$x^r$ occurs in the expansion of ${\left( {{x^3} + \frac{1}{{{x^4}}}} \right)^n}$ provided -

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

If the maximum value of the term independent of $t$ in the expansion of $\left( t ^{2} x ^{\frac{1}{5}}+\frac{(1- x )^{\frac{1}{10}}}{ t }\right)^{15}, x \geq 0$, is $K$, then $8\,K$ is equal to $....$

In ${\left( {\sqrt[3]{2} + \frac{1}{{\sqrt[3]{3}}}} \right)^n}$ if the ratio of ${7^{th}}$ term from the beginning to the ${7^{th}}$ term from the end is $\frac{1}{6}$, then $n = $

The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$ is $....$

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