1. iQyu
Qyu & f : A → B esa Qyu gS ;k ugha bldh tkap ds fy, fuEufyf[kr ijh{k.k
djrs gSa&
(i) A ds çR;sd vo;o dk f- ds vUrxZr B esa çfrfcEc fo|eku gS ;k ughaA
(ii) A ds çR;sd vo;o dk f- ds vUrxZr B esa ,d vksj dsoy ,d çfrfcEc
xf.kr fo|eku gksuk pkfg,A
Qyu&Øfer ;qXeksa ds leqPp; ds :i esa & Qyu f Øfer ;qXeksa (a, b) dk leqPp;
gSA tcfd
(i) a leqPp; A dk vo;o gksA
(ii) b leqPp; B dk vo;o gksA
(ii) f ds fdlh Hkh nks Øfer ;qXeksa esa çFke lnL; ,d ls ugha gksA
(iii) A dk çR;sd lnL; fdlh u fdlh ;qXe dk çFke lnL; vo'; gksA
Qyu ds çdkj & Qyu f : X → Y ,dSdh Qyu dgykrk gS ;fn X ds
egÙoiw.kZ lw=k fHkUu&fHkUu vo;oksa ds Y esa fHkUu&fHkUu çfrfcEc fo|eku gksA ;fn x1, x2, X ds
dksbZ nks vo;o gks vkSj
x1 ≠ x2 ⇒ f(x1) ≠ f(x2), f(x1) = f(x2) ⇒ x1 = x2 rc Qyu ,dSdh gksxkA
(i) cgq,dSdh Qyu & Qyu f : X → Y cgq,dSdh Qyu dgykrk gS ;fn X
ds fdUgha nks vo;oksa ds çfrfcEc Y esa leku gks] vFkkZr~ f : X → Y cgq,dSdh gksxk
;fn x1 ≠ x2 ⇒ f(x1) ≠ f(x2)
(ii) vkPNknd Qyu & Qyu f : X → Y ,d vkPNknd Qyu dgykrk gS
;fn Y ds çR;sd vo;o dk X esa çfrfcEc fo|eku gksA nwljs 'kCnksa esa f dk
ifjlj = f dk lgçkUrA
(iii) vUr{ksZih Qyu & Qyu f : X → Y vUr{ksZih Qyu dgykrk gS ;fn Y
Rajasthan Knowledge esa de ls de ,d vo;o ,slk gks ftldk çfrfcEc X esa fo|eku ugha gks vFkkZr~
IT shapes future CorporationLimited Y esa de ls de ,d vo;o ,slk gks ftlds fy, f–1(y) = φ rc Qyu vUr{ksZih
(A Public Limited Company Promoted by Govt. of Rajasthan)
gksrk gS] nwljs 'kCnksa esa f dk ifjlj ≠ f dk lgçkUrA
(2)
çfrykse Qyu & ;fn f : X → Y ,dSdh vkPNknd gks rks f dk çfrykse f–1 dqN egÙoiw.kZ dks.kksa ds f=kdks.kferh; vuqikr
: X → Y esa Qyu gS tks fd çR;sd vo;o y ∈ Y ds laxr x ∈ X ftlds fy, (Trigonometrical Ratios for Some Special Angles)
f(x) = y çfrykse Qyu dgykrk gSA 1º 1º
 7 15º 22 18º 36º
fo"ke ,oa le Qyu 2 2
(i) fo"ke Qyu & ,d Qyu f(x) fo"ke Qyu dgykrk gSA ;fn f(–x) = 4 2 6 3 1 1 5 1 1
sin  2 2 10  2 5
–f(x) lHkh x ds fy, fo"ke Qyu dk xzkQ foijhr iknksa esa lefer gksrk gSA 2 2 2 2 2 4 4
(ii) le Qyu & ,d Qyu f(x) le Qyu dgykrk gSA ;fn f(–x) = –f(x) 4 2 6 3 1 1 1 5 1
cos  2 2 10  2 5
lHkh x ds fy,A le Qyu dk xzkQ y-v{k ikfjr lefer gksrk gSA 2 2 2 2 2 4 4
125  10 15
f=kdks.kehfr; iQyu ,oa f=kdks.kferh; vuqikr tan   3  2   2  1 2  3 2 1 52 5
5
,d nwljs ds inksa esa f=kdks.kferh; vuqikr
(Trigonometrical Ratios in Terms of each Other) lacaf/kr dks.kksa ds f=kdks.kferh; vuqikr
(Trigonometrical Ratios of Allied Angles)
sin  cos  tan  cot  sec  cosec 
f=kdks.kferh; vuqikr
sin  sin  1  cos 2 
tan  1 sec2   1 1 sin  cos  tan 
1  tan  2
1  cos  2 sec  cosec  lacaf/kr dks.k
1 cot  1      sin  cos   tan 
cos  1  sin 2  cos 
;k     
1  tan 2  1  cot 2  sec 
 90      cos  sin  cot 
2 
sin  1  cos 2  1
tan  tan  sec2   1 
1  sin 2  cos  cot   90    ;k      cos   sin   cot 
2 
cot 
1  sin 2  cos  1
cot 
1 2
cosec  1 180    ;k      sin   cos   tan 
sin  1  cos 2  tan  sec2   1 180    ;k       sin   cos  tan 
1  cot 2  cosec 
;k  3   
1 1
sec  1  tan 2  sec   270      cos   sin  cot 

