UPSC NDA II 2024 Mathematics Paper

Concept:The sum of the expansions of $(8 + 3\sqrt{7})^{20}$ and $(8 - 3\sqrt{7})^{20}$ yields an even integer, and the fractional parts add up to 1.Explanation:Let $(8 + 3\sqrt{7})^{20} = U + V$ where $U$ is an integer and $0 < V < 1$.Let $(8 - 3\sqrt{7})^{20} = W$ with $0 < W < 1$.Adding these two equations gives:$U + V + W = (8 + 3\sqrt{7})^{20} + (8 - 3\sqrt{7})^{20}$.This sum expands using the binomial theorem: all terms with odd powers of $\sqrt{7}$ cancel, leaving only even-power terms. Each such term is an integer multiple of powers of $(3\sqrt{7})^2 = 63$, so the entire sum is an even integer.Thus $U + V + W$ is an even integer.Since $U$ is an integer, $V + W$ must also be an integer.Because $0 < V < 1$ and $0 < W < 1$, their sum lies strictly between 0 and 2. The only integer in that interval is 1.Therefore $V + W = 1$.Answer:$1$ (Option D)