®
EJJ (Mhvmbgjh) (MH\MBGJH)41=< 41=; EJJ
44-1:-41=< Ojoiry Dmsjh ]ujstkibs & Wicutkibs HMXJ5 4>-1:-41=;
QMQJT
- 4 | WZDEJGX 5 OMXFJOMXKGW QMQJT-4 5 KBWXTZGXKIBW XI GMBHKHMXJW GMBHKHMXJW
]ujstkib pmpjr-4 fms tfrjj (17) pmrts 5 Qfyskgs, Gfjokstry mbh Omtfjomtkgs.
Jmgf pmrt fms m titmc il jknftjjb il jknftjjb (=6) qujstkibs hkvkhjh kbti tfrjj kbti tfrjj (17) sjgtkibs (Wjgtkib-=, Wjgtkib-4 mbh Wjgtkib-7).
Xitmc buodjr il qujstkibs kb ]ujstkib kb ]ujstkib Qmpjr-4 mrj 5 Lklty Liur (:8) mbh Omxkouo Omr`s mrj Ibj Fubhrjh Jknfty Wkx (=6<).
Xypj il ]ujstkibs mbh Omr`kbn Wgfjojs WJGXKIB = (Omxkouo Omr`s 5 74)
Xfks sjgtkib gibtmkbs JKNFX (16) qujstkibs. (16) qujstkibs.
Jmgf qujstkib fms LIZT iptkibs IBJ IT OITJ XFMB IBJ il IBJ il tfjsj liur iptkib(s) ks(mrj) girrjgt mbswjr(s).
Lir jmgf qujstkib, gfiisj gfiisj tfj iptkib(s) iptkib(s) girrjspibhkbn ti (mcc) (mcc) tfj girrjgt mbswjr(s). mbswjr(s). Mbswjr ti jmgf qujstkib wkcc dj jvmcumtjh mggirhkbn ti tfj licciwkbn omr`kbn sgfjoj. Lucc Omr`s
5
+8 Kl ibcy (mcc) tfj girrjgt girrjgt iptkib(s) ks (mrj) (mrj) gfisjb.
Qmrtkmc Omr`s 5 5
+7 Kl +7 Kl mcc tfj liur iptkibs mrj girrjgt dut IBC[ tfrjj iptkibs mrj gfisjb.
Qmrtkmc Omr`s 5 5
+4 Kl +4 Kl tfrjj ir oirj iptkibs mrj girrjgt dut IBC[ twi iptkibs mrj gfisjb mbh ditf il wfkgf mrj girrjgt.
Qmrtkmc Omr`s 5 5
+= Kl += Kl twi ir oirj iptkibs mrj girrjgt dut IBC[ ibj iptkib ks gfisjb mbh kt ks m girrjgt iptkib.
Ujri Omr`s 5 Bjnmtkvj Omr`s 5 5
1 Kl bibj il tfj iptkibs ks gfisjb (k.j. tfj qujstkib ks ubmbswjrjh). ubmbswjrjh).
‗= Kb mcc itfjr gmsjs.
WJGXKIB 4 (Omxkouo Omr`s 5 =6)
Xfks sjgtkib gibtmkbs WKV (1<) qujstkibs. (1<) qujstkibs. Xfj mbswjr ti jmgf qujstkib ks m BZOJTKGMC \MCZJ. \MCZJ.
Lir jmgf qujstkib, jbtjr tfj girrjgt buojrkgmc vmcuj il tfj mbswjr uskbn tfj oiusj mbh tfj ib-sgrjjb vkrtumc buojrkg `jypmh kb tfj pcmgj hjsknbmtjh ti jbtjr tfj mbswjr. Kl tfj buojrkgmc vmcuj fms oirj tfmb twi hjgkomc pcmgjs, trubgmtj/riubh-ill tfj trubgmtj/riubh-ill tfj vmcuj ti X^I hjgkomc X^I hjgkomc pcmgjs.
Mbswjr ti jmgf qujstkib wkcc dj jvmcumtjh jvmcumtjh mggirhkbn ti tfj licciwkbn licciwkbn omr`kbn sgfjoj 5 Lucc Omr`s
5
Ujri Omr`s 5
+7 Kl IBC[ tfj girrjgt buojrkgmc buojrkgmc vmcuj vmcuj ks jbtjrjh. 1 Kb mcc itfjr gmsjs. gmsjs.
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
WJGXKIB 7 (Omxkouo Omr`s 5 =4)
Xfks sjgtkib gibtmkbs X^I (14) Ckst-Omtgf (14) Ckst-Omtgf sjts.
Jmgf Ckst-Omtgf sjt fms X^I (14) Ouctkpcj Gfikgj ]ujstkibs.
Jmgf Ckst-Omtgf sjt fms twi cksts cksts 5 Ckst-K mbh Ckst-K mbh Ckst-KK. Ckst-KK.
Ckst-K fms Ckst-K fms Liur jbtrkjs Liur jbtrkjs (K),(KK), (KKK) mbh (K\) Ckst-KK fms Ckst-KK fms Wkx jbtrkjs Wkx jbtrkjs (Q),(]), (T ), (W), (X) mbh (Z).
LIZT iptkibs LIZT iptkibs mrj nkvjb kb jmgf Ouctkpcj Gfikgj ]ujstkib dmsjh ib Ckst-K mbh Ckst-K mbh Ckst-KK mbh Ckst-KK mbh IBC[ IBJ il IBJ il tfjsj liur iptkibs smtkslkjs tfj gibhktkib ms`jh kb tfj Ouctkpcj Gfikgj ]ujstkib.
Mbswjr ti jmgf qujstkib wkcc dj jvmcumtjh mggirhkbn ti tfj licciwkbn omr`kbn sgfjoj 5 Lucc Omr`s
5
+7 Kl IBC[ tfj iptkib girrjspibhkbn girrjspibhkbn ti tfj girrjgt giodkbmtkib ks gfisjb. gfisjb.
