifu 6o qrrrru#rufiuffi4d'ue-rorirarir 1x; qlorhirnfioun-u
rior r,,= t(",v) lx = ouns y < o)
uu'jr nruT fintu 4 r?iu (0 , -1) uns (o , -2) d.:fiq'i6'ue-usTrnfirrnfioun-uCJ
fraz rr: {(",r)l* < ouns y=o}
nu'jr surfintu r, rdu 1-t, 0) unr G2, 0) d.ror'lilfiqio"uqi'utqrnu dria"'uEium-rnrir(llcJ
uHfioun-u
rios r-: t(*,v)1" < ouns v . o)3 '' /
n:-r'jr erur8nlu r, uiu (-1, -1) un: (-1, -zy d.:fiqjfuel-usTrnfi'rrnfiourYu,u
ila+ ro= t(",u)1" = outo y=o)
nu'jr nrurEnlu ro tau(0, 1) unt 1o,oy d.:firid-'ud-um-rnfirrnfiouriu
ln ufr u il 610 %t#inlunx1 il i
19. rans nrf,norao{'2 307 eia rinisr nr :'[a ufi s s,r
9r/
?to 17
9v
AOU TTA 4
ooA
n1alfr9tlj0.l f 9t0 (0)q
9,/
lJ0 1 fior LT{T'ry 6[run'lr f(x) - 0 6tlqroi'eLLfl% x olnnrlrnh^lu'ir
sYA,t
o
oiriu ri'r$ouuo.tfliln1T f(x) - 0 firfru.: 2 enqau
cr,/ A A o e-
1J0 2 llo rilT'tv Tqrilu 6o Lsala.:dru'lua3{ (R)
do 3 fisr Lvlltu t:ud'to.: t 6o [-r,*) (a1nnilill)
e,/Jaa_/&,JA
110 4 On Ly.lTlr Aqql'lflO 910 x=0 91.11{1,[ Fl'l9l'lqqnO f (0)
uqqq
rsafl o.:dru?%o3{fi fr eir rurnn'it 1
- l*l+t
I
D :R'
r
d'.rriu x # o
.ll
.. lxl
y1
a,/ &
9l.l1,[1,b y
R - (1,@)
r
tv
0f ou rra 2 410
oirloufl'rr uil.lrfju z eiru
eier,ufi t qirueirloun'r:riuoi'u +o u'ryr
'iAaAa'/;8g'/6il?%11 2 noQ1nffi urnl.lTn?{41nff?TTtuRilnnun.lu'lu 45
A'/
aqaoLtnuxfi'209tqq
9r/
sa 18
9v
?lo 19
_ l-l
ovrririlurAotd'
lv
gtou va 4
fior:n*r y
alY
o.rt,tl* y - 1
rflaooln T
0
0
1
aT9,'rrT
10 n LntS,JaTttTn a0.lo'lun1 [ou6['lT
aT9./rr
20 n Ln rilar#q'lil qi'0.:qir usirle u6r'tr
AaTiac,,rrT
on 15 fl Ln L8.JATil tUn 0 A o.lQ'l Ugt'l L9tU6['lT
r?ruoi'0.:qil ueirTq u6r'rr tuairufd 2lfluutu G0
alY49,/rrTdra
O.]ltU fltlT'lnqt0.l0'lUFl'l [qUfl'lTt]J1,{1.,:1.[ 40 + 370 - 410 U'lYl
af
n Ln rilar
10 X 6 = 60 U'l?l
20X8 = 160 U'l?]
15X 10 = 150 U'lYl
+ 160 + 150 = 370 U'lyl
la rfr s il a o ?bt#intuerere.r i
20. rilns arflsrgT asr{ 2 308 eirrinisrnr:-[a rfiser
lv
ra 20
lv
aau so 1 16
8 rtJsr:
Sior:rHr A eco
nr
ilu'j1 tJu EcD riluruilil1n tfiotorn rA ?Jn'ru Gqq
e.t &
er{uu ilil e6c = 30o (r{rlnrutu:il A :rruri'ulsi 180')q
fior:r*r A nso lryu'ir sin 3oo
n?lilu1?30.,1 AB
n?1ilU'l?110.,1 AD
g
X
.'.x=16ty
e,/ {J t e./
9t.,tu1+ Ft?'ltJU'l?110{ AD tyl'lnu 16 tilaT
195 $Jnr
1
2
D
ila zt nau ila z 16s",6
300 urn:
E
fior:r*r A nro
Itu'ir tan 600 -
x:
X:
9t?1il U'l?1J0.,1 AE
U1?1J0',1 EDn?'1il
300
X
300
Js
300
Js
,.6
Js
- 1ooA tiletT
lnrfisruaaut#intrrnerrui
21. 7
r6rnEJ Frrf,srgfran{ 2 309 aiT rinirr nr:-ta rfi s a,r
fior;rHr A BCD
Itu'ir tan 60o =
e'ilruurrto': BC
Fl?'luU'l?fl0.1 DC
.'. y- ryxf =
"/s '..6
d'.:riu:snsvirt:rn'irt:J:snren:tiunti'trt = x+y = roo"6 + 65rA = ros.6 lilel:
ilazz srou ria + vl
Ft y LrlAl*d.,: fi rin 13 u uil Tv u'r tr* 20o/o t o .:ld'er v u%1.[ c[r n'j r 6 o Pso
H..roi'rLL14il.r P -
r ( N+1)
r 100
-
80(16+1)
- 13.6
100
Pao od:vu'irrqi'luurirfd 13 LLns 14 riu6oa{:ru'jr.: 89 LLfls 93(u
. 93-89 x
14 -19 13.6 - 13
X
+
0.6
2.4:x_&
O.ll*U Pao = 89 + 2.4 = 91.4
