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|z| + z = 3 + i (or 2 + i): find |z|

The same question, asked 4 times: May 2022 S1, 11 May 2023 S1*, 15 May 2023 S1*, 26 Apr 2025 S2. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    May 2022 S1

    138
    Let z be a complex number such that ∣z∣+z=3+i|z| + z = 3 + i, i=−1i = \sqrt{-1}, then ∣z∣|z| is equal to

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  2. Question 2

    11 May 2023 S1

    102
    Let Z be a complex number such that ∣Z∣+Z=2+i|Z| + Z = 2 + i (where i=−1i = \sqrt{-1}), then ∣Z∣|Z| is equal to

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  3. Question 3

    15 May 2023 S1

    110
    Let zz be a complex number such that ∣z∣+z=2+i|z|+z = 2+i, where i=−1i=\sqrt{-1}, then ∣z∣|z| is equal to

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  4. Question 4

    26 Apr 2025 S2

    119
    Let zz be the complex number such that ∣z∣+z=3+i|z| +z= 3 +i where i=−1i=\sqrt{- 1}, then ∣z∣=|z| =

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