UPSC NDA I 2022 Mathematics Solved Paper

Concept:The principal argument of a complex number is its angle with the positive real axis, measured in the principal range $(-\pi, \pi]$.It is found by checking the quadrant of the complex number and using $\tan^{-1}(y/x)$ with appropriate sign adjustments.Explanation:First, write $\frac{1}{1+i}$ in standard $x+iy$ form.Multiply numerator and denominator by the conjugate $(1-i)$:$\frac{1}{1+i} \times \frac{1-i}{1-i} = \frac{1-i}{1 - i^2} = \frac{1-i}{1 - (-1)} = \frac{1-i}{2} = \frac12 - \frac12 i$.So, $x = \frac12 > 0$ and $y = -\frac12 < 0$.Since $x>0$ and $y<0$, the complex number lies in the fourth quadrant.For the fourth quadrant, the principal argument is given by $-\tan^{-1}\left|\frac{y}{x}\right|$.Here, $\left|\frac{y}{x}\right| = \frac{1/2}{1/2} = 1$.Thus, the argument is $-\tan^{-1}(1) = -\frac{\pi}{4}$.Therefore, the principal argument of $\frac{1}{1+i}$ is $-\frac{\pi}{4}$.Answer:$-\frac{\pi}{4}$Hence, option B is correct.