MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b + a b + a b + + a b + + a b o ( ) a b Now, puttig = i the biomial theoem o ( + ) = + + + + + + ( ) Pactice Poblems : Usig biomial theoem, idicate which umbe is lage () o Fid ( + ) 6 + ( ) 6 Hece o othewise evaluate ( + ) 6 + ( ) 6 Show that 9 + 8 9 is divisible b 64, wheeve is a positive itege 4 Usig biomial theoem, pove that 6 5 alwas leaves emaiig whe divided b 5 Popeties of Biomial Theoem : (i) The umbe of tems i the epasio is + (ii) The sum of the idices of ad i each tem is (iii) The biomial coefficiets (, ) of the tems equidistat fom the begiig ad the ed ae equal, ie =, = etc 4 Some impotat tems i the epasio of ( + ) : (i) Geeal tem : ( + )th tem of ( + ) is T + = (ii) Middle tem/(s) : (iii) If is eve, thee is ol middle tem, which is if is odd, thee ae two middle tems, which ae thad th tems Numeicall geatest tem i the epasio of ( + ), N Let T ad T + be the th ad ( + )th tems espectivel T = ( ) T + = th tem Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
MB Now, T T T T oside,,, Pactice Poblems : 5( ) Show that the middle tem i the epasio of ( ) is, whee N! Show that the coefficiet of the middle tem i the epasio of ( + ) is the sum of the coefficiets of two middle tems i the epasio of ( + ) Fid the value of, if the coefficiets of ( + 4)th ad ( )th tems i the epasio of ( + ) 8 ae equal 4 If the coefficiet of ( )th, th ad ( + )th tems i the epasio of ( + ) ae i the atio : : 5, fid ad 5 The d, d ad 4th tems i the epasio of ( + ) ae 4, 7 ad 8 espectivel Fid the values of, ad 6 Fid the coefficiet of 5 i the poduct ( + ) 6 ( ) 7 usig biomial theoem 7 Fid the tem idepedet of i the epasio of 6 8 Fid the coefficiet of a 4 i the poduct ( + a) 4 ( a) 5 usig biomial theoem 9 The sum of the coefficiets of the fist thee tems i the epasio of,, m beig a atual umbe, is 559 Fid the tem of the epasio cotaiig m Show that the geatest coefficiets i the epasio of 5( ) is! 6 Epess ( ) ( ) as a polomial i 6 If a, a, a ad a 4 be a fou cosecutive coefficiets i the epasio of ( + ), pove that a a a a a a 4 a a a [Aswes : () 6 (4) = 7, = (5) =, = ad = 5 (6) 7 (7) 5/ (8) 48 (9) 594 ] 5 Multiomial Theoem As we kow the Biomial Theoem ( )! ( )!! puttig =, = theefoe, ( )!!! Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
MB Total umbe of tems i the epasio of ( + ) is equal to umbe of o-egative itegal solutio of + = ie + = + = + I the same fashio we ca wite the multiomial theoem ( ) k k!!!! k k k Hee total umbe of tems i the epasio of ( + + + k ) is equal to umbe of o-egative itegal solutio of + + + k = ie + k k Pactice Poblems : (i) the middle tem i the epasio of (ii) the coefficiet of ad 7 i the epasio of 5 4 Fid the coefficiet of 5 i the epasio of the poduct ( + ) 5 ( ) 7 [Aswes : () (i) 5 6 (ii) 65, 65 () 7] 5 8 6 Popeties of Biomial oefficiets : ( + ) = + + + + + + () () The sum of the biomial coefficiets i the epasio of ( + ) is Puttig = i () + + + + = () o () Agai puttig = i (), we get + + + ( ) = () o ( ) () The sum of the biomial coefficiets at odd positio is equal to the sum of the biomial coefficiets at eve positio ad each is equal to ie, + + 4 + = + + 5 + = (4) Sum of two cosecutive biomial coefficiets + = + (5) Ratio of two cosecutive biomial coefficiets ( ) (6) ( ) Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
