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I C.S.£ Pr'l'-20011 I ol Ill

MATHEMATICS 11

3.

3.

4

5

I r r u, is a seriCl> of posrti ve lemJS, t hen a con\'crgencc l:l-1 )" u, rmplies U1e

corwergence-of r u. b. con1 ergence of I:u, impliCl> lhe

com erge of l:(-1 )"u, c conl•ergence or t( -I)" u,, tmplles the

dJ vergence on: u,, d. di\!t!t'getlCtl Of :!: U, 101plies Jhe

divergence ofl:H 1' u" The lntegrating factor 1>f a homogeneous ~quallon a(x,y) dy + h(x. y)dx =O,l ls

a. 1 a~- bv ~11 ax+hy • •

b

d.

I ..,.-..:....-. ax - b\ " n hx• t~\' ~

j --'--. ax • b'' " U ar +by •

I .,..--;__. bx + B) ;t 0 l>x ~ ay

·rhe smgulnr soluuon of the driTerentlaJ equauon txp- ,,f- p' - I rs a. xz--4- , l= l b. xa-y~= I . ' C. :("+V" = I d. X'- 2~ 1= 1 Whlcb one of lhe following rs NOT n solution of the drrfe(t'Jltinl equa1ion

[tly )

1

~xdy - y=tl'l dx clx

a. x2+ 4y=(J b. r = , + l c. ,·+ x- 1 cl ;-2 -~x.: t) The solution of the diCfer~nliru ettu•liorl J 'r ~ - , = g • (g rs a const:ml , s =tl and - = ~- ~

u " ·nen 1 = 0) is a. s = gt b. s = ut t Y, gt2

gl1 c. s~-

2

a Nr)ne Of the alxl\'e

6,

7

8.

In is a parameler • I hen the orthogQilal tmj0C!Orie!; of !be qtrdioid r s k tl· ~OS e) IS

a. r =C(I r cos9) b. r = c(l -sinS) c, rll-cose)=c d. r( l.-sin 9)=c The a~is of the light circul ar cylinder wilb

'd' . 'I , • ,, . gw mg c1rc _e. x-+y-+z-t'• x- y + lF 3 1s a :<.-l = y + l =?.-1 b - x : y =z

c. ~ = - y=.r.

d. ' = y = - z Malch l1sl I with J1sl 1.1 and select UJe. correc1 answer ustng the codes gil'ell below lbe lisiS: Lisl - 1 (DiOerent ialllti U'l~on)

Jy X A. -=-

Jx Y dy v

B. = -:-. dt .r

c ~ ,. _ 1

D. t(V =1: J.r .r

Lr.~t - II (Gepmelncnl meamng of the solnlron) I. Family of curves hal'lng slope at Bn) '

po1111 l•"- ~) equal 10 uegative of 1he slope .of line joinmg ong1n and the point (X, y)

2. Family of curves llrthogonnl 10 the line joming ongin and the poinl (x, y )

3. Family of straight Unes through origm. 4.. Famrly of s1rrugb1 lines hallng slope ­

I Code!i

A a~ 1 b. 2 ·c. 2

d.

B 2

c 4 3 4 3

D 3 4 3 4

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?,

1(),

11

12.

t:t.

14.

Tf D= .!!..._ and z = lng x. then the Jr

.l'y "'' diffen:nlial eqWllion •-- • 2--"- 6>< . d.t1 J.~

bccoma a_ 0(0·1 )Y - 6e' h. D<D·l)v= 6eh

c D<D+I)y = 6e"­d, D(D+ IJy = 6e' l'he uniJ veclor parnllel to tlte resuliant of

tlt¢ve<:lor 2 , t4 J ·S ~ ""d r ., 7.1 • ~ f Is

··~ Ji• 6£-ll

1

II 11 .. ] 2J> ,. e It 1l SA

I l t 2 J"81 .. tf a partJole aolcd upon by lhe fi>rc-os

11 = x1 j+ x~] •x,; . F, =,1ti • y2j-< Y>t - ' + ' + • 1'1 = Zli Z:j 711; ,

i~ in equilihrium. l~en .l1 X1 =)'r= 1 1; x1 -= Yt=-~h 'I(, =y$ ~l

h. ~'•""~'~; y.=y:=y3;t,;:'l.t=73 c. Xt-y1~t1=0; Xl+,Yl-Zl=ll: ll-rtyt+~•=ll

tL xt- )i=r.,: x~+r,=z1: x,+y.=t. Tf tlte pairs of sltaightlint:!l

., ., ., 'l_ fit l\··2pxy·y·- o a.nd x··2qKy·y - , ""' such lllal caoh pair bi~=~ the nugle between Ute oilier p~ir , Uton n. Jlq= I b. pq = · I c. p >q=l d, p+q=- 1 A re.:rangle ARCD is fanned ~y lh" hnes ~ = o. x - b • • Y =c and y - d. l'h~ "'lubt,ion of'ilS di•goual AC i.'l " · (d I c)x+(• • l l)y be· ad = 0 b. fd - C)X< (3• b))' - be - 3d = 0 c (d - ~)l' ·' (• + l))y + bc + •d = O

