JEE Questions Binomial
Transcript of JEE Questions Binomial
-
8/16/2019 JEE Questions Binomial
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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
-
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(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
-
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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
.
-
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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
-
8/16/2019 JEE Questions Binomial
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(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
-
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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
-
8/16/2019 JEE Questions Binomial
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
-
8/16/2019 JEE Questions Binomial
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
=
(
~
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f
~
R
t
t
s
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[
·
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k
.
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.
(
l
)
A
s
;
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m
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t
h
a
t
\
h
e
r
e
s
u
lt
h
o
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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
)
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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
}
,
~
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