Ujri Omr`s 5 Bjnmtkvj Omr`s 5
1 Kl bibj il tfj iptkibs ks gfisjb (k.j. tfj qujstkib qujstkib ks ubmbswjrjh). ubmbswjrjh).
‗= Kb mcc itfjr gmsjs.
Mbswjrkbn ]ujstkibs 5
Xi sjcjgt tfj iptkib(s), usj tfj oiusj ti gckg` ib tfj girrjspibhkbn duttib(s) il tfj iptkib(s).
WJGXKIB-= gckg` ib tfj duttib(s) il tfj gfisjb iptkib(s) mnmkb ir gckg` ib tfj Xi hjsjcjgt tfj gfisjb iptkib(s) lir tfj qujstkibs il WJGXKIB-= gckg` Gcjmr Tjspibsj duttib Tjspibsj duttib ti gcjmr mcc tfj gfisjb iptkibs.
Xi hjsjcjgt tfj gfisjb iptkib lir tfj qujstkibs il WJGXKIB-7, WJGXKIB-7, gckg` ib tfj duttib il tfj gfisjb iptkib mnmkb ir gckg` ib t fj Gcjmr Tjspibsj duttib Tjspibsj duttib ti gcjmr tfj gfisjb iptkib.
Xi gfmbnj tfj iptkib(s) il m prjvkiuscy mbswjrjh qujstkib il WJGXKIB-= mbh WJGXKIB-= mbh WJGXKIB-7 lkrst WJGXKIB-7 lkrst hjsjcjgt ms nkvjb mdivj mbh tfjb sjcjgt tfj bjw iptkib(s)
Xi mbswjr qujstkibs il WJGXKIB-4 usj WJGXKIB-4 usj tfj oiusj ti gckg` ib buodjrs (mbh/ir syodics) ib tfj ib -sgrjjb vkrtumc buojrkg `jypmh ti jbtjr tfj buojrkgmc vmcuj kb tfj spmgj privkhjh lir mbswjr.
Xi gfmbnj tfj mbswjr il m qujstkib il WJGXKIB-4 lkrst WJGXKIB-4 lkrst gckg` ib tfj Gcjmr Tjspibsj duttib Tjspibsj duttib ti gcjmr tfj girrjgt mbswjr mbh tfjb jbtjr tfj bjw buojrkgmc vmcuj.
IBC[ lir rjvkjw (k.j. wktfiut mbswjrkbn kt). Gckg` ib tfj Omr` lir Omr` lir Tjvkjw & bjxt duttib. Xi omr` m qujstkib IBC[ lir
Xi omr` ks qujstkib lir rjvkjw (mltjr mbswjrkbn kt), gckg` ib Omr` lir Tjvkjw & Bjxt duttib - tfj mbswjrjh qujstkib wfkgf ks mcsi omr`jh lir rjvkjw wkcc dj jvmcumtjh.
Xi smvj tfj mbswjr gckg` ib tfj Wmvj & Bjxt duttib, Bjxt duttib, tfj mbswjrjh qujstkib wkcc dj jvmcumtjh.
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
QMTX-KKK 5 OMXFJOMXKGW WJGXKIB = (Omxkouo Omr`s 5 74) (16) qujstkibs. Xfks sjgtkib gibtmkbs JKNFX (16) qujstkibs. IBJ il tfjsj liur iptkib(s) ks(mrj) girrjgt mbswjr(s). Jmgf qujstkib fms LIZT iptkibs IBJ IT OITJ XFMB IBJ il gfiisj tfj iptkib(s) iptkib(s) girrjspibhkbn ti (mcc) tfj tfj girrjgt mbswjr(s). mbswjr(s). Lir jmgf qujstkib, gfiisj Mbswjr ti jmgf qujstkib wkcc dj jvmcumtjh mggirhkbn ti tfj licciwkbn omr`kbn sgfjoj.
5
Lucc Omr`s
+8 Kl ibcy (mcc) tfj girrjgt girrjgt iptkib(s) ks (mrj) (mrj) gfisjb.
5 Qmrtkmc Omr`s 5
+7 Kl +7 Kl mcc tfj liur iptkibs mrj girrjgt dut IBC[ tfrjj iptkibs mrj gfisjb.
5 Qmrtkmc Omr`s 5
+4 Kl +4 Kl tfrjj ir oirj iptkibs mrj girrjgt dut IBC[ twi iptkibs mrj gfisjb mbh ditf il wfkgf mrj girrjgt.
5 Qmrtkmc Omr`s 5
+= Kl += Kl twi ir oirj iptkibs mrj girrjgt dut IBC[ ibj iptkib ks gfisjb mbh kt ks m girrjgt iptkib.
Ujri Omr`s 5
1 Kl bibj il tfj iptkibs ks gfisjb (k.j. tfj qujstkib ks ubmbswjrjh).
5 Bjnmtkvj Omr`s 5
=.
‗= Kb mcc itfjr gmsjs.
Xfrjj ckbjs C= , C 4, C7 mrj nkvjb dy C = 5 r ? k , C4 5 r ? º e + ` , C7 5 r ? k e ` . ^fkgf il tfj licciwkbn pikbts ] gmb dj tm`jb ib C 4 si tfmt tfj pikbts Q ib ckbj C=, pikbt ] ib C4 mbh pikbt T ib C 7 mrj gicckbjmr. ˌ
ˌ
ˌ
ˌ
ˌ
ˌ
rfu esP``,m C= , C 4, C7 dc kz h`e nW lh lh C= 5 r ? k , C4 5 r ? º e + ` , C7 5 r ? k e ` . luJu jsm cs cs h`Wuc` lgZbq ] esP`` C4 ke ly0` t` chr` nW ltccs lh esP`` C= ke lgZbq Q, esP`` C4 ke lgZbq ] rL`` es P`` C7 ke lgZbq T cmesP` n`s&
ˌ
= e 4
(M) ` ˌ
Mbs. Wic.
