If the fourth term in the binomial expansion of $\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^{6}$ is equal to $200$, and $x > 1$, then the value of $x$ is

If the $1011^{\text {th }}$ term from the end in the binomial expansion of $\left(\frac{4 x}{5}-\frac{5}{2 x }\right)^{2022}$ is $1024$ times $1011^{\text {th }}$ term from the beginning, then $|x|$ is equal to

The sum of the real values of $x$ for which the middle term in the binomial expansion of ${\left( {\frac{{{x^3}}}{3} + \frac{3}{x}} \right)^8}$ equals $5670$ is

If the coefficients of ${x^2}$ and ${x^3}$ in the expansion of ${(3 + ax)^9}$ are the same, then the value of $a$ is

Find the middle terms in the expansions of $\left(3-\frac{x^{3}}{6}\right)^{7}$

If the coefficients of $(r-5)^{th}$ and $(2 r-1)^{th}$ terms in the expansion of $(1+x)^{34}$ are equal, find $r$