2. suppose w = f(z) is a function of complex
variable z, then it may happens that = f() ; that
is the reflection of z in the real axis corresponds
to the reflection of w in the real axis, for
example , if
w = f(z) = z + + 1
then = f() = = () + () + 1 = + + 1
= () + () + 1
= + + 1,
and f() = + + 1
Hence = f(),
on the other hand , for the function
w = f(z) = + + 1 , we have
= f() =
= () + () + 1
= () + + 1
f(z) = () - i + 1
Hence
4. ekuo }kjk vkfndky ls gh Kku dk lap;
fd;k tkrk jgk gSaA izR;sd u;h ih<h dsk
iqjkuh ih<h }kjk dqN Kku lkekftd
fojklr esa izkIr gksrk gSA Kku dh ;g
ijEijkRed J`a[kyk gh f”k{kk gS ftlds
}kjk ekuo us viuh ekufld] vk/;kfRed
vkSj lkekftd izxfr dh gSA f”k{kk us gh
ekuo dks i”kq Lrj ls Åapk mBk;k gS
vkSjJs’B lkaLd`frdizk.kh cuk;k gSA
5. Lakdh.kZ vFkZ esa f”k{kk
dk rkRi;Z iqLrdh;
Kku vkSj fy[kus&i<+us ls
fy;k tkrk gSA O;kid n`f’V ls
f”k{kk dk rkRi;Z lHkh izdkj
ds Kku ds laxzg rFkk ekuo
ds pgqaeq[kh fodkl ls fy;k
tkrk gSA
6. egkRekxka/kh %&& ßf“k{kkls esjkvfHkizk;
cPp ds “kjhj]eu vkssssSj vkRek
esa fo|eku loksZRrexq.kksdk
lokZxh.kfodkl djukgssssSAÞ
%& ßf”k{kkeuq’; dh leLr lgt“kfDr;ks
dk LokHkkfod lkeatL;;qDr vkSj
izxfr”khy fodkl gSaaAÞ
izks-
isLVykWth
7. ifjokj dzhMk- lewg tkfr ,oa O;kolkf;d lewgksa
}kjkHkh vukSipkfjd:ils f”k{kkiznku djusdk dk;Z
fd;k tkrk gS vkSj vkfne lektksa esa rks os vkt Hkh
egRoiw.kZ f”k{k.k laLFkkvksa ds :i esa gSa]
fdUrq vk/kqfud ;qx esa f”k{kk iznku djus dk dk;Z
f”k{k.k laLFkkvksa tSls] dkWyst ,oa fo”ofo|ky;ksa
vkfn ds }kjk vkSipkfjd :i ls fd;k tkus yxk gSA ;s
laLFkk,a izR;{k :i ls vkSipkfjd vkSj O;ofLFkr <ax ls
f”k{kkiznku djrh gSA
9. f”k{kk ds izdkj
f”k{kkdsfofHkUuizdkjgSa& 1-
oS;fDrd,oalkewfgdf”k{kk&tcfdlhvdsysO;fDrdks
f”k{kknhtkrhgSrks mlsoS;fDrdf”k{kk,oatcdbZ
O;fDr;ksadks,dlkFk tSlsf”k{kkesa]f”k{kk nhtkrhgS rks
mls
2-izR;{k vkSjvizR;{k f”k{kk &tcv/;kidf”k{kkiznkudjus
okykizR;{k :Ik lsghlh[kusokysdksfdlhfo’k;dkKku
iznkudjrkgSrksmlsizR;{k f”k{kk dgrsgSA vizR;{k f”k{kk
eslh[kusokykvU;O;fDr;ksadkvuqdj.kdjrkgSA
10. vkSipkfjd,oavukSipkfd f‘k{kk%&
ikB‘kkyk ls ysdj fo‘ofo|ky; rd dhf’k{k.k laLFkk,a
vktvkSipkfjd :i ls ,oafo‘ks”khd`r f’k{kknsusdk
dk;Z djrh gSA vk/kqfud ;qx es f’k{kkvkSipkfjd :i ls gh
iznku dh tkrh gSA vkfne lektksesaf’k{kk vukSipkfjd
:i ls ifjokj]iMks+l]xks=laxBuksa vkfnds }kjknh tkfr
gS A
11. lkekU;rFkk fof'k"V f'k{kk & tcfdlh
lkekU;Kkuds fy, f'k{kkiznku dh tkrh gS rks og lkekU;
f'k{kkdgykrh gSA nwljh vksj fdlh fof'k"V Kku ls
lEcfU/kr f'k{kknh tkrh gS rks mls fof'k"V f'k{kkdgrs
gSA] tSls ]fpfdRlk]dkuwu]m|ksx vkfnA
udkjkRed,oaldkjkRedf'k{kk&bls
fu"ks/kkRed,oafu'p;kRedf'k{kkHkhdgrsgSa Atcfdlh
iwoZfuf'prm’s';dsvuq:Ik f'k{kkiznkudhtkrh gSrksmls
ldkjkRedf'k{kkdgrsgSa vkSjtcfcukfdlhiwoZfuf'pr
m’s';dsf'k{kknhtkrhgS rksmls udkjkRedf'k{kkdgrs
gSaA
