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

Question

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

Accepted Answer

$\begin{gathered} {\sin }^{-1}(\frac{3x-22}{2x-19})+{\log }_{e}(\frac{3{x}^{2}-8x+5}{{x}^{2}-3x-10}) \ -1\le \frac{3x-22}{2x-19}\le 1 \ \frac{3x-22}{2x-19}+1\ge 0\frac{3x-22}{2x-19}-1\le 0 \ \frac{3x-22+2x-19}{2x-19}\ge 0\frac{3x-22-2x+19}{2x-19}\le 0 \ \Rightarrow \frac{5x-41}{2x-19}\ge 0\frac{x-3}{2x-19}\le 0 \ x\in (-\infty,\frac{41}{5}]\cup (\frac{19}{2},\infty)x\in [3,\frac{19}{2}) \end{gathered}$

$\begin{gathered} \Rightarrow \, x\in [3,\frac{41}{5}]\, \ \frac{3{x}^{2}-8x+5}{{x}^{2}-3x-10}>0 \ \frac{(3x-5)(x-1)}{(x-5)(x-2)}>0 \ \Rightarrow \, x\in (-\infty,-2)\cup [1,\frac{5}{3}]\cup (5,\infty)\ldots \end{gathered}$

Taking intersection of individual domains

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

$\begin{gathered} \Rightarrow \, \alpha =5\beta =\frac{41}{5} \ \Rightarrow 3\alpha +10\beta =15+82 \ =97 \end{gathered}$

$	herefore \,$ Option (4) is correct

Solutions · 1