JEE Main 2025 — 24 January, Morning Shift — Question 5

$f(x)=\frac{4.2^{x}+16}{2.2^{2 x}+16.2^{x}+32}$

$f(x)=\frac{2\left(2^{x}+4\right)}{2^{2 x}+8.2^{x}+16}$

$f(x)=\frac{2}{2^{x}+4}$

$f(4-x)=\frac{2^{x}}{2\left(2^{x}+4\right)}$

$\mathrm{f}(\mathrm{x})+\mathrm{f}(4-\mathrm{x})=\frac{1}{2}$

So, $\quad f\left(\frac{1}{15}\right)+f\left(\frac{59}{15}\right)=\frac{1}{2}$

Similarly $\quad f\left(\frac{29}{15}\right)+f\left(\frac{31}{15}\right)=\frac{1}{2}$

$\begin{aligned} & \mathrm{f}\left(\frac{30}{15}\right)=\mathrm{f}(2)=\frac{2}{2^{2}+4}=\frac{2}{8}=\frac{1}{4} \\& \Rightarrow 8\left(29 \times \frac{1}{2}+\frac{1}{4}\right) \end{aligned}$ Ans. $\boxed{118}$