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I Physics I. ~ doubly ionizetl L·'- 1011 h1 Jlr<lUnd stale

ah<mhs YLb eV of eJtetg)'- Find the inLr=e m angUlar momentum of electron. (Take h = 6.63 x w-"' J-s) (a) 2.11x lO--s4J-s (b) 3.10x l0-34 J-s

(t} l.O'ix w->• J-> (d) 4.22~ JO"" J-'

2. Eiecmc conduction in a sc,llmnducmr takes place due m (a) electron' only (l>) holes on!)' (c) bmh electrons and buies (d) neither elcctmn< nor holes

3. A 1>1tC whose cros'->eHion is 4 rom' i> >tr<tched by 0.1 mm hy a cert:Jin weight. How far Wlil a wire uf the same material and length stretch, if iL> "'""'"sectional ar<:a ;, 8 mm' and

the same weight is attached? (.1) 0 5 rom (b) l.D rom (c) 0.05 nuu (d) 0.06 mm

4. A thin ring of ma" 2 7 kg and radius 8 crn rutdlc> at.nur an :cis through ~' e<ntte and 1-'t1P<'I1diculal to the plone of the ring at LS rev/,, Cokul•tc lhe ldnetic en.,-g_y of Ll1e 1~ng.

(o) 0.763 J (c) 1.5 J

(b) 0.345 J (til Zero

S- A body h"' " c_~"rge of - 2~C If it has 2.Sx 10" protorui, thon how many eleciron< <-~e 00<1} has?

(a) 1.2Sx 1013 (b) 2.5x 1013

(c) 3.75~ 10" (d) None of lhese

6. The graph olin (R!R0h<=-' ln.-1 (R- ra<lius

o£ a nude"' and .<_ = its mass nun:ber) ;, (a} a straight line {b) a parabola (c) an ellip<e (d) Nono of the abo>e

7. Ftgure shOVI~ the ellipticol poth of a planet Rhmlt the >UD. 'lllc "'" shaded pam have equal area. ill 1 and t, be t:1c Lime token by the planet Lu g\J from a ro b <U!J frum ' to d r<:>pc"C"\-d)', then

(a)l 1 <t 2 (b),, =t, (c) t1 >t2 (dJ Jn.<uffictem infurrmtinn to deduce the

relation betweCll.t- and r, a. When light pa,;s« from one mc.Iwm to other,

then "hidt of the following may ctumge"l (") Velocity and frequency (b) Frequency and rn\our (o) Velocity, wavclellgth and colour (d) Velocir;-and wavelength

9. Molar heat rapao1ty of a ga; depc~~' on {a) the <tate of the go• (b) the process {c) nature uf g.~s (d) nature of gas on<! the process

10. Two horlies at different temperatures are mOO m a C'llorirnet<r. Wluctl of the followin;: quantitie; ren!aln conserved? (~) SUlll of the t<mperoture of the rwu \>odic> (b) Total hca( olthe ""'" botlk> (c) Total internal energy of the two bod1os (d) tntcrn"l energy of e>ch hody

11. A p<rtick has been project<d at cerroin angle e wirh the htHizuntol, llnd the value of 9 for which the particle hil5 the ground in >uch "

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manner that initial and firull velomy vector> ore or 90'-(!l) 600 (c) 15'

(b) 30" (d) Foranyvaluco[~

1~. A particle is moving in a moighr lme 'vith corutantvelocity J m/s. Atl ~ 0, a forO<'""' Otting on the particle in a d1rcction jl<tpendirular to the direcrion of its initJal m~tion which causes an aceeleratio~ of 1 rnJs'. D8terrnine the magrtitude of partick's

velocity att 7 3s (a) 3 m/s

(c.l :J.J2nvs [b) 6rn!s (d) b1'3 rn/s

13. Three guns are aimed at the centre of a drdc. They are mounted on the circle_ 120" aparL They lire in a timed sequence, such that the three bullet> collide at the O<ntre and mash inro a stationary Jump. T'lvo of the bulle" have identical Iru!S3es of 4.50 g each and speeds of v1 and v 2 . The third butler hos a ma~< of 2 00 g and a speed of 575 m/s Find titc unknown speed<. (a) 200 rn/s each (bJ l45 rn/s and 256 rn/> {c) ~36m/> and 320 rn/s (d) None of the above

t~-~~~-~'-=~:~t';; ~}~~~=; ·~;1 ff tilmj, ~'llt ,fl' (R•""'"), F.ach of

·iti.iM,~-~0~~-d<<ma<iv< mo.,.,, oo(Y "" ':''?~~iii.;);, ,;,,,-,;;o,l(ot< /<;V!';M .. ~. t/'< <p>l~O ,/t.,ici; . · . ./ll-!IIJ~1Uoi""M'<l>',:rc~aoa:(ii.l'~k.-'<>W'. -_ : ' ' (iiJ : ~ fili iN,; *~roo!<"!ll "' .ru.,: ~~"-'=Jti! "· - : --~~~-::'PI<i~nj ... St•~<rmhU.

. "11>1: S~!rtt l » ~ ~"""-!/ "Job;i._ :: (4-::~;.t!~fi!I~Slai<m~llli~{r""., , , - W "·~'"""'" k"""' '.j'tm'em<nt lfH tiu<; S!'-"'""'~' II ~

·- ''lli<.~~;rili!""-"''SJiJ"';ru,;;"_' ' 14. Statement 1 , The fundament• I umt5 ofwlocity

of lighh< 3 x 10' rn/s and acceleration_ Due ''-'

gravi<y i• 10 m/s' and !he m.s; of proton i•

1.67 X 10-"k~.

S"""m•nt ll : The value of time m <uch a

'Y"""" is 3 x to" '·

15. Stalement I ' Angle of rw'-'-"' il equal to che angle of limiting friction.

Statementll: When ):he body "J"" " the pomt of motion, the fo,-, of friction in this Slagc r; called li!!Utin~ friction.

16. Mark the correct oprion

(a) ln electJn<totic< llre.e i> HU muUun of charge ot ill in cond<!L"':ots bulk

(b) hr olemu>t•ti<>, there is a motion of charge pm11de in mndllcr<><'< l>ulk

(c) In elecrrwmtics ond c:urr<ct cleclnoty there is a net motio~ of charge particles in the bulk ~1 tk matcrial of the oo~duccor

rd)ln electtosmtic.• and mrront •lecmcity there i> no net motion of cltarge porrid"' in rhe bulk oflh.e matetidl of the conductnr

17_ A dtild pu_;hcs a toy Jx,, 4.0 m along the floor by means of a force o~ fr 1\: oi1rrch'~ Oowr.word rlt ""dTIJ(]e of 3T to the homonmL How tntt<h wurk <toe:, the child do > (a) l8.0J [b) 19.2J (<") 1~.~ J (d) !-;one of the'"

1S. Currrnt passes through a soluliol1 of sod1um chloride. In 1.00 s, 2.6~x JU" :-Ia' ions amve at the llegat,ve ckctrodc arul 3.92~ 1016 cl­ions arrive at pmitive electrode. Determine the Cl1m·oc •ml the clire<tion in whiclt it is flowing. tal 1 OOh rnA, pCK<itiw to nogabv~ electrode (Ll 10.50 rnA, po<iOve m neg.1ri"e cleorode (c) 1.056 mA, negative to ?Ositive elecuodt (d) lO 56 to,\, ocg,.ivc to positive elecrrode

19. Mark rhe rorroc: opuon r<g.<c<.ling Seeback .~,

(a) The cemperaturt" nf j1.nction at which thermo emf i' m:'-<irm:m, JS called the neutral temperature

]b) '!he tem]>e<artll'(' of llle hot junction ot whkh thermo emf changes !ts sign. is c.illcd the inveJSlon tempe1amr<

