Question

Let S=\left\\{a+b\sqrt2:a,b\in Z\right\\},T_1=\left\\{(-1+\sqrt2)^n:n\in \N\right\\},\ \text{and}\ T_2=\left\\{(1+\sqrt2)^n:n\in\N\right\\}. Then which of the following statements is (are) TRUE ?

• A. $\Z∪T_1∪T_2⊂S$
• B. $T_1∩ (0,\frac{1}{2024})=Φ$, where Φ denotes the empty set.
• C. $T_2∩ (2024,\infin)\neΦ$
• D. For any given a, b ∈ Z, $\cos(\pi(a+b\sqrt2))+i\sin(\pi(a+b\sqrt2))\in\Z$ if and only if b = 0, where $i=\sqrt{-1}$

Answer 1

Answer: (A), (C), (D)

$\Z∪T_1∪T_2⊂S$

Answer 2

The correct option is (A):$\Z∪T_1∪T_2⊂S$, (C) : $T_2∩ (2024,\infin)\neΦ$ and (D) : For any given a, b ∈ Z, $\cos(\pi(a+b\sqrt2))+i\sin(\pi(a+b\sqrt2))\in\Z$ if and only if b = 0, where $i=\sqrt{-1}$.