UPSC NDA I 2024 Mathematics Paper

Concept:Check the differentiability of $f(x)$ for $x<0$, continuity of $g(x)$ at $x=0.0001$, and derivative of $g(x)$ at $x=2.5$.Explanation:Statement 1: For $x<0$, $f(x)=|x|+1=-x+1$, a linear function. Linear functions are differentiable everywhere. Hence, Statement 1 is correct.Statement 2: $g(x)=[x]-1$, where $[x]$ is the greatest integer function. $[x]$ is continuous at all non-integer points. Since $x=0.0001$ is not an integer, $g(x)$ is continuous there. Hence, Statement 2 is correct.Statement 3: For $2 \le x < 3$, $[x]=2$, so $g(x)=2-1=1$, a constant. Derivative of a constant is $0$, not $1$. Hence, Statement 3 is incorrect.Answer:Only statements 1 and 2 are correct. Thus, option $A$ is correct.