(D)
ˌ
`QT =
=+
=
cko
b
?
(G) ` e ˌ
ˌ
ˌ
ˌ
ˌ
(H) ` ˌ
= e 4
ˌ
=
=
ck b
=
=
gmbbit tm`j vmcuj il = mbh 1
= 7 4 ......... 7 b
> / 7
b
= 7 4 ......... 7 b
...... 4 4 (bm b)4 (bm =) (bm 4) =
=
(M) ‗; (MD)
b
7
Wic.
= = = b> / 7 ...... 4 4 4 ( bm = ) ( bm 4 ) ( bm b )
cko
b
7
Mbs.
ˌ
ˌ
(MH) Q(, 1, 1) ](1, , =) T(=, =, ) Q]
4.
`
ˌ
=
(D) 6 = b
b
r`s m hs ? :8, r`s (G) >
cmF``lir j`u nW& (H) ‗<
= / 7
r b r =
? :8, tfjb pisskdcj vmcujs m ks/mrj -
4 = b4 = b ......... (bm =)4 (bm 4)4 b (bm =)4
?
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # =
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW =
=
7 8 / 7 8 V 1
x= / 7 hx
1 =
?
m x 1 =
hx
(m x )
4
1
= m
=
=
7 8
?
= m
m =
7
m =
? :8
=
8 :8
= m(m =)
= >4
m ? 6 ir m ? ‗ ; 7.
Cjt l(x) ?
skb x x
4
, x 2 1.
Cjt x= 3 x4 3 x7 .... 3 xb 3 ..... dj mcc pikbts il cigmc omxkouo il l(x) mbh y= 3 y 4 3 y7 3 ...... 3 yb 3 ....... dj mcc tfj pikbts il cigmc okbkouo il l(x) tfjb girrjgt iptkib ks/mrj (M) |xb ‗ yb| 2 = lir jvjry b (D) x= 3 y=
(G) xb 4b, 4b
j`u` l(x) ?
sk b x x
=
lir jvjry b
(H) xb += ‗ xb 2 4 lir jvjry b
4
, x 2 1.
4
j`u` x= 3 x 4 3 x 7 .... 3 x b 3 ..... l(x) hs CL``uf0 olQp"D lgZbq nW nW rL`` y= 3 y4 3 y7 3 ...... 3 yb 3 ....... l(x) hs CL``uf0 lulJu"D lgZbq nW r`s cnf hL`u hL`u nW& (M) |xb ‗ yb| 2 =, kz (D) x= 3 y= kzT0sh b hs ly,
=
4
(G) xb 4b, 4b Mbs.
kzT0sh b hs ly, , kz
(H) xb += ‗ xb 2 4 , kzT0sh b hs
ly,
(MGH)
x tmb x 4
4x gis x
Wic.
l'(x) ?
x
1
8
=
=
y=
Lir l'(x)
+ y= Okb
x= :/4
4
4
‗
4
7
+
‗ x= Omx
y4 Okb
‗ x4 Omx
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # 4
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
8.
= Q ? 1 1
= 4 1
=
4 ] ? 7
4 1 x
x
x
8
1
x
mbh T ? Q]Q ‗= tfjb wfkgf mrj girrjgt <
4 x x (M) hjt T ? hjt 1 8 1 + 6 lir mcc x T x x :
1 (D) lir x ? = tfjrj jxksts m ubkt vjgtir k e ` lir wfkgf mrj T 1 1 ˌ
= (G) lir x ? 1 kl T m ? < d
ˌ
ˌ
= m tfjb m + d ? : d
(H) tfjrj jxksts m rjmc buodjr x sugf tfmt Q]? ]Q
= Q ? 1 1
= 4 1
=
4 ] ? 7
4 (M) hjt T ? hjt 1 x
4 1 x
x
x
8
1
x
x
8
1
x
x
rL`` T ? Q]Q ‗= r`s luJu jsm cs h`Wucs hL`u cT0 nWm& <
+6, cF`f x T hs ly, :
1 (D) x ? = hs ly, ,h dh`dU clb'` clb'` k e ` n`sx`_ ltchs ly, T 1 1 ˌ
= (G) x ? 1 hs ly, 0lb T m ? < d (H) ,h Mbs. Wic.
ˌ
ˌ
= m , r`s m + d ? : r`s m d
i`Crlih cmP0` x dc kzh`e nW lh Q]? ]Q
(MG) hjt T ? hjt Q Þ hjt ] Þ hjt (Q ‗=) hjt T ? hjt ] ? 8(=4 ‗ x4) ? 86 ‗ 8x4 Biw
4
x
x
1
8
1
x
x
:
? 8(=1 ‗ x4) ? 81 ‗ 8x4
4 hjt T ? hjt 1 x
x
x
8
1
x
:
+ 6 x T
Iptkib (M) ks girrjgt Mt x ? =, hjt ] ? 86 ‗ 8 ? 88 ? hjt T Djgmusj hjt T 1 si
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # 7
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
T ?
1 1 1
? ? ? 1
Wi k e ` ks bit m ubkt vjgtir Iptkib (D) ks wribn = = = 4 1 1 Q ? 1 4 4 , ](x ? 1) ? 1 8 1 1 1 7 1 1 < T ? Q ] Q ‗= ˌ
ˌ
ˌ
4 8 < < = ? 1 6 =4 1 < 1 1 =6 1 4 = 4 / 7 T ? 1 8 8 / 7 1 1 < 8 = (T ‗
‗4m + ‗6 +
7
6 7
4d 7
7
=4 = 4 ? 1 < 1 4 1
7 1
< 48 1
4 / 7 =
=
4 8 / 7 1 1
m d
? 1
d?1
? 1
d?7
m?4 (G) girrjgt Q] ? ]Q m=4 lir ditf mrj smoj x + 8 + x ? 4 + 4x + 1 Iptkib (H) ks wribn
:.
