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c ,S.t;.p., 2D07 I ol ICI

MATHEMATICS r J Let X = l u • n IS a posili1·e mteger. n. s 50},

If A - ln e :< ,n is evcrnf and. B= fue X:n "" mutuploof7J , then what is the number of elements in lhe smallest subset of X cQntntning both A and B? ~ 28 b. 2!1

c. 32 d. 35

2. Let A = [te N.Il and l are relatn•el) pcimel and B= 1t.: N. t s 241. Wtmtis lhe number of elements in A("' B 'I a. [() b. H c. 7 cL 4

3 t.et 7, cos(:r1 8) +isinf1t/ ~J and

A= rz" . u e NJ which 011e of the follo\\1liJ; is correct n. A is 1101 a linim set b. A contams 12 non-real comple'

numbers o. The number of elements in A 1S 16

d. A contains no mlegers -1 Which ooe of the follou ing is correct? The

eqn~tiQn '' 221ox 1 (2Ml' ~ >{261~1 ~ u

u Has no mull1ple roots. b, Has e'actJ~ one real root. c. I~ as no non-real roots, d. Has no integral roots

5. \Vl\at is the sum of the roots of the equnnon

((•-2)" H}(tN-3f +~) >=l)'/

n. 5 b. JO

0. l3 d. 18

6 Let rn be ll positive integer, m ~l. If ~t, ,~, ... _<.t,. 11re the roots of the equation x•-1 =\1. then what is the equation whose

roots are ~' ~u.,+ •>1 +,.·H<. -(m - t)<.t1 lit '='tt,; 1"% +-..--+ n ... - (m- I)txl.;

(3,.: 1t1 - +~1 +O.j,1 -f j r4m - (111 - l)t.t-1! ll.:,t1 ~ ,£<,.1 - tm - 1}~,1

3. x~11 + m111 =- H b. s"- ( ""'1)'" = II

c. x• +(m - l)'" = II

d, S-111 -(m - 1) 81 = 0

7 u· ·«di.y are tl1e roots of tl1e equation -"'- p~ 1 + qx - r = tt tlJon 11'h!U is lhc vnlue of I a'P? a. pq + 3r b. pq + r c, pq - 3r d. q2/r

R, Let G be !U1 infinite cyclic group· and H is its subgroup. Which one of the fol lowing IS

correcrl a. JJ •s not necessnnly cycho. b H is finite. c. H IS infintle d, H IS not necessaril) nbelr:m

9 l,et G ; f e} be a group w1lh no subgroup other lhenle~ and G T11en 11bieb one of lhe following fs correcfl a. GIS an mfinne cyclrc group. b. G is a fmile cyclic group, c. 0 1s an abelian non-cyclic grou~,>.

d. G is neither abelian nor cyclic. 1 (I Which one of tl1~ following group is <-'Yclic'l

3. Zu "z,, b. z,, - z., c. z. • z23 • Zo d. z"' ~ z1, • z.;,

II Which one of tl1e following is a group'/ a. tN. •)11hereacOb = aforaUo.be N b (Z, *)where a.• b ; n- bJ'oraiJ n.b.:Z c. (Q, •)IIherea•t> = abn forall o.beQ ct. ( R. •) where a + b = :1 .,. b + I for all

u.be R.

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11. ~umitler the group (R• R• t, where< R*=R-10} illld ro. b) • (c,d)"(ac,bc-d) \Vhnt an. th• identity element nnd the in,,._,., of (n, b) resp<-ctively'l a. (l.O) nnd(a 1.bn 1l b. (0. J) rutd (a1• ba 11 c. (0. l) nnd (o1

• • bn·t) d (1 . U) ond (al. · b"'1)

l3. Whlch one of ~1.: follQIIlng. •Uitetnenls is euttet:t'! a.. AbclU.n groups _may havt.: uou-~abc-linn

subgroups.

b. N011-abclfan groups may have abelian subgroup~.

c. Cyclic gtQUJl" may havg non•eycJic ;,ubgJ'OUilS·

d. Noq-.:yclic gmups C311MI ho1 e C)'lllic subgroul"-

14. U:t o-=-(1,3,5.7, 11)12..J.~)E S11 Wh.tl i~ the smollec~l 11ositive integer n. •uci• that ri' =rs"? n. 3 b. 5 <:. 7

d. II 1 S. Let (R. • 1 be ao abelian group. If

JUUUiplicatiou(•) •~ dcfmcd on R by sdtins ul' 0 for all u.. b. e R. Utcn which o,ne of th~ follo11 ins .11ll!lem.:nts ;. """""''? a. (R. ·) is ltOl n ring.

b. t R.- ) is a ring. hut nut eommutativo,

.:. (R.- ·) .- a commutnlive ring, btd ha~ no unity ..

d. (R, ~.·) i• nola field.