(c) IFA" A, ond e, denote the temperarure of rold juncoon_ 11n-cr.<iM t~mperanrre and neutral temperohlrt' rt'<pcctiwry, then

A,-A.-~e,-e"

(d) All of the above

20. An alternacing <tmcnt having p<ak value 14 A io used to heat a meml wire. To produce the same hca1 ulg dfuct, a conm:om curronl i can be n<e<1, whe.., r is (o)l4~

(c] 7 A (b) about 20 A (d)al>omlO~

21. ~" ,.,,h >Rtellite of rr~'lSs M ei!'cic>< the •ard! wl!]\ sped u. lly how murh .1<1<'< it> momenrum c1Lange as it goes halfwoy around the C<U1h7 (/gHOI' me ocJ< Ui,, roW< ion)

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10due~~n-~-~~t1<·eJ emr'--~ 11. Due to ('han~1ng magnetic !iehl, if /; i<

induced eleon io ~eld, then, ~ J F dl

llL Fmm abov" twu '~1Lome:nts f E dl ~- -~ . "'

(a) All the tb"" storements are <Ilwayo tru~ (b) Only] aPd II are comccl (.;) .>Jl rlu-ee aro "~'"'6 td) None of the obove

24. Comidt•x two light 5otu-res of w<>,elengtlt ).1 and ). , (). 1 > )_ 2 ) which "'~ cmtttm,o; n, nd n, photor.s r"pectiV«ly '"a gtven time. 1\<Stnrte equal [J<lWer for both the ,oun:e;, then

(;1) "' > "' fbJ "' < "' (c) ": ~ n2 (d) Cant>oy onyLiltrtg

:l.S. Fo"' cell< each of ernt 2.0 V and mternal resiscance mare <-"<lnnwed "' 'hvwn in tbc fi)';lll<'. Find J, I, ,, n d r ,.

(a} 0.81\,04A,0.4A (b) 0.0 A, 0.25 A, 0.25 A (c) (8/l:l)A, (4/131 A, (4113)11 (d) Noneofthoabove

26. M"'k the rotrecr optic-r. (a) Amp<''''' low 'ta'"' thet ~uR of B rlrrou~h

ar.y clo>ed ,Stttfar<e ".U., times tho l'Urrent posoing through the area bouruled bt clo.sed >urface

{b) Gou"'' klw for magnetic fie](] m

magne:os!atlcs serve, tlle sanl<' pmpose "' Gauss's low for elecrru: ~eld m cloc\ ro.<tatle.l

(c) Gau"'' Ia" fur 01agnenc field sti!e> lh"' <he Hu~ of 8 through any clo>cd ,ur[ac< IS

>OO' ~- -~~oo~ -3QOJ - l!OO~

Calculate tho heat of the cycle and efficiency of cycle_ (a) 400J, 10<)% (c) '-()0 J. 66.6?%

(b) 600J, 66.67% (d) 600 J, 100%

26, The dnnen>iunal formula for torque is [ML'r2), same a< tha< of work or ene<g)l, rn

~0-

propel Sl unit io (a) ,iuulc (c) N-m

(b) either N-m or ,ioule (d) Noneofthese

The penod of o<Cillation of a simple pendulum

" gi,·en by { ~ 21! v';, wh= lis ab<Jur 100 un

and " kn<>wn to have l mm acturacy. The period il about 2'- Tl>e time of 100 oscillanons is measured b) " stop watch of leaot count O.l ,_ The perc€Iltrtge ermdn g is (o) 0 l% (b) l% (c) 0.2% (d) 0.8%

The ~aviwtional potennal energy of a body of

'""" m at the earth's surface - mgR, lt> ;'J"ovitotio.nal potennal energy at a height R, froon earth's ;nrfu« Wlli be (Here, R, is the radius of the earth) (a) - 2mgR,

(<) ~ mgR,

(b) 2mgR,

' fdJ - 2 mgR,

---i--200V

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31. Which plate I•J form an isolaL<d >}~tem (o) plate l and pk1te 4 sepamcly (b) plate 2 and plate 3 separHely (e) plates 1 and 4 jointly (dJ plate1:.! and~ )mntly

~2- n,rge.s on l>"th the capdcitor> in ;teady state will t>c on 6 "-"' !ir>t (a) 4DO ~C, 4DO JLC (b) 700~C, 250JLC (cj H!lii)JC, ~Sil p( (d) 300~C 4i0J.LC

33. Suppose q,, q, and q, t>c the magrurudes of ch:uges fLo;,n (""" ,w[Lchc> .';, S, ami S, dftcr they are clo,ed_ Then (a) q,-q, arui q2 = 0

iblq,=•h=~ (c)q, =q,=2q, {d)q,=q,=~,

34. The "'""'"' breokd<m11 vol~age of a Zener diode foo wl•kh ol "followins grdph .,,;t; is

I! Chemistry 36. Wlllcl• of the fullmving sal,; is colourless?

(a) CdCI,

(b) Cu50, · SH,O (e) Mn50 1-7H,O (d) Ni504-7H,O

37. Drawback of DDT a< po>-ric,de is that (a) it bccomo•> mdfe<·ti;-e after some drne (b) lt" le."" cllcclive Lhan other> (o) it is not eO<ily/rapidly dogradOO in ""u"' (d) its high oost

38_ RCOKIJ, + 4Na0!-i + B.r,---> RNH1 + 2NoBJ

+ Na,C03 ' 2H00

Ro"non "' >aid (a) flo>fmann-bromarnide reaction (b) Sclunidt rco<tlOo (o) Curtiu_, reaction {d) Bedcrna:m reacrio11

l9. Vvl!ich of :he following i.s not the cxttnsrve propcrt)• I

(•) Gibb> free ene1gy (b) EnLkolpy

(c) Entropy {d) Vhcosity

(>) w

t ' ~.

' " ' ' ~ ' ~2

' ;'j':-,_~ci-c-c-c-c-,, t· 1234sareg x

'"P"' '"'"'9" IV)------

(c) !~'jv ,, (h) 6V

(d) 1_~1· ,, In four oomplctt rovoluMn of :he ('.ap, the <1o<tancP n-"velled on the pitch scale is 2 mm. lf there are 50 divisions on the cirwlar ><ale, then the loastcount of the screw gauge is (a) 0.001 ruru (b) O.Ql mm (c) 0.10 trun (d) l.O mm

4{). Wllicll o/ U1e lollowtllg ratclaw; ikl> m o>erall order of 0.5 for "'"c"tior, tllvolving subsmne<s x,_l"mJz? (a) Rate=k{C,)(C,J(C,)

{b) Rate - kCC, ['-' (C, J'·' (C, )'--'

{cl Rate=k(C,)'·'cc,:r'cc,f'

(d) Rate- k(C,)(C_, )0' I (C,)2

41. The. m"hule of ortho hydrog<n is distingllished from pam IJydrogon by (a) two electrons moving in oppm1ce

ilire<.tioru

"' two electrons mm,in.s m llie ,_, di«ction' ,,, '" proton> rc~'olving ,, oppo,,ite dir<:c'tiQruJ

'" ~· ptotuno revoh;ng m ,,, ,_, directiolls

42. WhiCh of the followmg do-e' JLUl cuutdin l'---0----P bond I (a) Isohyp<>pho;pbo.-ic add (b) Diphmphoros acid (cJ fli]>hosphorit· a<id Cd) H;vorho>phoric acrd

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Predominant pro<!uct is Q

(a) &Me '"' (c) (f<~ (d)

Mo

""-I)~·:.'~~: No. c~s_,~:'lti-~ ,tW,~- ~' :.'-'li#'iWill,i(rMitrOpli •. :,·._ .' :-- -·:-.:,-·. ·:. :; ,: , .-__j ·Ill ~rJilln· 1"11>\r- -"'""""• -ot!,- Uniie<~e! ·on: \OJlill'-ti®. · .~n:!O,>'~lll<h >.:Vl<i!';l.....<.· o;._<l'<l··~li. ~: ·-llil~:;orelfw o¢~<Wr- . ._ , .. -,• ·;·-*---' .. : -..a.;.f pq-;: %. S€lect incorrecr >tatemem (s).

