Maht Fundation Grade 11.pdf

7/25/2019 Maht Fundation Grade 11.pdf

1/195

lwm pl:n nigEsn BisidbriBaabR&tKNitviTa nig BaNiCkm

CMBUkTI1

sIVtcMnYnBit-sIVtnBVn-sIVtFrNImaRt

CMBUkTI2GnuKmn_Gics,NgEsl nig elakarIt

CMBUkTI3

GnuKmn_RtIekaNmaRtCMBUkTI4

maRTIs nig edETmINg


2/195

eroberogeday

Esn Bisidnig lwm pl:n


3/195

GkRtYtBinitbeckeTs

1-elak lwm qun

2-elak Esn Bisid3-elakRsITuy rINa

4-elak Tit emg

5-elak RBwm suxnit

6-elak pl bunqay

GkRtYtBinitGkraviruT

elak lwm miK:sir

karIkMBTr nig rcnaRkb

kBaa lIKUNaka

GkniBn nig eroberog

elak Esn Bisidnig elak lwm pl:n

rkasiTi


4/195

Garmkfa

esovePA kMEnlMhatfakTI11EdlRbiymitGksikakMBugkanenAkgdenrYmman CMBUkTI1 sItcMnnBit sItnBn nigsItFrNImaRtCMBUkTI2 GnKmnGics NgEsl elakartCMBUkTI3 GnmnRtekaNmaRtCMBUkTI4 maRTsngedETmNg

kgCMBUknImYy@xMJTneRCIserIsyklMhatmkeFIVkMEnyagek,ak,ay enAslCMBUkTI5 lImIt ng edrIevCMBUkTI6RbblIetnig CMBUkTI7si tixMJTnwgeFIVkMEnenAkgPaKeRkay.

CaTIbBab xMJTGkeroberogsUmeKarBCUnBrdlGksikakgRKbmCdan sUmmansuxPaBlRCJaQasv nig TTYln

C&yCMnkgRKbParkic.

tdMbgfTI10 sPa qaM2009

GkniBnnig eroberog

lwm pl:n nigEsn BisidTel : (017) 768 246


5/195

1-sIVtcMnYnBit

2-sIVtnBVn

3-sIVtFrNImaRt


6/195

sVItncMnYnBit CMBUk1 emeronTI1

lMhat

1-sresrbItYbnabnsIVtxageRkam k x....,20,15,10,5 ....,55,30,14,5,1

2-sresrtYnsIVtrabGsxageRkam

kn

)1(a

n

n

= cMeBa 8n1

x cMeBa2nn )2(b = 5n1 3-sresrbYntYnsIVtrabGsxageRkam

k2

n2

nn

)1(a

= xnn

1b

2n

Kc

X2n

n )2n()1( 1nn

n 2)1(d

4-kMnttYTIncMeBaRKb INn nsIVtxageRkam k x...,13,11,9,7,5 ...,5,3,1,1,3

K X...,4,7,10,13,16 ...,9

5,

7

4,

5

3,

3

2

5

-bgHajfasIVt Edl)a(

n 3n

1a

n CasIVtcu .

6-bgHajfasIVt Edl)b( n3n

1n2bn

= CasIVtekIn .

- 1 -


7/195


7-sikaPaBmUNUtUnnsIVtEdlkMntdUcxageRkam

ksIVt EdlINnn )a( n1

an=

xsIVt EdlINnn )a( 5n4na 2n KsIVt Edl*INnn )a( )nln(.4nan XsIVt Edl)a( n 2n1n aa = nig 2a0= 8

-bgHajfasIVt Edl)a(

n n1na 2

n

CasIVtTal.9-etIsIVtxageRkamCasIVtTaleT ?

ksIVt Edl*INnn )a(

nn

1n2na

2

23

n

=

xsVIt Edl*INnn )b( n

bn

=

KsIVt Edl)c( n3n1ncn

= 10-etIbNasIVtxageRkammYyNaCasIVtTalelITaleRkam

nigCasIVtTal? rYcrkeKalnsIVtnImYy@ .

k 1n;nn

1n2na

2

23

n

=

x 2n;4n3n

ea

2

n

n

- 2 -


8/195


11-eKmansIVt Edl CacMnYnKtehIy)a( n n 1a0=

nig .1aaa n2

n1n

kbgHajfasIVt Edl CacMnYnKtCasIVtekIn .)a( n n xbgHajfacMeBaRKb .na,n n

- 3 -


9/195


dMeNaRsay

1-sresrbItYbnabnsIVtxageRkam k ....,20,15,10,5

eyIgBiniteXIjfa

5u51520u

5u51015u

5u5510u

5u

34

23

12

1

=

ehtuenbItYbnabnsIVtenKW

355305uu

305255uu

255205uu

67

56

45

=

=

=

dUcen .35,30,25,20,15,10,5

x ....,55,30,14,5,1 eyIgBiniteXIjfa

22

23

21

22

1

3u3514u

2u215u11u

=

- 4 -


10/195


2

4

2

5

23

24

5u53055u

4u41430u

ehtuenbItYbnabnsIVtenKW

204641408uu

14049917uu

9136556uu

278

267

256

=

=

=

dUcen .204,140,91,55,30,14,5,12-sresrtYnsIVtrabGsxageRkam

kn

)1(a

n

n

= cMeBa 8n1

ebI ena1n= 1a1

ebI ena2n=21a2=

ebI ena3n=3

1a3

ebI ena4n=4

1a4=

ebI ena5n= 51

a5

ebI ena6n=6

1a6=

- 5 -


11/195


ebI ena7n=7

1a7

ebI ena8n= 81a8=

dUcen8

1;

7

1;

6

1;

5

1;

4

1;

3

1;

2

1,1 .

x cMeBa2nn )2(b = 5n1 ebI ena1n= 2a1= ebI ena2n= 1a2=

ebI ena3n=2

1a3=

ebI ena4n=4

1a4=

ebI ena5n=

8

1

a5=

dUcen8

1;

4

1;

2

1;1;2 .

3-sresrbYntYnsIVtrabGsxageRkam

k2

n2

nn

)1(a

=

eKn16

1;

9

1;

4

1;1 .

- 6 -


12/195


xnn

1b

2n

eKn201;

121;

61;

21 .

K 2nn )2n()1(c eKn 4;1;0;1 .X 1nnn 2)1(d

eKn 8,4;2;1 .4-kMnttYTIncMeBaRKb INn nsIVtxageRkam k ...,13,11,9,7,5

eyIgeXIjfa 50.25

54.213

53.211

52.29

51.27

dUcentYTInKW 5n2Un x ...,5,3,1,1,3

eyIgeXIjfa 30.23

- 7 -


13/195


34.25

33.23

32.21

31.21

dUcentYTInKW 3n2Un K ...,4,7,10,13,16 eyIgeXIjfa 160.316

164.34

163.37

162.310

161.313

dUcentYTInKW 16n3Un

X ...,95,74,53,32 eyIgeXIjfa

30.2

20

3

2

=

33.2

23

9

5

32.2

22

7

4

31.2

21

5

3

=

=

=

- 8 -


14/195


dUcentYTInKW3n2

2nUn

=

5-bgHajfasIVt Edl)a( n 3n 1an CasIVtcu .eKman

3n

1an

naM[4n

1a 1n

eyIgn3n

1

4n

1aa n1n

0)4n)(3n(

1aa

)4n)(3n(

)4n()3n(aa

n1n

n1n

=

dUcen CasIVtekIn .)b( n

- 9 -


15/195


7-sikaPaBmUNUtUnnsIVtEdlkMntdUcxageRkam

ksIVt EdlINnn )a(

n

1an=

eyIgmann

1an= naM[

1n

1a 1n

eyIgn 0)1n(n

1

n

1

1n

1aa n1n n1n

eyIgseg;teXIjfacMeBa }1,0{n= ena 0aa n1n < ehIy 0aa:2n n1n > .dUcen minEmnCasIVtmUNUtUn .)a( n

- 10 -


16/195


KsIVt Edl*INnn )a(

)nln(.4nan

eKman )nln(.4nan

naM[ )1nln(41na 1n eKn )nln(4)1nln(41aa n1n

1n

nln41aa n1n

edayRKb eKman1n 11nn

eKTajn INn;aa n1n

dUcen CasIVtmUNUtUn .)a( n8-bgHajfasIVt Edl)a( n n1na 2n CasIVtTal.

- 11 -


17/195


eKmann1n

1n1na

2

2n

=

eyIgeXIjfakalNa nxitCitGnnenan1n

12

xitCitsUn ehIyebI 0n= ena 1a0= .eKTaj cMeBaRKb1a0 n INn .dUcen CasIVtTal.)a( n

9-etIsIVtxageRkamCasIVtTaleT ?

ksIVt Edl*INnn )a(

nn

1n2na

2

23

n

=

ebI ena1n= 2a1= ehIynn

1n2n2

23

xiteTArk

GnnkalNa nxitCitGnn.dUcen CasIVtTaleRkamehIy2CacMnYnTaleRkam.)a( n

xsVIt Edl*INnn )b( n

bn

=

ebI ena1n= 1b ehIyn

xitCitsUnebI n

eKTaj nb0 .dUcen CasIVtTal.)b( n

- 12 -


18/195


KsIVt Edl)c( n3n

1ncn

=

ebI ena0n= 3c0= ehIy 3n 1n xitCit1kalNaxiteTACitGnn.n

eKn naM[ CasIVtTal.3c1 n )c( n10-etIbNasIVtxageRkammYyNaCasIVtTalelITaleRkam

nigCasIVtTal? rYcrkeKalnsIVtnImYy@ .k 1n;

nn

1n2na

2

23

n

=

ebI ena1n= 1a1= ehIynn

1n2n2

23

xiteTArk

GnnkalNan

xitCitGnn.dUcen CasIVtTaleRkamehIy1CacMnYnTaleRkam.)a( n

x 2n;4n3n

ea

2

n

n

ebI ena2n=6

ea

2

2= ehIy4n3n

e2

n

xiteTArk

GnnkalNa nxitCitGnn.

dUcen CasIVtTaleRkamehIy)a( n6

e2

CacMnYnTaleRkam.

