The sum of coefficients of integral powers of $x$ in the binomial expansion of $( 1 - 2 \sqrt { x } ) ^ { 50 }$ is
(1) $\frac { 1 } { 2 } \left( 2 ^ { 50 } + 1 \right)$
(2) $\frac { 1 } { 2 } \left( 3 ^ { 50 } + 1 \right)$
(3) $\frac { 1 } { 2 } \left( 3 ^ { 50 } \right)$
(4) $\frac { 1 } { 2 } \left( 3 ^ { 50 } - 1 \right)$

If $A$ and $B$ are coefficients of $x^n$ in the expansions of $(1+x)^{2n}$ and $(1+x)^{2n-1}$ respectively, then $\frac{A}{B}$ equals:
(1) $1$
(2) $2$
(3) $\frac{1}{2}$
(4) $\frac{1}{n}$

If the number of terms in the expansion of $\left(1 - \frac{2}{x} + \frac{4}{x^2}\right)^n$, $x \neq 0$, is 28, then the sum of the coefficients of all the terms in this expansion, is:
(1) 64
(2) 2187
(3) 243
(4) 729

The sum $\sum_{r=1}^{9} \frac{10!}{r!(10-r)!}$ is equal to:
(1) $2^{10} - 2$
(2) $2^{10} - 1$
(3) $2^9$
(4) $2^{10}$

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) $\frac { 4 } { 5 }$

The value of ${}^{21}C_1 - {}^{10}C_1 + {}^{21}C_2 - {}^{10}C_2 + {}^{21}C_3 - {}^{10}C_3 + {}^{21}C_4 - {}^{10}C_4 + \ldots + {}^{21}C_{10} - {}^{10}C_{10}$ is
(1) $2^{21} - 2^{11}$
(2) $2^{21} - 2^{10}$
(3) $2^{20} - 2^{9}$
(4) $2^{20} - 2^{10}$

The sum of the co-efficient of all odd degree terms in the expansion of $\left( x + \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } + \left( x - \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } , ( x > 1 )$ is
(1) 2
(2) - 1
(3) 0
(4) 1

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 $x ^ { 2 }$ in the expansion of the product $\left( 2 - x ^ { 2 } \right) \left\{ \left( 1 + 2 x + 3 x ^ { 2 } \right) ^ { 6 } + \left( 1 - 4 x ^ { 2 } \right) ^ { 6 } \right\}$ is
(1) 107
(2) 108
(3) 155
(4) 106

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 $\sum _ { r = 0 } ^ { 25 } \left\{ \left( { } ^ { 50 } C _ { r } \right) \left( { } ^ { 50 - r } C _ { 25 - r } \right) \right\} = K \left( { } ^ { 50 } C _ { 25 } \right)$, then $K$ is equal to
(1) $2 ^ { 25 }$
(2) $2 ^ { 25 } - 1$
(3) $2 ^ { 24 }$
(4) $( 25 ) ^ { 2 }$

If $\sum _ { i = 1 } ^ { 20 } \left( \frac { { } ^ { 20 } C _ { i - 1 } } { { } ^ { 20 } C _ { i } + { } ^ { 20 } C _ { i - 1 } } \right) ^ { 3 } = \frac { k } { 21 }$, then $k$ equals
(1) 200
(2) 100
(3) 50
(4) 400

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 third term in the binomial expansion of $\left( 1 + x ^ { \log _ { 2 } x } \right) ^ { 5 }$ equals 2560, then a possible value of $x$ is
(1) $4 \sqrt { 2 }$
(2) $\frac { 1 } { 8 }$
(3) $2 \sqrt { 2 }$
(4) $\frac { 1 } { 4 }$

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