n 2 x n k x 2 k k 1...knitvitüacmuvijbipbelak - 3 - !@> ekdwgfa cosa , cosb , coscmn p np pm m n...

2

21

1

1

n

knkk

nk k

kk

xx

yy

i

KN³kmμkarniBnæ nig eroberog

lwm plÁún nig Esn Bisidæ

KN³kmμkarRtYtBinitübec©keTs

elak lwm qun elak Gwug sMNag elak nn; suxNa elakRsI Tuy rINa elak RBwm sunitü elak pl b‘unqay elak Titü em:g

KN³kmμkarRtYtBinitüGkçraviruTæ elak lwm miKÁsir

karIkMuBüÚTr½ kBaØa lI KuNÑaka elak Gwug sMNag

ii

Garmáfa esoePAKNitviTüaCMuvijBiPBelakPaKTI7 EdlGñksikSakMBugkan; enAkñúgédenH EckecjCabIEpñkEdlEpñkTI1 CakRmglMhat;eRCIserIs EpñkTI2 CaEpñkdMeNaHRsay nig EpñkTI3 CalMhat;Gnuvtþn_ . ral;bRbFanlMhat;nImYy²enAkñúgesovePAenH eyIgxJúM)aneRCIserIsykEt lMhat; NaEdlmanlkçN³Bi)ak mkeFVIdMeNaHRsayKMrUy:agek,aHk,aybMput . eKalbMNgénkareroberogcgRkgKWedIm,ITukCaÉksarCMnYysRmab;Gñk sikSakñúgRKb;mCÄdæan nig müa:geTotedIm,IcUlrYmelIksÞÜyvisy½KNitviTüakñúg RbeTskm<úCaeyIg[kan;EtrIkceRmInqab;rhs½Rsbtamsm½yviTüasaRsþTMenIb. esovePAenHminl¥hYseKhYsÉgenaHeT . kMhusqÁgedayGectnaR)akd CamanTaMgbec©keTs nig GkçraviruTæ . ehtuenHeyIgxJMúCaGñkniBnæ rgcaMCanic©nUv mtiriHKn;EbbsßabnaBIGñksikSaRKb;mCÄdæanedaykþIrIkrayedIm,IEklMGesovenH [kan;EtmansuRkitüPaBRbesIreLIgEfmeTot . CaTIbBa©b; xJMú)aTGñkniBnæsUmeKarBCUnBrGñksikSaTaMgGs;mansuxPaBl¥ manR)aCJaQøasév nig TTYl)aneCaKC½yCanic©kñúgkarsikSa .

)at;dMbg éf¶TI 25 ExFñÚ qñaM2010 GñkniBnæ nig RsayRCav lwm plÁún Tel : 017 768 246 Email: [email protected] Website: www.mathtoday.wordpress.com

KNitviTüaCMuvijBiPBelak

- 1 -

kRmglMhat;

!> cMeBaHRKb; IRx cUrRsayBa¢ak;fa ³ 22|sincos| baxbxa @> cMeBaHRKb;cMnYnBit xcUrRsaybBa¢ak;fa

bab

xa

x

1cossin 44

)0,0(; ba #> eK[GnuKmn_ 9)cot(tan12)cot(tan4)( 22 xxxxxf Edl )

2,0(

x . cUrrktémøtUcbMputénGnuKmn_ )(xf . $> eK[GnuKmn_

x

xy

sin45

cos3

cUrrktémøtUcbMput nig témøFMbMputénGnuKmn_ y . %> eK[GnuKmn_ 89)2(1243),( 2222 yxxyyxyxyxf cUrkMNt; x nig y edIm,I[GnuKmn_ ),( yxf mantémøtUcbMput


- 2 -

rYckMNt;témøtUcbMputenaH . ^> eKmansmIkar 032 xx manb¤stageday 1x nig 2x cUrKNnatémø 5

23

1 419 xxA &> eK[GnuKmn_ cbxaxxf 2)( Edl IRcbaa ,,,0 . cUrRsayfaebIsmIkar xxf )( Kμanb¤skñúgsMNMucMnYnBitenaH smIkar xxff )]([ k¾Kμanb¤skñúgsMNMucMnYnBitEdr . *> edaHRsaysmIkar ³ xxxxx 126324 232 (> edaHRsaysmIkar ³

33

3 23 2

1952

)13(4)13(10

x

xx

!0> eK[smIkar 099999 xx cUrbgðajfasmIkarenHmanb¤s 0x CacMnYnBitEtmYyKt; rYcRsayfa

0x CacMnYnGsniTan . !!> eKdwgfa b

tan xa

.

cUrRsayfa 4 4cos x sin x 1

a b a b


- 3 -

!@> eKdwgfa m n pcosa , cosb , cosc

n p p m m n

cUrKNnakenSam ³ 2 2 2

2 2 2

sin a sin b sin cM

2 2cosa sin a 2 2cosb sin b 2 2cosc sin c

!#> eKmanRtIekaN ABC mYyEdl BC a , AC b , AB c . tag R nig S erogKñaCakaM nig épÞRkLaénRtIekaN ABC enH . cUrRsayfa ³

2 2 2cos A cosB cosC R(sin A sin B sin C)

a b c 4S

!$> kñúgRKb;RtIekaN ABCcUrRsayfa ³

2(sin A sinB sinC)(1 cos A)(1 cosB)(1 cosC)

2

!%> eK[RtIekaN ABC mYy . k> cUrRsaybBa¢ak;fa

2a1 cos A

2bc

Edl a , b , c CaRCugRtIekaN ABC . x> TajbBa¢ak;fa 1

(1 cos A)(1 cosB)(1 cosC)8

!^> eK[RtIekaN ABC mYy . k> cUrRsaybBa¢ak;fa 2 2 2sin A sin B sin C 2sinBsinCcos A x> TajbBa¢ak;fa

2 2 2sin A sin B sin Ccot A cot B cot C

2sin AsinBsinC


- 4 -

!&> eK[RtIekaN ABC mYymanmMu A ,B,C CamMuRsYcEdlepÞógpÞat; smPaB 1 1 1 1

cot A cot B cot C2 sin A sinB sinC

.

cUrRsayfa ABC CaRtIekaNsmgS½ ? !*> tag R CakaMrgVg;carwkkñúg nig S CaépÞRkLaénRtIekaN ABC mYy . k> cUrRsayfa

22R(cot A cot B)(cot B cotC)(cotC cot A)

S

x> ebI ABC CamMuRsYcenaHcUrTaj[)anfa ³ 1cos AcosBcosC

8 .

!(> eK[RtIekaN ABC mYymanRCug BC a , AC b , AB c carwkkñúgrgVg;mYymanp©it O nig kaM R . tag S nig OBCS CaépÞRkLaén OBC nig ABC erogKña . snμtfa A ,B,C CamMuRsYc . k> cUrRsayfa 2

OBC

1S a cot A

4 .

x> cUrTajbgðajfa 2 2 2a cot A b cot B c cot C 4S . K> cUrTajbgðajfa

2

abcacos A bcosB ccosC

2R .


- 5 -

@0> eK[RtIekaN ABC mYy . D CacMNucmYYénRCug [BC] Edl BAD nig DAC . cUrRsaybBa¢ak;fa BC sin sin

AD sinB sinC

?

@!> bgðajfaebI c,b,a tagerogKñaCatYTI r,q,p énsIVútnBVnþmYyenaHeK)ansmPaB 0c)qp(b)pr(a)rq( . @@> eK[ctuekaNe)a:g ABCD mYymanépÞRkLaesμI 1 . cUrRsaybBa¢ak;fa ³ AB BC CD DA AC BD 4 8 ? ¬Austria-Poland, 1985¦ @#> eK[RtIekaNBIrmanRCug a ,b ,c knøHbrimaRt a b c

p2

nigkaMrgVg;carwkeRkA R . cUrRsaybBa¢ak;fa 2 pR 1

5 2aR bc 2


- 6 -

@$> eKyk I nig O Cap©itrgVg;carwkkñúg nig carwkeRkAén ABC . rgVg; A carwkkñúgmMu A vab:HeTAnwg AB , AC nig BC erogKña Rtg;cMnuc K ,M nig N erogKña . ebIsinCacMnuckNþal P énGgát; KM sßitenAelIrgVg;carwkeRkA énRtIekaN ABC enaHcUrbgðajfabIcMnuc O , I , N rt;Rtg;Kña ? @%> cUrRsayfakaerénRKb;cMnYnKt;RtUvmanTRmg; k4 b¤ 14 k RKb; *INk . @^> cUrRsayfaebI 22 ba Eckdac;nwg 3 enaH a Eckdac;nwg 3 nig b Eckdac;nwg 3 . @&> cUrbgðajfa 44 n CacMnYnbzmEtkñugkrNI 1n . @*> eKdwgfa 0160321002004008 manktþabzm 250000p . cUrkMNt;ktþabzmenaH .

@(> cUrkMNt;EpñkKt;én

610

1

1

k k ?

#0> cUrkMNt;KUmYyéncMnYnKt;viC¢man ba , edaydwgfa ³ )(:)( baabi Eckmindac;nwg 7

777)(:)( babaii Eckdac;nwg 77 (IMO 1984 )


- 7 -

#!> eKGnuKmn_ xxf 2121

1)(

cUrKNna )1

(...)2

()1

(n

nf

nf

nfS

#@> cUrRsaybBa¢ak;facMnYn ³ n n n nA 113 168 141 427 Eckdac;nwg 7 Canic©RKb;cMnYnKt;FmμCati n. ##> eK[ m nig n CaBIrcMnYnKt;viC¢manEdl 1 m n . cUrkMNt;témøtUcbMputén m n edIm,I[cMnYn m22 nig n22 manelxBIrxÞg;cugeRkaydUcKña . #$> eK[ 2f (n) 4n 7n 34 Edl n IN * cUrbgðajfaKμan n EdleFIV[cMnYn f (n) Eckdac;nwg 121 eT. #%>eK[ 2f (a) 16a 50a 13 Edl a kMNt;RKb;témø a EdleFIV[cMnYn f (a) Eckdac;nwg 49 #^> kMNt;RKb;cMnYnKt;viC¢man x edIm,I[ 3 2x 10x 115x 237 Eckdac;nwg 343 . #&> cUrRsayfaRKb;cMnYnKt;viC¢man n eK)an

237

134)(

n

nnf

CaRbPaKsRmYlmin)an ?


