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IAS Prelims 2009

I.

2.

J

4.

.s.

H OI\ illM) urbi tral') constants does the geneml equation of l1 quadr::nic cone \lith a gJ I'CLl conditioo havc'l a. J b. 4 c. 5 d one of the abo,·e e. b What IS the equat10n of a C} lmder \\ hose genor:llors are parallel Lo 01e O.'(is of z. and wllich pusses tluough the ci rete 11 hose ccnti e IS \IL o. O) and radJus ::1 and lvme m the ~ -plane'' · -

2 ' ( J_ X y- - ll'l.l' =: I

b x1 + yJ -Jax= O

c. r + )•2 -4a~= o

cl ;:- 1 y: - ax= 0

e. a Lei i:i =i +2j e3k . li ={ -3j n k and

c = j - k What is i orthogonal tO ti artd sarisfi

x•b=h-c~ a 3i-k b - Ji+k

d i-2j.k e, c The ve.ctor h = the sum of a._,~-

is" o be wntten us

1 arallel to ii = 1 ' J perpend1cul nr to a \\ll1at

- (i.._j)n

c. 2l(i ~ i)l l d. None of the abel\ e. e. d

Whatistl.{n {6 (ii· J)}}~{!ualt o'! a. ( a,J) [t~- tl.l7J

7

8.

b ( h J)[ ci .~:, ,7] c (!i.a)[o,ci.J] d (d .. 5Jft;.<l til e b

The position ' ettors ii. points A. R C. D respec

the

arm of n vector parallel to ii = 1i - j d :1 vector ortlwgonnl to ii is

a [(3t$<J)i -(t t.JW)J]+ [l::-{ 31-/iii)}i -{t +(11-liO)} i -Jk J

b. l (312)i- (J/ 2) i]+

[ ( t/ 2) i + (J / 2)j - 3k] c [p t4)i-- (1/4) l]+

[(514)1 +(514)j - 3k]

d. (3i - j) t {~i' ~ 1.i-3k) e b lf J (x+ l)+t(r- l) =2f (x) and j{O)=V.

then wh~t tS f(n) where ll eN ?

a nf (t)

b. [ f'( t)J c tl tl 11

e. a If f(.() = l / ( 1 -K).g (I') = /[J ( ~)J aod

1/( .~) = f[g ( ~·)] . then 1\'h:tl is

I (x)~ (x)h(x) equnl to 'I

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10.

11

l2

13.

14.

h. (l

c.. I d. 2 e. a

2n+l I ;n•l What is lhl: value of lim • 'I

U -- t zH , ~If

a. I 3 b. e. ~

·' d. '1•

<.:. c I I •

lf F(.rJ : 2 J{4rJ 2F'(')}ur . tht:n X \ ~

1~hJl is F'!4)cqunlto 'l a. 32'9 b. li-t:; c. (j.IIQ

d. tnl3 e. a

Lct /(.t) = x f ~] . .r 0. where

3.

e. b \\llal is lhe ma.~imum l'aluc of I. for \\hich (cos " + sin x) + 11-:os x 5in X) I. !> 7 a I 0 b 8 c. tO d. Nnnc of the a hove e. c

15.

16.

17.

20.

"lltc e;x:c.entricit) of the:: uClni<l

( .r -J)2 tun1 a+ /sec::1 a= l. o ,. c~ ! rr 2

as a function of n. a. is incr(;.lsing b. is dc:cre3sing c a const:tnt d. doe.< not exist e. h lf A denotes the area between the cat-.·nary

.1· ccnsh ( .r c). x xt nnd an•l

the x ·:~.otis. ~nd • stands the inlerYening arc. tb In? d I (cs) b. cs e. c~ d. •

the lo llnwing dilfcr~ntinl

represents the orlhogon:tl --v·o.r~"""" ... -$ of the f:mtily of cur~·e~ ,ory-~1·1

·c/y-y.;{y = lJ xJy • yd'l: ~ 0

c. xd~-Y«•· = ll d. xdr+ yi{t' =0 c. c Wbat is i,he solution. of Ute dilfercntinl

. dt• x(2 lnx• 1) ..:q un bon - ·- = ?

