1.Algebra Booster

6.14 Algebra Booster
64. Find the co-efficient of x 49 in the polynomial
Ê C1ˆÊ 2 C2ˆÊ 2 C3ˆ Ê 2 C50ˆ
Á
x - x - 2 x -3 x -50 ,
Ë C ˜
¯
Á
Ë C ˜Á
¯Ë C ˜
¯ Á
Ë C ˜
¯
0 1 2 49
where C r
= 50 C r
. [Roorkee-JEE, 2001]
65. Find the sum of
C0 C1 C2
+ + + .
1.2.3 2.3.4 3.4.5
66. Find the sum of the series
1 1.3 1.3.5
1 + + + + .
3 3.6 3.6.9
x
e
2
n
67. If B0 B1x B2x Bn
x
1 - x = + + + + , prove that
1
Bn- Bn-1
=
n!
Ê 2n
ˆ Ê 2n
ˆ
68. If a = Á and b =
,
n= 1Ë(2n
- 1)! ˜
¯
Á
n=
1Ë(2n
+ 1)! ˜
¯
value of ab.
69. Find the sum of
1+ 2 1+ 2+ 3 1+ 2+ 3+
4
1 + + + + .
1! 2! 3!
( ,4)
70. Find the value of  ÊCn
ˆ e
Á =
Ë Pnn ( , ) ˜
¯ 24
.
  find the
n=
4
Ê
3n
3n
2
x ˆ Ê
-
x ˆ
71. If a = Á , b=
,
Ë(3 n)! ˜
¯
Á
Ë(3n-
2)!
˜
¯
n=
0
  and
n= 0 n=
0
Ê
3n-1
x ˆ
c = Â Á
Ë(3n-1)!
˜
,
¯
prove that a 3 + b 3 + c 3 – 3abc = 1.
72. Find the sum of
1 1 1
+ + +
1.2.3 3.4.5 5.6.7
.
73. Find the sum of
Ê1 1ˆ 1 Ê1 1ˆ 1 Ê1 1ˆ
1
1 + Á + + + + +
Ë
˜ 2 3
2 3¯ Á
4 Ë
˜
4 5¯ Á
4 Ë
˜
6 7¯
4
+ .
74. Find the sum of
Ê 2 -1ˆ Ê3- 2 2ˆ Ê5 2 - 7ˆ
1 + Á + Á ˜ +
Ë
˜
2 2 ¯ Ë 12 ¯ Á
Ë
˜
24 2 ¯
Ê17 - 12 2 ˆ
+ Á
+
Ë
˜
80 ¯
.
75. Find the sum of
5 7 9
+ + +
1.2.3 2.3.4 3.4.5
.
76. Find the sum of the series
2 1 2 5 1 2 5 8 1
1 + ◊ + ◊ ◊ + ◊ ◊ ◊
2 2
3 2 3 6 2 3 6 9 2
+ .
LEVEL IV
(Tougher Problems for JEE-
Advanced)
2
n
2n
1. If (1 + x+ x ) =Â a x , prove that
r
r = 0
a 0
+ a 3
+ a 6
+ … = a 1
+ a 4
+ a 7
+ …
a 2
+ a 5
+ a 8
+ … = 3 n – 1
2
n
2n
2. If (1 + x+ x ) =Â a x , prove that
r
r = 0
1
1 2 1 (3
n
a + a + + an-
= -an
),
2
2
n
2n
3. If (1 + x + x ) =Â a x , prove that
r
r = 0
a 0
a 1
– a 2
a 3
+ a 4
a 5
– … = 0.
r
r
r
4. Find the co-efficient of x 50 in the expression
(1 + x) 1000 + 2x(1 + x) 999 + 3x 2 (1 + x) 998 + … + 1001x 1000 .
5. Let
2 2
n
n+
4
(1 + x ) (1 + x)
=Â a x . If a 1
, a 2
, a 3
be in AP,
k
k = 0
find the value of n.
6. Find the co-efficient of x 18 in the expansion of
(1 + x)(1 + x + x 2 )…(1 + x + x 2 + … + x 2n ).
7. In the expansion of
(1 + x)(1 + x + x 2 )…(1 + x + x 2 + … + x 2n ),
prove that the sum of the co-efficients is (2n + 1)!.
8. Prove that the value of
n
C 0
+ n + 1 C 1
+ n + 2 C 2
+ … + n + k C k
= n + k + 1 C n + 1
.
9. If (1 + x + 2x 2 ) 20 = a 0
+ a 1
x + a 2
x 2 + … + a 40
x 40 , prove
that a 1
+ a 3
+ a 5
+ … + a 37
= 2 19 (2 20 – 21).
10. Prove that the co-efficient of x n in the expansion of
Ê
2 3
n
x x x x ˆ
Á1 + + + + +
Ë 1! 2! 3! n !
˜ is 2 n
¯ n !
.
11. If (1 + x) n = C 0
+ C 1
x + C 2
x 2 + … + C n
x n , prove that
 Â
0£< i j£
n
2
k
2 2n
2n
i - j = + n -
( C C ) ( n 1) C 2 .
12. If (1 + x) n = C 0
+ C 1
x + C 2
x 2 + … + C n
x n , prove that
Ê n-1 1 2n
ˆ
 Â( i + j) CC i j = nÁ2 - Cn˜.
Ë 2 ¯
0£< i j£
n
13. If (1 + x) n = C 0
+ C 1
x + C 2
x 2 + … + C n
x n , prove that
 Â
0£< i j£
n
2n
2n
r
r
r= 0 r=
0
n j n
j i
C C = 3 - 1, i£
j
r
14. If Âa ( x- 2) = Â br( x-3)
and a k
= 1, for all k ≥ n,
show that b n
= 2n + 1 C n + 1
.