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Nuclear Physics Notes For A-Level By Hubbak

Apr 06, 2018

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  • 8/2/2019 Nuclear Physics Notes For A-Level By Hubbak

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    _| lv S`f| 5

    @ ojcht ^bvsfos

    Oi`|c`|s ^hec

    H. \bc @ ojcht H|im Mhss @ mdct h`l H|imfo @ mdct Fsi|i~cs Hj~b h Lcohv Dc| h Lcohv Ehmmh Lcohv @ ojcht Cx h|fi`s H|imfo Mhss S`f| " & Cf`s|cf`%s Mhss-C`ctev Cx h|fi`Df`lf`e C`ctev h`l Mhss Lcaco|

    X cs|fi`s ait ^tho|foc @-W O tqc Df`lf`e C`ctev ~ct @ ojci` O tqc @ ojcht Afssfi` @ ojcht A sfi` T |bctaitl%s Hj~bh-^ht|fojc Lcajco|fi` Cr~ctfmc`|

    861

    D. Thlfiho|fqc Lcohv Cr~i`c`|fhj Lcohv Bhja-Jfac* \8

    4

    Lcohv Oi`s|h`| j \bc Dcox ctcj "Dx&Dhoketi `l Thlfh|fi`Thlfh|fi` Flc`|fafoh|fi` Lcajco|fi` ia Hj~bh* Dc|h h`l Ehmmh Thlfh|fi`s dv Mhe`c|fo Afcjls Lcajco|fi` dv Cjco|tfo Afcjls Oji l Obhmdct Thlfifsi|i~c Sscs

    Thlfiho|fqc Cr~ctfmc`| _hac|v

    4>1

    F )

    Tcqfsfi` h`l Crhmf`h|fi` ^tho|foc X cs|fi`s 481

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    861 @ ojcht ^bvsfos

    H. \BC @SOJCHT H\IM

    Mhss @ mdct h`l H|imfo @ mdct

    @ ojci` mdct "mhss ` mdct& H fs |bc ` mdct ia ` ojcht ~ht|fojcs "f.c. `c |ti`s + ~ti|i`s& f` |bc ` ojc s.

    ^ti|i` ` mdct "h|imfo ` mdct& W fs |bc ` mdct ia ~ti|i`s f` |bc ` ojc s. Ait h`c |thj h|im* |bc ` mdct ia ~ti|i`s ufjj cx hj |bc ` mdct ia cjco|ti`s.

    Uc utf|c |bcsc ` mdcts hs aijjius7

    @ ojci` mdct R

    ^ti|i` ` mdct

    ubctc R fs |bc obcmfohj svmdij ia |bc cjcmc`|; c.e.

    84

    o 1

    Crhm~jc

    Efqc |bc oim~isf|fi` f` |ctms ia |bc ` mdcts ia `c |ti`s* ~ti|i`s h`l cjco|ti`s aitchob ia |bc aijjiuf`e `c |thj h|ims7

    4?3

    "h& S. 674

    4?<

    "d& \b. 6>

    < "o& Bc.

    74

    8 "l& B.

    8

    H`suct

    "h& 64 ~ti|i`s 4?3 - 64 = 8

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    +

    @ ojcht ^bvsfos 865

    Crhm~jc

    Objitf`c bhs h` h|imfo mdct ia 85 h`l h` h|imfo ucfeb| ia ?3.

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    Ubc` h ` ojc s cmf|s h` hj~bh7

    F|s ` ojci` ` mdct lcotchscs dv <

    F|s ~ti|i` ` mdct lcotchscs dv 4

    Dc|h Lcohv |h ~ht|fojcs htc cjco|ti`s |thqcjjf`e h| bfeb s~ccls. \bcv htc ~til ocl dv h obh`ef`e f`|i h ~ti|i`h`l "ja 8.cjco|ti`. \bc ~ti|i` fs tc|hf`cl dv |bc ` ojc s

    h`l |bc cjco|ti` fs cmf||cl atim |bc ` ojc s hs h dc|h ~ht|fojc.

    > > Dc|h ~ht|fojcs htc utf||c` f` cx h|fi`s hs g? it c.

    -8 -8

    862 @ ojcht ^bvsfos

    Ubc` h ` ojc s cmf|s h dc|h ~htjfojc7 F|s

    ` ojci` ` mdct s|hvs |bc shmc F|s

    ~ti|i` ` mdct f`otchscs dv i`c

    Ehmmh Lcohv

    Ehmmh thvs htc cjco|timhe`c|fo thlfh|fi` cmf||cl atim `s|hdjc ojcf. \bcv c`hdjc`s|hdjc ` ojcf |i jisc crocss c`ctev.

    Ubc` h ` ojc s efqcs i | ehmmh thvs |bctc fs `i obh`ec f` cf|bct |bc ~ti|i` it` ojci` ` mdcts.

    @ ojcht Cx h|fi`s

    \ui fm~it|h`| ~if`|s7

    \bc s m ia |bc ` ojci` ` mdcts i` di|b sflcs ia |bc cx h|fi` m s| dc cx hj.

    \bc s m ia |bc ~ti|i` ` mdcts i` di|b sflcs m s| hjsi dc cx hj* c.e7

    Crhm~jc

    441

    Th------- 22 %.

    444 T`

    21

    Af`l |bc ~ti|i` h`l ` ojci` ` mdct f` (Ybc aijjiuf`e tcho|fi`s. Ssf`e |bcsc mdcts*s eecs| ubh| chob ~ht|fojc mfeb| dc.

    8< "h& @

    5

    < 85 f + Bc -------- > + R

    4 *2 8.

    "d& ? ?

    B Bc + R 8 4

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    4< <

    "o& @h + Bc 88 4

    45

    Me + R 84

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    H`suct

    8 "h& R = h ~ti|i` ~

    8

    %.* .

    @ ojcht ^bvsfos 866

    >

    "d& R = c it gG f.c. h dc|h ~htjflc -8

    "o& R = h ~ti|i` | Z 88%*Z& H|imfo Mhss S`f| " &

    I`c h|imfo mhss `f| "8 hm & fs |bc mhss ia 8| j 4|b ia h ohtdi`-84 h|im.

    84 I`c h|im ia ohtdi`-84 bhs mhss " Y . 1 >4 R 8>4?&e

    Bc`oc 8 = 8

    -R 84 84

    "1.>4 R 8 >4?&e

    = 8 .11 * R 8 >- 45 ke

    Cf`s|cf`%s Mhss-C`ctev Cx h|fi`

    H mhss* m ke* fs cx fqhjc`| |i h` hmi `| ia c`ctev* C gi jcs* efqc` dv |bc cx h|fi`

    C = mo;j *

    ubctc o fs |bc qcjiof|v ia jfeb|* f.c. ?.> r 8>2 ms -8

    Bc`oc 8 S 77778.11 R 8>;-45 R "? R 8>2&4 = 8.-8> G

    Hjsi 8 = 8.-8>

    Cjco|ti`fo obhtec

    8.-8> = 8.1> R 8>-86

    = 6 . ?8 R 8>2 cQ

    Bc`oc 8 = 6?8 McQ

    Df`lf`e C`ctev h`l Mhss Lcaco|

    \bc mhss ia h ` ojc s fs hjuhvs jcss |bh` |bc mhsscs ia f|s oi`s|f| c`| ~ht|fojcs* h`l |bclfaactc`oc f` |bc mhsscs fs ohjjcl |bc mhss lcaco|.

    Mhss Lcaco| = Mhss ia oi`s|f| c`| ~ht|fojcs~ti|i`s & Mhss ia ` ojc s "`c |ti`s h`l

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    \bc hdiqc cx h|fi` mhv dc scl fa |bc mhsscs ia |bc cjco|ti`s mhv dc `cejco|cl. Ait mitchoo th|c uitk f| fs `cocsshtv |i |hkc |bcm f`|i hooi `|* h`l |bc cx h|fi` |bc` dcoimcs7

    4>> @ ojcht ^bvsfos

    Mhss Lcaco| = Mhss ia oi`s|f| c`| ~ht|fojcs - Mhss ia h|im "cjco|ti`s* `c |ti`sh`l ~ti|i`s&

    \bfs tcl o|fi` f` mhss fs cx fqhjc`| |i h` hmi `| ia c`ctev ohjjcl |bc df`lf`e c`ctevubfob fs |bc c`ctev tcjchscl ubc` h ` ojc s fs aitmcl atim f|s oi`s|f| c`| ~ht|fojcs. F`itlct |i s~jf| ~ |bc ` ojc s f`|i f|soi`s|f| c`| ~ht|fojcs* |bc df`lf`e c`ctev ui jl`ccl |i dc s ~~jfcl.

