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# Class X Maths Formula Guide

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### Class X Maths Formula Guide

1. 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 5lHkh 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. 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 rrjg 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 jkfk 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) vfnk eSfVªDl & A= [aij] ,d vfnk 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  jlkjf.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. 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 vfufpr 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 fufpr lekdyu ds xq.k/keZ (9) (10) f  x, y dy  1 ;k dy F  y     dv  dxdx 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 lfnk (viii)  f  x  dx   f  a  b  x  dx lfnk ;k ØkWl xq.kuQy& ekuk a rFkk b nks lfnk 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 lfnk gSA (11) (12)
4. 4. lfnk xq.kuQy ds xq.kuQy f=kfofe; funsZkkad T;kfefr         (i) a  b  b  a  i.e. a  b  b  a  funsZkkad& 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~ lfnk gksrk gSA vuqikr esa foHkkftr djrk gS] rks  (vi) rFkk ds ry ds yEcor~ bdkbZ lfnk 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 lfnk   a b gksrk ab ¼vUr foHkktu½gSA 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 lfnk rhu ijLij yEcor~ js[kkvksa ds vuqfnk 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 lfnk 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 gksa0, 0 gksrh gSA blh çdkj y rFkk z-v{k dh fnDdksT;k,a Øek% (0, 1, 0) rFkk (0, 0, ;fn A rFkk B ijLij viothZ ?kVuk,a ugha gks rks1) 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 Øek% (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)