fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES

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1 fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

2 GATE-8 (PHYSICS).. rry on mr h.. For rbitrry mtris E, F, G nd H, if EF FE = thn Tr (EFGH) is qul to () Tr (HGFE) Tr (E), Tr (F), Tr (G), Tr (H) Tr (GFEH) Tr (EGHF). An unitry mtrix i i b d is givn, whr, b,, d, α nd β r rl. Th invrs of th mtrix is () b i d i b i d i i i b d b i d i. Th url of vtor fild F is xˆ. Idntify th pproprit vtor fild F from th hois givn blow. () F zxˆ zyˆ 5yzˆ F zyˆ 5yzˆ F xyˆ 5yzˆ F xˆ 5yzˆ.4 A rigid body is rotting bout its ntr of mss, fixd t th origin, with n ngulr vloity nd ngulr lrtion. If th torqu ting on it is nd its ngulr momntum is L, th rt of hng of its inti nrgy is () L L L.5 A ylindr of mss M nd rdius R is rolling down without slipping on n inlind pln of ngl of inlintion θ. Th numbr of gnrlizd oordints rquird to dsrib th motion of this systm is () 4 6 fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

3 .6 A prlll plt pitor is bing dishrgd. Wht is th dirtion of th nrgy flow in trms of th Poynting vtor in th sp btwn th plts? () Along th wir in th positiv z- xis Rdilly inwrd ( rˆ ) Rdilly outwrd (r ˆ) Cirumfrntil ( ˆ) xˆ ẑ rˆ ŷ.7 Unpolrizd light flls from ir to plnr ir-glss intrf (rfrtiv indx of glss is.5) nd th rfltd light is obsrvd to b pln polrizd. Th polrizd vtor nd th ngl of inidn i r () prpndiulr to th pln of inidn nd i = 4 o prlll to th pln of inidn nd i = 56 o prpndiulr to th pln of inidn nd i = 56 o prlll to th pln of inidn nd i = 5 o.8 A finit wv trin, of n unspifid ntur, propgts long th positiv x- xis with onstnt spd v nd without ny hng of shp. Th diffrntil qution mong th four listd blow, whos solution it must b, is x () x, t r, t v t v t m x i x, t t t r, t.9 Lt dnot th ground stt of th hydrogn tom. Choos th orrt sttmnt from thos givn blow (), L J x L y S, L. S x S y fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

4 . Thrmodynmi vribls of systm n b volum V, prssur P, tmprtur T, numbr of prtils N, intrnl nrgy E nd hmil potntil µ, t. For systm to b spifid by Mirononil (MC), Cnonil (CE) nd Grnd Cnonil (GC) nsmbls, th prmtrs rquird for th rsptiv nsmbls r () MC: (N, V, T); CE: (E, V, N); GC: (V,T, µ) MC: (E, V, N); CE: (N, V, T); GC: (V,T, µ) MC: (V,T, µ); CE: (N, V, T); GC: (E, V, N) MC: (E, V, N); CE: (V,T, µ); GC: (N, V, T). Th prssur vrsus tmprtur digrm of givn systm t rtin low tmprtur rng is found to b prlll to th tmprtur xis in th liquid-to-solid trnsition rgion. Th hng in th spifi volum rmins onstnt in this rgion. Th onlusion on n gt from th bov is () th ntropy of solid is zro in this tmprtur rgion th ntropy inrss whn th systm gos from liquid to solid phs in this tmprtur rgion th ntropy drss whn th systm trnsforms from liquid to solid phs in this tmprtur rgion of tmprtur th hng in ntropy is zro in th liquid-to-solid trnsition rgion. Th rdil wv funtion of th ltrons in th stt of n = nd l = in hydrogn tom is R r xp, / is th Bohr rdius. Th most probbl vlu of r for nd ltron is () 4 8. Th lst two trms of th ltroni onfigurtion of mngns (Mn) tom is d 5 4s. Th trm ftor of Mn 4+ ion is: () 4 D / 4 D / F 9/ D 7/ fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 4

