Student s Name : Class Roll No. ADDRESS: R-1, Opp. Raiway Track, New Corner Glass Building, Zone-2, M.P. NAGAR, Bhopal

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1 FREE Dowload Stud Package fom website: wwwtekoclassescom fo/u fopkj Hkh# tu] ugha vkjehks dke] foif s[k NksMs qja e/;e eu dj ';kea iq#"k flag ladyi dj] lgs foif vusd] ^cuk^ u NksMs /;s; dks] j?kqcj jk[ks VsdAA jfp% ekuo /kez izksk l~q# Jh jknksmklth egkjkt STUDY PAKAGE Subject : Mathematics Topic : Biomial Theoem Ide Theo Shot Revisio Eecise (E to 8) Assetio & Reaso Que fom ompt Eams Ys Que fom IIT-JEE 7 Ys Que fom AIEEE Studet s Name : lass Roll No : : ADDRESS: R-, Opp Raiwa Tack, New oe Glass Buildig, Zoe-, MP NAGAR, Bhopal : (7), , wwwtekoclassescom R OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

2 FREE Dowload Stud Package fom website: wwwtekoclassescom Biomial Theoem Biomial Epessio : A algebaic epessio which cotais two dissimila tems / etc ( ) is called biomial epessio Fo eample :,,, Statemet of Biomial theoem : If a, b R ad N, the ; (a b) a b a b a b a b a b o (a b) a b Now, puttig a ad b i the biomial theoem - o ( ) ( ) Solved Eample # : Epad the followig biomials : (i) ( ) (ii) Solutio (i) ( ) ( ) ( ) ( ) ( ) ( ) 9 7 (ii) Solved Eample # : Epad the biomial up to fou tems Solutio Self pactice poblems Wite the fist thee tems i the epasio of Epad the biomial 8 As () () 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 { } Solved Eample # : The umbe of dissimila tems i the epasio of ( ) is (A) (B) () (D) Solutio ( ) [( ) ] ( ) Theefoe umbe of dissimila tems i the epasio of ( ) is Some impotat tems i the epasio of ( ) : (i) Geeal tem : ( ) ( )th tem is called geeal tem T Solved Eample # Fid (i) 8th tem of ( 8) (ii) 7th tem of 9 OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

3 FREE Dowload Stud Package fom website: wwwtekoclassescom Solutio (i) T 7 7 () 7 (8) 7 (ii)!! 7! 7th tem of () (8) 7 T As 9!!! As Solved Eample # : Fid the umbe of atioal tems i the epasio of (9 / 8 / ) Solutio The geeal tem i the epasio of ( ) / / 9 8 is T 9 8 The above tem will be atioal if epoet of ad ae iteges It meas ad must be iteges The possible set of values of is {,,,, } Hece, umbe of atioal tems is As (ii) Middle tem (s) : (a) If is eve, thee is ol oe middle tem, which is th tem (b) If is odd, thee ae two middle tems, which ae th ad th tems Solved Eample # : Fid the middle tem(s) i the epasio of (i) Solutio (ii) (iii) (i) (ii) 9 a a Hee, is eve, theefoe middle tem is th tem It meas T 8 is middle tem 7 9 T 8 7 As 9 a a 9 9 Hee, is odd theefoe, middle tems ae th & th It meas T & T ae middle tems a T 9 (a) 9 89 a 7 As 8 a T 9 (a) 9 a 9 As α b Tem cotaiig specified powes of i a ± β Solved Eample # 7: Fid the coefficiet of ad 7 i Solutio: Let ( )th tem cotais m T ( ) 7 ( ) (i) fo, (T ) T ( ) Hece, coefficiet of is As (ii) fo 7, 7 7 OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

4 FREE Dowload Stud Package fom website: wwwtekoclassescom (iv) (T ) T 7 ( ) Hece, coefficiet of 7 is As Numeicall geatest tem i the epasio of ( ), N Let T ad T be the th ad ( )th tems espectivel T ( ) T Now, oside T T T T ase - Ι Whe is a itege (sa m), the (i) T > T whe < m (,,, m ) ie T > T, T > T,, T m > T m (ii) T T whe m ie T m T m (iii) T < T whe > m ( m, m, ) ie T m < T m, T m < T m, T < T oclusio : Whe is a itege, equal to m, the T m ad T m will be umeicall geatest tems (both tems ae equal i magitude) ase - ΙΙ oclusio : Whe (i) T > T whe < is ot a itege (Let its itegal pat be m), the (,,,, m, m) ie T > T, T > T,, T m > T m (ii) T < T whe > ( m, m, ) ie T m < T m, T m < T m,, T < T Whe geatest tem is ot a itege ad its itegal pat is m, the T m will be the umeicall Solved Eample # 8 Fid the umeicall geatest tem i the epasio of ( ) whe Solutio Let th ad ( ) th be two cosecutive tems i the epasio of ( ) T T ( ) ( ) ( ) ( )! ( )!! ( ) ( )! ( )! ( )! Eplaatio: Fo, T T T > T T > T T > T OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP) T T Fo >, T < T T < T T 7 < T

