UPSC NDA I 2025 Mathematics Paper

Concept:The argument (or amplitude) of a complex number is the angle it makes with the positive real axis. The conjugate of a complex number is its mirror image across the real axis, so its argument is the negative of the original argument.Explanation:Let $z = r(\cos\theta + i\sin\theta)$ with $r>0$ and $\theta$ as its principal argument. Then $amp(z) = \theta$. The conjugate is $\bar{z} = r(\cos\theta - i\sin\theta) = r(\cos(-\theta) + i\sin(-\theta))$. Hence $amp(\bar{z}) = -\theta$. Adding them gives $\theta + (-\theta) = 0$. This holds for any non-zero complex number, regardless of which branch of argument is chosen, as long as consistent principal values are taken.Answer:$0$, which corresponds to Option A.