BINOMIAL THEOREM & ITS SIMPLE APPLICATION

Transcription

1 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 MB Sllabus : BINOMIAL THEOREM & ITS SIMPLE APPLIATION Biomia Theoem fo a positive itegal ide; geeal tem ad middle tem; popeties of Biomial coefficiets ad simple applicatios.

2 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 MB O N E P T S Statemet of Biomial theoem : If, R ad N, the ( + ) = ( ) 0 a b ( + )th tem of ( + ) T + = Now, puttig = i the biomial theoem, we get ( + ) = ( ) Pactice Poblems :. The umbe of iatioal tems i the epasio of ( /6 + /8 ) If A ad B ae the coefficiet of i the epasio of ( + ) ad ( + ) espectivel, the A = B A = B A = B oe. The atio of the coefficiet of tems a ad a i the biomial epasio of ( + a) will be : a : : oe 0. The tem which idepedet of i the epasio of 0. The epasio [ + ( ) / ] + [ ( ) / ] a polomial of degee [Aswes : () b () b () d () b () c] Some impotat tems i the epasio of ( + ) : (i) Middle tem o tems : (ii) If eve, thee ol oe middle tem, which if 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.

3 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 MB Now, T T. T T oside Pactice Poblems :. The tem that will be middle i the epasio of ( + ) 6 9th tem 0th tem th tem th tem. The umeicall geatest tem i the epasio of () () () () 7. Which tem umeicall geatest i the epasio of ( + ) 9 if 6th tem 7th tem 8th tem oe. If eve positive itege, the the coditio that the geatest tem i the epasio of ( + ) ma have the geatest coefficiet also [Aswes : () b () c () b () a] oe of these Popeties of Biomial oefficiets : ( + ) = () () The sum of the biomial coefficiets i the epasio of ( + ) Put = i () we get = () Agai put = i (), we get, ( ) = 0 0 ( ) 0...() 0..() () The sum of the biomial coefficiets at odd positio equal to the sum of the biomial coefficiets at eve positio ad each equal to i.e., = = () Sum of two cosecutive biomial coefficiets : + = + () Ratio of two cosecutive biomial coefficiets :

4 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 ( ) (6) ( ) MB Pactice Poblems :. The sum of coefficiets i the epasio of ( + ) 6 will be 0 6. Sum of the last 0 coefficiets i the epasio of ( + ) 9 whe epaded ascedig powe of The sum of the last eight coefficiets i the epasio of ( + ) 6 oe of these. If ( + ) 00 = the the value of k ad k will be whee 0... k ad... k ad ad ad ad 0 0. If ( + ) = c 0 + c + c...c the c c c c c... will be 0 c c c ( ) ( ) 8 ( ) ( )( ) equal to oe 7. If ( ) c the 0 c c 0 c c c... c ( )! ( ) + ( ) ( )! ( )! [Aswes : () c () c () c () d () c (6) b (7) a] Biomial Theoem Fo Negative Itege o Factioal Idices If R the, ( ) ( )! ( )( )!... ( )( )...( ) f...!

5 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 Pactice Poblems : MB. If ve small so that its squae ae heighe powes ae eglected the the value of will be ( ) ( ). If (a + b) =... the (a, b) (, ) (, ) (, ) oe. / ca be epaded b biomial theoem if < < > >. The coefficiet of i the epasio of ( ) ( ) +. The coefficiet of i the epasio of ( +...) ()!! ()! (!) ()! (!) oe of these 6. If > the ( + ) will be I the epasio of the coefficiet of will be + oe 8. The coefficiet of i the epasio of ( )( ) [Aswes : () d () a () d () b () b (6) d (7) a (8) c] oe of these Time savig tips : ( )( ). Numbe of tems i ( z).. Numbe of tems i ( a) ( a) if odd if eve. No thee cosecutive biomial coefficiets ca be i G.P. o H.P.. The tem idepedet of i ( ) p q p q! p!q!

6 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 MB 6 INITIAL STEP EXERISE. Which of the followig ot equal to the middle tem i the epasio of? (, ) P(, )!...( )! oe of the above. If the atio of the 7th tem fom the begiig to the sevet tem fom the ed i the epasio of, the oe of these. The coefficiet of 99 i ( + ) ( + ) ( + )... ( + 99) oe of the above. If the ft thee tems i the epasio of ( + a) ae 79, 790 ad 07 espectivel, the the value of oe of these. If the fouth tem i the epasio of log equal to / 6 equal to 00 ad >, the oe of the above 6. The coefficiet of the tem idepedet of i the epasio of ( + + ) The coefficiet of 6 i the epasio of ( + ) The coefficiet of the tem idepedet of i the epasio of / / / 0 9. The total umbe of dsimilia tems i the epasio of ( ) ( )( ) 6 ( ) 0. If 0,,,..., 0 ae the coefficiet of 0,,,..., 0 i the epasio of ( + ) 0 ad if ! = k., the the value of k (0!) 6 8 oe of these. If the sum of the coefficiets i the epasio of ( + ) 66, the the geatest coefficiet i the epasio oe of the above. If the thid tem i the epasio of log 0 0 6, the ma be 0 0 / both ad

