Exercise Set 3.1

8/14/2019 Exercise Set 3.1

1/22

BVB^OK_B _B] ?,8

8,

?, &h+ 9 &? =$ 7 6+ 9 &8$ 8+&b+

9 &> ?$ 2 # 7$ = >+ 9 &2$ 8>$ 5+

2, &h+ Db}Z 9 &v$ {$p+ nb }fb klk}khd zakl} ae }fb mbsk~bm ubo}a~ hlm hssrib }fh} }fks ubo}a~

fhs }fb shib dblg}f hsu, _klob fhs }fb shib mk~bo}kal hsu 9 &=$ >$ 8+$ wb fhub

}fb bxrh}kal

9 &? v$ : {$ 2 p+ 9 &=$ >$ 8+ZX

ZX

Z Z8 >

Z Z8 >

6;

v

&?$ =$ 2+

{

p

v

{

p

&?$ =$ 2+

v

& ?$ =$ 2+

{

p

v

p

{

&?$ :$ ?+

&h+ &o+

&b+ & `+


2/22

Ke wb bxrh}b oaizalbl}s kl }fb hnaub bxrh}kal$ wb an}hkl

v 9 8$ { 9 >$ hlm p 9 =

]frs$ wb fhub earlm h ubo}a~ wfkof sh}ksbs }fb gkubl oalmk}kals, Hl{ zask}kub

ird}kzdb c wkdd hdsa wa~c z~aukmbm }fb }b~iklhd zakl} ~bihkls vbm h} X, ]frs$Z

oardm nb hl{ zakl} &? =c$ >c$ c 2+ wfb~b c < :,

&n+ Db}Z 9 &v$ {$p+ nb }fb klk}khd zakl} ae }fb mbsk~bm ubo}a~ hlm hssrib }fh} }fks ubo}a~

fhs }fb shib dblg}f hs u, _klob ks azzask}bd{ mk~bo}bm }a u 9 &=$ >$ 8+$ wb

fhub }fb bxrh}kal

9 &? v$ : {$ 2 p+ 9 &=$ >$ 8+

Ke wb bxrh}b oaizalbl}s kl }fb hnaub bxrh}kal$ wb an}hkl

v 9 7$ { 9 >$ hlmp 9 5

]frs$ wb fhub earlm h ubo}a~ wfkof sh}ksbs }fb gkubl oalmk}kals, Hl{ zask}kub

ird}kzdb c wkdd hdsa wa~c$ z~aukmbm }fb }b~iklhd zakl} ~bihkls vbm h} X, ]frs$

Z oardm nb hl{ zakl} &? # =c$ >c$ c 2+ wfb~b c < :,

7, Db} v 9 &v8$v

>$v

?+, ]fbl

>r u # v 9 &5$ >$ =+ &=$ :$ 6+ # &v8$v

>$v

?+

9 &8: #v8$ > #v>$ 8> #v?+

Al }fb a}fb~ fhlm$

7v #w 9 7&v8$v

>$v

?+ # &5$ 8$ =+

9 &7v8

# 5$ 7v>

8$ 7v?

=+

Ke wb bxrh}b }fb oaizalbl}s ae }fbsb }wa ubo}a~s$ wb an}hkl

7v8

# 5 9v8

8:

7v>

8 9v>

# >

7v?

= 9v?

# 8>

Fblob$ v 9 &6!?$ 8!>$ 6!?+,

ZX

ZX

ZX

ZX

ZX

ZX

;: Bvb~oksb _b} ?,8


3/22

;, _rzzasb }fb~b h~b sohdh~s o8$ o

>$ hlm o

?wfkof sh}kse{ }fb gkubl bxrh}kal, Ke wb bxrh}b

oaizalbl}s al na}f skmbs$ wb an}hkl }fb eaddawklg s{s}bi ae bxrh}kals0

>o8

?o>

# o?

9 :

;o8

# >o>

# 7o?

9 2

5o8

# o>

# 2o?

9 =

]fb hrgibl}bm ih}~kv ae }fks s{s}bi ae bxrh}kals ohl nb ~bmrobm }a

]fb }fk~m ~aw ae }fb hnaub ih}~kv kizdkbs }fh} :o8

# :o>

# :o?

9 8, Odbh~d{$ }fb~b ma la}

bvks} sohdh~s o8$ o>$ hlm o? wfkof sh}kse{ }fb hnaub bxrh}kal$ hlm fblob }fb s{s}bi kskloalsks}bl},

88, Wb wa~c kl }fb zdhlb mb}b~iklbm n{ }fb }f~bb zakl}s A 9 &:$ :$ :+$Z 9 &>$ ?$ >+$ hlm X 9