Pactice Poblems : Pove the followig idetities : + + 4 + = + + 5 + = + + 5 + + ( + ) = ( + ) + + ( ) = (e) + + + + = (f) + + + + ( + ) = + MB 4 (g) (h) = (i) 4 5 (j) + 5 + 8 + + ( + ) = ( + 4) 7 Biomial Theoem Fo Negative Itege o Factioal Idices If R the, ( ) ( )! ( )( )! ( )( )( ) f! Remaks (i) The above epasio is valid fo a atioal umbe othe the a whole umbe if < (ii) Whe the ide is a egative itege o a factio the umbe of tems i the epasio of ( + ) is ifiite ad the smbol caot be used to deote the coefficiet of the geeal tem (iii) The fist tems must be uit i the epasio, whe ide is a egative itege o factio (iv) The geeal tem i the epasio of ( + ) is T ( )( )( )! (v) Whe is a atioal umbe othe tha whole umbe the appoimate value of ( + ) is + ( ad highe powes of ca be eglected) (vi) Epasio to be emembeed ( < ) ( + ) = + + + ( ) + ( ) = + + + 5 + + + ( + ) = + 4 + + ( ) ( + ) + ( ) = + + + 4 + + ( + ) + Pactice Poblems : Fid the coefficiet of 6 i the epasio of ( ) 5/ Fid the coefficiet of i the epasio of esult holds ( ), metioig the coditio ude which the ( ) 55 [Aswes : () ] 6 Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
MB 5 SINGLE ORRET HOIE TYPE The epasio [ + ( ) / ] 5 + [ ( ) / ] 5 is a polomial of degee 5 6 7 8 Give positive iteges >, > ad the coefficiets of ()th ad ( + )th tems i the biomial epasio ( + ) ae equal The equals to + oe of these If i the epasio of ( + ) m ( ), the coefficiet of ad ae ad 6 espectivel, the m is 6 9 4 4 If is a odd atual umbe the equals / ( ) / oe of these 5 The total umbe of dissimila tems i the epasio of ( + + + ) is ( )( ) 6 4 ( ) 4 6 The coefficiet of 5 i the epasio of ( + ) + ( + ) + + ( + ) is 5 5 9 5 6 6 5 + 5 7 If is positive, the fist egative tem i the epasio of ( + ) 7/5 is 5th tem 8th tem 6th tem 7th tem 8 Sum of the coefficiet of the tems of degee m i the epasio of ( + ) ( + ) ( + z) is ( m ) ( m ) m m 9 The coefficiet of k i the epasio of E = + ( + ) + ( + ) ( + ) is k + k + k + oe The coefficiet of i the epasio of ( + ) 8 is 476 496 56 58 The coefficiet of i the epasio of is!!!!!!! The coefficiet of 4 i the epasio of ( + + + ) is 4 + + 4 + 4 + + The coefficiet of 5 i the epasio of ( + ) 5 ( + ) 4 is 6 6! 59 oe 4 If ( ) 6 = a + a + a a, the the a ad a 6 a = 6 6 ad a 6 = a ot possible ad a 6 = 6 6 a 6 ot possible ad a = 6 6 a = ad a 6 = 6 6 5 The tem idepedet of i the epasio of ( ) is 7 4 th tem 5 th tem 6 th tem oe 6 The coefficiet of 4 i the epasio of ( ( ) ) is oe 7 The tem idepedet of i the epasio of / / / 5 6 4 is Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
8 If ( + ) = c + c + c c the the value of k ad k c c4 whee c k ad 5 c c c5 k 4 6 ad ad ad ad 9 If ( + ) = c + c + c c the the value of c + c + c c MB 6 c c c5 5 The value of is equal to 4 6 6 If ( ) c the c c c c c c c c 99 c oe If ( + ) = c + c + c c the the value of c c + c c c ( )! ( )! ( ) + ( ) ( )! c c c 7 The coefficiet of i the epasio of If ( + ) = c + c + c c the c c c c c c c c ( ) ( ) I the epasio of ( ) 8 ( )( ) the coefficiet of 4 4 4 + oe c c c 5c is equal to c c 4 c 5 oe 4 If ( + ) 5 = c + c + c c 5 5 the c + c + c 4 4 c 5 equal to 4 4 4 4 oe ( ) is! ( )!( )! ()! ( )!( )! ()! ( )!( )! oe 8 If is eve positive itege, the the coditio that the geatest tem i the epasio of ( + ) ma have the geatest coefficiet also is 4 4 4 oe of these Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
