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Consider the set S consisting of all complex numbers z such that the magnitude of z² + z + 1 equals 1. Which of the following statements holds true?

1. The condition z + 1/2 is less than or equal to 1/2 is satisfied for all z in S.
2. The magnitude of z is at most 2 for every z in S.
3. The condition z + 1/2 is greater than or equal to 1/2 is satisfied for all z in S.
4. The set S contains exactly four distinct elements.

Correct answer: The magnitude of z is at most 2 for every z in S.

Solution

The condition |z² + z + 1| = 1 defines a locus in the complex plane. By analyzing the geometry of this locus, it is determined that the magnitude of z is at most 2 for all points in the set S.

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