If the number of integral terms in the expansion of $\left( 3 ^ { \frac { 1 } { 2 } } + 5 ^ { \frac { 1 } { 8 } } \right) ^ { n }$ is exactly 33, then the least value of $n$ is
(1) 264
(2) 128
(3) 256
(4) 248

In the expansion of $\left( \frac { x } { \cos \theta } + \frac { 1 } { x \sin \theta } \right) ^ { 16 }$, if $l _ { 1 }$ is the least value of the term independent of $x$ when $\frac { \pi } { 8 } \leq \theta \leq \frac { \pi } { 4 }$ and $l _ { 2 }$ is the least value of the term independent of $x$ when $\frac { \pi } { 16 } \leq \theta \leq \frac { \pi } { 8 }$, then the ratio $l _ { 2 } : l _ { 1 }$ is equal to:
(1) $1 : 8$
(2) $16 : 1$
(3) $8 : 1$
(4) $1 : 16$

If the sum of the coefficients of all even powers of $x$ in the product $\left(1 + x + x ^ { 2 } + \ldots + x ^ { 2n} \right) \left(1 - x + x ^ { 2 } - x ^ { 3 } + \ldots + x ^ { 2n } \right)$ is 61, then $n$ is equal to

The coefficient of $x^4$ in the expansion of $\left(1 + x + x^2 + x^3\right)^6$ in powers of $x$, is

The natural number $m$, for which the coefficient of $x$ in the binomial expansion of $\left( x ^ { m } + \frac { 1 } { x ^ { 2 } } \right) ^ { 22 }$ is 1540, is

The sum of all those terms which are rational numbers in the expansion of $\left( 2 ^ { \frac { 1 } { 3 } } + 3 ^ { \frac { 1 } { 4 } } \right) ^ { 12 }$ is:
(1) 89
(2) 27
(3) 35
(4) 43

If the greatest value of the term independent of $x$ in the expansion of $\left( x \sin \alpha + a \frac { \cos \alpha } { x } \right) ^ { 10 }$ is $\frac { 10 ! } { ( 5 ! ) ^ { 2 } }$, then the value of $a$ is equal to:
(1) - 1
(2) 1
(3) - 2
(4) 2

If $n$ is the number of irrational terms in the expansion of $\left( 3 ^ { 1 / 4 } + 5 ^ { 1 / 8 } \right) ^ { 60 }$, then $( n - 1 )$ is divisible by :
(1) 26
(2) 30
(3) 8
(4) 7

The value of $\sum _ { r = 0 } ^ { 6 } \left( { } ^ { 6 } C _ { r } \cdot { } ^ { 6 } C _ { 6 - r } \right)$ is equal to:
(1) 1124
(2) 1324
(3) 1024
(4) 924

For the natural numbers $m , n$, if $( 1 - y ) ^ { m } ( 1 + y ) ^ { n } = 1 + a _ { 1 } y + a _ { 2 } y ^ { 2 } + \ldots + a _ { m + n } y ^ { m + n }$ and $a _ { 1 } = a _ { 2 } = 10$, then the value of $m + n$, is equal to:
(1) 88
(2) 64
(3) 100
(4) 80

If the fourth term in the expansion of $\left( x + x ^ { \log _ { 2 } x } \right) ^ { 7 }$ is 4480, then the value of $x$ where $x \in N$ is equal to:
(1) 2
(2) 4
(3) 3
(4) 1

If ${ } ^ { 20 } \mathrm { C } _ { \mathrm { r } }$ is the co-efficient of $x ^ { \mathrm { r } }$ in the expansion of $( 1 + x ) ^ { 20 }$, then the value of $\sum _ { \mathrm { r } = 0 } ^ { 20 } \mathrm { r } ^ { 2 } \left( { } ^ { 20 } \mathrm { C } _ { \mathrm { r } } \right)$ is equal to:
(1) $420 \times 2 ^ { 18 }$
(2) $380 \times 2 ^ { 18 }$
(3) $380 \times 2^{19}$
(4) $420 \times 2 ^ { 19 }$

Let $[ x ]$ denote greatest integer less than or equal to $x$. If for $n \in N , \left( 1 - x + x ^ { 3 } \right) ^ { n } = \sum _ { j = 0 } ^ { 3 n } a _ { j } x ^ { j }$, then $\sum _ { j = 0 } ^ { \left[ \frac { 3 n } { 2 } \right] } a _ { 2 j } + 4 \sum _ { j = 0 } ^ { \left[ \frac { 3 n - 1 } { 2 } \right] } a _ { 2 j + 1 }$ is equal to :
(1) 2
(2) $2 ^ { n - 1 }$
(3) 1
(4) $n$

