If the $6^{th}$ term in the expansion of the binomial $\left[ \frac{1}{x^{8/3}} + x^2 \log_{10} x \right]^8$ is $5600$,then $x$ equals to

The coefficient of $x^4$ in the expansion of $\frac{(1-3 x)^2}{(1-2 x)}$ is equal to

The middle term in the expansion of $(1 + x)^{2n}$ is

If in the expansion of $(a-2b)^{n}$, the sum of the $5^{th}$ and $6^{th}$ term is zero, then the value of $\frac{a}{b}$ is

Let $\alpha > 0$ be the smallest number such that the expansion of $(x^{2/3} + 2x^{-3})^{30}$ has a term $\beta x^{-\alpha}$,where $\beta \in \mathbb{N}$. Then $\alpha$ is equal to $.............$.

If the coefficient of the $3^{\text{rd}}$ term from the beginning in the expansion of $\left(ax^2 - \frac{8}{bx}\right)^9$ is equal to the coefficient of the $3^{\text{rd}}$ term from the end in the expansion of $\left(ax - \frac{2}{bx^2}\right)^9$,then the relation between $a$ and $b$ is: