MHTCET 17 Apr 2026 Shift 2 Paper

Concept:An infinite nested radical with the same repeating expression can be replaced by the variable $y$ itself, giving a simple equation.Explanation:Since the entire expression inside the square root repeats infinitely, we can write the whole radical as $y$.So, $y = \sqrt{\cos x^2 + y}$.Squaring both sides gives $y^2 = \cos x^2 + y$.Now differentiate both sides with respect to $x$:$$2y \frac{dy}{dx} = -2x \sin x^2 + \frac{dy}{dx}$$Bring the $\frac{dy}{dx}$ terms together:$$(2y-1)\frac{dy}{dx} = -2x \sin x^2$$Therefore,$$\frac{dy}{dx} = \frac{-2x \sin x^2}{2y-1}$$Comparing with $\frac{dy}{dx} = \frac{f(x)}{2y-1}$, we get $f(x) = -2x \sin x^2$.Now integrate $f(x)$:$$\int f(x)\,dx = \int -2x \sin x^2\,dx$$Using the substitution $u = x^2$, $du = 2x\,dx$, we obtain:$$\int f(x)\,dx = \cos x^2 + C$$Answer:$\cos x^2 + C$, which is Option C.