UPSC NDA II 2015 Mathematics Paper

Concept:The domain of a function is the set of all real $x$ for which the function is defined.Explanation:The function is $f(x) = \frac{1}{\sqrt{|x| - x}}$.For the square root to be defined and real, the expression inside it must be positive: $|x| - x > 0$.This simplifies to $|x| > x$.Now, recall the definition of absolute value:If $x \ge 0$, then $|x| = x$, so $|x| > x$ becomes $x > x$, which is false (strict inequality cannot hold).If $x < 0$, then $|x| = -x$, so $|x| > x$ becomes $-x > x$, or $2x < 0$, which is true for all $x < 0$.Thus, the condition holds only when $x < 0$.Also, the denominator involves a square root, so it must not be zero; $|x| - x = 0$ only when $x \ge 0$ (since then $|x| = x$), so no further restriction.Therefore, the domain is all real numbers less than zero: $(-\infty, 0)$.Answer:$(-\infty, 0)$ (Option B).