UPSC NDA 2 2021 Mathematics Solved Paper

Concept:The differential equation of a family of curves is formed by eliminating arbitrary constants.The degree is the highest power of the highest derivative after the equation is made polynomial in derivatives.Explanation:Step 1: A circle touching both coordinate axes in the first quadrant has center at $(a, a)$ and radius $a$.Its equation is $(x-a)^2 + (y-a)^2 = a^2$.Step 2: Differentiate with respect to $x$: $2(x-a) + 2(y-a)\frac{dy}{dx} = 0$.Simplify to $x + y\frac{dy}{dx} = a\left(1+\frac{dy}{dx}\right)$.Step 3: Solve for $a$: $a = \dfrac{x + y\frac{dy}{dx}}{1+\frac{dy}{dx}}$.Step 4: Expand the circle equation: $x^2 + y^2 + a^2 = 2a(x+y)$.Substitute $a$ into this expanded form.Step 5: Multiply both sides by $\left(1+\frac{dy}{dx}\right)^2$ to eliminate fractions.The resulting equation contains only the first derivative $\frac{dy}{dx}$, and its highest power is $2$.No higher‑order derivatives appear; so the degree is $2$.Answer:Option B. 2