(2)

(3) (4)

iQyuiQyuiQyuiQyuiQyuQyu & 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

fo|eku gksuk pkfg,AQyu&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'; gksAQyu ds çdkj & Qyu f : X → Y ,dSdh Qyu dgykrk gS ;fn X ds

fHkUu&fHkUu vo;oksa ds Y esa fHkUu&fHkUu çfrfcEc fo|eku gksA ;fn x1, x

2, X ds

dksbZ nks vo;o gks vkSjx

1 ≠ x

2 ⇒ f(x

1) ≠ f(x

2), f(x

1) = f(x

2) ⇒ x

1 = x

2 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 x

1 ≠ x

2 ⇒ f(x

1) ≠ f(x

2)

(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 dkifjlj = f dk lgçkUrA

(iii) vUr{ksZih Qyu & Qyu f : X → Y vUr{ksZih Qyu dgykrk gS ;fn Y

esa de ls de ,d vo;o ,slk gks ftldk çfrfcEc X esa fo|eku ugha gks vFkkZr~Y esa de ls de ,d vo;o ,slk gks ftlds fy, f–1(y) = φ rc Qyu vUr{ksZihgksrk gS] nwljs 'kCnksa esa f dk ifjlj ≠ f dk lgçkUrA

çfrykse Qyu & ;fn f : X → Y ,dSdh vkPNknd gks rks f dk çfrykse f–1

: X → Y esa Qyu gS tks fd çR;sd vo;o y ∈ Y ds laxr x ∈ X ftlds fy,f(x) = y çfrykse Qyu dgykrk gSAfo"ke ,oa le Qyu(i) fo"ke Qyu & ,d Qyu f(x) fo"ke Qyu dgykrk gSA ;fn f(–x) =

–f(x) lHkh x ds fy, fo"ke Qyu dk xzkQ foijhr iknksa esa lefer gksrk gSA(ii) le Qyu & ,d Qyu f(x) le Qyu dgykrk gSA ;fn f(–x) = –f(x)

lHkh x ds fy,A le Qyu dk xzkQ y-v{k ikfjr lefer gksrk gSA

f=kdks.kehfr; iQyu ,oa f=kdks.kferh; vuqikrf=kdks.kehfr; iQyu ,oa f=kdks.kferh; vuqikrf=kdks.kehfr; iQyu ,oa f=kdks.kferh; vuqikrf=kdks.kehfr; iQyu ,oa f=kdks.kferh; vuqikrf=kdks.kehfr; iQyu ,oa f=kdks.kferh; vuqikr,d nwljs ds inksa esa f=kdks.kferh; vuqikr

(Trigonometrical Ratios in Terms of each Other)

22

2 2

2

2 2

22

2

2

2 2

sin cos tan cot sec cosec

tan 1 sec 1 1sin sin 1 cos

sec cosec1 tan 1 cos1 cot 1

cos 1 sin cossec1 tan 1 cot

sin 1 cos 1tan tan sec 1

cos cot1 sin

1 sin cos 1 1cot cot

sin tan1 cos sec

2

22

2 2

22

2 2

cosec 11

1 1 1 cot cosecsec 1 tan sec

cos cot1 sin cosec 1

1 1 1 tan seccosec 1 cot cosec

sin tan1 cos sec 1

dqN egÙoiw.kZ dks.kksa ds f=kdks.kferh; vuqikr(Trigonometrical Ratios for Some Special Angles)

1º 1º7 15º 22 18º 36º

2 2

4 2 6 3 1 1 5 1 1sin 2 2 10 2 5

2 4 42 2 2 2

4 2 6 3 1 1 1 5 1cos 2 2 10 2 5

2 4 42 2 2 2

125 10 15tan 3 2 2 1 2 3 2 1 5 2 5

5

lacaf/kr dks.kksa ds f=kdks.kferh; vuqikr(Trigonometrical Ratios of Allied Angles)

sin cos tan

sin cos tan

90 cos sin cot2

90 cos sin cot2

180 sin cos tan

180 sin cos tan

3270 cos si

2

f=kdks.kferh; vuqikr

lacaf/kr dks.k

;k

;k

;k;k

;k

n cot

3270 cos sin cot

2360 2 sin cos tan

;k

;k

xf.krxf.krxf.krxf.krxf.kr

egÙoiw.kZ lw=kegÙoiw.kZ lw=kegÙoiw.kZ lw=kegÙoiw.kZ lw=kegÙoiw.kZ lw=k

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(5) (6)