&7$ =$ 8+, Db}Vnb h zakl} al }fb dklb }f~argfZ hlm X hlm db} } &wfb~b } ks h zask}kub$

~bhd lrinb~+ nb }fb ubo}a~ wk}f klk}khd zakl}Z hlm }b~iklhd zakl}V, La}b }fh} }fb dblg}f

ae}

ks } }kibs }fb dblg}f ae , ^beb~~klg }a }fb gr~b nbdaw$ wb sbb }fh}

hlm

X

A

ZV

}ZX

AZ ZX AX

# 9

AZ } ZX AV

# 9

ZX

ZX

ZX

> ? 8 :

: > > 8

: : : 8

Bvb~oksb _b} ?,8 ;8


4/22

]fb~bea~b$

&h+ ]a an}hkl }fb ikmzakl} ae }fb dklb sbgibl} oallbo}klg Z hlm X$ wb sb} } 98!>, ]fks

gkubs

&n+ Law sb} } 9 ?!=,]fks gkubs

8?, X 9 &7$ ?$ 8;+

87, ]fb ubo}a~ r fhs }b~iklhd zakl} X wfkof ks }fb ikmzakl} ae }fb dklb sbgibl} oallbo}klgZ8

hlmZ>,

AZ8

AZ> AZ8

&AZ> AZ8+

A

X Z>Z8

AZ8AZ>

8

>

AV

9 # 9

8

=> ? >

?

=7 = 8

>?

=

;

=

8

=& $ $ + & $ $ + $ $

AV AZ AX

9 #

9 #

8

>

8

>

8

>> ? >

8

>7 = 8& $ $ + & $ $ +

99

;

>

8

>

8

>$ $

AV AZ } AX AZ

} AZ } AX

9 #

9 #

& +

& +8

;> Bvb~oksb _b} ?,8


5/22

8;, Gbaib}~kohdd{$ gkubl = lalpb~a ubo}a~s h}}hof }fb }hkd ae alb }a }fb fbhm ae hla}fb~ hlm

oal}klrb rl}kd hdd = fhub nbbl s}~rlg }agb}fb~,]fb ubo}a~ e~ai }fb }hkd ae }fb ~s} ubo}a~

}a }fb fbhm ae }fb dhs} alb wkdd nb }fbk~ sri,

v

{p

w

v # { # p # w

Bvb~oksb _b} ?,8 ;?


6/22


7/22

BVB^OK_B _B] ?,>

8, &h+ u 9 &=> # &?+>+8!>

9 2

&o+ u 9 S&2+> # :>Y8!>

9 2

&b+ u 9 S&7+> # >> # &8+>Y8!>

9 2=

?, &h+ _klob r #u 9 &?$ 2$ 7+$ }fbl

r #u 9 S?> # &2+> # 7>Y8!>

9 6?

&o+ _klob

>r 9 S&=+> # => # &5+>Y8!>

9 > 87

hlm

>r 9 > S>> # &>+> # ?>Y8!>

9 > 87

}fbl

>r # >r 9 = 87

&b+ _klob w 9 S?> # 5> # &=+>Y8!>

9 58$ }fbl

2, &h+ c 9 8$ d 9 ?

&n+ la zasskndb sadr}kal

7, _klob cu 9 &c$ >c$ 2c+$ }fbl

cu 9 Sc> # =c> # >2c>Y8!>

9 tct ?:

Kecu 9 =$ k} eaddaws }fh} tct ?: 9 = a~ c 9 =! ?:,

$ $8

ww 9

?

58

5

58

=

58

;2


8/22

;, &n+ E~ai Zh~} &h+$ wb claw }fh} }fb la~i aeu!u ks 8, Nr} keu 9 &?$ =+$ }fbl u 9 2,Fblob r 9u!u 9 &?!2$ =!2+ fhs la~i 8 hlm fhs }fb shib mk~bo}kal hs u,

88, La}b }fh} z z: 9 8 ke hlm ald{ kez z

:> 9 8, ]frs

&v v:+> # &{ {

:+> # &p p

:+> 9 8

]fb zakl}s &v$ {$p+ wfkof sh}kse{ }fbsb bxrh}kals h~b `rs} }fb zakl}s al }fb szfb~b ae ~hmkrs

