QJEEM24Macf75: JEE Main 2024 Mathematics - Class 12 JEEM24Macf75 Solution: If the domain of the function is , then is equal to:

$\begin{aligned}& \sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _e\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right) \\& -1 \leq \frac{3 x-22}{2 x-19} \leq 1 \\& \frac{3 x-22}{2 x-19}+1 \geq 0 \text { and } \frac{3 x-22}{2 x-19}-1 \leq 0 \\& \frac{3 x-22+2 x-19}{2 x-19} \geq 0 \text { and } \frac{3 x-22-2 x+19}{2 x-19} \leq 0 \\& \Rightarrow \frac{5 x-41}{2 x-19} \geq 0 \text { and } \frac{x-3}{2 x-19} \leq 0 \\& x \in\left(-\infty, \frac{41}{5}\right] \cup\left(\frac{19}{2}, \infty\right) \text { and } x \in\left[3, \frac{19}{2}\right)\end{aligned}$

$\begin{aligned}& \Rightarrow \quad x \in\left[3, \frac{41}{5}\right] \quad \text{.... (1)}\\& \text { and, } \frac{3 x^2-8 x+5}{x^2-3 x-10}>0 \\& \frac{(3 x-5)(x-1)}{(x-5)(x-2)}>0 \\& \Rightarrow \quad x \in(-\infty,-2) \cup\left[1, \frac{5}{3}\right] \cup(5, \infty) \ldots\end{aligned}$

Taking intersection of individual domains

$x \in\left(5, \frac{41}{5}\right]$

$\begin{aligned}& \Rightarrow \quad \alpha=5 \text { and } \beta=\frac{41}{5} \\& \Rightarrow 3 \alpha+10 \beta=15+82 \\& =97\end{aligned}$

$\therefore \quad$ Option (4) is correct