The value of $-{ } ^ { 15 } C _ { 1 } + 2 \cdot { } ^ { 15 } C _ { 2 } - 3 \cdot { } ^ { 15 } C _ { 3 } + \ldots - 15 \cdot { } ^ { 15 } C _ { 15 } + { } ^ { 14 } C _ { 1 } + { } ^ { 14 } C _ { 3 } + { } ^ { 14 } C _ { 5 } + \ldots + { } ^ { 14 } C _ { 11 }$ is equal to
(1) $2 ^ { 14 }$
(2) $2 ^ { 13 } - 13$
(3) $2 ^ { 16 } - 1$
(4) $2 ^ { 13 } - 14$

Let $\left( 1 + x + 2 x ^ { 2 } \right) ^ { 20 } = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \ldots + a _ { 40 } x ^ { 40 }$, then $a _ { 1 } + a _ { 3 } + a _ { 5 } + \ldots + a _ { 37 }$ is equal to
(1) $2 ^ { 20 } \left( 2 ^ { 20 } - 21 \right)$
(2) $2 ^ { 19 } \left( 2 ^ { 20 } - 21 \right)$
(3) $2 ^ { 19 } \left( 2 ^ { 20 } + 21 \right)$
(4) $2 ^ { 20 } \left( 2 ^ { 20 } + 21 \right)$

If $\sum _ { k = 1 } ^ { 10 } K ^ { 2 } \left( { } ^ { 10 } C _ { K } \right) ^ { 2 } = 22000 L$, then $L$ is equal to

Let the coefficients of third, fourth and fifth terms in the expansion of $\left( x + \frac { a } { x ^ { 2 } } \right) ^ { n } , x \neq 0$, be in the ratio $12 : 8 : 3$. Then the term independent of $x$ in the expansion, is equal to $\_\_\_\_$ .

If $\left( \frac { 3 ^ { 6 } } { 4 ^ { 4 } } \right) k$ is the term, independent of $x$, in the binomial expansion of $\left( \frac { x } { 4 } - \frac { 12 } { x ^ { 2 } } \right) ^ { 12 }$, then $k$ is equal to

If $\sum_{k=1}^{31} \left({}^{31}\mathrm{C}_k\right)\left({}^{31}\mathrm{C}_{k-1}\right) - \sum_{k=1}^{30}\left({}^{30}\mathrm{C}_k\right)\left({}^{30}\mathrm{C}_{k-1}\right) = \frac{\alpha(60!)}{(30!)(31!)}$, where $\alpha \in R$, then the value of $16\alpha$ is equal to
(1) 1411
(2) 1320
(3) 1615
(4) 1855

If the constant term in the expansion of $\left( 3 x ^ { 3 } - 2 x ^ { 2 } + \frac { 5 } { x ^ { 5 } } \right) ^ { 10 }$ is $2 ^ { k } \cdot l$, where $l$ is an odd integer, then the value of $k$ is equal to
(1) 6
(2) 7
(3) 8
(4) 9

$\sum _ { i , j = 0 , i \neq j } ^ { n } { } ^ { n } C _ { i } { } ^ { n } C _ { j }$ is equal to
(1) $2 ^ { 2 n } - { } ^ { 2 n } C _ { n }$
(2) $2 ^ { 2 n - 1 } - { } ^ { 2 n - 1 } C _ { n - 1 }$
(3) $2 ^ { 2 n } - \frac { 1 } { 2 } { } ^ { 2 n } C _ { n }$
(4) $2 ^ { n - 1 } + { } ^ { 2 n - 1 } C _ { n }$

The coefficient of $x^{101}$ in the expression $(5 + x)^{500} + x(5 + x)^{499} + x^2(5 + x)^{498} + \ldots + x^{500}$, $x > 0$ is
(1) ${}^{501}C_{101} \times 5^{399}$
(2) ${}^{501}C_{101} \times 5^{400}$
(3) ${}^{501}C_{100} \times 5^{400}$
(4) ${}^{500}C_{101} \times 5^{399}$

If $p$ and $q$ are real numbers such that $p + q = 3 , p ^ { 4 } + q ^ { 4 } = 369$, then the value of $\left( \frac { 1 } { p } + \frac { 1 } { q } \right) ^ { - 2 }$ is equal to (if the full expression were available).

If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of $\left( x ^ { n } + \frac { 2 } { x ^ { 5 } } \right) ^ { 7 }$ is 939, then the sum of all the possible integral values of $n$ is

Let for the $9 ^ { \text {th } }$ term in the binomial expansion of $( 3 + 6 x ) ^ { n }$, in the increasing powers of $6 x$, to be the greatest for $x = \frac { 3 } { 2 }$, the least value of $n$ is $n _ { 0 }$. If $k$ is the ratio of the coefficient of $x ^ { 6 }$ to the coefficient of $x ^ { 3 }$, then $k + n _ { 0 }$ is equal to $\_\_\_\_$ .