9 The sum of the atioal tems of ( /5 + ) is 7 85 97 oe The umbe of tems which ae fee fom adical sigs i the epasio of ( /5 + / ) 55 is 5 6 4 oe If sum of coefficiet i the epasio of ( + ) is 496 the the geatest biomial coefficiet is 6 7 8 oe MB 7 8 If ( + ) = a + a + a a the a + a + + a 9 The sum of the last eight coefficiets i the epasio of ( + ) 5 is 6 5 4 oe of these!( )!!( )! 5!( 5)! is equal to 4 is ( )! oe (!) The umbe of tems i the epasio of (a + b + c) + ( )( ) oe 4 The coefficiet of i the epasio of ( + + ) is ( ) + 5 The coefficiet of 5 i ( + + ) / is 5 6 oe of these 6 The coefficiet of, zt ad zt i the epasio of ( + + z + t) 4 ae i the atio : : 4 4 : : : : 4 4 : : 7 If ( + ) = c + c + c + c c the the value of c + c + c 4 + = k ad c + c + c 4 + = k k = k = 99 k = k = 98 k = k = k k c d c 4 a 5 c 6 c 7 b 8 d 9 c a ANSWERS (SINGLE ORRET HOIE TYPE) a d a 4 b 5 b 6 b 7 c 8 d 9 c a c a b 4 d 5 d 6 a 7 b 8 a 9 d b a a c 4 b 5 d 6 a 7 a 8 a 9 c 4 a Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
OMPREHENSION TYPE ompehesio- oside the followig epasio of ( + + ) = a + a + a + a + + a Let S = a + a + a 6 S = a + a 7 S = a + a 5 + a 8 the S : S : S : : : : : : : : The value of a a + a a ( ) a is k a [ ( ) a ] The value of k is ½ ½ oside the followig statemet : 4 5 (i) (ii) The The value of a l a + a ( ) a is zeo whe is ot multiple of The value of ompehesio- a l a + a ( ) a is ( ) m m whe is multiple of both ae coect ol (i) is coect ol (ii) is coect both ae icoect If a, b, c, d be fou cosecutive coefficiets i the biomial epasio of ( + ) the a b b c c d,, ae i a b c AP GP HP oe a b c,, ae i a b b c c d AP GP HP oe 6 (bc + ad), (ac b d), (b c) ae i AP GP HP oe MB 8 EXERISE BASED ON NEW PATTERN ompehesio- Let P be a poduct give b P = ( + a ) ( + a )( + a ) Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857 ad let S = i S a i aia j,s aia jak i j i jk ad so o, the it ca be show that P = + S + S + + S 7 The coefficiet of 99 i the epessio of ( ) ( ) ( ) ( ) must be 99 495 55 55 8 The coefficiet of i the epessio ( ) ( ) ( ) ( ) must be 5 9 The total umbe of tems i P must be (A) (B) () (D) + MATRIX MATH TYPE Matchig- olum - A k k k k (k )(k ) olum - B (p) ( ) (q) k k k k Matchig- () (s) ( )( ) ( ) ( )( ) Match the colum A with thei suitable tem idepedet of give i colum B olum - A olum - B (A) ( + + + ) (p) 6 (B) (q) 68
() ( / + 4 / + 4) 5 () 84756 (D) / / / ( 4 ) 9 (s) 85 Matchig- ( ) If the fid olum - A = a + a + a + + a + olum - B (A) a + a + a + (p) / (B) a a + a (q) () a + a + () (D) a + a + a 5 + (s) Matchig-4 olum - A olum - B (A) The coefficiet of 4 i (p) 664 the epasio of ( + + ) (B) The coefficiet of i (q) 6 (7 + + + + 4 + 5 ) 8 () The coefficiet of 7 i () 6 the epasio of ( + ) (D) The coefficiet of 4 z 5 (s) 9 i the epasio of ( + z + z) 6 MULTIPLE ORRET HOIE TYPE p m k = a 4 + a + a + a m k If p the a 4 a 4 a 4 a 4 Let ( + + + + 4 ) = a + a + a + + a 4 4 The a = a = a = 55 a + a = 65 MB 9 The tem idepedet of i the epasio of ( + ) is, if is odd ( ) if is odd ( ) if is eve oe of these 4 If ( + ) = a +a + a a the a + a = a + a + a 5 Let a + < a a = a + oe of these f () the 6 f() is a polomial of the sith degee i f() has eactl two tems coefficiet of is 48 coefficiet of 6 is 64 6 The value of i the epessio ( thid tem i the coefficiet is () 6 is 5/ oe log 5 ) if the 7 The umeicall geatest tem i the epasio of ( ) 9 whe = is d tem 4th tem 5th tem 6th tem 8 I the epasio of whe =, it is kow that the 6th tem is the geatest tem The possible positive itegal values of is 49 5 57 59 6 Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