- 13 -


19/195


- 14 -

11-eKmansIVt Edl CacMnYnKtehIy)a( n n 1a0=

nig .1aaa n2

n1n

kbgHajfasIVt Edl CacMnYnKtCasIVtekIn .)a( n n xbgHajfacMeBaRKb .na,n n> dMeNaRsay

kbgHajfasIVt CasIVtekIn )a( n

eyIgman 1aaa n2n1n eyIgn INn,01aaa 2nn1n

dUcen CasIVtekIn .)a( n xbgHajfacMeBaRKb na,n n

eyIgman 01a0 >

31831131691aaa

2131391aaa

131aaa

22

23

12

12

02

01

>

>

>

tamlMnaMeneyIgGacsnidanfaRKb .na,n n>


20/195

sVItnBVn CMBUk1 emeronTI2

lMhat

1-sresrrUbmnntYTInsIVtnBVnxageRkam k x....,37,26,15,4 ....,11,7,3,1,5 K ...,23,16,9,2 X ...,27,22,17,12,7

2-sresrRMtYdMbUgnsIVtnBVnEdlkMntedaytY

k 10U;1U 21 = x n37Un 3-eKmansIVtnBVn ...;7;2;3;8 .kKNna nig17U 43U

xetItYTIbun

annsIVtes

Inwg 3224-kMntcMnYntYnsIVtnBVnxageRkam k 102....,,8,13,18 x 171...,.,1,4,9

5

-kgcMeNamsVItxageRkam etIsIVtNamYyCasIVtnBVn?k )1n(613Un

x nn1n0 U4

1UU;3U

- 15 -


21/195


6-eK[ CasIVtnBVnebIeKdwgfa )U( n

k 17U;9U 4535 =

. KNna ?40U x

2

13U;

2

7U 137 = . KNna ?0U

K 18S,12U 122 = . KNna nigd;U1 6U7-eK[sIVtnBVn nig sIVtnBVn...,12,7,2 ...,8,5,2

etIkgcMeNam121

tYnsIVtTaMgBIrenmanbun

antYEdlmantmesIKa ?8-bgHajfasIVt EdlmantYTI kMnteday )U( n n

4

3n2Un

=

CasIVtnBVnekIn .

9

-eK[RMcMnYnCasIVtnBVn. KNnacMnYnTaMgenaebIeKdwgfaplbUkvaesI125 ehIyplbUkBItYdMbUgesI .3810-ebIplbUk 9tYdMbUgnsVItnBVnesInwg 162ehIyplbUk

tYdMbUgnsVItnBVnesI288 .12 KNnaplbUk 30tYdMbUgnsVItnBVnen ?11-rkbIcMnYntKaCasIVtnBVnedaydwgfaplbUkbIcMnYnenesInwg 36ehIyplKuNvaesInwg 1428.

- 16 -


22/195


12-KNnaplbUksIVtnBVnxageRkam

k 146....642

x )20(...859095100 K 1721....185189193

X )3(.....4

33

2

14

4

15

13

-salRbCMumYyragctuekaNBay ehIytamRCugeRTtnsalRbCMuenaeKerobekAGICa 15CYr EdlCYrmuxeKbMputerobn 25ekAGIehIyCYrTIBIrerobn ekAGICYrTIbI27

erobn 29ekAGI.eKerobekAGItamlMnaMxagelIrhUtdlCYeRkay .KNnaekAGIsRmabGg:yTaMgGsEdlmanenAkgsalRbCMu.

- 17 -


23/195


1

-sresrrUbmntYTInnsIVtnBVnxageRkam

dMeNaRsay

k ....,37,26,15,4 tamrUbmn d)1n(UU 1n eKman nig4U1= 11415UUd 12 =

eKn7n1111n114)11()1n(4Un

dUcen 7n11Un .x ....,11,7,3,1,5 tamrUbmn d)1n(UU 1n eKman nig5U1= 451UUd 12 eKn U 9n44n45)4()1n(5n dUcen 9n4Un .K ...,23,16,9,2 tamrUbmn d)1n(UU 1n

eKman nig2U1 729UUd 12 eKn 5n77n72)7()1n(2Un dUcen 5n7Un .

- 18 -


24/195


X ...,27,22,17,12,7

tamrUbmn d)1n(UU 1n

eKman nig7U1= 5712UUd 12 = eKn 2n55n57)5()1n(7Un dUcen 2n5Un .

2

-sresrRMtYdMbUgnsIVtnBVnEdlkMntedaytYk 10U;1U 21 = tamrUbmn d)1n(UU 1n eKman nig1U1= 9110UUd 12 = eKn 8n99n91)9()1n(1Un

ebI ena1n= 189U1 = ebI ena2n= 10818U2 = ebI ena3n= 19827U3 = ebI ena4n= 28836U4 =

ebI ena5n= 37845U5 = dUcenRMtYdMbUgnsVItnBVnKW .37;28;19;10;1

- 19 -


25/195


x n37Un

ebI ena1n=

437U1

ebI ena2n= 167U2 ebI ena3n= 297U3 = ebI ena4n= 5127U4 =

ebI ena5n= 8157U5 =

dUcenRMtYdMbUgnsVItnBVnKW 8;5;2;1;4

.3-eKmansIVtnBVn ...;7;2;3;8 .kKNna nig17U 43U

tamrUbmn d)1n(UU 1n

eKman nig8U1 583UUd 12 = eKn 13n55n58)5()1n(8Un dUcen 72138513)17(5U17 = nig 2021321513)43(5U43 = .

xetItYTIbunannsIVtesInwg 322 eyIgn 32213n5 = naM[ 67n= dUcentYTI67nsIVtEdlesInwg .322

- 20 -


26/195


4-kMntcMnYntYnsIVtnBVnxageRkam

k 102....,,8,13,18

CadMbUgeyIgRtUvkMntrktYTInnsIVtensin .tamrUbmn d)1n(UU 1n eKman nig18U1= 51813UUd 12

eKn23n55n518)5()1n(18Un

edaRsaysmIkar 10223n5

naM[ 25n=

dUcensIVtxagelImancMnYn tY.25 x 171...,.,1,4,9 tamrUbmn d)1n(UU 1n

eKman nig9U1 594UUd 12 = eKn 14n55n59)5()1n(9Un edaRsaysmIkar 17114n5 = naM[ 37n= dUcensIVtxagelImancMnYn tY.37

5-kgcMeNamsVItxageRkam etIsIVtNamYyCasIVtnBVn?k )1n(613Un eyIgn 7n66n613Un

- 21 -


27/195


nig 13n67)1n(6U 1n

eKman 6)7n6()13n6(UU n1n =

efrdUcen )1n(613Un CasIVtnBVn

x nn1n0 U4

1UU;3U

eKman nn1n U4

1UU

eKTaj nn1n U41UU minefrdUcenvaminEmnCasIVtnBVneT .

6-eK[ CasIVtnBVnebIeKdwgfa )U( n k 17U;9U 4535 = . KNna ?40U

tamrUbmn d)1n(UU 1n eKn d34UU 135 )1(9d34U1 = ehIy d44UU 145 )2(17d44U1 = eFIVplsgrvagTMnakTMng nig eKn )1( )2(

naM[8d10 54

d=

tam eKTaj)1(5

91

5

13645

5

4349U1 =

- 22 -


28/195


eyIgn 135

439

5

91d39UU 140 =

dUcen .13U40= x

2

13U;

2

7U 137 = . KNna ?0U

tamrUbmn INn;ndUU 0n

eKn

d13UU

d7UU

013

07

=

=

2

13d13U

2

7d7U

0

0

eFIVplsgeKn 3d6 naM[2

1d=

ehIy 02

7

2

7d7

2

7U0 =

dUcen .0U0= K 18S,12U 122 = . KNna nig ?d;U1 6U

tamrUbmn d)1n(UU 1n nig2

)UU(nS n1n

=

eKn 12dUU 12 naM[ d12U1

ehIy 18)UU(6S 12112 = 3UU 121 = eday d11UU 112 ena 3d11U2 1 = eKTaj 3d11)d12(2

=

naM[ 3d=

- 23 -


29/195


ehIy 15d12U1 .

bnabmk 0)3(515d5UU 16 =

dUcen 0U;3d;15U 61 = .7-eK[sIVtnBVn nig sIVtnBVn...,12,7,2 ...,8,5,2 etIkgcMeNam 121tYnsIVtTaMgBIrenmanbunantYEdlman

tmes

IKa ?tag nig erogKaCatYTI nigtYTIqnsIVt)a( p )b( q p nBVn nig sIVtnBVn...,12,7,2 ...,8,5,2 Edl .*INq;p eyIgn 3p5)1p(52ap

ehIy 1q3)1q(32bq cMeBa nig121p1 121q1 ebI qp aa= ena 1q33p5 )1q(3)1p(5

eKTajn

=

=

k51q

k31p smmUl

1k5q

1k3p

cMeBaRKb .INk eday nig121p1

121q1

- 24 -


30/195


eKTajn

1211k51

1211k31

24k0

40k0

naM[ 24k0 }24;...;2;1;0{k= .dUceneKGacsnidanfakgcMeNam 121tYnsIVtTaMgBIrenman tYEdlmantmesIKa .25

8-bgHajfasIVt EdlmantYTI kMnteday )U( n n

4 3n2Un = CasIVtnBVnekIn .eyIgman

4

5n2

4

3)1n(2U 1n

==

eyIgn 14

1n2

4

5n2UU n1n == efr

eday01UU n1n >

naM[ CasIVtnBVnekIn .)U( n

9-eK[RMcMnYnCasIVtnBVn. KNnacMnYnTaMgenaebIeKdwgfaplbUkvaesI125 ehIyplbUkBItYdMbUgesI .38

tag CaRMcMnYnEdlRtUvrk54321 a;a;a;a;a

tambMrabeKn =

=

125aaaaa

38aa:)S(

54321

21

tag dCaplsgrYmnsVItnBVn 54321 a;a;a;a;a

ena d4aa;d3aa;d2aa;daa 15141312

- 25 -


31/195


RbB&nsmIkar GacsresrCa )S(

smmUl

=

=

125d10a5

38da2

1

1

=

=

25d2a

76d2a4

1

1

eFIVplsgeKn 51a3 1= naM[ 17a1= ehIy 43438a238d 1 =

eKTaj

=

=

=

=

331617a

291217a

25817a

21417a

5

4

3

2

dUcen CaRMcMnYnEdlRtUvrk .33;29;25;21;1710-ebIplbUk 9tYdMbUgnsVItnBVnesInwg 162ehIyplbUk

tYdMbUgnsVItnBVnesI288 .12 KNnaplbUk 30tYdMbUgnsVItnBVnen ?tag CasIVtnBVnenehIy CaplbUkntYdMbUg*INn;n )u( nS

nsVIten .

tamrUbmn 2)uu(n

S n1

n

= eKn 162

2

)uu(9S 919 == naM[ 36uu 91 =

- 26 -


32/195


eday ( dCaplsgrYm )d8uu 19

eKn 36d8u2 1 =

118d4u1 =

dUcKaEdr 288)uu(6S 12112 = naM[ eday48uu 121 = d11uu 112 eKn 248d11u2 1 =

eKnRbB&n ==

48d11u2

18d4u

1

1

smmUl

=

=

48d11u2

36d8u2

1

1

eFIVplsgeKn 12d3 naM[ 4d= ehIy 21618d418u1 = .eKn )uu(15S 30130 eday 118)4(292d29uu 130 = ehtuen 1800)1182(15S30 = .

