KVPY Sample Paper KVPY Stream-SX Model Paper-29

Let\[f\]be a differential function such that \[f'(x)=7-\frac{3}{4}\frac{f(x)}{x},(x>0)\]and\[f(1)\ne 4.\]Then\[\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,xf\left( \frac{1}{x} \right):\]

A) exists and equals\[\frac{4}{7}.\]

B) exists and equals 4.

C) does not exist.

D) exists and equals 0.

Correct Answer: B

Solution :

\[f'(x)=7-\frac{3}{4}.\frac{f(x)}{x},x>0\]

\[\therefore \] \[f'(x)+\frac{3}{4x}f(x)=7\] (Linear)

\[f(x).{{e}^{\int{\frac{3}{4x}dx}}}=\int{7.{{e}^{\int{\frac{3}{4x}dx}}}}+c\]

\[f(x).{{x}^{3/4}}=\int{7.{{x}^{3/4}}}+c\]

\[=7\frac{{{x}^{7/4}}}{\frac{7}{4}}+c\]

\[\therefore \] \[f(x)=4x+c{{x}^{-3/4}}\]

\[\therefore \] \[f\left( \frac{1}{x} \right)=\frac{4}{x}+c{{x}^{3/4}}\]

\[\therefore \] \[\underset{x\to {{0}^{+}}}{\mathop{Lt}}\,xf\left( \frac{1}{x} \right)=\underset{x\to {{0}^{+}}}{\mathop{Lt}}\,4+c{{x}^{7/4}}=4.\]