csir net examination mathematical sciences june 2012.pdf
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CSIR NET EXAMINATION MATHEMATICAL SCIENCES JUNE 2012.pdfTRANSCRIPT
1636
Arinr ,k.F.-T )..s.
4
2012 (I) 71Prff- f4---v77
=
ifim r g f To- rraw riwry 71. crqr (20 VT/ 'AP -I' 40 'pr 'B' + 60 .giv 'C' Tff .10004 Sq7 CMCCM MAW) 31,PY 'A' ET sarw-diT 15 .3#7. Vf37 'B' 25 sm-91 num' qr,,r 4C1 120 sirq71 c e rat ftift ,sue ripqr r?' .ecr ricr .UW 70- ttr* !?irPT 'A' # 15, 1177T 'B' 4 25 mu
'C' 6 20 ocr.e) . 45'; q7;.0 ziO Mr* 1 2, OWN cf7 ,3rdi aur Tar / artriT qw ark rft 467 fcsi"k
Of -4q gRum TEr s.* i910 gi ete TR . # .LE 4Ye fTi frWreke !ii. grycw ,F7F-4 P4.) WY' 1Rrr?. g ,M ff-VF Tel7 rPr
r a5-41 craithifter- 7-4 riofq 1 3_ rig Fe av mr9 r airwr qiw. 3767 WIT REF
giqime d6T g'7 r 14¢ AfarKr my* e-FoIR4 Lrrew4 t 4, WV 4rcr* a.T/A/V., fivr-45r t-ff Ow eft, Pazgr w3• stf *
dAry tqfArr rfa MTV WTRT Zfr gtzwail JW4k70). t fik grg OUT? VAIN' 4" A'Kri TW2r). air yft twovr4 tl,reirq NTT W-4 17?' 0-40%1 0-471)` 7Fet rri94;` Jrtfecf ;ea WV Mfff? &FM' trft .1%.2/0? dr/T4ret
yreiPir arKit-fa 1ft .r.reArt. Fl ?met
5. oirir 1.rekt Wir9' 2 .5stIF ;1.7313 1 ArN7 crzi7 3 4+1 5 MOT 'C' # AF#W P19. 4.75 ,31-47. Wq. ,r oTgro. um.p. airp.Lunr.r0 2,014/451 w7.1- 'A' @ 0.5 at4F rINT VR. 'B' C7.., 0,15 2-4- #114wzr 40e.r.r 7irtrr 'C' vMerg't MrtirfFi'M rvarir
_ gritt 'A' mr. Vrg 'B' Frq * MTV 04671 'g g r # 41Nreir C.171; req .fr& arartir frarcrhy Fry I ‘3,74, it.) In-4 0. sin. wr qq)63.4 Fa.
rye r IFT77'C' Jr.aW Tr'ar "cr4F"T "tri;T ?MP IT vet Et F-40 e 377 r gr#41:7 ZIT l' 741 a5r qilet V7j9. .4'777t AtEr Prqr Fret ,O4KNIT acr 27.47 ffef 46-T4 147 ai7n7W sQc qe unr77
7, .111. P77# ,§7 C )0;1' .57117 W-e# g= or? 4raggre 4v7 $.yr ..-.7 ctt:Er# rtes' arzr)727 Z12; RT Kfltir
S. XP-Zrgc ktrrte ZIT VW r141 kNfi 47 ...Tar Tet .Gten rear 9, .47-4kr epypro.e. m.7# wfti 51%1.74 V7 val.? 4 vi-MvIT? dagrzr Mr'? 10. 0,,eg r,.?ck, 04.4).1 Qc-f# 7er # I 11. a e5 ̀a-** Itt.)-41 ?ifctrel VIFF OH 1
,*? rJ
ff.6-4 : 3 :00 Fi21. 200
awaw# gnr idt pie 03-dar 0. .ef run.' t 1
ct * 8-RFOR 5007 R0112---4 A14-1A
5107 RDA 2-4 A4-1 B
L The area of the sliuded region in cm2 is
4. +21 3.
3
Bev 4.71. *-4.wa zNO 2 4. 0
( A. 2. (ff -2)
(5-
4. (21r +2) 2 4
( PART A
a_
erwAve AVM' .?7% * .RT3T * *NT wiirrir ?ill* e .3#7 Vnth !ff. 10-00 t WT7 at V"?"a 4 *
2. The angles of a right-ariglecl triangle shaped garden arc in arithmetic pa-agression and the smallest side is 10.00 in. lie total length of the fencing of the garden in in is
1. 60.00 2. 47.32
3. 12.68 4. 22.0
1, 60.00
2. 4732
12.63
4. 2168
3, Mr 4 q:.4r &vet. 470 ArElyri a f M777 AB 61 / AQ = 2AP ff). f477 # Alm gta-eir i=re
If
AL is the diameter of the s,4nicircle as shown in the diagram. IC AQ = 2A13 then which of the following is correct?
L. LAPE=-.LAQB •
2. ZAPB=2..zAQB 3. Z APB - QB
4. Z APB = -Z
4. ifirOthrl) e wRy A 4.rfroordwr 25% rigeauzic 4614-wetgrrefttraftwBO5rfaTe 50% R04 24 e ritZ '? A aV Bcr trfrrerwiti, FrIFFT e e B 3itv WiF A .t.r.eaen qg5- WI" .3177ET *IT
I. 1.44 2. 1.72
3. 1.90 4. 1.25
1. .4 APB 1 = - .4AQR
APB =2Z AQB 3. LAPS = AQB
4. = 4 A 051
4. The rabbit population in community A increases at 25% rnr year while that in 13 increaSeS at 50% per year. Tillie picseig populations of A and B arc equal, the odio of the ►wriber of tho rabbits in B to that in A Lifter 2 yifIrs. will be
L 1.14 2. 1.72
3. 1.% 4. 1,25
4
5_ 02 -V IT, 9-c-a4i 4,1 4 *a. 'LW f1 .377173 tree 1 5- ur-if 150°0. .07 .1.r4 q?
ira -173.1 4 47W ffc.r7 71414
41=11.0 Zr6- vif 777 ;q !../75 9F1 ari* ,X4 efM ;rm. a.?` ORR' TT FR)-FPfir-Y 15I? rF.' FFIVI J7 re. 4.--fw 4.re CI N17
440. 4r.-7 NRITT,F ,F.);77
inoire4; each of 0, and H2 are in two ScParate cootainers, each of volume go and at 150 GC and L atmosphere. The two are made tO Knit in a third concauler to form water vapour Mai H! is exhausted., 115,1,en the temper me of the mixture iri the Third. container was restored to 150 `C„ its presbure became l atmosphere. 'the volume of the third container must he
I 2. 5Y„'4
.3. 3 4. 217D i. Pro 2. 51i/4
3. .3 V0 +2 4, 2 Y. ;,
,51" 4.1777 Wr... vr-q-R 6, ur-4.,r n I dird. rep T4-;..irZ-q7 air 773 4 e = 77 cpw rrld in' 4" uy4) AY-AP? .G7 .1 14r.q. word, g ,fi'mu
4st a 7i--gFER1-6:Tf 07 Or
], 161.5fEr.r trW4 Tar PrI1797 Wr4
2. Ptgrre w774 r F4T {Ile.?? 71,7
3. P.'orur a"..` FT17 FrirPf 4. .0..8irtv arY.4 zirmpl;:zef 4'37 Fsfrq
Helium and argon gases in tWQ separate containers are at the SaM.0 temperature and so have different root-mean-square (tins.) velricitica. The two are mixed in a third container 'keeping the tame temperature_ The velocity of the helium atoms in the mixture i5
1. more than what it was before mixing, I. Less than what it was before mixing_ 3. to what it was before Mixing. 4. equal to that of argon atoms in the mixture.
7_ Fir .-.3iv4.7 17'~ &Ay., ..7trzettr A-41" WWI t, .4).A vr,
(a) uKa4, rr 7r r Ur? e (h) Om-7.aq .75). Rrar; (C) grtv trr (d) yappe r a1.7 '-gr Tj"
.7-451/4Tr 4 4 ah-o/VF-4 .Farq ere gye ?
1. (ii) 2. (a) (e)
3 . (a) otf (b) 4. (a) al? (d)
8_ 100 um zro-risf.qifi Oft X•51.120 401 crqf
err
crq t 1 5.0.'4, Cr? 60
TN 4Fe# JI7ThFir
.7.ra. S ?,F.P7 X dirfir' 344 4.1 k71
1. 10 2, 8
3. 20 4. 80
9_ 7w fi.44r wo4 iriAlfrata r gt2.7T urzi wft
1 . per .00m. tre e)77 2. 11-4 Jac w. 4*45° crz err 3, .31T-FTY7 ire AT 4. Erthr* ..91fih,i ER enr
7, The mincrui talc is used in the manufacture of soap because. it
(a) gives hulk to the product (b) kills bacteria (c) gives fragrance (.111 is SOO grid does not scratch the skin
Which or the above statements Wan correct?
I_ (d)
2. (a) and (c)
3- ,(a) arid 05)
4. (a) and (d)
8. 10() g or an inorganic compound X -51420 wntaining a volatile impurity was kept in an ovcn at 154'C for 60 minutes. The weight of the residue alter heating is 8 g. The percentage of impurity in was
1. 10 2_ 8
3. 20 4 - SO
9. On a etrtairrniglit the moon in its waning phase was a half-moon. At TniiiniOit the moon will he
1. on the eastern hOriZ,Ort. 2. at 45° angular height above the eastern horizon. 3_ at the zenith, 4. on the western horizon.
1. 31X1
3. 2400
2_ 150
4..
11, DiSpilteCknellla VE LT5.14 1.1113e curve for a body is shown in the fripro. Select Oho graph that cOrri -LealY shows the variation of the velocity with lirr
CP
10.
Splint Nrow..t
CAI
6
10. yree 1.1' 5 1171 ) tiCh.1 RIMY War A. i Prrhew RM * 'WA.) RTATFEred -46 ..AM4e.e q3eirai .Rkoc-r Re lie 0 0 ilw779.
so wkr 75," farheft lepft 70- 04,4 ?
10, A gm-m.114)m is irradiated in a nualr-ar motor for 5 days. Ten days stiff irrational .), the activity of the chromium radioisotope in the iroln5rom is 600 disiritogi'alions. 1W hour- What is the aokr.virY of chromium. radioisotope 5 Sys after irradiation if its half life is 5 days?
1. 300 2, 150
3. 2400 4. 12120
11. Arvu 14-vr PPP 45-7 41w fim ziY ? al .? ow MIF .R4 Or 47 Ham. qi!9
k"
l.
HF
DI
P P
Lim dac
12. 1 2.
Sony e•rx
Y : .1.
55411 1•N rt —0
IS:•
fila A 4 4,..17 )47e fikm e 4.t errs h! gari 3.0 kw.mit". 7 qe,....r..1kr4
ereqrar F107 uire 12r3 etre 11.74 aehr ..Tdrzt; 4:Jr.iftY r .3174' 0.4 fflr.vz
4Tbir.r e ) 4 witc17 057 (
L
3. 3.i .0r.a7
2. 2.g
4. 15 Atilt? .
urraq Gsounqi
0 0 o •CF 0
o 0 cc 0 0 0 O 0
0 0 n 0 0
6
I. kg
3. 3.1 kg
2. 2,9 kg
4. 3.5 kg
I .