11-rkbIcMnYntKaCasIVtnBVnedaydwgfaplbUkbIcMnYnenesI

nwg 36ehIyplKuNvaesInwg 1428.tag dx;x;dx CabIcMnYnEdlRtUvrktamsmtikmeKGacbeg;ItRbB&n

- 27 -


33/195


=

=

1428)dx(x)dx(

36)dx(x)dx(

naM[

=

=

1428xdx

36x323

=

5d

12x

dYcen CabIcMnYntKaena .17,12,7 7,12,1712-KNnaplbUksIVtnBVnxageRkam

k146....642

eyIgn CasIVtnBVnmanplsgrYm2146;....;6;4;2 nigtYTImYyesI .tYTInnsIVtKW2 n2)1n(22un = ebI ena146un= 146n2= 73n= dUcen 5402

2

)1462(73146....642 =

=

x )20(...859095100 eyIgn )20(,....;85;90;95;100 CasIVtnBVnmanplsgrYm 5nigtYTImYyesI100.tYTInnsIVtKW 105n5)1n(5100un

ebI ena20un 20105n5 25n= dUcen 1000

2

)20100(25)20(...859095100 ==

- 28 -


34/195


K 47251721....185189193

X 227

)3(.....4

3

32

1

44

1

5 =

13-salRbCMumYyragctuekaNBay ehIytamRCugeRTtnsalRbCMuenaeKerobekAGICa 15CYr EdlCYrmuxeKbMputerobn 25ekAGIehIyCYrTIBIrerobn ekAGICYrTIbI27

erobn 29ekAGI.eKerobekAGItamlMnaMxagelIrhUtdlCYeRkay .KNnaekAGIsRmabGg:yTaMgGsEdlmanenAkgsalRbCMu.

25ekAGI

15CYr

cMnYnekAGITaMgGsEdlmanenAkgsalRbCMuKWCaplbUk 15tYdMbUgnsIVtnBVnEdlmantYTImYy 25u1= plsgrYm 2d=

- 29 -


35/195


- 30 -

ehIy 532)115(25u15 = .

ebI CacMnYnekAGIenaN 5852)5325(15

N == ekAGI.


36/195

sVItFrNImaRt CMBUk1 emeronTI3

lMhat

1-kMntpleFobrYmnsIVtFrNImaRtnigsresrbYntYbnabeTotnsIVtnImYy@xageRkam k x....,500,100,20,4 ....,4,22,2,2

2-eKmansIVtFrNImaRt ....,54,18,6,2

kKNnatYTI7 xetIcMnYn CatYTIbunannsIVt ?39366 KKNnaplbUk 15tYdMbUgnsIVtFrNImaRt .3-kMntcMnYntYnsIVtFrNImaRtxageRkam

k 8192....,,8,4,2,1 x

256

5,....,

2

5,5,10,20

4-bgHajfasVItxageRkamCassIVtFrNImaRtEdlmantYTInkMntdUcxageRkam

k x1n

n )3(7u

n

1n )4(8

1

U

5-eK[ CasIVtFrNImaRt ebIeKdwgfa )u( n k 1458U;8U 73 = . KNna qnig ?10U

- 31 -


37/195


x 1280U;20U 63 = . KNna qnig ?1U

K . KNna qnig ?21S;3U 31 =

7S6-KNnaplbUksVItFrNImaRtxageRkam

k8

3...122448

x 6561....313

1

K244

128.....469

X n21n25432 31

2

1....2

1

3

1

2

1

3

1

2

11

7-KNnaplbUksIVtFrNImaRtGnntYEdlman

nig9U2 3

1U5= .

8-KNnapleFobrYmnsIVtFrNImaRtGnntYEdlman

nig5U1= 15S= .9-plbUkBIrtYdMbUgnsIVtFrNImaRtesInwg 9nigplbUksIVtFrNIImaRtGnntYesInwg 25.KNna qnig ebIpleFob1U 0q>

10-sresrCacMnYnsniTanEdlmanTRmgba

nUvcMnYnTsPaKxYbxageRkam k 7.0 x 34.2 K 452.0

- 32 -


38/195


11-tYTI nsIVtnBVnesI14nig tYTI12esI .5 35

KNnaplsgrYmtYTI1nig tYTIn.12-plbUkn tYdMbUgnsIVtmYykMnteday .n nnS 2n

KNnatYTI nsIVt nigbgHajfatYnsIVtenCasIVtnBVn.5

13-tYTI nsIVtFrNImaRtesI nigtYTI6esI3 38

3 .

KNnapleFobrYm tYTI1

nigplbUk8

tYdMbUg .14-KNna aedIm,I[bIcMnYn )1a(,)3a(

nig )2a4(

CabItYtKansIVtFrNImaRtmYy .15-eK[bYncMnYnCasIVtFrNImaRt . KNnatYTIBIr nig tYTIbInsIVtebIeKdwgfa tYTImYyesI6ehIytYTIbYnesInwg 162.

16-eKTijpmYytm dula . GkTijRtUvbgRkCa600030 dMNakkal Edlbeg;ItnCasIVtFrNImaRtmanpleFobrYm

esInwg10

9 . kMntRkEdlRtUvbgtamdMNakkalnImYy@.

17-eKmansIVtnBVn nig kMnteday)U( n )V( n 3U0= nigplsgrYmesI3ehIy 3V0= nigplsgrYmesI .2 kkMnt nig CaGnuKmn_n n.nU nV

- 33 -


39/195


xeKmansIVt kMnteday)W( nn

nn

V

UW=

.

bgHajfa CasIVtekIn ..a )W( n kMntnedIm,I[.b

28

1WW 1nn = .

18-kMntplsgrYm nigtYTImYy nsVItnBVnekIn1U *INnn )U(

ebIeKdwgfa

=

=

4UUU

98UUU

321

23

22

21

KNnatYTI10nigplbUk 10tYdMbUgnsIVt .19-tYdMbUgnsIVtFrNImaRtesI ehIymanpleFobrYmesI3 r Edltmdacxatn rtUcCag 1.

plbUk tYdMbUgesI39

8 nplbUk 6tYdMbUg .

KNnaplbUkGnntYnsIVtFrNImaRt .20-1 edaRsaykg smIkarIR )E(04x8x3: 2

=

2 eKman CasIVtFrNImaRtekInEdlmantYINnn )U( nig nsIVtCacemIynsmIkar .3U 4U )E(

k-TajrkpleFobrYmnsIVten .x-KNnatYTImYy nsIVt .1U

- 34 -


40/195


dMeNaRsay

1-kMntpleFobrYmnsIVtFrNImaRtnigsresrbYntYbnabeTotnsIVtnImYy@xageRkam k ....,500,100,20,4

pleFobrYmnsIVtenKW

5100500

20100

420q

= dUcen .5q=

x ....,4,22,2,2 pleFobrYmnsIVtenKW

222

42

222

2q =

dUcen 2q=

.2-eKmansIVtFrNImaRt ....,54,18,6,2 kKNnatYTI7 xetIcMnYn CatYTIbunannsIVt ?39366 KKNnaplbUk 15tYdMbUgnsIVtFrNImaRt .

- 35 -


41/195


dMeNaRsay

kKNnatYTI7 tamrUbmn 617 quu

eday nig2u1= 32

6

u

uq

1

2=

eKn 145832u 67 =

dUcen .1458u7= xetIcMnYn CatYTIbunannsIVt ?39366 tYTInnsIVtenKW 1n1n1n 3.2quu = ebI ena39366un= 393663.2 1n =

eKTaj91n 3196833 =

naM[ 91n = 10n= dUcencMnYn CatYTI10nsIVt .39366

KKNnaplbUk 15tYdMbUgnsIVtFrNImaRt

tamrUbmn 1q1q

.uu...uuS

15

1152115

=

13

13

132S 15

15

15

.

- 36 -


42/195


3-kMntcMnYntYnsIVtFrNImaRtxageRkam

k 8192....,,8,4,2,1 tamrUbmntYTInnsIVtFrNImaRtKW 1n1n quu

eday 21

2

u

uq;1u

1

21 =

eKn *INn2u 1nn

ebI ena8192un=131n

281922 =

eKTaj naM[131n = 14n= .dUcensIVtxagelImancMnYn 14tY.

x256

5,....,

2

5,5,10,20

tamrUbmntYTInnsIVtFrNImaRtKW1n

1n quu

eday2

1

20

10

u

uq;2u

1

21 =

eKn *INn2

120u

1nn

ebI 2565

un=

ena 2565

2

120

1n

=

122 1n

=

eKTaj naM[121n = 13n= .dUcensIVtxagelImancMnYn 13tY.

- 37 -


43/195


4-bgHajfasVItxageRkamCasIVtFrNImaRtEdlmantYTIn

kMntdUcxageRkam k 1nn )3(7u eday 1nn 37u ena n1n 37u

eKn 337

37

u

uq

1n

n

n

1n=

=

efr

dUcen nu CasIVtFrNImaRtmanpleFobrYm .3q= x n1n )4(

8

1U

eday nn 48

1u

ena 1n1n 48

1u

eKn4

u

uq

n

1n=

efrdUcen nu CasIVtFrNImaRtmanpleFobrYm .4q=

5-eK[ CasIVtFrNImaRt ebIeKdwgfa )u( n k 1458U;18U 73 = . KNna nig ?q 10U

tamrUbmn1n

1n quu

eKn

=

=

1458q.uu

18q.uu

617

213

- 38 -


44/195


eKTaj18

1458

q.u

q.u

u

u2

1

61

3

7= 81q4

=

eKTaj .3q maeTot 373619110 q.uq).qu(quu =

3936631458u 310 dUcen }39366;3;39366;3{u;q 10 x 1280U;20U 63 = . KNna qnig ?1U

tamrUbmn 1n1n quu

eKn

=

=

1280q.uu

20q.uu

516

213

eKTaj20

1280q.uq.u

uu

21

51

3

6= 64q3 =

eKTaj .4q=

maeTot eKTaj213 q.uu=4

5

4

20u

21 =

dUcen 45

u;4q 1= .