J_,· su1 y + ycos y

a. y siny = x: ln x t

b. (} ' .1•7

) COS I'= :r2 1n X ' X ( '

c. ysin l· ~ cosr = x - ( .? 2) Jc

d. :-Jone of the abo\'e e. a Consider thdollo" ing stat~ments. l. If A i• an invertible square rnatri~. then

adjiAT) - Iadj A)T. 2. u· A is au in~·.:rtible squ.•rc matri~ of

order n. then adj (adj A) -I AI..-~ A. Which of the 3bov<! statement~ i< nre currco:t'l 3 . I only 1>. 2 only c. Both 1 and 2 d. Ncill:u::r !nor 2 e, c: TI1e difierential

(y- 2x.J )cL't - .t( 1- .\)' ) c{l' ~ 0 become~ www.examrace.com

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2 1,

22.

23.

2~.

exact on muliiplic:orion with which on.: of the following'! a. x

' b. x· ' c. I (lc"l

d. l 'x e. c \\ h:lt i~ the general ~nlut inn of the ..:quation 6y"-r 5y '-Gy-.,- ~

a. y-= , 1e • ...:3c' ~ (x 6} (5 ~6)

b. .1 - .:1e" 1 r c~e 1

' ~ ( .r: 6) ( 5 ~6 )

c. y :"lcl" , c3e- 1" 1 (.r16) , (S 36)

d, v= Gje3'

1+c,c· lr l +(h 36)+(1 6)

e. b Wh.1t iR the di!l'tlralltal cqu.etion corresponding to the 13m ily of curves

' y : k{x - kt. whcu: 1.: •s an arhtlr:~ry

constants?

( c/p )l 2 I d~· ) ,

3 . - ·- I 4.\y - · 8y• - 0 dx 1 .lx

h. ( '1l' )

3

- 4-w( tly ) 1 S:v2 o d.t tlx

c. ( dy )J - x1 ( dy ·)l 1 21•[ ely ] dx ) 1 dr cJx

d. None of the :thovc e. h \Vhal is lh.: equation or t l . c:u tangent 3t an point (X. y tan- 1

( l.r1 Jy ) weth

p3sses through ( I 2)

3 .

b.

' d1• v· x - · -· = y?

c/:r: X

• - I )

25

26

28.

c. In (~ )-(.~ ) c·

( t

d. lnx + ~ = c ' )' J

.;, b The only .:uJVe lo r which sulmormal it of Cl)n~f.:lnt length ~' a. d rcle b. dlips<: c. h YJJCThOIJI d parahola e. d Con~ ider tbc folio'' ·

I

2

.f

~

"

fo llowing

4 5

~) 5 6

-l 5 ;) 3 6

l' =l~ 2 3 ~ 5 61 ;> 2 -l 3 6 J

permutlllioos

\VIu.:b of Ute follnn ing i. are Cl>tre.:t'/

I a 6 r 1

1 6 2 . I' : (J

1111

::.elect !be correc1 ans"er usmg tho code givcm below: a. I only b. 2 on I~ c. Doth I and 2 J ~either I and 2 e b l .t:l 'L• denote the maltrplicative proup of nun-2eru campi<:.'- numbers. Let G1 be the: cyclic subgroup g<'ner;~ted by I t i nnJ 0 : be the cyclic 1ubgroup genC11ltcd by

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29.

(I • t)1..fi . \\<ltkh uno of tho fullowing iM

correc.t? a. Both 0 1 and G, nrc mlin1Le groups b. G1 is finite. hut G1 is infinite gmup c. G: is finite. but G 1 is infinite group d. Rotll lrt and G2 are lin it.: SI'OUilS e. c Cons1der Ute follotnnu statements: I. Lc:i (R.. ~. ·) be a ring. IJ (R.. + ) is a

c~ dk group. then R i~ 1 commutative ring. 34.

2. U' an int.:gral domaan is iniiOII.:. tlu.:n it cannot be a tic:ld .

Which of the above statement~ " :n <!

correct? a. l only b. 2 onl~·

c. Both I nnd 2 d. Nejthcr l and 2 e. a

30. Let G he a group and ld n,: G il O(n) - n and k is <Ill\' integer. Then whidt on..: of l.he follnwing is correct'' a. O(n' ) n only h. O (ak) _ n

c. 0(a' ) c. n nnl) c.J . O(nk) ~ n

e. d 3 1, l,;et R be :a linite ring •lith unity

be !he l.:asl posit ive Integ, ..ic.:-~....,.; mn 0 tor all nr- R ~nd lot -...... positive inlc.:g~T s uch th• n. - ( one ofthc folio" ing is co t7 a. m and n may no xist b. m and n ex is 1 ..

c. m and n ·t_ m d. m and n e. d

32 l <!t , 111 o integral c.Jomdllb. l et • . b)l,, .- R1.bo ll2 } and

,c/) s (u 1 c.b d).

