1.Algebra Booster

6.20 Algebra Booster
(a) 0
(c) (–1) n (n + 2)
(b) (-1)
(d) none
n+ 1
2
n
2 n
n-
2
 k Ck
= n( n+ 1)2 , n>
1.
k = 0
[IIT-JEE, 1986]
19. Prove that
[IIT-JEE, 1986]
20. Prove by the mathematical induction that
(2 n)! 1
£
2
for every positive integer n.
(2 n)! ¥ ( n) (3n+
1)
[IIT-JEE, 1987]
21. Let
2 1
R (5 5 11) n +
= + and f = R – [R] where [,] denotes
the greatest integer function, prove that
Rf = 4 2n + 1 [IIT-JEE, 1988]
22. Prove that
m
C 0
◊ n C k
+ m C 1
◊ n C k – 1
+ m C 2
◊ n C k – 2
+ … + m C k
◊ n C 0
= m + n C k
,
where m, n, k are positive integer and P C q
= 0, p < q.
[IIT-JEE, 1989]
23. Prove that
C 0
– 2 2 C 1
+ 3 2 C 2
– … + (–1) n (n + 1) 2 C n
= 0,
where n > 1 and C r
= n C r
[IIT-JEE, 1989]
Ê
7 5 3
24. Prove that
n n 2 n n ˆ
Á + + - ˜ is an integer for
Ë 7 5 3 105¯ every positive integer n. [IIT-JEE, 1990]
25. The product of n positive numbers is unity. Then their
sum is
(a) a positive integer (b) divisible by n
1
(c) equal to n + (d) never less than n
n
[IIT-JEE, 1991]
26. Prove that for any non-negative integer sm, n, r and k
k
Ê( r + m)!
ˆ ( r + k + 1)! È n k ˘
 ( n- m)
Á
m=
0
Ë k!
˜ = ¥
¯ k! Í -
r + 1 r + 2˙
Î
˚
[IIT-JEE, 1991]
27. If the sum of the co-efficients in the expansion of
(a 2 x 2 – 2ax + 1) 51 vanishes, the value of a is
(a) 2 (b) –1 (c) 1 (d) –2
[IIT-JEE, 1991]
28. If
2n
2n
r
r
r= 0 r=
0
r
Âa ( x- 2) = Â br( x-3)
and a k
= 1 for all k ≥ n,
show that b n
= 2n + 1 C n + 1
.
[IIT-JEE, 1992]
29. The co-efficient of x 53 in the expansion of
100
100
( 3) 100 - m m
 Cm
x-
2 is
m=
0
(a) 100 C 47
(b) 100 C 53
(c) – 100 C 53
(d) – 100 C 100
[IIT-JEE, 1992]
30. The expansion ( x+ 3 5
x - 1) + ( x- 3 5
x - 1) is a
polynomial of degree
(a) 5 (b) 6 (c) 7 (d) 8
[IIT-JEE, 1992]
31. The value of
C 0
+ 3 ◊ C 1
+ 5 ◊ C 2
+ 7 ◊ C 3
+ … + (2n + 1) ◊ C n
is equal to
(a) 2 (b) 2 n + n ◊ 2 n – 1
(c) 2 n (n + 1)
(d) none
[IIT-JEE, 1993]
k
r-13n
3n
32. Prove that  (- 3) C2r
-1=
0, where k = and n
r = 1
2
is an even positive integer. [IIT-JEE, 1993]
33. The largest term in the expansion of (3 + 2x) 50 , where
x = 1/5 is
(a) 5th (b) 51th (C) 7th (d) 6th
[IIT-JEE, 1993]
34 Let n be a positive integer and
(1 + x + x 2 ) n = a 0
+ a 1
x + a 2
x 2 + … + a 2n
x 2n ,
prove that
2 2 2 2 2
0 - 1 + 2 - 3 + + 2n=
n
a a a a a a
[IIT-JEE, 1994]
35. Let n be a positive integer. If the co-efficient of 2nd,
3rd and 4th terms in the expansion of (1 + x) n are in AP,
find n.
[IIT-JEE, 1994]
No questions asked in 1995 and 1996.
n
3!
r Ê Cr
ˆ
36. Prove that = Â (-1)
r + 3
2( n + 3) Á
r = 0 Ë C ˜
.
r ¯
[IIT-JEE, 1997]
37. The sum of the rational terms in the expansion of
1 1
( 22
35
)
10
+ is... [IIT-JEE, 1997]
n
Ê 1 ˆ
38. If an
= ,
Á n
r = 0Ë
C ˜
r ¯
n
Ê r ˆ
Á n
r = 0Ë
C ˜
r ¯
(a) (n – 1)a n
(b) na n
na
(c) n
2
(d) none
[IIT-JEE, 1998]
39. If in the expansion of (1 + x) m (1 – x) n , the co-efficients
of x and x 2 are 3 and –6, respectively, then m is
(a) 6 (b) 9 (c) 12 (d) 24
[IIT-JEE, 1999]
40. For a positive integer, let
1 1 1
an ( ) = 1 + + + + , then
2 3 2n
- 1
(a) a(100) £ 100 (b) a(100) > 100
(c) a(200) £ 100 (d) a(200) > 100
[IIT-JEE, 1999]
41. Let n be any positive integer. Prove that
ÊÊ2n
- kˆˆ
m ÁÁ
Ë k ˜
¯˜
Ê2n - 4k
+ 1ˆ
n-
2k
 Á ˜ ¥ ¥ 2
2k k
k = 0ÁÊ
- ˆ˜
Á
Ë2n - 2k
+ 1˜
¯
ÁÁ
k ˜˜
ËË
¯¯