(a) The dissociation is a redox reaction

(b) Tlyb<idisation ~<>from 'l'1d oo sp'd'

(PCr;J and st?lPCr.; 1 (c) Stn1cture changes from tngonal

brpyrnmidal to tctrahedrol (Pd:';) and

octahedro.J (PClOl

(d) None of the ""'"'c 1S inrorrect

46. Number of lone pain around phosphorus in PC15, PC14 anoi PCIO are respectively

(a) 0,1, 2 (h) !l, 0, 0 (c)l,2,.~ {d)0,2,1

47. l'l'hich of the fotlowing s<atemento is inoorrect about elimination reactton ?

yY"' I __) .!'"-I<OH~ Product

(a) Reaction follows E 1 [t'lth

(b) ~ate -k rBromideJ1 LKOH]' (c) Reaction l> srerempectfk (d) Reaetion is srereoselective

cnffillcaJ enet-gy [o) corwer5ion ,, chCIIl!COl cllcrgy imo

thermal energy (d) ronver<ion • chemiclll energy il\10

electrical energy so. The maximum number of ele<rron• in a

mbshell bavi...g the s>U)jC "'ILoe of spin

quantum numDer is given by

(,]1+2 (b)21+1 (cJ l(l+ll idl -./la-11

51_ What weight of Oinlllonia conrnins the some numDer of moleru!es as 11.2 L ni' morgen at S'fP I

{a) J7g (b) 8.5g (c}l6g {d}llg

!;2_ A comptrundfurmod llyelemen"A and B has a

cubic strucrure in which A atoms are at the

comers of the cube and B atoms ar~ al tlrc face centre>. 'Ihc furruulo lot Lbe compowld i>

(a] A,B, (b) M<, (c) A0B (d) A)l,

53, Which of the following define< enthalpy of OO)Ull<}O 1

(a) n,om ____, H•(aqJ+ Olf Coq)

(b) NH,CI(s}-------7 NH:(oq)-'- CJ (aq)

(c) CuS0 4 5H,0{>)+ 1!10 ------>CuS04{aq) (d) r.uso,(.<J+ 51-tzO_____, cuS04 SH,O(.<J

54. The vapour pressure of pure li~uid solvco\< A L<

0.80 atm. INhen a non volatile substance B is added 10 the solvent, lts vapour p<"SS\\1'<' drnp<

to 0.60 atm. What is the mole fraction of

component /l in the solution ?

(a) 0.75 (b) 0.25 {c) 0-40 (d) 0.60

55. Adsorption occurs not due to

(a) LJnbahmced force at surfacr molecules (b) unutili<od free ,a(enCJes at surface (c) increa>«< entropy at •urface (d) van der Waals' attraction arsurl'aco

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&6. Sometime! yellow turbidity appea" while passing H§ gas even m the absenee of !I group radicals. This is because of (a) <ulphur is J>l"'f''l[ in the mi:rture as impurity (OJ IV group radkol> ore pn""Pltated as

sulphides (c) Lhe oxidanon of H,S gas by some acid

radicals (d) Ill group radicals are precipitated as

hydr<utid ..

5?. Which of the following complex ions will not show oprical acrivity' (') [Pt(Br)(d)(D(NO,)(Py)NH,J

(b) ci<-[Co(en),CI2]•

(c) [Co(on)(NH,),CI,]'

(d) [Cr(NH,J,CI,t

58. The magnetic Moment o/a transition merol ion is fnund ro be 'i.92 BM_ The number of unpaired eleotrons pr.,ent m it is (a)2 (b)3 (c)4 (d)S

59. Zwlner Jon (amino actdl s<abllises into couju~a!_e a<id at (a) low pH (b) high pH (c) intermediate pH (d) Pl (looelectric point)

(a) 1, 2 addition of HCI followed by tautomensm

(b) I, 2 addition followed by reduction (c) I, 4 addition followed byrnutomerism (d) l, 4 addilion followed by oxJdation

rn, CH,OH I H,o''

61. !! CH,-C-"" A rn,oNa y~·., w

A and B are CH, CH3 I I

(a) CH,-C~at,, CH,----C-CH, I i I I

180H OH OH 0CH,

CH3 CH, I I

(b) Ol,-c-Cfl,, Ql1-C-CH2 I I I I OH ,,OH 0H OCH,

(c) Jloth (a) and (b) ar< rorrect (d) Nooo of the above is correct

=•· 62 Ol,-c ... c-CH, ~ producr

(a} CH3-CH2-at,-CH3 (b) CH, v CH3

(c) CH,~at, (dl CH,-CH, -C -c-CH,-CH_,

7:~;~~"7~tf;·~~~re~: . -'liOI"'"""''L~{ !lrOm!o.'nri «~, :41<!1 .of ~~d!OJ .... ~11(_41~':"-~ 0'1Y~'

. '<if>'l!~ld:s~_;Yonlii>'i<,O> ;ldM:t/1>1 ~:#'~-­

. -fr<m>.,.~lOh:ll');t<J~@),p..tr~;:' ," .­A<fi :".$tii(.im<nil!uio.; 0\o.e.,.,;rp~rooO; ~rl'li ·:::' i;;,:tdio_~:~~fo:~t - :.- . · lhl:~lfs:tr...;Srmom;;.l'h.IIM-'!1 _ .. < _:­·r~:~~~~~~.ru~"""'.

: {d}_. =z~~:ts:!!::rw~~~~* 63. Snrb:CIOI>tl : pH of 10 M HCI aque<>U& solution

i> less than L Statement 11 ' pH is negativt. logarithm of W concentration.

64. Starement I : l!thyl xanthate i> u.ed as a oolle.::ror in lrmh floatation pro=s. statement n : Collecton tlepress tile O""t<ltion propeny of one of the components of the ore and thus, help in the separation of differen1: mineral• present in the same on::.

6.5. StttemeJ~tl : The atomic radii of rhe elements of the oxygen family ore smaller than the atomic radii of the oorresponding elements of the nitrogen famil)'.

s-...t II : The memben of the oxygen faJruly""' more electronegative and thus, have lowu values of nuclear <:barge than those of the nitrogen family.

.. _ 0' -~'''crnC!:,~c~"'~c·"·c.. ..,. , ? Product is (H) o,nn,l-f,n

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"' y mo

6?. A mixture of formic >oid and rn<al!c ac!d is

heated w:ith ron.-,ntrated H,;S04. The gaseous

product is pa:lsed into KOH solution where the

volume decreases by l/6th. Tite molecular

proportion of the organk acids, formic acid ond

oxalic odd in the mixture i<

(a)4.l (b)l:4

(C)l:2 (d)2:1

61!. In the preparannn of p-uilro acetanilid<: from

anlllne, nirratiun i> not done by nitnlting

mixture (a mi~ture of cone. H,SO 4 and e<>nc.

