UPSC NDA 2 2016 Mathematics Solved Paper

Concept:The domain of a function is the set of all real $x$ for which the function is defined.For $f(x) = \frac{1}{\sqrt{|x| - x}}$, the denominator must be nonzero and the expression under the square root must be positive.Explanation:We require $\sqrt{|x| - x} \neq 0$ and $|x| - x > 0$.Solve $|x| - x > 0$: If $x \ge 0$, then $|x| = x$, so $|x| - x = 0$ which is not allowed.If $x < 0$, then $|x| = -x$, so $|x| - x = -x - x = -2x > 0$ (since $x < 0$).Thus the condition holds only for $x < 0$.Hence the domain is all real numbers less than $0$, i.e. $(-\infty, 0)$.Answer:A. $(-\infty, 0)$