MathElmi11P2 Book

7/23/2019 MathElmi11P2 Book

1/144

UL bU UIM`U bL qL' bL WOU bL

wbI qBO rU rOJ( b Wb wM

uHR*

U??????O{UdwU e?'

uU _ n?BKwUMB wLKF

5DK WwUF rOKF WOd


2/144

5DK W w wUF rOKF WOd d wb UF s b UNb w uU _ nBK UO{Ud U fb

UF d?

hL( u rOF Z?UM* W?M' fOz

5U ZUM* ed U db

ZUM* ed

hL( u dL ud d

WOMH dzb

Ud bz d?UL UL G rOLB WUDK Vu; b

UL ou r??????OL?B

UL UL WIM* WFD rOLB qbF

u UNL bOCM

rK dL 5I 5% u?G dd%

UO{Ud UNM* wMu odHU??H UMN jO{ UO UIMbF 5D

d??U UUO WH?OK wK bL qOK wK

p? ??A qz q??I bL bL bL

d od

bF u qOL U d

lO vNM `U bL UIM tU fOwUF qOL dU$ bF

WIM* WOd WFD

? ZUM* edwUF rOKF WOd u WuH l?D uI lOL

WMb* ed WN s 5LO vK U bUF* UuOB* w ZUM* ed fU uHK 5DK tK

[email protected] uMF www.pcdc.edu.ps WOdJ WHB


3/144

r?bI

UuL oOI WOMODKH WOuB) wd UNM l{ d{ wUF rOKF WOd

WOMu UO UM ULNUU bF wMODK UNM UM uFA 5 tUJ cQ v wMODKH VFAWb* WdA uLK WOLM wU ou WOdI1b rOI aOd UU wMODKH VFAKuK WOL) WD) U UN w

wu c lL* b oI UN s w rOKFK WOzd WKOu t w UNM* WOL sLJrKF* bO v_ _ rKFK jOu bB* t_ UNM* dUM b wb* UJU WU WUM uUdO d_ rKF WOK; WUI uU( Xd rKF qzU s dO v WU{ VUD

bU* jzUu snB VJ wMODKH UNMLK UND s WU WKd* oOD UF c u dWKd* V iF duD v WU{U wMI wMN* WOU uKF wLKF tdH uU _WOd uJ UN UI UF w uU wU nB UNM V UNFO WOU_vK UOU u qLF uHBK UF rOKFK WOb* VJ lOL b XKL b wUF rOKFqd* ZUM* WOKOK% WO1uI U qL wdJ rOKF UJA U w WO WOM lOu

wMODKH rOKF WOu 5% ud duD WKu* UuL UOI YU* lOL w UU UI Ub v dA b_ uHBK e$ w rKF* W WOb* VJ bFwIDM qN uKQ X{d UuKF UF s tOK qLA U0 rKF rOKF WOKL w WOU eO.uI VOU WDA_ qzUu fb ozdD qB W{ dR sLC WuM d dOu

uc* WOL) WD) U l

uuI sc ULKF* 5LKF* 5ud WUA0 UuM Ud UNOIM VJ WFd r w duD qbFK WKU WOd& UF WFd v v_ s UFD u d UNbbI0 e wMODKH wb* UJ WLO UO( UN wuuMJ wLKF bI w dOG lsc WOb* VJ b U w 5BB* s sJ2 b d WUA s uN s tO c Uw ZUM* ed UN WONM0 WFd* s WFu UOKLF s rOKF w UdudOOG ub

bOu vK qLF c su wd w d nOQ wUULEM* UR* v dbI dJA qe bI UNF wUF rOKF WOd

ZUM* dA* wU* UNLb UJOK WuJ WU WIbB WOdF b WObs wU wMu qLF c U$ w XU w WOMu Wud UHJU dH u UL

V yq eOL* ruN rNUA vK rdJA WOb* VJ bS uI w Wud UK qLF Uu 5UA* sd; 5HR* d ZUM*ed Wu ZUM* U' qLA tFuoOD UM bO* s WOb* VJ d w 5UA* 5FUD 5Fd* 5Ud 5LLB*

wUF rOKF WOd

ZUM* e?d

WOLKFK YULK WUF

U???O


4/144

WbI

bF 5*UF tK bL(nBK UO{Ud U s wU e' WKD UMzUM_ ULKF* 5LKF* UMze bI UdUNM* WD sL{ UO{Ud Y* WbF* WCdF uD) o wUMB wLKF uU _

_ wMODKHqU K* UOU* Wd* b_ w b l vK UJ c qL bI

UK* sb( WdE oOuWuL s bF UEM lOu WOKO bu UMb Wd* b_ WU) bu wH

UNOK UOKLF Wd* b_ u vK dF - ? Wd* b_ WuL v WOIOI( b_WOK* WODI uBU UNKO9

vK eOd d WKK* WOU* uNH UMbK* UOU* WU bu wdI UMb UL WOIOD UNOL_ WOzUN WObMN WObMN WOU( K* UOU*

wU_ dUF nB w WKD UNOK dF w Ud dD ULJ w{Ud bFK WHK UOOd UMb sb( WdE oOu qU WFU bu w

Vu `O Q sb( WdE UMb UL dU* bF vK uIwUOI rd uNH vK eOd - WOzUB UMOF WOUOI U_ WMU bu w

wUOI rd jO wUOI rd p w U0 U qJA UF wUOI rd U qJAWOzUB UMOF WdEM WO_ U* d - pc lK s WuL* dO WIdD `d*

ddI w ULN U WuU WKd* w wLKF U* U sc WKD WOuM UMI XU bIiF Ud uD WK_ iF uD) WKUJ OBH v QK rK u; d WIdbu 5 bu bu uM wIDM* qK WOLKF Wb vK UMd l UdEM ozUI(s UuL b bM q WUN w WuM* qzU* sUL s VUM* bF UUO WHK*5uH* WKDK UFOA dI* bu lOL' d UJ WUN w WOU{ WDA_ sUL

qzU* q oLF vK reH

w rNM U s UJ d w uLU sc bO* w Oe e qJ dJA bIWUM* bF U UJ c u WFd* ZUM* ed w bI w qLF W dUMb w oOD l{u UJ l{ bF UE* s be* vIK UMd vK sbR tOK

tK U WOMODKH

oOu w tK

uHR*


5/144

W??dL? b?????_ WOKO b_ ? Wd* b_ WuL

UNBzUB Wd* b_ vK UOKLF 5d sb WL od* bF Wd* b WOFOd c' `OB buK WOOFJ c' Wd* b WOK* WODI uB WU sU9

WU)bu

K* UOU* WOU* uNH K*

WbF WOU( UOU* WOU( WKK* uL WObMN UOU* UNuL WONM* WObMN WKK* WOzUN WObMN WKK* w{Ud dI WU sU9

WUb

u

U???????L wU_ bF b q?????????U o????????Ou sb( WdE u( wMOF UCH u( I dA* UL eO WdE WU sU9

WM

U

bu

WU WFd UO wzd oK q ZUd bU b dC U wU{ oK UKDB* WLzU l???????d?L?

WOUOI U WOUOI U

lK s WuL* wUOI rd Wu* W ]d* WOUOI U WU sU9

WFUbu

U???u?L


6/144

W????dL b??????_

WU)bu


7/144

(Complex Numbers)WdL b??_bONL

b_ r WOFOD b_ WuLL b b s WHK UuL WIU u_ w X

b_ WuL YO WOIOI b_ WuL dO WOM b_U WOBWOM dO

WU p s UNM l WuL v WuL q lOu duD XU WU XE ULw q WUFLK bu sJ w q UN bu WUFLU UFL q vUuLL w lu dL cJ UNO WUFL c q s sJLK v lOu vb UL

WIU UuLL qJ lOu w w WOIOI b_ WuL v UMKu v WbF

tFd wIOI b bu t_ w q UN bu dH q WUF S p lWdL b_ WuL vL l WuL v lOu d d vb c d_ UlOL q WOU ?edU UN ed w bb WuLL sO b WUFL c q s UMMJL wb_ WuL S d WOU s WOU s c UNM q W XU ULN b dO UFL

UdO WObMN WOzUeOH uKF UO{Ud w WLN UIOD w uK U U du?WdL

wL WUFL oI edU t ed bb b U vK ULKF oH dA U dI wWOKO bu bb bF

WOKO b

ndF

tFd wIOI dO b u wMF cWDu WOKO b XOL YO U qJA q b_ U

L VU wIOI b _ wFOd c WU sJ b_ c


8/144

WOB u

WOB uI s dOF sJL tS UIU pF d w f_ sOu oI bF dUL j uB bFK WHKL

WOU WF_ dUIL b uJ bFK WHKL WOB uIS U tu`O b YO bIL S U qJA

cJ uJ qLUVu wIOI b YO U qJA

L S sOU sOOIOI sb U WE

U

q

u jQ wK UL q WLO b

bIL

bIL


9/144

sULu jQ wK UL q WLO b

u jQ wK UL V

[

WOU WK_ s V r UL b qL.{ } sJ dC WOKL X tIKG WuLL q

vK WOFOL dC WOKL q

WOKLF cN bU dBM bu q ? w dOE dBM w dBM qJ bu q

WOKb vK dC WOKL q ? x UEM n UMMJL U

WOU{ sULdH sO

bIL WLO bQ XU

X


10/144

wKO e wIOI e Vd b u bF LwK UL ? WdL b_ WuL ndF sJL UM s

? WdL b_ WuLndF

YO uB vK tU sJL b uVdL bFVdL bF s wKO e VdL bF s wIOI e vL

qJA Vd b u wIOI b S uB vK tU sJL q wIOI b ULWdL b_ WuL WOIOI b_ tuL sO tF `{u wU

ndF

{ YO } ? WdL b_ WuL

?WdL b_ WuL

WOKO b_

WOIOI b_

WuLL bFWOBWOMWOM dOWOIOIWOKOWdL

[

V????b

WuLLK wLM bF U wU b s VUML dH w U l{


11/144

tUFL q

Uq

WdL b wUO qOL

Vd WDu sOF bF c S c uB vK bO WIdD VJ Vd b qtLU UUOb wUJb uL w WDIM U tKOL sJL c WOIOI b_ s

UUOb w WDIMU wNM q_ WDI s b c wUOI WDIMU qL VdL bF L UL qJA w UL tLU

qOL c WO wUJb wb uL vLrUFK ULdJ bU u WdL b_ uLwd uL vK oKDArgand bU wdH

r wI_ uL vK oKD wKO uL r czbMwIOI uLWOFOd WUFL q

UF uUIU YO ? WOFOd WUFL q WOHO WIU uM w XLKF bI

UU fO UOIOI b ? eOLL u WU w p ?

WOzUNM uK UD U lOL w `KB UF uUI S WdL b_ WuL suJ bF UWO WK_ w `{u u UL WOFOd WUFLK

dH WUFL q U

?

q

wIOI uL

uLwKO


12/144

WOU b_ s

UOUO q

?

WOU WdL _ s q tKL c VdL bF V ?

Vd u vK WOU WdL b_ s V

?

?w WOU UFL s q

?

