Q37 / 100 · BITSAT 2024 (Memory Based Paper 2) · Mathematics

If $z_1, z_2, \ldots . . z_n$ are complex numbers such that $\left|z_1\right|=\left|z_2\right|=\ldots . .=\left|z_n\right|=1$, then $\left|z_1+z_2+\ldots . .+z_n\right|$ is equal to

  • A
    $\left|z_1 z_2 z_3 \ldots . . z_n\right|$
  • B
    $\left|z_1\right|+\left|z_2\right|+\ldots . .+\left|z_n\right|$
  • Correct answer
    C
    $\left|\frac{1}{z_1}+\frac{1}{z_2}+\ldots . .+\frac{1}{z_n}\right|$
  • D
    n

Solution

$\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=\ldots \ldots .=\left|z_n\right|=1$ $\Rightarrow z_1 \bar{z}_1=z_2 \bar{z}_2=z_3 \bar{z}_3=\ldots \ldots=z_n \bar{z}_n=1$ $\overline{\mathrm{z}}_1=\frac{1}{\mathrm{z}}, \overline{\mathrm{z}}_2=\frac{1}{\mathrm{z}_2}, \overline{\mathrm{z}}_3=\frac{1}{\mathrm{z}_3}, \ldots \ldots, \overline{\mathrm{z}}_{\mathrm{n}}=\frac{1}{\mathrm{z}_{\mathrm{n}}}$ Now, $\left|z_1+z_2+z_3+\ldots \ldots .+z_n\right|$ $=\left|\bar{z}_1+\bar{z}_2+\ldots \ldots+\bar{z}_n\right|$ $\left|\bar{z}_1+\bar{z}_2+\bar{z}_3+\ldots .+\bar{z}_n\right|=\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}+\ldots \ldots+\frac{1}{z_n}\right|$

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

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