Q50 / 160 · TS EAMCET 2023 (14 May Shift 2) · Mathematics

If $z=x+i y$ is a complex number such that $z \bar{z}+\bar{z} z^3=350$ and $x, y$ are integers, then $|z|=$

  • A
    $\sqrt{41}$
  • Correct answer
    B
    5
  • C
    25
  • D
    $\sqrt{13}$

Solution

$\mathrm{Z} \overline{\mathrm{Z}}^3+\overline{\mathrm{Z}} \mathrm{Z}^3=350$ $\begin{aligned} & \mathrm{Z} \cdot \overline{\mathrm{Z}}\left(\overline{\mathrm{Z}}^2+\mathrm{Z}^2\right)=350 \\ & |\mathrm{Z}|^2\left((x-i y)^2+(x+i y)^2\right)=350 \\ & |\mathrm{Z}|^2\left(x^2-y^2-2 x y i+x^2-y^2+2 x y i\right)=350 \\ & 2 \cdot|\mathrm{Z}|^2\left(x^2-y^2\right)=350 \\ & |\mathrm{Z}|^2\left(x^2-y^2\right)=175=25 \times 7 \\ & \therefore|\mathrm{Z}|^2=25 \Rightarrow|\mathrm{Z}|=5\end{aligned}$

Hence, the correct answer is option: 'B'.

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