' ' · (d 1- c~~ + (;1 + h)y t- be +ad =II The eq11ntlon i.sl·l0xy t- 12y'-.-Sx· t6y· 3=0 will represent a 11air of m aight lm~~ if '· eqWJls n • .I

15 .

16.

17.

18,

19.

20.

21.

2 Ill Ill b, 2 c. 3 d. -1 I rtr. G) M t pol:1r coordinate.. the~

.!.=3 sin a • 4 co. thnd • !.= 3 cos G· ~. sin (:l ore equ31ioil! or r

u. lin"" p••l'ondicutur 1(1 eacl1 oll.tcr b. ellipse cutting each odter orthogoD• lly c. lines poraUcl to each other d. on ellipse nud • hyperbola respoclively The C{jiLltion of the cit<>le wluch touches the #Xi• of x is a. x1+y1-2hx- 2ky- ~~) b. x:+l =~2 c. x' +l =h1

d. x' + y'-- 2~-2~y +i=O Tilt: CltUatillll Of llto ~irolc: \Vhnse (Jiam.,lc:r is lbc cummoo ch<1rd of lhecirciCJ> ,.z. , ; 3x+ J.y~ I =0 nod x!_.._),l - 3x-t 4)'4..2 ::(] i!l :1. 2.-.z~2r~+6.x+6y + J>= 0 h. 2.x'- 2)·1"'(ix + 6y + ~:(1 c. 2-'>' -+ 2f ~6x '" 3 =0 d. None ofthe ubove 'lne str;~il)ht line y= mx + c is n(lrmal lo the: drclc (x·a)1 " (Y· b )1: ~ if •· • - bm ~ e b. c = nm t b C., b .: UJ.U •hJ:

d. Jl : ~'m ' ll

Tiu: rodl.:nl axis oftbc eirclcs x1•~ and~~ 2Y· 3y = S is a, :xz+ }.Z + x - 3y- 5 =0 b. 2,'( - 3y- s ;(} c. 2.x T 3y- 5 : O d. :.1+ yl+x - 3y =0 Th~ p•fr of ci.cle x2-·/ ·2nl< + t =() and x'• y·~2by -c ~ 0

a. "'"' mtlto~onal b. hnve radical n~i• ox I by I c • 0 c_. inlt.."fflccl ut the origin d. arc cuoceno·ic

If the •ugl~ between lbe lin"'! joining the end point' or minor n i; of on d lip!ie with

it~ foci is f ,lhcn tl1e eccentricity af the

ellipse is

•. !.

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I d. ~r.

22. rhe stmight line 2x• y - 11 = 0 .a a normal to !be pAr.!bolo

"· );. -lx 1

h. )J= 4x

c. ,,~ !!..x • J

d. /-Jx t3 ~ line pa•~inglhrough the puinl ( I • ./, -2j

iK pmlld l.n the pi;m<>t (ix r 2)•-2? + ::t =ll •nd X-. 2y - (iz +4 -C), 11Je t:ttu:•liQtl (>( th~ liuo in 1flhl$lillllll is

.s-• 1 ) + I 1. ~ n. s:~=-.,-

.,_ :!..:.!. ,.. ~- · =~ · It 19 ,

c. !::..!. = ·' +" -.:::!. s 19 5

J t• l 1 1 1 ::!' ' -~-~»~

.!4. 01e l!<lUatiou uf ilu: coue. whos~> vertf:'C i~ lhu vrigiu and the b.uc th~ ein:lc • x - u. y:~:~ b1• ill given by

, . l ... ., a_ 3"'(y ..... z _)= b-:i .. b. h~cy'.,.,().,~s:

c. a'(y'*x')= b~: ... '1 l ., ...

d. b'(y-+x )=aY 25. fhe equation of Ute rlghl ctrCUiar cylinder

with a.'tis . x 2v - z and radius .j is. A. sx'+Si>8z1-4~2yzl4n o b. 5x1+8y1

j 8zc1-4.xy+2yz•-br U e. 5xh·Sy1+5z1-I"Y --4yz 8n 14•1 () d. 5x1+~; Sz~+4yz-t 87-X·•l-<y-144~0