16. Con&ider the fuUowiug assertion.• 11tc cb•raot.cdstic of th.~ ring (Z. • ~) i.• z.cro.-

jj. F<lf' every cotnll<~itc 11umbar, n, Zn, tl•~ ring of rcitduc: olossc:s ntoilulo n, i.• • field.

iii, Z5. the ring of Te$idue cl .. ~s~ moduln 5, is •n mt<:gr.tl domoin.

il•. ! he ring of all complt~ numbers is :l

t:iehl. Which ofUtt> bOO\'e ll$St:rtiQI1~ nrc ~ottoet7

a. i. iii •nd iv b Lihmd iii c. i.. ii and i\•

~or tn d. i. iii ~nd iv

17. Let I' be a finite field with n ~lemenlll. What iH the pnssi~lc val11e qfn'l .,, 1

b. 36 0. 37

d. 125 18. If 'R is a l'inite inl.egral domain with n

elem.:ttL then wbat i.> the nUtllber of lnwrtibel ulcm.,uts unttr nutltiplfcallon In R'l 1;11 l b .. It

c. u 1 d. (n.'2 ] whcrt: H l.1lte brnckct .llmctiou

19. If Q. R. C ore rcspoctivdy lht litlds of rational numbcf>l, rc•l numbc~ 3nd cm~plex nnrnbers, ~'"" which cme of tl)e following olg*ruic Sl{\lttur"s ill nvt ~ yeciqr sp:tc~1 n, R ClVer the field Q. h Roverthdield R, "· CJ over lh• tiold R. •I (' Qver the field C,

20. Let "' = (3. 2. · 1). y = (2, 4, 1). z = (4, \), ·~ ) ond 11 -(10. -1, ·3) be vectors in R:\. il rtill 1 cclot ~pace. \Vl•icli on<> of tho follo11 iog Is correct? o., 2't z = \v,y t z -- w b. 2x y z,. y ..,._ 2z • w

c. :t"•z • w.2."Cty z. d .. y f 2z \\T, X y z_

21. lfV is ~~~ rool vector spoc.: of ~u m>ppings from R to R, ~:(f~VIrt·x):f(.'<)l and V,=-{f« V I f.( :<)-fl~)}. lhon which one of lh~ following i.s Cf)m:cl? IS. Nelther V 1 God V, h a Sltbsapce of V. b. V 1 i.• n ~Uhl!J>acc of V. hut. V: ;., not G

Rul:>,apace ofV. c. V1 '"not o subspace of V. hut \11 ;, ,,

suln pact: or v. d. Botl• Vt and v: ace subs5t>ces .,rv.

22. l.el F[x I he the ring or pOI)'nurnial. m One> variable J( 11\'Cl' O l'i'c:Jd F wilh Ule rclali<Jn x' - (), for o fixed n EN Whnl i• th e ~imension of F[x] over i''' a. I b n - I c. n d. inllrlite

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23. Which nne of the fo llowing i• co•1'ect 7 l'he st:t S = {a ~ ih, c + ul} L• o basis tb•· the ve~tJJupaoc c nvcr R to' a. ad - be ~O

b. ad + be - 0 <:.. od -+ be ~ 0 <L od - b.-: = \)

'24. Let V be the vecto•· sp•ce ol' all 2 2 mntriccs nva• the iJtld It of real number<

J t ~] and BL0 3

. If v -• v il /l l ineor

t rJn.~ro.rmalion defined by T(t\l = AB - BA. thC!I whot is the dimension of the kernel of 1'? n. I b. 2

d. 4 2 5. \\lhat ls the ronl;- of the line.r tr.m~fom1oUon

1 R1 -->R d~lined by T(x,y,?.)=(y iJ 2)'1

Il. ~

b, 2

c. 1 d. ()

29. Comider Lhe vcctor•pnce C uwr R und lot T C C' be n linenr trrtru~fnnnrotion given by 1'(z) = 1 . fhen which one 1.1f the folluwing is cotre<:t? a. 'T is one-nne. but not onto. b. 'T is onto. but no I ono><Jno. c. ·r is one-one as well ns onto. tl T is neitlter one-one nor onto.