- 8 -

#*> eKdwgfa ab

tan3 . cUrRsaybBa¢ak;fa ³

3 23 23 2

4

3 2

4

ba

1

b

sin

a

cos

#(> eKmansIVútcMnYnBit )a( n kMNt;eday ³ )

2

11....()

4

11)(

2

11(a n2n

cUrKNnalImIt nnalim

.

$0> eK[plbUk

222222n )1n(

1

n

11....

4

1

3

11

3

1

2

11

n

1S

cUrKNna nnSlim

.

$!> eKmanes‘rIGnnþmYy ....2

n.....

2

3

2

2

2

1)

2

n( n

2

3

2

2

22

1nn

2

cUrRsayfaes‘rIxagelIenHCaes‘rIbRgYm . $@> cMeBaHRKb; *INn eKmanplbUk ³ 1n432n 2).1n2)(1n2(

3n2....

2.7.5

9

2.5.3

7

2.3.1

5S

k> kMNt;cMnYnBit a nig b edIm,I[eK)ansmPaB 1nn1n 2).1n2(

b

2).1n2(

a

2).1n2)(1n2(

3n2

x> KNna nS nig nnSlim

.


- 9 -

$#> cMeBaHRKb; *INn eKmansIVútcMnYnBit )a( n kMNt;eday ³ 3a0 nig n2nn21n

3

1a

3

11a

k> KNna na CaGnuKmn_én n . x> KNnalImIt nn

alim

. $$> eK[ a nig b CaBIrcMnYnBitEdl

2ba0

cUrRsayfa bcos

abatanbtan

acos

ab22

$%> KNnaplbUk ³ n

n23

n2n

1nn Cn.....C9C4CS Edl

)!pn(!p

!nCp

n

$^> eK[GnuKmn_ 2x)x(f k> sikSaGefrPaB nig sg;Rkab )c( énGnuKmn_ f . x> eKBinitü n cMnuc n321 A...,,A,A,A sßitenAelIRkab )c( Edl )22,2(A1 nig )a,a(A 1nnn RKb; *INn . cUrkMNt;kUGredaenéncMnuc nA . K> cUrKNna nOA CaGnuKmn_én n rYcKNna n

nOAlim

.


- 10 -

$&> eK[RtIekaN ABC mYy . emdüanénRtIekaNEdlKUsecj BIkMBUl C,B,A kat;rgVg;carwkeRkARtIekaNerogKñaRtg; P,N,M . tag cba ,, CaRCug nig cba mmm ,, Caemdüanén ABC . cUrRsayfa

9

16

222 224

4

224

4

224

4

bac

m

cab

m

cba

m cba $*> eK[ M CaTIRbCMuTm¶n;énRtIekaN ABC . ebIbnÞat; AB b:HeTAnwgrgVg;carwkeRkAénRtIekaN AMC enaHbgðajfa 32

sin

1

sin

1

CBMCMA

$(> eK[ 0,, cba . cUrRsaybBa¢ak;fa ³

2

9)(4)(4)(4 3 333 333 33

ba

bac

ac

acb

cb

cba %0> eK[ 1,1 ba nig 1x . cUrRsayfa xxx baba

2

log2loglog


- 11 -


- 12 -

lMhat;TI1 cMeBaHRKb; IRx cUrRsayBa¢ak;fa ³

22|sincos| baxbxa dMeNaHRsay RsayBa¢ak;fa ³

22|sincos| baxbxa

tagviucTr½ ),( bau

nig )sin,(cos xxv

yk CamMurvagviucTr½ u nig

v .

tamniymn½yplKuNsáaEl ³

cos||.||.

vuvu

eKTaj ||.||

.cos

vu

vu

eKman xbxavu sincos.

1||;|| 22 vbau


- 13 -

eK)an 22

sincoscos

ba

xbxa

cMeBaHRKb;mMu 1cos1: IR b¤ 1|cos|

eK)an 1sincos22

ba

xbxa

dUcenH 22|sincos| baxbxa .


- 14 -

lMhat;TI2 cMeBaHRKb;cMnYnBit xcUrRsaybBa¢ak;fa

bab

x

a

x

1cossin 44

)0,0(; ba dMeNaHRsay RsaybBa¢ak;fa

bab

x

a

x

1cossin 44

tamvismPaB SchwarzCauchy eK)an ³

baxx

bx

ax

bx

ax

bx

ax

22244

222244

)cos(sincossin

)(cos)(sincossin

eday 1cossin 22 xx

dUcenH bab

xa

x

1cossin 44

.


- 15 -

lMhat;TI3 eK[GnuKmn_

9)cot(tan12)cot(tan4)( 22 xxxxxf Edl )

2,0(

x . cUrrktémøtUcbMputénGnuKmn_ )(xf . dMeNaHRsay rktémøtUcbMputénGnuKmn_ )(xf eKman

9)cot(tan12)cot(tan4)( 22 xxxxxf tag 2cottan2cottan xxxxt eK)an 2cottan)cot(tan 2222 xxxxt b¤ 2cottan 222 txx eK)an 8)32(912)2(4)( 22 tttxf eday 2t enaH 13432 t eK)an 781)( xf dUcenHtémøtUcbMputén )(xf esμInwg 7 .


- 16 -


xx

ysin45

cos3

cUrrktémøtUcbMput nig témøFMbMputénGnuKmn_ y . dMeNaHRsay rktémøtUcbMput nig témøFMbMput eKman

xx

ysin45

cos3

eK)an xxyy cos3sin45 b¤ )1(5sin4cos3 yxyx tamvismPaB SchwarzCauchy eK)an ³

)2(169|sin4cos3| 2yxyx tam )1( nig )2( eK)an 2169|5| yy b¤ 22 16925 yy

1

992

2

y

y

eKTaj)an 11 y . dUcenH 1;1 maxmin yy .


- 17 -


89)2(1243),( 2222 yxxyyxyxyxf cUrkMNt; x nig y edIm,I[GnuKmn_ ),( yxf mantémøtUcbMput rYckMNt;témøtUcbMputenaH . dMeNaHRsay kMNt; x nig y

89)2(1243),( 2222 yxxyyxyxyxf GnuKmn_enHGacsresr ³

5)3(4)2(3)6(),( 222 yxxyyxf edIm,I[GnuKmn_ ),( yxf mantémøtUcbMputluHRtaEt ³

03

02

06

y

x

xy

naM[ 3,2 yx

dUcenH 3,2 yx ehIytémøtUcbMputén ),( yxf KW 5 .


- 18 -

lMhat;TI6 eKmansmIkar 032 xx manb¤stageday 1x nig 2x cUrKNnatémø 5

23

1 419 xxA dMeNaHRsay KNnatémø 5

23

1 419 xxA eKman 1x nig 2x Cab¤sénsmIkar 032 xx eK)an )1(031

21 xx nig )2(032

22 xx

tam )1( eK)an 312

1 xx

343)3(

3

1113

1

12

13

1

xxxx

xxx

tam )2( eK)an 322

2 xx

2119

12)3(7127

)127(

)963(

)96(

)3(

25

2

2222

25

2

225

2

2225

2

22

225

2

22

42

xx

xxxxx

xxx

xxxx

xxxx

xx


- 19 -

yk 34 13

1 xx nig 2119 25

2 xx CMnYskñúgkenSam 5

23

1 419 xxA eK)an ³ )2119(4)34(19 21 xxA

141)(76

84765776

21

21

xx

xx

eday 121 xx eK)an 21714176 A dUcenH 217419 5

23

1 xxA .


- 20 -

lMhat;TI7 eK[GnuKmn_ cbxaxxf 2)( Edl IRcbaa ,,,0 . cUrRsayfaebIsmIkar xxf )( Kμanb¤skñúgsMNMucMnYnBitenaH smIkar xxff )]([ k¾Kμanb¤skñúgsMNMucMnYnBitEdr . dMeNaHRsay karbgðaj eKman xxf )( smmUl xcbxax 2 b¤ )(0)1( 1

2 Ecxbax smIkar )( 1E Kμanb¤skñúgsMNMucMnYnBitluHRtaEt ³

)1(04)1( 21 acb

mü:ageToteKman ³ xcxbfxafxxff )()()]([ 2

]1)(][)([

)(])([])([ 22

baxxafxxf

xxfxxfbxxfa

smIkar xxff )]([ smmUl ³ 0]1)(][)([ baxxafxxf


- 21 -

eday 0)( xxf CasmIkarKμanb¤s ¬ smμtikmμ ¦ eK)an 01)( baxxaf

)(01)1(

01)(

222

2

Ebacxbaxa

baxcbxaxa

DIsRKImINg;énsmIkar )( 2E KW ³ )1(4)1( 222

2 bacaba

)2(]44)1[(

]444)1[(22

22

acba

bacba

tam )1( nig )2( eKTaj)an 02 dUcenHebIsmIkar xxf )( Kμanb¤skñúgsMNMucMnYnBitenaH smIkar xxff )]([ k¾Kμanb¤skñúgsMNMucMnYnBitEdr .


- 22 -

lMhat;TI8 edaHRsaysmIkar ³ xxxxx 126324 232 dMeNaHRsay edaHRsaysmIkar xxxxx 126324 232 smIkarmann½yluHRtaEt 01263 23 xxx smmUl 0)42(3 2 xxx smmUl 0]3)1([3 2 xx naM[ 0x . tag 42,3 2 xxbxa eK)an 42 xxba . smIkarGacsresr abba 2 elIkCakaer abba 4)( 2

ba

ba

abbaba

0)(

422

22

eKTaj 423 2 xxx b¤ 0452 xx eday 0 cba dUcenH 4;1 21 xx .


- 23 -

lMhat;TI9 edaHRsaysmIkar ³

33

3 23 2

1952

)13(4)13(10

x

xx

dMeNaHRsay edaHRsaysmIkar ³

33

3 23 2

1952

)13(4)13(10

x

xx

smIkarGacsresr ³

0)13(419)52()13(10 3 2333 2 xxx EckGgÁTaMgBIrénsmIkarnwg 3 2)13( x eK)an ³

0413

13)52(

13

13.10 3333

2

3

x

x

x

x

tag 3

13

13

x

x

t eK)an 04)52(10 3323 tt

2333

333

3323

)52(25544

5825544

4.104)52(


- 24 -

eKTajb¤s 333

33

1 54

102

1025252

t

33

3

3

33

2 21

105

1025252

t

-cMeBaH 3

54

t eK)an 33

54

13

13

x

x

b¤ )13(4)13(5 xx 93 x b¤ 2x .