6 7< 8
x
Bi vmcuj jxkst
Cjt l5 T T dj m lubgtkib wj smy tfmt l fms l (f) l (1) Qripjrty = kl c ko jxkst mbh ks lkbktj, f 1 |f|
Qripjrty 4 kl
c ko
l (f) l (1)
jxkst mbh ks lkbktj. Xfjb wfkgf il tfj licciwkbn iptkibs ks/mrj girrjgt9 f4 (M) l(x) ? x|x| fms pripjrty 4 (D) l(x) ? x4/7 fms pripjrty = (G) l(x) ? skb x fms pripjrty 4 (H) l(x) ? |x| fms pripjrty = 0lb ,h ]yu l 5 T T dc kzh`e nW lh ]yu l f1
xq.`/`jU = h`s kzbl'`Ur her` nWnW 0lb xq.`/`jU 4 h`s kzbl'`Ur her` nWnW 0lb (M) l(x) ? x|x|, xq.`/`jU 4 h`s
ck o f1
ck o f1
l (f) l (1) |f| l (f) l (1) f4
kzbl'`Ur her` nW
(G) l(x) ? skb x, xq .`/`jU 4 h`s
li|j`u n`s rL`` rL`` lul'Q`r n`s ,
kzbl'`Ur her` nW
li|j`u n`s rL`` rL`` lul'pr n`sM r`s luJu jsm cs cT0 hL`u nWnW& (D) l(x) ? x4/7, xq xq.`/`jU = h`s
xq .`/`jU = h`s (H) l(x) ? |x|, xq
kz bl'`Ur her` nW kzbl'`Ur her` nW
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # 8
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
Mbs. Wic.
(DH) (M)
l(V) ? x|x| l (f) l (1)
cko
f4
f1
(D)
f1
(G)
|f| skbf 1
cko
f
f 1
(H)
<.
f4 / 7 1
cko
4
| f | 1
cko
|f|
f 1
? c k o f1
f | f | 1 f4
hijs bit jxkst
? 1 hijs bit jxkst
? 1
Lir bib-bjnmtkvj kbtjnjr b, cjt b
=
4
sk b b 4 skb b 4
l(b) ? ` 1
`
b
`
=
sk b b 4 4
`
` 1
Mssuokbn gis ‗=x tm`js vmcujs kb P1, _ wfkgf il tfj licciwkbn iptkibs ks/mrj girrjgt9 (M) Kl ? tmb (gis ‗= l(<)), tfjb 4 + 4 ‗ = ?1 (D)
c k o l(x)
b
(G) l(8) ?
=
?
4
7 4
(H) skb(> gis ‗= l(:)) ? 1 lhcf vY.``Tjh kw.``½h b hs b
=
ly,, j`u` j`u` 4
sk b b 4 skb b 4
l(b) ? ` 1
`
b
`
=
sk b b 4 4
`
` 1
j`u`
gis ‗=x
, P1, _ js js j`u ysr` nW_ r`s luJu luJu jsm cs = 4 ‗ (M) Kl ? tmb (gis l(<)), tfjb + 4 ‗ = ?1 (D)
c k o l(x)
b
(G) l(8) ?
?
h`Wucs hL`u hL`u cT0 nW&
= 4
7 4
(H) skb(> gis ‗= l(:)) ? 1 Mbs.
(MGH)
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # :
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
b
` = ` 4 skb sk b b 4 b 4 ` 1
b
Wic.
l(b) ?
b
` = b 4
?
4` 7
` 1
b
4 skb
4 sk b 4
` 1
gis b 4 gis b 4 ` 1
= b 4
4 `
b 7 b = skb b 4 b 4 (b =) gis (b 4) skb b4 ? b = gis . skb b 4 (b =) skb b4 b 7 (b =) gis (b =) gis gis gis b 4 b 4 (b 4) b 4 ? ? gis (b =) = b4 b 4 gis
(M)
? tmb(gis ‗= l(<)) ? tmb|gis ‗= (gis
tmb
(D)
8
?
c k o l(b)
b
= tmb
6 4
= ?
4 =
4
?= b 4 7
(G)
l(8) ? gis
(H)
skb(>gis ‗= l(:)) ? skb(>gis ‗= (gis
<
6
4 + 4 ‗ = ? 1
(M) girrjgt
6
? gis
6
)| ? tmb
4 tmb
4
(D) Kbgirrjgt
girrjgt
>
)) ? skb ? 1 girrjgt
(M), (G), (H) girrjgt
x
>.
Cjt l 5 T T dj nkvjb dy l(x) ? (x ‗ =)(x ‗ 4)(x ‗ :). Hjlkbj L(x) ?
l (t) ht, x 1 . Xfjb wfkgf il tfj 1
licciwkbn iptkibs ks/mrj girrjgt 9 (M) L(x) 1, x (1, :) (D) L(x) fms twi cigmc omxkom mbh ibj cigmc okbkom kb (1, ) (G) L(x) fms m cigmc omxkom mt x ? 4 (H) L(x) fms m cigmc okbkom mt x ? = x
0lb l 5 T T dc kzh`e nW lh l(x) ? (x ‗ =)(x ‗ 4)(x ‗ :). 0lb 0lb L(x) ? l ( t ) ht, x 1 luJu jsm cs cT0 hL`u 1
n`sx`An`sxsm9 (M) L(x) 1, x (1, :) b`s CL``uf0 olQp"D rL`` ,h (D) L(x) , (1, ) jsm b`s (G) L(x) h` x ? 4 ke CL``uf0 olQp"D j`u nWM (H) L(x) h` x ? = ke CL``uf0 lulJu"D j`u nWM Mbs.
CL``uf0 lulJu"D eP`r` nWM
(GH)
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # <
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
Wic.
l(x) ? (x ‗=) (x ‗ 4) (x ‗ :) x
l (t)ht
L(x) ?
