csir net december 2012 mathematical science with with answer key
TRANSCRIPT
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tff11 'C' it 20 \m7W Gift OTh1 lift vm.ofl 1
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'iF1 'C' ~ ..,;, v.- lit fl>ll - o;; a# im'J - W tl IM:;c !1R1 ~ I W#
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JA1 ;'11!1 <rWu/1 rt J1r;' .,.W., grn<J "' W1$IJf t I
8. ~ 05) ~ ~ "" q'"if W ~ fbtf >/~? '511 >JI 'ltf Hftrn mftt/ I
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(i);lfl llY'f •Y lii?Mil Ill •NJ i/ J/rvf/ """""' !1'1d #'II I
"" ... ...................... .. .-if ;m 'Ill '1'/ ormtiT8 "' if ~ ll<ill1 ( I
.............................. S/07 RD11Z-4AI+-1A
S/07 R0/12-V.H-1 B
( '1171 A )
I. (><- J2) 2. (n -2)
3. (·: -~) 4. (H2)
2. ~ ~ ~"'iifm &i<ljftt d; 'iTf'/l <t Jim Y71lfrr1 W;fi >f f. Jilv m<l u'n!l :fi11 10-00 41<? # 1 iff'( <til ,. .r; ilj1'T ~ 41<? 'i ;j
I. 60.00
3. 12.68
2. 47.32
4. 22.68
3. im1 'r <mfil "''Ff' ti1mt "'fvm 1P1 u;m AB tJ rrfif AQ = 2AP rh f.rr.r if W ~ ~ WI 3?
~ A •
I. LAPB=~LAQB 2. LAPB~2LAQB
3. LAPB= LA(}B
4. LAPB=!._LAQB 4
3
( PART;J
I. (;;- J2) 2. (:r-2)
3. (;-~) 4 . (H2)
2. 'l'hc <m,gks of a right·angiOO triangle shaped g;~.rdcn arc in arithmclic progrc.ssion nnd the smalkst side is 10.00 m. llll~ total length of 1hc fenc ing of the f,Wdcn in m i$
I. 60.00
3. 12.68
2. 47.32
4. 22.6$
3. AD is the d iameter of the scmicir'Cic: ~s shown itt the diagram. If AQ = 2Al' then which oflhe following is comet?
• • I. L APli=~L AQll 2. L APB =2LAQ8 ). LAPB= LAQB
4. LAPB = !._LAQB . 4
4. rmtmf $ '{Ui ~A i1ft W<ui&Jt 25% rrflrtTd 4. "'? •~ 1l "'flf) t ""'lilt WV B <til ~ ~ SO% kliri# t I oR "1.1' A >!lr D .,;} q;/'IT'f ,;:;td&n~ WTTR '# ffl ~ iTfi ;mnq 1711.!! B $/k W~t< A ;j; ~ <it fflFii 'liT &JWii itTf1 :
The robbil popula1ion in cooJ.munJiy A i.ncr<.'3.'~l> 3t 25% per year while that in B increases at 50% per ycat. If the present J)OJ)u l stion.~ of A and B arc tqt.•a1, 1he ra1io of 1he m1mbc1' of the rabbits in a ·to that in A after 2 )·e<•ri> will be
I. 1.44
3. 1.90
. 2. 1.72
4. 1.25
I. 1.44
3. 1.90
2. 1.72
4. 1.25
•
'
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,; -1rd tr >i1 fRi Rr H • ~ ?W w ~,.... ~' ;;rrtft f + " ,f ,.
~·1<1 \fl.-,·~ "fo? $ P!.J.''(;J' ;n rft'qlfN I 50° ff rrr Nmr \fm1 ~ m '5r? .. •,r m ' ~J!tff.'~('f If~ Cif/(11 (!I I r~11( G.'f;{ ;p, Si.-.'uN i;'r•tl cll/td
1'? ., .. $J'./~
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6. trfkP <! .. '-!J-:f:i ~?. ~ $lP1 wif * ~..t: m-'IN «R" t 1 sa ~:::..J Jt/ .. ;-:r,r;-p trr ~ 6
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4
r-,.u ~~ t3eh of 0; and H1 Ill: in N-'0 St"p2t8tt C()tl(3:inctS, c.xh of \OlUnv V0 3r.d at l SO <-c 300 I acmosphen\ 'The C\\n Jrt" nl3de to ua.::;. 1n a lhm:! c<>•':alf'l~r to rorm water vapQUr unlil H! ts exhausted. \\'he-n the tcmpc:r..uuce of the mixturt in 1he lhirtl cont:unt<r w:t.s n:~lored to 150 "C, ltl; prtssurt became l Olmo~ph~::rc:. 'llle votum~ of th>! 1]1ird comnincr mUlll b'l)
L v. 2. 5Vc/4
3. 3V0r2 4. 2Y,-.
Hclium mel :lJt06 PSf5 in '"'0 ~ (:(l;'lUinm
.ue at the $:!me ~~~~~ and so tuve diff~r~11 root-mc-m-sqtaart (r.m.s.) ~·c&oaliCS. 'The t\\'0 3tC'
rni'\~ iu ._ thud C'OfiQU'KI' keepirlg dlc S3mr
temp:n1urc. 'Ol:- r.rrt.) ~·clOCIC)' of the helium atoms ut th-e mi:uure i$
1. mf>rc 1h3.ll what h w:.ll before mixing. 2. l<:s..~ than what it Wt\S before mix ins. 3. .equ~l to wh:u h was before mixing. 4. t-quul 10 lh:.u ofat&On lltoms in the mixtutc:,
7. WiJ'1 ',J} iJ.t(;N';'r if tmfl·,ifi ~ il't1 IJ'qflt'f R1m CfmT # ;;rf;ftJ; ~ 7. lllc mincrJI talc is used'" the manufacture of soap
because it
(0) J1<iR "" """"" .,.,; t {b)~ .. .,j~~ (C) 'JW ti!7 ~ (d) :pm~ t .Jh? r•/IH :R ~ ~ I
:r-rj"'' ~ I) u#r/11/;r,m'!"""' ..-1) ~/.~ >
I. (d)
3. (•) "" (b)
2. (!!) Jllol (o)
4. (n) """(d)
(a} grvcsbull; 10 lllO pn>du<l (b) kills bo<lello (c) gn-.s fr.lgr>n .. (d) is sot\ 3nd does not scr.uch l.hc $kin
Which of the atxwc sr.uen1cnts Isfan; <::orre<::t?
I. (d)
3 . . (a) and (b)
2. (a) Md (c)
4. (n) and (d)
R. 100 Wf -Jit.'iri0ti l <l!fint; X·51h0 i.W ~~· ~ 8. lllr<tfflt-f J'rJf~ g. ~ ~ li l SO"W. li'f !10 PM~ U<Ji (i$1 ~r.n I # qwf ;t Q'tilln' J.~
~00 & of an inor&nnic compound X·SH~ wntainint;: o. \'O):nite mtpu.rity wa~ k.epl m "n ,we-n at ISG "C for 60 minut-es. Th: wcaghl or t.hc residue after h<:3tirtg is S g. The petenJt3ee of unpunly in X was ITi1 ¥>f s WI ~~ I 1 x ~ Jt§f; ri s;fllw
<1:
I. 10
3. 20
2. 8
4. 80
9. '!11f RTi• v;f!) >r ?f"'""'" iii! t'>' :Jo1!-tll: '·7 1 ow ljf;ll wf!: q/t w I. ~·~ 101/ilvJ <{{ em 2. 7;41 IS!ir.r iill45° .. ;Jofro ~~"' ~)qr 3. J,TI./iltl ;I; i~Nb'Z') rn ll:11f
4 wtnt ~w t/l rim"
1. iO
3. 2()
2. 8
•. so
9. On a tercain t'l ighc the mnnn ln 1t.s wa.,ing ph3Se w.lS s h3Jf-moon. At midniQI111hc moon will be
1. on llle eas(em hori1.on. 2. at456 angubr J:ei&h1 ~bove the eastern hori:ron. 3. at the 7.enilh. 4 . on the we!\tcm horir.on.
10. ~ ~ , ~ ..... "" s f?'l 11'0 ~-t/i,u ~ f1I'R'IT # I Jtl%rt=J., ~ tO ~ ~ r<'l f/1 llhl ~ ~ oil ~ 601)
~ l1fi} ~ t I ~ ;f; S ft.r m: ~~ •• oil ..a.- Rio'~/ tMt 1Ift ~ 3Mgsfrlt'
1.300
3. 2400
2. ISO
4. 1200
11 . (/'iff 1'1«$ 1/1 """""' #?;;: """ <;;; .,.. fW # <flfil >f:JIII~ t I ~W '1/W "'' 'J{itil oil fit ;w filvs W ~" tm wr11 lh .. ~' qf?<rf:r Htf. wt.J mmt
! • ' . ~ .' o .
' • . . . ' ' ' • ' . . ,. .
• . . • ' ... ' '
.. .. f f
' • ' ' '• ....
.. .. g • '• ' •• ' ' ... ""'
11. t_ r .... r-=- ~
~-u - =10\ ,_. - , 0 \
"' ,.
10
JO. A £!:m~Wil1o ill itradLatcd in ~ nuclear reacror for 5 days. Ten da)'l after irradl.llion, lhc :activity of the (hromi~o~m r1dioisorope ll1 the scmston: is 600 dWn.ICVJhOns per ho~.a. Wb:n is tbe aetivity of ehronuum NCIIICM$0topc 5 days 11\er irr3d.iatioo if Us halfhfc ISS <lays?
l.lOO l. 2400
2. ISO
4. 1200
J 1. Displacement ... mus lime eurve fot a body is shown in the f'igul'(. Scloct tho groph fl'mt corrt."C.tly shows the v3ritliCin ofdH: vdocily with litne
' ' ! • •' ' ' . ·' ' ' ·. • . . ' . .
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I. ;~.o ,,)en 2. 2.9 .L?t;m
3. 3.i .;m.lil 4. ~.$ ~1)'!
u. fZ'i, ~ i1i ;:).!}' t¥ ~ ~ qt.' :; r,t; :m: '*~ ..,fi~ £! f.t; 'mf: rr.nril ;ft\:8 7'i"ft t I F' ~ ~~i r-Hl} ti ~~· mr. wer ~ g:;ri wr?f: f. fiJi mit t*m ;r:tt ~ , ¢;-'<m .... t yuri ~ m arm ~~ f).~ • ;)1(1 ~} ~
L 'ttl>' ~·:1 2. IFl/ d~'Jrf
J.. if:R ,..-Jf 4. r.;tlllfl~
I . erri! ;.;.r& ;:<)1<1 ;6 q=l \t a!if :::r-ft * ~ff, f.1 ;,;m; F
2. ffTi"i c;$ t:~r.s tt~ ~ E «of qr-;) if 'i~r;f;r itl vmrr t
3. ;.rrt.r<t· ;i rrr:.:wf ;nN ut;! <GT fFWi m: Jltai {:
4. <Jii'mlt' t(f{;' ~ 1j<: ;r(f rm~
15. ;;,'J~~ ftrn i{ t:r,ki JtJ'i1'7 fff( rt't?<!'tl trn A. B. C •il7 D q <;w J!Rll m. R !J1fl t 1 :t•h.'/} :.U(J ti; ?f.!ffl if fi!rr.:! if W (fit="(ffl 1/KR mft t•
I. A m;r;rr(Y( i! IJ'~'W up; ;;;rm. 8, C. 0 J1k R :f 1
2. R ~ ~ tp.;"J,-~ ;r.;m A,B, C .J<~Df1 3. n >W1fi11i t 3m~ elY, !tr'f(tc, s. A 3)''?' R ~·~ ~.A it"'~· f. :;rn;/; W.: iSf(t,o; R, 13. C #iY D tJf
'
6
The spring balance in Fig. A n.•<tds O.S kg and the J>an balance in fig. Breads 3.0 kg, ll1~ iron bl()l:'k suspended fmm tin:; spriJ1g, bakm::c is pani :tl~y
irnmcrscd in th:: water _in lh t:: be.-.ker (Fig. C). Thespring b:tbnce now re~s 0.4 kg. 'fhc rca.d ing on che- pan balance in Fig. Cis
I. 3.0kj; 2. 2.9 kg
3. 3.1 kg 4. 3.5 kg
J3. TI1e ends of a r(l~ are fixed to two pegs. such d\3t the rope r~mains .$lack A pettcil is plact'd ag:Ainst the rope al\d mo\·~d. such th:H the rope always remains taut. 'fhe sha.pe of the curve traced by th<; p::m:il would be a parl of
L a cir~l¢ 2. ~~~ ~Uipw
J .. 3 square •• a triangle
14. During ice skating. the blades of1hc ice skater's shoes exert pressure on the icc. Ice sk<lt:.-r can efficicnlt}' skill¢ because
l. ice ge(.S <:on~·encd lo \\~i~c as ihe pressure exerted 011 il increase.~.
2. ice g<:ts convcr.ccl to water as the pr~S!:>utC CXC11Cd Oil it d~crt'<ISCS.
) , lhe density of ice ill cont.'lCI with the bhdcs decreases.
4. blades do tlO! ptlltlrate into icc.
15. Four s~'dim~ntruy rocks A. B, C and D arc intmded by :t.Jl i.gr\eoos ruck R as shov.·n in the cross·scction diagrum. Which of the foiiO\\'ing is corroc:t aboul !heir ages'!
l. A is the youn~est followed by D. C, 0 and R. 2. R is the )'Oltngest followed by A, 8, C and 0 . 3. D is the youngest followed by C, D. A and R. 4. A is the youngcs1 followed by R, B. C and 0 .
16. ""'"'" ""~' """' """'" "'!I' ~ t ~.,. fi<1r.1 /In h~ ¢1 '1"/ t I
I
~
Stra.-.-
I . !imml ~ ""' i!ffl Jfi'QI'I<tl1f'{!/;r; fiPijir .mr t 1
2. /ilyrrt17 fir~ If<' "'" ,..ht:>f,, """ ~ ,.., t I
3. fJlwffm Q:;t w Wa JJPH p #:PNf • HM .. .,
Jt(1 r:rrm t , 4. RVJMfff r.H ifl cs>u r;~m -. ~~ {q<IJCf t) rirlffl
t I
17. <"vii Hit >(): ~ Rl 'Wr't IK f.l•it >iimfi) <a Wli1f/1 r;t;1 ftl :fti It;} ~ !iF8 if t:lfit:1 r;:zi t I
,., """ il """' ~ ....... , '
:II
I •• ..
