UPSC NDA 2 2017 Mathematics Solved Paper

Concept:A function $f(x)$ is continuous at $x = a$ if $\lim\limits_{x \to a} f(x) = f(a)$.Explanation:1. For $f(x) = x + x^{2}$:$\lim\limits_{x \to 0} f(x) = 0 + 0 = 0 = f(0)$.Thus, it is continuous at $x = 0$. Statement 1 is correct.2. For $f(x) = x + \cos\left(\frac{1}{x}\right)$:As $x \to 0$, $\frac{1}{x} \to \infty$, so $\cos\left(\frac{1}{x}\right)$ oscillates between $-1$ and $1$.Hence, the limit does not exist.Therefore, the function is discontinuous at $x = 0$. Statement 2 is correct.3. For $f(x) = x^{2} + \cos\left(\frac{1}{x}\right)$:As $x \to 0$, $x^{2} \to 0$, but $\cos\left(\frac{1}{x}\right)$ still oscillates.Thus, the limit does not exist, and the function is not continuous at $x = 0$. Statement 3 is incorrect.Answer:Only statements 1 and 2 are correct, so the correct option is A.