Binomial Theorem Theory E

8/12/2019 Binomial Theorem Theory E

http://slidepdf.com/reader/full/binomial-theorem-theory-e 1/17

CDRN_

"cdlasniucdrpnysaos.al" 4

Falhcadm RnghrgcFalhcad m gxprgssahl :

Dly dmkgfrdao gxpr gssahl wnaon ohltdals twh bassacamdr tgrcs as odmmgb falhcadm gxpr gssahl.

Ehr gxdcpmg : x + y, x

6

y + 6xy

4

, ? ‟ x, 4x6

+ ?/4? )4x(

4

gto.

Rgrcalh mhky usgb al falhcad m tnghrgc :

Edothradm lhtdtahl : hr l! as prhlhulogb ds edothradm l dlb as bgealgb ds

l! 3

0lae 94

Llae 94.6.?)........6l)(4l(l

Lhtg : l! 3 l . (l ‟ 4)! 9 l L

Cdtngcdtaodm cgdlalk he lOr: Rng tgrc lO

r bglhtgs lucfgr he ohcfaldtahls he r tnalks onhhsgl

erhc l bastalot tnalks cdtngcdtaodmmy, lOr 3

!r )!r l(

!l

, l L, r [, 0 r l

Lhtg : Htngr sycfhms he he lOr drg

r

ldlb O(l, r).

Prhpgrt ags rgmdtgb th lOr

:

(a) lOr 3 lO

l ‟ r

Lhtg : Ae lOx

3 l Oy

Gatngr x 3 y hr x + y 3 l

(aa) lOr + lO

r ‟ 4 3 l + 4O

r

(aaa)4r

l

r l

O

O

3

r

4r l

(av) lOr

3r

ll‟4O

r‟4 3

)4r (r

)4l(l

l‟6Or‟6

3 ............. 34.6).......6r )(4r (r

))4r (l().........6l)(4l(l

(v) Ae l dlb r drg rgmdtavgmy pracg, tngl lOr as bavasafmg fy l. Fut ohlvgrsg as lht lgogssdramy trug.

_td tgcglt h e f alhcadm tnghrgc :

(d + f)l 3 lO0 d lf0 + lO

4 d l‟4 f4 + lO

6 d l‟6 f6 +...+ lO

r d l‟r fr +...... + lO

l d 0 fl



CDRN_


wngrg l L

hr (d + f)l 3

l

0r

r r lr

l fdO

Lhtg : Ae wg put d 3 4 dlb f 3 x al tng dfhvg falhcadm gxpdlsahl, tngl

hr (4 + x)l 3 lO0

+ lO4

x + lO6

x 6 +... + lOr xr +...+ lO

l xl

hr (4 + x)l 3

l

0r

r r

l xO

Gxdcpmg # 4 : Gxpdlb tng ehmmhwalk falhcadms :

(a) (x ‟ ?)7 (aa)

26

6

x?4

_hmutahl : (a) (x ‟ ?)7 3 7O0x7 + 7O

4x2 (‟ ?)4 + 7O

6 x ? (‟ ?)6 + 7O

? x 6 (‟?)?

+ 7O2 x (‟ ?)2 + 7O

7 (‟ ?)7

3 x7 ‟ 47x2 + 10x? ‟ 6;0x6 + 207x ‟ 62?

(aa)

26

6

x?4

3 2O

0 + 2O

4

6

x? 6

+ 2O6

66

6

x?

+ 2O?

?6

6

x?

+ 2O

2

26

6

x?

3 4 ‟ 5x6 +

6

6;x2 ‟

6

6;x5 +

45

>4x>

Gxdcpmg # 6 : Gxpdlb tng falhcadm

60

6

y?

?

x6

up th ehur tgrcs

_hmutahl :

60

6

y?

?

x6

3 60O

0

60

?

x6

+ 60O

4

41

?

x6

6

y?+ 60O

6

4>

?

x6

6

6

y?

+ 60O?

4;

?

x6

?

6

y?

+ ....

3

60

?

x6

+ 60.

4>

?

6

x41y + 410 .

45

?

6

x4> y6 + 4420

42

?

6

x4; y? + .....

_gme prdotaog prhfmgcs

(4) [ratg tng earst tnrgg tgrcs al tng gxpdlsahl he

5

?

y6

.



CDRN_

"cdlasniucdrpnysaos.al" ?

(6) Gxpdlb tng falhcadm

76

x

?

?

x

.

Dlswgrs : (4) 52 ‟ 52y +?

>0y6 (6)

62?

x40

+6;

7x; +

?

40x2 + ?0x + 6x

4?7+ 7x

62?.

Hfsgrvdtahls :

(a) Rng lucfgr he tgrcs al tng falhcadm gxpdlsahl (d + f)l as l + 4.

(aa) Rng suc he tng albaogs he d dlb f al gdon tgrc as l.

(aaa) Rng falhcadm ohgeeaoaglts (lO0, l O

4 ..........lO

l) he tng tgrcs gquabastdlt erhc tng fgkallalk dlb

tng glb drg gqudm, a.g. lO0 3 lO

l, lO

4 3 lO

l‟4 gto. { lO

r 3 l O

l‟r }

(av) Rng falhcadm ohgeeaoaglt odl fg rgcgcfgrgb watn tng ngmp he tng ehmmhwalk pdsodm‑s Rradlkmg

(dmsh ilhwl ds Cgru Prdstrd prhvabgb fy Palkmd)

Vgkdrbalk Pdsodm‑s Rradlkmg, wg lhtg tng e hmmhwalk :

(d) Gdon rhw he tng tradlkmg fgkals watn 4 dlb glbs watn 4.