HN03) becau;o

I Mathematics

I, The locus of the point z >Rti>fy!ng Re G) ~ 1<,

where k is a non-zero real nurnhex, i>

(o] a straight line (b) a cirdc

(c) an ellip>c (d) a hyperbola

~- If A=[ cos'« cos" sin"

and B =[ cos'~ cos~ sin~

=asin<>l

sin'n J =~ sm~l sin'~

are twu matnces such that the produ01 AB is

nullmatrix, therHX -~is

(a) 0 (b) rr,ultJple oh

(c) an odd multiple of;

(d) None of the obOV<

3. The image of the pomt x~3y=7is

(3, B) in the line

(II} u. 4) (b) (4, I)

(c) (-1,- 4) {d) {--4,-1)

pre.s.<ure nf 1400 ToiT (exc!UQing l!le Vllj><lur

pre>!ure of water) and comainin;r 68.5% l!, by

mlume, find the volume uf H., measmed at

STP, which w:itl dis>Oive in JL of water_

(a) 16 mL lb) 12 mL

(c) 23 mL ldJ 121 mL

70. A mixture Qf 0.50 mole of H1 and 0.50 mole of

S02 i; introduced into o 10.0 l container ar

25"C. The container ha• a 'pinholc'leok. Afrer a

petiod of time the portio) pre"""'e of H., ;n the

remaining rnixurre

(a) exceeds that of so, (b) is equal. ro that of SO,

(c) ISl.,nhanthatofSO"

(d) is the same"' in the~"',) m1xture

'" ' x+x'+----+x"-n 4. '"eva ueof hm ---- i.l

"~' x-I

(a) n (b) n+l

' (c) n(n+J; (d) n(n-1)

' ' 5. If j(X)=.<in_,l(~) then f(x) ''

1 + x2 '

ditkrent:able on (o) [-I, I] (b) R-{-1,1}

{c) R-(-1,1) (d) Non<Gfthe""

6. The distance borween the Origin and the

<angent to ~e ourve y = <? + .-' """"" at the

~oint " = 0, is

' (a)~

" ' (c)--./l

' (b) --

" (d) _;_ ,, 7. General solution of the

C-J3 -1J.<in o + (,·'3 + 1J ""' e = 2 is equation

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"~ >"" ,a, "'""-4+ 12

(b) n~+C-tT%+ 1n2 " " (c) 2mr ± 4

-12

(d) n1t+(-1)"%-f2

s. lhe numbor of orrang<mentS of the letters of the woTII BA.'IANA in which the two N'• do not appem adJacently, i> {a} ~0

(c) 80 (b) 60 (d) 100

9. If o cirdc of constant radius 3k f"'""' lhro"l!h the origin anii Ille€ts the """' at A and !!, the locrn of rhe oentmJd nl ~AOB is Cal x'+x"~k" Cbl x'+y0 =2k0

(c) x0 +y'=3k' (d) Noneofthese

10. The t.lngenl at Llre ~oint (r" ,Y, )to the parat>ola y' = 4e>e meots the parabola y 2

- 4a (;< - b) ot

Q oml R, then the mid point of QR is (a) (.c,.y1 } (b) (x1 +b,y1) (c) (x1 < b,y1 ~b) (d) (x1 -b,y1 -b)

11. lf h and F, be the feet ot the J"'r,>eruli<:ular5, fiDm the foci S1 aud 52 uf an ellipse x' y' 5 + 3 ~ 1 on the tangent at any point P on

the elltpse, then CS1hl (~,!',)is equal co (a)2 (b)3 (o)4 (d)'i

12. The crillcal point> uf the funttion j(x) = (x - Zf" (2x + 1) '"'

(a) land2 (b) lan<l-1!2 (c) -1 ""d 2 (d) 1

13. 'lhnalueof( ~(tan- 1 (±JJ<b:is C•J ~ (b) r.

' ' (d) None of these

~" 4 4t a, band c be thm:: nl>II-z.ero >=tors, no two of whidt are co~moor. If the vector

1 + 2 b is collinear ~dtl1 1 and b + 31 is

collinearWlth li, them/ + zli 1

(b) ),. b (d) 0

Thr<'<' six faced dke are thrown together. The prolmhihcy rhar tlle sum of the numbers appear:ng on the die£ is k [3 s k ~ 8), is

k' k(k-l) (a) 432 (b) 432

(c) d k(k-l)(k-2)

( ! -~32

,..~~ ~ :~tbt>W: ·16 •n{tr.~;,:~M o~ flw "-.foaii<>'int~""''<'f" .. ·.·.·. :. c·:· " :· :' ·,c•_:,_;:: ,· ., i.erJ:: ..(,.;,yh.,aruni;;~n~ 1!¥ y =j'{xi•ilmilmif ~.~~:,;;;,_,";;,. (l"joi:dv.l ~ cai<~ s.irJ~i. ii\eo ~

· ,:,o,ru · · ~, ; , ~~iq""' · t.in<tro~: i" , li .:.; · 'k. , ·.i.kh " ;uwt · f("J ·=·Y ·>;!.!.'l~}~ ;, ?'" E!A;;;,;rj: ~:it t!'i<i'&ii:~dm; _be:i.;;,.s.,Oi~ ·iii.s, . . .. , .. ' ," . · ·• .::.::. :::~::o: ... fi'cf' :'R~.A41f.W:>il< t,<;.i:6)f~iff1 ... · lf.:o;,"k.iill;f. Of·.,, ;,.,.;;, .W!li•il~·rit(rti.~efo.O: ·l,~ :~ u,.n:',- ."""'die."="''-':~-~ Of '<!(afl.i,c't\9ri:· ! ") . , .. .-~ .... ' ... ' . ,

16. If ~ ~ x ~ -~, <hen >in 1 (sin xJ"' equal to

(n)x (b)x-2~

(c)2n-x (d)-x

' (a) 2

{O) 3R

'

value ,,

!3. Ne1'!tion of·'Ram t< 10 d"" X or Rashrui is in do., XII";,

(o) Ram is not in da« X and Raslmri is not in doS<XTT

(h) Ram IS not in do" X b11t Ram;, class in XII (o) E1ther Ram J> not rn cla"X or Ram "no! in

cio:.; XII (d) None of the aUov~

19. The alg,.hr~io sum of the dcv1ation of 20 obscr;-allotts mearured from 30" 2. 'lbe mean uf deviad011 is (a) 2S.S

(c) 29.6 (b) 30.1

(d) 30 .. 1

fhe ba'<' uf a cliff is circular. From the eJ:tre1n11 '"' <f" diamOier of Lhc b..,. of angles of elevation of tl1e top uf the diff are 30" and oo·. If the he!ghL uf the cliff be 500"" Ll1cn the d!omeLe< ul Llte base of the cliff i<

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lf.f(x)=,m' ~+""'lx 1 ~)

Trosxmo(x~~J _ I .I', _

on~ ~ 1 1 • l, then go{(x) "' c~cwl to '4)

{a) 0 (\,) I

[c) ""'o (d) Noncoftltc"'

The Rrea of tho quaartlateral formed by tllo tangeut> althe end po1nt.1 oflatusrecturn w the

- x' y' elitpbC <) I s· -lis

(«) z: sq unic (b) 9 ;q unit

(c) -¥; >q unit (d) 27 sq unit

'lhe potu\' ( 0. {). (l, 3) and (82. 30) ""

venice< of (a) an obtu"" dnoled tria11gle (b) "" ocULO ongled tnangle

(c) a nght angled tnangle (d) '-lone of the above

Tbc cemre of the circle pd,illl; thwugh the point (0, 1} .md touching the curre y = x' at

point (2, 4) J5

I 16 271 Cal ,-s' 10

( 16 53)

(c) -5'16

il equal'"

' (a) 2

(c) 2_ '

(b) (_16 5~1 ' 7, 10;

(d) None oft~«<

' cos 8

28 Suwose A,. A,, _ .,A_., ""' tllirty sru each with fure element> ano:l Rv R,, ____ , R" are n >ets

each wltir three clements such that

"' " u il, = u B, =S. u- """~~~merit of S boelongs ·-] ,_, 10 cxaetly t<:n of A, '• and exacrly 9 of the 8,',, then the value of n IS

