JEE Main 2026 — 24 January, Evening Shift — Question 16

$\lim _{\mathrm{t} \rightarrow \mathrm{x}} \frac{2 \mathrm{t} \mathrm{f}(\mathrm{x})-\mathrm{x}^{2} \mathrm{f}^{\prime}(\mathrm{t})}{-1}=3$

$x^{2} f^{\prime}(x)-2 x f(x)=3$

$\frac{\mathrm{dy}}{\mathrm{dx}}-\frac{2 \mathrm{y}}{\mathrm{x}}=\frac{3}{\mathrm{x}^{2}}$

I.F. $=\mathrm{e}^{-\int \frac{2}{\mathrm{x}} \mathrm{dx}}=\mathrm{e}^{-2 \log _{\mathrm{e}} \mathrm{x}}=1 / \mathrm{x}^{2}$

$\mathrm{y} \cdot \frac{1}{\mathrm{x}^{2}}=\int \frac{3}{\mathrm{x}^{4}} \mathrm{dx}$

$\frac{y}{x^{2}}=-\frac{1}{x^{3}}+c \Rightarrow y=c x^{2}-\frac{1}{x}=f(x)$

$\mathrm{f}(1)=2=\mathrm{c}-1 \Rightarrow \mathrm{c}=3$

$f(x)=3 x^{2}-\frac{1}{x}$

$f(2)=12-\frac{1}{2} \Rightarrow 2 f(2)=23$