Binomial Theorem (Solved MCQs and Notes)

Notes 33 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

231
Unit - 6
Binomial Theorem
Important Points
• Cr = nCr =
 
n
r
=
 
!
! !
n
r n r
• For x, y

R and n

N
(x+y)
n
=
n
C
o
x
n
y
o
+
n
C
1
x
n-1
y + n c
2
x
n-2
y
2
+---+ n c
n
x
o
y
n
=
0
n
r 

nc
r
x
n-r
y
r
e.g
10
0r 

10c
r
2
10-r
(-5)
4
=(2-5)
10
= 3
10
(r+1) th term in expansion of (x+y)
n
is
Tr+1 = nc
r
x
n-r
y
r
• For x,y

R and n

N
(x-y)
n
= n C
o
x
n
y
o
- nc
1
x
n-1
y
1
+ nc
2
x
n-2
y
2
+---+(-1)
n
nc
n
x
o
y
n
nCr =
0
n
r 

(-1)
r
nc
r
x
n-r
y
r,
(r+1) th term in expansion of (x -y)
n
is Tr+1 = (-1)
n
nC
r
x
n-r
y
r
• Number of terms in exprression of (x+y)
n
or (x-y)
n
is n+1.
for each term, the sum of power of x and y is n
• MIDDLE TERM(S)
Page 1
232
If n is even then middle term is
n
1
2
 

 
 
th term It is given by
n
2
n
2
xnC
If n is odd then middle term is
n+1
2
 
 
 
th term and
n+3
2
 
 
 
th term. These are given by
(n-1) (n+1) (n+1) (n-1)
2 2 2 2
(n-1) (n+1)
2 2
n C x y and n C x y
• The Greatest Co-efficient
• If n is even, the greatest co-efficient in the expansion of (x+y)
n
is
n
2
n
C
• If n is odd, there are two greatest co-efficient in the expansion of (x+y)
n
These are
(n-1)
2
n
c
and
(n+1)
2
n
c
• The Greatest Term
If x>o,y>o,n

N then to find the greatest term in the expansion of (x+y)
n
find
K=
(n+1)y
x+y
.
if k is an integer then the expansion of (x+y)
n
has two greatest terms is, Kth and (k+1)
th terms.
if k is not an integer then the expansion of (x+y)
n
has just one greatest term and it is
given by T

+1.where

=[K].
• (x+y)
n
+ (x-y)
n
= 2 {nC
0
x
n
y
o
+ n C
2
x
n-2
y
2
+nC
4
x
n-4
y
4
+---+
1
2
r 
 

 
 
 
 
th term
Page 2
233
(x+y)
n
- (x-y)
n
= 2 {nC
1
x
n-1
y + n C
3
x
n-3
y
3
+---+
1
2
n 
 
 
 
 
 
th term}
e.g
   
9 9
9
1+ 2x + 1- 2x has +1=4+1=5
2
 
 
 
terms.
• (1+x)
n
= nC
o
x
o
+ nC
1
x
1
+ nC
2
x
2
+---+ n C
n
x
n
; n

N
- Co-efficient of x
r
in expasion of (1+x)
n
is =


n
r
= n C
r
- Co-efficient of (r+1)th term in expansion of (1+x)
n
is =


n
r
= n C
r
• (a
1
+a
2
+---+
a
m)
n
=
1 2
1 2
2 m
n
......
n! n !...n !
m
n
n n
m
a a a


where n
1
, n
2
, n
3
, ......n
m


N and
n
1
+ n
2
+ n
3
+.....+n
m
= n
Numbers of terms in expansion = (n + m - 1)
C
(m-1)
e.g. Number of terms in expansion of (x+y+z)
n
=
(n+1) (n+2)
2
sum of coefficients in
(x+y+z)
n
= 3
n
• Some properties of the Binomial coefficients
(1) n Co + n C
1
+n C
2
+......+n C
n
= 2
n
(2) n Co + n C
2
+ n C
4
+......+ = 2
n -1
(3) n C
1
+ n C
3
+ n C
5
+.....+n C
n
= 2
n
-1
(4) If a
o
, a
1
, a
2
, .......... a
n
are in A.P then
a
o
n c
o
+ a
1
n c
1
+ a
2
n c
2
+ .......+ a
n

n
c
n
= (a
o
+ a
n
) 2
n-1
(5) n C
o
- n C
1
+ n C
2
-.......+ (-1)
n
n C
n
= o
Page 3
234
(6) If three consecutive binomial co-efficients n C
r-1,
n C
r,
n C
r+1
are in A.P then r =
 
