BS Maths.pdf

7/25/2019 BS Maths.pdf

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BUSINESS MATHEMATICS

- 1 - Prepared by LIM PHALKUN

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!-niymnyTMnak;TMngeTVFatu fBIsMNMuE eTAsMNMuF CaGnuKmn_kalNaRKb;Fatu x nsMNMuE manrUbPaB y y:ageRcInnsMNMuF .

eKkMnt;sresr FE:f

xfyx:f

EdnkMnt;nGnuKmn_ fKWCasMNMunFatuedIm EdlmanrUbPaBtamGnuKmn_ feKkMnt;sresr xfy/Fy,ExD .

]TahrN_ cUrrkEdnkMnt;nGnuKmn_ xlne3x2xfy 2x2

GnuKmn_enHGackMnt;)anluHRtaEt0x

. dUcenH ,0D

.2-edrIevnGnuKmn_k> niymny ]bmafaeKmanGnuKmn_ fkMnt;elIcenaH I ehIy 0x CacMnYnBitenAkgcenaH I nig h CacMnYnBitxusBIsUnEdl hx 0 enAkgcenaH I .cMnYnedrIevnGnuKmn_ fRtg;cMnuc 0x ebIman CalImItnpleFobrvagGRta

kMeNIn 00 xfhxfy

CamYynwg hxhxx 00

kalNa x

xiteTArksUn . eKkMnt;sresr h

xfhxflim

x

ylimx'f 00

0h0x0

.

]TahrN_ eKmanGnuKmn_)x(V

)x(U)x(fy .

eRbIniymnycUrbgajfa)x(V

)x(U)x('V)x(V)x('U)x('f'y

0

2

0000

00

.

(MONOVARIABLE FUNCTION)


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eK)anh

)x(f)hx(flim)x('f 00

0h0

)x(V

)x(U)x('V)x(V)x('U

)x(V

)x(U).x('V

)x(V

1.)x('U

)x(V)hx(V

)x(U.

h

)x(V)hx(V

)hx(V

1.

h

)x(U)hx(Ulim

)x(V)hx(Vh

)x(V)x(U)hx(V)x(U)x(V)x(U)x(V)hx(Ulim

)x(V)hx(Vh

)hx(V)x(U)x(V)hx(Ulim

h

)x(V

)x(U

)hx(V

)hx(U

lim

0

2

0000

0

2

0

0

0

0

00

000

0

00

0x

00

00000000

0h

00

00000h

0

0

0

0

0h

dUcenH

)x(V

)x(U)x('V)x(V)x('U)x('f'y

0

2

0000

00

.

]TahrN_ eKmanGnuKmn_ xa)x(fy Edl 1a,0a .edayeRbIniymnycUrbgajfa aln.a)x('f'y 0x00

eK)an h)x(f)hx(f

lim)x('f00

0h0

aln.a

h

1alim.a

h

1aalim

h

aalim 000

00

xh

0h

xh

x

0h

xhx

0h

BIeRBaH alnh

1alim

h

0h

.

dUcenH aln.a)x('f'y 0x00 .

x> rUbmnedrIevnGnuKmn_sMxan; nig rUbmnRKwH

!> 1nn xan'yaxy Edl IRa @>

x2

1'yxy

#>2x

a'y

x

ay Edl IRa


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$> xx e'yey ...7182818.2e CaeKalelakarItenEB %> axax ea'yey Edl IRa ^> aln.a'yay xx Edl 0a .

&> x1

'yxlny *>

bax

a'ybaxlny

(>aln.x

1'yxlogy

a Edl 1a,0a .

!0> 'w'v'u'ywvuy !!> 1nn u'un'yuy !@>

u2

'u'yuy

!#> u'vv'u'yv.uy !$>

2v

u'vv'u'y

v

uy

!%>2v

'v'y

v

1y

!^> uu e'u'yey !&> alna'u'yay uu !*> ulnu'vuv'u'yuy 1vv

]TahrN_ cUrKNnaedrIevnGnuKmn_ xxln1

ey

Edl 0x .tamrUbmn UU e.'U')e( eK)an

x

xln1

2x

xln1

2x

xln1

e.x

xln11e.x

)xln1()'x(x)'xln1(e')x

xln1('y

dUcenH x

xln1

2e.

x

xln'y

.


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]TahrN_ cUrKNnaedrIevnGnuKmn_x4x2

x

xln)x(fy

eK)anx4x2

x

xlnlnyln

rUbmn VlnUln

V

Uln,UlnVUln V

xln)xln(ln)x4x(yln2

eFIVedrIevelIGgTaMgBIrnsmPaBenHeK)an

x

1

xlnx

1xlnxlnln)4x2(

y

'y

y.x

1

xlnx

1xlnxlnln)4x2('y

dUcenH x4x2)x

xln(.

xlnx

xln1)xlnxln)(ln2x(2)x('f'y

K> edrIevbnbnab;

]bmafa fCaGnuKmn_manedrIevTIn elIcenaH I .EdlehAfaGnuKmn_edrIevTI n nGnuKmn_ )x(fy KWCaGnuKmn_kMnt;tageday

xfdx

fdy )n(

n

n)n( Edl *INn .

X> rUbmnedrIevTI

nnGnuKmn_plKuN

]bmafaeKmanGnuKmn_ v.uy Edl u nig v CaGnuKmn_BIr .eKman u'vv'u'y ! - !

''uv'v'u2v''u''y ! - @ - ! '''vu''v'u3'v''u3v'''u'''y !- # - # - !

)4()3()3()4()4(

uvv'u4''v''u6'vu4vuy ! - $ - ^ - $ - ! ----------------------------------------------------------dUcenHeK)anrUbmnTUeTA

)n(nn)1n(1n

n

)2n(2

n

)1n(1

n

)n(0

n

)n()n( uvCv'uC.........''vuC'vuCvuCv.uy


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3-Gnuvtn_edrIevnGnuKmn_k> GnuKmn_cMNayma:CINl ( Marginal Cost Function )

]bmafaeKmanGnuKmn_cMNaysrub ( Total Cost Function )mYykMnt;eday

)x(TCTC Edl x tag[brimaNplitplEdlRtUvplit .EdlehAfa FunctionCostinalargM KWCaGnuKmn_edrIevn FunctionCostTotal eKkMnt;tageday )x('TC

dx

)x(dTC)x(MCMC .

]TahrN_ ]bmafaeKmanGnuKmn_cMNaysrubmYykMnt;eday 100x15x12x4)x(TC 23 Edl x CabrimaNplitplEdlRtUvplit.

k> cUrkMnt;rk Marginal cost functionx> cUrkMnt;rk Marginal costRtg;cMnuc units20x .

dMeNaHRsayk> rk Marginal cost functioneKman 100x15x12x4)x(TC 23

eK)an 15x24x12')100x15x12x4()x('TC)x(MC 223 dUcenH )5x8x4(3)x(MC 2 .

x> kMnt;rk Marginal costRtg;cMnuc units20x eK)an 33541520.2420.12)20(MC 2 .mannyfa ebIeKplitbEnm ! ktaTI@! enaHeKRtUvcMNaybEnm$##% ktarUbiyvt.x> GnuKmn_cMNUlma:CINl ( Marginal Revenue Function )

]bmafaeKmanGnuKmn_cMNUlsrub ( Total Revenue Function )mYykMnt;eday )x(TRTR Edl x tag[brimaNplitplEdlRtUvlk; .