3. QV ;sof 4,
2.cw. clmqrr 1_ a cire]e 2. all etlipS4
4. a lriangle .77 fttp•
3. a squwe
The spring balance in Fig. A reads 0.5 kg and the pan balance. in Pig. reads 3.0 kg. The irunb.loek. suspended (OM the spring balance is partially immersed in the water in the beaker (Fig. C}. The spring balance pow [email protected] 0.4 kg. The reading ore the pan balance in Fig. C is
13, qw y7-41 f#0. er.a?-el 4 FITT
4.1W 1i4.; yTell rraiTU 0.4* 7,5' 'YAW .'`c9*
TRCI.) 4-.SW Tra W? re. 70 wft FifErr 1 Oir?ra •o:r yo4-0 ..47# .E771" .517r gt.,7rdl
.3TP,' tell 2
13, The ends an rope arc fixed to rim" pegs, such than the rope remains slack, A rieu(Al is placed against the rope and moved, such that the rope aiways remains taut. The shape of tile Curx traced by the pencil would be a part of
14. ut;,; ;Fr 14,29 i7 awf oct frr azuv f ectF fek6-7
R17. 7-1710 qm'zr T e
1. 4.R4. e.7.4a U-0 'd
ro ..717Er
2. 7374 4'0 roe. i --e# ecg qpi? 4 gReif
3. 7:.? '4.4 114-4"d 2fd kr.;fif TAT FRa* tie" QM- g 4. glH•70 VOr g74 9";47: Mt.
15.. wg.rrAP-zor r4-41. 3,vie PT? were eff A. B.. C at? 13 # zi.57 R gift
3433 4; 4447 Prq Ottft.T MR' Fe
1.. -ferdrere v4- arm 7P711/! 8,C,D.3/kR 2. R LS'crq C .*D 3. D Erfrivivy e 4ref Virg' C, 8, A ale R 4. A Fri'r mird.; Niilt B, C aim D
14. During ice skating., the blades of the Ice skater's shoes exert pressure on the ice. lee skater can effIcientty skaie because
1. ice gets converted to water as the pressurc exerted on it increases.
2. ice his converted to water as the pressure exerted on it decreases.
3. the density of ice contact with the blade3 decreases.
4- blades do poi penetrate into ;DC,
15. Four sedimentary rocks A. B, C and D are intruded by an i.gneous ruck R as shown in the cross-section diagram. Which of the following is correct about their ages?
G round turrace
1. A is the youngest followed by 3, C, D and R. 2, R is the youngest followed by A, 8, C and D. 3. I) is the youncst followal by C, B, A and R. 4. A is the youngest followed by R, B, C and D.
7
16. Tv' u"reF 7k- '). .1* MOP figicr Virot Eff 4-itvro .11±47 ;NI-ft' RI -Troi #
rai[une
Strain
ri-3-.4 4 4 Kit. rpm 4FER. mEir ?
16. Tilt strain, in a solid subjeockrl to con'jnOCOS sh,e-ss is plotted.
fa Uwe •
:71
3_ giv-d-67 iRiMirdarft Oaf t I
2, rr r R371%70-11' 1 :0410r i
3. RIF-6w 6.M. tiff ohT r astqf
aFf
4, Row w zr•ler 4t)J
17. cv Ay p%, Py ,;‘1-42.r.-1 zieriw.# wiry- 747 qy 04f J'Z'* 0.77 4•17 WY 6 ir Jt-j. 773:17f Far a7 *71 ?
Which of the following stallements is me?
. The sol id deforms clasiicallytiEE the point of failure.
2. the solid deforms plaztically till the point of failure.
3. The solid comes back to •CiFiginal 5hapc and size.
on failure:_ 4. The solid is pffrriptberitly deformed. on feihu.
Growth of an .orgzurism was monitored at leg tar . inicryaL3 of time., and i5 shown in the graph bcicrw. Around which time is the rale of. gvowth pxrol
6.)
arm Di aril
1, 7-64 Rfrq q). ,247WIRir 2. 20 ATI 3, 20 30s eft
4. 30# 40
1 1 I 24 30 41 trr r‘d toll
1. Close to day 10 2. On da.y 2.0 3. 'Between days 20 and 30 4. Between days 30 and 40
18. cirt-r (R) YiO• FIF irri,rw.r) coW
eia•rPn• ,Q4411 TRY R-17 ?;977.7 Fe#
..114 kcfrfoi
(1) 04' OW Fs' (r) AA. (t) tit
yeirawrt ye-11.G
+ 'rJ .rfmr eru P,
IS_ A 21:n.? plant with Red seeds .(both dominitnt bait;) was crossed with a dwarf piant with white seeds. Lf the segE.eg.ating progeny produced equal Dumber of tail Fa and dmarf i.vbil. plams 7 wEnd yi.01.1 Id he the gortoryim of the parents?
. x TtR.P. likrx (Tr
3, TTRR x iier 4. ITRR x TtHe
1- Tift.r TtRR 2. TtRr t z rr 3_ TTRR 4_ TTRR x TtR.r
8
19 ir cft..? 4.1E# e.r. '40 “7.14? rre, qrfflivewt
vur uz.r 19. Three sunflower plants were placed in oandittorks as
indicated below.
OM' A: f r4f9 6-47 5 1r B RTE57117 N.4 Ntrir-r !r'$;
citig C : 41.0# fritM1 FVf
Plant A : still air Plant B : moderately turbulent air Plant C: still air in the dark
4..ror Ttir7 e
1. 9'4). 13 > -* A 2. rfilz A yltWZffri 4-7 > fi all 6.• 3. 4h C
cr? l ioymni-r :47 = 91' A 4' qirrriym
4, C 4.,Pow.i.re 177 >.1:170 A CR 47001-1
> F3 414.We'. TF
20. 29c -7' P JP/ err
1..'irrhic,h of the following statements is con rd.?
1. Transpiration rate of plant B > that of plant A. 2. Transpiration rat of plant A > that of plant B. 3. Transpiration rate of plant C = that of plant A. 4. Transpiration rate of plant C > that of plant A >
that ofplant B.
20. Which of (ha following is indicated by the =corn. panying diagram?
a I.
a b
a
L = at( 1.—I.5) for :id< t
2. a b 0 derqd O S
3. (a+b)2 = a2 2ah
4. > b 7-? MFP'i —a —b
a
E . 4. ail + ab2 . = 4.10 —b.) for IW1
2. a a3 > .133
3. 0+02 cr 2,b Tb
4. a > b implies •-a < -b
9
r PART B
21. e"0" 'MATFIElvIATICT AITM
re 7174: 274' twin) ?
21. The number of words that can be fanned by permuting the kners Dr 'MATHEMATICS' is
2. 4989.500
4, 8! 2. 4989600
$040
3. 11! . 5040
3. 11! 4_ g!
22. 50,000 451- triPiftlf 3177rck asp? e ? 22. The•num.ber of positive divisors of 50,000 is
1. 20 2.
3_ 40 4_ 50
I. 20 2. 30
3. 40 4. 50
23. 4FY .f& A u• nXn r, rg 4,7 GV-677 TR-7-41 lir Or Ye ?
23_ Let A, B be nxn real matrices. Which of the following statements is correct?
476tir (A-i-B) - (A) -I- Wfft (B). 2 Mfei (A+B) wffli (A) + %Ida (B).. 3_ ucrd (A+B) = virfq (.7,lia (A), krroici .
(13)). 4. Wit (A+B) = rf1 (wiiir (A), ma
(B)).
1. rank (A+B) = rank (A) + rank (8). 2 rank (A-Fl3) 5 rank (A) + rank (W.
rank (A=B) = min (rank (A), rank (B)) . 4. rank (A4-0 = max (rank (A), rank (13)
24, 7774 fds
{1 -.Plc fonr.40,1/.0t1 fiz) =
for xerli .1.11
for x 10.1111) 24. Let fdx„I =11- "
0 for x41,1,11
th Then
107/ [0,11Ere /:frra . rb70 riNPV -i•ca
p.m/ t 2. (4). • [0,1]V. itiorogpf sr.13P7,?ff
3. i9139. .t<0,1]*.1%le .04:(x. 0t f
4. 74-hr e[0, 11 l Urn fix) w siifflum
25. Trezri et"
;Oft itcf-cm- 2. sedwrqfw
3_ orrffrO ?itzr g 4. MIZIWINTT WitzT1
1. Ifni fix) defines 3 continuous function on
[O, 11- 2 . {fdl converges uniformly on [0,1 3. 11111_6(X)- 0 for an rc
4. liar ft, (x) exists for all xe(0,I).
25. The iIwnber Vrie is
1. a rational number. 2. a transcendental number_ 3_ an irrational number. 4. an imaginary number.
10
1. 0.
1, I 1. a 2, 1 47.
3. -1. 4. 2
3. -1 4, 2
26. ;77 . VW W7 Nrg"lir stir I A Oct q#P-Hr 26. Let P be a priraiiive cube !Doi cif unity. Define
.( 0 A=
Nar v = y ? , .1.9E ,ft 11. k. 7f4'nna.
ure vTw 7 .7f-ko w=
ILL') ff). 1w1A
A =( 0
° Lid
For a vrt.Lor V n • '0{. V;I.}ER. 3 ckffic
! p1.4 . ',3fhtTC 1,1 1S tTa(lEplaSe of v. if 44 =
(I ,1) then evals
c- ' 0
27 . 1:171 M at, a ).); Ai LE 3, 2. 3, 4 .1., a z + = 6). M opm.Y fP:54.1
al -I-
1. 8 2_ 9
3. 10 4_ 12
28. (3 8)2D?L
3r,45 I
1. 6 2.
3, 4 4,
29. 4,reOn' dab PPM' Tgi .11VET Dxii (n 2) cAR: e; 7 Wrerrr Male A OA a" 0 e 77a7ir .N.57
I. (u.2 -Fn-4).12 (n 2-n+4)..2
73. (n2-Fn-3).2 4. (n 2-n+3)./2
30. .574 / = [0,1]CR
(X.. = .19r=rum yc rO.
q..4' Vx.) mida
p(x) u:erf7 e TYV WtRi. N =
O.:57 731 g ? 3. 74.4e' u(); 1 6eref e grj. twasi = 0.x =
1L *gr. ..3{UM:Pftr le 2
4. 9(x) e
Lel M = (at, a2. 01): E ] 1.2 3 1 4.1, a l a2
= 6}.11grt 11x: number of' clitmeniE apt M is
. 2. 9
3. 10 4. 12
The 1.k El. digit ur (38)' °'
] . 6 2. 2
3. 4 4. S
29. The dimension of the vemr spa:. oi`a1I symmer.ric
maiden A = (a.0 of' onler nxT1. (n with real cntrics, All - 0 bud rate 'Zero is
{.n 2 -+11-4).12.. 2. (11 2-n--4Y2
3. (n.2 -1.n-3 .y2. 4. (r) 2 - •E0 3)12
] , $
L.
30 Let I = For x It = dirt x F 1) = Ulf y I .1.
(x) is d iscon I i nuous $oniewlicrc
2 9(x) i5 171311.1.1.33110LJS !xi( not c.....ntinuinisly
differentiable exactly 3t X 7 0. 0(X) is MraillUCIU2 Oen but not coniinnEvnsly di f fercnliahle cxactly x 7c. ID and at = 1.
4. cp(x) is f1inion;i3b1 rrn 2.
L 0*k.rw.g7We7rig 2. 0t zi7" .P71 76ff etO7 3. 1 tu•FiFIZTrrearit 4, 1e zrY .rintar
, 1 re 32. W0'4 L., — 4-3- 457 .4? ,f
1 n
W• Tow err : 1 (2n+] )
7T 7r 2,
4_
33. xfg? = iqx,y) i v(x,
-FaTimr-4 Gc-d7f. C — C
y)
1. C ri7 v AWer 4r-orr rEel Ort? ,e.f/tiVo VOW.' f
2 C4r9rie .d.ary cr?f ottTog*:-. ePTE i
3 , C v Tzgr q'f +I? ..47
Krffi 4. u arzighT7,9v Ref je
34. z)11 f: x Gtr wirea-5 4
a # SFIT nrci, (V, W) x rik -4 (FL 1()E! ' X R FarAw
3rd-47iW DAV., W) P17, if1' fhir OrdE g
1 . f (V, K) VI) 2 J(14, TO 3. f (V, H) 1f (W , K) 4. f (1-11 V) +.f (W, K)
35.. dr.004-Fq3 Tariff Irrek Mit Tr6-19-45 gc14 4rg
uRn wtrr? N Alre7 WM' 1:1frF4ir (50(x) = p (x + 1), peN i air (I, x., it', x 3 )
w1- eFiv itai7 qq.• * 57T sirare S ,3767. g2hTf arif
11
31. riT4 fiP an sin rrio LEM al , az , . tkr TfErw _ 31, Let an = sin Tan. For tree sequence a.1,. the suprernum is
32.. Using [he. fa re( that
3,
1 Y
z
12
8
72
6
@ 1 equals
t(211
2.
