Visa Matematika

YLD PDWHPDWLND/ L Qlnlfd Xjohl f

l

ll

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l{46 6 7 9 9 9 : ; 43 44 44 45 46 47 48 49 49 56 5< 63 66 6< 74 75 76 77 79

4 VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1

6

415

416

417

418

ly

VDGU]DM

5 OLQHDUQD DOJHEUD514

515

516

PDWULFH L GHWHUPLQDQWH 1 1 1 1 1 1 1 1 1 1 1 51414 Pdwulfd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51415 Ghwhuplqdqwd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51416 Pdwullq udqj 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 f 51417 Umhdydqmh vxvwdyd olqhduqlk mhgqdgded 1 1 1 } 51418 Ymheh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 } YHNWRUVND DOJHEUD 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51514 Xvpmhuhqd gxlqd l yhnwru 1 1 1 1 1 1 1 1 1 1 1 } 51515 Yhnwru x suryrnxwqrp nrruglqdwqrp vxvwdyx 51516 Yhnwruvnd pqrhqmd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 } 51517 Ymheh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 } DQDOLWLFND JHRPHWULMD 1 1 1 1 1 1 1 1 1 1 1 1 51614 Sudydf x survwrux l x udyqlql 1 1 1 1 1 1 1 1 1 51615 Udyqlqd x survwrux 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51616 Phxrgqrvl wrdnd/ sudydfd l udyqlqd 1 1 1 1 f 51617 Qhnh nulyxomh l sorkh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51618 Sroduql/ flolqgulql l vihuql nrruglqdwql vxvwdyl f 51619 Ymheh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 }

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

7 +h, Mhvw/ l wr lvwlqlw> +i, Qlmh/ mhu qlmh l}mdyqd uhhqlfd1 f f E / F / 111 Mhgqrvwdyqrvwl udgl/ dnr mh vxg D lvwlqlw slvdw hpr D 6 Vxgryh hpr r}qddydwl yholnlp +pdor qdjqxwlp, odwlqvnlp vorylpd D/ f f @ A +l f f Sulpmhu 41414 Nrmh rg vomhghlk uhhqlfd +ql,vx vxgrylB

7

SRJODYOMH 41 VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1@ B

f f lwdwl= wdx d mhvw wh,/ d dnr mh qhlvwlqlw slvdw hpr D qlmh wh,1

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41415

Rshuludqmh vxgrylpd

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EA B

A A B

B B B

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EA B

A A A

B A B

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EA B

A B A

B A B

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E,

EA B

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=

4141 RVQRYH PDWHPDWLFNH ORJLNH

8

+ly, Hnylydohqflmd1 Dnr vx D l E vxgryl/ rqgd D / E +lwdpr= d yulmhgl f +mhvw, rqgd l vdpr rqgd dnr eh yulmhgl +mhvw,, r}qdxmh vxg nrml mh lvwlqlw f wrqr rqgd ndg red vxgd D/ E lpdmx lvwx lvwlqrvqx yulmhgqrvw1 ]qdnrp f f f / +lwdpr= hnylydohqwqr, r}qdhqx rshudflmx qd}lydpr hnylydohqflmrp1 +Fhvwr vh }d D / E ndh l= D mh qxgdq l gryromdq xymhw }d E > D mh } } hnylydohqw+dq,qr E 1, Sulsdgqd wdeolfd lvwlqrvqh yulmhgqrvwl mhvwD +D/

E,

EA B

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=

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D =D

A B

B A

f Sulpmhu 41415 L}mdyqd uhhqlfd { vh mh urglr sulmh | qlmh vxg1 Rqd srvwdmh vxgrp lp vh }d qhsr}qdqlfh +ydulmdeoh, { l | xyuvwh rguhhqh rvreh1 Wdnr/ f sulpmhulfh/ }d { @ Duklphg l | @ Jdxvv grelydpr lvwlqlw vxg Duklphg vh mh urglr sulmh Jdxvvd1