(f) Dly gltry al d rhw as tng suc he twh gltrags al tng prgogbalk rhw, hlg hl tng accgbadtg mget dlb

tng htngr hl tng accgbadtg raknt.

Gxdcpmg # ? : Rng lucfgr he bassacamdr tgrcs al tng gxpdlsahl he (4 ‟ ?x + ?x 6 ‟ x?)60 as

(D) 64 (F) ?4 (O) 24 (B) 54

_hmutahl : (4 ‟ ?x + ?x6 ‟ x?)60 3 Y(4 ‟ x)?S60 3 (4 ‟ x)50

Rngrgehrg lucfgr he bassacamdr tgrcs al tng gxpdlsahl he (4 ‟ ?x + ?x 6 ‟ x?)60 as 54.

Kglgrdm tgrc :

(x + y)l 3 lO0

xl y0 + lO4

xl‟4 y4 + ...........+ lOr xl‟r yr + ..........+ lO

l x0 yl

(r + 4)tn tgrc as odmmgb kglgrdm tgrc dlb bglhtgb fy Rr+4

.

Rr+4 3 l

Or xl‟r

yr

Lhtg : Rng r tn tgrc erhc tng glb as gqudm th tng (l ‟ r + 6) tn tgrc erhc tng fgkalalk, a.g. lOl ‟ r + 4

xr ‟ 4 yl ‟ r + 4

Gxdcpmg # 2 : Ealb (a) 6>tn tgrc he (7x + >y)?0 (aa) ;tn tgrc he

1

6x

7

7

2x

_hmutahl : (a) R6; + 4

3 ?0O6;

(7x)?0‟ 6; (>y)6; 3!6;!?

!?0(7x)? . (>y)6;



CDRN_


(aa) ;tn tgrc he

1

x6

7

7

x2

R5 + 4

3 1O5

51

7

x2

5

x6

7

3

!5!?

!1 ?

7

x2

5

x6

7

3 ?x

40700

Gxdcpmg # 7 : Ealb tng lucfgr he rdtahldm tgrcs al tng gxpdlsahl he (14/2 + > 4/5)4000.

_hmutahl : Rng kglgrdm tgrc al tng gxpdlsahl he 40005/42/4 >1 as

Rr+4

3 4000Or

r 4000

2

4

1

r

5

4

>

3 4000O

r 6

r 4000

?

6

r

6

Rng dfhvg tgrc wamm fg rdtahldm ae gxphlglt he ? dlb 6 drg altgkgrs

At cgdls 6

r 4000 dlb 6

r

cust fg altgkgrs

Rng phssafmg sgt he vdmugs he r as {0, 6, 2, ............, 4000}

Nglog, lucfgr he rdtahldm tgrcs as 704

Cabbmg tgrc s :

(d) Ae l as gvgl, tngrg as hlmy hlg cabbmg tgrc, wnaon as

tn

6

6l

tgrc.

(f) Ae l as hbb, tngrg drg twh cabbmg tgrcs, wnaon drg

tn

6

4l

dlb

tn

4

6

4l

tgrcs.

Gxdcpmg # 5 : Ealb tng cabbmg tgrc(s) al tng gxpdlsahl he

(a)

426

6

x4

(aa)

1?

5

dd?

_hmutahl : (a)

426

6

x4

Ngrg, l as gvgl, tngrgehrg cabbmg tgrc as

6

642tn tgrc.

At cgdls R> as cabbmg tgrc

R>

3 42O;

;6

6

x

3 ‟

45

261x42.

(aa)

1?

5

dd?



CDRN_


Ngrg, l as hbb tngrgehrg, cabbmg tgrcs drg

6

41tn &

4

6

41tn.

At cgdls R7 & R

5 drg cabbmg tgrcs

R7 3 1O

2 (?d)1 ‟ 2

2?

5

d

3

>

4>1d4;

R5 3 1O

7 (?d)1 ‟ 7

7?

5

d

3 ‟

45

64d41.

Gxdcpmg # ; : Ealb tng ohgeeaoaglt he x?6 dlb x ‟4; al

47

?

2

x

4x

.

_hmutahl : Mgt (r + 4)tn tgrc ohltdals xc

Rr + 4

3 47Or (x2)47 ‟ r

r

?

x

4

3 47O

r x50 ‟ ;r (‟ 4)r

(a) ehr x?6 , 50 ‟ ;r 3 ?6

;r 3 6> r 3 2, sh 7tn tgrc.

R7 3 47O

2 x?6 (‟ 4)2

Nglog, ohgeeaoaglt he x?6 as 4?57

(aa) ehr x ‟4;, 50 ‟ ;r 3 ‟ 4;

r 3 44 , sh 46tn tgrc.

R46

3 4 7O44

x ‟4; (‟ 4)44

Nglog, ohgeeaoaglt he x ‟4; as ‟ 4?57

Lucgraodmmy krgdtgst tgrc al tng gxpdlsahl he d + f l, l L

Falhcadm gxpdlsahl he (d + f) l as ds ehmmhws : ‟

(d + f)l 3 lO0 d lf0 + lO

4 d l‟4 f4 + lO

6 d l‟6 f6 +...+ lO

r d l‟r fr +...... + lO

l d 0 fl

Ae wg put ogrtdal vdmugs he d dlb f al VN_, tngl gdon tgrc he falhcadm gxpdlsahl wamm ndvg

ogrtdal vdmug. Rng tgrc ndvalk lucgraodmmy krgdtgst vdmug as sdab th fg lucgraodmmy krgdtgst

tgrc.

Mgt Rr dlb R

r+4 fg tng r tn dlb (r + 4)tn tgrcs rgspgotavgmy

Rr

3 lOr‟4

dl‟(r‟4) fr‟4

Rr+4

3 lOr dl‟r fr

Lhw,r

4r

R

R 3 4r 4r l

r r l

4r l

r l

fd

fd

O

O

3

r

4r l .

d

f

Ohlsabgr r

4r

R

R 4

r

4r l

d

f 4

r

4l ‟ 4

f

d



CDRN_


r f

d4

4l

Odsg - [ngl

f

d4

4l

as dl altgkgr (sdy c), tngl

(a) Rr+4

< Rr

wngl r = c (r 3 4, 6, ? ...., c ‟ 4)

a.g. R6 < R

4, R

?< R

6, ......., R

c < R

c‟4

(aa) Rr+4

3 Rr

wngl r 3 c

a.g. Rc+4

3 Rc

(aaa) Rr+4

= Rr

wngl r < c (r 3 c + 4, c + 6, ..........l )

a.g. Rc+6

= Rc+4

, Rc+?

= Rc+6

, ..........Rl+4

= Rl

Ohlomusahl :

[ngl

f

d

4

4l

as dl altgkgr, sdy c, tngl RR

c dlb R

c+4 wamm fg lucgraodmmy krgdtgst tgrcs (fhtn tgrcs drg

gqudm al cdklatubg)

Odsg -

[ngl

f

d4

4l

as lht dl altgkgr (Mgt ats altgkrdm pdrt fg c), tngl

(a) Rr+4

< Rr

wngl r =

f

d4

4l

(r 3 4, 6, ?,........, c‟4, c)

a.g. R6

< R4 , R

? < R

6, .............., R

c+4 < R

c

(aa) Rr+4

= Rr

wngl r <

f

d4

4l

(r 3 c + 4, c + 6, ..............l)

a.g. Rc+6

= Rc+4

, Rc+?

= Rc+6

, .............., Rl +4

= Rl

Ohlomusahl :

[ngl

f

d4

4l

as lht dl altgkgr dlb ats altgkrdm pdrt as c, tngl R c+4 wamm fg tng lucgraodmmy krgdtgst

tgrc.

Lhtg : (a) Al dly falhcadm gxpdlsahl, tng cabbmg tgrc(s) nds krgdtgst falhcadm ohgeeaoaglt.

Al tng gxpdlsahl he (d + f) l

Ae l Lh. he krgdtgst falhcadm ohgeeaoaglt Krgdtgst falhcadm ohgeeaoaglt

Gvgl 4 lOl/6

Hbb 6 lO(l ‟ 4)/6

dlb lO(l + 4)/6

(Udmugs he fhtn tngsg ohgeeaoaglts drg gqudm )



CDRN_

"cdlasniucdrpnysaos.al" ;

(aa) Al hrbgr th hftdal tng tgrc ndvalk lucgraodmmy krgdtgst ohgeeaoaglt, put d 3 f 3 4, dlb prhoggb

ds basoussgb dfhvg.

Gxdcpmg # > : Ealb tng lucgraodmmy krgdtgst tgrc al tng gxpdlsahl he (? ‟ 7x) 47 wngl x 37

4.

_hmutahl : Mgt r tn dlb (r + 4)tn fg twh ohlsgoutavg tgrcs al tng gxpdlsahl he (? ‟ 7x) 47

Rr + 4

Rr 47O

r ?47 ‟ r (| ‟ 7x|)r 47O

r ‟ 4 ? 47 ‟ (r ‟ 4) (|‟ 7x|)r ‟ 4

!r !)r 47(

)!47

|‟ 7x | !)4r (!)r 45(

)!47.?

7 .7

4(45 ‟ r) ?r

45 ‟ r ?r

2r 45

r 2

Gxp mdld t ahl :

Ehr r 2, Rr + 4

Rr

R6

< R4

R?

< R6

R2

< R?

R7

3 R2

Ehr r < 7, Rr + 4

= Rr

R5

= R7

R; = R5

dlb sh hl

Nglog, R2 dlb R

7 drg lucgraodmmy krgdtgst tgrcs dlb fhtn drg gqudm.

_gme prdotaog prhfmgcs :

(?) Ealb tng tgrc albgpglbglt he x al

16

x

?x

(2) Rng suc he dmm rdtahldm tgrcs al tng gxpdlsahl he (?4/7 + 64/?)47 as

(D) 50 (F) 71 (O) 17 (B) 407

(7) Ealb tng ohgeeaoaglt he x ‟4 al (4 + ?x6 + x2)

>

x

44

(5) Ealb tng cabbmg tgrc(s) al tng gxpdlsahl he (4 + ?x + ?x6 + x?)6l

(;) Ealb tng lucgraodmmy krgdtgst tgrc al tng gxpdlsahl he (; ‟ 7x)44 wngl x 3?

6.

Dlswgrs : (?) 6>.?; (2) F (7) 6?6

(5) 5lO?l

. x?l (;) R2 3 ‟

1

220Û ;> Û 7?.



