BINOMIAL THEOREM OBJECTIVE PROBLEMS in the expansion of ( 3 +kx ) are equal. Then k =

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wwwskshieduciocom BINOMIAL HEOREM OBJEIVE PROBLEMS he coefficies of, i e esio of k e equl he k /7 If e coefficie of, d ems i e i AP, e e vlue of is he coefficies i e,, 7 ems i e esio of e i AP he 7 7 em i e esio of y 7 y b y / c y d y is If e coefficies of,, ems of e i AP, e b 7 c d q q If d q be osiive, e e coefficies of d i e esio of will be Equl b Equl i mgiude bu oosie i sig c Reciocl o ech oe d Noe of ese 7 If e coefficies of em d em e equl i e esio of, e e vlue of will be 7 b c d he io of e coefficie of ems d i e biomil esio of will be c : b : : d Noe of ese wwwskshieduciocom

wwwskshieduciocom If coefficie of d ems i e esio of e equl, e vlue of is b c d I e esio of b, e coefficie of c d Noe of ese 7 he coefficie of i e esio of b will be b b b c b d b If A d B e e coefficies of i e esios of d esecively, e c A B b A B A B d Noe of ese If e coefficies of, d ems i e esio of e i AP, e b c d Noe of ese If e coefficies of, d 7 ems i e esio of be i AP, e 7 oly c 7 o b oly d Noe of ese wwwskshieduciocom

wwwskshieduciocom If e coefficie of em i e esio of b is, e is b c d m If e id em i e biomil esio of is, e e iol vlue of m is b / c d 7 If e coefficies of e b c d Noe of ese, d ems i e esio of e i AP, If coefficies of em d em e equl i e esio of, e e vlue of will be b c d m If occus i e esio of, e e coefficie of m is! m! m! b!!! m! c! m m!! d Noe of ese 7 If e coefficies of d i e equl, e is b c d oefficie of i e esio of is b c d wwwskshieduciocom

wwwskshieduciocom If e coefficies of secod, id d fou em i e esio of e i AP, e 7 is equl o b c d / If i e esio of m, e coefficie of d e d esecively, e m is b c d he coefficies of d - ems i e esio of e equl he 7 I e esio of, e coefficies of d ems e i e io : he is 7 he s d d ems i e esio of e equl he e vlue of is /7 / 7 he coefficie of i e esio of is b c d I e esio of e coefficie of is b c d 7 he middle em i e esio of is b c d 7 If e secod, id d fou em i e esio of e, 7 d esecively, e e vlue of b c d wwwskshieduciocom

wwwskshieduciocom he em ideede of i is 7 b c d 7 he em ideede of i e esio is b c d I e esio of e coefficie of d ems e esecively d q he q b c d he em ideede of i e esio of / b / will be c / d Noe of ese he gees coefficie i e esio of is!!! b {!} c!!! d!!! he coefficie of i esio of is b c d 7 he coefficie of i e esio of is b c d wwwskshieduciocom

wwwskshieduciocom If is eve osiive iege, e e codiio e gees em i e esio of my hve e gees coefficie lso, is < < b < < c < < dnoe of ese he em ideede of i e esio of is b c d Noe of ese he coefficie of i e esio of is b c d 7 b c d he sum o ems of e followig seies is c b d Noe of ese he vlue of is equl o b c d!!! b!!! c!! d Noe of ese wwwskshieduciocom

b / wwwskshieduciocom c d If, e b c d Noe of ese 7 If e sum of e coefficies i e esio of α α vishes, e e vlue of α is b c d If e sum of e coefficies i e esio of y z is e e gees coefficie i e esio of b c is d Noe of ese he sum of coefficies i e esio of y z is b c d Noe of ese he sum of coefficies i e esio of is b c d If S d, e S is equl o b c d If, e e vlue of will be b c d wwwskshieduciocom

k k wwwskshieduciocom k If k, fo k,,,,,, e k c b d he vlue of is b c d I e esio of, e sum of e coefficie of e ems is b c d b c d Noe of ese 7 oefficies of [ ] i e esio of b c d Noe of ese he esio of / biomil eoem will be vlid, if < b < c < < d Noe of ese I e esio of, e coefficie of will be b c d Noe of ese c b d wwwskshieduciocom

wwwskshieduciocom em i e esio of will be b! c d Noe of ese k k k b c d If <, e e vlue of will be! c b d 7 is equl o / b / c / d c b d he coefficie of i e esio of b c d Noe of ese will be / 7 oefficie of i e esio of is!!! b! c!! d!!! wwwskshieduciocom

wwwskshieduciocom / If <, e i e esio of, e coefficie of is b c d b c d Noe of ese 7 If is e coefficie of, i e esio of, e b c d 7 he umbe of ems i e esio of b c will be b c d Noe of ese 7 If,,, e e coefficies of y fou cosecuive ems i e esio of, e b c d 7 Le R d f R [R], whee [] deoes e gees iege fucio he vlue of Rfis b c d 7 If e ee cosecuive coefficie i e esio of e, d 7, e e vlue of is b c d 7 he gees iege less o equl o b 7 c d is wwwskshieduciocom

wwwskshieduciocom 7 Fid e vlue of 7 7 7 b c d 77 he umbe of iegl ems i e esio of is b c d 7 If,,, eese e ems i e esio of, e b c / / d 7 he umbe of o-zeo ems i e esio of is b c d b c d he vlue of is b c d divisible whee N by b by c by d All of ese he ol umbe of ems i e esio of fe simlificio will be b c d Noe of ese b 7 c d wwwskshieduciocom

wwwskshieduciocom BINOMIAL HEOREM HINS AND SOLUIONS d b 7 c y 7 y / y d ; / ; By e give codiio i!!!!!!!!!, Bu is o give Hece q q oefficie of is d coefficie of is q Bu q q q q, 7 c d Rio of coefficie of d is Bu I e esio of, e geel em is Hee, e eoe of is wwwskshieduciocom

wwwskshieduciocom he equied coefficie b Fo umbe of em, 7 7 b b hus coefficie of 7 is oefficie of b oefficie of i esio of i esio of!!!!!!!!!! : A B A B b oefficie of, d ems i esio of e,, he c oefficie of, d 7 Accodig o e codiio, Afe solvig, we ge 7 o c b!!! 7 m m! b We hve m m m m By hyoesis, wwwskshieduciocom

m m m m 7 7 7 c 7 7 wwwskshieduciocom oefficie of em i esio of d coefficie of em coefficie of { } em Accodig o quesio he o o c, his cois m, if m ie if m oefficie of m, m!!!! m m!!! m m!! b 7, will occu i d oefficies of d e equl 7 7 7 b I e esio of e geel em is Hee, eoe of is b Hece coefficie of is,, oefficie of,, e i AP!!!!!!!! wwwskshieduciocom

wwwskshieduciocom 7 c m m m m!! m m m m Give, m o m Hece m m m m m m m m m m 7m m m m m m m 7 c [ ] oefficie of i e give eessio oefficie of i [ ] oefficie of i [ ] d [ ] Oly ges fom oefficie of 7 b Middle em of is d i wwwskshieduciocom

wwwskshieduciocom wwwskshieduciocom 7 ii iii o elimie, 7 7 Now, Puig d i bove eessio, we ge d We hve Hece e equied em is 7 b I, I is ideede of 7 b em em q he, coefficie of q!!!!!! q q q b So e em ideede of

b Gees coefficie of is! {!} wwwskshieduciocom b oefficie of i esio of ie, coefficie of i esio of coefficie of i esio of Now, [ ] [ ] 7 d { } heefoe e coefficie of If is eve, e gees coefficie is / / heefoe e gees em / / / > / / / / d > / > / d < > d < > d < c As i Pevious quesio, obviously e em ideede of will be oefficie of wwwskshieduciocom

wwwskshieduciocom wwwskshieduciocom c Pu,,,, S S Now by lee c, u,, S S d d d i he iegl o e LHS, d by uig d Whees e iegl o e RHS of i o ems We kow Iegig fom o, we ge d } { d o

wwwskshieduciocom i ii Mulilyig bo sides d equig coefficie of i we ge e vlue of equied eessio o e coefficie of i!!! ick: Solvig covesely Pu d i fis em, give codiio i, Pu,, e ii Now check e oios!!! i Pu,, we ge!!! ii Pu,, we ge c ick : Pu, A, A, Which is give by oio c b We hve Diffeeiig bo sides wi esec o, we ge Puig, we ge 7 c he sum of e coefficies of e olyomil α α is obied by uig i α α wwwskshieduciocom

wwwskshieduciocom wwwskshieduciocom heefoe by hyoesis α α α Sum of e coefficie i e esio z y is 7 heefoe, gees coefficie i e esio of 7 is 7 o 7 becuse bo e equl o c Sum of e coefficies is obied by uig, z y so sum of e coefficies b We c obi sum of coefficies by uig i olyomil d We hve, S d, ] [ S S S S S ick: Pu, e eessio is equivle o Oly oio gives e vlue b k k k k k k k

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Hee odd ems ccel ech oe wwwskshieduciocom b heefoe c k k k u o ems u o ems If is elced by d is 7 d Le y y y! omig e ems, we ge y, y! Solvig,, y / Hece give seies Le e give seies be ideicl wi e esio of ie wi ; <! he, d Solvig ese wo equios fo d We ge d / / c Sum of e give seies wwwskshieduciocom

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wwwskshieduciocom 7 Sice, < < < <, 7 c Le e ee cosecuive coefficies be, d 7, so Ad 7 his gives d 7 b Le k f, whee k is iegl d f e fcio < Le ', < f' < f, Sice < < 7 he umeo is of e fom b b b b N 7 D f his is clely e esio of N D 77 b / ems would be iegl if d bo e osiive iege As,,,,,, Fo bove vlues of, is lso iege ol umbe of vlues of 7 b Fom e give codiio, elcig by i d i esecively, we ge i i i wwwskshieduciocom

Ad i i wwwskshieduciocom ii Mulilyig ii d i we ge equied esul ie 7 c Give eessio [ ] he umbe of o-zeo ems is c ovesely, So, b { } b Fom bove i is cle is divisible by b We kow { } heefoe, umbe of ems i esio of { } is c [ ] [ ] [ ] [ ] N N { } N wwwskshieduciocom

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