7 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 MB 7 FINAL STEP EXERISE. The umbe of tems i the epasio of ( + + ) 0 whe epaded i decedig powes of 0 0. The coefficiet of i the epasio of ( ) The umbe of tems which ae fee fom adical sigs i the epasio of ( / + /0 ) 6 7 oe of these. If ( + ) = 0, the the value of 0 + ( 0 + ) + ( ) ( ). ( + ). ( + ).. If a, a, a, a ae the coefficiets of a fou cosecutive tems i the epasio of ( + ), the a a a a a a a a a a a a equal to a. a a a a a 6. If T 0, T, T,...,T epeset the tems i the epasio of ( + a), the the value of (T 0 T + T T ) + (T T + T...) ( a ) ( + a ) (a ) oe of these 7. The digit at uit s place i the umbe If + =, the.. equals. 0. ( + ) ( + ) oe of the above 9. The coefficiet of µ i the epasio of [( + ) ( + µ) ( + µ)] If ( + + ) 0 = a 0 + a + a a 0 0, the a + a + a... +a 7 equals 9 ( 0 ) 0 ( 9 9) 9 ( 0 + ) oe of the above. The value of to ( + ) tems ( + ) ( + ). + ( + ). The iteval i which must be so that the geatest tem i the epasio of ( ) has the umeicall geatest coefficiet 6, 6, 6, 6,. If the ieth tem i the epasio of log 7 /8log ( ) 0 >, the equals equal to 80 ad log 0 log log e oe of the above. Give positive iteges >, > ad the coefficiets of ()th ad ( + )th tems i the biomial epasio ( + ) ae equal. The equals + oe of these. If the sum of the coefficiets i the epasio of (p p + ) vahes, the the value of p

8 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, If i the epasio of ( + ) m ( ), the coefficiet of ad ae ad 6 espectivel. the m Fo, =. c. a. b. b. b 6. c MB 8 ANSWERS (INITIAL STEP EXERISE) 7. b 8. a 9. c 0. b. c. d 8. If a odd atual umbe the 0 / 0 ( ) / oe of these equals 9. The coefficiet of i the epasio of ( + ) + ( + ) ( + ) The emaide whe 99 divided b If 0,,,..., ae coefficiet i the biomial epasio of (+ ) ad eve, the the value of ( ) 0 ( ) ()!! (!) (!) (!) (( )!). d. c. b. d. c 6. b 7. b 8. c 9. d 0. a ANSWERS (FINAL STEP EXERISE). c. b. b. d. c 6. c 7. d 8. a 9. c 0. b. b

9 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 MB 9 AIEEE ANALYSIS [00]. The coefficiet of i ( ) / 6 oe of these. If <, the the coefficiet of i epasio of ( ) + + AIEEE ANALYSIS [00]. If positive, the ft egative tem i the epasio of ( + ) 7/ th tem 8th tem 6th tem 7th tem. The umbe of itegal tems the epasio of 8 ( ) 6 AIEEE ANALYSIS [00/00]. The coefficiet of the middle tem i the biomial theoem i powes of of ( + ) ad ( ) 6 the same if equals /0 0/ / / [00] 6. The coefficiet of i epasio of ( + ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [00] 7. If s ad t equal to 0 0 [00], the 8. If the coefficiet of th, ( + )th ad ( + )th tems i the biomial epasio of ( + ) m ae i A.P., the m ad satf the equatio m m( + ) + + = 0 m m( ) + = 0 m m( ) + + = 0 m m( + ) + = 0 [00] t s 9. If the coefficiet of 7 i equals a b the coefficiet of 7 i a, the a ad b b satf the elatio a + b = a b = a ab = b [00] 0. If so small tha ad highe powes of ma be eglected, the appoimated as [00] 8 ( ) ( ) The value of ma be 6 6 [00]

10 MB 0 AIEEE ANALYSIS [006]. If the epasio i powes of of the fuctio a ( a)( b) 0 + a + a a +... the a b a b a b a b a. Fo atual umbes m, if ( ) m ( + ) = + a + a +... ad a = a = 0, the (m, ) (, ) (0, ) (, 0) (, ) [006] a b b a a b b a [006] AIEEE ANALYSIS [007]. I the biomial epasio of (a b),, the sum of th ad 6th tems zeo, the b a equals. The sum of the seies ANSWERS AIEEE ANALYSIS. d. b. b. a. a 6. b 7. c 8. d 9. c 0. d. c. a. a. b. d Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87

11 Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 MB TEST YOURSELF. The coefficiet of i the epasio of ( + ) ( + ) The umbe of tems whose values deped o i the epasio of oefficiet of the tem idepedet of i the epasio of The value of , ( ) ( ) ( ) ( ). If ( ), the value of ()! [( )!] ()! ( )!( )! [( ()! )!] oe of these. The geatest tem i the epasio of ( + ), whe = /, ( ) ( ) ( 0 ) oe of these 6. If so small that ad highe powes of ca be eglected, the the value of ( ) / (6 ) ( ) / d. a. b. b. a 6. b 7. d ANSWERS