99
ilo zg nou rTo 3 fi'uoruro.rrirurin#runr:nornrirufilosJff'r:rfi.:unsneiruei'loun-rmrsfirirrvirrYu4quuqg
y . - 9+15+17+18+20+20+25+24+ 37+35 ar u
1J0 1 nn x - = - 22ntnn:il. ?ltu.t 10
rio z nn fi#uto.rneirufiIornr:r1u = 35 -7 = 28ru .t qI
av '
qjl
fiffita.:ncirriloruar:nrfir = 37 - 9 = 28q.ua
rio g fio il-rud1tlta.ir'llounr:mu - 16 + 20
= ra ttn;ril'uo'ruro{riTousr:ufir = 20@ql
2oqlu
rio + nn fi'roruro.rfiTouar:vYrililo = 20qt4.u
aq,fv
o']uiluilTo.ltJ [qum:n{?luo = 204qt
t- 195
{3-
.J;
195
v_ ,.6
65.6 tilar
la rfi s il 6{ a ltt#inlun ?'l r,r i
22. [anEJ srrf,srqran{,2 310 drfin?mnr:'tnrfieril
9v,ty
zta 24 aou ua 2 40
9v
zta 25
drurueiTq unr:fi#runr:srfi u 20 fiInniru = 8 nu(u
acr8
Flglt:JU x100-40%
20
ty
aou ?lo1 2,4
ryr
ilarrafiofi 1 x _#
,i
6[? U tU U.1 tU lAtl'l 9l T0'l 1*
20 + 19 + 21 + 18 + 22 + 23 + 17
S-
(zvzo)'* (tvzo)'* (ta-zo)'* (zs-zo)'*
20
(n-zo)'(zz-zo)'*(zl-zo)'*
I
S - 4+ - 2
ilaernfioftz x-#
9t?'1il urj::Jr?u (,s',)
25+24+26+23+27+28+22
25
s- (zs-zs)'+ (24-2s) 1 (zr-lzs)'* (zz-zs)'* (za-zs)'* (zz-25)'(n-2s)'*
-4
1 a rd s sr a o u'[fin'[r+n er er i
23. rons nrfinrrran{ 2 311 atrrin?rrnr:'tn rfrsil
lv t,/
ra 26 aou ?to 2 B
oQ9'/Aa{-ra/
fl'll4l.tgt LHLnrq?T'l?l 4 t?l'lnu x
j
LnTqLilnU = 2.75
(g"z )t(z"g )+(rl.+ )+z* - 2.rs
Ig * 2+1+2 ]
6+6+ 4+2x : 2.75
I
16+ 2x = 22
x=3
-&vaA
fr.rl.rl.r [t1q a.]fl ou lr:rn 4 loirn:q g
9/ tYioa
?lA 27 AOU gA 3 nltilnutnlntun > ilr',Ud',tu > d']ltl*Uilp4
nt
ria U n fi rrivl't.tr?'t (Ri ght-skewed distri bution )ru.
nu'ir orufiuil < ris'soru < rirradutntntflqrop
r7o zB srou ila + fiqrouuuuiir:rosia.:finmrui uns:J:;nunr:nfrfiurn-unr:tdrin:rfl:6er.ll ,u
e l" - A y A oqrllls a d A u qyv o j
nn:*il3?1fl'tn6u11fln1a{Hqau?10sYt't[14 [qn?1ilnqtilu [nu?nun1TtTlJo:tn:o9tytoq(ll
v a e e 4 v t d A - q eu
nna 0{ n 0 ila 0u a 0{ iln?l ilTun;lJ:rflljn1Tiltn u? nunlT lxu gtrrnro aql ,u ,u
ln rfr s il 6[ a r+'[#in'[r,ta ?1 il i
![7
n'l:6tauo-orfionrinGsulr,lrsmfrsilurnrnrnfiarfiorz?rdnsrtuemnrivrEJ'1fr'EJrfislolnd
ufi riil sr fiuz ooigua rra Rrr,t il nr:6ntn zssz
t' a r o7 orfisrorragrf z?to6tou't51 07 Fl6un9r16{9t: 2
uasAv
aoufuarfiorifi zt frurrerru fl.c[.25s1 rxd'l 12.00 - 14.00 ll.
j jd'
fO - }['.lil6fn4........ ....1n1?111.:fl0uq
OA
nlafiu'tEI
u - !a&
1. rioaauo:iufiiivitraro 32 ila:?il 7 n#r uriroonriju 2 eou neuilr.udru 100 nsuur.{
-&91.1U
ool,rfi I fi + ila d'.:usirio 1 - 4 zianc 4 orcur.ru
norrfi 2 fiz}ria d'.ousirio 1 -28 fraac 3 ncurru
2. fraetouna$i{ r rio t - + rilurionouuuutri'fioairurnr tr{rirriroroufinnsio.r Torst#.u
nrtfisruaaocflrurario"leirJosr'eras '[:Jsroutun:cortr{i'lsrrur"t fi t n"odlri'uirr7.a"u{
4 aA { r v , o ct A- u A
gfnr*f ifinl:oou [u1i0.]gr?0u't.lGrlaoufloyi 25 uiln:39t'u*l']00u14il1?1 1
tt/ )
3. rTaaau oar,lfi zila I - zg rilurioflauuuurdonnau usinvrioocfirirqoutvirfion +
riroau 6a t. z. g. uns 4. tri'fior:nn16onrirEroufirfiu'ir nnsioofiaora3ornmsar,r(uq
i a o d yqea o a o r q i.
