PHYSICS. The figure shws fur pairs f plarizig sheets, see face-. Each pair is muted i the path f iitially uplarized light. The plarizig directi f each sheet (idicated by the dashed lie) is refereced t either a hriztal -ais r a vertical y ais. Rak the pair accrdig t the fracti f the iitial itesity that they pass, greatest first (A) (i) > (ii) > (iii) > (iv) (B) (i) > (iv) > (ii) > (iii) (C) (i) > (iii) > (ii) > (iv) (D) (iv) > (iii) > (ii) > (i) Fial itesity f light is give by Brewster's law I I cs ; where = Agle betwee trasmissi aes f plariser ad aalyser. Hece decreasig rder f itesity is (i) > (iv) > (ii) > (iii). The maimum itesity f friges i Yug's eperimet is I. If e f the slit is clsed, the the itesity at that place becmes I. Which f the fllwig relati is true? (A) I = I (B) I = I (C) I = I (D) There is relati betwee I ad I Suppse slit width s are equal, s they prduces waves f equal itesity say I. Resultat itesity at ay pit IR Ics Fr maimum itesity ma where is the phase differece betwee the waves at the pit f bservati. I I I (i) If e f slit is clsed, Resultat itesity at the same pit will be I ly i.e. Cmparig equati (i) ad (ii) we get: I I O I IO (ii). If the critical agle fr ttal iteral reflecti frm a medium t vacuum is, the velcity f light i the medium is (A) m/s (B).5 m/s (C) 6 m/s (D) sic si v.5 m /s m/s
. A presbypic patiet has ear pit as cm ad far pit as cm. The diptric pwer fr the crrective les fr seeig distat bjects is (A) D (B) D (C).5 D (D).5 D I this case, fr seeig distat bjects the far pit is cm. Hece the required fcal legth is f d(distace f far pit) = cm Pwer P cm.5 D f 5. I the fllwig circuit I ad I are respectively (A), (B) 5 ma, 5 ma (C) 5 ma, (D), 5 ma Sluti (D): Equivalet circuit ca be redraw as fllws 6. A wavefrm shw whe applied t the fllwig circuit will prduce which f the fllwig utput wavefrm. Assumig ideal dide cfigurati ad R R (A) (B) (C) (D) Sluti (D): The P-N jucti will cduct ly whe it is frward biased i.e. whe 5V is fed t it, s it will cduct ly fr rd quarter part f sigal shw ad whe it cducts ptetial drp 5 vlt will be acrss bth the resistrs, s utput vltage acrss R is.5 V. V.5 V
. Highly eergetic electrs are bmbarded a target f a elemet ctaiig eutrs. The rati f radii f ucleus t that f Helium ucleus is /. The atmic umber f ucleus will be (A) 5 (B) 6 (C) 56 (D) By usig / / R A R R A R A / / R A / A () R He A = 56 s Z = 56 = 6.. The pht-electrs emitted frm a surface f sdium metal are such that (A) They all are f the same frequecy (B) They have the same kietic eergy (C) They have the same de Brglie wavelegth (D) They have their speeds varyig frm zer t a certai maimum Sluti (D): Cceptual 9. The istataeus values f curret ad vltage i a ac circuit are i si t A ad e si(t / )V respectively. If the resistace is the the reactace f the circuit will be (A) (B) (C) / (D) V = i Z = Z Z = Als Z R X L () () X L L X. A cpper disc f radius. m rtates abut its cetre with revlutis per secd i a uifrm magetic field f. Tesla. The emf iduced acrss the radius f the disc is (A) V (B) V (C) mv The iduced emf betwee cetre ad rim f the rtatig disc is E BR. (.) vlt (D) mv. A square lp f side a hags frm a isulatig hager f sprig balace. The magetic field f stregth B ccurs ly at the lwer edge. It carries a curret I. Fid the chage i the readig f the sprig balace if the directi f curret is reversed (A) iab (B) iab (C) IaB (D) IaB
Iitially F mg IaB (dw wards) whe the directi f curret is reversed F mg IaB (dw wards) F IaB. A hmgeeus electric field E ad a uifrm magetic field B are pitig i the same directi. A prt is prjected with its velcity parallel t E. It will (A) G mvig i the same directi with icreasig velcity (B) G mvig i the same directi with cstat velcity (C) Tur t its right (D) Tur t its left Sluti (A): Here magetic frce is zer, but the velcity icreases due t electric frce.. I the circuit shw, the pit B is earthed. The ptetial at the pit A is (A) V (B) V (C) 6 V (D) 5 V Curret i the give circuit 5 i A (5 ) Ptetial differece betwee A ad B VA VB VA V VA V. A resistace f ad a wire f legth 5 metres ad resistace 5 are jied i series ad cected t a cell f e.m.f. V ad iteral resistace. A parallel cmbiati f tw idetical cells is balaced acrss cm f the wire. The e.m.f. E f each cell is (A).5 V (B). V (C).6 V (D). V V ir E l l l L e R E ( R R r) L l 5 E V (5 ) 5 h