(d) IS (c) 4-S

(b) 135 (d) 90

...- Direction~· ; !l~,;ij.;; """' ':!>i ~lf s;, ""'' '!'>•,-"" -~"""" .. lYP<- -nor~ -~""'>if iliw.- (/liiS!ioils <\JMlW_ '!"!>­- '"""?"-"" ' : Srrtl<n\irn- r 'fli.,.,...,i(i .,.,, 'S!)l"""'~'- 11 ~'(I:«<" i>f '·"'-" 1"""""" olio'""-' fi>Ur llltmli<CW< ihl>ihls,-~ <=<ifL<I,fubi>llli~,j,i,-, ~~.~"'"""'' ""k<tff><> "'_,: '~''"' )'n 1ll< ~ ( aJ,;M; f2 rube! fdl '-'! ril< P'•><!i<!clo''

("! ""'""""' l is "'""-Sl\1wm-n;:,. _"'!>\- ""'"'~-d.ll ~ . .-,.,.to fl"ll!-'I:!~Dl\for s;,w,mu. -

·11>/' s:rn..m.nt l ~·lni<, Sli<.\<""IW ~'""'fo!<. " {<I -"""""'"' t a,-<m., stLitim<nod~·""""· hti . Soi~ i .; ""~ s..,M:.tli Iiu,tnoi; 5limm''''t il' ,;

a "'' """·"'JJIa~Jb; q,;,.,,..,;, ·i.·, 29- Statement I : If e~uations ox' + bx + c ~ 0;

[o,b,«R) :md :?.x'+3>:-+4~0 hm'<' a

cu1Il!l1on root then a: b : c = 2, 3, 4.

Sta<ement II : Roots of 2x' + 3~ + 4 ~ 0 ore

illlilgi:rory.

~0. Statement!: Iiu + b·t c ~ l2;1a, b, c ~ 0). then moXJmum value of abc i5 64.

Stat<:ment II : M•<llnum value occurs ;vh<n a-b~L

31. Moh!menll: The termlndcpcndem ofx in the

cxpan•ion of ( :< ~ ~ + 2 )" i< "c" _

Statemenl.!!: ln • binomial tllparu;ion middle torm "ino:lependent ofx.

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j cos' x e"""" "' dx is equal to

• • • , 0 ' sm ~ Coo _cos x+o "--,-+c 4

(d) Noneoft:hese

A b<Jx cuntdiru 100 Lld<.eu, numbered l, 2 •... ,lOG. l"wo tickets are chosen at rnndom. lt" gi""n that the maximum numl>er on the nvo chOS<n tlckets is not more than 10. Then, the prob•biliry that the minimum number on themis5,is

(<!) !~

" . ) 12 lC J:§ (d) None of these

..,. PHYSICS AND CHEMISTRY

' {c) '· '" •• '" < ,,, •• '" '"' (c) u. ,,, u. ,,,

n. '" "· "' ... "' ~ . '" ... '" ... "' "· ,,, ~. '" " '" ~. "' " "' ... ,,, " '" "· "' ~. ~d) "· "' ... '" ... "' ... "' "· "' n. "' " '" ... '" ~. '" "· '"' "

,,, "· ,,, "· ,,, •• MATHEMATICS

), ,., ' ,,, •• ,,, •• ,,, •• "' '"' (a) u. (b) u. ,., "· (0) "

,,, "· ,., ... "' u. ,,, "

,,, " "' ... ,,,

l3. "' " ,,,

" [a)

34. Let p(xl~"o+o1 x'+a,x'+ .... +o.x'" be a polynom110l 1n a n<ol variable .:< wi<b O<"o ~"'<a,< .... <cr.. The fun«ion !'(rl m

'· "' "· "· "· ~. ... "· "· •• u. "· "·

(a) n<Oither a maximum nor a minimum IJ>) only on• maximum (c) only one minimum (d) only one maximum ;md only one mmunum

H [·] denotes the great- function, then the

I" ' integral 0

[x ] dJ< egmili

(b] Z+ ..;2 (d) None of these

,,, •• "' '· "' •• '" '"' " ,,, ... ,,, ... '"' '" "· (dj " "' "· ,,, ,,, ~. '" OL "' "· "' ,,, ... ,,, ~ '" ~. (c) ,,, ~. ,., "· ,,, ~. ,,, ,,, ~ ,., "

,,, "

,,, ,,, "· ,,, " (a) ~ . "' ,,, "· ,,,

"' •• ,,, '· ,,, • ,,, ,,, " '" ... "' "· "' "' "· ,.,

" '" " ,,, ,,, "· '" "· ,., "· "'

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Cllang< io angular momentum

" . > ~ • ~

~1-05x!0' 34 J-s

1. lo semicmd~crors, lhe charge carrlen are doctroru and holes, so conducllon Is due to 00.

3. If the matert.al and length are ;erne as well as the ,o-etching weigh~ then F, L andY are fixed

in the relot!on .>!. ~ !t -Then, , . 1J.L ~ A• AM~ constant

A1.>~. 1 nA,M, After •ubst!tut!ng the values, wo get

t. L, - o.os mm 4. For o thin ring , .

KE • :i/"' Here, I ~ mr' ~ 2. 7 (O.os)' • O.OJ 721<g-m'

OJ n 1.5 rev/ s• 3 ~ tad!• After rub>tituting the values, we get

KE~ 0.763 J

5. Let n be the number of ele<trons in the body. - 2X 10--< ~-n X L6 X lQ_,.

••

+ 2.5x IO" x !_6x 10-t9

n • 3.75 x 10"

R nR,A1"

3.~A11' '•

7. From kepler'< law, eqMI area will be swept in e<jnal time intervals_

3. When light 1s passing from one modiurn to another. then frequency Md ooimrr will not oilang<.

9. It depends on nature of gas, C1 is different for diffu:rent gases and it olso dof>"'\ds on process .

10. Due ro tem{ler1lture difference heaLLran,(cc i> taking place from one b<>dy to another, and no external source is rupplymg eneiJfl to the bodtes, thus total internal energy of the rwo OOdie• remain' conserved olthough individually it chang«.

11. From the diagram shoWTI

29 ~ 180° 90° 0 ~45"

1~. Al~ng initial drrcttion, v, ~ 3m;,_ Along perpeodkular d ir<"i on

1 ml•'t i L ___ ,,,

v ~U+lx:J~3m!< )' ,,----------,- •C

I vi~,,., 1 vy ~3v2m;s 13. lhre•- gtlll' are fired toward• the oentre of

circle as sh<><vn in figure.

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15.

SiTI<<, """1 fioo.l momentum 1f zero, a~d no oxteiT'.al forc< is "ctmg on rh< system. ~o total irutial momcmum should al:;o be zero.

Thrc" \'e<:tms, which are at OTI ongle of 120° leads ro zero resultant if and on!), if tbcy have same magnitude. So,

4.50 v, n 2.5 x 570- 4.Sv, After soh-in~,"" witl get

v1 and v, he .. ~20m/s.

fcl "[LI1 j" 3xlO' rn/s

[g] "[L"J' 'J

"!On1/s'

'""rLT-'l~r 2 [L'l 2

]

~-~xlO' "3x!O'; w

T"3xl07s

Al: the »>im uf" rough oontactwherc >!upping is about to occlll the twu for<cs acting on eadt object are normal reacti011 N and frictional

furce~-

Thm, limiting "ngle

"

... (!)

Now, at the point ohliding down the angle ol \nclmation of the plane with the honzontal is '-"lk<1 the angle or repose (u).

From h~;ttrt

~tmg cos a = mor sm u

ur '''"""~ "' ""llll\-'(j_t) .. (li)

from Eqs. (il and (jj)

)., "" In elec:1:Tc<!mtics ihcr< \s no net motion of ohorge portides b:tt electrons (frco) can move r~o<lomly tesulting in zero n<t motion. In

rurrent eleciri<ity. clem-ons move with '"""" 1\Ct ""locttyor we can;ay, nel motion \s there.