4.
2 , L
12
7t 2 --I
43
33. Let f C—C be a complex valued funenon of the formi(x•y) = u(x,y) i v(x, y).
Suppose that u(x, y) = 3:ey,
Then
1. feannot be holornolpNe on C for any choioe of V.
2 f is holornorpbie or, C far a saitable choice of V.
3, / is hotonborphie an C for art choices of v. 4. le is not differentiable.
34. Leif T2<2 x re—R be a bilinear map, Le., linear in
each variable separately. Then for (V, \I() E;2 2
x the derivative D f (v, 1,41) evaluated on {ft,
K)EIR :.R 2 is given by
1. f(V, K) + de (1-E, siV) 2 jr(14.1(,) 3. f(V, li) +/(W, K) 4. f (1.4, V) +/(W, K)
35 . Let N be the vector space ofall Teal polynomials of degree ar most 3. Define
S: N '—N by (SON) =p (x + 1), peN.
Then the matrix of S in the basis I , x, x? considered as column vectors,. is Oven br
1. 0 and it is attained. 2. 0 and it is not attained, 3. 1 and it is allain.ed 4_ 1 and it is not attained_
12
0 0 I I 1
2 0 0 0 I. 2 3 1 . 2
0 0 3 0 0 0 1 3
0 0 0 4_ 0 0 0
1 1 2 3 0 0 0
1 I 2 3 I 0 0 0 3. 4.
2 2 2 3 0 1 0
'3 3 3 3_. 0 0 1 U -
36. igrif F, 8.91473w Frr,17 f'W L:0 .ru A= F 'x'=1 T x"-r.L.k 7 Proft w:07ice? ..3W a
A $ wcirtr,g 7e,7 Wffiff :
1 2. 2
3. 2
4. 6
0 0 0 I. I 1
0 2. 0 0 I. 2 3 2
0 0 3 0 0 0 1 3
0 0 0 4 0 0 0 ]
I 2 3- 0 0 0 0-
] i 2 3 4_
Q 0 1 2 2 2 3 1 0 0
3 3 3 3 0 1
36_ F be a 11.&d elements and A = {xe P I x 7 = and :&E for AC riziturdt numbers k Then Ehc
number of u.trrients. in Al is
2.
3. 3
4. 6
34, Effff 4,1-9fe ± 3-x (.3-024. 74# ft g
717
1. lz; . 3 2_ kl
1,:rr- 11.45 4_ 12.-1!.f.sori
37. The: power series. 2 3 -1 (3.-0 2 " converges if
E. 1E1 3 2 .
3. I e--,.15 4. V-1 I 5,.13
3$. -6•0 0 = 20 it: iratod- 7-4 0 2y EMZIr .41 Fri .31W 4 97 e j f J?
Er ;row fry g f 0!7e. wy Wro t .7
zrra-raw 2 F',F. e
3, Ff rf ,V7r .E't STrocriwo 9-67 I
7.f)' 971.7 I
Cure tder the group C = where Q and Z are ihe
gribUpS MrAtiOnai limbers and integers respevively. L n be i pusitive integer.. Then is aim a cyclic snhgrnup of .. .order n?
L nut neccssari iy.
2 a um i<31.1e one. 3_ cs F but not ntxes&-u-i4r a unique Otte,
4. never
39, m71'n f(..r) 2c 1-21.2 -1. I 7u. g(x) x M•
I. Rx) g(x) Awiramft I 2 1(x) vilroTfAr ra-3 . g(x)
(Y-) Yarro-eus177 Letr-E1 A:0 4. u• 305-47ErTM
39, . Lc; 105 =x3 +23?-F1 end x) = 2x2 + x +2.111en
ever
L f() and 0:(2c) are irreducible
2 r(x) is irreducible, but g(x)15 not. • g(x) is irralue0ale, but gx) is not, 4_ neither ffy„) rtur {x) is Irreducible_
1 3
40. : 2) 0 Zoo M1.7 argalr arda 1.3..koepr ttr
'fitall A : 401. The number of nort-trivia1 ring homomorphisms
from inn to Zaol is
1. 2 3 3, 4 4, 7
1, I 2 3 3, 4 4. 7
41. m/Far = y(1). y(0) = 1 we f: R-R ?OFF ii:Icra I rg gff irtifflo qpy
-TEE I F# ,F•i-arr: irR 1
2. R %119 -47 IrR IOW I
3 gr? Tv 1 44. Al We nig ;r6I sl-dr
4. 0mfZre wgF1 Tr IP ,Fitre-r6c.
era'q -eT4 ' , f Wikcrvq-e7
41 , C':' cider the initial value prabkm
f(t) y( t), y(0) = 1
where t. R-R is OnntinIVOUIS. Then. this initial
value pi-oh/ern has
L infinitely many solutions for sum .c. 1.
2„ a unique solution yi EL
3. no solution in 2 for some 1. 4, a solution in an interval containing 0, but mat an
R for some f.
42. ,774 f 7a.rrqw '07 4u 1 (t) 30)=0 t ER Oft rHY *iprgi
V g efl V
I. 2 0 hff 7irm941 0141 Wi
2. Atv. I 0= Trar447 77,17t g 3. ttl'vff Toi ijusf u l If crifcv
4_ ft-a e 4 -41 UF) ARTift ORM I
{1 e 41 Z
-vi , f>0. x€
0 .10.xeR .3?-- 11...P0 erg. PT/. 0.47 .3ftweir k. rrweug:
1(x, xeR, IGR}.
2. {(x, t) xeR, t > q-eT• Trprfr
((x,,t) xg2, E < Te,
3. gx, xeR, teRrii{(0,0)).
4. ffx, t) ; x€R, t > -1),
42. Let V be the set bEall bounded solutions of the ODE u"(1) -440 + 3u(t) = O.( Gia
Then V
1, is a real ,rector space of dimensicat 2. 2. i$ a neat vector space of dimension 1,
3. contains only the trivial familial) u 4. contains exactly two functions.
43. The function
0 .4 r; U(x,1)= „It -
o .10,xe[Pt.
is a Solution athe heat equati WI in
1, {(x, x€2, te2).
{(x, xeR„ l > 0) but not in the set
1. (x,,t) keR, t <01.,
3. I(x,t): xeLR, teT_WI{(010))..
4. *1 0 xeR. t>-1).
>0, E
1?2 2. 2
4. 4 a, 312
14
44_ 5.x afft7r ei4Awvi
*R" X }111= 0
e ;
1. .154) X G yR t 014 g
2. ?reP x yGR Oa g
3. 1741 y < 0 e,s .11W *-r0q
4_ Yi-69 xcEt„ y <0 Ser 174-efir0'7. e I
45. Wife a ?.1.107.m1 er4g tr.Vra' .3.r .a7. 0' w.r . l='.'71
4 0 Ter. W.) W..9' 0 75'47r
1. iir(-NEr ..o#7? # 2 t-cr wiTirew Wfrrifte .4:4-4 Of*
.347r 1.)11.14L',1 .3177;17 e RFE,rim 3ravE7 atthfet t.IR rr -.stly PA.Lov.. Sr
4. 91-AR. sfov ..iegii7V1 aviKreiT ATI% RM. 00. Er,WP
46..
e. 1 111...)(;111)=
3 Y(3X Y)thr; y(3) = 4 -2 , .1)(1)=1
WYTERVWF7T rh,A Hol
E. 77 WVer Tiir 2ft i? 2 zfpr-,-0..p.F.T.we 3_ NFU eics.vr ut # 4_ a.74.1 grY Rh'
t: 47. q1320,5 WITFR i9n)Wi'DV 0(4= + 0(04.
ier4 77P-P17 .31*' IVA I) e •
44_ The second order PDE - + :eu -
is
L elliptic for all ;14 y E
2_ parabolic for all xGR, yc
3_ elliptic for al x.E y < 0-
4. hyperbolic for all x y < D.
45. Consider a second order 'ordinary differential Equatim (ODE') and its finite difference reprdscniation. Idatlify which of the following statenterris is correct.
'The finite difference represcittnion is unique. 2 The finite difference representation is unique
for sonw ODE. Thew is no unique finite difference scheue for the ODE
4. The uniqueness of a finite di flerencil. scheme can not be deicrmined.
46. The variational problem of extremizing the functional
RA 3
X))= v(3x - y)411 . y(3) = 4- y.0) = 1 2 1
has
1. a unique solution. 2 exactLy two solutions. J. an in fin i number solittians. 4. no solution.
47. For the linear integral equation eu#
000 = ;C - 93GL-)th,r,
resolvent kernel R(x,. is
3_ 3(2 4, .1 1. 112 2, 2
48. " Erg 4154 C H = - We.
e. Wzr
_ q c*, p
I q p
3, q c*. P 4. g O , p 4=C•
413. If the Hamiltonian or a dynamical system is given
by I-1 = pq - then as
1. q p
q - 0, p 3. q p - 0 4. q) 1 p oa
49_ feu*o ibri4 F l (t) F2(t) d mr47/ gqRr W-77. f1(1) rr 13(0- et J..ro.ro zr7 T 1 7 Tz Okft-pf ;DIM WtRilZ )11(0= 312 ha° t >
0 ei
L mft t >0 F 1(t) 2 F2(t). 2 arn#P t >1 43' ft PO< PM). 3, E(ri) < B(T ). 4. mit > 0 *Przt fl(t)<
59. .5773 f Xi, X2, 11:0,1) NTRYT 70-1767
MtF ' cal 4/1"144.ce r I q2* f ri
tW -1). SH =X,2 +.)0+---+X. f
SfiNuf (S ) 7c
.S.,1 LIS 21
1. 4 2. 6
3. 1. 4_ 0
51,4)14' (X, ; /I a 0 )v -41 eft aore,rf ?Fit' RFET vikenur gazisfirr wow Wfif trw TO*
51trqr ill 1 7114 tar Aiare;FT aog-4,-.94, 4E1 r YiTOW N*HT
E gegei KciEr 4'e'q TiOr 2 vw arTkl RI7-7 7lezt ureq 3. 47W'Y' ?Frer R.ay go =re t I 4, d;t- Y ?WO af-eq• TT* Reef
52. qFV .1W X Y iqfliF rtrfagrF w we 0* Y Prift # r FrA: ft' J= Y g
V= X- Y th
49. The hard rates of two life time variables T 1 and Ti With respective c.d.f..s F,(t) tind F 2(t) and p.d..f.s f'1(t) and fAi), are WO 7e 3e and 112(0 = 4t1., t > 0 respectively. Then
1. Fl(t) F2(t) for all t > 0_ 2 Fl (t) < F2(t) for all t > 1. 3- Ea') < WO- 4. fi (t) fz(t) for all t
50_ Let X11, Xz, • - - be N(I,1) random variable., Let
S.= Xi + X: + • for Then
Var(S is
4 F fr
4 Z. 6
3. 1 4_ 0
51. Lei fX„ 21 0 } be a Markov chain on a finite state space S with stationary transition probability matrix. Suppasc that the chain is not irreduzihte, Then the Markov chain
1. admits infinitely many stationary distributions. 2 admits a unique stationary distribution. 3_ may not admit any stationary distribution, 4. cannot #Idinit exactly two stationary
distributions.
52. Suppose Xand. V arc independent raudom variables where 1/is sylrenetric about 0. Let U= 1? and
X- Y, Then
I. U wkrtt Zrrwrergei 2 UT 42'• f at-q wqrq. 3. U 007 0* 0-4 ,ffirri5 +R 4 re g I 4. V ff eie ffrciq Wilqa g
1. U and Vast always independent. 2 U and Y have the same distribution. 3. U is always symmetric about 0_ 4. V is always symmetric about 0.
