If the domain of the function is and the domain of the function is , then is equal to : [2025]

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Solution

Q.
If the domain of the function $\log_{5} (18 x - x^{2} - 77)$ is $(α, β)$ and the domain of the function $\log_{(x - 1)} (\frac{2 x^{2} + 3 x - 2}{x^{2} - 3 x - 4})$ is $(γ, δ)$ , then $α^{2} + β^{2} + γ^{2}$ is equal to : [2025]

1 195

2 174

3 186

4 179

Ans.
(3)

Let $f_{1} (x) = \log_{5} (18 x - x^{2} - 77)$

$∴ 18 x - x^{2} - 77 > 0$

$\Rightarrow x^{2} - 18 x + 77 < 0$

$\Rightarrow (x - 11) (x - 7) < 0$

$\Rightarrow x \in (7, 11) ∴ α = 7, β = 11$

Also, let $f_{2} (x) = \log_{(x - 1)} (\frac{2 x^{2} + 3 x - 2}{x^{2} - 3 x - 4})$

$∴ x - 1 > 0 \Rightarrow x > 1, x - 1 \neq 1 \Rightarrow x \neq 2$ ,

$\frac{2 x^{2} + 3 x - 2}{x^{2} - 3 x - 4} > 0 \Rightarrow \frac{(2 x - 1) (x + 2)}{(x - 4) (x + 1)} > 0$

$\Rightarrow x \in (4, \infty) ∴ γ = 4$

Now, $α^{2} + β^{2} + γ^{2}$ = 49 + 121 + 16 = 186.

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