"' 'b_ Wotk- p, co• 37'

"(6)(4) (0.30)

"1?.2.!

o (n _+n _J, Curt~'"=:!= '" n

' ' (2 68 X JO"" + 3.92 x 1Q 10 j (1.6 K !0-!0)

1.00 =l0.56mA

D\r<cdon of eurrenr is from p05itiv>3 dc<trndo ro negative electrode a> No- ore po<itively charged while Cl- an- neg>tiwly charged_

19. All arc 1'<\Mc charactetistics of S<ehaek effect.

20. 1,-~J==l"''"'

" H ,0,

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L _____ _jc,;lctlonlos•

Let in "bove e>rnmple, if blocks are- movin~ reJativt'ly then ""ic friction is pl'<'sent ""' hoth the blods will rnuve rogether and hence w1,is po,;nve for bigger block_

The third statement is trUe only for statirnlary

~· '""~ H~JI [·.r,~ll Power~---;-=---

nme l!f. !It

" ~-__,. ~, - ~,

~ ~-1 ' . ' n, >n, Wnt< KVl fur CDH:

' 1, (2+ 2+ l + l)-4-1, (2- 2+ 1 + 1)+ 4= o =* 61,=61,

___ (i)

!llld 12 +1,=11 ... (ii)

Apply KVL for EFAB

1,(2+2+1+1)-4+1,(5)=0 ... (ilil lifter solving the f,qs. (i), (ii) and (hi), we get

11 = 0.5A, I 2 = 0.2511,/3 = 0.251\

26. Ampere's law slates J B d/ = ~,J- equivalem ofGanss's law for elecrnc held in electm<;tatie<

Gauss's law fur magnetic field states jB dS = 0.

Here perc<ntage error in 1 mrn 0.1

I= 100 ,,;;-" 100 =TOO X 100 ~ 0.1%

" ' 0 l anu% errorlll T = 2 < lOO X 100 =0.05%

:. %error ing = %error ml + 2(% error In T J = 0.1 + 2x 0.05= 0.2%

w. l!.u-u,-u, _ mgh _ l7'.gll, _ mgH, -;---,.---;n-;---,-

+R +R ' ' L' 2 ~<l.U+L' 1

u,~"'f<-.,~.

' L':=··;;:mgR,

Platos 2 and 3 ai€ joined rogether bm they ore neither mnnected w any of the terminal of the battery nor to any other «>l1!'CC of charge. So, they jointly from "" isulared systeiiL Befu.re closing the switch the charges shown m ftgure (a).

''e

I•J s.~o~c wh<n mi<e'les a<ewen

<OO 0

O,e '"" (liOO+<l)r.o.C ' ,,;.,~

(0) ' '· when

"'"''"'"' =clu•eti 200 v

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Let qWC) cha'!l"' goes from the battery. Then

c!mrges on both wii!!IIfSI'•s• by q. Applyjng Kirchhoffs see<>nd law in figure (b)

afte:r <l<»inl! the 'witch

200- (~~0_+ ql)-(150 + ql ~" ', 6 ' 3

Charge on 6 ~r is 700 ~c •nd !JJl 3 1!f is 250~C.

33. From all the three switthes 100 IIC of charges will Dow.

I Chemistry

36. CctCl2 has Cd2+ ion (4d101 whlch i5 culourle"-

37. DDT-a pesticide, ;, not e.asily and rapidly

degraded in narure.

0 0 II II

36. R --C -l\1!2 + Br,_--; R --C- NHBr-----> -~'

0 II •• _...... ..

\!! -C -NL Br--t R-N=C =0 ~ ..

~:;: J "•' RNH 1 + C0 2

This reattion i:l called HofmaM-bromamide re.actlmL

39. Viscosity is an intenstV€ property

4{L Rorc~klcxf'cc,r'cc.t'

Order ~ •urn of allexponemlal <enn ~1.5-e-(-1)~0.5

41. In 01oho hydrogen, proton revolve in the sanle d1oeceun.

4J. llypnphosphonc acid (H,P,o,)

~ ~ , ___ , 1\ 1\

1!0 01!1!0 OH

l>ohypopl=phoric arid lH.,P,O,) 0 0 " II

1!--- --D-P-01! I I OH 01!

The ouput voltage ,; constant at some input wltage catled the reverse breakdown volrase ofa Zener diode H<re lt is 6 V.

Pitch~ j

·~ ~-,-=G Sl!llD

Least <Otmt =

~==~Pitch Numtlei of division on oi...,ular scale

"~ =---so =!J.Olmm

Diphosplwrus acid (Pyrophosphorus acid,

ll,P,O,l-

~ IT H-P---0 -P-H

I I

'" '" DiphOIJ>h<>ri~ R,P,O,l

(Pyrqlbosphorio

Solubility of 12is greatly inorell>ed by adding tU salt RI in water soluli~n-

' CHC!B.-2 + Me,COK-----+ CC1Br,--.,CC1Rr

Chlorobromo carbene (; CC!Br) undergoes addition reacnon. The reactions, fo which oner<actant {$ oridi&ed while dw. other i< l'€duced, are ealled redo»: reactions.

" +S ..,; 2PC15 ,....... 1'\;l": + PClO

In the above reaction Lhe oxidation Slate of P remruns the .. me .,;en after dWociation, thus it is not a rodoY '""ction. (every disliOCiation reac-tion i> not a redox roacnon)

ZP<:l, .-= PCli + PCl, .,, ' ., .,,,, (tn.,ool h;p,...m ;o. ,) [ "'"" ""'"ll ·· "'"'b"""O

PCI5 =

"

a " \j /j---,"

"

(no lone pair of electron)

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(no lone pair of electron)

47. 3" halide !11 alroholic KOH prefers £ 1 path

Y)"~,oo•yY

ll (mo"' """'') pro<luot i> tb< ma)OT proow:r and the==~ S<O<eoselecm"eOO<

""'""'P""Iic

48. OJ, =Oi---C = N

' ' ' prop-2-orte-1-nitril<!

49. Galvanic oeU is electrochernioal cell in which

emf i• geneTated at r~~ "'" of reducrlon­oxido~ott

SO. Nwnber of orbitals in a sub-shell~ 21 + I. Eoclt of these oan bave one electron hav'..ngthe same value of spin.

Sl. Since 11.2 L ofO, ga> contain half mole i~ N/2

rmlecoles, hence Mlf mole of NH, will weigh molerularweight 17 •

~ 2 ~-yfto.5~

52. !;umbel of corner atom (A)= 8 x ~ = 1

Number offace OOJttre atrun (B) ft 6 x ~ = 3

Hen<e, formula of the compound Is AB,.

5 •. p=X,,p' A,=%0 =~:~~::~0.7-'i x, = 1-0.75 = 0.25

55. Entropy (magmtude of "'nrlnmness] decreases at surface during adsOl1JUOn.

56. Turbidity is due to oolloidal fumted, "' a rosult ofd1e oxidation ofii1Sby wme n'tdising agent

(radi""l') P""""' m th~ nuxture.

H§ + [OJ ----> H,P + S

57. For optical omvity at least one bide,ltate ligancl :iliould be there.ln (a) all ligands are different, hence chiral. Thus, [Cr('lH3 )4Cl2 r will not

exh!bit optical activity. ... ~

'"·

Awlying ~<!' m ~·n(n + 21 I' •'ff m 'i.92 , . " obtained from n = '> R ---CH-coo· ___, R -CH- OOOH

I ... I +NH, +NH,

It is naturally predominating ''' acidic wlution to avoid ionisation nl -COOH.

61. A in the acidic medium, appears"" S"l type

product"' 3" "''00<-.ation is more stable thafs ~,

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"' I CH, -C-CH, forms am! A i> . .