Yrdo ril EJir o'ln a u [o s? un? lTo1t6tosir ze uS0 air n'ir rsu't u [u? { n a iln a o{ n1Ttilq
oivu
nTeo,lr*st'loauoa$Yi 2 0{0? 0 u'r.l
4. nr:yr o rfl 1r tvivr otusirrio sou uns#r rut#ro4a o 6 o dr ur r*1o 1 To u r6ozr o
s. #rmirria aou uaen:eo1r*dr n au o a no1 n#oo aauTo s r6os1o
ffia chutiltlodu#z I#ao fr ayitilal{au
vt
a?o El'10 nlT9rOU
(0) 2 + 3 = ?
1. 2
2.3
3.4
4.5
s,' X o i A
1J0U nlnoUytnnno 5qt
6..:1:-t q ou o1'r ud on + tunrsen :grir a ou d".rfi
(0)
o@@o
tila s#o6[ouTo cr o1o'l:elo'lrn1 rnrfi'o
la l8 s N 6[ a %'[#in'[r,ra ?1 ilr i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-1-320.jpg)
![eJ A A v
iuarfi nHfr 21 or,trtnil v{ .G7.2551
289
or orrfisrcf1cfn{ 2 ra#T {d 1
1.
aor,ufi 1 (z?o 1 - 4 ilaae 4 a&ur,tr,t)
g,/OOAIA'l
n 1 fl 'l 14 }[q a1 u? l,bfl 1l'l Tn ? 0..1 tflr 9l a 1 { t] 9l ..1
9l'l T'l..l
ts9t U A B C ANB Anc BNC AJBJC
6A
orurl{flillfrn 60 17 20 24 8 7 I 40
'iruruarur8nturryn (AJB) nc tvirn-u tvirto
2. 1?i'a" tlYtullorfd n totiiroiutntnrf,q ri'r au = 31 rtns an = 5a, ttd'l a',o rviT ri'utvirtgl
5q ,u A i
rnuilatililnunu.trruot#lrj t fl ogfi:rnrtvirriu c(t) u't?l le'ilid C(t) - O.7k(0.9)t-1
A e, , s u X q ' q - X g'..1a----.-: v
rfio r riJuritnloi'r d'r6o:n-lnrirurlu:rnr soo,000 u'tm unel6trsurnfliluil:'lgt'ltmlnu 405,000
, - Xqu v Aa,
ulYr o{14',1?1Tnnilil [T3J1 lln? nu
j i ua o ct -X
gn:'t{ uo n uolnm rufr t oI tn:q taa ilt 0Iun Hn :*ro'l1*?1a 40 o1.[ tljuo.]t,t
!
A
rnTorilaEl
I
A
a'l'liln
1.5 - 1.9 4
2.0 - 2.4 6
2.5 - 2.9 14
3.0 - 3.4 12
3.5 - 3.9 4
u A v v 6 & Aa i q u I 4-'-.--'-q-
Soungto.:nltruflgsnruerultm:ig0.tgt*yiiln?1ilnfl{flq tuql:'].]fl't.lul{u iln']tYl'l[9t.ilq
3.
4.
U
'[n 16 s il 6[ o r,l'[#in'[un ?1 EJ i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-2-320.jpg)
![a,/ A dd ar
??tolytag?l 21 fi%?1ail y{ .F1.2551
290
or orrfisrctlrfm{ 2 urirfi 2
A.vtt
oouri 2 (ra 1 -28 ?raa& 3 acuilr,t)
. u A y A oo e a a a i,, t y A
1. untTUl{fi0.1141t.1il0'tu?u 50 nu asqo{[:ul.lR1:*10u14T0nl191ruilu0u'l{1t8u141*.]R1:*'l
e v a i a nrr*rrfi:Juluncirudf iru?u 30 nu 6uunr:*r6uorirrrdutata6sunr:srfidun'runr:uuYrrTUu s r q 0'11*?u 30 nu t:u1*R1t*'101*0u't.ttou?fi:a s r
, a ao y o v a ia a aA
0 u't.J [9] u ? tjo't u? u 42 Ftu un? 01 u? uun t: u l{Yt tTU uR1 t*'10 uil n n u
1. 12 2. 18 3. 20 4. 28
2. ritalbqril LLU ufl o.r flru n1T lil ud'.,: fi'(u
3
1 =M4
13 +23
f +23 +33
= (r'Xr')
= (r'Xo')
4
4
4
YqYqgtutatte,6l
n1 lTnl: t14 tu a il n ttuu 01J1,[u un? t{ nfl il AB ilfi1 tYt''l nu rYrl toqq{
1. n'+2n+1 2. n'+2n 3. n'+n 4. n'+1
3. rirnuo
- <3 v a cr u d
[]tst 1) fl0?[nu.]flno?ultililnT1,[9ttiluflq?una'{q
- <X y a dt v d
2) flA?[n U.]nno?uuilu't.tt1{9ttljufla? 2 11
nr n n:dturiotq nil ril q sil r{ n.lq
udctud
1. fl91? 2 lj1flntril$ltuuflq?Un
v6:adrv6:
2. flgt? 2 1J1U't.t5U9tt:jl.lflq?Un
u6'acru6'
3. fl9?Unnn5u9t[UlAfl9? 2 1'.1
l,l 'au f j c, u
4. lililfl9? 2 lj','lYtrillrflAiun
4. tri," , b lflud,ru?uoi.rto .l riotqfisr
1. d'r a > b rrd'r a" bu
3. d'ro<a< l rrd'r a2<a
'[n rfr E il 6t al,t'l#intua ?1 il ;
9,/ 9,/
2. nla<bUn?-a<-b
sl cr,/ a+b
4. n1a>bLbn? a>
2
b](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-3-320.jpg)
![ilrarfi nEft zt #urrnil il .Fr. zss1
29t
or nrflsrsrasr{ 2
tvi
14?$'114 3
2. Curro{nnrrqruro.r 2+^.6 6o 2-,.6$
t.l lt*l . o) rfJurryq'jl..:
s. riotq fio
1. 2rr rijudru?t*aarrnuv
/ ^2
3. *2-3x+1-[*-gl
." 2)
v
LLn? zunnil x$1
_ 13
4
9,t Cr o
n1 x LLnf x LUl.tFl'lA0U1J0.lfltln1T
12
d r r v 9,/ 6l
tJgt'l L?]'lnul,0 L9t
1. -1 2. 1
4.
