JEE 2023 — Mathematics PYQ

$5f(x)+4f\left(\frac{1}{x}\right)=\frac{1}{x}+3$

$x\to \frac{1}{x}$

$5f\left(\frac{1}{x}\right)+4f(x)=x+3$

$(1)\times 5-(2)\times 4$

$$\Rightarrow f(x)=\frac{5}{9x}-\frac{4x}{9}+\frac{1}{3}$$

$$\Rightarrow 18{\int }_1^{2}f(x)=18{\left[\frac{5}{9}\ln x-\frac{4}{9}\frac{x^2}{2}+\frac{1x}{3}\right]}_1^2$$

$$\Rightarrow 10\ln 2-6$$