CDRN_

"cdlasniucdrpnysaos.al" >

Vgsumt : Ae l)F D( 3 + e, wngrg dlb l drg phsatavg altgkgrs, l fgalk hbb dlb 0 = e = 4, tngl ( + e)

e 3 il wngrg D ‟ F6 3 i < 0 dlb D ‟ F = 4. Ae l as dl gvgl altgkgr, tngl ( + e) (4 ‟ e) 3 il

Gxdcpmg # 1 : Ae l as phsatavg altgkgr, tngl prhvg tndt tng altgkrdm pdrt he (; + 2 ? )l as dl hbb lucfgr..

_hmutahl : Mgt (; + 2 ? )l 3 + e .............(a)

wngrg & e drg ats altgkrdm dlb erdotahldm pdrts rgspgotavgmy.

At cgdls 0 = e = 4

Lhw, 0 = ; ‟ 2 ? = 4

0 = (; ‟ 2 ? )l = 4

Mgt (; ‟ 2 ? )l 3 e .............(aa)

0 = e = 4

Dbbalk (a) dlb (aa)

+ e + e 3 (; + 2 ? )l + (; ‟ 2 ? )l

3 6 YlO0 ; l + lO

6 ; l ‟ 6 (2 ? )6 + ..........S

+ e + e 3 gvgl altgkgr (e + e cust fg dl altgkgr)

0 = e + e = 6 e + e 3 4

+ 4 3 gvgl altgkgr

tngrgehrg as dl hbb altgkgr.

Gxdcpmg # 40 : _nhw tndt tng altgkgr just dfhvg ( ? + 4)6l as bavasafmg fy 6l + 4 ehr dmm l L.

_hmutahl : Mgt ( ? + 4)6l 3 (2 + 6 ? )l 3 6l (6 + ? )l 3 + e ..........(a)

wngrg dlb e drg ats altgkrdm & erdotahldm pdrts rgspgotavgmy

0 = e = 4.

Lhw 0 = ? ‟ 4 = 4

0 = ( ? ‟ 4)6l = 4

Mgt ( ? ‟ 4)6l 3 (2 ‟ 6 ? )l 3 6l (6 ‟ ? )l 3 e . ........(aa)

0 = e = 4

dbbalk (a) dlb (aa)

+ e + e 3 ( ? + 4)6l + ( ? ‟ 4)6l

3 6l Y(6 + ? )l + (6 ‟ ? )lS 3 6.6l YlO0

6 l + lO6 6 l ‟ 6 ( ? )6 + ........S

+ e + e 36l + 4 i (wngrg i as d phsatavg altgkgr)

0 = e + e = 6 e + e 3 4

+ 4 3 6l + 4 i.

+ 4 as tng altgkgr just dfhvg ( ? + 4)6l dlb wnaon as bavasafmg fy 6l + 4.

Gxdcpmg # 44 : _nhw tndt 1l + ; as bavasafmg fy >, wngrg l as d phsatavg altgkgr.

_hmutahl : 1l + ; 3 ( 4 + >)l + ;

3 lO0

+ lO4

. > + lO6

. >6 + ....... + lOl

> l + ;.

3 >. O4

+ >6. O6

+ ....... + Ol

. >l + >.

3 >, wngrg as d phsatavg altgkgr

Nglog, 1l + ; as bavasafmg fy >.



CDRN_


Gxdcpmg # 46 : [ndt as tng rgcdalbgr wngl 711 as bavabgb fy 4?.

_hmutahl : 711 3 7.71> 3 7. (67)21 3 7 (65 ‟ 4)21

3 7 Y21O0 (65)21 ‟ 21O

4 (65)2> + .......... + 21O

2> (65)4 ‟ 21O

21 (65)0S

3 7 Y21O0 (65)21 ‟ 21O

4 (65)2> + ...........+ 21O

2> (65)4 ‟ 4S

3 7 Y21O0

(65)21 ‟ 21O4(65)2> + .......... + 21O

2> (65)4 ‟ 4?S + 50

3 4? (i) + 76 + > (wngrg i as d phsatavg altgkgr)

3 4? (i + 2) + >Nglog, rgcdalbgr as >.

Gxdcpmg # 4? : Ealb tng mdst twh bakats he tng lucfgr (4;) 40.

_hmutahl : (4;)40 3 (6>1)7 3 (610 ‟ 4)7

3 7O0 (610)7 ‟ 7O

4 (610)2 + ........ + 7O

2 (610)4 ‟ 7O

7 (610)0

3 7O0 (610)7 ‟ 7O

4 . (610)2 + .........7O

? (610)6 + 7 Û 610 ‟ 4

3 D cumtapmg he 4000 + 4221

Nglog, mdst twh bakats drg 21

Lhtg : [g odl dmsh ohlomubg tndt mdst tnrgg bakats drg 221.

Gxdcpmg-42 : [naon lucfgr as mdrkgr (4.04)4000000 hr 40,000 8

_hmutahl : Fy Falhcadm Rnghrgc

(4.04)4000000 3 (4 + 0.04)4000000

3 4 + 4000000O4 (0.04) + htngr phsatavg tgrcs

3 4 + 4000000 Û 0.04 + htngr phsatavg tgrcs

3 4 + 40000 + htngr phsatavg tgrcs

Nglog (4.04)4000000 < 40,000


(>) Ae l as phsatavg altgkgr, prhvg tndt tng altgkrdm pdrt he (7 7 + 44)6l + 4 as dl gvgl lucfgr..

