MHTCET 11 Apr 2026 Shift 2 Paper

Concept:To identify the family of curves, solve the differential equation by separating variables and integrating. The form of the resulting equation reveals the curve type.Explanation:The given differential equation is:$$2x \frac{dy}{dx} - y = 3$$Rearranging the terms gives:$$2x \frac{dy}{dx} = y + 3$$$$\frac{dy}{dx} = \frac{y+3}{2x}$$Separating the variables:$$\frac{dy}{y+3} = \frac{dx}{2x}$$Integrating both sides:$$\int \frac{dy}{y+3} = \int \frac{dx}{2x}$$$$\ln|y+3| = \frac{1}{2}\ln|x| + \ln C$$Combining the logarithms:$$\ln|y+3| = \ln(C\sqrt{x})$$Removing the logarithm:$$y + 3 = C\sqrt{x}$$Squaring both sides to eliminate the square root:$$(y+3)^2 = C^2 x$$Let $Y = y+3$ and $X = x$. Then:$$Y^2 = C^2 X$$This matches the standard parabola form $Y^2 = 4aX$.Hence, the solution represents a family of parabolas.Answer:The correct option is C: parabola.