-cMeBaH 3

21

t eK)an 33

21

13

13

x

x

b¤ 13)13(2 xx 33 x b¤ 1x . dUcenH 2;1 xx .


- 25 -

lMhat;TI10 eK[smIkar 099999 xx cUrbgðajfasmIkarenHmanb¤s 0x CacMnYnBitEtmYyKt; rYcRsayfa

0x CacMnYnGsniTan . dMeNaHRsay bgðajfasmIkarmanb¤sEtmYyKt; tagGnuKmn_ 9999)( 9 xxxf Edl IRx eK)an IRxxxxf 0)1(999)(' 88 naM[ )(xf CaGnuKmn_ekInCanic© . dUcenHRkab )(c tag f kat;GkS½ oxx' EtmYycMnucKt; . dUcenHsmIkar 099999 xx manb¤s 0x EtmYyKt;. bgðajfa 0x CacMnYnGsniTan ]bmafa 0x CacMnYnsniTanenaH

q

pxo Edl 1),( qpGCD

eK)an 0999)(9)( 9 qp

qp b¤ )111(9 89 pqqp

edayGgÁTIBIrEckdac;nwg q enaH p Eckdac;nwg q ¬pÞúyBIkarBit¦ dUcenH 0x CacMnYnGsniTan .


- 26 -

lMhat;TI11 eKdwgfa b

tan xa

.

cUrRsayfa 4 4cos x sin x 1

a b a b

dMeNaHRsay Rsayfa

4 4cos x sin x 1

a b a b

eKman btan x

a naM[ 2 b

tan xa

eday cos xtan x

sin x

eK)an 2

2

sin x b

cos x a b¤

2 2 2 2cos x sin x cos x sin x 1

a b a b a b

eKTaj 2cos x 1

a a b

naM[

4

2

cos a(1)

a (a b)

ehIy 2sin x 1

b a b

naM[

4

2

sin b(2)

b (a b)

bUksmPaB (1) nig (2) GgÁ nig GgÁeK)an ³ 4 4

2 2 2

cos x sin x a b a b 1

a b (a b) (a b) (a b) a b

dUcenH 4 4cos x sin x 1

a b a b

.


- 27 -

lMhat;TI12 eKdwgfa m n p

cosa , cosb , coscn p p m m n

cUrKNnakenSam ³ 2 2 2

2 2 2

sin a sin b sin cM

2 2cosa sin a 2 2cosb sin b 2 2cosc sin c

dMeNaHRsay KNnakenSam ³ eKman 2 2sin a 1 cos a (1 cosa)(1 cosa) nig 2 2 22 2cosa sin a 1 2cosa cos a (1 cosa) eK)an

2

2 2

sin a (1 cosa)(1 cosa) 1 cosa

2 2cosa sin a (1 cosa) 1 cosa

ehIy 2

2

sin b 1 cosb

2 2cosb sin b 1 cosb

nig 2

2

sin c 1 cosc

2 2cosc sin c 1 cosc

eK)an 1 cosa 1 cosb 1 coscE

1 cosa 1 cosb 1 cosc

m n p1 1 1n p p m m nm n p

1 1 1n p p m m n

n p m p m n m n p1

m n p


- 28 -

lMhat;TI13 eKmanRtIekaN ABC mYyEdl BC a , AC b , AB c . tag R nig S erogKñaCakaM nig épÞRkLaénRtIekaN ABC enH . cUrRsayfa ³

2 2 2cos A cosB cosC R(sin A sin B sin C)

a b c 4S

dMeNaHRsay tamRTwsþIbTkUsIunUs nig sIunUsGnuvtþn_kñúgRtIekaN ABC eK)an ³

2 2 2a b c 2bccos A b¤ 2 2 2cos A b c a

a 2abc

a b c2R

sin A sinB sinC b¤

a 2R sin A

b 2R sinB

c 2R sinC

nig abcS

4R

eK)an 2 2 2 2cos A 4R (sin B sin C sin A)

a 8RS

b¤ 2 2 2cos A R(sin B sin C sin A)

(1)a 4S

dUcKñaEdr 2 2 2cosB R(sin A sin C sin B)

(2)b 4S

nig 2 2 2cosC R(sin A sin B sin C)

(3)c 4S

bUksmPaB (1) ,(2) &(3) eK)an ³ 2 2 2cos A cosB cosC R(sin A sin B sin C)

a b c 4S

Bit .


- 29 -

lMhat;TI14 kñúgRKb;RtIekaN ABCcUrRsayfa ³


2

dMeNaHRsay RsaybBa¢ak;fa


2

tagRCug BC a , AC b , AB c nig a b cp

2

CaknøHbrimaRt

yk RCakaMrgVg;carwkeRkA nig S CaépÞRkLarbs; ABC . tamRTwsþIbTkUsuInUseKman 2 2 2a b c 2bccos A eK)an 2 2 2a (b 2bc c ) 2bc(1 cos A) eKTaj

2 2(b c) a (b c a)(b c a)1 cos A

2bc 2bc

eday a b cp

2

enaH a b c 2p nig b c a 2(p a)

eK)an 4p(p a) 2p(p a)1 cos A

2bc bc

dUcKñaEdr 2p(p b) 2p(p c)1 cosB , 1 cosC

ac ab

eK)an 2

2

8p .p(p a)(p b)(p c)(1 cos A)(1 cosB)(1 cosC) (1)

(abc)

tamrUbmnþehr:ug abcS p(p a)(p b)(p c)

4R


- 30 -

eKTaj 2 2

p(p a)(p b)(p c) 1(2)

(abc) 16R

ykTMnak;TMng (2) CMnYskñúg (1) eK)an ³ 22

2

8p 1 p(1 cos A)(1 cosB)(1 cosC) . (3)

16R 2 R

tamRTwsþIbTsIunUs a b c2R

sin A sinB sinC

eKTaj a b c 2p psin A sinB sinC (4)

2R 2R R

tam (3) nig (4) eK)anTMnak;TMng ³ 2(sin A sinB sinC)


Bit .

smÁal; ³ eKGacsYrbEnßmeToteday[RsaybBa¢ak;fa ³ 23

cos A cosB cosC sin A sinB sinC1

3 2

edayeRbIvismPaBmFümnBVnþ mFümFrNImaRteK)an ³ 3

1 cos A 1 cosB 1 cosC(1 cos A)(1 cosB)(1 cosC)

3

b¤ 3

cos A cosB cosC(1 cos A)(1 cosB)(1 cosC) 1

3

eday 2(sin A sinB sinC)


dUcenH 23

cos A cosB cosC sin A sinB sinC1

3 2


- 31 -

lMhat;TI15 eK[RtIekaN ABC mYy . k> cUrRsaybBa¢ak;fa

2a1 cos A

2bc

Edl a , b , c CaRCugRtIekaN ABC . x> TajbBa¢ak;fa 1


dMeNaHRsay k>RsaybBa¢ak;fa

2a1 cos A

2bc

tamRTwsþIbTsIunUseKman 2 2 2a b c 2bccos A tamvismPaBmFümnBVnþ mFümFrNImaRteKman 2 2b c 2bc eKTaj 2a 2bc 2bccos A 2bc(1 cos A) dUcenH

2a1 cos A

2bc .

x> TajbBa¢ak;fa 1(1 cos A)(1 cosB)(1 cosC)

8

tamsRmayxagelIeKman 2a

1 cos A (1)2bc

RsaydUcKñaEdr 2b

1 cosB (2)2ac

nig 2c

1 cosC (3)2ab

KuNvismPaB (1) , (2) , (3) GgÁ nig GgÁeKTTYl)an ³

1(1 cos A)(1 cosB)(1 cosC)

8 Bit .


- 32 -

lMhat;TI16 eK[RtIekaN ABC mYy . k> cUrRsaybBa¢ak;fa 2 2 2sin A sin B sin C 2sinBsinCcos A x> TajbBa¢ak;fa ³


2sin AsinBsinC

dMeNaHRsay k>RsaybBa¢ak;fa 2 2 2sin A sin B sin C 2sinBsinCcos A tag a , b , c CaRCugRtIekaN ABC nig R CakaMrgVg;carwkeRkARtIekaN. tamRTwsþIbTsIunUs a b c

2Rsin A sinB sinC

eKTaj a 2R sin A

b 2r sinB (1)

c 2R sinC

tamRTwsþIbTkUsIunUs 2 2 2a b c 2bccos A (2) yk (1) CMnYskñúg (2) eK)an ³

2 2 2 2 24R sin A 4R (sin B sin C 2sinBsinCcos A) sRmYl 24R kñúgGgÁTaMgBIrénsmPaBeK)an ³

2 2 2sin A sin B sin C 2sinBsinCcos A Bit


- 33 -

dUcenH 2 2 2sin A sin B sin C 2sinBsinCcos A . x> TajbBa¢ak;fa ³


2sin AsinBsinC

tamsRmayxagelIeKman 2 2 2sin A sin B sin C 2sinBsinCcos A

eKTaj 2 2 2sin B sin C sin A

cos A2sinBsinC

eday cos A

cot Asin A

eK)an 2 2 2sin B sin C sin A

cot A (i)2sin AsinBsinC

RsaydUcKñaEdreKTTYl)an 2 2 2sin A sin C sin B

cot B (ii)2sin AsinBsinC

nig 2 2 2sin A sin B sin C

cot C (iii)2sin AsinBsinC

eFIVplbUksmPaB (i) , (ii) & (iii) eK)an ³ 2 2 2sin A sin B sin C

cot A cot B cot C2sin AsinBsinC

Bit

dUcenH 2 2 2sin A sin B sin C

cot A cot B cot C2sin AsinBsinC

.


- 34 -

lMhat;TI17 eK[RtIekaN ABC mYymanmMu A ,B,C CamMuRsYcEdlepÞógpÞat;smPaB

1 1 1 1cot A cot B cot C

2 sin A sinB sinC

.

cUrRsayfa ABC CaRtIekaNsmgS½ ? dMeNaHRsay Rsayfa ABC CaRtIekaNsmgS½ tag a , b , c CaRCug nig S CaépÞRkLaénRtIekaN ABC eKman 1 1 1

S bcsin A acsinB absinC2 2 2

eKTaj)an 2S 2S 2Ssin A , sinB , sibC

bc ac ab

eK)an 2 2 2

2 2 2b c a

cos A b c a2bccot A2Ssin A 4Sbc

ehIy 2 2 2 2 2 2a c b a b c

cot B , cot C4S 4S

eK)an 2 2 2a b c

cot A cot B cot C (1)4S

mü:ageTot 1 1 1 bc ca ab(2)

sin A sinB sinC 2S

tamsm μtikmμ 1 1 1 1cot A cot B cot C (3)

2 sin A sinB sinC


- 35 -

yksmIkar (1) & (2) CMnYskñúg (3) eK)an ³

2 2 2a b c ab bc ca

4S 4S

b¤ 2 2 2a b c ab bc ca TMnak;TMngenHsmmUl 2 2 2(a b) (b c) (c a) 0

eKTaj a b 0

b c 0

c a 0

naM[ a b c .

edayRtIekaN ABC manRCugbIes μIKñavaCaRtIekaNsmgS½ . smÁal; ³ eKGacRsayfa


2sin AsinBsinC

ehIysm μtikm μ 1 1 1 1cot A cot B cot C

2 sin A sinB sinC

eKTaj)ansmIkar ³ 2 2 2sin A sin B sin C 1 1 1 1

2sin AsinBsinC 2 sin A sinB sinC

2 2 2

2 2 2

sin A sin B sin C sin AsinB sinBsinC sinCsin A

(sin A sinB) (sinB sinC) (sinC sin A) 0

eKTaj sin A sinB 0

sinB sinC 0

sinC sin A 0

naM[ sin A sinB sinC

b¤ A B C enaH ABC CaRtIekaNsmgS½ .


- 36 -

lMhat;TI18 tag R CakaMrgVg;carwkkñúg nig S CaépÞRkLaénRtIekaN ABC mYy . k> cUrRsayfa

22R(cot A cot B)(cot B cotC)(cotC cot A)

S

x> ebI ABC CamMuRsYcenaHcUrTaj[)anfa ³ 1cos AcosBcosC

8 .

dMeNaHRsay Rsayfa

22R(cot A cot B)(cot B cot C)(cot C cot A)

S

KUskMBs; aAA' h én ABC . tag BC a , AC b , AB c . kñúgRtIekaNEkg ABA' & AA'C eKman

BA' A'Ccot B ; cot C

AA' AA'

eK)an 2

a

BA' A'C a acot B cot C

AA' h 2S

. S CaépÞRkLa ABC

dUcKñaEdr 2 2b c

cot C cot A , cot A cot B2S 2S

A

B C A'


- 37 -

eK)an 2 2 2

3

a b c(cot A cot B)(cot B cot C)(cot C cot A)

8S

eday abcS

4R enaH abc 4RS

eK)an 2 2

3

16R S(cot A cot B)(cot B cot C)(cot C cot A)

8S

dUcenH 22R

(cot A cot B)(cot B cot C)(cot C cot A)S

.

x> Taj [)anfa ³ 1cos AcosBcosC

8

man 22R

(cot A cot B)(cot B cot C)(cotC cot A) (i)S

ebI ABC CamMuRsYcenaH cot A 0 , cot B 0 , cot C 0 tamvismPaB mFümnBVnþ mFümFrNImaRteKman ³ cot A cot B 2 cot Acot B , cot B cotC 2 cot Bcot C

cot C cot A 2 cotCcot A

KuNvismPaBxagelIenH GgÁ nig GgÁ eK)an (cot A cot B)(cot B cotC)(cotC cot A) 8cot Acot BcotC (ii) tam (i)&(ii) eKTaj)an

22R8cot Acot Bcot C

S

Et 3

2abc 8R sin AsinBsinCS 2R sin AsinBsinC

4R 4R

eK)an 2

2

2R8cot Acot Bcot C

2R sin AsinBsinC

dUcenH 1cos AcosBcosC

8


- 38 -

lMhat;TI19 eK[RtIekaN ABC mYymanRCug BC a , AC b , AB c carwkkñúgrgVg;mYymanp©it O nig kaM R . tag S nig OBCS CaépÞRkLaén OBC nig ABC erogKña . snμtfa A ,B,C CamMuRsYc . k> cUrRsayfa 2

OBC

1S a cot A

4 .

x> cUrTajbgðajfa 2 2 2a cot A b cot B c cot C 4S . K> cUrTajbgðajfa

2


2R .

dMeNaHRsay k> Rsayfa 2

OBC

1S a cot A

4

tag K CacMNuckNþalénRCug [BC] enaH [OK] [BC] . eKman BOC

BOK BAC A2

.

A

B C

O

K


- 39 -

kñúgRtIekaNEkg OBK eKman ³ OK 2OK 2OK

cot BOK cot ABK BC a

eKTaj 1OK acot A

2 .

RkLaépÞénRtIekaN OBC KW 2OBC

1 1S BC.OK a cot A

2 4 Bit

dUcenH 2OBC

1S a cot A

4 .

x> Tajbgðajfa 2 2 2a cot A b cot B c cot C 4S tamsRmayxagelIeKman 2

OBC

1S a cot A

4

dUcKñaEdr 2OCA

1S b cot B

4 nig 2

OAB

1S c cot C

4

eday OBC OCA OABS S S S eK)an 2 2 21

S (a cot A b cot B c cot C)4

dUcenH 2 2 2a cot A b cot B c cot C 4S . K> Tajbgðajfa

2


2R

eKman 2 2 2 abca cot A b cot B c cotC 4S

R eRBaH abc

S4R

eday 2 2aa cot A .acos A 2Racos A , b cot B 2RbcosB

sin A

enaHeK)an 2


2R .


- 40 -

lMhat;TI20 eK[RtIekaN ABC mYy . D CacMNucmYYénRCug [BC] Edl

BAD nig DAC . cUrRsaybBa¢ak;fa BC sin sin

AD sinB sinC

?

dMeNaHRsay RsaybBa¢ak;fa BC sin sin

AD sinB sinC

tamRTwsþIbTsIunUsGnuvtþn_ kñúg ABD& ADC eKman BD AD

sin sinB

b¤ sinBD .AD (1)

sinB

ehIy DC AD

sin sinC

b¤ sinDC .AD (2)

sinC

bUkTMnak;TMng (1) & (2) eK)an sin sinBD DC .AD

sinB sinC

eday BD DC BC enaH sin sinBC .AD

sinB sinC

dUcenH BC sin sin

AD sinB sinC

.

A

DB C


- 41 -

lMhat;TI21 bgðajfaebI c,b,a tagerogKñaCatYTI r,q,p énsIVútnBVnþmYyenaHeK)ansmPaB

0c)qp(b)pr(a)rq( . dMeNaHRsay karbgðaj ebI c,b,a tagerogKñaCatYTI r,q,p énsIVútnBVnþ )u( n enaHeK)an

d)1r(uuc,d)1q(uub,d)1p(uua 1r1q1p

eKman 0c)qp(b)pr(a)rq( b¤ 0cqcpbpbraraq

b¤ 0)ab(r)ca(q)bc(p eday

d)pq(ab

d)rp(ca

d)qr(bc

enaH 0)pq(rd)rp(qd)qr(pd b¤ 0)prqrqrpqpqpr(d 00 Bit dUcenH 0c)qp(b)pr(a)rq( .


- 42 -

lMhat;TI22 eK[ctuekaNe)a:g ABCD mYymanépÞRkLaesμI 1 . cUrRsaybBa¢ak;fa ³ AB BC CD DA AC BD 4 8 ? ¬Austria-Poland, 1985¦ dMeNaHRsay RsaybBa¢ak;fa ³ AB BC CD DA AC BD 4 8 tag AB a , BC b , CD c , DA d nigGgát;RTUg AC m , BD n . tagmMu AIB CID .

A

B

C

D

I


- 43 -

RkLaépÞénctuekaN ABCD kMNt;eday ³ ABCD ABC ACD

1 1S S S absinB cdsin D (1)

2 2

b¤ ABCD ABD BCD

1 1S S S adsin A bcsinC (2)

2 2

bUkTMnak;TMng (1) nig (2) eK)an ³ ABCD

12S (adsin A absinB bcsinC cdsin D)

2

edayeKman sin x 1 RKb;cMnYnBit x enaHeKTaj)an ABCD

12S (ad ab bc cd)

2

b¤ ABCD

1S (a c)(b d)

4 eday ABCDS 1

eKTaj)an (a c)(b d) 4 tavismPaB AM GM eKman ³ a b c d 2 (a c)(b d) b¤ a b c d 4 (i) müa:geTot ³


- 44 -

ABCD IAB IBC ICD IADS S S S S

1(IA.IB IB.IC IC.ID ID.IA)sin

21

(AC.IB AC.ID)sin21 1

AC.BDsin AC.BD2 2

eK)an ABCD

1S mn

2 eday ABCDS 1

eK)an mn 2 EttavismPaB AM GM eKman ³ m n 2 mn 2 2 8 (ii) bUkvismPaB (i) nig (ii) eK)an ³ a b c d m n 4 8 Bit dUcenH AB BC CD DA AC BD 4 8 .


- 45 -

lMhat;TI23 eK[RtIekaNBIrmanRCug a ,b ,c knøHbrimaRt a b c

p2

nigkaMrgVg;carwkeRkA R . cUrRsaybBa¢ak;fa 2 pR 1

5 2aR bc 2

dMeNaHRsay RsaybBa¢ak;fa 2 pR 1

5 2aR bc 2

tamvismPaBRtIekaNeKman ³ AB AC (AH BH) (AH HC) 2AH BC eKTaj)an AB AC BC b c a

AH p a2 2

H B C

A

a

b c


- 46 -

tag S CaépÞRkLaénRtIekaN ABC enaHeK)an ³ 1 1

S BC.AH a.AH2 2

b¤ 2SAH

a

eKTaj)an 2Sp a

a b¤ 2S a(p a) Et abc

S4R

ehtuenH abc

2. a(p a)4R

b¤ bc 2pR 2aR b¤ bc 2aR 2pR naM[ pR 1

(i)bc 2aR 2

mü:ageTot]bmafa pR 2

2aR bc 5

Bit

eK)an 4aR 2bc 5pR (1) eday abc 1

S a.AH4R 2

b¤ bc 2R.AH tam (1) eK)an 4aR 4R.AH 5pR b¤ 5p

a AH (2)4

kñúgRtIekaNEkg ABH nig AHC eKman ³ 2 2 2AB AH BH nig 2 2 2AC AH HC b¤ 2 2 2c AH BH nig 2 2 2b AH HC enaH 2 2 2 2b c AH HC AH BH


- 47 -

tamvismPaB Minkowsky eKman ³ 2 2 2 2 2 2

1 1 2 2 1 2 1 2x y x y (x x ) (y y ) ehtuenH 2 2 2 2b c 4.AH (BH HC) 4.AH a b¤ 2 2a b c a 4AH a b¤

2 2a 4AH ap (3)

2

tam (2) nig (3) eyIgnwgRsay[)anfa ³ 2 25 a 4AH a

a AH ( )4 2

Bit

b¤ 2 23a 8.AH 5 4.AH a b¤ 2 2 2 29a 48a.AH 64.AH 100.AH 25a b¤ 2 216a 48a.AH 36.AH 0 b¤ 2(4a 6AH) 0 Bit ehtuenH pR 2

2aR bc 5

(ii)

tamTMnak;TMng (i) nig (ii) eK)an ³ 2 pR 1

5 2aR bc 2

Bit


- 48 -

lMhat;TI24 eKyk I nig O Cap©itrgVg;carwkkñúg nig carwkeRkAén ABC . rgVg; A carwkkñúgmMu A vab:HeTAnwg AB , AC nig BC erogKña Rtg;cMnuc K ,M nig N erogKña . ebIsinCacMnuckNþal P énGgát; KM sßitenAelIrgVg;carwkeRkA énRtIekaN ABC enaHcUrbgðajfabIcMnuc O , I , N rt;Rtg;Kña ? dMeNaHRsay RsaybBa¢ak;fabIcMnuc O , I , N rt;Rtg;Kña

A

B

C

PQ

K

MN

I

O

8


- 49 -

yk Q CacMnucRbsBVTIBIrrvagbnÞat; PM CamYyrgVg; (ABC) tag KP PM x nig PQ y . eyIgman KB p c nig MC p b . tamRTwsþIbTsV½yKuNéncMnuc K nig M eFobnwgrgVg; (ABC) eKman KP.KQ KA.KB nig MP.MQ MC.MA b¤ x(x y) p(p c) nig x(x y) p(p b) bUksmIkarBIrenHeK)an 22x p(2p b a) pa b¤ 2 1

x pa2

ehtuenH 2 2MK 4x 2pa . tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN AKM eKman ³

2 2 2MK MA KA 2MA.KAcos A eday MA KA p nig 2MK 2pa enaHeK)an ³

2 2 2 2 A2pa 2p 2p cos A 4p sin

2 b¤ 2 A

a 2psin2

Et A (p b)(p c)sin

2 bc

eK)an 2p(p b)(p c)a

bc

b¤ abc 2p(p b)(p c) (1)


- 50 -

yk D CacMeNalEkgén I elI BCehIysnμtfa N' CaRbsBV rvag (OI) CamYy [BC] nig tag E CacMnuckNþalén BC . ]bmafa N' N enaHeK)an N'C NC p b eday BD p b enaHeK)an ED EN' b¤ E CacMnuckNþal énGgát; [DN'] . RtIekaNEkg N'DI nigRtIekaNEkg N'EO manmMu DN'I rYmvaCaRtIekaNdUcKña . ehtuenHeK)an DI DN' 2DE

2EO EN' EN'

b¤ DI 2.EO eKman BOC

BAC EOC2

.

kñúgRtIekaNEkg EOC eKman OEcos EOC cos A

OC

eKTaj OE OC.cos A naM[ DI 2.OC cos A yk DI r nig OC R ¬kaMrgVg;carwkkñúg nig eRkA ABC ¦ eK)an r 2R cos A b¤ r

cos A2R

tamrUbmnþépÞRkLaén ABC eKman abc

S pr4R

nigRTwsþIbTkUsIunUs 2 2 2b c a

cos A2bc


- 51 -

eK)an 2 2 2 2

Sb c a 2Sp

abc2bc abcp2S

b¤ 2 2 2 24S ap(b c a ) Et S p(p a)(p b)(p c) eK)an 2 2 24p(p a)(p b)(p c) ap(b c a ) b¤ 2 2 24(p a)(p b)(p c) a(b c a ) eday 2 2 2 2 2b c a (b c) a 2bc 2bc (b c a)(b c a)

2bc 4(p b)(p c)

enaH 4(p a)(p b)(p c) 2abc 4a(p b)(p c) b¤ abc 2(p a)(p b)(p c) 2a(p b)(p c) b¤ abc 2p(p b)(p c) (2) tamTMnak;TMng (1) nig (2) bBa¢ak;fakar]bma N' N Bit dUcenHbIcMnuc O , I , N rt;Rtg;Kña .


- 52 -

lMhat;TI25 cUrRsayfakaerénRKb;cMnYnKt;RtUvmanTRmg; k4 b¤ 14 k RKb; *INk . dMeNaHRsay karbgðaj eyIgdwgfaRKb;cMnYnKt;viC¢man manrag n2 b¤ 1,12 nn eK)an 22 4)2( nn manrag k4 Edl 2nk ehIy 1)1(4144)12( 22 nnnnn manrag 14 k Edl )1( nnk . dUcenHkaerénRKb;cMnYnKt;RtUvmanTRmg; k4 b¤ 14 k RKb; *INk .


- 53 -

lMhat;TI26 cUrRsayfaebI 22 ba Eckdac;nwg 3 enaH a Eckdac;nwg 3 nig b Eckdac;nwg 3 . dMeNaHRsay karbgðaj snμtfa a nig b CacMnYnEckmindac;nwg 3 enaHcMnYnBIrenHRtUvman rag 23,13,23,13 qbqbpapa Edl ...,2,1,0...;,2,1,0 qp . eK)an 1)23(3)13( 22 pppa ehIy 1]1)23([3)23( 22 pppa eKTaj)anfa 0,132 xxa nig 0,132 yyb eK)an 2)(322 yxba Eckmindac;nwg 3 EdlpÞúyBI smμtikmμ . dUcenHeKGacsnñidæanfaebI 22 ba Eckdac;nwg 3 enaH a Eckdac; nwg 3 nig b Eckdac;nwg 3 .


- 54 -

lMhat;TI27 cUrbgðajfa 44 n CacMnYnbzmEtkñugkrNI 1n . dMeNaHRsay karbgðaj eKman 2224 4)2(4 nnn

]1)1[(]1)1[(

)22)(22(22

22

nn

nnnn

cMeBaHRKb; 1n eKman 11)1( 2 n nig 11)1( 2 n enaH 44 n minEmnCacMnYnbzm . ebI 1n enaH 544 n CacMnYnbzm . dUcenH 44 n CacMnYnbzmEtkñugkrNI 1n .


- 55 -

lMhat;TI28 eKdwgfa 0160321002004008 manktþabzm 250000p cUrkMNt;ktþabzmenaH . dMeNaHRsay kMNt;ktþabzm edayeRCIserIs 2,1000 ba eK)an ³

543223450160321002004008 babbababaa

k

bababababa

ba

ba

.250501.1002.4

)998004)(1002004)(1002(

))()((

2

2222

66

Edl 250000k enaH 250501p .


- 56 -

lMhat;TI29

cUrkMNt;EpñkKt;én

610

1

1

k k ?

dMeNaHRsay kMNt;EpñkKt;

tag 0,1

)( 2

1

xxx

xf

eK)an 0,021

)(' 2

3

xxxf enaH )(xf CaGnuKmn_cuH . cMeBaH 0k eK)an

kxdx

k

k

k

11

1 1

eKTaj

610

1

1610

1

11610

1

1x

dxx

dxk k

k

kk

ehIy

610

1

610

2

1610

1

11

1x

dxkk kk

eKTaj

1610

1

610

1

610

2

11

kk kxdx

k

eday

610

1

610

2

11

1

kk kk


- 57 -

ehIy

610

1

1610

1

1

1000

11

kk kk

nig 1998)110(22 3610

1

610

1

xx

dx

eK)an

610

1

610

1

1

1000

11998

11

kk kk

b¤ 19991

1000

11998

610

1

k k

dUcenHEpñkKt;én

610

1

1

k k KW 1998 .


- 58 -

lMhat;TI30 cUrkMNt;KUmYyéncMnYnKt;viC¢man ba , edaydwgfa ³

)(:)( baabi Eckmindac;nwg 7 777)(:)( babaii Eckdac;nwg 77 .

(IMO 1984 ) dMeNaHRsay kMNt;KUmYyéncMnYnKt;viC¢man ba , ³ tamrUbmnþeTVFajÚtuneK)an ³

222777 ))((7)( bababaabbaba tambRmab;eKman )(:)( baabi Eckmindac;nwg 7 dUcenHedIm,I[ 777)(:)( babaii Eckdac;nwg 77 luHRtaEt 22 baba Eckdac;nwg 37 . eKman 3222 7)( bababa enaH 19 ba edayeFIVkarsakl,gCMnYs 18,1 ba enaHeK)an ³

32222 7343181811 baba dUcenHKU 18,1 ba CacemøIy.


- 59 -

mü:ageTottamRTwsþIbTGWEl ³ ebI 1),( naGCD enaH )mod(1)( na n . eKman ))(( 2233 babababa ehtuenHedIm,I[ 22 baba Eckdac;nwg 37 luHRtaEt

)7(mod 333 ba nig ba Eckmindac;nwg 7 . eKman 9837677)7( 2233 . ebI c CacMnYnKt;viC¢manEckmindac;nwg 7 enaHeK)an )7(mod1)( 3398 c . dUcenHedIm,I[ )7(mod1 33 a luHRtaEt 98ca ]TahrN_ ³ -ebIeKyk 2c enaH )7(mod182 398 ehtuenH )7(mod1182 33398 dUcenH 1,18 ba CacemøIymYy . -ebIeKyk 3c enaH )7(mod3243 398 ehtuenH )7(mod13243 33398 dUcenH 1,324 ba CacemøIyepSgmYyeTot .