26. l'he: gcnernl solution of U1o equntion

(2l>T 1)). = 0, IF- '! i~ ( A,H are arhit'rnry (l..~

constant<) tL r = (A+BX)<:,. b. ;. = (A T·Bxl•:"'~ ;:. y =IA+B)e"'~ d. FAe4:t

27. Assertion (M: Ute diJTer<Jiltiotion equation d't· d•• t -'-+II-~ ""~'~ f:c:-, So•-~' is • line:tr dx' ilx i

O<llUtion

~ ul Ill Re:\>on (R)I Every liNt degree eqwnion [, a linear .:t,uation a. 1300t 1\ :ind R 311: lrUo and R f~ tho­

cum:.:! e.~plu•uttion ill' A b. 13otJ, A •nd R arc true but R is nut o

correct e:q>lnnation of A c. A iotrue but R is false d, A i• f•l•e ~ul R i~ true

23. A.~~fl'tion (A) l'he funtlion

J (X) ; .!. •'' ! ,1 .!. ··' • • bi'IS ~n extreme .. J z

valu<r -at x = I. Reason (Rj: }"( 1)=0 a. Both .'\ ond R ""' true ond R ~ lhe

eom:ct el(plllnlltion ,)1 A b. BoUt A and R are b'ue hut R i~ mil A

com:ct c::.xpltmoti<>n of .\ c. ,\ i~ true but R i~ tal<c d. A i• fnl•e but R i$ true

29. Assertion (A)I Moment of ~ time F

which is represented m lc:ngth and dirc.:ti<m b)' tile line A13 Ab()ut u (>Oint C is equal to twico the >11ln of the tnnu~lc

ABC. R.ea~OTI (R): t\ re11 of n tri:m!!le i~ the nmducl ofllasc and • ltitutle. a. B<1th A and R "'" ltuc und R LS the

com:ct c:q>lanation Qf A b. Both A and R are true but R i.!! 1101 •

eurrect explanollon of A c. A i~ till~ hut R '"- r.,lse d. A i~ fnli<e but R i$ trut;

30. A.ssenion (A): Three. equ~t Ioree ~ctlng cl<lcl. wise ~lung Ute sidCIIQf on cq.uil:tl<'fal lriongle nre in equilibrium. Reason (R): n~ three• :1cting ut n (IOint llre m equilibrium iff th<:lle c•n be repre•entJ:d in m~gnirud~ and dirc:ctinn by U1c $ide;! of • lri>nglc. u. Botb 1\ und R ora !rut$ anJ R i.s Ill<!

tqm:ct e~pl3nntlon of A b. Both A Mel R. are U'ue but R ·~ not •

c.orreel C~'J)Ianation uf A c. A i• true but R i.i false d. A io f'ulse hut R is tro1e

31. 11tcCCfllroid ofn ri$id bod) i.p ''- poml wh.ich can be mad.e to Uc on or

out.<idc ~tc body b~· chongin!l the coordinatesystenL

h. Pi.xed point in >'JlJCe reg:.r(l)ox< nr the nri.;mU>tion ofOte il<I4Y-

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c. Point on 1<'hich Ute bQd) c~n b~ b~lanced by a roree when it i• subjtlllcu It) e>(tcm•l furces

tl Uni4uc 1'""'1 f"ctcd ~>ilh respc:<;l lo Ooo bndy.

n. 1f two bodie§ A nrod B rot ""'t ho\lin• muss e. in the mtio 1,: 1 '"" oubjecJQ<I t~ Ioree.• In Ute ratio l: l feor oqunl uorcrvnl• of 1 ime .tbon "· llli!ir mnlllcttts will be eq ua] " · lheir w tocities will be cquol .:.. their o:n~:rgie.~ wlll be equal 'L none o~ lhe nbovc

;13. Consider the following HDtemcnts: If a .Jlarllcle move:; under rhe roctlun of' a Ioree l{r) ltlwnrds a liXed poinl r hcing lhc dlswuee of the 1•••til:l~ fro111 tbe fi>:etl point , lltt:n che I. Lineru- rn<Jmentii!Jl ,,r ll•e< particle will

be IXIn•l•nt 2.. Angul.or mnmenrmn oJ tbe 11nrticlc

about the center of fore<: wlll ~m!Jlnt. Orbit~) paUt l'"ill ne•'<>mrily be~ p]J)ne CJJfVO