) , <:"'tl'~ ftoT ~ .J.'fflW'R
2. 20 >I R~ 3. 20 .. )0 ~ .. ""' 4. )03 .:0 ~ill '*'
18. NT« (R) >1/Jf/ or# r:<n <f,) (T) r/IU (1MT ff19 ~i+·~tmlt) at f'1i rttR (r) 1fr.it ;:m .W:t (t) ifh) ?t ~ t';i;1f1 rr.t1 P..7 J mt ~ ffrmt ;t fPWI fl/f!ll11 ~ r# tva :s ~ fPI;:: ~ W:r t);tJ
"' ;pi$ .,. "" ""' ....., ""' , •
J. TtRr x TtRR 2. 'J'U\r> urr 3. TTRR • ttn
4. TTRR x TlRr
7
16. The S-train iJl a solid subjected to continuous stress is plo«cd.
I
j
Strain--
Which of the following s1<ltemcntS it 1n1e'?
1. The .!tolid dcfomts daslically tilt the pomt of fadun:.
2. The solid deforms pl...,.,.uy om <h• point of failu~.
3. The.solid comes b::k to onjpnal sh:tpc: 3nd srxe on failure.
4, Th.c solid Is pemumentlydcformcd on failure.
11. Growth of an orpnism was mon1tor<:d ac regular int.ct'\'al$ oft~. and as ~-n in &hi: graph lxlow. Arotmd ~C'h lime rS me rate of growth 1,1:!0?
l!
I .,
I. CloS<to<by 10 2. ~d.ly20 3. lkf9.-un days 20 and 30 .:l. Rc1v.•ccn da~ 30 and ~0
.,
IS. A 'IOU pla.~n widt Hi!d s~-ds (bolh dominant lntils) was C:COS$ed wich ~ dw:nfplant w11h wh1tc seeds. II 1h.e segr~g:uin~J p:og-~tny p1oducW cqt.~3li'umbcr of ull nxl and d--.-;wf' while pfanb.. "h .. t -.ould bt lhot ::}."00t)'1)e of the p:rec\:s?
1. 'ftRrx T1RR :2. TtRr x urr 3. TTR R. )C ucr 4. TTRR w ·rcRr
_____ }
19. dJ:-, 1f!fdt rf.lh ~ ~ <:71ftft rn:i Q~JflmY if '{ffl If#/ fiT
qr., A . f:trn< P'7 't ~"\:! B ; ~~ flr-i if ~7i.i n; ~ ~zrC : J~t.l? l1 ~ 1:81 q
I. Q'h-i B 11ft WT~il7.''f ~ > ith~ A 2. 3. 4'iti C iJl QUUV"!' <:7 .a. rfN A fi'O llll'IQIH
<' -'· ~ C 1fh iiiu..rw.l <:i" > ;f;$ A fft WlN?H
v > *' 0 11ft qji/(4(;'7 "
• •
•
•
I. a., aid ali+ .. . = <d(l -h) for !bl<l
2. a > b il mlf'fli a!. > bl
J. • (tt-hr u-'l + 2ah t-b1
• a > b ~ dk<fti -a < -b
8
J9. "Jlvte sunflower pi:m;s wen: plaet<J in co!tdiuon!> ii.S
indicotod below.
Pla.nt A : stiU air Phtnl B : modcratdy turbultnt ~.lr P~t C : at~ll air in the: <btl:
Which of l.he following stncmcnls is ~orrt<et7
I. l 'ranrpirtuion rate of plan I 8 > that of plane A. :2. Trat1Spita.Hon rate or pl3n! A > lhat of plant B. l Tnanspin.tion r.tlc of plant C • tha1 ofpiwt A. 4. Tr.w.s:pirotion rt.t.e ofpl:snt C > IJwt()f pl~.nt A >
"'" o( pbn B.
20. Which of the foltowing i!> indkutt<t.l by l11e :~ccom· ,onying diagram?
• •
•
•
I. a + <1b + ub' ., .. . = al(l - b) for lbl<l
2. a> I> implies a3 > b3
3. {o•b)1 .. ol- ltJb Tb)
4, a > b implies - a < b
•
9
[ 'I7'T B ) [ PARTB I '
21. VfP: 'MATHEMATIC$' <6 .mm' It U0Tm1 11 lt. .~ m<: ·~It) Vfl "'"" If ?
The number or word~ chat can be formed by pcrcmning the letter$ of 'M/\TI lEMA TICS' is
I. 5040 2. 4989600
3. II! 4. 8!
22. so.ooo .. - ,,_ ~ 1'
I. 20 2. 30
3. 40 4. so
I. OJ7llt (ATB) = ..tit (A)+ V11iil (0). 2 om'i) (A+B) S tV1fi) (A)+~ (B). 3 . 01/li) (A +B) = "l"'!" { ...mt (A), 0!1l.l
(8)). 4. ~ (A+B)= ~{...mt (A),I1fl/il
(B)}.
24. 'IT'/ ,.
n .
I. 5040 2. 498%00
J. II! 4, 8!
TI.: number o( positi~ divisot5 of SO.OOO is
I. 20 2. 30
3. 40 4. so
Let A, B be: nxn 11~01 m~tric<:s. following statements is correct'!
Which of the
I. r.lnk (A+B) • mnk (A) • ronk (B). 2 rank (A+B) S: nmk (A) .... rank (8). 3. rank (A-B) • min {r>nk (A), rank (B)}. 4. rank (A•B) • mox {"'nk (,\),rank (B)I.
24. tet .r.r x J = {'o-"" for x o (0.11 nJ for x"'{lln,lJ
{l-nx
/.(.<) = 0 forxG{O.il nJ
for xo{l l n,lJ ffl rnen
1. limf.,(x), (O,I)'IHimr"""" 1111 qfNpn ·-I. lim f,,( x) defines o, con!lnuous fimccion on ·-· (0,1].
llmfl i I 2. {/.,} convet~ts uniformly on (0, I]. 2. (/;,}. (O,I]~o'7 q4'i'd'i,., ~ 1'rlff t 1
3. ~ xe[O,l]<t .I>\' /lmf.( xr 0; 1 -4. Wxe[O,I ] ;f; ~lim f.lx) l1f1 ~ ·-
3. limf.{.T) • Oforall:r, (O,lJ. -· 4. limf.(;r) e.,isu roullxe[O.I). -1 I
25. The number .fir•s is
I. qf"#1;f 'ff~ t I 2. ~frl 3. ~mrme, 4. ~#0211~ I
1. a rational number, 2. a tronsccndcntal number. 3. an irralional number. 4. an imaginary ''~unbcr.
(~-· Q) Aa . 0 <;
~ ,.= (\' • '':, v,)c;:i' ;t f5ii 1'"1, ,r, ~ • 'I ''!'""""' ........,. c· V vAv ""<"' " , " iF.1 '•J(·:;r: t' I :Nrn w • (l.l.l)ill!w!,
l. 0 <i: 11'l1'l F I 2. 1 ~ iPlr-1/ I
3. -1 "' '"""' I I ~- 2 * (rr;pJ t ,
I
10
26. Ltl (:be a primitive cube rom of un11y. l)etine
For :a vettor v • (v1• v:. "'')<(~'define
l '' l4=Jiv,.h•rl wht-re••1 islTan:.~ufv. I[ " ._ (1 ,1,1) 1hcn w1~ eqtt:tls
I. 0 l . ) 3. -1 <!, 2
27 .rrr~ ii5 ~ - i(a .. a:. a)): a;GO, 2, ,1, J}, a1 +a.;=6}. M ~ ~NIIJt~Y't/J~ if~'l I
3 1 + Zi . Lt:t M - <(a1, a!, a}): a;t= P , 2, J, 4~. a1 + a:+ 3} = 6}.'fhcn lh~: number ofclcmunt5 m M is
l. s ). 10
2 9
~- 12
28. (38)'0 " q;r :J.'til>t >ilK~ I
I. 6 2. 2
J . 8
29. <1¥P./2ih ~ ~ 'f"~""'""n(n~2) q,'fe tl~ 'fPo' ?r~ 30/d/ft A- (a .• ), n11 .. 0 It ~'11 11'#;: ~~ fiN! t . .
(n1-+n-4)!2 2. (n2- n+4)1l
3. (n'+n-3)•2 • . (n'-n+3)>2
l. $ 2. 9
3. 10 4. 12
28. Tht: I:L<~t diait uf (38):o11 is
l. <) 2. 2
3. 4 4. 8
l9 The d~1l110n or the .. -c:aor $pac(' o( all S)'tl'l:r.r~c munccoc A.,. (a,-.) ofO'l'der n)f;n (n ~ 2.) with real cntncs, a11 - (}and tmce 2cro 1S
l. { n'•n-4)12. 2. (n1- n+<l)l2
3. (n1<n-3)12. 4. (n' nl })ll
JO. '"~ f.k 1 ~ (O,qc::< 1 x<?. .t flM .,.~ ftlr 30 let I - (O.I JC:<. foe- x~::<. 1t1 'l(x) = dis•
o(.x)= :;,It (x..l) " tt"""'" {tx-y!: ycl}, n> (x. I)= tnf (llc )' : yel}. 'then
I. :.:tY? (o'JI.t ~X) .fmffCf ~ i I. o(:<} i:l disronlinuous iJomewhct'C ~·) ·(,
2 ~CR 9(x) r.mr f. "FIJ ~-m:; x = 0«1\' 2 9(Jt) '10 OOClliUUOUS 1.)1) 2 but llOt ~l:llliii U.CtiS!~·
"""'-~ '1'.6 I ' diff4.-re!'lltab!ee.ualy3tx ~ 0.
.l. ~W o(x) ~ l 'ff;J lR;Xtt.~ 't., 0 ~ x., 3. Q(x) lS IXIOIIUUOUS on :::_but no1
l W 'H."f(f ~~Nf·f:ll 'IJ;.": # 1 <:Otl1ln1.1<rli'IY dtffcrcnlr.~bfe u .,cd:r at x-
4. ~'IT l,'(X) ~fh.· 1..~ I 0 ilnd Ill .X - l.
• <;(x) i$ t.l iff~.:n.~u :i:tble on R '·
11
3J. 'TiFf fit; a.. • sin rt/n I JJ;7f a~. a: .... .,t ~ Vf:irq; .3~ Lot :In • sin ltln. For tne sc:quencc :.1, a:,.··
I. 0 t U <1lf R1'<r ~'tlrT t I the supremum is
2. 0 t 'I ~ lll'if'J{i iflm I 3. I f'lllf1R'lf#m/ I
4. I !. • q 11/Vf '1tf f!Ru I
l . 0 Mel 1t is anaincd. 2. 0 and 11 ;s noc. aua:incd. 3. I and k isaltrin:xl 4. I and tJ. is not attained.
32. Usin~ chc fl.ll:t lhat
-t- 1 "' ~ 1 , cqu•ls f;,r=6· ~(2n +l)"
,. , I I. 2. :.:....-1
12 12 I. •' 2. •' --1 12 12
•' ' 3. 4. ~-1 8 8
•' • 3. 4. .:!.__, 8 ij
33. 'R'f fit; Ax.y) • u(x,y) ~ o ,~x. y)n "" I'"' 33.
~ """'f: (-( lj I
l<tf. t-C b< • <O<q>kx >-alued funau>n or lhe fomofi:x,y) • u(x,y) •; '~'· y).
Sup~ thOl u(x, y) = 3x1y. 'llfl IW u(x, y) • 3x1y, 8)
I . ("' v '* I'GiiJ >fl .... T ""I ~ <(./ i.liT'R11 I
2 c.,. y • """' - 'R I t<l><l~ #'7T I
3. (({f v <8 ws8 rmrr w J t=)?.lOJiMw ;!tw I
4. u IW1iii<T•flq 'f{f ~ I
34. 01'!' fit; f: il<' X :<1-R ("X fii/fGtl1 •fl!fl;7 t. .,.;iq 8'f •• II $f(f1 if ~ t 1 m (V, IV) e!?.:' x ~~- fir# {H. K}e:<l x ~~ tn ~"it .,...,.,.. D f(V. II~ f.tGr <9 R>.n """ t •
I. J(V, K) ·•/(H, W) 2 /(H, I<} 3. /(V, fl) + /(W, K) 4. f (H, V) + f(W, K}
3s. .,...-.,., ) ""' 11711. ,..;; ""'R"' arm ~~~t >;R-n ~ N '""' 011') 1 S: N -N 1lfl ~ # (SpXx) • p (x + I ), peN 1 ol (1, x, x1, x'l ViT ffP,~ ~ v} fiQ 1f f.. STf 31/UI"I .;• 5 1fiT ~ ro lliKR ~~~ wmr i :
Then
I. fcannol be holomorphtc on ( for ""Y <h6it<: orv.
2 /is hobnorphicon C for a S\U1ablccboitt of v,
3. /is holornorphic on (for 111!1 choices ofv. 4. u i$ not difTeretuiablt-..
34. Lei {:: 2: x R~-R. be 3 bilineo.r msp, i.e .• linear 1n
ach \Wb~ ~kly. Then fot (V, W) c;:3.:
x :~:.the deri\'36\·c 0 f(V, W) c\'lkl3:ted on (H.
K)E~' >< R:1 is giva~ by
I. f(V, K) + / (K, II~ 2 / (II, K) 3. /(V, H)+ f(W, K) 4. /(11. V) ~ f(W. K)
35 . 1..c1 N be W v«t()f Space of all m1 pol}-nomiaJs of degree ot mos1 3. DefUlc
S: N -·Nby (Sp)(x) = r>(x+ l ), peN.
11ttn th~ m:;l(Ci.x Of S in llle b:btS p, X, X~, xh. (OC\Stdercd os ootumn \tt10t'S, IS ~vcn by:
f1 0 0 0 I I I
0 2 0 0 0 I 2 3 I. 2 I) I) l 0 0 0 I 3 0 0 0 ' , 0 0 0
I' 2 ~l 0 0 0 0 I 2 I 0 0 0 3. 4
l~ 2 2 3 0 I I) 0 3 • lj 0 0 0 ,
36 . .l(l'r.'' /#1' r.s ~ ;mr. tf-t~~ m 1 f!ii.A •lxeF x:::: t :r ~\.-1. k < 1 m m' ~ ~ <t
ffi.·) J I H) A ~ J(:tr;t'il> •If! <H'Gtli t :
I.
3. 3
... L; :S")
2. 2
•. 6
2. "'<,I) 4. ~~II £ J5
38. "'10 0 • 01.' ;f; .,. • • 'nm ""'Q c ~ l1f'170
r,_.f/M "II rf ~'?li{ 1ft ~ 1 I fl /ill R \~ EJ"';f(.~t,p fr~J / I u> 'Mf :Jf!,Q 11 i1i! I("$ q.$1 <Pm!Jffm t?
