UPSC NDA I 2018 Mathematics Solved Paper

Concept:For a complex number $z = x + iy$, the principal argument $\theta$ lies in the range $-\pi < \theta \le \pi$.It is found using $\theta = \tan^{-1}\left(\frac{y}{x}\right)$, adjusted for the quadrant in which the point $(x,y)$ lies.Explanation:Given $z = -1 - i$, we have $x = -1$ and $y = -1$.Both coordinates are negative, so the point is in the third quadrant.The reference angle is $\tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1) = \frac{\pi}{4}$.For the third quadrant, the principal argument is $-\pi + \text{reference angle}$ (or $\pi - \text{reference angle}$ but within principal range we use $-\pi + \frac{\pi}{4}$).Thus, $\theta = -\pi + \frac{\pi}{4} = -\frac{3\pi}{4}$.This value lies in the interval $(-\pi, \pi]$, as required for the principal argument.Answer:The principal argument is $-\frac{3\pi}{4}$, which corresponds to option C.