The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2 x^{3}+\frac{3}{x^{k}}\right)^{12}, x \neq 0$ is $2^{8} \cdot \ell$, where $\ell$ is an odd integer, is......

For the natural numbers $m, n$, if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots .+a_{m+n} y^{m+n}$ and $a_{1}=a_{2}$ $=10$, then the value of $(m+n)$ is equal to:

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 $.............$.

The coefficient of $x^8$ in the expansion of $(1 -x^4)^4 (1 + x)^5$ is :-

If the coefficients of $x^{7}$ in $\left(x^{2}+\frac{1}{b x}\right)^{11}$ and $x^{-7}$ in $\left(x-\frac{1}{b x^{2}}\right)^{11}, b \neq 0$, are equal, then the value of $b$ is equal to:

Arrange the expansion of $\left(x^{1 / 2}+\frac{1}{2 x^{1 / 4}}\right)^n$ in decreasing powers of $x$.Suppose the coeff icients of the first three terms form an arithmetic progression. Then, the number of terms in the expansion having integer power of $x$ is