- 60 -

lMhat;TI31 eKGnuKmn_ xxf 2121

1)(

cUrKNna )1

(...)2

()1

(n

nf

nf

nfS

dMeNaHRsay KNna S CadMbUgeyIgRtUvRsayfa ebI 1 qp enaH 1)()( qfpf eKman qp qfpf 2121 21

1)(;

21

1)(

qpqfpf 2121 21

1

21

1)()(

qpqp

qp

qp

qp

2222121

2121

2121

2121

2221

222

)21)(21(

222

ebI 1 qp enaH 0)(22222 qpqp

1222

222)()( 2121

2121

qp

qp

qfpf


- 61 -

eKman )1()1

(...)2

()1

(n

nf

nf

nfS

b¤ )2()1

(...)2

()1

(n

fn

nf

n

nfS

bUksmIkar )1( nig )2( eK)an ³

11.....1112

)]1

()1

([.....)]1

()1

([2

)1(

nSn

fn

nf

n

nf

nfS

n

dUcenH 2

1

nS .


- 62 -

lMhat;TI32 cUrRsaybBa¢ak;facMnYn ³ n n n nA 113 168 141 427 Eckdac;nwg 7 Canic©RKb;cMnYnKt;FmμCati n. dMeNaHRsay Rsayfa A Eckdac;nwg 7 tamrUbmnþ n n n 1 n 2 n 1a b (a b)(a a b ... b ) eK)an n n n nA (427 168 ) (141 113 ) 1 2

1 2 1 2

(427 168)m (141 113)m

259m 28m 7(37m 4m )

dUcenH A Eckdac;nwg 7 RKb; 1 2m , m IN * .


- 63 -

lMhat;TI33 eK[ m nig n CaBIrcMnYnKt;viC¢manEdl 1 m n . cUrkMNt;témøtUcbMputén m n edIm,I[cMnYn m22 nig n22 manelx BIrxÞg;cugeRkaydUcKña . dMeNaHRsay kMNt;témøtUcbMputén m n ³ edIm,I[cMnYn m22 nig n22 manelxBIrxÞg;cugeRkaydUcKñaluHRtaEt

n m22 22 Eckdac;nwg 100 . eKman n m m n m22 22 22 (22 1) eday m22 CacMnYnKU nig n m22 1 CacMnYness . ehtuenH m n m22 (22 1) Eckdac;nwg 100 4 25 luHRtaEt

m22 Eckdac;nwg 4 nig n m22 1 Eckdac;nwg 25 . cMnYn m22 Eckdac;nwg 4 luHRtaEt m 2 . müa:geToteKman 222 96 5 4 elIkCakaereK)an 4 222 (96 5 4)

4 2 2

2

2

22 (96 5) 2(96 5).4 4

(96 154) 25 6

25a 6 , a 96 154


- 64 -

eK)an 4p p p22 (25a 6) 25q 6 , q IN eday 6 5 1 enaHeK)an ³

pp p k

k 0

pk

k 2

6 (5 1) C(p,k)5

1 5p C(p,k)5

eKTaj p

4p k

k 222 1 5p 25q C(p,k)5

b¤ p

4p k

k 222 1 5p 25q C(p,k)5

eday p

k

k 225q C(p,k)5

Eckdac;nwg 25 ehtuenH 4p22 1

Eckdac;nwg 25 luHRtaEt 5p Eckdac;nwg 25 eBalKWeKRtUv[ p 5k , k IN * . dUcenH n m22 1 Eckdac;nwg 25 luHRtaEt n m 4p 20k eday m 2 enaH m n (n m) 2m 20k 4 24 dUcenHtémøGb,brmaén m n KW 24 .


- 65 -

lMhat;TI34 eK[ 2f (n) 4n 7n 34 Edl n IN * cUrbgðajfaKμan n EdleFIV[cMnYn f (n) Eckdac;nwg 121 eT. dMeNaHRsay karbgðaj eKman 2f (n) 4n 7n 34

2

2

(4n 4n 1) (11n 33)

(2n 1) 11(n 3)

eyIgeXIjfaedIm,I[ f (n) Eckdac;nwg 121 luHRtaEt n 3 nig 2n 1 Eckdac;nwg 11 naM[eKman 1 2q ,q IN * Edl

1n 3 11q nig 22n 1 11q . eK)an 1 22(n 3) (2n 1) 22q 11q b¤ 1 211(2q q ) 7 CasmIkarKμancemøIykñúgsMNMu IN * BIeRBaH GgÁTImYyénsmIkarEckdac;nwg 11 EtGgÁTIBIrminEckdac;nwg 11 . dUcenH Kμan n EdleFIV[cMnYn f (n) Eckdac;nwg 121 eT.


- 66 -

lMhat;TI35 eK[ 2f (a) 16a 50a 13 Edl a kMNt;RKb;témø a EdleFIV[cMnYn f (a) Eckdac;nwg 49 dMeNaHRsay kMNt;RKb;témø a eKman 2f (a) 16a 50a 13

2

2

(16a 8a 1) (42a 14)

(4a 1) 7(6a 2)

edIm,I[ f (a) Eckdac;nwg 49 luHRtaEteKman p , q Edl 4a 1 7p nig 6a 2 7q . eK)an 3(4a 1) 2(6a 2) 21p 14q 7 7(3p 2q) b¤ 3p 2q 1 eKGacsresr 3(p 1) 2(q 1) eKTaj p 1 2k nig q 1 3k RKb; k b¤ p 2k 1 , q 3k 1 eKTaj)an 4a 1 7(2k 1) naM[ 7k 3

a2


- 67 -

eday a enaH 7k 3

2

.

edIm,I[ 7k 3

2

luHRtaEt k CacMnYnKt;essKW ³

k 2 1 , enaHeK)an 7(2 1) 3a 7 2

2

dUcenH a 7 2 , .


- 68 -

lMhat;TI36 kMNt;RKb;cMnYnKt;viC¢man x edIm,I[ 3 2x 10x 115x 237 Eckdac;nwg 343 . dMeNaHRsay kMNt;RKb;cMnYnKt;viC¢man x tag 3 2f (x) x 10x 115x 237

3 2

2

(x 1) 7(x 1) 49(2x 5)

(x 1) (x 8) 49(2x 5)

edIm,I[ f (x) Eckdac;nwg 343 luHRtaEt ³ x 1 nig 2x 5 Eckdac;nwg 7 b¤ x 8 Eckdac;nwg 343 nig 2x 5 Eckdac;nwg 7 . -krNI x 1 nig 2x 5 Eckdac;nwg 7 enaHeKman 1 2q , q IN * Edl 1

2

x 1 7q

2x 5 7q

eK)an 1 22(x 1) (2x 5) 14q 7q

1 2

1 2

1 2

7 7(2q q )

1 2q q

2(q 1) (q 1)

eKTaj 1

2

q 1 k

q 1 2k

b¤ 1

2

q k 1

q 2k 1 , k IN


- 69 -

eKTaj)an x 1 7(k 1) b¤ x 7k 6 , k IN dUcenH x 7k 6 , k IN CacemøIy . -krNI x 8 Eckdac;nwg 343 nig 2x 5 Eckdac;nwg 7 eK)an 1

2 1 2

x 8 343m

2x 5 7m , m ,m IN *

eKman 1 22(x 8) (2x 5) 686m 7m 1 2

1 2

21 686m 7m

3 98m m

eKGacsresr 1 298(m 1) m 95 naM[ 1

2

m 1 k

m 95 98k

b¤ 1

2

m k 1

m 98k 95 , k IN

dUcenH x 343(k 1) 8 343k 335 RKb; k IN .


- 70 -

lMhat;TI37 cUrRsayfaRKb;cMnYnKt;viC¢man n eK)an

237

134)(

n

nnf

CaRbPaKsRmYlmin)an ? dMeNaHRsay bgðajfa )(nf CaRbPaKsRmYlmin)an tag )237,134( nnGCDd eK)an dyndxn .237,.134 Edl 1),( yxGCD . eK)an xdydnn 74)134(7)237(4

dxy

dxynn

).74(1

).74(91289228

eKTaj)an 1d enaH 1)237,134( nnGCD )ann½yfa 134 n nig 237 n CacMnYnbzmrvagKñaRKb; 1n dUcenH

237

134)(

n

nnf CaRbPaKsRmYlmin)an .


- 71 -

lMhat;TI38 eKdwgfa

ab

tan3 . cUrRsaybBa¢ak;fa ³

3 23 23 2

4

3 2

4

ba

1

b

sin

a

cos

dMeNaHRsay RsaybBa¢ak;fa ³

3 23 23 2

4

3 2

4

ba

1

b

sin

a

cos

eKman a

btan3 naM[ 3

sin btan

cos a

eKTaj 3 3

cos sin

a b

b¤ 2 2 2 2

3 3 3 3 3 32 2 2 2 2 2

cos sin cos sin 1

a b a b a b

eKTaj 2

3 3 32 2 2

cos 1

a a b

nig

2

3 3 32 2 2

sin 1

b a b

eKTaj 34 2

3 3 32 2 2 2

cos a

a ( a b )

nig

34 2

3 3 32 2 2 2

sin b

b ( a b )

bUksmIkarBIenHeKTTYl)an 3 23 23 2

4

3 2

4

ba

1

b

sin

a

cos

Bit .


- 72 -

lMhat;TI39 eKmansIVútcMnYnBit )a( n kMNt;eday ³

)2

11....()

4

11)(

2

11(a n2n

cUrKNnalImIt nnalim

.

dMeNaHRsay KNnalImIt nn

alim

yk

2

1q eK)an ³

)q1)......(q1)(q1(an22

n edayeRbIsmPaB

yx

yxyx

22

eK)an ³

1n2

1n2

1n2

n

n2

1n2

2

42

n

2

112

2

11

2

11

q1

q1a

q1

q1....

q1

q1.

q1

q1a

kalNa n enaH 1n2

2

1

xitCitsUnü

dUcenH 2alim nn

.


- 73 -

lMhat;TI40 eK[plbUk

222222n )1n(

1

n

11....

4

1

3

11

3

1

2

11

n

1S

cUrKNna nnSlim

.

dMeNaHRsay KNnalImIt nn

Slim

eKBinitü 22

2222

22 )1k(k

k)1k()1k(k

)1k(

1

k

11

2

2

22

2

22

22

22

2222

1k

1

k

11

)1k(k

11

)1k(k

1)1k(k

)1k(k

1)1k(k2)1k(k

)1k(k

k1k2k)1k(k

eK)an 1n

11)

1n

11n(

n

1

1k

1

k

11

n

1S

n

1kn

dUcenH 11n

11limSlim

nnn

.