(7) (8)

f=kdks.kferh; vuqikrksa ds dks.kksa ds eku(Trigonometrical Ratios for Various Angles)

lkjf.kdlkjf.kdlkjf.kdlkjf.kdlkjf.kdr`rh; dksfV ds lkjf.kd dk eku

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

1 1 1 2 1 322 23 21 23 21 2211 12

32 33 31 33 31 321 1 1

a a a a a aa a

a a a a a a

22 23 21 23 21 2211 12

32 33 31 33 31 32

a a a a a aa a

a a a a a a

milkjf.kd ,oa lg[k.M(i) milkjf.kd

;fn 11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

rks a11

dk milkjf.kd 22 2311

32 33M

a a

a a blh

rjg 21 2312

31 33M

a a

a a lkjf.kd dk eku fuEu çdkj Kkr fd;k tkrk gSA

Δ = a11

M11

– a12

M12

+ a13

M13

;k Δ = –a21

M21

+ a22

M22

– a23

M23

;k Δ = a31 M31 – a32 M32 + a33 M33

(ii) lg[k.M & vo;o aij dk lg[k.M çk;% F

ij ls O;Dr fd;k tkrk gS]

tksfd (–1)i+j Mij ds cjkcj gksrk gS tgka M vo;o a

ij dk milkjf.kd gSA

;fn11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

rks 1 1 22 2311 11 11

32 33F 1 M M

a a

a a

1 2 21 2312 12 12

31 33F 1 M M

a a

a a

lkjf.kd ds xq.k/keZ &(i) fdlh lkjf.kd dh fdlh iafDr ¼LrEHk½ dks fdlh la[;k ls xq.kk djus ij

lkjf.kd dk eku Hkh ml la[;k ls xq.kk gks tkrk gS vFkkZr~

ka kb kc a b c ka b c

p q r k p q r kp q r

u v w u v w ku v w

(ii) fdlh lkjf.kd dh fdlh iafDr ¼LrEHk½ dk çR;sd vo;o ;fn nks inksa dk;ksx gks rks ml lkjf.kd dks mlh dksfV dh nks lkjf.kdksa ds ;ksxQy ds :i esaO;Dr fd;k tk ldrk gS vFkkZr~

a b c a b c

p q r p q r p q r

u v w u v w u v w

rFkk

a b c a b c b c

p q r p q r q r

u v w u v w v w

(iii) ;fn fdlh lkjf.kd dh fdlh iafDr ¼LrEHk½ ds çR;sd vo;o esa fdlhnwljh iafDr ¼LrEHk½ ds laxr vo;oksa dks fdlh ,d dh jkf'k ls xq.kk djds tksM+s

;k ?kVk;sa rks lkjf.kd dk eku ugh cnyrkA vFkkZr~

a b c a b c b c

p q r p q r q r

u v w u v w v w

nks lkjf.kdksa dk xq.kuQynks lkjf.kd ftudh dksfV nks gS dk xq.kuQy fuEu çdkj ifjHkkf"kr gS&

1 1 1 1 1 1 1 2 1 1 1 2

2 2 2 2 2 1 2 2 2 1 2 2

a b m a b a m b m

a b m a b a m b m

nks lkjf.kd ftudh dksfV rhu gS dk xq.kuQy fuEu çdkj ifjHkkf"kr gS&

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

a b c m n

a b c m n

a b c m n

1 1 1 2 1 3 1 1 1 2 1 3 1 1 1 2 1 3

2 1 2 2 2 3 2 1 2 2 2 3 2 1 2 2 2 3

3 1 3 2 3 3 3 1 3 2 3 3 3 1 3 2 3 3

a b c a m b m c m a n b n c n

a b c a m b m c m a n b n c n

a b c a m b m c m a n b n c n

lefer lkjf.kd;fn fdl lkjf.kd ds çR;sd vo;o ds a

ij fy, a

ij = a

ji ∀ i, j gks rks mls lefer

lkjf.kd dgrs gSA

vFkkZr~ a h g

h b f

g f c

fo"ke lefer lkjf.kd;fn fdl lkjf.kd ds çR;sd vo;o ds aij fy, aij = – aji ∀ i, j gks rks mls fo"ke

lefer lkjf.kd dgrs gSA

vFkkZr~

eSfVªDleSfVªDleSfVªDleSfVªDleSfVªDleSfVªDl ds çdkj(i) iafDr eSfVªDl & A=[a

ij]

m×n ,d iafDr eSfVªDl gS ;fn m = 1

(ii) LrEHk eSfVªDl & A=[aij]m×n ,d LrEHk eSfVªDl gS ;fn n = 1

(iii) oxZ eSfVªDl & A=[aij]

m×n ,d oxZ eSfVªDl gS ;fn m = n

(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

ds fy,(vi) fod.kZ eSfVªDl & ,d oxZ eSfVªDl A–[a

ij]

m×n ,d fod.kZ eSfVªDl gS ;fn

aij = 0 tc i ≠ j

(vii) vfn'k eSfVªDl & A= [aij] ,d vfn'k eSfVªDl gSA ;fn 0

iji j

ak i j

tgkaK vpj gSA

(viii) bdkbZ eSfVªDl & ,d oxZ eSfVªDl A=[aij] ,d bdkbZ eSfVªDl gSA ;fn

1

0iji j

ai j

(ix) f=kHkqtkdkj eSfVªDl(a) Åijh f=kHkqtkdkj eSfVªDl& ,d oxZ eSfVªDl [a

ij] Åijh f=kHkqtkdkj

eSfVªDl dgykrk gS ;fn aij = 0 tcfd i > j.

(b) fuEu f=kHkqtkdkj eSfVªDl& ,d oxZ eSfVªDl [aij] fuEu f=kHkqtkdkj eSfVªDldgykrk gS ;fn a

ij = 0 tcfd i < j.

(x) vO;qRØe.kh; vkSj O;qRØe.kh; eSfVªDl&;fn lkjf.kd |A| = 0 ⇒ vO;qRØe.kh;;fn lkjf.kd |A| ≠ 0 ⇒ O;qRØe.kh;

(9) (10)

(11) (12)

eSfVªDl dk ;ksx ,oa O;odyu;fn A[a

ij]

m×n rFkk [b

ij]

m×n nks leku dksfV dh eSfVªDl gks rks mudk ;ksx A + B

og eSfVªDl gS ftldk çR;sd vo;o eSfVªDl A rFkk B ds laxr vo;oksa ds ;ksxds cjkcj gSA vFkkZr~ A + B = [a

ij + b

ij]

m×n

vfuf'pr lekdyuvfuf'pr lekdyuvfuf'pr lekdyuvfuf'pr lekdyuvfuf'pr lekdyuekud lw=k

(i) 1

11

nn x

x dx c nn

(ii)1

logedx x cx

(iii) x xe dx e c (iv) loglog

xx x

ee

aa dx c a e c

a

(v) sin cosxdx x c (vi) sin sinxdx x c (vii) tan log sec log cosxdx x c x c (viii) cot log sinxdx x c

(ix) sec log sec tan log sec tan log tan4 2

xxdx x c x x c c

(x) cosec log cosec cot log cosec cot log tan2

xdx x x x x c c

(xi) sec tan secx xdx x c (xii) cosec cot cosecx xdx x c (xiii) 2sec tanxdx x c (xiv) 2sec cotco xdx x c

(xv) 1

2 22 1

tanx

dx ca ax a

(xvi) 2 2

1 1log

2

x adx c x a

a x ax a

(xvii) 2 2

1 1log

2

a xdx c x a

a a xa x

(xviii)1 1

2 2

1sin cos

x xdx c c

a aa x

(xix) 2 2 1

2 2

1log sinh

xdx x x a c c

ax a

(xx)2 2 1

2 2

1log sinh

xdx x x a c c

ax a

(xxi)2

2 2 2 2 1sin2 2

x a xa x dx a x c

a

(xxii) 2

2 2 2 2 1sin2 2

x a xx a dx x a c

a

(xxiii) 2

2 2 2 2 1cos2 2

x a xx a dx x a h c

a

(xxiv)1

2 2

1 1sec

xdx c

a ax x a

(xxv) 12 2 2 2

sin sin cos sin tanax ax

ax e e be bxdx a bx b bx c bx c

aa b a b

(xxvi) 12 2 2 2

cos cos sin cos tanax ax

ax e e be bxdx a bx b bx c bx c

aa b a b

(xxvii) 1f ax b dx ax b c

a

lekdyulekdyulekdyulekdyulekdyufuf'pr lekdyu ds xq.k/keZ

(i)

b b b

a a a

f x dx f t dt f u du

(ii)

b b

a a

f x dx f x dx

(iii) b c b

a a c

f x dx f x dx f x dx a c b

(iv)

0 0

a a

f x dx f a x dx

(v)

0

2

0

;fn ¼le Qyu½

vkSj ;fn ¼fo"ke Qyu½

a a

a

f x f xf x dx f x dx

f x f x

(vi)

2

0 0 0

2 ¼lkekU; :i ls½a a a

f x dx f x dx f a x dx

0

2 2

20

a

f x dx if f a x f x

if f a x f x

(vii)

T T

0

T N¼;fn vkSj ½a n

a

f x dx n f x dx f x f x n

(viii)

b b

a a

f x dx f a b x dx

(ix)

h x

g x

df t dt h x f h x g x f g x

dx

vody lehdj.kvody lehdj.kvody lehdj.kvody lehdj.kvody lehdj.kvody lehdj.k dh dksfV rFkk ?kkr& vodyu lehdj.k esa fo|+eku

vodytksa dk mPpre Øe gh ml lehdj.k dh dksfV dgykrk gS rFkk vodylehdj.k esa mPpre vodyt dh ?kkr gh ml vody lehdj.k dh ?kkr

dgykrh gSA vody lehdj.k 23

33 xd y dy

y edxdx

dh dksfV 3 rFkk 1 ?kkr gSA

çFke dksfV o çFke ?kkr vody lehdj.k

(i) dy dyf x f x dy f x dx

dx dx nksuksa rjQ lekdyu djus ij

;kdy f x dx c y f x dx c

(ii)

dy dy dyf x g y f x g y f x dx c

dx dx g y

(iii) dy dv

f ax by c dxdx a bf v

(iv)

1

2

,F

, F;k

f x ydy dy y dv dx

dx f x y dx x v v x

(v) P Q Qpdx pdxdyy y e e dx c

dx

lfn'klfn'klfn'klfn'klfn'klfn'k ;k ØkWl xq.kuQy& ekuk a rFkk b nks lfn'k gS rFkk θ muds e/; dks.k

gS rc a × b = |a||b| sin θ n ;gka n, a rFkk b ds yEcor~ bdkbZ lfn'k gSA

(13) (14)

(15) (16)

lfn'k xq.kuQy ds xq.kuQy

(i) . .a b b a i e a b b a

(ii)

(iii)

(iv) ;fn 1 2 3ˆˆ ˆa a i a j a k

rFkk rks

(v) a rFkk nksuksa ds yEcor~ lfn'k gksrk gSA

(vi) rFkk ds ry ds yEcor~ bdkbZ lfn'k gksrk gSA rFkk ¼ a rFkk

;k rFkk ½ ds ry ds yEcor~ ifjek.k dk ,d lfn'k a b

a b

gksrk

gSA

(vii);fn ˆˆ ˆ, ,i j k rhu bdkbZ lfn'k rhu ijLij yEcor~ js[kkvksa ds vuqfn'k gS rks

;k

(viii) ;fn rFkk lejs[kh; gS rks

(ix) vk?kw.kZ % cy tks fcUnq A ij fcUnq B ds lksi{k dk;Zjr gS rks lfn'k

cyk?kw.kZ gksrk gSA(x) (a) ;fn ,d f=kHkqt dh nks vklUu Hkqtk,a rFkk gks rks bldk {ks=kQy

(b) ;fn ,d lekukUrj prqHkqZt dh nks vklUu Hkqtk,a a rFkk gks rks bldk

{ks=kQy

(c) ;fn ,d lekukUrj prqHkqZt dh nks fod.kZ rFkk gks rks bldk {ks=kQy

f=kfofe; funsZ'kkad T;kfefrf=kfofe; funsZ'kkad T;kfefrf=kfofe; funsZ'kkad T;kfefrf=kfofe; funsZ'kkad T;kfefrf=kfofe; funsZ'kkad T;kfefrfunsZ'kkad& nks fcUnqvksa rFkk ds e/; nwjh