27. J( r is • littear· tran.•foonalion frolll • real . ' veclor· space R • to • roal w clor •puce R· J<ucb llul 1'(x, y) = (x - y. y - x, • -l<). then what is the nullity qf1'1 ... 0 b. I

c. 2 d. 3

If n '"

A -[•oo9 - - sm 9

'I

•• l~ne &in rta] smnO cosnO

[-oosl14l sinnl'l] h. ''" ne coone

1!. IcOllne >in ne ] !do nil - co:;n&

d. [ Ct>;n6 Sin n!l] -~•n n!l cosn!.t

3 o! 10

29. If A ond B an: •rmm~tric ,nntrices of the ~:orne order, '""" whrch """or''"' fi>llowing ~~ llOL"'Jrrt:Ct 1 a. A -+ B i~ a symmetric matrix. h. AB-BA is 11 symmclnc maui,~

c. AB + BA is a symmctrrc mellix.

d. A - A7 and B +- aT ore symmetri" moLriccs.

30. If A=[! =~1 $•lisGcs ll.>el]latr~ equation

A' - KA + 21 = 0, I hen what is I he vali•e of ld ~. 0

IJ. I c. 2 d. 3

;:t I. Wh~l is lite Vfilue of the detcnninonl D...~-c b' rc=

c+ ~ c' 1-11 'I ~-~. .1 + l>'

a. (a - b)(b - c)(" - 3) b. (a -t b)( b + c)(l! ~ a)

c. abc d. 11' b <c

32. Under wltieh " "" of llu: Jbllowiug

3~.

""rr~li~ dor~l~ili:lsys:,::c of.~:::: I J u - ~ % ll

wlulion'l

•· For all u " R b. •= s c. FQrall n ~oZ.

d. ;j = 8 Consider the equations 2x t 2y = 1 :tnd :!x - v = 1 over ZJ. \Vl •al is lhc solution or (x. y)7 o. ( I. l) bulnOI (2. 0)

b. (2.0) hU11JIJI ( I, I)

c. BoU1 ( I, I) and (2, 0)

d. (1'2, 0)

Wltich l>nc ul' the r<> lluwinl!. is eorrcct? For difli::t-.:ol 1'31Ut8 or n Ull U b, U1c ~traighl line

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given by xu - 2b) Yl • - 3h) = :o b p:tt<se.• lhmugh

u. A coojug~e poinl

b. A Ji.xed JlOIOl

c. The origin

d None oflhese 31. The line 3:< - 2y = 2.1 meets the y ... ,xl~ al A

and U1e x-a:<:is ~~ B. a11d perpend1oular biscclor uf AB meets iloo tine tluough (0. •I) parol lei to !be ;x-axis al C.. What is lhe are• of the triangle AB<:"I a. 91 SCI unit b. ~ 1 SCI unit c. 61 sc1 unit

d. 4 htt unit

36, Consider the t'ollowing-stalemenos s, ; The equation

37.

38.

39.

._,..•- 2h•y- by'-~X -lt}· - r. =II repr<:!'tTlUi

• p.lJI' or strajght hne.

S:; Thee<JUalion u "-t2hxy .,. b.Y' =I) .ll\\ay3 represenls a pair of •ltaigln lines p:~~~•ing lhroug.h lhe origin,

Which one oft he fo llowing is correct'!

•· If s, is true. :;, Is always true. b. If S1 is nol lnw. then Sz 1s also nourue.

~;t.. S, i< • lw•:rll lnte •nd 5 1 implies S, if c = 0.

d. BoUJ S1 arid :>: unply e;1cb oibet. \VItal is ibe anglco between tl1e two tangent~ drawn from ( L 0) to tl1c eurvo: y!- 4:t OJ

3. 30° b. 4'" ·)

c. 60•

d. <)0•

If~ cu·de ami lhe reewugubr hypel'bola xy c1 meet in f-our poi.nt8 lt, 1!!- t3 and 4. th..:u

what ;, t1t 1(3t.. equal to? :1 I b. - J "- 2 d. 2 lfP. Q. A. B lire\ I, 2, 5), (·2. 1. 3).(4, -1. 2), (2. I, -~I respectively. then what ill lhe projt:<.1loo o( PQ on ..\131 a. 3 b. 712 c. 4-ll 9/2

~ nl ltJ 40. W1Jal 1~ the C<JU>tion o( the pfune

whidt bist:c!S ihe line juining the point• (3. -2. l ) oud ( l. ~-. ·3)nl rigbl angl""? ~· J<- 3y I 2.z t- 3 • 0

b. 3x- 2y z • 3 • U c. x .,. 4y- 3z '2~ 0

d. x - 3y t- 2.z + 2 = 1)

~1. Whal is 1!1" equation of the pl•ue \vluch passes Uu ough the z-~xis and is perpendicular to the lone J<-• =Y42=z-3 7 ehllll sme tl a. ~+ytanl:l=H

b. :f~ XIJlnij "'ll

C. XCI>SO · ~~iol9:cl

d. :<smll-vcos&="

~2. A s1tllighl line I. on the XY·plAnc hl~cctl! the ~nSf.: botw·¢en OX und OY Wh•l on:. lho tliro!liun cc~in~ ofl ,1

a, (1 .P,,1 ../2.o) b. (m . ../3' 1 o) c. (itU.I)

d. (~ ' 3,:! • 3..1 n) 43, Wbol. i, the cqu.~tion of lite cone with vertex

at origin and pMstng througl1 the ctrole " '+y' =4.~=21

~~-

4 5.

