Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

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1 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 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

2 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

3 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] 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) 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

4 Pactice Poblems : Pove the followig idetities : = = ( + ) = ( + ) + + ( ) = (e) = (f) ( + ) = + MB 4 (g) (h) = (i) 4 5 (j) ( + ) = ( + 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 ( < ) ( + ) = ( ) + ( ) = ( + ) = ( ) ( + ) + ( ) = ( + ) + 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

5 MB 5 SINGLE ORRET HOIE TYPE The epasio [ + ( ) / ] 5 + [ ( ) / ] 5 is a polomial of degee 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 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 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 The coefficiet of i the epasio of is!!!!!!! The coefficiet of 4 i the epasio of ( ) is 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 = 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 / / / is Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857

6 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 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 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 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 oe of these Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857

7 9 The sum of the atioal tems of ( /5 + ) is oe The umbe of tems which ae fee fom adical sigs i the epasio of ( /5 + / ) 55 is oe If sum of coefficiet i the epasio of ( + ) is 496 the the geatest biomial coefficiet is 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 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

8 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 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

9 () ( / + 4 / + 4) 5 () (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 ( ) 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 ( ) = 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 Eistei lasses, Uit No,, Vadhma Rig Road Plaza, Vikas Pui Et, Oute Rig Road New Delhi 8, Ph : 9695, 857

10 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 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 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

11 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

12 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 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