- 39 -


45/195


K . KNna qnig ?21S;3U 31 = 7S

tamrUbmn 21)1q.(31q1q

.uS

2

13 =

= eKTaj .6q

=

mageTot )16(5

3

16

16.3

1q

1q.uS 6

66

17

=

=

dUcen )16(53

S;6q

6

7

.6-KNnaplbUksVItFrNImaRtxageRkam

k8

3...122448

tamrUbmntYTInnsIVtFrNImaRtKW 1n1n quu

eday 21

48

24

u

uq;48u

121

=

eKn1n

n2

148u

ebI8

3un= ena

8

3

2

148

1n

=

eKTaj 71n2

1

2

1

=

naM[ 8n=

dUcenplbUkxagelICaplbUk 8tYdMbUgnsIVtFrNImaRt .

- 40 -


46/195


eyIgn211

2

11

488

3

...122448

8

8

765

256

125696

=

dUcen 876583...122448 = .

x3

295246561....31

3

1=

K243

1261

244

128.....469 =

X n21n25432 31

2

1....

2

1

3

1

2

1

3

1

2

11

n1n 4

11

3

2

9

11

8

9

7-KNnaplbUksIVtFrNImaRtGnntYEdlman

nig9U2 3

1U5= .

tamrUbmnq1

uS 1

Edl CatYTImYy nig qCapleFobrYm .1u

- 41 -


47/195


eKman27

1

9

3

1

qu

q.u

u

u

1

41

2

5

27

1q3 naM[

3

1q

ehIy 273

1

9

q

uu 21 =

=

dUcen 481

3

11

27

S =

= .

8-KNnapleFobrYmnsIVtFrNImaRtGnntYEdlman nig5U1= 15S= .

tamrUbmn q1u

S 1

naM[ 32

15

51

S

U1q 1 =

dUcen

3

2q= .

9-plbUkBIrtYdMbUgnsIVtFrNImaRtesInwg 9nigplbUksIVtFrNIImaRtGnntYesInwg 25.

KNna qnig ebIpleFob1U 0q> tambMrabeKman

>

=

0q,25S

9UU 21

- 42 -


48/195


eKn

=

=

25q1

U

9q.UU

1

11

=

)q1(25U

9)q1(U

1

1

eKTaj 9)q1(25 2 =

naM[25

16

25

91q2 = eday 0q>

eKTaj5

4q= ehIy 5

5

4

1

9

q1

9u1 =

=

dUcen 5u;5

4q 1= .

10-sresrCacMnYnsniTanEdlmanTRmgb

a nUvcMnYnTsPaKxYb

xageRkam

k 7.0 eyIgn ....007.007.07.07.0

9

7

9.0

7.0

1.01

7.0=

dUcen9

77.0 = .

x99

23234.2 = K

495

224452.0 =

- 43 -


49/195


11-tYTI nsIVtnBVnesI14nig tYTI12esI .5 35

KNnaplsgrYm tYTI1nig tYTI .n tag CasIVtnBVnen)u( n

eKman

naM[

=

=

35u

14u

12

5

=

=

35d11u

14d4u

1

1

eFIVplsgeKn 21d7 naM[ 3d=

bnabmk 21214d414u1 = ehIy 1n3)1n(32d)1n(uu 1n dUcen 1n3u;2u;3d n1 .

12-plbUkn tYdMbUgnsIVtmYykMnteday .n nnS 2n

KNnatYTI nsIVt nigbgHajfatYnsIVtenCasIVtnBVn.5 - KNnatY 5U eyIgman n321n U...UUUS

ehIy 1nn1n3211n USU...UUUS

eKTajn1n1n SSU

ehtuen 10)44()55(SSU 22455 = dUcen .10U5=

- 44 -


50/195


- bgHajfatYnsIVtenCasIVtnBVn

eKman n1n1n SSU

eday nnS 2n naM[ 2n3n1n)1n(S 221n eKn 2n2nn2n3nU 221n

eKTajn22)1n(2Un =

eday 2n22n2UU n1n = efrdUcen CasIVtnBVn.)U( n

13-tYTI nsIVtFrNImaRtesI nigtYTI6esI3 38

3 .

KNnapleFobrYm tYTI1nigplbUk 8tYdMbUg .

tamsmtikmeKman

=

=

8

3q.UU

3q.UU

516

213

eKn8

1q

U

U 3

3

6=

naM[2

1q= .

eday eKTaj213 q.UU= 12

2

13

qUU

2231 =

=

- 45 -


51/195


mageTot 256255

24211

2

11

.12q1

q1

uS

8

8

18

=

=

dUcen32

765S;12U;

2

1q 81 = .

14-KNna aedIm,I[bIcMnYn )1a(,)3a( nig )2a4(

CabItYtKansIVtFrNImaRtmYy .KNnatmaedIm,I[bIcMnYn )1a(,)3a(

nig )2a4(

CabItYtKansIVtFrNImaRtmYyluRtaEt )2a4)(3a()1a( 2

0491564'

05a16a3

6a12a2a41a2a

2

22

>

=

eKTajs 53

78a;

3

1

3

78a 21 ===

dUcen 5a;3

1a 21 = .

15-eK[bYncMnYnCasIVtFrNImaRt . KNnatYTIBIr nig tYTIbInsIVtebIeKdwgfa tYTImYyesI6ehIytYTIbYnesInwg 162.

- 46 -


52/195


KNnatY nig2u 3u

tambMrab CasIVtFrNImaRt .4321 u;u;u;u eday ehIy6u1= 162q.uu 314 =

eKTaj 276

162

u

162q

1

3= naM[ 3q=

dUcen 183.6q.uu 12 =

nig .543.6q.uu 22

13 =

16-eKTijpmYytm dula . GkTijRtUvbgRkCa600030 dMNakkal Edlbeg;ItnCasIVtFrNImaRtmanpleFobrYm

esInwg10

9 . kMntRkEdlRtUvbgtamdMNakkalnImYy@.

tag Ca RkEdlRtUvbgtam654321 U;U;U;U;U;U dMNakkalnImYy@ . tamsmtikmeKn

=

=

10

9q

)1(00030U...UU 621

tam eKsresr)1( 00030q1 q1.U6

1 =

eKTaj61 q1

)q1(30000U

= eday 9.010

9q =

- 47 -


53/195


eKn 088.64026$)9.0(1

)9.01(30000U

61 =

=

ehIyedayeRbI q.UU...,;q.UU;q.UU 562312 = eKTaj 479.57623$U2=

765.37806$U;516.42007$U

018.46675$U;131.51861$U

65

43

=

=

17-eKmansIVtnBVn nig kMnteday)U( n )V( n 3U0= nigplsgrYmesI3ehIy 3V0= nigplsgrYmesI .2 kkMnt nig CaGnuKmn_n n.nU nV


nn

V

UW=

.

bgHajfa CasIVtekIn ..a )W( n

kMntnedIm,I[.b28

1WW 1nn = .

dMeNaRsay

kkMnt nig CaGnuKmn_n nnU nV

tamrUbmnn33ndUU 0n

ehIy n23ndVV 0n dUcen 2n3V;3n3U nn

- 48 -


54/195



nn

V

UW=

.

bgHajfa CasIVtekIn.a )W( n eyIgman

3n2

3n3

V

UW

n

nn

=

eKn5n2

6n3

3)1n(2

3)1n(3W 1n

=

=

eday 3n23n3

5n2

6n3

WW n1n

=

0)3n2)(5n2(

3

)3n2)(5n2(

15n21n618n21n6 22

>

=

dUcen CasIVtekIn .)W(

n

kMntnedIm,I[.b28

1WW 1nn =

eKman)3n2)(5n2(

3WW n1n

naM[28

1

)1n2)(3n2(

3WW 1nn =

eKTaj 30028416';071n8n4 2 = naM[smIkarKanskg IN.

- 49 -


55/195


18-kMntplsgrYm nigtYTImYy nsVItnBVnekIn1U *INnn )U(

ebIeKdwgfa

=

=

4UUU

98UUU

321

2

3

2

2

2

1

KNnatYTI10nigplbUk 10tYdMbUgnsIVt .dMeNaRsay

- KNnaplsgrYm dnigtYTImYy 1U

eday CasIVtnBVnekInenaplsgrYm321 U;U;U 0d > ehIy 231 U2UU = 2321 UUUU = eKTaj ( eRBa4U2= 4UUU 321 = )

eKman

eday

=

=

4UUU

98UUU

321

23

22

21 4U2=

eKn

=

=

8UU

82UU

31

23

21

eKman 312312321 UU2)UU(UU

eKTaj2

)UU()UU(UU

23

21

231

31

=

9UU

28264UU

31

31

=

- 50 -


56/195


eKnRbB&n

=

9UU

8UU

31

31

naM[ nig CassmIkar1U 3U 09X8X2 = eday enacab 9X;1X 21 = eKTaj 9U;1U 31 = dUcen 9U;4U;1U 321 =

eKn 5)1(4UUd 12 = dUcen nig1U1 5d= .-KNna nig10U 10S

eyIgn 44451d9UU 110 =

ehIy 215)441(52)UU(10

S 10110 ==

dUcen 215S;44U 1010 = .19-tYdMbUgnsIVtFrNImaRtesI ehIymanpleFobrYmesI3 r Edltmdacxatn rtUcCag 1.

plbUk tYdMbUgesI3 98

nplbUk 6tYdMbUg .KNnaplbUkGnntYnsIVtFrNImaRt .

- 51 -


57/195


dMeNaRsay

KNnaplbUkGnntYnsIVtFrNImaRt tamrUbmn

r1

US 1

smtikm 3U1=

ehIy )U...UU(9

8UUU 621321

)r1)(r1(9

8r1

r1r1U

98

r1r1U

333

61

31

=

eKTaj8

11

8

9r3 = naM[

2

1r=

dUcen 62

11

3S =

= .

