If z is a unimodular complex number, then arg z2+z¯=

z in a unimodular ⇔|Z|=1Let z=cos⁡θ+isin⁡θ, then z¯=cos⁡θ−isin⁡θarg⁡z2+z¯=arg⁡(cos⁡2θ+isin⁡2θ+cos⁡θ−isin⁡θ)=tan−1⁡sin⁡2θ−sin⁡θcos⁡2θ+cos⁡θ=tan−1⁡tan⁡θ2=θ2=12arg⁡z

If z is a unimodular complex number, then arg z2+z¯=

z in a unimodular ⇔|Z|=1Let z=cos⁡θ+isin⁡θ, then z¯=cos⁡θ−isin⁡θarg⁡z2+z¯=arg⁡(cos⁡2θ+isin⁡2θ+cos⁡θ−isin⁡θ)=tan−1⁡sin⁡2θ−sin⁡θcos⁡2θ+cos⁡θ=tan−1⁡tan⁡θ2=θ2=12arg⁡z

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If z is a unimodular complex number, then arg $(z^{2} + \bar{z}) =$

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$\frac{1}{2} \arg z$

$- \frac{1}{2} \arg z$

$\arg z$

$- \arg z$

answer is A.

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Detailed Solution

z in a unimodular $\Leftrightarrow | Z | = 1$
Let $z = \cos θ + isin θ$ , then $\bar{z} = \cos θ - isin θ$
$\begin{array}{l} \arg (z^{2} + \bar{z}) = \arg (\cos 2 θ + isin 2 θ + \cos θ - isin θ) = \tan^{- 1} (\frac{\sin 2 θ - \sin θ}{\cos 2 θ + \cos θ}) \\ = \tan^{- 1} [\tan \frac{θ}{2}] = \frac{θ}{2} = \frac{1}{2} \arg z \end{array}$