6. (*+ 2)' +(* -1)' - o
1
4.
2
1
3.
2
4tt3. (-m , -3] oo)
T . riotqlilurrqrriraoura.l
1. [-a , s] 2.
B. riotouijurrynrirqo
1 t"lt(x <rl
3 t"l- >2 Hto
U? 0.1
x' -2x( ts
[-s , 3]
lx-sl <2
2 t-l -T<x(-s)
4. {-l->r Hto *<3}
9.
x( -2
3
cr,t 6l A I
rio LqF ooirr o.: (16) o
1. 1 2.
1 o. riato riluoirr o.l
1. atb*
9
3. aob-3
3.7
atb -o
I
a'ba
to.rdrfiu turioter
2. 1,3,
4, 1,3,
114.
t-
{"tb'
g
(a-1b-')'
2.
4.
11. an _
1. 1
3. 1
2n*1 + (-1)" I
dt dA
ruu il01{?1 n
3
,3, 5,7 ,9
, 3 ,7 ,17 ,
7 , 15 , 31
5, 11 ,21
ln rfi s er aau'[#in'[r,unrT e,r i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-4-320.jpg)
![iuarfi n$fi zt #r,rerail T{ .Fr. zssl
292
or orrfisrrrran{ 2
tvi
u}[1m 4
12. ri'r sn u?ruzunu?nro.l
d
n 1^lOl.tLLTnT 0.1 0l,bnTilq
1+1a'+1a
248
1
'l-t-
2n
m'lt- ,^')
1
' ,n+'
1
2n
S
urd'r n
Sn+r
i_
t Lt 9,/ 6l
t?l'tnulj0 [o
1
1-
2n
2.1.
4.3.
,l- 1
z^"
t 1l2t 1- |
L 2")
t+. 'lunrmoonnrcl.r z nnnforunTu"doo#o nmrudrovrilusorruornr:nffirunuanrorusi'rururnn'ir,uruq
A t v A 9q
14T0 [yt']nu 10 n010to
tg. d'rfinuuolnn:.rtyrvl.r6.:arrgr 2g1u ornousurfi.:un:nr::ri'1 fl"ru ttarolnilnrar::d'qqq
A A A r v o u A I u & a d q tA eq
n.ttTU.t[14il 3 fl',tut[n'] 0'11*'tulfluYt'l]iluqnq1{nuil{l43.J9r0']nn:.ltY'lT{9n{[5u.1lililna?J0Lo.l
1. 3 2.4 3.5 4.6
2.
15. rita
1.
1
tl-
I .. n+1
z
Fr ?'r e.r #ru rnTufluriot q r [J ufi .: ri'd'u
1. 11- t(*,v)lx - o LLnv y < o)
3. 13 - t(*,v)lx < o Ltnvy < o)
17. ri'rHt,ton:lulto.t y = f(x1 eYtgrJ
T
6
1
1.
3
5
4.
36
1
3.
12
Tq uilt,tfl or r rvirri'urioter
* -3,3) 4. t.l. =
R)
(x-3Xx+3)
3t'
2. 12- t(*,v)lx < 0 LLnv y - 0)
4, 14- t(*,v)lx - o LLnv y - o)
rletFl?'ta.t#ufi'uf r - {,*,v)lv -
L
t.l.*3I 2 t.l.*-3)
x'-g
lx
16.
9,/ 51 9,/
lJ 0 [qnnq 0.](u
dojdraao
1. 6[ilnlT f(x) - 0 tlgl'lA0UyltlJ1A011A?%AT.] 3 A11,[?1.[
2. Tqrilr,lto.r f 6orir.: [-r,+.o)
dAo^
3. LT1,[410.1 f 9l0LS9|1J0.1A1}[?1,{QT.1
oJA
4. n1n1fr9t1J0.t f O0 f(0)q
1 n rfi s il a o ?r'[#in'Lr,u or er e.r i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-5-320.jpg)
![iua'rfi nl.ffr zl fiinraal y,t .Gr. zss1
293
or orfisrcrran{ 2
tvi
14%',11'l 5
18 td'
1.
3.
