If domain of the function log e ( 6 x 2 + 5 x + 1 2 x - 1 ) + cos - 1 ( 2 x 2 - 3 x + 4 3 x - 5 ) is , then 18 ( 2 + 2 + 2 + 2 ) is equal to ____________. [2023]

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Solution

Q.
If domain of the function

$\log_{e} (\frac{6 x^{2} + 5 x + 1}{2 x - 1}) + \cos^{- 1} (\frac{2 x^{2} - 3 x + 4}{3 x - 5})$ is $(α, β) \cup (γ, δ],$ then 18 $(α^{2} + β^{2} + γ^{2} + δ^{2})$ is equal to ____________. [2023]

Ans.
(20)

For domain, $\frac{6 x^{2} + 5 x + 1}{2 x - 1} > 0$

$\Rightarrow \frac{(3 x + 1) (2 x + 1)}{2 x - 1} > 0 \Rightarrow x \in (\frac{- 1}{2}, \frac{- 1}{3}) \cup (\frac{1}{2}, \infty)$    $...(A)$

and

$\Rightarrow \frac{2 x^{2} - 1}{3 x - 5} \geq 0 and \frac{2 x^{2} - 6 x + 9}{3 x - 5} \leq 0 \Rightarrow 3 x - 5 < 0$

$\Rightarrow x \in [\frac{- 1}{\sqrt{2}}, \frac{1}{\sqrt{2}}] \cup (\frac{5}{3}, \infty) ...(B)$

$x < \frac{5}{3}$ $...(C)$

$∴$ $A \cap B \cap C \equiv (\frac{- 1}{2}, \frac{- 1}{3}) \cup (\frac{1}{2}, \frac{1}{\sqrt{2}}]$

$So, 18 (α^{2} + β^{2} + γ^{2} + δ^{2}) = 18 (\frac{1}{4} + \frac{1}{9} + \frac{1}{4} + \frac{1}{2})$

   $= 18 + 2 = 20$

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