The ratio of the coefficient of $x ^ { 15 }$ to the term independent of $x$ in the expansion of $\left( x ^ { 2 } + \frac { 2 } { x } \right) ^ { 15 }$ is:
(1) $7 : 16$
(2) $7 : 64$
(3) $1 : 4$
(4) $1 : 32$

The term independent of $x$ in the expansion of $\left(\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right)^{10}$ is
(1) 210
(2) 310
(3) 4
(4) 120

The number of terms in the expansion of $( 1 + x ) ^ { 101 } \left( 1 - x + x ^ { 2 } \right) ^ { 100 }$ in powers of $x$ is
(1) 301
(2) 302
(3) 101
(4) 202

The coefficient of $x ^ { 1012 }$ in the expansion of $\left( 1 + x ^ { n } + x ^ { 253 } \right) ^ { 10 }$, (where $n \leq 22$ is any positive integer), is
(1) ${ } ^ { 253 } C _ { 4 }$
(2) ${ } ^ { 10 } C _ { 4 }$
(3) $4 n$
(4) 1

If the coefficients of $x ^ { - 2 }$ and $x ^ { - 4 }$, in the expansion of $\left( x ^ { \frac { 1 } { 3 } } + \frac { 1 } { 2 x ^ { \frac { 1 } { 3 } } } \right) ^ { 18 } , ( x > 0 )$, are $m$ and $n$ respectively, then $\frac { m } { n }$ is equal to
(1) 27
(2) 182
(3) $\frac { 5 } { 4 }$
(4) $\frac { 4 } { 5 }$

If $n$ is the degree of the polynomial, $\left[ \frac { 2 } { \sqrt { 5 x ^ { 3 } + 1 } - \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 } + \left[ \frac { 2 } { \sqrt { 5 x ^ { 3 } + 1 } + \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 }$ and $m$ is the coefficient of $x ^ { n }$ in it, then the ordered pair $( n , m )$ is equal to
(1) $\left( 8,5 ( 10 ) ^ { 4 } \right)$
(2) $\left( 12,8 ( 10 ) ^ { 4 } \right)$
(3) $\left( 12 , ( 20 ) ^ { 4 } \right)$
(4) $\left( 24 , ( 10 ) ^ { 8 } \right)$

If $n$ is the degree of the polynomial,
$$\left[ \frac { 1 } { \sqrt { 5 x ^ { 3 } + 1 } - \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 } + \left[ \frac { 1 } { \sqrt { 5 x ^ { 3 } + 1 } + \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 }$$
and $m$ is the coefficient of $x ^ { n }$ in it, then the ordered pair ( $n , m$ ) is equal to
(1) $\left( 12 , ( 20 ) ^ { 4 } \right)$
(2) $\left( 8,5 ( 10 ) ^ { 4 } \right)$
(3) $\left( 24 , ( 10 ) ^ { 8 } \right)$
(4) $\left( 12,8 ( 10 ) ^ { 4 } \right)$

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

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) $\alpha - \beta = -132$

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 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

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

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 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

The coefficient of $x^{-6}$, in the expansion of $\left(\frac{4x}{5} + \frac{5}{2x^2}\right)^9$, is $\_\_\_\_$.

The absolute difference of the coefficients of $x ^ { 10 }$ and $x ^ { 7 }$ in the expansion of $\left( 2 x ^ { 2 } + \frac { 1 } { 2 x } \right) ^ { 11 }$ is equal to
(1) $13 ^ { 3 } - 13$
(2) $11 ^ { 3 } - 11$
(3) $10 ^ { 3 } - 10$
(4) $12 ^ { 3 } - 12$

The coefficient of $x ^ { 5 }$ in the expansion of $\left( 2 x ^ { 3 } - \frac { 1 } { 3 x ^ { 2 } } \right) ^ { 5 }$ is
(1) $\frac { 80 } { 9 }$
(2) 9
(3) 8
(4) $\frac { 26 } { 3 }$

The constant term in the expansion of $\left( 2 x + \frac { 1 } { x ^ { 7 } } + 3 x ^ { 2 } \right) ^ { 5 }$ is $\_\_\_\_$.

Let $\alpha > 0$ be the smallest number such that the expansion of $\left(x^{\frac{2}{3}} + \frac{2}{x^3}\right)^{30}$ has a term $\beta x^{-\alpha}$, $\beta \in \mathbb{N}$. Then $\alpha$ is equal to $\underline{\hspace{1cm}}$.

Let $[ t ]$ denote the greatest integer $\leq t$. If the constant term in the expansion of $\left( 3 x ^ { 2 } - \frac { 1 } { 2 x ^ { 5 } } \right) ^ { 7 }$ is $\alpha$ then $[ \alpha ]$ is equal to $\_\_\_\_$

Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3}x^{\frac{1}{3}} + \frac{1}{2x^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{n}{m}\right)^{\frac{1}{3}}$ is:
(1) $\frac{4}{9}$
(2) $\frac{1}{9}$
(3) $\frac{1}{4}$
(4) $\frac{1}{4}$

If the constant term in the expansion of $\left( \frac { \sqrt [ 5 ] { 3 } } { x } + \frac { 2 x } { \sqrt [ 3 ] { 5 } } \right) ^ { 12 } , x \neq 0$, is $\alpha \times 2 ^ { 8 } \times \sqrt [ 5 ] { 3 }$, then $25 \alpha$ is equal to :
(1) 724
(2) 742
(3) 639
(4) 693

Let $\alpha , \beta , \gamma$ and $\delta$ be the coefficients of $x ^ { 7 } , x ^ { 5 } , x ^ { 3 }$ and $x$ respectively in the expansion of $\left( x + \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } + \left( x - \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } , x > 1$. If u and v satisfy the equations $\begin{aligned} & \alpha u + \beta v = 18 \\ & \gamma u + \delta v = 20 \end{aligned}$ then $u + v$ equals :
(1) 5
(2) 3
(3) 4
(4) 8

Where $m$ and $n$ are integers,
$$\left(x^2 + 2y\right)^7$$
In the expansion of this expression, if one of the terms is $mx^ny^2$, what is the sum $m + n$?
A) 56
B) 64
C) 72
D) 86
E) 94