(1) Ae (; + 2 ? )l 3 + , wngrg as d phsatavg altgkgr dlb as d prhpgr erdotahl tngl prhvg tndt

(4 ‟ ) ( + ) 3 4.

(40) Ae l as d phsatavg altgkgr, tngl snhw tndt ?6l + 4 + 6l + 6 as bavasafmg fy ;.

(44) [ndt as tng rgcdalbgr wngl ;40? as bavabgb fy 67 .

(46) Ealb tng mdst bakat, mdst twh bakats dlb mdst tnrgg bakats he tng lucfgr (>4)67.

(4?) [naon lucfgr as mdrkgr (4.6)2000 hr >00

Dlswgrs : (44) 4> (46) 4, 04, 004 (4?) (4.6)2000.

_hcg stdlbdrb gxpdls ahls :

(a) Ohlsabgr tng gxpdlsahl

(x + y)l 3

l

0r

r l O xl‟r yr 3 lO

0 xl y0 + lO

4 xl‟4 y4 + ...........+ lO

r xl‟r yr + ..........+ lO

l x0 yl ....(a)

(aa) Lhw rgpmdog y ‟ y wg kgt



CDRN_


(x ‟ y)l 3

l

0r

r l O (‟ 4) r xl‟r yr 3 lO

0 xl y0 ‟ lO

4 xl‟4 y4 + ...+ lO

r (‟4)r xl‟r yr + ...+ lO

l (‟ 4)l x0 yl ....(aa)

(aaa) Dbbalk (a) & (aa), wg kgt

(x + y)l + (x ‟ y)l 3 6YlO0

xl y0 + lO6

x l ‟ 6 y6 +.........S

(av) _uftrdotalk (aa) erhc (a), wg kgt

(x + y)l ‟ (x ‟ y)l 3 6YlO4 xl ‟ 4 y4 + lO? x l ‟ ? y? +.........S

Prhpgrt ags he f alhcadm ohg ee ao aglts :

(4 + x)l 3 O0

+ O4x + O

6x6 + ......... + O

r xr + .......... + O

lxl ......(4)

wngrg Or bglhtgs lO

r

(4) Rng suc he tng falhcadm ohgeeaoaglts al tng gxpdlsahl he (4 + x)l as 6l

Puttalk x 3 4 al (4)

lO0 + lO

4 + lO

6 + ........+ lO

l 3 6l ......(6)

hr

l

0r

lr

l 6O

(6) Dkdal puttalk x 3 ‟4 al (4), wg kgt

lO0 ‟ lO

4 + lO

6 ‟ lO

? + ............. + (‟4)l lO

l 3 0 ......(?)

hr

l

0r

r lr 0O)4(

(?) Rng suc he tng falhcadm ohgeeaoaglts dt hbb phsatahl as gqudm th tng suc he tng falhcadm ohgeeaoaglts

dt gvgl phsatahl dlb gdon as gqudm th 6 l‟4.

erhc (6) dlb (?)

lO0

+ lO6

+ lO2

+ ................ 3 lO4

+ lO?

+ lO7

+ ................ 3 6l‟4

(2) _uc he twh ohlsgoutavg falhcadm ohgeeaoaglts

lOr + lO

r‟4 3 l+4O

r

M.N._. 3 lOr + lO

r‟4 3

!r )!r l(

!l

+

)!4r ()!4r l(

!l

3 )!4r ()!r l(

!l

4r l

4

r

4

3)!4r ()!r l(

!l

)4r l(r

)4l(

3!r )!4r l(

)!4l(

3 l+4Or

3 V.N._.

(7) Vdtah he twh ohlsgoutavg falhcadm ohgeeaoaglts



CDRN_


4r l

r l

O

O

3

r

4r l

(5) lOr

3r

ll‟4O

r‟4 3

)4r (r

)4l(l

l‟6Or‟6

3 ............. 34.6).......6r )(4r (r

))4r (l().........6l)(4l(l

Gxdcpmg # 47 : Ae (4 + x) l 3 O0

+ O4x + O

6x6 + ............. + o

lxl, tngl snhw tndt

(a) O0

+ ?O4

+ ?6O6

+ .......... + ?l Ol

3 2l.

(aa) O0

+ 6O4

+ ?. O6

+ ........ + (l + 4) Ol

3 6l ‟ 4 (l + 6).

(aaa) O0

‟6

O4+

?

O6 ‟

2

O?+ ......... + ( ‟4)l

4l

Ol

3

4l

4

.

_hmutahl : (a) (4 + x)l 3 O0

+ O4

x + O6x6 + ........... + O

lxl

put x 3 ?

O0

+ ? . O4

+ ?6 . O6

+ .......... + ?l . Ol

3 2l

(aa) Cgtnhb : Fy _uccdtahl

M.N._. 3 lO0

+ 6. lO4

+ ? . lO6

+ ........ + (l + 4). lOl.

3

l

0r

)4r ( . lOr 3

l

0r

r lO.r +

l

0r

r l O

3 l

l

0r

4r 4l O +

l

0r

r l O 3 l . 6l ‟ 4 + 6l 3 6l ‟ 4 (l + 6). VN_

Cgtnhb : Fy Baeegrgltadtahl

(4 + x)l 3 O0 + O4x + O6x6 + ........... + Olxl

Cumtapmyalk fhtn sabgs fy x,

x(4 + x)l 3 O0x + O

4x6 + O

6x? + ........ + O

l xl + 4.

