JEE MAIN - Mathematics (2023 - 25th January Morning Shift - No. 20)
Let $$S = \left\{ {\alpha :{{\log }_2}({9^{2\alpha - 4}} + 13) - {{\log }_2}\left( {{5 \over 2}.\,{3^{2\alpha - 4}} + 1} \right) = 2} \right\}$$. Then the maximum value of $$\beta$$ for which the equation $${x^2} - 2{\left( {\sum\limits_{\alpha \in s} \alpha } \right)^2}x + \sum\limits_{\alpha \in s} {{{(\alpha + 1)}^2}\beta = 0} $$ has real roots, is ____________.
Answer
25
Explanation
$$
\begin{aligned}
& \log _2\left(9^{2 \alpha-4}+13\right)-\log _2\left(\frac{5}{2} \cdot 3^{2 \alpha-4}+1\right)=2 \\\\
& \Rightarrow \frac{9^{2 \alpha-4}+13}{\frac{5}{2} 3^{2 \alpha-4}+1}=4 \\\\
& \Rightarrow \alpha=2 \quad \text { or } \quad 3 \\\\
& \sum_{\alpha \in \mathrm{S}} \alpha=5 \text { and } \sum_{\alpha \in \mathrm{S}}(\alpha+1)^2=25 \\\\
& \Rightarrow x^2-50 x+25 \beta=0 \text { has real roots } \\\\
& \Rightarrow \beta \leq 25 \\\\
& \Rightarrow \beta_{\max }=25
\end{aligned}
$$
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