WUFLK WF_ c b

dH WUFL qoKDL b qu b d bIL qu b UA

dH

WKuD WLIU

bIL qu b UNM

UNM

Uq

sUL


13/144

sOd sb U

UNBzUB WdL b vK UOKLF

ndF

? S ? U

U sOOIOI b UNM

UNM

Uq

U

q

WUFL UII sOK sOOIOI s q WLO b

lLU

ZM iuFU

? vK lL WOKL

ndF

? S ? U ?


14/144

U

q

UOUO p q bQ U

sONL rd UOUO p qOL

sbF L ? { u qLJ

uLLK L ? tL uJO

WDIM w ? WDIM rd s k

Uq

WLO b

? vK dD WOKL

ndF

? S ? U ?

? vK lL WOKL hzUB

WO hzUB s ?vK lL WOKL oIWOKb WOU WOFOL WOU G WOU

dOE dBM qJ bu bU dBM bu ? UEM n UMMJL UL p s oI

?


15/144

? vK dC WOKL hzUB

WO hzUB s ?vK dC WOKL oIWOKb WOU WOFOL WOU G WOU

lL WOKL vK dC WOKL u bU dBM bu

p s oI

? vK dC WOKL

ndF

? ? xS ? U Uq

WLO bx x x x

lu WOU bU UNH WOM vK uB UMMJL t k

Uq

WOD qu v bIL qK bIL

U

q

qIM c wbMN qu uU bFK wFL dOEM VU U v bF

u bFK wFL dOEM

qIM c wbMN qu k UL qJA sq WDI w UJF u wFL dOE v bF bIL q WDI u


16/144

wK UL q w sOOIOI b

?

u vK wK UL V

?

?

dD ZU UOUO b

WUFLK q s=O WOFOd WUFL q UF uUI b b dH

quF v qK

sUL

WOU{ sUL

WUFL oI

VdL bF s=O

rK sOOIOI b

Wd U fJ bIL q WDI u UNKL w Vd b d{ WOKL `{ WU UI


17/144

sOd sb WL (Conjugate)odL bF ndF

u od S U ULNO wKO e U w jI UHK sOIdL sOdL sbF

Uq

VdL bF od b

WLO b

odL bF hzUB

p s oI tH VdL bF u VdL bF od od S Ud b U

S YO U u ULNd wIOI b u sOId sb uL wMF c

L wKO b

S YO U wIOI b u sOId sb d{ qU wMF c

L

S U

p vK U j ULNOId uL U sOd sb uL od wMF c

U j udU p s d ULNOId d{ qU U sOd sb d{ qU od

p vK

U

q


18/144

dH WUFL q

q WuLUId Ud b UL WUFL cN sOdL sc k

UHK UOIOI c UN uJ WOIOI b UNUF uJ w WOFOd WUFL S U tuUId Ud c UU UOIOI c

U

q

UNc b w WOIOI UFL WOFOd WUFL bd c u tId S WUFLK c u UL

w Uc w WOFOd WUFL rKF

WUFL c w sc uL sc d{ qU

w WuKDL WUFL

Uq

Ub

WO b_ s q od b

dH WUFL q


19/144

sOd sb WL

UIL qF UIL od w UIL j s dC Q sOd sb WL WOKL d UMMJL YO U SUOIOI b

S

U b

WLO b

U

q

Uq

VdL bFK wdC dOEM uKILVdL bF u VdL bFK wdC dOEM uJF S U

YO


20/144

u vK wK UL q WLO b

X

UNc b U w WOIOI UFL WOFOd tUFL u

oI r

bQ U

uJ v

wQ UL qJ wdC dOEM b

?

? WUFL q

sUL

ULN oI sd sc bQ dH WUFL c b U

UId

UId ? ? sbF X

sOId dO sc sO WUFL q UF uUI b

tId v Vd b qIM c wbMN qu u U

WOU{ sUL


21/144

UL WUFL ULNM q oI UOIOI b bu tS Uu UOIOI b U t rKF

sOOFOd c UOL WUFL ULNM q oI UOFOd c UC t bu tS U t WOU WK_ w sOM

WdL b WOFOd c

Uq

Uq

bFK WOFOd c bS bFK sOOFOd sc b u dH

bFK WOFOd c bS YO bFK sOOFOd sc b u dH

tM

WUFL s

ZM WUFL w iuFU

UNM

UL bFK UOFOd c


22/144

bFK WOFOd c bS YO bFK sOOFOd sc b u dH

WUFL s

ZM w iuFU

UNM _ W{ud c

UL bFK UOFOd c uJO

Uq

WO WdL b_ s qJ WOFOd c b

VdL bFK UOFOd c UL sO

sUL

wU{ sdLWOFOd WUFLK q WuL bQ Ud b U

dH


23/144

ÒB buK WOOFJ c hzUB

UId UdL c dH W c uL

sOdL sc d{ qU d VdL c sOdL sc s ld

hzUB c W s oI WuN pMJLS tOK uJ d c S edU sOdL sc b_ Ue

dH

(The Cube Roots of 1)ÒB buK WOOFJ c? WUFL q ÒB buK WOOFJ c U

UNM

UNM w ÒB buK W WOOFJ c

UId Ud d wIOI b c c b k

V????b

{_ U YK uJ UN sO bU u w ÒB buK W WOOFJ c sO


24/144

sO

U

q

Uq

u jQ WO dUIL s V

?

?

u jQ WO dUIL s V

UEM sO r UL b qL WOKb d uJ

X

Uc w w WOU Wb s WUF u

X

sUL

wU{ sdL XQ

XU

dH


25/144

VdL bF qOL sJL t UM bI

WE s VdL uL w tDIM b UL qJA

? U ? U S tOK

? U ? U WOKL uBUUO vL VdL bFK WODI uBU? U ? UuB vL

edU t ed t WIKDL WLOI VdL bF UOI uD vL sJL pc q_ WDI s WDIM bF s U u

UU uJ VdL bF WF b lK{ l UNFMB w ? WuL We vL

? U

? U ? U YO

UHUC bQ d s Ub nK Vd b qJ UF s wzUN b bu t kVdL bFK WOU WFvL dHU VdL bF WF b c

U

q

uBU UNM V r WO WdL b s qJ WOU WF UOIL b WODI

UOI ? U

? U

UL qJA k U ld ld w ?

U

U

WdL b (Polar Form)WODI uB

?

?


26/144

WOUJb uB v WO WdL b_ u

U U U U

U U

U U

U

q

WODI uBU b q V WO WdL b WOU_ WF UOIL b

?

WOUJb uB v WO WdL b_ s u

U U U

U

uBU WOU_ tF tUOI c bF V

WF UOI b r sO U

WF UOI b U

sUL

UOI ? U ? U

UL qJA k ? WOU WF

U U

?


27/144

WOU{ sUL

WOU WdL b_ s WOKL uB vK VU U

XQ U

YO XQ U


28/144

WUFL oI w WOIOI rO b

uB vK bIL V

?w WOU UFL s q dH

c b bF Q ULK ?

VdL bF UOIL WOU u s X

?

X dH ? WUFLK c U dH ?

bFK wFOd c sO

UNc b w WOIOI UFL WOFOd WUFL b

b U U U

wb uL w UNMO WODI uBU OB buK WOOFJ c V

XQ `OB buK WOOFJ c w XU

WU sUL


29/144

bF UL ULNd{ qU ULNuL b

VdL bFK WOFOd c b

dH

WUFL q

WOU{ sULWOU UFL s ?w q

XQ XU

Uc w WOFOd WUFL u

dH XQ

XU

XQ `OB buK WOOFJ c w XU


30/144

?K?L U?????OU?L

WU bu


31/144

(Sequences)UFUL UOUL

WOUL uNH

bONL

d w_ d b dO d q Ud s WHK Uu UIU XWOUL vL Ud s d Uu b bu c w UdO wM d wLUuKUL_ c s sOu vK eOd rO WHK Wb UL WO vK UO_ s dO w dNE WFUL

WObMN WOUL vL d WOU WOUL vL ULbUO{Ud w WdE WOU s UNOL v WU{U dO WOKL UIOD UN U qJA UOUL

WOdFL WLE_ s UdO

UwU b w UL U U w rNU XU dA s uJ n

VUD rWF

cJ WdL _ s WuL u vK b c qOL sJL

tU Ud qL WdL _ s WuLL c YO

b s WOze WuL tU c d c

uB U WOUvL s WOze WuLu wU b edU t ed u _ b YO WOUL bvLcJ edU t ed

tS cJ r r UM UNU bM b VOd UMe U uB vK dB qJA ucL WOUL WU UMMJL

cJ u b UC WOUL WU UMMJL UB w


32/144

sJOWuL b WuL WOB b_ lOL WuL tU c d c

w Ub WOU UC vL s WOze

U

UF Ub w WOUL s v_ WL b VwK U vK qB UF b w b_U s iuFU

ndF

WuL WONM dO WOU vL WuL WOB b_ WuL U tU d wWOULWONM WOU vL uB vK s WOze

U

q

UF Ub w WOUL s lU ld sb s q WLO b

wK U vK qB sbFU s iuFU

U tu

(General Term)WOULK UF b

b wuM b tOd c b vL WU bU jL o WOUL b vu U tu edU t ed WOULK UF

Uq

WO_ b_ WOU q WdF bU UN fO WLNL UOUL iF

WOIOI UOU vL s WOze WuL UNM q b w UOULU UMUL dBIM


33/144

WOU UOUL s qJ UU b V

?

[ [ [

wK UL WU q w UU b b sJL tOd b q WLO sO jdU

[ ?

U

q

wK UL WdFL WOULK v_ W b V

vDF

vDF wK U vK qB b_U s iuFU

U

q


34/144

UF Ub w WOUL UOUO q U

qWdL _ sOF qLO wd qLO wI_ suL rd

qL c UL qJA k

WOU UOUL s qJ v_ WL b b

U

?

w b d b

?

WOU UOUL s qJ UU b V

U U

U U?

qzU sUL

WOULK wUO qOL

tS uB vK UNM WONM WOze WuL tU Ud WOIOI WOUL XU ULwU UL w `C UL WKBHML UIM s WONM WuLL wUJb uL vK UOUO UNKOL UJU

UOze OL ucL WOUL

UIM WONM dO WOUL bdI WdL WKLLc rOIL j s d d

tUF


35/144

WOULK UF b V

UNO WOU

sO

WOUL w jLM nA

UNO dUF b V r

UF Ub w WOUL w sO

WOU{ sUL

UF Ub w WOUL UOUO q

Ub w WOUL w UdzUE U UF Ub w WOULK v_ W b sO ZM U ULNM q w ld b b r

UF

U WOUL w YU b qF w WLO b UNO WOU

(Fibonacci Sequence )wUuU WOU WOUL c vL


36/144

US WKUH U UMb uB vK UNU sJL WOUL UME

WKK vL ZUM S lLWOULU WDdL WKKL w ULMO WOU w L

ucLWKKL L ULO dI U ed b sJL dB uB KL WUJ

uB vK UNU sJL

w WLO s dOG dOGL UMKF uLL e U UF b UM UM k

d q w Ue w WLO d v

WKKL WU c w k

uB vK VJ WKKLp vK WbK ed UM WONM dO

(Series)KL

WKKL WU w ed b

u WKKLK UF b

WKKL WU w ed b

u WKKLK UF b

U

q

U

q

U sOOU sOKKL s q uL U

U


37/144

S tOK Ub uL u WONML WKKL uL

U U U U U

uLL e hzUB

uL* ed WOU hzUB) W s oI sJ1 ? YO ? ?

