JEE Main 2025 (Online) 29th January Evening ShiftQuestion 1
If the domain of the function $\log_5\left(18x-x^{2}-77\right)$ is $(\alpha,\beta)$ and the domain of the function $\log_{(x-1)}\left(\frac{2x^{2}+3x-2}{x^{2}-3x-4}\right)$ is $(\gamma,\delta)$, then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is equal to :
Functions
Solution
$18x-x^2-77>0\Rightarrow(7,11)$, so $\alpha=7,\beta=11$. For the second, $x>1,x\ne2$ and $\frac{(2x-1)(x+2)}{(x-4)(x+1)}>0$, which for $x>1$ gives $x>4$, so $\gamma=4$. Then $49+121+16=186$.