3 Pallt will alwa)~ he ~lose;!

Which uf lh~ abCJ\;, sL1tcmt:flt' are •'\lrrect'l n. I and 2 b. h nd 3 c. 3 ~nd+ tl 2 aod4

34. Keeping the ' "" u1 fu~us. lhc planet~ <lese rib<: :J, Jli1r:,lWJ)US

b. ~llip•~ c. c•rcles d. hyperhn Ins

;l5. ron~id.er lhe foii<!Wiog ~tmements : A particle i~ .:XccutiiJg simrl• hannonil: motion io • $lra igltl linu. 1f tbc p<:riod uf osciU~tion t8 T, und umplitude of osci lint ion is A. tben I. $peed ls gre.1tc~t when lhe o~celerotoon

2. Spco<l i;. lc><L "hen the accelcrahutl i~ lea!l

3 , A depends on 1'. b11l does nol dcp•'lld on initial condition

.L A is independent of 'I' but depends on inltial conditions

Which of the above SUJtL'\llt:nl< ure ~IJJTc~t'? .1. land 2

b. 2 and 3 o, ~ and4

d, I and 4

.1 ul Ill

36. If tbe velod1)' at lh~ max•mum heigltt ot' • pmjectile h half or iL' initial velocit)' 'u" U1on its borlzmual range will be

:t .),;1

ZJJ

b. l g lr1 J-.. 1

c.

d. ~ 17, U' • bn<ly is projected al such ~n • ngle that

t}l~ borizonl~ l r~ng• is >t timC!I Uu:- gr.,le~t hcigbL lhllll tho atlgle of11rojeclion ~ • . 456

