Let S= : _2 (9^ 2 -4 +13 )- _2 (5 2 3^ 2 -4 +1 )=2 . Then the max… - Vidyadip

Question

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

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Answer

b
$\log _2\left(9^{2 \alpha-4}+13\right)-\log _2\left(\frac{5}{2} \cdot 3^{2 \alpha-4}+1\right)=2$

$\Rightarrow \frac{9^{2 \alpha-4}+13}{\frac{5}{2} 3^{2 \alpha-4}+1}=4$

$\Rightarrow \alpha=2 \quad \text { or }$

$\sum_{\alpha \in S} \alpha=5 \text { and } \sum_{\alpha \in S}(\alpha+1)^2=25$

$\Rightarrow x^2-50 x+25 \beta=0 \text { has real roots }$

$\Rightarrow \beta \leq 25$

$\Rightarrow \beta_{\max }=25$

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