3, t:"r fR'f) J.~ ~ 11fl Jt,ciiW>rf1 "lf'if t 4. ili'ft '1t.1 I
I. I(X) ~ S(X) .n>(f'CI"f/q t I
2 l(x) Jf«;<t>?"T• d '''-g g(x) o!/1 ' 3. &'(~) JiS!jitON::t ~ W'!7 1\x) :rt:· 1
~. ~ fR I(X)Z.V. 'f ~i g(x) .,;(t§a;'llf,zo t I
12
I 0 0 0 I
0 2 0 0 0 I 2 3 I. 2 0 0 3 0 0 I) I 3 0 0 0 4 0 0 0 1
2
~1 0 0 0 0 I 2 I 0 0 0 3. 4.
2 2 2
~J 0 I 0 0
3 3 3 I) 0 0
36. lm Fbe: fieldofS<iemenosondA • {xeFJx'= I and x'~l foe :tfl n:nur•l numbtrS k < 11. Then che m• mb~c of e!em..:nl$ •n A is
I. 2. 2 l ) .!. 6
3 7. The powtr scrie11 f J." {z- t i " COilvccges tf
I. ~1~3
3. ~· II <.J3
•••
l 1:1<,/)
4. 1:-IJS J5
J.S. CO:'I.$Lder L~ group a • tb'Z ~o~,·hc:~ Q And z are lhc
i.'J'WPS of nu.ioo11 r.umbets and inttam ~tively. Let n be: 01 positive uut"et, '0\en is there \l C)'clic linhzrour of ardcr n'!
I. na1 ncccs.s.srily, l )'(S. 3 ur'l~ Ont.
3. )'H. but no1 ne<CUMlly~ urtiQUC' one. ~. n.wet
l . f(s)and s(x)tarl!' irrt:duciblc 2 ('(x) is jrr~duc1b lt.:. bul &'(x) is noa. l. a(x) is irreduribl~>~. but f(x) is noL .;, ~&her f'(x) nor if a.) rs lm."dcabk
13
40. Zc1JJ " Zut) 1r.. <t &!Ji9 il&17 "6W>Ji'ttu <1ft 40. 'l16IIJ , '
lbc number o( non~cnv1sl ring hocnomorphjsms from :z.,,, to~ is
I. I 2 3 3. 4 4. 7
41. flldilor '"" - y'(t) = l(t) )\ t), y\0) = t <m r: 111:-11\""" t .. mm 1m ;u- ~"' _ .. I , 'fjf1 1' iff fM "'""'' liS" ffl t)d i! I 2. II\ .. 1!11' ., tl'l Smt t I
3. R 'lr 'IJU ( Iii 1M 76'1'! l[H 'ltl E1oT 1
4. o 1.<1 ~ ,; 1!11' .;hroo; ii ""' m m t <1RJ 'IJU f Iii !itt R .,. 'It/ 1
42. '!PI Ia. ""'"".,_ w>II<>M u"(t)-4uXt) + 3u(t) • 0, t •li< ill w<li '*~ .,U.., ~ v 1 unv
I. la'll 2 rtll Vl' 'lf!l'itf# .trn wrl<i: t 1
2. f'J'IJ I rt/1 Vl' rmrt~or .trn 'l1'lk t I
3. 111'tnt VW !JfU r&((it u=O rrfJ 3P.ffif;:e iiRffl t I
4. lfl>r-5/i/f 111 t/ """";r;) 3Rifil<: l/m11 # I
{
I -.• • ~fie" . r>O. xe IR
43. "'""' u(.•.t) • 01
.tSO .. reR ~ w>f.!:sftl' 1lf1 p,,., JrifRm ~ ~ tt:i.:f t :
I. ((x, t) : xER. tG~}-
2. ((X, l): XE:i, l > 0} <TRJ ,....""'
((X,l) : XG:i, t < 0} 0 'It! I
). {(x,t) : xER, tER}\{(0,0)1.
4. {(x. t) : xeR, t >-I}.
I. I 2 ) 3. 4 4. 7
41. Consider the )nitial value problem
y'(t) • f(t) y\t), y\0) = I
where f: ~-!R it: continuous. Then this initial r..luc problem has
I. infiniccly m3t\y solutions for some f. 2. a umquc sofuliOn i.n !.It
J. no soluhon tn R for some f. 4. :a 101uoon in o.n incerr.al (Ontainirrs 0, bul not on
!Horsomo (.
·U.. Lc1 V be lhe set of all bounded solutio-:l.Softhe ODE u"(t) - 4u'(t) • Jl>(t) =O.t <::!
Tllcn Y
1. is~ rc::.l ,·ector spa~ of dimension 2. 2. is G rcaJ \'CtiOr Space of dunc:n~ion 1.
J. conU.IUl ~ only lhl! l'ri vit~ l ful\ction u=:O. 4. comains exactly two functions.
43. The function
{
1 L -,.,~It • l >0. XE IR
tt( X ,I) : ,JOt/ .tSO,xeR
is o1l ~Judon o( lht ~I eqwbon in
I. ((X. I): xGl, teR}.
2. ((X, I): Xf'!i, t>O)butnotintb:sct
l(x,t) : xol. 1 < 01.
), ((x, t): xeR. tE::!I\((0,0)1.
4, ((x,t): xoB. t > - 1}.
44. 'f!dt ff:}f~ tf, J.'ifii<n 3,'11"11 ~4F&ro
u,. yu., ~ x'u = 0
#
I. ir.Jl XG~~. y~Jl ;j; fir.) ?i~a ~ I
2. ~;;I} '<ti ~. yeR 1$ (i:td VNC'ffi:;</; & J
'3. ii".:t x~ .i . y < 0 lr" .Qrl) E:t~ # I
4. fr.f} XC l'(. y < 0 m ftrll ~"' 1? I
45. <;!ifi q,t.Q ;)'; f[f5 ?JM7flf st7• .. .,~r lr4Pir'i~ ~ ~ ~ SG1 ~ rl tm ;; fJut~ I ~ .. fl 1'"' fi<~ <l>'f'l IR q;ll1'f I
I , olfl/1)/f ~ i~Uf J(iR1 F ! 2 T!f1 " " r.ro ~ fp!r.1rfQb"' J; [;?rt u."'flrrr ~~Ji'FVt~ I
3. """"' - ~~ ,;; ~ ~"""' ~.Yr'R~~# t
4. qml'm sffl Jt(i)?flvrtrr wl Ji'~'" l.}ul.'m l f l·ft tt) U4l(/t I
I / (vi<)) • f, y(3x -y)dx;
l. '!fll JFRI Mi f.tm f. I
Z >fl.,_,. ?) tJ 1!<'1 p:?il (! I
3. .)('<'!~ rrGIIT if'; Fl' .-Ill # I
-' • t/i ~ Jl1li! liifJ ;:1;J1 I
47. <ib•> ""'""' ~ ;ix)=.•~ [ ' ~(C)d( . •• '11 lffd flf.J'<P J>'k R(x, ~; I) f ·
]. l"l
3. ~tl
l . 2
•• 4
14
4-4. The: ~coond otd¢r I' I")E
o, - yu.. + x'u 0
is
l. ollipti¢ fo~~ll xG!!L ye~.
2 f!3rn001!c tbr all xc3, yc:::: L~.
l . cU!pltC for all '(fl 'l. y < 0.
J . h)'J)I!'rt>ohc for:all xeR, y <O.
45. Cons,(kr a second order ordmary differenu~J E<;\'..,1101'1 (00 l:') and its finite difference rqxnc:ntabon. h.k'l\tlfy '4ilich or dte folloo.:mg StOlCMC:IUS i& CQIITC'CI.
t. ·rite finite (li(f¢tCJlCC n::prcscn1n1ion is unique. 2 !'he flmtc difl(fcnce represcrHJ1101l is \lniquc
(or son-.: 001:. ). ·n-.m :s no amquc: Cimlc dttT:m-.c ~ foe'
th~OO£. 4, 111e UOiQllCilC~~ of~ finite <hfl!:l'CII{C Scheme
enn no1 be d::Jcnuincd.
46. ·1 he nrra.;::omJ probkm of c:dn:mrA.,g the
funtttooal
l l ( y (.<)) = f. y(J ' - >•)</.<;
M.•
1. :t umqoe soluuon. 2 exactly tv-·o soluuons. J, nn ullinilc number of !:>Oiuttons. 4. no solution.
I . 112 2. 2
J. 312
48. % tt!frrrtm fr'i $t ~~ H = pq - q~ tt omti 48. If du: J[;un.il toniatt ut'"a dym:micai •)'Ster\l is given
f.(,') iUI! ·-. by H: pq - <l·. thc:nns t--
I.
" "· ' •• 4.
q- - .p - (10)
4 -0. I'- 0 (( · ~ . p- 0
4 - o. p - ~
1. q - · <».p- 00
2. q - o. p- 0 J. q- -.p - 0 J , ll -(l, p --
49. ff<;41 t;::r o;wr F,(t) >r F,(t) r{1l rnfir.lmr """' '"""' f,(t) • f.(o)$ ol - WT, • T, oll ;:f:i&"( ITftr;!j/ iJ;tm; )l,(t) = 31~ II h1(t) • 4~, 1 > 0 C I~~
1. .,n, > o ttl f«ii F,(<l~ P,(o). 2 !70h I> I ttl /i;>1 F1(l) < f.(l). 3. E(To) < e(T,). 4. w>l) l > 0 41 f,rz) f,(l) < f,( t).
SO. "'" /tl X,, x, . ... o\~1, 1) <6 3fjm< mkr.m W<ftl lW 6 #til t.,t'J2NI "' { I 'li'l It; o ~ !.P fi:rd S •X,l+X!+···+ X~ -' 1 ;r) .. - " ('
g1w:(S) lim ... .., ~ t: _,.. ll
I. 4
3.
2. 6
4. 0
51 . 'fP. f.N {X,,:" ~ o J vw 'lMitH Mnon ~tifh s 'R ..,., 7/ifj"'f ~ 3l"''J! '""' ""' 'fl18lo .;.,., t I ... ~ ~ fWifl JRO!f- 1t.l f: I ffl ~ rrr.r.'frr ~·l!IFtT
I. ,f: Jf'f<Kf' lf§if '({f'U ifc.t ;ngt f. I 2 <1>1 ViG A .jpRI '«16 { tk:"-t ~/fiJI 1 I
J. 01!1 oat ># """' - 11$1 ·rt!l # I 4. ;t U/q;- ;JM l:t fffN dc-r JIII/I ;rtt 'I I
51. 'l1'f f!t; X • Y <I - ~ ., #. om O>l &-ff fff'lf Y Wl~ 8 I lfF'I' fi5 U = X -+ Y' lJ
V=X- Y m
I. U >V ~''""'' t 2 (JJtY ;it-~~~'fPrRt I
3. U (P)m O;l o:T-ff <fl'll <1'111rtf d I
4. YKlknorteAiiff'II~Z I
15
49. The ~~rd rates of two life time \'tmbies T1 aM T1 wid' respectivt e.d.r..s F,~l) and F:(t) and p.d.f.s f1(t) and f:(l), arc h1(1) ~ 31 and h~(t) = 4tJ. t > 0 ccsp=ctively. Then
l. F1(t) l' F2(t) for sll t > 0. 2 F0(t) < F,(t) fonll t> I. 3. E(T,) < E(T,). 4. fo(l) < f1(t) for nll t > 0.
SO. Cct X,, Xz, ··· be: i.i.d.l•t(l.l) random VJriable$. Let
S.o:-Xf•X:+···rX~ for , ~ 1. Then
lim Yar(S.) is _.,. II
l. 4
3. l
2. 6
4. 0
St. 1.<:« {X.. : 11 f! 0} lx: :a. M:nkov ch::un on a finite $talle space S with Jtuionary k:imkJOn p«)bobility n'-11nx. S\tpp<>sc: thm the ~hain 11> nQt in-educible:. TI1en the M:ukov dta.i1~
1. admi1s mfinitcly nl.,ny St:llion:try diStl'lbutions. 2 admits a unique ;iii<Hionory dlstnbution. 3. rn:ay nOr ad:lua anyst.u.omry <b.suibution. 4. cannot:tdmit (:103Ct1y two smtiV11al}'
distributions.
52. Suppose X 2nd }' .ve ind<:pc:ndent r.u1dom v.'u;ablcs wh<:reYi~eymmc:tricaboutO. LetU=X-t Y and r • X- r Then
I. U and V are always itt<h:pcndent. 2 U and V hotve &he-~nc dtSCribution. 3. U is 111"'li:)'S symmetric about o. 4, Vis a.IWA)';ii SymmetriC #bMII 0.
53 . <:0 'g'llit ;; at •r-f.lt11<w "'" 'II flfil >tm!llrT>ff ;f,l 53. ""' 11/) ~ f.;r.r 2 :oa mGm # ~ ~ t; :rdJ t 1 ~ "~Pr-r $) rrtr'fli~ :
6,nsider the following 2 X 2. bble O( froque~'ICi:es of '\'OC.·cr prc(en:nt"C:S to "''0 parues cbSS.~fk"d by l_!Onder, Ill :ill election. IdentifY lh~,:: CQITCCl ~t<ltcme-JU:
16
..-rT\ ttl'" ~ ff.:p. .:e e-c f1fJi m ~ t m ~ JOYV-flJ..J M,
ISO •:zo 120 280
2 -,! Tf &·j'flfktfa <5 tm:m ~ mt rl ·-:f:'t<ffl-.r. 0 f, I
3. {~ot </ Wl mrf!m :;fF t : 4, !(H J fil ;;'f;ff ~ r:t~ (' $t ~ ~i't
;' I
$4 t1'l"'f fir.\': • • {·,···.X. n (~ 2), N(p. <1:) R:r1 ~ 54.
f1Jt.~r6q tt~tG '{;(1 ~ ~ ~1"1 ~· ij"'f-#C< p
... -'11 O<o:<r.J..'ifm~? 1Jfm",f!J; .. .. . ~...,___ _...._ (7' , ;; IJ .. ,~T F ~ C' 4) SC07~c~<f .,...,,JW.H fl
tr,.-,n;pf 1j,;r,J'"i ~ ~$';7ft J/!4ot<( ~ / Rll " ¥~~~-) :
0:1;, ;:t;i 'f1'7RV! (jl~.\fl'l./C ;6 r:?n"1 r$ Vf11N
t I
2 a-~.\,1{'" ;7f) gM.,l tt u;l,b ;r.r .rrm~ .JI'fhvr F I
3. 6/\11cr. <Iff go?T # U~u. tm ~ li'! 'tllr. 1$q $ I
4 a:.r:;; u(2V11'£ m· <t j~ .:rf 'liW ~q,.., • I
55 "'"' ,. pet - x .. x,. ···.X. ""' """""" <17:~ 55. liM PrliH'f Mtt~ M ~ ifRrt t-~ j I ~ ~olfr 'IT-t' f); Plf{ qro \fit) JltrYf--"'' m llJj;?
rwv ~lf'lf ilfFPl r,. r!····.Y .. '" I w~ -41 mivrrf'J ~~fl W 1Jfi:l/ f. I ¥Pi' flJ; 1(1• ,. X 'if r 't.l"'> ~ !fP'trr 'H'J'Qfa :!f x~; fh1 :~(lli'Pr~ rt"'t) 1,•11 •• ,,,,v.N ! v1f R) = gli1rt 'H'J'"•'~' ~· y,;lf} t~>? ;;;~tlit1,·t •hi 1;)rr~ ;.'t t rt)
I. P(llr-Ro·> Oi> .!.. 2
2 I
P(Rr-Rx>O)> - . 2
), E(Rx) = E(Rr).
