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 ?

Updated On: Aug 22, 2024

\(\Z∪T_1∪T_2⊂S\)
\(T_1∩ (0,\frac{1}{2024})=Φ\), where Φ denotes the empty set.
\(T_2∩ (2024,\infin)\neΦ\)
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 \(i=\sqrt{-1}\)

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The Correct Option is A, C, D

Solution and Explanation

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 \(i=\sqrt{-1}\).

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