8 wk}f obl}b~ &v:$ {

:$p

:+3 }fh} ks$ }fb{ h~b hdd }fb zakl}s wfasb mks}hlob e~ai &v

:$ {

:$p

:+

ks 8,

8?, ]fbsb z~aaes h~b ea~ ubo}a~s kl ?)szhob, ]a an}hkl z~aaes kl >)szhob$ `rs} mbdb}b }fb ?~m

oaizalbl}, Db} r 9 &r8$ r

>$ r

?+ hlmu 9 &u

8$u

>$u

?+, ]fbl

&h+ r #u 9 &r8

#u8$ r

>#u

>$ r

?#u

?+

9 &u8

# r8$u

># r

>$u

?# r

?+ 9u # r

&o+ r # : 9 &r8

# :$ r>

# :$ r?

# :+

9 &:# r8$ : # r

>$ : # r

?+

9 &r8$ r

>$ r

?+ 9 : # r 9 r

&b+ c&dr+ 9 c&dr8$ dr

>$ dr

?+ 9 &cdr

8$ cdr

>$ cdr

?+ 9 &cd+r

82, _bb Bvb~oksb ;, Bxrhdk}{ aoor~s ald{ wfbl r hlmu fhub }fb shib mk~bo}kal a~ wfbl alb ks}fb pb~a ubo}a~,

87, &h+ Kev 4 8$ }fbl }fb zakl} v dkbs klskmb }fb ok~odb a~ szfb~b ae ~hmkrs alb wk}f obl}b~ h}

}fb a~kgkl,

&n+ _rof zakl}s v irs} sh}kse{ }fb klbxrhdk}{ v v: < 8,

;5 Bvb~oksb _b} ?,>


9/22

;7

BVB^OK_B _B] ?,?

8, &h+ r u 9 &>+&2+ # &?+&7+ 9 88

&o+ r u 9 &8+&?+ # &2+&?+ # &=+&?+ 9 :

?, &h+ r u 9 &5+&>+# &8+&:+# &=+&?+ 9:, ]frs }fb ubo}a~s h~b a~}fagalhd,

&n+ r u 9 8 4 :, ]frs ks an}rsb,

2, &h+ E~ai Z~andbi =&h+$ wb fhub

w>

9 r w8

9 r 9 &5$ >+

&o+ E~ai Z~andbi =&o+$ wb fhub

w>

9 &?$ 8$ 7+ &85!8?$ :$ 6:!8?+ 9 &22!8?$ 8$ 88!8?+

8?, Db}w 9 &v$ {$p+ nb a~}fagalhd }a na}f r hlmu, ]fbl r w 9 : kizdkbs }fh}v #p 9 : hlm

u w 9 : kizdkbs }fh} { #p 9 :, ]fh} ksw 9 &v$v$ v+, ]a }~hlsea~i kl}a h rlk} ubo}a~$ wb

mkukmb bhof oaizalbl} n{ w 9 ?v>, ]frs bk}fb~ &8! ?$ 8! ?$ 8! ?+ a~ &8! ?$

8! ?$ 8! ?+ wkdd wa~c,

]fb iklrs skgl kl }fb hnaub bxrh}kal ks bv}~hlbars nbohrsb k} {kbdms hl hlgdb ae >!?,

87, &n+ Fb~b

M 9 #

#9= > 8 2 >

= 8

8

87> >

& + & +

& + & +


10/22

8;, Ke wb srn}~ho} Bxrh}kal &**+ e~ai Bxrh}kal &*+ kl }fb sadr}kal }a Z~andbi 86$ wb an}hkl

r #u> r u> 9 =&r u+

Ke wb }fbl mkukmb na}f skmbs n{ =$ wb an}hkl }fb mbsk~bm ~bsrd},

>8, &h+ Db} k 9 &8$ :$ :+$` 9 &:$ 8$ :+$ hlm c 9 &:$ :$ 8+ mbla}b }fb rlk} ubo}a~s hdalg }fbv$ {$

hlmp hvbs$ ~bszbo}kubd{, Keu ks }fb h~nk}~h~{ ubo}a~ &h$ n$ o+$ }fbl wb ohl w~k}bu 9 hk # n`