b. t~n42 c. C()t'lz d. llln' 1 1,

~~~. M~lcb ll5t 1 with Ll~t ll and •eLect ll1e correct BOSI>'~r uslng Lhe •'Odes sivcn below lite lists:

3!1.

t. i. t -1 (Elcp""'siOIIij ""' Nfm'tlu to plouc poi[IJ' eoord.in•ies (r, 0) ) A. r

13. .!. r' 0 ~

C. r~+ 2r,. Lid- U IL;,l of dynaJrut.'ll quantities)

l. .1-\..ngulnr momcntwn 2 A component of accultrntion 3. A con'tll()ncnl of veloclty 4 .A real velocity

A D r "· 3 I 4 b. 3 4 2 .:.. I 3 2 d.. 4 J MMcl1 list With list II an,J sclccl tile eth:1"ct diJS\\CI li'inl!, lho litldC$ siVOTI

below the lists: Ust - 1 A. The vel<1city required to llllow escape

of a body from the grnvitational Altrnction of e<~rlll .

B. RC!Iu ltnnt of• lbrce ~od a couplo

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C. Titc velocity is neco:,,~ary for au artilieial sateiHte to revolve a'Ound the enrth in a drculnr alrbiL

lJ. Work d1mu hy -!1 foa=- in moving a particle from one posil ioo to ~nolhur is

List -H I. 8 kilometers pur sec 2. Conrle J I (, ldlnmele)'!< per o-ee ~ . Vector c1uantit) 5. A t"orcc: (;, Scalar qu;t.nlity 7, II kilometl.'11i per sec C'od!!$

t\ B C D ~ l 2 4 3 b. 7 2 3 6 c:, '1 5 l (i

1L 3 S T 4

40 tr a panicle moves in a ciNular r•fh of mtlius r 11'ith a ~on•Lill l k~d, then its a. velocity il; perpendiculnr to the rndiua

vcetor and nccclctlllion i> radiaU) lnward§

h velocity i'< perpendicular lu the r.dillll "V~1nr and nccelern1.ion is rJdialls uul\lardg

.:. velocity is alons the lildius vector and llccelcratipn is 1'3dia0y outWIU'd>

d. vciocity i$ ~lfing thl) mdiij• Vel)lot and ac.:eleration i$ rodilllly inward5

41 A pani~le· $tarting from rest moves w•lh cunstant acceleration, J r il ltavd~ a disiJinoc S 1 an the fjr.u two scwnd5, S! in tho oe.xt two second.• and S 1 in the next two gecond$. then n. !lt;S,; !h = I I · 2 b. S,::)J:S·, = I · 2 ·~ c. s, . s. :s, : i ; 3 : 3 d. S1 : S:: s, = l ; 5 : 9

42. lUeradix o.f Utc bin~ry numb<:rsy.IJ:m i• ,1, l6 h. 2 .... 10 d. 8

4~. The hex3d<:<lim3l numiM' F1i' ~~ e<tuivo lo:nt to the decimol numbor n. 4Q80 b. (;25

~. 552

d. 255 -14. The (JCt11 sumH 670 ;. 255, 267 + 53 1 nnd

671 + 265 ar" ~pectiWI} a. I 020, 1145lrnd I J 56 b. 1020. ILSG and 1145 c. 1145 • I 02iland 1156 d. 114:5 , Ll56 und 1021)

-IS. A combination of hils u.ed to '"l'""'cnt 4 chrtf'ncter is kuown os ;a

a. byte b. binory digit c.. word d. code

-16. In > flow chon. A porallelogram hc1x L~ used

..J8.

-Ill.

50.

a. o t<.'1'111inol b(l11. b. a pro"'"'>ing bol> c. on in1111t or oulpul l>o~ d. o connector bOl\

The value Pf (- l~ ,.,fir . •=..J-:i i~ a. I h. 2 c, 2'~

d 241

I . l I I 2 cos 1:1~ < .._ - ~nd cos '!! = I y + - ,

;r y !hen lhe value o( cos lU • 4•• IV1 11 he

.'( l' :i, - t- . .L...

v ;t

[I' y] b. 2-- -) ' ....

c. ..!.l..,. . ...!..] ~ ...,.

I d . ..,,. __ • .cy

I I 'I he expre..,.ion (:~-t J?s6'fi + ~7-/K6p t.

equivalent Jo a. 3 b. 9 c, 18 d. 27 J2 a11d .,fj. nre nlgebra ic owr Z. A

ll01)11omi~h satil!Gecl by ../2 r ../3 werZ is B. X~- 5x1,. {; f 0 b. x"·3x1+5x- G = I)

" · "·~ '2"2+ 3JCt 6- 0

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'L x•· I Ox'~ l :j,l 51. If I he HC'F of two <tUntlmfic e~lttll$lvns 'i&

(Jt+2) ond lhG I.CM is x3~2.~·-x·'2 • 1hen lhe <:xpo't:s•ioll ore rt, x2·3ll 1 2 ond x1+K·'2 h. .Iii. ~ <4--Jx1 2 and x2--'· .x+-2 ~. .~1+3x•l •nd ~-x - 2 d. x?--3x• 2 ::.nd ;><_