1  sin 2  cos  cot  cosec2   1  2 
1 1 1  tan 2  sec   3 
 270    ;k      cos  sin   cot 
cosec  1  cot 2  cosec 
sin  tan   2 
1  cos 2 sec2   1
 360    ;k  2     sin  cos   tan 
(3) (4)

2. f=kdks.kferh; vuqikrksa ds dks.kksa ds eku (ii) lg[k.M & vo;o aij dk lg[k.M çk;% Fij ls O;Dr fd;k tkrk gS]
(Trigonometrical Ratios for Various Angles) tksfd (–1)i+j Mij ds cjkcj gksrk gS tgka M vo;o aij dk milkjf.kd gSA
a11 a12 a13
;fn   a21 a22 a23
a31 a32 a33
a a23
rks F   1
11 11  M11  M11  22
a32 a33
lkjf.kd F   1
1 2 a
 M12   M12   21
a23
12
r`rh; dksfV ds lkjf.kd dk eku a31 a33
a11 a12 a13
lkjf.kd ds xq.k/keZ &
(i) fdlh lkjf.kd dh fdlh iafDr ¼LrEHk½ dks fdlh la[;k ls xq.kk djus ij
  a21 a22 a23
a31 a32 a33 lkjf.kd dk eku Hkh ml la[;k ls xq.kk gks tkrk gS vFkkZr~
ka kb kc a b c ka b c
11 a a23 1 2 a a23 13 a21 a22 p q r  k p q r  kp q r
   1 a11 22   1 a12 21   1
a32 a33 a31 a33 a31 a32 u v w u v w ku v w
a a23 a a23 a21 a22 (ii) fdlh lkjf.kd dh fdlh iafDr ¼LrEHk½ dk çR;sd vo;o ;fn nks inksa dk
 a11 22  a12 21 
a32 a33 a31 a33 a31 a32 ;ksx gks rks ml lkjf.kd dks mlh dksfV dh nks lkjf.kdksa ds ;ksxQy ds :i esa
milkjf.kd ,oa lg[k.M O;Dr fd;k tk ldrk gS vFkkZr~
(i) milkjf.kd a b c  a b c   
a11 a12 a13 p q r  p q r  p q r
a a23
;fn   a21 a22 a23 rks a11 dk milkjf.kd M11  22 blh u v w u v w u v w
a32 a33
a31 a32 a33
a b c a b c  b c
a a23 p q r  p q r   q r
rjg M12  21 lkjf.kd dk eku fuEu çdkj Kkr fd;k tkrk gSA rFkk
a31 a33 u v w u v w  v w
Δ = a11 M11 – a12 M12 + a13 M13
;k Δ = –a21 M21 + a22 M22 – a23 M23 (iii) ;fn fdlh lkjf.kd dh fdlh iafDr ¼LrEHk½ ds çR;sd vo;o esa fdlh
;k Δ = a31 M31 – a32 M32 + a33 M33 nwljh iafDr ¼LrEHk½ ds laxr vo;oksa dks fdlh ,d dh jkf'k ls xq.kk djds tksM+s
(5) (6)
;k ?kVk;sa rks lkjf.kd dk eku ugh cnyrkA vFkkZr~
