UPSC NDA I 2022 Mathematics Solved Paper

Concept:An exponential function $f(x) = 10^x$ is defined for all real numbers, is continuous everywhere, and is differentiable at every point, including $x = 0$.Explanation:Statement 1: The domain of $f(x) = 10^x$ is the set of all real numbers because a positive base can be raised to any real exponent.Thus, the domain is $(-\infty, \infty)$ — correct.Statement 2: Exponential functions are continuous on their entire domain.Since $f(x)=10^x$ is an exponential function, it is continuous for all $x \in \mathbb{R}$ — correct.Statement 3: Differentiability at $x=0$ requires that the derivative exists at that point.The derivative of $f(x)=10^x$ is $f'(x) = 10^x \log_e 10$ (where $\log_e$ denotes natural logarithm).At $x=0$, $f'(0) = 10^0 \log_e 10 = 1 \cdot \log_e 10 = \log_e 10$, which is a finite real number.Hence, the derivative exists, so $f$ is differentiable at $x=0$ — correct.All three statements are correct.Answer:Option D: 1, 2 and 3 only.