·'· P (Rv ~ Rx)= l.
I. If lhett is no 'USOC'latiOn bet" C¢11 patty aod
gcoda. ~ cxp«ttd fr«qunnes are
ISO 420 120 2SO
l The chi.-!qu.are Slll115bC (or ll:~tlng no 3 SSOCilliOil !SO.
3. Gend-er and 'part)' are not a&,Oclatcd. 4. nolh m::~1es aJld f.:mulc.s ~quaJty pn:fet pany c.
Lc! X;, X:. ···.X, be n (~ 2) 11.d. oh!>eT\'tniMs from N(f.l. a~) distribuliOil, where -crl~ JJ < cca.nd 0 < q! < :tl are W'l~'n puamc;ers. let
U~and Of.,. (,.,. dencMt the nwimum likclthood 3:n:d urut"onnl)• mmtmum Y.Vlanet u~ tsiH'lltlles of q: r~pt\:11\~l)'. kttcnllfy the corr«t stateu.ent:
I.
2
3.
4,
U,~t~l.' hOl$ th¢ S8nl( \1\'IMIJnCC :1$ that Of a~'IM/~,
cr.~tt.r. ha:o taq.ocr v;lti u.nc~ lh:m '""' (II' q~"' 1.'! • •
a.~t;,c has smaller mean squttrcd cnor than th:r.t
Of 0}. 411 OJ!,
.. .. . ~ <T.~tuand at~,.., ht\'C the !t.mlc mean squared mot.
S~ 1!!11 W~ h'ne ud obkzY.b>nS X~o X."- • ••• X. with z notm::!J d1$tribut10n. Suppose fUrdlet' ~r we h::~ve M indtpenlknt Ul of obsef''~tton:s Y1•
Y:.~··.Y. which :~.re :ll:£.0 i.i.d. wnh the sam.: nonn~f dis;ribution. l.ct R,1 • th¢ !ium of 1.he r':lr\ks of the ;r s when they are ranked in 1hc combit~cd se1 of X aud Yvalt.•es. (!f'ld Rr • lhC .sum ot'thc rank$ of the r~ iu th;; COJlll)ittcd ;)CCI. 1111.)11
I. I
P(Rx- Rr;;. 0) > 2
•
2 I
P(R,-R.>O)>-2
3. E(R.,J = E(R,).
•• P {R.,. = Rx}= I.
56. ('#/ ~!lllf<lfj il'flsr.P: Pt~'lf Y e p X+c w f.l7!tl 1 tff.j f.t; i9 , X.::: X<~ '17 ,, iltrvil' (Y11 ~). I
56. Co~:asider a simple linetlr rcarcssi<1n :n<Kkl
• 1 .... ,. u x =.!. f x, ~ Jllflrfitt r " ,. ,._, "i""" 11'1 ll1'WffT t I i!T I1l"'<!ftt Yo till .....,...
JR
t' = /)X+ t: l.c:t ~be lhc l c:.~st liquares predictor of Y 3 ( X • X11 b::~sed on 11 observruions (Y.. ~Y.). i = 1.
· d X- 1 ~" ... ,n an = - L"''' 11 (R1
Then !he $tanda.rd error ofihe ptcdicOOT ~
4
I. ;;rq X W Xo F FZOf 't ffl iStf itift t 1
2 ""' x 'II ·'•'l!.., t m "'tiroq# t, 3. vm 0 a} W'6 x0 JiTW # fit 3ffW; Etcf1 t 1
4. VJ4 0 zy qro ·'Co MFfF t 8t <1J7T irift g J
51. ~q;; m r1 I, 2, ... , N ti&JI41~ N ~;J;? Jhffif'Z t 1 N <6'1 1J.Rf mn~ .mrm t 1 wW ~· W wrm m fir.rr rt<!i immVT ?if'ii~ n ~-if WI f!fit;rW ftt/;t;m1 1flJT 1 '!R fW Xto X!o··.X.. ~ .. r .. ~ .w mwr t VJt iSm m. ~ .. ·• n i ' i1R f.'rtliffi' *' rntJ vmT : 1 f.!rr.r fi 11 m N wr :;,"'lf1ffll.,.,., t ?
I. 2i<-l "'* X;~(x, + ... ..,x.)
2. 2X+I
3. - l 2X+-
2
4. - l 2X--2
58 . ..-~ >il ~ R; = ,) """" wl J!llq A q
8 """' r#r( '" ~ ~ ;t?t f. 'f'll il<7fil<>; rr1tem ft1i1n >trJt ~ mt;fcuu> nr~ W yt n UJfilfr r.Tflli;r R,-i/ •W I i!llq A P UJfilfr >il
~ \<'<l' <) '!fll 'Tllt 'f'll !I"' it """" ""' ll itiP/ B rub ?fi'1 i; I rrR' fi5 :fN) '1i7ft WiifJJ41 f((1i mm m f. 1 ~ tJt rrfiWl t!J fitm ~ ~ '1l1 ""'"" W{7{ orft:iJ ?
1. 3f1R JHwn;;qm 1Wft ii!T YT<Iirft c· m ~ - flftrrn ;!t- fl'fl?fV!
2. 3f1R fi7Wfl':lJ<TI woft Vfl fPGi# t dt ~ ;!t -'1t~truf
3. .-t 1lfimf i61?1'1'11iM-it'l'lf< 'ltki't 4. 41<zlt&<Mn w vl'kful
59. 11Ft fit; wx.2:0~ x1~0~Jfx1+~C:3 'I x1+2x2 4:4WT wrMR q;«7 ~ 1 ffl f.rq ii W
'""' '"' 'Fffl t ?
]. sx. + 1:<, 7fi1 r]l?li{(('f 'if'J 21 i 'q \S'fflliT
wt Wlfilo "l'**'1 'lt.r t 1 2 . Sx1+ 7x2 WT ~ 7J!!l1 17 t <1 :::mw. <lfrt
- """""' 'lfl t I 3. sx,+7x, '"'""""' \\<"12 1 3 11 ::Jfl11f/
"l'**'1 '!!"' 17 t I 4. Sx1+7x~ tm., W <itt r;fffTm ~ ~ 1 ;h
\/Wlrof I
S/07 RD/U-4 A~2A
17
I. d1:Crease.sas.romovts.away from X. 2 increa.ses as.t0 moves aw:t>' from Jf. 3. increases ~s X6 mo\·es closer to 0. 4. dt.ocrc.t~Ses a<; :co mo\'cs closer to 0.
57 .. A box contains N tickets which are 1\\Jil\bered l. 2 ..... N. Tit<: value ofN is howt\'tr, unknown. A ~impl~ r.mdom .sampl~ of n tic.kels is drawn without replacement from the box. Let X,, X?, ... ,Xn be numbers on the tickets ob1ain.cd in the
lu, 2"1, .... 1)1~ draws rcspcctiv¢1y. \Vhich of the
follo\ving: is an unbiased estimator ofN?
- - I ) I. 2X-1 where X = -(X,+ . .. +X .. N
2. 2i+l
3. - l 2X+-
2
4. - l 2X- -2
58. (n a clinical trial 11 randomly chosen persons were enrolled to elCamine whether two d ilfcrcnt skin creams, A and B, have different effects on the h~mllln body. Cream A \\:as :lpplied to onC' of the ro.ndomly chosen anus ot' each person, crc.:un n to the other anu. Which statistical ttst is 10 be used to examine the-diff<:r<.·nce? Assume th:u the response measured is a conti.nuous \•ari3b1e.
1. 1\o:o·sample: t-test if nonnality can be assumed. 2. Paired t--test if nom1ality can b<: assumed. 3. 'l\.~o·o·samplc Kolmosor~w-Sminl<W test. 4. Test for randomness.
59. Suppose that the \':!riablcs x,1 ~ 0 and x1~ 0 S3tisfy t11c constraints x,+xl ~ 3 :llld x1+2x1 ~ 4. Which of the followi•.lg is true?
1. The maximum value of 5x1 + 7x~ is 21 and it does not hrwe ;·my fmite-minlmum.
2. The ntininnun "'aloe of 5x1+7x2 is 17 al\d it docs not ha\•c any finite maximum.
3. The maximum value of Sx,+7x1 is 2l Md its minimum vs.lue is 17.
4. Sx1+7x1 neither has a fmi1e maxinmrn nor a fini1e minimum.
60. 3M "''fl rdrl ). > 0 t1 #in rrfir Jl > 0 Ui % W Mil ""'f< INrW ~'!Pl. ill; X(t) lffl'i61 1!11 ~l'm if I 'll!fmr X(t) ~
L Wtl ' --JL ;;;, c;;rffl. yfj};w t I
2. w fi! 'lfl1 i--~· iii11J<r """' ofib711 g 1 3. ~ r;ffl I. 7.i rrrc7 :rliT ~ ;tt vr-r-; •. ~
)lfft;rlf !. I
4. WA Tfft! ~a' 't({flf :rfrr .! ;fit VFF!- 71ffl ,_ ~
ilfilnlo' !. 1
18
60. let X(() be the number of~ustomcrs in an \\1(/Mil queuing sys1cm with arrivnl mlc /. > 0 and service (ate J.' ::. 0. TI\e pro~ss X(t) is a
I . Polsson proce.ss with rate i ... ~t . 2. pure birth process Witll birUl rate A-~t.
3. birth and death p-roccs$ w ith birth r~l¢ ), Md de:ub l'3tc ll·
4 . birl.h and dc:!th JUQC.CS$ with birth rate ~<Uld
death rate .! . ~·
,,
S/07 RD/12--4 AH--28
19
( 'Wtl'ort C J
~ 1/Unit I
61. 'lrn'1 ftx) • co$(1x- 51) + ''" (lx - 31) •lx + 10; - Oxl + 4)1 "' fm? l!fi!St~ 1 r f.rr.t ~'II '111~1 31QiPR·fW ':fi1 t ?
3. .<~-to 4. x • O
61. Ccnsidt:r 1lle function
ftx> = C<n(lr- St) +sin(..-- 3D+ lr + 1011 - (IQ - 4)'.
A1 which of the followiJ~ poinb is r.ruu diiTeren1iablc?
I. X- 5 3. x - - 10 4. :t ... 0
I. ((z.y) :lrJ S I. bi~ 21
). ((z.y):.1+3ysS)
2. ((z.y) :IriS 1.1)\' S 2)
~. {(Z.)') :z' s/- S)
62. Which of the following subst1s orR' are compoc1?
I. {(<.y) :~•1 ~I, b•l> 2)
3. {(>.y):x'+ Jy'ss)
2. {(.T.yJ :Ixl s J,b·l'~2)
4. {(.T.y) :x' ~~ + Sl
I. d;{.g)= sup{l/(x)-g(z)l :xe(O.IJ}.
2. d;f. g)= inf{( f(x)-g(.t) :xe (O,IJ}.
I
3. dci. g)- flr<-•> -g(.<lld•. 0
' 4. a(t; g)= sup{ H•J • s~•)l : xo (0.1]}+ jlf(x) • g(.<)ldx .
•
20
1,3 \\~oich of thcfollowmg are metrics on C = ({: [0. l J -> R is a continuous function}
•·'1/. J<) • <opllf(.<) - sc(x)l :.e!O,I]i.
2 •11/.ZJ• infllf(x) - g(x)l:xe!O,I]i.
I
oNf f)~ f f(x) • g(.<~ott . •
' .: o!lfg) $tol>! lf(x) · g,•)l : x <:[O,l ]l+ Jlf(x)·g(x)ld.< ·
'
• 1. U A J ~-~~ Tfl1'Pfm 1f1im1 ; I
·-· ~ "
2 . un AJ':OI'7'0J"/)1J g I n'd 1 ~l
6-t. I'OC'cXh J • I. 2, 3 ..... ktA, be * fuU:c .sec con~aan~na atlca.u 1W0d1shntt doe~es. 1"'hm '
l . U A1
h a ctXXttab&e ut.
' ' • ~. n A, IS,JilCOU013blc.
' '
( I)"'' 1. I •·., - )L·v·run ·-?Q'),
"· ' )"' .l. lll; - >ciJP.l n - '>«> .
65. \\'htdl nt the followmg 1s/;m:: <:omx:f?
I. ( ~r I+; ->e b n - ) tO.
3.
, (I+-.;.. J' -teas r.-t-(1().
•• z. UTI ... , '"""""'nable.
~ljd
• 4 . . UA1 is unco-m~o .
2.
4.
2.
••
•••
(t+...L)" -;c Qfll 11- >«: . n + l
( •• _!_ r -...~" ~~~<1) . n+l
(•·;;rJ' -uas n~co.
21
l. Jog~s10fX+losr l'l>llx,y>OIItr.nt 1 2 2
2. !!l e" -t~
e 2 S l ,.h,y>O VI~ 1
3. . ~s•inx+siny l'lt/t >O ot ltR sm 2 2
x.y ~ · 1
4. (x;;fsmax{l./j!Nix,y>OnH~ 1 ¥/tR 1
lllhicb oflhe followin&..,... '""'?
l~x•ysloax+logy fo<tllxy>O. "b 2 2 . I.
2.
J.
••
!!1. e' +c" e 1 s
2 forall.r,y>O.
. .X+ y Ssinx+siny fi II O stn2 ~ ora x,y> .
(•;;J' Smox{x'.y'j foroll x.y>Oundallk > 1.
