03. statistics

8/12/2019 03. Statistics

1/24

kflnsdoukfrpdysnes.nl

Nkpirtflt Xinlts

Rdm wirb bftf kmfls nlgirkftnil (nts mxfet bnetnilfry

kmflnlc ns: cnvml gfets). Utftnstnefa bftf frm ig twi

typms :

(n) Xrnkfry bftf

(nn) Umeilbfry bftf

Sdml fl nlvmstncftir eiaamets bftf dnksmag wntd f

bmgnlntm pafl ir bmsncl nl dns (dmr) knlb, nt ns efaamb

Xrnkfry bftf.

Bftf wdned frm lit irncnlfaay eiaametmb rftdmr i`tfnlmb

grik puànsdmb ir ulpuànsdmb siurems frm oliwl fs

Umeilbfry bftf.

Fgtmr eiaametnil ig bftf, tdm nlvmstncftir dfs ti gnlb

wfys ti eilbmlsm tdml nl tfùafr girk nl irbmr ti

stuby tdmnr snamlt gmfturms. Uued fl frrflcmkmlt ns

efaambXrmsmltftnil ig bftf.

Pfw bftf wdml put nl fsemlbnlc ir bmsemlbnlc irbmr

ig kfclntubm ns efaamb fl frrfy ir frrflcmb bftf.

Rdm luk`mr ig tnkms fl i`smrvftnil ieeurs nl tdm

cnvml bftf ns efaamb grmqumley ig tdm i`smrvftnil.

Eafssms/eafss nltmrvfas frm tdm criups nl wdned faa tdm

i`smrvftnils frm bnvnbmb.

Uuppism eafss-nltmrvfa ns 68-08, tdml 68 ns efaamb aiwmr

anknt flb 08 ns efaamb uppmr anknt ig tdm eafss

Knb-vfaum ig eafss-nltmrvfa ns efaamb Eafss-kfro

Eafss-kfro ;0

ntankuppmrntankaiwmr

Eafss-kfro ; aiwmr anknt +0

6

(bnggmrmlem `mtwmml tdm uppmr flb aiwmr anknts)

Ng tdm grmqumley ig gnrst eafss nltmrvfa ns fbbmb ti tdm

grmqumley ig smeilb eafss flb tdns suk ns fbbmb ti

tdnrb eafss flb si il tdml grmqumlenms si i`tfnlmb

frm oliwl fsEukuaftnvm Grmqumley (e.g.).

Rdmrm frm twi typms ig eukuaftnvm grmqumlenms

(f) amss tdfl, (`) crmftmr tdfl

MTFKXAMU

Mx.6 Cnvml `maiw frm tdm fcms ig 0= stubmlts ig eafss

NT nl f sediia. Xrmpfrm f bnsermtm grmqumley

bnstrnùtnil.

6=, 67, 67, 62, 63, 63, 67, 6=, 6=, 67, 67, 63, 6=, 67,

67, 62, 67, 6=, 62, 6=, 67, 67, 6=, 62, 6=.

Uia. Grmqumley bnstrn utnil ig fcms ig 0= stubmlts

Fcm Rfaay kfros Grmqumley

62 2

6= 467 68

63 5

Ritfa 0=

Mx.0 Girk f bnsermtm grmqumley bnstrnùtnil grik tdm

giaaiwnlc seirms:-

Uia. 6=, 64, 67, 08, 0=, 02, 0=, 08, 67, 6=, 64, 64, 67, 02,

6=, 08, 04, 58, 03, 67, 02, 0=, 08, 64, 04, 03, 0=, 02,

02, 64, 64, 0=, 08, 67, 6=, 08, 03, 04, 0>, 67.

5EDFXRMP

UR RNURN U


2/24


Grmqumley Bnstrnùtnil ig Ueirms

Wfrnftm Rfaay kfros Grmqumley

6= 2

67 7

64 7

08 7

02 =

0= =

03 5

04 5

0> | 6

58 | 6

Ritfa 28

Mx.5 Rdm wftmr tfx `naas (nl rupmms) ig 58 diusms nl f

aiefanty frm cnvml `maiw. Eilstruet f criupmb

grmqumley bnstrnùtnil wntd eafss snzm ig 68.

58, 50, 2=, =2, 32, 34, 684, 660, 77, 37, 44, 28, 62,

08, 6=, 5=, 22, 77, 3=, 42, >=, >7, 680, 668, 44, 32,

660, 62, 52, 22.

Uia. Dmrm tdm kfxnkuk flb knlnkuk vfaums ig tdm

vfrnftm frm 660 flb 62 rmspmetnvmay.

Pflcm ; 660 62 ; >4.

Nt ns cnvml tdft tdm eafss snzm ns 68, flb

snzmEafss

Pflcm;

68

>4; >.4

Ui, wm sdiuam dfvm 68 eafssms mfed ig snzm 68.

Rdm knlnkuk flb kfxnkuk vfaums ig tdm vfrnftm

frm 62 flb 660 rmspmetnvmay. Ui wm dfvm ti kfom

tdm eafssms nl sued f wfy tdft gnrst eafss nleaubms

tdm knlnkuk vfaum flb tdm afst eafss nleaubms

tdm kfxnkuk vfaum. Ng wm tfom tdm gnrst eafss fs

62-02 nt nleaubms tdm knlnkuk vfaum 62. Ng tdm afst

eafss ns tfoml fs 682-662, tdml nt nleaubms tdm

kfxnkuk vfaum 660. Dmrm, wm girk eafssms `y

mxeausnvm kmtdib. Nl tdm eafss 62-02, 62 ns

nleaubmb ùt 02 ns mxeaubmb. Unknafray, nl itdmr

eafssms, tdm aiwmr anknt ns nleaubmb flb tdm uppmr

anknt ns mxeaubmb.

Nl tdm vnmw ig fìvm bnseussnil, wm eilstruet

tdm grmqumley bnstrnùtnil tfàm fs giaaiws:

@naa (nl rupmms) Rfaay kfros Grmqumley

62-02 2

02-52 0

52-22 5

22-=2 5

=2-72 | 6

72-32 0

32-42 =

42->2 5

>2-682 5

682-662 2

Ritfa 58

Mx.2 Rdm kfros i tfnlmb `y 28 stubmlts ig eafss NT

nl fl mxfknlftnil frm cnvml `maiw :

64, 4, 60, 7, 4, 67, 60, =, 05, 0,67, 05, 0, 68, 08, 60,

>, 3, 7, =, 5, =, 65, 06, 65, 6=, 08, 02, 6, 3, 06, 67,

65, 64, 05, 3, 5, 64, 63, 67.

Xrmsmlt tdm bftf nl tdm girk ig f grmqumley

bnstrnùtnil usnlc tdm sfkm eafss snzm, ilm sued

eafss `mnlc 6=-08 (wdmrm 08 ns lit nleaubmb)

Uia. Rdm knlnkuk flb kfxnkuk kfros nl tdm cnvmlrfw bftf frm 8 flb 02 rmspmetnvmay. Nt ns cnvml

tdft 6=-08 ns ilm ig tdm eafss nltmrvfas flb tdm

eafss snzm ns sfkm. Ui, tdm eafssms ig mqufa snzm

frm

8-=, =-68, 68-6=, 6=-08 flb 08-0=

Rdus, tdm grmqumley bnstrnùtnil ns fs cnvml

ulbmr :

Grmqumley Bnstrnùtnil ig Kfros

Kfros Rfaay kfros Grmqumley

8-= 7

=-68 68

68-6= 4

6=-08 4

08-0= 4

Ritfa 28


3/24


Mx.= Rdm eafss kfros ig f bnstrnùtnil frm :

23, =0, =3, 70, 73, 30, 33, 40, 43, >0, >3, 680 Bmtmrknlm

tdm eafss snzm, tdm eafss anknts flb tdm trum eafss

anknts.

Uia. Dmrm tdm eafss kfros frm ulngirkay spfemb. Ui,

tdm eafss snzm ns tdm bnggmrmlem `mtwmml fly twi

eilsmeutnvm eafss kfros

Eafss snzm ; =0 23 ; =

Sm oliw tdft, ng f ns tdm eafss kfro ig f eafss

nltmrvfa flb d ns nts eafss snzm, tdml tdm aiwmr flb

uppmr anknts ig tdm eafss nltmrvfa frm f 0

d

flb f +0

drmspmetnvmay..