5 .4 Th ohrn lngth of lsr light is: () Dirtly proportionl to th lngth of th tiv mdium Dirtly proportionl to th width of th sptrl lin Invrsly proportionl to th width of th sptrl lin Invrsly proportionl to th lngth of th tiv lsing mdium.5 Mtlli monovlnt sodium rystllizs in body ntrd ubi strutur. If th lngth of th unit ll is 4 x 8 m, th onntrtion of ondution ltrons in mtlli sodium is () 6. x m.5 x m.56 x m.5 x m.6 Th plot of invrs mgnti susptibility ntifrromgniti smpl orrsponds to () vrsus tmprtur T of n TC T TC T TC T TC T.7 Aording to th qur, th K + mson is omposd of th following qur: () u u d u u s s u fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 5

6 .8 An O 6 nulus is sphril nd hs hrg rdius R nd volum 4 V R. Aording to th mpiril obsrvtions of th hrg rdii, th volum of th 54 X 8 nulus, ssumd to b sphril, is () 8 V V 6.75 V.89 V.9 A ommon mittr trnsistor mplifir iruit is oprtd undr fixd bis. In this iruit, th oprting point () rmins fixd with n inrs in tmprtur movs towrds ut-off rgion with n inrs in tmprtur movs towrds th sturtion rgion with drs in tmprtur movs towrds th sturtion rgion with n inrs in tmprtur. Undr norml oprting onditions, th gt trminl of n n-hnnl juntion fild fft trnsistor (JEET) nd n n-hnnl mtl oxid smiondutor fild fft trnsistor (MOSFET) r () both bisd with positiv potntils both bisd with ngtiv potntils bisd with positiv nd ngtiv potntils, rsptivly bisd with ngtiv nd positiv potntils, rsptivly..75 rry two mr h. os sin. Th ignvlus of th mtrix r sin os () i whn θ = 45 o i ± sin th mtrix is unitry i whn θ = o whn θ = o. If th Fourir trnsform F[δ(x )] = xp( iπ v) will orrspond to () δ(x ) δ(x + ) onstnt [δ(x ) + iδ(x + )] [δ(x ) + δ(x + )] fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 6

7 . If I dzlnz C fizis, whr C is th unit irl tn ntilowis nd Ln(z) is th prinipl brnh of th Logrithm funtion, whih on of th following is orrt? () I = by rsidu thorm I is not dfind sin Ln(z) hs brnh ut I dzlnz I C.4 Th vlu of i i z dz is () πi πi ( + i)π d y dy.5 Considr th Bssl qution (v= ), y. Whih on of th following dz z dz sttmnts is orrt? () Eqution hs rgulr singulr points t z = nd z = Eqution hs linrly indpndnt solutions tht r ntir Eqution hs n ntir solution nd sond linrly indpndnt solution singulr t z = Limit z, tn long x xis, xists for both th linrly indpndnt solutions.6 Undr rtin rottion of oordint xs, rn- tnsor v ( =,, ) trnsforms ; ording to th orthogonl trnsformtion dfind by th rltoins v v v v v v ; v v. Undr th sm rottion rn- tnsor T,,b would trnsform suh tht () T, T,T, T, T, T, T, T, T, T, ( T, T, T, T, ) fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 7

8 .7 Th Lgrngin of systm is givn by q () hrmoni osilltor n nhrmoni osilltor qq q. It dsribs th motion of dmpd hrmoni osilltor systm with unboundd motion 8.8 Th momnt of inrti tnsor of rigid body is givn by Th mgnitud 8 of th momnt of inrti bout n xis n,, is () 6 5 8/.9 A hoop of rdius R is pivotd t point on th irumfrn. Th priod of smll osilltions in th pln of th hoop is R R () g 4g R g 9R 7g. A mss m is onstrind to mov on horizontl fritionlss surf. It is st in irulr motion with rdius r nd ngulr spd by n pplid for F ommunitd through n inxtnsibl thrd tht psss through hol on th surf s shown in th figur. This for is thn suddnly doubld. Th mgnitud of th rdil vloity of th mss () inrss till th mss flls into th hol drss till th mss flls into th hol rmins onstnt boms zro t rdius r whr < r < r F fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 8