5. A capacitr f capacitace 5 F is cected as shw i the figure. The iteral resistace f the cell is.5. The amut f charge the capacitr plate is (A) C (B) 5 C (C) C (D) 5 C I steady state cditi. N curret flws thrugh lie (). Hece ttal curret.5 i A (.5) Ptetial differece a crss lie () = ptetial differece a crss capacitr Vlt S, charge capacitr = 5 = C 6. Figures belw shw regular heags, with charges at the vertices. I which f the fllwig cases the electric field at the cetre is t zer (A) (B) (C) (D) 5
Electric field at a pit due t psitive charge acts away frm the charge ad due t egative charge it act s twards the charge.. A thi spherical cductig shell f radius R has a charge q. Ather charge Q is placed at the cetre f the shell. The electrstatic ptetial at a pit p a distace R frm the cetre f the shell is (A) (q Q) R Sluti (D): Electric ptetial at P k. Q k. q Q V R / R R (B) q R Q R (C) Q q R R (D) Q q R R. A trasverse wave is described by the equati Y Y si ft. The maimum particle velcity is fur times the wave velcity if Y (A) (B) Y (C) Y (D) Y Cmparig the give equati with y a si( t k), We get a = Y, = f, particle velcity (v ma ) particle a Y f ad wave velcity (v) wave (v ) v ma Particle Wave Y Y f f. f f k / k. Hece maimum 9. A cylidrical tube, pe at bth eds, has a fudametal frequecy f i air. The tube is dipped vertically it water such that half f its legth is iside water. The fudametal frequecy f the air clum w is (A) f / (B) f (C) f / (D) f 6
pe clsed As l v l pe v v v l l / l clsed clsed pe pe l pe, i.e. frequecy remais uchaged.. A ma havig a wrist watch ad a pedulum clck rises a TV twer. The wrist watch ad pedulum clck per chace fall frm the tp f the twer. The (A) Bth will keep crrect time durig the fall. (B) Bth will keep icrrect time durig the fall. (C) Wrist watch will keep crrect time ad clck will becme fast. (D) Clck will stp but wrist watch will fucti rmally. Sluti (D): Fucti f wrist watch depeds up sprig acti s it is t effected by gravity but pedulum clck has time perid, l T. Durig free fall effective accelerati becmes zer, s time perid cmes g ut t be ifiity i.e. the clck stps.. O ivestigati f light frm three differet stars A, B ad C, it was fud that i the spectrum f A the itesity f red clur is maimum, i B the itesity f blue clur is maimum ad i C the itesity f yellw clur is maimum. Frm these bservatis it ca be ccluded that (A) The temperature f A is maimum, B is miimum ad C is itermediate (B) The temperature f A is maimum, C is miimum ad B is itermediate (C) The temperature f B is maimum, A is miimum ad C is itermediate (D) The temperature f C is maimum, B is miimum ad A is itermediate Accrdig t Wei s law, m T cstat r y b Tr Ty Tb r TA TC TB. Efficiecy f a Cart egie is 5% whe temperature f utlet is 5 K. I rder t icrease efficiecy up t 6% keepig temperature f itake the same what is temperature f utlet (A) K (B) K (C) 6 K (D) K T 5 5..(i) T T T 6 T ' T Dividig equati (i) by (ii), T '..(ii) T 5 5 5 T K T '
. A small sphere f mass m is drpped frm a great height. After it has falle m, it has attaied its termial velcity ad ctiues t fall at that speed. The wrk de by air fricti agaist the sphere durig the first m f fall is (A) Greater tha the wrk de by air fricti i the secd m (B) Less tha the wrk de by air fricti i the secd m (C) Equal t mg (D) Greater tha mg I the first m bdy starts frm rest ad its velcity ges icreasig ad after m it acquire maimum velcity (termial velcity). Further, air fricti i.e. viscus frce which is prprtial t velcity is lw i the begiig ad maimum at v v. Hece wrk de agaist air fricti i the first m is less tha the wrk de i et m.. A thi metal disc f radius r flats water surface ad beds the surface dwwards alg the perimeter makig a agle with vertical edge f the disc. If the disc displaces a weight f water W ad surface tesi f water is T, the the weight f metal disc is (A) rt W (B) rtcs W (C) rtcs W (D) W rtcs Weight f metal disc = ttal upward frce = upthrust frce + frce due t surface tesi = weight f displaced water + T cs ( r) = W + rt cs T 5. Tw plaets have the same average desity but their radii are R & R. If accelerati due t gravity these plaets be g ad g respectively, the g R g R (A) (B) g R g R (C) g g R R (D) g g R R Sluti (A): g R g GR. If = cstat the g R 6. The frce actig a bdy mvig alg -ais varies with the psiti f the particle as shw i the fig. The bdy is i stable equilibrium at (A) (B) (C) bth ad (D) either r Whe particle mves away frm the rigi the at psiti frce is zer ad at, frce is psitive (repulsive i ature) s particle mves further ad des t retur back t rigial psiti.