# oc, Fblob$ }fb hlgdb nb}wbblu hlm k ks gkubl n{

sklob k 9 8 hlm k ` 9 k c 9 :,

>?, N{ }fb ~bsrd}s ae Bvb~oksb >8$ wb fhub }fh} keuk

9 &hk$ n

k$ o

k+ ea~ k 9 8 hlm >$ }fbl oas

k9

$ oas k

9 $ hlm oas k

9 , Law

>2, La}b }fh}

u &c8w

8# c

>w

>+ 9 c

8&u w

8+ # c

>&u w

>+ 9 :

nbohrsb$ n{ f{za}fbsks$u w8

9u w>

9 :, ]fb~bea~bu ks a~}fagalhd }a c8w

8# c

>w

>ea~ hl{

sohdh~s c8

hlm c>,

>7, &h+ ]fb kllb~ z~amro} v { ks mblbm ald{ ke na}f v hlm { h~b ubo}a~s$ nr} fb~bu w ks h

sohdh~,

&n+ Wb ohl hmm }wa ubo}a~s a~ }wa sohdh~s$ nr} la} alb ae bhof,

&o+ ]fb la~i aev ks mblbm ald{ ea~ v h ubo}a~$ nr} r u ks h sohdh~,

&m+ Hghkl$ }fb ma} z~amro} ae h sohdh~ hlm h ubo}a~ ks rlmblbm,

u u8 > 8 >hlm h~b a~}fagalhd 9

#

uu uu :

8 > 8h h n nn o o

h h n n o o

> 8 >

8 >

8 >

8 >

8 >

8 >

8

:# 9

# #uu uu uu uu uu

oas oas oas

uu>

8 > 8 > 8 >

:9

# # 9oas oas oas ::

ok

kuu

k

kuu

hk

kuu

oas 9 9# #

9uu kk

uu kk uu

h

h n o

h

> > >

;6 Bvb~oksb _b} ?,?


11/22

>;, Ke$ ea~ kls}hlob$ r 9 &8$ :$ :+$u 9 &:$ 8$ :+ hlmw 9 &:$ :$ 8+$ wb fhub r u 9 r w 9 :$ nr}

u w,

?8, ]fks ks `rs} }fb Z{}fhga~bhl ]fba~bi,

r

r # uu

Bvb~oksb _b} ?,? ;;


12/22


13/22

BVB^OK_B _B] ?,=

8, &h+

&o+ _klob

wb fhub

&b+ _klob

u >w 9 &:$ >$ ?+ &=$ 8>$ 8=+ 9 &=$ 8:$ 87+

wb fhub

?, &h+ _klob r u 9 &7$ 8$ ?+$ }fb h~bh ae }fb zh~hddbdag~hi ks r u 9 2;,

&o+ _klob r hlmu h~b z~aza~}kalhd$ }fb{ dkb al }fb shib dklb hlm fblob }fb h~bh ae }fbzh~hddbdag~hi }fb{ mb}b~iklb ks pb~a$ wfkof ks$ ae oar~sb$ r u,

rr uu ww& > + 9

$ $> 8

8: 87

? 8

= 87

? >

= 88:== 22 >>

9 & $ $ +

& +rr uu ww 9

9$ $

; 5

5 7

= 5

> 7

= ;

> 5&& $ $ +>7 =: =>

rr uu 9

9 $ $ & $> 8

> ?

? 8

: ?

? >

: >

= ;; 5$ +

uu ww 9

9 $ $ & $ $

> ?

5 7

: ?

> 7

: >

> 5?> 5 +=

8:8


14/22

7, Ofaasb hl{ lalpb~a ubo}a~w wfkof ks la} zh~hddbd }a r, Ea~ kls}hlob$ db}w 9 &8$ :$ :+ a~

&:$ 8$ :+, ]fblu 9 r w wkdd nb a~}fagalhd }a r, La}b }fh} ker hlmw wb~b zh~hddbd$ }fblu 9 r w wardm nb }fb pb~a ubo}a~,

Hd}b~lh}kubd{$ db}w 9 &v$ {$p+, ]fblw a~}fagalhd }a r kizdkbs >v ?{ # 2p 9 :, Law

hsskgl lalpb~a uhdrbs }a hl{ }wa ae }fb uh~khndbsv$ {$ hlmp hlm sadub ea~ }fb ~bihklklg

uh~khndb,

;, &b+ _klob &r w+ u 9u &r w+ ks h mb}b~iklhl} wfasb ~aws h~b }fb oaizalbl}s aeu$r$ hlmw$ ~bszbo}kubd{$ wb kl}b~ofhlgb ^aws 8 hlm > }a an}hkl }fb mb}b~iklhl} wfkof

~bz~bsbl}s r &u w+, _klob }fb uhdrb ae }fks mb}b~iklhl} ks ?$ wb fhub &r w+ u 9 ?,

88, &h+ _klob }fb mb}b~iklhl}

}fb ubo}a~s ma la} dkb kl }fb shib zdhlb,

82, N{ ]fba~bi ?,=,>$ wb fhub

&r #u+ &r u+ 9 r &r u+ #u &r u+

9 &r r+ # &r &u++ # &u r+ # &u &u++

9 : &r u+ &r u+ &u u+

9 >&r u+

87, &h+ ]fb h~bh ae }fb }~khlgdb wk}f skmbs hlm ks }fb shib hs }fb h~bh ae }fb }~khlgdb

wk}f skmbs &8$ >$ >+ hlm &8$ 8$ 8+ wfb~b wb fhub iaubm H }a }fb a~kgkl hlm

}~hlsdh}bmN hlm O hooa~mklgd{, ]fks h~bh ks 8>&8$ >$ >+ &8$ 8$ 8+ 98>&=$ 8$ ?+ 9

>5!>,

8;, &h+ Db} r 9 9 &=$ :$ >+ hlmu 9 9 &?$ >$ =+, ]fbl }fb mks}hlob wb whl} ks

&=$ :$ >+ &?$ >$ =+!&?$ >$ =+ 9 &=$ >>$ 6+ ! >;, 9 > 8=8! >;HN

HZ

HO

HN

9

8 > 8

? : >

2 = :

85 :

8:> Bvb~oksb _b} ?,=


15/22

>8, &n+ Alb ubo}a~ l wfkof ks zb~zblmkordh~ }a }fb zdhlb oal}hklklgu hlmw ks gkubl n{

l 9w u 9 &8$ ?$ ?+ &8$ 8$ >+ 9 &?$ 8$ >+

]fb~bea~b }fb hlgdb nb}wbbl r hlm l ks gkubl n{

Fblob }fb hlgdb nb}wbbl r hlm }fb zdhlb ks gkubl n{

9 >

,5;6> ~hmkhls &a~ =:8;+

Ke wb fhm kl}b~ofhlgbm }fb ~adbs ae u hlm w kl }fb ea~irdh ea~ l sa }fh}

l 9 u w 9 &?$ 8$ >+$ }fbl wb wardm fhub an}hklbm 9 oas8 >,>5;

~hmkhls a~ 8?:,::2>, Kl }fks ohsb$ 9 >,

Kl bk}fb~ ohsb$ la}b }fh} ih{ nb oaizr}bm rsklg }fb ea~irdh

>2, &h+ N{ ]fba~bi ?,=,8$ wb claw }fh} }fb ubo}a~u w ks zb~zblmkordh~ }a na}fu hlmw,Fblobu w ks zb~zblmkordh~ }a bub~{ ubo}a~ kl }fb zdhlb mb}b~iklbm n{u hlm w3ia~baub~ }fb ald{ ubo}a~s zb~zblmkordh~ }au w wfkof sfh~b k}s klk}khd zakl} irs} nbkl }fks zdhlb, Nr} hdsa n{ ]fba~bi ?,=,8$ r &u w+ ks zb~zblmkordh~ }au w ea~ hl{ubo}a~ r : hlm fblob irs} dkb kl }fb zdhlb mb}b~iklbm n{u hlmw,

&n+ ]fb h~gribl} ks oaizdb}bd{ skikdh~ }a Zh~} &h+$ hnaub,

>;, Keh$ n$ o$ hlm m dkb kl }fb shib zdhlb$ }fbl &h n+ hlm &o m+ h~b na}f zb~zblmkordh~ }a

}fks zdhlb$ hlm h~b }fb~bea~b zh~hddbd, Fblob$ }fbk~ o~ass)z~amro} ks pb~a,

oas ,9

8 r l

r l

;

8=

9

9

oas oas

,

8 8 ;

8=

: 67>5

rr ll

rr ll

~~hmkhls a~& , +=; ;;

Bvb~oksb _b} ?,= 8:?


16/22

?8, &h+ ]fb ~bxrk~bm uadrib ks

9 >!?

??, Db} r 9 &r8$ r

>$ r

?+$u 9 &u

8$u

>$u

?+$ hlmw 9 &w

8$w

>$w

?+,

Ea~ Zh~} &o+$ wb fhub

r w 9 &r>w

? r

?w

>$ r

?w

8 r

8w

?$ r

8w

> r

>w

8+

hlm

u w 9 &u>w

?u

?w

>$u

?w

8u

8w

?$u

8w

>u

>w

8+

]frs

&r w+ # &u w+

9 &Sr>

#u>Yw

? Sr

?#u

?Yw

>$ Sr

?#u

?Yw

8 Sr

8#u

8Yw

?$ Sr

8#u

8Yw

> Sr

>#u

>Yw

8+

Nr}$ n{ mblk}kal$ }fks ks `rs} &r #u+ w,

Ea~ Zh~} &m+$ wb fhub

c&r u+ 9 &cSr>u? r?u>Y$ cSr?u8 r8u?Y$ cSr8u> r>u8Y+

hlm

&cr+ u 9 &cr>u

? cr

?u

?$ cr

?u

8 cr

8u

?$ cr

8u

> cr

>u

8+

]frs$ c&r u+ 9 &cr+ u, ]fb kmbl}k}{ c&r u+ 9 r &cu+ ih{ nb z~aubm kl hl hlhdagarswh{,

