UPSC NDA I 2018 Mathematics Solved Paper

Concept:The modulus-amplitude form of a complex number $z = x + iy$ is $z = r(\cos \theta + i \sin \theta)$, where $r = |z| = \sqrt{x^2 + y^2}$ is the modulus and $\theta = \arg(z)$ is the amplitude.Explanation:Let $z = \sqrt{3} + i$. Represent it as $z = r(\cos \theta + i \sin \theta)$. Compare real and imaginary parts: $r\cos\theta = \sqrt{3}$ …(1) $r\sin\theta = 1$ …(2) Square and add (1) and (2): $r^2(\cos^2\theta + \sin^2\theta) = (\sqrt{3})^2 + 1^2 = 4$. Since $\cos^2\theta + \sin^2\theta = 1$, we get $r^2 = 4$, so $r = 2$ (modulus is positive). Now divide (2) by (1): $\dfrac{r\sin\theta}{r\cos\theta} = \dfrac{1}{\sqrt{3}}$, i.e., $\tan\theta = \dfrac{1}{\sqrt{3}}$. Hence $\theta = \dfrac{\pi}{6}$ (since $\sqrt{3} > 0$ and $1 > 0$, $\theta$ lies in first quadrant). Therefore, the modulus-amplitude form is $z = 2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)$.Answer:$2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)$ which corresponds to option B.