Verified Solution • 2025–26 Session moderate

QJEEM23Ma4e57: JEE Main 2023 Mathematics - Class 12 JEEM23Ma4e57 Solution: Let . Then the maximum value of for which the equation has real …

Let S={α:log⁡2(92α−4+13)−log⁡2(52. 32α−4+1)=2}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 β\beta for which the equation x2−2(∑α∈sα)2x+∑α∈s(α+1)2β=0{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 ____________.

Step-by-Step Verified Solution

amp;log⁡2(92α−4+13)−log⁡2(52⋅32α−4+1)=2amp;⇒92α−4+135232α−4+1=4amp;⇒α=2 or 3amp;∑α∈Sα=5 and ∑α∈S(α+1)2=25amp;⇒x2−50x+25β=0 has real roots amp;⇒β≤25amp;⇒βmax⁡=25\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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