JEE Main 2024 — 1 February, Shift 2 — Question 13

$f(\mathrm{x})=\frac{\sqrt{\mathrm{x}^{2}-25}}{4-\mathrm{x}^{2}}+\log _{10}\left(\mathrm{x}^{2}+2 \mathrm{x}-15\right)$

Domain : $\mathrm{x}^{2}-25 \geq 0 \Rightarrow \mathrm{x} \in(-\infty,-5] \cup[5, \infty)$

$4-x^{2} \neq 0 \Rightarrow x \neq\{-2,2\}$

$\mathrm{x}^{2}+2 \mathrm{x}-15>0 \Rightarrow(\mathrm{x}+5)(\mathrm{x}-3)>0$

$\Rightarrow \mathrm{x} \in(-\infty,-5) \cup(3, \infty)$

$\therefore \mathrm{x} \in(-\infty,-5) \cup[5, \infty)$

$\alpha=-5 ; \beta=5$ $\therefore \alpha^{2}+\beta^{3}=150$