' "" '"• I CH,ITH, "OH OH

II oppeors as SK2 type product as 1° ;, less

>terically hindered that's why

'"· ' CH, ----{;- CH., foiTils and II i:; - I - • ue

6~. 2CH0N,~:l:Cl-l, 63. TI!<ul~llcally che pH should b< negative. But

practically it i< not f>OSSible beeaUM' cff<O«ive concentration of H,o• ions will not be equal tu

IQ Min >udl a ooncemrared solutio-n. Uence, pH is approximately zero.

64. ColltttOrs •dsorb themselves polar group grains of ores and thus, derive ore intu the frmh.

65. In a period, on moving fmrn left to right, the numt>er nf <hell< remain constant but number of eloctrons incr""-'<S ie, effective nuclear charge l_z) increases

atomic radii ~ ~ Thu>. atumic radti of elemonts of VI group ""' smalle< than the ewmic md1i of corresponding clcmene< of group V.

66. Reaction is Diels-Alder reacdon.

0 u II

+ CH,=CH-C-CH,­dlenoplnle

Zn. !l,O

67. Sappos• mole of HCOOH ~ ~

Mole of!-1[,04 ~y HCOOH-t H2S0 4 __, COi H;O + H,SO,

' ' H2C20 4 + H,SQ 4 -->CO+C0 2

Y y -' +H,O+H,SO,

Total ga=us prodllct> ~ x -' 2y Only CO, 1< ah,orbed by 1\0H, y mole

Hence,- Y H~ '

"'2y~6)' x~4y

" " 0

' ' 68. The nitration of arulino is difficult to oarry out wit.':L nitrating mixmre, smce - NH, g.ronp get oxidised v-hich i< nnt requLred. So, tlw amino group is fn-.r l"mec"ted by acylallnn to form aceronilide which i> then "'<rated to give p-n>lro acetanilide ., a major product.

69. p(lJ2 )~(1400Torri(0.6S.\) ~ 9SO Torr= 959 1 /6Q ann ~ J .26 arm

Acmrdmg to Hem}''s l'w

"Nnount of gas •b,orbed is directly proporMn•l "' p~'>ssure •

Hence, -'-=!·26 •"!' IBmL 1 a.m

V-23ml

"· Jmti•l O.E mol - w, 0.5 mol

after a period of time H2 being lighter, effu"' faorer and hcnc'<', in larger amol!Ilt. So, rema!nir.g hydrogen m"-'' bo lo«or.

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Which is an equation of a circle.

Hence. the required h:m is a drde

2. Since, AB-o

[ co,•" lUH< >!fi "]

cm<J.<in" 'in'«

'[ '"''' '"''""'],[" ",] c<"~ >in~ sin'~ 0

4 'l'"" Gt eo<~ COS (U - ~) ""' fl sio " cos(« - ~~

oosusirr~oos(~.-~Jl_'i" "] sirr<I>in~ms(<I-~)j 0 0

-> ccs(«-~l-0

So. (St -~lis an odd mnlnpl~ of%·

3. Le!Q (x1, y 1)be !he imo!(e Gf the point P (3, 8) in the line x + 3y - 7. Then, PQ " petpe•rdiculor Ln gfven lin~.

' " I ', -·-·-~ --1=-1 x 1-3 , 3,

'=> :>v::,- y, = 1 ... (i) ,,, . ,,...,. fx1 +3 y 1+8) hl"' """ pomt o "'< !S 0 • " w cuues ' ' . Oil the line<, 3y -7.

x,+3 ,•y1+8)·• 2 + 't 2

='> x1 +3+3y1 -24=14 = x1 +3y1 +13=0 ... (ii) On sohirrg Eq<. (i) and (ii), we g<t

x,=-l,y1--4

!Ienre, doe coordino<es of poull Q "'~ (-I, - 4).

-tim --• • . rx-l (X2

-1°) i·c''"''f'"'' •~' X -1 X 1 x [

(x":-1")j ,., -1+2+3+4+ ... +"

n("+l)

' 5. Since, j(x)-sin-•(-"-,)

l+ X

l-~·~~"· -'-"-·-~! j' (x)-' o-x'l' V'(,::.J

(l+x'J 2(1-x'J

,•'(l x"i (I+ x'i' 2(1->2)

(l+x2)jl x'l

if fxf>l

j' {x)doc' not cxi>t f<>rl x-1 ~I ie, x = ± 1

Hence, j{.<)" differentiable on R - { ·1, 1}.

&. !"he equanon nf giveTl rurve is

y - e"' - x'

Atx=O,y-e0+0-1

On iliffer=tidtin); E4. (i) w.r.t. x, we ge<

:i;' =2!'" + 2x

=> (d;j =2e0+0-2 \ '(~!)

... (i)

The eq<ldllOn uf Lhe ld<ltt.Hl "' puim [0. 1) is

y-1=2(x-Ol

2x-y +1~0

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om:. R<quired distanc"<O ~ length of perpendirular

from point (0, 0) w:t< - y + 1 = 0

j'0-0+11 l =~71= /S

7. 'Tile gwen equallon is

U3- t)sin ~ + h'3 +I) rose= 2

Let (,13 - 1) = r ;in ct and (>13 + 1) = 1 """' ,,~,,"' r=,,(,3 J) +(-'3+lJ -2~2

,/3- l and ton n ~ ~- -=tan (60"- 45°) = tan 15"

,'3- 1

~ 0:=150-~2

.-. csilw,in8+roose<ros6=2

z,12ctJ>({I-a)=2

' " CO>(E «)= .rz=C<>S4

' ' Q-12= 2n~±4 . ~ ~ " => =~nlt=4-12

a. Required number of amUJ8ements

6! 51 = 2131 -Ji- 60-20 =40

9. let fue coordinates of A and B be (a, 0) and

{0, I>J respettively. Cleatly, t>OA!I is a right

angled triangle, the hyporeno"" iltl is a

diameter of the cirde.

' l

""''1/~ ,."'""·""''':fc::=:::7:,c,,.,.,, • ,

' ~~~ = 2 (3k) = 6k

Now, 0.42 +OR'- All'

,.,(i)

Let («, ~)be the cootdinates of the centrot'd of . ' .l.OAB, thenn = 3 , ~- 3

From F.q. (i),

(:hf+(3~i=36k'

"" a'+~'=4k'

Thus, tlle locus of cennoid of l>OAB is

x'+y'-4k'.

10. The eqmttioo of the tang~t to y' = 4...: at

p<Jint (.<1, y 1) is Y.l'! ~ 2 a {x + x1).

or 2=- Y.V1 + 2ax1 ~ 0 ... m

I~ {h, k) t.. the mid J>Oint of QR- Then, th~

equation of QR ;,

'"'"- 2o [X+ h)- 4ob~ e -4<I(h + 0)

(·.·r~s,l

.., 2ax-i<y+k' 2<Ih~O ... (ii)

Clearly. Eq•. (i) and {ii) repre.ent the = ~'-

1 _!!. _ 2ox1 1-k-k'-2ah

_, k ~ y1 and F- :?nh ~ 2ax,

y1' - :?nh ~ :?nx 1

__., 4ax 1 - 2ah = 2ax, => h~x1 !fence, th• nud point of QR w (x, y,).

li. We know that the product of perpendicut.rs

from two feci of an etlipse upon any tang.nt i:!

equal to b" if, 3.

12. Givon that, f{>:) ~ (x - 21"" (2x + 1)

=> f'{x)-i (x -2T'" (2x + 1) + 2(x- 2/''

Clearly, f' (x) i:! not definod at x r 2, S<:> "~ 2

is a o;ritioal point and fm another crittca[ point,

putj'(x)~u .