67. f : I a, b I -+ R !l'lf >l1l 'lllf'l t lfll'fllll-1'11111 !Wr d
I. w>fl as c < d S b '*/$Tv- J!(x)dlt• O ~~ [ 5 0? 1
' 2. w>fla ScS b •~f<f~•m f t(x)dx=O m[=Ot 1
• d
3. w» a sa< d s b '* f<f~ "'"'" jJ(x)d.r =0 m '11'""'"""' '1tf t fil; [= o it 1
' '
4. ""' 11 S c S b Ill tm!- jJ(x)dx=O lf) zr~ Jlr.mru; '1tf t fil; [= 0 ill •
67. Let}': (a. b)_,. R b( a niiC-;lSUnlblc ru,lction. TI1en d
I. If Jf(x)d.tc O for•lla,c<d,bch<nj•Oo.e.
' '
2. If JJ(.<)dx•O ro .. Jio s c s b. lhenf• oa.• . •
22
• 3. If /f(x)dx =0 for 311 a ~c <d ~ b, doe-s not neecwnly imply l)~l f-= 0 4.¢-
' '
rr J!(>)d.t:O ro. ana sc s b does DO< r.«<ss;;l\ly imply that/~ o .... •
1. <~.-r;fttn;, n. ~'ifi t , 2. do· '{frlr if 8, /itrpr ~ I
3. d:·fftrn if Bo Ryir 'lit t I 4 . d:·'ffr" If n, flvrr ~1!1 t 1
6S. fot X. (XJ. X:, •..• x,.) andy= (yJ, )':'>· · ··'·~) Ul ~· le1 d ,cx.y>·(tlxi-Y,r)flp to; I 'S p < «>, ,., and d.tr.y) • mu 11-rJ~ :j • I. 2.. .. .-j. 1.<1 B, • {xoR": clo (;c. 0) <II . I Sp <: m.
W'hteh or lbe followitlg are cocretl?
1. 81 IS open m the d,..-metric.
J. 81 1< not open in lhe d:·metric.
I . (0, 0) ""fordmr 8 1
2. 81 is Optl'l Ul the d.._·metric.
4 , 81 IS not open in the d',-mctric.
2, (0, 0) «/ - t <1 (0, 0) W <r.ft o'l:<i;-Jr.11111<W oR Jlf1<rrrl # I
3. (0, 0) "' f "''"''fr.flo ? '"'!! Jlif',.'fl'il Df(O, 0) '!/""-.,'<!)" <r!h /. 4. (0. 0) 'R 1-*q t ""'] ~ Dj(O, 0) •]riiWII>J t
I. [is d-uous at (0. 0). 2. /•1 continuous at (0. 0) and aJI dittc:rion31 dmnti\U cxW at (0. 0). 3 f" dofT<te,.;able 21 (0, 0) bu< the ~<riVllt;~ D/(0, 0) is 1121 uwc•Jble . 4, fu (h(fctet~ciablc l'll (0. 0) and the de:i\'<1tive Dj{O, 0) as invertible.
23
I. 11/11® :=sup{!l(.<)l : xe[O, ! ]}. I
2. 11/lh := ~f(x)idx . 0
3 11/11 ~· := 11/llo> + Jl{l ll + 11{0)1. I
4. llfllz = fl/(< >f d\- . 0
70. i1le Sp3.-te qo, I] of continuous functions on [0 .. lj is complete with respect to the norm
I.
2.
3.
4.
11/11. :=supJV(x)J : x e JO, l JJ. 1
ll.(lh = flJ(x)id.<. 0
11111 ~· := 11/1. + V(l)l + VlO)I·
urn,· J~f(.• >I' <l• .
71. •r-'1 fil; o .... , (r) = {(x. y} : ~'-a)'+()'- b)'< r) 1 R ~) f.lq ~~ < ~ <fi/W W~
~/1'
l. D(Ohl(l) U {(I, 0)) U Da.Ql( I)
3. D(o.oJ(l) u {(l,O)} V D&2~1)
71. lei D~) (r) = {(x, y): (x - a}2 + (y - b)1 < r}. "Which of the following subsets of R are
connected?
I. D(om(l) U{(l, 0)} U D.,,,,(J)
3. D(o,O)(I) u {(1,0)} u D(o.»(l)
l. X~# (["(if]~ 7ft~ I
3. X~ <17Im ~ I
2. D(Mj(l) U O(>.OJ(I)
4. D<o.oJ(I) U D<• '1(1}
24
72. LC't X • {x • {0.1): x J.l / n,n e: N} be g~xcn ches:u~x.e topolo8)'. Then
I Xos coor.ccled but no1 ~ 2. X is ntlllltf 001q>0« not c:onoecud
). X 11 ~~.and conr..«tcd. 4. X 1s ~·but noc comectcd-
73. P, .. , ~· If " ll'J1:r e s;•J~lli-f:rfii1ct t ?
I. [~ ~] 2. [~ ~] 3. [~I ~I] 4. [~ ~]
73. Which ofi)IC follow1ng m~utccs are positive dcfini1e'!
I .
.l.
l.
•• [0 .-
4 0 J
H. 1fi'q fir.J:.!C·nll ~natJr.') ~ ~ Y :SA~ '?J:im ff&w (fiQI"ffRV ~ I ~~ arw'fk Yo
C: ~~A W .Jid4d V ~ JIG!,~ t I 'lr-T fi'li k ~ t:Nt (V11) < 11 1{11 'lf·Y ,"ii, f~:!J ; .. c.R ,;; fir:~
A: • A,..t I tl~
7~.
I. ..t. 1. 2. ln>~M A "ll,j' J. ; .. A ;m qm 11r:f J1"f)!;:rn.rfi:;n 1'ff.1 t 1
4, ~~w J'f{lU Olfffllt~ V1c ,; ~ ~ ::fi~ ,rF- Y1 $ f!rn A.r • 0
I tl A be a nOII•?'(n.> Jint.'lr lrtmsformalion on a tl'ill ~clot' ~pace V of dmtclmon "· L<t th:; >llb>IXI"'~ 11., c. Vbe the image of V und::-r A. L~·1 k = dnn Jl111 <, :md suppose (hl1 for some A.c:t .. r: )A. The-n
I. ;. • I ) _ dttA•i.~
l ; iJ the ool) t~g,tnvaluc of A . .: lhcn: rt a nontn,"taJ ~< Ytc Y web thea...t.r = 0 for 1l1 xc V,.
'
25
75. 'TFf filr C '1'5 n x n Ql<lf~<i $ITUf!' f. I 'fA /ilf W, {I, C, C ..... C"') iJ1<1 fil"ln '1'1' ~ wr/1<: t 1 ~ Wit?<: w lh? /irt;r t •
I . 2n 2. lllfir.1; W lllfir.1; II
4 . ;,f.;;;; :# &ftr.p 2.n
75. Let c be all )( ll ((."31 matrix. Let Vi' be the vec«>r SJX'Ce spa.nned by {1. c. <:: •... , ct.}. The di.me.lsion of the vector space W is
I. 2n
3. ,,:
1. v, n v,.
3. v, + v, = {x+ y: xoV,,yoV,).
2. at most 11
4. at most 2n
2. v,u v,. 4. V1 IV, = (x< V, and y~ V,).
76. let v •• Y2 be subspaces of a vector spaceY. Which of the following is oe«ssarily a subsf)3ce of Y?
1. v,n v,.
77. •!R filr N '1'1' 3 x 3 ~"""!if~_, 'J"l t N' =0. f'rr.r ii 'It i1lf'f m/'lt WI ~/t >
I. N '1'5 fi/;nof-J!tU!1 W f1'ffiCI 'It/ t. 1
2. N '1'1' fimr,f-J!fU{1 il 'l1'ffi'l t I
3. N'51'1'/i ~~~~~~I 4. N >/;liP. ~ """'~erf'r;n l~ '$ 1
77. Let N be~ nonzero 3 x 3 matrix with the property N-l • 0 . Which of the foUowing isl~re true?
1. N i~ not similar to a diagona.l m;HriX, 2. N is similar to a diagonal matrix. 3. N h3s Olle 1\0n~uro eigenve(tor. 4. N has. thn.-c linC:.'3rl)' indCpt.'Tldj,."flt eigenvectors.
78. 'TF¥ fili x,yGC" 1/(x,y)=Sup(l•"'x+e''}t :&,q>E Rj '17 fim7 1 f'rr.r if 'It >ir.J-171/W •••
I. f(x. y) "!xi' +IIY~ 4 2l(x.yl
3. f(x,y) =HI' +IIYI' • 2i(x,y}l
2. f <x. y) =l~<f +IYII' + 2Re(x,y).
4. f(x,y) >jjxjj' +!>il' +2!(x,y}l-
26
18. IC'I ~.)~c:'. C"onOO.cr /(.t.y)=Sup~c·1t·"-~>i: :O.tp• ~}. V."'nthorthc roJJowwg
•••
I. /(.T,)') ,;14' +Jyr +2,(x.yl
3. /(.r.y) =11.<1' +I)JI' ~ 1 i{x. y~.
2. f(x,y)~l-<f' •(y 1 +2 Rc(x.y) .
• . f(x. y ) >ll·rl'•l>fl'+2;{x.rX·
l(<1iiff .1 1/U n i 1 II
79. !iP'f If II oit-•- ril <fj«>• C(O. I) if mP< /J > (<rnr4o'·'ll•l4> fiiF;•ai~T/11 rt ~ < (0. I) '1<
.,...,r:l1J ~ - """" vi) o'lft:)
{f<C'[O, 1) :f<., ·~w 5} 2. (f6CIO. I] :jlO) • o: I
4. U'c(10, I] : JJ(x)<l<~5l •
19. \Vhd\ ofdN:: fotlowing. seu attdeose in qo. II (ahe $f>Ue o(r'('at , ... ~continuous (w><I>Onl on (0, I) wnb resp«< 10 "'P'"""" oopology)•
I. {/<C(O.I]:fi•apolynomial) 2. l{cC(O.li:JIO)•O)
l. (feqo. I) :;<W)~ O}
I
4. if•C(O. I) : fJCx)<lx=S) 0
80. 1flil illr f'. •C.-. c. ,Hrt l«iif(.!.)=-"- i61 71'11'1R llmll ~311 '!Ill •M•trf11f<R 'ffl'l 8 iRIO n 2n +l
rrr t:rmrm< d '
I. J{O) • ill
3. }{2). J/4
2. 2""-2 "N/01 ~r,; ~~ g I
4. ~)m q,)J l1'"tt1FJ!vr fPrR ~~<WI t 1
80. Let/: C-• C be a mtromorphic function 11UI)'t1C 910 sattsf)'ing 1(.!.)=~ for "2: 1. n 2n.a.l
Then
I. ./{0) . 112
l. 1{2). 114
l. f hu a Stmpk pole at z = -2
4. no such mt'r0f1'1(.)rphic f~etioc~ t::Usu
27
I. f 11>1 IIT'Rifil;n '11'1 ff</1 ~ I 2. [fMt I
3. fcO . 4. f' \'IF ~ fi>Rf0 t I
81. LC1 f be on enlire function. lflm/ ~10, ihen
I. Ro / i.s conS1ant 2. I is const:~nt
3. f•O 4. f ' is a nonzero constant
82. '<P'f fW f : D-+ El [(0)=0 • /(112)•00: "'"'.M.,.1(• t. ;n!' D= (:: 1:1 <I}. f'lq .j' >I
""" " "'" '1<tl t ?
1. If' (112)1;; 413 2. 1/' (0)1 S I
3. If' (112)1;; 4/3 and 1/' (0)1 s I 4. f(:) • :, :e D
82. I..C1/: ll_. 9be bolomorplla< ' 'ilh /(O) =Oand/(l f2)•0. ,...,. :&= (z: 1:1 <I). Wbi<hof
the foUO'NI.ng statements~ eo""'?
I. 1/' (112)1,;; 413 2. 1/' (0)1 £ I
3. 1/' (112~ s 4/3 and If' (0)1 $ I 4. f(:) • r, ze I)
83. z e ••lyfN,.zeCai/IIV~"" '
I.
2.
3.
4.
lll"• (z.,C:y>O},
iii'• {:eC:y<O},
L .. • {zt:C:x>O},
L'• (:eC:x<O}.
f(:)= 2z+l 5z+3
[-J' l1h !1\ ;t .,;w • [~' 01) If' rm< rtlirr.d'!rif 1JlWI1 f. I
ff fhi lt I'd UiW 11 H" ~ H'" 1$ iR'tR Ylitfb~d f1mf1 I I
tr•t L. iJ ww ilK~ L $ a;w lfklA~u n'U i 1
l!'ilt L' il ;;w w H'W L',;; <P17 Rll!lillihl """'t 1
H' • (:eC:y>O},
It" • (:oC:y<O}.
L' • (:<C :x>O},
L t:cC:x<O}.
I. mnp~ nr onto M' and !i·r (I!HO [-r. 2. tn<lp.$ 1r OlllO H' nnd )['onto l•C
3. 111;11)$ u· QOlO L' and [-j onto rt.-.
4. m.ap~ Oi' onto G; and HI or.to L'.
84. :•OWII!?n' /(:}=cxr( z ) "" I-eos =
I. !"" J;v*• M/P.t"T t I
2. ~ Jl"Rf>< t I
3. \'4'< JOf.r"Jrli ~ ~ I
28
~. : - o .t MR am j(') d! ~ ;~«R!.,., >l y.v.y,. •"' w-.rw • """'- f1l1f: .,; Ill
84. At:: • 0. the function /(=)= exp( z ) has t -<::OSZ
I . 1\ •.:ln'.('IV1lh lc sitl!)'UIQnty. 2. :>pol<:. 3. an cllsc:ntial ~ingul:tril)'. J , !he l.nurent exp:m.~ion ofJtz) ~ound: = 0 h:I.S infmttely m:any po:~oi tive and negatiV¢
J)QWCf'$ O( Z.
85. 'fl-1 (.), It • Q !x )II om I. I + x'"" ;#;o '1<1 >n<•l /. I .r-1 .Q< R h ;fi/ ""'d'J"f" )' t I
al R Vl ""'~~· t
I.
J
>) 1 l1!1o 1rretlue1ble 0\'¢1' R.
J. >"'- y + l IS UTeductbk O\'er R.
2. • )'"" • y • l
4 l•l·r • •
2. l + y + 1 is irredocJble ovtt R.
4, y' + );. '"y + l i~ u-rcductblc over R.
I. Sint'. Q .,.lflof/>1/. 1
3. s;.-• •. Q .. -~,
29
2. Co. n/17. Q '17- I 1
4. .Ji ~,J; .Q(x)w <ll<i!/r1 t 1
·, 86. Wh1ch of the following Is Lruc'J
I. Sin'fisalgcbr.llcoverQ. 2. Cosit/171Salgtbraic ovcrQ.
3. Sin~ I isol~o • .,.,.Q. •. .Ji +.fi IS algebr:ri<O>uCl(• ).
87. m"f f>;j(x) - :?+x1 +.t +l ~·g(x) =.-.'• 1 nltQ[.<]ff
I. '1/'mT>< """ ¥1'1f1< (/{.< ), g(x)) = x + I. 2. Jfilm'iJIPP/10(1/f U(x),g(,t))=..' - 1. 3. "'JifR fPIPIIId (/{x), g(x)) = x' + ...' + x' + I. 4 . ~ H•Nor# (/{x). g(x)) = x' ~ x' + :J + x' ~ I .