5 ad so o Hece, T ad T ae umeicall geatest tems ad both ae equal Self pactice poblems : Fid the tem idepedet of i The sum of all atioal tems i the epasio of ( / / ) is (A) (B) 9 () 9 (D) Fid the coefficiet of i ( ) Fid the middle tem(s) i the epasio of ( ) 7 Fid the umeicall geatest tem i the epasio of (7 ) whee As () 8 7 () B () 9 8 OMPLEX NUMBERS / Page of FREE Dowload Stud Package fom website: wwwtekoclassescom () (7) T Multiomial Theoem: As we kow the Biomial Theoem ( )! ( )!! puttig, theefoe, ( )!!! 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 oegative itegal solutio of k k ie k Solved Eample # 9 Fid the coeff of a b c d i the epasio of (a b c d) ()! Solutio (a b c d)!!!! ( a) ( b) ( c) (d) we wat to get a b c d this implies that,,, coeff of a b c d is ()!!!!! ( ) ( ) As Solved Eample # Solutio I the epasio of 7 ( )! 7!!! ( ) () 7 fid the tem idepedet of The epoet is to be divided amog the base vaiables, ad 7 i such a wa so that we get Theefoe, possible set of values of (,, ) ae (,, ), (9,, ), (7,, ), (,, ), (,, ), (,, ) Hece the equied tem is ()! ()! ()! (7 ) 9!!! 7 ()!! 9!!!! 7 ()! 7!!! ()! 7!! 7!!! ()!!!! 7 7 ()!!! ()!!!!!!! 7 7 ()!!!! ()! 8! ()! ()!! 8!!! 7!!!! As Self pactice poblems : 8 The umbe of tems i the epasio of (a b c d e f) is (A) (B) () (D) 9 Fid the coefficiet of z i the epasio of ( z) 9 Fid the coefficiet of i ( ) TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

6 FREE Dowload Stud Package fom website: wwwtekoclassescom As (8) (9) 9!!!! Applicatio of Biomial Theoem : (i) If () 9 ( A B) Ι f whee Ι ad ae p ositi ve iteg es, beig od d a d < f < the (Ι f) f k whee A B k > ad A B < If is a eve itege, the (Ι f) ( f) k Solved Eample # If is positive itege, the pove that the itegal pat of (7 Solutio Let (7 ) Ι f (i) whee Ι & f ae its itegal ad factioal pats espectivel It meas < f < Now, < 7 < < (7 ) < Let (7 ) f (ii) < f < Addig (i) ad (ii) Ι f f (7 ) (7 ) [ 7 7 ( ) ] Ι f f eve itege(f f must be a itege) < f f < f f Ι eve itege theefoe Ι is a odd itege Solved Eample # Show that the itege just above ( ) is divisible b fo all N Solutio Let ( ) ( ) ( ) Ι f (i) whee Ι ad f ae its itegal & factioal pats espectivel < f < Now < < < ( ) < Let ( ) ( ) ( ) f (ii) < f < addig (i) ad (ii) Ι f f ( ) ( ) [( ) ( ) ] [ ( ) ] Ι f f k (whee k is a positive itege) < f f < f f Ι k ) is a odd umbe Ι is the itege just above ( ) ad which is divisible b (ii) hekig divisibilit Solved Eample # : Show that 9 7 is divisible b 8, whee is a positive itege Solutio 9 7 ( 8) λ whee, λ is a positive itege, Hece, 9 7 is divisible b 8 (iii) Fidig emaide Solved Eample # What is the emaide whe 99 is divided b Solutio: () 9 ( ) 9 [ 9 () 9 9 () () 9 9 () ] [ 9 () 9 9 () () ] [ 9 () 9 9 () () ] (k) 8 (whee k is a positive itege) (k ) 8 Hece, emaide is 8 As (iv) Fidig last digit, last two digits ad last thee digits of the give umbe Solved Eample # : Fid the last two digits of the umbe (7) Solutio (7) (89) (9 ) (9) (9) (9) (9) (9) (9) (9) 9 A multiple of 9 Hece, last two digits ae 9 As Note : We ca also coclude that last thee digits ae 9 (v) ompaiso betwee two umbes Solved Eample # : Which umbe is lage () o,? Solutio: B Biomial Theoem () ( ) () othe positive tems othe positive tems othe positive tems, Hece () >, OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