?2, &h+ Ansb~ub }fh} r u ks zb~zblmkordh~ }a na}f r hlmu$ hlm fblob }a hdd ubo}a~s kl }fbzdhlb wfkof }fb{ mb}b~iklb, _kikdh~d{$w 9u &r u+ ks zb~zblmkordh~ }a na}fu hlm}a r u, Fblob$ k} irs} dkb al }fb dklb }f~argf }fb a~kgkl zb~zblmkordh~ }au hlm kl }fbzdhlb mb}b~iklbm n{ r hlmu,

8

58 ? > > : ? > ? 8 > ? ? 8 ? : > 8& $ $ + && $ $ + & $ $# # # ++

& $ $ + & $ $ +

?

8

5

= = ? 5 8: =

& +

9 & +

8:= Bvb~oksb _b} ?,=


17/22

&n+ E~ai }fb hnaub$u w 9 :, Hzzd{klg Zh~} &m+ ae ]fba~bi ?,7,8$ wb fhub

w 9u &r u+ 9 &u u+r &u r+u

sa }fh}

r w 9 &u u+&r r+ &u r+&r u+

9 u>r> &r u+>

?7, ]fb bvz~bsskal r &u w+ ks odbh~d{ wbdd)mblbm,

_klob }fb o~ass z~amro} ks la} hssaokh}kub$ }fb bvz~bsskal r u w ks la} wbdd)mblbmnbohrsb }fb ~bsrd} ks mbzblmbl} rzal }fb a~mb~ kl wfkof wb oaizr}b }fb o~ass z~amro}s$

k,b,$ rzal }fb wh{ kl wfkof wb klsb~} }fb zh~bl}fbsbs, Ea~ bvhizdb$ &k `+ ` 9 c ` 9 knr} k &` `+ 9 k : 9 :,

]fb bvz~bsskal r u w ih{ nb mbbibm }a nb hoobz}hndb nbohrsb }fb~b ks ald{ albibhlklgerd wh{ }a klsb~} zh~bl}fbsks$ lhibd{$ r &u w+, ]fb hd}b~lh}kub$ &r u+ w$mabs la} ihcb sblsb nbohrsb k} ks }fb o~ass z~amro} ae h sohdh~ wk}f h ubo}a~,

Bvb~oksb _b} ?,= 8:2


18/22


19/22

BVB^OK_B _B] ?,2

2, &h+ La~ihd ubo}a~s ea~ }fb zdhlbs h~b &=$ 8$ >+ hlm &7$ ?$ =+, _klob }fbsb ubo}a~s h~b la}

ird}kzdbs ae alb hla}fb~$ }fb zdhlbs h~b la} zh~hddbd,

&n+ La~ihd ubo}a~s h~b &8$ =$ ?+ hlm &?$ 8>$ ;+, _klob alb ubo}a~ ks }f~bb }kibs }fb

a}fb~$ }fb zdhlbs h~b zh~hddbd,

7, &h+ La~ihd ubo}a~s ea~ }fb zdhlbs h~b &?$ 8$ 8+ hlm &8$ :$ >+, _klob }fb kllb~ z~amro} ae}fbsb }wa ubo}a~s ks la} pb~a$ }fb zdhlbs h~b la} zb~zblmkordh~,

88, &h+ Hs kl Bvhizdb 5$ wb sadub }fb }wa bxrh}kals skird}hlbarsd{, Ke wb bdkiklh}b {$ wb

fhubv # 7p # 8> 9 :, Db}$ sh{$p 9 }$ sa }fh}v 9 8> 7}$ hlm srns}k}r}b }fbsb uhdrbs

kl}a }fb bxrh}kal ea~ bk}fb~ zdhlb }a gb} { 9 =8 >?},

Hd}b~lh}kubd{$ ~bohdd }fh} h mk~bo}kal ubo}a~ ea~ }fb dklb ks `rs} }fb o~ass)z~amro} ae

