Complex Numbers 5 Question 6
6.
If $\omega(\neq 1)$ be a cube root of unity and $\left(1+\omega^{2}\right)^{n}=\left(1+\omega^{4}\right)^{n}$, then the least positive value of $n$ is
(a) 2
(b) 3
(c) 5
(d) 6
(2004, 1M)
Show Answer
Answer:
Correct Answer: 6. (b)
Solution:
- Given, $\left(1+\omega^{2}\right)^{n}=\left(1+\omega^{4}\right)^{n}$
$\Rightarrow$ $(-\omega)^{n}$ $=\left(-\omega^{2}\right)^{n}$
$\left[\because \omega^{3}=1\right.$ and $\left.1+\omega+\omega^{2}=0\right]$
$\Rightarrow$ $\omega^{n}$ $=1$
$\Rightarrow$ $n$ $=3$ is the least positive value of $n$.
