The multiplicative inverse of $\mathrm{z}$ is

The multiplicative inverse of $\mathrm{z}$ is
  1. $\frac{1}{z+\bar{z}}$
  2. $\frac{z}{|\bar{z}|}$
  3. $\frac{\bar{z}}{|z|^2}$
  4. $\frac{1}{\bar{z}}$

Solution

Let multiplicative inverse of $\mathrm{z}$, be 'A' hence $\mathrm{z} \cdot \mathrm{A}=1$ $\Rightarrow \quad A=\frac{1}{z}=\frac{1 \cdot \bar{z}}{z \cdot \bar{z}} \quad$ (where $\bar{z}$ is conjugate of $z$ ) $\Rightarrow A=\frac{\bar{z}}{z \bar{z}}=\frac{\bar{z}}{|z|^2} \quad\left\{\because z \bar{z}=|z|^2\right\}$ Hence option (c) is correct.

Asked in: AP EAMCET 2023 (19 May Shift 1)

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