f } Ghqlflmd 41415 L}mdyqx uhhqlfx nrmd vdgul mhgqx lol ylh qhsr}qdqlfd l nrmd srvwdmh vxgrp ndg vyh wh qhsr}qdqlfh srsulph rguhhqh yulmhgqrvwl/ qd}lydpr rwyruhqrp uhhqlfrp lol +orjlnlp , suhglndwrp1 f fXreldmlor vh rgqrv phx qhsr}qdqlfdpd x suhglndwx r}qddydwl yholnlp f f voryrp1 Wdnr vh suhglndw l} suhwkrgqrjd sulpmhud r}qdxmh v S +{> | ,/ jgmh f mh S rgqrv vh mh urglr sulmh/ d ydulmdeoh { l | vplmx elwl elor nrmh rvreh +xpuol lol wmhohvqr lyl omxgl,1 Srqhndg/ mhgqrvwdyqrvwl udgl/ r}qdnx rgqrvd } phx qhsr}qdqlfdpd udelpr l ndr r}qdnx flmhorjd suhglndwd1 Dnr mh S suhglndw/ rqgd mh +;{,S qryl suhglndw= ]d vydnl+r, { mh S 1 ]qdn ; r}qdxmh qhrguhhqx }dpmhqlfx +}d, vydnl+r,/ d qd}lydpr jd xql0 f yhu}doqlp nydqwlndwrurp1 Qdgdomh/ dnr mh S suhglndw/ rqgd mh + }> zj1

f f Ghqlflmd 41514 Uhl hpr gd mh\

[ +}qdn

\ srgvnxs lol glr vnxsd [ / r}qdnd= f } lwdpr= mh vdgudq x ,/ rgqrvqr/ gd mh [ qdgvnxs vnxsd

4151 VNXSRYL L UHODFLMH

:

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rg [ 1 Qd sulpmhu/ dnr mh [5

Ghqlflmd 41515 Vydnlp vnxsrp [ mh srvyh rguhhq qmhjry w}y1 sduwlwlyql f vnxs / r}qdnd= 5f +lwdpr= gyd qd lnv ,/ hohphqwl nrmhjd vx vyl srgvnxsryl

f @ i>> i{j> i|j> i}j> i{> |j> i{> }j> i|> }j> [ j=

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rqgd mh

41515

Rshuludqmh vnxsrylpd

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9

9

Xqlmd/ suhvmhn l ud}olnd1

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;

SRJODYOMH 41 VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1

41516

Uhodflmh

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vylk xuhhqlk sduryd +{> | , hohphqdwd { 5 [ l | 5 \ 1 Gdnoh/ [ \ @ f { 5 [ a | 5 \ j1 X voxdmx [ @ > b \ @ > vwdyomdpr [ \ @ >1

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9

9

Ghqlflmd 41518 Elqduqrp uhodflmrp qd vnxsx [ vpdwudpr vydnl srg0

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4151 VNXSRYL L UHODFLMH[B

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Ghqlflmd 41519 Qhnd mh

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Wyugqmd 41514 Dnr lqpxp vnxsd Grnd}1

D srvwrml/ rqgd mh mhglqvwyhq1

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plq D @ lqi

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D srvwrml/ rqgd mh mhglqvwyhq/ r}qd0

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SRJOD YOMH 41


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4171

UHDOQL EURMHYL

4 q> s 5 Q mh p+q . s, @ pq . ps l +p . q,s @ ps . qs1 Qdsrnrq/ srwsxqrp lqgxnflmrp +sr p, qlmh whnr grnd}dwl lvwlqlwrvw rylk wulmx wyugqmd= +;p> q 5 Q, p . q 9@ q> +;p> q> s 5 Q, p . q @ p . s , q @ s> +;p> q> s 5 Q, pq @ ps , q @ s1 Vdgd suhod}lpr qd xuhlydqmh vnxsd Q1 Qhnd mh p> q sdu sulurgqlk eur0 mhyd1 Ghqludpr= p ? q +lwdpr= hp mh pdqml+h, rg hq, dnr srvwrml s 5 Q f wdndy gd mh p . s @ q1 L} suhwkrgqlk lqmhqlfd surl}od}l gd mh s mhglqvwyhq l f srwsxqr rguhhq eurmhylpd p l q/ sd r}qdxmhpr s q p +lwdpr= hq f f pdqmh hp lol hq plqxv hp,1 Qdgdomh/ rlwr mh gd p ? q sryodl p 9@ q f f +gdnoh/ ? qlmh uh hnvlyqd uhodflmd,/ wh gd p ? q a q ? p qh prh elwl lvwlqlw } vxg1 Uhodflmx qd Q/ }d nrmx hpr grnd}dwl gd mh xuhdmqd uhodflmd/ ghql0 f udpr ndnr volmhgl= p q / +p ? q b p @ q,1 Mhgqrvwdyqr mh surymhulwl gd vh udgl r sduflmdoqrp xuhdmx/ wh gd/ }d vydnl p> q> s 5 Q/ p q , p . s q . s/ wm1 uhodflmd mh xvnodhqd +nrpsdwleloqd, vd }eudmdqmhp1

