Let S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1= \{(-1+\sqrt{2})^n: n \in \mathbb{N}\} and T_2= \{(1+\sqrt{2})^n: n \in \mathbb{N}\}. Then which of the following statements is (are) TRUE? [JEE Adv 2024 P1]

\mathbb{Z} \cup T_1 \cup T_2 \subset S

T_1 \cap (0, \frac{1}{2024}) =ϕ, where ϕ denotes the empty set

For any given a, b \in \mathbb{Z}, \cos (\pi(a+b \sqrt{2}))+i \sin (\pi(a+b \sqrt{2})) \in \mathbb{Z} if and only if b=0, where i=\sqrt{-1}

Complex Numbers

Section: Mathematics

Question : 4 of 36

Marks: +1, -0

Let S={a+b√2:a,b∈ℤ},T1={(−1+√2)n:n∈ℕ} and T2={(1+√2)n:n∈ℕ}. Then which of the following statements is (are) TRUE?

[JEE Adv 2024 P1]

ℤ∪T1∪T2⊂S
T1∩(0,‌
1
2024
)=ϕ, where ϕ denotes the empty set
T2∩(2024,∞)≠ϕ
For any given a,b∈ℤ,cos(π(a+b√2))+isin‌(π(a+b√2))∈ℤ if and only if b=0, where i=√−1

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