11, x' y' I z' - -1 b. x',-y' -7:' -U

~- ~t't-r-r-l

o ..•. ~r '2' :~

ff ~.b,; arc rwn-ze:m \c::d.ur$ ~uth. Uukl

{; ~~ ;.;,fb ·~ Uum wruch one of !he­following is correct'/

a. a and h ~re colline:tr

b. a and c nte collinear. - -c. b and c nre collino:.'lr.

d. Non.e t>f the<c:. Con•idor lh.c rollo" lng slotomonts

S 1: ~. b. o Qrc non·Zet'O, non ooplllllllr vectors~

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~ " b c' =- . - . are non coplonar. Jnbcl

Which one of th" fo llowins i~ L'Orrcct?

u. S 1 implies S1 but~ docs nOl imply S t,

I), s , doe;; not im11ly St hut S1 implits s,. .::. S, implies S, and lh impliCll S, , d. 5 1 docs Ml imt!ly St and S1 dnos !lUI

imply s., ~6. \Vhnt is the volume of the lctrahetlron witlt

vurticcsat(O, U, O).(L 1.1),!2 Ll):tnd (L 2, I)? • . 1/6 b, 113

c. 1:

47. I f r ••ti;>lies the equ;lli~n

f ·li~2j+L)=i-t<. then for any !.Clllar m.

what is ; "ll""j to• a. i-m(i-2)+ k) h. j 1 m(1.zj' k) .:. k · mti +l]-k) d. i- k+m(lT 2j+-k)

48. For the triangle OB('. one '""""" 0 i• the origin l\nd the position vector.. of Ute oUt<r

vertices B ~nd C ;tre b •nJ ;;- n:spectivel} and o, 1!, c ara the lengths of the-side.\ BC. OB. lnd OC n:spcclivcly, Wbnl is lh" po• ition Vll\:_lor nf Ute inc<ntr.: of Ute trbngl~

OBC'I•e::J[

c. cb+bc ~~b-e

~ -bb-cc d. a-b-<

4?. Wbot i< the rhJil!c ol tbo functiOtl f(x)=IOl;.ji<~n~t-eou.+ M l t ~~?

a. J I. 2]

b, JO. II

50.

51.

s Ill Ill c. ( 1, 2)

d. (0. I)

If lim(x 3~'u x - >t lcsinhx) cxisis then ,...,.. -~ -~~~ t x;r -1~1

what i~ the '"'lne of k'/ •. - 1 h. 2 c. 3 d. d

i stn(a + 2)x-'- <tn x} "· X 4_ ()

lf f(x)= b, J<:O is

{!><-Jx Y''-~" ~~".± J

" > ()

COntinu(ttJS at .~ ~ 0. th~n what a.-.: the 1'3lllc:!

of" und h rt$t>ccti\lely? 0. - J. -I b. 1.-1

~. 2. I

d. - 2. l 32. Lei I(x) • x1l!j for x.flx) i> lliffetenliable at

tho origin if u is e<1ulll to wltich od~ t>f the following'/

53.

~- 1 b. 0 c.. anyrc-.tl number d. any posili\rc integer \Vbnl J!l the mu..Unum y-;.<Wt 1

X\jOS-;t.0<"X ,._ 1f'l

•• -J./5 I to

b. 3./3 I ~

c. -3111>

~. J../3 I I ~

v~luo of

54. 1\ latch List - I w ilh l..i~t-11 and $elect the ~OtToct unswcr u.s ing the code gjvon below tbi: lisl•

Li•tl

A. lltc function x• 6x1 36.< .j '7

in~'tC:l$c when B. Tite .function •' 6xJ. 36s 7 ••

ma;timumnl

c. 'fbe function X 6x1 l6x 7 1.. minimum :n

D. Tho l'unc.lioo " 6x 36:< 1

1iill! I. X- ·2 2. x ~ 6

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55.

