Let $\mathrm{S}=\{\mathrm{\alpha }:{\log }_2(9^{2\mathrm{\alpha }-4}+13)-{\log }_2(\frac{5}{2}.3^{2\mathrm{\alpha }-4}+1)=2\}$ . Then the maximum value of  $\beta$ for which the equation  ${\mathrm{x}}^2-2{\left(\sum _{\mathrm{\alpha }\in \mathrm{s}}\mathrm{\alpha }\right)}^2\mathrm{x}+\sum _{\mathrm{\alpha }\in \mathrm{s}}{(\mathrm{\alpha }+1)}^2\mathrm{\beta }=0$ has real roots, is …..  

$\begin{array}{l}{\log }_2\left(9^{2\mathrm{\alpha }-4}+13\right)-{\log }_2\left(3^{2\mathrm{\alpha }-4}\cdot \frac{5}{2}+1\right)=2\\ \therefore \frac{9^{2\mathrm{\alpha }-4}+13}{3^{2\mathrm{\alpha }-4}\cdot \frac{5}{2}+1}=4\end{array}$
${\text{Let\hspace{0.17em}3}}^{2\mathrm{\alpha }-4}=\mathrm{t}$
$$\begin{gathered} \Rightarrow {\mathrm{t}}^2+13=10\mathrm{t}+4 \\ \Rightarrow {\mathrm{t}}^2-10\mathrm{t}+9=0 \\ \Rightarrow \mathrm{t}=9,1 \\ \Rightarrow \mathrm{\alpha }=3,2 \end{gathered}$$
Now equation will become
$$\begin{gathered} {\mathrm{x}}^2-50\mathrm{x}+25\mathrm{\beta }=0 \\ \Rightarrow 2500-100\mathrm{\beta }\ge 0 \\ \Rightarrow \mathrm{\beta }\le 25 \end{gathered}$$