Aiwmr anknt ig gnrst eafss nltmrvfa

; 23 0

=; 22.=

Flb, uppmr anknt ig gnrst eafss nltmrvfa

; 23 +0

=; 2>.=

Ui, gnrst eafss nltmrvfa ns 22.= 2>.=

Unknafray, wm i`tfnl tdm itdmr eafss anknts fs cnvml

ulbmr :

Eafss kfros Eafss anknts

23 22.=-2>.=

=0 2>.=-=2.=

=3 =2.=-=>.=

70 =>.=-72.=

73 72.=-7>.=

30 7>.=-32.=

33 32.=-3>.=

40 3>.=-42.=

43 42.=-4>.=

>0 4>.=->2.=

>3 >2.=->>.=

680 >>.=-682.=

Unlem tdm eafssms frm mxeausnvm, si tdm trum eafss

anknts frm sfkm fs tdm eafss anknts.

Mx.7 Rdm eafss kfros ig f bnstrnùtnil frm 07, 56, 57,

26, 27, =6, =7, 76, 77, 36. Gnlb tdm trum eafss anknts.

Uia. Dmrm tdm eafss kfros frm ulngirkay spfemb. Ui,

tdm eafss snzm ns tdm bnggmrmlem `mtwmml fly twi

eilsmeutnvm eafss kfros.

Eafss snzm ; 56 07 ; =.

Ng f ns tdm eafss kfro ig f eafss nltmrvfa ig snzm

d, tdml tdm aiwmr flb uppmr anknts ig tdm eafss

nltmrvfa frm f 0

dflb f +

0

drmspmetnvmay..

Dmrm d ; =

Aiwmr anknt ig gnrst eafss nltmrvfa

; 07 0

=; 05.=

Flb, uppmr anknt ig gnrst eafss nltmrvfa

; 07 +0

=; 04.=

Gnrst eafss nltmrvfa ns 05.= 04.=.

Rdus, tdm eafss nltmrvfas frm:

05.= 04.=, 04.= 55.=, 55.=54.=, 54.=25.=,

25.=24.=, 24.==5.=

Unlem tdm eafsss frm girkmb `y mxeausnvm kmtdib.

Rdmrmgirm, tdmsm anknts frm trum eafss anknts.

Eukuaftnvm Grmqumley

F t f à m w dn e d b n sp a fy s t dm k fl l mr n l w dn e d

eukuaftnvm grmqumlenms frm bnstrnùtmb ivmr vfrnius

eafssms ns efaamb f eukuaftnvm grmqumley bnstrnùtnil

ir eukuaftnvm grmqumley tfàm.

Rdmrm frm twi typms ig eukuaftnvm grmqumley.

(6) Amss tdfl typm (0) Crmftmr tdfl typm

MTFKXAMU

Mx.3 Srntm biwl amss tdfl typm eukuaftnvm grmqumleyflb crmftmr tdfl typm eukuaftnvm grmqumley.

Grmqumley

628 62= 68

62= 6=8 60

6=8 6== 64

6== 678 =

678 67= =

67= 638 54

638 63= 00

63= 648 08

Dmncdt (nl ek)


4/24


Uia. Sm dfvm

Dmncdt

(nl ek)62862=

62=6=8

6=86==

6==678

67867=

67=638

63863=

63=648

Grmqumley 6 8 60 64 5= 2= 54 00 08

Dmncdt

Amss tdfl

typm

6 2= 6 =8 6 == 6 78 6 7= 6 38 6 3= 6 48

Eukuaftnvm

grmqumley68 00 28 3= 608 6=4 648 088

Dmncdt

Crmftmr

tdfl typm

6 28 6 2= 6 =8 6 == 6 78 6 7= 6 38 6 3=

Eukuaftnvm

grmqumley 088 6>8 634 678 60= 48 20 08

Mx.4 Rdm bnstflems (nl ok) eivmrmb `y 02 efrs nl 0

diurs frm cnvml `maiw :

60=, 628, 604, 684, >7, 62>, 657, 660, 42, 605, 658,

608, 685, 4>, 7=, 685, 62=, >3, 680, 43, 73, 34, >4,

607

Pmprmsmlt tdmk fs f eukuaftnvm grmqumley tfàm

usnlc 78 fs tdm aiwmr anknt ig tdm gnrst criup flb

faa tdm eafssms dfvnlc tdm eafss snzm ig 6=.

Uia. Sm dfvm, Eafss snzm ; 6=

Kfxnkuk bnstflem eivmrmb ; 62> ok.Knlnkuk bnstflem eivmrmb ; 7= ok.

Pflcm ; (62> 7=) ok ; 42 ok.

Ui, luk`mr ig eafssms ; 7

7.=

6=

42

Rdus, tdm eafss nltmrvfas frm 78-3=, 3=->8,

>8-68=, 68=-608, 608-65=, 65=-=8.

Rdm eukuaftnvm grmqumley bnstrnùtnil ns fs cnvml

`maiw :

Eafss Rfaay G rm qumle y Eukuaftnvm

nltmrvfa kfros grmqumley

78-3= 0 0

3=->8 2 7

>8-68= 7 60

68=-608 0 62

608-65= 7 08

65=-6=8 2 02

Mx.> Rdm giaaiwnlc tfàm cnvms tdm kfros seirmb `y

534 stubmlts nl fl mltrflem mxfknlftnil :

Kfros Li. ig stubmlts

8-68 5

68-08 60

08-58 57

58-28 37

28-=8 >3

=8-78 4=

78-38 5>

38-48 60

48->8 60

>8-688 7

Grik tdns tfàm girk (n) tdm amss tdfl smrnms, flb

(nn) tdm kirm tdfl smrnms.

Uia.(n) Amss tdfl eukuaftnvm grmqumley tfàm

Kfros i tfnlmb Luk`mr ig stubmlts

(Eukuaftnvm grmqumley)

Amss tdfl 68 5

Amss tdfl 08 6=

Amss tdfl 58 =6

Amss tdfl 28 603

Amss tdfl =8 002

Amss tdfl 78 58>

Amss tdfl 38 524

Amss tdfl 48 578

Amss tdfl >8 530

Amss tdfl 688 534

(nn) Kirm tdfl eukuaftnvm grmqumley tfàm

Kfros i`tfnlmb Luk`mr ig stubmlts

(Eukuaftnvm grmqumley)

Kirm tdfl 8 534Kirm tdfl > 53=

Kirm tdfl 6> 575

Kirm tdfl 0> 503

Kirm tdfl 5> 0=3

Kirm tdfl 2> 6=2

Kirm tdfl => 7>

Kirm tdfl 7> 58

Kirm tdfl 3> 64

Kirm tdfl 4> 7


5/24


Mx.68 Gnlb tdm uloliwl mltrnms (f,`,e,b,m,g,c) grik tdm

giaaiwnlc grmqumley bnstrnùtnil ig dmncdts ig =8

stubmlts nl f eafss :

Eafss nltmrvfas Grmqumley Eukuaftnvm

(Dmncdts nl ek) grmqumley

6=8-6== 60 f

6==-678 ` 0=

678-67= 68 e

67=-638 b 25

638-63= m 24

63=-648 0 g

Ritfa c

Uia. Unlem tdm cnvml grmqumley bnstrn utnil ns tdm

grmqumley bnstrnùtnil ig dmncdts ig =8 stubmlts.

Rdmrmgirm,

c ; =8.

Grik tdm tfàm, wm dfvm

f ; 60, ` + 60 ; 0=, 60 + ` + 68 ; e,

60 + ` + 68 + b ; 25,

60 + ` + 68 + b + m ; 24 flb

60 + ` + 68 + b + m + 0 ; gLiw, ` + 60 ; 0= ` ; 65

60 + ` + 68 ; e

60 + 65 + 68 ; e _ ` ; 65[

e ; 5=

60 + ` + 68 + b ; 25

60 + 65 + 68 + b ; 25 _ ` ; 65[

b ; 4

60 + ` + 68 + b + m ; 24

60 + 65 + 68 + 4 + m ; 24

_ ` ; 65, b ; 4[

m ; =

flb, 60 + ` + 68 + b + m + 0 ; g

60 + 65 + 68 + 4 + = + 0 ; g

g ; =8.

Dmlem, f ; 60, ` ; 65, e ; 5=, b ; 4,

m ; =, g ; =8 flb c ; =8.

Mx.66 Rdm kfros iut ig 68 i`tfnlmb `y 50 stubmlts

frm : 0, 2, 5, 6, =, 2, 5, 4, >, 3, 4, =, 2, 5, 7, 3, 2,

3, >, 4, 7, 2, 0, 6, 8, 8, 0, 7, 3, 4, 7, 6.