3. >< < -lorx " 6 ... - 2 c; ~ 6

Codcsi .\ B c D

b. 4 2 ] 3 c. 3 1 2 4 d. 3 2 1 4

c. 4 1 ? 3 (f .t. 2b 0 0. ih<'ll tho equat.lon 3ax2 2bx ~ c = 0 has at least on" reo I root lyiog t...aween wh.ich oflhefollowiJog'l a I) ond I

!), I and 2 c. 0 lfnd 2

d. none of these. 36. Under which one of the fo llowing

c:ondit.lon:s does tlo~ fuu.:tiuu {(x) & l(x')m •in(lfln1, >< ~ !l.n >0 n11d 110) ~ O. hnvc • dum 'II live nl >< \1'/ :a.. tn 2- l ' ~

b. m 0 c. m 1

:

d~ m? l ~

57. tfthc kngUo to the cm" c f(x ) , x= any point {c. f(~)} i~ pMnllelt~ Uoo j,;inins the pvinlli {n. fta)} and lb. Jtb)t <m the cuTVe, then \\ hi~h Or1c ufthe ~IIC>Wing i• coow.:t? u. "' c, b are in AP b. a, c. b ure in GP e. ~- c. h andn HP d. a, c. b do oot follow delintlc ~"'~oenoo.

Sl). Whnt •~ the mnxionurn ~rca 11f the recllln[l,le '' h(l$e siJc$ pass fluvugb llu> aogubr poinJS nfa given rectangle of sld&~ ·a· o.nd ·b • ?

• • (a I b)012

b. (a • b)1

e. (a~ b1)/2 d. (u' • b1

)

59. \Vhnt is tbc ubscis!fll of lbc point • l wWclt lbe lllngcnl LO UJo curve y - ,} is parnllel to the chord joining the e.weiJlities of the curve in Ute int.:rvnl 10. II? a. .... b ill ll'e) ~. ln (e - l) d. I 'e

60 Whnl is th~ subncmnol ~I X= ll o 'Z on Ooe c.tUVt2 )' =- X ~tin X ?

!loll Ill

"· I b. ! ll

c. ll l u. 2

61. \VWcb one of lhe following is cortccl 'I The inclined asymptotes of the curve x' - ><y· - :l.'<Y - lx- y =II are t.hemselv.,. o. perpendicuJ:tr b. pnmllel

~- ln~lined .ot on .ltl&le n ' J . d. lndincd a l J1J1 nnglc n I 4

62 Which nne of the lollm1 inS pertaining In the f•ngenl •I any point oon th ecuTVc xu" +yu.s ::q~0 is CO"["RCt?

•· •um of ils inlertcpl<l mode with the coordinate h;(es is e<m$1~n~

h. It cn~l""es a triangle uf con<t:ml .oreo "'i lh lh~ coord in~R: a.•es.

c. l.<mgtl1 of jlS p01tion mtereepted bclwc~n tbc. coordinate u.xC>~ .is consilio!.

J . It alway$ passes throujih the orjgin. 6~. \Vhol U. the least ob;olut~ l"dlue o( tloe mdius

ofuxplonaLion cUC\•aturc for U1e curw y ln s'l a. J./i l> ' ./3 -.. c. ./i.fi d. :Nit 1

04. Whull.\. the v:tlue of f'" ~sm " +l"''S'I dx? o f :unx.roosx

n~ o b. it' ..

c. 411

d. 211

65. 111e maximum value of tls}. where

f("-1 "' J' son {J<O-x)}<l.~ OCCU!l! nl 11 W~b on~

oftJoe followin& points? a. x = 0 b. 'I = I C. X =-1 d. none of these

(16, What iR the volume of~lllid gcner.uec!, when lbe area of lhe ellip5e (x1 111) ... (y·1 4)~1

(in th~ ll•·d o.tuadr•ot) i~ l"loh•lld about y­nxis? :r. 1671

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b, 12 n

c. Sn d. Gn

o7. Jr j'xull-~)'ds: f' .. · r l - xl'$ then whot • ••

68.

i& p equn l to?

a. 2n b. Ill

C.. Ol -i tt

d. ~DIIn

Wh~t 1S 1!1~ area of the region bound~ b~ the curve 2y :,~ -3~- l y' •nd !he l!·n.'l.is'J

a. 12514$ ~q uph

b. 4 ~'I unit c.. 3 ~Ct unit d.. 1 2~/U sq un11

. x' - y• G9. \Vha11s the~·alneor hm--1

.-yJt"-y'

7(l.