9 . For simpl hrmoni osilltor th Lgrngin is givn by L q q. If P iq p, q nd H(p, q) is th Hmiltonin of th systm, th Poisson brt {A(p, q), H(p, q)} is givn by () ia(p, q) A * (p, q) ia * (p, q) ia(p, q) i. A pln ltromgnti wv is givn by ( xˆ yˆ)xp{ i( z )}. At givn t lotion, th numbr of tims nr vnishs in on sond is () An intgr nr whn δ = nπ nd zro whn δ nπ, n is intgr An intgr nr nd is indpndnt of δ An intgr nr An intgr nr whn δ = nπ nd zro whn δ nπ, n is intgr nd is indpndnt of δ. A diltri sphr is pld in uniform ltri fild dirtd long th positiv y-xis. Whih on of th following rprsnts th orrt quipotntil surfs? () fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 9

10 .4 A rod of lngth L with uniform hrg dnsity λ pr unit lngth is in th xy-pln nd rotting bout z-xis pssing through on of its dg with n ngulr vloity s shown in th figur blow. ( rˆ, ˆ, zˆ ) rfr to th unit vtors t, is th vtor potntil t distn d from th origin O long z-xis for d << L nd J is th urrnt dnsity du to th motion of th rod. Whih on of th following sttmnts is orrt? () J long rˆ ; long ẑ ; Z d ω J long ˆ ; long ˆ ; P d d J long rˆ ; long ẑ ; d y J long ˆ ; long ˆ ; L d x rˆ r os.5 A irulr dis of rdius on th xy pln hs surf hrg dnsity. Th ltri dipol momnt of this hrg distribution is () 4 x ˆ 4 x ˆ 4 x ˆ 4 x ˆ At tim t =, hrg distribution ( r, t) is givn by () t ( r, t) ( r,) xp ( r, t) ( r,) t / t t ( r, t) ( r,)xp ( r, t) ( r,)xp sin t fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

11 .7 A nonrltivisti hrgd prtil movs long th positiv x-xis with onstnt positiv lrtion ˆ x. Th prtil is t th origin t t =. Rdition is obsrvd t t = t distnt point (, d, ) on th y-xis. Whih on of th following sttmnts is orrt? () Th rdition is unpolrizd Th rdition is pln polrizd with polriztion prlll to th x-xis Th rdition is pln polrizd with polriztion prlll to th xy pln long lin inlind to th x-xis. Th rdition is lliptilly polrizd.8 For physil systm, two obsrvbls O nd O r nown to b omptibl. Choos th orrt implition from mongst thos givn blow: () Evry ignstt of O must nssrily b n ignstt of O Evry non-dgnrt ignstt of O must nssrily b n ignstt of O Whn n obsrvtion of O is rrid out on n rbitrry stt of th physil systm, subsqunt obsrvtion of O lds to n unmbiguous rsult Obsrvtion of O nd O, rrid out on n rbitrry stt of th physil systm, ld to th idntil rsults irrsptiv of th ordr in whih th obsrvtions r md.9 An xt msurmnt of th position of simpl hrmoni osilltor (SHO) is md with th rsult x x. [Th SHO hs nrgy lvls E n (n =,,,..) nd ssoitd normlizd wv-funtions n ]. Subsquntly, n xt msurmnt of nrgy E is md. Using th gnrl nottion Pr (E = E ) dnoting th probbility tht rsult E is obtind for this msurmnt, th following sttmnts r writtn. Whih on of th following sttmnts is orrt? () Pr(E = E ) = Pr(E = E n ) = for som vlu of n Pr(E = E n ) (x) Pr(E > E ) > for ny E n fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