i.e. the equilibrium is t stable. Similarly at psiti frce is zer ad at, frce is egative (attractive i ature) S particle retur back t rigial psiti i.e. the equilibrium is stable.. I the figure shw, a blck f weight N restig a hriztal surface. The cefficiet f static fricti betwee the blck ad the surface s.. A frce f.5 N will keep the blck i uifrm mti, ce it has bee set i mti. A hriztal frce f N is applied t the blck, the the blck will (A) Mve ver the surface with cstat velcity (B) Mve havig accelerated mti ver the surface (C) Nt mve (D) First it will mve with a cstat velcity fr sme time ad the will have accelerated mti F R. mg. N i.e. miimum N frce is required t start the mti f a bdy. But l s applied frce is ly N. S the blck will t mve.. A particle f mass m mvig with hriztal speed 6 m/sec as shw i figure. If m M the fr e dimesial elastic cllisi, the speed f lighter particle after cllisi will be (A) m/sec i rigial directi (B) m/sec ppsite t the rigial directi (C) m/sec ppsite t the rigial directi (D) m/sec i rigial directi Sluti (A): Speed f separati = speed f apprach. Speed f heavy particle after cllisi remais same as m/s. v 6 v m/ s 9. A circular plate f radius R/ is cut frm e edge f thi circular plate f radius R ad mass M. The mmet f iertia f remaiig prti abut a ais thrugh O perpedicular t plate f plate is: y R O R/ (A) MR (B) MR (C) MR (D) 5 MR 9
I remaiig = I whle I cut MR Icut Mass f cut circle = M/ Nw, R O R/ M R M R Icut MR MR MR (Parallel ais therem) 6 MR MR Hece I remaiig MR. A massless rpe is wrapped arud a rig (with a grve alg its circumferece) havig a radius R ad mass m. The rig is allwed t mve dwward. The liear accelerati f the rig is: (A) g/ (B) g/ (C) g/ (D) g Mg T = ma a ad TR = II R mr a T = ma R mg ma = ma a = g/ a mg T
CHEMISTRY. The iizati eergy f hydrge atm (i the grud state) is kj. The eergy required fr a electr t jump frm d rbit t the rd rbit will be (A) /6 (B) 5 (C). (D) 5/6 Sluti (D): E E EZ kj I.E fr H-atm E.Z 5 5 Thus; E E E.Z kj 9 6 6. The majr prduct f the fllwig reacti is: CH CH HCl H (A) (C) CH HC H Cl CH CH H Cl (B) (D) HC H CH CH H Cl It is simple case f Markwikff additi. The reacti prceeds as: CH Cl. Air is cmpressed i a vertical cylider by the weight f a pist. Whe a weight is added t the pist, the vlume f gas decreases frm 5mL t ml. If ather weight f same magitude is added, the vlume f the gas will decrease by: (A) ather ml (B) less tha ml (C) mre tha ml (D) isufficiet ifrmati Let, the iitial pressure f gas=p 5p p p.5p.5p atm is icrease i pressure due t etra weight. It weight is dubled, etra pressure =.5p ad ttal pressure will be.5p. 5P.5p.V V.mL Hece d decrease i vlume is 66.6 ml ml