Baeegrgltadtalk fhtn sabgs

(4 + x)l + x l (4 + x)l ‟ 4 3 O0

+ 6. O4x + ? . O

6x6 + ....... + (l + 4)O

lxl.

puttalk x 3 4, wg kgt

O0

+ 6.O4

+ ? . O6

+ ...... + (l + 4) Ol

3 6l + l . 6l ‟ 4

O0

+ 6.O4

+ ? . O6

+ ...... + (l + 4) Ol

3 6l ‟ 4 (l + 6) Prhvgb

(aaa) Cgtnhb : Fy _uccdtahl

M.N._. 3 O0

‟

6

O4+

?

O6 ‟

2

O?+ ........ + (‟ 4)l.

4l

Ol

3

l

0r

r )4( .4r

Or l

34l

4

l

0r

r )4( . l + 4Or + 4

4r

4lr

l OO.4r

4l

34l

4

Yl + 4O

4 ‟ l + 4O

6 + l + 4O

? ‟ .............+ (‟ 4)l . l + 4O

l + 4S

3 4l

4

Y‟ l + 4

O0 + l + 4

O4 ‟ l + 4

O6 + ......... + (‟ 4)l

. l + 4

Ol + 4 + l + 4

O0S



CDRN_


34l

4

3 V.N._. , salog 0O)4(...OOO 4l

4ll6

4l4

4l0

4l

Cgtnhb : Fy Altgkrdtahl

(4 + x)l 3 O0

+ O4x + O

6x6 + ...... + O

l xl.

Altgkrdtalk fhtn sabgs, watnal tng macats ‟ 4 th 0.

0

4

4l

4l

)x4(

3

0

4

4l

l

?

6

6

404l

xO.....?

xO6

xOxO

4l

4

‟ 0 3 0 ‟

4l

O)4(.....

?

O

6

OO l4l64

0

O0

‟6

O4+

?

O6 ‟ .... .... .. + (‟ 4)l

4l

Ol

3

4l

4

Prhvgb

Gxdcpmg # 45 : Ae (4 + x)l 3 O0

+ O4x + O

6x6 + ........+ O

lxl, tngl prhvg tndt

(a) O0

6 + O4

6 + O6

6 + ...... + Ol

6 3 6lOl

(aa) O0O6 + O4O? + O6O2 + .......... + Ol ‟ 6 Ol 3 6lOl ‟ 6 hr 6lOl + 6

(aaa) 4. O0

6 + ? . O4

6 + 7. O6

6 + ......... + (6l + 4) . Ol

6 . 3 6l. 6l ‟ 4Ol

+ 6lOl.

_hmutahl : (a) (4 + x)l 3 O0

+ O4x + O

6x6 + ......... + O

l xl. ........(a)

(x + 4)l 3 O0xl + O

4xl ‟ 4+ O

6xl ‟ 6 + ....... + O

lx0 ........(aa)

Cumtapmyalk (a) dlb (aa)

(O0

+ O4x + O

6x6 + ......... + O

lxl) (O

0xl + O

4xl ‟ 4 + ......... + O

lx0) 3 (4 + x)6l

Ohcpdralk ohgeeaoaglt he xl,

O0

6 + O4

6 + O6

6 + ........ + Ol

6 3 6lOl

(aa) Erhc tng prhbuot he (a) dlb (aa) ohcpdralk ohgeeaoaglts he xl ‟ 6 hr xl + 6 fhtn sabgs,

O0O

6 + O

4O

? + O

6O

2 + ........ + O

l ‟ 6O

l 3 6lO

l ‟ 6hr 6 lO

l + 6.

(aaa) Cgtnhb : Fy _uccdtahl

M.N._. 3 4. O0

6 + ?. O4

6 + 7. O6

6 + .......... + (6l + 4) Ol

6.

3

l

0r

)4r 6( lOr 6 3

l

0r

r .6 . (lOr )6 +

l

0r

6r

l )O(

3 6

l

4r

l. . l ‟ 4Or ‟ 4

lOr + 6lO

l

(4 + x)l 3 lO0

+ lO4

x + lO6

x 6 + .............lOl

x l ..........(a)

(x + 4)l ‟ 4 3 l ‟ 4O0

x l ‟ 4 + l ‟ 4O4

x l ‟ 6 + .........+l ‟ 4Ol ‟ 4

x0 .........(aa)

Cumtapmyalk (a) dlb (aa) dlb ohcpdralk ohgeeoaglts he xl.l ‟ 4O

0 . lO

4 + l ‟ 4O

4 . lO

6 + ........... + l ‟ 4O

l ‟ 4. lO

l 3 6l ‟ 4O

l

l

0r

4r 4l O . lO

r 3 6l ‟ 4O

l

Nglog, rgquargb succdtahl as 6l. 6l ‟ 4Ol

+ 6lOl 3 V.N._.

Cgtnhb : Fy Baeegrgltadtahl

(4 + x6)l 3 O0

+ O4x6 + O

6x2 + O

?x5 + ..............+ O

l x6l

Cumtapmyalk fhtn sabgs fy x

x(4 + x6)l 3 O0x + O

4x? + O

6x7 + ............. + O

lx6l + 4.

Baeegrgltadtalk fhtn sabgs



CDRN_

"cdlasniucdrpnysaos.al" 4?

x . l (4 + x6)l ‟ 4 . 6x + (4 + x6)l 3 O0

+ ?. O4x6 + 7. O

6 x 2 + ....... + (6l + 4) O

l x6l ........(a)

(x6 + 4)l 3 O0

x 6l + O4

x 6l ‟ 6 + O6

x 6l ‟ 2 + ......... + Ol

........(aa)

Cumtapmyalk (a) & (aa)

(O0

+ ?O4x6 + 7O

6x2 + ......... + (6l + 4) O

l x 6l) (O

0 x 6l + O

4x6l ‟ 6 + ........... + O

l)

3 6l x6 (4 + x6)6l ‟ 4 + (4 + x6)6l

ohcpdralk ohgeeaoaglt he x6l,

O0

6 + ?O4

6 + 7O6

6 + .........+ (6l + 4) Ol

6 3 6l . 6l ‟ 4Ol ‟ 4

+ 6lOl.

O0

6 + ?O4

6 + 7O6

6 + .........+ (6l + 4) Ol

6 3 6l . 6l‟4Ol + 6lO

l. Prhvgb

Gxdcpmg # 4; : Ealb tng succdtahl he tng ehmmhwalk sgrags ‟

(a) cOc

+ c+4Oc

+ c+6Oc

+ .............. + lOc

(aa) lO?

+ 6 . l+4O?

+ ?. l+6O?

+ ......... + l . 6l‟4O?

_hmutahl : (a) Cgtnhb : Wsalk prhpgrty, lOr + lO

r‟4 3 l+4 O

r cO

c + c+4O

c + c+6O

c + .............. + lO

c

3 c

4c4c

4c OO

+ c+6Oc

+ .............. + lOc

{ cOc

3 c +4Oc+4

}

3 c

6c

4c

6c OO

+ .................. + lOc

3 c+?Oc+4

+ ............. + lOc

3 lOc+4

+ lOc

3 l+4Oc+4

CgtnhbcO

c + c+4O

c + c+6O

c + .......... + lO

c

Rng dfhvg sgrags odl fg hftdalgb fy wratalk tng ohgeeaoaglt he xc al

(4 + x)c + (4 + x)c+4 + ......... + (4 + x)l

Mgt _ 3 (4 + x)c + (4 + x)c+4 +.............. + (4 + x)l

3

x

4x4)x4( 4clc

3

x

x4x4 c4l

3 ohgeeaoaglt he xc al

x

x4 4l

‟

x

x4 c

3 l + 4Oc + 4

+ 0 3 l + 4Oc +4

(aa) lO?

+ 6 . l+4O?

+ ? . l+6O?

+ .......... + l . 6l‟4O?

Rng dfhvg sgrags odl fg hfdtalgb fy wratalk tng ohgeeaoaglt he x? al

(4 + x)l + 6 . (4 + x)l+4 + ? . (4 + x)l+6 + ........... + l . (4 + x)6l‟4

Mgt _ 3 (4 + x)l + 6 . (4 + x)l+4 + ?. (4 + x)l+6 + ........... + l (4 + x)6l‟4 .....(a)

(4 + x)_ 3 (4 + x)l+4 + 6 (4 + x)l+6 + ............. + (l ‟ 4) (4 + x) 6l‟4 + l(4 + x)6l ....(aa)

_uftrdotalk (aa) erhc (a)

‟ x_ 3 (4 + x)l + (4 + x)l+4 + (4 + x)l+6 + .............. + (4 + x) 6l‟4 ‟ l(4 + x)6l

3x

4)x4()x4( ll

‟ l (4 + x)6l

_ 36

ll6

x

)x4()x4( +

x

)x4(l l6

x? : _ (ohgeeaoaglt he x? al _)

x? :6

ll6

x

)x4()x4( +

x

)x4(l l6

Nglog, rgquargb succdtahl he tng sgrags as ‟ 6lO7 + lO

7 + l . 6lO

2



CDRN_


Gxdcpmg # 4> : Prhvg tndt O4 ‟ O

? + O

7 ‟ ........ 3 6l/6 sal

2

l.

_hmutahl : Ohlsabgr tng gxpdlsahl (4 + x)l 3 O0

+ O4

x + O6

x 6 + ...... + Ol x l ....(a)

puttalk x 3 ‟ a al (a) wg kgt

(4 ‟ a)l 3 O0 ‟ O

4a ‟ O

6 + O

?a + O

2 + ....... (‟ 4)l O

l a l

hr 6l

6

2

lsala

2

lohs

3 (O0 ‟ O6 + O2 ‟ .......) ‟ a (O4 ‟ O? + O7 ‟ ...... ) ....(a a)

Gqudtalk tng acdkaldry pdrt al (aa) wg kgt O4 ‟ O

? + O

7 ‟ ........ 3 6l/6 sal

2

l.


(42) Prhvg tng ehmmhwalk

(a) O0

+ ?O4

+ 7O6

+ ............. + (6l + 4) Ol

3 6l (l + 4)

(aa) 2O0

+

6

26

. O4

+

?

2?

O6

+ .............. +

4l

2 4l

Ol

3

4l

47 4l

(aaa) lO0

. l+4Ol

+ lO4

. lOl‟4

+ lO6

. l‟4Ol‟6

+ ........... + lOl

. 4O0

3 6l‟4 (l + 6)

(av) 6O6

+ ?O6

+ ......... + lO6

3 l+4 O?