I. g.e.d.(ltx).i(l))•x+ I. 2. s-<.d.(A,t).a(,<))•x'-1. 3. l.e.m. (/{x), 8(1)) • x' + x' + x' + I. 4. l.e.m. (/{x).~o(,<))=x'+x' +?+x'+ I.
I. HcZ(G). 2. H =Z(G).
3. G If H lmT'll"' I 1 4 , H <'dl J/TO#r ~ t 1
88. For 3ny sn>vp G of order 36 and any subQrOup J/ ofG order 4,
I. HcZ(G). 2. H= Z(G).
J. II is nonnalm G. 4 , /lu an abcli•n group.
89. ;,R. ffti G a'F- S, x SJ ffi1 f¥i!rt: llm1T 1. 1 n)
I. G <m" 2. 11m <l'ffl'f,l! fTf11'<F'J t. I 2. G .. ).l11!1 'IW!'f.l! ;an- t I
). G lli1 "'" "'!J"' 1ffll'll"' """7.1' 4 I 4. a ., ~"" """""' 'J'fO'J.l' llllft n '"' t
30
89. 1.<1 0 dcn<Kc the groupS. • S,. Then
I. a 2-Sylow•ub-ofG is ,_,l
J. 0 h>S o nontnvb1 nomu1 "'bsnr"P·
2. a >.Syk>w subpip of G 1s nomul.
.1. c; has a ~l subpoup oi o.rdef n.
90 ~ iW X ("fJ' 'l'lfi'I(R Gl.,.&.fith ~ t 1 ~:, fJ; ,\,.A:_. 1\ ,, X • lf'{iit ~~ I "'t ~ ""'f'fi i 1 It'! X <1'1 I'"' OiJi1 ••aQ'" 'if"ldriJ <6<« f•4tll mfl t m fix)= o, liR x.A, J•l.l.).
1. ~ (J im m o m 1 t:r.t rn ~ , 2. 'i/;'ff lt ;cq n,, ,,,, al q ~ t/t ;:7 m.;:;:-~ u} N¥R (.)<=l tri II 1
3. t11, tr:. fi1 ~ ~1 iff(fl/4<1; :ffd! ;/; fc:n) I
4. ,.,;1 :'/11 ~ tf,, A~ f[if A) # w f!l1l ~ ~ I
90 LUl Xb~ ~ riOI'Il'llll H~usdorff sp:!.ce. Let Ah Az, A) be: ct~~d subitC$ of X \\•hich :HI! Jl3irwisc <li~JOint. 11lcn there oiW3)'S .:xi~ ISs continuous real valued CVIlCiiOn/ Oil X S\ICh th:u
91.
Jiy) u,.it'n.At.i- l.2,3
l ,rf each 3, l$1!'tthcr 0 or I. 2 .rr QC l~stlwo of th.;: numbers a 1. ~.a~ are 4.-qu~l. 3. ((lr all re~l wlue~ o( u1 • a~.Q). J Otliy 1f one amcY.1i; lhc stU At, A~ and A1 1S empty.
Y [>·,(x)] u) >',(.r)
Vl5ili UJ.Il: nit Ill
I. y,(x}-+ oo ~ Yl(x} - ) 0 w: x-> a>.
2. }'o(<) > 0 \<i y,(.t) > 0 or• x - > <IJ.
J. )'I(,\} > <0 \'4 )'~(.\'} ->-(f) WI X - > - <0,
•1. y o(r). )':(x) -> -oo iirll x -t -<10.
91. (.'QO$Jdc:l the SY"k:mofOOE
.!.r.,r. Y(O). [ 2 -J
tl'f -1
"!>= A-(1 2 )lnd Y=[y,(.r)]· Then 0 - 1 y,(.<)
I. J',(l')-• 0'>3nd)~(X)-t0 3SX~~.
2. )'l(')..,.O«~ndn(x)~O~s .~· ~ 'i'J.
31
) . rA')-*O')aod):(.~)-+ ru.r-.-« . • 1, y,(x).,l-'t(.Y)-+ · tiJ as."-> ..u.;,
9Z. 0/t<il>v ""' .,...,. y" + i.y = 0; )'(0) = o. y(l) = 0 ... /1;11 ~"' '-"" ~ ""' " .. Jlfktm t. ~,_} (0. I ) >Y r;.r. Jil;l~ "''"' ffrnf 8. vii
I.
).
2.
4.
92. l·or the bol.lnd:rry value proble-m
93.
!N.
Y' + l.y • 0; y(O) • O. y(l) • O. l~ cxisb an cigen~tue A ft>r •"uch tho:rc C'JOrtC!;ponds on Cttcnf'hnction 111 (0, 1) thllt
I . OOc:; 1101 ch3tt.ge JtgJ1.
3 . is po$AIJ\'t.
I. ""1"' 2. ~
2. cl\:mg<:s sjgn.
4. n negative.
3. ii.IIWA11i 4.
t/2 >' 'ThC' solution o(lhe boun<bsy vaNe problem --;-+ y-a:cosecr, " O<x<-
dx· 2
y(O)=O. y(; )=Ois
I. convex 2. conc:we 3. neg:ui\'e 4.
I. Wtft x &(!(. YER It Rr1i \"• tWIt I
2. ((X, y) Elit' : (X, y) ~ (0, 0)) '1\' '-"" """" ~ f. I
3. {(x. y) eR': (x, y) • (0. 0)1 ""1"" ~ r<r t 1
!R(i"qi}l
posiLive
4, {(x, y) el<': (x, y) • (0, 0)) '" 'Ill' .,...., ffl 6. W"J 1rR Jl'dm; t I
ot(.<.y) = .T,
! ....
I. a soluctOn (o( :aU xeR. y•R
32
2. an unoqu• solution ot l(t, y) •~: (x, y) • (0, 0)1
3. a bowlded oolut"'n'" !(x. y) eR1: (x. y) • (0. O)l
4. an unoquc sotuuon m II•. y) tR': (x. >) • (0. 0)}, but the so!u1"'" •• unboond..-.1.
u, - uu =O. O<.t<Jf and 1>0} u(O.r) -u(.T .r)-0, 1 >0
u(J.,O) = s.tn.l'<t$1D2-'. OS:xS r.
I. .,ft x£(0, ::).1 r..~ .,(x,r) - • 0 <illt _,"'
2. W'f1xG (0. x)~ f?n) t' ,.(x, r) -t 0 \iffl t -+G>
3. .tc (0. r.). 1 >0 Ill IM c'11(t, 1) 'll1! 'litq """' ! 1 4. •eDxe(O, ~)•> Rnl •''u(x.r) -> 0 1lfil 1 _,. «>
95. Let ., be a soJuHon or lhc he;u cq1t:11ion
u, - 11"~ =0, O<.x<r. ond t>O} u(O,I) • ot(/1,1) • 0. 1>0
11(x,O) = $in,\'+Sin2x, O~ .t$1:
l. tr{x,J)-+ Oas t -+ oo roc:~ ll xe (O, ,or). 2. lu(x, 1) -> 0 os 1 -> "' foroii .<C (0. If). 3. tl•t(x,l) is <1 bounded funellon for :ui (0. JT), t > 0. 4. e:\J(x.l) _, 0 :a,.;t -+~for all xfi(O, tr).
96. 1iR Jr;}; u flftTffrr. IP. wwm
u' + ~~~· • /(1), I
u'(O)=o, u(l)•b
1e(O.I} }
41
A.H
• 33
<61 Tf'l I u! +IS I It~ ~ ,.., ~ >{>, )') • u(Jx' +I)~'>' g(x, y) •
!(Jx'+y').•tv ~-lf'll-v,..+v,=g {cx,y):x'+/<1)>1} v(x,y)=O {;T,y):.'+/•I)Q?
I. a>O~ h >O
3. a=O ~ b=O
u• +~u' = f(t), IG (0,1) }
u'(O) =a, u(l)~h
2. a >Oub•O
4. u<Ott b•O
Define for.i +-I ~ l. v(x. }''). u( ,I xz +;) a:td ~'(:c. y) = r( J,t2 + l ) . I hen v i$ 3 solution
oflhe POE
vu-rvn=8 in {C.t,y):x2t /<l}};r
>(x,y) = 0 "" {~<.}') :x' + y 1 =I}
I. a>Oandb >O
3. a • Oandb=O
2. (t>Onndh • O
4. a<Oondh • O
91. >tF fro SJif t ~ If" aqR {)>p ~ [UTM) <§••·•·»• l1ofl C ... <mot rr>/1 {.)""" m>PJ 'Ff ,; PFr f. I 1'1'-1 ~f&J; rmTrfl Q?·fi/rn7 I
2x1 t-3r2 -:r) = S
4x1 +4x, -3x, = 3 (I)
-2x1 ~3x1 -xl: 1
rrl """"' ( 1)
I, {!l6 U'fM /f ~~ 11!1 WI 1fl1rth ~ rrr-iJ <tC Tl'fii.'f:r-1Jll •fll # 11JJ'if~ iJNi6 t:Wit1f1 Y~ 'Ff il fiR 'f,f/ f I •
2. ~~ ! lJTRifih U'rM ,y \'l'h'rifid aft :raJ :;(; ~ I
S/01 RD/114 AK-3A '
34
>. ~ JM • ~' "'' :17 ~ t miff<>~~"' m ~ ~~·r fJ"" <1 .'~=? r,, ~. l'T~tS 4, .. ClJ-aRi1<#17ll~#{Til'l~{ <1>7r.<"(t) ~7tf)(Rff. I
97. (jt\~1 du1 an upper tnanguLY rr.atrix (l;~ is ilt\"Miblt tf and onty rf alllC$ di:atontl clcrn.:t~'.lo ~ drff~n:nt from.t~ro. eo::sider th¢ I1~CM t.)"(te!":"
I.
' -·
2t,+.h;-.lj -5
h.•4tr- 3x.; ~3
2..t1 • hz x~ ;.; J
(I)
(,ln be hl\lbiOifl'ICd 1 nt~ an UT~I hut is not rnvcmhlc l'!ccau&e the diagonal ~:tur•..:s uftl•c UTM one n(l l ,Jjff(:tC!ll fH)Ill ?.CtO. h uwcrublc ~hough ':mno: b:: tr:wsfonucd in1o an U'l~l.
3. CJn be trunstOnmoJ tulo au UT~f becau~e above du•uon:ll c.nhJ.::s :.u~ all dtiY'cr<.'m from {1,!11),
can be truus.fonned into M UT~I ~od lhe solution of;he UTM 1~ tile wtution or(l ).
2.
.1
J
~(.f)
Y.(.t)- \
2 2. g!.•)-(.
.T
It\) .l• l -2 '"' 0 (1)
I ~· t Y J.,o(\ l. su ll•..tt all) !i~t>d ;wint of:.,'(.\'} is a ::.l'1utiou -.>((I). 1'111..'11
,
i,'( \") \
)
t( 't • "": 2. g( -r)""-i .._ ..=. :'t"~JXlo.<;<ih!ce~ T
X! -:r-:! ~C\)•t- - .A. ;0. KcR :s3•VK .. •bli!!<hOtC.: K ~-
-\'( \) -- '~ - 2. J:l-') -I- .=. <lit' lht' (,.'lily po~tbl.; ~111.11"1,!)
,(
I '
S/07 R0/12-4 AH-3B
' ~>9. """"""wl/1lmr ji(Ji)=.t fK~<,()9(()d( 0
<il!!f 2 .,.. .,... II.
r:0 K(;r,()={cosxsin(, ror OSx<( cos(smx, ror ( i!.xSn
35
t. 'l'l' '1fMt'l1 'IR i'PmiT #'(x)-}{.1.) ¢(x)=0, p(K)= 0, /(0)• 0 1111""' ff 011<11 t <iriff .!{-<) - I 1111" 'lftlJJ>o ""' - ""
I. VW j{A) • 0, 'M JJ"<PW CCI I I
3. ""'j().) < 0. lOii ffl ;;if I
99. 1'he integral equ:a1ion
' p(x)=). fK(x,()o>(.;')d(
•
2. OR J{J.) > 0. 31''" f#!zli <il ffl t I
4 • "" ). > I , 'l'l' .,.,., r<l t I
{
COSX$iR,, forO~x<( where A: i~ a J);;\nunetcr, snd K(x,()= ·
cos,slru, for( S.t·~;r
lc>ds 10 a bounobry v•lue problem jl'(x)- j{.l) t'(x) = 0, 9 (•)" 0. ~(0)= 0. wh<:r< ftl) ;. ....,.,. Then ohe boundary , ... ue pmblcm hu
I. 'unique so!tllion whenj(J.) • 0. 2. infinite number or solutions wh<.>n ;t;.) > 0.
3. no solution \1/henj{).) < 0. 4, a unique sohnion when l > I.
100. lftf(f.llll l(z(x.y)) • [{ (: )' •( ~ J-2z }t.rdy em 0 ~"" t/NI' f.. ~ qf11fl•lf w-I
~ x ~ 1.- !~ y ~ l, fl z • o I It Df.Pi:Ft$N _, Mm lSI uf"'~ ea z • i:O(''f,y) p. ... ~~ N
I. z0 = I, a1~, (x. y) • 01111 a, ~ f. \'If D w '"""' ;. r::•rnrm: ""*t I! 1
•=I
2. z.= a,;,(x,y)+a.-9.{x,)•),;;nofa, <F1 a,~~ <F1;. • Q1 .t """'~ ~rJtlf.1
J. zo = a(.{x. y)om a fl<""" t r:O D "'~ ''"" f. 1
4. :o=(x2- J)(y'-1)116.
36
I 00. An app:Cl'lultat-: S<ll' •liOn 1 • 1., (x. :;) 10 lhc problcn1 of exrremi:cing «he fuJ)cti<ln.'ll
/(:(.<.y))- rl(~)' •ll ~)' -?: ldui;, •' l <> i)y
\\t.~.·rc r> 1:. ~l..: ~~uJ:~. I s Jl. ~ 1.- 1:< y s l. ~~11. • 0 on the bou.nd:uy ofdlc square, is of the :Om1
" I. =•- La,(. (x.y). "f.ctc a. arc cort\IMlS and fuoc••ons ~ 3tc ll!te3tly tndepcOOI!'tlt
' •. '
' Ill I>. ~ ~ o 6-(l,)') .. a4'(1C. J'). v.hcr.: a »''d U11JC ttlft!iU.'lb. and~ md ~love wntJnuoo:;. putt.~;) tk•m ~~~\~
:. '= ad(t. ) ') "btrc at' 3 C(IRCUru :tnol 6 ·~ CO.'ttlnoou5 1n 0
:. -<.t1 - iH,:-IJ'I(t.
l .~ ... ~-( Allrt ~~Nft flcKI ::tl ~ 'r t l .. •: •M .:..'lll~.Jr' r.t .. 'llWi $ 3.~ ~ m $ 1Wr1 1tm :rtf \ilTiil. 1ffitie-t RtP1
\-"~~J; ;~ w ft:r4 ~ :rrtt mr , ,\, ,!f.'tr<·r }',):Jtt {'(fll .f't <t1·1~ol·\'lfl ff1 Jr.)~~ .t t ••• --lC-1 .,.r .f,rfM f-1'711 ~JrR!'f r-t:<JJ~ 1Ft J,'J'T1'fl t 1
! I l:ml:iton 's rrlndplc rollon ~; ((('Ill\ th(' J)' A lembCil' ~ llflnciplc. 2. I hutnllvn's prua·tplc 1\ IXJ1 munlly <~i>l)lic~blc to nonho!onom•c system. unless a
relation oo•utcchut! 1h~ ~hllC•·etUI:\I t of gau:mh"-'C'd cooulirutlCl\ is gt\·en. 3. J la•mlhm '1- p•·i~\l!tpl...- th llow .. rrn1u 1 .11gr.m~..:: 's ~qu:\l lflW~.
