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

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 $\_\_\_\_$ .

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 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 the coefficients of the middle terms in the expansion of $\left( \frac { 1 } { \sqrt { 6 } } + \beta x \right) ^ { 4 } , ( 1 - 3 \beta x ) ^ { 2 }$ and $\left( 1 - \frac { \beta } { 2 } x \right) ^ { 6 } , \beta > 0$, respectively form the first three terms of an A.P. If $d$ is the common difference of this A.P., then $50 - \frac { 2 d } { \beta ^ { 2 } }$ is equal to $\_\_\_\_$.

Let the sum of the coefficient of first three terms in the expansion of $\left( x - \frac { 3 } { x ^ { 2 } } \right) ^ { n } ; x \neq 0 , n \in N$ be 376. Then, the coefficient of $x ^ { 4 }$ is equal to:

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2} - \frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3l}$ is $2^\alpha \beta$ where $\beta < 0$ is an odd number, then $|\alpha l - \beta|$ is equal to $\_\_\_\_$.

If the coefficients of $x^{7}$ in $\left(ax^{2} + \frac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax - \frac{1}{3bx^{2}}\right)^{11}$ are equal, then
(1) $729ab = 32$
(2) $32ab = 729$
(3) $64ab = 243$
(4) $243ab = 64$

If the coefficients of $x$ and $x ^ { 2 }$ in $( 1 + x ) ^ { p } ( 1 - x ) ^ { q }$ are 4 and $-5$ respectively, then $2p + 3q$ is equal to
(1) 60
(2) 69
(3) 66
(4) 63

Let $\left( a + b x + c x ^ { 2 } \right) ^ { 10 } = \sum _ { i = 10 } ^ { 20 } p _ { i } x ^ { i } , a , b , c \in \mathbb { N }$. If $p _ { 1 } = 20$ and $p _ { 2 } = 210$, then $2 ( a + b + c )$ is equal to
(1) 6
(2) 15
(3) 12
(4) 8

Let the coefficients of three consecutive terms in the binomial expansion of $( 1 + 2 x ) ^ { \mathrm { n } }$ be in the ratio $2 : 5 : 8$. Then the coefficient of the term, which is in the middle of these three terms, is

If the co-efficient of $x ^ { 9 }$ in $\left( \alpha x ^ { 3 } + \frac { 1 } { \beta x } \right) ^ { 11 }$ and the co-efficient of $x ^ { - 9 }$ in $\left( \alpha x - \frac { 1 } { \beta x ^ { 3 } } \right) ^ { 11 }$ are equal, then $( \alpha \beta ) ^ { 2 }$ is equal to

If the coefficients of three consecutive terms in the expansion of $( 1 + x ) ^ { n }$ are in the ratio $1 : 5 : 20$ then the coefficient of the fourth term is
(1) 2436
(2) 5481
(3) 1827
(4) 3654

If the coefficient of $x^7$ in $\left(ax - \frac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax + \frac{1}{bx^2}\right)^{13}$ are equal, then $a^4 b^4$ is equal to:
(1) 11
(2) 44
(3) 22
(4) 33

Let $a$ be the sum of all coefficients in the expansion of $\left( 1 - 2 x + 2 x ^ { 2 } \right) ^ { 2023 } \left( 3 - 4 x ^ { 2 } + 2 x ^ { 3 } \right) ^ { 2024 }$ and $b = \lim _ { x \rightarrow 0 } \frac { \int _ { 0 } ^ { x } \frac { \log ( 1 + t ) } { t ^ { 2024 } + 1 } dt } { x ^ { 2 } }$. If the equations $c x ^ { 2 } + d x + e = 0$ and $2 b x ^ { 2 } + a x + 4 = 0$ have a common root, where $c , d , e \in R$, then $d : c : e$ equals
(1) $2 : 1 : 4$
(2) $4 : 1 : 4$
(3) $1 : 2 : 4$
(4) $1 : 1 : 4$

${ } ^ { n - 1 } C _ { r } = \left( k ^ { 2 } - 8 \right) ^ { n } C _ { r + 1 }$ if and only if:
(1) $2 \sqrt { 2 } < k \leq 3$
(2) $2 \sqrt { 3 } < \mathrm { k } \leq 3 \sqrt { 2 }$
(3) $2 \sqrt { 3 } < \mathrm { k } < 3 \sqrt { 3 }$
(4) $2 \sqrt { 2 } < \mathrm { k } < 2 \sqrt { 3 }$

Suppose $28 - p,\ p,\ 70 - \alpha,\ \alpha$ are the coefficient of four consecutive terms in the expansion of $(1 + x)^n$. Then the value of $2\alpha - 3p$ equals
(1) 7
(2) 10
(3) 4
(4) 6

If the coefficients of $x ^ { 4 } , x ^ { 5 }$ and $x ^ { 6 }$ in the expansion of $( 1 + x ) ^ { n }$ are in the arithmetic progression, then the maximum value of $n$ is:
(1) 7
(2) 21
(3) 28
(4) 14

Let $0 \leq \mathrm { r } \leq \mathrm { n }$. If ${ } ^ { \mathrm { n } + 1 } \mathrm { C } _ { \mathrm { r } + 1 } : { } ^ { n } \mathrm { C } _ { \mathrm { r } } : { } ^ { \mathrm { n } - 1 } \mathrm { C } _ { \mathrm { r } - 1 } = 55 : 35 : 21$, then $2 \mathrm { n } + 5 \mathrm { r }$ is equal to:
(1) 50
(2) 62
(3) 55
(4) 60

If the term independent of $x$ in the expansion of $\left( \sqrt { \mathrm { a } } x ^ { 2 } + \frac { 1 } { 2 x ^ { 3 } } \right) ^ { 10 }$ is 105 , then $\mathrm { a } ^ { 2 }$ is equal to : (1) 2 (2) 4 (3) 6 (4) 9

If in the expansion of $( 1 + x ) ^ { \mathrm { p } } ( 1 - x ) ^ { \mathrm { q } }$, the coefficients of $x$ and $x ^ { 2 }$ are 1 and $-2$, respectively, then $\mathrm { p } ^ { 2 } + \mathrm { q } ^ { 2 }$ is equal to :
(1) 18
(2) 13
(3) 8
(4) 20

For some $n \neq 10$, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $(1+x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+x)^{n+4}$ is:
(1) 20
(2) 10
(3) 35
(4) 70

Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1 + x)^{2\mathrm{n}-1}$. If $2\mathrm{A} = 5\mathrm{B}$, then n is equal to:
(1) 22
(2) 20
(3) 21
(4) 19

Let $n$ be a natural number. Given that the arithmetic mean of all coefficients in the expansion of
$$\left( x ^ { 3 } - \frac { 2 } { x ^ { 2 } } \right) ^ { n }$$
is 0.2, what is the coefficient of the $x ^ { 2 }$ term in this expansion?
A) 12
B) 16
C) 24
D) 32
E) 40

Let $a$ be a positive real number,
$$\left(x + \frac{a - 7}{x}\right)^{13}$$
In the expansion of this expression, the coefficient of the $x^{11}$ term is $\frac{234}{a}$.
Accordingly, what is $a$?
A) 9 B) 12 C) 13 D) 15 E) 18