(ct.b .(<·.d) : (ac.N/) Whidt onc of th u

follow ing is correct? a. ~ R1 is also an intcgrnl domain. hut

not necessaril\• n field

b. R1 R1 ncud ~01 be: ~~~ intcgr:1l clomnin

c. R1 Rl is no t a ring

d. R1 R1 is a lidd

e. b

36.

3 7

Which one of tlte li•llowing slalcmcnt< i~ not correct? a. Eve1y Jlltegral dnmn)n D hn'

ch~r3Cicristic w ro or a prime b If R is a ri11g such that al a. lo r nil

a ':R.. tlu.:n a- b- 0 implic.:s a - b. c. If R. is a ring in "hich xJ=x, f<1r all

, ,- !{_ llten R is 3 cnmmutal ivc ring nf

d

• - 1,2.3 . .....

6.'et:~~' be the vc:ctor sra.:~: of all mnpJ'inj!S · n IR' to 1!1. and le t

'• = {f -: 1"1/ ( - .t) = n~·Jl :md

,..1 = {/ E r 1 j ( ..1) = -/ (x)} · Wh1ch one

of the following is con-eel'/ a. NeitllE'I" " • and v~ jq Sllbspacc or v b. V 1 is a subspace of V hut \11 ls not a

~ubspa.cc oi'V c. V 1 i~ not o suhspnce of \ but \1 ~ is B

subspace ofV tl . Both V1 and V ~are subspaccs o l' \' e. d \\"!tat is the dimension of the suh;;pace generated by tlte vectors ( 2. 1.~.3). (2.L20). (0_0.2.3) and (4.263) m

I =R* ? a I b. 2 c. 3 d. -1

e b Let r: 12 l -> ~ l be n hnc:ar mapl>ing defmed by ft:~.b.c) % (a_ J i b, 0) for all

(a.b.c ) .:i13 . \Vb.1l is the matrix of the

linear mapping with respect to st:mdard ba.sis te, . c~.~}?

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3&

39.

r: II

Il l 3. I 0

0 II

[:

0 :I b. I

I (I

r: 0 u

c. I 0

() I

r: u

"] d. I (I

\ n () n e. d Let T ::l' transl'orrnalion represen tation

wdercd lusl~

nul lity ot'T 'I ~ . (I

b. I c. 2 d . .3

b

2.

. ~· with

I I

is I)

I

2

d. Neitht.:r I no r 2 e. ...

be whose matrix •·e.pcet to ~t!lndard

0 ()

I 0 , Whnt

0

nf j~

4U. A pnnicle is projected vcnicJIIy up from n poin~ TI1c particle is seen tD Jl3&8 Lhrough a point on the p11lh at times lt. •~ "hih.: moving up and down rcspcclively. \\'hat is the velocity of the pmjection'l

;I g(Ti ~ ti)

,, + ':

h. g(r1 - r~)

2

c. g{f, - 1!) d. :-:one of the abo"" , .. b

•• 4 I \\' b:Jt is lhe value of l101 _•_.....,.~

~ 0 b. I '2 c 2 d. e e. h

42. \\ 'hich one of lllltl:jJI

43.

into an equivalent

..:. ~ ... latch tist I willl list ll and select tha l!orrect :ttl~wer using tbe code ghe•1 belrm the lists: l.isi I (Hjm·ise opetati(ln) A. 10111 Al\0 I LIOI B. 10001 OR 11001 C. 111010 I XOR J 10 I I Li•t 11 (Out pol) J. 10110 2. 010111 3. Hllfll ..J. 1100 I ) . 000 10 Code:<i:

A a. ..J b. 3 1!. ' _, d. ..J e . ..

B ('

1 5 ..J j

..J 2 1 2

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44.

45.

46.

47.