7 FREE Dowload Stud Package fom website: wwwtekoclassescom Self pactice poblems : If is positive itege, pove that the itegal pat of ( ) is a eve umbe If (7 ) α β, whee α is a positive itege ad β is a pope factio the pove that ( β) (α β) If is a positive itege the show that is divisible b 7 What is the emaide whe 7 is divided b Fid the last digit, last two digits ad last thee digits of the umbe (8) Which umbe is lage () o 8 As () 8 (),, () () 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 fom () ad () () Sum of two cosecutive biomial coefficiets!! LHS ( )!! ( )! ( )!! ( )! ( )! ( )! ( )!! RHS () Ratio of two cosecutive biomial coefficiets () ( ) ( )( )( ( )) ( ) ( )( ) Solved Eample # 7 If ( ) c, the show that (i) (ii) ( ) ( ) (iii) ( ) Solutio (i) ( ) put (ii) Ι Method : B Summatio (iii) LHS ( ) ( ) ( ) RHS ΙΙ Method : B Diffeetiatio ( ) Multiplig both sides b, ( ) Diffeetiatig both sides ( ) ( ) ( ) puttig, we get ( ) ( ) ( ) Poved Ι Method : B Summatio! ( ) ( )! ( )! ( ) OMPLEX NUMBERS / Page 7 of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP) LHS ( )

8 FREE Dowload Stud Package fom website: wwwtekoclassescom ( ) ( ) [ ( ) ] [ ( ) ] RHS { ( ) } ΙΙ Method : B Itegatio ( ) Itegatig both sides, with i the limits to ( ) ( ) ( ) Poved Solved Eample # 8 If ( ), the pove that (i) (ii) o (iii) ( ) Solutio (i) ( ) (i) ( ) (ii) Multiplig (i) ad (ii) ( ) ( ) ( ) ompaig coefficiet of, (ii) Fom the poduct of (i) ad (ii) compaig coefficiets of o both sides, o (iii) Ι Method : B Summatio LHS ( ) ( ) ( ) ( ) ( ) (i) ( ) (ii) Multiplig (i) ad (ii) ad compaig coeffciets of Hece, equied summatio is RHS ΙΙ Method : B Diffeetiatio ( ) Multiplig both sides b ( ) Diffeetiatig both sides ( ) ( ) ( ) ( ) (ii) Multiplig (i) & (ii) ( ( ) ) ( ) ( ) ( ) compaig coefficiet of, ( ) (i) ( ) Poved Solved Eample # 9 Fid the summatio of the followig seies (i) m m m m m m m (ii) Solutio (i) Ι Method : Usig popet, m m m m m m m m m m m m m m { m m m m } OMPLEX NUMBERS / Page 8 of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

9 FREE Dowload Stud Package fom website: wwwtekoclassescom m m m m m m m m ΙΙ Method m m m As m m m m m m m The above seies ca be obtaied b witig the coefficiet of m i ( ) m ( ) m ( ) Let S ( ) m ( ) m ( ) m m [( ) ] ( ) m : S (coefficiet of m i S) m ( ) ( ) ( ) ( ) m : Hece, equied summatio of the seies is m As (ii) The above seies ca be obatied b witig the coefficiet of i ( ) ( ) ( ) ( ) Let S ( ) ( ) ( ) ( ) (i) ( )S ( ) ( ) ( ) ( ) ( ) (ii) Subtactig (ii) fom (i) S ( ) ( ) ( ) ( ) ( ) [( ) ] ( ) ( ) ( ) ( ) ( ) S : S (coefficiet of i S) ( ) ( ) ( ) : Hece, equied summatio of the seies is Self pactice poblems : 7 Pove the followig (i) ( ) ( ) m As (ii) (iii) ( ) ( i v ) 8 Biomial Theoem Fo Negative Itege O Factioal Idices If R the, ( ) ( )! ( )( )! ( )( )( )! Remaks:(i) (ii) The above epasio is valid fo a atioal umbe othe tha a whole umbe if < Whe the ide is a egative itege o a factio the umbe of tems i the epasio of (iii) ( ) is ifiite, ad the smbol caot be used to deote the coefficiet of the geeal tem The fist tems must be uit i the epasio, whe ide is a egative itege o factio ( ) if <! ( ) ( ) < if! (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) Epasios to be emembeed ( < ) (a) ( ) ( ) (b) ( ) (c) ( ) ( ) ( ) (d) ( ) ( ) Solved Eample # : Pove that the coefficiet of i ( ) is Soltio: ( )th tem i the epasio of ( ) ca be witte as T ( )( )( )! ( ) OMPLEX NUMBERS / Page 9 of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