. ·. ~ (x - :.rr''' (2x + 1)+ 2 [X - 2'F' ~ o

=> 2x+l=-3[X-2) => X -I lienee, I and 2 ore two crit:iC>ll points of f(x).

13. Smce, !taCL·1 (~J-~coC1 x~- 1 .,.1 x'

)', ! (em'(~))~ =-C ~--:x,<h --2f~ -__!____,dx~-2[t=" 1 x)~

H>

• • '

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omQ..+6)1=(2)l+l)lt

~+b=Oand2tJ.+lrO

h-6,~=-~ __. __, ..... _,. a+2b+6c=U

15 n(S]=6'=6x6x6=216

n(E) =Cocllkient nf x' in(x + x' + ... + x•)'

= Ca<fficient of x' _, in ( \-:_ x:)' = Coefl'icicnr of x'-' inn- x•f (I - xl '

= CoefficteJJt of x' -' in (I - x)-"

(·.·Hk$.8) _,_,_,_,~

- -,.. 3 ·I

='·'c, .. lk-lllk-~

' i'!ertre, the probability of tho requiie<l event

.. Solution< }Or Q. II<>. 16 ta 17.

'.

~- 3•~ ~Slt ' ulvtll, ·2 =X~ 2

:.sin-1(sin x}= x- 2~

17- Sinoe, 2tan _, x =" - •in-"L 1 !" ;r"). X ~ 1

2<an-1x+>in-'( ~ ,) ,.,

•• 18. (.et p; Ram ;, in class X

a:u:l q · Rasbmi is in class XII. Given propooition is pv q.

,_20-~0+_.! - 20 20

=3<1+0.1m3(),1

'"' ]niiAEC, tan 60"=d •

=-> d1 =5J:m

and lnallliC, tan so"= ~00

' "" d,=SOOJ3m

Required diameter, ;oo c

AB "'"d, + d, =--:fi+ 504h3

500 2000 ~-,]3(1+3)=-./3 m

... (i)

21. We know that! z- z11 »I• - •,1 rtl'""'""ts the regloD on riJ!ht side of perpendicular bisertoi

of"' and o,. Th"", the given inequality I< - 21 > i • - 41 represen"' !be region oo right •ide of

Jl'!'l"""llcular b~ of 2 •nd 4-,. Re(>}0>3andlm(rl~R.

:1.2. /(x)~sln'x+sin'(x+~)

+cos"""' (x+ ~J

-om' x + ( siu rcos ~ + ""' r •in j)' +"" x( <:OS X""'~- sin x sin ~J

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om

2 sin' ;r 3 ms' " 2--/J = sm x ~ -,- , --,- - + 22 sm X cos _,

'"'' :.< - ,'3 +-2

--oosx"nx 2 , " 2 1 ' "n X o CO<S X 00S X

~'m x< -,-, --,-•-,

Ssin'r 3ros"x 12cos1 x --:r- + --, 5 - , 5 ~ 4 (,;n•x+<>" x)~ 4

:. gof(x)=g[f{x)]=g m=l 23. Given <QUation of elllp<e 1S

' ' :':__+L =I

' ' 0<=-./a'-- b'

=,'9-5~2 Hy S)'llllnetry, tl1e quodril•teral ;, a rhombu,_

'

~-,. so, wtH! area is four times the "'''a of the light angled triangle forrned t>y the r.ongent ond aJUS in the!" quadrant.

=> ~qo.atio.LJ oftangenl at ( 2, ~) "

' ' ' :gx+3 s=l

~+-1:=1 912 3

.-. !lrea of quad~ktteral JIBC!J

~ 4 (area of !I 1\0B)

=4x~x~x3 =27 sq unit

24. The '"'" of tnangle (ormed by the verhces . '1 ! 0, 0 '(I, 3) and (82, 31J) \ ,,

• ' ' ' ' 30 I

= 1[-%(1- 82), 1 (3G- 24{i)J

=~[<.:lfl-215]=0 Thus, tMe points are collinear.

25, Let ~1e centre b< O(h, k). So, OA' =08 2

and {>lope of OA) (slope of tangeJ\t at A)=- 1

8 (0, 1)

~-­

~2.4)

' (!>- 2)2 ~ (k- 4l' = h' ~ (k-ll'

4h + M-19=0 H Also, siope otoA = h _

2

... (i)

11I1d equ&LiOil of Tflflg"'" at (2, 4) toy ~x' is y+4~2x 2,n.,Jo~is4_

k-::__±_4=-l (·: m,m,=-lJ ,_, 4k--16=-h+2

www.exam

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omSin<o, rhe line; in•ersect, •herefore they nmsr

have a point in COJtli'T>an. ie, :<-1 y•l z-1 --o -~- -=~

2 3 4 ::<-3y-k£ -,-~--2=1 =f1

~ <=2l+Ly=3l.-l,"=4/,~J

(say)

(;ay)

and x =~ + 3, y = 2~ + k, 2 =~ore otmrc-

2A+l=u +3, JJ..-1 =2~· k

and 4~+1=f1

On "'Ivins these. we get

' ~=-;;:•nd"=-5

k=:JI.-2"-1

=3(-~)-2(-51-l

' " =2

Sinc-e, each A, ha> 5 dement>.

'" :£ n(ii,)=5X30-J50 ,_, - (r)

11m distinct elemffitt: m S and each dement of

S belon&' to exattly )0 of A;s. ,owe have

" l: nlA,) = lDm _, From F.q<. (i) and (ill. v:e get

10,-, = 150

3n~9m

Henrc,

.. (ii)

... (rii)

lfrom Eq. (iii)]

'"·

If one root i5 onmmon, then other root lS also

oommon .

. Roo"' art COll)Ugate (u, b, < e R).

Hellce, both cquatlorn are rdeittic"I.

a:b:c=2.:3:4

Hence, Statement I i< true.

Since, AM2GM

a+~+c"-(abcf"

E<o(abc)1"

' ah: 5 64

Also_ ma,in1um value of abc Is 64 only wh011

a=b~c

H51CC, option (d);, ru<ruL-

''"r­Now.rx+--+21 = ,:, \ .< ) •

~"t>(xfl-c

Fur rndep<ndent of x, 21 - r = 0 => r- 21

T,, '' = "'c,l

lienee, SIO<ement T is true.

In a bmomia! e"!""l'i"" (-<- a)' (¥ly) middle

term is independeru ui xwhich is possible onlv

when.;·a=l ·

Henoe, it l< "ot neu:ssary thattnidddle term is

lndependt"t of x.

32. Let I~ I ous' x e1« "" dx

~ I '""' '"'111 , .rr m.n=t => -slnxd::<=<ll

r~-f•'rlr

'4 "''4"' =-4+c=---4-+'

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om

·--.-"- r-v""' --,·c;- 9 (Since, in the cbosen rtcl:eto OJle number is S and !he rest of !he numben ""'Y be 6, 7. 6, 9, 10.)

~4. GWen. p(x) ~a, + o, x' + a,x4 + .... + a,x"' ... (i) where a,><>,_1 ><>,_, >· .. ><1; >a, >"o > 0 On differentiating Eq. (i) w.r.t. x, we get p' (r) ~ 2 a,x+ 4a,r' + ... + 2 na,x'"-1

~ 2r (<It + Z<t,x2 + .. _ + na,x"'- 2 ) ... {il)

where (o, + Z<t,x' + ... + na,x""-'l > 0,

\;f X dl

k, p'(xl changes sign from (--vel to ( +v.l at X- 0. '!'Mrefure, p (:r)atrains minimum at x - 0. Hence, it has only one nUnimurn at X - 0.

" I ,r, j" 3.'1, J0

" {x"Jok~J, Ook+J, ld>:;- ,.-, 2d>:

-o+[r>f"+2[xJ!~ -.J2-1+2(l.S-J2) --/2-l+3-2J2 ~2-../2