2 2 22 1 2 1 2 1P Q x x y y z z

ewy fcUnq ls fcUnq 1 1 1, ,x y z dh nwjh

;fn fcUnq 1 1 1P , ,x y z rFkk dks feykus okyh js[kk dks fcUnq

vuqikr esa foHkkftr djrk gS] rks

1 2 2 1 1 2 2 1 1 2 2 1

1 2 1 2 1 2; ;

m x m x m y m y m z m zx y z

m m m m m m

¼vUr foHkktu½

rFkk 1 2 2 1 1 2 2 1 1 2 2 1

1 2 1 2 1 2; ;

m x m x m y m y m z m zx y z

m m m m m m

¼cká foHkktu½

;fn 1 1 1P , ,x y z rFkk dks feykus okyh js[kk dks fcUnq

vuqikr esa foHkkftr djrk gS] rks

vUr foHkktu ds fy, /kukRed fpUg rFkk cká foHkktu ds fy, _.kkRedfpUg ysrs gSA

PQ dk ek/; fcUnq 1 2 1 2 1 2. ,2 2 2

x x y y z z

,d f=kHkqt ABC ftlds 'kh"kZ rFkk gS]

dk dsUæd gSA

,d prq"Qyd ABCD ftlds 'kh"kZ rFkk

gS] dk dsUæd gSA

fnDdksT;k,a ,oa ç{ksi& x- v{k dh fnDdksT;k,a cos0, cosπ/2, cosπ/2 vFkkZr~ 1,

0, 0 gksrh gSA blh çdkj y rFkk z-v{k dh fnDdksT;k,a Øe'k% (0, 1, 0) rFkk (0, 0,

1) gksrh gSA

;k

fdlh js[kk PQ ds fnd~ vuqikr ¼tgka P rFkk Q Øe'k% (x1, y

1, z

1) rFkk (x

2, y

2,

z2) gS½ x2 – x1, y2 – y1, z2 – z1 gksrs gSaA;fn a, b, c fnd~ vuqikr rFkk l, m, n fnd~dksT;k,a gS rks

çkf;drkçkf;drkçkf;drkçkf;drkçkf;drkçkf;drk dh xf.krh; ifjHkk"kk& ;fn A dksbZ ?kVuk gS rks

AP A

A

dh vuqdwy fLFkfr;ksa dh la[;kdh dqy fLFkfr;ksa dh la[;k

m

n

10 P A ] P A 1 1 P An m m

n n

∴ P A P A 1

?kVuk ds fy, la;ksxkuqikrA ds i{k esa la;ksxkuqikr = m : (n – m)

A ds foi{k esa la;ksxkuqikr = m : (n – m) : m

çkf;drk dk ;ksx fl)karfLFkfr & 1 : tc ?kVuk,a ijLij viothZ gksa

;fn A rFkk B ijLij viothZ ?kVuk,a gks rks

fLFkfr & 2 : tc ?kVuk,a ijLij viothZ ugha gksa;fn A rFkk B ijLij viothZ ?kVuk,a ugha gks rks

P A B P A P B P A B

;k P A B P A P B P A B

çkf;drk dk xq.ku fl)karfLFkfr & 1 : tc ?kVuk,a Lora=k gks;fn A

1,A

2,…,A

n Lora=k ?kVuk,a gks rks P(A

1,A

2,…,A

n)

1 2P A P A P An

;fn A rFkk B nks Lora=k ?kVuk,a gks rks B dk ?kfVr gksuk A ij dksbZ çHkko ughaMkyrkA blfy,

P A/ B P A rFkk P B/ A P B

rc P A B P A P B ;k P AB P A P B

fLFkfr & 2 : tc ?kVuk,a Lora=k u gks] nks ?kVuk,a A rFkk B ds ,d lkFk ?kfVrgksus dh çkf;drk A dh çkf;drk rFkk B dh çfrcaf/kr çkf;drk ¼tc A ?kfVr gkspqdh gks½ ds xq.kuQy ds cjkcj gksrh gS ¼;k B dh çkf;drk rFkk A dh çfrcafèkrçkf;drk ds xq.kuQy ds cjkcj gksrh gSA½ vFkkZr~

P A B P A P B/ A ;k P A B P B P A/ B ;k

P A B P A P B/ A ;k P B P A/ B