UPSC NDA 2 2021 Mathematics Solved Paper

Concept:The absolute value of a function is always greater than or equal to the function itself, so its integral is also greater than or equal to the original integral.Explanation:We know that for any real number, $|f(x)| \ge f(x)$. This inequality holds at every point $x$ in the interval $[-2, 0]$. Therefore, by the property of definite integrals, we have:$\int\limits_{-2}^{0} |f(x)| \, dx \ge \int\limits_{-2}^{0} f(x) \, dx$.The right-hand side is given as $k$.Thus, $\int\limits_{-2}^{0} |f(x)| \, dx \ge k$.Equality occurs if $f(x) \ge 0$ for all $x$ in $[-2, 0]$; otherwise the integral is strictly larger.Answer:greater than or equal to $k$ (Option D).