a b c a  b   c b c vFkkZr~
p q r  p  q   r q r
u v w u  v   w v w
eSfVªDl
nks lkjf.kdksa dk xq.kuQy eSfVªDl ds çdkj
nks lkjf.kd ftudh dksfV nks gS dk xq.kuQy fuEu çdkj ifjHkkf"kr gS&
(i) iafDr eSfVªDl & A=[aij]m×n ,d iafDr eSfVªDl gS ;fn m = 1
a1 b1 1 m1 a  b  a1m1  b1m2 (ii) LrEHk eSfVªDl & A=[aij]m×n ,d LrEHk eSfVªDl gS ;fn n = 1
  1 1 1 2
a2 b2  2 m2 a2 1  b2  2 a2 m1  b2 m2 (iii) oxZ eSfVªDl & A=[aij]m×n ,d oxZ eSfVªDl gS ;fn m = n
nks lkjf.kd ftudh dksfV rhu gS dk xq.kuQy fuEu çdkj ifjHkkf"kr gS& (iv) ,dy eSfVªDl & A=[aij]m×n ,d ,dy eSfVªDl gS ;fn m = n = 1
(v) 'kwU; eSfVªDl & A=[aij]m×n ,d 'kwU; eSfVªDl gS ;fn aij = 0 lHkh i rFkk j
a1 b1 c1 1 m1 n1
ds fy,
a2 b2 c2   2 m2 n2
(vi) fod.kZ eSfVªDl & ,d oxZ eSfVªDl A–[aij]m×n ,d fod.kZ eSfVªDl gS ;fn
a3 b3 c3  3 m3 n3
aij = 0 tc i ≠ j
0 i  j
a11  b1 2  c1 3 a1m1  b1m2  c1m3 a1n1  b1n2  c1n3 (vii) vfn'k eSfVªDl & A= [aij] ,d vfn'k eSfVªDl gSA ;fn aij   tgka
k i  j
 a2 1  b2  2  c2  3 a2 m1  b2 m2  c2 m3 a2 n1  b2 n2  c2 n3 K vpj gSA
a31  b3 2  c3 3 a3 m1  b3 m2  c3 m3 a3 n1  b3 n2  c3 n3 (viii) bdkbZ eSfVªDl & ,d oxZ eSfVªDl A=[aij] ,d bdkbZ eSfVªDl gSA ;fn
lefer lkjf.kd 1 i  j
aij  
;fn fdl lkjf.kd ds çR;sd vo;o ds aij fy, aij = aji ∀ i, j gks rks mls lefer 0 i  j
lkjf.kd dgrs gSA (ix) f=kHkqtkdkj eSfVªDl
a h g (a) Åijh f=kHkqtkdkj eSfVªDl& ,d oxZ eSfVªDl [aij] Åijh f=kHkqtkdkj
vFkkZr~ h b f eSfVªDl dgykrk gS ;fn aij = 0 tcfd i > j.
g f c (b) fuEu f=kHkqtkdkj eSfVªDl& ,d oxZ eSfVªDl [aij] fuEu f=kHkqtkdkj eSfVªDl
fo"ke lefer lkjf.kd dgykrk gS ;fn aij = 0 tcfd i < j.
;fn fdl lkjf.kd ds çR;sd vo;o ds aij fy, aij = – aji ∀ i, j gks rks mls fo"ke (x) vO;qRØe.kh; vkSj O;qRØe.kh; eSfVªDl&
lefer lkjf.kd dgrs gSA ;fn lkjf.kd |A| = 0 ⇒ vO;qRØe.kh;
;fn lkjf.kd |A| ≠ 0 ⇒ O;qRØe.kh;
(7) (8)

3. eSfVªDl dk ;ksx ,oa O;odyu 1 1 ax
;fn A[aij]m×n rFkk [bij]m×n nks leku dksfV dh eSfVªDl gks rks mudk ;ksx A + B (xvii)  a 2  x 2 dx  2a log a  x  c  x  a 
og eSfVªDl gS ftldk çR;sd vo;o eSfVªDl A rFkk B ds laxr vo;oksa ds ;ksx 1 x  x
 dx  sin 1    c   cos 1    c
ds cjkcj gSA vFkkZr~ A + B = [aij + bij]m×n (xviii)
a 2  x2 a a
vfuf'pr lekdyu 1 x
 dx  log x  x 2  a 2  c  sinh 1    c
ekud lw=k (xix)
x2  a 2 a
xn 1 1  x
 x dx  loge x  c
1
x  c  n  1  dx  log x  x 2  a 2  c  sinh 1    c
n
(i) dx  (ii)
n 1 (xx)
x2  a2 a
ax
a x 2 2 a 2 1 x
x
dx   c  a x log e e  c
e
x
dx  e x  c
(iii) (iv)
log e a (xxi)  a 2  x 2 dx 
2
a  x  sin
2 a
c
(v)  sin xdx   cos x  c (vi)  sin xdx  sin x  c x 2 a2 x
(xxii)  x2  a 2 dx  x  a 2  sin 1  c
(vii)  tan xdx  log sec x  c   log cos x  c 2 2 a
(viii)  cot xdx  log sin x  c x 2 2 a2 x
(xxiii)  x2  a 2 dx  x  a  cos h 1  c
2 2 a
 x
(ix)  sec xdx  log sec  tan x  c   log sec x  tan x  c  log tan  4  2   c 1 1 x
dx  sec 1  c
  (xxiv)  a a
x x 2  a2
x
(x)  cosec dx   log cosec x  cot x  log cosecx  cot x  c  log tan  2   c
  eax eax   b 
(xxv)  eax sin bxdx  a sin bx  b cos bx   c  sin bx  tan 1     c
a2  b2 a 2  b2   a 
(xi)  sec x tan xdx   sec x  c (xii)  cosec x cot xdx   cosec x  c
eax eax   b 
(xiii)  sec2 xdx  tan x  c e
ax  a cos bx  b sin bx   c  cos bx  tan 1     c
 co sec
2
(xiv) xdx   cot x  c (xxvi) cos bxdx
a 2  b2   a 
a 2  b2
2 1 1 
x
(xv)  x2  a 2 dx  a tan  c
a (xxvii)
1
 f  ax  b  dx  a   ax  b   c
1 1 xa
(xvi)  x2  a 2 dx  2a log x  a  c  x  a  lekdyu
fuf'pr lekdyu ds xq.k/keZ
(9) (10)
f  x, y 
dy
 1 ;k dy F  y  
 
dv

dx
dx f 2  x, y  dx  x  F v   v x
b b b h x 
f  t  dt  h  x  f  h  x    g   x  f  g  x  
d
(i)  f  x  dx   f  t  dt  f  u  du (ix) dx  
a a a g x
b b
 f  x  dx    f  x  dx
vody lehdj.k
(ii)
a a vody lehdj.k dh dksfV rFkk ?kkr& vodyu lehdj.k esa fo|+eku
b c b vodytksa dk mPpre Øe gh ml lehdj.k dh dksfV dgykrk gS rFkk vody
(iii)  f  x  dx   f  x  dx   f  x  dx a  c  b lehdj.k esa mPpre vodyt dh ?kkr gh ml vody lehdj.k dh ?kkr
a a c
2
d3y  dy 
a a dgykrh gSA vody lehdj.k  3   y  ex dh dksfV 3 rFkk 1 ?kkr gSA
dx 3  dx 
(iv)  f  x  dx   f  a  x  dx
0 0 çFke dksfV o çFke ?kkr vody lehdj.k
a 
a dy dy
 f  x   f  x   dy  f  x  dx
 f  x  dx  2 f  x  dx  ;fn f   x   f  x  ¼le Qyu½