1 .... x-2 51. me plllynomiol

2: 1 S ' · d' ' '"I I • ' ' f' :< - x--+mx -rn 1,; t\''ISUJ c JY x-... 1

'" (m.n) : (2,3) b (m. n)= (S. -20) c. (m, n)= (·8. 20) d. (m.n)= (.l, 7)

53. lf u. . PAn: the roo IS uf lh~ .:qu.11iun .~1~a~ t fJ- \) . then (a. fj) l~ equal to a. 10. 0) b. (1.0) .:.. (Lit <1, < ~. m r o ,. llll y "'~' = o

5-l-. If a, p . y are the root~ of x'• q.'< +r 0 .

II ~ (A •

''"' .t..--18 p 'I u. 3 b, q t r c. qfJ

d. 3

55. rr IL. p. T ., the JUUL' nf l\.3-p.~!, qx-o· -o. UIC!ll tfot: no()1.' U( lhc CljlllltlHII X(l\+11)~• (px+d =O nre n. cx1

, J31• i

I I I h a' fJ 'r

1 1 ~- --(7' p" r' d. a'. fl", r'

56. fhe set {2. 4. C.. ~f is a group ,,;u, binary ovcr•litlll ~1o, t1'1ullip)icaliou modulo 10. The id.:ntity ~tcmo:u.Lofth.i. £ri'Uf' il'

·'· 2

" · 4 c. 6 d. 8

57. Lc1 (S. - , • ) b~ !he ~vstem nf ntm-neg:.tivc in(cgCf!", If we· tldirn; 11 """

biuory (lperntiol\ 1•1 inS a.~ x•l -- x y t xv. vx...y €:S.

()ill Ill

Utcn with rcspe<:l to tlti• new operotion (•). which one of the foiiPWinl! stntemcnls is I roc? A. (S."') C<)nlalns •n idcnlily el¢rn<:nt 0 b. (S, •1 cootoinu n itl~otity clcmcol I <>. (S ... ) do<:il nol eon111in •u identity

e lement d . (S,"') C<lntn iM nn identity element - !

58, I.e! Z denol<: the ~el qf integt-'111 I t tl' be • prirne numher ntul lt;l/,1={0, I)

le1 f:Z ~z. ~nd g:Z->Z1 be defined as foUows; ftn) ~ pn forany n€ Z g(n)= I of n is

- 0 ollll:tw!Be g(nj 1 lf n ;,. a pcrfc.oc:l squaro () otherwise

II we consider the composite functaon gof-. llten a. gof i.~ ont<> but f i• not onto b. gof1s one to one but !!- i~ no I Ma 10 one c. ~;or as mcre•smg bul g ls ool

dO<lr.::a•lng d. f nntl g l!ro belli on~ to 011e

59. The nwnb<!r of disjo in! cycfcg of length 2'

1 in Ut• penpolotion

a ( I 2 3 ~ 5 6 7 ) is 5276 3 ~1

a. 2 b. 3 "' 4 d. 5

@ , Let ( : G -> 1-1 he u J!TOUII hOmOtllOJlliU!Im from a group G lnto o group H. with k"'nol K Jf 1he order Qf G. R lllld K are 75, 45 11n<l 1 S re~p<>=ti·vel~ • then I he ord~;r uf lhe imAge rtGiis a. J h. 5 c. IS tl. 45

61. Coltllider ~1e following Klnl<men~~ wi~• regMd Ill S1, the symmttne group·ol' order 3; L There c:.~ ist twn c:lemen~-; x aod y in S,1

... ucl! ,h,H <~ >., ~)1/,v·. 2. Thcore an: four cle:men ls •• ti.fying xt,.

e and lhree dements •nilst)<ing yl-e.

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3. Tit ere is one cycJjc wbgroup of order 3 and three cyclic sub&fOup of order 2

Which oflhe above statemet~ IS ore C<lm:ct? ;t. I ond .1 b. 1 ami 2 c. 2 andJ

d. 1- 2-11111 3 6'2. Which one vt tbc I'Ollvll'ins ring• is ""

integral domoin1 n~ Z1o1 b. Z tlll

c, 7·lfo'l

,). :'.!!6

6.'i. Let t 1 l)e an arbil.r:~ry nQu-emvty set and S denote tbc set of all subset~ ol t '. F~>r ~­y " S .u we dt:!in c

X t Y {x:x eX llr x-= Y but !<<!..\:. Y) ond Xo Y - Xn Y ,U1crt

a. tS .. -, ol is ..a commulativo ring Without unHy

b (S, +,o) i> • aammut>tive ring wiU1 11uity ond X o X • X for •ILX: S

c, for X.. Y.ZE S d. S c.ttn be m<~de n ring if U and Ute null

NCt t GI'C added to S

~4. Let ~= tu- ../2 b: tl. /? .c. Ql, R i.~ a licld with respect le> ttslllll ~ddition 1md muJI.ipUootion. lf 3n element of Ra~~ lr-0. then tiS mulhplicative inverse is given by

1

•· (n+ ~)' I h .

., .[!i

.. <I .fi2 b - II' - 2JI I ()! 2b1

d. tJ • ..fi b a1 - 1./;1 u• - 21>1