EdlehAfa FunctionvenueReinalargM KWCaGnuKmn_edrIevn


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FunctionvenueReTotal eKkMnt;tageday )x('TRdx

)x(dTR)x(MRMR .

]TahrN_ ]bmafaeKmanGnuKmn_R)ak;cMnUlsrubmYykMnt;eday 2x02.0x40)x(TR Edl x CabrimaNplitplEdlRtUvlk;.

k> cUrKNna )1000(TR rYcBnl;BIlTplenH ?x> cUrkMnt;rk Marginal Revenue FunctionK> cUrkMnt;rk Marginal Revenue Function Rtg; units100x .

dMeNaHRsay

k> KNna )1000(TR rYcBnl;BIlTplenH

eKman 2x02.0x40)x(TR eK)an 000200002000040100002,01000.40)1000(TR 2 dUcenH 00020)1000(TR ktarUbiyvt. lTplenHmannyfakalNaeKlk;plitplcMnYn units1000 enaHeKnwgTTYl)anR)ak;cMnUlsrub 00020 ktarUbiyvt.x> kMnt;rk Marginal Revenue Function

eKman 2x02.0x40)x(TR eK)an x04.040')x02.0x40()x('TR)x(MR 2 dUcenH x04.040)x(MR

K> kMnt;rk Marginal Revenue Function Rtg; units100x eK)an 36440)100(04.040)100(MR .

mannyfaebIeKlk;bEnm !ktaeTot ktaTI!0!enaHR)ak;cMnUlnwgekIneLIgcMnYn #^ ktarUbiyvt.K> GnuKmn_R)ak;cMeNjma:CINl ( Marginal Profit Function )

]bmafaeKmanGnuKmn_R)ak;cMeNjsrub ( Total Profit Function )mYykMnt;eday


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)x(TPTP Edl x tag[brimaNplitplEdlRtUvlk; .EdlehAfa FunctionofitPrinalargM KWCaGnuKmn_edrIevn

FunctionofitPrTotal eKkMnt;tageday )x('TPdx

)x(dTP)x(MPMP .

]TahrN_ ]bmafaeKmanGnuKmn_R)ak;cMenjsrubmYykMnt;eday 750

2

x225x50)x(TP 2 Edl x CabrimaNplitplRtUvlk;.

k> cUrrk )8(TP rYcBnl;BIlTplenH .x> cUrrk Marginal Profit Function.K> cUrrk Marginal Profit FunctionRtg;cMnuc units20x .

dMeNaHRsay

k> rk )8(TP rYcBnl;BIlTplenHeK)an 96750

2

8225850)8(TP 2

dUcenH 96)8(TP ktarUbiyvt. lTplenHmannyfakalNaeKlk;plitplcMnYn * ktaeKnwgTTYl)anR)ak;cMenj (^ ktarUbiyvt.x> rk Marginal Profit Function.eKman 750

2

x225x50)x(TP 2

eK)an2

1

225x

50

2

1

225x2

')225x(50)x('TP)x(MP

22

2

dUcenH2

1

225x

50)x(MP

2

.

K> rk Marginal Profit FunctionRtg;cMnuc units20x eK)an 50.1

2

3

2

12

2

1

625

50

2

1

22520

50)20(MP

2

.

mannyfaebIeKlk;bEnm ! kta TI@! enaHeK)anR)ak;cMenj !>%0 ktarUbiyvt.


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4-brmakmnGnuKmn_( Optimization )enAkgRbtibtikarGaCIvkmeKeRcInCYbnUvbBaaenAcMeBaHmuxEdlRtUvedaHRsay

ehIyral;dMeNaHRsayTaMgenaHeKEtgcg;)anCanicnUvdMeNaHRsayEdlRbesIrbMput

solutionOptimal dUcCa - karkMnt;brimaNplitplEdlRtUvlk;edIm,I[R)ak;cMnUlsrub (Total Revenue )bR)ak;cMeNjsrub ( Total Profit )Gtibrma .-kMnt;brimaNplitplEdlRtUvplitedIm,I[R)ak;cMnaymFmkg1ktaGb,brmabBaaTaMgGs;enHeKGacbkRsay)antamKNitviTaedayeRbIbrmakmnGnuKmn_.

viFIkMnt;rkbrmanGnuKmn_manmYyGefr ]bmafaeKmanGnuKmn_ )x(fy .edIm,IkMnt;rkbrmanGnuKmn_enHeKRtUv

!-KNnaedrIevTImYy )x('f'y @-rkbsrbs;smIkar 0)x('f ]bmafavamanbs 0xx

#-KNnaedrIevTIBIr )x(''f''y rYcrktmelxn )x(''f''y 00 snidan -ebI 0)x(''f 0 enaHGnuKmn_ )x(fy mantmGtibrmaRtg; 0xx .-ebI 0)x(''f 0 minGacsnidan)an .-ebI 0)x(''f 0 enaHGnuKmn_ )x(fy mantmGb,brmaRtg; 0xx .

rMlwksmIkardWeRkTIBIr IRc,b,a,0a,0cbxax2

edIm,IrkcemIysmIkarenHeKRtUv

-KNnabrimaN ac4b2


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-ebI 0 smIkarmanbsBIrepSgKaKW

a2

bx

a2

bx

2

1

-ebI0

smIkarmanbsDbKW a2b

xx 21

.-ebI 0 smIkarKanbskgsMNMucMnYnBit .]TahrN_ ]bmafaeKmanGnuKmn_R)ak;cMnUlsrub 32 xx60x13500xTR Edl x CabrimaNplitplEdleKRtUvlk;.etIeKRtUvlk;plitplenHb:unanktaedIm,I[)ancMnUlsrubGtibrma ?

cUrkMnt;R)ak;cMnUlGtibrmaenaH ?dMeNaHRsay

-KNnaedrIevTImYy 2x3x12013500)x('TR -edaHRsaysmIkar 0)x('TR naM[ 013500x120x3 2

222 21044100)13500)(3(60ac4b

eKTajbs 090321060

x,503

21060

x 21

minyk -KNnaedrIevTIBIr x6120)x(''TR eday 0420506120)50(''TR naM[GnuKmn_mantmGtibrmaRtg;

50x mannyfaedIm,I[R)ak;cMnUlsrubGtibrmaeKRtUvlk;plitplcMnYn units50 ehIyR)ak;cMnUlsrubGtibrmaenaHKW 000,400)50(TR ktarUbiyvt.

]TahrN_ ]bmafaeKmanGnuKmn_R)ak;cMnUlsrub x12600)x(TR nigGnuKmn_R)ak;cMnaysrub 32 xx156000)x(TC Edl xCabrimaNplitpl .cUrkMnt;brimaNplitplEdlRtUvlk;edIm,I[)anR)ak;cMeNjGtibrma ?cUrkMnt;rkR)ak;cMenjGtibrmaenaH ?


