The user wants me to solve a JEE Advanced question about complex numbers.

The question defines a set S of complex numbers z that satisfy the equation $|z^2 + z + 1| = 1$.

I need to check which of the given four statements about the elements of S are true. This is a multiple correct question (MCQM).

Let's analyze the given equation: $|z^2 + z + 1| = 1$.

Step 1: Simplify the given equation

To analyze the properties of z, it is convenient to complete the square for the expression inside the modulus. Let's make a substitution w = z + 1/2. This implies z = w - 1/2. Substituting this into the expression $z^2 + z + 1$: $z^2 + z + 1 = \left(w - {1 \over 2}\right)^2 + \left(w - {1 \over 2}\right) + 1$ $= \left(w^2 - w + {1 \over 4}\right) + \left(w - {1 \over 2}\right) + 1$ $= w^2 - w + {1 \over 4} + w - {1 \over 2} + 1 = w^2 + {3 \over 4}$ The given condition $|z^2 + z + 1| = 1$ transforms to: $\left|w^2 + {3 \over 4}\right| = 1, \quad \text{where } w = z + {1 \over 2}$

Step 2: Analyze options A and C

These options relate to the magnitude of z + 1/2, which is |w|. We can use the triangle inequality on the transformed equation $|w^2 + 3/4| = 1$.

First, using $|a+b| \le |a| + |b|$: $\left|w^2 + {3 \over 4}\right| \le |w^2| + \left|{3 \over 4}\right|$ $1 \le |w|^2 + {3 \over 4}$ $1 - {3 \over 4} \le |w|^2$ ${1 \over 4} \le |w|^2 \implies |w| \ge {1 \over 2}$ Since w = z + 1/2, we have $|z + 1/2| \ge 1/2$ for all $z \in S$. This shows that statement C is TRUE.

To check statement A, let's find the upper bound for |w|. Using the reverse triangle inequality, $|a+b| \ge ||a| - |b||$: $\left|w^2 + {3 \over 4}\right| \ge \left||w^2| - \left|{3 \over 4}\right|\right|$ $1 \ge \left||w|^2 - {3 \over 4}\right|$ This inequality is equivalent to $-1 \le |w|^2 - 3/4 \le 1$.

• The left side: $-1 \le |w|^2 - 3/4 \implies -1/4 \le |w|^2$, which is always true as $|w|^2 \ge 0$.
• The right side: $|w|^2 - 3/4 \le 1 \implies |w|^2 \le 7/4$.

Combining our findings, we have $1/4 \le |w|^2 \le 7/4$, which means $1/2 \le |w| \le \sqrt{7}/2$. So, $1/2 \le |z + 1/2| \le \sqrt{7}/2$.

Statement A says $|z + 1/2| \le 1/2$ for all $z \in S$. This is false because |z + 1/2| can be greater than 1/2. For instance, we can check if the maximum value $\sqrt{7}/2$ is attainable. If $|w| = \sqrt{7}/2$, then $|w|^2 = 7/4$. This requires the equality condition in the reverse triangle inequality, which means $w^2$ and 3/4 are on the same line and $w^2$ is negative real number. Let $w^2 = -7/4$. Then $|w^2 + 3/4| = |-7/4 + 3/4| = |-1| = 1$. This is satisfied. So, there exists z in S such that $|z+1/2| = \sqrt{7}/2 > 1/2$ (e.g., $w = i\sqrt{7}/2 \implies z = -1/2 + i\sqrt{7}/2$). Therefore, statement A is FALSE.

Step 3: Analyze option B

This option relates to the magnitude of z. We can use the triangle inequality on z = (z + 1/2) - 1/2: $|z| = \left| \left(z + {1 \over 2}\right) - {1 \over 2} \right| \le \left|z + {1 \over 2}\right| + \left|-{1 \over 2}\right|$ From Step 2, we know the maximum value of |z + 1/2| is $\sqrt{7}/2$. $|z| \le {\sqrt{7} \over 2} + {1 \over 2} = {1 + \sqrt{7} \over 2}$ The statement is $|z| \le 2$. We need to check if $(1 + \sqrt{7}) / 2 \le 2$. $1 + \sqrt{7} \le 4$ $\sqrt{7} \le 3$ $7 \le 9$ This is true. Thus, the maximum value of |z| is less than 2. This means $|z| \le 2$ for all $z \in S$. Therefore, statement B is TRUE.

Step 4: Analyze option D

The statement is that the set S has exactly four elements. The set S is the locus of points z satisfying the equation $|z^2 + z + 1| = 1$. This equation defines a curve in the complex plane (a Cassini oval). A curve contains infinitely many points. To show this, we can find more than four points in S.

• z=0: $|0^2+0+1| = |1| = 1$. So, $0 \in S$.
• z=-1: $|(-1)^2+(-1)+1| = |1-1+1| = |1| = 1$. So, $-1 \in S$.
• z=i: $|i^2+i+1| = |-1+i+1| = |i| = 1$. So, $i \in S$.
• z=-i: $|(-i)^2+(-i)+1| = |-1-i+1| = |-i| = 1$. So, $-i \in S$.
• $z = -1/2 + i\sqrt{7}/2$: We found this point in Step 2. It also belongs to S. Since we have already found at least five distinct elements, the set S cannot have exactly four elements. The set is infinite. Therefore, statement D is FALSE.

Conclusion

Based on the analysis:

• Statement A is FALSE.
• Statement B is TRUE.
• Statement C is TRUE.
• Statement D is FALSE.

The correct options are B and C.