1 g. nirlqr u fl'rrrnd.,: 6 qor n nirlq u nr:riuqi'u +o u'r?l
6;t a't X
LlJ1,[q.llt
af
1O N LN Lil9ITLLTN
af
n Ln ril aTnv 6
20 fiInLelqrn-oulil fiInLilerTnr 8
a'/'TAAiA
T? 3.J n U n I L O U fl'l TS.1 9t O Q'l n Tr U y ?1'l .l ?1 Tn 1 .:
u']?1
u'l?1
r. = ft*,y) lt = l*l+ r) ti"'Irrur*ro.r r 6orryorot.iruruoSrnuud'r r:ud'ro.r r 6ariotel
rrqro.rriruruof.: 2. tonrotdruruo3tlrn
ooia,ratv
rsqro:aru?uor.t?urrnurRR?lnrarillnu 1 4. rrqro{,iruruottfifiriT rurnn'ir r
9'/
a/ a.r
Hnt0lnuil
aT
n Ln Lilqrnv 10 u.,t?l
^&auanrutr6fru:nornnurrfiunr::runfin#rfirriruuilt*reuvyr.r{ +s fiTnriler: srum6osd'orsiruir
Ti
Lq U fllTNU1YI
1. 370 4. 490
CE rt,t'ruriu BA
I ril9l:
2. 410 3. 450
ABD rijulnlilln $nvfiI
A
20. a1n3il ABD rfJunreJ Lufi uruilililln.ur
gt?'t3.JU't?lJA.: AD lilufi tS.tgtT
1. 16
2. B{3
3. 20
4. 166
zt. rinviattfiurnuradrfiuorjuuHrirzur (oqn) dr*orln:rr-urilnr,, 300 rilqr e.:rnqrrfiuGo oql tq / ct
--
6rv^vaqAuyuAru6a
tuilrlilnil 60" RUttlr?Tsqu lutil;toiurnuttilru{rffla.:ilTsR'lct']: (oqe) fl.ttnntfrut5o oq 4 , -'--"'
cte^vu:Y
riiurlruflru 6o' n:..llrul:sct-u ri'1il:rololtdrria.:orn:roi'uri'.tnrr, 195 rila: :susdr.::yn.jr.:q uq
:J:vnr qrr:fiuH #r zur ltuuu?ryo-u ) ufJ ufi ril er
A
656
1656
3006
4ssJi 195 rsJA:
'[ n ufi u aJ 6[ o t,t'L#in'Lua ?1 EJ ;
!
AA
?1uiluq
1.
2.
3.
4.](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-6-320.jpg)
![q/AdAqr
?.1ta11i nun 21 fi%?1Ftil ?rt .F1.2551
294
or orrf,not'16fa{ 2 uriT fd 6
22. ri'r sr r LLt,lt*6ro ur arfin 6u u 1 6 eru 6u{ a1 ru dr oi'uld'd'.:fi
56 60 65 69 73 73 75 78 83 85 87 87 89 93
a./4{9,rVl 9,/lVltA4{a'/,={
t+n LT u ua I { 610 u Ld'sr y rLtAurvirlH i 6.r e v il t,rn tr u uil :v tl'r tu 20o/o
1. 66.6 2. 67.0 3. 91.0
'[#rre{%R1nsir,r-'tu {dritnt,to'[# aourirn'rerz7a 23 un v 24
^ *u-tu rfler.:ri'narin#runr:; tfiInniru) to.t{Tou6r'rr 2tL9,llAR']1{A
nsirunr 1o Fru rijuoY.,rfi'q
A
?t CU.'0
at
rA
nniln0q
94 97
Yue./
? s .1 T-lAt gt'9t g LtUl*
$.1
n'j'l
4. 91 .4
IA
nnilT'luLtn3HRJ.,lqo,
4. 60
g1U
87
463
2023
5
0
1
2
3
q(U
9
578
0054
75
zs. folqfio
, f,
^
3 u v
' rrl a At I v af v
1. n1tilnulfl1nila18.]u'1fi1.lRflilR',]:rljA{nnilHtotufl1T14tu{ iln',]lYtlnu 22 nlnnTil.l 'u
av Y - - t gill t 9,l a A I I u
2. T.tflU10itll',tfiilnflilR'rTsl0innilt't loufl1:t1uun;nnilil [ouR]Tfiru{iln1tytlnilqflat
I
v6sv'9,ia.r^-A'tv
3. uI, ud'r1.[1a{il']14uRflilR1:r1J0{nRilil [9lufl1Tnru.]tlflrndilil toufl1:5't uiln1[yi''lnu49{otru
qa7--rf&artv
4. ilr'ud'tuttntd't uuu lt'u 0{ 11'114 1.[R fl lJ Rl:e1 0.1 H to u fi"'lT?l.1 20 fiil il nl tyt1 nilsql
e e u A 4 yu a e^ i
^
aY u vl r a af u
24. n1[0'r14u't?]outu't9t[14fltJR1TstouYt't{1a{ru [oufl'rTto:o.]uil illi']l{un tNti'lu zo ?l [nn:rulq 'u
a e , a A y o e'f , XAu , a ^J
j ! - - a
filn[nUOA{A1U[.]U[y{3Jttfl? O'lU?1*9.1[9lUfi"lTnnilUYl9l0.:0'lU[.:Uly{eJtilO.]01nillnilnflilR'ITJ[RU{q
a ct j t d 6d d
nqrilunLuOTrsuet
1. 35 2. 40
2s . rio ru n ro fd, d.: rJ :v n o u d'r u
$q
.)'/ i t 9v
lJ O TJ NTq?1 fl O.] :JTUN O U O? EJ
ruq
,ioAA,,
K?rlruu{ruurl'rqr16'rura.:Toilnroyr14u{ [[nrn?rruu:J::J:':utorriorunrofiaar6o@9tq{q
9', 6f o Qrt
lJ 0 [9tA1ilnl0U
1. 2,4 2. 2,7 3. 4,7
3. 55
20 19 21 18 22 23 17
25 24 26 23 27 28 22
'[ a rfi g il 6[ a ?bl#in'[ua ?1 il i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-7-320.jpg)
![7
iuarfi n fift zt #lmrail r{ .Gt.2ss1
295
or nrflsrgT asr{ 2
jvi
141,[1tt 7
o/d A dA
26, un tSuugt t,lr4 il.: n.: yr v Lil u urT u t* 4 itr Isr u fi dr t,r? t*H r.ir u fi q un r Lnro al il enyt.: qi otrj fi
AA
?T'IYI 1 2 3 4
IA
HU? UNA 3 2 1 2
tnTo C B A
27.
