KCET 2026 Mathematics Solved Paper

Concept:Use differentiation to convert the given integral equation into a simple first-order differential equation, then solve it by separating variables.Explanation:Start with the given equation:$$\int x f(x)\,dx + \frac{f(x)}{2} = 0$$Differentiate both sides with respect to $x$.The derivative of $\int x f(x)\,dx$ is $x f(x)$, and the derivative of $\frac{f(x)}{2}$ is $\frac{1}{2} f'(x)$.So we get:$$x f(x) + \frac{f'(x)}{2} = 0$$Multiply through by $2$:$$2x f(x) + f'(x) = 0$$Rearrange to write the derivative in terms of $f(x)$:$$f'(x) = -2x f(x)$$Separate the variables by dividing both sides by $f(x)$:$$\frac{1}{f(x)} \frac{df}{dx} = -2x$$This can be written as:$$\frac{1}{f}\,df = -2x\,dx$$Integrate both sides:$$\int \frac{1}{f}\,df = \int -2x\,dx$$$$\ln|f| = -x^{2} + C$$Exponentiate both sides to solve for $f(x)$:$$f(x) = C e^{-x^{2}}$$Since $C$ is an arbitrary constant, $f(x) = e^{-x^{2}}$ is the simplest matching form among the given options.Answer:Option C: $\displaystyle e^{-x^{2}}$