The sum of the co-efficients of all even degree terms in $x$ in the expansion of $\left(x + \sqrt{x^3 - 1}\right)^6 + \left(x - \sqrt{x^3 - 1}\right)^6$, $x > 1$ is equal to
(1) 26
(2) 32
(3) 24
(4) 29

If the fourth term in the Binomial expansion of $\left( \frac { 2 } { x } + x ^ { \log _ { 8 } x } \right) ^ { 6 } , ( x > 0 )$ is $20 \times 8 ^ { 7 }$, then a value of $x$ is
(1) $8 ^ { - 2 }$
(2) 8
(3) $8 ^ { 3 }$
(4) $8 ^ { 2 }$

If the fourth term in the binomial expansion of $\sqrt { x ^ { \frac { 1 } { 1 + \log _ { 10 } x } } } + x ^ { \frac { 1 } { 12 } }$ is equal to 200 , and $x > 1$, then the value of $x$ is
(1) 100
(2) $10 ^ { 4 }$
(3) $10 ^ { 3 }$
(4) 10

If some three consecutive coefficients in the binomial expansion of $( x + 1 ) ^ { n }$ in powers of $x$ are in the ratio $2 : 15 : 70$, then the average of these three coefficients is:
(1) 227
(2) 964
(3) 625
(4) 232

The coefficient of $t^4$ in the expansion of $\left(\frac{1-t^6}{1-t}\right)^3$ is
(1) 10
(2) 14
(3) 15
(4) 12

The total number of irrational terms in the binomial expansion of $\left( 7 ^ { \frac { 1 } { 5 } } - 3 ^ { \frac { 1 } { 10 } } \right) ^ { 60 }$ is
(1) 48
(2) 55
(3) 54
(4) 49

If $\alpha$ and $\beta$ be the coefficients of $x^{4}$ and $x^{2}$, respectively in the expansion of $\left(x + \sqrt{x^{2} - 1}\right)^{6} + \left(x - \sqrt{x^{2} - 1}\right)^{6}$, then
(1) $\alpha + \beta = 60$
(2) $\alpha + \beta = -30$
(3) $\alpha - \beta = 60$
(4) $\alpha - \beta = -132$

If the term independent of $x$ in the expansion of $\left( \frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 3 x } \right) ^ { 9 }$ is $k$, then $18k$ is equal to:
(1) 11
(2) 5
(3) 9
(4) 7

If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of $( 1 + x ) ^ { n + 5 }$ are in the ratio $5 : 10 : 14$, then the largest coefficient in the expansion is:
(1) 462
(2) 330
(3) 792
(4) 252

If the constant term in the binomial expansion of $\left(\sqrt{\mathrm{x}}-\frac{\mathrm{k}}{\mathrm{x}^{2}}\right)^{10}$ is 405, then $|\mathrm{k}|$ equals:
(1) 9
(2) 1
(3) 3
(4) 2

The coefficient of $x ^ { 7 }$ in the expression $( 1 + x ) ^ { 10 } + x ( 1 + x ) ^ { 9 } + x ^ { 2 } ( 1 + x ) ^ { 8 } + \ldots + x ^ { 10 }$, is
(1) 210
(2) 330
(3) 120
(4) 420

Let $\alpha > 0, \beta > 0$ be such that $\alpha^{3} + \beta^{2} = 4$. If the maximum value of the term independent of $x$ in the binomial expansion of $\left(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}}\right)^{10}$ is $10k$, then $k$ is equal to
(1) 336
(2) 352
(3) 84
(4) 176

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

If $\{ \mathrm { p } \}$ denotes the fractional part of the number p , then $\left\{ \frac { 3 ^ { 200 } } { 8 } \right\}$ is equal to
(1) $\frac { 5 } { 8 }$
(2) $\frac { 7 } { 8 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 8 }$

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

The coefficient of $x ^ { 256 }$ in the expansion of $( 1 - x ) ^ { 101 } \left( x ^ { 2 } + x + 1 \right) ^ { 100 }$ is:
(1) ${ } ^ { 100 } C _ { 16 }$
(2) ${ } ^ { 100 } C _ { 15 }$
(3) ${ } ^ { - 100 } C _ { 16 }$
(4) ${ } ^ { - 100 } C _ { 15 }$