12 .4 Considr th ombind systm of proton nd ltron in th hydrogn tom in its (ltroni) ground stt. Lt I dnotd th quntum numbr ssoitd with th totl ngulr momntum nd lt dnotd th mgnitud of th xpttion vlu of th nt mgnti momnt in th stt. Whih of th following pirs rprsnts possibl stt of th systm ( is Bohr mgnton)? () I =, I =, B = I =, = B B I =, = B.4 A prtil is pld in on dimnsionl box of siz L long th x-xis ( <x < L). Whih of th following is tru? () In th ground stt, th probbility of finding th prtil in th intrvl hlf In th first xitd stt, th probbility of finding th prtil in th intrvl is hlf. This lso holds for stts with n = 4, 6, 8, L L, 4 4 is L L, 4 4 For n rbitrry stt, th probbility of finding th prtil in th lft hlf of th wll is hlf In th ground stt, th prtil hs dfinit momntum fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

13 .4 An inlsti bll of mss m hs bn thrown vrtilly upwrds from th ground t z =. Th initil inti nrgy of th bll is E. Th phs trjtory of th bll ftr sussiv bouning on th ground is () p z p z O z O z. p z p z O z O z.4 A systm ontining N non-intrting lolizd prtils of spin nd mgnti momnt µ h is pt in onstnt xtrnl mgnti fild B nd in thrml quilibrium t tmprtur T. Th mgntiztion of th systm is, () Nµ oth Nµ sinh Nµ tnh Nµ osh fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

14 .44 Two idntil prtils hv to b distributd mong thr nrgy lvls. Lt r B, r F nd r C rprsnt th rtios of probbility of finding two prtils to tht of finding on prtil in givn nrgy stt. Th subsripts B, F nd C orrspond to whthr th prtils r bosons, frmions nd lssil prtils, rsptivly. Th r B : r F : r C is qul to () : : : : : : : :.45 A photon gs is t thrml quilibrium t tmprtur T. Th mn numbr of photons in n nrgy stt ε = ħω is () xp xp xp xp.46 Considr systm of N toms of n idl gs of typ A t tmprtur T nd volum V. It is pt in diffusiv ontt with nothr systm of N toms of nothr idl gs of typ B t th sm tmprtur T nd volum V. on th ombind systm rhs quilibrium, () th totl ntropy of th finl systm is th sm s th sum of th ntropy of th individul systm lwys th ntropy of mixing is N B n th ntropy of th finl systm is lss thn tht of sum of th initil ntropis of th two gss th ntropy of mixing is non-zro whn th toms A nd B r of th sm typ fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 4

15 fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: Considr systm of two non-intrting lssil prtils whih n oupy ny of th thr nrgy lvls with nrgy vlus E =, ε nd ε hving dgnris g(e) =, nd 4 rsptivly. Th mn nrgy of th systm is () 4xp xp 8xp 4xp 4xp xp 8xp xp 4 xp xp 4xp xp xp xp xp xp.48 Thr onsutiv bsorption lins t m, 77. m nd m hv bn obsrvd in mirowv sptrum for linr rigid ditomi molul. Th momnts of inrti I A nd I B r (I A is with rspt to th bond xis pssing through th ntr of mss nd I B is with rspt to n xis pssing through th ntr of mss nd prpndiulr to bond xis) () both qul to.855h gm m zro nd.855h gm m both qul to 6.47h gm m zro nd 6.47h gm m

16 .49 A pur rottionl Rmn sptrum of linr ditomi molul is rordd using ltromgnti rdition of frquny. Th frquny of two onsutiv stos () B, 4 B + B, + 4 B B, 4 B + B, + 4 B.5 Whih on of th following sttmnt is INCORRECT in vibrtionl sptrosopy with nhrmoniity? () Th sltion rul for vibrtionl sptrosopy is = ±, ±,. Anhrmoniity lds to multipl bsorption lins Th intnsitis of hot bnd lins r strongr thn th fundmntl bsorption Th frqunis of hot bnd lins r smllr thn th fundmntl bsorption.5 Th molulr sptr of two linr moluls O-C-O nd O-C-S r rordd in th mirowv rgion. Whih on of th following sttmnt is orrt? () Both th moluls would show bsorption lins Both th moluls would not show bsorption lins O-C-O would show bsorption lins, but not O-C-S O-C-S would show bsorption lins, but not O-C-O.5 Whn th rfrtiv indx µ of th tiv mdium hngs by µ in lsr rsontor of lngth L, th hng in th sptrl sping btwn th longitudinl mods of th lsr is ( is th spd of light in fr sp) () ( ) L L zro L ( ) fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 6