1
L (x) ? l(x) ? (x ‗=) (x ‗ 4) (x ‗ :)
‗
+
|
‗
|
=
|
+
:
4
L(x) fms twi cigmc okbkom pikbts mt x ? = mbh x ? :
L(x) fms ibj cigmc omxkom pikbt mt x ? 4
6.
= Cjt Q= ? 1 1 1 1 Q< ? 1 = = 1
1 = 1
1
1 , Q4 ? =
= 1 1
=
, mbh V ? 1 1
1 1 =
<
Q
`
` =
1
1 = 1 1 = 1 1 1 = = , Q7 ? = 1 1 , Q8 ? 1 1 = , Q: ? = 1 1 , 1 1 1 = = 1 1 1 = 1 4 = 7 = 1 4 Q X. ^fjrj Q X ks trmbspisj il omtrkx Q . Xfjb wfkgf il tfj ` ` ` 7 4 =
licciwkbn iptkibs ks/mrj girrjgt 9 (M) V ks m syoojtrkg syoojtrkg omtrkx
= (D) Kl V = ? =
= = , tfjb ? 71 =
(G) V ‗ 71 ks mb kbvjrtkdcj omtrkx (H) Xfj suo il hkmnibmc jbtrkjs il V ks =6.
= j`u` Q= ? 1 1 1 1 Q< ? 1 = = 1
1 = 1 1 = 1 = 1 = , Q7 ? = 1 1 , Q8 ? 1 1 = , Q: ? 1 = 1 1 1 = = 1 1 = 4 = 7 < = 1 4 Q X. tn`m 1 , mbh V ? Q` Q`X _ Q` h` kleirU v`I0wn nW ` tn`m Q ` = 7 4 = 1 1
1
1 , Q4 ? =
= 1 1
1
1
1 = 1
1
=
1
1
=
, 1
r`s luJu jsm cs h`Wucs cs hL`u
cT0 nW& (M) V ,h cjljr v`I0wn nWM = (D) 0lb V = ? =
= r`s ? 71 = , r`s =
(G) V ‗ 71 ,h I0qTÖj.`f0 v`I0wn n`sx`M U hs vi0i`sm h` 0`sx =6 n`sx`M (H) V hs lih.``s hs Mbs. Wic.
(MDH) Gcjmrcy
Q=
X
Q=
Q=
=
Q4 Q4X Q4= . . . Q< Q
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # >
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
4 mbh MX ? M, wfjrj M ? = 7
=
7
1
4
4
=
Zskbn liroucm (M + D) X ? MX + DX
VX ? Q=MQ=X
X ..... Q< MQ<
X
? Q=M XQ=X
..... Q< M XQ
V ks syoojtrkg.
= Cjt D ? = = VD ? Q=MQ=XD Q4 MQ4XD .... Q< MQ
? 71
= (V ‗ 71)D ? 1 fms m bib trkvkmc sicutkib D ? = =
|V ‗ 71| ? 1
V?
X
Q=MQ=
X ...... Q< MQ<
trmgj (x) ? tr (Q=MQ=X ) + ……..+tr
X
Q< MQ<
? (4 + 1 + =) + ….+(4 + 1 + =) ? 7 Þ < ? =6
WJGXKIB 4 (Omxkouo Omr`s 5 =6) (1<) qujstkibs. Xfj mbswjr ti jmgf qujstkib ks m BZOJTKGMC \MCZJ. \MCZJ. Xfks sjgtkib gibtmkbs WKV (1<) qujstkibs. Lir jmgf qujstkib, jbtjr tfj girrjgt buojrkgmc vmcuj il tfj mbswjr uskbn tfj oiusj mbh t fj ib-sgrjjb vkrtumc buojrkg `jypmh kb tfj
pcmgj hjsknbmtjh ti jbtjr tfj mbswjr. Kl tfj buojrkgmc vmcuj fms oirj tfmb twi hjgkomc pcmgjs, trubgmtj/riubh-ill tfj trubgmtj/riubh-ill tfj vmcuj ti X^I hjgkomc X^I hjgkomc pcmgjs. wkcc dj jvmcumtjh mggirhkbn ti tfj licciwkbn licciwkbn omr`kbn sgfjoj 5 Mbswjr ti jmgf qujstkib wkcc Lucc Omr`s
5
Ujri Omr`s 5
;.
Mbs. Wic.
+7 Kl IBC[ tfj girrjgt girrjgt buojrkgmc vmcuj vmcuj ks jbtjrjh. 1 Kb mcc itfjr gmsjs. gmsjs.
M sjt W ks nkvjb dy {=,4,7,8,:,<}. |V| ks m buodjr il jcjojbts kb sjt V. Wudsjts M mbh D il sjt W mrj gfisjb sugf tfmt jmgf jcjojbts il W ks jqumccy ck`jcy mbh = |D| 3 |M|. Xfjb tfj buodjr il irhjrjh pmkrs il (M,D) mrj ,h cjqQp0 W dc kzh`e nW lh W ? {=,4,7,8,:,<}. |V| lhcf lhcf cjqQp0 V js vi0i`sm hf cP0` h`s b'``Ur` nWM cjqQp0 W hs okcjqQp0 M rL`` D dc kzh`e pqus t`rs nW lh W h` kzT0sh vi0i cj`u kz `l0hr` cs pq u` t` chr` nW rL`` = |D| 3 |M| r`s Öljr 0qVj (M, D) hf cmP0` n`sxf& (=:47) Buodjr il irhjrjh pmkr (M, D)
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # 6
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW b
b
`
` 1
b
G` .`
4 b
` 1
1 fichs lir sioj pisktkvj kbtjnjr b, tfjb
=1. b
b
b
G` .`
` 1
b
G` .7
` 1
` 1 b
`
` 1
lhcf /`u`Tjh kw.``½``½h b hs ly,
b
G` .`
` 1
b(b =) 4 b 4b =
b?1
8
b
r`s 1 , r`s
b
G` .7
`
b
G`
` = h` j`u nW& ` 1
` 1
b 4b = b(b =)4b 4 8
8
ir
Gr
1
8(b + =) ‗ 4b ‗ b(b ‗ =) ? 1 8b + 8 ‗ 4b ‗ b4 + b ? 1 7b ‗ b4 + 8 ? 1 b4 ‗ 7b ‗ 8 ? 1 (b ‗ 8)(b + =) ? 1 b?8 8
:
r 1
1
b
b4 (b =) 6
r =
Wic.