<1 r-.~wl..,t t's Sl.!l.'\lttd l.tw uf mulion folluw11 fiutn ~1c ll<~ruilton's pritt<:ipl~ .
I. t•FU~-.~ ff4'o!•+<''lf'it•fl<l J,},. II~ ,lfil·fi~'l ?Jift;"J;fv} f. I 2. v;•/}-MVJ) f/~ tJ,·r NO:.<lJJ ••• :HINfi~Jtcl {:)Jfift~~Y tJiJ (({.PJ/ tJ} ~t;IN if I
J_ f(!,';,j{ l,Jo"'~ fl,,'>/f:l•h ¥01:,' ~ J{lfNI •lf.l ~. ~r>f•!( lifllr.At IJfit W4ttr--(t-it m 'fl'r; 'PI
'IR,.-..'f'Jf ~m it : 4 ~•ri/ ?mi•NI .Jl1P.'frl (~ t.•flfr;.fl IJJcN 'a/{ff-:}~~·cl '11/(t 'ifil ('ifi /i:tmtPI f:iifFf it I
I. '
3.
L:o~~J.a• •s~ ':s ('\IUJ.Iton" an: sct:onJ otdcr 'hift:n:nti:al t.-qUJtion.~ Toul number ot ~'!u.u.vol) ~ ~Ql:Jl kJ the ''~•mbec of 'l}!rlet:lliud ~oo."\la\3.\e$, LO\~inl::•:a.• l c. nol un:qut: in ll'i funtuon:JIIorm. buc the (onn of dtc Y~llif¢C's cqolCIOnol· mot tOn ~ bt~s~n'<'d l..al,Tiltlg!Oin flll':c-IKMl 1\ ::I q11~lr:mc fun~(!Ot'l n( ~rDh7ed \-"Ciocil)' When lhc j'Ol.('nlt.)lle'US~<i.
37
i"PQf I Unit IV
103. 'll'f t:IY F(x, y), G(x) ~d H(y) ifi'M (X, Y)1117 ory-«r- m ""'"- X "'1 oW1r riWI ~ '"R't <f.1 Y <Pl '3'l17f - """ !,6«< f. I qft>nftH ~ flly
{I ~ x sa u-- 1 'IR X>n " •{ I - I
I. ,.. ,.."'"' (U, V) = 0 Ill fi'll x • y <$ f<t~ f (x, y) • G(x) H(y) 2. N•r«r41 x • y ol ~ F(x, y) = G(x) H(y) nt ....,.,l'>l (U,V) = 0 3. oWl? U .. V oml!t ' •) X • Y w.nt d I
<1'. JI1R X <1 Y r.rrif:l #' fl) U :r V ~~ntr ! 1
103. lei F(x, y), G(x) and H(y) be lhc join< e.d.f. of (X, Y~ marsinal c.d.f. of X and m~r&in<~l c.d.f. ofY respectively. Define
. { ' if X So U = - 1 if X >a
and ~' •{ ' - I
where a and bare ftxed real numbers. Then
if Y Sb
if Y>b
I . If Cov(U, V) • 0 then F(x, y) = G(x) H(y) for 311 x andy. 2. lfF(x, y) • G(x)ll(y) for all x andy lhen Cov(U,V) • 0. 3. lfU and V aJe independent then X andY src independent 4. If X andY are independent then 0 and V ate independcnL
104. f.lnr Jf 'II .,,., 'II ;;iit;r11 >Jrf[/qq; ..., X~ Y oil "'"""'ollo;)r 'liim .,.,; t ?
I. ?l>l)ae R ol RrQ P(X> •I Y >a) • P (X >a)
l . ria,b e R ol Rl>! P(X>a i Y < b) = I' (X >a)
3. X~Y~>rl/81 • 4, ffllo,b GRIG~ E[(X - a)(Y - b))=E(X -a) E(Y - b)
104. Which. of the following conditions imply indepcnden<;e- of tht random variables X and Y?
I. p(X> a 1 Y> a) • P (X> a) foralla e R.
2. p(X >a 1 Y <b) • P (X> a) foroll n, b G R.
3. X and Yare-uncotrelated. 4. E!(X - a)(Y - b!] = ~(X-a) G(Y - b) for all a, b G R.
lOS. :J/'iTP..1 ?'Pd~ s = ~ t ;2,3.4,5} ~ ~ w~1 t;F/Vi!ifft P viT ;ft't! ,7p;r 1'f'1i'! ff. <JR9 f'N ;r,<mu 11~<11 w R~•rl
rO~I 0 0.2 0.1 0
0 0 0
P=l 0.7 0 0.1 0.2 0
0.2 0 0.7 0.1 0
0 o.s 0 0 0.5
I.
1 0~. ('ons!dcr :t Markov chsin with St:Jic spsc-c S-= { l ,2,3 ,4,5} and stalionary tnlrL~ition probability m~\trix (• gtwn by
' 0 I 0 0 .2 0.7 0
0 I 0 0 0 I' = 0.7 0 0.1 0.2 0
0.2 0 0.7 0.1 0
' 0 0.5 0 0 0.5
l.ct p~n) b::th<: {i,J)rh d cmcm(lf/J'"
Th.:n
l.
'
, L tim p~~~ ; -t .
1 ·1-" ,. (0.25. ().25. 0.25, 0.25. o; l$:! Sl"liOil:lt)' dis.ttibutioo for Ole MatkO\' ch~il\,
106.
106.
39
}. f Pt.><«> . .... 4. lim pf:) = J/3.
-~~
(i)
(ii)
(iii)
~;.[; x E R ~,., fi."l)d? u fi nrtt u(-x)- -u(x)
.u (-1. I) <I Ri'~ 11(x) =0 I
m11 H :R <1 f<'l';; ~~·~s JiJf 2 "
1. /:tlliflf'lih "fr1 ~ ~ # I
2. 1"./1 X ,; /iiPJ /(.t) > 0 \'1/ j ~if I
3. R '{'{ f '{'!' ;nfiJ;mrT "''"' ... .,. f. I
I . .!.,,~ let R(.t)=-z.--e l for x • Rand u be il continuous func-tion on R su<h that
'o/2:<
(i) 1>(-x) = -o(x~ ro ... IJ X • R, """u ...,_,.,
M u{%) =0 f« xe (- 1. 1).
(ou) Jii(.T)fs -;/ • rorall x< R. 2 2tl'e
l..ctj(x} = &'(.\') + tt(x}. for all X 4 R. then
I. f \:;l.t\ take nct;alivc v;;alucs. 2. /(:r) > (l for ;dl.t and[ i!; Ml integrable. 3. f is a p.rob~bility dc:1~U)' ful'lc:tion on R.
4 , I is an intcg,robk f111KC10n
107. ""'fit; X, X,, ... """'urf(I>F#; w I. <l1<f X.. -n l1 3n (n = 1, 2 .. .. ) <i -<t-<r ~ '""""''""
~ t I 'Uif F.; N = I, 2. .. . r6 flr6 S N = );, ± :!,_ f'1' S, <liT •1:'1 ,_ P, If I 7PJt •• -1 vN ... d n
~ Cf>~ :rFP$ 'iUTI1fRI 411fib:c:r, fr? ¢ i'lt;.r rtiFPf itfJ f.)f{r(J 11m11 t I f.rq t) ~} (Jft.l ... m/tf 111.1 t/l >
40
lim f:V(O)S<I>(Ql ... _. 2. lim ,.::.,(0)~<1>(0) .\' .... <6
l. Inn r:,.(I)S<I>(I) ·"'-"'
107. l ... X,, x,, .. be: tndcprndcnl r.lOdom variabiC$ With x. b«<& umlomlly <ltstnbutcd be\Ween n o.nd3n.n= l,2,.
lOR.
' Lee S \ ~~ t X" ror :-; -= 1, 2 •... .md let 1:~ be the distrlb,,tion function ofS~. Also I~ <1> vN ... 1 "
denote the di.stnbudon function of a St<Uldard 1sorm3t r1ndom vamble. Whicl) of (he f(lllowing lslart lmc-?
I.
).
2. lim F,(O)~<l>(O) ~-'-"" .
4. lim Fy(l) ~ <l>(l) o\'-."'1 '
2. x, • 2x,o.t ~ w4<1 * , I -(X,·2X,),Or#~'JM'f.r~<rt I 2
I OR. Sup1xw:c X1 h:a$ density J;(x) = ~ e-... :o ,x>O and X: h:.ls density fz (x)•~((ltnl ,x>O and
X., X, :.re ind~.":~ndcnl. Th¢n
' ••
4.
x . t 2X: Is sotncicnl tOr e. 1 -(X,~ 2X2)i~ t.mbh•~d for 0. 2
J 09. '111 .9t t:'li~ .,.,e. n ( ~ 2 H*~>??"iH'd': ~ fW W 1ife7f ~X~. Xr .... .X. #. t:t i(<P. ~ .s;;;:r
~~ ... <J') m '*""' ~ _, < p <ov{'O' O<o' <oo n.t"""", I ot
1 2
),
4
0: ~1 ,J.~ l().~ ~ <f lR 3Ffiir-ro ~I I o: _, \.,..,,.,;;a ~ ;;(Rq ~l:ia ~ ~ o1 It~ r;.~ .JtW'iiP. 1Jft ~ it llf11 ir-1 1J<Cq ~ tRb f I
o· It .~ rr.if.tm 3Gl1i8"1 fll 'f'.st'1IJ'ffl' "l'f.."1 rmm ~ lit* """*' wm ·"'~~~ Ol ~ f0fi'J ~ ~-!.6l ·o'd -3-~ tfi ~ ~ •7' '(,'lft {fm Jl~ $(\It~ ;pf I{1VII ~ iJ,,, /, I
'
41
109. SuppOSe that we ha\ .. n ( ~ 2) I.Ld. observations X,. X,, ••• .X.eac:h •·llh a...,.,... N(}J. a') di$triburion. "~~ < p < co:aod 0 < q 1 <oo s.Te both unknoWIL The-n
I. che maximum likdihood eslimalt of cf is 311 \lnbi~ esrinute for cl. 2. the uniformly minimum variance unbiastd estim:ue of a2 has amaller nl(:M squared
error than dtc: nuximum likelihood estimate of~. 3. both the maximum liktlihood atimate 8Jld the unifomtly minimurn vnriancc ~timate
of ti are asymptotically consistent es-timates 4. for any unbiased estim:U<! of 0:, there is. another estimate o( a1 whh a $-Tmller mean
squ.ued error.
110. ""'~ x, x,. ... .x,. .v-.. .m to- l'I.&HI 1 >~'""' , . ..a.,.,~ 11 ~RroT t. vm-«><o <"' ~ .J!~ Jm1"R I I tit
I. Rfimf '""l 0 .., ~ ;JI;<rR'f t I
2. 1lf<Rd 7fi;rn;l. e• ~ w~-~ 1 , 3. gfimf 'li"l; 0 ltf1 ~il'fl'lr'flr: "J:'11f'f """"~ ._., 'Ttl t I
4. gf(r<"~f ¥fi){iff1. e (fiT e:¥w•n-l<ov: ~ !fflFJ1 3AfiRrr ~ 7ftf 1 ,
110. Let x1, x~ ... . .x!j be i.i.d. obscr\'li.tions from~ unifotm distrib\llion on U\c interval (0 - %. e + !IS ) whcre-oo < 0 <co is on unkno\1.11 paramt.ter. Then the
1. sample mean isM unbiased estimate fo. 0. 2. ~Je rnedi:.t is an unbiased estim3.te for 0. l. ~ae mean is not che uniformly minimum "-ari3aoe mbiJ.SCd cswmte fur 0. 4. sunpk mcdizln d not i.ht uniformly minimom mancc: unb~ tsCWl~ for 9.
Ill. ""'f.l;x """""' if(x;I.)•A<""", x > O, ;;ffl 1->0 JIWi'ft 1..-. k <X~ k+l, k ~ 0, 1,2, .. .. ?,X »Hil~~ >I Y= kfil<mrt I Y '* m >IY,. Y,, .. . , Y, 'f'l'~
I ' • 1>flmf muiliff t 1 ,-:r filr l' =-L;r, 1 ni .< w 3ITf["' fitfrurmmw A t: , ; ...
l. 2. . I A= -= + 1 y
3. 4. ~~--"""'I
111. S'UppoS¢ X h3s domry f (x; ).) • A,e--4, x. > 0, where i. > 0 is Ufl.ktlown. X is d.iscretized lo gi\'e Y """ k if k < X s. k + 1, k • 0, 1, 2. . . .. A tsndom sample V 1, Y.: • ... , Y11 is available-
from the dislribmion of v·. Let P I:J!. f,r, . "lben the mcLhod of momc1ll$ estimator j of i. 11 t .. 1
is
I. : } I'.=-=
y 2.
42
• 1 J..--+1
1'
) . • ( 1 ) ; .. ;; loll + y a. the: so.me os tlte tn3.ximumlik~l ihood c~t1n«1tor
112. ?."iff Q.'M;2ili x i /({V Oft~ n#irt, &uf'i[ ((x- 0) = f {0 - X).~ ffifl1 ~ <PWS
~ r rt ~ 714 (~N:w:m. ;:<T:j;i ~ tt ;;m;, t:tM f. X 1• x~ ....• x,. 'rdls-111 lt:9 • o o;:rn •
u,.:&>o~~~fit$~J:/i'Jqf.., s.~L ~ CXJ.VIif
'"'
{
1, mil »0
f<:3 (.<) = O,llfll x• O w fi#;H 1
- l,<rfil « 0
'iR fi); z..,"fr:lfli ~ •r.r W7 v;v.fl 100(1 - a)=' mRPrr1! I_ ;;;rn 0 < Q. < t I nt f:'lr.r if N """ i11/ i; ;q!) t/f. ,
I. oft 8=0, "' lim rfs.>.Jn:.} =1 ·~ .
2. '* 0=0, ~~ lim r(s">fn•.}=a .. ....... ) . mil 0>0, u) lim P{S,> ,I;;za} = I
Jl • ' "1
•• <Tfil 0>0, ffl limP{ S.,.>,r,;="} =a ·-·· 112. let X1. ~ .... .X .. be l.l.d. obsotr,·:l(iO:lS frorn a d!:Stnb\&CIOn with continuous pt~hly dc:nS:it)"
function f whieh is symmrtnc around 0 i.e. 1\x- S) = f (0- x) ft,r (IJI real :c.
