NDA 1 2024 Maths, question 34

Solution.

Answer: (a) $1$

1. $(x + y)^{2n+1}(x - y)^{2n+1} = (x^{2} - y^{2})^{2n+1}$, which has two middle terms.
2. Their sum is $(-1)^{n}\binom{2n+1}{n}x^{2n}y^{2n}(x^{2} - y^{2})$.
3. For nonzero $x, y$ this is zero only when $x^{2} = y^{2}$, so $\frac{x^{2}}{y^{2}} = 1$.

Trap: $x = 0$ also makes the middle terms zero, but then the ratio is 0; the question means nonzero $x$ and $y$.