(i)
dx dx
nksuksa rjQ lekdyu djus ij
a
(v) 0
 vkSj ;fn f   x    f  x  ¼fo"ke Qyu½
0 
  dy   f  x  dx  c ;k y   f  x  dx  c
dy dy dy
 f  x g  y   f  x g  y  
2a a a
 f  x  dx  c
(vi)  f  x  dx   f  x  dx   f  2a  x  dx ¼lkekU; :i ls½ (ii) dx dx g  y 
0 0 0
dy dv
 f  ax  by  c   
a  bf  v  
a  (iii)  dx
2 f  x  dx  if f  2a  x   f  x 
 
dx


0
 if f  2a  x    f  x 
0 
 (iv)
an T T
(vii)  f  x  dx  n  f  x  dx ¼;fn f  x  T   f  x  vkSj n  N ½ dy
 P y  Q  y e
pdx
  Q e
pdx
dx  c
(v)
a 0 dx
b b lfn'k
(viii)  f  x  dx   f  a  b  x  dx lfn'k ;k ØkWl xq.kuQy& ekuk a rFkk b nks lfn'k gS rFkk θ muds e/; dks.k
a a
gS rc a × b = |a||b| sin θ n ;gka n, a rFkk b ds yEcor~ bdkbZ lfn'k gSA
(11) (12)

4. lfn'k xq.kuQy ds xq.kuQy f=kfofe; funsZ'kkad T;kfefr
       
(i) a  b  b  a  i.e. a  b  b  a  funsZ'kkad& nks fcUnqvksa rFkk ds e/; nwjh
(ii)
 PQ   x2  x1 2   y2  y1 2   z2  z1 2
(iii)

ewy fcUnq ls fcUnq  x1 , y1 , z1  dh nwjh
(iv) ;fn a  a1iˆ  a2 ˆ  a3 k
j ˆ rFkk rks
 ;fn fcUnq P  x1 , y1 , z1  rFkk dks feykus okyh js[kk dks fcUnq
(v) a rFkk nksuksa ds yEcor~ lfn'k gksrk gSA
vuqikr esa foHkkftr djrk gS] rks

(vi) rFkk ds ry ds yEcor~ bdkbZ lfn'k gksrk gSA rFkk ¼ a rFkk
m x  m2 x1 m y  m2 y1 m z  m2 z1

x 1 2 ;y  1 2 ;z  1 2
  m1  m2 m1  m2 m1  m2
;k rFkk ½ ds ry ds yEcor~ ifjek.k dk ,d lfn'k   a b gksrk
ab ¼vUr foHkktu½
gSA m1 x2  m2 x1 m y  m2 y1 m z  m2 z1
ˆ j ˆ
rFkk x  m1  m2
;y  1 2
m1  m2
;z  1 2
m1  m2
(vii) ;fn i , ˆ, k rhu bdkbZ lfn'k rhu ijLij yEcor~ js[kkvksa ds vuqfn'k gS rks
;k ¼cká foHkktu½
(viii) ;fn rFkk lejs[kh; gS rks ;fn P  x1 , y1 , z1  rFkk dks feykus okyh js[kk dks fcUnq
(ix) vk?kw.kZ % cy tks fcUnq A ij fcUnq B ds lksi{k dk;Zjr gS rks lfn'k vuqikr esa foHkkftr djrk gS] rks
cyk?kw.kZ gksrk gSA
(x) (a) ;fn ,d f=kHkqt dh nks vklUu Hkqtk,a rFkk gks rks bldk {ks=kQy
vUr foHkktu ds fy, /kukRed fpUg rFkk cká foHkktu ds fy, _.kkRed
fpUg ysrs gSA