4
(<.41)
b(b =) b4 4 8
Mbs.
G` .`
b
==.
b
` 1
b
Wic.
G`
` = jqumcs.
`
b
Mbs.
b
Gr =
r 1
:
?
4
:
:
=
7=
:
? <.41
sjg ‗=
= =1 > ` > sjg sjg (` =) wkcc dj 4 =4 8 ` 1 =4 4
sjg ‗=
= =1 > ` > sjg sjg (` =) h` j`u n`sx`& 4 =4 8 ` 1 =4 4
(1.11) Jvmcumtj
sjg ‗=
Nkvjb jxp ?
= =1 > ` > sjg sjg (` =) 4 =4 =4 =4 8 ` 1
sjg
=
= =1 > ` > ` gis jg sjg =4 =4 =4 4 8 ` 1
=1 = =1 =1 = = 4 = = = = ? sjg ? sjg ? sjg ‗= (=) ? 1 ? sjg ` = 8 ` 1 > 4 ` = ( = ) ` 1 ( =) ` 1 skb skb ` < <
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # ;
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
/ 4
=4.
(
Kl ?
1
7 gis
/ 4
0lb ?
1
Mbs.
7 gis :
( skb gis )
r`s 4 h` h, r`s
j`u n`sx`&
(1.4:) / 4
Wic.
h, tfjb 4 ks jqumc ti
skb gis ):
K?7
1
sk b
/ 4
gis
:
/ 4
( 1
7 sk b h
gis 1
/ 4
skb gis ):
1
7
? 7
7 gis h
(
4
/ 4
gis h
? 7
1
skb
:
skb h ( gis sk b ):
/ 4
4K ? 7
( 1
h skb gis )8
sjg 4 h tmb =)8
cjt tmb ? t4 sjg4h ? 4tht
=7.
4 7
4tht
(t =)
8
1
7
4(t =)
4
= = ht 7 8 7 ( t = ) ( t = ) 1
=
7
( t =)
1
7 = = 4 4
Lkvj pjrsibs M,D,G,H & J mrj sjmtjh kb m gkrgucmr mrrmbnjojbt. Kl jmgf il tfjo ks nkvjb m fmt il ibj il tfj tfrjj giciurs rjh, dcuj & nrjjb, tfjb tfj buodjrs il wmys il hkstrkdutkbn tfj fmts sugf tfmt tfj pjrsib sjmtjh kb mhemgjbt sjmt njts hklljrjbt giciurjh fmts ks k`¡p I0lHr0`sm M,D,G,H rL`` J h`s iaÛ`f0 Öj js gWD`0` t`r` nWM 0lb 0l b kzT0sh I0lHr h`s rfu emx`s ³y`y_ ufy` rL``
ne`¿ jsm cs ,h emx hf \`s kf bf t`rf nW_ r`s \`slk0`sm h`s lhrus lhrus refh`sm cs cs g`¡\` t` chr` nW tglh k`c&k`c gWDs I0lHr0`s hs m hs k`c lF`Zu&lF`Zu em x hf \`slk0`¡ n`s& Mbs. Wic.
(71.11) Omxkouo buodjr il fmts usjh il smoj giciur mrj 4. Xfjy gmb bit dj 7 itfjrwksj mtcjmst 4 fmts il smoj giciur mrj gibsjgutkvj. Biw, Cjt fmts usjh mrj T, T, N, N, D (^fkgf gmb dj sjcjgtjh kb 7 wmys. Kt gmb dj TNNDD ir TTNDD mcsi) Biw, buodjrs il wmys il hkstrkdutkbn dcuj fmt (skbncj ibj) kb : pjrsib jqumc ti : Cjt dcuj fmt nijs ti pjrsib M. M D (Dcuj)
J
G
H Biw jktfjr pisktkib D & H mrj lkccjh dy nrjjb fmts mbh G & J mrj lkccjh dy Tjhs fmts Ir D & H mrj lkccjh dy Tjh fmts mbh G & J mrj lkccjh dy Nrjjb fmts 4 wmys mrj pisskdcj Fjbgj titmc buodjr il wmys ? 7 Þ : Þ 4 ? 71 wmys
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # =1
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
=8. =8.
Cjt
m
? 4 k e ` & d k 4 e ` dj twi vjgtirs. Gibskhjr m vjgtir g m d , , T. ˌ
ˌ
ˌ
ˌ
ˌ
ˌ
priejgtkib il
0lb
g
Kl tfj
ib tfj vjgtir (m d) ks
, tfjb tfj okbkouo vmcuj il (g (m d)). g jqumc ti
7 4
? 4 k e ` rL`` d k 4 e ` b`s
clb'` lb0s nq, nW j`u` ,h clb'` g dc kzh`e nW lh g m d , , T. 0lb clb'` g h` clb'` (m d) ke kz{`sk 7 4 nW r`s (g (m d)). g h` Z0wurj j`u n`s n`sx`&
m
ˌ
ˌ
ˌ
ˌ
ˌ
Mbs.
(=6.11)
Wic.
g
ˌ
(4k e ` ) + ( k 4 e ` )
ˌ
ˌ
ˌ
ˌ
ˌ
ˌ
g (4 ) k ( 4) e ( )` ˌ
ˌ
ˌ
g . (m d )
|md|
?
7 4
;( + ) ? =6
+ ? 4
(g m d) . g ? (m d m d) . (m d) ? <4 + < + < 4 ?