65. lf W t 10, 111, •~ I and ltr,. fbt. 0. b3} lito two sub!paces o!' )::'. where F" is a vectors ~pace of dimen.•ion n. 'fhen the •ub•pate W1 W: i~ t~umorphie II> the \-<:Cil'r space "- F' h. F2

c. F' tL 2F'

66. TJ lite $01 of nil lriplt• (x, x1, XJ) of rent

"""'~"'" R ((lrm • vector NpitCe v.h then • s~ksp•ce dcvou:d by " ver1icol pl>nc y = s

7 ul Ill can nb Obt'lined by ~ linc::tr combin•lion oft.he sets a. ( I. I.O) ~nd(O. O. I) b. ( I. I. il) •nd ( 1. (1. ()) ~ (I. 0. 0) :tnd (0. J.OJ d. ( 1.0, ll4nd(O,O, J}

67. [o R3, c<111~idcr tbe foii<H• ing ~littenouul.( ahout tin: ""~el E={(l. (\, 0),(0, I, Ol,t 1, L I), (L I, Oll I. F. "' a lincorly dependent•et 2 .'\ny three vectnn< qf F. are ljpearly

indepeodeol

3. Any frou r- """~"~ Qf F. '""' lio.,.rly tl opt.'tl dc:n L

Wltic1t ol'lhc abova :\llllcments ~re com:cl? a, L,2and3

b. I and 2 c. I and 3 d. h nd .>

6S. lfT:R:-'tR iK • lmc:or mnp for which T( l, 1)=3 and T(O. I )= -2. lhen T(a, h) iJi ettual to •, So-1.b b. 2a- 5b c. 2o + Sb

d. 5o -'" 2b 69. TI1e ronk. of tho linear ~n$lbntlrtlion T:R3

- > R" defined by T(x:. y; z) = (y, t), 7.)ls a, 3 h 2 c. I d. u

70. lf P,t(R) L< ll!c set nf pol~•oom.ia.b •1f order u over R. then motcl1 lirt l •(itlt li>t II and select the correct answer using. the codes given l>elow the lists: List -l (Linear transform ations)

A, T;lt'~ R' "I'( I\. y. z)={s + y. y~<!:l A. 1':R3->P~R) r(a,

b,e)-{o+b) 1 (b+c)x• i!lt~ C. I':Pl(R)--> l'1(R) 'l'( p(:t))'>~p(ll)

D. f :R1--> R3 T(x. yl=(x.x+y,yl List n (~·ratrict'S)

I. (i, :J 2 [~~::]

0 I ' I

u {1 J

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n.

73

74.

75.

• .>. [~ I til)

·L [~ t iJ u

Code.•: A B c D

... l ~ l 3 b. 2 3 4 ~. 3 4· l d. 3 1 4 2

If A=G t a=r~ 9 "

~]. ~len in order th~l AB=O • the \'alues of x and y will b~ respectively ll.. 6ttnd - l b. Good 1 c. 6 nod -3 d. -5 ond - 14 If A i• a 2 '2 non-singular •qwtr<: .mo!rix. then Adj(adj. A) i• n, A1

b A ~. A''

d. Non~ of Ut" abuvl:

(f A= l·: ~ ~ ~] .then del (A'1) is equal 0 J J 0 I ~ I 6

to u. 2 b. liZ c. -112 'L -2

~~y~tl~: tlu~ ~.~"i::;r~:~~~:; ;.oo~ Qf

r,~ I cu

b. "'!':

If A- I 1 J ~nJ D - od.i A, t.hctl the [

I 11 l - 1 3

vnlue o£ B"'( t.he elemcnlm 13 belonging to second row and 3'~ column) is a 4 b 5

c. - 7 d. - 4

76. ·n,e system of et1untion~ :;;- y +3z= 4 x +z =2 x-y - z- 0 h~~£

a·. • unique sulution b. {inilcly nrany solutiOns

e. iulinitcl.\1 m>n}' SI)!U(iuf\.•

d. no soMion

77. ll' [ (x)'-2'< + 1 , ihcn 1(.!.1 ;~ ·'·

I a. --l.r+ I

b. j --tJ

' c. I

I

d. l.t ..

8 All Ill

78. I ~t~hm ~ --2

l.hc'tl/(><) i•

~. eontinuous nl .!t• ~t b. nol dcimcd ot x •' c. d.isoo·nt.inuous ::Jt x ~:.

d. continuom for all.Jt, 0 -s: x c 1 79. ConsidcT tltu following conditions·

1. / (x) i.s wclldcr.ncd al x - n 2 tunf(x) must =ist .... 3, f (x) i.s continuouo

4. [ lxl " 0 ot x =• Which Q( tl>e<e condit.lon< are (let:ess•ry for a functlonj{.!t) to be derivable at 3 point x a oflts domoln ? a. land 2 b. 3 nnd4 o. L2nnd 3 d. I, 2 and~

So. [f ~. ('- 12. 01cn tbe value uf ~ at(2 . 3)

' 1S

a. - -q'-'_-'~"'"'"'1 :.9"' <I TJ~(\4

b. 9 -!<If 7lil II ... los;&.l

li10J1V c. - 1+1<'!8

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d. l-IOlt9 .!+ &o.(!:S

81. 1\latch li61 l wilh l i~l n und select (he ~Qrrecl • tlli\\IC1' ~ing Ute ll()des given below lite ~18:

