UPSC NDA I 2015 Mathematics Paper

Concept:We assume the square root of $i$ as a complex number $x + iy$, then square both sides and compare real and imaginary parts to solve for $x$ and $y$.Explanation:Let $\sqrt{i} = x + iy$, where $x$ and $y$ are real numbers.Square both sides: $i = (x + iy)^2 = x^2 + i2xy + (iy)^2 = x^2 - y^2 + i2xy$.Compare real parts: $x^2 - y^2 = 0$. Compare imaginary parts: $2xy = 1$. From $2xy = 1$, we get $y = \frac{1}{2x}$. Substitute into $x^2 - y^2 = 0$: $x^2 - \frac{1}{4x^2} = 0$. Multiply by $4x^2$: $4x^4 - 1 = 0$, so $x^4 = \frac{1}{4}$. Thus $x = \pm \frac{1}{\sqrt{2}}$. Using $y = \frac{1}{2x}$, we get $y = \pm \frac{1}{\sqrt{2}}$, with the same sign as $x$. Therefore $\sqrt{i} = x + iy = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} = \frac{1+i}{\sqrt{2}}$.Answer:C. $\frac{1+i}{\sqrt{2}}$