- 74 -

lMhat;TI41 eKmanes‘rIGnnþmYy ....

2

n.....

2

3

2

2

2

1)

2

n( n

2

3

2

2

22

1nn

2

cUrRsayfaes‘rIxagelIenHCaes‘rIbRgYm . dMeNaHRsay plbUkedayEpñkrbs;es‘rIenHKW n

2

3

2

2

22

n 2

n...

2

3

2

2

2

1S

BinitüGnuKmn_ cbkak)k(f 2 EdlRKb; *INk eKman ³

1kkk

2

2

)1k(f

2

)k(f

2

k

b¤ 1k

2

k

2

k

2

2

c)1k(b)1k(a

2

cbkak

2

k

b¤ cbbkaak2akc2bk2ak2k2 222 back)a2b(akk2 22 eKTaj 2a ehIy 4b0a2b nig 6bac ehtuenH 1k

2

k

2

k

2

2

12k8k2

2

6k4k2

2

k

eK)an

n

1k1k

2

k

2

n )2

12k8k2

2

6k4k2(S

n

2

2

6n4n6

eday 62

6n4n6limSlim n

2

nnn

dUcenHes‘rIGnnþxagelICaes‘rIbRgYm .


- 75 -

lMhat;TI42 cMeBaHRKb; *INn eKmanplbUk ³

1n432n 2).1n2)(1n2(

3n2....

2.7.5

9

2.5.3

7

2.3.1

5S

k> kMNt;cMnYnBit a nig b edIm,I[eK)ansmPaB 1nn1n 2).1n2(

b

2).1n2(

a

2).1n2)(1n2(

3n2

x> KNna nS nig nnSlim

.

dMeNaHRsay k> kMNt;cMnYnBit a nig b edIm,I[eK)ansmPaB 1nn1n 2).1n2(

b

2).1n2(

a

2).1n2)(1n2(

3n2

eK)an )1n2(b)1n2(a23n2 ba2n)b2a4(3n2 eKTaj

3ba2

2b2a4 naM[ 1b,1a

dUcenH 1a nig 1b . x> KNna nS nig nn

Slim

tamsRmayxagelIenHcMeBaH 1a nig 1b eK)an ³

1nn1n 2).1n2(

1

2).1n2(

1

2).1n2)(1n2(

3n2


- 76 -

b¤ 1pp1p 2)1p2(

1

2)1p2(

1

2)1p2)(1p2(

3p2

eK)an

n

1p1pn 2).1p2)(1p2(

3p2S

1n

n

1p1pp

2).1n2(

1

2

1

2)1p2(

1

2)1p2(

1

dUcenH

nn 2).1n2(

11

2

1S .

ehIy 2

1

2).1n2(

11lim

2

1Slim nnnn

.


- 77 -

lMhat;TI43 cMeBaHRKb; *INn eKmansIVútcMnYnBit )a( n kMNt;eday ³

3a0 nig n2nn21n3

1a

3

11a

k> KNna na CaGnuKmn_én n . x> KNnalImIt nn

alim

. dMeNaHRsay k> KNna na CaGnuKmn_én n eKman n2nn21n

3

1a)

3

11(a

n2n

1n

nn21n

n2nn21n

3

11

1a

1a

)1a()3

11(1a

)3

11(a)

3

11(1a

eK)an

1n

0kk2

1n

0k k

1k

3

11

1a

1a

tamsmPaB ))((22 b¤

22

edayyk 1 nig k23


- 78 -

eK)an k2

1k2

k2

3

11

3

11

3

11

ehtuenH )

3

11

3

11

(3

11

1n

0kk2

1k21n

0kk2

n2

n2

3

11

2

3

3

11

3

11

ehIyeday 2

1a

1a

1a

1a

1a n

0

n1n

0k k

1k

eKTaj )3

11(

2

3

2

1an2

n b¤ n2n

3

34a

dUcenH n2n3

34a

x> KNnalImIt nnalim

eK)an 4)3

34(limalim n2nnn

eRBaH 03

3lim n2n

dUcenH 4alim nn

.


- 79 -

lMhat;TI44 eK[ a nig b CaBIrcMnYnBitEdl

2ba0

cUrRsayfa bcos

abatanbtan

acos

ab22

dMeNaHRsay Rsayfa

bcos

abatanbtan

acos

ab22

tagGnuKmn_ xtan)x(f Edl )2

,0[x

eK)an xcos

1)x('f 2 .

eday )x(f CaGnuKmn_Cab; nigmanedrIevelIcenøaH )2

,0[x

enaHtamRTwsþIbTtémømFüm enaHman )b,a(c Edl ³

)1(ab

atanbtan

ab

)a(f)b(f)c('f

ehIycMeBaH ]b,a[x Edl 2

ba0

eKman acosxcosbcos enaHeKTaj bcos

1)x('f

acos

122

yk cx eK)an )2(bcos

1)c('f

acos

122

tam )1( nig )2( eKTaj bcos

abatanbtan

acos

ab22

.


- 80 -

lMhat;TI45 KNnaplbUk ³

nn

23n

2n

1nn Cn.....C9C4CS Edl

)!pn(!p

!nCp

n

dMeNaHRsay KNnaplbUk ³

nn

23n

2n

1nn Cn.....C9C4CS

tameTVFajÚtun nnn

33n

22n

1n

0n

n xC...xCxCxCC)x1( eFVIedrIevelIGgÁTaMgBIrénsmIkarenHeK)an ³

1nnn

23n

2n

1n

1n xnC....xC3xC2C)x1(n KuNGgÁTaMgBIrnwg x eK)an ³

nnn

33n

22n

1n

1n xnC....xC3xC2xC)x1(nx eFVIedrIevelIGgÁTaMgBIrénsmIkarenHeK)an ³

1nnn

22n

1n

2n1n xCn...xC4C)x1(x)1n(n)x1(n yk 1x CYskñúgsmPaBenHeK)an ³

nn

23n

2n

1n

2n1n Cn...C9C4C2).1n(n2.n nn

23n

2n

1n

2n Cn...C9C4C2)]1n(nn2[ dUcenH 2n

n 2).1n(nS


- 81 -

lMhat;TI46 eK[GnuKmn_ 2x)x(f k> sikSaGefrPaB nig sg;Rkab )c( énGnuKmn_ f . x> eKBinitü n cMnuc n321 A...,,A,A,A sßitenAelIRkab )c( Edl )22,2(A1 nig )a,a(A 1nnn RKb; *INn . cUrkMNt;kUGredaenéncMnuc nA . K> cUrKNna nOA CaGnuKmn_én n rYcKNna n

nOAlim

.

dMeNaHRsay k> sikSaGefrPaB nig sg;Rkab )c( énGnuKmn_ f eKman 2x)x(f mann½yluHRtaEt 2x > EdnkMNt; ),2[D > TisedAGefrPaB -edrIev

2x2

1)x('f

RKb; 2x eK)an 02x2

1)x('f

naM[ f CaGnuKmn_ekInRKb; ,2x . -lImIt

2xlim)x(flim

xx


- 82 -

-taragGefrPaB x 2

)x('f )x(f

-sMNg;Rkab 2xy:)c( x> kMNt;kUGredaenéncMnuc nA eday 2xy:)c()a,a(A 1nnn

0

2 3 4 5-1-2

2

3

4

-1

0 1

1

X

Y

2xy:)c(


- 83 -

enaHeK)an 2aa n1n eKman )22,2(A)a,a(A 211 eK)an 21 2

cos22a

322 2

cos22

cos2222a

]bmafavaBitcMeBaHtYTI n KW 1nn 2cos2a

eyIgnwgRsayfavaBitdl;tYTI 1n KW 2n1n 2cos2a

eKman 2aa n1n eday 1nn 2cos2a

enaH 2n1n1n 2cos22

2cos2a

Bit

dUcenH )2

cos2,2

cos2(A 2n1nn

K> cUrKNna nOA CaGnuKmn_én n rYcKNna nn

OAlim

eK)an 2

1n2

nn aaOA eday 1nn 2

cos2a

nig 2n1n 2cos2a

dUcenH 2n2

1n2

n 2cos

2cos2OA

nig 22OAlim n

n


- 84 -

lMhat;TI47 eK[RtIekaN ABC mYy . emdüanénRtIekaNEdlKUsecj BIkMBUl C,B,A kat;rgVg;carwkeRkARtIekaNerogKñaRtg;

P,N,M . tag cba ,, CaRCug nig cba mmm ,, Caemdüanén ABC . cUrRsayfa

9

16

222 224

4

224

4

224

4

bac

m

cab

m

cba

m cba dMeNaHRsay Rsayfa ³

9

16

222 224

4

224

4

224

4

bac

m

cab

m

cba

m cba tamvismPaB SchwarzCauchy eK)an ³

2222

2222

224

4

224

4

224

4

)(

)(

222 cba

mmm

bac

m

cab

m

cba

m cbacba

eyIgnwgRsayfa 3

4222

222

cba

mmm cba


- 85 -

eyIgman AMCABC ¬mMusáat;edayFñÚrrYm AC ¦ ehIy C'MAB'AA ¬mMuTl;kMBUl ¦ eKTaj)an B'AA nig C'MA CaRtIekaNdUcKña . eK)an

a

2

m4

a

'AA

'CA.B'AM'A

M'A

B'A

'CA

'AA

ehIy a

2

a m4

amM'A'AAAM

RsaydUcKñaEdr b

2

b m4

bmBN nig

c

2

c m4

cmCP .