10 FREE Dowload Stud Package fom website: wwwtekoclassescom ( ) ( )( )( )! ( )( )( )! ( )! ( )( ) ( )!! ( ) ( )! Hece, coefficiet of is ( )!! Poved Solved Eample # : If is so small such that its squae ad highe powes ma be eglected the / ( ) ( ) fid the value of / ( ) / / / ( ) ( ) Solutio / ( ) / 9 / As 8 Self pactice poblems : 8 Fid the possible set of values of fo which epasio of ( ) / is valid i ascedig powes of 9 If!!, the fid the value of The coefficiet of i ( ) is (A) (B) 7 () 97 (D) As (8), (9) () OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

11 FREE Dowload Stud Package fom website: wwwtekoclassescom Shot Revisio BINOMIAL EXPONENTIAL & LOGARITHMI SERIES BINOMIAL THEOREM : The fomula b which a positive itegal powe of a biomial epessio ca be epaded i the fom of a seies is kow as BINOMIAL THEOREM If, R ad N, the ; ( ) This theoem ca be poved b Iductio OBSERVATIONS : (i) The umbe of tems i the epasio is ( ) ie oe o moe tha the ide (ii) The sum of the idices of & i each tem is (iii) The biomial coefficiets of the tems, equidistat fom the begiig ad the ed ae equal IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE : (i) Geeal tem (ii) Middle tem (iii) Tem idepedet of & (iv) Numeicall geatest tem (i) The geeal tem o the ( ) th tem i the epasio of ( ) is give b ; T (ii) The middle tem(s) is the epasio of ( ) is (ae) : (a) If is eve, thee is ol oe middle tem which is give b ; T ()/ / / / (b) If is odd, thee ae two middle tems which ae : T ()/ & T [()/] (iii) (iv) Tem idepedet of cotais o ; Hece fid the value of fo which the epoet of is zeo To fid the Numeicall geatest tem is the epasio of ( ), N fid T Put the absolute value of & fid the value of osistet with the T iequalit T T > Note that the Numeicall geatest tem i the epasio of ( ), >, N is the same as the geatest tem i ( ) I f, whee I & ae positive iteges, beig odd ad < f <, the If ( A B) (I f) f K whee A B K > & A B < If is a eve itege, the (I f) ( f) K BINOMIAL OEFFIIENTS : (i) (ii) (iii) ² ² ² ² ( )!!! ()! (iv) ( ) ( )! REMEMBER : (i) ()!! [ ( )] BINOMIAL THEOREM FOR NEGATIVE OR FRATIONAL INDIES : If Q, the ( ) ( ) ( )( ) Povided <!! Note :(i) Whe the ide is a positive itege the umbe of tems i the epasio of ( ) is fiite ie ( ) & the coefficiet of successive tems ae :,,, (ii) Whe the ide is othe tha a positive itege such as egative itege o factio, the umbe of tems i the epasio of ( ) is ifiite ad the smbol caot be used to deote the oefficiet of the geeal tem (iii) Followig epasio should be emembeed ( < ) (iv) (a) ( ) (b) ( ) (c) ( ) (d) ( ) The epasios i ascedig powes of ae ol valid if is small If is lage ie > the we ma fid it coviiet to epad i powes of, which the will be small APPROXIMATIONS : ( ) ( ) ( ) ( ) ² If <, the tems of the above epasio go o deceasig ad if be ve small, a stage ma be eached whe we ma eglect the tems cotaiig highe powes of i the epasio Thus, if be so small that its squaes ad highe powes ma be eglected the ( ), appoimatel OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