}fb la~ihd ubo}a~s ea~ }fb }wa zdhlbs$ k,b,$

&7$ >$ ?+ &?$ 8$ >+ 9 &7$ >?$ 8+

]frs ke wb ohl lm h zakl} wfkof dkbs al }fb dklb &}fh} ks$ hl{ zakl} wfasb oaa~mklh}bs

sh}kse{ }fb bxrh}kals ea~ na}f zdhlbs+$ wb h~b malb, Ke wb sb}p 9 : hlm sadub }fb }wa

bxrh}kals skird}hlbarsd{$ wb gb}v 9 8> hlm { 9 =8$ sa }fh}v 9 8> 7}$ { 9 =8

>?}$p 9 : # } ks alb sb} ae bxrh}kals ea~ }fb dklb &sbb hnaub+,

8?, &h+ _klob }fb la~ihd ubo}a~s &8$ >$ =+ hlm &>$ =$ 6+ h~b zh~hddbd$ sa h~b }fb zdhlbs,

&n+ _klob }fb la~ihd ubo}a~s &?$ :$ 8+ hlm &8$ :$ ?+ h~b la} zh~hddbd$ lbk}fb~ h~b }fb

zdhlbs,

87, _klob }fb zdhlb ks zb~zblmkordh~ }a h dklb wk}f mk~bo}kal &>$ ?$ 2+$ wb ohl rsb }fh} ubo}a~

hs h la~ihd }a }fb zdhlb, ]fb zakl})la~ihd ea~i }fbl {kbdms }fb bxrh}kal >&v # >+ # ?&{ 8+

2&p 7+ 9 :$ a~ >v # ?{ 2p # ?5 9 :,

8;, &h+ _klob }fb ubo}a~ &:$ :$ 8+ ks zb~zblmkordh~ }a }fb v{)zdhlb$ wb ohl rsb }fks hs }fb

la~ihd ea~ }fb zdhlb, ]fb zakl})la~ihd ea~i }fbl {kbdms }fb bxrh}kalp p:

9 :, ]fks

bxrh}kal oardm `rs} hs wbdd fhub nbbl mb~kubm n{ klszbo}kal$ sklob k} ~bz~bsbl}s }fb sb}

ae hdd zakl}s wk}f vbmp hlmv hlm { h~nk}~h~{,

8:7


20/22

>8, H la~ihd }a }fb zdhlb ks l 9 &2$ >$ 8+ hlm }fb zakl} &?$ 5$ 7+ ks kl }fb mbsk~bm zdhlb,

Fblob$ hl bxrh}kal ea~ }fb zdhlb ks 2&v ?+ >&{ # 5+ # &p 7+ 9 : a~ 2v >{ #p ?= 9 :,

>2, Ohdd }fb zakl}sH$N$ O$ hlmM$ ~bszbo}kubd{, _klob }fb ubo}a~s 9 &8$ >$ =+ hlm 9

&>$ 8$ >+ h~b la} zh~hddbd$ }fbl }fb zakl}sH$N$ hlm O ma mb}b~iklb h zdhlb &hlm la} `rs}

h dklb+, H la~ihd }a }fks zdhlb ks 9 &:$ 8:$ 2+, ]fb~bea~b hl bxrh}kal ea~ }fbzdhlb ks

>{ p # 8 9 :

_klob }fb oaa~mklh}bs ae }fb zakl} M sh}kse{ }fks bxrh}kal$ hdd ear~ zakl}s irs} dkb kl }fb

shib zdhlb,

Hd}b~lh}kubd{$ k} wardm sreob }a sfaw }fh} &ea~ kls}hlob+ hlm h~b zh~hddbd$ sa }fh} }fb zdhlbs mb}b~iklbm n{H$ N$ hlm O hlm H$ M$ hlm O h~b zh~hddbd,

_klob }fb{ fhub zakl}s kl oaiial$ }fb{ irs} oaklokmb,

>7, La~ihds }a }fb }wa zdhlbs h~b &=$ >$ >+ hlm &?$ ?$ 5+ a~$ skizdke{klg$ l8

9 &>$ 8$ 8+ hlm

l>

9 &8$ 8$ >+, H la~ihd l }a h zdhlb wfkof ks zb~zblmkordh~ }a na}f ae }fb gkubl zdhlbs

irs} nb zb~zblmkordh~ }a na}f l8

hlm l>, ]fh} ks$ l 9 l

8 l

>9 &8$ 2$ ?+, ]fb zdhlb wk}f }fks

la~ihd wfkof zhssbs }f~argf }fb zakl} &>$ 8$ 2+ fhs }fb bxrh}kal

&v # >+ # 2&{ 8+ # ?&p 2+ 9 :

a~

v # 2{ # ?p 86 9 :