. I the fllwig reacti; Aldehyde Alchl HCHO t BuOH CHCHO MeOH The best cmbiati is: (A) HCHO admeoh HCl Aldehyde Alchl Acetal (B) t HCHO ad BuOH t (C) CH CHO ad BuOH (D) CHCHO ad MeOH Sluti (A): O OH HCl C R' OH R C H R H OR' Reactivity decreases with steric hidrace i bth aldehyde ad alchl. 5. The ph f the sluti prduced whe a aqueus sluti f strg acid ph 5 is mied with equal vlume f a aqueus sluti f strg acid f ph is: (A). (B).5 (C).5 (D). Sluti (A): 5 V V. [H ] V ph.96 6. Fr the dissciati equilibrium NO (g) NO (g), which represets crrectly the variati f degree f dissciati ' ' with D/d; where D is vapur desity f NO ad d is vapur desity f equilibrium miture. (A) D/d (B) D/d (C) (D) D/d Sluti (D): mn O D NO mmi dm (NO NO ) If d D r D d Als if icreases, D d icreases. D d D/d. What is the chage i iteral eergy whe a gas ctracts frm ml t ml uder a cstat pressure f 5 trr, while at the same time beig cled by remvig J heat? [Take: ( L atm) = J ] (A) J (B) J (C) 6 J (D) J
E q w 5 P atm V.L 6 w (. ).L atm J E J. The hydride that is NOT electr deficiet is: (A) AlH (B) BH 6 (C) SiH (D) GaH GaH,AlH ad BH 6 are the hydrides f grup- (s p ), whereas SiH is a hydride f grup. S, BH 6 AlH ad GaH are electr deficiet hydrides. But SiH is a electr precise hydride f grup- (s p ), i.e. these hydrides ca have the required umber f electrs t write their cvetial Lewis structures. 9. Fr the reacti, A B C D, the statemet t crrect is: (A) The rate f disappearace f B is e furth the rate f disappearace f A (B) The rate f appearace f C is half the rate f disappearace f B (C) The rate f frmati f D is half the rate f csumpti f A (D) The rates f frmati f C ad D are equal da db dc dd dt dt dt dt. M (CO) is a ragametallic cmpud due t the presece f: (A) M C bd (B) M M bd (C) C O bd (D) M O bd Sluti (A): M (CO) is a rgametallic cmpud due t the presece f M C bd.. The crrect decreasig rder fr acid stregth is :- (A) NOCH COOH NCCH COOH FCHCOOH ClCH COOH (B) FCHCOOH NCCH COOH NOCHCOOH ClCH COOH (C) NOCH COOH FCHCOOH CNCH COOH ClCH COOH (D) CNCH COOH ONCH COOH FCHCOOH ClCH COOH Sluti (A): All the give cmpuds are -msubstituted acetic acid derivatives ad the - substitutis have bee made by strg-i grups/atms. Mre pwerful the I grup, strger will be the acid. -I pwer f differet grups is as fllws: NO CN F Cl Thus, the crrect decreasig rder fr acid stregth is NOCH COOH NCCH COOH FCHCOOH ClCH COOH
. Dry air was passed successively thrugh a sluti f 5 gm f a slute i gm f water ad the thrugh pure water. The lss i weight f sluti was.5 gm ad that f pure slvet is. gm. The mlecular weight f the slute is (A).5 (B).5 (C).5 (D) e f these Sluti (A): P P s lss i weight f water chamber P s lss i weight f sluti chamber P Ps w Ps N m W. 5.5 m. The majr prduct f the fllwig reacti is: () KOH (aqueus) () CrO /H () HSO / (A) (B) (C) (D) The rad map f the give reacti is as fllws:. The cmple that has highest crystal field splittig eergy ( ), is : K [C CN ] (B) [C(NH ) 5(HO)]Cl (A) 6 (C) K [CCl ] (D) [C(NH ) 5Cl]Cl Sluti (A): All the cmple give are the ctahedral cmplees f C(III) ecept K [CCl ], which is a tetrahedral cmple f C (II) (sp -hybridised). We kw, t t 9 S, the ctahedral cmplees (a, b, c) have higher values tha that f tetrahedral, K [CCl ], Nw, fr the cmplees, a, b ad c, the magitude f ligad stregth, which is based their psitis i the spectrchemical series. Cl H O NH CN Hece, K [C(CN) 6] will have the highest value.
