MHTCET 16 May 2024 Shift 1 Paper

Concept:For a function to be continuous at a point, the value of the function at that point must equal its limit at that point.Explanation:Since $f(x)$ is continuous on $\left[0, \frac{\pi}{2}\right]$, it is continuous at $x = \frac{\pi}{4}$.Therefore, $f\left(\frac{\pi}{4}\right) = \lim_{x \to \frac{\pi}{4}} f(x)$.Substitute: $\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{4x - \pi}$.As $x \to \frac{\pi}{4}$, both numerator and denominator tend to $0$, so apply L'Hospital's rule.Differentiate numerator: $\frac{d}{dx}(1 - \tan x) = -\sec^2 x$.Differentiate denominator: $\frac{d}{dx}(4x - \pi) = 4$.Thus, $f\left(\frac{\pi}{4}\right) = \lim_{x \to \frac{\pi}{4}} \frac{-\sec^2 x}{4}$.At $x = \frac{\pi}{4}$, $\sec^2\left(\frac{\pi}{4}\right) = 2$.So, $f\left(\frac{\pi}{4}\right) = \frac{-2}{4} = -\frac{1}{2}$.Answer:$-\frac{1}{2}$ (Option A).