F rr f y t dm b ft f f lb g ir k t dm g rm qu ml ey

bnstrnùtnil

Uia. Fl frrfy ig tdm cnvml bftf ns prmpfrmb `y

frrflcnlc tdm seirms nl fsemlbnlc irbmr fs

giaaiws :

8, 8, 6, 6, 6, 0, 0, 0, 5,5,5, 2,2,2,2,2,

=,=, 7,7,7,7, 3,3,3,3, 4,4,4,4, >,>.

Grmqumley bnstrnùtnil ig tdm kfros ns sdiwl

`maiw.

Kfros Rfaay kfros Grmqumley

8 || 0

6 ||| 5

0 ||| 5

5 ||| 5

2 =

= || 0

7 |||| 2

3 |||| 2

4 |||| 2

> || 0

Mx.60 Xrmpfrm f bnsermtm grmqumley bnstrnùtnil grik

tdm bftf cnvml `maiw, sdiwnlc tdm wmncdts nl oc

ig 58 stubmlts ig eafss WN.

5>, 54, 20, 26, 5>, 54, 5>, 20, 26, 5>, 54, 54 26, 28,

26, 20, 26, 5>, 28, 54, 20, 25, 2=, 25, 5>, 54, 26, 28,

20, 5>.

Uia. Rdm bnsermtm grmqumley bnstrnùtnil tfàm gir tdm

wmncdt (nl oc) ig 58 stubmlts ns sdiwl `maiw.

Smncdts (nl oc) Rfaay kfros Grmqumley

54 | 7

5> || 3

28 ||| 5

26 | 7

20 =

25 || 0

2= | 6


6/24


Mx.65 Rdm eafss kfros ig f bnstrnùtnil frm 40, 44, >2,

688, 687, 660 flb 664. Bmtmrknlm tdm eafss snzm

flb tdm eafssms.

Uia. Rdm eafss snzm ns tdm bnggmrmlem `mtwmml twi

eilsmeutnvm eafss kfros. Eafss snzm ; 44 40

; 7. Liw 40 ns tdm eafss kfro ig tdm gnrst eafss

wdism wnbtd ns 7.Eafss anknts ig tdm gnrst eafss

frm 40 0

7flb 40 +

0

7n.m. 3> flb 4=. Rdus, tdm

gnrst eafss ns 3>-4=. Unknafray, tdm itdmr eafssms frm

4=>6, >6 >3, >3685, 685 68>, 68> 66= flb

66= 606.

Mx.62 Rdm eafss kfros ig f bnstrnùtnil frm 65, 63, 06,

0= flb 0>. Gnlb tdm trum eafss anknts.

Uia. Rdm eafss kfros frm 65, 63, 06, 0= flb 0>.

Rdm eafss kfros frm ulngirkay spfemb.

Eafss snzm ; bnggmrmlem `mtwmml twi eilsmeutnvm

eafss kfros

; 63 65 ; 2

Dfag ig tdm eafss snzm ;0

2; 0

Ri gnlb tdm eafssms ilm dfs ti su`trfet 0 grik

flb fbb 0 ti mfed ig tdm eafss kfros.

Dmlem, tdm eafssms frm

66 6=

6= 6>

6> 05

05 03

03 56

Unlem tdm eafssms frm mxeausnvm, tdm trum eafss

anknts frm tdm sfkm fs tdm eafss anknts. Ui tdm

aiwmr eafss anknts fs wmaa fs tdm trum aiwmr eafss

anknts frm 66, 6=, 6>, 05 flb 03. Rdm uppmr eafss

anknts fs wmaa fs tdm trum uppmr eafss anknts frm 6=,

6>, 05, 03 flb 56.

Mx.6= Eilvmrt tdm cnvml snkpam grmqumley smrnms nlti f :

(n) Amss tdfl eukuaftnvm grmqumley smrnms.

(nn) Kirm tdfl eukuaftnvm grmqumley smrnms.


8-68 5

68-08 3

08-58 60

58-28 4

28-=8 =

Uia.(n) Amss tdfl eukuaftnvm grmqumley smrnms


Amss tdfl 68 5

Amss tdfl 08 68 ( ; 5 + 3 )

Amss tdfl 58 00 ( ; 5 + 3 + 60 )

Amss tdfl 28 58 ( ; 5 + 3 + 60 + 4 )Amss tdfl =8 5= ( ; 5 + 3 + 60 + 4 + = )

(nn) Kirm tdfl eukuaftnvm grmqumley smrnms


Kirm tdfl =8 8

Kirm tdfl 28 =

Kirm tdfl 58 65

Kirm tdfl 08 0=

Kirm tdfl 68 50

Kirm tdfl 8 5=

Mx.67 Eilvmrt tdm giaaiwnlc kirm tdfl eukuaftnvm

grmqumley smrnms nlti snkpam grmqumley smrnms.


Kirm tdfl 8 28

Kirm tdfl 68 57

Kirm tdfl 08 0>

Kirm tdfl 58 67

Kirm tdfl 28 =

Kirm tdfl =8 8

Uia. Unkpam grmqumley bnstrnùtnil tfàm

Kfros Grmqumley

8-68 2 ( ; 28 57)

68-08 3 ( ; 57 0>)08-58 65 ( ; 0> 67 )

58-28 66 ( ; 67 = )

28-=8 = ( ; = 8 )

Mx.63 F tmst wfs eilbuetmb nl f eafss ig 58 stubmlts

wntd kfxnkuk kfros 0=. Rdm kfros i`tfnlmb frm

cnvml `maiw. Eilstruet f grmqumley bnstrnùtnil

tfàm wntd f eafss snzm ig =, 04, 03, 08, 06, 63, 64,

64, 68, 60, 4, 3, 6=, 68, 66, 0=, 6>, 06, 63, 60, 64,

2, 4, 60, 63, >, =, 3, 60, 64, 03

Uia.Kfros Rfaay Luk`mr ig

Utubmlts

8 = N 6

= 68 =

68 6= 4

6= 08 >

08 0= NNN 5

0= 58 NNNN 2

Ritfa ; 58


7/24


Mx.64 Rdm kfros i`tfnlmb `y 5= stubmlts nl f eafss frm

cnvml `maiw. Eilstruet tdm eukuaftnvm grmqumley

tfàm :

Kfros Luk mr ig

i`tfnlmb stubmlts

8 6

6 0

0 2

5 2

2 5

= =

7 2

3 7

4 5

> 0

68 6

Uia.

Kfros Grmqumley Luk`mr ig stubmlts

8 6 6

6 0 5 (;6 + 0)

0 2 3 (;6 + 0 + 2)

5 2 66 (;6 + 0 + 2 + 2)

2 5 62 (;6 + 0 + 2 + 2 + 5)

= = 6> (;6 + 0 + 2 + 2 + 5 + =)

7 2 05 (;6 + 0 + 2 + 2 + 5 + = + 2)

3 7 0> (;6 + 0 + 2 + 2 + 5 + = + 2 + 7)

4 5 50 (;6 + 0 + 2 + 2 + 5 + = + 2 + 7 + 5)

> 0 52 (;6 + 0 + 2 + 2 + 5 + = + 2 + 7 + 5 + 0)68 6 5= (;6 + 0 + 2 + 2 + 5 + = + 2 + 7 + 5 + 0 + 6)

Ritfa ; 5=

Mx.6> Rdm bnstrnùtnil ig fcms (nl ymfrs) ig 28 pmrsils

nl f eiaily ns cnvml `maiw.

Fcm Luk`mr ig

(nl ymfrs) Xmrsils

08-0= 3

0=-58 68

58-5= 4

5=-28 7

28-2= 2

2=-=8 =

(f) Bmtmrknlm tdm eafss kfro ig mfed eafss

(`) Sdft ns tdm uppmr eafss anknt ig 2td eafss

(e) Bmtmrknlm tdm eafss snzm

Uia.(f) Eafss kfros frm

0

0=08 ,

0

580= ,

0

5=58 ,

0

285= ,

0

2=28 ,

0

=82=

; 00.=, 03.=, 50.=, 53.=, 20.=, 23.=

( ) Rdm giurtd eafss nltmrvfa ns 5=28. Nts

uppmr anknt ns 28

(e) Rdm eafss snzm ns 0= 08 ; =

Mx.08 Giaaiwnlc ns tdm bnstrn utnil ig kfros ig 28

stubmlts nl f eafss. Eilstruet f eukuaftnvm

grmqumley bnstrnùtnil tfàm.

Kfros Lu k mr i g s tub ml ts

8-68 5

68-08 4

08-58 >

58-28 6=

28-=8 =

Uia.

