jee questions binomial

8/16/2019 JEE Questions Binomial

1/9

ll

F

ILL

IN

THE BL

NKS

1, T

he

""m

uf l

he

co

"f fici

em s

of the p

lj·n

omial

1

1

x - 3x

' )

2103

;, . .

. .

2.

The

coeff

icient

of

x'' ' in th

e po

lynom

ial

(x-I )

(X .. 2)

.. (x

-1 0

0)

i

s.

(Il

l 98

2)

(Ill 1

982; 2M

)

(I ll

9

83 ; 2

M)

l .cl

n

be a

po sitive in teg er.

If th e

co eff ioicnts

of

2nd, 3 rd. and

4th

terms

in the

expansi on

of

(I

+X)

are

in

A.P . th en

the

'a lue

lf(

ll =

J ~ I

+ & x +

2 - 1 x

+

....

h o n a

.

. . and

n

~

Ol

1

1 i.< .

B OB

JECT

IVE Q

UEST

IONS

-

Only r

me

r

plion

i• co

rrect:

'

G

i,en

po

Slti\c

i n l ~ ~

r

>

1 n>2

a

nd

the

cw

fl idc

nt

o

f

(3

t) th

and

(r +2)

th t

erm•

in th e

bino

mial

expan

si on of

tl +xl

2

'

arc

e q ua

l. TI1

en:

(I ll

1

9801

( a ) n ~ 2 r

(O)n-2r+1

c

)n -3

r

td)no

neo[

thc>

c

'

"

1he

cocftl

coen t

ot ,

\ ' in :

: -

3

,

·

is :

• 2

)

lilT

198 3

; IM

)

'

'

a)

-

25

6

(c)

26 3

'

'

b )

.v

~ 5 9

lf C,

sm

nds

fo r ' 'C

, t

he n th

e sum

o

f

il

e serie

o

lr:

X

i } lC

; -2

C,

2

+

.iCi

'

... 1 (-1 )

'' {n

+l lC

;']

"

here

n

i

s an

even p

os it l

'e m tt

gcr.

is equ

al to:

la

)l-1)

" '

2

(

n12)

(c )

(

l)" ' '

(n l l)

l

hc ex p

re ssio

n

(I

l l 19 8

6;

2M

]

(d

)

no

ue

of h

e


2/9

(o) c , +l

Ic) c,

(bJ ''c,

+1

(dl c, 1-2

Jf" C , ~ ( k ' - 1 ) 1 ' , . , t h e n k :

(a)

( - ~ > . - 2 j

c . .

(c) l -v3.D]

(b)[2.c


3/9

NS

WER

S

fJ

Fill in t

he blanks

1. - I

2.-

5050

3

. a = 2 , n

~ 4

l l O

bjective

Question

s (Only o

ne option

)

1. (

a) 2

(a)

J.

(a)

8.(bl

9.(b)

lO

.(c)

[ I

Subjec

t ive Ques

Uons

• c C c

,

l (2

- 1)

1

FIL L

IN

THE

BLANKS

4 . n ~ 7

4

(c)

l l.

(d)

1

.

S

um of c

oefficie

m s 1> ob

tained

by l i n g

'

r

e.,

(1

·1

~ - l

Thu

s sum nfth

e c(>O

fficionts

of th

e

p

olynomi

al

( I ~

x

-

3x'

)

is -1

The

toefficiC

T11 of x

' in

(x

-

1)

(

x .. z ...

(x-lOO

)i<

- ( J+2o

3+

..... . +11

)0)

~

-

J

OO

1

+ l O U

) ~ -

50(101

)

= -

50 .10

3. He

re,

th u

s

(

1+aSion

of

1

•

x

)" is

c

,

No

z r c , J

~ c ,

c,

~ o n d i t i

=>

2

n

n-1)

'

(

accordin

g

, , ; - · ~ · '

n-1

=1+

+ n

Jn+2

c\n -6=

6+n

2

-3n+2

n' -

9 n + l 4 ~

U

n -7

n -2n+l

4=11

n

(n -7) -2

(11 -7

)= 0

(n-2)

(n-7)=

0

n=2

or

n=7

£l VC

{}

be

lt C,

is

true f

ar n

;;

3

e f o r ~ " ? io

the un.m

er.

Jr

-1

=

r

+

lo

r ln {31· -

I ) o(r

+ 1)

2r=

2 or 2 n ~ 4

r=)

n ~

2 r

r> l

.


4/9

2. The

gene

ral

te

rm

r

"

3 ;

10

~ - - ,

I

"

X

I

~ I )

' C

_.2 . '_,X

LO- l •

'

z1 U

-r

for

coeff

icient

ofx', w ~

hav

e 10

-

3

r 4

3r=6 i lr

r=2

( '

J'

: . Coe

ff icient of

x

4

in l::_

- 2

_ = (

-1 )

2

.

1

°C,.

____ _

2 x

2

)

1°

9x

45 405

256 256

3. (a

) We hav

e

'·

g -lei

+

Jeff -4Cj +

..... •

(-1

)' (n + J)

c;

{'

·

' ··

' •. c'

\ <

J C'J

,-. ._

_,-.._,

-

,

.... +

-

"

-{ C

1

2

-2

Cf +3

Cf - .....

+(-l)"n

C;)

I

'_ I

I

= (

- l )

; ~ ( ~ ·

;-]c

(-I)

2

n

I 2)

( ,

I 2)1

=(-

1)'/2

'

'

\

-

i l

n

I

(n/2)

1

(nl2)

l

~ 2 )

+(-l) ' (n+l)C ,J '

=

2(nl2

) (n/2)

(-IJ"; '

n

(n+l)

fl

(n

/2J'(n/

2) 2

=(-IJ

"' '(f l+1

)

W

e Jmuw

th at

la+b)

' +(

a

-

b) '=

5

C

0

a +

5

C

1

a

4

b

•

+

'c,a-'b

1

+

'c,a'h'

+

'C,ah'

+ 'c,b

'

1

0

a'- 'cf

a'b

'

• ' ,,,

'

'

2

a

-

a

+

'c,a

b

4

-

'c,

'

=2[a

5

+1

0u

3

b

2

+ l0ab

4

]

[x

+(r

3

-1 )

1

'

2

J'• [

x - (x ' -

tJ '" ' 'J

5

= 2 [x

5

O . ~ c x ·

) - l O . ~ ( x

-1)

2

]

Theref

ore,

the given

~ p r e s s i o

is a

polyno

m ial of

de

gree 7

.

Hence

, (c) i

s

th

e n•w

er

;

"

Letb

=

'

"

"

n -(

n -r )

'

·c,

r=

O

C

,

n -r

=na

-E--

' =

o

"C

-"- '

:.

2b=na

~ b=

-a

"

' "

T

herefore

, (c) is th

e an s we

r

[

·:

C = c

I

'

"

-

'

E. W

ehavc

••

) ~ ( l -

=

[I

+mx+ ( ~ - )

·I . .. J

[ l-m

+ l(n2

-l) x '- ...

. J

[

m(

m-1) n

(n -l l

l ,

l+

(m-n)x

+

2

+

2

-- - -mn

X

+ .

t

orm cunta

in ing

pow

er

of x 2:

3.

N

ow,

m n ~ 3

.. . ( )

{

cocfficie

m

oh

=

3 g iven)

'

'

n d -

m(m-1

) >-n(n

-T)-m

n=-6

'

'

c

m(m-

I)+ n(n-

I ) - lmn=

- 12

m

2

-

m+n

2

-n-2m

n=-1

2

•

(m-

fl)

- (m

+ "

) = -

12

=:.

m

1 1 = ~ 1

2 ~ 2 1

... (2)

Solvi

ng I I) a

nd (2)

m

=

12.

therefore (

L )

;,

the answer.

(;H

,:,J ·

. ,J

fH

:,Jl

t ,

){

,l;

·

[ ')·

f"''

]·["'

')

r-

1 r

So. an

swer

is (d )

Gh·cn

T

5

+

r,

=

0

[·.- c, +

"c ,_

, =" '

'c, (tU·.Il

1Uio)J

=:

.

C,a

-'b

4

- C,a

-

5

h

5

-0

E.=

c,

=n-4

b

"c

4

s

Therefor

e, (b) LS

tile anower

9. Acco

rd ing

to giv n co

ndition,

T =

c,

and

r , ,

1

-T

,=21

=:.

