Mathematics - Limits Question with Solution | TestHub

$\begin{array}{rl}& \underset{t\to x}{\lim }\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1\\ & \Rightarrow \underset{t\to x}{\lim }\frac{10t^{9}f(x)-x^{10}{f}^{\prime }(t)}{9t^8}=1\end{array}$ $\begin{array}{rl}& \Rightarrow 10xf(x)-x^2{f}^{\prime }(x)=9\\ & \Rightarrow x^2{f}^{\prime }(x)=10xf(x)-9\\ & \Rightarrow f^{\prime }(x)=\frac{10f(x)}{x}-\frac{9}{x^2}\\ & \Rightarrow \frac{dy}{dx}-\frac{10}{x}y=-\frac{9}{x^2}\\ & \Rightarrow y\cdot \frac{1}{x^{10}}=\int -\frac{9}{x^2}\cdot \frac{1}{x^{10}}dx\\ & \Rightarrow \frac{y}{x^{10}}=\frac{9}{11x^{11}}+c...(1)\end{array}$ $\begin{array}{rl}& \because \mathrm{f}(1)=2\Rightarrow \frac{2}{1}=\frac{9}{11}+\mathrm{c}\Rightarrow \mathrm{c}=\frac{13}{11}\\ & \therefore \mathrm{f}(\mathrm{x})=\frac{9}{11\mathrm{x}}+\frac{13}{11}{\mathrm{x}}^{10}\end{array}$ $\Rightarrow$ Option (2) is correct.