20-1 edaRsaykg smIkarIR )E(04x8x3: 2=

2 eKman CasIVtFrNImaRtekInEdlmantYINnn )U(

nig nsIVtCacemIynsmIkar .3U 4U )E(

k-TajrkpleFobrYmnsIVten .x-KNnatYTImYy nsIVt .1U

- 52 -


58/195


- 53 -

dMeNaRsay

1 edaRsaykg IRsmIkar)E(04x8x3 2 =

2

3

24x;

3

2

3

24x

41216'

21 ===

=

dUcen2x;

3

2x

21 =

.2k-TajrkpleFobrYmnsIVten

eKman 2U;3

2U 43 = (eRBa 43 UU< sIVtekIn)

eKman nig213 q.UU= 314 q.UU=

eKn 33

22q

q.Uq.U

UU

21

3

1

3

4=

dUcen .3q= x-KNnatYTImYy nsIVt 1U

eday naM[213 q.UU=27

2

3

3

2

q

UU

223

1 =

dUcen27

2U1= .


59/195

1-GnuKmn_Gics,NgEsl2-GnuKmn_elakarIt


60/195

GnuKmn_Gics,NgEsl CMBUk2 emeronTI1

1

-KNnatmnkenam

lMhat

k 16 x 225 K 144X 3 3)2y(008.0

g 2)x6(

c 3 27 q 3 64 C3 3x64 2

-sresrkenamxageRkamCaTRmgGics,NgEsl k 4 10x=

x k568=

K 8 h500a=

X x yb

=

3-sRmYlkenam k4 625 x5 1 K 5 32

X532

1

g6 6x c4 4)b7(

q7 7y C5 5)2x(

4-rkEdnkMntnGnuKmn_k x)x(f

=

x 3 x)x(f = K 8x2)x(f

X x34)x(f g 2x3)x(f c 1x)x(f 2

q5x3x2

x)x(f

2

C2xx

3x)x(f

2

=

Q|x|x

1x)x(f

=

- 54 -


61/195


5-KNnakenamxageRkam

k 3.2 x 33

5.3 K 444

72.9.8 X b10.a3 g ax.ax c

5

y.

x

6

6-sRmYlkenam

k 724

116

2

35.4

3

2 3

x 33 33 y543

xyx16y2x

7-eRbobeFobk 3 25 nig 3 312 x3 2nig 12 45K6 5nig 8 8

8-sRmYlkenam

k 22223 3223

dc.dc

dcd3dc3c

x 1xxx

K 222222 q2pq3q

1

q8pq6p

q3p.

ppq4q3

p

9-sresrkenamxageRkamCaTRmgraDIkal

k 3

1

8 x 4

1

x K 5

1

y X 2

135 )ta( g 4

133 )yx( c 4

153 )yx(

- 55 -


62/195


10-edaRsaysmIkar

k x322x=

1)1x(

3x4x2=

K 2)32()32( xx=

X 65613 )2( x=

g c

q C981 )4(

x

=

13 )3( x

=

0)6(5 x=

)9(2)4(3 xx

4)32()32( xx=

11

-edaRsayvismIkark x12x > 13x K16

12 x2

X g1xx 816 < x52x5 1632 c256

14 1x3

>

12-edaRsayvismIkar

k x125.016x

>

82x

K 31

3.27 x1x

13-mIgsynykkmYycMnYneTAepJIenAFnaKarmYyedayTTYlnGRtakarRk kgmYyqaM. ryeBl qaMeRkaymk%6 7

KatndkRkBIFnaKarenavijedayTTYlnRksrubcMnYndula . etImIgsymanRkedImcMnYnbunan ?300

- 56 -


63/195


14-mnusmakmanGayu qaMnykRk eroleTAepJI30 00050

enAFnaKarmYyedayTTYlnGRtakarRk kgmYyqaM.%4 etIKatmanksrubcMnYnbunanebIbcb,nKatmanGayu65qaM?15-mnusmaknTijrfynfImYyeRKOgkgtm dula .00045 rfynenacuf CaerogralqaM.%15

rktmrfynkgryeBl7

qaMeRkay .

- 57 -


64/195


1

-KNnatmnkenam

dMeNaRsay

k 4)4(16 2 = x 15)15(225 2 = K 12)12(144 2 =

X )2y(2.0)2y()2.0()2y(008.0 3 333 3

g |x|6x6)x6( 2 = c 3)3(27 3 33 = q 4)4(64 3 33

C x4)x4(x64 3 3

3 3

2-sresrkenamxageRkamCaTRmgGics,NgEsl k 4 10x

=

naM[ 10x4=

x k568 = naM[ 8k56 = K 8 h500a= naM[ h500a8 =

X x yb= naM[ ybx =

- 58 -


65/195


3-sRmYlkenam

k 5)5(625 4 44

=

x 115 K 2)2()2(32 5 55 =

X2

1

2

1

32

15

5

5

g |x|x6 6 = c |b|7)b7(4 4 = q yy7 7 = C 2x)2x(5 5

4-rkEdnkMntnGnuKmn_k x)x(f = GnuKmn_mann&ykalNa 0x

dUcen [;0[D

x 3 x)x(f = GnuKmn_enGackMntnCanicRKb IRx

dUcen IRD = K 8x2)x(f

GnuKmn_mann&ykalNa 08x2 naM[ . dUcen4x [;4[D

- 59 -


66/195


X x34)x(f

GnuKmn_mann&ykalNa

naM[

0x34

. dUcen ]34

;]D

3

4

x

g 2x3)x(f uKmn_enKann&yCanicelIkElgEtGn 0x= .

dUcen }0{D =

c 1x)x(f =

GnuKmn_enGa2

ckMntnCanicRKb IRx

endUc IRD =

q5x3x2

x)x(f

2

uKmn_mann&ykalNa

=

05x3x2

0x

2Gn

0x

2

5x;1x

dUcen [;2

5][

2

5;0[D

C 2xx3x

)x(f =

2

uKmn_mann&ykalNa

02xx

03x

2Gn

- 60 -


67/195


2x;1x

3x

dUce 1n [;2][2;1][;3[D Q

|x|x

1x)x(f

=

mn_mann&ykalNaGnuK

0|x|x

01x naM[ 0x >

dUcen [;0]D 5-KNnakenamxageRkam k 6323.2 = x 3333 15535.3 =

K 2623672724 9872.9.8 444 = =

X ab30b10.a3b10.a3 = g 22 ax)ax)(ax(ax.ax

cx5

y6

5

y.

x

6

5

y.

x

6=

Ylkenam k

6

-sRm2

2

3235

3

472

4

116

25.4

3

32 33

- 61 -


68/195


33 33 y543

xyx16y2x x

0y2xy2x2y2x 333=

beFob7-eRbo k 3 25 nig 3 312 eKman 33 3 3 2.25 2505

=

33 33

24831.231ehIy 2 = dUcen 33 31225 > x 3 2nig 12 45eKman 123 12 4 1622 =

dUcen12 452

6

8-sRmYlkenam

dc

1

dc

dc

dc.dc

3dc3c

22

3 3 dcd2222

322

=

k

- 62 -


69/195


x )xx

2 xx()1x(x

)1xx(x

1x

x=

222222K-

q2pq3p

1

q8pq6p

q3p.ppq4q3

p

q4p

p

q4p

p

)q2p)(qp()q4p)(q2p(

q3p.

)pq3)(qp(

p

resrkenamxageRkamCaTRmgraDIkal9-s

k 33 88= 1

44

1

xx=x

551

yy=

K

352

135 ta)ta(X

4 334

133 yx)yx( g =

4 5341

53 yx)yx( =

c

- 63 -


70/195


10-ed

k

aRsaysmIkar

322x=

22=

naM[ 5x5x=

lkxN x( 1)1x 3x4x2 =

1x 1x0

IkarsmmUlsm

=

3x4x

11x

2 eKTajs 3 3x;1x;0x 21 =

K 2)3 x = 2()32( x

32

1 eKman 1)32)(3 = 32( 2

IkarsmmUl 2)32(

1)32(x

x=

sm

1)32(2x

0

1)32(2 xx2

=

eKTaj

0)32( =

1)32( x = dUcen .0x=

- 64 -


71/195


X 13 )2( x 656

3x

x

8)2(

22

8233

x

=

=

=

eng

aj

dUc 3x=981 )4(

x

=

12x2

22

24

3

2x2

1x

2

=

=

=

=

34 1x

eKT 21

.x

caMgBIrnwg

0)6(5)9(2)4(3 xxx =

EckGg:T x9 eKn

0233

25

2 xx2

=3

tag 03

2t

x

>

=

mans02t5t3 2 =32t;1t 21 =

-cMeBa eKTaj1t1= 13

2 x

=

0x=

- 65 -


72/195


-cMeBa3

2t2=

3

2

3

x

=

2eKTaj

1x=

dUcen 1x; 2 .

)

0x1= =

qeKTaj ( Kans

0)3( 313x

=

03x=

C 4)2( x=

32()3 x

aneKm32

1= 32

IkarsmmUl 4)32(

1)32(

xsm x =

01)32( 2(4)3 xx2 =

tag 0)32(t 2

x

>

ajs314';01t4t =

eKT 32t;32t 21 -cMeBa ena 32)3232t1 ( x

naM[ .2x

=

-cMeBa 32 ena 32)32( x t2= = .naM[ x 2

- 66 -


73/195


11-edaRsayvismIkar

naM[M[k x

2>

x>

0

21=

0 x 0x3 na31

= 0x

K 4x2 216

12

=

naM[ 2x

4x2

sX mmUl1xx 816 < 3x3x4 22 <

ajeKT 3x3x4 3 x

Ulsmm 41x3 44 >

ajeKT 41x3 x 1

yvismIkar12-edaRsa k smmUl 2125.016x

>

3x4 2>

eKTaj 3x4 4

3x>

envismIkarenmancemIykgx 82

x

eday 02:IRx x

>

dUc .IR

- 67 -


74/195


K3

1

eKTaj3

2x <

13-mI kkmYy nGRtakarRk %6 kgm

gsyny cMnYneTAepJIenAFnaKarmYyedayTTYlYyqaM. ryeBl qaMeRkaymk

IFnaKarenavijedayTTYlnRksrubcMnYn7

KatndkRkB 300dula . etImIgsymanRkedImcMnYnbunan ?

tamrUbmn t)i(pA= naM[1 tAp= )i1(

eday 7t;06.0%6i;300$A qaM

- 68 -


75/195


- 69 -

eKn 517.199$p =)06.01(

3007

.

akmanGayu qaMnykRk eroleTAepJIarmYyedayTTYlnGRtakarR kgmYyqaM.

nYnb manGay

14-mnusm 00030 50 enAFnaK k %4 etIKatmanksrubcM unanebIbcb,nKat u65qaM?

tamrUbmn t)i1(pA eday =

=

=

=

35t

04.%4i 50p

0

000

eKn erolnTijrfyn Ogkgtm ula .

uf CaerogralqaM.

5.197304)04.01(50000A 35 =

15-mnusmak fImYyeRK 0d0045

rfynenac rktmrf%15

ynkgryeBl 7qaMeRkay .cemIyRMBIrqaMeRkaymkrfynmantm 14426$


76/195

GnuKmn_elakarIt CMBUk2 emeronTI2

lMhat

1-rksmIkarnGnuKmn_RcasrbsGnuKmn_xageRkam k x K3x4y 1x3y 2 3y3x 22 =

X g c6xy 1xy2 = 14

y

9

x 22=

q x5

y=

C 2xy

2-eK[GnuKmn_ 2x)x(f . cUrkMntrkGnuKmn_Rcas )x(f 1 rYcsgRkabrbsva .

3-sgRkabnGnuKmn_xageRkam k xxlogy 5 xlogy 7 K

X gxlogy 10

xlogy 15 xlogy 5.0 c xlogy 3.04-bMElgCaTRmgelakarIt

k x10000104 = 2831

= K 21641

=

X 251

5 2 =

g 641

4 3 =

c 210 3010.0 =

q Cyx a = ckA=

- 70 -


77/195


5-sRmYl

k x K2log3

3

10log5

5

10

a alog X g caq qlog 64log4 1.0log10 Q 16log 2 j 1log106-edaRsaysmIkarelakarIt

k

x3xlog3 = 216logx =

K 1)9xlog(xlog =

X 1xlog2 g2

1xlog9 =

c3

1xlog8 = q xlogxlog 3 =

C 2)3x(log)3x(log 44 =

Q 11xlog 2

5 =

j 33 43 2 2logxlogxlog =

7-edaRsayvismIkarelakarIt k )3x(logx2log 42 x )7x(log)5x3(log 33

K 11x

xxlog

2

2

4 3

x

15

1

15

g 3x

04.05

1

1x

0x

eKTaj naM[2x16=

416x =

K 1)9xlog(xlog = lkxN

>

>

09x

0x0x >

eKn 10log)9x(xlog = eKTaj 010x9x2 = eday 0cba =

X eKTaj1xlog2 2

12x 1 =

g2

1xlog9 = eKTaj 39x 2

1

=

- 81 -


88/195


c3

1xlog8 = eKTaj 28x 3

1

=

q xlogxlog 3 = lkxN naM[

>

0xlog

0x1x

eKn xlogxlog3

1=

0xlog9xlog2 =

0)9xlog(xlog = eKTajs .910x;1x

=

C 2)3x(log)3x(log 44 =

lkxN naM[

>

>

03x

03x3x >

smIkarGacsresr

525x

25x

4)3x)(3x(

2)3x)(3x(log

2

2

4

=

=

=

eday (lkxNxagelI)3x > dUcen .5x=

- 82 -


89/195


Q 11xlog 25 =

smmUl 51x2

=

62x

24x

251x

2

2

=

=

j 33 43 2 2logxlogxlog = lkxN 0x

eKn 33 43 2 2logx.xlog =

4

2x

8

1x

2x

2

33 6

=

=

7-edaRsayvismIkarelakarIt k )3x(logx2log 42 lkxN 0x >

eKn )3x(log2

1x2log 22

1x4

303xx4

)3x(log)x2(log

)3x(logx2log2

2

22

2

22

- 83 -


90/195


eday enaeKTaj0x > 1x0

dUcen CasMnMucemIysmIkar .]1;0]x x )7x(log)5x3(log 33 lkxN

3

5x>

eKn 12

57x7x5x3

dUcen [;3

5]x

K 11x

xxlog

2

2

eKTaj108

6977logx

108

69773 3

x ==

- 86 -


93/195


X 3xx3x2 2122

0)12)(12(

012)12(2

x3x

3x3xx

=

=

eday eKn012x>

3x012 3x

11-ebI . bgHajfazyx )ab(ba = )yx(zxy tag 0t)ab(ba zxx

>

eKTaj

=

=

=

tlogz

tlogx

tlogx

ab

b

a

eKmanblogalog

1

)ab(log

1tlogz

tttab

yx

xy

y

1

x

1

1

tlog

1

tlog

1

1z

ba

=

=

dUcen )yx(zxy .12-ebI cba 1232 = . bgHajfa )b2a(cab

tag0t1232 cba >

eKTaj

=

=

=

tlogc

tlogb

tloga

12

3

2

- 87 -


94/195


eKman2log23log

1

)2.3(log

1tlogc

tt2

t12

b2aab

a

2

b

11

tlog

2

tlog

11c

23

=

=

dUcen .)b2a(cab 13-ebI 23a nig 23b .

bgHajfa 1b1 1a1 1 =eKman 1)23)(23(ab =

a

1b=

eKna1

a

a

11

1

b1

1

=

ehtuen 1a1

aa1

1b1

1a1

1=

dUcen 1b1

1

a1

1=

.

14-ebI .ab11ba 22 =

bgHajfa)

3

balog(2blogalog

=

?eKman ab11ba 22 =

ab9bab2a 22=

- 88 -


95/195


ab9)ba( 2 = ab3

ba 2

=

eKn blogalog)ablog(3

balog

2

ebI eKTajba > )3

balog(2blogalog

= .

15-ebI nigaMlogx = bMlogy = .

bgHajfa baab

Mlogxy

eKmanylogxlog

1

)xy(log

1Mlog

MMMxy

ba

ab

b

1

a

1

1Mlog

Mlog

1

Mlog

1

1Mlog

xy

yx

xy

=

=

dUcenba

abMlogxy

.

16-edaRsaysmIkarelakarIt k 25logxlog)3x2(log2 1021010

lkxN2

3x03x2

>

- 89 -


96/195


eKn 25logxlog)3x2(log100log 102101010

624x;224x

41216';012x8x

x25)3x2(100)x25log()3x2(100log

21

2

2

2

10

=

=

=

=

x 13)xlog8(log6 8x = lkxN 1x;0x

tag naM[xlogt 8 8logt1 x

smIkarsmmUl 13)tt

1(6 =

25144169;06t13t6 2=

eKTajs 34

6

513t;3

6

513t 21 ===

-cMeBa eKTaj3t= 5128x3xlog 38 =

-cMeBa3

4t= eKTaj 168x

3

4log 3

4

8 =

dUcen 512x;16x 21 =

K 25xlog9log 9x = mans 81x;3x 21 = X 19logxlog 2x9 = smIkarKans .

- 90 -


97/195


17-edaRsayvismIkarGics,NgEsl

k2x3x

12525

7

6x6x9x2

55 )2x3(3x2

x 273x >

3x33 3x

> K tag082.64 xx > 02t x >

0)4t)(2t(

08t6t2

>

>

naM[ 4t2t0 >

eKTaj 1x220 x < 2x42x > dUcen [;2][1;]x

X 4 3

x

15

115

4

3x515 4

3x

g3

2x04.0

5

1 3x

>

- 91 -


98/195


c 2x9

1003.0 x

18-edaRsayvismIkarelakarIt k 1xlog 2x3

smmUl

2x3x

12x30

2x3x

12x3

smmUl

1x

1x

3

2

x1

1x

dUcen [1;3

2[x .

x tag015xlogxlog 224 < 0x;xlogt 4 > eKn 015t2t2 <

5t30)3t)(5t(

- 92 -


99/195


eKn [ ]4

32)13(log)13(log x4

x4 <

tag )13(logt x4

2

3t

2

1t0)1t2)(3t2(

03t8t4

04

3t2t

4

3)2t(t

2

2

>

>

==

=

eKn oooo 48.28436052.75x52.75x4

1xcos

=

X 0ysin

ycosycos3 =cos3 0ycoty =

0)1ysin3(ycos =

3

1ysineKTaj 0ycos

- 152 -


160/195

smIkarnigvismIkarRtIekaNmaRt CMBUk3 emeronTI3

dUcen y ooo 47.199y;52.340y;90 = .

g 2sin2 01x =

o

oo

o

210sinx2s

s

in

)30180sin(x2in

30sinx2sin

2

1x2sin

=

eKTaj Zk;108.k15x180.k105x oooo eday o0 o360x dUcen o ;285 oo 195xx;105x = c xsin6 xcos5xsin8 22 =

03

2)1x(sin

03

5xsin2xsin

05xsin6xsin3)xsin1(5xsin8xsin6

2

2

2

22

=

=

=

smIkarKans .q t5x2cot2 oo 80.201x80.21x4.0x2an =

- 153 -


161/195


C 08ytanysin3 =

3

1ycos3ycos0)1ycos3)(3y(cos

03ycos8ycos3

03ycos8ycos3

0ycos8)ycos1(3

0ycos8ysin3

08ycos

ysin

.y

2

2

2

2

=

=

=

=

=

eKTajQ

sin3

oo52.250y;47.109y =

5ysin7ycos3 2

3

1ysin2ysin0)1ysin3)(2y(sin

02ysin7ysin3

5ysin7ysin33

=

5ysin7)ysin

2

2

2

=

j

j

1(3

eKTa oo 47.199y;58.340y =

0)2

zsin21(2

zcos02

zcos22

zcot

=

eKTaj2

1

2

zsin0

2

zcos =

- 154 -


162/195


dUcen z = ooo 300z;30z;180 = .

aj

.

d xtan( 050tan)30 oo

=

eKT ooo 180.k30x0)30xtan( eday 0 o360x

dUcen oo 210x;30x =5

-eK[

cos2sin3x

ign y

sin2cos3

mnmMuRsYckrkt cMeBa x y=

eKn s2co3cos2sin3

co5sin

ins

Zk;180.k7.785tan

s

oo

CacMnYnefrcMeBaRKbtmn

efr .

kaN

xbgHajfa2 yx 2

eKn 22 sincos3()cos2sinx 22 )23(y

13)cos(sin13 22 = dUcen yx 1322 =

6

-eK[RtIe manABC Atan

AcosBsin

sin BcosA=

.Ca Ekg .bgHajfa AB RtIekaNC

- 155 -


163/195


dMeNaRsay

CaRtIekaNEkgbgHajfa ABCeKman Atan

Assin

Bssin=

coB

coA

00 90C90BA0)BAcos(

0BsinAsinBcosAcos

AcosAsinBsinAsinBcosAcosAcosAsin

Acos

Asin

AcosBsin

BcosAsin

=

=

=

dUcen CaRtIekaNEkgRtg .ABC C

7-eK[RtIekaN ABCman Aos2Bsin

Csin=

c .rkRbePT ABC.

dMeNaRsay

eKman AcosBsin2CsinAcos2Bsin

Csin=

eday )BAsin()BA(sinCnsi eKn AcosBsin2)BAsin( =

2

BA0)BAsin(0AcosBsinBcosAsin

AcosBsinAcosBsinBcosAsin

=

=

=

dUcen CaRtIekaNsmtkMBUl .ABC C

- 156 -


164/195


9-Gg;tFAB ckrgVgmYyEdlmankaMesInwg cm2E CaBIrGg;t.