? <(4 ‗ 4 + 8) ? okb vmcuj =6
WJGXKIB 7 (Omxkouo Omr`s 5 =4) (14) Ckst-Omtgf sjts. Xfks sjgtkib gibtmkbs X^I (14) Ckst-Omtgf Jmgf Ckst-Omtgf sjt fms X^I (14) Ouctkpcj Gfikgj ]ujstkibs.
cksts 5 Ckst-K mbh Ckst-K mbh Ckst-KK. Ckst-KK. Jmgf Ckst-Omtgf sjt fms twi cksts Ckst-K fms Liur jbtrkjs Liur jbtrkjs (K),(KK), (KKK) mbh (K\) Ckst-KK fms Ckst-KK fms Wkx jbtrkjs Wkx jbtrkjs (Q),(]), (T ), (W), (X) mbh (Z). Ckst-K fms LIZT iptkibs mrj nkvjb kb jmgf Ouctkpcj Gfikgj ]ujstkib dmsjh ib Ckst-K mbh Ckst-K mbh Ckst-KK mbh Ckst-KK mbh IBC[ IBJ il IBJ il tfjsj liur iptkibs smtkslkjs LIZT iptkibs tfj gibhktkib ms`jh kb tfj Ouctkpcj Gfikgj ]ujstkib. Mbswjr ti jmgf qujstkib wkcc dj jvmcumtjh mggirhkbn ti tfj licciwkbn omr`kbn sgfjoj 5
Lucc Omr`s
5
+7 Kl IBC[ tfj iptkib girrjspibhkbn girrjspibhkbn ti tfj girrjgt giodkbmtkib ks gfisjb. gfisjb.
Ujri Omr`s 5
1 Kl bibj il tfj iptkibs ks gfisjb (k.j. tfj qujstkib qujstkib ks ubmbswjrjh).
Bjnmtkvj Omr`s 5
‗= Kb mcc itfjr gmsjs.
Omtgf tfj gicuob 5 Cjt tfj gkrgcj G=5 x 4 + y4 ? ; mbh G45 (x ‗ 7) 4 + (y ‗ 8) 4 ? =< kbtjrsjgt mt tfj pikbts V mbh [. Wuppisj tfmt mbitfjr gkrgcj G75 (x ‗ f)4 + (y ‗ `)4 ? r4 smtkslkjs tfj licciwkbn gibhktkibs. (k) gjbtrj il G7 ks gicckbjmr wktf tfj gjbtrjs il G= & G4, (kk) G= & G4 ditf ckj kbskhj G7 mbh (kkk) G7 tiugfjs G= mt O mbh G4 mt B Cjt tfj ckbj tfriunf V mbh [ kbtjrsjgt G 7 mt U mbh ^ mbh cjt m giooib tmbnjbt il G = & G 7 dj m tmbnjbt ti tfj pmrmdicm x4 ? 6y Xfjrj mrj sioj jxprjsskibs nkvjb kb tfj licciwkbn cksts5 Ckst-K Ckst-KK (K) 4f + ` (Q) < cjbntfil U^ (KK) (]) < cjbntfil V[ (KKK) (K\)
Mrjm il OUB Mrjm il UO^
(T) (W) (X) (Z)
: 8 4= :
4 < =1 7
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # ==
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
j`u` iaÛ` G=5 x4 + y4 ? ; rL`` iaÛ` G4 5 (x ‗ 7)4 + (y ‗ 8)4 ? =< lgZbq V rL`` [ ke kz lrQBsb hers nWM j`u` ,h iaÛ` G7 5 (x ‗ f)4 + (y ‗ `)4 ? r4 luJu kzlrgZ/``sm h`s cZrq cZrq"\ her` nW& (k) G7 h` hsZbz G= v`We G4 hs hsZbz `sm hs c`L` cmesP` nWM (kk) G= rL`` G4 b`su`sm_ G7 hs vZbe oklCL`r nWM (kkk) G7 , G= h`s lgZbq O rL`` G4 h`s lgZbq B ke Ck'`U her` nWM 0lb V rL`` [ cs xqteus i`yf esP`` G7 h`s U rL`` ^ ke kzlrQBsb herf nW v`We G= rL`` G7 hf oF`0lu"D Ck'`U esP`` keiy0 x4 ? 6y hf F`f Ck'`U esP`` nWM Ckst-K 4f + `
(K)
Ckst-KK (Q)
U^ hf yJg` dU
(KK)
V[ hf yJg` dU
OUB h`
{`s?`]y
(KKK)
UO^ h`
{`s?`]y
(K\)
(])
(T) (W) (X) (Z)
=:.
< <
: 8 4= : 4 <
=1 7
^fkgf il tfj licciwkbn ks tfj ibcy girrjgt giodkbmtkib9
luJu jsm cs h`Wuc` ljy`u ljy`u cnf nW& Mbs.
(M) (K)-(W) (D)
(D) (KK)-(])
(G) (K)-(Z)
=<.
^fkgf il tfj licciwkbn ks tfj ibcy Kbgirrjgt giodkbmtkib9
(H) (KK)-(X)
luJu jsm cs h`Wuc` ljy`u ljy`u xyr nW& Mbs.
(M) (K\)-(Z) (G)
Wic.
=: ti =<
(D) (KKK)-(T)
(G) (K\)-(W)
(H) (K)-(Q)
^ B (7,8)
[
• •
Q
G V
I O
U
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # =4
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
()
4r ? OB ? 7 +
7
4
4
8
+ 8 ? =4 8
gjbtrj G il gkrgcj G 7 ckjs ib y ?
7
r?<
x
8 f 7
Cjt G f,
=4
IG ? OG ‗ IO ? 4
f
`?
8 7
f?
f
;
4
‗ 7 ? 7
? 7
=6
=4
+
:
f?
; :
? <
:
Jqumtkib il ckbj U^ G= ? G4 7x + 8y ? ; Hkstmbgj il U^ lrio (1, 1) | ; | ; ? : 74 84 4
; Cjbntf il V[ ? 4 7 ? : Hkstmbgj il U^ lrio G 7; :
7
=4
8 4
: 8
;
4
4
Mrjm il UO^ ?