?8, Ke$ ea~ kls}hlob$ wb sb} } 9 : hlm } 9 8 kl }fb dklb bxrh}kal$ wb an}hkl }fb zakl}s &:$ 8$ ?+

hlm &8$ :$ 2+, ]fbsb$ }agb}fb~ wk}f }fb gkubl zakl} hlm }fb ib}fams ae Bvhizdb >$ wkdd

{kbdm hl bxrh}kal ea~ }fb mbsk~bm zdhlb,

??, ]fb zdhlb wb h~b daacklg ea~ ks `rs} }fb sb} ae hdd zakl}sZ 9 &v$ {$p+ srof }fh} }fb mks}hlobs

e~aiZ }a }fb }wa vbm zakl}s h~b bxrhd, Ke wb bxrh}b }fb sxrh~bs ae }fbsb mks}hlobs$ wb

fhub

&v # 8+>

# &{ # =+>

# &p # >+>

9 &v :+>

# &{ # >+>

# &p >+>

a~

>v # 8 # 6{ # 85 # =p # = 9 ={ # = =p # =

a~

>v # ={ # 6p # 8? 9 :

MO

HM

NO

HN

NO HN

NO

HN

8:6 Bvb~oksb _b} ?,2


21/22

?2, Wb ofhlgb }fb zh~hib}b~ kl }fb bxrh}kals ea~ }fb sboalm dklb e~ai } }as, ]fb }wa dklbs wkdd

}fbl kl}b~sbo} ke wb ohl lm uhdrbs aes hlm } srof }fh} }fbv$ {$ hlmp oaa~mklh}bs ea~ }fb

}wa dklbs h~b bxrhd3 }fh} ks$ ke }fb~b h~b uhdrbs ea~s hlm } srof }fh}

=} # ? 9 8>s 8

} # = 9 5s # 7

8 9 ?s # 2

]fks s{s}bi ae bxrh}kals fhs }fb sadr}kal } 9 2 hlms 9 =!?, Ke wb }fbl srns}k}r}b } 9 2

kl}a }fb bxrh}kals ea~ }fb ~s} dklb a~s 9 =!? kl}a }fb bxrh}kals ea~ }fb sboalm dklb$ wb lm

}fh}v 9 87$ { 9 8$ hlmp 9 8 ks }fb zakl} ae kl}b~sbo}kal,

?7, &h+ Ke wb sb}p 9 } hlm sadub ea~v hlm { kl }b~is aep$ }fbl wb lm }fh}

?;, &n+ N{ ]fba~bi ?,2,>$ }fb mks}hlob ks

=8, &h+

&n+

&o+

=2, &h+ La~ihds }a }fb }wa zdhlbs h~b &8$ :$ :+ hlm &>$ 8$ 8+, ]fb hlgdb nb}wbbl }fbi ks

gkubl n{

]frs 9 oas8 &>! 5+ ?2822>,

oas$ $ $ $

9& + & +

# #9

8 : : > 8 8

8 = 8 8

>

5

m

m

m

9

9

9

?:

88

?6>

88

: sklob zakl} ks al }fb dkllb

M 9& + # & + & +

# # & +9

> 8 ? > = 8 8

> ? =

8

>;> > >

v } { } p }9 # 9 988

>?

7

>?

=8

>?

8

>?

$ $

Bvb~oksb _b} ?,2 8:;


22/22

=7, Ke wb srns}k}r}b hl{ uhdrb ae }fb zh~hib}b~sh{ }:kl}a ~ 9 ~

:# }u hlm }

:kl}a ~ 9 ~

: }u$

wb odbh~d{ an}hkl }fb shib zakl}, Fblob$ }fb }wa dklbs oaklokmb, ]fb{ na}f zhss }f~argf }fb

zakl} ~:

hlm na}f h~b zh~hddbd }au,

=;, ]fb bxrh}kal ~ 9 &8 }+~8

# }~>

ohl nb ~bw~k}}bl hs ~ 9 ~8

# }&~>

~8+, ]fks ~bz~bsbl}s h dklb

}f~argf }fb zakl}Z8 wk}f mk~bo}kal ~> ~8, Ke} 9 :$ wb fhub }fb zakl}Z8, Ke} 9 8$ wb fhub}fb zakl}Z

>, Ke : 4 } 4 8$ wb fhub h zakl} al }fb dklb sbgibl} oallbo}klgZ

8hlmZ

>, Fblob

}fb gkubl bxrh}kal ~bz~bsbl}s }fks dklb sbgibl},

88: Bvb~oksb _b} ?,2

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Exercise Set 3.1