QJEEAdv24Ma4ab3: JEE Advanced 2024 Mathematics - Class 11 JEEAdv24Ma4ab3 Solution: Let , and . Then which of the following statements is (are) TRUE?

$\begin{aligned}\text{(A)} \quad & (-1+\sqrt{2})^{\mathrm{n}}=\mathrm{m}+\sqrt{2} \mathrm{n}, \mathrm{m}, \mathrm{n} \in \mathbb{Z} \\& (1+\sqrt{2})^{\mathrm{n}}=\mathrm{m}_1+\sqrt{2} \mathrm{n}_1, \mathrm{~m}_1, \mathrm{n}_1 \in \mathbb{Z} \\& \Rightarrow \mathbb{Z} \cup \mathrm{T}_1 \cup \mathrm{T}_2 \subseteq \mathrm{S}\end{aligned}$

but $b \sqrt{2} \in S$ for negative $b \in \mathbb{Z}$ .

So $\quad \mathbb{Z} \cup T_1 \cup T_2 \subset \mathrm{S}$

$\begin{aligned}\text{(B)}\quad& (\sqrt{2}-1)^{\mathrm{n}}=\frac{1}{(\sqrt{2}+1)^{\mathrm{n}}}<\frac{1}{2024} \\& \Rightarrow 2024<(\sqrt{2}+1)^{\mathrm{n}}, \exists \mathrm{n} \in \mathbb{N} \\& \Rightarrow \mathrm{T}_1 \cap\left(0, \frac{1}{2024}\right) \neq \phi\end{aligned}$

$\begin{aligned}\text{(C)}\quad& (1+\sqrt{2})^{\mathrm{n}}>2024, \exists \mathrm{n} \in \mathbb{N} \\& \Rightarrow \mathrm{T}_2 \cap(2024, \infty) \neq \phi\end{aligned}$

$\mathrm{(D)}$ $\quad \sin (\pi(a+b \sqrt{2})=0) \Rightarrow b=0, a \in \mathbb{Z}$ .

$\Rightarrow$ Options (A), (C), (D) are Correct.