JEE Advanced · Mathematics · 3. Complex Numbers

Let $S$ be the set of all complex numbers $z$ satisfying $|z^{2} + z + 1| = 1$ . Then which of the following statements is/are TRUE?

A $|z + \frac{1}{2}| \leq \frac{1}{2}$ for all $z \in S$
B $|z| \leq 2$ for all $z \in S$
C $|z + \frac{1}{2}| \geq \frac{1}{2}$ for all $z \in S$
D The set $S$ has exactly four elements

Verified Solution

Answer & Solution

Correct Answer

(C) $|z + \frac{1}{2}| \geq \frac{1}{2}$ for all $z \in S$

Step-by-step Solution

Detailed explanation

z^{2} + z + 1 = e^{i θ} θ \in (- π, π]

\Rightarrow z^{2} + z + 1 - e^{i θ} = 0

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let S be the set of all complex numbers z satisfying z2+z+1=1. Then which of the following statements is/are TRUE?

Step-by-step Solution

Let $S$ be the set of all complex numbers $z$ satisfying $|z^{2} + z + 1| = 1$ . Then which of the following statements is/are TRUE?