Let S = \ a + b√(2) : a,b ∈ Z \, T_1 = \ (-1+√(2) )^n : n ∈ N \, and T_2 = \ (1+√(2) )^n : n ∈ N \. Then which of the following statements is (are) TRUE?

For any given a,b ∈ Z, cos ( π ( a + b√(2) ) ) + i sin ( π ( a + b√(2) ) ) ∈ Z if and only if b=0, where i = √(-1).

T_1 ∩ ( 0, 1/2024 ) = φ, where φ denotes the empty set.

Let S = \ a + b√(2) : a,b ∈ Z \, T_1 = \ (-1+√(2) )^n : n ∈… — JEE Advanced 2024 Mathematics PYQ

JEE Advanced 2024 · Paper 1 · one or more correct

Let

S = \left\{ a + b\sqrt{2} : a,b \in \mathbb{Z} \right\}

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

, and

T_2 = \left\{ \left(1+\sqrt{2}\right)^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.
$T_2 \cap (2024, \infty) \neq \phi$
D.
For any given $a,b \in \mathbb{Z}$ , $\cos\left( \pi \left( a + b\sqrt{2} \right) \right) + i \sin\left( \pi \left( a + b\sqrt{2} \right) \right) \in \mathbb{Z}$ if and only if $b=0$ , where $i = \sqrt{-1}$ .

Show answer

Correct options: A, C, D

[JEE Advanced 2024 · Paper 1]

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