Complex Numbers
nth Roots of Unity – Equation in Powers
Complex Numbers_PYQ
Grade 11
Question:
If $\omega$ ($\neq 1$) be a cube root of unity and $(1 + \omega^2)^n = (1 + \omega^4)^n$, then the least positive value of $n$ is
Step-by-Step Solution
Key Concept: Using $1+\omega^2=-\omega$ and $1+\omega=-\omega^2$ reduces the equation to $\omega^n=1$, which holds iff $3\mid n$.
**Step 1: Simplify using 1+ω+ω²=0**
$1+\omega^2 = -\omega$ and $1+\omega^4 = 1+\omega = -\omega^2$ (since $\omega^4=\omega^3\cdot\omega=\omega$).
**Step 2: Set up the equation**
$(-\omega)^n = (-\omega^2)^n \Rightarrow \omega^n = \omega^{2n} \Rightarrow \omega^n(\omega^n-1)=0$.
**Step 3: Solve**
Since $\omega\neq 0$: $\omega^n=1 \Rightarrow 3\mid n$. Least positive $n=3$.
Correct Answer: 2