Gg;t

.Uc .

eKdwgfa ACs;atFEdlmanmMupites

I .KNna o

60

kRkTLapceRmokfas AOB.

O

B

A xRkTLapRtIekaN AO B KRkTLapGg;tfast

dMeNaRsay

kRkTLapceRmokfas AOB

eyIgn 21 R2

1S eday

360;cm2R o

=

dUcen 21 cm2

S3

4.3

.2

1 ==

.

pRtIekaNxRkTLa AOB

eyIgn 202 s.OA2

1S= cm360sin.2.2.

2

1AinOB. =

;tfastUcKRkTLapGg

)cm(33

2 eyIgn SSS 213

=

2 .

- 157 -


165/195


10-KNna

k 960sin o

=

)60180sin(240sin)3602240sin(

ooooo

2

360sin

o

x 1215cos oooo 135cos)3603135cos(1215cos =

2

245sin)4590cos(

ooo

K2360sin)360860sin(2940sin o ooo =

2

3

6cos)

62cos(

6

13cos ==X

g 1ta 4

tan)54

tan(4

19n

=

2

3

3sin)63sin(3

17

sin ===

cNna 11-K

)2

3sin()sin()

2 k sinsin (

0cossincossin = .

x 21)1)(1()1)(21(c )3cos(25sin47tan320os 0)cos()2sin()2

2cos()

2sin( =K

- 158 -


166/195


12-eKman tantantantantantan

Edl 2,,2


167/195


tag A

010cos10cos

10cos60cos210cos

)70cos50(cos10cos

70cos50s10cos

oo

ooo

ooo

000co

=

dUcen .0A=

tagB

2

4

3]

2

1110cos2[

2

110cos

]120cos20cos[2

110cos

70cos50cos10cos

o2o2

ooo2

ooo2

=

tag222

C

2

3)20cos20(cos

2

1

2

3

)20cos120cos220(cos21

23

)140cos100cos20(cos2

1

2

3

2

140cos1

2

100cos1

2

20cos1

70cos50cos10cos

o0

000

000

000

o2o2o2

=

=

- 160 -


168/195


14-bgHajfa

k

n .

)Acos1)(Asin1(2)AcosAsin1( 2

tag A=

2)AcosAsin1(

)Acos1)(Asin1(2

Acos)Asin1(2)Asin

Acos)Asin1(2Asin22

AcosAcos)Asin1(2AsinAsin21

AcosAcos)Asin1(2)Asin1(

Acos)Asin1(

22

22

2

1(2

dUce A)cos1)(Asin1(2)AcosAsin1( 2

AcosAsin

1AtanAcot

=

x

eKman AcosAsinAsin AcosAtanAcot

AcosAsin

1

AcosAsin

AsinAcos 22

=

=

AcosAsin

1AtanAcot

=

dUcen .K = 22 sintan)sin)(tansin(tan

tag A )sin)(tansin(tan

- 161 -


169/195


=

2222

2

2

2

tansin1tan1sin

1cos

1sin

1cos

11cos

sin

sincos

sinsin

sinA

.X

1

cos

dUcen

22 sintan)sin)(tansin(tan

cotcossin1sin

11

tag )sin1)(sin

11(A

=

=

=

cotcossin

cos

sin

sin1

sin

sin1)(sin1(

22

cotcossin1sin

11 .

IkarxageRkam

dUcen

15-edaRsaysm

c4 01xcos4x2cosxos3

=

cos40)1xcos2)(1xcos2(

02xcos4xcos2x

01xcos4)1xcos2(xcos4

2

23

23

=

=

=

- 162 -


170/195


0xcos2( = )1xcos2)(1xcos2)(1

eKTaj Zk;k23x3cos2xcos 1

=

Zk;k24

3x

4

3cos

2

2xcos

Zk;k24

x4

cos2

2xcos

=

==

16-keK[

ncMnYnefr nig .eKman

eKTaj

n qAsinp)AsiAcos2()AsinA(cosAcosAsin2 222

cUrrktm p q

AcosAsin21)AsinA(cos 2

( AcosAsin4Asin34)AsinAcos2 22

3Asin3qAsinp 22

dUcen p 3q;3 .xeK[ xsinba)xcot2xtan5(xcosxsin 2cUrrktmncMnYnefr nig b.a

tag A )xcot2xtan5(xcosxsin si xn32xcos2xsin5 222

dUcen a .3b;2 =

- 163 -


171/195


17-edaRsayvismIkarxageRkam

2

1

3

2

cos

2

1

3 edayxcos

k

3

2cos)xcos(

3

eKTaj Zk;k23

2

3xk2

3

2

x Zk;k23

k2

x 1)6

xtan(

)4

(tan)6

xtan(

aj k4

3xk

4

6 eKT

Zk;k12

11xk12 dUcen .

smIkarxageRkam

18-edaRsay

0cos2sin2 =2

2

2

4

232t;2

4

232

t1 =

1t2 2

8162;02t2

)1t1(cost;0cos2cos2

0cos2)cos1(2

2

2

2

2

=

=

=

=

- 164 -


172/195


4 eKn cos

2

2cos

=

dUcen Zk;k24 .19-eK[ manmMu CamMuRsYc .

gHajfaABC 3

kb Ctan.Btan.AtanCtanBtanAtan = xtag . bgHajfa 33TCtanBtanAtanT ?

aydMeNaRskbgHajfa Ctan.Btan.AtanCtanBtanAtan =

eKman ena CBA CBA eKn )Cn()B taAtan(

CtanBtanAtanCtanBtan

)(CtanBtanAan

CtanBtanAtan1

BtanAtan

BtanAtan1t Atan =

dUcen Atan Ctan.Btan.AtanCtanBtan = .faxbgHaj 33T

eKman CtanBtanAtanT eday CamMuRsYcenaC;B;A 0Ctan;0Btan;0Atan >

- 165 -


173/195


- 166 -

tamvismPaB eKman GMAM

33

27T =

27T

T27

T

T3

T

CtanBtanAtan3

CtanBtan

2

3

3

3

dUcen

Atan

33CtanBtanAtanCtanBtanAtanT


174/195

1-maRTIs2-edETmINg


175/195

maRTIs nig edETrmINg CMBUk4 emeronTI1

lMhat

1-Tinn&ykgePsCdanmYybgHajfa enAfsuRkeKlkkaehVnEkv EtRkUcqa 125Ekv nig sura Ekv .160 210

enAfesAreKlkkaehVn 145Ekv EtRkUcqa 130Ekvnigsura Ekv ehIyenAfGaTitlkkaehVn 120Ekv275

EtRkUcqa 155Ekv nig sura Ekv .325 cUrbgHajBtmanenCamaRTIs nig RblMdabnmaRTIsen .2-KNnamaRTIsxageRkam

k x

52

73

01

64

15

26

105

98

03

42113

981

K

28

42

57

31

14

23

X

125

3613

523

132

3-eK[maRTIs

=

513

15D;

103

94C;

97

149

129

B;

83

416

213

A

- 167 -


176/195


KNna

k xBA3

CAB

K D4C5C2

X g)AB(3A D7)C3(2 c )CD(2A 4-KNna

k

56

47

12

132

2412

x 156

5

29

423

5-eK[

c36

20

10

21

0a

7b

02

10

rktm nig c.b,a6-eKdwgfatmsMbuRtcUlTsnaBiBNmYysnwksRmabkUnekg

erol mnuscas erol nigCnbreTs 10000 erol2000 5000 cMnYnsMbuRtEdlnlkenAcugsh_kngeTAenmansegbkg

taragxageRkam

- 168 -


177/195


kUnek

g mnuscas CnbreTsfesAr

205 160 70fGaTit

310 200 65

ktagtmsMbuRtCamaRTIsCYrQrC

xtagBtmankgtaragxagelICamaRTIs N KKNnaplKuN . etIFaturbsmaRTIs tag[GIV?CN CN7-eKmanmaRTIs

=

= m9

412C;

3

10

k

4

1

B;30

24A

kKNna nig2A 1A EdlCamaRTIsRcasn A xrktmk edIm,I[ CamaRTIsktaBA KrktmmedIm,I[edETmINg esInwgedETmINgn .A C

8-eK[maRTIs

=

11

13B;

45

12A . KNna

k nig xBA 1BA 1111 AB,B,A

- 169 -


178/195


KetIGkseg;teXIjyagNacMeBa 1BA nig 11AB

9

-rkmaRTIs Edl X k

=

9

11X

23

25 x

11

132X

11

11

10-rktm nig Edlepgpata;q;p b

k

1

4

q

p

13

310 x

=

b

15

1

a

12

34

K

=

03

51

21

12

1b

a1

11-kgsmIkar

=

32

10

21

32

01M cUrRblMdabnmaRTIsM

nigrkmaRTIs .M

- 170 -


179/195


dMeNaRsay

1-Tinn&ykgePsCdanmYybgHajfa enAfsuRkeKlkkaehVnEkv EtRkUcqa 125Ekv nig sura Ekv .160 210

enAfesAreKlkkaehVn 145Ekv EtRkUcqa 130Ekvnigsura Ekv ehIyenAfGaTitlkkaehVn 120Ekv275

EtRkUcqa 155Ekv nig sura Ekv .325 cUrbgHajBtmanenCamaRTIs nig RblMdabnmaRTIsen .dMeNaRsay

bgHajBtmanenCamaRTIs nig RblMdabnmaRTIsen

tag ACamaRTIsbgHajGMBIBtmanen .eKn CamaRTIskaerlMdab

=

325155120

275130145

210125160

A 33 .