The n~<.:mOr) n~cC:);s time of diJ:ll;n::n l. memo!") de\ icas ;Jmmgcd 111 a~c.:nding order is: a. Cache. R. \M, hard dis'-. magnetic IJi'~ b. RAJ\l. hml diRI., mQgn~tic tape, cache c. M:t!,~lctic tape. hat'\! disk. RJ~L o.:adtc d. J lard disk, RAM .. cache, magncltc tape e. a Which one of the following i>nli>..:rtics is used 10 ~>lnrc: a binary digit in a hurd dask'/ 3 , m~lrical b. !'llagnetio c. Magnt!lo-optical d. Ophe;ol e. b What is lhc hcxadc.:.imal CijUI\ ~lent of th~ de<: im~ I num lx:r ~ 18 L9'! a. 05JA b. A35A c. A3SB d. !\35('

IS J 7ef0,

a. who:n all inp ut~ at.: :r.CI'!J

b. when all inp u t~ n:re 011c c.. when :~ny Qlle input is 7.ero d. wh..:n any one input is one

"· b

Dir ectio n~: The following four (4) ' f,:t

of l" o slal<.m<.:niJ;, one labelled as .. Lh-.., ..,,.....,,._ (A) ' nod the vther as ·Reason examine these two statement~ ca and sdcd

111,1 the g11'en cud~-s . • 4K here D,). 31 and DJ

t , q(x) p(X• ·' >· r a. p. y are IJJe roots of

- f 9. o t (R): If o i., a 1oot of p(x) 0. tlu:n s ot of q(x) 0

th A .md R nn: indivulu:oll) true anJ R is the correct explanation of A

b. Both .\ :tnd R arc individually true but R i• not the com:ct c::xplanat ion of' A

c. A i~ LTu.: but R is false d. A is f3l~e but R is true e. d

49.

50.

5:!.

As$<•rt ipu (A) The ~ulutinn c•l'

d2

v "'' ,. - ·- - 3- ·- - :!r =e •~ d··2 d\· .

' = c1e~ -"f ~el.r - xe11'.

Rt>ason (!l) : 1lle particu la1 integr81 nf ltD))•- e"' is x[f'(all Je"' when f(a) 0. 3 . Both . \and R are individuaU} (me .md

I~ is the cun'I!Ci ~pt.1naiton of A b Both .-\ and R ~re indi\;dual true but

R is nut the corr"'"t <..:xpla.n f i\ c. .-\ is true but R IS f11lse d.. i\ is false but R is tnu:

b

e l'

Let(~ ~ {;{; x is a positive rea l number! .

For any xo:G, let ..{;: denote U1o po~iti1 c

square root of x. For any x. y£Li. define

I"•J' ,frY \sserlinn (i\) ; • i;; a commutnth hinal)

OJ>"ration un G and if x • :V \ • z lor an)

x. y. 7• G.then .Y - z. Reason (R); (G. •) IS a group. a. Both A and R arc .indivu.lualJ} tlllc and

R ~· Ute correct expl3n3hoo of A b. Both A and Rare indi\'idually true but

R j, nollhc corred cxplan.ttion u l A c A is trve hut R is false d. A is false but R is true e 1.'

Tn o like parallel forces f 1 and F! .1ct pn n rigid body at t.\ and B. If F1 and h nrc interchanged in posilton. the pnint of application of lhe resu lt;tnt ~~ Jisplaccd through how much distan.::.:'/

(Pj - P; )a J .

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53.

5-t

55.

56.

(FJ F,)u h.

(l'j - F:)

( fl l ~<; ) t l c.

(F( - FJ}

(fi +Fj )ll tl ..

21 r;-1-; )

e. b along AR where An i~ equal to 11

ForcL"S of rclahw ma~itudc5 5 .. I.. I.. 3 a.:l along the sld.:Jt AB. AC' .. CO and ,\0 ~pcctivcl'• of a "l""rc ABCO of side length Cl. \\ b~t IS the equation or tho: line of ac11on along \\ h1ch the ~iogle re~utunt acts? a. x - r : a b. 2(x- y) u

c. x-y = 2<•

b.

c.. "" a

d. ol thll

e . A squan: on four legs plac .. '<l fCj!pec te middle point>J of 1~ sideo;f.f'.,._o3oe I c \\eight or the t.1hl" and \\ e grea test \\eight that can be put

""'''...::"ll:c the comers withollt llJ)Sclting the then nhich one of the following ;,

corra:r~ a. \V' - \\' b.. \\ ' - 2\\"" c. W' - 3W d. \\i' - -1\\' e. a What i~ tho distan..:c of ~'lltr<.> ur gravi ty from ori&iu of the \'OIUmc formed by tho: revolution of th~ portion ur the p:U'ahol;l

51.

