If sum of the coefficient of the first, second and third terms of the expansion of ${\left( {{x^2} + \frac{1}{x}} \right)^m}$ is $46$, then the coefficient of the term that doesnot contain $x$ is :-

The coefficient of $x^9$ in the expansion of $(1+x)\left(1+x^2\right)\left(1+x^3\right) \ldots . .\left(1+x^{100}\right)$ 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 $a^3 + b^6 = 2$, then the maximum value of the term independent of $x$ in the expansion of  $(ax^{\frac{1}{3}}+bx^{\frac{-1}{6}})^9$ is, where $(a > 0, b > 0)$

The greatest value of the term independent of $x$ in the expansion of ${\left( {x\sin \theta  + \frac{{\cos \theta }}{x}} \right)^{10}}$ is

If the ${(r + 1)^{th}}$ term in the expansion of ${\left( {\sqrt[3]{{\frac{a}{{\sqrt b }}}} + \sqrt {\frac{b}{{\sqrt[3]{a}}}} } \right)^{21}}$ has the same power of $a$ and $b$, then the value of $r$ is