NDA 2 2025 Maths, question 13

Solution.

Answer: (b) $2$

1. $\frac{1 - i}{1 + i} = -i$ and $\frac{1 + i}{1 - i} = i$.
2. So the product is $(-i)^{2m}(i)^{2n} = (-1)^{m}(-1)^{n} = (-1)^{m+n} = 1$, which needs $m + n$ even, so $m - n$ is even.
3. The smallest positive even value is 2, for example $m = 3$, $n = 1$.

Trap: This takes $m$ and $n$ as integers, as the question intends with powers of $i$.