Eafss Grmqumley Eukuaftnvm

nltmrvfa Grmqumley868 5 5

6808 4 66 ( ; 5 + 4 )

0858 > 08 ( ; 5 + 4 + >)

5828 6= 5= ( ; 5 + 4 + > + 6=)

28=8 = 28 ( ; 5 + 4 + > + 6= + =)

Ritfa ; 28

Mx.06 Rdm eafss kfros ig f bnstrnùtnil frm 0=, 5=, 2=,

==, 7= flb 3=.

Bmtmrknlm tdm eafss snzm flb eafss anknt.

Uia. Eafss snzm ; Rdm bnggmrmlem `mtwmml tdm eafss

kfros ig twi fbjfemlt eafssms.

; 5= 0= ; 68

Sm lmmb eafssms ig snzm 68 wntd eafss kfros fs

0=, 5=, 2=, ==, 7=, 3=

R dm e af ss a nkn ts g ir t dm g nr st e af ss f rm

0= 0

68flb 0= +

0

68

n.m. 08 flb 58

Gnrst eafss ns, tdmrmgirm, 0858

Unknafray, tdm itdmr eafssms frm 58 28, 28 =8,

=8 78, 78 38, 38 48

Mx.00 Cnvml `maiw ns tdm eukuaftnvm grmqumleybnstrnùtnil tfàm sdiwnlc tdm kfros smeurmb `y

28 stubmlts.

Kfros Luk`mr ig

stubmlts

@maiw 08 =

@maiw 28 68

@maiw 78 0=

@maiw 48 50

@maiw 688 28


8/24


9/24


Amt T `m tdm kmfl ig tdm vfaums 5, 2, 7, 4, 62. Rdml,

T ;=

5=

=

624725

; 3

Liw, suk ig tdm bmvnftnils ig tdm vfaums 5, 2, 7,

4, 62 grik tdmnr kmfl T ; 3 ns cnvml `y; (5 3) + (2 3) + (7 3) + (4 3)

+ (62 3) ; 2 5 6 + 6+ 3 ; 8.

Mx.07 Rdm kmfl ig 28 i`smrvftnils wfs 678. Nt wfs

bmtmetmb il rmedmeonlc tdft tdm vfaum ig 67= wfs

wrilcay eipnmb fs 60= gir eikputftnil ig kmfl.

Gnlb tdm eirrmet kmfl.

Uia. Dmrm, l ; 28, T ; 678

Ui, T ; nxl6

678 ; nx28

6

nx 28 ; 7288

Nleirrmet vfaum ig nx ; 7288Liw,

Eirrmet vfaum ig nx; Nleirrmet vfaum ig nx Nleirrmet ntmk

+ Eirrmet ntmk

Eirrmet vfaum ig nx ; 7288 60=+ 67= ; 7228

Eirrmetkmfl

;28

7228

l

xigvfaumEirrmet n ; 676.

Mx.03 Rdm kmfl ig 68 luk`mrs ns 08. Ng = ns su`trfetmb

grik mvmry luk`mr, wdft wnaa `m tdm lmw kmfl 9

Uia. Amt x6, x0,...,x68 `m 68 luk`mrs wntd tdmnr kmfl

mqufa ti 08. Rdml,

nxl

6T

08 ;68

x...xx 6806

x6+ x0 +...+ x68 ; 088 ....(n)

Lmw luk`mrs frm x6 =, x0 =,..., x68 =. Amt

'T `m tdm kmfl ig lmw luk`mrs.

Rdml,

68

)=x(...)=x()=x('T 6806

68

=8088

68

68=)x...xx('T 6806

_]snlc (n)[

'T ; 6=.

Mx.04 Rdm kmfl ig 67 luk`mrs ns 4. Ng 0 ns fbbmb ti

mvmry luk`mr, wdft wnaa `m tdm lmw kmfl 9

Uia. Amt x6,x0,x5,...,x67`m 67 luk`mrs wntd tdmnr kmfl

mqufa ti 4. Rdml,

nxl6T

4 ;67

x...xx 6706

x6+ x0 +...+ x67 ; 67 4 ; 604 ...(n) Lmw

luk`mrs frm x6+ 0, x0+ 0, x5 + 0,..., x67+ 0. Amt 'T

`m tdm kmfl ig lmw luk`mrs. Rdml,

67

)0x(...)0x()0x('T 6706

67

50604

67

670)x...xx(

'T

6706

_]snlc(n)[

6867

678'T

Mx.0> Ng x6, x0,...,xl frm l vfaums ig f vfrnf`am T sued

tdft

l

6n

n )0x( ; 668 flb

l

6n

n )=x( ; 08. Gnlb tdm

vfaum ig l flb tdm kmfl.

Uia. Sm dfvm,

l

6n

n )0x( ; 668 flb

l

6n

n )=x( ; 08

(x6 0) + (x0 0) +...+ (xl 0) ; 668

flb (x6 =) + (x0 =) +...+ (xl =) ; 08

(x6 + x0 +...+ xl) 0l ; 668 flb

(x6 + x0 +...+ xl) =l ; 08

l0x

l

6nn

; 668 flb

l

6nn

08l=x

U 0l ; 668 flb U =l ; 08

Rdus, wm dfvm

U 0l ; 668 ....(n)

flb U =l ; 08 ....(nn)

Uu`trfetnlc (nn) grik (n), wm cmt

5l ; >8 l ; 58


10/24


Xuttnlc l ; 58 nl (n), wm cmt

U 78 ; 668 U ; 638

l

6n

n 638x

Kmfl ;5

63

58

638x

l

6 l

6n

n

Dmlem, l ; 58 flb kmfl5

63.

Frntdkmtne Kmfl ig ]lcriupmb BftfFrntdkmtne kmfl ig rfw bftf (wdml grmqumley ns

lit cnvml) :Rdm frntdkmtne kmfl ig f rfw bftf ns i`tfnlmb `y

fbbnlc faa tdm vfaums ig tdm vfrnfàms flb bnvnbnlc

tdm suk `y titfa luk`mr ig vfaums tdft frm fbbmb.

Frntdkmtne kmfl

( x ) ;

l

x...xx l06 ;

l

6

l

6n

nx

Rdm sykìa

l

6n

nx bmlitms tdm suk x6+ x0+ ... + xl.

MTFKXAMUMx.58 Lmmtf flb dmr giur grnmlbs smeurmb 7=, 34, 40, >2

flb 36 kfros nl f tmst ig kftdmkftnes. Gnlb tdm

fvmrfcm (frntdkmtne kmfl) ig tdmnr kfros.

Uia. Frntdkmtne kmfl ir fvmrfcm

;=

36>240347= ;

=

5>8; 34

Dmlem, frntdkmtne kmfl ; 34

Mx.56 Rdm kfros i`tfnlmb `y 68 stubmlts nl pdysnes

iut ig 28 frm 02, 03, 0>, 52, 50, 6>, 07, 5=, 64, 06.

Eikputm tdm kmfl ig tdm kfros.

Uia. Kmfl ig tdm kfros ns cnvml `y

68

06645=076>50520>0302x

;68

07=; 07.=8

Mx.50 Rdm kmfl ig 08 i`smrvftnils wfs giulb ti `m

2 3. @ ut a ft mr n t w fs b ns ei vm rm b t df t i lm

i`smrvftnil 77 wfs wrilcay tfoml fs 47. Gnlb

tdm eirrmet kmfl.

Uia. Dmrm, l ; 08, x

; 23

Sm dfvm, x ;l

x

l

6n

n 23 ;

08

x

l

6n

n

l

6n

nx ; 23 08 ; >28.

@ut tdm seirm 77 wfs wrilcay tfoml fs 47.

Eirrmet vfaum ig

l

6n

nx ; >28 + 77 47 ; >08

Eirrmet kmfl ;08

>08 ; 27

Mx.55 F efr iwlmr ùys pmtria ft Ps. 08.88, Ps. 02.88

flb Ps. 0=.88 pmr antrm gir tdrmm sueemssnvm ymfrs.

Eikputm tdm fvmrfcm eist pmr antrm ig pmtria wdml

dm spmlbs Ps. 60888 il pmtria mfed ymfr.

Uia. Dmrm, tdm fkiults ig pmtria puredfsmb nl tdrmm

sueemssnvm ymfrs frm8808

60888

.antrms,

8802

60888

.antrms

flb880=

60888

.antrms rmspmetnvmay n.m. 788 antrms, =88

antrms flb 248 antrms rmspmetnvmay.

Ritfa fkiult ig kilmy spmlt il pmtria nl

5 ymfrs

; Ps 60888 5 ; Ps 57888

Fvmrfcm eist pmr antrm ig pmtria

;puredfsmbpmtriaigfkiultRitfa

pmtriailspmltkilmyigfkiultRitfa

;antrms248=88788

57888Ps

)( ;

antrms6=48

57888Ps

; Ps 00.34 pmr antrm (fpprix.)