58.

6(1,

J' ' - 4{!X cut offlryx r h ;ihiJul the axis of x'? a. It~ b. b 3 1!. h/2 d. 21V3 e. d ,\ pu.:..:.: oJjce slides down a 45~ inclind in

'"ice ih.; time it takt:~ to ~I ide ~ fiictionless -15 incline. \\' coctlici.mt of friction hem th the incline? a. l 3 b.. 1 '2 c. 2.3 d .. ' •

• b. ~s·

c. 110" ll . 90° e. c At a distance x from a centre of force. I he velocity \. of a particle.. moving in a

straighl line is gi\'en by :r = 112 ..,. \\here a. b are eon.,brtL• and It II, \\ hidt one or the following 1s CQm:ct? a. The accderat1on is invcr.o.cl~

proportional to x and lhl' lon:c is allractiYe.

h. 11te acceleration t• mv~el)

proportional to x and the Ioree is repulsive.

t:. "llte acceler~tion is inversely

proportiunal to ./X aml the force i• nttractiVl.:.

d. The acce]ep l iou

proportional to .[X repul'iive.

"· a

• . I I~ lltVCrt;C y

and the force is

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6 1.

62

i\ bullet lln:d into u tnrg.:t lo~t:s 2. 3 of i ~ vtlocity after pcnc.:lrating 1' 12 ern. it will come to res1 after 1x:netrnting further <1. I em

b 11\ Scm C. 1/96 C:ITI

d. I 19llcm e. C'

A parti le ill projected at 3n nnglc: of 75" and after 5 'ccond~ ot a.ppcan. to ha\.: ao ele\ at ion of 45• from the point of projection \\'hnt is th.: imtia l velocity or projection?

a.. 5o.fi g

b. 5g .fi c. 5g d. lOg ~. a \\'hc:re g i~ tl1 t: acccler:atiun due tu gr3Vit) .

63. [f we n;~umc U1at the Earth 1s a homogeneous sphere ot radius a. tho: attraction on n pnrtidc in~iuc: the E:ortb produces an ac~ch:rntion which v;orics directly a~ th<l d ls1uncc from lh..: centre nf the Eurth and tho.: nccl.lll!nltion on thu surface is g. then what time would • t)nr1 icle tnkc to move dll\~11 a smcw straight tuhc li·rn11 the ~ur1"3ce of th •

IH.

65.

to i Is ceo !ro?

a . (2~r)~ur ~

b. (r. 1 2) ~a g

c. lfh d. (4r.lh • e. b

' lc uf Ilia>& 01 os ~~~·o surf~.:e of tJ1 e e.Jrtl1 with

·0<1i~~~ll nglc v. to thu \ c:rtie31. llto as~ • ed to be a 5ph..:n: of mass ~ I

ilrll_...li'u~ R. wh31 i~ the o:m:tg} 1 iately otfier the launch?

a ,,,.l 2

b. (nn•1 2) - mgR

c. (mv' 2) - mgR

d. mgRcosa e. b If a non-singular squan: mutrix A s:ltL~ lics

theequ;~tion /+. 1+ .11 + ...... + . 1 = ()then what is A· ' equal to?

68.

69.

70.

:1 , .1• I

b. ~· c - .-J''' (\. A' e. h lf the 'et of equationi

(b +c)x-(c-o )y-(u + b) (),

c.T+<!)I+b = O. <.1\'+by+ c =ll 01.re con.<i\tenl then which one ofdt following is COITC\!t? 1. a-b.,-c=O 2. .1=b=. ~. a-b-c=O

given belo\\ : a. lor 2 h. •

cos/) - sin 0 ·,, , tJ1cn which nno

sinO cosO ;

d1e fo llowing i~ corrccl? a, .-l11 .,l~- r11 + 11~1 b. il... "'I" - t l0 - ·1.._

c. ( ..I, r = ~~.s l'or any pusitil't: integer n

d. .4, = ( {J ) I

e c W T: ].3

• R' b~ a tin~r lmnsfunuatiun

gi\'en by(x,y.;)