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dMeNaHRsay

eKman CostTotalvenueReTotalofitPrTotal eK)an 6000x12600x15xxx156000x12600)x(TP 2332

-KNnaedrIevTImYy 12600x30x3)x('TP2

-edaHRsaysmIkar 0x'TP naM[ 012600x30x3 2 KNna 22 195380253780022512600315' eKTajbs 070

3

19515x,60

3

19515x 21

minyk

-KNnaedrIevTIBIr 30x6)x(''TP .

eday 03060660''TP naM[GnuKmn_mantmGtibrmaRtg; 60x .dUcenHedIm,I[R)ak;cMenjGtibrmaluHRtaEteKlk;plitplcMnYn units60 .R)ak;cMenjGtibrmaenaHKW 000,48060TP ktarUbiyvt.]TahrN_ ]bmafaeKmanGnuKmn_cMNaysrubkgkarplitmYykMnt;eday

2500x29

xxTC

2

Edl x CabrimaNplitplEdl)anplit .

etIeKRtUvplitplitplenHb:un

anktaedIm,I[cMnaymFmGb,brma ?cUrrkR)ak;cMnayGb,brmaenaH ?

:SOLUTION -eKRtUvplitcMnYn units150 -cMnaymFmGb,brmakg!ktaKW 33.35 ktarUbiyvt.

5-GaMgetRkalminkMnt;( Indefinite integrals )k-niymny GaMgetRkalminkMnt;nGnuKmn_ )x(f KWCasMNMunRBImITIvTaMgGs;

nGnuKmn_ )x(f EdleKkMnt;sresr c)x(Fdx).x(f .Edl c CacMnYnefr ehIy )x(f)x('F .


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x-rUbmnGaMgetRkalminkMnt;sMxan;!> cx

1n

1dx.x 1nn

Edl IRc,1n

@> cxlnx

dx

#> cx1

xdx

2 $> cx2

x

dx

%> cedx.e xx ^> cea

1dx.e axax Edl 0a .

&> calna

dx.a

xx

Edl 1a,0a .K-lkNHnGaMgetRkalminkMnt;

!> dx)x(fkdx)x(fk @> dx).x(hdx).x(gdx).x(fdx.)x(h)x(g)x(f

X-rUbmnGaMgetRkalbrGefr!> ]bmafa dx).x(fI ebIeKtag dt).t('dx)t(x

eK)anrUbmn dt).t('.)t(fdx).x(fI .@> ]bmafa dx).x('.)x(fI ebIeKtag dx).x('du)x(u

eK)anrUbmn du.ufdx).x('.)x(fI .

g-rUbmnGaMgetRkaledayEpkeKman dvuduv)vu(d eKTaj duv)vu(ddvu

vduuvvdu)vu(dudv


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dUcenH duvuvdvu .

]TahrN_ KNnaGaMgetRkal dx.x

1xx3x2x4I

2

245

eK)an

dx.

x

1

x

13x2x4I

2

23

cx

1|x|ln

3

x2

4

x4

x

dx

x

dxdxx2dxx4

34

2

23

dUcenH cx

1|x|lnx

3

2xI 34 .

]TahrN_ KNnaGaMgetRkal

1x

dx.x4I

4

7

tag dx.x4dt1xt 34 nig 1tx 4 eK)an

c|t|lntt

dtdtdt.

t

11

t

dt)1t(

1x

)dxx4.(xI

4

34

dUcenH c1xln1xI 44 .

]TahrN_ KNnaGaMgetRkal xlnx

dxI

eKman xln

xdx

xlnx

dxI tag

x

dxduxlnu

eK)an c|xln|lnc|u|lnudu

I

dUcenH c|xln|lnI .

]TahrN_ KNnaGaMgetRkal dx.e)1x4(I x2

tag

x2x2 e

2

1v

dx4du

dx.edv

1x4u

eK)an dx.e2e)1x4(2

1dx4.e

2

1e1x4

2

1I x2x2x2x2


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ce1x42

1cee1x4

2

1 x2x2x2

dUcenH ce1x42

1I x2 .

]TahrN_ dx.exI x2 tag

dx.edv

xux

2 naMeGay

xev

dx.x2du

dx.xe2ex xx2

tag

dx.edv

x2ux

naMeGay

xev

dx.2du

ce2x2x ce2xe2ex

dx.e2xe2ex

x2

xxx2

xxx2

]TahrN_ KNnaGaMgetRkal dx.xlnxI 3

tag

dx.xdv

xlnu3

naMeGay

4x4

1v

dx.x

1du

cx16

1xlnx

4

1

dx.x4

1xlnx

4

1

dx.

x

1.x

4

1xlnx

4

1

44

34

44

6-Gnuvtn_GaMgetRkalminkMnt;k> kMnt;rkGnuKmn_ CostTotal edaysal;GnuKmn_ CostinalargM

]bmafaeKmanGnuKmn_ CostinalargM kMnt;eday xMCMC .GnuKmn_ CostTotal kMnt;eday dx.xMCxTC .

]TahrN_ eKmanGnuKmn_ CostinalargM kMnt;eday 42

xxMC Kitkg1Ex

sRmab;karplit ehIycMNayefresInwg !00 ktarUbiyvt.


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cUrrkGnuKmn_ CostTotal sRmab;ExnImYy ?-tag xTC Ca GnuKmn_ CostTotal eK)an kx4

4

xdx).4

2

x(dx.xMCxTC

2

-edaycMNayefres

I !00 mannyfa 1000TC naM[ 100K .dUcenH 100x4

4

xxTC

2

.

x> kMnt;GnuKmn_ ofitPrTotal edaysal;GnuKmn_ ofitPrinalargM ]bmafaeKmanGnuKmn_ ofitPrinalargM kMnt;eday xMPMP .GnuKmn_ ofitPrTotal kMnt;eday dx.xMPxTP .

]TahrN_ eKmanGnuKmn_ ofitPrinalargM kMnt;eday x230xMP .Edl x CabrimaNplitplEdl)anlk;. eKdwgfaebIeKlk;TMnij !% ktaenaHeKnwg)anR)ak;cMenj @% ktarUbiyvt.cUrkMnt;rkGnuKmn_ ofitPrTotal ?

eK)an kxx30dx.x230dx.xMPxTP 2

eday 2515TP naM[ 200K . dUcenH 200x30xxTP 2 .

K>kMnt;rkGnuKmn_ venueReTotal edaysal;GnuKmn_ venueReinalargM

]bmafaeKmanGnuKmn_ venueReinalargM kMnt;eday xMRMR .GnuKmn_ venueReTotal kMnt;eday dx.xMRxTR

]TahrN_ eKmanGnuKmn_ venueReinalargM mYykMnt;eday 2x3x808000xMR Edl x CabrimaNplitplEdl)anlk;.

cUrkMnt;rkGnuKmn_ venueReTotal ebIeKdwgfa kalNaeKlk;plitpl $0 ktaeKTTYl)anR)ak;cMnUlsrub !(@ 000 erol .


