JEE Main 2023 (Online) 25th January Morning ShiftQuestion 1
Let $S=\left\{\alpha:\log_2\left(9^{2\alpha-4}+13\right)-\log_2\left(\frac52\cdot3^{2\alpha-4}+1\right)=2\right\}$. Then the maximum value of $\beta$ for which the equation $x^{2}-2\left(\sum_{\alpha\in S}\alpha\right)^{2}x+\sum_{\alpha\in S}(\alpha+1)^{2}\beta=0$ has real roots, is
Sets, Relations and Function
Answer$25$
Solution
With $t=3^{2\alpha-4}$: $\frac{t^2+13}{\frac52t+1}=4\Rightarrow t^2-10t+9=0$, $t=1,9$, so $\alpha=2,3$. Then $\sum\alpha=5$, $\sum(\alpha+1)^2=25$, and $x^2-50x+25\beta=0$ has real roots iff $2500-100\beta\ge0$, i.e. $\beta\le25$. Maximum $\beta=25$.