Mx.52 Gnlb tdm kmfl ig tdm giaaiwnlc luk`mrs : 60, 62,

63, 0=, 68, 66, 08, 4, 6=, flb 64.

Uia. Kmfl ;68

646=40866680=636260

;68

6=8; 6=

Mx.5= Rdm kmfl ig =, 3, p, 66, 6=, 63, flb 08 ns 60,

gnlb p.

Uia. Kmfl ;3

08636=66p3=

60 ;3

p3=

p + 3= ; 60 3 p ; >


11/24


Mx.57 Ng x bmlitm tdm kmfl ig x6, x0, ..., xl, sdiw tdft

l

6n

n xx )(

Uia. l

xxxx l06

...

; x6 + x0 +... + xl ; lx (n)

;(x6 x ) ; (x6 x ) + (x0 x ) + ...

... + (xl x6)

; (x6 + x0+ ... + xl) lx

; lx lx

; 8 (grik (n))

Mx.53 Ng tdm kmfl ig = i`smrvftnils ns 6= flb tdft ig

flitdmr 68 i`smrvftnils ns 08, gnlb tdm kmfl ig

faa 6= i`smrvftnils

Uia. Amt gnrst gnvm i`smrvftnils `m x6,...,x=

Kmfl ;=

xxx =06 ...

6= ;=

xxx =06 ...

x6 + ... + x= ; 3= (n)

Amt lmxt tml i`smrvftnils `m y6 + . . . + y68.

Kmfl ;68

yy 686 ...

08 ;68

yy 686 ...

y6 + ... + y68 ; 088 (nn)

Rdm kmfl ig faa 6= i`smrvftnils wnaa `m

6=

yyxx 686=6 )...()...(

;6=

0883= (grik (n) flb (nn))

; 64.55

N g f v fr nf tm T t fo ms v fa um s x6,x 0...xl wntd

eirrmspilbnlc grmqumlenms g6,g0,g5,...,glrmspmetnvmay,

tdml frntdkmtne kmfl ig tdmsm vfaums ns

T ;l06

ll0066

g...gg

xg...xgxg

ir T ;L

xg

l

6n

nn

, wdmrm L ;

l

6n

ng ; g6+g0+...+ gl

Mx.54 Gnlb tdm kmfl ig tdm giaaiwnlc bnstrnùtnil :

x : 2 7 > 68 6=

g : = 68 68 3 4

Uia. Efaeuaftnil ig Frntdkmtne Kmfl

xn

gn

gnx

n2 = 08

7 68 78

> 68 >8

68 3 38

6= 4 608

28gL n 578xg nn

Kmfl ;28

578

g

xgT

n

nn

; >.

Mx.5> Gnlb tdm kmfl ig tdm giaaiwnlc bnstrnùtnil :

x : 68 58 =8 38 4>

g : 3 4 68 6= 68

Uia. Efaeuaftnil ig Kmfl

xn gn gnxn

68 3 38

58 4 028

=8 68 =88

38 6= 68=8

4> 68 4>8

gn; L ; =8 gnxn ; 03=8

Kmfl ;=8

03=8

L

xg nn

; ==.

Mx.28 Gnlb tdm vfaum ig p, ng tdm kmfl ig giaaiwnlc

bnstrnùtnil ns 3.=.

x : 5 = 3 > 66 65

y : 7 4 6= p 4 2

Uia. Efaeuaftnil ig Kmfl

xn gn gnxn

5 7 64

= 4 28

3 6= 68=

> p >p

66 4 44

65 2 =0

L ; Ugn; 26 + p Ug nxn; 585 + >p

Sm dfvm, gn ; 26 + p, gnxn ; 585 + >p


12/24


Kmfl ;n

nn

g

xg

3.= ;p26

p>585

3.= (26+ p) ; 585 + >p

583.=+3.=p ; 585 + >p

>p 3.=p ; 583.= 585

6.=p ; 2.= p ; 5

Mx.26 Gnlb tdm knssnlc grmqumlenms nl tdm giaaiwnlc

grmqumley bnstrnùtnil ng nt ns oliwl tdft tdm

kmfl ig tdm bnstrnùtnil ns 6.27.

Luk`mr i g

feenbmlts (x) : 8 6 0 5 2 = Ritfa

Grmqumley (g) : 27 9 9 0= 68 = 088

Uia. Amt tdm knssnlc grmqumlenms `m gt flb g0

Efaeuaftnil ig Kmfl

xn gn gnxn

8 27 8

6 g6 g6

0 g0 3=

5 0= 0g 0

2 68 28

= = 0=

L ; 47 + g6 + g0 gnxn ; 628 + g6 + 0g0

Sm dfvm : L ; 088

088 ; 47 + g6 + g0 g6 + g0 ; 662 ....(n)

Fasi,

Kmfl ; 6.27

6.27 ;L

xg nn

6.27 ;088

g0g628 00

0>0 ; 628 + g6 + 0g0

g6 + 0g0 ; 6=0 ....(nn)

Uiavnlc (n) flb (nn) wm cmt g6 ; 37 flb g0 ; 54.

Mx.20 Ng tdm kmfl ig tdm giaaiwnlc bftf `m >.0, gnlb tdm

vfaum ig p.

x 2 7 3 p + 2 60 60

g = 7 2 68 4 3

Uia. Rdm tfàm ns rmwrnttml fs `maiw :

x g g.x

2 = 08

7 7 57

3 2 04

p+2 68 68p + 28

60 4 >7

62 3 >4

Ritfa 28 564 + 68p

Liw, Kmflg

xgx

.;

28

p68564 .

>.0 ;28

p68564 .

564 + 68.p ; 574

68p ; =8 p ; =

Mx.25 Rdm kfros ig 58 stubmlts frm cnvml `maiw, gnlb

tdm kmfl kfros.

Kfros Luk`mr ig Utubmlts

68 2

66 5

60 4

65 7

62 3

6= 0

Uia.

x g gx

68 2 28

66 5 55

60 4 >7

65 7 34

62 3 >46= 0 58

g ; 58 gx; 53=

Kmfl ;g

gx

;

58

53=; 60.=


13/24


Criupmb Grmqumley BnstrnùtnilRdmrm frm 5 kmtdibs gir efaeuaftnil ig kmfl :

6. Bnrmet Kmtdib

0. Fssukmb kmfl bmvnftnil kmtdib

5. Utmp bmvnftnil kmtdib.

Bnrmet Kmtdib gir Efaeuaftnil ig Kmfl

knb -v fau m grm qu ml ey g nxn

x6 g6 g6x6

x0 g0 g0x0

: : :. . .

xo go goxo

Ritfa L

o

6n

nn xg

Feeirbnlc ti bnrmet kmtdib

x ;

o06

oo0066

g...gg

gx...gxgx

;

n

o

6n

nn

g

gx

;L

6 o

6

nn xg

_L ; g6 + g0 + ... + go[

MTFKXAMUMx.22 Efaeuaftm tdm kmfl gir tdm giaaiwnlc bnstrnùtnil :

Wfrnfàm = 7 3 4 >

Grmqumley 2 4 62 66 5

Uia.

x g gx

= 2 08

7 4 24

3 62 >4

4 66 44

> 5 03

Ritfa L ;g ; 28 g x ; 046

Kmfl ;g

xg

;

28

046; 3.80=

Mx.2=

Knb-vfaums 0 5 2 = 7

Grmqumlenms 2> 25 =3 54 65

Knb vfaums ig eafss nltmrvfa frm cnvml wntd tdmnr

grmqumlenms. Gnlb tdm kmfl `y bnrmet kmtdib.

Uia.

Knb-vfaums Grmqumlenms (gn) gnxn

0 2> >4

5 25 60>2 =3 004

= 54 6>8

7 65 34

Ritfa L ;gn; 088 gnxn; 305

@y bnrmet kmtdib.

Kmfl ;n

nn

g

xg

;

088

305; 5.76=

Mx.27 Gnlb tdm kmfl ig tdm giaaiwnlc grmqumley

bnstrnùtnil :

Eafss Nltmrvfa Grmqumley

68 58 >8

58 =8 08

=8 38 58

38 >8 08

>8 668 28

Uia.

Eafssnltmrvfa

g Knb vfaum (x) g x

68 58 >8 08 6488

58 =8 08 28 488

=8 38 58 78 6488

38 >8 08 48 6788

>8 668 28 688 2888

g ; 088 g x ; 68888

Kmfl ( x ) ;

g

xg

;

088

68888; =8

Mx.23 Gnlb tdm kmfl ig tdm giaaiwnlc grmqumleybnstrnùtnil :


6= 0= 78

0= 5= 5=

5= 2= 00

2= == 64

== 7= 6=


14/24


Uia.

