UPSC NDA I 2016 Mathematics Solved Paper

Concept:Order is the highest derivative; degree is the highest power of that derivative after removing radicals.Explanation:The given solution is $y = cx + c^2 - 3c^{3/2} + 2$, where $c$ is a parameter.Differentiate with respect to $x$: $\frac{dy}{dx} = c$.Substitute $c = \frac{dy}{dx}$ back into the original equation:$y = x\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2 - 3\left(\frac{dy}{dx}\right)^{\frac{3}{2}} + 2$.Rearrange to isolate the term with a fractional exponent:$3\left(\frac{dy}{dx}\right)^{\frac{3}{2}} = \left(\frac{dy}{dx}\right)^2 + x\frac{dy}{dx} - y + 2$.Square both sides to eliminate the radical (so the equation is free from fractional powers):$9\left(\frac{dy}{dx}\right)^3 = \left[\left(\frac{dy}{dx}\right)^2 + x\frac{dy}{dx} - y + 2\right]^2$.The highest derivative appearing is $\frac{dy}{dx}$, hence the order is 1.In the squared equation, the highest power of $\frac{dy}{dx}$ is 4 (from $\left(\frac{dy}{dx}\right)^2$ squared), so the degree is 4.Answer:Order 1, Degree 4 → Option D.