e u A &u "
I e r.lll y r v y a I 6t ctit r t v e q
n1untTUilnuila0{nlTYt1tn:otilnulfi [9tm1nu 2.75 t1'19t0.1flo]l?{17t 4 tlt totn:otm1nulja[91
q er o u cr - !
(fl1l4ilglt14fl'l6i1OUn3[t1*Ut:JUO.:U tnTO A = 4 , B = 3 , C = 2 , D = 1 llfli F = 0)
1. A 2. B 3.C 4. D
ATAe''e)ie/
a 6r Lil L[n T il ry{ u nq { n n :gilu nl Tn Tv 0't u lJ 0 .,]
{ 0 il n yt tri n r .t T ?'l$
Aa/O.rd.tg,/51
u 9t?1 u 6fil y,l uSTgH?'l.,] Ft'l n n1 { nl illl 0 lgt
Aa/,iA
1. 0'l1,tuu3.J4:' d9
e,/riaA
2, ilr'uo1u > gtltilnutnljnilad:, d9
,iAa"A
3.9t1tflnutn1jnila d.l, oag
,iAa./A
4. 9lltilnutn1nilA = il5'U011,[ = 0'll,[Uuu
(rig ht-skewed d istri bution I rir.: u ud
o,oQ-tAA
n n uilv 6['t 9t cu ?1 6[9t ?1
arq
zg. tunr:iir:r o nm r 6 o rfi urfi m n-u rdo r "nr:t"i'u-n:[fl :6 n " Io uldttu u dr:r o
llw a a A e t A a a yq
ag ton?1ilfl I tl4umnna o:u't [Ton 0 n ofl 0 to
1.'irurur{rnrruhinr:rurn rfi ul:J
v ao A - - e
2. 9t 0.1tJFl','ln']il tn u?nunnt*ilyila0u
"
J v a d e ll rl
3. n1n1 il tR U? nUO?1 ufi9t rfi ua a{ lil rill.[Ft''ln'Iill.i''l
4. flqr ou uu u iir:r o qio r fi nm eri unsil:s nu nr:nlrfi ur fiu nr:tfu-n:ror:6 erqtql
**
-[a
tfr er il 6[ o lt'[#in'[ua ?1 ar {](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-8-320.jpg)
![rilns arfing.,rasr{ z 298 airrlnirrnr:'tn rfierru
iror:'rs#fio6fou imnrf,srcf'rc[ ni z
nar {d 1
ila 1 aou 13
ornTon ri n (A J B J C) - 40
9v
?lo 2 aou 39
6[aT a - a +(n-1)d$n1
ornTonri a = 31
8
a+7d:311
ornTonri a = 5a
92
a.+ 8d - 5(a +d)1'1 '
-4a + 3d = 0
1
Ox+ ;4a + 28d = 124
1
@.@ ; 31d - 124
.'.d:4
Lt?lu d - 4 tuO
?1+28 = 31
"'
a1 =3
01n oro = ?1 + 9d
?ro = 3 + (9)(4)
.'. dto = 39
ln rfr g e.r a a ut#intnn rr ar i
f,qr
n (A JB JC) = n(A) +n(B) + n(C) -n(A r)B) -n(A nC) -n(B -tC) +n(A nB nC)
40 - 17 +20 +24 8 7- 9+x
X-3
.'. n Itn JB) nc ]
= (z-*) + x + (g-*)
= (t-s) + 3 + (g-s)
= 13
o
@
@
@q](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-10-320.jpg)
![rcr6rrJ Frrfisrflran{ 2 299 eir rln irr nr :'[ n u8 ar sr
?lo 3 srou 2
;ntrari t - o
5oo,ooo - o.zk (o.o1o-,
7
500,000 - g
n
500,000x9 - k
7
rJ rurTnrt en lyirrYu 405,000
C(t) - 405,000
o']n C(t) _ 0.7k (0.9)t-1
o
405,000- O.Tx500,000xYx(O.o)t-1
7
o.e = (o.oy,-,
1 - t-1
t= 2
tv
?to 4 aou 60
d
LNTq LilN U
1.5 - 1.9
2.0 - 2.4
2.5 - 2.9
3.0 - 3.4
3.5 - 3.9
A
Ft?1iln
4
6
14
12
4
A
Ft?'.liln6[yfr8J
4
10
24
36
40
founvto.rer?'rarfrnynru#ruvt'yrfiifin?1ilfin.:nq -
24
x 100 - 60$t 40
-[n
rfr s il a a ltt#in'Lr,ror er er i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-11-320.jpg)
![7
rons nrf,srnrasr{ z 301 dtlnirrnr:'tnrfisEr
tv
?ro 2 grou 3 n'+ n
13
?JA
i,'i-1 [ ;
('.