Whruhp 41717

Suyr/ 4 4 mhu mh 4 @ 41 Qdgdomh/ 4 ? 4 . 4 @ v+4, sd mh 4 v+4,1 Vdgd vh lqgxnflmrp odnr f srndh gd mh 4 v+q, }d vydnl sulurgdq eurm q1 Exgxl gd mh v^Q` @ Qqi4j } +y1 Whruhp 41714,/ wr mh 4 q }d vydnl q 5 Q1 Wlph mh wyugqmd grnd}dqd1 Vdgd grnd}xmhpr gd mh srwsxql xuhdm/ wm1 gd }d vydnl sdu p> q sulurg0 qlk eurmhyd yulmhgl p q lol q p1 Gryromqr mh x}hwl elor nrml p 5 Q l grnd} suryhvwl srwsxqrp lqgxnflmrp sr q 5 Q1 ]d q @ 4 mh wyugqmd lvwlqlwd/ wrqlmh/ 4 p mhu mh 4 @ plq+Q> ,1 Suhwsrvwdylpr gd mh wyugqmd lvwlqlwd }d f elor nrml q sd grndlpr qmh}lqx lvwlqlwrvw }d q . 41 Dnr mh p q rqgd mh sr } wudq}lwlyqrvwl/ }erj q q . 4/ l p q . 41 Dnr mh sdn q ? p/ rqgd srvwrml f s wdndy gd mh q . s @ p1 Wr sryodl/ }erj 4 s l xvnodhqrvwl uhodflmh v rshudflmrp ./ gd mh q . 4 q . s @ p1 Wlph mh l ryd wyugqmd grnd}dqd1 Suhrvwdmh grnd}dwl mhglqvwyhqrvw1 Suhwsrvwdylpr gd mh l xuhdmqd uhodflmd qd Q/ nrmd mh xvnodhqd vd }eudmdqmhp l }d nrmx mh 4 @ plq+Q> ,1 Wuhed grnd}dwl gd mh/ }d vydnl sdu p> q sulurgqlk eurmhyd/ p q rqgd l vdpr rqgd ndg mh p q1 Qhnd mh suyr p q1 Dnr mh p @ q rqgd/ }erj uh hnvlyqrvwl xuhdmqh uhodflmh/ prud elwl p q1 Dnr mh p 9@ q/ wm1 p ? q/ rqgd srvwrml s }d nrml mh p . s @ q1 Suhwsrvwdylpr surwlyqr/ wm1 gd qh f yulmhgl p q1 Exgxl gd mh xuhdmqd uhodflmd/ prud elwl q p/ d }erj q 9@ p }dsllpr wr ndr q ? p1 Surl}od}l gd mh p . s ? p wr vh surwlyl f lqmhqlfl p p . 4 p . s x +Q> ,/ }d vydnl sdu p> s 5 Q1 +Wyugqmd p v+p, @ p . 4/ p 5 Q/ vh odnr grnd}xmh lqgxnflmrp/ d p . 4 p . s mh srvomhglfd xvnodhqrvwl uhodflmh vd }eudmdqmhp1, Reudwqd lpsolndflmd f p q , p q vh grnd}xmh qd lvwl qdlq1

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53

SRJOD YOMH 41


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Qdsrphqd 41714 +d, Sulplmhwlpr gd vpr x grnd}x jruqmhjd whruhpd xsr0

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4171

UHDOQL EURMHYL

54

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vnxsd sulurgqlk eurmhyd1

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55

SRJOD YOMH 41


Vdplpr xnudwnr jodyqh lqmhqlfh r vnxsx Q= } f Srvwrml wrqr mhgdq vnxs sulurgqlk eurmhyd Q @ i4> 5> 6> > q> j +ehv0 f nrqddq , v elqduqlp rshudflmdpd . +}eudmdqmh , l +pqrhqmh ,> nrmh vx f } dvrflmdwlyqh l nrpxwdwlyqh/ . mh glvwulexwlyqd v re}lurp qd / l v glvnuhwqlp xuhdmhp nrml vh greur srqdd suhpd . l l x nrmhpx mh 4 @ plq+Q> ,= Grelyhqx vwuxnwxux hpr r}qdlwl v +Q> .> > ,1 f f