53_ 1r ;1' pL44.0.0 cize qatrand` wit ocr. ,5.715e41 P4Ir 2 FrirMkr 4 Alf.sum zreiNRT 7* wPrq 45-1 riffe
Mir 13 t,q 4
5- t-o• ' , 200 00 000
MEI
100 100 400 cf 300 • 700 1000
53. Consider the foltciwing 2 x 2 table of frequencies of voter preferesim to two parties classified by gender, in an election. Identify the mereet statement:
Gender Party B Party Crfotai Male 200 400 600 'Pernale 100 300 400
` Total 300 700 1000 J
E 6
L . P -r z .&;7a 7a31e r ?it if Tel xel.tkr ,11;;. q.7 :
LSO 420 120 .280
r
•
ir7i Jar/fttill reitem wW.-u4 1Pd4.;..e 0 e
3. (-r'rii 4p. T .k
4, 5-617 ..7"tY 3747". ff-3 17*{ C rt?IeT W?#
54.E /• X... X:. • 11 2), NIA c2) a.21.7
kerchyrev F;re'3 '.11Deff TWE, er. <
▪ r 0' ci < crirViNF IFF
.4 '"1.. 6' 2 2411-4W77 jr'lft7 7 747 te , FIR Zf9:16.1̀ 41ff ..5.17wrq . 0.0 ▪ craTrte... :
657 97;717 16 1.2.,..nve."17" 6-479.
Lunw 7 Se"? .# ,5 Sit•t'•'?
t
3 - g'"? 4 L.L.r. 07 3.2 ar# '71' 71 467 *
4.. 2
Ar(l E Cr„„vr * 4.4 ;via/ mir• g I
55, .74f CrW Veq
Vrei FFitNifAr• 4iiefr ,'fir.,-Ard ;1T-4: A Fm? gm wed .cror.urrRi ee .a. zird
?veil, MDT xl.gc-41 .7 Y1 , Yz.. • - • e 911
W4241779' Kitf7tW gl r JO' X Er wrgry # XR:o iso &AV Il)
O1. g wi R5 = giefff u5r.-47.i. i YNY 051 a:11;77. Fr)
P(Rx-Rr> 0)› -] 2 1
2 P(.1? Rx > 0I> - - 2
3. E{Rx) = E(Ry). 4. P (Ry = Rx)= I -
56. tre; gill Vilcirf.RM PrIg Y= X+c 71-0 r ,cv cfc, , X - ire tir?' ri
= --• .11.1 "77 = -1 •? ..5.11E.reA7 Y C.3
rPf riffirem r JET 4 3-k•
1. if there is no association between party and gender, the rrxrtected frequencies are
Igo 420 120 2.80.
The chi-square statistic for testing no
associatiOn LS 0. 3. Guider and 'party are not a.sociatccl. 4. 1:30th males and femakts equally prefer party C.
54. Let X., X-, be n 2) i.i,r1. ohservoticals from
N{,u, ef 2) distribution, where < Az< G'C' and
0 < ‹. %' are unknown parameters. Let
and danore the maximum iikeith0.0d
and uniformly minimum variance 'unbiased estima: of 17 2 respectively. klentiry the correct
stalernent:
I . .170:416. has the Same variance as that ot
2 - mu E h as larger variumn thirst that of 6"/.2...m.. L.I.
3. er,421.4c has smaller mean squared error than that
of. 6 ?..4.4.1.A •
4. arzs arid ert2. ,,,,.vx have the .same mean squared error ,
55. Suppose chat vie have Lid. observations X I , .•12, • • •,
i th a normal distribution_ Suppose nether that we have an independent Hi of observations
• which are also lid, with the same normal
distribution. Let RA. = the sum of the ranks of the X's when they are ranked in the combined set of X and Y values, and Rr = the sum ne the ranks or the n. in the combined set. Tiler'
L. 1
2
2 PVT- >
3- BRA = E(R/4. 4. P (Rv = Rx)= 1 -
56. Consider a simple linear regression model Y = .X+ Let bc the least squares predictor of
Vat X based On n observations ( = 1,
Jr
and y 1
the standard error of the predictor
4
17
I . -47 .17 4xo vEreart45-4 eargy
2 m' umr. t .sviRt. 2;1F erk 3. CR 0 xa arlIFT t-.1 .3{.aw FM! ti 4. ar4 0 4. r477,- amr 11' W3ir *Jr
57. Tir; /nth I, 2, , N fiesiamd N fa-47 stefte t N met Atm Hirvi+# # urrre •?.1 cfw wThrrew zeriT4ro n fame acq
F445121 1-.57 siT4 .1*
cry azrvet *Ai /FR pt n f4zGral 4' 974 vifi er f479 ,
eq' N ac orm e ?
• 2X-1 Fc.=-1 (X, X.)
58- 2IF tart 747 C"). 3rePT u mtir A 41. B WO'? 91r altcr4 4,WIT7 t 40"51W 17-4V7 *VT zl-ftw F4r e y-4 ir meau rniqa 07-4 .07 A 57 240Fi a• zirfAiw y* tr4F yQ'T • Rfta. # lTO' a- WI, zrzt
gtdff TT?' *iv Afvf. • Trnbrdtes flik1"7 Zfr7 %.14i4)q 4577f 4.44 ?
1. am xrfrPTIT 67-4t .07 4d g . ;F-g.tz.rET
2. am ofonieeri qr- 4t ?i95-- J -q•&'w
gfikFf dkirhAlq-N314 , cavort 4_ anirai, ....07 457 crIkly
I. deuvasts as xo MOVCS away from X . 2 increases as..4 moves away fromX . 3. increases as xe. moves Moser to O. 4 deCrell_SCS as Xo. moves closer to 0_
57. A box contains N tickets which are numbered 1.
2. N. The value of N is however, unknown. A simple random sample of n tickets is drawn without replacement from the box. Le( X I ,
be numbers on the tick ets obtained in the
2r1 nth draws respectively. Which of the following is an nobiate,d estimator of N?
1. 2-1 where R.-I
2. 2X+1
2.2 +-1-- 2
4. 21- 1 2
53., In a clinical trial n randomly chosen persons were enrolled to examine whether two different skin creams, A and E. have wa
s effects on the
human body. Cream A was applied to one of the randomly chosen arms &f each person, cream B to
the other arin. Which statistical test is to be used to examine the difference? Assume that the response measured is a continuous variable.
1. 'Nola:11le t-test ifnonnality can be assumed. Z. Paired [-test if normality can be assumed, 3. Two-sample Ko]rno8orov•Smitnov test. 4. Test for randomnes.s,
59. orit i XL '20 x, f.dot ...V x 1 +7,2 3 +2x2 4-wr b-3vrEiTrq. w-iFt Frt 147r
alq wr ?
1. 5x, + 7.xl Tztrff+T VW 21 t afresr
*P! rii*i'Ff -rfr1*1 g 2. 534. 1 -V1x2 rar. Rpm?' 7gRir 17 g• eic11. 451
.1 a Lcid'i qLet g I 3. 5xl-Vh'.2 WF /RI 21 * %Arm.,
P1 17 t 4. 5x. 1 --77r..z r 9' 45)-zi thrArt -eglcvt t 7 4
Frinpr
54, Siippase that the variablezx. 0 arid 0 satisfy the constraints x i +x.t 3 and x.-E-2x2 4. Vellich of the following is true?
1. The maximum value of 5x, + 7x, is 21 and it does not have any finite minimum.
2. The minimum value of 5x 1 -1-7x2 is 17 and it does not have any finite maximum.
3. The maximum value of 5x,+.7x2 is 21 and its minimum value is 17.
4, 5x 1 +7x2 neither has a finite maximum nor a finite minimum.
S1 7 RIDil 2-4 AFF-ZA.
18
60. Let X{i) be the number Drcusicimus is Pr'. MI MI] qu.6.i lig sy5(em with arrival raic > D and service rate Ft 0. The process X{C) is a
L. PoL55on prucms with rale 2_ pure birth process with birth rake 3_ birth and death process with birth rate A and
death talc p. 1
4_ birth and death pruces.s. with birthrate —and
death rare — Ft
60. .31;9169. YeD . > 0 1,41 i F.6.Grte mKgrk? X(i) A64 ;it
3ZW :Et I .Erl%74.7 X(1.) cr4F
rikt WRY ria72..rr
7.9.7 In% gu. aq7 U # I 3. Trey_ TrFcr Ter pt --
e I
4. 7:77 37P: —3
7 37P7 7ift 1 WW-7377 11
99r7:If e
3107 R1?.P17-4 AI-1-2B
19
Tart
cY*717 IfUnit I
61. kInIFI fix) 15 CO* — 5D + sin — 31) + I- - + 4) 2. cr? lam .56707, i fF a#
1. x= 5
x"3 3, x=-10 4. x = 0
61, Consider the function
Ax)= cos@ - 51) + (ix- 3 + ix 4- 2011 - + 4V.
At which of the fed 10WillIg points is fad differentiable?
C. x - 5 2. x 3. x -10 4. x 0
62. *14z;:i ucr-tuwn.Y .gty.641 ?
{(x.Y):Ix11•1./.1a 2 )
2. (x, y) 1.1,v1 2 2
3. (ix, y) r 2 + s 5
4 . = x2 y2 ÷ 5)
h oC the followin g subsets or R2 ace compact?
{(x., :14 1,1y1L.- 2
2. 1(x, y) tx1 5 1, [03 2
{(X, Ji•3 +
(r. y):x1 1"1 +5
drXg)= sup11 f (x)— g(x) I :x e PAM
2. ce(4 g..1= infflf (x) g(x) ; x e[0,1]) .
3, flf(x) g(x)I&
4. off, g) = sup{ 11(x) -141_01 :x ji f(x) g(x)Icix .
0
6.3 Valid.' of the following are metrics On C = {f: [0, 1] R is a continuous funztion}
aVg)= suP Px)—g(x)I :xe[0.1] .
2. g) = 4)1{1 f(x)-8(41 :x LCD .
3. (ICC = (x) - g(x)lin
supl (x) - g(x)1 x e [0,1)) -I- 0
64. j = 2, 3„ „ 974 a- Ai E)47 .1.1z2V.d. f,' 4, Wfi 3IT4V 2r
ce 1.1 n
1 - I U Aj .r.45. q IYf '9 a 7776177 ;' 1 2. Uil A) afFtw-4tv 6' j-1 ri=1 1=1
0 .:0 3. nifi ,3117711-11 g 1 4. U A i 317G7tal g .1r
i-I /-='..
6d- fi.or each) = 1, 2. 3...., let 4 be a finite Set containing at least two distinct elements. Then 34 IA n
E. U A i • iii a cutirtiabl. set. 2. un A) is thicotmtable. 1 , i N=1/=1
cr. 3. 11 Ai is uncouniablc. 4. 0 Ai is lancountible .
j=1 n.1
(•5. 4'. W9. RV 81 ?TA i :7
\.13
1 - 1 ""F I "—) e C'rc5' --'1N 2, I 14- 1— 7-4 LI n +1
I 4- — I —> e r‘tiu n . 4. I. + —r WVri —)`ad' - n
65. Winch of the fallowing istarc corm!?
1. (1 -I-- —)..eas IJ ••.) oa . ( 2. l V
1+-1 —/Fe. as it—§-0,-
•II
1 r
3. (1+ as . 4. 1-i- — —)E ?1—>G0 -
21
66, R9' et in). Fre 0`./e
1_ log x-F y logx+Togy fillit.x,,y>0 7#&7,' 1
2 2
2.
e ge' + e in 0 x, y > 0 fa'.1-4. i
2
3. sin '' 4 y ssiax 4- sin -!. Fet x, y > 0 - At .1
2 2 tic+ ilk
4. 1` 2
° 51-riagx' of°. 77tft x, p• 0 N.' k 2 i 41::' i i
615. Which of the following isiare true?
1. 1.... x +y log r+ log y for all 2.,y > 11 —4' 2
ei -Fe" 2. e 2 S for all x,y > II
2
3. sin .x+ y S sinx-rsiny for all x., y > 0,
2 ? x + Y)
I 4. max W,.,i, } for all x.,y > 0 and all k 2 ] _
2
67. f , I R 74.7 ko oar-- Wfa 141
tfr b 31177 jf(x)dx=o X f= 0. ;
2, atro* aScSb Fri% Mv w•ffir If(x)dr= 0 4f=0 a
3. FPft ase<dsb* arm if (x)dx = 0 .F-it 71,
4, ;;;?" = 0 74 a c5 .1W -sPF if (x)dx =0 ey) vzr air P774=1
67.. Lett ; [a. ti) L be a measurable function. Then
1. If if(x)cir =I) for all a s c< d s b thenf= 0 a. e,
2. If 11(x)dir=0 kir all aczb,. then f= 0 a.a. a
0
22
il 3- If ff(x)thc=0 for al] s <d b, dots not necessarily imply tlsat f
4. If if.(2)etic .=.0 for al1 . 5 Z. dims riot necessarily imply that f = 0 tr-s.
68. r 4:x = y_ 1 13# IW
F
dp(x, y) = — y) I :j= 1, e 11174f J =
1. .TER" %. Ef., Gt., .0) 1 gp pW7R. FVO ?;z9 ?