17 ,.5 Th primitiv trnsltion vtors of th body ntrd ubi ltti r xˆ yˆ zˆ b xˆ yˆ zˆ, xˆ yˆ zˆ riprol ltti r () xˆ yˆ ; yˆ zˆ ; C xˆ zˆ. Th primitiv trnsltion vtors, nd C of th xˆ yˆ ; yˆ zˆ ; C xˆ zˆ xˆ yˆ ; yˆ zˆ ; C xˆ zˆ xˆ yˆ ; yˆ zˆ ; C xˆ zˆ.54 Th strutur ftor of singl ll of idntil toms of form ftor f is givn by S hl f xp i x h y z l whr (x j, y j, z j ) is th oordint of n tom nd j j j j hl r th Millr indis. Whih on of th following sttmnts is orrt for th diffrtion ps of body ntrd ubi (BCC) nd f ntrd ubi (FCC) lttis? () BCC: (); (); () BCC: (); (); () FCC: (); (); (4) FCC: (); (); (4) BCC: (); (); () BCC: (); (); () FCC: (); (); (4) FCC: (); (); (4) fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 7

18 .55 Th ltti spifi ht C of rystllin solid n b obtind using th Dulong Ptit modl, Einstin modl nd Dby modl. At low tmprtur >> B T, whih on of th following sttmnts is tru ( nd A r onstnts)? () Dulong Ptit: C xp ; Einstin: C = onstnt; Dby: T T C A T Dulong Ptit: C = onstnt; Einstin: C ; Dby: C xp A T / Dulong Ptit: C = onstnt; Einstin: C xp T T T ; Dby: C A / T Dulong Ptit: C ; Einstin: C xp A T T ; Dby: C = onstnt.56 A linr ditomi ltti of ltti onstnt with msss M nd m (M > m) r oupld by for onstnt C. Th disprsion rltion is givn by M m C Mm C M m Mm Whih on of th following sttmnts is inorrt? 4C sin Mm () Th toms vibrting in trnsvrs mod orrspond to th optil brnh Th mximum frquny of th ousti brnh dpnds on th mss of th lightr tom m Th disprsion of frquny in th optil brnh is smllr thn tht in th ousti brnh No norml mods xist in th ousti brnh for ny frquny grtr thn th mximum frquny t = /.57 Th inti nrgy of fr ltron t ornr of th first Brillouin zon of two dimnsionl squr ltti is lrgr thn tht of n ltron t th mid-point of sid of th zon by ftor b. Th vlu of b is () b = b = b = 4 b = 8 fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 8

19 .58 An intrinsi smiondutor with mss of hol mh nd mss of n ltron m is t finit tmprtur T. If th top of th vln bnd nrgy is E nd th bottom of th ondution bnd nrgy is E, th Frmi nrgy of th smiondutor is v () E F E E v h BT ln 4 m m E F BT E 4 v E m ln m h E E E v h F BT ln 4 m.59 Choos th orrt sttmnt from th following m E F BT E 4 v E m ln m h () Th rtion p p n prod irrsptiv of th inti nrgis of K + nd K - Th rtion p p is forbiddn by th bryon numbr onsrvtion Th rtion is forbiddn by strngnss onsrvtion Th dy K π π - prods vi w intrtions 6. Th following givs list of pirs ontining (i) nulus (ii) on of its proprtis. Find th pir whih is inpproprit. () (i) N nulus; (ii) stbl nulus (i) A sphroidl nulus; (ii) n ltri qudrupol momnt (i) 8 O 6 nulus; (i) U 8 nulus; (ii) nulr spin J (ii) Binding nrgy = 785 MV (pproximtly).6 Th four possibl onfigurtions of nutrons in th ground stt of 4 B 9 nulus, ording to th shll modl, nd th ssoitd nulr spin r listd blow. Choos th orrt on () s p ; J s p p / / / / / ; J 4 s p ; J s p p / / / / / ; J fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 9

20 .6 Th mss diffrn btwn th pir of mirror nuli 6 C nd 5 B is givn to b MV/. Aording to th smi-mpiril mss formul, th mss diffrn btwn th pir of mirror nuli 9 F 7 nd 8 O 7 will pproximtly b (rst mss of proton m p = 98.7 MV/ nd rst mss of nutron m n = MV/ ) ().9 MV/ (.9 +.5) MV/.86 MV/ ( ) MV/ 6. A hvy nulus is found to ontin mor nutrons thn protons. This ft is rltd to whih on of th following sttmnts? () Th nulr for btwn nutrons is strongr thn tht btwn protons Th nulr for btwn protons is of shortr rng thn thos btwn nutrons, so tht smllr numbr of protons r hld togthr by th nulr for Protons r unstbl, so thir numbr in nulus diminishs It osts mor nrgy to dd proton to (hvy) nulus thn nutron bus of th Coulomb rpulsion btwn protons.64 A nutrl pi mson (π o ) hs rst-mss of pproximtly 4 MV/ nd liftim of τ s. A π o produd in th lbortory is found to dy ftr.5 τ s into two photons. Whih of th following sts rprsnts possibl st of nrgis of th two photons s sn in th lbortory? () 7 MV nd 7 MV 5 MV nd MV 75 MV nd MV 5 MV nd 5 MV.65 An.. voltg of V rms is pplid to th primry of : stp-down trnsformr. Th sondry of th trnsformr is ntr tpd nd onntd to full wv rtifir with lod rsistn. Th d.. voltg ppring ross th lod is () 6 44 fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

21 .66 Lt I nd I rprsnt msh urrnts in th loop bd nd bfb rsptivly. Th orrt xprssion dsribing Kirhoff s voltg loop lw in on of th following loops is, () I 5I = 5 I + I = I 5I = 5 I + I =.67 Th simplst logi gt iruit orrsponding to th Booln xprssion 5 d b I I 5 V f Y P P is: () P Y P Y P Y P Y.68 An nlog voltg V is onvrtd into -bit binry numbr. Th minimum numbr of omprtors rquird nd thir rfrn voltgs r () V V V,,, 4, 4, V V,,, V V V V 4V 4,,,, 5 5, 5 5 V V V 4,,,, V 4, 5 fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

22 .69 Th following iruit (whr R L >> R) prforms th oprtion of V R R V V () OR gt for ngtiv logi systm AND gt for positiv logi systm R L V NAND gt for ngtiv logi systm AND gt for ngtiv logi systm.7 In th T typ mstr-slv JK flip flop is shown long with th lo nd input wvforms. Th n output of flip flop ws zro initilly. Idntify th orrt output wvform. CLK Input CLK J J K K Input () fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

23 Common Dt ustions fizis Common Dt for ustions 7, 7 nd 7: A bm of idntil prtils of mss m nd nrgy E is inidnt from lft on potntil brrir of width L (btwn < x < L) nd hight V s shown in th figur (E < V ). V x V x For x > L, thr is tunnling with trnsmission offiint T >. Lt A, A R nd A T dnot th mplituds for th inidnt, rfltd nd th trnsmittd wvs, rsptivly..7 Throughout <x<l, th wv-funtion () n b hosn to b rl is gnrlly omplx is xponntilly dying is zro.7 Lt th probbility urrnt ssoitd with th inidnt wv b S. Lt R b th rfltion offiint. Thn () th probbility urrnt vnishd in th lssilly forbiddn rgion th probbility urrnt is TS for x>l for, x<, th probbility urrnt is S ( + R) for, x>l, th probbility urrnt is omplx.7 Th rtio of th rfltd to th inidnt mplitud A R /A is () - A T /A T in mgnitud rl ngtiv numbr A T E A V E fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit:

24 Common Dt for ustions 74 nd 75: Considr two onntri onduting sphril shlls with innr nd outr rdii, b nd, d s shown in th figur. Both th shlls r givn mount of positiv hrgs..74 Th ltri filds in diffrnt rgions r ) for r ; rˆ for r b 4 r for b r ; E 4 r rˆ for r d rˆ for r ; for r b 4 r b d rˆ for b r ; 4 r 4 r rˆ for r ; E for r b 4 r rˆ for r d for b r ; 4 r rˆ for r d for r ; E for r b rˆ for b r ; 4 r 4 r rˆ for r d.75 In ordr to hv qul surf hrg dnsitis on th outr surf of both th shlls, th following onditions should b stisfid () d = 4b nd = d = b nd = d = b nd > d > b nd = fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 4

25 Lind Answr ustions.76 to.85 rris two mrs h. Sttmnt for Lind Answr ustions 76 nd 77: Considr th β- dy of fr nutron t rst in th lbortory..76 Whih of th following onfigurtions of th dy produts orrspond to th lrgst nrgy of th nti-nutrino? (Rst mss of ltron m =.5 MV /, rst mss of proton m p = 98.7 MV / nd rst mss of nutron m n = MV / ) () In th lbortory, proton is produd t rst In th lbortory, momnt of proton, ltron nd nti-nutrino ll hv th sm mgnitud. In th lbortory, proton nd ltron fly-off with (nrly) qul nd opposit momnt. In th lbortory, ltron is produd t rst.77 Using th rsult of th bov problm, nswr th following. Whih of th following rprsnts pproximtly th mximum llowd nrgy of th nti-nutrino? (). MV.8 MV.5 MV. MV Sttmnt for Lind Answr ustions 78 nd 79: Considr two dimnsionl ltron gs of N ltrons of mss m h in systm of siz L x L..78 Th dnsity of stts btwn nrgy ε nd ε + dε is () 4L m d h 4L m d h 4L m h d 4L m d h.79 Th ground stt nrgy E of th systm in trms of th Frmi nrgy E F nd th numbr of ltrons N is givn by () F F F F 5 fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 5

26 Sttmnt for Lind Answr ustions 8 nd 8: Th rt of lo in spship Suryshti is obsrvd from rth to b of th rt of th 5 los on rth..8 Th spd of spship Suryshti rltiv to rth is () Th rt of lo in spship Ashgng is obsrvd from rth to b 5 of th rt of th los on rth. If both Ashgng nd Suryshti r moving in th sm dirtion rltiv to somon on rth, thn th spd of Ashgng rltiv to Suryshti is () Sttmnt for Lind Answr ustions 8 nd 8: 4 5 Th following iruit ontins thr oprtionl mplifirs nd rsistors. R R R V R R V V b R 8 7 V 5 6 R R V V b R R V V R R fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 6

27 .8. Th output voltg t th nd of sond oprtionl mplifir V is () V = (V + V b + V ) V = (V + V b + V ) V = (V + V b + V ) V = 4 (V + V b + V ).8 Th output V (t th nd of third op mp) of th bov iruit is () V = (V + V b + V ) V = (V + V b + V ) V = (V + V b + V ) Zro Sttmnt for Lind Answr ustions 84 nd 85: Th st V of ll polynomils of rl vribl x of dgr two or lss nd with rl offiints onstituts rl linr vtor sp V { + x + x :,, R}.84 For f(x) = + x + x V nd g(x) = b + b x + b x V, whih on of th following onstituts n ptbl slr produt? () (f, g) = b b b (f, g) = (f, g) = b b b (f, g) = b b b b b b.85 Using th slr produt obtind in th bov qustion, idntify th subsp of V tht is orthogonl to ( + x): () {f(x): b ( x) + x ; b, R} {f(x): b ( x) + x ; b, R} {f(x): b + x ; b, R} {f(x): bx + x ; b, R} fizis /o Annd Institut of mthmtis, 8-B/6 Ji Sri Nr IIT, Huz Khs, Nw Dlhi, PIN- 6 (INDIA) Phon: , Wbsit: 7