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Let the complex number $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ be such that $\frac{2 \mathrm{z}-3 \mathrm{i}}{2 \mathrm{z}+\mathrm{i}}$ is purely imaginary. If $\mathrm{x}+\mathrm{y}^{2}=0$,

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  • 22 October, 2026 at 6:00 pm #12381
    UPLIFTJEE
    Keymaster

    Let the complex number $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ be such that $\frac{2 \mathrm{z}-3 \mathrm{i}}{2 \mathrm{z}+\mathrm{i}}$ is purely imaginary. If $\mathrm{x}+\mathrm{y}^{2}=0$, then $\mathrm{y}^{4}+\mathrm{y}^{2}-\mathrm{y}$ is equal to
    [JEE (Main) 10 April 2023 S I]
    (A) $\frac{3}{2}$
    (B) $\frac{4}{3}$
    (C) $\frac{2}{3}$
    (D) $\frac{3}{4}$

    22 October, 2026 at 6:00 pm #12382
    UPLIFTJEE
    Keymaster

    Solution

    $\frac{2 \mathrm{z}-3 \mathrm{i}}{2 \mathrm{z}+\mathrm{i}}$ is purely imaginary : $\frac{2 \mathrm{z}-3 \mathrm{i}}{2 \mathrm{z}+\mathrm{i}}+\frac{2 \overline{\mathrm{z}}-3 \mathrm{i}}{2 \overline{\mathrm{z}}-\mathrm{i}}=0$
    $\mathrm{z}=\mathrm{x}+\mathrm{iy} \Rightarrow 4 \mathrm{x}^{2}+4 \mathrm{y}^{2}-4 \mathrm{y}-3=0$ Given that $\mathrm{x}+\mathrm{y}^{2}=0 \Rightarrow \mathrm{y}^{4}+\mathrm{y}^{2}-\mathrm{y}=3 / 4$

    22 October, 2026 at 6:00 pm #12383
    UPLIFTJEE
    Keymaster

    Answer

    D

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