CHAPTERS 5-7 BOOKLET-2

Transcription

1 MATHEMATIS XI HAPTERS -7 BOOKLET-

2 otets: Pge No hpte Bioil Theoe 7-8 hpte Stight Lies 8- hpte 7 Sequees Seies -

3 Bioil Epessio A lgei epessio osistig of two tes with ve o ve sig etwee the is lle ioil epessio Fo eple :,,, q p et Bioil Theoe fo Positive Itegl Ie The ule whih powe of ioil e epe is lle the ioil theoe If is positive itege, the ie, i Hee,,, e lle ioil oeffiiets!!! fo Ipott Tips The ue of tes i the epsio of e The epsio otis eesig powe of iesig powe of The su of the powes of i eh te is equl to The ioil oeffiiets,, equiistt fo egiig e e equl ie, Su of o tes su of eve tes Soe Ipott Epsios Replig i i, we get, η ie, ii The tes i the epsio of e ltetivel positive egtive, the lst te is positive o egtive oig s is eve o o Replig i equtio i we get, ie, This is epsio of i seig powe of Replig i i we get, ie, ] [ ] [ The oeffiiet of th te i the epsio of is The oeffiiet of i the epsio of is Bioil Theoe hpte 7

4 Bioil Theoe Note : If is o, the equl to, oth hve the se ue of tes If is eve, the hs tes hs tes Eple: 8 8 Solutio: Eple: ovesel The vlue of will e Solutio: We kow tht, [ ] Eple: [ ] [8 ] 98 The lge of 99 is [IIT 98] Noe of these 99 Both e equl Solutio: We hve, i ii Eple: 98 7 Suttig, 99 > Hee > 99 Su of o tes is A su of eve tes is B i the epsio of, the [Rjsth PET 987] AB AB AB Noe of these A B Solutio: Geel Te Siill, A B ii Fo i ii, we get AB Tik: Put i The, A B opig oth sies A, B Optio LHS AB, RHS ie, LHS RHS The fist te The seo te The te The thi te Let T eote the th te so o th is the te fo egiig i the epsio of T i This is lle geel te, euse givig iffeet vlues to, we eteie ll tes of the epsio I the ioil epsio of I the ioil epsio of, T, T hpte 7

5 Bioil Theoe egiig I the ioil epsio of, T Note : Ipott Tips I the ioil epsio of I the epsio of, N T T The oeffiiet of The oeffiiet of Eple: Solutio: Eple: i the epsio of i the epsio of, the p th te fo the e is If the th te i the epsio of p is fo ll R the p /, p,, th p te fo We hve T T p p i lel, RHS of the ove equlit is iepeet of, Puttig i i we get p p Hee p /, p Noe of these If the seo, thi fouth te i the epsio of e, 7 8 espetivel, the the vlue of is [Kuukshet EE 99; DE 99, ] Solutio: It is give tht T, T 7, T 8 Eple: 7 Now, T T T 8 T 8 To eliite, Now T T T 8 77 i 7 iii T T T T T T 7 T Puttig i ove epessio, we get T The th te fo the e i the epsio of 9 is 7 8 Solutio: th th te fo the e 9 te fo the egiig i the epsio of Eple: 8 ii Noe of these 9 9 T T If T i the epsio of T i the epsio of e equl, the T T [Rjsth PET 987, 9] T T Solutio: T T 9 T hpte 7

6 Bioil Theoe T T give ; T T Iepeet Te o ostt Te Iepeet te o ostt te of ioil epsio is the te i whih epoet of the vile is zeo Eple: 9 oitio : [Powe of ] [Powe of ], i the epsio of The te iepeet of i the epsio of Solutio: T Eple: Solutio: Eple: Solutio: Eple: [ ] [IIT 9; BIT Rhi 99; Ktk ET ; UPSEAT ] 8 / will e Noe of these The te iepeet of i the epsio of is [EAMET 989] We kow tht, Oviousl, the te iepeet of will e Tik : Put i the epsio of We wt oeffiiet of Noe of these i opig to equtio i The, we get ie, iepeet of Optio : ; Put ; The The oeffiiet of Fo oeffiiet of 7 i the epsio of will e [IIT 97; Rjsth PET 99] 7, 7 7 ; T 7 If the oeffiiets of seo, thi fouth te i the epsio of e i AP, the 9 7 is equl to / Solutio: T, T, T e i AP the, Eple: Solutio: Eple: O solvig, 9 7 The oeffiiet of i the epsio of [AMU ] is [EAMET 99; UPSEAT ; P ET ] Noe of these We hve So oeffiiet of If A B e the oeffiiet of i [ ] i the epsios of espetivel, the [MP PET 999] A B A B A B Noe of these hpte 7

7 Bioil Theoe Solutio: A oeffiiet of Eple: Solutio: B oeffiiet of B i ii we get, The oeffiiet of i i A B i the epsio of!!!!!!!!! oeffiiet of oeffiiet of i i ii is [AIEEE ] oeffiiet of i oeffiiet of i i [ ] [ ] Nue of Tes i the Epsio of e epe s : { } te te te te Totl ue of tes Siill, Nue of tes i the epsio of Eple: If the ue of tes i the epsio of z is, the Noe of these Solutio: Give, totl ue of tes 9 8 Eple: 7 Solutio: The ue of tes i the epsio of [ ] is [Rjsth PET 98] 8 We hve [ ] [ 8 ] ; Nue of tes 8 Mile Te The ile te epes upo the vlue of Whe is eve, the totl ue of tes i the epsio of oe ile te ie, th te is the ile te T / / is o So thee is ol Whe is o, the totl ue of tes i the epsio of is eve So, thee e two ile tes ie, th th / e two ile tes T T Note : Whe thee e two ile tes i the epsio the thei ioil oeffiiets e equl Eple: 8 Bioil oeffiiet of ile te is the getest ioil oeffiiet The ile te i the epsio of is [BIT Rhi 99; Rjsth PET ; P ET 99] Solutio: is eve so ile te T Eple: 9 T T T The ile te i the epsio of is [P ET 998] hpte 7 7

8 Bioil Theoe!! Solutio: Sie is eve, so ile te T T T To Deteie Ptiul Te i the Epsio α!!!!!! α α I the epsio of ± β, if ous i T, the is give α β α β Thus i ove epsio if ostt te whih is iepeet of, ous i T the is eteie α α α β Eple: α β 9 The te iepeet of i the epsio of is [Rjsth PET 99; AMU 998] 7/ 7/8 7/ 7/ Solutio: 9, α, β The Hee, T 7 8 Eple: Solutio: 7 If the oeffiiet of i 7 is equl to the oeffiiet of i the [MP PET 999; AMU ; P ET ] / 7 Fo oeffiiet of i ;, α, β, oeffiiet of i T i 7 7 fo oeffiiet of i ;, α, β, 7 ; 7 oeffiiet of i T 7 ii Fo equtio i ii, we get Getest Te Getest oeffiiet Getest te : If T T e the th th tes i the epsio of, the T T T Let ueill, T e the getest te i the ove epsio The T T o T o i Now sustitutig vlues of i i, we get f o whee is positive itege f is ftio suh tht < f < Whe is eve T is the getest te, whe is o T T e the getest tes oth e equl Shot ut etho : To fi the getest te ueill i the epsio of hpte 7

9 Bioil Theoe i lulte ii If is itege, the T T e equl oth e getest te iii If is ot itege, thee T [ ] is the getest te, whee [] eotes the getest itegl pt Getest oeffiiet i If is eve, the getest oeffiiet is / ii If is o, the getest oeffiiet e Ipott Tips Fo fiig the getest te i the epsio of we ewite the epsio i this fo Getest te i Eple: Solutio:, Eple: Solutio: Eple: Getest te i The lgest te i the epsio of, whee is th 8 th 7 th th, Now getest te i itege T T[ ] T T [ ] T T 7 e ueill getest tes The getest oeffiiet i the epsio of is [BIT Rhi 99]!!! [!] is eve so getest oeffiiet i is!!!! [!]!!! The itevl i whih ust lie so tht the getest te i the epsio of hs the getest oeffiiet is, Solutio: Hee the getest oeffiiet is >, > Popeties of Bioil oeffiiets I the ioil epsio of,, < > Noe of these Hee the esult is whee,,,, e the oeffiiets of vious powes of lle ioil oeffiiets, the e witte s,,, Hee, i The su of ioil oeffiiets i the epsio of is hpte 7

10 Bioil Theoe Puttig i i, we get ii Su of ioil oeffiiets with ltete sigs : Puttig i i We get, iii Su of the oeffiiets of the o tes i the epsio of eve tes eh is equl to Fo iii, we hve iv ie, su of oeffiiets of eve o tes e equl is equl to su of the oeffiiets of Su of pout of oeffiiets : Replig i i we get Fo ii iv, v so o Multiplig i vi, we get Now opig oeffiiet of o oth sies We get, Su of sques of oeffiiets : Puttig i vii, we get 7 vi vii Eple: The vlue of is equl to [Ktk ET ] Solutio: We hve!! Tik: Fo, Whih is give optio [ ] Eple: The vlue of is equl to [IIT 99] Noe of these Solutio: We hve Eple: 7 If S [ ] [ ] Tik: Put i give epsio Whih is give optio t The hpte 77 t S [ ] is equl to [AIEEE ]

11 Bioil Theoe Solutio: Tke, the, Eple: 8 Solutio: t t S S t t S S S If The [MNR 99; DE 99; AMU 998; Rjsth PET 999; Ktk ET 999; UPSEAT 999] Puttig, we get ; i Agi puttig, we get ii Aig i ii, we get, [ ] Eple: 9 If, the Solutio: Eple:! We hve!!! /! Tik : Put,,,, S S Solutio: Puttig i Eple: Solutio: Eple: If sts fo Whih is give optio,! ; Fo,! 9 9! [BIT Rhi 987] I the epsio of, the su of the oeffiiet of the tes is [Rjsth PET 99, 97; Kuukshet EE ] 8, the equie su of oeffiiet If the su of oeffiiet i the epsio of α α vishes, the the vlue of α is [IIT 99; P ET 988] The su of oeffiiet of poloil α α is otie puttig i α α hpothesis α α α positive itege, is Theefoe, the su of give seies /! /! [! ] whee is eve hpte 78 [IIT 98]

12 Bioil Theoe Solutio: Eple: / / We hve [ ] [ ] / / / / / /!! Theefoe the vlue of give epessio! / /!!! If, the the vlue of Solutio: Tik: Put will e [MP PET 99; Rjsth PET 997; DE 99; IIT 97; AMU 99; EAMET ] the epsio is equivlet to Whih is give optio Use of Diffeetitio : This etho pplie ol whe the ueils ou s the pout of ioil oeffiiets Solutio poess : i If lst te of the seies levig the plus o ius sig e, the ivie if q e the quotiet e the eie ie, q q The eple i the give seies ultiplig oth sies of epsio ii Afte poess i, iffeetite oth sies, wt put o o i o i et oig to give seies iii If pout of two ueils o sque of ueils o thee ueils o ue of ueil the iffeetite twie o thie Eple: [Rjsth PET 99; MP PET ] Solutio: We kow tht, i Diffeetitig oth sies wt, we get Puttig, we get, Eple: If is itege gete th, the Solutio: We hve ] [ ] ] [ ] [ [ We kow tht ; Put, The iffeetitig oth sies wt to, we get Put, [ ] [] [IIT 97] Use of Itegtio : This etho is pplie ol whe the ueils ou s the eoito of the ioil oeffiiets Solutio poess : If, the we itegte oth sies etwee the suitle liits whih gives the equie seies i If the su otis,,, with ll positive sigs, the itegte etwee liit to ii If the su otis ltete sigs ie, the itegte etwee liit to iii If the su otis o oeffiiets ie,,, the itegte etwee to iv If the su otis eve oeffiiets ie,,, the suttig ii fo i the iviig v If i eoito of ioil oeffiiets is pout of two ueils the itegte two ties, fist tkig liit etwee to seo tie tke suitle liits hpte 79

13 Bioil Theoe Eple: [Rjsth PET 99] Solutio: osie the epsio i Noe of these Itegtig oth sies of i withi liits to we get, [ ] [] ; Eple: 7 [MP PET 999; EAMET 99] Solutio: It is le tht it is epsio of Itegtig wt oth sies etwee the liit to [ ] Eple: 8 The su to tes of the followig seies is Solutio: Noe of these i The itegl o LHS of i t t t puttig t, t t t Whees the itegl o the RHS of i to tes Tik : Put i give seies Whih is give optio A Ipott Theoe If A B I f whee I e positive iteges, eig o f < the A B K > A B < I f f K whee Note : If is eve itege the A B A B I f f Hee LHS I e iteges f f is lso itege; f f ; f f Hee I f f I f f A B A B A B K hpte 8

14 Bioil Theoe Eple: 9 Let R f R [R] whee [] eotes the getest itege futio The vlue of Rf is [IIT 988] Solutio: Sie f R [R], R f [R] [ ] f [ R], whee [R] is itege Now let f' [ ], < f' < the ' f [ R] f [ ] [ ] [ ] Hee Hee Rf Itege K N K Eve itege f f' eve itege [R], ut < f' < f Theefoe, f f' f f' R f' Multioil Theoe Fo positive itegl ie If is positive itege,,! Whee,!!!!,, e ll o-egtive iteges sujet to the oitio,, The oeffiiet of i the epsio of The getest oeffiiet i the epsio of is is Whee q is the quotiet is the eie whe is ivie q!!!!!! [ q!] If is ve itege,,, the oeffiiet of i the epsio of is!!!!! Whee, e ll o-egtive iteges sujet to the oitio: The ue of istit o issiil tes i the ultioil epsio Eple: Solutio: Eple: Solutio: The oeffiiet of i the epsio of is! is [EAMET ] 8 8 8! oeffiiet of i the epsio of is!!! whee The possile vlue of, e show i gi The oeffiiet of!!!!!!!!!!!! 8 Fi the oeffiiet of i the epsio of Noe of these z z z I this se, z,, z ; z ; z The,, z! Theefoe the oeffiiet of i the epsio of!!! hpte 8

15 Bioil Theoe Bioil Theoe fo Ie Stteet : tes up to!!! Whe is egtive itege o ftio, whee < <, othewise epsio will ot e possile If <, the tes of the ove epsio go o eesig if e ve sll stge e ehe whe we eglet the tes otiig highe powe of i the epsio, the Ipott Tips Epsio is vli ol whe < < ot e use euse it is efie ol fo tul ue, so The ue of tes i the seies is ifiite will e witte s If fist te is ot, the ke fist te uit i the followig w,, if < Geel te : T! Soe ipott epsios: i!! ii!! iii!!! iv!!! Reple i iii : Reple i iv : Reple i iii : Reple i iv : Geel te, T, Geel te,! T, Geel te, T e Reple i iii :! Geel te, T /! f Reple i iv : Geel te, T!, Geel te, T! Eple: Solutio: / To ep s ifiite seies, the ge of shoul e [AMU ], /, e epe if < ie, if ie, if < [, ], < < ie, if, hpte 8

16 Bioil Theoe Eple: If the vlue of is so sll tht highe powe e eglete, the Solutio: Give epessio e witte s / / / 8 8, whe, e eglete is equl to [Rookee 9] Eple: If 8 the the vlue of is [IIT 98; P ET 99, 99],,,, Solutio: We kow tht!! Eple: 8!!!! opig oeffiiets of oth sies we get, 8,! oeffiiet of!! i the epsio of / o solvig,, 9 [Kuukshet EE ]!!!!!!!! Solutio: oeffiiet of!!!!! Eple: Solutio: The oeffiiet of i is oeffiiet of oeffiiet of oeffiiet of i i oeffiiet of i i [ ] [ ] [ ] ] oeffiiet of i [ ] oeffiiet of i [ ] Eple: Solutio: We kow tht! Eple: 8 Hee, 8 8, 8 8 Noe of these If is so sll tht its two highe powe e eglete k the k hpte 8 / / / [EAMET 99] [Rjsth PET 99]

17 Bioil Theoe Solutio: / / k / / / / 7 / / k!!!! Highe powe e eglete The Eple: 9 The ue oot of is Solutio: Eple: Solutio: k ; k ; k!! 9 Noe of these / / We hve [ ] ; [ ] The oeffiiet of i the epsio of oeffiiet of is tes [ [ ] ] Noe of these Tik: Put,, fi the oeffiiets of,, opig with the give optio s : oeffiiet of is [ ] Thee / Fou oseutive tes o oeffiiets 7 ; Whih is give optio If oseutive oeffiiets e give: I this se ivie oseutive oeffiiets pi wise We get equtios the solve the If oseutive tes e give : I this se ivie oseutive tes pi wise ie if fou oseutive tes T T T e T, T, T, T the fi,, λ, λ, λ s the ivie λ λ λ λ solve T T T Eple: If,,, e the oeffiiets of fou oseutive tes i the epsio of, the 7 [IIT 97] Solutio: Let,,, e espetivel the oeffiiets of,,, tes i the epsio of The,,, Now, Soe Ipott Poits Psl's Tigle : th th th th hpte 8

18 Bioil Theoe Metho fo fiig tes fee fo il o tiol tes i the epsio of Psl's tigle gives the iet ioil oeffiiets Eple : q N, pie ues : Fi the geel te / p / N / p N / q N N p T Puttig the vlues of N, whe iies of e iteges Note : Nue of itiol tes Totl tes Nue of tiol tes Eple: The ue of itegl tes i the epsio of 8 is [AIEEE ] Solutio: T 8 Fist te 8 itege fte eight tes, ie, 9 th te itege th otiuig like this, we get AP, st,9 7 ; T Eple: The ue of itiol tes i the epsio of th is Solutio: T 8 As e o-pie T will e tiol if is ultiple of 8 is ultiple of lso,,9 ;,, i But So, is to e ultiple of 8, 8,,,9 ii oo tes i i ii e,,, 88 8,,, give tiol tes The ue of itiol tes 97 8 q hpte 8

19 Stight Lies Defiitio The stight lie is uve suh tht eve poit o the lie seget joiig two poits o it lies o it The siplest lous of poit i ple is stight lie A lie is eteie uiquel oe of the followig: Two iffeet poits euse we kow the io tht oe ol oe stight lie psses though two give poits The tigooetil tget of the gle tht lie kes with the positive ietio of the -is i A poit give ietio Uique lie though two give poits Thus, to eteie lie uiquel, two geoetil oitios e equie Slope Giet of Lie tilokwise sese is lle the slope o giet of the lie The slope of lie is geell eote Thus, t θ Slope of lie pllel to is is t Slope of lie pllel to is is t 9 Slope of the lie equll ilie with the es is o Slope of the lie though the poits A, B, is Slope of the lie, is o o Slope of two pllel lies e equl 7 If e the slopes of two pepeiul lies, the Note : e efie s θ t fo < θ π If thee poits A, B, e ollie, the Slope of AB Slope of B Slope of A π θ tke i the se oe Eple: The giet of the lie joiig the poits o the uve, whose sisse e, is [MP PET 997] Solutio: The poits e,, O Y o X Uique lie though give poit with give ietio O Y Ifiite ue of lies though give poit X Y O Y º º º º X Ifiite ue of lies with give ietio B θ X X A O Y Y B θ X O A X Y Hee giet is hpte 8

20 Stight Lies Eple: Slope of lie whih uts iteepts of equl legths o the es is [MP PET 98] Solutio: Equtio of lie is Hee slope of the lie is Iteept fo : If stight lie uts -is t A the -is t B the OA OB e kow s the Equtios of Stight lie i Diffeet fos Slope fo : Equtio of lie though the oigi hvig slope is Oe poit fo o Poit slope fo : Equtio of lie though the poit, hvig slope is Slope iteept fo : Equtio of lie o-vetil with slope uttig off iteept o the -is is The equtio of lie with slope the -iteept is iteepts of the lie o -is -is espetivel The iteepts e positive o egtive oig s the lie eets with positive o egtive ietios of the ooite es I the figue, OA -iteept, OB -iteept Equtio of stight lie uttig off iteepts o is is espetivel is Note : If give lie is pllel to X is, the X-iteept is uefie If give lie is pllel to Y is, the Y-iteept is uefie Two poit fo: Equtio of the lie though the poits A, B, is I the eteit fo it is gives s: is the equtio of lie X ' X ' A Y θ O B A O, O Y Y ' Y Y ' A L B, X B X X Nol o pepeiul fo : The equtio of the stight lie upo whih the legth of the pepeiul fo the oigi is p this pepeiul kes gle α with -is is os α siα p Y Y B P p α O A Y X hpte 87

21 Stight Lies 7 Setil o peti o iste fo of the lie : Equtio of lie pssig though, kig gle θ with the positive ietio of -is is whee is the iste etwee the poit P, A, osθ siθ, The ooites of poit o this lie e tke s osθ, si, kow s peti o-oites, is lle the θ Equtio lie pllel to -is o pepeiul to -is t iste fo it pete Note : Equtio of -is Eple: X' O Y Equtio of -is Equtio of lie pllel to -is o pepeiul to -is t iste fo it X ' O Y Y Y Equtio to the stight lie uttig off iteept fo the egtive ietio of the is of ilie t to the positive ietio of, is [MP PET ] Solutio: Let the equtio of the stight lie is Hee t Hee, the equie equtio is Eple: The equtio of stight lie pssig though, uttig iteept equl i gitue ut opposite i sig fo the es is give [Rjsth PET 98; MP PET 99] Solutio: Let the equtio e But it psses though,, hee Hee the equtio of stight lie is Eple: The equtio of the stight lie pssig though the poit, kig iteept o the o-oites es whose su is, is [AIEEE ] hpte 88 X X X ' θ O Y Y ', A P, θ X

22 Stight Lies Solutio: Let the equtio of lie is, whih psses though, The Hee equtio is Eple: Let PS e the ei of the tigle with veties,, Q, ± P R7, The equtio of the lie pssig though, pllel to PS is [IIT Seeig ] Solutio: S i poit of QR,, Slope of PS ; The equie equtio is Equtio of Pllel Pepeiul lies to give Lie Equtio of lie whih is pllel to is λ Equtio of lie whih is pepeiul to is λ The vlue of λ i oth ses is otie with the help of itiol ifotio give i the pole Eple: 7 The equtio of the lie psses though, pllel to the lie, is [Rjsth PET 98, 99] Solutio: The equtio of pllel lie to give lie is λ This lie psses though poit, λ λ Hee, equie lie is Eple: 8 A lie psses though, is pepeiul to the lie Its -iteept is [IIT 99] Solutio: The equtio of lie pssig though, pepeiul to is o Puttig i this equtio, we oti So -iteept Eple: 9 The equtio of lie pssig though, π pepeiul to osθ siθ siθ osθ siθ osθ is [EAMET ] osθ siθ siθ osθ Solutio: Equtio of lie, pepeiul to siθ osθ is It is pssig though, π Hee, siπ osπ k / k π π si θ os θ k hpte 89

23 Stight Lies osθ siθ osθ siθ Eple: The equtio of the lie isetig pepeiull the seget joiig the poits, 8, 8 is [Ktk ET ] Solutio: Equtio of the lie pssig though, 8, 8 is 8 i 8 Nol fo : To hge the geel fo of lie ito ol fo, fist tke to ight h sie Now equtio of lie to it is λ ii This lie psses though the ipoit of, 8, 8 ie,, 7 Fo ii 7 λ λ 9, Equtio of lie is 9 Geel equtio of Stight lie its Tsfotio i St fos Geel fo of equtio of lie is, its Slope iteept fo:, slope iteept o -is is, Iteept fo :, iteept is iteept is / / ke it positive, the ivie the whole equtio like, whee osα, siα p Seletio of o-oite of Poit o Stight lie If the equtio of the stight lie e, i oe to selet poit o it, tke the o-oite oig to ou sweet will Let λ ; the λ o λ ; λ λ, is poit o the lie fo el vlue of λ If λ is tke the the poit will e, Siill suitle poit e tke s, If the equtio of the lie e the poit o it e tke s, λ whee λ hs el vlue I ptiul, is oveiet poit o it whe λ If the equtio of the lie e the poit o it e tke s λ, whee λ hs el vlue I ptiul, is oveiet poit o it whe λ Eple: If we eue 7 to the fo os α siα p, the the vlue of p is [MP PET ] Solutio: Give equtio is 7, Diviig oth sies 7 7, 7 7 p Poit of Itesetio of Two lies hpte 9

24 Stight Lies Let e two o-pllel lies If, e the o-oites of thei poit of itesetio, the Solvig these equtio, we get,, Note : Hee lies e ot pllel, the hve uequl slopes, the, I solvig ueil questios, we shoul ot eee the o-oites, we solve the equtios ietl Geel equtio of Lies though the Itesetio of Two give Lies If equtio of two lies P Q, the the equtio of the lies pssig though the poit of itesetio of these lies is Q Vlue of λ is otie with the help of the itiol ifotio give i the pole P λ o λ ; Eple: Equtio of lie pssig though the poit of itesetio of lies, pepeiul to 7, the its equtio is [Rjsth PET ] Noe of these Solutio: The poit of itesetio of the lies is, 7 7 The slope of equie lie 7 7 Hee, Equtio of equie lie is, 9 7 Eple: The equtio of stight lie pssig though poit of itesetio of the stight lies 7 hvig ifiite slope is Solutio: Requie lie shoul e, λ 7 i λ λ 7λ As the equtio ii hs ifiite slope, λ λ 7λ ii λ λ λ Puttig λ i equtio i, We hve 7 Agle etwee Two o-pllel Lies Let θ e the gle etwee the lies itesetig t A whee, t α α θ β θ α β t β O give ove, ut Y A β θ α B X [UPSEAT ] tθ tα t β tα t β hpte 9

25 Stight Lies θ t Agle etwee two stight lies whe thei equtios e give : The gle θ etwee the lies is give, t θ i oitio fo the lies to e pllel : If the lies pllel the, e ii oitio fo the lies to e pepeiul : If the lies e pepeiul the, iii oitios fo two lies to e oiiet, pllel, pepeiul itesetig : Two lies e, oiiet, if Pllel, if Pepeiul, if Itesetig, if Eple: Agle etwee the lies is [Rjsth PET ] Solutio: 9 8 tθ tθ θ t t o Eple: To whih of the followig tpes the stight lies epesete 7 elogs [MP PET 98] Solutio: Pllel to eh othe Ilie t to eh othe Hee, ; Pepeiul to eh othe 7 Hee, lies e pllel to eh othe oiiet pi of stight lies Equtio of Stight lie though give poit kig give Agle with give Lie Sie stight lie L kes gle θ α with -is, the equtio of lie L is t θ α stight lie L kes gle θ α with - is, the equtio of lie L is t θ α whee tθ Hee, the equtio of the stight lies whih pss though give poit O α θ L α θ θ L X hpte 9

26 Stight Lies, ke give gle α with give stight lie e ± tα t α Eple: The equtio of the lies whih psses though the poit, e ilie t to the lie,, o [IIT 97; MP PET 99] I isoseles tigle AB, the ooites of the poit B o the se B e espetivel,, If the Noe of these Solutio: The equtio of lies pssig though, is i Eple: 7 The slope of the give lie is So, t o ± O solvig, we get o Puttig the vlues of i i, the equie equtio is equtio of the lie AB is Solutio: Slope of B AB A, AB AB, the the equtio of the lie A is [Rookee ] ±, But slope of AB is ; Hee is the giet of the lie A Equtio of the lie A is o A Lie equll ilie with Two lies Let the two lies with slopes e equll ilie to lie with slope the, Note : Sig of i oth kets is se B, θ θ A, Eple: 8 If the lies e equll ilie to the lie, the [ISM Dh 97] 7 7 Solutio: If lie e equll ilie to lies with slope Equtios of the isetos of the Agles etwee two Stight lies hpte 9 ± 7 7,the ± 7

27 Stight Lies give, The equtio of the isetos of the gles etwee the lies e ± Algoith to fi the iseto of the gle otiig the oigi : Let the equtios of the two lies i To fi the iseto of the gle otiig the oigi, we poee s follows: Step I : See whethe the ostt tes i the equtios of two lies positive o ot If ot, the ultipl oth the sies of the equtio to ke the ostt te positive Step II : Now oti the iseto oespoig to the positive sig ie, This is the equie iseto of the gle otiig the oigi Note : The iseto of the gle otiig the oigi es the iseto of the gle etwee the lies whih otis the oigi withi it To fi the ute otuse gle isetos Let θ e the gle etwee oe of the lies oe of the isetos give i Fi the this iseto is the iseto of ute gle the othe oe is the iseto of the otuse gle t θ If tθ <, If tθ >, the this iseto is the iseto of otuse gle othe oe is the iseto of the ute gle Metho to fi ute gle iseto otuse gle iseto i Mke the ostt te positive, if ot ii Now eteie the sig of the epessio iii If, the the iseto oespoig to sig gives the otuse gle iseto the > iseto oespoig to sig is the iseto of ute gle etwee the lies iv If, the the iseto oespoig to sig < give the ute otuse gle isetos espetivel Note : Bisetos e pepeiul to eh othe If, the the oigi lies i otuse gle if > <, the the oigi lies i ute gle Eple: 9 The equtio of the isetos of the gles etwee the lies e [Oiss JEE ] ± Noe of these Solutio: The equtio of lies e The equtio of isetos of the gles etwee these lies e ± ± Tkig ve sig, we get ; Tkig ve sig, we get Hee, the equtio of isetos e, Eple: The equtio of the iseto of the ute gle etwee the lies 7 is Solutio: [IIT 97, 98; Rjsth PET ] Biseto of the gles is give 7 ± 9 i 77 ii Otuse iseto Aute iseto P, L L hpte 9

28 Stight Lies Let the gle etwee the lie 7 i is α, the Hee 9 is the iseto of the ute gle etwee the give lies tα < α < Diste of poit fo lie : The legth p of the pepeiul fo the poit Legth of Pepeiul is give p Note : Legth of pepeiul fo oigi to the lie is, to the lie Legth of pepeiul fo the poit, to the lie os α siα p is osα siα p Diste etwee two pllel lies : Let the two pllel lies e Fist Metho : The iste etwee the lies is Seo Metho : The iste etwee the lies is i λ if the e o the se sie of oigi ii λ if the oigi O lies etwee the λ, whee Thi etho : Fi the ooites of poit o oe of the give lie, pefel puttig o The the pepeiul iste of this poit fo the othe lie is the equie iste etwee the lies Note : Diste etwee two pllel lies O, k k is k Diste etwee two o pllel lies is lws zeo Positio of Poit with espet to Lie Let the give lie e osevig poit is,, the i If the se sig is fou puttig i equtio of lie,, the the poit, is situte o the sie of oigi O, o hpte 9

29 Stight Lies ii If the opposite sig is fou puttig i equtio of lie,, the the poit, is situte opposite sie to oigi Positio of Two poits with espet to Lie Two poits,, e o the se sie o o the opposite sie of the stight lie oig s the vlues of e of the se sig o opposite sig Eple: The iste of the poit, fo the lie is [MP PET ] Solutio: p Eple: The iste etwee the lies 8 is [AMU 979; MNR 987; UPSEAT ; DE 999] 7 Solutio: Give lies 8, iste fo the oigi to oth the lies e Eple: lel oth lies e o the se sie of the oigi Hee, iste etwee oth the lies e, 7 7 Noe of these, If the legth of the pepeiul w fo oigi to the lie whose iteepts o the es e e p, the p p Solutio: Equtio of lie is Pepeiul iste fo oigi to give lie is p p p [Ktk ET ] p p p Eple: The poit o the -is whose pepeiul iste fo the lie is, is[rjsth PET ; MP PET ] ±, ±, ±, Noe of these Solutio: Let the poit e h, the h ± h ± h ± Hee the poit is ±, hpte 9

30 Stight Lies Eple: The vete of equiltel tigle is, the equtio of its se is The legth of its sies is [UPSEAT ] Solutio: AD AD t o BD / BD BD Noe of these B BD ouet Lies Thee o oe lies e si to e ouet lies if the eet t poit Fist etho : Fi the poit of itesetio of two lies solvig the siulteousl If the poit stisfies the thi equtio lso, the the give lies e ouet Seo etho : The thee lies, e ouet if, Thi etho : The oitio fo the lies P, Q R to e ouet is tht thee ostts,, ot ll zeo t the se tie e otie suh tht P Q R Eple: If the lies, e ouet, the [IIT 98; DE ] Noe of these Solutio: Hee the give lies e,,, The lies will e ouet, iff Eple: 7 If the lies, e ouet, the equls [Rjsth PET 99; MP PET 997; EAMET ] Solutio: If these lies e ouet the the itesetio poit of the lies, is,, whih lies o the thi lie Hee, Eple: 8 The stight lies whee, will e ouet, if poit is [Rjsth PET ],,, Noe of these Solutio: The set of lies is, whee Eliitig, we get The pss though the itesetio of the lies ie,, ie,, Refletio o the Sufe Hee The, IP Iiet R PN Nol to the sufe PR Reflete R IPN NPR Agle of iiee Agle of efletio hpte 97 B I Iiet o A, D Nol N R Reflete θ θ α α Tget P Suf

31 Stight Lies Ige of Poit i Diffeet ses The ige of poit with espet to the lie io : The ige of A, with espet to the lie io e B h, k is give, h k A, B h, k The ige of poit with espet to -is : Let P, e poit P, its ige fte efletio i the -is, the Y O is the i poit of P P P, The ige of poit with espet to -is : Let P, e poit P, its ige fte efletio i the -is Y the O is the i poit of P P The ige of poit with espet to the oigi : Let P, e poit P, e its ige fte efletio though the oigi, the P, X O O O Y Y O is the i poit of P P O P, P, X P, X X N O P, Y M X The ige of poit with espet to the lie : Let P, e poit P, e its ige fte efletio i the lie, the O is the i poit of P P Y P, O P, º X X O Y hpte 98

32 Stight Lies lie The ige of poit with espet to the lie t θ : Let P, e poit P, e its ige fte efletio i the tθ the os θ si θ O is the i poit of P P si θ os θ Y P, O t θ P, X θ X O Eple: 9 The efletio of the poit, i the lie is [EAMET 99],,,, Solutio: Let Q, e the efletio of P, i the lie The the poit R, lies o 9 i Also PQ is pepeiul to Theefoe 9 ii Solvig i ii, we get, Eple: The ige of poit,8 A i the lie 7, is [Rjsth PET 99],, 8,, 8 Solutio: Equtio of the lie pssig though, 8 pepeiul to 7 is The itesetio poit of oth the lies is, Now let the ige of,8 A e A, The poit, will e the ipoit of A A 8 Hee the ige is, Soe Ipott Results Ae of the tigle foe the lies,, is Ae of the tigle e the lie with the o-oite es is Ae of the hous foe the lies ± ± is Ae of the pllelog foe the lies ;, is h The foot of the pepeiul h, k fo, to the lie is give k, Hee, the ooites of the foot of pepeiul is Ae of pllelog p p A, whee p p e the istes etwee pllel sies θ is the siθ gle etwee two jet sies Y hpte 99

33 Stight Lies 7 The equtio of lie whose i-poit is, i etwee the es is 8 The equtio of stight lie whih kes tigle with the es of etoi, is Eple: The ooites of the foot of pepeiul w fo, to the lie,,, is [Rookee 99] Solutio: Applig the foul, the equie o-oites is,, Eple: The e elose withi the uve is [Rjsth PET 99, 97; IIT 98; UPSEAT ] Solutio: The give lies e ± ± ie,,, These lies fo quiltel whose veties e A,, B,,, D,, Oviousl ABD is sque Legth of eh sie of this sque is Hee, e of sque is sq uits Eple: If,,,, e oth i GP with the se oo tio, the the poit,,,, [AIEEE ] Lie o stight lie Lie o ellipse Lie o ile Ae veties of tigle Solutio: Tkig o-oites s,,,, Aove o-oites stisf the eltio, the thee poits lie o stight lie Eple: A sque of sie lies ove the -is hs oe vete t the oigi The sie pssig though the oigi kes π gle α < α < with the positive ietio of -is The equtio of its igol ot pssig though the oigi is osα siα siα osα osα siα siα osα osα siα siα osα osα siα siα osα π Solutio: o-oites of A osα, siα ; Equtio of OB t α Eple: π A to OB ; slope of A ot α π Equtio of A, siα ot α osα siα osα os siα [AIEEE ] The ue of itegl poits itegl poit es oth the ooites shoul e itege etl i the iteio of the tigle with veties,,,, is [IIT Seeig ] 9 Solutio: The ue of itegl solutio to the equtio < ie, < 9 Nue of itegl o-oites 9 8 9, B Y O B π/ α A X O, A, hpte

34 Sequees Seies Itoutio Sequee : A sequee is futio whose oi is the set of tul ues, N If f : N is sequee, we usull eote it < f > < f, f, f, > It is ot eess tht the tes of sequee lws follow eti ptte o the e esie soe epliit foul fo the th te Tes of sequee e oete os Eple :,,,,, 8, is sequee A seies is fiite o ifiite s the ue of tes i the oespoig sequee is fiite o ifiite Seies : B ig o suttig the tes of sequee, we get seies If t t, t, t, is sequee, the the epessio t t t t is seies, Eple : is seies Pogessio : A pogessio is sequee whose tes follow eti ptte ie the tes e ge ue efiite ule Eple :,,, 7, 9, is pogessio whose tes e otie the ule : eotes the th te of the pogessio T, whee T Pogessio is il of thee tpes : Aitheti pogessio, Geoeti pogessio Hoi pogessio Howeve, hee we hve lssifie the stu of pogessio ito five pts s : Aitheti pogessio Geoeti pogessio Aithetio-geoeti pogessio Hoi pogessio Miselleous pogessios Defiitio A sequee of ues < > Aitheti pogessioap t is si to e i itheti pogessio AP whe the iffeee t t is ostt fo ll N This ostt is lle the oo iffeee of the AP, is usull eote the lette If is the fist te the oo iffeee, the AP e epesete s,,,, Eple :, 7,, 7,, is AP whose fist te is oo iffeee Algoith to eteie whethe sequee is AP o ot Step I: Oti the th te of the sequee Step II: Reple i to get Step III: lulte hpte 7

35 Sequees Seies If is iepeet of, the give sequee is AP othewise it is ot AP A itheti pogessio is lie futio with oi s the set of tul ues N t A B epesets the th te of AP with oo iffeee A Geel Te of AP Let e the fist te e the oo iffeee of AP The its th te is T p th te of AP fo the e : Let e the fist te e the oo iffeee of AP hvig tes The p th th te fo the e is p te fo the egiig Ipott Tips Geel te T is lso eote l lst te oo iffeee e zeo, ve o ve ue of tes lws elogs to set of tul ues If T k T p of AP e give, the foul fo otiig T is If pt p qt q of AP, the T p q p th te fo the e T p T T k k Tp Tk p k If p th te of AP is q the q th te is p, the T p q T p q If the p th te of AP is q the q th te is p, the its pq th te is If T p q, the it will fo AP of oo iffeee p fist te p q Eple: Solutio: Let T e th te of AP whose fist te is oo iffeee is If fo soe positive iteges,,, T T, the equls [AIEEE ] T T i ii p Sutt ii fo i, we get, s Theefoe Eple: The 9 th te fo the e of the seies 8 is 8 Solutio: 8 9 th te fo e t t t 9 9 Eple: I eti AP, ties the th te is equl to 8 ties the 8 th te, the its th te is [AMU 99] Solutio: We hve T 8 T8 Let e the fist te oo iffeee espetivel { } 8{ 8 } hpte 7

36 Sequees Seies Eple: Solutio:, ie Hee th te If 7 th th te of AP e espetivel, the its 8 th te is Let e the fist te e the oo iffeee of the give AP, the T i 7 T ii Noe of these Fo i ii,, T Tik: 8 T Tk k Tp Tk p k T8 T 8 7 Eple: If < > is itheti sequee, the 7 T T 7 7 T8 p equls Solutio: Let e the fist te the oo iffeee The Eple: p p p p p p p T 8 89 p The th te of the seies 7 7 e equl, the the vlue of is Solutio: th te of st seies 7 7 th te of seies we hve, 7 Seletio of Tes i AP Whe the su is give, the followig w is opte i seletig eti ue of tes : Nue of tes Tes to e tke,,,,,,,,, p [Kel Egg ] I geel, we tke,,,,,,,,, i se we hve to tke tes ie o ue of tes i AP A,,,,,,,, i se we hve to tke tes i AP Whe the su is ot give, the the followig w is opte i seletio of tes Nue of tes Tes to e tke,,,,,,,,, hpte 7

37 Sequees Seies Su of tes of AP : The su of tes of the seies { } is give S [ ] Also, S l, whee l lst te Ipott Tips The oo iffeee of AP is give S S whee S is the su of fist two tes the fist te, whe > The su of ifiite tes, whe < If su of tes S is give the geel te T S S, whee S is su of tes of AP Su of tes of AP is of the fo A B oeffiiet of ie A If fo the iffeet AP s S S T A B S If fo two AP s the T D S Soe st esults f, the φ ie quti epessio i, i suh se, oo iffeee is twie the T f T φ A B D Su of fist tul ues Su of fist o tul ues Su of fist eve tul ues If fo AP su of p tes is q su of q tes is p, the su of p q tes is { p q} If fo AP, su of p tes is equl to su of q tes, the su of p q tes is zeo If the p th te of AP is q th te is, the su of pq tes is give S pq pq q p Eple: 7 7 th te of AP is, the the su of fist tes is [Ktk ET ] 8 Solutio: S { } { } T7 Eple: 8 The fist te of AP is oo iffeee is The su of its tes will e [MNR 978; MP PET ] 8 Solutio: S [ ] [ ] Eple: 9 The su of the fist thi te of AP is the pout of fist seo te is, the fist te is 8 Solutio: Let,,, e AP S is the su of fist te o [MP PET ] hpte 7

38 Sequees Seies Also, Fist te Eple: If S eotes the su of the fist tes of AP, the Solutio: Eple: S S S S is equl to S S S S { } { } { T {8 } If the su of the fist tes of,, 8 is equl to the su of the fist tes of 7, 9,, the is equl to Solutio: We hve, { } { 7 } Eple: Solutio: } [IIT Seeig ] If the su of the tes of AP is ties to the su of its tes, the the tio of fist te oo iffeee is [Rjsth PET 98] : : : : Let e the fist te the oo iffeee The, {{ } { } 9 8, : : Eple: wokes wee egge to fiish piee of wok i eti ue of s wokes oppe the seo, oe wokes oppe the thi so o It tkes eight oe s to fiish the wok ow The ue of s i whih the wok ws oplete is [Kuukshet EE 99] Solutio: Let the wok ws to e fiishe i s Wok of woke i Now the wok will e fiishe i 8 s Wok oe Su of the ftio of wok oe 8 to 8 tes Eple: Solutio: 8 8 8{ 7} , 8 Hee wok oplete i s If the su of fist p tes, fist q tes fist tes of AP e, z espetivel, the z q p p q is p q 8z pq pq p We hve, the fist te, the oo iffeee, { p } p p z Siill, q q hpte 7

39 q p p q q p q p z p q q p } { } { q p p q q p q p p q } { } { [{ q p p q q p pq q p pq Eple: The su of ll o ues of two igits is [Rookee 99] Solutio: Requie su, 99 S Let the ue of o tes e, the S l S Eple: If su of tes of AP is T, the [Rjsth PET 99, 9; DE 999] 7 8 Noe of these Solutio: S S T } { 7 Eple: 7 The su of tes of the seies 7 is [UPSEAT ] Solutio: 7 S 7 ] [ ] 7 [ Eple: 8 If,,, e i AP, the is [AMU ] Solutio: S As,,,,, e i AP, ie s ] [ S S Aitheti Me Defiitios i If thee qutities e i AP the the ile qutit is lle Aitheti e AM etwee the othe two If, A, e i AP, the A is lle AM etwee ii If A A A A,,,,,, e i AP, the A A A A,,,, e lle AM s etwee Isetio of itheti es i Sigle AM etwee : If e two el ues the sigle AM etwee Sequees Seies hpte 7

40 Sequees Seies ii AM s etwee : If A, Ipott Tips A,, A, A, A e AM s etwee, the A, A,, Su of AM s etwee is equl to ties the sigle AM etwee ie A A A A If A A e two AM s etwee two ues, the A, A Betwee two ues, Su of AM' s Su of AM' s If ue of tes i seies is o, the ol oe ile te eists whih is th te If ue of tes i seies is eve the thee e two ile tes, whih e give th A Eple: 9 Afte isetig AM s etwee 8, the su of the esultig pogessio is The vlue of is [MP PET ] 8 9 Noe of these Solutio: Thee will e tes i the esultig AP, A, A,, A, 8 Eple: Su of the pogessio 8 8 AM s etwee 9 e 7,,,,,, Noe of these Solutio: Let A, A, A e thee AM s The, A, A, A, 9 e i AP 9 oo iffeee Theefoe A 7, A, A Eple: If,,,, e, f e AM s etwee, the e f is equl to 8 Noe of these Solutio: Sie,,,,, e, f e si AM s etwee Popeties of AP Theefoe, e f f If, e i AP whose oo iffeee is, the fo fie o-zeo ue K R, i ± K ± K, ±, will e i AP, whose oo iffeee will e, K ii K, K, K will e i AP with oo iffeee K iii,, will e i AP with oo iffeee /K K K K The su of tes of AP equiistt fo the egiig the e is ostt is equl to su of fist lst te ie A te eept the fist te of AP is equl to hlf of the su of tes equiistt fo the te ie k k, k < hpte 7 7 th te

41 Sequees Seies If ue of tes of AP is o, the su of the tes is equl to pout of ile te ue of tes If ue of tes of AP is eve the AM of ile two tes is AM of fist lst te If the ue of tes of AP is o the its ile te is AM of fist lst te 7 If,,,, e the two AP s The ±, ±, ± oo iffeee, whee e the oo iffeee of the give AP s 8 Thee ues,, e i AP iff 9 If T, T T e thee oseutive tes of AP, the T T T If the tes of AP e hose t egul itevls, the the fo AP e lso AP s with Eple: If,,,, e i itheti pogessio, the [MP PET 999; AMU 997] Solutio: 7 Eple: Solutio: I AP the su of the tes equiistt fo the egiig the e is se is equl to the su of fist lst te 7 9 If,, e i AP, the,, will e i [DE ; MP PET 98; Rookee 97] AP GP HP Noe of these,, e i AP,,, will e i AP [Diviig eh te ] Eple: If log, log log e i AP, the [MP PET 998; Ktk ET ] / log log Solutio: As, log, log log e i AP Theefoe log log log As ot e egtive, hee o log Geoeti pogessiogp Defiitio A pogessio is lle GP if the tio of its eh te to its pevious te is lws ostt This ostt tio is lle its oo tio it is geell eote 8 Eple: The sequee,,, 8, is GP, euse, whih is ostt lel, this sequee is GP with fist te oo tio 9 The sequee,,,, 8 is GP with fist te oo tio Geel Te of GP We kow tht,,,,, is sequee of GP Hee, the fist te is the oo tio is hpte 7 8

42 Sequees Seies The geel te o th te of GP is It shoul e ote tht, T T T T T p th te fo the e of fiite GP : If GP osists of tes, p th te fo the e te fo the egiig p Also, the p th te fo the e of GP with lst te l oo tio is Ipott Tips If,, e i GP o If T k T p of GP e give, the foul fo otiig T is If,, e i GP the o T T k k Tp T k o pk l p Let the fist te of GP e positive, the if >, the it is iesig GP, ut if is positive less th, ie < <, the it is eesig GP Let the fist te of GP e egtive, the if >, the it is eesig GP, ut if < <, the it is iesig GP If,,,, e i GP, the the e lso i otiue popotio ie Eple: The ues,, will e i [AMU 98] AP GP HP Noe of these Solutio: lel Eple:,, e i GP If the p th, q th th te of GP e,, espetivel, the q p pq is equl to [Rookee 9,, 7; P ET 99, 9] pq Solutio: Let,,,, e GP th p, q, Now, q p pq p q q p p q q p pq q p pq pq q p pq p q q p pq Eple: 7 If the thi te of GP is the the pout of its fist tes is [IIT 98; Rjsth PET 99] Noe of these Solutio: Give tht The pout of fist tes [ ] Eple: 8 If,, e i GP, the the fouth te is [MNR 98, 8] 7 7 Solutio: Give tht,, e i GP Theefoe,, hpte 7 9

43 Sequees Seies Now fist te, seo te, the th 8 te Puttig, we get 7 T 8 Whe the pout is give, the followig w is opte i seletig eti ue of tes : Su of Fist Tes of GP If e the fist te, the oo tio, the su S, < S, > S, Seletio of Tes i GP Nue of tes S of fist tes of GP is give Tes to e tke,,,,,,,,, Whe the pout is ot give, the the followig w is opte i seletio of tes Nue of tes Tes to e tke,,,,,,,,, Eple: 9 Let e the th te of the GP of positive ues Let oo tio is α β β α α α β β α β, suh tht α β, the the [IIT 99] Solutio: Let e the fist te, the oo tio of the GP The, α β 99 hpte 7

44 Sequees Seies Eple: 99 α 98 β α β Thus, oo tio α β The su of fist two tes of GP is eve te of this seies is twie of its pevious te, the the fist te will e Solutio: We hve, oo tio ; Let e the fist te, the Su of Ifiite Tes of GP Whe <, o < < S If, the S oes t eist [Rjsth PET 988] Eple: The fist te of ifiite geoeti pogessio is its su is The [IIT Seeig ] Solutio: Eple: < < < < > Aoig to the give oitios,, eig the oo tio Now, < ie < < < < < < > > ie < <, < < li e is [AIEEE ] e e e e / / / / / / / / / / Solutio: li e li e li e e e e li [ e e e e ] Eple: Put h, we get h / / e li / e / e e e e e li e li li li / / / / e e e e h e li h e h fo e li e e h h e The vlue of is [MNR 98; UPSEAT ] hpte 7

45 Sequees Seies Solutio: Eple: Solutio: If,, e i AP,, <, e lle GM s etwee z The,, z shll e i [Ktk ET 99] AP GP HP Noe of these z Now,,, e i AP,, e i AP,, e i HP Theefoe,, z e i HP Geoeti Me Defiitio : i If thee qutities e i GP, the the ile qutit is lle geoeti e GM etwee the othe two If, G, e i GP, the G is lle GM etwee ii If, G, G, G, G, e i GP the G, G, G, G Isetio of geoeti es : i Sigle GM etwee : If e two el ues the sigle GM etwee ii GM s etwee : If G, G, Ipott Tips, G, G, G e GM s etwee, the G, G,, G Pout of GM s etwee is equl to th powe of sigle geoeti e etwee ie G G G G GM of is / If G G e two GM s etwee two ues is The pout of geoeti es etwee is / /, G G If GM s isete etwee the Popeties of GP If ll the tes of GP e ultiplie o ivie the se o-zeo ostt, the it eis GP, with the se oo tio hpte 7

46 Sequees Seies The eipol of the tes of give GP fo GP with oo tio s eipol of the oo tio of the oigil GP If eh te of GP with oo tio e ise to the se powe k, the esultig sequee lso fos GP with oo tio k I fiite GP, the pout of tes equiistt fo the egiig the e is lws the se is equl to the pout of the fist lst te GP ie, if,,, e i GP The If the tes of give GP e hose t egul itevls, the the ew sequee so foe lso fos If,,,, is GP of o-zeo, o-egtive tes, the log,log,log,log, is AP vie-ves 7 Thee o-zeo ues,, e i GP iff 8 Eve te eept fist te of GP is the sque oot of tes equiistt fo it ie T T p T p ; [ > p] / 9 If fist te of GP of tes is lst te is l, the the pout of ll tes of the GP is l If thee e qutities i GP whose oo tio is S eotes the su of the fist tes, the the su of thei pout tke two two is S S Eple: The two geoeti e etwee the ue e [Kel Egg ] Solutio: Let G Eple: 8 G e two GM s etwee the ue G, G The GM of the ues,, is [DE ] / Solutio: GM of Eple: 7 Solutio: Defiitio If,, e i AP, e i GP, the : : is : : : : : : : : Give,,, e i AP,, e i GP So [ ] Put i, we get Theefoe : : : : Hoi pogessiohp A pogessio is lle hoi pogessio HP if the eipols of its tes e i AP hpte 7

47 Sequees Seies St fo : Eple: The sequee,,,,, is HP, euse the sequee,,, 7, 9, is AP 7 9 Geel Te of HP Eple: 8 The th te of HP is 8 th te is the its th te is [MP PET ] If the HP e s,,, the oespoig AP is,,, T of AP is T of HP is I oe to solve the questio o HP, we shoul fo the oespoig AP Thus, Geel te : T o Solutio: Let,,, e HP th te 7 i Siill, 7 ii Fo i ii,, th te 7 T of HP T of AP Eple: 9 If the oots of e equl, the,, e i [Rjsth PET 997] AP GP HP Noe of these Solutio: As the oots e equl, isiite { } { } { } [ ] Thus,,, e i HP Eple: If the fist two tes of HP e the the lgest positive te of the pogessio is the th te 7th te th te 8th te Solutio: Fo the oespoig AP, the fist two tes e ie 7 hpte 7

48 Sequees Seies oo iffeee 7 9 The AP will e,,,,,, The sllest positive te is, whih is the th te The lgest positive te of the HP will e the th te Hoi Me e isete etwee suh tht the Defiitio : If thee o oe ues e i HP, the the ues lig etwee the fist lst e lle hoi es HM s etwee the Fo eple, /, /, /7, /9 e i HP So /, / /7 e thee HM s etwee /9 Also, if, H, e i HP, the H is lle hoi e etwee Isetio of hoi es : i Sigle HM etwee ii H, HM of o-zeo ues,,,, is give H iii Let, e two give ues If ues H, H, H sequee, H, H, H H, is HP, the H, H, H e lle hoi es etwee Now,, H, H, H, e i HP,,,, e i AP H H H Let D e the oo iffeee of this AP The, th te T D D Thus, if hoi es e isete etwee two give ues, the the oo iffeee of the oespoig AP is give D Also, H Ipott Tips D, D,, D H H whee D If,, e i HP the If H H e two HM s etwee, the H H Popeties of HP No te of HP e zeo If,, e i HP, the hpte 7

49 If H is the HM etwee, the i H H ii H H H iii H H H H Eple: The hoi e of the oots of the equtio 8 is [IIT 999] 8 Solutio: Let α β e the oots of the give equtio β, 8 αβ Hee, equie hoi e 8 8 β α αβ Eple: If,, e i HP, the the vlue of is [MP PET 998; P ET ] Noe of these Solutio:,, e i HP,, e i AP Now, Eple: If,, e i HP, the whih oe of the followig is tue [MNR 98] Noe of these Solutio:,, e i HP, optio is flse, optio is flse optio is flse Aithetio-geoeti pogessioagp th Te of AGP Sequees Seies hpte 7

50 Sequees Seies If,,,, is AP,,,, is GP, the the sequee,,,,, is si to e ithetio-geoeti sequee Thus, the geel fo of ithetio geoeti sequee is,,,, Fo the set we oti tht the th te of this sequee is [ ], e ithetio-geoeti sequee The, Also, let,,,, is ithetio-geoeti seies Su of AGP Su of tes : The su of tes of ithetio-geoeti sequee,,,, is give S { } [ ], whe, whe Su of ifiite sequee : Let < The, s it lso e show tht s So, we oti tht S, s I othe wos, whe < the su to ifiit of ithetio-geoeti seies is S Metho fo Fiig Su This etho is pplile fo oth su of tes su of ifiite ue of tes Fist suppose tht su of the seies is S, the ultipl it oo tio of the GP sutt I this w, we shll get GP, whose su e esil otie Metho of Diffeee If the iffeees of the suessive tes of seies e i AP o GP, we fi th te of the seies the followig steps : Step I: Deote the th te T the su of the seies upto tes Step II: Rewite the give seies with eh te shifte oe ple to the ight Step III: B suttig the lte seies fo the foe, fi T Step IV: Fo T, Eple: S e fou ppopite sutio 7 is equl to [DE 999] S 9 hpte 7 7

51 Sequees Seies Solutio: 7 S S S o suttig S S / Hee S / Eple: Solutio: 99 Su of the seies is [IIIT H ; Kel Egg ] Let S i S 99 ii Equtio i Equtio ii gives, S upto 99 tes S 99 Eple: The su of the seies tes is [Rjsth PET ] Solutio: S to tes S to tes 9 7 T o suttig 99 T to tes S S Noe of these 7 Eple: 7 The su of tes of the followig seies will e [IIT 9] Noe of these Solutio: S S to tes T o suttig T S T Eple: 8 The su to tes of the seies 7 is [IIT 9] Noe of these hpte 7 8

52 Solutio: o suttig tes to T S S tes to 8 T T S Miselleous seies Speil Seies Thee e soe seies i whih th te e peite esil just lookig t the seies If δ γ β α T The T S δ γ β α δ γ β α δ γ β α Note : Su of sques of fist tul ues Su of ues of fist tul ues V Metho To fi the su of the seies Let e the oo iffeee of AP The Let S T eote the su to tes of the seies th te espetivel S T Let V ; V V V T ] } { [ ] [ } { V V T, V V T S Sequees Seies hpte 7 9

53 S Eple: If,, e i AP, the If S T Let V, V T T V V ]} [ ] {[ V V T } { V V V V T S } { Eple: } { } { Eple: 9 The su of is [MP PET ] Solutio: S ; Fo, the vlue of Eple: A seies whose th te is, the su of tes will e [UPSEAT 999] Solutio: t S Eple: If t t t, the t is Solutio: t t t t t t ] [ t t t t S S S t Eple: The su of the seies 7 7 is [MNR 98; UPSEAT ] Sequees Seies hpte 7

54 Sequees Seies 9 Solutio: S Eple: The su of the seies to tes is [Kuukshet EE 998] Solutio: T S to tes S [ ] [ ] Popeties of Aitheti, Geoeti Hoi es etwee Two give Nues Let A, G H e itheti, geoeti hoi es of two ues The, A, G H These thee es possess the followig popeties : A G H A, G H A G A G i G H G H ii Fo i ii, we get A G H Note tht the equlit hols ol whe A, G, H fo GP, ie G AH AH G Hee, G AH The equtio hvig s its oots is A G The equtio hvig its oots is hpte 7

55 Sequees Seies A G A G The oots, e give A ± A G If A, G, H e itheti, geoeti hoi es etwee thee give ues,, the G the equtio hvig,, s its oots is A G H A /, G H G A, G H The equtio hvig,, s its oots is G A G H Reltio etwee AP, GP HP the If A, G, H e AM, GM, HM etwee, the If A, A e two AM s;,g G G H H A H A H H e two HM s etwee two ues G e two GM s, H A whe G whe / H whe Reogiztio of AP, GP, HP : If,, e thee suessive tes of sequee The if, If, If,, the,, e i AP, the,, e i GP, the,, e i HP If ue of tes of AP/GP/HP is o, the AM/GM/HM of fist lst tes is ile te of seies If ue of tes of AP/GP/HP is eve, the AM/GM/HM of ile two tes is AM/GM/HM of fist lst tes espetivel If p th, q th th tes of GP e i GP The p, q, e i AP 7 If,, e i AP s well s i GP the 8 If,, e i AP, the,, will e i GP ± Eple: If the AM, GM HM etwee two positive ues e equl, the [Rjsth PET ] > < hpte 7

56 Sequees Seies Solutio: AM GM Eple: Solutio: Eple: Solutio: GM HM Thus AM GM HM So Let two ues hve itheti e 9 geoeti e The these ues e the oots of the quti equtio [AIEEE ] A 9, G e espetivel the AM GM etwee two ues, the the quti equtio hvig its oots s the two ues, is give A G ie 8 If,, e i HP, the [UPSEAT ],, e i AP,, e i HP,, e i GP Noe of these,, e i HP,, e i AP,, e i AP,, e i AP,, e i AP Eple: 7 If,, e i GP, the log, log, log e i [Rjsth PET ] Solutio: AP GP HP Noe of these,, e i GP log, log, log e i AP log, log, log e i HP Eple: 8 If A, A ; G, G, H Solutio: A A H H log, log, log e i AP H e two AM s, GM s HM s etwee two qutities, the the vlue of Let e the two ues A, A A A H H / / / / G, G H, H / / A A H H A A H H GG H H is [Rookee 98; AMU ] / / / / G G H H 9 hpte 7

57 A A 9 H H 9 H H G G H H A A Eple: 9 If the tio of HM GM of two qutities is :, the the tio of the ues is [Rjsth PET 99] : : : Noe of these Solutio: Let e the ues HM, GM / GM HM,, 8, 8 ± ± Rtio of ues : 9 : o 9 : : 9 o : 9 Eple: If the AM of two ues is gete th GM of the ues the tio of the ues is :, the the ues e [Rjsth PET 988],,, Noe of these Solutio: Let e the ues AM GM Also, : The ues e, Eple: If the tio of AM etwee two positive el ues to thei HM is :, the : is Noe of these Solutio: We hve, / / Let, ± ± osieig ve sig, Hee, Applitios of Pogessios Sequees Seies hpte 7