Complex Numbers
Modulus of a Complex Number
MJAT None
Grade 12
Question:
If $|z^2 + z + 1| = 1$, then which of the following statements is/are TRUE?
A) $|z+ 1/2| \le 1/2$ for all $z \in S$
B) $|z| \le 2$ for all $z \in S$
C) $|z+ 1/2| \ge 1/2$ for all $z \in S$
D) The set $S$ has exactly four elements
Step-by-Step Solution
Key Concept: The key idea here is that we can use the quadratic formula to solve the equation $z^2 + z + 2 = 0$.
We start by noting that $|z^2 + z + 1| = 1$ implies $z^2 + z + 1 = \pm 1$. We then consider the two cases separately.
$$
z^2 + z + 1 = 1 \quad \text{and} \quad z^2 + z + 1 = -1
$$
For the first case, we have $z^2 + z = 0$, so $z(z + 1) = 0$. This gives us $z = 0$ or $z = -1$.
For the second case, we have $z^2 + z + 2 = 0$. Using the quadratic formula, we get $z = \frac{-1 \pm \sqrt{1 - 8}}{2} = \frac{-1 \pm \sqrt{-7}}{2}$.
Therefore, the solutions are $z = 0, -1, \frac{-1 \pm \sqrt{-7}}{2}$.
Correct Answer: A, B, C