') ]
(
^')(
t')
ornTovrri
9v
aou ?to
n4qt 1
q
13 +23 +33+... ..+n3
[;
(n*')
]
n2 (n + 1)2
4
n' (n + 1)2
{ n2 (n+1)2
n(n+1)
n2+n
4
-
(AB) 2
4
-
(AB)2
4
= AB2
-AB
-AB
-AB
lv
?ro 3 2 #qr'5' 2 "u1u'r.rrfiqriju#sr{un
rna 2q
at cX <rt
fln? rnuf $nFt?uurJ
tllsr 1 LLnr 2
I
e.t 6'
fln?un
frr{ 2 rj
ffn{ 2 11
A
HTO
n:rJq
vd
5[n?un
vd
6[n?un
A
I
la rfi n ar a a r,rt#in'[nn m ar #
ffn{ 2 11](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-13-320.jpg)
![r$nEJ srrfisrFflan{ z 302 rir rin irr nr :'[ a ufr s, &r
ty 91,/
uo 4 aou ?ra 2
9/A
lJ 0 2 9.lO
rir
9'/ 9,/
nl a<bLtn?-a<-b
o
!1 {T1r nlilUql a < b
9,/
e't 9/
-1 u'l9ltil?l.] 2 lj'l.l .'. -a > -bq
9y lr,/ .2
?ro 5 aou ?ro 3 x
nn tllT'ly$
_3x+1 :
[._:)'
_+
n rfJuonrrnuv dlr,hu zn qi'oruflluoarrnuyoiru1.
2.
3.
nn Lil{r'rv (r*,t6 ) x (r-,t6 ) -
fie L?lrry ( *
.
=X
3 2 13
_t
z) 4
2 3x+
4. nn Ly,trlr I r" I
913
44
1
aou z?ag -I2
(x+z)'+("-r)'- o
^2
+ 4x + 4 + *2 2x + 1 _ o
2x2 + 2x + 5 _ 6
2x2 + 2x 1 _ 0
^2
3x
tv
sa6
-2tln
zunsil,l"l
'l'. Jo)
r-'- fr)
[ 4 ,[ 4 )
d'lgn (Conjagate) to{ff1fi'urravfi'u
e./ 3vr . ey aa o c'vl e,/
?tilt1EJ t?ta 101,[ Lila 0{ u6[9t.:rlytrfl Lc{
-
nlaau il Loaytilu6 q
2
4
A 6l e'/
[1 lT'lY Ltl S [U6[AT x -il
(t/t d 9'/ Vl 9'/
nilnunqo{ Lq 1 uY,
U
13
44
1
2
-b +
la rfi s il a a ?b'[#5ntr,rn r,r ar {](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-14-320.jpg)
![-
rans nrflErarasri 2
303 air r['n ? g'r n 1 :-[ n tfi s e.r
lv
sa7
ila 8 nail ila1
I
lx
-2
2+5
3
aau z?a 1 t-3 , sl
*2 2x
^2 2x - 1s
(*-s) (*+3)
Lsetrirqou6o [-3 , 5]
H
@-30s@
t.lt ( x <rI
5l
x-5
X
X
lv
so I aau ila+ 11
3
(,'0 );
3
( 24't;
)
23 (-
8+3 =
3)
11
-27
1 n rfr cr au a o r,u'[#in'[u er rr ar i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-15-320.jpg)
![7
rilns nrfinorasr{ 2 30s eirrlnirrnr:'tn tfieril
z?a 12 aau
1
1-
2nlv
?to 1
1-
nu'ir 1+ I +
2
1
fl
1
+
4
[;)
m
[;.1
1
+
I
ldra
... + tulAol,[nTilLTIJ1nfUA
aq
n
f=
a1
1r
H'l
S n
Sn*t
a1 ( ,-rn*1
1r
1
11
^n*1z
9,/ ly
ga 13 aou zta 4 6
:-- -Y-. e..-^.r L' &
dr ur uL 6[lAyl'.t.1 Yl LLq n q1.] n u?l { 14 u ql
1 ,n*1
aulu't + t,tnrc[?::ri'q
1
2
(
'-'^ )
1 l.1)'*'
2)
1
1
2n
1 rn
[;)'
2
X
nr.,rtyn{ -+ ou5'u'lqq
cy
6 f flU?]'l.l
d
l,$FITfl?TTFI -+ rfru.:tHil
'[ n rfi s e.r a o ut#inlu o rr a.r i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-17-320.jpg)
![rans nrf,norao{'2 307 eia rinisr nr :'[a ufi s s,r
9r/
?to 17
9v
AOU TTA 4
ooA
n1alfr9tlj0.l f 9t0 (0)q
9,/
lJ0 1 fior LT{T'ry 6[run'lr f(x) - 0 6tlqroi'eLLfl% x olnnrlrnh^lu'ir
sYA,t
o
oiriu ri'r$ouuo.tfliln1T f(x) - 0 firfru.: 2 enqau
cr,/ A A o e-
1J0 2 llo rilT'tv Tqrilu 6o Lsala.:dru'lua3{ (R)
do 3 fisr Lvlltu t:ud'to.: t 6o [-r,*) (a1nnilill)
e,/Jaa_/&,JA
110 4 On Ly.lTlr Aqql'lflO 910 x=0 91.11{1,[ Fl'l9l'lqqnO f (0)
uqqq
rsafl o.:dru?%o3{fi fr eir rurnn'it 1
- l*l+t
I
D :R'
r
d'.rriu x # o
.ll
.. lxl
y1
a,/ &
9l.l1,[1,b y
R - (1,@)
r
tv
0f ou rra 2 410
oirloufl'rr uil.lrfju z eiru
eier,ufi t qirueirloun'r:riuoi'u +o u'ryr
'iAaAa'/;8g'/6il?%11 2 noQ1nffi urnl.lTn?{41nff?TTtuRilnnun.lu'lu 45
A'/
aqaoLtnuxfi'209tqq
9r/
sa 18
9v
?lo 19
_ l-l
ovrririlurAotd'
lv
gtou va 4
fior:n*r y
alY
o.rt,tl* y - 1
rflaooln T
0
0
1
aT9,'rrT
10 n LntS,JaTttTn a0.lo'lun1 [ou6['lT
aT9./rr
20 n Ln rilar#q'lil qi'0.:qir usirle u6r'tr
AaTiac,,rrT
on 15 fl Ln L8.JATil tUn 0 A o.lQ'l Ugt'l L9tU6['lT
r?ruoi'0.:qil ueirTq u6r'rr tuairufd 2lfluutu G0
alY49,/rrTdra
O.]ltU fltlT'lnqt0.l0'lUFl'l [qUfl'lTt]J1,{1.,:1.[ 40 + 370 - 410 U'lYl
af
n Ln rilar
10 X 6 = 60 U'l?l
20X8 = 160 U'l?]