,, h ln v 1 - In'"

b 1- lny

I r lny

"--!+loy l+lny

d. 1 · loy I lny

X~ .s3n X COS.X

rf fC~<)- (I - 1 u where I' i~ ~

p p· p·

constaut.. Ute:n '' bal i5 Ute value of ~ {f(x))

nt x : O']

~. p b. p • p' c. p - p·' d. [ndepcndent of p

71. Whot ore Ute order and cl"!9"'e respective!) ol' the dilferentiaJ l!CjUOttOn nf the f.1milv of t'urve~ }~:Jcb<+ ./C}. where c ;,·an ;trhitr:u-,· conslanrl n. I, 1 b 1, 2 c. I. 3 d. '2, I

7 ul Ill 72. \VJ1ot is the oofutiou o( lhe

73.

d-)' dv ditlerentiat eqnotion - . -2- · ""~Y "'"· d,· ~~. .. with tho< givc11 conditimu ) (tl) - 0 ond y '(O)= 1?

11. V= i! • CmiX

b~ y=e~" !ttn ~

c. y:(oosx- •m .'<)~ ..

d. y :Sfn :<

Wh31 iii lhc ~olution of

equation (1• e'")dx•e"'(l

•• X+ye~"' =~

h. Y t-Xe:tJ' =c

1!. X - "JI!"fi'f -C

d. n<Jn~ 11fthesc

Ute differential

~\)yell " ,.

74, 1'hc smgulor $(llut1on or lhe dinerenlial cquatioll y pll + f1p) will bo QblJJined b)• elirniunting p between !he equation ) : P.'~ ~ l' p) and wl!ich one of the 1911nwing "'l'Ull.•on~ 1

df tt. Jt+--11

dp

dv elf b. - · - •t -

dp dp

c. dy - p d.' Jy <If

d. - - p f -d.~ ~~·

15. Consider 1he ·loUowiuj! •Lotem~lliS m~ ••espocl dv of ll1c difT.,rt:ntin1 cquotiOo lX\' - : V X

- tl't -

iJi

i. 'l'he difl'ermtia1 eqoation is a homogeneous equation

ii. The .wrve repNsented by ~1e differential tquntion is a f.'lnlily of circles.

liJ. Tb~ dilTcrcntU.J cquauoo of iJ..;

oribogonal tr'!]ectnri .. is ely -~ d.• x' y '

\VJtkb lli!C Of the fl) ltOWiiJ& is C(IITCCI1

a. i and ii only b. i and iii only c. it and ili only d. i.lJ and tii

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76. Whol arc the urthogonal trajectories or d•<:

•Y~rem 1•l'~un·c:s (dr)L -~'' dx x

u. 9n~ cl ± :b<Vl b. 9a(y c)! ~ 2x?JJ

c. 9n()' "i ~.'C2

d. 9~()' ~)1 '" 4..c1

77. From • squa(~ lnmina ubcd sbos~ cliagou~ ls meet nl 0. the triangle AOB is cul and the l'cmaiJiins p:>rt is hung up ut D. In lh<;: fl<>Sitinn nr equilibrium. how much Jngle doC$ DC m•J...c \\ illl Ute Vc11ioal?

7ll.

~- tlln (7 9)

h~ tun-1{ :' IJ)

"- ~s·

d. :w

_L A pillar UD IS lo be

pulled down by tying a rope of leugth I = AB to some point D of Ut¢ pllliot ond then polling tho rope with a force F ~• •hown on the abnve figu"'- F will have ma.'<m>um momenl aboul 0 wb<n 0 8 equ•l• 10 which one or th.c fallowing 1

tL .J3t h. 11 ./2

"· J3J d. 1! ..{3

79. A D A force F. having magnitude of 10 dyne. is applied 1111 lhc eonlor C of a n:c~Jutgulnr p!Jtc .<\BCD. M

<h 0 \\11 in tl1c nhovc iigurc. If AB - 8 em,

AD = 12 Clll. tlten whnl is the moment of f about N!

a, '21)(-2 + •..fil I ll Jllm

b. ~()( 1 I 3..{3) • 10" Nm

c. !0(2 - !./i1 Il l ' '1/m

d, ~0(2 3./31 10' 1 Nm

8 ul Ill SCJ. A heavy svhericsl bnll .. r weight \\

i~ on • smnollt mclittcd plapc ( u. = onglc u( in~linntlon of Ute plan< lo l11c borlzonlllf). A tim:u of m"gnilude P is applied Uttuuglt lhc ce-ntre qf lhe boll in otder to· m•inloin the h:oll ~t rest What iK I he vnlut; or P'l

a. P =W./1 I - •u b. l'= Wco.~

~. P = W stn!X

n. P~ w.Jrlt-.-,.-. -!.l

ltl. Thu " ci~hl of~ triooguiM lnmln• ABC i.s 9 g Wh4l i$ O>e a<)dilional woighl IQ be placed >l A s<.> lh,,l U\e new centre or grovily di\llde. Ut~ median through A in tile mtlo 3 : 47 •. '4 b. 3g 1!, ~&

d. S& 82 Two ~phercs of radti 6 ~m. 3 em •ro finnly

uniled. 11•o two ~beo an: tolid 3nd of llle same mnt,.,.;al. What i;. the distnnce of the centtc or ~·vily of tl1e wlwl• body lrom lhe centre nf 1l1c la•·ser ~'J'lhcu;7 a, I em b. 1cou t:.. 3 ~tu

d. -1 ·~n 83. U th.<>angl~ of friction IS :t. lhcn who I is Uie

ycntcsl hcigbl nl wlticll 11 parl.i~le iUIQ test inside • hollow sphereofrndi\J$ a7 ::t. .t.si'n ).