1_ r 2. aci:;. .8.2
3, ci2-00 i .?p• 7-gr
4, cireco 182 AV
± )11
RFT X = and y = (y,r, Z.:if Lot d .„.(x y) = I for <
and d..(x, y) ma..1 fk.—).51:j. , 3,
Let = (xER" : Or, 0) < 05.
Which of the following are correct? .
R I is open in the tt-mutriC. 2. B is ope55 in the ek-metric.
3. 3 1 is not open in the .12-metric. 4, .32 is not open. in the 4-metric.
69. jilx„ = x +x4 , 4y 1-y4 ) &-7?.r 0.1Wf 7?'
_ oil)) cry f Ffeffr
2_ 0, a) 77./ yitu ex (0, 0) ?5* ,m*E-4r * 3_ (131, Et) ERf ,7/4qrri'VAr g ArATArci IVA F) Wavivk Ter g 4. (0, WI 471-46—dtg q-e7 DA% 0) ,25I rYPoilel lAr
69.C n ider the map f:2.R.2 .1.3.2 defined by
_1(.4.. = (7x -F 4y + Then
I. f is discontinuous at (0, 0). 2. fis otatinuots at (0, 0) and all directional derivatives exist at (0, 0). 3, f is differentiable at (0, 0) but the dcrivallin DM. 0) is 1521 4, f is difircroltiabk al (0, 0) and. rivatiree DA0. 0) is invertible.
23
70- Va. iirvorfre vrv)' u# 777* C10., fli7 mi.* al" Mrzir yrif
;= suP HUI 10, 1D.
2. Ilf ;= jlf(x)Idx , 0
3. Ilf lif16+ 14 1 )1 4- ROA-
111f16 = iitilf(x)11 d0
IC. The spaoe. CIO, 1] orcorairmoits flanaionsSri [1:P., 1 ] is complete with respect to the norm
II III := stip rx)I : xe I 0, 1]).
2_ := - o
3. := III 1/(1)1 VOA.
1 2
4. lIf16 1,111/(4 etc
0 71. av4 ALLI,.) (r) = : — 42)' + — < 17.3u9. Ul; r Tarral 4. aft wrf0 ?3.4
Do0-01( 1 ) U 0)) U .0%24 I.)
2, DA.0.41) U El.(2,0)(1)
3_ Ato.){1) u O)} u Do.2){ I.)
4 . D4.0..09(1.) U tf,1)(E
71. Let = .1(x, — + (y — 12) 2 <pi_ "'Which of the following s.ubsets of R
conneeled?
L Dem° ) f(1 0)} u Da,ny(1)
2. 45,0)0 ) u A2,0 )(1)
3. Doxi y(I.) IJ 1:0) .13 Dem(1)
4_ ..D.L.Q0)(1) U .4E% 2)0)
72. T4.197 X ={x (0,11.1 li.rr,.12 041 a-WIN, UrriTqlt TIArae
k X 11{;bg 41'91 2_ ArFF#Ti-47,. 9. thwqr
3, r 41. WIPT 4_ X ?io- 91F.:pr Te I
p 21 [0
4
4]
[1
2
2]
1
[0 4 0
1
2 1 2.
4.
13. V.Prtic]i of the following mokiec-s are positive- dein ite?
L. I2 1
[L 2 2.
[ 4 — 3. 4-
—1 4 . 1
24
72, Let _kr=tx E [4:1..,1.]: X .LC in, ?? Pf be glA•cri thu: subspate topology.
E- Xis corm:med. but not compact. 2- Xit neither conpael or ocaneeted,
Xis compact and conneeled. 4_ Xis c-otripac.1. but tint connected.
73. .P.Neg aine # W,•1. ?
74, TF.-? YVIRd V A c?7 .e'Dker-rit've r > gr, uvicymt- y
v: V W' ? 414 k (Vu ) FrA.' !Fa
= Eg
], t = I .
2, 7:INNV A = . A w' .c)TT 7ra. 0537r.re.6174.1 ITO
4. ..1,p arTur v.orpo 11E= MI ;.V.f.' Pri Ax =
741. f.cE A be a ii ri-m-re. iTglEisformitLiQn cm a real 11C1.01" coact Y of dtmensum n. Let the subspace VD c V be IIE imp a of V under ,L LGt k = < 2116 suppose ghat for some
ACE, /2 2 Then
I . J = .
3., is the only .eigenv.aluc ofA 4. Them is ;:5 nontrivial subspace Y Fc V such thEa.ty= 0 for all x
25
75. i C T-45M x n wen. AirlAr ung f W C, ..... 3Ter 94.11]. •P4F walr R7Vt. wft mitePYOfter:
I. 2n 2. sift ;31$* 77
3. n2
4. arPaer .5M; 231
75, Let C be a it X PI real matrix. Let W be the vector space spanned by {LC, 0,-, el- The dimension of the vector space W is
2n 2. at most A!
3. r2 4. at most 2n
76. UPI' a' Pi VelM WRIT Tiai'fitr V 2idl WPM' AT8' W.7 ,31FETKIWE' .re weepy.f
11 V2. 2- V U V2 ,
3, VL = {X 1-y : xEY3 , ye. V2) 21. Yi p = Ore Y3 and ye V21.
76. Let P, , P.2 be subspaces of a vector space V. Which of the following is necessarily a subspace of I'?
1. Vi n r12. 2. V, U Y.
3. Prk -F = (x y : V,), G1v2}. 4. VI N.V2 Ixc V, and ye N..
77. ciw 3 x 3 r.,(4ohe mraig ?E*tio,r t = Q. ftR ??7,/# wift */# 7
I. N 14-4-4armer WIEN1 1? Wei t
2, N Poor -stiv # 3, N To' FpkavdirPreatore taw * 4. \r ?IN. NW. wffi7 arol4wrenta iyAw
77. Let N be a nonzero 3 x 3 matrix with the proptrtyM - 0. Which of the following Ware true?
1. N is not similar to a dingotiai matrix, 2. Nis similar to a clisprial matrix. 3. N has one non-zero eigenvecior. 4. N hay throe linearly independent eigenvcetors.
78, 310' x, ye e° if(x,y)sup{lex+eyl l : Ei Tr Qqtrq RO' OV-WT/#
7.71,1
1, f(x,Y) 1.'1 1 + or 4 21(X1 2. f Or, Y) = 11-1 4'11Yr + 2 Re (x, -
3. f (x, Y) +112 4-11Y12 4 2 1(X,y)I. 4_ fix, Y) > 11x112 ± fr11 2 +21(x4
26
yeC". COn5ickr f(XJ')= S Up {V119;1: = 61, 0, which, or th.c fcilIowui
islarC 004-Mel?
I r Y} 114 4 fL)11' 2 14
f (x,y) -11x1r + 2 R(x.Y) -
3_ f Y) = 1143 + HI' + 21(x,4
4' f (x 'Y ) > 114 +1,11 2 + 21(x ,
celmez 111.Arni t
79. A7q. d uF*-R 63W71 (ED, L] r NrE 19:riqe110 0 azrzT 4 [0, upar4,17,6 1R137# kkid0 ipc14 . 4 Zr).eiffi
I_ ffcCiO, ]1 !fir krOrF,
{f clo, 1] :go) =0:
3. ife(10, : j(0) 0}
4. VG C[0. I] ; fi(x)dx= 5
79, Which of the following seu are dense in C[O, 1] spar. or real valmel corrlitwoiss furiclions on [O., I] wilt' Tupelo! bo sup-norm Lop:Aug:Fr
1._ ffeCt0. 13 :his a polynamiall 2. op go. 1 :fio) =o1
3. tA OQ. 1 1 :AO) 7t
4. C[0, 1 ] fir ( X )41' )
( gi). 71" : M711: c— C..n . 1.0.&"4 f = ." W7 711PIFgq" c.11.7.Er gar feziv MirAW g5-67 g wir 0 n 2.?,1 ÷ 3
EN' hrdlcr?
3 . AO) 1f2
2_ z -2 of r wrei57 sirR'dv *
3. j12) = ]14
4_ li?r PlUf IFF7 0—dF:r qe.
80. Li./-: C—) C be a ineromorphic function analytic al. 0 saiisfying f ( 1 ='PI for Al I, n 21H- l
Mat
]. AO) =
2- j' Ins a simple petc at z = —2
3, iik'2). - 1/4
4_ 110 such marornurphie funetEori exists
27
81. 774 AFf ?ireg clecat. gricrq * r..317.1 -7f d,r wpr
L f arvo,Q0 RFT tter( * 1 2. f t
3. f = 0. 4, f Tr-6e Threw
81. Let f be an entire fraction. If Int f 0, then
1. Ref is constant
2. f is constant
3. f
4. f' is a nnnzero constant
82. 7171 Rff: D f(0) = f(Li2) = wEr t ue 13 = 1.71 .<1) I49'
Ifto.•eir-4.weg?
3. If' (112)1 413 2. If (0)1
3. if' (1i2)] 541'3 and If (0)11 4. f (z) = z, 111
82. Leff: 03:. be hi:4=w* with f(0).= 0 and f (I.2) = 0, where DI, =12= <1). Which of
the following =lune= are corrcra
1 . If' (112)1 48
if' (I/2gs 413 and (0)I s I
83. zmx+iymir 2.12Cz#A2.) err = oFf
= tzeC y >
w = 4z€C:y <0),
L4 = (zieC :x>0),
Lma {?eC x 01_
f (z). 2z+1
5z + 3
2. If' (0)1 1
4, f(r) = z, zGEI
1. ri+ .4.5P ar wirT EraTvir kw-44d amir
kr w urN. c-r* I 36 47 A1 c7 Vrecif g
3. it L+ wcr-yr 121.1 Wv?. sedrequ c,irri
4. WO' L. wgrz 1474 ROWleff . ZIT WIT
28
For zGC .0r the enren z =x P iy,..defme
=
= (zed; y< 0),
L .(F,QC:_v<
2z +1 The funetion J
5z + 3
L. maps rrr . onto or aluE 43t1t011 -,
2. rnaps E onto H - and H" tivo
3- maps W orao 17 and a 0%*
4_ maps re CniE.O. IL - and lir onto [..7.
2 = 0 47 IT-07 f(:)= 6xn(— t z
1. .3M,Er Pr .RM g f
2. vw 3+ 10 _ aftrpi 11;112.1-0
4_ z. — 0 -YR?. Er777M Wfre 415-k9ME1 4,7-dva Ei-9-07-47 Erinr Z e
84- At z = CI, the function .1(z)= exp( )1 s I - cos
. rcalorabie 5inp.lanLy_
2_ pale, 3. an cs5cntia1 4, VW- Laurent expansion OFftz) arourd z = 0 has infinitely many positive and negaiivq•
powers az.
85. AFT-; Jar R. Q ;741,1 L I + Krvir 7-Pef g rirff a a xT x TreTrfarze y 21' atvgamft g
1. y2 2_ y2 y
3_ —y, .E.I 4 - Yi 1
S5_ Let R Q I, ....vii(70 I is she gencroced t y L -r x '_ Let y to the oset of x in R. Then
]. 1 is irreducible over R. 2. y + y 1 is irreducible over R.
,3. — y is Irreducible over R.. 4, ).3 y -F 1 i5 over R.
29
86. Alq # oti.-TIT Ref ?
I . in7a , Q qz ,4274:14 2_
3. 4. Sin-I 1, Q .0.43941 I
86. Which of IlL following is true?
1. SiEd is algebraic. arm- Q. 2.
3. Sin- ' I is algebraic over Q. 4,
COS 707 ,Q airF gir *
,11+..,Q100r7 .4`
Cos rril 7 is algebraic over Q.
Nri + J is algebraic ovor Q(z)_
87. ye fix) +2 + + g(x) = 4 I 1 al- [x]#
I. MiTir ATir VIWIF y(x), g(x)) = x 1 • - 2. MP?' Wag (f(x), g(x)) = - 1 . 3. &jt tomsreird (fix), ex)) = + + x2 + 1,
4. mimw trwrgrzi gfrzy) = + x3 1 a + 1.
Leti(x) +3? x -F I and g(x) = x= + Thtn in Q[4.
. g.c.d. (fix), x I. 2. &c.d. (0), g(x))= Xt - 1. 3. 1..c.m. g(.4) +Jr' X2 + 1. 4. Le.rn. x), 4'(x)) = + #1+ 1.
88. )./-t 36 .0 V/W 0 kPrio UtriFir H G 451k 4 Mzit
c Z(G}. 2, 11 ;T. Z(G),
3. G 11 irarrim 1 4. H qZri 4/14* q97j'a' te
88. Far any group G of order 36 and any subgroup H of G order 4,
I. li Z(0. 2. It = AG),
3. 11 is normal
4. If is an iiheWn group.
$9. q14 Q UV' x S1447 RIZire Ofrtf
C. Q w2.ftt erTreF wirimsrgi
2. G r 3- of ti crffF :401 t I
G YFw irfiTTIW wirer r * 1 4. C. T,7 .4747 Frelf- q J J 72 w.
30
t..ct iJ cl.ctiote the group S. .5). Then
1. a 2-..ylow sub up f G is necrial. 2_ a 3-Sylow subgroup of G i5 Torrrtal-
3 . G has a roncriwiall noTrual sub up.. 4. (.* has a nonnal giabp-oup OR* fl.
90 .19.; X 4.:-w t•rorekri qr/ A. 4 , A.1, X f crf 17:47.116 e irrf .7.1-4747, gr I al X 4 '?W .orefraprX717F73 c.crq Y6111 t jr,X) = x trr , L
= , 2,3-
L rir12, 7 0z71 O'w7 611 i
2. it al? a 1 , cll. tft e YieGinzie nooR .e) 1 cr? 3. 4777f, .1*4
frofP PV .e1T;.1.41i ,, if k. vu7 F#
90 Le[ X be a Timm! ilausdorli gam. Let A i , A. bd subsets. of X which an: paimke disjoimir Then ilrmic always exisis a ounLinueluS real waked Cunctionfon X 5 114 that
Ni al if 3
L . iff each a; is either 0 or 1. 2. iff as 1ca9.1 Iwo of the nLi nibcr5 4.13 an Nand..
far all um] values ofrr,, CI er3.
4. only if Orld among. the se[s A. ,. A 2 and .43 empty.
qr-O-W YlIl1Jni# 1(11
91, mc.r.r.,E7 WirPirtur x Y = A Y. Y(.1:1) r Zire: A . 2 cru. . 0
Y33
[
( ?# 3 1 2 (X)
1. .vr(x) —› 01....2(x) x CO.
2. y (x) 0 Qv: Mx) x
3. yict) —> cc N ). 2 (x) —to w el. ..1...3{x),.y2(x) — —›
91. Consider the SySleIll of ODE [ 2 1
—Y,= , Y(0) —1
where A=( 1 2 )
and y=_ Then o —1 Y2(.0
L yi(x) —> and Mx} 0 Is x —> =Q,
.3:1( 491 —1, 0 §Incl —1, 9 es .v —}
31
3_ •air.0 0, and h(x) –) a$ 4. .1 1 1(x) , Y2(x) -44 as
92. CritiNT 7177 717(FIT y" + = 0; y(0) = 0, y(1) = 0, Pr?' i5uT viN airRthilw I1R
,74-fierz A )re-6.zrii. (0, 1)• ,.$0eirsftx Wff" lin'ff
Reg Tgl 2. eK(Int;
3_ qr.t.ivico ; 4_ Wrgr
92. For the boundary varue problem
+ w°1:1 ; y(0) 11). Y( 1 ) 7-7 0.
there exiSIS an eigenvaille A for which there corresponds an civntniiction 'm (0, E) that
I _ does not change sign 2. charges sign.
3. is positive. 4_ is negative.
if 91 4,fd,r fir x1,6,4 .x —+ v =co.= X; 0 <X < yi(0)..-. 0, yH -z t.t aiu e
d 2 2
. ai ipr 2. 13~ w I wairK0 4. LF/PrW
d 2 y 91 The solution Of the boundary value p.robkm
the —+ y cosecs; 0<x <—
- 2
3'(0)= 0. ELI2r
= is
1.. convex 2. 4..nmeave 3. ilegatiwc 4_ posilive
.4. Wit um?' ;cm,. +yuy = 0
u(x,y) = + y2 =I* OW'
1. ?Pt xe yER .FAt cro .e
2. ((,y' : y) (0, 0)) gir eTW e 3. ((x. ER2 Ot, (01, OP ;T- 75074- e
4. {(xi y) a t (x• y) (0. 17T VW 30=-4 tt. ire/ re %WV/kr g
.32
94. The Cauchy problem
= 0
u(x,y) = On X2 y-
ha.s
1. a solution for all XR, yell.
2. an unique SCilkikt1 in 10, y) e'er' y) {Di
3. a bouixled solution in i(x, y) (x, y) 4t (0, 0))
4. an unique solution in “x. y) €R y) x U, 0)1, but the solution Ls unhomvini.
95.. JO" mair rl cfl
ze — = 0 ,
0<.(‹i and >01
U(0,1) .=/103V 0, r>0 r
ircz,0) = Sinx+sin 2x, Oxli•
crw
Mit x (0, ff)ei PL JAZ t) -* 0 WV —)- 2. snit xc (0, z) ,i' Az) [2u(x, —> 0 Ww 3. ,kG (0, 2*, t >0 1 r co*, g) 147w Ova- wffvel 4. yr.19xE(0, 7r)* e 2Iu(x, 6--)Ow
95. Let u be a solution of the heat equation
41.1 =°, 0<x<n• are t>0
14(01,1)-ii(ff,i)=0, 0.0
tr(x, 0) = sin x sin 2x, 0
Then
ef(x, 0 —1 , 0 as —> x. For all x 00, _a). 2. ?u(x, 0 as i—)-to for all _re (0, xi). 3_ ergx, is a bounded function For se (0, ir), i> 4_ e'uf..r, al as r x For all meg), ir).
96- # 4Rt-10.w ai'77 UsiF; 17
FA.+u,=f(t
), e (0,1
)
/41).g.-b
33
asr eg f x + y2 S I .0 iitr4 refiliritFF v(x,, y) U(47473010 r y)
f (4,77-01 .31-4E5u Few .? Dr
vm +v.y.y = g ((x,y) ;x2 + y 2 .<1}#
v(x...y) = y) :x 2 -t y 2 =11 rt?
45T ?Pr Strff
I. a0 4r b>0
2. a>04 r b=0
3. a =0 41" b =
4, rd<Oir b=0
96. Le 24' be Sol'utron of the boundary value ppublem
24 .1 4--1u"=f (1), e (0,1)
re (0)=a, 14(1).b
Define for xl +).(2 1. vex, y) = u( ..?..-72)and g(x, f( x T y' ) , 1hor, u is a soitnion
of the PDE
v.,,-Fv.,,,,.. g in {(x,y) :x 2 + ,y1 <1} if
v(x„ y) = 0 on {(x,y) :orz y'2 =I}
a >0 arid b
3. a 0 end b = 0
2. r > 0 and li. =0
4, a <0 and .5 =0
97, M AV PP ftw .5777F (UM 7w -tuft e ,Jtla 7?* ;yr r f 7 `t&0 Jet t77-,ez,4 I
2xi . 1. 3x2 = 5 4x1 t4x 3x3 = 3
{I)
—2x1 +3x2 t
rr3' mq,
i. T45 UTM r ri.LefrOrr F:7' .0' me* g re-qu ww9-47 1-61, wrft ,y4 Afkre 107 # RR( e GVP'l Zrnda UTIVE V9 ?Prat al -7-4 13.4,L ti
S/07 R01112-4AH-3A
34
3, Lam. 3t: 17r. F A M7 WTI). g 4dia pRw . vpi R-3 e UTM rfkg' 11,11 -M Z;77 (L) r 'ft
97_ that an upper trian8ular matrix (UTM) jS itiverliblt if and only if all its Eliapttal demtrns are di f.frtni. from ro. orftsi.der the system
2.r i -Fly., -1 3 = 5
•4.x. -L 4x, = 3
(11 i. -•xs
I:- h.:11 sy:;corn 1)
1, bc` trail$1:ror=j into. an 1.1TM but is not invertible becouse she diagonal
con- ios of ihu UTM are nUl .1.1.ifl-Qren1 fa 0177 %Qui,
2, is JJsrre rxtblc ihough .canno!. b iransforrned into an UTM 1.
3. can be irunsrlarnicl into all UTM because above diagonal entries arc all .1 ,L__ereni from
Zer
4. call be iransfbrrne...1 inco. an UTM acrd die solildi -in of the UTM •k.1k1.41. uiion LIFO ).
98. t.r..717 = X 2 - .t• - 2=0.... 61" I x = ex) cii4 g(x) .;P
Puri P;.-..-s• (1) of re? g(x) rE Vle?-77217
- -- 2 g.Y, 1- x 6607= 4Th-FEIT 7-47 N:207:211; J
1.11
) - 2. g +— :P r
-x- 2 g(..r) x K ;to, K R. Era; 6tri 7J Writ
4.
9a Cc-Insider Liv: Function
-- - 2 =
- g1:11, su chi any tiNed :Juin" oi.fg(x) is a so.liaLiort 01(0. Then
1. mGI. red, rr j a pinithiC ChC.ICC wileiLti is positivil. g.11
- 2, g .11 -I + — are po%ihte
-x--2
(Ai =x- - K =0_ c R is a possible choice_
2 .7 g.(.0=x - g(x )= I r — ihe only possible cho.k.:es..
5107 RDPI 2-4 A14-312.
35
7C
41 ,124i7c1 fe Mut 000.2 5K(x,4)0(41d;
uffif gr4U
cosxsin, for 0.x<c cr4 K(x,.4)=.
cos c sin x, for ; sxs
A vr AN, gm. wit e(x) -1t) 0(.0=0 (aj= 0, 001r 0 61' .30 4. Loaf
ak C99*17 ?NM KEr
/WI = 0, cr45' 6U. i 2. wirj(et) >0, aregihr ffazi7 3_ < 0, 01 tmr 7611 4, mr..z.›iperw .3p/RF FR.
99- The integral equation
=2:11qx, c)OgNir
cosxsin, for Osx‹ ,4" ....eftcrc A is a parionetr..y, and K(x,;)=
cosCsinx, for 4" Sx
leads Ito a boundary value problem (x) - fi2) = 0, 45 ()= 0, 0(0)= 0, whare jtA) is
kncivai. 'rhea the bC141ThdarY Val= probiern has
1. a lin ique solution when j(),)= 0_ 2_ infinite number of solutions Nvhen j(A) O.
y. DU. lu/ion when jtA.) < 0. 4. a unique solution wean 1 > I.
