Question

Let $z$ be a complex number such that $z^{3}+iz^{2}-iz+1=0$ where $i^{2}=-1$. Then $|z|=$

• A. 2
• B. $\frac{1}{2}$
• C. 1
• D. $\frac{1}{4}$
• E. 3

Hint:

$|z^n| = |z|^n$ is a useful property for complex equations.

Answer 1

The Correct Option is C

Step 1: Analysis
Factor the equation: $z^2(z + i) - i(z + i) = 0$. $(z^2 - i)(z + i) = 0$.

Step 2: Analysis
Case 1: $z = -i$. Then $|z| = |-i| = 1$. Case 2: $z^2 = i$. Then $|z^2| = |i| \implies |z|^2 = 1 \implies |z| = 1$.

Step 3: Conclusion
In both cases, $|z| = 1$. Final Answer: (C)