1.Algebra Booster

Binomial Theorem 6.13
(ii)
Ê 1 1 1ˆ
A1
= ( n)! ¥ Á1+ + + +
Ë
˜
2 3 n¯
40. If (1 + x) n = 1 + A 1
x + A 2
x 2 + … + A n
x n and
(1 + x) n + 1 = 1 + B 1
x + B 2
x 2 + … + B n
x n , prove that
B r
= A r
+ A r – 1
.
n
Ê 1 ˆ
41. If a = Â Á n
Ë C ˜ , find the sum of
¯
42. If
43. If
n
n
ÂÂ
i= 0 j=
0
n
r = 0
r
 Â
Ê
Á
1 1
+
n n
0£< i j£
n Ë Ci Cj
n
n n r
r
r = 0
ˆ
.
˜
¯
(1 + x)
=Â C x , find the value of
( C + C ).
i
j
n
n n r
r
r = 0
(1 + x)
=Â C x , find the value of
 Â
( C + C ).
i j
0£< i j£
n
n
n n r
r
r = 0
n
n n r
(1 x)
Cr
x
r = 0
n
44. If (1 + x)
=Â C x , find the value of
45. If
46. If
 Â
0£< i j£
n
n
0£< i j£
n
n
n
ÂÂ
i= 0 j=
0
( CC ).
+ =Â , find the value of
( CC ).
i
j
n
n n r
(1 + x)
=Â C x , find the value of
 Â
r = 0
(( i ¥ j) CC ).
i
r
j
Ê
n n
n j
ˆ
47. Let Fn ( ) = ÁÂÂ
( Cj◊
Ci)
˜ , where i £ j, find the
Ëi= 0 j=
1 ¯
value of F(10).
n n n n
48. Find the value of ÂÂÂ Â (1) .
i= 0 j= 0m= 0 p=
0
49. If n is any positive integer, prove that
n 2 n 2 n 2 n 2 (2 n)!
C0 + C1 + C2
+ + Cn
= .
( n)! ¥ ( n)!
[Roorkee-JEE, 1983]
50. In the binomial expansion of (1 + y) n , where n is a natural
number, the co-efficients of the 5th, 6th and 7th
terms are in AP. Find the value of n.
[Roorkee-JEE, 1984]
8
Ê
1/8
x 1/8
51. Find the terms in the expansion of x - ˆ
Á +
Ë
˜
2 ¯
which does not contain x.
[Roorkee-JEE, 1985]
i
j
52. Find the co-efficient of x n in the series
2 2
( a + bx) ( a+ bx) ( a+
bx)
1 + + + +
1! 2! 3!
[Roorkee-JEE, 1985]
53. If A be the sum of the odd terms and B be the sum
of even terms in the expansion of (x + a) n , prove that
A 2 – B 2 = (x 2 – a 2 ) n [Roorkee-JEE, 1986]
53. Find the sum of the co-efficients in the expansion of the
binomial (5p – 4q) n , where n is a positive integer.
[Roorkee-JEE, 1987]
54 Find the sum of
3 ◊ n C 0
– 8 ◊ n C 1
+ 13 ◊ n C 2
– 18 ◊ n C 3
+ … upto (n + 1) terms
[Roorkee-JEE, 1988]
Note No questions asked in 1989
55. Find the co-efficient of x 50 in the expansion of
(1 + x) 1000 + 2x(1 + x) 999 + 3x 2 (1 + x) 998 + … + 1001x 1000 .
[Roorkee-JEE, 1990]
56. If n be a positive integer and C k
= n C k
, find the value of
n
2
3 Ê Ck
ˆ
k
Á
k = 1 ËC ˜
k - 1 ¯
 .
[Roorkee-JEE, 1991]
57. Determine the value of x in the expansion of
log10x
5
( x+ x ) , where the 3rd term is 10,000.
[Roorkee-JEE, 1992]
58. Find the value of the expression
5
47 47-
j
C4 C3
j = 1
+ Â . [Roorkee-JEE, 1993]
x
59. Find the value of x for which the 6th term in the expansion
of log(10-3 ) 5 ( x-2)log3
m
( 2 + 2 ) is equal to 21 and
the co-efficients of 2nd, 3rd and 4th terms are the 1st,
3rd and 5th terms, respectively of an AP.
[Roorkee-JEE, 1993]
3
60. Given that the 4th term in the expansion of
Ê xˆ
Á2
+
Ë
˜
8 ¯
has the maximum numerical values, find the range of
values of x for which this will be true.
[Roorkee-JEE, 1994]
Note No questions asked in 1995.
61. Let
2 2
n
n+
4
(1 + x ) (1 + x)
=Â a x . If a 1
, a 2
and a 3
be in
k
k = 0
AP, find n. [Roorkee-JEE, 1996]
62. In the expansion of (x + a) 15 , if the 11th term is the GM
of the 8th and 12th terms, which term in the expansion
is the greatest. [Roorkee-JEE, 1997]
63. Find the largest co-efficient in the expansion of
(1 + x) n , given that the sum of the terms in the expansion
is 4096. [Roorkee-JEE, 2000]
k
10