83.

84.

~5.

Li~L -J fl'uncli<m) A. ,r: .. B.-

' c. 2"

I D . .5 List -11 (Detivath•el I, 2•l Jog 2. 2.-c

2. 1...!. - - .. J

1

3 . ~-· -· ) •'

4. ,r.

* Codes. ,\ B

n. 4 2 b. 3 4 .:, I 3 ~~ 4 3

C D 3 I 1 2 2 4 I 2

[f ltxl ~ log. ( log " ), !ben t'(e) is equal lo

•- l

b. 0

• 1 cl. 0

; ~•' .J:;)ls .:qu•l ill

n. J>Jl 2r. ~

b. sio 2..[;

c. <n~Z£ 11:"

d. c;;JS 2f.

ThL! deriv•tiv~

respect IJ'J :< is n. 0 b. 112

<1. 2

of I 1'1n ,f<l\.'QII l'J

lllrt ·I "'"';r-"Jtn l' W iUt

C'~l1!!ider 1he following ~lntemeptS :

87.

89.

IJ o!JO 1. Rolle's llu:orem ensures lhtit there is a

point on (he cwvc. Lite t.1ngcnl at which ;,; parallel to the ;oc ax is.

2. L•grangc·, mean vlllue lhcoro."m c:rulwes lhul then! iH • point on ihe cur\le , the l~ngent ni which i• parnllel l.ll llte )' a!rt~.

3. Cauchv's mc•n value l.b.:orem can lx:­dcduc~ from Logrongc'8 mean \':tlue thC!lrem

4, Rolle's mean ' 'alae theorem con bo deduced from J~,grange '• mean value lhcl)rcm

\Vbich or lite "bll\ 'c Mtalcnlcnt(s) iS/Ilre co m.'ct? a. I and 4 b. 2 an.J4. c. I ~ion« d. I, 3 :~nd -1-The v alue ol' .fl •in x - cos x wi ll he great ... wh.:n IC iB equal I() a. fl

b. •

d . .2 • .,f..t) n'; ~1: -Lr z = ton A , ~•en ,... • ....,. ,. "' i'r

~. (I

"· ta{~) ¢. )(}'

d. xV llte derivative or

" · '/1 - r IS

a. I

Jt ,.>

b. $ln-1 x c c--0.'\-1 X

d. I If x'- 4x'- 2x-7.,. o(x·l)'+b(x·l J'+c[x· L)'-td(x-J )'~e(.x-1)'+1: tlten Ute voluts of a, b,oo, d , " and r wiil be r<>speeL.iveJy a. I. 5, U,20 18 and 0 b. l. 5. 14. 22. liJ ""d 0 c:. 1, 24, 12, 20, l S.nd 1 d. I, 14 , 22, 5. 11) ond 1)

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9()

93.

The pmieulnr intcsrul11f (D~+ 11 y : ,."'­ls

e n. -(Sx - 4)

2$ l

~ b.

25 (4x- 51

~ c. -14~- S) 2) e•

d. - (5x-4l 2..5

' lbo curve y - x3 3x' Ox ~ 9 bas " point of infl~mn t1l a. x = -l b. x = 1 c.. x = - 2 d. x - 2 Mllt~h list l witb ll and select the ~orr<:et ans" a ming the codes gn·cn below tho li•t.: I ist-1 ((.'u1ves) A. Cnb1cal parabola B. Catenary C. Astl'oid D, Serni-"ob1c;ol p•raboln u.t n (EquatioQS) I. y=., cos h (Xi c)

2. vl • a.xl • I 3. -y s. IIX'

.4 . ~!IJ - y 'l>J = n!ll

Codes

b.

A 3 3 2 ~

B 4 .1

1

c I .1 ~ 1

D 2 2 l 3

lf [x] denolos the grool""t intcg .. T NOT l .

~re:tier than 't. tlten ~·l<b os equal to •

•• 0 b. 1 c. 2 d. 4 J •1• equals

(.• I l)J (• I II

a. um'1(x •2)111 • c b. lou '(:~ 1)

111 • c

"- 2 tJm'1(lc t 2)1'1 ~ c

d. 'l ion '1(x • 1)'11~c

95.

96.

97.

98.

9\),

Tite 10 oJ 10

of .huo j-1- -t.....!...., - 1- + .... +..!..) is cquollo ·~ ll t l IH! tJt- 3 )u

a. l.ogc:2 b. Jog~~:3

c. log,!. l

d. log, .!. l

• d. •ff(m •Yl' 0

The Utt: boundod by the curve x ~ u CO'> 1 1 .)' =• $ln3

t equal• u. ...!..n.:t!

l

b ..!.-x .. l • 3 •

3 ! c. - na •I

d. l n11! g

The pcrjntoter nf the cardioid r - ·~ I • t!<l• ~)is a, 6n b. 4a ~ ~a

d l2a

tr ou= ./iii- [,I. v .. = .r,;;-;t n', then

a. ~tr11 convery;es bul iu .. d1veraes -· _, b. i u, diverges \lUI £,,, convcrgC'I

... r-1

c. !:u~ tand ~"" botb converge .... .,.. d. tu, and i •.., both diverge _, ""''

I (1(1, Which one or U1c l'ollo11 ing l.<lSlll dnos NOT givc:.ab$olutc C<onVfl1'!,"'11tt: uf J seo'ies ? a, Rool test l1. ComJ130sOn le>1

c-. Killio t<st d. LeibnitZ lOlii

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