b. o(l cos1..)

c. ~ trul?..

d. •0 - liiJl.).l

:~~u:· !14. .~ B 1\w poinl$ A and B

hnve velocitiel< u 1 nod u1 ns shc111n ill !he figu~ obl)ve, lf AB = d, whal i~ U1c 11ngular v~locity of A n:JatiVQ 111 1:1'1 3. ( u1 <'1,>1l.1 - u,c<>:;~L )ld

b. tu1co•!.l ; u •O><I.,)I d

c. ('u,iln~1 - u siuu ) / d

(1, t u1• inu,.,_ '"" ' m") d

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SS. 1'wo p;~rticleo ure projected vcrGca ll~ UpWaTil< Jrum a pfaee :ll an tnlC!Vaf of 2 socottds. If Utt .Gn<t and tho soo()nd p•1ticl~ attain the n:spcctivc grcntcst hoigltl> H1 and n situultaooousty, tblltl whkb ouu of tltu foUowing ;. con:cct ?

a. .pi; =(.jt{. ,fi8) h. .foL-t./H I ./2i) e; ~JI, m~- :.zs d. JH I Ii -l

86. A rarticte or unit mass 1S U1)05tmined to move in a smooth clrcul3.r pAth ul' ,ndim> a witlt ronsi.Ont spoe<l. lf now an additional radiol fore<!- ttf magnitude P octs on the partido, h,ow does the lcinelio enerl!Y (E) o( the pllttlcle cltang~7 11. I~ chong"" by Pa./2

h. F. oh~ngC'I by .f"1Pa. c. E ohnngcs by Pd 14

d. E cltaoge8 b)' 2P Ct

87. A ~month bcU\'Y bead moves :tlong ~ wil'O. which is bent 'a• • circl.r of radius (X in a verticlll pl3nt:. 'l'he ~!<lad ~totll\ l'rom the re:<l positioo '' here I he radi~ t<1 it mak<8 "" unj!lc of 60" wiUt the vertical. Whol ;. tho 'cloeil) nf the bend when it rcoches the lowe:sr t><linl ( the wirei.s fi,xocl •n •r•<:<:l'1 ~. J>sa b. J'4."J. c. 1,J;ci d. ~5gf1.

8$. A particle is projected wiUt 'elocity v at~~~ angle ( ~S"l to tho ltorizoolltl :tnil !Clldu.:.; u point on the borizonllll dlslllnl R fmm the poml of 1>rojection. Whllt L' U1o greatest hctgbt (h) all~ inctl tlunug ~10 paih uf I he pfojtctilu 7

v•r rg'R'J]) • . h = - 1- ., 1 --lg ~ ' v•

b. h= ~[1+ ~ ~~-(~JIJ

~. h= ~~ l~ -~ i ~ -(lfJl)

89.

~0.

~1.

92

9nl lll

A paniole Is executing simple ham.onic motion and its tli.splncetnent Irotn i.ts mean position IS given by .,.'< ~cas{ru --.k).. where

l denotes the Urn.: and n, n. k at'(< pasili"" con•tants. Under w1hat condilion will the speed oflbe por1itle be maximum~/

a. t =l2p+ l)lr :n. tl befug,an mtegor

n. t =('lp+l)ni Zn - (1; n) p 1Jein8 :m

integer

c. 1 " (2p • 1 )~ an ' ~~; a} r being :t11

integer d. t p1r/ n (k n). p bcmg :uuntcgcr

A. portkle who,e w~ight OJJ IJ1d surf•~ of the earth is \V. falls to the surfoce of tl!~ earth fmm a heigh! !Xll13llo the dinmeter 2R of tlte e.1rth. What is the work dnoe by U1e

eatth •• iltlJ'Ol<Liol1?