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:SOLUTION 32 xx40x8000xTR .K>kMnt;rkGnuKmn_plitplsrubEdl)anplitkgryHeBl t edaysal;GnuKmn_

TinplBlkm

enAxN t

ebIeKsal;TinplBlkm

manGnuKmn_ tWW enAxN t enaHGnuKmn_plitplsrubEdl)anplit)ankgryHeBl t kMnt;eday

dt.tWtTP .

7-GaMgetRkalkMnt; ( Definite integrals )k-niymny GaMgetRkalkMnt;nGnuKmn_ )x(f kgcenaH b,a kMnt;eday

]x.)(f[limdx).x(f kn

0kk

b

an

Edl kn

abax k

1kkk1kkk x,x,

n

abxxx,

.

x-RTwsbT LeibnizNewton ebI )x(f Cab;kgcenaH b,a nigmanRBImITIv )x(F kgcenaH b,a enaH

GaMgetRkalkMnt;nGnuKmn_)x(f

kgcenaH b,a

kMnt;eday b

a

b

a )a(F)b(F)x(Fdx).x(f Edl )x(f)x('F ).

K-lkNHnGaMgetRkalkMnt; !>

a

a

0)x(f

@> b

a

b

a

dx.)x(fkdx.)x(fk

#> b

a

b

a

b

a

b

a

dx).x(hdx).x(gdx)x(fdx.)x(h)x(g)x(f

$> c

a

b

c

b

a

dx).x(fdx).x(fdx).x(f

%> b

a

a

b

dx).x(fdx).x(f


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X-rUbmnGaMgetRkalkMnt;brGefr!> ebIeKman

b

a

dx).x(fI ebIeKtag dt).t('dx)t(x

cMeBaH b,ax RtUvnwg 21 t,tt eK)anrUbmn

dt).t('.)t(fdx).x(fIb

a

t

t

2

1

@> ebIeKman dx).x('.)x(fIb

a

ebIeKtag dx).x('du)x(u

cMeBaH b,ax RtUvnwg ,u eK)anrUbmn

du).u(fdx).x('.)x(fIb

a

g-rUbmnKNnaGaMgetRkaledayEpk

b

a

b

a

b

a duvvudvu

7-Gnuvtn_GaMgetRkalkMnt;k-tmmFm ( Average Value )

tmmFmnGnuKmn_ )x(f kgcenaH b,a kMnt;eday

b

a

dx.xfab

1 .

x-GtnyFrNImaRty

x0

)x(fy:)c(


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y10x10xy26y10x171980)y,x(TP 22 Edl x CabrimaNplitplRbePTTImYy nig y CabrimaNplitplRbePTTIBIr.etIeKRtUvlk;plitplnImYyb:unanktaedIm,I[)anR)ak;cMenjGtibrma ?-KNnaedrIev 10x26y20'TPand10y26x34'TP yx

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plitplTIBIrcMnYnunits20

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1>KNnaedrIev z,y,x'f,z,y,x'f,z,y,x'f zyx

2>edaHRsayRbBnsmIkar

0)z,y,x('f

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]TahrN_ rktmbrmanGnuKmn_ 490z12y6x4zyxZ 222 -KNnaedrIev 12z2'Z,6y2'Z,4x2'Z zyx

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!> 25y8xz12xy4z4y2x13)z,y,x(fZ 222


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@> 2005z6y4xy4zy2x4z,y,xfZ 222 #> 2004y2x2yz2xz2zy2x3)z,y,x(fZ 222

]TahrN_ eK[GnuKmn_cMNaysrubkgkarplitTMnijbIRbePTkMnt;eday

50000yz6y360x320zy18x4)z,y,x(fTC 222 Edl z,y,x CabrimaNplitplRbePTnImYy .etIeKRtUvplitTMnijRbePTnImYyb:unanedIm,I[cMnayGs;ticbMput ?cUrrkcMNayGb,brmaenaH ?-KNnaedrIev y6z2'Z,z6360y36'Z,320x8'Z zyx

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eday 0576D,0288D,08D 321 naM[GnuKmn_mantmGb,brmaRtg; 60z,20y,40x ehIyR)ak;cMNayGb,brmakgkarplitenaHKW

00040$500007200720012800360072006400

50000)60)(20(6)20(360)40(320)60()20(18)40(4TC 222min

X-rebobkMnt;rkbrmanGnuKmn_manlkxN

]bmafaeKmanGnuKmn_Edl yandx epgpat;lkxN c)y,x(g .edIm,IrkbrmanGnuKmn_enHeKGacedaHRsay)antamBIrrebob

!> rebobTImYyeKRtUvTaj y BIsmIkar c)y,x(g ]bmafa )c,x(y rYcykeTACYskgGnuKmn_


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eK)an )c,x(,xfZ CaGnuKmn_manmYyGefrEdleKGacrk)antamrebobEdleyIg)ansikSarYcmkehIy .

@> rebobTIBIr

edayeRbIemKuNrbs;LaRkg; sMultiplierLagrange EdGnuvtn_dUcteTA -begItGnuKmn_ y,xgcy,xf,y,xF -KNnaedrIev ,y,x'F,,y,x'F,,y,x'F yx

-edaHRsayRbBnsmIkar

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x

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yx

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sndan -ebI 0D enaHGnuKmn_manGtibrma.-ebI 0D enaHGnuKmn_manGb,brma.

]TahrN_ eK[GnuKmn_cMnUlsrubBIkarlk;plitplBIrRbePTkMnt;eday yxx8000)y,x(TR 2 Edl y&x CabrimaNplplRbePTnImYy

ehIyeKdwgfa 040yx etIeKlk;plitplRbePTnImYyb:unanedIm,I[eK)anR)ak;cMnUlGtibrma ?rkR)ak;cMnUlenaH ?eKman yxx8000)y,x(TR 2 eday 040yx enaH x40y eK)an )x40(xx8000)x(TR 2 32 xx40x8000 -rkedrIevTImYy 2x3x808000)x('TR -edaHRsaysmIkar 0x3x808000)x('TR 2 216024001600' eKTajbs 0

3

200

3

16040x,40

3

16040x 21

minyk

-KNnaedrIevTIBIr x680)x(''TR naM[ 0320)40(680)40(''TR


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mannyfaGnuKmn_enHmantmGb,rmaRtg; 40x .dUcenHedIm,I[eKTTYl)anR)ak;cMnUlGtibrmaeKRtUvlk;plitplRbePTTImYycMnYn units40 nig plitplRbePTTIBIrcMnYn units80 .

]TahrN_ eK[GnuKmn_R)ak;cMenjsrubBIkalk;plitplBIrRbePTkMnt;eday xy200)y,x(fTP Edl y&x CabrimaNplplRbePTnImYy

ehIyeKdwgfa 30yx etIeKlk;plitplRbePTnImYyb:unanedIm,I[eK)anR)ak;cMenjGtibrma ?rkR)ak;cMenjenaH ?-begItGnuKmn_ yx30xy200),y,x(F

-KNnaedrIev yx30'F,x'F,y'F yx -edaHRsayRbBn 15yx

0yx30

0x

0y

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101

110

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mannyfaedIm,I[eK)anR)ak;cMenjGtibrma eKRtUvlk;plitplTImYycMnYnunits15 nigplitplTIBIrcMnYn units15 ehIyR)ak;cMenjenaHKW 25$TPmax .


