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જો વિધેય $\log _e\left(\frac{6 x^2+5 x+1}{2 x-1}\right)+\cos ^{-1}\left(\frac{2 x^2-3 x+4}{3 x-5}\right)$ નો પ્રદેશ $(\alpha, \beta) \cup(\gamma, \delta]$ હોય, તો $18\left(\alpha^2+\beta^2+\gamma^2+\delta^2\right)=......$
$20$
$21$
$22$
$23$
Solution
$\frac{6 x^2+5 x+1}{2 x-1} > 0$
$\frac{(3 x+1)(2 x+1)}{2 x-1} > 0$
$-\frac{1}{2}-\frac{1}{3} \quad \frac{1}{2}$
$x \in\left(\frac{-1}{2}, \frac{-1}{3}\right) \cup\left(\frac{1}{2}, \infty\right)…..(A)$
$x \in\left[\frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right] \cup\left(\frac{5}{3}, \infty\right)……….(B)$
$x < \frac{5}{3}……..(C)$
$A \cap B \cap C \equiv\left(\frac{-1}{2}, \frac{-1}{3}\right) \cup\left(\frac{1}{2}, \frac{1}{\sqrt{2}}\right]$
$\text { So } 18\left(\alpha^2+\beta^2+\gamma^2+\delta^2\right)=18\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{4}+\frac{1}{2}\right)$
$=18+2=20$