Eafss

nltmrvfa

g Knb vfaum (x) g x

6= 0= 78 08 6088

0= 5= 5= 58 68=8

5= 2= 00 28 4482= == 64 =8 >88

== 7= 6= 78 >88

g ; 6=8 g x ; 2>58

Kmfl ( x ) ;

g

xg

;

6=8

2>58; 50.4 7ir 50.43 (fpprix.)

Mx.24 F survmy wfs eilbuetmb `y f criup ig stubmlts

f s f p f rt i g t dm n r m lv n ri lk ml t f wf rml mss

pricrfkkm, nl wdned tdmy eiaametmb tdm giaaiwnlc

bftf rmcfrbnlc tdm luk`mr ig paflts nl 08 diusmsnl f aiefanty. Gnlb tdm kmfl luk`mr ig paflts pmr

diusm.

Luk`mr

ig paflts

8 0 0 2 2 7 7 4 4 68 6 8 6 0 6 0 6 2

Li. ig

diusms

6 0 6 = 7 0 5

Sdned kmtdib bnb yiu usm gir gnlbnlc tdm kmfl

flb wdy 9

Uia. Luk`mr i g

paflts

Luk`mr i g

diusms (g)

Knb vfaum x g x

8 0 6 6 6

0 2 0 5 7

2 7 6 = =

7 4 = 3 5=

4 68 7 > =2

68 600

66 00

60 62 5 65 5>

g ; 08 g x; 670

Kmfl ;g

xg

;

08

670; 4.6

Dmrm, wm dfvm usmb bnrmet kmtdib `mefusm

lukmrnefa vfaums ig x flb g frm skfaa.

Mx.2> Gnlb tdm kmfl ig tdm giaaiwnlc bnstrnùtnil `y

bnrmet kmtdib.

Eafss

nltmrvfa

8 68 66 08 06 58 56 28 26 =8

Grmqumley 5 2 0 = 7

Uia.Eafss

nltmrvfa

Grmqumley

g

Knb vfaum

x

g x

8 68 5 =.8 6=.8

66 08 2 6=.= 70.8

06 58 0 0=.= =6.856 28 = 5=.= 633.=

26 =8 7 2=.= 035.8

g ; 08 g x ;=34.=

Kmfl ;g

xg

;

08

=.=34; 04.>

Mx.=8 Gir tdm giaaiwnlc bnstrnùtnil, efaeuaftm kmfl

usnlc faa tdm suntfàm kmtdibs.

Unzm ig Ntmk 6 2 2 > > 67 67 03

Grmqumley 7 60 07 08

Uia.Unzm ig ntmk Knb vfaum

(xn)

Grmqumle

y (gn)

gnxn

6 2 0.= 7 6=

2 > 7.= 60 34

> 67 60.= 07 50=

67 03 06.= 08 258

gn; 72 gnxn; 424

Kmfl ;n

nn

g

xg

;

72

424; 65.0=

Fssukmb Kmfl Kmtdib

Frntdkmtne kmfl ; f +

l

6n

n

l

6n

nn

g

bg

Litm : Rdm fssukmb kmfl ns edisml, nl sued f

kfllmr, tdft

6. Nt sdiuab `m ilm ig tdm emltrfa vfaums.

0. Rdm bmvnftnil frm skfaa.

5. Ilm bmvnftnil ns zmri.

Sironlc Puam :

Utmp 6: Ediism f luk`mr 'f' grik tdm emltrfa

vfaums ig x ig tdm gnrst eiaukl, tdft wnaa `m

iur fssukmb kmfl.

Utmp 0: I`tfnl bmvnftnils bn`y su`trfetnlc 'f' grikxn. Srntm biwl dmsm bmvnftnils fcfnlst tdm

eirrmspilbnlc grmqumlenms nl tdm tdnrb

eiaukl.

Utmp 5: Kua t np a y t d m grmq um l en m s i g sme i lb

eiaukl wntd eirrmspilbnlc bmvnftnils bnnl tdm tdnrb eiaukl ti prmpfrm f giurtd

eiaukl ig gnbn.

Utmp 2: Gnlb tdm suk ig faa tdm mltrnms ig giurtd

eiaukl ti i`tfnlgnbn flb fasi, gnlb tdmsuk ig faa tdm grmqumlenms nl tdm smeilb

eiaukl ti i`tfnl gn

.


15/24


MTFKXAMUMx.=6 Rdm giaaiwnlc tfàm cnvms tdm bnstrnùtnil ig titfa

diusmdiab mxpmlbnturm (nl rupmms) ig kflufa

wiromrs nl f enty.

Mxpmlbnturm

(nl rupmms)688-6=8

6=8-088

088-0=8

0=8-588

588-5=8

5=8-288

288-2=8

2=8-=88

Grmqumley 02 28 55 04 58 00 67 3

Uia. Amt fssukmb kmfl ; 03=

Mxpmlbnturm

(nl rupmms)

Grmqumley

(gn)

Knb

vfaum (xn)

bn;x03= gnbn

6886=8 02 60= 6=8 5788

6=8088 28 63= 688 2888

0880=8 55 00= =8 67=8

0=8 588 04 03= 8 8

588 5=8 58 50= =8 6=88

5=8 288 00 53= 688 0088

288 2=8 67 20= 6=8 0288

2=8 =88 3 23= 088 6288

gn; 088 gnbn;

63=8

n

nn

g

bgfx

; 03= +

088

63=8; Ps 077.0=

Mx.=0 Efaeuaftm tdm frntdkmtne kmfl ig tdm giaaiwnlc

bnstrnùtnil :


8 =8 63

=8 688 5=688 6=8 25

6=8088 28

088 0=8 06

0=8 588 02

Uia. Amt fssukmb kmfl ; 63= n.m. f ; 63=

Eafss Knb vfaum

(xn)

bn;xn63= grmqumle

y gn

gnbn

8=8 0= 6=8 63 0==8

=8688 3= 688 5= 5=88

6886=8 60= =8 25 06=8

6=8088 63= 8 28 8088-0=8 00= =8 06 68=8

0=8588 03= 688 02 0288

gn; 648 gnbn; 23=8

Liw , f ; 63=

n

nn

g

bgfx

; 63= +

648

23=8; 63= 07.5>

; 624.76 fpprix.

Mx.=5 Efaeuaftm tdm frntdkmtne kmfl ig tdm giaaiwnlc

grmqumley bnstrnùtnil :

Eafss

nltmrvfa=8 78 7838 3848 48>8 >8 688

Grmqumley 4 7 60 66 65

Uia. Amt fssukmb kmfl ; 3= n.m., f ; 3=

Eafss grmqumley

gn

Knb vfaum

(xn) bn;xn3= gnbn

=878 4 == 08 678

7838 7 7= 68 78

3848 60 3= 8 8

48>8 66 4= 68 668

>8688 65 >= 08 078

g ; =8 gnbn

; 6=8

f ; 3=,gnbn; 6=8,gn ; =8

Kmfl (x ) ; f +n

nn

g

bg

; 3= +

=8

6=8; 34

Mx.=2 Rdnrty wikml wmrm mxfknlmb nl f dispntfa `y f

bietir flb tdm luk`mr ig dmfrt `mfts pmr knlutm

wmrm rmeirbmb flb sukkfrnsmb fs giaaiws. Gnlb

tdm kmfl dmfrt `mfts pmr knlutm gir tdmsm wikml,

ediisnlc f suntfàm kmtdib.

Luk`mr ig dmfrt

`mft s pmr knlut m

Grmqumley

7= 74 0

7436 2

3632 5

3233 4

33 48 3

48 45 2

45 47 0

Uia. Amt fssukmb kmfl f ; 3=.=

Li. ig

dmfrt `mfts

Li. ig

wikml g

Knb vfaum

(x)b ; xf gb

7=74 0 77.= > 64

7436 2 7>.= 7 02

3632 5 30.= 5 >

3233 4 3=.= 8 8

3348 3 34.= 5 06

4845 2 46.= 7 02

4547 0 42.= > 64

g ; 58 gb ;60

Kmfl ;g

gbf

; 3=.= +

58

60; 3=.= + 8.2 ; 3=.>


16/24


Utmp Bmvnftnil Kmtdib

Bmvnftnil kmtdib efl `m gurtdmr snkpangnmb il

bnvnbnlc tdm bmvnftnil `y wnbtd ig tdm eafss nltmrvfa

d. Nl sued f efsm tdm frntdkmtne kmfl ns rmbuemb

ti f crmft mxtmlt.