    \bc cx h|fi` ait h ~issfdjc afssfi` tcho |fi`

    4?3 8 S + `

    64 %% >

    63 8?6 Mi + Jh +

    > 5 aG -8

    Fa |bc mhsscs ia |bc cjco|ti`s mhv dc `cejco|cl* ohjo jh|c* sf`e |bc aijjiuf`e lh|h* |bcc`ctev tcjchscl ~ct h|im ia afssfi` ait |bc hdiqc tcho|fi`7

    Mhss ia ` = 8.>>6 X F

    8?6 Jh = 8?2.6>1

    35 %

    4?3 = 4?3.>4 r 8>4? miG-8

    H`suct

    \bc cx h|fi` fs lfqflcl f`|i |ui ~ht|s h`l |bc |i|hj mhss i` chob sflc ia |bc cx h|fi`fs ai `l7

    4?3

    64

    Tcho|h`|s

    4?3.>6

    >

    8?6

    35 Jh 8?2.6>1

    8 4 ` 4.>82

    >

    4?1.>3? 4?3.2?

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    .. . -7;77.

    @ ojcht ^bvsfos 4>8

    Mhss lca co | == 4?1.>3? - 4?3.2? = >.44? >.4?? R 8.11 R 8>-45 ke = ?.5> R 8>-42 ke

    \bfs oittcs~i`ls |i h` c`ctev* C* efqc` dv7

    C = mo4 ?.5> R 8>-42 R "? R 8> 2&7( ?.?? %R 8X-88 g

    \bc hdiqc c`ctev fs |bc c`ctev tcjchscl ~c*t S h|im.

    4?3.>4 r 8>4? h|ims %:

    8 e ia S oi`|hf`s 1.>4 R 8>4?

    4?3.>48 h|ims

    Bc`oc c`ctev tcjc scl dv 8 e = ?.?? r 8>- 88 r 4.31 r 8>48

    = 2.3 R 8>8> G

    {XSC_\FI@_ AIT ^THO\FOC 0 %

    8. \bc fsi|i~c ia irvec` >-85* bhs h mhss ia 85.>>>262 h`l |bc mhss ia h ~ti|i` fs 8.>>28< * ohjo jh|c |bc df`lf`e c`ctev ia|bc irvec` ojc s f` McQ. "8 = 6?8 McQ &

    4. Ohjo fh|c |bc df`lf`e c`ctev ~ct ` ojci` f` |bc bcjf m* Bc* ` ojc s efqc` |bcaijjiuf`e lh|h7

    Mhss ia ~ti|i` = 8.>>28< Mhss ia `c |ti` = 8.>>262 Mhss ia bcjf m ` ojc s = >83

    "8 = 6?8 McQ &

    ?. H |bitf m h|im ia mhss 4?4.>?2 lcohvs dv |bc cmfssfi` ia i`c hj~bh ~ht|fojc |ih thlf m h|im ia mhss 442.>?8 . Fa |bc hj~bh ~ht|fojc bhs h mhss ia >? * bium ob c`ctev* f` G*fs tcjchscl l tf`e |bfs ~tiocss9

    "8 = 8.11 r 8>-45 ke h`l |bc s~ccl ia jfeb|= ? r 8>2m s-%-8&

    Ssc |bc aijjiuf`e lh|h ait vi t ohjo jh|fi`7

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    4>4 @ ojcht ^bvsfos

    _~ccl ia jfeb| = ?.> r 8>2m s-8

    4 Mhss ia B = ?.?-45 ke

    8

    ? B 3.>>2 R 8>-45 ke 8

    < Bc= 1.1-45ke

    4

    8 ` = 8.153 r 8>-45 ke

    >

    3. L tf`e h a sfi` tcho|fi`* fsi|i~cs ia bvltiec` "lc |ctf m h`l |tf|f m& a sc|iec|bct |i aitm h bcjf m ` ojc s h`l h `c |ti` hooitlf`e |i |bc aijjiuf`ecx h|fi`7

    4 ? B + B

    < 8 Bc + `

    8 8 4 >

    Fa |bc c`ctev tcjchscl ~ct h|im l tf`e |bc a sfi` ~tiocss fs 4.22 r 8>-84 G*ohjo jh|c |bc mhss ia |bc bcjf m ` ojc s ~til ocl f` |bc tcho|fi` sf`e |bcaijjiuf`e lh|h7

    4 Mhss ia B ` ojc s %((7777?.?-45 ke

    8

    ? Mhss ia B ` ojc s = 3.>>2 r 8>-45 ke

    8

    8 Mhss ia ` = 8.153 r 8>-45 ke

    >

    _~ccl ia jfeb| ?.>> r 8>2m s-8

    @-W Ostqc

    Jiik h| |bc Afe tc 5.8 ubfob fs h ~ji| ia |bc ` mdct ia `c |ti`s hehf`s| |bc ` mdct ia~ti|i`s ait hjj k`iu` fsi|i~cs7

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    @ ojcht ^bvsfos 4>70

    @ mdct ia`c |ti`s

    81> 8?- cmf||cts 8

    @

    84.>

    8>>

    2>

    4.>

    >--------- -------- ----------J--------- @ t`dctia > 1>

    Afe tc 5.8

    2> ~ti|i`s W

    _|hdjc ojcf aijjiu h o tqc. H| afts|* jfeb| fsi|i~cs bhqf`e jcss |bh` 4> ~ti|i`s aijjiu|bc s|thfeb| jf`c @ = W* si |bcv bhqc |bc shmc ` mdcts ia ~ti|i`s h`l `c |ti`s.

    Hs |bc h|imfo mhss f`otchscs* ait |bc ` ojcf |i dc s|hdjc |bcv `ccl |i bhqc h etch|ct` mdct ia `c |ti`s |bh` ~ti|i`s. Hdiqc |bfs htch* ubc` |bc ` ojcf dcoimc jhtec*|bcft f`s|hdfjf|v f`otchscs h`l |bcv lcohv dv `ceh|fqc dc|h cmfssfi`. \bcsc ojcf bhqc h m ob etch|ct ` mdct ia `c |ti`s h`l si oi`qct| simc ia |bcsc f`|i ~ti|i`s uf|b |bc cmfssfi` ia h `ceh|fqc dc|h ~ht|fojc.

    \bc ` ojcf bhqf`e |ii mh`v ~ti|i`s oim~htcl |i `c |ti`s oh` ehf` s|hdfjf|v dv h~ti|i` obh`ef`e f`|i h `c |ti`* uf|b |bc cmfssfi` ia h ~isf|fqcjv-obhtecl dc|h ~ht|fojcohjjcl h ~isf|ti`.

    `lf`e C`ctev ~ct @sojci` Ostqc

    _f`oc h smhjj ojc s bhs i`jv h smhjj df`lf`e c`ctev oim~htcl |i h jhtec ` ojc s* f`itlct |i oim~htc df`lf`e c`ctefcs ait lfaactc`| ` ojflcs f| fs dcs| |i oim~htc |bcdf`lf`e c`ctev ~ct ` ojci`. \i uitk |bfs qhj c i | uc lfqflc |bc |i|hj df`lf`e c`ctevait |bc ` ojc s dv |bc ` mdct ia ` ojci`s.

    Ait mis| ` ojflcs |bc df`lf`e c`ctev fs dc|ucc` 5 h`l 6 McQ ~ct ` ojci` d | simc ia|bc jfeb|ct ` ojcf bhqc df`lf`e c`ctefcs smhjjct |bh` |bcsc qhj cs. \bc s|hdjc ` ojcf htc`cht |bc |i~ Ia |bc eth~b sf`oc |bc etch|ct |bc df`lf`e c`ctev |bc etch|ct |bc c`ctevubfob bhs dcc` jis| f` aitmf`e |bc ` ojc s* h`l |bc jiuct |bc c`ctev ia. |bc ` ojc s*ubfob f` | t mhkcs f| mitc s|hdjc. Afe tc 5.4 sbius |bc df`lf`e c`ctev ~ct ` ojci`o tqc7

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    4M @ ojcht ^bvsfos

    fI

    Df`lf`e c`ctev ~ct ` ojci` 2

    "McQ )` ojci`&

    1

    <

    4

    > ---- ---- ---- ---- -----J---- @ mdct ia > 2> 84> 81> 4>> 4 ` ojci`s

    Afe tc 5.4

    @ ojcht Afssfi`

    @ ojcht afssfi` .f`qijqcs h bchqv ` ojc s s~jf||f`e f`|i |ui jfeb|ct ` ojcf uf|b |bc tcjchsc ia h jhtec hmi `| ia c`ctev. Ubc` h S- 4?3 ` ojc s fs f`qijqcl f` afssfi`* hdi | 822 McQ ia c`ctev fs tcjchscl. Mis| ia |bfs c`ctev ~tiqflcs |bc kf`c|fo c`ctev ia |bc

    afssfi` athemc`|s h`l |bc t s| ia f| fs efqc` i | hs |bctmhj c`ctev.

