PUNE MCA Exam 2014 Question Paper with answers for online practice

Concept:Differentiate the given integral equation to obtain a simple differential equation.Explanation:Given $\int x f(x) \, dx = \frac{f(x)}{2}$.Differentiate both sides with respect to $x$:$\frac{d}{dx} \left[ \int x f(x) \, dx \right] = \frac{d}{dx} \left[ \frac{f(x)}{2} \right]$.Left side becomes $x f(x)$ (by Fundamental Theorem of Calculus).Right side becomes $\frac{f'(x)}{2}$.Hence $x f(x) = \frac{f'(x)}{2}$ which gives $f'(x) = 2x f(x)$.Solve: $\frac{df}{dx} = 2x f \Rightarrow \frac{df}{f} = 2x \, dx$.Integrate: $\ln|f| = x^2 + C \Rightarrow f(x) = k e^{x^2}$.Now check the original equation: $\int x \cdot k e^{x^2} dx = \frac{k}{2} e^{x^2} = \frac{f(x)}{2}$ holds for any constant $k$.Among the options, option D is $\frac{e^{x^2}}{2}$ which matches $k = \frac{1}{2}$.Answer:$\frac{e^{x^2}}{2}$