12 FREE Dowload Stud Package fom website: wwwtekoclassescom This is a appoimate value of ( ) 7 EXPONENTIAL SERIES : (i) e ; whee ma be a eal o comple & e Limit!!! (ii) a l a l a l a whee a >!!! Note : (a) e!!! (b) e is a iatioal umbe lig betwee 7 & 8 Its value coect upto places of decimal is (c) e e (d) e e!!!!! 7! (e) Logaithms to the base e ae kow as the Napieia sstem, so amed afte Napie, thei iveto The ae also called Natual Logaithm 8 LOGARITHMI SERIES : (i) l ( ) whee < (ii) l ( ) whee < (iii) ( ) l ( ) < REMEMBER : (a) l (b) e (c) l 9 (d) l EXERISE - Q Fid the coefficiets : (i) 7 i a b (ii) 7 i a b (iii) Fid the elatio betwee a & b, so that these coefficiets ae equal Q If the coefficiets of ( ) th, ( ) th tems i the epasio of ( ) 8 ae equal, fid Q If the coefficiets of the th, ( ) th & ( ) th tems i the epasio of ( ) ae i AP, fid / / Q Fid the tem idepedet of i the epasio of (a) (b) 7 Q Fid the sum of the seies ( ) up to m tems Q If the coefficiets of d, d & th tems i the epasio of ( ) ae i AP, show that ² 9 7 Q7 Give that ( ²) a a a ² a, fid the values of : (i) a a a a ; (ii) a a a a a ; (iii) a a a a a Q8 If a, b, c & d ae the coefficiets of a fou cosecutive tems i the epasio of ( ), N, a c b pove that a b c d b c 8 log Q9 Fid the value of fo which the fouth tem i the epasio, is log 7 Q Pove that : Q (a) Which is lage : (99 ) o () (b) Show that >, N, > 7 Q I the epasio of fid the tem ot cotaiig Q Show that coefficiet of i the epasio of ( ²) ( ) is Q Fid the coefficiet of i the epasio of : (i) ( ) (ii) ( ) 8 OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

13 FREE Dowload Stud Package fom website: wwwtekoclassescom Q Fid umeicall the geatest tem i the epasio of : (i) ( ) 9 whe (ii) ( ) whe Q Give s q q² q & S q q q, q, pove that s s s S Q7 Pove that the atio of the coefficiet of i ( ²) & the tem idepedet of i is : 9 Q8 Fid the tem idepedet of i the epasio of ( ) Q9 I the epasio of the epessio ( a), if the eleveth tem is the geometic mea of the eighth ad twelfth tems, which tem i the epasio is the geatest? Q Let (²)² () a K K K If a, a & a ae i AP, fid Q If the coefficiet of a, a, a i the epasio of ( a) ae i aithmetic pogessio, pove that ( ) ( )( )( )( Q If J ( )( )( )( ) Q Pove that K si Kcos( K) si K ), pove that J J Q The epessios,,, ae multiplied togethe ad the tems of the poduct thus obtaied ae aaged i iceasig powes of i the fom of a a a, the, (a) how ma tems ae thee i the poduct (b) show that the coefficiets of the tems i the poduct, equidistat fom the begiig ad ed ae equal (c) show that the sum of the odd coefficiets the sum of the eve coefficiets ( )! Q Fid the coeff of (a) i the epasio of (a² b c) 9 (b) z i the epasio of (a b cz) 9 (c) a b c d i the epasio of (a b c d) Q If a ( ) b( ) & a k fo all k, the show that b i k Q7 If P k () i the pove that, kpk () P i k Q8 Fid the coefficiet of i the epessio of : ( ) ( ) ( ) ( ) ( ) ( ) Q9(a) Fid the ide of the biomial if the 9th tem of the epasio has umeicall the geatest coefficiet ( N) (b) Fo which positive values of is the fouth tem i the epasio of ( ) is the geatest (7)! Q Pove that is divisible b 7 ( )! Q If the d, th, th & th tems i the epasio of ( ) be espectivel a, b, c & d the pove that b ac a c bd c Q Fid fo which the (k ) th tem of the epasio of ( ) is the geatest if ad >, > Q If is so small that its squae ad highe powes ma be eglected, pove that : (i) ( ) ( ) / ( ) / / ( ) / ( ) 7 (ii) / 7 ( ) ( ) /7 ( ) 7 ( ) 7 ( 8 ) Q (a) If the pove that (b) If!! Q If p q eal ad >, show that the fid the value of ² ( )p ( )q p ( )p ( )q q / o OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