5. "E " differs by. vlts if the ph is chaged frm X t Y, keepig the MO /M,H ccetratis f all ther species equal. What is the umerical differece betwee X ad Y? (A) 5 (B) 6 (C) (D) e f these Sluti (A): 6. Accrdig t mlecular rbital thery, which f the fllwig is true with respect t Li ad Li? (A) Bth are ustable (B) Li is ustable ad Li is stable (C) Li is stable ad Li is ustable Sluti (D): (D) Bth are stable. The prduct frmed i the reacti f cumee with O fllwed by treatmet with dil. HCl are: (A) (B) (C) (D) The give prcess is cumee prcess (Hck prcess) t sythesize phel ad acete idustrially. I Hck prcess, Ph grup migrate ad release HO. The radius (R) f atms frmig a clsed packed structure i the give figure is The maimum radius f atm ccupyig the ctahedral hle (H) is: 9 (A). cm (B) (C)..5 cm H cm. 5
(D).5 cm Sluti (A): Give radius sphere is R( cm). Let radius f vids be r. The ctahedral vid. I the triagle ABC AC (AB) (BC) (R) (R) [R r R] R R r R r R R r R r.. R 9 r.r.. 9. The majr prduct btaied i the fllwig reacti is : (A) (B) (C) (D) Rate f acetylati: NH OH because N-bases are strger tha O-bases. Size f N-atm is larger tha O-atm ad at the same time, N atm is less electregative tha O-atm. S, N-atm f the NH grup ca date its le pair f electr (Lewis basicity) mre easily tha that f O atm f the OH grup. 6
5. Hall-Herult's prcess is give by: (A) CrO Al AlO Cr (B) Cu (aq.) H (g) Cu(s) H (aq) Cke, 6K (C) ZO C Z CO (D) AlO C Al CO Sluti (D): Hall-Herult s prcess is a electr-reducti prcess by which pure alumia ( AlO ) is reduced t crude Al. I this prcess, electrlysis f a fused miture f AlO,Na [AlF 6] (crylite) ad CaF (flurspar) is carried ut at carb cathde ad graphite ade. The verall reacti is represeted as: AlO C Al CO 5. Which is the mst suitable reaget fr the fllwig trasfrmati? OH CH CH CH CH CH CH CH CH CH CH CO H (A) alkalie KMO (B) I /NaOH (C) Tlle's reaget (D) CrO /CS OH I /NaOH CH CH CH CH CH CH CH CH CH CH CO H CHI 5. Amg the fllwig cmpuds, which e is fud i RNA (A) (B) (C) (D) Sluti (A): RNA ctais, adeie (A), guaie (G), cytsie (C) ad uracil (U).
5. The chlride that CANNOT get hydrlysed is: (A) CCl (B) SiCl (C) SCl (D) PbCl Sluti (A): The cmpuds give are the tetrahalides (MCl ) f grup elemets. Fr the hydrlysis, (uclephilic substituti) f MCl the ature f the M Cl bd shuld be as Here, M ca be Si, S ad Pb because they have vactat d-rbital. But carb is a member f secd perid (=, l=, ), it des t have d-rbital (l=). S, CCl will t be hydrlysed. PA PB 5. Fr a ideal biary liquid sluti with which relati betwee X A (mle fracti f A i liquid phase) ad Y A (mle fracti f A i vapur phase) is crrect, X B ad Y B are mle fracti f B i liquid ad vapur phase respectively: (A) XA YA (B) XA YA X (C) A Y A XB YB (D) X A,Y A,XB ad Y B cat be crrelated ' PA PA XA PM YA ' PB PB XB PM YB PA XA YA P X B B YB A A A X B B YB P X Y P 55. Which f the fllwig cmpuds reacts with ethylmagesium brmide ad als declurizes brmie water sluti: (A) (B) (C) (D)
56. c.c. f.5 M NaOH sluti is added t 5 c.c. f.5 M H SO sluti. Mlarity f NaOH i the resultig sluti will be (A).6 M (B).5 M (C). M (D). M Sluti (A): Meq. f HSO.5 5 5meq. Meq. f NaOH.5 5 meq. HSO is limitig reaget s meq. f NaOH left after reacti=5-5=5 meq. 5 MNaOH.6M 5 5. A metal cmbusti i ecess air frms X. X up hydrlysis with water yields HO ad O alg with ather prduct. The metal is: (A) Li (B) Mg (C) Na (D) Rb Sluti (D): Metal (A) is rubidium (Rb). I ecess f air, it frms RbO (X) X is a superide that have O i. It is due t the stabilisati f large ai by large catis thrugh lattice eergy effects. RbO (X) gets easily hydrlysed by water t frm the hydride, HO ad O. The reacti ivlved are as fllws: Rb O RbO (superide) RbO HO RbOH HO O (X) 59. The majr prduct f the fllwig reacti is: CN (X) 5. Number f electrs havig m fr magesium atm is: (A) (B) (C) 6 (D) Sluti (D): 6 Mg :s,s,p,s p-rbital has m,, ; s rbitals have m O (i) DiBAL H (ii) HO O CHO CHO CHO (A) (B) (C) (D) O OH O CHO O CH NH OH OH 9