1
n+ n+2
2
(7) Co-efficient of x
n
in (1+x)
2n
=2 {co efficient of x
n
in (1+x)
2n-1
}
(8) co-efficient of
1
x
in expansion of (1+x)
n

1 (2n)!
1
(n-1)i(n+1)!
n
x
 
 
 
 
• To find remainder when x
n
is divideel by y, try to express x or some power of x as
ky

1 and then apply binomial theorem.
Page 4
235
Question Bank
1. If coefficients of x
7
and x
8
are equal in expansion of
2
3
n
x
 

 
 
then n = _______
(a) 55 (b) 56 (c) 54 (d) 58
2. The constant term in expansion of
9
2
3 1
, o is
2 3
x
x
x
 
 
 
 
 
(a)
5
18
(b)
7
18
(c)
5
17
(d)
7
17
3. Coefficients of middle terms in expansion of
7
3
2
3
x
 

 
 
 
are...
(a)
560 280
- , -
27 81
(b)
560 280
, -
27 81
(c)
560 280
- ,
27 81
(d)
560 280
,
27 81
4. Middle term in expansion of
12
2
3xy
x
 

 
 
is_____
(a) 14370048 y
6
(b) 14370024 y
6
(c) 43110144 y
6
(d) 43110124 y
6
5 If middle term is Kx
m
in expansion of
12
1
x
x
 

 
 
then m =_____
(a) -2 (b) -1 (c) 0 (d) 1
Page 5
236
6. Co efficient of middle term in expansion of
8
3
5
x
x
 

 
 
 
=_____
(a)
14
625
(b)
70
625
(c)
14
125
(d)
70
125
7. Index number of middle term in expansion of
2
1
4
n
a
a
 
 
 
 
 
is ______
(a)
n
1
2

(b)
n+1
2
(c) n+1 (d)
n+3
2
8. In the expansion of (x+y)
13
the co efficients of 3 rd term and ____th terms are equal.
(a) 12 (b) 11 (c) 8 (d) 13
9. In the expansion of (x-y)
10
, (co efficient of x
7
y
3
) - (co-efficient of x
3
y
7
) =_____
(a) 10 C
7
(b) 2 .10 C
7
(c) 10 C
7
+ 10 C
1
(d) 0
10. In the expansion of
35
2 1
5 3
a b
 
 

 
 
a

b, the number of terms in which the power
of a and b are integers are ______
(a) 1 (b) 2 (c) 3 (d) 4
11. 6 th term in the expansion of
1 0
8
2
8
3
1
+ x l o g x
x
 
 
 
 
 
is 5600 then x =_____
(a) 2 (b)
5
(c)
10
(d) 10
12. If p and Q are coefficients of x
n
in expansion of (1+x)
2n
and (1+x)
2n-1
then_____
(a) P = Q (b) P = 2 Q (c) 2P = Q (d) P + Q = 0
Page 6
237
13 The constant term in expansion of
10
2 1 1
3 3 2
1 1
1
x x
x x
x x
 


 
 
 
 
 
 
is ______
(a) 210 (b) 105 (c) 70 (d) 35
14 If w

1 is quberoot of 1 then
100
r=0
100

c
r
(2+w
2
)
100-r
w
r
=____
(a) -1 (b) 0 (c) 1 (d) 2
15. The Co-efficient of x
3
in (1-x+x
2
)
5
is
(a) -30 (b) -20 (c) -10 (d) 30
16. It coefficients of middle terms in expansion of (1+

x)
8
and (1-

x)
6
are equal then

=__
(a)
2
7
(b)
2
7

(c)
3
7

(d) None of these
17. (1-x)
m
(1+x)
n
= 1+a
1
x + a
2
x
2
+......; m,n

N and a
1
=a
2
=10 then (m,n) =______
(a) (45, 35) (b) (35, 20) (c) (35, 45) (d) (20, 45)
18. The expansion of
5 5
3 3
1 1x x x x
   
    
   
   
is a plynomial of degree
(a) 5 (b) 6 (c) 7 (d) 8
19. The Co-efficient of x
r
in expansion of S =(x+3)
n-1
+ (x+3)
n-2
(x+2) + (x+3)
n-3
(x+2)
2
+...+ (x+2)
n-1
is.................... .
(a) nCr (3
r
- 2
r
) (b) nCr (3
n-r
-2
n-r
)(c) 3
n-r
- 2
n-r
(d) 3
n-r
+ 2
n-r
20. Number of terms in expansion of
   