    Afssfi` ia th`f m fs dti eb| hdi | dv sjiu |bctmhj `c |ti`s dcf`e hdsitdcl dv |bc` ojc s |i aitm h` `s|hdjc S- 4?6 ` ojc s ubfob s~jf|s f` |ui* uf|b |bc tcjchsc ia |uiit |btcc ahs|-miqf`e c |ti`s.

    a ojcht A sfi`

    A sfi` f`qijqcs |bc gif`f`e |iec|bct ia |ui jfeb| ` ojcf |i aitm h `cu bchqfct ojc s.Ait crhm~jc* |bc fsi|i~cs ia bvltiec` "lc |ctf m h`l |tf|f m& oh` a sc |iec|bct |iaitm bcjf m hooitlf`e |i |bc aijjiuf`e tcho|fi`7

    4 ? B + B

    < 8 Bc + `

    8 8 4 >

    I` |bc ~til o|fi` ia i`c bcjf m h|im f` |bfs uhv* hdi | 85.1 McQ ia c`ctev htctcjchscl. Qctv bfeb |cm~cth| tcs htc `cclcl |i ~til oc a sfi`.

    T |bctaitl%s Hj~bh-^ht|fojc Lcajco|fi` Cr~ctfmc`|

    Cqflc`oc ait |bc crfs|c`oc ia |bc ` ojc s uhs ~tiqflcl dv T |bctaitl%s cr~ctfmc`|ubfob f`qcs|feh|cl |bc soh||ctf`e ia hj~bh ~ht|fojcs dv |bf` eijl aifj.

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    @ ojcht ^bvsfos 4is

    Crhm~jc

    "h& _|h|c |bc tcs j|s ia cr~ctfmc`|s i` |bc soh||ctf`e ia hj~bh ~ht|fojcs dv h |bf` aifj. Cr~jhf` |bc fm~it|h`oc ia |bcsc tcs j|s.

    "d& F` s ob cr~ctfmc`|s cr~jhf`7

    "f& Ubv |bc aifj sbi jl dc |bf` h`l ~if`| i | h`v lfaactc`ocs f` |bc tcs j|s aitaifjs ia lfaactc`| mh|ctfhjs

    "ff& Ubv |bc hj~bh ~ht|fojcs f`oflc`| i` |bc aifj sbi jl dc f` h `httiu ~hthjjcjdchm

    "fff& Biu |bc soh||ctcl hj~bh ~ht|fojcs mfeb| dc lc|co|cl

    "o& H` hj~bh ~ht|fojc |thqcjs lftco|jv |iuhtls h ` ojc s ubfob mhv dc hss mcl |itcmhf` s|h|fi`htv. Lcsotfdc f` x hjf|h|fqc |ctms biu* l tf`e |bc mi|fi`* |bckf`c|fo c`ctev h`l |bc ~i|c`|fhj c`ctev ia |bc hj~bh ~ht|fojc mhv qhtv uf|b |bcsc~hth|fi` dc|.;qcc` |bc hj~bh ~ht|fojc h`l |bc ` ojc s.

    H`suct

    Q$ "h& F` |bc cr~ctfmc`| h |bf` dchm ia hj~bh ~ht|fojcs uhs lftco|cl h| h |bf` ~fcoc ia eijl aifj. \bc h~~hth| s uhs cqho h|cl iuf`e |i |bc aho| |bh| hj~bh ~ht|fojcs i`jv bhqc h th`ec ia 3 cm f` hft. \bc hj~bh ~ht|fojcs* ha|ct f`|ctho|f`e uf|b |bc aifj* uctclc|co|cl hs ajhsbcs ia jfeb| i` h sof`|fjjh|fi` lc|co|it ubfob oi jl dc miqcl hti `l i` h | t`|hdjc. Afe tc 5.? sbius |bc htth`ecmc`|7

    I";~ht|flcsi toc

    eijl aifj *.-sof`|fjjh|fi`

    lc|7co|it

    Afe tc 5.?

    \bc tcs j|s ia |bc cr~ctfmc`| uctc7

    Mis| ia |bc hj~bh ~ht|fojcs uc`| s|thfeb| |bti eb |bc aifj uf|b jf||jc it `ilcqfh|fi`.

    _imc ia |bc hj~bh ~ht|fojcs uctc soh||ctcl |bti eb h`ejcs ia dc|ucc` 4> h`l 6>.

    H acu "hdi | 8 f` 2>>>& uctc lcajco|cl dv mitc |bh` 6>. \bcv uctc|bctcaitc soh||ctcl dhok |iuhtls |bc f`oflc`| dchm.

    \bc fm~it|h`oc ia |bcsc tcs j|s fs |bh| |bcv7

    _biucl |bh| mis| ia |bc h|im uhs cm~|v s~hoc.

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    8

    4>1 @ ojcht ^bvsfos

    F`lfoh|cl |bh| |bctc uhs h oi`oc`|th|fi` ia mhss h`l ~isf|fqc obhtec h| |bcoc`|tc ia |bc h|im. \bc jhtec h`ejc soh||ctf`e uhs l c Ji cjco|tis|h|fotc~ jsfi` dc|ucc` |bc ~isf|fqcjv-obhtecl hj~bh ~ht|fojc h`l ` ojc s.

    \bc hj~bhs ubfob %di `oc% dhok bhqc mc| |bc ` ojc s %bchl-i`%. \bc aho| |bh| |bfsbh~~c`s qctv thtcjv f`lfoh|cs biu smhjj |bc ` ojc s m s| dc.

    "d& "f& Hj~bh~ht|fojcs htc hdsitdcl dv h |bf` sbcc| ia ~h~ct. F| fs `cocsshtv ait |bceijl |i dc |bf` si |bc hj~bhs oh` ~hss |bti eb. Lfaactc`| mh|ctfhjs mhv dc

    scl d | |bcv sbi jl bhqc h bfeb h|imfo ` mdct "|i f`otchsc |bc obhtec i`|bc ` ojc s& h`l dc hdjc |i dc dch|c` f`|i |bf` sbcc|s.

    "ff& @httiu dchms htc scl si |bc soh||ctf`e h`ejc oh` dc mchs tcl hoo th|cjv.

    "fff& Ubc` hj~bh ~ht|fojcs bf| h sotcc` oih|cl uf|b wf`o s j~bflc* h ajhsb ia jfeb|

    ohjjcl h sof`|fjjh|fi` fs ~til ocl. \bc sof`|fjjh|.fi` lc|co|it fs mi `|cl i` h | t`|hdjc h`l |bc hj~bhs soh||ctcl h| qhtfi s h`ejcs mhv dc lc|co|cl f` h afrclf`|ctqhj.

    Hs |bc hj~bh ~ht|fojc h~~tihobcs |bc ` ojc s* f| jiscs kf`c|fo c`ctev iuf`e |i f|bhqf`e |i li uitk hehf`s| |bc tc~ jsfqc aitocs. Kf`c|fo c`ctev fs jis| h`l cjco|tfo~i|c`|fhj c`ctev fs ehf`cl. Fa |bc hj~bh ~ht|fojc h~~tihobcs h eijl ` ojc s bchli`* |bc` |bc hj~bh ~ht|fojc ufjj s|i~ h| |bc lfs|h`oc ia ojiscs| h~~tihob ubctc f|bhs wcti kf`c|fo c`ctev h`l |bc` miqc dhok f` |bc lftco|fi` f| ohmc* `|fj hjj |bcs|itcl ~i|c`|fhj c`ctev fs obh`ecl f`|i kf`c|fo c`ctev.

    D. THLFIHO\FQC LCOHV Cr~i`c`|fhj Lcohv

    \bc ` mdct ia ` ojcf ia |bc itfef`hj cjcmc`| lcohv cr~i`c`|fhjjv uf|b |fmc hooitlf`e|i |bc eth~b sbiu` f` Afe tc 5.

