If $z$ and $w$ are two complex numbers such that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$,then
- A$\bar{z}w = i$
- B$z\bar{w} = \frac{-1 + i}{\sqrt{2}}$
- C$z\bar{w} = \frac{1 - i}{\sqrt{2}}$
- D$\bar{z}w = -i$
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If $a = \frac{1 - i \sqrt{3}}{2}$, then the correct matching of List-$I$ with List-$II$ is:
List-$I$ List-$II$ $(i)$ $a \bar{a}$ $(A)$ $-\frac{\pi}{3}$ $(ii)$ $\arg \left(\frac{1}{\bar{a}}\right)$ $(B)$ $-i \sqrt{3}$ $(iii)$ $a - \bar{a}$ $(C)$ $2i / \sqrt{3}$ $(iv)$ $\operatorname{Im}\left(\frac{4}{3a}\right)$ $(D)$ $1$ $(E)$ $\pi / 3$ $(F)$ $\frac{2}{\sqrt{3}}$
| List-$I$ | List-$II$ |
| $(i)$ $a \bar{a}$ | $(A)$ $-\frac{\pi}{3}$ |
| $(ii)$ $\arg \left(\frac{1}{\bar{a}}\right)$ | $(B)$ $-i \sqrt{3}$ |
| $(iii)$ $a - \bar{a}$ | $(C)$ $2i / \sqrt{3}$ |
| $(iv)$ $\operatorname{Im}\left(\frac{4}{3a}\right)$ | $(D)$ $1$ |
| $(E)$ $\pi / 3$ | |
| $(F)$ $\frac{2}{\sqrt{3}}$ |
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