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?

Correct answer: A. ℤ T_1 T_2 S (1) aligned & (-1+ 2 )^ n =m+ 2 n, m, where, n ∈ ℤ & (1+ 2 )^ n =m_1+ 2 n_1, where, ~m_1, n_1 ∈ ℤ \& ℤ T_1 T_2 S aligned but b 2 ∈ S for negative b ∈ ℤ . So ℤ T_1 T_2 S (2) aligned & ( 2 -1)^ n = 1 ( 2 +1)^ n 2024, where, n ∈ ℕ & T_2 (2024, ∞) ≠ φ aligned (4) (π(a+b 2 )=0) b=0, where, a ∈ ℤ Options (1), (3), (4) are Correct.

JEE Advanced 2024Paper 1

Exam: JEE Advanced
Year: 2024
Paper: Paper 1
Subject: Mathematics
Chapter: Complex Number

JEE Advanced 2024 · Paper 1 · Mathematics · Complex Number

Let S=\ a+b 2 : a, b ∈ ℤ\ , T_1=\ (-1+ 2 )^n: n ∈ ℕ\ , and T_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?

Options

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 $T_2 \cap(2024, \infty) \neq \phi$
DFor 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 $i=\sqrt{-1}$ .

Correct answer

A. $\mathbb{Z} \cup T_1 \cup T_2 \subset S$

Step-by-step solution

(1)

\begin{aligned}& (-1+\sqrt{2})^{\mathrm{n}}=\mathrm{m}+\sqrt{2} \mathrm{n}, \mathrm{m}, \text{ where, } \mathrm{n} \in \mathbb{Z} \\ & (1+\sqrt{2})^{\mathrm{n}}=\mathrm{m}_1+\sqrt{2} \mathrm{n}_1, \text{ where, } \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

\mathbb{Z} \cup \mathrm{T}_1 \cup \mathrm{T}_2 \subset \mathrm{S}

(2)

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

(3)

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

(4)

\sin (\pi(a+b \sqrt{2})=0) \Rightarrow b=0, \text{ where, } a \in \mathbb{Z} \Rightarrow

Options (1), (3), (4) are Correct.

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