If $a=\frac{1-i \sqrt{3}}{2}$, then the correct matching of 'List-I from List-II is List-I List-II (i) $a…

If $a=\frac{1-i \sqrt{3}}{2}$, then the correct matching of 'List-I from List-II is List-I List-II (i) $a \bar{a}$ (A) $-\frac{\pi}{3}$ (ii) $\arg \left(\frac{1}{\bar{a}}\right)$ (B) $-i \sqrt{3}$ (iii) $a-\bar{a}$ (C) $2 i / \sqrt{3}$ (iv) $\operatorname{Im}\left(\frac{4}{3 a}\right)$ (D) 1 (E) $\pi / 3$ (F) $\frac{2}{\sqrt{3}}$ correct match is (i) (ii) (iii) (iv)
  1. $\begin{array}{llll}D & \text { E } & \text { C } & \text { B }\end{array}$
  2. $\begin{array}{llll}\mathrm{D} & \mathrm{A} & \mathrm{B} & \mathrm{F}\end{array}$
  3. $\begin{array}{llll}\text { F } & \text { E } & \text { B } & \text { C }\end{array}$
  4. $\begin{array}{llll}D & A & B & C\end{array}$

Solution

Given, $a=\frac{1-i \sqrt{3}}{2}=\frac{1}{2}-\frac{i \sqrt{3}}{2}$ $\therefore \quad \bar{a}=\frac{1}{2}+\frac{i \sqrt{3}}{2}$ (i) $\begin{aligned} a \bar{a} & =\left(\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)\left(\frac{1}{2}+\frac{i \sqrt{3}}{2}\right) \\ & =\left(\frac{1}{2}\right)^2-i^2\left(\frac{\sqrt{3}}{2}\right)^2 \\ & =\frac{1}{4}+\frac{3}{4}=1\end{aligned}$ (ii) $\arg \left(\frac{1}{\bar{a}}\right)=\tan ^{-1}\left(\frac{-\sqrt{3}}{2} \times \frac{2}{1}\right)=-\frac{\pi}{3}$ (iii) $a-\bar{a}=\left(\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)-\left(\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)=-i \sqrt{3}$

Asked in: AP EAMCET 2007

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