CN CHO O (i) DiBAL H (ii) HO OH O CHO 6. Hw may g f KCl wuld have t be disslved i 6 g HO t give % by mass f sluti? (A) 5g (B).5g (C).5 (D).5g Sluti (A): mass f slute w % by mass r r w 5g mass f sluti (w 6)
MATHEMATICS 6. If A ad B are tw matrices with ad elemets respectively, the the prbability that the prduct AB is pssible is (A) (B) Let A: umber f matrices pssible with elemets. The pssible rders f matri A are:,, 6, 6,, A 6 Let B: umber f matrices pssible with elemets. The pssible rders f matri B are:,,,,,, 5, 5 B Hece S 6 (C) Let E: umber f pssible rders f matri A & B such that the prduct AB is pssible. The pssible rders f matri A & B such that the prduct AB is pssible are:,, 6,,, 5,, Hece E. Therefre the required prbability is. 6. Which f the iequalities belw is TRUE? (A) si si si (B) si si si (C) si si si (D) si si si Sluti (D): 5 & si si si si si si 66 si66 9 si9 Sice fr,, y si is a icreasig fucti, therefre 9 5 66 si9 si5 si66 si si si (D) 6. If the circles y a by c & y b ay c, where c, have eactly e a b pit i cmm the the value f is c (A) (B) (C) (D) Sluti (A): Fr the circle y a by c : Ceter a, b C Radius r a b c a b c Fr the circle y b ay c : Ceter b, a C
Radius r b a c b a c Sice the circles have eactly e pit i cmm hece distace betwee the ceter must be equal t the sum f the radii. C C r r a b c b a c a b a b c b a b c a b ab a b b a a b a b a b c a c c a b ab a b c c a b 6. Suppse that w ad z are tw cmple umbers such that bth iw ad iz are differet real umbers. The slpe f the lie cectig w ad z i the cmple plae is (A) (B) (C) (D) Let w i iy ad z iy As per questi: iw R & iz R i iy R & i iy R y i y & Im y i Im y & y y y y y y y y Slpe y y 65. Number f regular plyg that have itegral iterir agle measure (i degrees) is (A) (B) (C) (D) Let regular plyg be sided; Sum f iterir agles Oe iterir agle 6 6 Fr iterir agle t be itegral, Z 6 Z 6 Fr t be a iteger, must be a divisr f 6. 6 5 Hece umber f divisrs f 6 are Hece there are pssible values f. But sice cat be r therefre umber f pssible values f are. 66. Frm a pit P utside f a circle with ceter O, taget segmets PA ad PB are draw, where A & B are pits f ctact. If the legth f chrd AB is AO PA 6 (A) 6 (B) (C) (D)
As per questi: 6 AO PA I the figure give belw: Legth f chrd AB, where Q is midpit f AB. Legth f taget PA PB L, Radius AO BO R, PO = d R L 6 R L d RL 6 RL 6 RL d (i) Area f APO R L PO AQ R L d LR d d d, (usig (i)) AB r Cr is r (A) 9 9 (B) 9 (C) r Cr r Cr rr r C r r r r rr Cr r Cr r r r r Cr r Cr r r r r 9 9 9 9 9 6. The value f 9 (D) 9 6. If a & b are tw uit vectrs ad is the agle betwee them, the si is (A) a b (B) As per questi: a, b a b a b a.b a b (C) a.b (D) 9 a b a b a b cs, a, b cs cs a b si a b si si a b si si a b
69. Csider the set A,, ad the relatis R,,,,,,,, R,,,,, R,,,,,, R 5,,,,, the (A) Ris refleive (B) R is symmetric ad R 5 is trasitive (C) R is refleive ad symmetric bth (D) R is trasitive A relati R i a set A is called: a. Refleive, if a,a R fr every a R. b. Symmetric, if a,a R a,a R c. Trasitive, if a,a R ad a,a R,a R,a,a where a,a A a where a A. Verifyig pti (A): R is t refleive sice, ad, R. Hece pti (A) is t crrect. Verifyig pti (B):, R, R, R is symmetric, sice, R, R ad, R, R. R 5 is trasitive. Hece pti (B) is crrect. Verifyig pti (C): R is t refleive sice, &, R. Hece pti (C) is t crrect. Verifyig pti (D): R is t trasitive, sice, R but, des t R. Hece pti (D) is t crrect. R,. If the statemet r : p ~ p q the r is lgically equivalet t (A) p ~ q p (B) p ~ q ~ p (C) q p p (D) q p ~ p Sluti (D): p q ~ q ~ p, (law f ctra psitive) Therefre p ~ p q ~ ~ p q ~ p p ~ q ~ p q p ~ p., (De-Mrga s law). The arithmetic mea ad mde f a data are ad respectively, the the media f the data is (A) (B) (C) (D) Sluti (A): Media Mea Let media = M. Mde M M M M. The umber f pssible tagets which ca be draw t the curve 9y 6 which are perpedicular t the straight lie 5 y is (A) zer (B) (C) (D) Sluti (A): 5 Slpe f the lie : 5 y is.
