MHTCET 11 Apr 2026 Shift 1 Paper

Concept:In an infinite nested radical, the repeating part inside the root equals the whole expression $y$ itself.Explanation:Since the pattern repeats infinitely, rewrite the given equation as $y = \sqrt{\tan x + y}$.Squaring both sides gives $y^2 = \tan x + y$.Differentiate both sides with respect to $x$:$$2y \frac{dy}{dx} = \sec^2 x + \frac{dy}{dx}$$Group the $\frac{dy}{dx}$ terms together:$$(2y - 1)\frac{dy}{dx} = \sec^2 x$$Hence, $\frac{dy}{dx} = \frac{\sec^2 x}{2y - 1}$.At $x = \frac{\pi}{4}$, we know $\tan \frac{\pi}{4} = 1$.Substitute into $y^2 = \tan x + y$ to get $y^2 - y - 1 = 0$.Since $y$ represents a square root, it is positive, so $y = \frac{1 + \sqrt{5}}{2}$.Now, $2y - 1 = 2\left(\frac{1 + \sqrt{5}}{2}\right) - 1 = \sqrt{5}$.Also, $\sec^2 \frac{\pi}{4} = 2$.Therefore, $\frac{dy}{dx} = \frac{2}{\sqrt{5}}$.Squaring this result, $\left(\frac{dy}{dx}\right)^2 = \frac{4}{5}$.Answer:The correct option is C: $\frac{4}{5}$.