General and middle terms

In the expansion of $\left(9x-\dfrac1{3\sqrt x}\right)^{18}$, $x>0$, if the term independent of $x$ is $(221)k$, then $k$ is equal to:

If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left(\dfrac1{x^3}-x^4\right)^n$, $x\ne0$, is zero, then the value of $n$ is ________.

The coefficient of $x^2$ in the expansion of $\left(2x^2+\dfrac1x\right)^{10}$, $x\ne0$, is :

If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x)^{26}$ and $(1-\alpha x)^{28}$, $\alpha\ne0$, are equal, then the value of $\alpha$ is:

Let $a_0, a_1, \ldots, a_{23}$ be real numbers such that

$$\left(1 + \frac25x\right)^{23} = \sum_{i=0}^{23}a_ix^i$$

for every real number $x$. Let $a_r$ be the largest among the numbers $a_j$ for $0 \le j \le 23$. Then the value of $r$ is _____.

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^5$ in the expansion of $\left(ax^2 + \dfrac{70}{27bx}\right)^4$ is equal to the coefficient of $x^{-5}$ in the expansion of $\left(ax - \dfrac{1}{bx^2}\right)^7$, then the value of $2b$ is _____.

The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^2(5+x)^{498}+\dots+x^{500}$, $x>0$, is

If the sum of the co-efficients of all the positive even powers of $x$ in the binomial expansion of $\left(2x^3+\dfrac3x\right)^{10}$ is $5^{10}-\beta\cdot3^9$, then $\beta$ is equal to ______.

If the coefficient of $x^{10}$ in the binomial expansion of $\left(\dfrac{\sqrt x}{5^{1/4}}+\dfrac{\sqrt5}{x^{1/3}}\right)^{60}$ is $5^k\cdot l$, where $l,k\in\mathbb N$ and $l$ is co-prime to 5, then $k$ is equal to ______.

If the sum of the coefficients of all the positive powers of $x$, in the Binomial expansion of $\left(x^n+\dfrac2{x^5}\right)^7$ is 939, then the sum of all the possible integral values of $n$ is ______.

The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2x^3+\dfrac3{x^k}\right)^{12}$, $x\ne0$ is $2^8\cdot l$, where $l$ is an odd integer, is ______.

The term independent of $x$ in the expansion of $\left(1-x^2+3x^3\right)\left(\dfrac52x^3-\dfrac1{5x^2}\right)^{11}$, $x\ne0$ is :

If the constant term in the expansion of $\left(3x^3-2x^2+\dfrac5{x^5}\right)^{10}$ is $2^k\cdot l$, where $l$ is an odd integer, then the value of $k$ is equal to :

Let the coefficients of $x^{-1}$ and $x^{-3}$ in the expansion of $\left(2x^{\frac15}-\dfrac1{x^{\frac15}}\right)^{15}$, $x>0$, be $m$ and $n$ respectively. If $r$ is a positive integer such that $mn^2={}^{15}C_r\cdot2^r$, then the value of $r$ is equal to ______.

For two positive real numbers a and b such that $\dfrac1{a^2}+\dfrac1{b^3}=4$, the minimum value of the constant term in the expansion of $\left(ax^{\frac18}+bx^{-\frac1{12}}\right)^{10}$ is :

If the maximum value of the term independent of $t$ in the expansion of $\left(t^2x^{\frac15}+\dfrac{(1-x)^{\frac1{10}}}t\right)^{15}$, $x\ge0$, is K, then 8 K is equal to ______.

Let for the 9$^{\text{th}}$ term in the binomial expansion of $(3+6x)^n$, in the increasing powers of $6x$, to be the greatest for $x=\dfrac32$, 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 :

Let the coefficients of the middle terms in the expansion of $\left(\dfrac1{\sqrt6}+\beta x\right)^4$, $(1-3\beta x)^2$ and $\left(1-\dfrac\beta2x\right)^6$, $\beta>0$, respectively form the first three terms of an A.P. If d is the common difference of this A.P., then $50-\dfrac{2d}{\beta^2}$ is equal to ______.

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $\left(\sqrt[4]2+\dfrac1{\sqrt[4]3}\right)^n$, in the increasing powers of $\dfrac1{\sqrt[4]3}$ be $\sqrt[4]6:1$. If the sixth term from the beginning is $\dfrac\alpha{\sqrt[4]3}$, then α is equal to _______.

The coefficient of $x^{1011}$ in the expression $(1+2x)^{2022}+2x(1+2x)^{2021}+(2x)^2(1+2x)^{2020}+\dots+(2x)^{2022}$ is