Therefre slpe f the required taget is m 5 5 y Let y m c is taget t the hyperbla. 9 c a m b c 9 5 c 6 9 c 6 c 5 5 5 c 6 c c 5 Hece there is such taget pssible. y. If a taget havig slpe, f the ellipse, a,b is rmal t the circle a b y the the maimum value f ab is equal t (A) (B) (C) (D) Sluti (A): y Equati f taget havig slpe, f the ellipse a b y a b (i) is is: y Ceter f the circle y,. Sice taget is rmal t the circle therefre the taget must pass thrugh ceter f the circle. a b a b a b 6 a b / a b 6 / ab a b, AM GM ab ab Hece maimum value f ab is.. Let f ad g f & f f 6 g are tw ctiuus fuctis satisfyig the fllwig cditis: R the the graph f g is (A) symmetrical abut the lie (C) symmetrical abut the pit, (D) t symmetrical As per questi: f f 6, (i) R Ad g f f g (ii) Frm (i) & (ii): g g, R 6 g g, R Replacig by : g g g g g g Hece graph f a b (B) symmetrical abut the pit, g is symmetric abut the pit,. 5
5. If f d g, (where is cstat f itegrati) the f d iverse f f ) (A) f g C (B) f gf (C) f g C (D) f g C I f d Let f t t f / d f tdt / I t f t d dt tf / t dt dt I II f t f tdt tf t g t f g f C t f / t t C dtdt f f f g f C C is equal t ( f dy 6. The curve which satisfies the differetial equati ad which passes thrugh, is d y (A) a pair f lies thrugh, (B) a hyperbla with eccetricity (C) a hyperbla with eccetricity dy ydy d d y ydy d ydy d y C y C Sice the curve passes thrugh,, therefre C C Hece the required curve is y. y y y y y, which is a hyperbla f the frm. a b y y Cmparig with : a b a & b. b a e e e e e (D) a ellipse with eccetricity is 6
. Fr the parabla y y which f the fllwig is / are crrect (i) Verte at, ad ais parallel t -ais (ii) Verte at, ad ais is parallel t -ais (iii) Verte at, ad fcus at, (iv) Verte at, ad ais is parallel t y-ais (A) (iii) & (iv) (B) ly (ii) (C) (i) & (iii) (D) ly (i) y y y y y y y y y y Abve is a parabla f the frm y a, where the rigi has bee shifted t,. Hece verte,. Als sice parabla is f the frm y a, therefre ais f parabla is parallel t -ais. Ais f the abve parabla is y. Fr the abve parabla a, hece fcus,,. dy y. If the sluti f differetial equati ca be epressed as P y Qy C where C d y is cstat f itegrati ad P, Q are c-prime umbers the P Q (A) (B) (C) (D) 5 Sluti (D): dy y dy y dy y d d y y dy y d dy y dy yd dy yd dy y dy y dy d y y y dy d y y C y y C y C y y y C y Cmparig the abve with P, Q P Q 5 P y Qy C : cs cs is/are si cs (A) (B) (C) (D) ifiitely may 9. The umber f itegral value(s) i the rage f f Sluti (A):
f cs cs cs cs si cs 9 si cs cs cs cs cs cs cs 9 cs cs 9 cs cs 9 cs cs 9 cs cs 9 cs cs cs cs cs 9 cs cs 9 cs cs 6 cs cs Hece rage f f is, 6.6 9 &.5 cs cs f 6 cs 6 9 f. 6 Hece umber f itegral value i rage f f is zer.. I a regular heag havig cetre at rigi whse e verte lies at, is such that the prduct f the cmple umbers represetig the ther five vertices is a ib, the the value f a b is (A) (B) 6 (C) 9 (D) Sluti (A): I cmple frm Z i. 9 The cmple umbers Z,Z,Z,Z5 & Z6 ca be btaied by rtati f cmple umber Z abut rigi 5 i aticlckwise directi by a agle f,,, & respectively. i / Z ie, i / Z ie, i / Z ie, i / Z 5 ie & i5 / Z 6 ie. Prduct P ZZZZ5Z6 i i/ i/ i/ i/ i5 / ie ie ie ie ie i 56... 5 5 i5 i e ie ie ics5 isi5 i i i Cmparig P i with a ib : a,b a b.