2-KNnamaRTIsxageRkam

k

=

=

16

62

27

5624

7135

0216

52

73

01

64

15

26

- 171 -


180/195


x

105

98

03

42113

981 (KanRbmaNviFIbUk)

K

28

42

57

31

14

23

=

=

43

32

251874

432213

X

125

3613

523

132

=

=

2412

10155

3562153

9118332

3-eK[maRTIs

=

513

15D;

103

94C;

97

149

129

B;

83

416

213

A

KNna

k

=

92479

1412948

126399

97

149

129

83

416

213

3BA3

=

332

2397518

- 172 -


181/195


x (KanRbmaNviFIbUk )CAB

K D4C7D4C)52(D4C5C2

=

=

=

=

5073

678

20705221

4632028

2052

420

7021

6328

513

154

103

947

X A4B3A3B3A)AB(3A

=

=

533

68754813

32271221

16524827

84361427

83

416

213

4

97

149

129

3

g D7C6D7)C3(2

=

=

=

25109

6111

3591

735

6018

5424

513

157

103

946

c ( minGaceFVIRbmaNviFIn ) .)CD(2A

- 173 -


182/195


4-KNna

k

56

47

12

1322412

=

=

=

911

640

51226214

101612122824

)5)(1()4)(3()1)(2()6)(1()7)(3()2)(2(

)5)(2()4)(4()1)(12()6)(2()7)(4()2)(12(

x 156

5

29

423

68155315241087

15)6)(4()5)(2()29)(3(

=

5-eK[

c36

20

10

21

0a

7b

02

10

rktm nig cb,a

c36

20

10

21

0a

7b

02

10

5c,3b,1a

141c3

6b2

1a

1c36

01

14b2

0a=

=

=

=

- 174 -


183/195


6-eKdwgfatmsMbuRtcUlTsnaBiBNmYysnwksRmabkUnekg

erol mnuscas erol nigCnbreTs 10000 erol2000 5000 cMnYnsMbuRtEdlnlkenAcugsh_kngeTAenmansegbkgtaragxageRkam

kUnekg mnuscas CnbreTs

fesAr 205 160 70fGaTit310 200 65

ktagtmsMbuRtCamaRTIsCYrQr C

eyIgn

=

10000

5000

2000

C

xtagBtmankgtaragxagelICamaRTIs N

eyIgn

=

65200310

70160205N

KKNnaplKuN CN

=

10000

5000

2000

65200310

70160205CN

- 175 -


184/195


=

=

=

1712000

1910000CN

650000100000062000

700000800000410000CN

)10000)(65()5000)(200()2000)(310(

)10000)(70()5000)(160()2000)(205(CN

FaturbsmaRTIs tag[RkcMnUlsrubBIkarlksMbuRtCN

sRmabfesArnig fGaTit .7-eKmanmaRTIs

=

=

m9

412C;

3

10

k4

1

B;30

24A

kKNna nig2

A 1

A

EdlCamaRTIsRcasn A eKn

=

=

90

1416

30

24

30

24AAA

2

tamrUbmn

ac

bd

bcad

1A 1

Edl 3d;0c;2b;4a =

ena

=

=

3

10

6

1

4

1

40

23

12

1A 1

- 176 -


185/195


xrktmk edIm,I[ CamaRTIsktaBA

eyIgman

=

=

1032k41

3

10

k4

1

30

24BA

edIm,I[ CamaRTIsktaluRtaEtBA6

1k0

3

2k4 .

KrktmmedIm,I[edETmINg esInwgedETmINgnA C

eKman 1230

24)Adet( =

36m12m9

412)Cdet(

eday 1236m12)Cdet()Adet( =

eKTaj .2m 8-eK[maRTIs

=

11

13B;

45

12A . KNna

k nigBA 1BA

=

111

15

11

13

45

12BA

=

6

5

6

116

1

6

1

511

11

6

1BA

1

- 177 -


186/195


x 1111 AB,B,A

=

=

=

=

=

65

611

6

1

6

1

3

2

3

53

1

3

4

2

3

2

12

1

2

1

AB

2

3

2

12

1

2

1

31

11

2

1B

3

2

3

53

1

3

4

25

14

3

1A

11

1

1

KetIGkseg;teXIjyagNacMeBa 1BA nig 11AB

eyIgeXIjfa 111 ABAB = .9-rkmaRTIs Edl X

k

=

911X

2325

tag

=

2

1

x

xX

- 178 -


187/195


eKn

=

9

11

x

x

23

25

2

1

=

9

11

x2x3

x2x5

21

21

eKTajn

=

9x2x3

11x2x5

21

21 naM[ 3x,1x 21

dUcen .

=

3

1X

x

11

132X

11

11

tag

=

dc

baX

eKn

11

132

dc

ba

11

11

22

26

dbca

dbca (minGac)

dUcenKanmaRTIs NabMeBjlkxNeneT .X10-rktm nig Edlepgpata;q;p b

k

1

4

q

p

13

310

1

4

qp3

q3p10

- 179 -


188/195


eKTaj 2q,1p1qp3

4q3p10

= .

x

=

b15

1a

1234

=

b

15

1a2

3a4

eKTaj 7b;3ab1a2

153a4=

=

=

K

=

03

51

21

12

1b

a1

=

03

51

2b1b2

a21a2

eKTaj

=

=

=

=

02b

31b2

5a21

1a2

( RbB&nK

ancemIy ) .

11-kgsmIkar

=

32

10

21

32

01M cUrRblMdabnmaRTIsM

nigrkmaRTIs .M

- 180 -


189/195


- 181 -

dMeNaRsay

lMdabmaRTIs MkgsmIkar

=

32

10

21

32

01M bBaakfamaRTIs Mman

lMdab .32

tag

=

fe

dc

ba

M

eKman

=

f3f2e

d3d2c

b3b2a

32

01

fe

dc

ba

eKTaj 32

b2b3 nig 3

1

a1b2a =

3

1d1d3 nig

3

2c0d2c

1f3f3 = nig 0e2f2e =

dUcen

=

1031

32

3

2

3

1

M .


190/195


lMhat

1-KNnaedETrmINg

k734

123

432

x322

112

131

K 144153

321

2-KNnaedETrmINgedayeRbIkarBnattammIn&r

k263

014

211

x250

134

111

K325

140113

3-brimaRtctuekaNEkgmYymanRbEvg . eKdwgfaRbEvgTTwgcm86 dg elIsRbEvgbeNay . rkvimaRtnctuekaNeday2 cm2

eRbIkaredaRsaytamviFIRkam&r .4-eKlkEpBuRTa kg50 kg kg1 mantm3500erol . buEnedayBuRTaenmanEpFMnigEptUcenaEptUceKlkEt erolkg3000 kg1

- 182 -


191/195


nigEpFMlkf5000erolkg .kg1

etIEpBuRTaFMmanbun

anKITLRkam nig tUcbun

anKITLRkam ?5-eKdwgfaplbUknbIcMnYnesInwg 20cMnYnTI1esInwgplbUkncMnYnTI nig TI esInwg dgcMnYnTI1. rkcMnYnnImYy@ ?2 3 3

6-brimaRtRtIekaNmYymanRbEvg . eKdwgfaRCugEdlEvgCagm18

eKes

Inwg2

dgRCugEdlxICageK . RCugmYyeTotCamFmnBVnnRCugEdlEvgCageK nig RCugxICageK .rkRbEvgRCugnImYy@nRtIekaN ?

7-edaRsayRbB&nsmIkar

k

=

=

6y3

1x

2

1

0yx

4

3

x

=

2yx2

6yx

K X

=

y615x3

5y2x

=

=

=

8z2yx

15zy2x4

14z2y3x2

g c

=

=

5z2y7x3

10zyx4

10z3y2x

=

=

1z6y2x4

4z3y7x

2z3yx2

- 183 -


192/195


dMeNaRsay

1-KNnaedETrmINg

k34

234

74

133

73

122

734

123

432

=

3747534

)89(4)421(3)314(2

x22

12

32

123

32

11

322

112

131

=

312245 =

K 4453

314

132

14

15

144

153

321

=

11196141 =

2-KNnaedETrmINgedayeRbIkarBnattammIn&r

k 6314

223

04

26

01

263

014

211

524282 =

- 184 -


193/195


x50

34

20

14

25

13

250

134

111

10381 =

K25

40

35

10

32

143

325

140

113

5520530 =

3-brimaRtctuekaNEkgmYymanRbEvg . eKdwgfaRbEvgTTwgcm86 dg elIsRbEvgbeNay . rkvimaRtnctuekaNeday2 cm2 eRbIkaredaRsaytamviFIRkam&r .

tagx

CabeNay nig CaTTwg ( KitCa )y cm

tambMrabeKnRbB&n

=

2xy2

86y2x2

bnabBIedaRsayeKn cm15y,cm28x = .4-eKlkEpBuRTa kg50 kg kg1 mantm3500erol . buEneday

BuRTaenmanEpFMnigEptUcenaEptUceKlkEt erolkg3000 kg1 nigEpFMlkf5000erolkg kg1 .etIEpBuRTaFMmanbunanKITLRkam nig tUcbunanKITLRkam ?

- 185 -


194/195


tag xnig erogKaCaTmnEpBuRTatUc nig BuRTaFM(KitCa )y kg

tamsmtikmeKnRbB&n

=

350050y5000x3000

50yx

bnabBIedaRsayeKn kg5.12y;kg5.37x =

5-eKdwgfaplbUknbIcMnYnesInwg 20cMnYnTI1esInwgplbUkncMnYnTI nig TI ehIycMnYnTI esInwg dgcMnYnTI1.2 3 3 3

rkcMnYnnImYy@ ?tag erogKaCacMnYn TI1TI2TIz;y;x 3

tambMrabeKnRbB&n

=

=

x3z

zyx

20zyx

bnabBIedaRsayeKTTYn 30z,20y;10x = .6-brimaRtRtIekaNmYymanRbEvg . eKdwgfaRCugEdlEvgCagm18 eKesInwg 2dgRCugEdlxICageK . RCugmYyeTotCamFmnBVnnRCugEdlEvgCageK nig RCugxICageK .

rkRbEvgRCugnImYy@nRtIekaN ?tag aCaRCugxIbMput bCaRCugEvgbMput nig CaRCugTIbI.c tambMrab )1(18cba =

- 186 -


195/195


nig)2(a2b= )3(2

bac

=

eKnRbB&n

=

=

=

0c2ba

0ba218cba

bnabBIedaRsayeKTTYln m6c,m8b,m4a = .7-edaRsayRbB&nsmIkar

0yx3

Documents

Maht Fundation Grade 11.pdf