JEE MAIN - Mathematics (2025 - 4th April Morning Shift - No. 16)
Let $f, g:(1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x)=\frac{2 x+3}{5 x+2}$ and $g(x)=\frac{2-3 x}{1-x}$. If the range of the function fog: $[2,4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta-\alpha}$ is equal to
56
2
29
68
Explanation
$$\begin{aligned}
& g(2)=4, g(4)=\frac{10}{3} \\
& f \text { is decreasing in }\left(\frac{10}{3}, 4\right) \\
& \therefore \quad \alpha=f(4)=\frac{1}{2} \\
& \beta=f\left(\frac{10}{3}\right)=\frac{29}{56} \\
& \frac{1}{\beta-\alpha}=\frac{1}{\frac{29}{56}-\frac{1}{2}}=56
\end{aligned}$$
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