Kmfl (x ) ; f + dg

ug

n

nn

Sironlc Puam :

Utmp-6: Ediism f luk`mr 'f' grik tdm emltrfa

vfaums ig x(knb-vfaums)

Utmp-0: I`tfnl un ; d

fx n

Utmp-5: Ku at n pa y t d m grmq u ml ey g n w nt d t d m

eirrmspilbnlc un ti cmt gnun.

Utmp-2: Gnlb tdm suk ig faa gnunn.m.,gnun

Utmp-=: ]sm tdm girkuaf x ; f +n

nn

g

ug

.d ti cmt

tdm rmqunrmb kmfl.

MTFKXAMUMx.== Ri gnlb iut tdm eilemltrftnil ig UI0 nl tdm fnr

(nl pfrts pmr knaanil, n.m.ppk), tdm bftf wfs

eiaametmb gir 58 aiefantnms nl f emrtfnl enty flb ns

prmsmltmb `maiw :

Grmqumley

8.88 8.82 2

8.82 8.84 >

8.84 8.60 >

8.60 8.67 0

8.67 8.08 2

8.08 8.02 0

Eilemltrftnil ig

UI0(nl ppk)

Gnlb tdm kmfl eilemltrftnil ig UI0 nl tdm fnr.

Uia. Amt tdm fssukmb kmfl f ; 8.68.

Eilemltrftnil

ig UI0

(nl ppk)

Grmqumley

gn

Knb vfaum

xngnun

8.88 8.82 2 8.80 0 4

8.82 8.84 > 8.87 6 >

8.84 8.60 > 8.68 8 8

8.60 8.67 0 8.62 6 0

8.67 8.08 2 8.64 0 4

8.08 8.02 0 8.00 5 7

gn; 58 gnun; 6

82.8

68.8xu nn

@y stmp bmvnftnil kmtdib

Kmfl ; f +n

nn

g

ug

d ; 8.68 + 82.8

58

6

; 8.68 8.8865; 8.8>43 ; 8.8>> ppk

Mx.=7 Rdm wmmoay i`smrvftnil il eist ig anvnlc nlbmxnl f emrtfnl enty gir tdm ymfr 0882088= frm cnvml`maiw. Eikputm tdm kmfl wmmoay eist ig anvnlcnlbmx.

Eist ig Anvnlc nlbmx Luk mr ig wmmos

6288-6=88 =

6=88-6788 68

6788-6388 08

6388-6488 >

6488-6>88 7

6>88-0888 0

Uia. Amt fssukmb kmfl `m 63=8 n.m., f ; 63=8

Eist ig

anvnlc nlbmx

Grmqumley

gn

Knb vfaum

xngnun

62886=88 = 62=8 5 6=

6=886788 68 6==8 0 08

67886388 08 67=8 6 08

63886488 > 63=8 8 8

64886>88 7 64=8 6 7

6>880888 0 6>=8 0 2

gn; =0 gnun

; 2=

688

63=8xu nn

@y stmp bmvnftnil kmtdib

Kmfl (x ) ; f +n

nn

g

ug

d

; 63=8 + 688=0

2=

; 63=8 47.=2 ; 6775.27Dmlem, tdm kmfl wmmoay eist ig anvnlc nlbmx

; 6775.27Mx.=3 Gnlb tdm kmfl kfros grik tdm giaaiwnlc bftf `y

stmp bmvnftnil kmtdib

Luk`mr ig st ubmlts

@maiw 68 =

@maiw 08 >

@maiw 58 63

@maiw 28 0>

@maiw =8 2=

@maiw 78 78

@maiw 38 38

@maiw 48 34

@maiw >8 45

@maiw 688 4=

Kfros


17/24


Uia. Amt fssukmb kmfl ; == f ; ==

Eafss

nltmrvfa

Grmqumley

gn

Knb vfaum

xngnun

8 68 = = = 0=

68 08 2 6= 2 6708 58 4 0= 5 02

58 28 60 5= 0 02

28 =8 67 2= 6 67

=8 78 6= == 8 8

78 38 68 7= 6 68

38 48 4 3= 0 67

48 >8 = 4= 5 6=

>8 688 0 >= 2 4

gn; 4= gnun; =7

68

==xu nn

Dmrm, f ; ==, d ; 68,gn ; 4=, gnun; =7

Kmfl (x ) ; f +n

nn

g

ug

d ; == + 684=

=7

; == 7.=> ; 24.26

Dmlem, kmfl kfro ; 24.26.

Mx.=4 Gnlb tdm kmfl fcm ig 688 rmsnbmlts ig f eiaily

grik tdm giaawnlc bftf :

Luk`mr ig pmrs ils

Crmftmr tdfl 8 688

Crmftmr tdfl 68 >8

Crmftmr tdfl 08 3=

Crmftmr tdfl 58 =8

Crmftmr tdfl 28 0=

Crmftmr tdfl =8 6=

Crmftmr tdfl 78 =

Crmftmr tdfl 38 8

Fcm nl ymfrs

Uia. Amt fssukmb kmfl f ; 5=

Fc m

(nl ymfrs)

Luk`mr ig

pmrsils

Knb vfaum

xn gnun

8 68 68 = 5 58

68 08 6= 6= 0 58

08 58 0= 0= 6 0=

58 28 0= 5= 8 8

28 =8 68 2= 6 68

=8 78 68 == 0 08

78 38 = 7= 5 6=

Ritfa gn; 688 gnun

; 28

68

5=x

u n

n

Dmrm, f ; 5=, d ; 68

x ; f + dg

ug

n

nn

x ; 5= + 5668

688

28

Dmlem, tdm kmfl fcm ; 56 ymfrs :

Mx.=> Rdm giaaiwnlc bnstrn`utnil sdiw tdm bfnay pieomt

faaiwflem ig ednabrml ig f aiefanty. Rdm kmfl

pieomt faaiwflem ns Ps. 64.88. Gnlb tdm knssnlc

grmqumley g.

Eafss-

Nltmrvfa6665 656= 6=63 636> 6>06 0605 050=

Grmqumley 3 7 > 65 g = 2

Uia. wm dfvm,

Eafss-nltmrvfa

Grmqumley Knb vfaumx

g x

66 65 3 60 42

65 6= 7 62 42

6= 63 > 67 622

63 6> 65 64 052

6> 06 g 08 08g

06 05 = 00 668

05 6= 2 02 >7

g ; 22+g g x

;3=0+08g

Kmfl x ;g

gx

64 ; g22 g083=0

64 (22 + g) ; 3=0 + 08g

3=0 + 08g ; 3>0 + 64g

0g ; 28 g ; 08

Dmlem, tdm knssnlc grmqumley ns 08.

Mx.78 Rdm frntdkmtne kmfl ig tdm giaaiwnlc grmqumley

bnstrn`utnil ns =8. Gnlb tdm vfaum ig p.

Eafss-

Nltmrvfa808 0828 2878 7848 48688

Grmqumley 63 X 50 02 6>

Uia.

Eafss-

nltmrvfa

Grmqumley

(g)

Knb vfaum

(x) g x

8 08 63 68 638

08 28 X 58 58 X

28 78 50 =8 6788

78 48 02 38 6748

48 688 6> >8 6368

g ; >0 + X g x ;

=678 + 58X


18/24


19/24


Dmrm, tdm luk`mr ig i`smrvftnils l ; > (ibb)

Kmbnfl ; Wfaum igtd

0

6>

i`smrvftnil

; Wfaum ig =td i`smrvftnil ; 5>.

Mx.72 Rdm kmbnfl ig tdm i`smrvftnils 66, 60, 62, 64,

x + 0, x + 2, 58, 50, 5=, 26 frrflcmb nl fsemlbnlc

irbmr ns 02. Gnlb tdm vfaum ig x.

Uia. Dmrm, tdm luk`mr ig i`smrvftnils l ; 68. Unlem

l ns mvml, tdmrmgirm

Kmbnfl ;0

li`smrvftni60

lli`smrvftni

0

l tdtd

02 ;0

li`smrvftni7li`smrvftni= tdtd

02 ;

0

)2x()0x(

02 ;0

7x0

02 ; x + 5 x ; 06.

Dmlem, x ; 06.

Mx.7= Gnlb tdm kmbnfl ig tdm giaaiwnlc bftf : 6>, 0=, =>,

24, 5=, 56, 58, 50, =6. Ng 0= ns rmpafemb `y =0, wdft

wnaa `m tdm lmw kmbnfl.

Uia. Frrflcnlc tdm cnvml bftf nl fsemlbnlc irbmr, wm

dfvm 6>, 0=, 58, 56, 50, 5=, 24, =6, =>

Dmrm, tdm luk`mr ig i`smrvftnils l ; > (ibb)

Unlem tdm luk`mr ig i`smrvftnils ns ibb.