9 If ( + + ) = a + a + a + + a the the value of a + a + a + + a is a a + a + a is a + a + + a is a + a + a 5 + + a is If ( + + ) = a + a + a + + a the choose the coect statemet a a + a a + + a = a a a a a + a a 4 + a a = a + a i = a i oe of these Assetio-Reaso Tpe Each questio cotais STATEMENT- (Assetio) ad STATEMENT- (Reaso) Each questio has 4 choices (A), (B), () ad (D) out of which ONLY ONE is coect (A) (B) () (D) Statemet- is Tue, Statemet- is Tue; Statemet- is a coect eplaatio fo Statemet- Statemet- is Tue, Statemet- is Tue; Statemet- is NOT a coect eplaatio fo Statemet- Statemet- is Tue, Statemet- is False Statemet- is False, Statemet- is Tue Let ( + ) 6 = a + a + a + + a 6 6 MB STATEMENT- : a + a + a 6 + a 6 = ( 5 + ) STATEMENT- : a + a + a + + a 6 = 6 ad a + a + + a 6 = 5 STATEMENT- : The coefficiet of the middle tem i ( + ) is equal to the sum of the coefficiet of the two middle tems i ( + ) STATEMENT- : + = + STATEMENT- : If is a positive itege the + 4 is divisible b 5 if > STATEMENT- : It ca be show b usig the biomial theoem 4 STATEMENT- : If be a eve positive itege the!!!! 5! 5!!!! STATEMENT- : If is eve the + + 5 + + = 5 STATEMENT- : () 5 > () 5 + (99) 5 STATEMENT- : It ca be show b usig the biomial theoem (Aswes) EXERISE BASED ON NEW PATTERN OMPREHENSION TYPE c a a 4 c 5 a 6 b 7 c 8 c 9 d MATRIX MATH TYPE [A-s, B-, -q, D-s] [A-s; B-; -q; D-p] [A-p; B-p; -p; D-s] 4 [A-q; B-s; -p; D-] MULTIPLE ORRET HOIE TYPE a, b, c, d c, d a, c 4 a, b, c 5 a, b, c, d 6 a, b 7 b, c 8 a, b, c, d 9 a, b, c, d a, b, c ASSERTION-REASON TYPE B A A 4 A 5 A Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
MB INITIAL STEP EXERISE (SUBJETIVE) Give that the 4th tem i the epasio of has a maimum umeical value, fid 8 the age of values of fo which this tue If a, b, c, d be fou cosecutive coefficiets i the biomial epasio of ( + ) the pove that b b c ac if > (a b)(c d) If the geatest tem i the epasio of ( + ) has the geatest coefficiet if ad ol if, ad the fouth tem i the epasio of k is 4, the fid the value of mk 4 If,,, deote the coefficiets i the epasio of ( + ), pove that ( + ) ( + )( + ) = (e) (f) (g) (! ) ( )!!! + + + + + + =!!! + + + + = ( + ) + 4 = ( )!( )! ()! m (h) ( ) 4 7! 47( ) 5 If a ad b ae two comple umbes, the fid the sum of ( + ) tems of the seies a (a + d) + (a + d) (a + d) + 6 If ( + ) = I + f whee I ad ae positive iteges ad < f <, show that I is a odd itege ad ( f) (I + f) = 7 If (9 + 45) = I + f, ad I beig positive iteges ad f is a pope factio, show that (I )f + f is a eve itege 8 Show that [( + ) ] + is divisible b + fo all N 9 If =, pove that : + + = ( ) If (66 + 4) + = P, pove that the itegal pat of P is a eve itege ad P f = + whee f is the factioal pat of P If ( + + ) = a + a + a + 4 the fid the value of a + a + + a 8 Show that ( ) ( ) Show that k k si(k) cos[( k)] si() 4 If ( + ) = + + + +, pove that i j 5 Pove that p + q = ( i j) ( ) p q = pq + p if 6 Show that the oots of the equatio a + b + c = ae eal ad uequal, whee a, b, c ae the thee cosecutive coefficiets i a biomial epasio with positive itegal ide Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857
MB FINAL STEP EXERISE (SUBJETIVE) If ( + ) = + + + + the pove that : i j (i j) i j i j i j i i i i ij j j Show that the HM of + ad + + is times of Also show that ( ) Fid the sum of the seies 7 If ( + ) = + + + + ( N) the k show that k ( ) ( ) k 8 Show that 5 k 9 4! 59(4 ) ( ) 4 9 If ( + + ) = a + a + a + + a, N Show that a 4 equals to ( )( 5 ( )( )) 57 ( ) 4 Show that 7 5 4 upto m tems ( ) 5 Give that S = + q + q + + q ad q q q whee q pove that + + + S + + S + + + + S = 6 Let k ad be positive iteges ad put S k = k + k + k + + k show that m + S + m + S + + m + m S m = ( + ) m + ( + ) Show that ( ) ( ) Show that m! (m )! ( ) (m )! ( ) (m )! (m )(m )(m ) (m )! If >, the (i) ab (a )(b ) + (a )(b ) + ( ) (a )(b ) = (ii) abc (a )(b )(c ) + (a )(b )(c ) + ( ) (a )(b )(c ) = ANSWERS SUBJETIVE (INITIAL STEP EXERISE) 64 64,, 5 9 ( ) ANSWERS SUBJETIVE (FINAL STEP EXERISE) ( m m ( ) ) Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857