<
=
= 4
4
<
4
:
4
?
48 < :
<
= U^ ? 4
. BO
>4 < :
. U^ (IO + IQ) ?
Mrjm il OUB Mrjm il UO^
=
48 <
4
:
. 7
; ? :
46 6 < 4:
: 8
?
Wcip il tmbnjbt ti G= mt O ?
:
:
cjbntfil U^ ? cjbntfil V[
Mrjm il OUB ?
48
<
?
Cjbntf il U^ ? 4
(\)
7
? 7
:
4
()
:f
=4
4f + ` ?
()
=<
4
7 = ? ‗ 8 / 7 8
Jqumtkib il tmbnjbt y ? ox ‗ 7 = o4 y ? ‗ y?
7 8
x ‗ 7
7 x 8
‗
=
=: 8
; =<
x ? ‗
8y ‗ : 7
….(k)
tmbnjbt ti x4 ? 8(4)y ks
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # =7
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
x ? o y +
4
….(kk)
o
Giopmrj (k) mbh (kk) o ? ‗
8 7
mbh
4 o
? ‗ :
?
=1 7
OMXGF X[QJ Cjt l(x) ? skb( gisx) mbh n(x) ? gis(4skbx) dj twi lubgtkibs hjlkbjh lir x 2 1. Hjlkbj tfj licciwkbn sjts wfisj jcjojbts mrj wrkttjb kb kbgrjmskbn irhjr V ? {x 5 l(x) ? 1}, [ ? {x 5 l (x) ? 1} U ? {x 5 n(x) ? 1}, ^ ? {x 5 n(x) ? 1}
Ckst - K
Ckst - KK
7 , , 8 , > 4 4
()
V
(Q)
()
[
(]) mb mrktfojtkg prinrjsskib
()
U
(T) bit mb mrktfojtkg prinrjsskib
(\)
^
(W) ,
> =7 , < < < 4 (X) , , 7 7 7 (Z) , < 8
0lb l(x) ? skb(gisx) rL`` n(x) ? gis(4skbx) b`s ]yu x 2 1 hs ly, kleF``l"`r nW_ r`s luJu cjqQp0`sm h`s b'``Ud0s ltuhs vi0i v`e`snf Öj jsm lyP`s nW& V ? {x 5 l(x) ? 1}, [ ? {x 5 l (x) ? 1} U ? {x 5 n(x) ? 1}, ^ ? {x 5 n(x) ? 1} -K
- KK
7 , 8 , > , 4 4
()
V
(Q)
()
[
(]) cj`Zre
Es.`f js nW
()
U
(T) cj`Zre
Es.`f js unfm unfm nW
(\)
^
> =7 , (W) , < < <
4 , (X) , 7 7 7 < 8
(Z) ,
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # =8
| EJJ (MH\MBGJH) 41=; | HMXJ 5 4>-1:-41=; | QMQJT-4 | OJOIT[ DMWJH | OMXFJOMXKGW
=>.
^fkgf il tfj licciwkbn ks tfj ibcy girrjgt giodkbmtkib
luJu jsm cs h`Wuc` ljy`u ljy`u cnf nW&
Mbs.
(M) K\ ‗ (Q), (T), (W) (D) KKK ‗ (T), (Z) (G) KKK ‗ (Q), (]), (Z) (H) K\ ‗ (]), (X) (M)
=6.
^fkgf ks tfj ibcy girrjgt giodkbmtkib
luJu jsm cs h`Wuc` ljy`u ljy`u cnf nW&
Mbs. Wic.
(M) K ‗ (]), (Z) (D) KK ‗ (]), (X) (G) KK ‗ (T), (W), (H) K ‗ (Q), (]) (H) (=> ti =6) l(x) ? 1 skb ( gisx) ? 1 gisx ? b gisx ? b gisx ? ‗=, 1, =
V? {b, (4b + =)
4
} ? {b
4
, b}
l'(x) ? 1 gis( gisx) ( ‗ ‗ skbx) ? 1
gisx ? (4b + =) =
gisx ? b + gisx ? ½
= 4
4
ir x ? b
ir x ? b
[ ? 4b
n(x) ? 1
ir x ? b
4
7
,4b
4 7
, b, b K
gis (4 skbx) ? 1
skbx ?
4 skbx ? (4b + =)
4
4b = = 7 , 8 8 8
U ? b sk b =
n'(x) ? 1
= 8
, b skb
=
7 8
, b K
‗skb(4 skbx) (4 gisx) ? 1
4 skbx ? b ir x ? (4b + =) skbx ?
b 4
? 1, ½
= 4
4
, ½ = ir x ? (4b + =)
4
^ ? b, (4b =) , b , b K 4 <
Tjsibmbgj Jhuvjbturjs Ckoktjh TJNKWXJT TJNKWXJTJH JH & GITQITMXJ ILLKGJ 5 GN Xiwjr, M-8< & :4, KQKM, KQKM, Bjmr Gkty Omcc, Efmcmwmr Efmcmwmr Timh, @itm (Tme.) (Tme.) - 74811: Qf.Bi. 5 1>88-4>>>>>>, 1>88-4>>>>11 | 1>88-4>>>>11 | Xicc Lrjj 5 =611 4:6 :::: | LMV Bi. 5 +;=-144-7;=<>444 5 +;=-144-7;=<>444 | Xi @ @biw o oirj 55 sos TJWI mmt :<<>>
^jdsktj 5 www.rjsibmbgj.mg.kb | J-omkc 5 gibtmgtArjsibmbgj.mg.kb | GKB 55 Z61714TE411>QCG14814; Xfks ssicutkib w wms h hiwbcimh l l rio T Tjsibmbgj E EJJ ((MH\MBGJH) 4 41=; W Wicutkib p pirtmc
QMNJ # =: