JEE Main 2026 — 22 January, Evening Shift — Question 14

Let $\mathrm{x}^{2}-10 \mathrm{x}+85=\lambda$

∴ Domain for first term $\begin{gathered} \lambda>0 \end{gathered}$ $\begin{gathered} \ 7-\log _{2} \lambda>0 \Rightarrow \lambda<2^{7} \end{gathered}$ $\begin{gathered} \ \log _{5}\left(7-\log _{2} \lambda\right)>0 \Rightarrow \lambda<2^{6} \end{gathered}$

∴ from (1), $0<\lambda<2^{6}$

$0<\mathrm{x}^{2}-10 \mathrm{x}+85<64$ $\begin{gathered} \Rightarrow \mathrm{x} \in(3,7) \end{gathered}$

domain for second term $-1 \leq \frac{3 x-7}{x-17} \leq 1$

$$\begin{gathered} \Rightarrow \mathrm{x} \in[-5,6] \end{gathered}$$

From (A) & (B),

domain of function will be ( 3,6 ]

$\Rightarrow \alpha=3, \beta=6$

$\Rightarrow \alpha+\beta=9$