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4171

UHDOQL EURMHYL

56

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Qhnd lwdwhom grndh ryh mhgqrvwdyqh wyugqmh= f } +l, Vnxs ] mh hnylsrwhqwdq vnxsx Q/ gdnoh/ m]m @ Cf > +ll, Xuhhql vnxs +]> , qhpd qdmpdqmhjd hohphqwd1

41715

Udflrqdoql eurmhyl

Surpdwudmpr gluhnwql surgxnw ] Q @ i+p> q, m p 5 ]> q 5 Qj l qd qmhpx elqduqx uhodflmx ghqludqx sudylorp=+p> q,

/ @ Odnr vh surymhul gd mh ud}uhgehqd uhodflmd/ sd vh vnxs ] Q sr qmrm flmhsd f vnxs vylk wlk qd glvmxqnwqh ud}uhgh ^+ ,` ] Q1 R}qdlpr voryrp ud}uhgd/ wm1 ,` m @ i^+ 5 ] 5 Qj1 Qd vnxsx ghqludpr elqduqh uhodflmh . +}eudmdqmh, l +pqrhqmh, l uhodflmx +pdqmh lol mhgqdnr, }p >q

+

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57

SRJOD YOMH 41


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t

t> = t2 @ 6 ? f hpx mh +p> q,

4171

UHDOQL EURMHYL

58

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rqgd/ dnr mh

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59

SRJOD YOMH 41


T1 X wx vyukx/ vmhwlpr vh suhuh}d

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5;

SRJOD YOMH 41


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4171

UHDOQL EURMHYL

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Ghqlflmd 41717

Nrpelqdflmrp q0wrj uhgd l u0wrj ud}uhgd

vpdwudpr / grsxwdmxl l voxdm u @ 3/ wm1 f f ^4> q`3

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Sulpmhu 41715 Srsllpr vyh nrpelqdflmh guxjrjd ud}uhgd vnxsd

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Whruhp 417148 Nduglqdoql eurm

o N? vnxsd vylk nrpelqdflmd q0wrj uhgd l ? &

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Ghqlflmd 41718

Ydulmdflmrp q0wrj uhgd l u0wrj ud}uhgd^4> q`

vpdwudpr vyd0 f 3 / grsxwdmxl l

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Sulpmhu 41716 Srsllpr vyh ydulmdflmh guxjrjd uhgd vnxsd

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Nrurodu 41716 Nduglqdoql eurm

o Y? vnxsd vylk ydulmdflmd q0wrj uhgd l u0wrj +q u . 4, +q,o 1

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4171

UHDOQL EURMHYL

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Ghqlflmd 41719

Shupxwdflmrp v srqdyomdqmhp +q

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Whruhp 417149 Nduglqdoql eurm

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yomdqmhp +q @ q . . q& ,0wrjd uhgd mhvw

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Ghqlflmd 4171:

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SRJOD YOMH 41


Ghqlflmd vh qd rlw qdlq suhqrvl qd vydnl nrqdql vnxs [ @ i{ > > {? j1 f f f Sulwrp jryrulpr r nrpelqdflmdpd v srqdyomdqmhp u0wrjd ud}uhgd vnxsd [ lol r nrpelqdflmdpd u0wrjd ud}uhgd pxowlvnxsd P [ +qu> u { > > u {? ,1

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f Sulpmhu 41718 Srsllpr vyh nrpelqdflmh v srqdyomdqmhp wuhhjd ud}uhgd

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Qdsrphqd 41716 Sulplmhwlpr gd vh x Ghqlflml 4171: qlwd elwqr qh pl0

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Whruhp 41714: Nduglqdoql eurm N?o vnxsd vylk nrpelqdflmd v srqdyomdqmhp

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uhgd qd}lydpr u @ 3 0 sud}ql vorj1

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4171

UHDOQL EURMHYL

6 u { > > u? {? , qdg vnxsrp f @ i{ > > {? j/ lp vh rguhgl ud}uhg u plqiu m l 5 ^4> q`3 j1 f Sulpmhu 41719 Srsllpr vyh ydulmdflmh v srqdyomdqmhp wuhhjd ud}uhgd vnxsd@

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Nrurodu 41717 Nduglqdoql eurm

vnxsd vylk ydulmdflmd v srqdyomdqmhp q0wrj uhgd l u0wrj ud}uhgd mhvw srwhqflmd qo 1Y?o

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