Numerically greatest term in the expansion of $(3 x-4 y)^{23}$ when $x=\frac{1}{6}$ and $y=\frac{1}{8}$ is

Let $K$ be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[3]{3})^{6144}$. If the coefficient of $x^P(P \in N)$ in the expansion of $\frac{1}{(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}$ is $\alpha_p$, then $\alpha_k-\alpha_{k+1}-\alpha_{k-1}=$

If $\frac{3 x+1}{(x-1)^2\left(x^2+1\right)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C x+D}{x^2+1}$, then $2(A-C+B+D)=$

If $\tan \left(\frac{\pi}{4}+\frac{\alpha}{2}\right)=\tan ^3\left(\frac{\pi}{4}+\frac{\beta}{2}\right)$, then $\frac{3+\sin ^2 \beta}{1+3 \sin ^2 \beta}=$