"''c

,,-

'c

, =

21


5/9

(n + )

(

n)(n- 1 - ~

n(n - l )(n-2)

-21

•

•

n

(n

-

I )

- -

· (n.,.

IJ- n -2)]=21

0

n(

n-1)·3 _

21

•

n{n-1)=42

Ther.JO

re, (b);, th e

= '-

10.

j'

l D

{

20

_1

is the coeftlci

+ t"(l +

t'J" + 1

21

{1

+ 1

2

)

12

}:

36 ' L '

(negle

cting/ (1 I I )-)

D

SU JECT

IVE QU STIONS

1.

w ~ k n o w

n e e = { -

- {- -cc

_{--{--cc

.; ' L

. L . ' ' L L

'

_ . _ , ~ ,

• -0 _;-U • -

1-0

2

2

' c '

'-)'

l' h '

LL c c =

- - - ~ =2' -'

__

_ ~ · -

' 'l

'

1

2

2(n )

2

=

cooffLcrent

of

r ( c ,

+

c ,

+

=

coeffLcient

oft'

"';, (2 +

12

C,,) .

12 l lere , ,_ 'c, =

(k' -3 )

C,+,

n - J c

~

c k - 3 ) - ~

n lc

'

r-rl '

-)

(S ince. n;,

r=

- -

:< ; l and

n

r> O)

OOj

JO

"'

' '

,.,c ,,,,, ,

n

,,.. ,

.,,,

1

,s, T

·: · · ' ' T .. .

+ 'n•isn


6/9

0

-

~

-

{

2

H

I

-

(l

+

q

)

"

T

]

}

+

.

..

.

7

(

q

;

I

J

.

1

)

.

.. (2

)

C

,

•

•

+

.

..

+

.

s

.

'•

•

'

7

"

2

)

-

1

)

c

.

I

-

J

+

..

. u

p

t

o

m

t

e

rm

s

,

•

,

'

8

(

\

"

3

1

f

] -2 :)

+[1--:J,

+ll-ij

+

..

ll ptomte rms

r

I

\

,

.

_

+

.

...

u

p

t

o

m

te

r

ll

l>

\ 8

.

'

2 '

I

(

2

J

4

.

c

~

-

z

c

,

+

J

c

,

..

..

+

(

-

l)

"

(

n

+

O

'

c

,

'

'

I

;

C

-

IJ

(

r

+

l

)"

"C

,

f}-

I

)'

(

r

•

c

,

·=

=

:f

:<

-

1

)

r

'

'C

,

+

2

f

(

-

)

' r

c

,

•=

•

=

0

'

+

}

)

-l

)

'

."

C

,

'

'

'

=

f

m

__

_1

_

~

o

''

'

"

'

n

w

,

l:

b

,

(

x

-

3

)

'

'

=

'

"

"

'

'

D

i

ff

i:

n

:n

ti

at

in

g

b

o

th

s

id

e.

v

,=

' -

-

•

'

.,

.

+

.

,+

.

..

•

·

:

J ,

~

I

V

r

<

:

n

)

=

·

·

•

'c

+

"

'

1

C

'

)+

'

c

n

+

l

•

'

+

n

,

.,

,

,

c

,

+

.

..

+

'

c

+

'

c

=

, •

•

•

.

n

•

n

'

H

+

I


7/9

6.

Since

n

i ;

a

n eve

n pos

it ive

integ

er , w

e Ca l

WTitc

n=2

m;m

=l ,2

,3,.

wit

h

k

=

3

" = J

(lm ]

=3 m

'

'

T11c

refore

,

; "

c'""

,

""

S =

(->_ l

C

1

+(-3

) C

,

(

-'

)l m-1

0"',.

+

,

F

ro m

the bin

om ia

l expa

nsion

, w

e write

(

1 +xlm

=

6

'"C

0

., '' "

C

1

x + 'r

nc,x

' +

. _

--1 I)

' c

''

c

'

' '

.

.

<

,.,

_,x

+ ,m x

..

.

(1 - x)

' '"

+

c, ( --x

)+

+ _

.

+

(

-:x)'m

-l

f-

... (

3)

eq

ua tion

(

3)

fr

om e

quati

on (2 )

, we

g

e l

..

-(I )

R

epbcm

g.-.:

b y 1/

x

.

we obt

ain

.. .