tag 222 CPBNAMT

A

B C

M

'A


- 86 -

2c

4

2b

4

2a

42222

c2

b2

a

2

c

2

c2

b

2

b2

a

2

a

m

c

m

b

m

a

16

1

2

cbammm

)m4

cm()

m4

bm()

m4

am(T

eday )cba(4

3mmm 2222

c2

b2

a

eK)an

2c

4

2b

4

2a

4222

m

c

m

b

m

a

16

1)cba(

4

5T

tamvismPaB SchwartCauchy eK)an )cba(

3

4

mmm

)cba(

m

c

m

b

m

a 2222

c2

b2

a

2222

2c

4

2b

4

2a

4

eKTaj)an ³ )cba(

3

4)cba(

12

1)cba(

4

5T 222222222

naM[ 3

4222

222

cba

mmm cba Bit

dUcenH 9

16

222 224

4

224

4

224

4

bac

m

cab

m

cba

m cba


- 87 -

lMhat;TI48 eK[ M CaTIRbCMuTm¶n;énRtIekaN ABC . ebIbnÞat; AB b:HeTAnwgrgVg;carwkeRkAénRtIekaN AMC enaHbgðajfa 32

sin

1

sin

1

CBMCMA

dMeNaHRsay bgðajfa 32

sin

1

sin

1

CBMCMA

tamvismPaB SchwarzCauchy

CBMCMACBMCMA

sinsin

4

sin

1

sin

1

G

A

M

B

C

'M


- 88 -

eyIgnwgRsayfa 3

2CBMsinCAMsin

yk G CacMNuckNþalénRCug ]AB[ . tag c,b,a CaRbEvgRCug AB,CA,BC ehIy cba m,m,m CargVas;emdüanKUsBIkMBUl C,B,A erogKña . eday A CacMnucb:HrvagbnÞat; )AB( CamYyrgVg;p©it 'M enaHeKman 22 GC

3

1GC.GMGA ¬ eRBaH GC

3

1GM ¦

eday cmGC,2

c

2

ABGA eK)an 2

c

2

m3

1

4

c

tamRTwsþIbTemdüaneKman 4

c

2

bam

2222

c

eKTaj )4

c

2

ba(

3

1

4

c 2222

b¤ 222 c2ba .

ehtuenH 4

c3

4

c

2

c2m

2222

c b¤ c2

3mc

ehIy 4

b3

4

ac2b2

4

a

2

cbm

22222222

a

naM[ b2

3ma ehIy a

2

3mb .

tag S CaRkLaépÞénRtIekaN ABC ehIy 'BB,'AA Caemdüan . eK)an ³


- 89 -

CAMsinAC'.AA4

1CAMsinAC.AM

2

1SCAM b¤ S

4

1S

2

1CAMsinbm

4

1S C'AAaCAM

eKTajJ)an bm

SCAMsin

a

ehIy am

SCBMsin

b

eK)an am

S

bm

SCBMsinCAMsin

ba

Et Csinab2

1S ehIy b

2

3ma nig a

2

3mb

eK)an )1(Csinab3

baCBMsinCAMsin

22

tamRTwsþIbTkUsIunUs Ccosab2bac 222 eKTaj Ccosab2ba

2

ba 2222

eRBaH 222 c2ba

ehtuenH )2(Ccosbc4ba 22 ykTMnak;TMng )2( CMnYskñúg )1( eK)an ³

3

2C2sin

3

2

3

CcosCsin4CBMsinCAMsin

3

2CBMsinCAMsin Bit

dUcenH 32sin

1

sin

1

CBMCMA .


- 90 -

lMhat;TI49 eK[ 0,, cba . cUrRsaybBa¢ak;fa ³

2

9)(4)(4)(4 3 333 333 33

ba

bac

ac

acb

cb

cba dMeNaHRsay RsaybBa¢ak;fa ³

2

9)(4)(4)(4 3 333 333 33

ba

bac

ac

acb

cb

cba eKman )cb(bc3)cb(cb 333 tamvismPaB GMAM eKman bc2cb eKTaj

2

2

cbbc

naM[ 3)cb(4

3)cb(bc3

eK)an 33333 )cb(4

1)cb(

4

3)cb(cb

eKTaj 3 33 )cb(4cb b¤ 3 33 )cb(4acba

naM[ )1(1)(43 33

cb

a

cb

cba

dUcKñaEdr )2(1)(43 33

ac

b

ac

acb


- 91 -

nig )3(1)(43 33

ba

c

ba

bac

edaybUkTMnak´TMng )3(,)2(,)1( eK)an ³

ba

c

ac

b

cb

aT

3 Edl

ba

bac

ac

acb

cb

cbaT

3 333 333 33 )(4)(4)(4

eyIgnwgRsayfa 2

3

ba

c

ac

b

cb

a

tag

pba

nac

mcb

eK)an pnmbaaccb naM[

2

pnmcba

eKTaj

2

pnmc

2

pnmb

2

mpna

eK)an p2

pnm

n2

pnm

m2

mpn

ba

c

ac

b

cb

a


- 92 -

3

n

p

p

n

p

m

m

p

n

m

m

n

2

1

ba

c

ac

b

cb

a

tamvismPaB GMAM eK)an 2

n

p

p

n;2

p

m

m

p;2

n

m

m

n

eK)an 2

33222

2

1

ba

c

ac

b

cb

a

dUcenH 2

9

2

33 T Bit .


- 93 -

lMhat;TI50 eK[ 1,1 ba nig 1x . cUrRsayfa xxx baba

2

log2loglog

dMeNaHRsay Rsayfa xxx baba

2

log2loglog

eKman a

xx

a log

1log nig

bx

xb log

1log

tamvismPaB SchwarzCauchy eK)an ³

baxx

xxba loglog

4loglog

eyIgnwgRsayfa xba ba

xx 2

log2loglog

4

b¤ 2

log2loglogba

ba xxx

cMeBaH 1a nig 1b eyIgman b.a2ba ( vismPaB GMAM b¤ b.a

2

ba


- 94 -

naM[ )loglog(2

1)

2(log ba

baxxx

b¤ )1()2

(log2loglogba

ba xxx

mü:ageToteKman ³ baba xxxx log.log2loglog

)2()loglog(2

1loglog

)loglog()log(log2

loglog.log2log)log(log2

2

2

baba

baba

bbaaba

xxxx

xxxx

xxxxxx

tamTMnak;TMng )1( nig )2( eyIgTaj ³

)2

(log2loglog

)2

(log4)loglog(

)2

(log2)loglog(2

1

2

2

baba

baba

baba

xxx

xxx

xxx

dUcenH xxx baba

2

log2loglog


- 95 -

!> eK[

2,0

.

cUrbgðajfa 1cos

cos

sin

sin2

4

2

4

?

@> eK[GnuKmn_ x

xxxf

sin2

cossin1)(

RKb; IRx bgðajfa 1)(3

1 xf .

#> eK[ 1,, cba . cUrRsaybBa¢ak;fa ³

cbaac

c

cb

b

ba

a acb

9logloglog 222

$> edaHRsaysmIkar ³ 0)1(213)1( 3 223 243 22 xxxxxx %> eKmanGnuKmn_ ³ 9)cot(tan2cottan)( 22 xxxxxf Edl

20

x . cUrrktémøtUcbMputénGnuKmn_enH ?


- 96 -

^> cUrbgðajfaebI k CacMnYnKUenaH 12 n

k Eckdac;nwg 22 n cMeBaHRKb;cMnYnKt;FmμCati n . &> cMnYnKt; n mYynwgehAfa Good ebIeyIgGacsresr ³ kaaan ....21 Edl kaaa ...,, 21 CacMnYnKt; viC¢man ¬mincaM)ac;xusKña ¦ EdlepÞógpÞat; ³ 1

1....

11

21

kaaa

.

bgðajfaebI n KW Good enaH 82 n nig 92 n k¾ Good .

*> RKb; 1n bgðajfa 2!

!2

1

4

n

n

n

n

.

(> cUrbgðajfaebI n CacMnYnKt;FmμCatimYyenaHeK)an ³

30325

345 nnnn k¾CacMnYnKt;FmμCatiEdr .

!0> bgðajfa 24

13

2

1....

2

1

1

1

nnn RKb; 1n .

!!> cUrbgðajfa nnn 45 35 Eckdac;nwg 120 RKb; INn !@> cUrbgðajfa 3579 496 nnnn Eckdac;nwg 8640 RKb; INn .


- 97 -

!#> eK[ n CacMnYnKt;viC¢man . cUrbgðajfa ³ )2)....(2)(1( nnn Eckdac;nwg n2 ? !$> bgðajfa 124 22

nn Eckdac;nwg 7 Canic©RKb; INn !%> bgðajfaebI 22 ba Eckdac;nwg 7 enaH a Eckdac;nwg 7 nig b Eckdac;nwg 7. !^> cUrkMNt;RKb;cMnYnKt;viC¢man a nig b ebIeKdwgfa ³ 8511322 ba nig 1764),( baLCM !&> RKb; 1n cUrRsayfa ³ 1)2187,13( 233 nnnnnGCD !*> ebI 1),( nmGCD cUrRsayfa

!!.

)!1(

nm

nm CacMnYnKt; . !(> ebI 1n nig 1)6,( nGCD enaHbgðajfa

)!2(!

)!42(

nn

n

CacMnYnKt; .

@0> cUrRsayfa 504

39 nn CacMnYnKt;ebI INn .


- 98 -

@!> RKb;cMnYnBit x nig y cUrRsayfa ³ yxyx

21

1

41

1

41

1 @@> RKb; IRx bgðajfa ³

3

4

cos1

1

sin1

122

xx

@#> RtIekaN ABC mYymanmMukñúgCamMuRsYc . cUrRsayfa 6

cos

1

cos

1

cos

1

CBA

@$> kñúgRKb;RtIekaN ABC cUrRsayfa ³ 4

sin

1

sin

1

sin

1222 CBA

@%> RKb;cMnYnBitviC¢man cba ,, cUrRsayfa ³

cbaabc

c

cab

b

bca

a

1

888 222 @*> eK[ cba ,, CaRCugrbs;RtIekaNmYy . cUrRsayfa ³

abccbacbacbabba

3

)(

1

)(

1

)(

1222

@(> bgðajfa 11

cos2sin)1(2

2

a

xaxa RKb; IRx .


- 99 -

#0> cUrRsayfa 20

7

9sin

3

1

#!> RKb; )1,1(x bgðajfa 21

1

1

144

xx

#@> eK[ zyx ,, CacMnYnBitviC¢man .

cUrRsayfa 2

3

zx

z

zy

y

yx

x

##> eK[ cba ,, CaRCugrbs;RtIekaNmYy . cUrRsayfa ³

abcbacabcacbabccbaabc 3

1

)(8

1

)(8

1

)(8

1333

#$>ebI cba ,, CabIcMnYnBitviC¢manEdl 1 cabcab cUrRsayfa ³

abc

ac

cb

ba

16

16

16

1333

n 2 x n k x 2 k k 1...knitvitüacmuvijbipbelak - 3 - !@> ekdwgfa cosa , cosb , coscmn p np pm m n...

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