Rdmrmgirm.

Kmbnfl ; Wfaum ig

0

6>tdm i`smrvftnils

Kmbnfl ; vfaum ig =td i`smrvftnil ; 50.

Dmlem, Kmbnfl ; 50

Ng 0= ns rmpafemb `y =0, tdml tdm lmw i`smrvftnils

frrflcmb nl fsemlbnlc irbmr frm :

6>, 58, 56, 50, 5=, 24, =6, =0, =>

Lmw kmbnfl ; Wfaum ig =td i`smrvftnil ; 5=.

Mx.77 Gnlb tdm kmbnfl ig tdm giaaiwnlc bftf

(n) 63, 03, 53, 65, 64, 0=, 50, 52, 05

(nn) 02, 53, 6>, 26, 04, 50, 0>, 56, 55, 06

Uia.(n) Rd m se irm s w dm l f rrf lc m b n l f sem lb n lc

irbmr frm

65, 63, 64, 05, 0=, 03, 50, 52, 53

Dmrm, tdm luk`mr ig seirms l ; > (ibb)

Kmbnfl ;0

6>t ; t= ; 0=

(nn) Rdm seirms wdml frrfcmb nl fsemlbnlc

irbmr frm

6>, 06, 02, 04, 0>, 56, 55, 52, 53, 26.

Ritfa luk`mr ig seirms ; 68, wdned ns mvml. Ui

tdmrm wnaa `m twi knbbam-tmrks wdned frm t= ; 0>flb t7 ; 56.

Kmbnfl ;0

tt 7=;

0

560> ; 58

Mx.73 E fa eu af tm t dm k mb nf l g ir t dm g ia ai wn lc

bnstrn`utnil

Smncdt (nl oc) Luk mrig submlt

27 5

23 0

24 2

2> 7

=8 =

=6 0

=0 6

Uia. Rdm eukuaftnvm grmqumley tf`am ns eilstruetmb

fs sdiwl `maiw :

Smncdts Luk`mr ig Eukuaftnvm

xn stubmlts grmqumley

gn

27 5 5

23 0 =

24 2 >

2> 7 6=

=8 = 08

=6 0 00

=0 6 05

Dmrm, l ; 05, wdned ns ibb

Kmbnfl ;0

605t ; t60

; 2>

(n.m. wmncdt ig tdm 60td stubmlt wdml tdm wmncdtsdfvm `mml frrflcmb nl irbmr)

Mx.74 Gnlb tdm kmbnfl ig tdm giaaiwnlc bftf :

(n) 4, 68, =, 3, 60, 6=, 66

(nn) 60, 62, 68, 3, 6=, 67

Uia.(n) 4, 68, =, 3, 60, 6=, 66

Rdmsm luk`mrs frm frrflcmb nl fl irbmr

=, 3, 4, 68, 66, 60, 6=

Rdm luk`mr ig i`smrvftnils ; 3 (ibb)


20/24


21/24


Smncdts Li. ig kml Eukuaftnvm Xribuet

x g grmqumley g.x

=2 7 7 502

=7 2 68 002

=4 = 6= 0>8

78 = 08 588

70 7 07 530

75 = 56 56=

72 0 55 604

30 7 5> 250

48 6 28 48

Ritfa 28 027=

Kmfl ; g

xg

.; 28

027=

; 76.70=

Dmrm, Li. ig seirms ; 28 (mvml)

Kmbnfl ;0

tt 0608;

0

7078 ; 76

Kibm ; 5 kmbnfl 0 kmfl

; 5 76 0 76.70=

; 645 605.0= ; =>.3=

Rdus, kibfa wmncdt ; =>.3= oc

Mx.32 Ng tdm dmncdts ig = pmrsils frm 622 ek, 6=5 ek, 6=8ek, 6=4 ek flb 6== ek rmspmetnvmay, tdml kmfl

dmncdt ns -

(F) 6=8 ek (@) 6=6 ek

(E) 6=0 ek (B) Lilm ig tdmsm

Uia. Kmfl Dmncdt ;=

6==6=46=86=5622

;=

378; 6=0 ek.

Mx.3= Frntdkmtne kmfl ig tdm giaaiwnlc grmqumley

bnstrn`utnil :

x : 2 3 68 65 67 6>

g : 3 68 6= 08 0= 58

ns -

(F) 65.7 (@) 65.4

(E) 62.8 (B) Lilm ig tdmsm

Uia. Rdm cnvml grmqumley bnstrn`utnil ns -

xn gn gnxn

2 3 04

3 68 38

68 6= 6=8

65 08 078

67 0= 288

6> 58 =38

ng ; 683

nn xg ; 6234

x ;

n

nn

g

xg;

683

6234; 65.46

Mx.37 Rdm kmfl nleikm ig f criup ig pmrsils ns

Ps.288. Flitdmr criup ig pmrsils dfs kmfl

nleikm Ps.248. Ng tdm kmfl nleikm ig faa tdm

pmrsils nl tdm twi criups ticmtdmr ns Ps.258,

tdml rftni ig tdm luk`mr ig pmrsils nl tdm criups:

(F)5

2(@)

2

=

(E)5

=(B) Lilm ig tdmsm

Uia. x ;06

0066

llxlxl

6x ; 288, 0x ; 248, x ; 258

258 ;06

06

ll

)248(l)288(l

58l6 ; =8l0

0

6

l

l;

5

=

Mx.33 Rdm kmfl ig f smt ig luk`mr ns x ng mfed luk`mr

ns nlermfsmb `y , tdml kmfl ig tdm lmw smt ns-(F) x (@) x +

(E) x (B) Lilm ig tdmsm

Uia. x ;l

x.......xx l06 ;

l

x n

nx ; lx

Lmw kmfl ;l

)x( n ;

l

lx n

; x +


22/24


Mx.34 Rdm luk`mr ig ruls seirmb `y 66 pafymrs ig f

erneomt tmfk ig sediia frm =, 6>, 20, 66, =8, 58, 06,

8, =0, 57, 03. Rdm kmbnfl ns-

(F) 06 (@) 03

(E) 58 (B) Lilm ig tdmsm

Uia. Amt us frrflcm tdm vfaum nl fsemlbnlc irbmr

8, =, 66, 6>, 06, 03, 58, 57, 20, =8, =0

Kmbnfl K ;

td

0

6l

vfaum

;

td

0

666

vfaum ; 7td vfaum

Liw 7td vfaum nl bftf ns 03

Kmbnfl ; 03 ruls.

Mx.3> Kibm ig tdm bftf 5, 0, =, 0, 5, =, 7, 7, =, 5, =, 0,

= ns -

(F) 7 (@) 2 (E) = (B) 5

Uia. Unlem = ns rmpmftmb kfxnkuk luk`mr ig tnkms,

tdmrmgirm kibm ig tdm cnvml bftf ns =.

Mx.48 Ng tdm vfaum ig kibm flb kmfl ns 78 flb 77

rmspmetnvmay, tdml tdm vfaum ig kmbnfl ns-

(F) 78 (@) 72

(E) 74 (B) Lilm ig tdmsm

Uia. Kibm ; 5 Kmbnfl 0 kmfl

Kmbnfl ;5

6(kibm + 0 kmfl)

;5

6(78 + 0 77) ; 72

Mx.46 Kmfl ig 0= i`smrvftnils wfs giulb ti `m 34.2.

@ut aftmr il nt wfs giulb tdft >7 wfs knsrmfb 7>.

Rdm eirrmet kmfl ns

(F) 3>.02

(@) 3>.24

(E) 48.68

(B) Lilm ig tdmsm

Uia. Kmfl x ;l

x

ir x; lx

x; 0= 34.2 ; 6>78

@ut tdns xns nleirrmet fs >7 wfs knsrmfbfs 7>.

eirrmet x ; 6>78 + (>7 7>) ; 6>43

eirrmet kmfl ;0=

6>43; 3>.23

Mx.40 Ng x ns tdm kmfl ig x6, x0,....,xl tdml kmfl ig

x6 + f, x0 + f, .....,xl+ f wdmrm f ns fly luk`mr

pisntnvm ir lmcftnvm ns -

(F) x + f (@) x

(E) fx (B) Lilm ig tdmsm

Uia. Sm dfvm x ;

l

x....xx l06

Amt 'x `m tdm kmfl ig x6 + f, x0 + f,...., xl + f

tdml

'x ;l

)ex(....)fx()fx( l06

;l

lf)....x....xx( l06

;l

x....xx l06 + f ; x + f


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