si. If d A the / cs d is equal t (A) A (B) A (C) A (D) A si A d d si d si d si d d I d II si cs si cs d d si si cs cs cs lim d d d Let y. y & y y d dy / cs A y y dy / cs y y dy / cs d f y dy f d, / / cs cs A d d A. A lady parks her car i a parkig lt ad parkig her car she bserves that there are w 5 cars i a rw. O her retur she fids that eactly 5 places are still ccupied. Assumig that she des t parked her car at either eds, the prbability that bth the eighburig places f her car are empty, is 5 (A) (B) (C) (D) 9 6 9 Sluti (D): Out f 5 cars, e car is her car, i.e., cars were there ther tha her car. Whe she came back 5 cars were there, i.e., cars were there ther tha her car. I ther wrds ut f cars, cars were there, i.e., ut f cars, cars left t create vacat places. Out f cars vacat places ca be created i Let E: The evet that bth the eighburig psiti are vacat. C ways, hece S C. C ways, hece C If eighburig places f her car are vacat, it ca be de i E. (Sice tw eighburig cars must have t be mved ut ad ut f remaiig ther cars, cars must be mved s that ttal f cars are mved ut t create vacat places)! E C P E!!!!!! 9! 9 5. S C!!!!!! 9!! 9
. Each side f triagle ABC is divided it equal parts. The rati f area f heag UVWXYZ t the area f ABC is 5 (A) 9 (B) (C) (D) Area f heag UVWXYZ = Area f triagle ABC (Area f triagle AUV + Area f triagle BZY + Area f triagle CWX) (i) c b Area f AUV sia bcsi A Area f ABC 9 9 c a Area f BZY sib ac sib Area f ABC 9 9 a b Area f CWX sic ab sic Area f ABC 9 9 Frm (i): Area f heag UVWXYZ = Area f ABC ( Area f ABC + Area f ABC + Area f ABC ) 9 9 9 Area f heag UVWXYZ = Area f ABC Area f ABC 9 Area f heag UVWXYZ = Area f ABC Area f ABC Area f heag UVWXYZ = Area f ABC Area f heag UVWXYZ Area f ABC r. I the figure shw, radius f the circle C be r ad that f C is (where P ad Q beig ceter f C & C respectively) where PQ r the legth AB is (A) r (B) r (C) r (D) r Sluti (D):
I PAC & QBC : PAC QBC 9 PCA QCB, (vertically ppsite agles) Sice tw agles are equal therefre all the three agles are equal sice sum f agles i a triagle is. PAC ~ QBC PA QB PC AC QC BC r r / PC QC PQ r PC r & QC r AC BC PC PC QC QC AB AC CB PC PA QC QB r r r r r r r r r r r 5. Let S k : 5... k k the which f the fllwig is true? (A) priciple f mathematical iducti ca be used t prve the frmula. (B) Sk Sk (C) S k Sk (D) S is true S k : 5... k k Fr k : S :, which is t true, ad hece priciple f mathematical iducti ca t be used t prve the frmula. S k : 5... k k Sk : 5... k k k Sk : 5... k k k Sk : k k k Sk : k k k Sk : k k Hece Sk is true wheever k Sk Sk. S is true. is 6. The sum f the terms f the series a a a a a a... a a a a a a (A) (B) (C) a a a a a a a a (D) a a
Sluti (A): Let S a a a... a... a. a. a. a S... a a a a S a a a... a a S... a a a... a terms a a S a a a a S. a a. I a ABC si Asi B C si Csi A B the which f the fllwig terms will frm a AP. (A) ta A,taB,ta C (B) si A,si B,si C (C) cs A,cs B,cs C (D) e f these siasib C sicsia B si B CsiB C si A BsiA B, A B C, if si B C si B C si A B si A B si B si C si A si B si B si A si C si A,si B,si C are i AP. si A, si B, si C are i AP si A, si B, si C are i AP. cs A,cs B,cs C are i AP........ If lim... k the k 5 (A) 6 (B) (C) (D) 9............ lim lim...... lim lim...... lim 5 lim......... lim... lim lim... lim.. lim..
r r r r lim r lim r lim 5 5 5 d d d 5 5 5 k k k 9. The largest iterval lyig i, fr which the fucti f cs lg cs is defied, is (A), (B), (C), (D), Sluti (D): f is defied if ad cs r ad,. 9. Let f : R R be a fucti defied by f mi,. The, which f the fllwig is true? (A) f fr all R (B) f is t differetiable at (C) f is differetiable everywhere (D) f is t differetiable at, mi f R f