COMEDK 2025 Morning Solved Paper

Concept:The function is discontinuous at $x=1$ when the limit as $x \to 1$ does not equal the function value at $x=1$.Explanation:For $x \neq 1$, the expression is $\frac{1-x^m}{1-x}$, which is a $\frac{0}{0}$ indeterminate form as $x \to 1$.Apply L'Hôpital's rule: differentiate numerator and denominator with respect to $x$.$\lim_{x \to 1} \frac{1-x^m}{1-x} = \lim_{x \to 1} \frac{-m x^{m-1}}{-1} = m$.For continuity at $x=1$, we require $\lim_{x \to 1} f(x) = f(1)$.Given $f(1) = 2m - 1$, so $m = 2m - 1$ implies $m = 1$.Thus the function is continuous only when $m = 1$.For discontinuity, $m \neq 1$.Answer:Option D: $m \neq 1$.