Q45 / 120 · NDA 2024 (Phase 2) · Mathematics
Paragraph:
Let $(8+3 \sqrt{7})^{20}=\mathrm{U}+\mathrm{V}$ and $(8-3 \sqrt{7})^{20}=\mathrm{W}$, where U is an integer and $0 < \mathrm{V} < 1$.
Question:
What is $\mathrm{V}+\mathrm{W}$ equal to?
Question: What is $\mathrm{V}+\mathrm{W}$ equal to?
- A$8$
- B$4$
- C$2$
- Correct answerD$1$
Solution
We have, $(8+3 \sqrt{7})^{20}=\mathrm{U}+\mathrm{V}$....(i)
and $(8-3 \sqrt{7})^{20}=\mathrm{W}$....(ii)
Here, $\mathrm{o} < \mathrm{W} < 1$
Now, adding equation (i) \& (ii), we have,
$\begin{aligned}
& \mathrm{U}+\mathrm{V}+\mathrm{W}=(8+3 \sqrt{7})^{20}+(8-3 \sqrt{7})^{20} \\
& =2\left[{ }^{20} \mathrm{C}_0 \cdot 8^{20}+{ }^{20} \mathrm{C}_2 8^{18} \cdot(3 \sqrt{7})^2+\ldots .+(3 \sqrt{7})^{20}\right]
\end{aligned}$
R.H.S. is an even number.
Also, $\mathrm{o} < \mathrm{V} < 1, \mathrm{o} < \mathrm{W} < 1$ and U is an integer
$\Rightarrow \mathrm{V}+\mathrm{W}$ is an integer $\Rightarrow \mathrm{V}+\mathrm{W}=1$
So, $V+W=1$
Hence, the correct answer is option: 'D'.