(2)

N

'

'

'

2

ow,a(i-aj+a;:-a

3

.

term

indep

en de n

t oh

- in

I

,

• I

au +a

1

x+

u

2

x '

..

. +a

,,,-

[ a ·

- ~

< ~ -

r=,

.I R--

., if +

': l=2

andl

l= rtl

3

N

ow, z

6

' =[r

{w:

.G

+isin

0} ]

6

"'

=

r'

'

(

cos e

+

-

i

sin

G)'m

=

r

6

"'

{co

s 6m

l ; s

in 6m

OJ

Agai

n

Z

=r(c

os l

-sin l

l)

an d

(zJ""

=r""

(oos

6m l

-i,in

6mll

)

"""

N

ow, e

qu atio

n ( 4)

beco

mes

2

i ,/J

2

6

rn si n

6m 0

. j j

r

"" '(2 i

s in6 m

0)

c

2

r o J3

=Oas

m

EZ , a

ld ~

11:

/J

___

(

4)

.

. . (5)

••

'

lhus,

-

"

2

"

=

(l+x

+r ' )

(

x x+l)

"

,

l(x '

x

2

]

'

=( I

+2-.:

2

+x

1

-

x

2

)

x

'

{I

+x'

x'l"

'

of

the term indepondel\t

in

)

(l

)

)

-

x

+x

x

'

c i e n

ofx

'"

in

(I -'-

x

1

-1- x' )

« ~ e f

i c i e

o

f

1"

in

1 +

1

+

I

2

l"

·c

W c

ha>e

L

t-1

)'.-

-.-'-

'="

>

oc,

n

.3

"

=

LH

J'

{n -

r ) '(

r ,J )

~

~ ~ '

'

= -

i

{ 1

/.(n+

J)

{n

+l) (

n 1 2

)(n

t 3)

c-D (n

r) (r

+W

. HJ

'

(n+l)

(n+2

)(n-t

·3) , .o

·" '

c

'

'

" +

3

~

{-n

' .". ,

c,

(n

+l)(n

12)(

n t-3)

,_,

-

3

)

c

a

n

b

e

w

r

i

t

t

e

n

a

s


8/9

-

3

n

+

\

)

n

+

2

)

n

+

3

)

~

-

-

3

o

-

-

l

+

n

+

3

)

-

~

3

)

n

+

2

)

l

2

-

3

'

_

_

(

n

+

2

)

(

2

-

n

-

3

)

(

n

-

l

)

(

n

+

2

)

(

n

+

3

)

2

-

3

2

(

n

+

3

)

9

.

T

h

i

•

r

e

s

u

lt

c

a

n

b

e

p

r

o

v

e

d

i

n

d

u

c

t

io

n

.

o

r

m

~

O

(

' -')

( 'J

.

L

[

2

~

:

k

)

~

-

~

:

:

>

·

-

,

.

=

r

~

r

·

[

'

;

)

L

.

H

.

S

.

=

(

2

:

(

1

=

(

~

'

f

~

R

t

t

s

.

[

·

:

k

.

.

.

(

l

)

A

s

;

u

m

i

n

g

t

h

a

t

\

h

e

r

e

s

u

lt

h

o

l

d

s

f

o

r

a

l

l

n

o

n

-

n

e

g

a

t

i

v

e

i

n

«

:

g

e

r

s

m

<

n

.

T

h

a

t

;

,

,

a

s

s

u

m

<

:

l

ln

.

t

(

-

1

~

k

)

2

n

-

4

k

+

t

\

n

-

»

•

=

C

k

l

n

-

2

.

1

:

+

t

(

:

)

,

-

,

~

~

)

2

"

.

,

.

.

.

..

(

2

)

12n

-

2

m

n

-

m

.

.

.

(

3

)

N

o

w

.

L

.

J

-

.S

.

o

f

(

[

,

_

'

J

'

t

>

~

:

]

-

2

_

,

.

1

'2

n

-

(

m

+

(

m

i

-

l

)

)

C

n

-

4

(

m

+

l

)

+

l

]

• _

x

z

•

-

l

{

M

T

I

)

{

2

n

-

2

{

m

+

I

)

1

I

}

,

~

[

:

)

_

2

,

_

o

_

,

,

_

(

~

=

~

)

e

n

-

:

-

\

(

2

n

-

2

m

-

l

(

2

)

)

n

m

(

n

m

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