AP EAMCET · Maths · Complex Number
For how many natural numbers ' \(n\) ' such that \(1 \leq n \leq 2021\) is \(\left(\frac{1+i}{1-i}\right)^n=1\) ?
- A 504
- B 505
- C 506
- D 503
Answer & Solution
Correct Answer
(B) 505
Step-by-step Solution
Detailed explanation
It is given that, \(\begin{aligned} \left(\frac{1+i}{1-i}\right)^n=1 & \Rightarrow\left(\frac{1+i^2+2 i}{1-i^2}\right)^n=1 \\ \Rightarrow \quad\left(\frac{2 i}{2}\right)^n=1 & \Rightarrow i^n=1 \end{aligned}\) If \(n\) is a integral multiple of 4, then \(i^n=1\)…
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