100. um, 4zi 42(x,y)) 11[(—°- 42-1 - 22 ltrely 0-61 D cr4F rekT f4u4' riffiftqr CrY -1 ex N
1, F/z 3.4 0g. .;; Wqrrb....ve nrririttff E UNWree ee'v Tz? e
1. =EC c • 7 qi fit177-W cr4" D rre ,ixeir;" TWEIEn rviciAr is/ i=l
= cr ce20c al 0 .rx1 FATAT v. -Warm .3.907w .5TWRTG1 g
3_ z.0 = aAx, ARe.42.• fAreretv cr2i D Er? ► f-Am
4. zo (x2 - I) (y2 -0116.
35
100. An appro);irni ......te &anion = , :r') to Ihe problem aFextrernizing the functional
= jjf 62 12 ?1 1 2 — 2: idrdy,
°;' 1.5 is Liu squaze.. y S t, z = C on the boundary Draw square, is of the
L. 2 0 E ai0(x,,y),wheru aro 4:011Slarit5 albj liarICIi01 15. alr linearly sndepandimt
L
sn I). RI 0 1 (x, v), wherc and ix, arc anti A imd love
ecninnuous pavial 626 yatives.
= where cx. is ❑ 1:onslant awl 's continuous in D
10] _ JW7q 4 77T.It tAr
LOni*Z-rLi. AD1Ftt 19i7e /c7 30-17610 t ). wrc LAI7 Meigirt7tr:WMT.
may},' W.;:. 71 -47er YlVdr. .114ZP1
VI a 19IX1' RTZI EMT
PT::LL'=r l?? -;10' yt (4 -677fi aFT ,27.97*
71e.7-1. 7FT eitif 7i.j.gurti
101.. Which of die foqowirkg ithre correct?
L. .lainilltan's principle follows from the ryAickikbellig prineiplq. 2, 1 I i.1111ili.1)31'S priiteiPle is 7:431 totally applicable to rionbolonomic sysEcm. unless
re Imion eermeel.ing. Ulu dillrereatials of gerkerali7vd coard inatcs is given. 3. •amilton's. prie_cipie Col loves from. 4. t.ZewLort's setUL11-5 law f motion foituws kW]) FEELLill.011'S
I. 02- zjt;) . 11.7T Firi.crRO cf.7.6:4 f
L . i.1117(1:51i' t71-47?"4 aitZir ,YiTertf e 2. .Y..? QiyoNci cri% Vfotri, g I
. Id; nrY %hi? "-F 1147 7-e-T msf—vft 91-9
R 4174.117 Wi177 2?"
4_ aT;ird.31 .30W.9 4 zvd.1.74:1 tgR7 arern:4rf ava p.7 iid5Ereft qcoq
102. Let I. denote the La.r.,rangian of a system. 'Then the
I . equations are conn...3 order di Efti -ential equations. Total number orei.?0,w;00.5 is oquai to the number of generalized coordinates._
3. Lagangian. L L nit LiTif..1kr in it; rum:lions! form, but the form of the [..a.g.ong`s equation of motion cf be preseryed. La.6..rangian runiAion quadrat is functrm elf generalized ve !achy when the pi:geniis]
37
vort) Unit IV
103. F(2 F y), 0(x) vrt I-1(y) ow (X, 'Ow writr ?;.747ft 4F-e/' weft X 4igr own. TO* itetf ;5737 Krj 77. ow' riat FR .1. e rrfionrft I .07
= 1
1 eiRfr XScr
—1 499' X > =
{
21m. y5b
—I 4rfk
wiaq l rrrdryrfd fic .31.w g.;
1. .3117 wirerrgrets( (13, V) = 0 ;4, y .04 F(x, y) = G(x) II(y) 2. aPrf frtilt x y Ike F(x, y) = 0(x) My) rrT fre.sPoifq (UN) = 0 3.. arqvUavFt-dv ff) X 4rYtqr(v 4.. arip. X a. Y it' a? Krefire
103. Let F(x, y), e(x) and I-1(y) be the joint c.d.f. of (X., Y), marginal c.d.f. of X and marginal of Y teSpccfivdy. Define
u j I if X 151
1-1 if X > a
{ I if Yob and V.=
—1 if Y>b
where a and b are fixed real numbers. Then
L if Cov(U, V) = 0 then F(x, y) = G(x) 1l(y) for all x and y, 2_ If F(x = C(x)1-1(y) for all x and y then Coy (ITN) = 0. 3. If U and V are independent then X and Y arc independent_ 4. If X and Y are independent then U and V are independent.
104. knir aft vi gA-4fir zirfATIF vvXqy Nirilde Acor
1. .a4taeR041. .P(X>alY>a);4P(X>a)
2. iirilla i bER.0. P(X>arf<b)=P(X>a)
I X 4.• Y 47:ticencl TA i
4_ vr4ta, bGR A.2) ERX a) (Y —13)] = E (X —) E(Y — b)
104. Which of the following conditions imply independence of the random variables X and Y?
L p(X>alY>a)=P(X>a)forallaeR.
2. p(X>alY.c. b)=P(X>a)forailaihG
as
3. X and V a3. =correlated_
4. E (X — a} — = (X — a} E.(Y —10 FOT all a, b G R
105, = 11,2,1,4,5 ). V-4" FM" top sreM571 P ,fir 777 g Vjg '0761..7 . ard.' 1H' Pnea
f 0,] 0 0.2 0.7 0
0 i 0 0 0
[ P= 0.7 0 0.1 0.2 0
0.2 0 0.7 0.1 0
0 0. 0 0 0,5,
clt c711 (i, W7H7 .k—r4"17 .t? e •
"2 I.
✓ " —FV.
2. F MF.Ireti P13. A. ? (0.25. rJ.25, 0_25 % 0.25. 0) c ,r7 .77-ft7
3. n.2
4. lim f7;4't) = C 13 - FL')
105. CCIMid.CI a MO.TiMV 6113111 with 512te q)EICe S = (0,3,4,5} and Stlii0Tlary i.raLlS0.1.1:13) pr a matrix E 3 gi Yori. by
(0.1 0 0.2 117 0
1 0 0 0
P= O.? 0 0.1 0.2 0
0.2 0 0.7 0,1 0
0 0.5 0 0 0.5 .„
).41; 9 (;) br the y, j).th clement of
Ben
3
L, iii = I .
7 . tA25., 0.25, 0_25, 0:25. 0) is a $ialicinary.disfribm.i.ori for the Marko.. chain.
Lit
39
3. E F. <03.
4. Itrn p;(; ) =113, 174.4.
1 -1 N2
106- x R,* J1-4 774 g(x)=--re 2 iirea kt v wr r r S r: 42.17
9-,frx E R 22 . IIW# 2f—/C) -,mm(x)
x (-1, 1) $ u(x) = 0
x G Nt 4l ire(.016-.i712.7xv
vn.ft x e 1 f(x) = g(.4) rorx), ?e'
1. f Nrrisvi-rw F2. a „q„f,
2_ mit x 1k4f(X) > OPRirriVR14 j'
R cryfir47 grAchcfi giref eirt
4. fkra7 WiechN•PV triZ97 I
I -
I106. Let g = e 3 --.5 "- for x e R. and 1.1 be a continuous function On Rich that
2 .7r
(i) tr(-x)=-10). fora]] x R, and u non-2)2m
= 0 for x (-1, 3),
(iii) loran xe R _ 242yre
Le fix) = Cbr R. Then
L. /Can take negatilie varues_ 2. f Cr) 0 for all x and f is runt integrable.
3_ f is a probability density ruction Un R
4_ f is an integrable function.
107. Tr# ft X.,. X..1 , -477/F 41. we X., -n 41' 2n = l„ 2, — ) Afixf ierhem -ko: •,--../6( X -
.51 0C/ 70 Ar̀, Iskr, 1, 2, 1%4 S,.= cm" S, a.4ni.IFRq ffarif ho as
4:11 {7W .1913-0 weirgri las,0 atm. ;um' *k jiWrc. 92 L'ff
u ,./4.;.
1. 432 r T J.7117 arinTr4u7J 2. (32 wr Wrdir
5.3c11? ,F77 rrzf ;Inv 3_ 13 1 ...A wAriAti
• e I 4_ E R5.0? 4 Jr-fg.i'd
Wri i?
108,. 1774 lei X, fcir g
wroz.Fr c95 Tf.rw air-o-a7 * PAW rf Ro4
3.1TO-NR. 02 Wf ,311T /
7.hrrO#1 !Nem. 317-6Wq. '7 CT 2 Ata-av winfaar ‘31M5
erit airrerf firripiTni:. =Tux!' .PW717 .3R7d7Z. uvebri;'
40
FN (0) S. 11T? 2. iirn FAO) Iv)
F,i (l)(1)(1)
4. lim F.6.(1)L O(1)
107. Let X I , _._ be independent random variabies and 3n, n = 1, 2,—
Let S.., for = 1, 2, and lel F?.;
denote the distribution function of a following isiare true?
L Rtn Fh: (C)L": do) 2. Lim F(0)?. O(P) Ae—)54
3. Um F.(1) 4(1) 4. lim Fv(1)?..4:13 (1) '
with X„ being uniformly distributed btiween
by the distribution fungliOn of AiSCI. le( 0
standard not mal. random variable_ Which of the
-2.v19, '>0
1. x, 4 x2 ,617.)ft4 rim* r 2. + 2X2 ,0* kegiil 11
3. X} 4 2X2 ,0* f 4,
Itzif l 2X2) e
1 2 Suppme X has density ji(x)= — e
_ .0 ,.r >0 and X• has density /3 (x)=-
0
e..2x
,x > 0 and 0
X X2 are independent That
L . X i + iS sufficient for 0_ 2. X, .1- ?X:. is sufficient for 6
3, X, 1- 2X2 is complete for 0. 4. 2
(XI 2X) is unbiased rer B.
1019. "Vie Er6r0 iirfi 3) { 2 )'%F:ii?PinU:' V*rcraisW # 4R11 r413-7 XL, XR,.. —X i , E47 ciW. 7W' ‘Wrq
NW, a') 4'e .7 * 2E 7e-7; < < =C. o < 7 <ce,5 .a.MTC7
41
109. Suppose that we have n ( a 2 ) i,i4. observations X I , each with.a common N(p, 0 1) distribution, ..„vhere ‹. to and O.< 0. 4 <co lire both unkralWn. Then
1. the muhnurn likelihood estimate of 43 .2 is an unbiased estimate for 02 . 2. the uniformly minimum variance unbiased estimate ofd has Smaller Mean squared
error than the maximum likelihood call:in:We of o2, 3, both the maximum likelihoue.1 estimate and the unieormly minimimn Variance intimate
crl are asymptotically consistent estimates for any unbiased estimate of a z ,. there is another estimate 010'2 with a smaitor mean squared error.
110+ R74 a XL, X,2,--,Xu 0,07U [ 19 + # 74' 741-6n77 * fi e 7°- < ° <1)5 Kruarm Argo * I 0
1_ A a5r ariftrlu q t r 172-<zif qgidor. areffef .31700.{ t I
3, sh0 1. 0.1 Tr.,7 0 wy. krolvew: werirve ,31171}747 g 4_ give! ;diz:rwi, ay. yo.efrefrirg -ronl g-eirar arlf47Fir sn'aFFR' vEff I
110. Lef X1 , X.2,., ,,X•5 be i,i,d. observations from a uniform distribution on the interval (01 — 3, 0 -I- I-A I where flo < 0 < eo is en unknown parameter. Then the
1. sample mean is an unbiased estimate for 0. 2. sample median is an unbiased estimate for O. 3. sample mean is not the uniformly minimum variance unbiased estimate for 0.
sample median is not the nnifonnty minimum variance unbiased estimate for 0.