= (2TA x - y . 2x 3.r =l for ~II

(.r,y.z) · ii!'J. \\lticltoncuflhclullo\ling

is correct'! 3 . f is onto. but not one-one b. 1' os one-one. bot not onto c. T is onto anJ 0111.:-onc d. ] i; neither one-one nor unto e. C'

\'ltat is the locus of th.: mid-point of a focal chord of a parabni.YI 3 . Circk h Parabola .;. Ellip:~ce

d. 1-l}perbola t:. b The diameters of a circle of area 154 sq units lie alonp. U1e lines 2\ - 3)' 5 and www.examrace.com

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71.

12

73.

U .

75.

~x - 4y 7 . \\'hat i~ th C•tuntio!l of the

circle? a , x'- y 1

- 2.>: - 2y (o2

b. x1 1'

1 2 t 2•·= .n c. x' - .v' - 2r + 2y 47

d. J? · _l'l 2T I 2 _1' - 62

e. c \\"hat os the locus of the mid-point\ of the chords of lhc circle T

1 •·

1 -1 "bich subtends a righ t angle at tl1e orig in'! a. x + y - 2

b. J?- r 1 - 1

c. .\'1 1'

1 = 2

d. \' + y = l e. c Th~ limiting casu of an ellipse of 11 hicb tl1e eCCo:Jllricity tend• to zero IS a. Straightline b. C'ircle c. P;trab()ht d. 121Upsc

h

a. h.

c. h.\'+ ~)' -= h' r k' d. hv - kx = O

I. ~) respecti\·ely. V'llll1!1\tll1 cusines of PQ?

3 ---Jr) 5·0·-i)

d. ( --E.u-Ji) e. tl If f ( X. y. ::') - (I Ill a h 0 1110!!UileOUS

e<ruilli(JJI in x.. \ . t., then whid1 of lh~o: folio\\ ing i~ correct?

~ . f ( .~. y. -: ) : 0

76.

77

79.

80.

h. f( - x,-y,-: ) 0

~. ((rx, tJ•, r::) = O Vr -=:it

d. Ooth (b) and (c) .:. d Which is lhc:: n:.ture of the: intur<cction of

tlu: set uf' planes x ~ ' '·' ' + ( b- ' ) ::- - J - 0.

.T- br- (c-a )= + d = O ami

ncy.(a•b): d=(l '1

a. They me.>~ at a point b. The) l(>nn a trianguJJ c. They pass through !:!!!""" d. They are al eq

origin e t:

\\'hal is lhe po l>l~ Ill.: point (2., I )7

3 .

tangent at any

' ). ' 5 )I= X i .X i X I

with x-ruris, then following is correct'/ a . I A. I~ l

b. I ..t I· .J3 ~. 0 A. ' J d. A. -= (0.1 )

e. b

point of the CUI'\ c makes acutCl nn&lu which ono of rho

The normal to the cun>e at P(x.. yl mCCb

the '\·axis ~~ G. If the di~l~ncc of G li'<Jm the origin is twict: the abscissa of P. lhcn "hat is the nature of tlte curve? ~ C'ircle b. Parabola c. Ellipse d. Uyperbola e. d \\lml 1s the number of asymptotes of 01u

-.-unc y=e·• tlon( :r.> )?

J . t1

b. 2 t: I d. u e. "

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81.

82.

83.

S5.

\Vhal Is 1111.: ligtuc 3S)'mplotes of

X1 )'

1 c' ( x' - i) 'I J . A t)ar:,llelogrnm b. A rectangl.: c. A trinngl" d . A squnre e. d

fnm1cu by :111 lhl.l 1hc ~UC\'"

Lei f (.~) .\ 1

• 11 - 2 and (l •.c ) "

g{x) : (J.J~{. ..... .n lim~-s)(x l th'n \\h3t

is Jx 'g (x)c.L,- equallo''

(t+nx• )' ' ' a. ... c

n I

I )' I ' \ I • '" b. f c

n(n - 1)

( . )'"' ., I I TIS c. +c

n-. I

( • )1~!1 .. , l iiL~

d. I C 11(11 + 1)

e. b rr [:'I.] denote& lh!: grcah:sl integer fu 1

of x, then wh~l is the vn l I ' J {[x] l ln (J • s ) tJ

a. - I '2 b. u c.. d. e. a \\lu i

• cnclo~ed b) !he x - ( m:os t) .,J;.

