If $n$ is positive integer and a complex number with unit modulus is a solution of the equation $Z^n + z^{-1} = 1$, then the value of $n$ can be
Step-by-Step Solution
Key Concept: Use polar form of complex numbers and calculus to optimize expressions involving complex arguments
Given $z = r(\cos\theta + i\sin\theta)$, we find $Im\left(\frac{z}{(1+z)^2}\right) = A(1 + \cot)$. Expanding the imaginary part and simplifying yields $A = \frac{r\sin\theta}{(1+r\cos\theta)^2 + r^2\sin^2\theta}$. Taking the derivative $\frac{dA}{d\theta}$ and setting it to zero: $\frac{dA}{d\theta} = \frac{1}{2r}[\sin(4\theta) - 0]$. This equals zero when $\sin(4\theta) = 0$, giving $\theta = \frac{\pi}{4}$ as a critical point. Verification shows the minimum occurs at $\theta = \frac{\pi}{4}$.
Correct Answer: 4