If the domain of the function [ ^-1((3 x-22)/(2 x-19))+ _e((3 x^2-8 x+5)/(x^2-3 x-10))] is…

Question

If the domain of the function $$\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$$ is $$(\alpha, \beta]$$, then $$3 \alpha+10 \beta$$ is equal to:

Accepted Answer

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

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

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

Solutions · 1