12. f”k{kk ds mn~ns”;%&
1Û Hkk’kkdsfy[kus]]cksyus,oaO;kdj.krFkk xf.krdkKku
iznkudjukA
2Û tfVylaLd`frdksle÷kusdsfy,KkuiznkudjukA
3Û cPpsesalkekftdvuqdwyudh{kerk iSnkdjukA
4Û vkfFkZdvuqdwyudsfy,izf”k{k.knsukA
5Û laLd`fresalq/kkj,oao`f)esa;ksxnsukA
13. f’k{kk dhlkekftd fu;U=.kesHkwfedk
%&
f’k{kkdsfofHkUuizdk;kZsesals,degRiw.kizdk;Zlekt
esa
fu;U=.kdsvU;lk/ku O;fDrdBksjrkcjrldrsgS] n.M]ncko
,oacnysdhHkkouklsdkeysldrsgS]fdUrqf’k{kkO;fDr
esrdZ,oafoosdiSnkdjrhgS] mlesvkRefu;U=.k
dh‘kfDriSnkdjrhgSftllsfdogLo;aghmfpr,oa
vuqfprdks/;ku esj[kdjlkekftdfu;eks]ifzrekuks ,oa
dkuwuksdkikyudjrkgSA
14. Lkekthdj.k%&
Lkekthdj.kdkrkRi;ZO;fDrdksviuhlaLd`fr ,oalektdk
Kkuiznkudjmls lektdh,dIkzdk;kZRedbdkbZcukukgS
lekthdj.kdhizfdz;kds}kjk ghO;fDrdsO;fDrRodkfodk-l
gksrkgSA fo?kfVrO;fDrRookysO;fDrlslektdsvuq:i
vkpj.kdjusdhvis{kkughdhtkldrhAf’k{kkds}kjk gh
O;fDrdkslektdsewY;ksaekun.Mkasfu;eks]dkuwukas]
vkn’kksZ]vk- fndkKkudjk;ktkrk gSAlekt}kjkfu/kkZfjr
,oaLohd`r fu;eksdkikyudjuslslektesfu;u=.k],drk,oa
le#- irk cuhjgrhgSA
15. ckSf}d fodkl%&
f‘k{kkdk,dizeq[k dk;ZekuoKku ,oacqf}dkfodkldjuk
gSAcgqrle;igysgSyohfVlusdgkFkk fd Seuq”;tUels
vKkuhgStMcqf}ugha]f’k{kk mUgsatMcqf}cukrh
gSA^^ ;gckr mlle;BhdFkhtcizkphu,oavkfnelektks
es f’k{kk:f<xrfunsZ’knsrhFkhvkSjO;fDr dksfcukrdZ
,oav- ykspukdsmUgsLohdkjdjukIk<rk FkkfdUrqvkt
dsoSKkfud,oaizkS|ksfxd;qxesa;gckrlghughgSAvkt
dhf’k{kkrdZ ijv/kkfjr gSA
16. uSfrdxq.kks dk fodkl%&
f”k{kkekuoesuSfrdxq.kkstSls] lg;ksx]lfg’.kqrk]n;k]
bZekunkjhvkfn dkfodkldjrhgSAf”k{kkdk,ddk;Zpfj=-
fuekZ.kHkhgSAlnxq.kksa,oa lnpfj=dsvHkkoesekuo
esi”kq-izo`fRr;kacyorhgkstkrhgSaA og funZ;h]fueZe]
bZ’;kZyq]vR;kpkjh,oapfj=ghucutkrkgSA ,slhn”kk es
O;fDRkijfu;U=.kj[kuklektdslkeus,dleL;k cutkrhgSA
uSfrdxq.kksals;qDrO;fDrghfu’Bkoku,oadrZO;ijk;.k
gksrkgSA ,slsO;fDrghlektesaO;oLFkk],drk,oalaxBu
cuk,j[kuses;ksx nsrsgaS Ablizdkjf”k{kkekuoesa
uSfrdxq.kksdkfodkl,oapfj=dkfuekZ.kdjlkekftd
fu;U=.kcuk,j[kusesa;ksxnsrhgSA z z
18. f”k{kk ds leL;kvksa dk fuokj.k
%&
1-orZekuvkSipkfjdf”k{kkiz.kkyhesaizpfyrijh{kk
iz.kkyhesaifjorZufd;ktk, A
2-Nk=,oav/;kidksads chp ?kfu’BrkiSnk dh tk,
ftllsdhv/;kidksadkNk=ijuSfrd ncko,oa cukjg
ldsA
3-f”k++{kkdsoylkfgfR;dKku rd ghlhferugksdj
O;kogkfjdthoulsHkh lacfU/krgksuh pkfg,A
19. izR;sd O;fDr dks f”kf{krcukdjf”k{kkmldh
ckSf}d {kerk esa o`f} djrh gSA
nwljs“kCnksa esa] f”k{kklektesa vKku dks
nwjdjrh gSA
f”k{kkdk nwljkegRoiw.kZlekftddk;Z
fofHkUulaLd`fr;ks ,oa :fp;ksa ds yksxksa esa
le>dk fodkldj muesaO;kIr Hkzedks nwjdjuk
gSaA blls Vdjko] ruko ,oa la?k’kZ dh fLFkfrls
efqDr feyrhgS vkSjlekftdlfg’.kqrk c<rh gSA
![ckyksn
Hkkuq”kadj
egyokj
yrk lkgw ] dapu ;kno
fu”kk ;kno] lquhrk lko
iz;ksfxddk;Z 2014 & 15
fo’k; & f”k{kk dk egRo](https://image.slidesharecdn.com/student-150701123055-lva1-app6892/85/Student-1-320.jpg)