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lMhat;mandMeNaHRsaylMhat;TI1

eKmanGnuKmn_cMNaysrubnkarplit plitplBIrRbePTA

nigB

kMnt;eday 65y10x10xy26y10x17)y,x(fTC 22

Edl y&x CabrimaNplplRbePTnImYy .k-KNnacMNaym:aCINleFobnwg x .x-KNnacMNaym:aCINleFobnwg y .

K-etIeKRtUvplitnUvplitplnImYyb:un

anedIm,I[cMNayGs;ticbMput ?dMeNaHRsayk-KNnacMNaym:aCINleFobnwg x eK)an 10y26x34

x

TC'TC x

x-KNnacMNaym:aCINleFobnwg y

eK)an 10x26y20yTC

'TC y

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-edaHRsayRbBn 20y,15x010x26y20

010y26x34

-KNna 2bac eday 20''TC,26''TCb,34''TCa yyxyxx eK)an 034a,04676680 naM[GnuKmn_mantmGb,brma .dUcenHedIm,I[cMNayGs;ticbMputeKRtUvplit plitplRbePTTImYycMnYn !%nigplitplTIBIrcMnYn @0 .


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lMhat;TI2]bmafa )y,x(TC CaGnuKmn_cMNaysrubnkarplitnUvplitplBIrRbePT .eKdwgfa tcosinalargM eFobnwg x KW y8x2

x

TC

nig tcosinalargM

eFobnwg y KW y2x8yTC ehIycMNayefres

Inwg 50$ .cUrkMnt;rkGnuKmn_cMNaysrub )y,x(TCTC ?dMeNaHRsayrkGnuKmn_cMNaysrub )y,x(TCTC eKman y8x2

x

TC

eK)an )y(Kxy8xdx).y8x2(TC 2 naM[ )y('kx8'TC y eday y2x8'TC

y

TCy

eKTaj)an y2x8)y('kx8

naM[ y2)y('k naM[ cydy.y2)y(k 2 eK)an cyxy8x)y,x(TCTC 22 cMNayefresInwg 50$ enaH 50$c

dUcenH 50yxy8x)y,x(TC 22 .lMhat;TI3]bmafa )y,x(TCTC CaGnuKmn_cMNaysrubnkarplitnUvplitplBIrRbePTEdl y&x CabrimaNplplRbePTnImYy .eKdwgfaDIepr:g;EslnGnuKmn_cMNaysrubenHkMnt;eday

)6y8x4(dx).6y4x4(dTC ehIycMNayefres

Inwg 100$ .k-cUrrk tcosinalargM eFobnwg x nig tcosinalargM eFobnwg y .x-cUrkMnt;rkGnuKmn_cMNaysrub )y,x(TCTC .


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dMeNaHRsayk-rk tcosinalargM eFobnwg x nig tcosinalargM eFobnwg y tamTMnak;TMng )6y8x4(dx).6y4x4(dTC

eKTaj)an 6y4x4xTC'TC x Ca tcosinalargM eFobnwg x nig 6y8x4

y

TC'TC y

Ca tcosinalargM eFobnwg y.

x-kMnt;rkGnuKmn_cMNaysrub )y,x(TCTC eday

)y(kx6xy4x2dx).6y4x4(TC6y4x4

x

TC 2

eK)an)y('kx4'TC y

eday6y8x4

y

TC'TC y

eKTaj 6y8x4y'kx4

cy6y4dy).6y8()y(k6y8)y('k 2 eKTaj cy6y4x6xy4x2)y,x(TCTC 22 eday cMNayefresInwg 100$ naM[ 100$c .

dUcenH 100y6x6y4xy4x2)y,x(TC22

.lMhat;TI4eK[GnuKmn_R)ak;cMenjBIkarlk;plitplBIrRbePTkMnt;eday

y100x280xy4yx5)y,x(fTP 22 Edl y&x CabrimaNplplRbePTnImYy .

k-Find Maginal Profit eFobnwg x Rtg;cMnuc 25,20 .x-Find Maginal Profit eFobnwg y Rtg;cMnuc 25,20 .K-kMnt;brimaNplitplRbePTnImYyedIm,I[eKTTYl)anR)ak;cMenjGtibrma?cUrrkR)ak;cMenjGtibrmaenaH ?


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dMeNaHRsayk-Maginal Profit eFobnwg x Rtg;cMnuc 30,25 eK)an 280y4x10)y,x('f

x

)y,x(f'TC xx

Rtg;cMnuc 30,25 eK)an 150280)30(4)25(1030,25'f'TC xx x-Maginal Profit eFobnwg y Rtg;cMnuc 30,25 eK)an 100x4y2)y,x('f

y

)y,x(f'TC yy

Rtg;cMnuc 30,25 eK)an 60100)25(4)30(230,25'f'TC yy K-kMnt;brimaNplitplRbePTnImYyeKman 100x4y2'TC,280y4x10'TC yx

-edaHRsayRbBn 30y,40x0100x4y2

0280y4x10

-KNna 2bac eday 2''TCc,4''TCb,10''TCa yyxyxx eK)an 010a,041620 naM[GnuKmn_manGtibrma.

dUcenHedIm,I[eKTTYl)anR)ak;cMenjGtibrmaluHRtaEteKlk;plitplRbePTTI1cMnYn units40 nig RbePTTI2cMnYn units30 .lMhat;TI5GnuKmn_tRmUvkarnplitplBIrRbePT YandX kMnt;erogKaeday

y280pandx60p 21 ehIyGnuKmn_cMNaysrubplitplTaMgBIrKW

xy220)y,x(TC Edl y&x CabrimaNplplRbePTnImYy .kMnt;brimaNplitplRbePTnImYyedIm,I[eKTTYl)anR)ak;cMenjGtibrma?cUrrkR)ak;cMenjGtibrmaenaH ?


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dMeNaHRsaytag )y,x(TP Ca FunctionofitPrTotal eK)an )y,x(TC)y,x(TR)y,x(TPTP

20y80x60xy2y2x

xy220y2y80xx60

)xy220(y)y280(x)x60()y,x(TCypxp

22

22

21

-KNnaedrIev 80x2y4'TC,60y2x2'TP yx

-edaHRsayRbBnsmIkar 10y,20x080y4x2

060y2x2

-KNna 2bac eday 4''TCc,2''TCb,2''TPa yyxyxx eK)an 02a,0448 naM[GnuKmn_manGb,brma .dUcenHedIm,I[eKTTYl)anR)ak;cMenjGtibrmaluHRtaEteKlk;plitplRbePTTI1cMnYn units20 nig RbePTTI2cMnYn units10 .