YO

?

?

q

XLK sOIdD b

dU iuF v_ WIdD

uLL e hzUB WOU WIdD

U

q


38/144

U

WKKL uL b

u

WKKL uL b

XQ wU UND UF s WuL XU

dH

WOU{ sUL

qzU sUL

WOULU WDdL WKKL ed bU V

uLL e bU WOU KL s V

XU ? ? ?

oI

WOU KL s q uJH V

XLK

bIL WLO U uLL e u bUuUIU vDF tuI s WOU bF d UuI jU r UNFDI w U_U WUL XU

uLL e bU v_ X wu w WuDIL UUL WKK V


39/144

WOU w Ub v_ WOU w Ub lDI vK v UO wLKF Y d_ U oK

UUL WOU S jLM cN uFBU UB dL S cJ WU WOU w b WOU w WuDILc vL Ub u UUbI U dU t oU b b q sO dH WOUL c w k

Wb WOU WOUUOUL s uMU tu

(Arithematic Sequencies)WbF WOU UOUL

ndF

U dU t oU b b q sO dH UNO uJ w WOUL wWbF WOU WOUL edU U t ed WOUL U vL UU bI WOUL U XU bI

wK ULO UdO s WOU UOUL eO

WO_ b_ WOU ? _ WOU WOUL

_ WOU dO WOUL

_ WOU dO WOUL ?

U

q

WOU WOULK UF b

uB vK UNU _ Ub w WOU WOUL WU sJL bUIU vDF WOULK wuM b UF b vL tOd c b S tOK

bU

U_ _ b YO u WOU WOULK UF b


40/144

r WOUL V UNO dA fU b WOU WOU w lU b U UNO sOL b b

UNU _ Ub w WOU WOUL w lU b b

WOU WOUL w WOUL b b u bF q b

uJO t WOUL b b u bF dH

UNM

WOUL b s b fO bF S UL

Uq

U

q

U

q


41/144

WOU U_sd sbK UOU UD uJ UNM j_ b S WOU WOU s WOU b W Uc t sUL sbFK wU ju u bF k WOU WOUL wH

t sUL sbFK wU ju u bF UL _ U tu

ZM sOUFL q w WOUL

sbF sO WOU U WF qWOUL S w WuKDL WOU U_ d vK

_ Ub Ub b WOU WOU w

U Ub sJ

UNM w WOUL

w WuKDL WOU U_

Uq

b_ qJA

ju c uJO sbF sO wU j U sJL sO b b U sJL tS sb U WOU WOU

b_ wL WOU WOU

WUM WOUL qJA YOt sUL sbFK UOU UD uJ UNM _ sbF sO WOU UU WOU WOUL w dU


42/144

sbF b ULNMO WM wU ULND b wU bF _ bF dH

ZM sOUFL q

UL bF

WOU WOUL s dUF b b

uF Ub U dA fU Ub UNU WOU WOU

b c W UL WOUL b b U

WOUL w VU b W b

WOUL U u U WLO UL U S _ Ub WOU WOU

sbF sO WOU U W q

b UNO WOU WOU

XQ ? sbFK wU ju u U

? ? ?

Uq

qzU sUL


43/144

b_ c b UNd{ qU UNuL WOU WOU uJ b W

VOd vK w WOU WOU s dO_ b u fU b dA wU b U WOUL c b b UL

XQ b wU u w ? XU

b wUu w ?

b wUu w ? ?

b_ l VUM b_ c XQ wU u w We rzU YK { XU

bQ WOU WOU ? ? b_ Xu

? WLO WOUL c b b

WOU{ sUL


44/144

wK UL WONM WOU WKK b s b lL UMMJL

dO_ Ub UNU _ Ub WOU WKK sJ

S ? edU WKKL c uLL Ue Ub b

?S b l{ fJF

?ZM sOUFL lL

b b ? ?

?

?

(Arithematic Series Sum)WOU WKKL uL

WKKL uL b YO WOU WKKL

?

?

Uq

bU

bUIU vDF UNU dO_ Ub _ Ub WOU WKK b s b uL

?

uLL bU UM w UUMb w WIdD bU uLL c U s UdI uM dA s u U wUL_ rUF sJL bI


45/144

b

UNO WOU WKK c

sJ

cJ d WIdD WKKL b b U sJL t k UNM b b

? sJ ?

WOU WKKL s b uL b

uJO

? ?

U

q

Uq

Ub v_ WOU w Ub WU lDI d UuI b UH s r jIb cJWU WOU w Ub WOU WOU w

WU WOU w r UNFD w WUL v_ fL wu w WuDIL UUL uL

_ v r q v ?WOU WOU w w v_ wu w WuDIL UULWOU

UNU

b

U

q


46/144

?

b ? ?

wu bF _ v r qB

wU dL dH s UM[ _ dL dH s UM v{UI jHM s VOIMK Wdd UNIL d dH nOUJ mK r cJ YU dL dH s UM [

[ [ w nOUJ WKK

[ UNO WOU WKK w[ ?

[ d dH nOUJ UM [

U

q

U

q

bUIU wDF WOU WKK s b uL U UNO dUF b b ?

? ?

b

b

b


47/144

WOU WKKL w b s b q bUU b UNuL uJO

b b dH

?

dH ? sJ

dH

dH dUML WUFL c b WMUL c q

[ dH dH

UL qJA UN{u bIL U S cWU b_ qLN [ WMUL q uJO

w ? WLO q

U

q

WOU WKKL s b uL b

Ub b U Ub _ Ub w WOU WKKL uL b

dA U dUF UNb uL Ub b w WOU WKKL uL b

UNuL uJO

WKKL s c eK fU b s b b r

UNOK w b WOUL uL UNO b uL w WOU WKKL U

t Ub c b WLO UL u WOU WKKL s UOU b uL U

vK WLI qI sO uBL WOB b_ uL b

qzU sUL

[


48/144

U Vu w b uL X

X WOU WOU _

WOUL q WOU WOU =uJ Ub UuKI XU (Harmonic)

WOIu WOU WOUL vL

WOIu WOUL w dAF b u U

WF uL b WF uLUNO w WOU WKKLK U_ _ b b b

wu tMJ WOU WOU uJ UD vK b vK oH UM mK ob s h d

ULNF scK dO_ _ sODI s q bI b sb YK tOK wI sO jI l bF Ue tO VB c p bF r v_ WU w d tO XdQ u w UOL VB WOHM

d u w uJ WU r bF UN WIU WU s WU q w d

WOU{ sUL


49/144

t oU b b q sO dH w UN eOLL WHB Ub WOU UOUL UIU UMdF

b q sO WM Q eOL dO WOIOD WOL UN d u s UOU UM UU bI U dUWObM UOU UOUL s uM c wL UU bI U dU t oU b

(Geometric Sequences)WObMN UOUL

ndF

WM WU W dU t oU b b q sO WM UNO uJ w WOUL wWObMN WOUL edU UN ed WObMN WOUL U vL WU

WObM WOU UOUL s

?

UF Ub w WOUL

WObM WOUL XU

WObM WOUL XU

WObM dO WOUL UL

?

WObM dO WOUL UL

U

q

(General Term)WObMN WOULK UF b uB vK UNU _ Ub w WObMN WOUL WU sJL

bUIU wDF tOd c b S tOK

bU u UNU _ Ub w WObMN WOULK UF b


50/144

U

q

WLO UL WObM WOU qJA UOLJ XU S WObM WOUL UL

ZM wU dCU

UNM

WObMN WOUL b s tLO c b W U

u tLO c b dH

sU b u tLO c b

U

q

UNI w WM U s U WM w U uJ YO UuM VcK rM U hUM

WU WM w U bI UL rGv_ WM w rML U U S [ [ [ [ w U WOU

[ UNO WObM WOU w

[

WU WM w rML U V rG [

Uq

UNO U b b UNU _ Ub WObM WOU

Uq


51/144

(Geometric Means)WObMN U_

ndF

WObM WOU ? uJ YO ? bF p u sbFK wbMN ju

Uq

sbFK wbMN ju u U? u sbFK wbMN ju dH

S tOK WObM WOU ? uJ ndF s

? ?

?

? UL sbFK UObM UD bu

d_ ULb UOU UMb uJ k

sO WObM U W qWOUL uJ WuKDL WObMN U_ w dH

UNO WObM WOU

ULb UObM UOU bu WObMN U_ uJ d_

WObMN U_ uJ

Uq

bU ? S sbFK wbMN ju u ? U

dU ndF s WOU bUI U pMJ1


52/144

bF UL wbMN ULND wU ULND Uu bwU _ bF dH

b v_ WUFL s

ZM WUFL wd lOd

dH UNM

uJ UbM uJ UbM

UL bF

Uq

s WOL w UdOJ b U S WU nB q sOM bu `B UdOJ dUJWU w UdOJ b `B b r UU WMU WU w b VOK

UU dA WU UU WMU WU w UdOJ b

UU nBM WMU WU w UdOJ b UU WFU WU w UdOJ b

w UdOJ b WOU S tOK UNU _ Ub WObM WOU w

WM_ dUM Ub b U

b b lU b dA WU WU w UdOJ b uJO

b

U

q


53/144

WOUL s U b b

sU Ub fU Ub WObM WOU WOUL V U_ _ b b

ld _ sb uL WObM WOU s wU _ sb uL WOUL cN U_ _ b b

b_ c UL UNd{ qU UNuL WObM WOU qJA b W

WObM WOU qJA UOLJ qF w WLO b

bF UL wbMN ULND ULNMO dH Uu b

XQ WObM WOU ? Xu

?

?

qzU sUL

ULN wbMN ju U s d sOuL sbFK wU ju X

u sO dDI vK ? uLF UNM e uI vK U WDI ? dz nB w dD wu sO wbMN ju u ?

WFU wDF WOU WOU s b uL U WLO UL WObM WOU qJA WOUL c w b XU ?

WOU{ sUL


54/144

U sJL tS UNU _ Ub WONM WObM WKK XU

wK UL WKKL cN ? uLL ?S w dCU

? uJ sOUFL dD

? ? ?

?

UNuL WONML WObMN WKKL

WKKL s b W uL b UNO WObM WKKL

?

?

Uq

Uq

WKKL uL b UNO WObM WKKL

vK WLIU

bU

bUIU vDF UNU _ Ub WObM WKK b s b uL

?


55/144

b

UNO WObM WKK w

uJ vK WLIU

?

?

?

U

q


56/144

WKKL s v_ b WF uL b

wuM Ub w WKKL s v_ b W uL b

b

u b b U_ _ Ub w WONML WObMN WKKL uL X WKKL w dO_ b YO ?

