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If $|z| = 1, z \neq 1,$ then value of $a r g (\frac{1}{1 - z})$ cannot exceed

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a

π/2

b

π

c

3π/2

d

2π

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detailed solution

Correct option is A

As |z|=1, z≠1, z=cos⁡θ+isin⁡θ,−π<θ≤π, θ≠0. Now ω=11−z=11−cos⁡θ−isin⁡θ =(1−cos⁡θ)+isin⁡θ(1−cos⁡θ)2+sin2⁡θ=(1−cos⁡θ)+isin⁡θ2(1−cos⁡θ)=12+i2cot⁡θ2=12+i2tan⁡π2−θ2This shows that w lies on the line x =1/2 and −π/2

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Suppose $z$ is a complex number such that $z \neq - 1$ , $| z | = 1,$ and $\arg (z) =$ $θ$ Let $ω$ $= \frac{z (1 - \bar{z})}{\bar{z} (1 + z)}$ ,than $Re (ω)$ ) is equal to

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