'~~k·a of n .:irclc of rndiWi J:J. . . -~ I;

b. Th.: ;uca olthc clhpse -+- - I u" h1

c. The area of o rectangle of $id ~ a and b d. None of th.: above e. c The length or the upper half ponion nf th~ cardiuitls r = ( I <.:OS 8) is bist.:t.:lcod by

1vhich one of the following lines? a. f) tr l (,

87

90

9 1

b. (/= ;7'4 c (} : r. J d. fl = lT ' 2 ...:. c What is the perimeter of the curv" r - 2cns0'1

~. ~~r

d. S;r e. a The I.in<: scgm~nl revol\ cd about the ·i col) e. \\ 'h:u is tl~ l;ll I su

a .fin b. ;r .fi c. d.

c.

L.d r (.~) ,1) <'-~ - ltlx "" th:ll

f (l ) - -6 andf(2) = 0. hy the mean vn luc

th.:orem. there exist~ 3 numbc•· y i11 the

open iuleC\'l!J (0.1) such lb.at f'( I • r) - 6 .

111e \·alue of y turns OUllO be 3 , Gn:ato:r th:m 3 ~ b. Between 1 2 md 3 -1 c. Rei\\ een 1 .t and 1 2 d. Less tltan 11-1 e.. b

l6sin' U equals to \\ hich on.: of the following'? a. sin50+5sin3U T lOsinU b. ~in~0- 5.si n3B 1 IOsin f.l c. sin 50 5sin3fJ lOsin 9 d. s in5&+ 5~in38 - 10sin0

e. h If a.fl.r are the rooL< or the etrunlion

x3 - 2.r +5=U. tlteu ~\hat is the vn luc of

(a - !1)( a - 1)- ( fJ - y )( jl - a) ­

(y- q)( y- fl) '' www.examrace.com

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92.

J , 2 b. 4 c. s d. 6 e. {I

IT a.p.,,S m : the root! ul th.: cqu~tion

px•-~ ;- rx' - s.~ • 1 II. then what ·~

the value of L ( rl/1) 1

'

3 . r 'p b. p r c. r I d. None of the above ... c

93. u· p represents the vari3blc point z Jnd if 2: - I 1- 2 : Uten "hal is the locus of P?

~- Circ le b. SLraight line c. Ellipse d. llyperbo ln e, b

9-i. \\1tal is the equiva lent binat·y numb~T uf de<:imaJ ( l 1.8 I 25 )10?

a . 1011.1101 b, 1010.1101 c. 101 1,1110()1 I d. ItO 1.10001 e. a

95, Let f be :1 111:11'P'"S friJlll the .~.:l X l Let A 1 and AJ he the arbitrary su c; s f and l.:l B1 nnd BJ b~: the arb ·

of Y. \\1tlch onc of the fum· tg nccessatily hold?

•• _r( .. ~.n..~,l =-/ ~ln . b. I (.-1, U.-!1 ) • )Uf ( ·I,) c. r' IB, B,)n1 •(n, ) d {"( '(8,)UJ '(B,)

% . the value uf

n) isin(/1' n)

[ l+cos( ll' u) - tsin(n n)r ~~J-ure

11 • Z and n .. I ']

3. 1 b 2

97.

98

c. 1 (l - 2 e. c Wba! is tho vulu~: or a For "hidt thu

equations .l• err I = 0 and

x 1 + .JX' + l ~o have a common root'! J .

b. c d.

a. b.

hegr<>L!p (0.+ 6 ) whereG - {0, I. 2,.3.

~- 5} :mu +-6 is mldi tion modu lo 6. what i~

th~ inverscof(2 t 6 :<-1 '• 4)?

a. 0 b. 2 c. ~

J. 5 e. c

llltl. Consider Ute following ~talement.s: I. "J I, [3). r 511 IS a subgrOU(l of l pI •

12]. [3]. [4]. [5]. 1611 und~T

muJliplic.:uioo modulo 7. 2 lf G is a group of urd.:r 5. 11 ha• no

proper subgroup. \\'hic.h of the sblements given 3hove i'l 3fC

curra:l? a l only b 2 onlv c. Both I and 2 d. :"--eitho::r 1 nor 2 e. b

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