Question

For any real number $r$, let $A_r=\{e^{i\pi rn}:n\text{is a natural number}\}$be a set of complex numbers. Then

• A. $A_1,A_{\frac{1}{\pi }},A_{0.3}$ are all infinite sets.
• B. $A_1$ is a finite set and $A_{\frac{1}{\pi }},A_{0.3}$ are infinite sets.
• C. $A_1,A_{\frac{1}{\pi }},A_{0.3}$ are all finite sets.
• D. $A_1,A_{0.3}$ are finite set and $A_{\frac{1}{\pi }}$is an infinite sets.

Answer 1

The correct option is D $A_1,A_{0.3}$ are finite set and $A_{\frac{1}{\pi }}$is an infinite sets.

$A_1=e^{i\pi n}=\cos n\pi +i\sin n\pi =\cos n\pi$
So $A_1$ is a finite set.
$A_{\frac{1}{\pi }}=\cos n+i\sin n$
So $A_{\frac{1}{\pi }}$ is an infinite set.
$A_{0.3}=\cos \left(\frac{3n\pi }{10}\right)+i\sin \left(\frac{3n\pi }{10}\right)$
So $A_{0.3}$ is a finite set.