• Consider the test J ~: 0 • 0 \'S H,. : 0 > 0 nnd the: ~ign test statisticS" = L sign(Xr) whe;e
,·:.1
fl. if .»0 sign(.<) =1 0. if x=O. 1..oo z, be the._, 100(1- ")lh p=cntikoflbesoancbrdoom,.1
-1, rf x<O
d1$ltibmion where 0 < o. <. 1. Wbich of the t'Ollowing i&'an: ~et?
1. lfO ~o. then llmPis,>fnz,,} ~ L "''"~ 1
'
43
3. If 8>0, then lim r(s. > $.z,J =I. -~~
113. .,., ~ x .. x, .. .. x ... N(O, a')."' ~ 10 w f.r>I>'T<IT '"" ~"' •TTfllnr •filffl * I e •> ... 9-N(O,
r>. r .. 20 :n ftrnf , ;;F}. fi';; x =..!..f. X1 m a ;~. ~ ;fc:i i5l ~ 8 ifO Cff((ff t : 10 (•! •
• - • 20% I. &=X 2. 8=-21
3 8,;x if x~ o 4.
113. Suppc>se X., X:, ... Xu> is a random sample fTom N(9, ol), ¢ = 10. Consider fl)C priof tOt(;),
e-N(O, ..'),., = 20. Let X=_!... I; x,. Then tlle mode &or o •• posterior distribution for 0 JO (•I
satisfies:
I. B=X 2. , 2ox fi=-
21
3. o~x i f x .... o 4. 8~X ifXso
114. (X. Y) l1'l fii>U 1fil f.tt:r rff'J yYcff tR ~: (0, 1), (I , 2), (2, 3), (3, 2), (4, I) 1 ;it
I.
2. 3. 4.
9 X 'T7 Y <6f "'J'f"f- IT'/ '1/Wt; ""''17/W t Y = - I s Y >n X W ~-iPI fflNi WIT<mVT t X a 2 X <r Y •h4t<nsr 'lTffl;iq 'J"1ftR 0 t ! X l.1 Y <i: ;/r.; (1;1 ~ TjVlfrfi + I t I
ll4. Consider the following five observations on (X, Y): (0, 1), (1, 2), (2. 3), (3. 2), (4, 1). Theo
1. The least-square linear regression of Yon X is V e ~-2. The least-square linear regression of X on Y is X= 2. 3. l11e con-elation coefOcient ~tween X and Y is 0. 4. The correlation. coefficient between X andY 'is + I.
44
II S. ""' !* .,. t,. ... C. ....... "" l'fli'9 ?1'1 <hi(O, ct') 6 "'J"" ~ o I Y,. Y ,, .•. , Y, $ iiR >f RdHW~~~rm#
Y, •p+ c1,Y,.,-p=p(Y. - JJ)+,ft-Jl ,,.,. i=l.2 .... ,n - l;
l • 'i!fYf./; T =- L Y;#O<p < l 1fo'>O 1 f17 n~ 2-*~
)/ ( - 1
I . T "f'P IlWMPV ;Wf -8 I
}. E(T) ~ ,u, """" (T) > o'in.
2.
••
T fh1 "Y"i1'1 J1 r::f r:rmr:1 a' lu ff I
T-N (p, 8') omo1 > cr11n.
115. Suppose tto th··· ~ an' u.d. :-..xo; cr'). Qlnsida' v •• y'lo····v .. defined by
r, • p.f. c,. r .. :- 11 = p(Y,-p) •H t, • ., 1=1. 2 • . ... n-1.
Let T =.!. :t >;. SuppO<e 0 < p < l ond o' > 0. Tit<n for"~ 2 It i:l
l.
}.
'l' has a nonm:l dislnlluliOtl.
E(T) = p, ,,..r("f') > a: tn.
2.
4.
T hils m~an J1 nnd V':'ltiancc fJt/n.
T-N (Jl. &2) where S::;. ~/n.
116. ~.. ~- if ([/$ ~ .. q,; 1# flrrf """" alii 'Ia! iii) "'3""' p 111 Jl'""" ii!J Rn1 '1'1 '-"'; ff.!Hrt ~ a't :m:<&..;;oQ"i A IS B JINlf .... J(IVT vfi:trof ~ ~1Jt! t '
l>~ A: 100 ~ W 'P' f11'fR1 111ff/W; ~ R'lf - (i;4 5'iiR t (SRSWORl ll<i11 t ~ M 11 x ;m l1ll Ill ti'rt 'fit ilit & p iii/ :;w ffl( J6llffl ii'W1f t ;
• p, . 2(10
l p, \"S -""""'""" _, ~ '"'g p; ->1/ I 2. P••P:iJFK.Jo~~f I 3. p. •1'1 ;f?.'f .:1.~~ .... ~ tt C71f IJ:illf ~ p1 :Si g:wg f lll d) #IJI14 'ffll1"11 I 4. J7l tl'!l 57 ;m ~-? <Jtd sc'.-. ,.,., rrtr<fl?.FJif uf1 Jij··m iPfr., $to) w If""' ;: p;. '* f1'ff'f"1l
tfliN tiM 1
116. Jn 1\ stn..,·cy to estimate the pro,,onion p of votes that & p<Orty will poll in an clt.'<:tion, $latisti~,;iaus A ~nd B follow different sumplin~ suat~gies as follows: St;),tistic:.i~ Sclcctl'i fi l'iirnple rnndom sample without ~phi.c~mtrll (SIU>WOR) of 200 vot1:1S, finds that x of them will \'Otc for the -pa.rty and estimate$ p by
'
45
·Stali~ticlan R: Divides the-voters> list into Male and Female lists, selects lOO ·from each list by SRSWOR, firtds that .t1. x2 respectively will vote for the party and estimates p by
x1 +x1 p,=zoo· Th~; number of vote~ in the (wo lists are the same. Then
1. P• is an unbiased estimate butp: i.s not. 2. PI and P2 are both unbiasod c:stimatc,s. 3. Pt and P'! &J'C· both unbiased estimates. but P2 has a smaller variMce than p1, or
the S.'lme v3riance as P• · 4. Variances of PhP! are the same only i f" the proportions olroate. and tem.ale
voters who vote for the party arc the ~me.
JJ7. I, 2, .... 5 l{ ~ 5 """"5"" l:t <'~"<: OW. f.J"'' """'~<a fii<Tti :
<!fey j : /l' 2, 3}; """)J: { l, 4, 5)
i't"T >i "*'1 ffl/W """I w(l l/f.?
I. 3/~ <f.<li! # I 2. ;;r{f o' r:;r. !).~ w IrrRfli t. r:rr. rnifi';;r. ~ ~ ;;f; ~ 'fftr.rr w;f.r.m ,J{'fiJ7(fl(5
(Gf ffffl11J Vi ff} 2d ?If 4ci /. 1
:>. 'hfililr<'RI. r.ffl11~ li<'lmr rt"' tt ll'f>r.1r'f t •"' w;tila 1'wr !fir{~ tfltritr 'ffl'l' 'ftf t 1 4. 1ft </; ruo; ~ - <ilfe '!.:" # I
111. Consider the fo11owing block design involving 5 treatments, labelled 1, 2, ... , s. and two blocks: Slock l: {1,2, 3); 8lockll: {l,4,S). Which of the following statements is/arc tn1c'?
I. The design is colUlected. 2. T'1e variance of lhe best linear unbiased estimator of ao elementary treatment
contrast is either 2<1 or 4cr2• where o2 is the varianc-e of an observation. 3. There is no non 4 triviallinear function of observations coJlected through the
design whose expectation is identically equal to zero. 4. The degree~ of freedom a!>$ociat~d wilh the enor is zero.
I . 3/fi~R;'f "1fT -:tid/ <::FPJ} fffl?Vl :8 JiliMf =afl t;} ~111711 I
2. ~y 161 :fi<<li ~ )RrFj Ti /&If rrtfl' l\} ~I I 3. """ """' w _., >t'ff'l t. fW'il 'fffOI'I ~n f1l; >fiW '..''" t 1
4. ~- "1"1 ""' 1'11T ""' """' 0 't/1 I r.ro I
46
1 Ut Suppo~o~ ll1ou ''e: hav~ 3 daLa set constst:ngof25 observt.lt<m1. where eoch value is e1lher 0 or L
TllC mean ofttt: dab cVlllot be': Ja!ger than the \'~riancc. 2. 1k mc:u of lh: cbt2: WX\OC. be Sttl3l~cr th3!:i the 'm~c~.
l The: me~ bctng same as the variance implies th:lc the: mt2n tS1.t'ro. J lhr \&r*'« Mil b: 0 rf and only if the :':"Qn i~ etlhct 1 or 0.
I 19. :z::r.m.1 ~ ~ "~l ~ 5. 4,11:1 - llo! S IS('(i 4xz-X1 S IS o) ~ ~ ::Rf W x :tO~ ;oc_.. 2:0 VY ~~ 1 fl!J•J ~ ff 411~-r 'HJ/?} tR'ZR ~ 1/f?
3'(1 • 2x: <~~t J,ofittli8'1 rfR t 25 r 2. 3x1 "'" 2x1 l1ii "~ rrr:r t ll i
3. ~'<1 I 2X: IM <~>tf qfWI';a Jt'R!tll?Pi "f#1 (.1 4. 3x,+ 2x: :m ttl!¢ ~ y.«r-t :rtf t J
119. ('on:o~elc.r thv \';iriab1es x , 2: 0 and X! ~ 0 s:uisfyint~ dtL~ con:ar.:unli x1 ·I• ~ ~ S, 4x, - x1 :S lS and 4x1 - x1 ~ IS. Which .,fthc follt,wing statements isfa~ OOI'r>ect'l
I. 111e m:u.tmum vain.: of3x1 + 2x: !s 25 2. 'l1le .uinunum value of3x1 t- 2X! is I I
'· :l>c1 <1 2x: has no f1m1e m.:nimum 4. Jxf+ 2>.1 hitS no fit Lite n:Uniuu.:m
110. ~ """ "',/ f"' V'fkll ~ 'IF¥ R; ~ """' '~fit 1lfit 12 fir.« 4 't"' lll1Rit ol ..,; # v> """' mr - 'ila ¢it 8 fll·ri: ,; "' >!>:~' <l 117 OJ!1iV # I ofl: _, 'lfll 20% ~ t i!t tunll ,JI.4IWI 4
I. ~ tf ~ ;;J; r.:uJ ?frF.8 ~ 41.kt 2 /. I
2. Y.,'f.th 4 ~~nr;r,1 ;J4 !G!~ rfrprj ii 1iOtJ 4 t I
3. ~ it tll4tb f;m Rffrrnrrrn :r.«"'T ?J'l!'ll it ;;.;pr 16 ~ t 1
4. Sf",'fA il Vftrrh tflY.'I fitd;IIJ )fl/1 i{J<:"l tflflr i/ ~ 24 ~ t I
120. In a S)'Siom with o smglc SC:.'\>'er. s~ppOSc thot t u$tOmerS nrrive m a Poisson rote of I person .::very 12 minlltCS and arc .scrvtoc'<J ut the Poisson r.ue of 1 scrvic(! eve!')' 8 m.imues. If •he orrivt~ l ~l<lnl<:renscs hy 20% 1hcn in the s1csdy st:uc
I.
' .. 3. J .
the lncrc:a~c in the average number of cus1omcrs in the syst<:m is 2. the mcrt:II$C nl the :l\'Cntgc: tMnbc!' oi cus•omcrs in lh<: t.y~1cn\ I~ 4 • the UIC-rcasc 111 the average tim:: spent by a custOtl<er in Lhe system itt 16 mimttcs. I he tncreasc in the average 1imc spent b)' a customer in tJ1c 'Yftcm '' 24 minute$,
•
II
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Q.No. CD ® 0 0 Q.No. CD ® 0 0 Q.No. CD ® 0 0 37 0 0 • 0 38 0 39 0 40 0 41 0 • 42 0 0 43 0 0 44 0 0 45 0 0 46 0 0 47 0 • 48 ••
49 0 50 0 @
51 0 0 52 0 ~
53 0 54 0 0 55 0 0 56 0 0 57 0 58 0 • 59 0 • 60 0 0
61 0 62 0 s3 e 0
0 0 0 0 0 0 0 0 • 0
0 0
~ 0 0 • 0 0 0 0 0 0 0 0 0 • 0 0 0 0
0 0
0 <0 0 0 0 0 0 0
0
0 @ • 0
64 • 0 • 0 0 OJ 0 0
s5 e 66 0 67
68 ~
69 0 70 •
71 •
72 0
• 0 • 0 ~ 0 • 0 • 0 0 • 0 0 ~ 0
0 0 0
73 • 0 • 0 74 0 @ 0 (I
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
0 • 0 0
~ 0 • 0
<t 0 • . ~ • • 0
0 • 0 0 0 0 ~
0 ~ () • • 0 @ 0 • 0 0 0 0 0 • 0 0 • • 0 • • G1!J
0 I)
0 0 • 0
0 0
• 0 0 0 0 0 0 0 0 @
0
• 0
• • 0 0 ® 0 0 0 0
97 0 0 • f)
98 0 •• 0 99 •ooe 100 •
101 •
102 •
103 0 104 0 105 0 106 0 107 •
108 0 109 0
t1
• • • 0 ~
0 0
• 0
• • • 0 0 0
• • • • \
0
• e • 0
• • 0
• •
110 • tll 0 • 111 0 0 • 0 112 0 • 0 113 0 • • • 114 • • • 0 115 • 0 • • 116 0 • • 0 117 • • • • 118 0 • • • 119 • • 0 0 120 • 0 • 0
II
rrz CD 0 -:::CD ro:IJ a.o s·= ... z :JC CD 3 0"0" 0 CD X_...., co en Cllc err (/)~·
-·CD ::I (") co,..() :TO co a.
-oCD ll>Ro ::+o ~ CD ::I ::I -co-. -·CD ~() ::I 0 ll>a. Cll CD .. (/)
:J 0 c a: