JEE Main 2025 — Functions Question with Solution

$$\begin{array}{rl}& f(x)={\cos }^{-1}\left(\frac{4x+5}{3x-7}\right)\\ & \Rightarrow -1\le \left(\frac{4x+5}{3x-7}\right)\le 1\\ & \left(\frac{4x+5}{3x-7}\right)\ge -1\\ & \frac{4x+5+3x-7}{3x-7}\ge 0\\ & \Rightarrow \frac{7x-2}{3x-7}\ge 0\end{array}$$

$$\begin{array}{rl}& x\in \left(-\infty ,\frac{2}{7}\right]\cup \left(\frac{7}{3},\infty \right)\\ & \&\frac{4x+5}{3x-7}\le 1\Rightarrow \frac{x+12}{3x-7}\le 0\end{array}$$

$\therefore$ Domain of $\mathrm{f}(\mathrm{x})$ is

$$\left[-12,\frac{2}{7}\right]\alpha =-12,\beta =\frac{2}{7}$$

$$g(x)={\log }_2\left(2-6{\log }_{27}(2x+5)\right)$$

Domain

$$2-6{\log }_{27}(2x+5)>0$$

$$\begin{array}{ll}\Rightarrow & 6{\log }_{27}(2\mathrm{x}+5)<2\\ \Rightarrow & {\log }_{27}(2\mathrm{x}+5)<\frac{1}{3}\\ \Rightarrow & 2\mathrm{x}+5<3\\ \Rightarrow & \mathrm{x}<-1\end{array}$$

$\&2x+5>0\Rightarrow x>-\frac{5}{2}$

Domain is $\mathrm{x}\in \left(-\frac{5}{2},-1\right)$

$$\begin{array}{rl}& \gamma =-\frac{5}{2},\delta =-1\\ & \begin{array}{rl}& |7(\alpha +\beta )+4(\gamma +\delta )|=|7(-12+\frac{2}{7}+4\left(-\frac{5}{2}-1\right)|\\ & |-82-14|=96\end{array}\end{array}$$