1.Algebra Booster

Binomial Theorem 6.19
3. Find the sum of the co-efficient of
(1 + 5x – 3x 2 + 4x 3 – 7x 4 + x 5 ) 2017 .
4. Find the degree of the polynomial
3 5 3 5
( x + x - 1) + ( x - x -1) .
5. If the 9th term in the expansion of
x-1 log
1
3 (25 + 7)
x-
- 1/8log 3(5 + 1) 10
{3 + 3 } is 180, find x.
6. Let
n-1Ê
n
C ˆ
r
Fn ( ) = Â Á ,
n n ˜
r = 0 Cr + C
find the value of F(16).
Ë
r+
1¯
10
7. If Â Ê r ˆ
Á
20,
10 =
r = 0Ë
C ˜
r ¯
find the value of Â Ê 1 ˆ
Á 10
r = 0Ë
C ˜
r ¯
.
8. If m be unit digit of (27 50 + 18 50 ) and n is the remainder
when 7 98 is divided by 5, find the value of (m + n).
n
Ê 1ˆ
9. If the number of terms in the expansion of Áx
+ 1 +
Ë
˜
x¯
is 17, where n ΠI + , find n.
10. If (r + 1)th term in the expansion of
10
Ê x+ 1 x-1
ˆ
Á
-
2/3 1/3 1/2
Ë x - x + 1 x-
x ˜
is independent of x, find
¯
r.
11. If the co-efficients of (2r + 4)th and (r – 2)th terms in
the expansion of (1 + x) 18 are equal, find r.
12. If m be the number of rational terms in the expansion of
13. If
1/5 10
+ and n the number of integral terms in the
1/3 12
( 2 3 )
expansion of (7 + 5) , find (m + n + 2).
Â
n
8
Êr
+ 2ˆ
n
Ê2 -1 ˆ
Cr
= Á ˜ ,
Ë
Á r + 1¯
˜ Ë 6 ¯
find the value of n.
r = 0
14. If m be the value of R{1 – R + [R]}, where R = (2 + 3)
and n the unit digit of 3 100 , find the value of (m + n + 4).
15. If the number (5 25 – 3 25 ) be divisible by m, where m is
the unit digit prime number and n the sum of the coefficients
of (1 + 3x 100 – 5x 201 ) 2018 , find (m + n).
Questions asked in Previous Years’
JEE-Advanced Examinations
1. Prove that
2 2 2 2 2 2 2 2 2 2
( n C0) -( n C1) + ( n C2) -( n C3) + + ( n C n )
= (–1) n ( 2n C n
) [IIT-JEE, 1978]
2. Prove that
2 2 2 2 n-1
C1 - 2C2 + 3C3 - - 2 nC2n
= (-1) nCn.
[IIT-JEE, 1979]
3. Given positive integers r > 1, n > 2 and the co-efficients
of (3r)th and (r + 2)th terms in the binomial expansion
of (1 + x) 2n are equal, then
(a) n = 2r (b) n = 2r + 1
(c) n = 3r
(d) None
[IIT-JEE, 1980]
10
4 The value of
5
47 52-
j
C4 C3
j = 1
+ Â is equal to
(a) 47 C 5
(b) 52 C 5
(c) 52 C 4
(d) none
[IIT-JEE, 1980]
5. Let u 1
= 1, u 2
= 1, u n + 2
= u n + 1
+ u n
where n > 1.
Use mathematical induction to show that
1
È
n
n
Ê1+ 5ˆ Ê1-
5ˆ
˘
u n = ÍÁ ˜ -Á ˜ ˙, for all n > 1.
5 ÍÎË 2 ¯ Ë 2 ¯ ˙˚
[IIT-JEE, 1981]
6. Prove that 7 2n + (2 3n – 3 )(3 n – 1 ) + 2 is divisible by 25 for
any natural number n. [IIT-JEE, 1982]
7. The sum of the co-efficients of the polynomials
(1 + x – 3x 2 ) 2163 is... [IIT-JEE, 1982]
8. The co-efficient of x 99 in the polynomial
(x – 1)(x – 2)(x – 3) … (x – 100) is...
9. The larger of 99 50 + 100 50 and 101 50 is...
[IIT-JEE, 1982]
10. If (1 + ax) n = 1 + 8x + 24x 2 + …, then
a =..., and n =... [IIT-JEE, 1983]
10
11. The co-efficient of x 4 in Ê x 3 ˆ
Á -
2 ˜
is
Ë2
x ¯
(a) 405/256 (b) 504/259
(c) 450/263
(d) none
[IIT-JEE, 1983]
12. If (1 + x) n = C 0
+ C 1
x + C 2
x 2 + … + C n
x, show that the
sum of the products of the C i
s, taken two at a time, is
2n-1
1 (2 n)!
represented by  ( CC i j) = 2 - ◊ .
0£< i j£
n
2 ( n)! ¥ ( n)!
[IIT-JEE, 1983]
13. Prove that n(n 2 – 1) is divisible by 24, by mathematical
induction where n is any odd positive integer.
[IIT-JEE, 1983]
14. Given S n
= 1 + q + q 2 + … + q n and
2
n
Êq + 1ˆ Êq + 1ˆ Êq
+ 1ˆ
Dn
= 1 + Á ˜ + Á ˜ + + Á ˜ , prove that
Ë 2 ¯ Ë 2 ¯ Ë 2 ¯
n + 1
C 1
+ n + 1 C 2
S 1
+ n + 1 C 3
S 3
+ … + n + 1 C n + 1
S n
= 2 n D n
[IIT-JEE, 1984]
15. If p be a natural number, prove that p n + 1 + (p + 1) 2n – 1 is
divisible by p 2 + p + 1 for every positive integer n.
[IIT-JEE, 1984]
16. Prove that 2.7 n + 3.5 n – 5 is divisible by 24 for all natural
number n. [IIT-JEE, 1985]
17. Find the sum of the series
n
r r r
rn
È 1 3 7 15
˘
 (- 1) Cr Í + + + + m terms .
r 2r 3r 4r
˙
Î2 2 2 2
˚
r = 0
18. If C r
stands for n C r
, the sum of the series
Ênˆ Ênˆ
2 Á ! !
Ë
˜
2¯ Á
Ë
˜
2¯ ¥
2 2 n 2
[ C0 + C1
+ + ( - 1) ( n + 1) Cn
],
n!
where n is an even positive integer, is equal to