UU UNuL uJO WKKL s Uc eK b r

W V dO_ Ub _ Ub WObM WKK b uL U UNM b

qzU sUL

WKKL uL b

b s v WKKL uL b

r v_ WOUL Ub uL b fU Ub YU Ub WObM WKK WQLK

U u q tO VB U U S U W w TK r

UU U d tF U ZdN c UL bI _ uO w tO V c UL bI V dU oU uO w tO VOB U

dO_ uO w tO V

u u u u WKKL uL b

WOU{ sUL


57/144

KL UMHM UNU U ULN WObM WKK b s uL U WOHO o ULO pF dU Ub b e UbM WObMN WKKL uL vK bM c w dF WONM dO WONM vWKKL c uL U WOUJ b WOzUN WObM WKK WKKL c wL b dO

UNU vK nu UF Ub w WObMN WKKL bM

WLO dOG b WLO dOG `{u wU b

WLO dO U YO WLO WLO ULK t UM sO b s

WIU WOzUN WObMN WKKL uL U sJL wUU b dO U

b WLO dOG `{u wU b S UF Ub w WObMN WKKL w U

WLO dOG

U U b dO bIL WLO XBI dO U WLO t b s sOS WKKL s b uL

[

?u WKKL s b uL

[

?

d d WLO ULK d d bF s dI WKKL uL wMF UL

(Infinite Geometric Series)WOzUN WObMN WKKL


58/144

Uq

WUN U v WKKL uL b UNU _ Ub WOzUN WObM WKKL

U uu WKKL uL S UL

?

UF Ub w WOzUN WObMN WKKL uL b

w WKKL

u UNuL uJO

UNO WOzUN WObM WKK wN

?

U

q

dI S sO uB U_ WLO XU t kU tu

bIL s ? WLO wUU dI b dO U U bM dHB sWOU bUI v UuI c

bU

YO u UNU _ Ub w WOzUN WObMN WKKL u


59/144

tM XDI c UH v ULz bd XU U UH s WOUD d XDIsJ q dJ UNFD w UUL uL b

UL wO{u qJA kqH v WDU w dJ UNFD w UUL uL

?

_ Ub WOzUN WObM WKK cUNuL uJO UNU

d [

[

?vK v bd w dJ UNFD w UUL uL

?

UNuL uJO UNU

_ Ub WOzUN WObM WKK c

d [

[

?

d ? ? WuDIL UULK wKJ uLL

U

q

U d v b dAF dJ u[ dAF dJ wUJ

[ [ UNO WOzUN WObM WKK uL

[[

[[

dA d v dJ qu u W s oI

WOU WK_ s `C UL UIOD iF w WOzUN WObMN WKKL uL b sJLUq


60/144

WOU WOzUN WObMN KL uL b

UF Ub w WOzUN WObMN WKKL?

WU u v WOU Wb WdAF uJ s u [ ?

uB vK YO bIL V

WKKL uL UL U b WOzUN WObM WKK s _ b U

{ UHBM XK d ld uJ t{ UHBM XK r r tFK{ u ld r

UFdL c UU uL U dLU cJ YU ld uJ wU ldL

qzU sUL

WOzUN WKKL uL b WFd b wUL s uJ UL qJA

qJA bU ZUM W s oI

wdC dOEM _ b WUN U v tOK w b uL YU b UNO WObM WKK WKKL s b W uL b fU bK

WOU{ sUL


61/144

w dF dUL dO Ud dUL Ud q Ud d iF dUF nB w XdF

w{Ud dI od s Ud w Ud d s bb WId bM csOUL k WU U WE s WU ULOLF v UMKu dOJH s u u dI b sOOU

qu sJL WOU WKK qL _ `O U w UM UIU d c WOU WKKL uL uU bU uLL v

S uUI w WLOI UM{u uK `O dO UM uN U w UM UUO fO uN wUU vK WLI qI b u

b_ s WuL UM w UF W WMd WO{U WId U v WU QA UM s

wU bL bLF c w{Ud dIWId Ud w WIdD c XOL b WOFOD

(Mathematical Induction)w{Ud dI

U

ZM

US U w b u w b u

w b u w b u

WOFOD rO lOL WO b wDF uUI ZM


62/144

U w{Ud dI b b YO

UbM WO UF oI

sL_ dD d_ dD UbM WO UF

UbM WO UF dH

UbM WO UF U bd

sL_ dD

WIU WO{dH s

d_ dD UbM WOB UFuKDL u lOL' WO UF

U

U

U S wFOD bF Wb U sJ w{Ud dI b UbM WO UF WO

UbM WO UNS UbM WO UF XU WO WO

WOFOD rO lOLWO UF S


63/144

w{Ud dI bU WOU UF W X

WO dO WO UF sO

dH

wU{ sdL

YO w{Ud dI bU sd XU

qzU sUL


64/144

w U t c b U c WObMN WOUL w b W U

WOU WOUL

VOd vK b_ c v nO{ UNuL b W s WuJ WOU WOU W WOK_ b_ w UL WObM WOU WUM b_ Xu

b WOUL v WKKL uL b

U d v [b dAF dJ u

b ? U S UNU _ Ub WOU WOU

YKL { u U r tU WOU WOU qJA t{ u We rzU YK

UNM b W b u WOU WKK s b uL U

WOU{ sUL

WOULK UF b V

WKKL s b uL b

WOU WOU w YO uB vK UF Ub WOU X

WObM WOU w YO uB vK UF Ub WOU X

UNO WOU

fU Ub b

WU sUL


65/144

WOUOI U_

WFU bu

sD

WKF

?WMb

dK

WOLJUOI b WFK


66/144

(Index Numbers)WOUOI U_

wUOI rd uNH

sOOLJ b W U od s sOOL sO WUIL d vK WOUOI U WOU_ dJH uI

Wu W v WM c qu rd_ v

sO WM S U w UM v U w UM s U WFK dF dOG U w UdF v U w WFK dF

U

w WFK dF U w UdF s U U w WFK dF wMF cUd sJL U w WFK c s WOLJ U w UdF s bIL U

U w UM mKL dA X U w UM mKL U w UdF v WMU U w WFK dF wUOI rd WIU WuL WM vL

WUIL WM WM vL U_ WM WM vL

bONLu lK UF q UJL Ue dOG UNLO dOG w duE s dOJ WOuO UMUO w budL UU b we U WOL WO_ W WKD b WAOFL nOUJ ULF

p dO D e U b sOFUe w dOGK WO d sOdU w dOG WUIL U dU vK dD c dOG l UOI

wUOI rdr tOK oKD c wzUB dRL _ uU b UJL UL w UuO WOUOI U_ d qF UL s dO w b dO WOUOI U_ wMODKH UB edL UN dU b lK UF_ wUOI rd r tOK oKD U u UB

vK dD w dOG bd U_ s uM c u WOzUB dA dUI bS U cMnOUJ u_ d UF_ UU vK uu d WOM d sO Ub lK UFdI c sOUK WODOD WUB Ub w UuL rU UL WOzdA uIM u WAOFLuM jOD w t bd qOb d_ wMODKH suL UC bOH WOMFL u dzb w

WKzUF WOeO w WHKL UH


67/144

U

b w sODK s WHK b w UMbU bM bM dF ju w b sO

U dN

WMbLbIetK fKUsOM

dF

S U_ UJ e Ud S

e w dF WMU bI w bM bM dF wUOI rd

WMU e w bM bM dF wUOI rd S U_ UJ tK Ud

, tK w dF

nd??FdF WuL WMu sOF UJ U w dF uM UC vL U WFK dF wUOI rd

UQ c d UJ U w UdF v sOFL UJL Ue w WFK

WUIL UJ U w WFK dF YO

U WFK dF wUOI rd udU U_ UJ U w WFK dF

U wM w sOFK UOL UF w b sO

b U_ WM w WM UU

U w WFK dF wUOI rd

U w WFK WOLJ wUOI rd

U w WFK vK UH WLOI wUOI rd

WFKrGd dFrG WOLJ

U tu


68/144

U

q

wUOI rd U sUM _ uU w rG WK iO Wud dF ju U

u uU w dFK WMU _ uU w iO Wud dF

_ uU w iO Wud dF U r

dFK wUOI rd

UM ,

q U w dF U w dF

U w WFK dF wUOI rd

U w WOLJ U w WOLJ


U w WFK vK UH WLOI wUOI rd

U w UH WLO U w UH WLO

U w UNOL WFK dF

U w UNOL WFK dF


69/144

sUL

u U w WOzcG uL UF_ wUOI rd wMODKH UB ed uAM w

UF cN uBIL U w U_ WM

WMU U w WFK dF u UF WMU U w WFK dF XLK

UF WMU U w WFK dF wUOI rd UL u UF

U w jHM s WOdF b iF U w b sO

WbWuFXuJdFdzeUOOU WOL

UOu qOd uOK,

,

,

WO WOUOI U_ s V U_ UJ XuJ UU

WuF U WOLJ wUOI rd

dze U WOLJ wUOI rd

wU w lK UOL UF w b sO

?

WFKrGd dFrG WOLJ

b U_ U U UU

U w WFK dF wUOI rd


U w ? WFK vK UH WLOI wUOI rd ?


70/144

lK s WuLL wUOI rd uM c s dUE wUOI rd d b X w dOG dUM b s dU uJ b

WAOFL nOUJ wUOI rd U YU S U w dUE c dUM lOL U s bUb lK UFQ rN YU vK S pKNL UF_ wUOI rd rU U dF c

UdAL WK r UNOK oKD U w Ub lK s WuLL c UM WOU N qL w

UNCF UL WU XOI UL lK iF UF S U WFOD pKNLK Ub lzUC WK

rd V UF UF_ u s b dOG c WU s dOFK d iF iH

UuK UM rd c U UF lK WuLL wUOI

u WOL_ fH lK lOL wDF t U sOF WFK qJ WOM WOL_ cQ uKt U sOF WFK qJ WOM WOL_ cQ uK jO wUOI rdZUM wUOI rd vL

uL `dL wUOI rdZUM wUOI rd vL b dOUF o WHKL lKK U wDF

jO wUOI rd U WIdD b

sOIdD VlK s WuLL jO wUOI rd

UDuL VM WId

WUM WOUOI U_ lOL wU ju U r b vK WFK qJ wUO r U vK uIjO wM wUOI rd vL U QAMO

lOL WId

w dC r U_ WM w UF_ uL vK WUIL WM w UF_ uL WL vK uI

UF jO wFOL wUOI rd vL U QAMO

nd??FlK pK WOUOI U wU ju u lK s WuL UF_ jO wM wUOI rd

WOUOI U_ uL

lK b t


71/144

nd??Fv sOF UJ U w lK UF uLL WuL WM u UF jO wFOL wUOI rd

U_ UJ U w UF_ uL

WUI* UJ U w UF_ uL U_ UJ U w UF_ uL

U t

wK UL U_ WM WMU U lK s UM W WOUOI U_ XU

UF jO wM wUOI rd u U

WOUOI U_ uL

lK b UF jO wM wUOI rd

U

q

U w lK UF_ UF uL w UbI U uLF t vK UM wMF c

U w UUFQ WUI


72/144

q

b U_ WM w WM UU

UF jO wM wUOI rd

UF jO wFOL wUOI rd

UF jO wM wUOI rd

,

UF jO wFOL wUOI rd

U wU w lK UF w b sO

?