14 FREE Dowload Stud Package fom website: wwwtekoclassescom EXERISE - Q Show that the itegal pat i each of the followig is odd N (A) ( ) (B) ( 8 7) () ( ) Q Show that the itegal pat i each of the followig is eve N (A) ( ) (B) ( ) Q If ( ) 7 pβ whee & p ae positive iteges ad β is a pope factio show that ( β) (p β) Q If deotes ( ) Q If P ( 8 7), N & [] the itegal pat of the fid the value of : ² [] ad f P [P], whee [ ] deotes geatest itege fuctio Pove that : P ( f) ( N) Q If ( ) Q7 Pove that if p is a pime umbe geate tha, the the diffeece ( ) N & F be the factioal pat of N, pove that NF ( N) p, whee [ ] deotes geatest itege cotais as facto ( N) Q8 Pove that the itege et above ( ) Q9 Let I deotes the itegal pat & F the pope factioal pat of ( ) deotes the atioal pat ad σ the iatioal pat of the same, show that Q Pove that ρ (I ) ad σ (I F ) is a itege, N EXERISE - (NOT IN THE SYLLABUS OF IIT-JEE) PROBLEMS ON EXPONENTIAL & LOGARITHMI SERIES Fo Q TO Q, Pove That :!!! Q Q e e!! 7!!!!!!! p p is divisible b whee N ad if ρ e Q e!!!!!! Q e!!! Q 7 e Q e² e!!! Q7 8 e Q 8 (e )!!!!!!!! Q9 log 7 e Q log 7 e Q l 7 Q l l Q l 7 7 Q l Q If whee <, the pove that!!! EXERISE - If,,,, ae the combiatoial coefficiets i the epasio of ( ), N, the pove the followig : OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

15 FREE Dowload Stud Package fom website: wwwtekoclassescom Q ² ² ² ² ( )! ( )! Q!! Q Q () () Q () () Q ( )( )( ) ( ) ( )! ( ) Q7 Q 8 ( )! ( )! Q9 o! Q o ( )! ( )! Q o ( ) Q o ( ) ( ) ( )!! ( )! Q o ( ) () Q o ² ² ² ² ( ) ² o ( ) / / accodig as is odd o eve Q If is a itege geate tha, show that ; a (a ) (a ) ( ) (a ) Q ( )² ( )² ( )² ( ) Q7 o ² ² ² () ² ( ) ( )!!! Q8 If a, a, a, be the coefficiets i the epasio of ( ²) i ascedig powes of, the pove that : (i) a a a a a a (ii) a a a a a a a a a o a (iii) E E E ; whee E a a a ; E a a a 7 & E a a a 8 Q9 Pove that : ( ) ( )! ( )! ( )! Q If () ², the show that the sum of the poducts of the i s Σ Σ i j take two at a time, epeseted b is equal to! i < j (!) Q [ ] Q ( ) / fo EXERISE - Q If (), the fid the value of : Q If ( ² p ) a a a ²a p p, the fid the value of : a a a p a p Q ² ² ² ² ()² () () Q ( ) Q Give pq, show that p q p[ ( ) p ] Q Show that ( ) whee deotes the combiatoial coeff i the epasio of ( ) Q7 ( ) ( ) Q8 Pove that, Q9 If () the pove that ; OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

16 FREE Dowload Stud Package fom website: wwwtekoclassescom Q Q Q ( ) ( ) ( ) ( ) 9! ( ) 9 ( ) ( ) ( ) ( ) Q ( ) ( ) ( ) Q ( ) Q If () ², the show that : ( ) ( )² ( ) ( ) ( ) ( ) ( ²) ( ) ( ) Q Pove that, ( ) Q7 If N ; show that! ( ) ( ) ( ) ( ) ( )! Q8 Pove that, ( )² ( )² ( )² ( )² [ ( )! ] Q9 If ( ) a, N, the pove that ( ) a ( ) a ( ) a ( < < ) Q Pove that the sum to ( ) tems of equals ( ) ( ) ( ) ( ) ( ) ( ) d & evaluate the itegal EXERISE - Q The sum of the atioal tems i the epasio of ( ) / is [JEE 97, ] Q If a, the equals [JEE 98, ] (A) ( )a (B) a () a / (D) Noe of these Q Fid the sum of the seies 9 [REE 98, ]!!!!! Q If i the epasio of ( ) m ( ), the co-efficiets of ad ae ad espectivel, the m is : [JEE '99, (Out of )] (A) (B) 9 () (D) Q(i) Fo, (A) (B) () (D) (ii) I the biomial epasio of (a b),, the sum of the th ad th tems is zeo The a b equals: [ JEE ' (Sceeig), ] (A) (B) () (D) Q Fo a positive iteges m, (with m), let m m Pove that m m m m m m Hece o othewise pove that, m ( m ) m m m m m OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

17 FREE Dowload Stud Package fom website: wwwtekoclassescom [ JEE ' (Mais), ] Q7 Fid the lagest co-efficiet i the epasio of ( ), give that the sum of co-efficiets of the tems i its epasio is 9 [ REE ' (Mais) ] Q8 I the biomial epasio of (a b), >, the sum of the th ad th tems is zeo The a b equals (A) (B) () (D) [ JEE ' (Sceeig), ] Q9 Fid the coeffciet of 9 i the polomial [ REE ' (Mais), ] whee m Q The sum ( )( m i ) i i, (whee ( q ) p if P < q ) is maimum whe m is (A) (B) () (D) Q(a) oefficiet of t i the epasio of ( t ) ( t ) ( t ) is (A) (B) () (D) oe [JEE, Sceeig out of ] (b) Pove that : K K K K K K ( ) K K K K [JEE, Mais- out of ] Q (K ), if K (A) [, ] (B) (, ) () (, ) (D) (, ] [JEE (Sceeig)] Q The value of is, whee (A) (B) Pat : (A) Ol oe coect optio () EXERISE (D) [JEE (Sceeig)] 7 I the epasio of the th tem is a:, (A) positive itege (B) positive iatioal umbe () egative itege (D) egative iatioal umbe If the secod tem of the epasio / a a is a / the the value of a (A) (B) () (D) The value of, is : (A) (B) () (D) oe P Q Let the co-efficiets of i ( ) & ( ) be P & Q espectivel, the Q (A) 9 (B) 7 () 8 (D) oe of these If the sum of the co-efficiets i the epasio of ( ) is, the the geatest tem i the epasio fo / is : (A) th (B) th () th (D) oe of these Fid umeicall the geatest tem i the epasio of ( )9, whe / (A) 9 9 (/) (B) 9 9 (/) () 9 9 (/) (D) 9 9 (/) 8 7 The umbes of tems i the epasio of a is a (A) (B) () (D) 8 The coefficiet of i the epasio of ( ) 8 is (A) 7 (B) 9 () (D) 8 9 ( ) ( ) ( ) ( ) whe witte i the ascedig powe of the the highest epoet of is (A) (B) () (D) is: OMPLEX NUMBERS / Page 7 of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

18 FREE Dowload Stud Package fom website: wwwtekoclassescom If ( ) 7 [] f, the ( f) (A) (B) () (D) The emaide whe is divided b 7 is (A) (B) () 8 (D) oe of these The last two digits of the umbe ae: (A) 8 (B) () 9 (D) The value of 9 is, whee (A) (B) () (D) The value of the epessio K K ( ) K K is (A) (B) () (D) If <, the the co-efficiet of i the epasio of ( ) is (A) (B) () (D) The umbe of values of ' ' satisfig the equatio, 9 9 is : (A) (B) () (D) 7 Numbe of elemets i set of value of fo which, is satisfied (A) elemets (B) elemets () 7 elemets (D) elemets 8 The co-efficiet of i the epasio of, ( ) ( ) ( ) is : (A) (B) 9 () (D) 9 If ( ) a a a a, the (a a a a a 8 a ) (a a a a 7 a 9 ) is equal to (A) (B) () 9 (D) oe of these The value of is equal to (A) ( 9) (B) () 9 ( ) (D) oe of these If,,, ae the Biomial coefficiets i the epasio of ( ) beig eve, the ( ) ( ) ( ) is equal to (A) (B) () (D) If ( ) a a a a, the a a a a 8 equals (A) 9 ( ) (B) 9 ( ) () ( 9 ) (D) oe of these o-efficiet of i ( ) is : (A) (B) () (D) is The sum of the coefficiets of all the itegal powes of i the epasio of ( ) (A) (B) () ( ) (D) ( ) If { } deotes the factioal pat of ' ', the 8 (A) 9/8 (B) 8/8 () /8 (D) /8 The coefficiet of the tem idepedet of i the epasio of is (A) 7 (B) () (D) 7 The coefficiet of i polomial ( ) ( ) ( )( ) is (A) (B) () (D) oe of these 8 I the epasio of ( ) ( ) ( z), the sum of the co-efficiets of the tems of degee ' ' is : (A) (B) () (D) 9 p p is equal to p (A) (B) () (D) If ( ) ², the show that the sum of the poducts of the i s take two at a time, epeseted b Σ Σ i j is equal to i < j! (A) (!) (B)! () (!) Pat : (B) Ma have moe tha oe optios coect I the epasio of ( z) (A) eve tem is of the fom k k z k! (!) (D)! (!) OMPLEX NUMBERS / Page 8 of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

19 FREE Dowload Stud Package fom website: wwwtekoclassescom (B) the coefficiet of 8 9 z 9 is () the umbe of tems is (D) oe of these is divisible b (A) (B) () (D) 7 EXERISE Fid the value of ' ' fo which the fouth tem i the epasio, log log is 7 I the biomial epasio of, the atio of the 7th tem fom the begiig to the 7th tem fom the ed is : ; fid Fid the tems idepedet of '' i the epasio of the epessio,( ) If i the epasio of ( ),the co-efficiet of is deoted b a, the pove that a a Show that the tem idepedet of i the epasio of is, Fid the coefficiet of a b c 7 i the epasio of (bc ca ab) 8 7 If ( ) a a a a, the calculate a, a, a 8 If ( ) ( ) p f, whee p is a itege ad f is a pope factio the fid the value of 9, N 9 Wite dow the biomial epasio of ( ), whe 8 Deduce that is divisible b, wheeve is a positive itege Pove that is divisible b Which is lage : (99 ) o () If,,,, ae the combiatoial co-efficiets i the epasio of ( ), N, the pove the followigs: (Q No - ) o ( ) ² ² ² ² ()² () () Assumig ' ' to be so small that ad highe powes of ' ' ca be eglected, show that, ( ) ( ) / (8 ) / is appoimatel equal to, 9 7 If ( ) to m tems k, the fid the value of k m 7 Fid the coefficiet of i the epessio: ( ) ( ) 999 ² ( ) q q q Give s q q² q & S, q, 9 pove that s s s S Show that if the geatest tem i the epasio of ( ) has also the geatest co-efficiet, the ' ' lies betwee, & Fid the emaide whe is divided b 7 If ( ² p ) a a a ²a p p, the fid the value of : a a a p a p ( )! Pove that, ( )² ( )² ( )² ( )² {( )! } If () ², the show that: ( ) ( )² ( ) ( ) ( ) Pove that p q pq p if p q ( ) ( ²) ( ) ( ) OMPLEX NUMBERS / Page 9 of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)

20 FREE Dowload Stud Package fom website: wwwtekoclassescom Ifa ( ) b( ) & a k fo all k, the show that b If a, a, a, be the coefficiets i the epasio of ( ²) i ascedig powes of, the pove that : (i) a a a a a a (ii) a a a a a a a a a (iii) E E E ; whee E a a a ; E a a a 7 & E a a a 8 7 If ( ) p p p p, the pove that : (a) p p p / cos π (b) p p p / si π 8 If () ², the show that the sum of the poducts of the i s take two at Q Q i j! a time, epeseted b is equal to i < j (!) ANSWER KEY EXERISE - (i) a (ii) a b (iii) ab Q Q o 9 Q (a) b (b) T 7 m ( ) m ( ) ( ) Q 7 (i) (ii), (iii) a Q 9 o Q o Q (a) (Pove that 99 some ive qt) Q k k k k 7 k Q (i) 99 (ii) Q (i) T 7 7 (ii) Q 8 7 Q9 T 8 Q o o Q (a) Q (a) 8b c ab c 7a b c 8a c ; (b) a b c ; (c) Q 8 ( ) Q 9 (a) (b) 8 < < Q k Q (a) Hit : Add to both sides & compae the RHS seies with the epasio () to get & (b) Q EXERISE - EXERISE - Q divide epasio of () both sides b & diff wt, put to get 99 Q Diffeetiate the give ep & put to get the esult p (p) Q 9 Itegate the ep of ( ) Detemie the value of costat of itegatio b puttig Itegate the esult agai betwee & to get the esult Q oside [() ( ) ] ² Itegate betwee & Q Multipl both sides b the ep () Itegate both sides betwee & Q Note that ( ) ² Itegate betwee & Q ( )! ( )! ( )! EXERISE - Q Q Q e Q Q (i) D (ii) B Q7 Q8 B Q9 Q Q (a) A Q D Q A EXERISE - 7 B A A D B A 7 A 8 A 9 B D B D B B A B B A D D D B AB A o 9 7 EXERISE f, if is eve ad f, if is odd p (p ) 8 7 a, a, a 88 7 OMPLEX NUMBERS / Page of TEKO LASSES, HOD MATHS : SUHAG R KARIYA (S R K Si) PH: (7)-, , BHOPAL, (MP)