Cum talhcadm tnghrgc :

Ds wg ilhw tng Falhcadm R nghrgc ‟

(x + y)l 3

l

0r

r l O xl‟r yr 3

l

0r !r )!r l(

!lxl‟r yr

puttalk l ‟ r 3 r 4 , r 3 r 6

tngrgehrg, (x + y)l 3 lr r 64

64!r !r

!l64 r r

y.x

Rhtdm lucfgr he tgrcs al tng gxpdlsahl he (x + y) l as gqudm th lucfgr he lhl-lgkdtavg altgkrdm shmutahl

he r 4 + r

6 3 l a.g. l+6‟4O

6‟4 3 l+4 O

4 3 l + 4

Al tng sdcg edsnahl wg odl wratg tng cumtalhcadm tnghrgc

(x4

+ x6

+ x?

+ ........... xi)l 3

lr ...r r i64

i64

!r !...r !r

!li64 r

ir 6

r 4 x...x.x

Ngrg thtdm lucfgr he tgrcs al tng gxpdlsahl he (x4

+ x6

+ .......... + xi)l as gqudm th lucfgr he lhl-

lgkdtavg altgkrdm shmutahl he r 4

+ r 6

+ ........ + r i 3 l a.g. l+i‟4O

i‟4

Gxdcpmg # 41 : Ealb tng ohgeeaoaglt he d6 f? o2 b al tng gxpdlsahl he (d ‟ f ‟ o + b)40

_hmutahl : (d ‟ f ‟ o + b)40 3 40r r r r 2?64

2?64

!r !r !r !r

)!40(2?64 r r r r

)b()o()f()d(

wg wdlt th kgt d6 f? o2 b tnas ac pmags tndt r 4

3 6, r 6 3 ?, r

? 3 2, r

2 3 4

ohgee. he d6 f? o2 b as!4!2!?!6

)!40((‟4)? (‟4)2 3 ‟ 46500



CDRN_


(aa) [ngl tng albgx as d lgkdtavg altgkgr hr d erdotahl tngl lucfgr he tgrcs al tng gxpdlsahl he

(4 + x)l as alealatg, dlb tng sycfhm lOr

odllht fg usgb th bglhtg tng ohgeeaoaglt he tng kglgrdm tgrc.

(aaa) Rng earst tgrc cust fg ulaty al tng gxpdlsahl, wngl albgx „l‑ as d lgkdtavg altgkgr hr erdotahl

(x + y)l

3

4y

xae .....

y

x

!6

)4l(l

y

x.l4y

y

x4y

4x

yae .....

x

y

!6

)4l(l

x

y.l4x

x

y4x

6

l

l

l

6l

ll

(av) Rng kglgrdm tgrc al tng gxpdlsahl he (4 + x)l as Rr+4

3!r

)4r l().........6l)(4l(l xr

(v) [ngl „l‑ as dly rdtahldm lucfgr htngr tndl wnhmg lucfgr tngl dpprhxacdtg vdmug he (4 + x)l as

4 + lx (x6 dlb nakngr phwgrs he x odl fg lgkmgotgb)

(va) Gxpdlsahls th fg rgcgcfgrgb (|x| = 4)

(d) (4 + x) ‟4 3 4 ‟ x + x 6 ‟ x? + .......... + (‟4)r xr + .........(f) (4 ‟ x) ‟4 3 4 + x + x6 + x? + .......... + xr + .........(o) (4 + x) ‟6 3 4 ‟ 6x + ?x6 ‟ 2x? + .......... + (‟4)r (r + 4) xr + ...........(b) (4 ‟ x) ‟6 3 4 + 6x + ?x6 + 2x? + ............. + (r + 4)xr + ...........

Gxdcpmg # 64 : Prhvg tndt tng ohgeeaoaglt he xr al (4 ‟ x) ‟l as l+r‟4Or

_hmutahl: (r + 4)tn tgrc al tng gxpdlsahl he (4 ‟ x) ‟l odl fg wrattgl ds

Rr +4

3!r

)4r l)......(6l)(4l(l (‟x)r

3 (‟4)r !r

)4r l)......(6l)(4l(l (‟x)r 3 !r

)4r l)......(6l)(4l(l xr

3!r !)4l(

)4r l)......(4l(l)!4l(

xr

Nglog, ohgeeaoaglt he x r as!r )!4l(

)!4r l(

3 l+r‟4Or Prhvgb

Gxdcpmg-66 : Ae x as sh scdmm suon tndt ats squdrg dlb nakngr phwgrs cdy fg lgkmgotgb, tngl ealb tng vdmug he

6/4

?/76/4

)x2(

)x4()x?4(

_hmutahl : 6/4

?/76/4

)x2(

)x4()x?4(

3 6/4

2

x46

?

x74x

6

?4

36

4

x

5

416

6/4

2

x4

36

4

x

5

416

>

x4 3

6

4

x

5

41

2

x6 3 4 ‟

>

x ‟

46

41x 3 4 ‟

62

24x



CDRN_

4


(4>) Ealb tng phssafmg sgt he vdmugs he x ehr wnaon gxpdlsahl he (? ‟ 6x)4/6 as vdmab al dsoglbalk

phwgrs he x.

(41) Ae y 37

6+

!6

?.4 6

7

6

+

!?

7.?.4 ?

7

6

+ ............., tngl ealb tng vdmug he y6 + 6y

(60) Rng ohgeeaoaglt he x400 al 6)x4(

x7?

as

(D) 400 (F) ‟7; (O) ‟41; (B) 7?

Dlswgrs : (4>) x

6

?,

6

?(41) 2 (60) O

Binomial Theorem Theory E

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Transcript of Binomial Theorem Theory E