111.;71# ft . X 47. vRilrIt f = .ems" , x > 0, 761 A>C1Wit'ffe rwrk k+1„k=. 0, 1 .... X 40 JW.44-01reor = k171m2r e Of en' Y1. Yh 74- 4911 '
1 " Oftlf g7W Irla7 e I RI* Fr =—EY, aj vizuf aor anwmg *:
n 1.1
A=141 -F1 4. 3.7 -rx-Fic 736E0717" affimzig * ?Fro
111. Suppose X has density f (x; x > 0, where X > 0 is unknown. X is discretized 10 give Y k if k < X S k + 1, k m 0, 1, 2,..„. A random sample YI„ Y,, is avaiLabie
firm the distribution of Y. Let r Then the method armaments estimator A of X
is
42
1,_
3 + = „1
Y 4. the. same as the maximum 1 ikelibe-od. eslimaior
112. mil I; eca co x * At Dar 3rNircra7 7r ;f1411, Sfe-I ri Rx. - 0) = 1 i0 — :,,t), ci, fire mr.Q.chdi 4--qw
ipol f d P2F0 .7..4 criktYAMICI; rqrf.:4' e>.. * .7,17 -)Erei . X 1 , X......X. l 91 k-P91 110:0 = 0 4raT7 .
ii,,. : 0 > 0 0 ag tlesw IA< 4.1v) .5: H .E ik-, (,:)..7-0 (-I
1, ea- x>0
ag (A) = Ocuft x =0 tf? NTT4'e . i
,..-1,ZIR x <0
Er4 xrplw Tr eirEfree Re7 F arrit9 10(1 (1 — .1•0 C26r it oil 0 -7-- < I I R.). 4 7:f 4517 # ?
1. Eta ,=0, I im .105„ > } =1
2. 7tz. 0.0, lirn Pfrn >J1Z.0.} = a R
3. wa. 650, ft 9im P{Sr > =-1
$>0, r lim P{S,,>47?2,.)
112. Lot X L. be i.i.d. obscrvations from a distribution with continuous ProbabilitY dolsitY function f which is symmetric around 0 i.e. fXx - 0) = f 0 — x fora]] mai X.
Ceasikr the test Ile : = 0 ! 0 > 0 and the sib, best statistic Sin =Zsign (X,.) where
II. if x > 0
sign (x) = O. if x=0 , Let zc , be the upper !Mt — rt)ih percentile of the standard non al
--1, if x[0
distribution where fl. -z a < I. Vilieh of 'Ina following iskru correct?
1_ Tf 0.0, then litn P > '1.1;70. } . -.1":6
2_ If =All, then liirn P > jrz =
3
43
3. If 0>0, then Lim P{.,5„>1,,cuer ) =1.
4. if 19 > 0, then Um /1.3,,, >4;;.2...) rr—Prt
113. 1117 XI, XI,— X 30, N(0. ce)„ ce 53 10 ?.)' AMU ;rte! arffkreir efrde 1 0 e 9--INT(CP, 10
r2 ) , T21.20 trf' .19.44. TM: .1% .17= =Z4K, a). H a iln-Pe 47F Tr 4'0.0 el VIVni:enril -e : iu 6.1
3. S if 2;0
20:k
2. 61 21
4, 0
I3. Suppose X1,. -.Xi¢ is a random sample from NO, 0), 02 = Q. Consider the prior for 0,
rr),T2 = 20, Let ,V=—y.x. Then the mode d of the posterior distribution for 10
sat isfles :
3. if XY 0
2. 20 kr
tar 21
4. if
114+ (X. 'to p.t. Tro trry 14Y r ; (0, 1), (1, 2), (2, 3), (3, 2), (4, f
1. X wY T7 durrri—r1 fiwsfziu = —9
5 2_ cr-( X 7r winr404 ex. 2 1 X 2ir Y e zmffer pew 0 ..t? i
4_ X 4.V 3//ra. E rfeeiiir yon + 1 t fi
114. Consider the fo6wing five observations on (X, Y): (0, 1), (1, 2), (2, 3), (3. 2), (4, 1). Then
1- The least-square linear regression of Y on X is Y = —9 5
2. The least-square linear regression of X QII Y is X = 2, 3_ The correlation ooeffieiont between X and Y is 0. 4_ The correlation coefficient between X and Y is in I.
44
HS. 74 tit; 4 1 , vh-trelfrgi: tkariA' } TAG, cr) argifF V 2 ci # tr rdeqTR7r /2r Ir4 e
= P Eb 4.1 (.r + =t, 2,.., - 1
ft; Tz-LY4 cr41:1<p<1 z ci>C1 r 112*.0-4. n
1. [ te7 til• 2. T Ei.57 M'uf I
3. WO ==p,. (T) in. 4. TV N (ft, 62) wi AL.
1.1.5.. Suppose e l , sT, are N(0; al). Consider V,. YN„....Y. defined by
P )113- =pit; -14+ 1111 ;2 Er, iri = i, av 1 .
I " Lei =— • LIPVI CFSe. < P 1 and 'DA > Then limn 2
P r i=1
1. 1 has a normal distribution_ 2. T has Man )2 and variance er".‘in.
3, 1;(1) _fir, ruff) > cr2in. 4. N (g, 62) where e> o2in,
116. kl'45' r-4F VFM/Ya4V al M7-47 VEY T, WPM P aMraq 6V A-el 74' ni &DTFaercreqk A ti fi arefq-3o7r r il lZ:rell
aikaroVei A: 200 WC-drat 9ThirRi zirro:Fw Yiktfir ler4 .57or e
(sRswoR). raw 07r# x x fitrz) tia R. cia: TOT Ms' Sivoctl oteof
P1=200
7u-ft. -400-413: T-d-dwIr0 wit 01. j-i5v F-4 trOgy) ei Mu 0 WM' r? # SRSVIPO.R. ZPOW rtkil ird-r eJ Arrqn: A? 74 Mr 9T7 4,721 p .7r Ter MM.?
7Fuff p = 200
4+-4 lic;-.4 ,q-a-zimsn v.?. 7757E 7,---qm fi)
pi gg 3fl7 —da- ,9 .f
2. /77, TM .17A97 Nreraq S 3. p, 71#.earemu 3774q11.7 4 p2 T P.rwv.1 p i a, ge.pir 47' Zff MITT g 4. Q77 Vd.r !OH 7M r vccir YV ,I.ovrrisli aygrr FAFF ei-o. tar ePi p2 iitr'm
F5Vir eN
116. ln sur..cy to estimate the proportion p of votes that a party will poll in an eloction., siaiisailiiariA A and 13 follow different sampling. strategies as follows: Statistician A: Scicom a rattilom sample without rc....pla.r.ement (SRSWCM.) of 200 voters, finds that x of thew will vote for the party and estimates p by
1 •
45
= t 200
Stntistielati Et Divides the voters' list into Male. and Female Has, selects 101)• from each list by SRSWQR., finds dig xL., i respectively will vote for .h party Arad estimates p by
_xI x2
The number of voters in the two lists are. the same. Then
1, pl. is an unbiased estimate but p 2 is not. 2. pi and p2 arc both unbiased estimaies. 3. pt and p, are both unbiased estimates, but p2 has a. smaller variance than joi , or
the same variance asp,. 4. Variances of p i , p2; are Ehe same only if the proportions of male and female
voters who vote for the party arc the same.
117. 5 -AM.' 5 !pew,. z?' Ktrzu ittgT kirtV -.WPM—KT 47 Fro:
Gr.w 1: {1,2, 3}; Erzw
q4 407 wr/R 4tFE.m. wei' gz . g. ?
1. AZW•I WOW. I 2. rY2iF *mar w spepob life ;17 sJgcote r. 4107T9r .3ff/47a .307-ow
.Erz Veff 2c5 zi'T 4 cr e ARire;c17.. ParfTet A—eti'reff 711. mrzirr-eri. grfr 7-4*47 (117 Zi';11-1 .n70- ORM rh77 Ter e
4_ wr2z qk—vig mri..ru g
117. Consider the following block d.esign involving 5 treatments, Labelled 1, Z. ..., 5, And two blocks= FA look T: (1., 2,3}; Block 11 V 4, 5). Which of the following statements isiate true?
1. The design is connected, 2. The variance of the best linear unbiased estimator of an elementary treatment
contrast is either 2o-2 or 402, where (5 2 is the variance of an cb5crwation. Third i 5 no non-1rivi81 linor function of observations collected through the design whose expectation is identically equal to Duo.
4. The. degrees of freedom aksorriated with the error is zero.
118. ?1 fi7 WU ?I—NW it f;161 25 skrai 1 1.- B.? T'w rffr 0 9 g
I . 91W Z:r9* WRIT eft`? ?MI/ 1
2. wp:15::7; 7 P:T.r.r wr* sravGy # rre1 wzlzri 3. zt# Ian W 7ifq A r e 4, gam yv *W' RITM 0 I Farr
46
Suppose that tine he a data s i consisdn of 25 ebsetwations.. Where caCh valve is either 0. DT t-
1. The minn of the data rannOL be laIL.ree than the variance. 2. The mean lathe data can of be smaller than the variartCe-
3. The mean being same as the variance im ties that the mean is m.o. 4. Tile variance will he U iFand only if the sr..can is either ] r T O
wu-i-he + 5, - 1.5 crP - x i s. 15 al Ta" 477/.• w x.4 0 CI VT
T.I7ff .1.951-7 .71: v,.?;-r VIVO zgoFF Fre, g/g 2
_ 2x i + Z7 de677 779" e 25 r
2. 3..x + 2 xi ?{7.f 6" 7Fir 61. 11.
3_ 3.N., w 21.1Vfig 70:1 4. 2)4 2 w IRRIFi 7rffl
U. Consider the variables x, .!4 2 0 SAlisfying the consiniints x, .1- :Kg 5. 4x, - xr S 15 and ax2
- Which or the Colko.ving staternencs ►Sf5M C.OrrCet7
1. The maximum rah= of 3 Ki is 25 2, The minimktm. value of 3x, .1- is 11
3, 3 i + 2;4 2 has no Fsnite maximum 4. ax iT 2x 2 ha.; no finite M.1337([1.1:ML
n. .ft177 izo., mord, 4. PO. .11W .dreq7 JIFFY 7rija gff 12 14-q-e- Ve WW1: 3tt.
0-9-4 x r izrry#ii7 0?). 14-rx• c-6' 7.137 ast 07125. ze.'" 31773irq" rrla 20%Wr e worn? aiTgrf
i7 SAX479 7r7-71. eiwr r era' 2 k _ Krisgc-ft ?;tzu q-d77 4
3_ .1;r6-774 1-4-ffrai ur-.77 777 7i1 ] 6 Plc. e 4. .crulFR wro firer Rauh. krO• WA. MrEr qTr7 24 /47c e
120, 11-141. system with a single: SLI-vcr, suppose that customers arrive at a Poisson rate of 1 person cvery 1 2 minatc-3 and arc serviced at the Pois5On rate of 1 service every 8 minutes. If The
arrival rite incruses by 20% then in ih.e steady ate
the increase in the avcra .sc number n€' easterners in the system is 2_ 2. the increase in the ....ver-..1.ge number of customers in the SystCrn is 4 -
3. the increase in the average time spent by a custuni.er iri the system is I& minion.
4. the increase in the groctage time spent by a customer in the system is 24 minutes.
1 1S.
] 9.
1 MATHEMATICAL SCIENCES I
I PART ' I
0 0 . 0 ~ 0 0 0 o o e o 0 0 . 0 0 0 0 . 0 0 . 0 a 0 0 0 0 0 0 . 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 . 0 0 0 0 0 a 0 0 . 0 0 0 . 0 0 , 0 0 0
3 . 0 0 I71 610. 0 0
0 . 0 0 0 0 0 . 0 0 . 0 . 0 0 0 o o e o
Bookletcode Medim -Code comecoder
O H F- 2
SIDE - 2
mwsd;fdqm INSTRUCTION FOR INVlGlUTOR
piPrnm&*&-?hisrz, n~;rrn*,rn*~** sMaii3wmdR*wi;tl
Cluck Roll Numbw, Boolrlaf Codr, Subjwt Coda, bntm Codr k Slandun of
Wndihll brtMl8Ilpnlng.
ANSWER COLUMNS
3m / ANSWER 0 0 0 0 0 . 0 . 0 0 . 0 0 0 0 0 . 0 0 0 . 0 0 . 0 0 . 0 0 e 0 0 0 0 . 0 m o o 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 .