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Tel : 017 768 246

lMhat;Gnuvtn_!> GnuKmn_cMNaynkarplitnUvplitplBIrRbePT A and B kMnt;eday

00015000y6000x4000yx)y,x(fTC 22

.Edl y&x CabrimaNplplRbePTnImYy .cUrkMnt;brimaNplitplEdlRtUvplitedIm,I[R)ak;cMNayGb,brma ?kMnt;rkR)ak;cMNaysrubGb,brmaenaH ?@> GnuKmn_cMNaynkarplitnUvplitplbIRbePT A , B and C kMnt;eday

00060yz6y360x320zy18x4)z,y,x(fTC

222

Edl zandy,x CabrimaNnplitplRbePTnImYy .cUrkMnt;brimaNplitplEdlRtUvplitedIm,I[R)ak;cMNayGb,brma ?kMnt;rkR)ak;cMNaysrubGb,brmaenaH ?#> GnuKmn_R)ak;cMenjBIkalk;nUvplitplbIRbePT A , B and C kMnt;eday

y160x80xz4zy4x5)z,y,x(fTP

222

Edl zandy,x CabrimaNnplitplRbePTnImYy .kMnt;brimaNplitplRbePTnImYyedIm,I[eKTTYl)anR)ak;cMenjGtibrma?cUrrkR)ak;cMenjGtibrmaenaH ?$> GnuKmn_tRmUvkarnplitplbIRbePT A , B , CkMnt;erogKaeday

z80pandy40p,x50p 321 ehIyGnuKmn_cMNaysrubplitplTaMgBIrKW z6y4x220)y,x(TC Edl z&y,x CabrimaNplplRbePTnImYy .kMnt;brimaNplitplRbePTnImYyedIm,I[eKTTYl)anR)ak;cMenjGtibrma?cUrrkR)ak;cMenjGtibrmaenaH ?


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%> ]bmafa )y,x(fTC CaGnuKmn_cMNaysrubrbs;plitplBIrRbePT .Edl y&x CabrimaNplplRbePTnImYy ehIycMNayefresI 60$ .eKdwgfa Maginal Cost eFobnwg x KW 40y6x10

x

TC

Maginal Cost eFobnwg yKW 40y4x6yTC cUrkMnt;rkGnuKmn_cMNaysrubenH ? rYcrkR)ak;cMNayGb,brmankarplit .


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kkaarrRR))aakk;;eeTTaallnniiggkkaarrRR))aakk;;ppbb

1-sBaaN nig RbePTkarR)ak;k>karR)ak;( Interest )karR)ak;KWCaR)ak;EdlGkeRbITun RtUvbg;eTA[mas;Tun edIm,IeRbIR)as;TunenH

kgryHeBlmYykMnt; eBalKWCaR)ak;cMeNjEdl)anmkBItwkR)ak;Edldak;[xIkgryHeBlmYykMnt;.

x>GRtakarR)ak;( Interest rate )

pleFobrvagkarR)ak;Edl)anbg;kgmYyqaM CamYynwgTunEdl)anxIehAfaGRtakarR)ak;.GRtakarR)ak;KitCaPaKry bTsPaK .

]TahrN_ GRtakarR)ak; 09,0%9i .

K>RbePTnkarR)ak;karR)ak;manBIrRbePTKW karR)ak;eTal (Simple interest) nig karR)ak;smas ( Compound Interest ).2-karR)ak;eTal ( Simple Interest )k>niymny

karR)ak;eTalKWCakarR)ak;EdlKitelIR)ak;edImEdl)anxI kgRKb;ryHeBl .karR)ak;rbs;ryHeBlnImYy min)anbUkbBalCamYyR)ak;edIm edIm,IykkarR)ak;enARKaeRkayeToteLIy .


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x>rUbmnKNnakarR)ak;eTalebIeKyksac;R)ak;cMnYn PV R)ak;edIm eTAeFIVvinieyaKkgryHeBl n xYb b RKaedayTTYl)anGRtakarR)ak; i enaHkarR)ak;EdlTTYl)anGacKNnatamrUbmnRKwH

inPVIn .]TahrN_1 elaksarI )anykR)ak;edImcMnYn 000,75$ eTAeFIVvinieyaKkgryHeBl#qaMedayTTYl)ankarR)ak; %20 kgmYyqaM.KNnakarR)ak;EdlelaksarITTYl)an ?

tamrUbmni.n.PVI

n

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k> &% f .x> % Ex .K> @ qaM& Ex . sisSedaHRsayxng .K>rUbmnKNnafGnaKt;( Future Value )

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Tel : 017 768 246

]TahrN_1 elak suKn )anykR)ak;eTAvinieyaKcMnYn 000,000,240 erolkgryHeBl @ qaM% Ex CamYynwgGRtakarR)ak; RbcaMqaM %12i .cUrKNnafGnaKttamkarR)ak;eTal ?

tamrUbmn )i.n1(PVFV eday 2n qaM% Ex )

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sisSedaHRsayxng .3-karR)ak;pb ( Compound Interest )k>niymny

karR)ak;pb KWCaRbePTkarR)ak;EdleRbIkgParkichirBavtryHeBlEvg .karR)ak;pbmannyfakarR)ak;rbs;RKamunRtUvbEnmCamYyR)ak;edImEdl)anykeFIVvinieyaKedIm,IykkarR)ak;enARKabnab; . kgkarR)ak;pb karR)ak;begItkarR)ak;x>rUbmnKNnafGnaKttamkarR)ak;pb

ebIeKykR)ak;edIm PV eTAeFIVvinieyaKedayTTYl)anGRtakarR)ak;RbcaMqaM i tamkarR)ak;pbkgryHeBl n qaM enaHenAcugqaMTIn R)ak;EdleKTTYl)an

bfGnaKtGacKNna)antamrUbmn n)i1.(PVFV

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Tel : 017 768 246

tamrUbmn n)i1(PVFV eday

08.0%8i

4n

000,000,75PV

eK)an ...)08.01(000,000,750FV 4

]TahrN_2 elak sava:t)anykR)ak;cMnYn 000,250$ eTAepJIkgFnaKarmYykgryHeBl$qaM (Ex CamYyGRtakarR)ak; %5i kgmYy .KNnafGnaKt;tamkarR)ak;pb ?

sisSedaHRsayxng .]TahrN_3 eKykR)ak;edImcMnYn 000,450$ eTAeFIVvinieyaKkgryHeBl

$ qaM# Ex .eKdwgfaGRtakarRbcaMqaMKW %12i .k-cUKNnafGnaKttamkarR)ak;eTal .x-cUrKNnafGnaKttamkarR)ak;pb .

sisSedaHRsayxng .

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X-rUbmnKNnaryHeBlvinieyaKtamkarR)ak;pbtamrUbmn n)i1.(PVFV eKTaj)an

PV

FV)i1( n

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I 790,602,23$FV

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m

i .


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dUcenHtmGnaKtkMnt;edaymn

m

i1.PVFV

.

kalNa m eK)an

mn

m m

i1PVlimFV

ebIeyIgtag x1

m

i

b im

x kalNa m enaH x eK)an n.i

n.ix

x

n.x.i

xe.PV

x

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x

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.