UM dFWFK


73/144

w WON lK s UM W dIU bu duKOJ dF w b sO

u_ s wU uU dN w

U w UF jO wFOL wUOI rd b

U WM WM bU

U WM WM bU UOU

WM WMU UF_ jO wFOL wUOI rd

WM w UF_ uL

WM w UF_ uL

,

WM WMU UF_ jO wFOL wUOI rd

WM w UF_ uL

WM w UF_ uL

,

U

q

rM r

u X

U r

W??M???? WFK


74/144

Vb

wM w lK l UF w b sO

UF WMU U UF_ jO wFOL wUOI rd b

WUI* WM UF uL

U_ WM UF uL jO wFOL wUOI rd

,

,

,

q

U

enOd,iOWMb

UHduKOJ

UUDrGqd

d dFUOI b WFK

b rG qd UH UO b UU UF jO wFOL wUOI rd b oU UL w

qd s b d uKOJ UUD UO b d uKOJ sWHKL lK w WbL UOI b vK u nu q ZM U


75/144

lOL WIdD uL VM WIdD uL wUOI rd wLO WIU WK_ s k

ULNM q U WId lu d c

jO wM wUOI rd s UU qN jO wFOL wUOI rd dF U tu

WbL bIM b uJ b lK s WuL UF_ jO wFOL wUOI rd U bM

WHK XU bu s tH uM v UNKu eK c b lK lOL w

lK b dF uK UL tOd w WKb lK UO bu jO wFOL wUOI rd dQ

sJL w uOF s c lK WuLL wUOI rd vK dO dOQ dR lK WOI UF_ WMU

WdL WOUOI U_ uK bU UdOQ hOKI

UN U u UL b WKUF lK lOL qUF wFOL wM tOuM jO wUOI rd U bM

uK v QK WFK qJ WO WOL UD l{u c `OB b U d c UNH WOL_

WdL WOUOI U_

WU UE

sUL

?

UM dFWFK

sD

WKF

?WMb

dK

WOLJUOI b WFK

w b s UF WMU U UF_ jO wFOL wUOI rd b

w b s UF WMU U UOLJ jO wM wUOI rd b


76/144

(Weighted Index No.)WuL W]dL WOUOI U_

wU w oUM w UMbU ULFK WdNA u_ juL UO wK ULO

s V U_ WM w WM UU

U w ULF u_ jO wM wUOI rd

U w ULF u_ jO wFOL wUOI rd

w b s UF WMU U UH WLOI jO wM wUOI rd V

?

u_ juWIDML

d d rG rG

UM ,UM ,q q

? d d

WFKbu bu dFUOLJ

lK WU UF ULN jO wFOL wUOI rd jO wM wUOI rd s q vK cR

v c d_ U sOF WFK qJ WOM WOL_ cQ b WKUF ULNOd w WKb

qJ wDF w WuL WdL WOUOI U_ XUJ hIM c vK VKG d WOUO U s Y

U_ WM vK bLF b sJL t YO WFKK WOM WOL_ qL sOF UOF o U WFK

UdN s `dL wUOI rd VOd WGO s d dN bI WUIL WM vK bLF

rd u U_ d Q `dL u Laspeyres Price Index UF wUOI dO r

U w pKNL UF_ wUOI rd VOd w wMODKH UB edL UN tb c

sODK


77/144

nd??FUu u U_ WM Q `dL wUOI rd u UF wUOI dO r

U_ WM qL YO

cJ V wM dO r

cJ V wFOL dO r

U

?

uLL

WFKdFUOLJ

q

s V lK UF_ U UOLJ UU

UF wM dO r

UF wFOL dO r

UF wM dO r

,

,

WUIL WM U_ WM w lK l s UFOL UOL UF sO w b


78/144

UF wFOL dO r

,

U

q

lK UOL UF w b qL

U_ WM WM UU

U w UF jO wFOL wUOI rd b

UF U UOLJ UU U w UF wFOL dO r

UF jO wFOL wUOI rd

[

lK wU UF_ WMU UdD dF dF c ? WFK dF UF wUOI rd dQ k

UF wFOL dO r

[

UF_UOLJUF_UOLJ

?

WFK


79/144

UAUB edL UN UbB w dAM b s WOUOI U_ UF_ s UUO lL WKD uI

vK sJL UL WOOd _ UUO lL WId UdAL WK vK wMODKH

www.pcbc.gov

WOdJ WHB s UUO c

sUL

UF_UOLJUF_UOLJ

?

WFK

UF_UOLJUF_UOLJ

?

WFK

lK l UOL UF wU b qL

s b U WM WM UU

U UF_ jO wFOL wUOI rd

UF U UOLJ UU U UF_ wM dO r

UF WMU U UF_ wFOL dO r V wU b vK bLF


80/144

WFKwUOI rdU_ WM UOL

UF

f

u

w b s wM dO r b

lK lOL wM dO r Q ULK w b w ? WFKK wUOI rd b

wK UL U v U s UF_ dOG

bIL UFD UF

bIL uL UF

bIL fL UF XBI

, fLK ,= uLK , UFDK WdL _ XU S

U_ WM UF WMU UF_ wM dO r b

WFKwUOI rdU_ WM UOL

?


81/144

WU sULWO dIH s dI qJ WOB WU d

U uM b w tK d WMU fKU w bM dF wUOI rd XLK w dH fH w fKU w bM d uKO dF S Ud tK w bM d uKO dF

u wUOI rd UNO V

p dO ?

U WFK dF wUOI rd U_ WM WM UU WFK UF_ WOUOI U_ X

u

p dO tdF sJL ?

sOO ? sOdIH s WU w b w UDFL UUO vK bL WFK

UFUOLJ

U UF WMU U UF_ jO wFOL wUOI rd ?

?

UF WMU U UF_ wFOL dO r

?

U_ b U_ WM w YO lUBL b U UOLJ WOUOI U_ qL wU b

U_ WM w WM UU lMBL U UOLJ WOUOI

WMwUOI rd

u lK lOL wM dO r Q ULK WLO b wU b s

WFKwUOI rdU_ WM UOL

?


82/144

U????????L????????????????

WMU bu


83/144

U

q

WuLL d????UM s????O s???? U r???? U???? Y????O t????MuJ s????JL s????OeM s???? U????uJ b???? r????{ } WuLL dUM sO s dAF r{ }

cJ WMJLL b_ lOL wKF UB d WOUO dA b sJL

v_ WKdLU r UO

WOU WKdLdAF r UOZUM bF

U WUM b_ lOL b U r UO WHK d UM k

v_ dDs WId q qUI dAF r UO sOHK sOId UM dAF U wL UO WHK d UMb uJO

sb WdE oOu qUbONL

b UOKL d v UNK U qzU UMNu U dOJ UO{Ud w WOU_ UNL s bF

UO d b WdF vK WHK V WF VOd d b WdF p s WHK dDs WOUL WMO UO d b WdF U dA wM sO s sO WL s uJ WK dJ oda h s uJ wzUB lL

U vK dFU bM bFK WHK UOOd vK ]dFM UdO qzUL c s WUwU_ bF bL vL

(Fundamental Counting Principle)wU_ bF b


84/144

U

q

dHK WHK d bI v sOM s dHK WHK d b dUL sJLWMbL U ` v sOM s dH hA c lOD WHK WId rJ ` v bI s

bIsWHKL dDK udU bI v sOM s WHKL dDK Ue

WQL UODF `{u wU jDL S udU ` v bI

dA bU bI WMbL d ` v sOM s dHK WHKL dD b UMMJL

cJ WOUO

bI WMbL d ` WMb v sOM WMb s dHK WHK WId bu bI v sOM s UI WHK d UM k

v_ WF_ dD s WId q qUI `v bIs UI WHK d UM bI WMbL d `v sOMs dHK WHK WId UMb uJ c

sOMbI`

v_ WKdLbI v sOMs

WOU WKdL`v bIs

WOzUNM WOM`v sOMs


85/144

wU uM vK tUO sJL c wU_ bF b UIU UL `{u

qd vK r sJL WOzcG Wu UO

d lQ r uK s nM UO d r UDK s nM UO UOUsOIdD r uK s nM UO UU

WId WOzcG Wu UO d b

WO WK_ s `C UL sOKd s d_ bL c rOLF sJL

U

q

r uK s sOHM UDK s UM uK s UM rUDL b bIu q s b nM s WuJ WOzc Wu WMJLL UO b

V d b b d_ uK bu d X WHK d tO bMU l U V U dJ UqU V

v_ dJ V d b WOU dJ V d b WU dJ V d b

WId dJ V d b

U l V v_ dJ V d b WOU dJ V d b WU dJ V d b

WId dJ V d b

wU_ bF bb U u v_ WKdL d d b U sOKd vK Wd WOKL d sJ

u UF sOKdLU WOKLF d d b S u WOU WKdL d d


86/144

WF s uJ bK fKL bMBK UMO fOzdK UzU UOz U sJL WId rJUC

edLfOzdfOzd VzUbMB sO

dD b UL fOzd ed UG d b UL fOzd VzU ed UG d b UL bMB sO ed UG d b

d W edL UG d b

U

q

U

q

wU b w UL qu W vK dOG WO w UdOJ uL nu

qUFuL(pH)W{uL W[ [ [ u d W

U{WCHM WDu bb

WHKL UNUuL quF c hH UdOJ W UN sJL WHK WId r W{uL W UO d b d W UO d b U{ b UO d b

WId UdOJ W d b WHKL UO d b


87/144

qzU sULtK v fKU s dHK WHK d tU U S r XO v fKU s dH h

U r XO v fKU s dH tMJL WId rJ r XO v tK s dHK UHK UIdtK d

d U s d u_ b s WIbK ub WU lOD WId rJ u WIb

uKF w WHK V UNML d WFUDL dI WIU w WuHL WUD rdJ sJL WId rJ V WuG V WOU V WOLK V sO s UUO r WBI WGK aU

WOBBbMN s uJ qL od suJ sJL WId rJ U sOOM sObMN Wd w qLF

qU wM

q w s q b suJ{ }U_ WuL s tMuJ sJL sOeM sb r sOOU sOU s

bu bF w rd dJ `L dJU `L r

d uJ YO U qJA w UL U d s WuJ Uu UO hOd VJ bB sJL hOd Wu r U dAF UEM U w U_ Ud WOdF WGK d wU_ d Q ULK U W sOd s WuJ bu WuK uJ YO VJL UbB

dJ sJL


88/144

dz jO vK lI UI wUL s sOHK sODI q sO qBO tL sJL dH dO UN r

v s UNHdF sJL w Ud lOL b UL { } { } XU WOLN UDDL Ud c `{

uK vK rNM UM V n w wd fL vK uK U WL sJL WId rJ sU

WOM vK uB UL U WOU d rEM d d wI

dJU UL b U bU tMuJ sJL UM sUuJUHKb r tMuJ sJL UM sUuJUOb r

tMuJ sJL UM sUuJ vK WLI qIb r ?tMuJ sJL UM sUuJUdb r

tMuJ sJL UM sUuJ s beb r ?

WOU{ sUL


89/144

WuL dUM VOd UN r w dD b WdF w tb wU_ bF bL UIOD r s

U UNOd U UO UeO UO{U w WHK V UMb U L WMJLL dD qJ UWOU VOd vK qB UMS WMJLL dD qJ vK_ UJLwU UJLYU UJL

UO{UUeOUOUO{UUOUeO

UOUO{UUeO

UOUeOUO{UUeOUO{UUOUeOUOUO{U

U VOd b b VOd c s VOd q wL

U tu

(Permutations)qU

ndFc lOL bF ed WuLL c dUMF VOd u dUMF s s WuJ WuLL qb

edU qU VOd

sJL WHK WId rJ b n w UF rNuu WOUL u c U WF U_ R nDB

dUM WF s WuJ WuLL WHKL qU w U_ UHD WHKL dDU_ UN nI w WF_ luL uB UMMJL U

cJ WF_

luL_wUYUlddD b

U

q

d q w UNFOL uQ WHKL UO_ s qU b vK WbK UC

ed b


90/144

U tu

d ? _ luL UG sJLd ? wU luL UG sJL

sOIdD YU luL UG sJL

b WIdD ld luL UG sJLWId dD lOL b

dJ WHK ULK suJ sODK WLK d lOL VOd sJL WId rJvMF WLKJK uJ dCU fO d

luL_wUYUldfUUdD b

WHK d dUM s WuJ WuL qU b U ULKJ bd ? _ UJL q sJLd ? wU UJL q sJL

d ? YU UJL q sJLd ? ld UJL q sJLsOIdD fU UJL q sJL

b WIdD U UJL q sJLWId sU_ lOL q d b

WLK WUM ULKJ b

U

q

U dUMF c VOd qU b S UdUM b WuL XU

uB vK tJ UMS dC qU WU w UB

(n factorial)

dC dI


91/144

ndFcJ dF edU t ed dC S Uu UO b U

ZU b

Uq

bFU wNM bFU b WOUL WOFOD b s d{ qU U dC

Uq

s=O sL_ dD

d_ dD

dCL e bU wK UL V

?

?

U

q


92/144

WLO UL U UdG Ud WOUL WOFOD b s d{ qU UdG WOU qu d{ qU u vK d_ dD VJ c

Ud uJO

Uq

WLO UL U

VU u id

U

q

Ub

b n w UF uHI UH W s WuJ WKzUF WUc u c sJL WId rJ

n w wd vK U fK sJL WId rJ

u j v bIL dB n w UM WF fK sJL WId rJ

UM sU UF _ fK n w UM WF fK sJL WId rJ U UF

WLO UL U


93/144

uQ WHKL dUMF s qU UMb U S UNFOL fO WHKL UO_ s WuL dUM iF VOd rN UO_ s dO w

w WMJLL VOd S d q w sOM sOM UNOd U UB UOdG UO{U uK w V

uK UBUO{U UB

UOdG UB

uK UOdGUO{U UOdG

UB UOdG

uK UO{UUOdG UO{UUB UO{U

UO{U uKUOdG uKUB uK

udU U d q w sOM sOM uQ WHK UO WF qU b

U tu

edLK U WO tO dNE sJL WId rJ WOUC U w sOIU dtH edL w UM q r tQ ULK v_ W

edL_wUYU dD b d ? _ edL q sJL

d? wU edL q sJLd? YU edL q sJL

WId Ub dD v_ W edL q sJL

udU W sOIUL sO s sOIU W VOd w dD c s WId q k

bFU b WOU WOFO b W d{ qU U UC k

uQ WHKL UO_ s qU b s dOFK UC ed b

U

q

d q w uQ WHKL UO_ s qU b vK WbK ed b


94/144

U tu

Ud UN WOUL WOFOD b s d{ qU U

d q w uQ UO_ s qU b wMF UdsU_ s UuB SUJL q sJL tS WOUsJL Ub dD _

Ub dD wU UJL q Ub dD YU UJL q sJL

cJ Ub dD wzd dO_ UJL q sJL

Ub dD sU_ lOL q sJL

WdE UOFO b YO

UJL_wUYUwzddD b

s q WLO b

U

q

Ud

uJ tOK WLI bILU dCU

UL

WO


95/144

WO

WO s

_

WLO UL U

v_ WIdD

VJ c ULd sOOU sOOFO sb d{ qU U sL_ dD

ULd uJO sOOU sOKU d{ qU u vK d_ dD

WOU WIdD

id U

WLO UL U Ud WOUL WOFOD b s d{ qU sL_ dD

Ud WOU qu d{ qU u vK bF VJ c uJO

U

q

Uq


96/144

nPr

U

bF qb Shift UH jGC nCr UH jGC

bF qb

UL UH jGC 1680u dI

X

s q WLO U WOLKF WU W b

wK UL uD uJ WOLKF WU W bU

U

bF qb Shift UH jGC

dCL UH UO X-1 UH jGC

UL UH jGC 3628800 u dI

U

q

wK UL q WLO V

?

s q WLO U WOLKF WU W b ?

sOO sOU s q w{

}

U_ WuL s tMuJ sJLUHKb r

dJU `L r UM s UuJ bF U

UM l s UuJ bF U

WUFL q

qzU sUL


97/144

sOOU sOU s q w V vK WHK V VOd sJL WId rJ sU UMOF UU qE

sbU UMOF UU qE

[ bF w WdAF WKUH l{u U_ l{u dOOG s tMuJ sJLb r

UOzuA VFKL w sO ed dO dzUD dJ uFK b W edL lu sJL WId rJ

WOU edL q sJL WId rJ qdL u sO b bL U

qdL u bL uJ UL U ?

d WF sJL WId rJ u UN Wb

X

WOU{ sUL

sO

WLO UL

U

sO

sOOU sOU s q w rOWLO b


98/144

(Combinations)oOuq w UNCF UNK uQ UO_ s WuL s UNMuJ sJL Wd UO w qU UMd

WKD s WOUL WM qOJA bM qB UL VOd UO d v U UO_ iF w dUdUM b WuL s uQ dUM s WuJ WOze WuL suJ UU sO s rUO rWOL UNO VOd uJ U cN a dUM

aU UOuuMJUO{UuKw V WL sO s V UO sJL WId rJw WMJLL UO lOL

aU uK UOuuMJ UO{U uK UOuuMJ UO{U UO{U uK

aU UO{U aU UO{U uKaU UOuuMJ UO{U UOuuMJ uK

aU UOuuMJ aU UOuuMJ uK

UIOuUO c s UO q vL U UO b

tH u UOuuMJ UO{U uK

UOU rN dO UO q w VOd k uKUO{U UOuuMJ tH u UOuuMJ uKUO{U

s b ed VJ iF b

Uq

U tundF

s WuL s UNMuJ sJL tH dUMF b UN WOze UuL Wd dO UO w oOuedU d q w uQ dUMF s oOu bF edd q w UNCF UNK uQ UO_

UOFO b YO u dI WdE


99/144

WO

p sd

Ud

Ub w oOu s oOu q dUM UM U S Wd dO UO w oOuUL qU s UMb ZMtS

uKDL u

WLO b

p r WId rJ sOLKF s WuJ WM qOJA d ULKF UNO Wb

WMK qOJA d b WId

p a{Ub_ sO XL WUB r d rNM q `UB Ub vI

UUBL bUUB

UNKL sOM q sO UUBL S ? udU WF_ Ub Ue ? ? ? WOU WOzeUuLL

UuL c

U

q

U

q

U

q


100/144

sL_ dD Ud

d_ dD

UU UdD

dUMF s oOu b U d q w uQ dUMF s oOu b uI WOM ctKUI dUM sO s dUM s WOze WuL suJ _ `{ c d q w uQWuLL WO WOze UuLL b WuLL fH s dUM s WOze WuL suJ dU

WuLLK WOU WOze UuLL b U WHK dUM s WuJ

WO W sO

sO ?

WO

WO

S U

Ub


101/144

sOU UU s WuJ WOLK WM UO d UU tO jK np r WId rJ

UUD UO d bWId

D UO d bWId

WKU WMK UO d b WId

WLO UL U

id

U

q

U

q


102/144

UdU XQ U sL_ dD

uKDL u d_ dD

s q WLO U

rOWLO UL

U

UL UNM U UNM

U

q

U

q


103/144

s q WLO U WOLKF WU W b

WOU uD lWOLKF WU b bU oOu b U

U bF qb nCr oOu UH jGC

bF qb UL UH jGC

W UA vK u dI

b UNH WIdDU Xu bN UB w WU W WOL k

wK UL q WLO b

?s q WLO U WOLKF WU W b

UNM _ uJ vK WK s WU U VUD sJL WId rJ WK s uJ U

b WUI vK UNM lI uL w UI wUL ucL UIM bU UNL sJL WLOI WFD r

ucL UIM s t uJ YO tL sJL UK r

wb qJA UD b U

U

q

qzU sUL


104/144

WUFL q rOWLO UL wU u w XU

s q WLO UL XLK bILj = bUI bU

UOKF WOFU UNA WKL s rNM W UC s UFU b w UM_ fKuJsOOU sOU w UC s WuJ W WO suJ sJL WId rJ

UOKF UNA WKL s jI b uC rC WON XU UOKF UNA WKL s q_ vK suC rC WON XU

WOU{ sULUuH d U qF UNMuJ sJL WHK U r WOU pOU wUL b

dUM s WuJ WuLL WOze UuLL lOL b U

d WMK c qOJA sJL WId rJ sOb sO s WOU WM qOJA d b VU UM Ub q_ vK UNO uJ

X WLO UL wbM u w U


105/144

(Binomial Theorem)sb WdEvK dUIL p WOHO Wd dUIL d{ WOKL X o

L Vu `O b YO

uB

sJL UDL l WIU W dUIL s qJ wzUNM uJHL kwK UL tH

udL _ s b be d_ dD w uJHL b b

sb( WdES Uu UO b XU

WOU WLl??{ w??? q??C??H?? u??F?? v sb WdE f

u?? r??K????L?? w??{U??d?? r??U??F??VU s s bL dJW wdJ

bG UL s d vw?? U????H????R???? b?????? t??????d??? s??? d?????? U?????? U w dH tUHRl??b???? w???U???J??? U??????U w qzU WuLiF WU WM d

uDqCH rUF cN uF ULYK q??U??L?? b???? U??M?? w??

UJU

uJH b b W u uJH b b L sL_ dD w uI tOWF u

w uL uJ YO uJHL w tOK c b v b s e ULMO hUM

vK w f

uJH w L uI t udL _ U UU b q w b q w sO_ uL w f VOduJH b UF L oOu u vK uJH b UF WU sJL

uB vK UNU sJL cVOdvK w

dO_ b qUF U _ b qUF YO uJH q b UF w qUL buqUF uJH w L cJ dO_ q b qUF U wU b qUF

cJ UU UL u dO_ b qUF u _ bsb WdE vL w WOU WdEM v UuI c


106/144

uJH U w sb WdE b

sb WdE bU uJH b

uJH b

U

q

U

q

U

q


107/144

sOdA sOeM d_ ZUM d sb uJH bU , WLO b, ,

,

,

,

, , , , , ,

, , , , ,

:Pascal's TriangleUJU YKrUF v W UJU YK rU dF YK s UNd sJL uJH b UF

wK UL dNE c dA lU dI w UJU wdH

U wU nB bF n w WKb q ? wNM ? b n q YKL c w kqJA w `O{u k dU oU nB w UN sOUL sOKbL uL

UJU YK bU uJH VuJ UJU YK s sU nB w UFL dI

Uq

Uq

w UFL w UFL

w UFL

w UFL


108/144

Ub

wK UL q uJH b

?

s q uJH w v_ WF_ b V

s qJ sOdA sOeM d_ WOdI WLO U w sb uJH b [ [

WOLKF WU W bU u s oI sb WdE bU sO

u jQ wK U V


109/144

Uq

uJH w qUF b u vK qLA c b dH

S tOK uKDL qUFL

wK U k uJH s U _ b

U wU b U YU b

U lU b cJ

U??F b??

uJH w fU b b fU b

ndF

edU t ed UF b uJH w tOd c b vL

ndF

u uJH w UF b

Uq


110/144

WE

S Wd XU tOdbUD b UM S WO XU uJH w

wu vK ULNOd sOD sb UM

uJH w sOD_ sb b

V????b

uJH w s wU b W bdH Wud tO uJ c b u s wU b

u s wU b dH

S s uK b UL

lU b u s wU b

uJH w j_ b b

c b uN YU b u j_ b uJO uJHL w b b

b tOK b tI

Uq

U

q


111/144

wK UL q uJH b

wK UL q WLO b sb WdE bU sOdA sOeM d_ [ uOK d_

s q uJH w sOD_ sb j_ b b

uJH w lU b b

WLO UL sdAF U b qUF U uJH w sU b qUF U

qzU sUL

U w sb WdE bU Uu UO b U vK WLI qI

UL `O b

vK WLI qI

U

q


112/144

uJH w s U b bu t X

WLO UL wU u w uJH w UF XU

u V uJH w sU b uJH w U b sO WM XU WLO UL w WOUM

rO WLO UL uJH w U

WLO UL uJH w dA ld b U dA YU b U

q WLO UL U tO fU b U uJH w YU b U s

WOU{ sUL

UbM WLO b WOU WOU uJ uJH w v_ W b XU

U w sb WdE bUUuUOb U vK WLI qI

sOF_ lU b qUF fH t uJH w b

uJH w vK u c b b

uJH w s wU b b


113/144

U) v R w U_ s Y WU d UJ rUF UbM UL rK b

UN b WOM e' sJ1 YO WOzuAF U s dF UF_ c UF_ WU2 bM VJw uIM WIU qd w tMOu UL uNH X b ZzUM pK lOL WuL0 rKF r WbrOUH* c r_ WFd WFd0 bM c

WuL bb lOD UMMJ UNzd q UNO bb lOD w Wd w WOzuAF WdWOze WuL q vL WdK wMOF dH WuLL c vL WdK WMJLL ZzUM lOL

UUwMOF dH ssOOF U ULvL U c u Wd WOUJ s dF b U qJ jd

U u d b sO WM UK wM dJvL U v UOKL uK sJL UL cdJ tM dI c bF vL UK wd ULvb U c Wd d d b

U ULvL b dO U Wd d d U bM UK wd UL wMdH wL UMS uu w UNH WdH wMOF dH dUM lOL UNO uJ w WU WU w

U dUM bdUM b

UdE UL czbM UK sOF ULEM UOMO Ud wMOF

(1)edUE tu WE b d bI WFD w Wd wMOF dH b

WUJ uB uN U u Wd c ZzU

UF bI WFD d d UI Wd wMOF dH bdM d

wMOF dH WdF w bU oU b'

eU q1 dM d vK b WDI bIM WFD vK WUJ uN k

ULNMO d YO

bI WFD

(2)e

(Sample Space & Events)u wMOF UCH


114/144

dH V UU sdUE sbF uL U sJO d d UI w WOzuAF Wd sJrdU UNO{u l dUM pc wMOF

k U* qJA kUIM UOb w dUM

_ ld wW{u*

w WU) u( iF UM s WOze WuL U( U0qO* U( q9 w

jI b dBM vK u c u jO U( tH UGO wMOF UCH u bR* U(

(3)e

U 5Nu vK sbF uL

_ d(

wUd(

UE

V????b

WOU U w dUM b b wMOF UCH V sdUE 5Nu WE b d 5eUL bI wFD w U_ qL% UUD fL s 5UD V

bu U( V u qK fM' V UH WF qOA WOzuAF Wd sJ

d_ vK b b


115/144

W`r dJ uJ UL V bMB s UOzuA d X UCO XO u U

U Wu* dJ

dHu Wu* dJ

UCO XO Wu* dJ

VJ dUM b

dUM b U( UL

(1)e

UL V d bI WFD XOIu vK uB(

d_ vK 5u vK uB(

wK UL dA od s dUM WU sJ1

(2)e

WU WOdWOU WOdv_ WOd

S rEM lu wMO UC dUM s dBM qJ uu Wd fH wDF wMO UC U

u d UCO d U d vK u bM

( )( )( )( )( )( )( )( )

UL uNH


116/144

u vK uB( d_ vK 5u vK uB(

oU w UNLKF w UL 5u rQ dc wK ULOdH U qO* U( UL dH

b U bR* U( UL bu v dH s U UL WLO

b u b UL u UL

UF ULNu UL u UL u UL U UL u UL S 5KBHM 5U( u WU w

UF ULNu UL u UL U u b u UL

(3)ed s d( uJ UL V WOdF WGK d s d UOzuA dOd WLK d s WKF

WK d vK uB( sJO d WLK d

BHM

_


117/144

(4)eUL [ e UdH UL [ u bI v vKO dH UL U WOU UL V [ 5Mb* v UdH

bI v UdH b q_ vK 5sb b v UdH UL

bI v vKO dH sJO

e v vKO dH [[

[[[

qzU? sU9

dH V UUD X U_ UNOK V UUD vK u bM WOU U( w wMOF

U wu vK UUD X

U l wu vK UUD X b WF UUD X

u u u dL u w UUD WuL s WUD X Wu* WUD uJ UL V iO

dL u iO dL u

dL XO qK fM' V UH_ qO Wd wMOF dH V UH WKzU b

WOU UL V uu UH W WKzUF b uJ

v qH sd 5KH WKzUF b uJ

q_ vK d qH WKzUF b uJ

[

[

[

[


118/144

W s W t_vK dUE UIM uL be UL V d U XOI

V bMB s UOzuA d X v s U UNOK V d vK u% WOI UL qL% dJ uJ UL

UO vK WLI qI

vK WLI qI

d bI d uFK WK d uFK UU b d uFK UU rNM UU tO n nB c s UOzuA UU dO WK

WOU UL VWK d VFK bI d VUD VFK 5FK s U VFK

WLO V [ [ [ U

u b u UL


119/144

dAL UL dH w ZuM lOL U dUM sO WF WdFq sOF U UL U VKDc w l b q U UU Q UuKF du UO_ iF w t dO WOzuAF WdK wMOFu dA u UL U sJL u UL vK dOQ U uu uJ b WU

U dUM U dUM sO WF WdF s

ndF

u dA u UL S YO wMO d w sOU U

cJ dF edU t ed

uJ rEML UL wMOF dH WU w

v_ WOd w jIM b be UL U 5OU 5d rEM d d UI bM U U sdUE sbF 5 oKD* dH XLK vK

= ()dUM b = dH

vK v_ WOd w jIM b be U

U sdUE sbF 5 oKD* dH U { }

U vK UNO _ jI* be w dUMWuL{ }

(Conditional Probability & Independent Events)u I dAL UL


120/144

S rEM UL wMOF dH U0 d q

bQ [ [ [ YO U U

fl

?

[ [

[ [[

[ [ [ [

[[

fl

fl

?

[ [[ [

[ [


121/144

sOU lUI UL

rKF

bUI vK qB WEL wU dC WOKL dS

WOU

bU

wu vK Ud tM X su 5d UCO d l vK bM u U

uJ UL U U uK wHK UdJ ? u ULUK UCO WOU UCO v_

WOU v_ v_ WOU v_

[

U* qJA dE??

u ULUK v_ WOU v_

dJv_

dJWOU

"uM

uK UHK UdJ ? UCO WOU u v_ u WOU UCO v_

BHM UU(


122/144

UNO U WL* eF UL [ u U w r uH UL U UUL VU [ u UNO eH r WL* eF UL [ u

eF U* w uH WL* eF U* c w uH

U* c w r* u U dHuJO WL* te U

fl

fl

[ [ [

U* qJA dE

fl

fl

fl

[ [ [

fl

fl

fl

[

[

[

[

[

[

fl

fl

fl

fl

u I UOUfO c dO d u UL w dOQ t U sOU b u WIU WK_ w UMEc uI WU c w d u UL vK Ub u dR u bu ULz WU bU

WKI u

ndF u UL w dR tu b u U U s qI U uI

YO UNM

I d UHdF bUI c cR

bU 5KI uJ


123/144

wu vK Ud tM X u b d UCO d l vK bM u UuJ UL U U l

u d_ UCO ULb UCO ULUK

WOU v_ UCO ULUK

WOU v_

u d_ UCO ULb UCO WOU u v_ u WOU UCO v_

WES WKI u XU

UNO oKD d q w bNK tU UL U S XU b vK UM q oKD U

d l bN vK qd oK UMjI v_ d* w bN VOB UL U jI dO_ v_ 5d* w bN VOB UL U

bN UB UL U ? bN TD U bN VOB U dH

XU d q w bN qd VOB UL

WKI u( WIU WO w WUS dQ U WO w WU UL

jI v_ d* w bN VOB jI dO_ v_ 5d* w bN VOB

bN UB ?


124/144


125/144

s dO w b t dO d U u dA sOF U u UL oU bM w UM

wMOF dH u s U s d uu q U u UL dQ WOzuAF UWMOF d du bM UL c U bM c w ZUFM

ndFdu jI bU WKU WOU) dO u??( UI

UF UOU UdAWKU u(

bU u( { }

sO ndF c s

UL qJA dE

YO wMO d w WKU bU u XU U bQ

S

dH UNM

(Bayes' Theorem)eO WdE


126/144

Total Probability Theorem wKJ UL WdE U wMO d w WKU bU u XU

S

Ud

U* qJA k

bU U0

Bayes' TheoremeO WdES U wMOF dH w bU WKU u XU

VJ* UUD s lD YO ? U W WU VJ w qLF Us [ [ [ XU S UUD) wU ? lD UNM lDUL UOzuA VJ* UUD b dO UD UN VOd vK ? UUD

tF b uJ UL UL QD t UD) U QD t uJ UL

UD) lD U dHUD) lD U UD) ? lD U

U* UD) w QD u U bU WKU u u( uJ

U* qJA dE

fl

[

[

?

fl

[

[

fl

[

[


127/144

[ [ [ [ [ [ [

[

[ [

uH UL U S t U wU t vK tUU s b d od VFK Ub U odH c VF [ u t U uH UL [ t vK

UNO uH UL U t vK uJ UL UL U* w U b t rK

odH V vK U* U dHodH V U U* U

odH u U

sbU 5KU 5U UU( uJ

[

[

[

[

[

tFKvK

tFK

U

[

uH

uH

b

uH

uHb

[ [ [ [

[ [ [

U* w U t vK U*

[[ [


128/144


129/144

sU9

dH s r U WM v WOMODKH UFU b w u UKD ubI sc WKD s U s sObIL l w_ dH s sObIL YK WFU XK S w_ dH s wU wLKF

UL VU UOzuA sObIL b dO wLKF dHF q b t rK wLKF dH s uJ uI uJ WO XU S dL c s nAJK U rL U bK w UJ s U d U

bI F v{d uO r sL bM UC WOU F v{d r sL bM WOU UU cN hHK bK c UJ b

WOU WOM uJ UL U

F UCd hA uJ UL UL WOU U WO XU v hB W XU S UL jI ULKF U Wb w wU_ wU nB fb uI qLF UL [ u WOLOKF WKO qLF UL U U hB

UL UL UOzuA nB c hB b dO [ U WOLOKF WKOWOLOKF WKO UNO qLF

WOLOKF WKO UNO XKLF WLKFLK WB( uJ

tM U wK WOU ZM WOU U wK v_ ZM WOU b tO uKUM UO_ lMB w VOd vK WU WOU v_ U w WUB dO UO_ W XU S WUUOzuA lMBL c U s fO dO

`U dO fOJ uJ UL U WOU U s uJ UL UL U(U U dO c fOJ rK

l sdL sOd vK u u b dL d l vK u UbM

Ud s V dL XU S UNu k bMB s d UOzuA V u dU wu vK Ud s V u XU U l wu vKtH uK s dJ uJ UL U

u d_ sdL UM uJ UL U bMB s uJ UL UL u dJ s jI b d rK ?

W [ U sO UM W XU w WDK Wb w U v uc W XU

sOUM w uc W bQ [ w WbL c w UM W [ uc sO UM


130/144

r U_ s tMuJ sJL UM WF s UuJ s dG b r dJU `L

U WHK V sO s V UO sJL WId rJ WL VJ sO s ULz sOF U WL VJ sO s fO sOF U

s q WLO UL U WLO UL U

dCL e bU wK UL V

?WOU WOU uJ uJH w dA U dUF lU b UF XU

WLO UL

uJH w s wU b uJH w qUF sO WM X w

b v

WKKL uL X U

uJH w qUF b

WU sUL


131/144

l wu vK d bMB s X s WLd d vK bM u jCU sOd rd qL w dJ uN UL U U

UUD dO UU b UNM q qL UUD Uu b UNM q qL UUD sO s

U l wu vKUU 5UD vK sbF d{ qU uJ UL U

Uu 5UD vK sbF d{ qU uJ UL U

d_ uK b bMB s dJ X dL d UCO d vK bM u U

WU W w dL d dNE UL U WFd W w WOU dLK dL d dNE UL U

sOu ULNM q qL sOU dO UOU WU UNM uI lD fL vK bM u d XOI r UOzuA b WFD bMB s X

u uN UL U WU dO Wu* WFDI uJ UL UL u dN t rK

bMB u u d UCO d vK bMB u UbM s d X r v s d XKI u d b UCO d vK

UCO Wu* dJ uJ UL U UCO WuIM* dJ uJ UL UL u XU Wu* dJ rK

uB s tH bF vK B UL U WLDM bI lD sOB s q wd


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Uc w WOFOd WUFL u

w v_ Wb s qu W v bIL qK

b v WKKL uL b

YO U `OB buK WOOFJ c w U bIL WLO bQ UOIOI b

b v WKKL uL b

WUFL q

? bUIU vDF Ub s uL WOU fU Ub b WObM WOUL sO

[ [ [ [ WKKL s b s uL U

sd wbM u w ? XU b u w ? XU

?

X

WK d od WO{U d v UU rb nB WK rOI WO{Ud WOd b p d sJL WId rJ rb UL od rb bI d od rb

sU _ fK YO bUI vK rNU U uK WMJLL dD b V

U Ue fK

WLO UL

U

uJH wqUF U uJH w qUF U

XQ

WU WFd


133/144

uJH w s U b bu t XQ Uu UO b XU

UbM s wU b bI W b r UNUHUC b XU

dH U U U U X U U U U X

U U U U U U U U U X

U U U X

sO lKC l UFMB ? dDI r[ tO { u ?

b VOd vK ULNOUOsdDI s q u

{_ u WU

?

r[

U wu vK UOzuA UUB V WHU UNM OUB vK bM u

UU _ uJ UL UL UHU wU UBL UU_ tN vK qL U dO d d h wd UU d d h wd

5HK 5L vK UM qB UL U

w b vK q_ vK ULb qB UL U

sOGK uLKJ WOdH uLKJ WeOK uLKJ U_ s U U WMb w UOzuA U_ b dO UF

WeOK$ rKJ tQ ULK WOdH rKJ UL U WOdH rKJ tQ ULK WeOK$ rKJ UL U

jI 5GK b rKJ UL U ?

XU

Uw sOU U

I


134/144

WOB WU e u dz l{ WO jI UNM b UU fL WOU qzUL s qJ

WKKL uL b ? dH ?

s qI w WdH WuL WOB b_ lOL d{ qU U ?

?

bIL WLO U ? ?

WLO U WF oI VdL bF ? ?

U U bIL WLOU p dO ? UU ?

YO

XU WO dO WOU UF

? ?

WLO b VOd vK w WOU WOU s v_ WF_ b XU ?

?

UO wzd oK

WKD s Vd s UU WO U s UL WuML qzUL s WuL c


135/144

? U WLO UVOd vK tOFK{ UHBM ld ? UL qJA w

p dO ? ?

U U bIL WLO U

U ? U U ? U U

cJ UMd `O b wIOI b _

bIL WLO U ? ?

U U bIL WLO UL XU p dO ? U U ?

U U U U bIL j U U ? U U U ? U

bu dz w WudL dzb VdL uL w b_ s WuL sO UL qJA u U bF uKI u b_ c b

? ? ? ?

w b_ c s W VdL uL w ld bM lI Wd b WFld bF u UL

?

?

?

?

?

?


136/144

w rO b S U U U U U ? ?

WLO U cJ WdF WOU ? ?

U U U U U bIL

U U ? U U

U ? U

lOL cJ WdF WOIOIb_ s WOU WLO U

? p dO ?

U U U U ? w

UbU ? WLO U ?

?

WOUL WuL WOB b_ s uLL tU sJL WOU WOB b_ s

? ?

wU ju U u UF Ub w WOUL pb WOUL c b s b _

? , ? ,


137/144

u q_ vK b u dNE UL S d WLEM bI WFD UI bM p dO

?

sdL sOM UCO d wUL vK u bM s U wu vK Ud dO u UCO v_ XU dL WOU uJ UL S u fL

?

nB s UU dO S WU UUM b UJ U U WU tO n bI u sOU UuJ UL S UOzuA

?

UCO dJ s wU w dL dJ s UCO dJ s _ w UbM uJ UL S b d tM X UOzuA sObMB b dO dL dJ s

u UCOp dO

?

d_ vK d uKF tu vK UI uN UL uJ d rEM d d UI bM

?

t_ vK u dNE UL uJ WOU d WLEM bI WFD UI Wd w u v_ WOd w u dN WuKF

?

S wMOF dHK sOKU sbU sOU U p dO ? I

S [ fl

[ U sOKI sOU U U

[

[ ?

[

[


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dC U e c w dF u WO{Ud UF s dOJ U w q ZUd b WO uD l bF dC U L dO dCL c WLO uJ UbM WU b

(Excel)q ZUd bU b dC UwU{ oK

W u bM dNE

WLzU s fxW d q ZUd

d r W bb qJ d ou jG{ W bb s FactFactorial ? UB w Fact WLK

dC wMF


139/144

vK bF dC WLO dNE

WUA

Wb jzU u bM dNE

UL qODL w bF V ou jG{ NumberWLKJ


140/144

qJA w UL Excel ZUd WU UL

WOzuA U bOuExcelZUd b

uU bU UUO WId rKF UUO WOzuAF WMOF XdF WOzUB UMOF Xw UL w UL Excel ZUd od s

c w Excel ZUd b b lL s UNL WOzuA WMO UO d UUO

uD

v s K UL lLL d s d qJ wD

UFL jd w WO WUFL V = Int(250*rand())+1

WUA w UL U ou jG{

ULwMF Integer WLKJ UB w Int WLKJ UB w Rand `O b

wzuA wMF RandomuB WO b wDF WUFL k

sbF sO WOzuA


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d v uL s dOG b_ k w uu* b_U UNH WUF* UMIuu* b_ s nK (A)uLF

uu* s UC nK (C) uLF wvK c b (F) uLF wWMOF UO WOzuA

U uN bM QH bU V UL bF WOK sOL qH vK +dNE uLF fH w WOK bILWOzuAF WMOF w dAF b_

UL WUA w UL UL

WOzuAF WMOF s UL _ bF dNE

UL WUA w UL

tL wzUB lL s UNL WOzuA WMO UO Excel ZUd bVb

bM UNH b_ dNE b WE

UL cN pIOD


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UKDBL W???????????L?zU?

Complex Numbers WdL b_ Imaginary unit WOKO bu

Conjugate

odL bFPolar form WODI uBTrigonometric form WOKL uBModulus UOILAmplitude (argument) WFCube roots of unity `OB buK WOOFJ c

Sequences and Series KL UFUL UOUL General term UF b

Series WKKLArithematic sequerce WOU WOULArithematic means WOU UAr. seq. sum WOU WKKL uLGeometric sequence WObMN WOULInfinite Geometric series WOzUN WObMN WKKLMathematical induction w{Ud dI

WOzUB UMOF WOUOI U_ Index Number wUO rBase period U_ sCompared period WUIL sPrice Index No. UF wUOI rdSimple relative price index UF jO wM wUOI rdSimple aggregate price index

UF jO wFOL wUOI rdWeighted Index `dL UOI rdStatistical Population wzUB lLSample WMOSimple Random Sample WDO WOzuA WMOStratified Random Sample WOI WOzuA tMOSystematic Random Sample WLEM WOzuA WMOMulti-stage Sample qdL bF WMOAccidental Sample WO{d WMOPurposive Sample WbB WMOQuota Sample WOBB WMOBias error eO QDSampling error WMUFL WbB QD

Permutations, Combinations and Binomial theorem sb WdE oOu qU Fundamantal counting principle wU_ bF bPermutations qUn factorial ndCCombinations oOuBinomial theorem sb WdEConditional probability dAL ULBays theorem eO WdE


143/144

1) Pre calculus second edition, Larson / Hostetler.

2) The Pre-calculus Problem Solver, Staff of Reseorch and Educa Him Associa, Dr. M. Fogiel, Director.

3) Precalculus Matrematics, A Functional Approach, 2ed. edition, James F. Connelly.

4) Modern Algebra and Trigonometry, 2ed. edition, J. Robison

5) Complete Business Statistics, Amir D. Aczel & Jayavel Sounderpandian, 5thEdition, 2002

6) Understandable Statistics, C.H. Brase & C.P. Brase, 7thEdition, 2003

7) www.uncwil.edu/courses/mat111hp/1zs/complex.html

l????????????????dL


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UJ qL W w uUAL

qOK jU bdU$ bF

u bL

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qOL uLd u UB

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bU d

d bU

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vO UL

wDOD XU

wu' qO

d UM

rU rOJ( bwU dL

u sLd b

d VO

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