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Solve an equation in z and conj(z) by putting z = x + iy, then evaluate a quantity over its solutions

This type of question, asked 7 times: 27 Jun 2022, 28 Jun 2022, 27 Jul 2022, 13 Apr 2023, 30 Jan 2024, 4 Apr 2024, 22 Jan 2026 S2. All 7 questions below, oldest first.

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  1. Question 1

    27 Jun 2022

    61
    The area of the polygon, whose vertices are the non-real roots of the equation Zˉ=iz2\bar{Z}= iz^{2} is :

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  2. Question 2

    28 Jun 2022

    178
    Sum of squares of modulus of all the complex numbers zz satisfying z‾=iz2+z2−z\overline{z}=iz^{2}+z^{2}- z is equal to

  3. Question 3

    27 Jul 2022

    90
    Let S={z∈C:z2+zˉ=0}S =\left\{ z\in C:z^{2}+\bar{z}= 0 \right\}. Then ∑z∈S(Re(z)+Im(z))\sum_{z \in S} (Re(z) + Im(z)) is equal to

  4. Question 4

    13 Apr 2023

    166
    Let S={Z∈C:zˉ=i(z2+Re(zˉ))}S =\left\{ Z \in C:\bar{z}= i\left( z^{2}+ Re(\bar{z}) \right) \right\}. Then ∑z∈S∣z∣2\sum_{z \in S} |z|^{2} is equal to

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  5. Question 5

    30 Jan 2024

    63
    If z=x+z = x + iy, xy≠0xy \neq 0, satisfies the equation z2+izˉ=0z^{2}+ i\bar{z}= 0, then ∣z2∣\left| z^{2} \right| is equal to:

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  6. Question 6

    4 Apr 2024

    75
    Let α\alpha and β\beta be the sum and the product of all the non-zero solutions of the equation (zˉ)2+∣z∣=0,z∈C(\bar{z})^{2}+ |z| = 0,z\in C. Then 4(α2+β2)4\left( \alpha^{2}+\beta^{2} \right) is equal to:

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  7. Question 7

    22 Jan 2026 S2

    54
    Let S={z∈C:4z2+z‾=0}S =\left\{ z\in C:4z^{2}+\overline{z}= 0 \right\}. Then ∑z∈S∣z∣2\sum_{z \in S} |z|^{2} is equal to :

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