(b) ;fn ,d lekukUrj prqHkqZt dh nks vklUu Hkqtk,a a rFkk gks rks bldk  x  x y  y2 z1  z2 
PQ dk ek/; fcUnq  1 2. 1 , 
{ks=kQy  2 2 2 
(c) ;fn ,d lekukUrj prqHkqZt dh nks fod.kZ rFkk gks rks bldk {ks=kQy ,d f=kHkqt ABC ftlds 'kh"kZ rFkk gS]
dk dsUæd gSA
(13) (14)
?kVuk ds fy, la;ksxkuqikr
A ds i{k esa la;ksxkuqikr = m : (n – m)
A ds foi{k esa la;ksxkuqikr = m : (n – m) : m
,d prq"Qyd ABCD ftlds 'kh"kZ rFkk çkf;drk dk ;ksx fl)kar
gS] dk dsUæd gSA fLFkfr & 1 : tc ?kVuk,a ijLij viothZ gksa
;fn A rFkk B ijLij viothZ ?kVuk,a gks rks
fnDdksT;k,a ,oa ç{ksi& x- v{k dh fnDdksT;k,a cos0, cosπ/2, cosπ/2 vFkkZr~ 1, fLFkfr & 2 : tc ?kVuk,a ijLij viothZ ugha gksa
0, 0 gksrh gSA blh çdkj y rFkk z-v{k dh fnDdksT;k,a Øe'k% (0, 1, 0) rFkk (0, 0,
;fn A rFkk B ijLij viothZ ?kVuk,a ugha gks rks
1) gksrh gSA P  A B   P  A   P  B   P  A B 
;k ;k P  A B  P  A   P  B  P  A B
çkf;drk dk xq.ku fl)kar
fLFkfr & 1 : tc ?kVuk,a Lora=k gks
fdlh js[kk PQ ds fnd~ vuqikr ¼tgka P rFkk Q Øe'k% (x1, y1, z1) rFkk (x2, y2, ;fn A1,A2,…,An Lora=k ?kVuk,a gks rks P(A1,A2,…,An)
z2) gS½ x2 – x1, y2 – y1, z2 – z1 gksrs gSaA  P  A1   P  A 2   P  A n 
;fn a, b, c fnd~ vuqikr rFkk l, m, n fnd~dksT;k,a gS rks
;fn A rFkk B nks Lora=k ?kVuk,a gks rks B dk ?kfVr gksuk A ij dksbZ çHkko ugha
MkyrkA blfy,
P  A/ B   P  A  rFkk P  B/ A   P  B
çkf;drk rc P  A B   P  A   P  B  ;k P  A B  P  A   P  B
çkf;drk dh xf.krh; ifjHkk"kk& ;fn A dksbZ ?kVuk gS rks fLFkfr & 2 : tc ?kVuk,a Lora=k u gks] nks ?kVuk,a A rFkk B ds ,d lkFk ?kfVr
gksus dh çkf;drk A dh çkf;drk rFkk B dh çfrcaf/kr çkf;drk ¼tc A ?kfVr gks
m A dh vuqdwy fLFkfr;ksa dh la[;k
P A   pqdh gks½ ds xq.kuQy ds cjkcj gksrh gS ¼;k B dh çkf;drk rFkk A dh çfrcafèkr
n A dh dqy fLFkfr;ksa dh la[;k çkf;drk ds xq.kuQy ds cjkcj gksrh gSA½ vFkkZr~
0  P  A  1 ] P A 
nm m
 1  1 P A P  A B  P  A   P  B/ A  ;k P  A B   P  B   P  A/ B ;k
n n
P  A B   P  A   P  B/ A  ;k P  B  P  A/ B  
∴ P A  P A   1
(15) (16)