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    @ ojcht ^bvsfos 4>5

    _f`oc |bc ` mdct ia ` ojcf tcmhf`f`e fs ~ti~it|fi`hj |i |bc ho|fqf|v* |bc eth~b f` Afe tc 5.3 mhv dc id|hf`cl7

    Ho|fqf|v )o~s. F`f|fhj ho|Gqf|v H >

    Bhja ia f`f|fhj ho|fqf|v Hi

    --4

    jfmc)s

    Afe tc 5.3

    Q Bhja-Jfac) \.8. 4

    \bfs fs |bc hqcthec |fmc ait bhja |bc ` ojcf itfef`hjjv ~tcsc`| |i lcohv.

    Crhm~jc

    H shm~jc ia h oct|hf` thlfiho|fqc ` ojflc bhs h bhja-jfac ia 2.> s. Ubh| fs |bc |fmc* f`scoi`ls* tcx ftcl ait scqc`-cfeb|bs ia |bc shm~jc |i lcohv9

    H`suct

    Fa 5)2 bhs lcohvcl* |bctc m s| dc 8)2 ia |bc itfef`hj ` ojflc jca|.

    Ha|ct 8 bhja-jfac |bctc ufjj dc 8)4 jca|* ha|ct 4 bhja-jfqcs bhja ia h bhja f.c. 8)< h`l ha|ct ? $- - .. %t

    bhja-jfqcs |bctc uffj dc bhja ia h x ht|ct* f.c. 8)2 jca|. ? bhja-Gfqc9 = ?* r 2 = -s

    Thlfiho|fqc lcohv fs h th`lim ~tiocss h`l f`qijqcs |bc s~i`|h`ci s cmfssfi` ia~ht|fojcs it thvs atim |bc ` ojc s ia |bc h|im.

    \bc th|c ia lcohv ia |bc ` oj7;;- " -l @& * it ho|fqf|v* fs ~ti~it|fi`hj |i |bc ` mdct ia

    thlfiho|fqc ` ojcf* @* tcmhf`f`e.

    q -l@ Bc`oc -.- h7 @ @i|c |bh| |bc mf` s sfe` fs f`|til ocl dcoh sc |bc ho|fqf|v

    %l| lcotchscs uf|b |fmc.

    F`|til of`e h oi`s|h`| ia ~ti~it|fi`hjf|v* j* ohjjcl |bc lcohv oi`s|h..FG.J |bc cx h|fi`dcoimcs7 ----- - ...

    Q -l@ --=j@

    l|

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    l@ >

    v h

    4>2 @ ojcht ^bvsfos

    l@ -= -j@

    ql|

    F`|ceth|f`e |bc hdiqc cx h|fi` efqcs7

    Ubctc @ = ` mdct ia ` ojcf tcmhf`f`e ha|ct |fmc | @ > = ` mdct ia ` ojcf itfef`hjjv ~tcsc`|

    | = |fmc j = lcohv oi`s|h`|

    H sfmfjht cx h|fi` |i |bc i`c hdiqc mhv dc utf||c` f` |ctms ia |bc ho|fqf|fcs hs aijjius7

    l@ l@ > *

    -= -- c - .. l| l| . ubctc l| = itfef`hj ho|fqf|v "f.c. h| | = >&

    l @| = . . ho|tqt| a|ct . |tmc |

    F| mhv hjsi dc utf||c` f` |ctms ia |bc mhsscs7

    t7-7

    777=7 $ 8

    j.7.7....... ... ... - -----Yg Ubctc m > = itfef`hj mhss h| | = >

    m mhss ha|ct |fmc |

    Crhm~jc

    H thlfiho|fqc si toc lcohvcl |i 8(842|b ia f|s itfef`hj ho|fqf|v ha|ct 3> lhvs. Ubh| fs f|sbhja-jfac9

    H`suct

    @%

    |-Gi

    8 --h`l @ = x c-j| 842 -| c

    \b s - 8- = c 842

    ->.16? R 3> \

    \hkf`e jiec ia di|b sflcs efqcs7

    - .16? R 3> \8

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    = -7t*-=

    @ ojcht ^bvsfos 4|f(G

    Bc`oc \ = 5.8 ~e ia |bc fsi|i~c uctc ~tcsc`|._biu |bh| |bc hmi `| ia |bfs fsi|i~c bhs dcc` tcl ocl |i h~~tirfmh|cjv ?.3 ~e.

    H`suct

    >.16? >.16? Lcohv oi`s|h`| 8

    42 4

    _f`oc |bc ` mdct ia h|ims fs ~ti~it|fi`hj |i |bc mhss* |i uitk i | |bc mhss |bc lcohvaitm jh fs scl7

    @ = @> c -j|* ubctc @ > fs |bc ` mdct ia h|ims itfef`hjjv ~tcsc`| h`l @ fs |bc` mdct ia h|ims ~tcsc`| ha|ct |fmc |.

    @ = 3 r c->.16? r 8< ) 42 = ?.3 ~e

    Crhm~jc 4

    \bc oi `| th|c ia h sij |fi` oi`|hf`f`e 8 e ia ohtdi` cr|tho|cl atim h` h`ofc`|lio mc`| fs 5>> ~ct bi t. Ubc` h sfmfjht sij |fi` oi`|hf`f`e 8 e ia ohtdi` cr|tho|clatim h jfqf`e ~jh`| uhs scl*.|bc oi `| th|c uhs 6>> ~ct bi t. Ubh| fs |bc hec ia |bclio mc`| fa |bc bhja-jfac ia ohtdi` fs 335> vchts9

    H`suct

    Ssf`e |bc cx h|fi` l @| = l@ > c-j|l|

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    48> @ ojcht ^bvsfos

    \bctc fs `i `ccl |i obh`ec |bc oi `| th|cs |i s- F sf`oc |bc shmc `f|s h~~cht i` di|bsflcs si |bc oi`qctsfi` aho|its ui jl oh`ocj i |.

    ^ ||f`e f` |bc ` mdcts efqcs 5>> = 6>> c- >.16GS335>

    >.5552 = c- >.16?|(335>

    \hkf`e jiec ia di|b sflcs efqcs ->.483? = - >.16?|(335>

    _ijqf`e efqcs hec ia lio mc`|* = 85?> vchts

    \bc Dcoxsctcj "Dx&

    H dcox ctcj fs |bc _j `f| ia thlfiho|fqf|v. F| fs |bc ho|fqf|v ia h thlfi` ojflc lcohvf`e h|h` hqcthec th|c ia i`c s~i`|h`ci s `% ojcht |th`sf|fi` ~ct scoi`l* si7

    8 tDx = 8 s -8 "

    Dhoketis`l Thlfh|fi`

    Dhoketi `l thlfh|fi` fs `h| thj thlfh|fi` ubfob fs hjj hti `l s h`l f| mhv dc mchs tclsf`e h Ecfect-M jjct | dc uf|b `i si toc ~tcsc`|. Ubc` mchs tf`e |bc thlfh|fi`

    atim si tocs f| fs `cocsshtv |i |hkc |bc hqcthec dhoketi `l oi `| iqct h 8 >-mf` |c~ctfil h`l |bc` mhkc |bc `cocsshtv hlg s|mc`| |i |bc si toc tchlf`es.

    Dhoketi `l thlfh|fi` itfef`h|cs f` h qhtfc|v ia uhvs; simc fs ~tcsc`| f` tioks f` |bceti `l h`l f` d fjlf`e mh|ctfhjs. M ob ia f| oimcs f` |bc aitm ia oismfo thvs atimi |ct s~hoc.

    Thlfh|fi` Flc`|fafoh|fi`

    ) Crhm~jc q%

    "h&. Thlfiho|fqc lcohv fs h th`lim ~tiocss f` ubfob |bc |fmc ia lcohv ia h` f`lfqfl hj ` ojc s oh``i| dc ~tclfo|cl. \bc lcohv ia h thlfiho|fqc s ds|h`oc aijjius h`cr~i`c`|fhj jhu uf|b h ucjj-lcaf`cl lcohv oi`s|h`|. Cr~jhf` biu |bcsc |uis|h|cmc`|s htc oi`sfs|c`|.

    "d& Vi htc efqc` h smhjj thlfiho|fqc si toc dcjfcqcl |i cmf| |ui lfaactc`| |v~cs iathlfh|fi`. Lcsotfdc |cs|s vi mfeb| ohttv i | |i flc`|fav |bc thlfh|fi`s ~tcsc`|.

    "o& \bc fsi|i~c ia dfsm |b ia mhss ` mdct 4>> bhs h bhja-jfac ia 3 r 8>? s. F| cmf|shj~bh ~ht|fojcs uf|b h` c`ctev ia 2.4 r 8>-8? ].

    "f& _|h|c |bc mch`f`e ia |bc |ctm bhja-jfac.

    Ohjo jh|c ait |bfs fsi|i~c7

    "ff& \bc lcohv oi`s|h`|

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    =

    @ ojcht ^bvsfos 488

    "fff& \bc f`f|fhj ho|fqf|v ia 8.> r 8>- d mijc ia |bc fsi|i~c

    "fq& \bc f`f|fhj ~iuct i |~ | ia |bfs x h`|f|v ia |bc fsi|i~c

    H`suct

    "h& Thlfiho|fqc lcohv fs h 4.t7(4... 4s;;*;;Y *ubfob mch`s f| fs fm~issfdjc |i lc|ctmf`cubc` h ~ht|fo jht ` ojc s ufjj lcohv. Uc oh`* biucqct* cso|fmh|c |bc ~tidhdfjf|v iah ` ojc s lcohvf`e f` h efqc` |fm%c f` |bc shmc uhv |bh| uc oh` cs|fmh|c |bc~tidhdfjf|v ia |btiuf`e h sfr i` h` `dfhscl lfc. \bc ~tidhdfjf|v ia |btiuf`e h sfr"f.c. 8)d|b& i`jv h~~jfcs fa h jhtec shm~jc fs |hkc`. \bc shm~jc ia ` ojcf `itmhjjvdcf`e oi`sflctcl fs jhtec si uc oh` dc ahftjv s tc ia |bc ~tidhdfjf|v.

    \bc th|c ia lfsf`|ceth|fi * l@)l|* lc~c`ls i` |bc ` mdct ia `lcohvcl ` ojcf~tcsc`|* @* f.c7

    l@ ot7.@

    l|

    H oi`s|h`|* ohjjcl |bc lcohv oi`s|h`| H.*fs |bc oi`s|h`| ia ~ti~it|fi`hjf|v f` |bfscx h|fi`.

    l@ \bc cx h|fi` `iu dcoimcs l| -..j.@

    "d& \bc |cs|s mfeb| dc7

    \bc dhoketi `l oi `| fs |hkc` uf|b `i si toc ~tcsc`|. \bfs fs s d|tho|clatim s dscx c`| tchlf`es |i id|hf` |bc oittco|cl oi `|s.

    \bc si toc fs ~jhocl ojisc |i h E-M | dc h`l |bc oittco|cl oi `| id|hf`cl "Afe tc 5.1&7

    :3om

    _i toc bijlct Hdsxtic t bijlct

    Ecfect- M jjct | dc

    Th|c mc tc| it sohjht

    Afe tc 5.1

    \bf` ohtl fs ~jhocl dc|ucc` |bc si toc h`l E-M | dc. H jhtec tcl o|fi` f`|bc oittco|cl oi `| th|c f`lfoh|cs |bc ~tcsc`oc ia h-~ht|fojcs.

    \bc ohtl fs `iu tc~jhocl uf|b hj mf`f m hdi | ? mm |bfok. H jhtec tcl o|fi`f` |bc oi `| th|c f`lfoh|cs |bh| g?uctc cmf||cl.

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    l@

    l@

    484 @ ojcht ^bvsfos

    Fa |bc oi `| th|c fs s|fjj bfebct |bh` |bc itfef`hj dhoketi `l thlfh|fi`* |bc` |bcsi toc m s| dc cmf||f`e v thvs.

    "o& "f& Bhja-jfac fs |bc hqcthec |fmc ait |bc ho|fqf|v |i ahjj |i bhja ia f|s itfef`hj qhj c.

    >.16? >.16? "ff&. Lcohv oi`s|h`|* j = --=

    \8 3 R 8>? 4

    = 8.?21 R 8>-< s-8

    "fff& Ho|fqf|v0* l| = -j@ ubctc @ fs |bc ` mdct ia thlfiho|fqc h|imsitfef`hjjv ~tcsc`|.

    F` 8 mijc |bctc htc 1.>4 r 8>4? h|ims* bc`oc f` 8 r 8>-1 mijcs |bctc htc 1.>4 r 8>4? r 8 r 8>-1 = 1.>4 r 8>85 h|ims.

    Bc`oc l| = -7-8.?21 r 8>-< r 1.>4 r 8>85

    2.?< r 8>8?s-8

    "fq& ^iuct Th|c ia lfsf`|ceth|fi` r C`ctev ia ~ht|fojc tcjchscl

    = 2.?< R 8>8? R 2.4 R 8>)

    = 12.< uh||s

    Lcajco|fi` ia Hj~bh* Dc|h h`l Ehmmh Thlfh|fi`s dv Mhe`c|fo Afcjls

    Ajcmf`e%s jca|-bh`l t jc "oiqctcl jh|ct& mhv dc scl |i ~tclfo| |bc lftco|fi`s "Afe tc 5.5&. Tcmcmdct |i sc |bc lftco|fi` ia |bc oi`qc`|fi`hj o ttc`|.

    o tv eics |bti eb`lcqfh|cl

    Lftco|fi` ia Oi`qcm7fi`cj o ttc`|

    afcjl

    Aitoc

    Ait8oc ) Lf |fi` ia a? Ac|l

    Lftco|fi` ia oi`qc`|fi`hj o ttc`|

    Afe tc 5.5

    Ehmmh thvs* bhqf`e `i obhtec* ~hss |bti eb `lcqfh|cl.

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    %. % @ ojcht ^bvsfos 48?

    Lcajco|fi` dv Cjco|tfo Afcjls

    Hehf`* ehmmh thvs tcmhf` `lcqfh|cl7

    a?- ~ht|fojc +

    8- % % % i"- ~ht| flc Afe tc 5.2

    _ mmhtv ia |bc ^ti~ct|fcs ia Hj~bh* Dc h h`l Ehmmh Thlfh|fi`

    h ~ 8%

    @h| tc

    Obhtec

    Mhss

    Hdfjf|v |i fi`fscehscs

    Hdsit~|fi` "_|i~~cl dv&

    Lcajco|fi` f`cjco|tfo afcjls

    Lcajco|fi` f`mhe`c|fo afcjls

    Oji l Obhmdct

    Bc ` ojc s

    +4

    >> hm

    Mclf m

    ? mm ia HF; 3 m ia hft

    \iuhtls + qc

    ^tclfo|cl dvAjcmf`e%s jca|bh`l t jc ait - qc~ht|fojcs

    Cjco|timhe`c|fothlfh|fi`

    @i obhtec

    @i mhss

    Jiu

    \bfok jchl

    @i lcajco|fi`

    @i lcajco|fi`

    --- .

    \bctc htc |ui |v~cs ia oji l obhmdct d | |bc i`c lcsotfdcl bctc fs |bc LFAAS_FI@\V^C. _cc Afe tc 5.67

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    48< @ ojcht ^bvsfos

    Acj| tf`e sihkcl f` hoi bif

    --------

    Djhok dhsc

    si toc

    3~i`sc |i kcc~ --ltv foc f` oi`|ho|

    uf|b dhsc

    Afe tc 5.6

    F|s i~cth|fi` fs hs aijjius7

    Iuf`e |i |bc jiu |cm~cth| tc `cht |bc dhsc* |bc hft f` |bfs tcefi` dcoimcss ~ctsh| th|cl uf|b hjoibij qh~i t.

    \bc s ~ctsh| th|cl qh~i t fs hdjc |i oi`lc`sc fa f| oh` af`l simc ~ht|fojcs |ioi`lc`sc i`.

    Ubc` |bc thlfh|fi` ~hsscs |bti eb |bc hft* fi`s htc ~til ocl ubfob ~tiqflc sf|csi` ubfob |bc qh~i t oh` oi`lc`sc. \bc |thfj ia fi`s |bctcaitc sbius ~ hs hsctfcs ia smhjj lti~jc|s ia jfx fl.

    "h& h-^ht|fojc \thoks

    _f`oc h-~ht|fojcs htc jhtec |bcv htc eiil fi`fscts h`l jcss jfkcjv |i dc lcqfh|cl.\bcv ~til oc |bfok* s|thfeb| h`l ucjj lcaf`cl |thoks "Afe tc 5.8> &7

    i:-si toc

    Afe tc 5.8>

    "d& a|-^ht|flc \thoks

    8?-~ht|fojcs htc cjco|ti`s h`l iuf`e |i |bcft smhjj sfwc |bcv htc ~iit fi`fscts h`lhtc chsfjv lcqfh|cl. \bc |thoks ~til ocl htc |bf` uhqv jf`cs "Afe tc 5.88 &7

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    F

    @ ojcht ^bvsfos 483

    *!si toc F

    -+---i Z j Z --

    Afe tc 5.88

    "o& v-Thv \thoks

    S`jfkc h it 8? ~ht|fojcs* v thvs m s| bf| |bc h|im lftco|jv f` itlct |i fi`fsc f|. \bfsfs qctv `jfkcjv si |bc |thoks htc `i| ojchtjv lcaf`cl "Afe tc 5.84 &7

    - -

    Thlfifsi|i~c Sscs

    Afe tc 5.84

    "h& F`l s|tfhj h`l Limcs|fo H~~jfoh|fi`s

    Crhm~jc

    _|h|c h`l cr~jhf` ubfob ia |bc thlfiho|fqc si tocs "hj~bh* dc|h it ehmmh& ui jldc |bc mis| s f|hdjc ait sc f`7

    "f& H limcs|fo smikc lc|co|it

    "ff& Crhmf`f`e |bc |bfok`css ia h |bfok s|ccj ohsf`e

    "fff& Mi`f|itf`e |bc |bfok`css ia hj mf`f m kf|obc` aifj l tf`e ~til o|fi`

    H`suct

    "f& \bc si toc ui jl `ccl |i ~til oc thlfh|fi` |bh| ui jl chsfjv dc hdsitdcl dv|bc smikc. Bc`oc |bc `ccl ait h` hj~bh si toc.

    "ff& Hj~bh h`l dc|h ui jl dc chsfjv s|i~~cl dv |bc |bfok s|ccj ohsf`e; biucqct*ehmmh thlfh|fi` ui jl |thqcj |bti eb. H`v ajhus f` |bc s|ccj ufjj dc sbiu`

    ~ hs obh`ecs f` f`|c`sf|v f` |bc ehmmh thvs lc|co|cl dcbf`l |bc s|ccj.

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    481 @ ojcht ^bvsfos

    "fff& Dc|h ~ht|fojcs htc hdsitdcl dv ? mm ia HF. Ait HF |bf``ct |bh` |bfs |bc` mdct ia dc|hs ~hssf`e |bti eb lcotchscs uf|b |bfok`css. \bfs mhv dc scl|i mi`f|it h`l oi`|tij |bc |bfok`css.

    "d& Mclfohj Sscs

    Fa h s ds|h`oc fs k`iu` |i ~jhv h ~ht| f` h ~tiocss f` |bc dilv* f| mhv dc ~issfdjc|i f`|til oc f`|i |bc svs|cm h thlfiho|fqc qctsfi` ia |bh| s ds|h`oc ubfob ufjjdcbhqc crho|jv jfkc f|s `i`-thlfiho|fqc oi `|ct~ht|. \bfs mc|bil c`hdjcs h lisc iathlfh|fi` |i dc h~~jfcl ait |bcth~c |fo ~ t~iscs. F` hllf|fi`* uf|b |bc hfl ia |bccmf||cl thlfh|fi`* ~ht|s ia |bc dilv mhv dc s| lfcl ait lfhe`is|fo ~ t~iscs.Thlfiho|fqc |thocts oh` dc scl |i tcqchj f`aitmh|fi` hdi | ~ht|s ia |bc dilv s obhs |bc djiil* tf`c it h` iteh` .

    Thlfiho|fqc Cr~ctfmc`| _hac|v

    \bc fi`fsf`e caaco| ia thlfh|fi` lhmhecs ocjjs f` |bc dilv h`l oh` jchl |i qhtfi s aitmsia oh`oct. \i ~tcq `| |bfs* |bc aijjiuf`e s|c~s sbi jl dc |hkc` ubc` ~ctaitmf`ethlfiho|fqc cr~ctfmc`|s7

    Thlfiho|fqc si tocs sbi jl `cqct dc bh`ljcl uf|b |bc af`ects. Hjuhvs bh`ljcsf`e |i`es.

    _i tocs sbi jl `i| dc ~if`|cl |iuhtls vi tscja it h`vi`c cjsc .

    O |s i` bh`ls m s| dc oiqctcl h`l bh`ls uhsbcl ha|ct sc. Hj|ct`h|fqcjv* sclfs~ishdjc t ddct ejiqcs.

    _i tocs sbi jl dc kc~| f` h jchl-jf`cl dir h`l hjuhvs tc~jhocl f` |bc dir ha|ctsc. \bc ubijc dir sbi jl dc jiokcl shacjv huhv.

    _bfcjls ia |bfok jchl-ejhss sbi jl dc scl ubc` sf`e bfeb-ho|fqf|v si tocs .

    TCQF_FI@ H@L CRHMF@H\FI@ ^THO\FOC XSC_\FI@_

    8. Oi`sflct |bc hj~bh ~ht|fojc* cjco|ti` h`l ~ti|i`. \bc aijjiuf`e oittco|jv ~jhocs|bc ~ht|fojcs f` itlct ia f`otchsf`e mhe`f| lc ia |bc obhtec |i mhss th|fi "X)m&7

    H Cjco|ti`* ~ti|i`* hj~bh ~ht|fojc

    D Cjco|ti`* hj~bh ~ht|fojc* ~ti|i`O Hj~bh ~ht|fojc* ~ti|i`* cjco|ti`L ^ti|i`* hj~bh ~ht|fojc* cjco|ti`C ^ti|i`* cjco|ti`* hj~bh ~ht|fojc

    4?2

    4. H ` ojc s ia S lcohvs dv cmfssfi` ia i`c hj~bh ~ht|fojc* 64

    aijjiucl dv |bc cmfssfi` ia |ui dc|h ~ht|fojcs. \bc tcs j|f`e ` ojc s aijjiuf`e |bcsc lcohvs bhs h|imfo ` mdct W h`l mhss ` mdct H. Ubfob i`c ia |bcaijjiuf`e sbius |bc oittco| qhj cs ia H h`l W9

    *.

    {

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    @ ojcht ^bvsfos 485

    H w H 4?2 22 D 4?< 6> o 4?< 64 L 4?< 22 C 4?> 26

    ?. F` Ecfect h`l Mhtslc`%s cr~ctfmc`|s |i f`qcs|feh|c |bc `h| tc ia |bc h|im* h qctv|bf` eijl aifj uhs dimdhtlcl uf|b hj~bh ~ht|fojcs. F| uhs ai `l |bh|7

    H Hjj hj~bh ~ht|fojcs uctc lcajco|cl atim |bcft itfef`hj ~h|b D @i hj~bh ~ht|fojcs uctc lcajco|cl atim |bcft itfef`hj ~h|b O @i hj~bh ~ht|fojcs uctc lcajco|cl |bti eb h`ejcs etch|ct |bh` h tfeb| h`ejc L H acu hj~bh ~ht|fojcs uctc lcajco|cl |bti eb h`ejcs etch|ct |bh` h tfeb| h`ejc C Mis| hj~bh ~ht|fojcs uctc lcajco|cl |bti eb h`ejcs etch|ct |bh` h tfeb| h`ejc

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    F

    a 482 @ ojcht ^bvsfos

    1. Fsi|i~cs ia h` cjcmc`| bhqc |bc shmc7

    8 Obcmfohj ~ti~ct|fcs 4 ^ti|i` mdct ? @ ojci` ` mdct

    H`suct7

    H fa 8* 4 h`l ? htc hjj oittco|D fa 8 h`l 4 i`jv htc oittco|O fa 4 h`l ? i`jv htc oittco|L fa 8 i`jv fs oittco| C fa ? i`jv fs oittco|

    5. H` c`ef`cct ufsbcs |i sc h thlfifsi|i~c |i |thoc h othok f` h ~f~c d tfcl hdi | hmc|tc dcjiu |bc eti `l. Ubfob i`c ia |bc aijjiuf`e fsi|i~cs sbi jl dc obisc`9

    Cmfssfi` Bhja-jfac H Hj~bh H acu bi ts D o

    Dc|h Dc|h

    H acu bi ts . _cqcthj vchts

    L Ehmmh H acu bi ts C Ehmmh _cqcthj vchts

    2. \bc bhja-jfac ia thlf m-441 fs 81>> vchts. Ubfob ia |bc aijjiuf`e s|h|cmc`|s fs)htcoittco|9

    8 \bc lcohv oi`s|h`| fs -< vcht -8 4 \bc atho|fi` ubfob tcmhf`s ha|ct 1> vchts fs 8)81 ? \bc atho|fi` ubfob tcmhf`s ha|ct 2>> vchts fs ?)<

    H`suct7

    H fa 8 h`l 4 i`jv htc oittco|D fa 4 h`l ? i`jv htc oittco|O fa 8 i`jv fs oittco| L fa ? i`jv fs oittco|

    6. \bc bhja-jfac ia h oct|hf` thlfiho|fqc fsi|i~c fs 4 lhvs. \bc f`f|fhj oi `| th|c ia hoct|hf` x h`|f|v ia |bfs fsi|i~c uhs mchs tcl hs 84 >>> oi `|s ~ct mf` |c. \bcoi `| th|c* f` oi `|s ~ct mf` |c* ha|ct 2 lhvs ui jl dc7

    H ?>>> D %83>> o 53> L ?53

    8>. "h& Lcaf`c |bc lcohv oi`s|h`| ia h thlfiho|fqc mh|ctfhj h`l sbiu biu f| fs tcjh|cl|i |bc bhja-jfac ia |bc mh|ctfhj.

    "d& H ehmmh thv cmf||ct* ubfob mhv dc oi`sflctcl |i dc h ~if`| si toc thlfh|f`e`faitmjv f` hjj lftco|fi`s* fs sf| h|cl 3> cm huhv atim h ehmmh thv lc|co|it.

    ubfob bhs h` caaco|fqc htch ia 1.> om4. \bc tcoitlcl oi `| th|c h| h efqc`|fmc fs ?4> s -8.

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    @ ojcht ^bvsfos 486

    "f& Cs|fmh|c h qhj c ait |bc ho|fqf|v ia |bc si toc h| |bfs |fmc.

    "ff& Ofqc` |bh| |bc bhjj-jfac fs ?> scoi`ls* ohjo jh|c h qhj c ait |bc ` mdct a thlfiho|fqc h|ims ~tcsc`| f` |bc shm~jc 84> scoi`ls dcaitc |bc mchs tcmc`|uhs mhlc .

    .88. \bc thlfiho|fqc lcohv th|c l@)l| ait h s~cofmc` ubfob oi`|hf`s @ h|ims ia h thlfiho|fqc fsi|i~c fs cjh|cl |i |bc lcohv oi`s|h`| *jhs aijjius7

    l@ = -*|@

    l|

    \bc lcohv th|c ait h shm~jc ia thli` ehs h| h ~ht|fo jht f`s|h`| fs ai `l |i ic 5.< r 8>? Dx "f.c. 5.< r 8>? lfsf`|ceth|fi`s -~ct scoi`l&. \bc lcohv oi`s|h`| |itthli` fs 84.1 r 8>-? s-8.

    "h& Cr~jhf` ubh| vi `lcts|h`l dv |bc lcsotf~|fi` ia thlfiho|fqc lcohv hs hth`lim ~tiocss.

    %

    "d& Ohjo jh|c h qhj c ait |bc ` mdct ia thli` h|ims ~tcsc`| h| |bc ~ht|fo jhtf`s|h`|.

    "o& Ubv fs f| ~issfdjc ait |bc qhj c ia @ ohjo jh|cl f` "d& |i lfaact atim |bc ho| hjqhj c h| |bh| f`s|h`| cqc` uf|bi | cttit f` |bc lh|h9

    "l& Lcaf`c |bc |ctm bhja-jfac h`l ohjo jh|c h qhj c ait |bc bhja-jfac ia thli`.

    "c& \bc c`ctev tcjchscl f` |bc lcohv ia i`c thli` ` ojc s fs 1.? McQ ubctc 8 McQ cx hjs 8.1 r 8>-8? g. Af`l |bc th|c ia c`ctev tcjchsc ait |bc thli` shm~jc hdiqc.

    84. H| |bc s|ht| ia h` cr~ctfmc`| h mfr| tc ia thlfiho|fqc mh|ctfhj oi`|hf`s 4>.> ]je ia hthlfifsi|i~c H*ubfob bhs h bhja-jfac ia 5> s* h`l .> ~e ia thlfifsi|i~c D*ubfobbhs h bhja-jfac ia ?3 s.

    "h& Ha|ct ubh| ~ctfil ia |fmc ufjj |bc mfr| tc oi`|hf` cx hj mhsscs ia chobfsi|i~c9 Ubh| fs |bc mhss ia chob fsi|i~c h| |bfs |fmc9

    "d& Ohjo jh|c |bc th|c h| ubfob |bc h|ims ia fsi|i~c H htc lcohvf`e ubc` |bcmhsscs htc |bc shmc.

    "Mijht mhss ia fsi|i~c H = 4?< e Hqiehlti oi`s|h`| = 1.> r 8>4? mij-8&

    8?. "h& "f& Utf|c liu` h` cx h|fi` cr~tcssf`e |bc jhu ia thlfiho|fqc lfsf`|ceth|fi`s*lcaf`f`e hjj svmdijs scl.

    "ff& Cr~jhf` biu |bc cr~tcssfi` fs h oi`scx c`oc ia |bc th`lim `h| tc iathlfiho|fqc lcohv.

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    ..

    "d& \bc eth~b f` Afe tc 5.8< sbius |bc oi `| th|c l c |i |bc lfsf`|ceth|fi` ia |bcohtdi`-8< f` 8 e ia ohtdi` |hkc` atim lchl ~jh`| mh|ctfhj ~ji||cl hehf`s| |bc hec ia |bc mh|ctfhj7

    83

    ....F.sC ja G F

    ff t.

    o7 778 >

    .X ...G . a( $o$7%G$% " 77j > ".&

    > 4>>> ciii 8>>>> 84.>>> Hec "vchts &

    Afe tc 5.8<

    "f& Id|hf` h qhj c ait |bc lcohv oi`s|h`| ia |bc fsi|i~c.

    "ff& >.4> e ia ohtdi`-8< id|hf`cl atim h` h`ofc`| dih| ehqc h oi `| th|c ia 8.6 oi `|s ~ct mf` |c. Cs|fmh|c |bc hec ia |bc dih|.

    84bi ts h`l lcohvs dv |bc cmfssfi` ia h ehmmh thv |i ~til oc h mitc s|hdjcqctsfi` ia |bc shmc fsi|i~c.

    F` i`c ohsc |bc ~h|fc`| suhjjius h x h`|f|v ia hjmis| ~ tc \o-66m uf|b h` ho|fqf|via?.5 r 8>Ds-8.

    "h& Cr~jhf` biu |bc cmfssfi` ia h ehmmh thv oh` jchl |i |bc ~til o|fi` ia h

    mitc s|hdjc fsi|i~c.

    "d& Ohjo jh|c |bc lcohv oi`s|h`| ait \o-66m h`l lc|ctmf`c |bc ` mdct ia \o-66mh|ims ~tcsc`| f` |bc mh|ctfhj efqc` |i |bc ~h|fc`|.

    "o& Lfso ss |bc hlqh`|hec ia sf`e h ehmmh cmf||ct th|bct |bh` h` hj~bh cmf||ctit h dc|h cmf||ct ait |bfs ~ t~isc.

    83. F` i`c thlfiho|fqc sctfcs |bc aijjiuf`e lcohv ~tiocsscs ioo t7

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    --*

    @ ojcht ^bvsfos 44> @ ojcht ^bvsfos 448

    4?2

    4?< \b

    4?<

    ^h 64 68

    "h& @hmc |bc lcohv ~tiocsscs ioo ttf`e f` "f& th`f m* "ff&|bitf m.

    "d& Lfso ss ubc|bct mhss fs oi`sctqcl f` |bcsc ~tiocsscs.

    81. Afe tc 5.83 sbius skc|obcs ia oji l obhmdct |thoks ~til ocl ubc` hj~bh ~ht|fojcsia |bc shmc c`ctev |thqcjjf`e atim jca| |i tfeb| htc soh||ctcl dv h|ims ia ehs f` |uioji l obhmdcts R h`l V. \bc soh||ctf`e fs |bc tcs j| ia cjhs|fo oijjfsfi`s. I`cobhmdct fs k`iu` |i oi`|hf` i`jv bcjf m h`l |bc i|bct i`jv `f|tiec`.

    Y*)

    F Oji l obhmdct R Oji l obhmdct V Afe tc 5.83

    "h& "f& Ubfob obhmdct oi`|hf`s |bc `f|tiec`9 Cr~jhf` vi t tchsi`f`e.

    "ff& Lthu h lfhethm |i sbiu |bc ~h|b ia |bc hj~bh ~ht|fojc hs f| h~~tihobcsh`l fs lcajco|cl dv |bc ` ojc s ia |bc h|im ia `f|tiec`.

    "fff& Lthu i` |bc shmc lfhethm |bc ~h|b aijjiucl dv h` hj~bh ~ht|fojc ubfobfs lcajco|cl |bti eb h smhjjct h`ejc.

    "d& "f& _|h|c |bc i |oimc ia h bchl-i` oijjfsfi` ia h` hj~bh ~ht|fojc uf|b hbcjf m ` ojc s ubfob fs f`f|fhjjv h| tcs|.

    "ff& Ubh| ui jl dc |bc i |oimc ia h bchl-i` oijjfsfi` fa h ~ti|i` uctcf`qijqcl f`s|chl ia |bc hj~bh ~ht|fojc9 "@i ohjo jh|fi` fs tcx ftcl.&

    H@_UCT_ \I XSC_\FI@_ AIT ^THO\FOC 8. \bc >-85 ` ojc s oi`|hf`s 2 ~ti|i`s h`l 6 `c |ti`s.

    Mhss lcaco| Mhss ia oi`s|f| c`| ~ht|fojcs - Mhss ia h|im

    "2 R 8.>>28< + 6 R 8.>>262& - 85.>>.8.8

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    4. Mhss lcaco| = Mhss ia oi`s|f| c`| ~ht|fojcs - Mhss ia h|im

    = "4 R 8.>>28< + 4 R 8.>>262& - >83>

    =

    >.>?54<

    >.>?45< R 6?8 R 8>1 cQ = ?.>5 cQ

    Df`lf`e c`ctev ~ct ` ojci`

    Df`lf`e c`ctev ait ` ojc s it h|im = @ mdct ia ` ojci`s

    ?.>5 =

    < = 5.14 R 8>1 cQ = 5.14 McQ

    ?. \bitf m = Thlf m + Hj~bh ~ht|fojc 4?4.>?2 = 442.>?8 + >?

    + C`ctev + C`ctev

    Bc`oc C`ctev = >.>><

    = >.>>< R 8.11 R 8>-45 R "? R 8>2&4 G

    = 3.62 R 8>-8? G

    -45&

    = ?.8 R 8>-46 ke

    @iu C = mo4 ubctc o fs |bc s~ccl ia jfeb|

    = ?.8 R 8>-46 R "? R 8>2&4

    = 4.56 R 8>-84 G

    3.. _7`oc C == mo4

    C = Mhss lcaco| f` ke r o4

    4.22 r 8>-84 = Mhss lcaco| f` ke r "? r 8 >2&4

    Efqf`e mhss lcaco| f` ke 4.22 R 8>-84

    = "? R 8>2&4 = ?.4 R 8X-46 ke

    Mhss lcaco| = "?.?-45 + 3.>>2 r 8>-45& - "8.153 r 8>-45 + M&

    ubctc M fs |bc mhss ia |bc bcjf m ` ojc s

    % Z Cx hjf`e h`l s i jqf` e ef qc s m h ss i a b|jf m ` ojc s7 Z M = 1.1-45 ke

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    @ ojcht ^bvsfos 444

    >

    @ ojcht ^bvsfos 44?

    Cx hjf`e h`l sijqf`e efqcs mhss a bcjf m ` ojc s7

    M = 1.1-45 ke

    H@_UCT_ \I TCQF_FI@ H@L CRHMF@H\FI@ ^THO\FOCXSC_\FI@_

    8. H`suct= H

    4. H`suct = o

    ?. H`suct= [&

    > = -< vcht-(

    . . "8&< 8 1> vchts fs < bhja-jfqcs. Atho|fi` tcmhf`f`e ufjj dc 4 = 81

    H`suct= H

    6. 2 lhvs fs < bhja-jfqcs* si |bc ho|fqf|v ia |bc si toc ufjj bhjqc ai t |fmcs* f.c.

    84 >>> |i 1 >>> |i ? >>> |i 83>>

    H`suct= D

    8>. "h& _cc |cr|.

    "d& "f& Tcact |i Afe tc 5.817

    Afe tc 5.81

    \bc si toc sc`ls i | thlfh|fi` iqct h s~bctc ia htch

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    -*=;--

    >.16?

    44< @ ojcht ^bvsfos

    Htch ia s~bctc < r ` r ">.3&4 = ` mj Htch ia lc|co|it = s omj = 1 r 8>-< t`J

    ` Bc`oc* htch ia s~bctc fs ----- = 34Gb |f`us FGtect |bh` |bc

    "E R 8>- = 8.12 r 8 "&1 s- F

    Bc`oc* ho|fqf|v ia si toc = 8.12 r 8>1 s -8

    >.16? >.16? "ff& Lcohv oi`s|h`|* j = = ---7?"&= >.>4 8$8 s- F

    4

    $ _f`oc ho|fqf|v cr7 ` * dct ia ` ojcf* @ uc oh` utf|c7

    H = H> c-.j| ub tc H fs |bc ho|fqf|v ubc` mchs tcmc`| uhs mhlch`l H > fs |bc ho|jZ"f|v 84> s dcaitc |bc mchs tcmc`| uhs mhlc.

    8.12 R 8>1 = H > c...7.% ->4?8R 84>

    Efqf`e Hi = 4.16 R \>5 l@

    = -j@ l|

    4.16 R 8>5 = ->.>4?8 @

    Efqf`e ` mdct ia ` ojcf* @ = 8.81 r 8>6

    88. "h& _cc |cr|.

    l@ "d&

    l|

    " = -j@

    5.< R 8>? = -84.1 R 8>-? @

    Efqf`e mdct ia ` ojcf* @ = 3.25 r 8>3

    "o& _cc |cr|.

    "l& Lcohv oi`s|h`|* = --7t*- -w

    84.1 R 8>-? = >.16?

    \8. 4

    16?

    Efqf`e b hFa -F fa c* \ = -

    >

    - -Y -

    -Y-

    = 33s

    -r- 84 1 8> ?

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    @ ojcht ^bvsfos 443

    l@ "c&

    l| 5.< r 8 "&? s -8 h`l l tf`e chob lfsf`|ceth|fi`

    1.? r 8.1 r 8>- 8? Gia c`ctev htc tcjchscl.Bc`oc th|c ia c`ctev tcjchsc

    5.< R 8>? R 1.? R 8.1 R 8>-S

    5.-6 U

    84. "h& jH --= I.II:G:G s-8 5>

    jD = >.16?

    = >.>$862s- F ?3

    Ssf`e m = m > c-j| ait shm~jc H efqcs

    t`H = 4> r 8>-d c->.>>66| h`l* ait shm~jc D*

    ms = r 8>-1 c-I.Ij62|

    Ha|ct h oct|hf` |fmc |bc mhsscs ufjj dc cx hj h`l t`H = me7

    4> r 8>-1 c->.>>66| = r 8>-1 c-I.(tj62|

    _ijqf`e dv |hkf`e jiec ia di|b sflcs efqcs7

    | =5>s

    Mhss ia chob fsi|i~c ha|ct 5> s = 4> r 8>-1 c- i.|G>66 r 5i

    = 8> R 8>-d e = 8>~e

    "d& l@ -= -j| l|

    = ->.>>66 R 8> R 8>-1 R

    = -4.3 r 8>8< s-8

    1 R 8>4?

    4?<

    Th|c ia lcohv ia H ubc` mhsscs htc |bc shmc

    = 4.3 r 8>8< 3 -8

    8?. "h& "f& _cc |cr|.

    "ff& _cc |cr|.

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    4

    >.16? >.16?

    4

    .

    441 @ ojcht ^bvsfos

    "d& "f& Atim |bc eth~b* bhja-jfac 3>>> vchts.

    >.16? Lcohv oi`s|h`|* .j = = 8.?21 R

    \-%$ 8>-< vcht- 8

    8.?21 R 8>-<

    = "?13 R 4< R 1> R 1>& = -84 s

    "ff& >.4 e ia ohtdi` efqcs af`hj oi `| th|c ia 8.6 o~m h`l 8.> e efqcs 6.3 o~m. Bc`oc* atim |bc eth~b |bc h~~tirfmh|c hec = >> vchts.

    8