15X 10 = 150 U'lYl
+ 160 + 150 = 370 U'lyl
la rfr s il a o ?bt#intuerere.r i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-19-320.jpg)
![7
r6rnEJ Frrf,srgfran{ 2 309 aiT rinirr nr:-ta rfi s a,r
fior;rHr A BCD
Itu'ir tan 60o =
e'ilruurrto': BC
Fl?'luU'l?fl0.1 DC
.'. y- ryxf =
"/s '..6
d'.:riu:snsvirt:rn'irt:J:snren:tiunti'trt = x+y = roo"6 + 65rA = ros.6 lilel:
ilazz srou ria + vl
Ft y LrlAl*d.,: fi rin 13 u uil Tv u'r tr* 20o/o t o .:ld'er v u%1.[ c[r n'j r 6 o Pso
H..roi'rLL14il.r P -
r ( N+1)
r 100
-
80(16+1)
- 13.6
100
Pao od:vu'irrqi'luurirfd 13 LLns 14 riu6oa{:ru'jr.: 89 LLfls 93(u
. 93-89 x
14 -19 13.6 - 13
X
+
0.6
2.4:x_&
O.ll*U Pao = 89 + 2.4 = 91.4
99
ilo zg nou rTo 3 fi'uoruro.rrirurin#runr:nornrirufilosJff'r:rfi.:unsneiruei'loun-rmrsfirirrvirrYu4quuqg
y . - 9+15+17+18+20+20+25+24+ 37+35 ar u
1J0 1 nn x - = - 22ntnn:il. ?ltu.t 10
rio z nn fi#uto.rneirufiIornr:r1u = 35 -7 = 28ru .t qI
av '
qjl
fiffita.:ncirriloruar:nrfir = 37 - 9 = 28q.ua
rio g fio il-rud1tlta.ir'llounr:mu - 16 + 20
= ra ttn;ril'uo'ruro{riTousr:ufir = 20@ql
2oqlu
rio + nn fi'roruro.rfiTouar:vYrililo = 20qt4.u
aq,fv
o']uiluilTo.ltJ [qum:n{?luo = 204qt
t- 195
{3-
.J;
195
v_ ,.6
65.6 tilar
la rfi s il 6{ a ltt#inlun ?'l r,r i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-21-320.jpg)
![rons nrfinrrran{ 2 311 atrrin?rrnr:'tn rfrsil
lv t,/
ra 26 aou ?to 2 B
oQ9'/Aa{-ra/
fl'll4l.tgt LHLnrq?T'l?l 4 t?l'lnu x
j
LnTqLilnU = 2.75
(g"z )t(z"g )+(rl.+ )+z* - 2.rs
Ig * 2+1+2 ]
6+6+ 4+2x : 2.75
I
16+ 2x = 22
x=3
-&vaA
fr.rl.rl.r [t1q a.]fl ou lr:rn 4 loirn:q g
9/ tYioa
?lA 27 AOU gA 3 nltilnutnlntun > ilr',Ud',tu > d']ltl*Uilp4
nt
ria U n fi rrivl't.tr?'t (Ri ght-skewed distri bution )ru.
nu'ir orufiuil < ris'soru < rirradutntntflqrop
r7o zB srou ila + fiqrouuuuiir:rosia.:finmrui uns:J:;nunr:nfrfiurn-unr:tdrin:rfl:6er.ll ,u
e l" - A y A oqrllls a d A u qyv o j
nn:*il3?1fl'tn6u11fln1a{Hqau?10sYt't[14 [qn?1ilnqtilu [nu?nun1TtTlJo:tn:o9tytoq(ll
v a e e 4 v t d A - q eu
nna 0{ n 0 ila 0u a 0{ iln?l ilTun;lJ:rflljn1Tiltn u? nunlT lxu gtrrnro aql ,u ,u
ln rfr s il 6[ a r+'[#in'[r,ta ?1 il i](https://image.slidesharecdn.com/ma2-130704101541-phpapp02/85/Ma2-23-320.jpg)