"' 2R\V b. 2\V ~ c. 4R\VI3 d. 3R\V '2 A floppy wi11t 1.44 !\JB ozapacil~ l!oltl ~tor6 lhc ntformntion equivalont to which 000 of the followin!!'l

"' 1.4-1 2" byw; h. 1.-1.1. 2 10 byl~:~< c. 1.-14 21nbytes

a. I .'14 I 0~ b~1'"' Under wh~l. ~onditiOt\$ of the Inputs " and H. will the ou1p111 in I he gates for operntloru OR And XOR IJCdiiT~-rmt? ""A = l , B= u b. A ~ o.n : 1 c. A =0. B = II d. A = l.B = l Step 1: get t\.. H Comment · t\(t, J' and iJl~)) ~rem n ;ond

n pmatrice~ Sl~p l: l lo m Do fnrj = l 10 p Oo C(t,j) ~ IJ

fDr k l lo n

Do C(t,Ji -X

s.~ fl 3: \)utput C'

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Comment ! l':f'(i,J) is the product matnx AB of th~ order m • p.

Wltnt is X in the obovc ~lgorithm'l

a. C'(i. j) I A(i, k)"B(k,J)

b. C(Lj) r A(l. k)• B(J,k J

c. A(t,k)• B(l;.j)

d, C'(t.j)~ A(i,J) •B(t.J)

94. What is tltc decimal equivalent of tile hexaclecimo I numbc:r FF ? .•. 225

b. us ~- 255 d" 256

95. WWch ouc of tlt<> cnllcd "eoincidence deteo1or" (l

•• OR gotc b. NAND !l-''" "" NOT gato

d. AND gate

96. Let n ::. 3.n be odd.

A.S>I!rtiou (A): For 01uy 1-l,!.. .. "n l if u,,c.., ... ,a ,, ""' the t'tlol.s of the cqu•tiotl x" -~- I :IJ then (I 1 «, '(l !1. ). .. (! a,, ) ~ I

Rt•asun (R.) : lf11, ,rt, ore the m1•1Jr<>i'th.: cqu:itiun.•, .,.• - '< - I :(), an<ltht'll

(l • cr,J(I • a.,} .(1 .. <>,.) = 1

u. J3olh A ""d R ore individually lruc ond It ,. lhc co.rrccl <ll<plruwtion or'A

b. Bolh ..\ ond R are individunllv lruo but R is the COTI"<cl explnnntion of A

¢. A 1s !rut hut R is f11 lse d. A is Ioise hut R i• ~rue.

97. Assertion (1\)r Then:: i• at least (me '"'clie !!fflUp 1)1' order 100 which ht~.' oniy S •ubgroup~.

R~sun {R) ~ A lit1~c cydk group of order m b11s • unique $ubgroup ofQtder n. wb;;ren is 11 diviSor or m.

10 ell Ill ~- Both A and R-are lndjvidunllv

tru« and R t~ the cotTt:et cxplr;n>tion nr A

b. Botlt A and R ar~ indhdduo lly true bul R iR t!tc ~n.,et Cl<l>lnnatiun nr A

c. A ls tru<> but R i!> f:ilie d. . U s fal!!c but R iu.ruc.

9 8. Assertion (A): Tho l\mc6on f (>.) - - "- is 1 tl xl

nm difforcntiabl" 111 s = 0, Rt•uson {R) t lsi nnd hcnQc; (J I lXI) i.~ n<ll cliffettntlablc ot 1< = 0. a. Both A nnd R nre individuolly lruc nnd

R i~ the CUITC:CI. e:<planation or A b. Hoth A and Rare mdhidlliiiiY Jru,.. bui

R i• Lite "on=! ""l>lanalion uf ~\ c. t\ is true but R IS fobc d. A ;, fill~~ but R i~ l;nle.

99. A-scrllon (A): 'rhc ruuction y : x'I-J is u

~iul!ul"r•olutiou or(dy )" _, ely T \ ~cl ~~,, d.'t

flcnson tRl : l'he .general solut ion or 1he giveu ec:1u~1iou is y """ t:..~ - e1 and th_e. given solution co.nuol be obt.ilitoo by assigning • dcfinit\!>-wlue to c m the gcncml solution.

• · BoUt /1. .end R .ere individu•lly true und R iR the ~oi1'<'Ct explonnlion or A

b. Hoth A Md R ore md"~duallv lru~ bul R '" !he .:om;ct eXplanation of A

c:. A is true bul R l~ til be

d. A is fnl!lc bLLl R is '"'""

10(). Assertion (A): r· C05 •• <d'<- ~1"' oolxdx ' ,, Rea•on (R) : 'Tho inlcgr:md i$ on oven functi<>o . "' Jloth A and R nre individually true nod

R i~ the .:otwet ~xplon11tion or A b. Bot.h A :md R ure mdividunltv true but

R r. lbc .:orrect cxplnnnlion of A tl. A i• true but -a [, fn L<c d. i; fnlsc but R is true.

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