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

The natural number $m$,for which the coefficient of $x$ in the binomial expansion of $\left( x^{m} + \frac{1}{x^{2}} \right)^{22}$ is $1540$,is

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 $(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}})^{10}$ is $10k$,then $k$ is equal to

If $n$ is even,then in the expansion of ${\left( {1 + \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}} + \dots} \right)^2}$,the coefficient of ${x^n}$ is

The coefficient of $x^{53}$ in the expansion $\sum_{m = 0}^{100} {^{100}C_m (x - 3)^{100 - m} \cdot 2^m}$ is

For $n \in N$,in the expansion of $\left(\sqrt[4]{x^{-3}}+a \sqrt[4]{x^5}\right)^n$,the sum of all binomial coefficients lies between $200$ and $400$ and the term independent of $x$ is $448$. Then the value of $a$ is