10 10
x + y + x - y is.................... .
(a) 5 (b) 6 (c) 7 (d) 8
Page 7
238
21. The co-efficient of middle term in expansion of (2x - 3y)
4
is.................... .
(a) -96 (b) 216 (c) -216 (d) 96
22. Constant term in expansion of
4
x 2
1+ -
2 x
 
 
 
is.................... .
(a) 5 (b) -5 (c) 4 (d) -4
23.
   
5 5
2+1 + 2 1 =________
(a) 58 (b)
58 2
(c) -58 (d)
58 2
24. If sum of Even terms are denoted by E and sum of odd terms are denoted by O in
expansion of (x + a)
n
then O
2
- E
2
=_______
(a) x
2
- a
2
(b) (x
2
- a
2
)
n
(c) x
2n
- a
2n
(d) None of these
25.
n
r
n r
r=o
4 _____C 

(a) 4
n
(b) 5
n
(c) 4
-n
(d) 5
-n
26. (10.1)
5
=_______
(a) 105101.501 (b) 105101.0501
(c) 105101.00501 (d) 105101.05001
27. If a,b

N, a

b then for r

N, a
n
-b
n
is divisible by ________
(a) a-b (b) b-a (c) both(a)and(b) (d) None of these
28. (1.1)
10000
______1000
(a) > (b) < (c) = (d) None of these
29. 10th term in expansion of
25
2
2
1
2x
x
 

 
 
is...........
Page 8
239
(a)
2
1760
x
(b)
3
1760
x
(c)
2
880
x
(d)
3
880
x
30. The 11th term from last, in expansion of
25
2
1
2x
x
 

 
 
is
(a)
10
15
20
2
25 C
x
(b)
10
15
20
2
-25 C
x
(c)
11
14
11
2
-25 C
x
(d)
11
14
11
2
25 C
x
31. 16th term and 17th term are equal in expansion of (2+x)
40
then x=______
(a)
17
24
(b)
17
12
(c)
34
13
(d)
34
15
32. 3
rd
term in expansion of
5
log
1
10
x
x
x
 

 
 
is 1000 then x=______
(a) 10 (b) 100 (c) 1000 (d) None
33.
6
1
1
1
1 2
lo g x
x x

 
 

 
 
 
has 200 is as 4th term and x>1 then x = ______
(a) 10 (b) 100 (c) 1000 (d) None
34.
1
1
3
3
10
1
log (5 1)
log 25 7
8
3 3
x
x


 

 
 

 
 
 
has 180 is as a 9th term then x =______
(a) 1 (b) 2 (c) 3 (d) 0
Page 9
240
35. If the sum of co-efficient of first three terms in expansion of
2
3
m
a
a
 

 
 
, m

N, a

o
is 559 then m = _____
(a) 10 (b) 11 (c) 12 (d) 13
36. Co-efficient of x
5
in expansion of (1+2x)
6
(1 - x)
7
is......................... .
(a) 150 (b) 171 (c) 192 (d) 161
37. Constant term in expansion of
10
2 1 1
3 3 2
x+1 x-1
-
x-x
x -x +1
 
 
 
 
 
is........................... .
(a) 190 (b) 200 (c) 210 (d) 220
38. If the ratio of co-efficients of three consecutive terms in expansion of (1+x)
n
is
1 : 7 : 42 then n = ______
(a) 35 (b) 45 (c) 55 (d) 65
39. If n is even natural number then middle term in expansion of
1
n
x
x
 

 
 
is............... .
(a)
n !
n
!
2
 
 
 
(b)
2
n !
n
!
2
 
 
 
(c)
2
2(n !)
n
!
2
 
 
 
(d)
2
n !
n
2 !
2
 
 
 
40. When 5
20
is divided by 48 then remainder is______
(a) 46 (b) 47 (c) 48 (d) 49
41. The remainder when 2
3n
-7n+4 is divided by 49 is ................ .
(a) 0 (b) 1 (c) 4 (d) 5
Page 10

Binomial Theorem (Solved MCQs and Notes)

Binomial Theorem (Solved MCQs and Notes)

Download this file to view remaining 23 pages