BIeRBaHtamrUbmnlImIt ....718281828.2e)x

11(lim x

x

.

dUcenHeK)anrUbmnfGnaKttamGRtakarR)ak;pbbnbnab;eRcIndgkgmYyqaM

minkMnt;n.iePVFV

Edl ....71828.2e CaeKalelakarItenEBr .]TahrN_ eKykR)ak; 000,50$ eTAepJIkgFnaKarmYyedayTTYl)anGRtakarR)ak;

pbbnbnab;eRcIndg minkMnt;kgmYyqaM 06.0%6i .etIkgryHeBl #0 qaMeRkaymk eK)anfGnaKtb:unan ?eRkayBIepJIryHeBl #0 qaMeRkaymk eKnwgTTYl)anfGnaKt

$.....71828.250000$FV3006.0

.


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Tel : 017 768 246

lMhat;Gnuvtn_1-elaksarI ykR)ak;cMnYn 000,000,10$ eTAepJIkgFnaKarmYyCamYy

GRtakarR)ak;RbcaMqaMKW %25i .k> cUrKNnafGnaKt;tamkarR)ak;eTal .x> cUrKNnafGnaKt;tamkarR)ak;pb .

2-elaksarIykR)ak; 000,000,40$ eTAepJIkgFnaKarmYyryHeBl ^ qaM( ExCamYyGRtakarR)ak;pbRbcaMqaM 05.0%5i . cUrKNnafGnaKt ?

3-eKykR)ak;edIm fbcb,n cMnYn 0C eTAeFVIvinieyaKkgryHeBl # qaMCamYyGRtakarR)ak;pbRbcaMqaM %9i . eKdwgfafGnaKtKW 290,950,12$ .

cUrKNnafbcb,n R)ak;edIm ?4-elaksarI )an[elaksava:txIR)ak;cMnYn 000,60$ tamGRtakarR)ak;eTalRbcaMqaM %8 .# qaMeRkaymkelaksarI)anRbmUlTaMgedIm TaMgkarBIelaksava:t

ehIyKat;)anykeTAepJIkgFnaKarmYyCamYyGRtakarR)ak;pbRbcaMqaM %6 .cUrKNnatmGnaKtbnab;BIelaksarIepJIkgFnaKar $ qaM ?

5-elaksava:t)anykR)ak;cMnYn 000,200$ eTAcgkar tamGRtakarR)ak;pbdUcteTA

+ %6i1 kg 1qaM kgryHeBl # qaMdMbUg

+ %8i2 kg 1qaM kgryHeBl $ qaMbnab;+ %10i

3 kg 1qaM kgryHeBl @ qaMcugeRkay .

KNnafGnaKtenAcugqaMTI( ?


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6-elak sarI)anykR)ak;cMnYn 000,40$ eTAepJIkgFnaKamYyCamYyGRtakarR)ak;pbRbcaMqaM %12i .etIelaksarIRtUvepJIFnaKar ryHeBlb:unanedIm,I[)anfGnaKtesInwg $dg

nR)ak;edIm .7-elaksarIykR)ak; 000,100$ eTAepJIkgFnaKarmYytamGRtakarR)ak;pb .

eKdwgfa ebIKat;epIJkgryHeBl *qaMKat;nwgTTYl)anfGnaKtesInwg$dgnfbcb,n. cUrKNnafGnaKtkgryHeBl % qaMeRkaymk ?

8-elaksarI)an[elak A xIR)ak;cMnYn 000,20$ tamGRtakarR)ak;eTal

RbcaMqaM %t .# qaMeRkaymk elaksarI)anRbmUlTaMgedIm TaMgkarBIelak A ehIyykeTA[elak BxIbnCamYyGRtakarR)ak;eTalRbcaMqaM )%4t( .@ qaMbnab;BIdak;[elak BxIelaksarIRbmUlR)ak;TaMgGs;TaMgedIm TaMgkar )ancMnYn 540,85$ . cUrkMnt;rkGRtakarR)ak; t ?9-eKykR)ak;cMnYn 000,000,800$ eTAepJIkgFnaKarmYyCamYyGRtakarR)ak; %6

kgryHeBl 12 qaMtamGRtakarR)ak;pbbnbnab; .cUrKNnafGnaKt ?

10-eKepJIR)ak;kgFnaKarmYy CamYyGRtakarR)ak; %8 kgryHeBl 25qaMtameKalneya)aypbkarR)ak;bnbnab; . eKdwgfafGnaKtesI 000,389,7$ cUrKNnafbcb,n?

11-eKykR)ak;cMnYn 000,600$ eTAeFIVvinieyaKryHeBl ^qaM CamYYyGRtakarR)ak;RbcaMqaM %8i . cUrKNnafGnaKt;tamkarR)ak;eTal ? tamkarR)ak;pb ?tamkarR)ak;pbbnbnab;?


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Tel : 017 768 246

12-elak A )an[R)ak;elak BxIcMnYn 000,20$ tamGRtakarR)ak;eTalRbcaMqaM %t .! qaMeRkaymk elak A )anRbmUlR)ak;TaMgedIm TaMgkarBIelak BehIy)andak;[elak CxIbneTottamGRtakarR)ak;pbRbcaMqaM %t .

# qaMbnab;BIdak;[elak C xIelak A RbmUlR)ak;TaMgedIm TaMgkarsrubTaMgGs;)ancMnYn 282,29$ . cUrKNnaGRtakarR)ak; t ?13-eKykR)ak;cMnYn ! 000 000 erol eTAepJIkgFnaKarmYy CamYYyGRtakarR)ak;

%8 kgmYyqaM .eKdwgfa FnaKar)anelIkTwkcitdl;GkepJIrbs;xn edayGnuvtn_eKalneya)ay karR)ak;pbCaeRcIndgkgmYyqaM minkMnt; .

cUrkMnt;ryHeBlvinieyaKedIm,I [)anfGnaKtes

I& #*( 000 erol .14-elak pln )anykR)ak; 000,80$ eTAdak;epJIkgFnaKar A edayTTYl)anGRtakarR)ak;pbRbcaMqaM %6 . @qaMeRkaymkelak pln)anRbmUlR)ak;TaMgedImTaMgkarBIkgFnaKar A ehIyykeTAdak;epJIkgFnaKar BtamGRtakarR)ak;pbRbcaMqaM %8 .

eKdwgfa FnaKar)anelIkTwkcitdl;GkepJIrbs;xn edayGnuvtn_eKalneya)aykarR)ak;pbCaeRcIndgkgmYyqaM minkMnt; .cUrKNnafGnaKtbnab;BIdak;epJIkgFnaKar BryHeBl#qaMmk .15-elakeBRC )anykR)ak;cMnYn 000,40$ eTAvinieyaKtamGRtakarR)ak; %12 kgmYyqaMpbbnbnab;.

k> cUrKNnafGnaKt EdlelakeBRC TTYl)anebIKat;vinieyaKryHeBl # qaM.x> etIelakeBRCRtUvvinieyaKkgryHeBlb:unaneTIbTTYl)anfGnaKtesInwg @

dgnR)ak;edIm fbcb,n .


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FnlaP b sMNgRbcaMRKa( ANNUITY )

1-niymny k> Annuities : KWCa esrInkarcMNay sg bbg;R)ak;snSMCaRbcaM

elIcenaHeBlmYyeTogTat;esIKa .cenaHryHeBlesIKanImYy ehAfa RKa bkal behAfaxYb EdlCaTUeTAKWmYyqaM.

FnlaPmYyGackMnt;)anluHRtaEteKsal;GMBI

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-FnlaPFm

ta bFnlaPbg;edImRKa KWCakarbUksrubTwkR)ak; Edl)anbg;TaMgGs;enAeBl)anbg;rYcPam .-FnlaPbg;edImRKa KWCakarbUksrubTwkR)ak;Edl)anbg;TaMgGs; enA

eRkaymYyRKabnab;BIkarbg;cugeRkay .


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Tel : 017 768 246

]TahrN_eK)andak;R)ak;snSMkgKNnImYyerogral;cugqaMnUvsac;R)ak;efrcMnYn400$ .cUrkMnt;rkcMnYnsac;R)ak;srubkgKNnIenHenAeBlKat;)andak;R)ak;elIkTI %

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tamrUbmn i

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n

1kkn321A FVFV......FVFVFVFV

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--------------------------------------------------------- 21n1n i1.AFV fGnaKtRKaTI 1n manryHeBl @

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1n

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n

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kA i1.AFV @


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Tel : 017 768 246

-ebIeKman AA......AAA n321 enaHrUbmn@ Gacsresr Ca

i

1)i1()i1(Ai1AFV

nn

1k

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dn

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.


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Tel : 017 768 246

]TahrN_ elaksarI)andak;R)ak;snSMkgKNnIrbs;Kat;cMnYn$00 000 000 erolerogral;cugqaMCamYyGRtakarR)ak;pb %12i .bnab;BIKat;)anbg;R)ak;elIkTI rYc Kat;BMu)anbg;R)ak;bEnm bkdkR)ak;enaHeT

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tamrUbmn dn

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A

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edIm,I sRmab;dkcMNay CabeNIrenAfeRkay .-KWCacMnYnTwkR)ak;EdlkMNt;sgCaRbcaM elIryHeBlesIKacMnYn n dg

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.


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Tel : 017 768 246

]TahrN_ 1 etIeyIgRtUvdak;R)ak;epJIkgFnaKarcMnYnb:unanenAf 2004/01/01 edIm,I[eyIgGacdkvij)ancab;taMgBIf 2005/01/01 mYyqaMcMnYn 1000$ cMnYn $ dg ?

eKdwgfaGRtakarR)ak;pbRbcaMq

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n eday

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edayeFIVkicsnamYytRmUv[elak BsgbMNulenHerogral;cugqaMnUvsac;R)ak;efrCamYyGRtakarR)ak; %20i kgmYyqaM .etIelak BRtUvsgbMNulenHkgmYyqaMb:unan ?tamrUbmn

i

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Tel : 017 768 246

4-mUlniFiTUTat;bMNul ( SINKING FUND )KWCamUlniFisnSMTukedayELk edayrdaPi)al-Rkumhun>>>>>>. l >.

EdlRtUv)aneKeRbI sRmab;TUTat; bMNul tamkMritmYykMnt;.

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4n

20.0%20i

000,200$FVA

dUcenH $.....

1)20.01(

20.0.

20.01

000,200A

4


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Tel : 017 768 246

lMhat;Gnuvtn_!-edIm,IbegInTun GaCIvkrmak;)andak;R)ak;snSMkgKNnIrbs;Kat;erogral;cugqaM

nUvcMnYnsac;R)ak;efr5000$

CamYyGRtakarR)ak;pbRbcaMqaM%20

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#-elaksarI)ansnSMR)ak;kgFnaKaredaybg;CaRbcaMkgmYyqaM1500$

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x> Kitdl;f2007/01/01

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^-If you invest $8,000 per period for the following number of


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Tel : 017 768 246

period ,how much would you have ?a / 7 years at 9 percentb / 40 years at 11 percent

7-You invest a single amount of $12,000 for 5years at 10 percent.At the end of 5years you take the proceeds and invest them for 12

years at 15 percent. How much will you have after 17 years ?8-Mrs.Crawford will receive $6,500 a year for the nex 14 yearsfrom her trust. If an 8 percent interest rate is applied, what is thecurrent value of the future payments ?9-At a growth (interest) rate of 8 percent annually, how long willit take a sum to double ? To triple ? Select the year that is closestto the correct answer .10-How much would you have to invest to day to receive:

a / $12,000 in 6 years at 12 percent ?

b / $15,000 in 15 years at 8 percent ?c / $5,000 each year for 10 years at 8 percent ?d / $40,000 each year for 40 years at 5 percent ?

11-Your grandfather has offered you choice of one of the threefollowing alternatives:$5,000 now , $1,000 a year for eight , or$12,000 at the end of eight years. Assuming you could earn 11 percentannually, which alternative should you choose ? If you could earn 12

percent annually, would you still choose the same alternative ?

12-elak A RtUv)anelak BbwgehIytulakar)antRmUv[elak AsgfCMgWciteTA[elak B tamlkxNdUcxageRkam -ebIbg;kgfenHKWRtUsgcMnYn 000,10$ .-ebIbg;sgCaerogral;qaMsRmab;%qaM KWkgmYyqaMRtUvbg; 500,2$ .-ebIbg;sgkgfenH 000,6$ rg;caM @qaMeTotKWRtUvbg;sg 600,5$ .

]bmafaGRtakarR)ak; %8 RtUv)aneKGnuvtn_. etIelak A KYeRCIserIsyklkxNmYyNa sRmab;bg;sgeTA[elak B ?


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Tel : 017 768 246

vviiPPaaKKeellIIbbnnSSMM

1>hVak;tUErl Factorial EdlehAfahVak;tUErlncMnYn n CaplKuNn n cMnYnKt;viCmandMbUg b

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]TahrN_ 1!1

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3628809.8.7.6.5.4.3.2.1!9

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.....n)1n)(2n()!3n(n)1n()!2n(n)!1n(!n

sMKal; 1!1!0 .


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Tel : 017 768 246

]TahrN_ eKeGay)!1n(

n......

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sMKal;tMerobminsareLIgvijKitBIrlMdab;BIeRBaH ABnig BACacMlas;xusKa .


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Tel : 017 768 246

]TahrN_ etIeKGactMerobtYGkSrEdlmanBIrBakxusKaecjBItYGkSrEdlmanbYnBakLOVE xusKa)anb:unanrebob ?

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Tel : 017 768 246

4>bnSMminsareLIgvij nCombinatio bnSMp FatukgcMeNam n FatuCatMerobminKitlMdab;EdlkMnt;eday

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eKtagbnSMsareLIgvijnp

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.


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]TahrN_cUrrkbnSMnBItYrGkSrecjBItYrGkSrbYn D,C,B,A EdlGkSrnImYyGacmaneRcIndg

tambMrab;bnSMntYGkSrCabnSMsareLIgvij .

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lMhat;Gnuvtn_

!> KNnaplbUk !n.n......!3.3!2.2!1.1Sn

.@> etIcMnYn !2005 manelxsUnb:un

an ?#> KNnatMl 2

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0

n


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#$> k>KNnaGaMgetRkal 1

0

n2

ndx.)x1(I

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BS Maths.pdf

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