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Complex Numbers question

2024 · Shift 1 · Q22

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PYQ Library
JEE Advanced
Mathematics
Complex Numbers
2024 · Shift 1 · Q22

Complex Numbers question

2024 · Shift 1 · Q22

JEE AdvancedMathematicsComplex NumbersMultiple correct+4 / −2

Let

S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}

, and

T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}

. Then which of the following statements is (are) TRUE?

A
$\mathbb{Z} \cup T_1 \cup T_2 \subset S$
B
$T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$ , where $\phi$ denotes the empty set.
C
$T_2 \cap(2024, \infty) eq \phi$
D
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 $i=\sqrt{-1}$ .

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