z is a unimodular complex number Statement 1: arg(z2+z¯)=argz Statement 2: z¯=cos(argz)−isin(argz)

|z|=1⇒z=cosθ+isinθ where θ=argz z¯=cosθ−isinθStatement -2 is truearg(z2+z¯)=arg{cos2θ+isin2θ+cosθ−isinθ}=tan−1sin2θ−sinθcos2θ+cosθ=tan−12cos3θ2sinθ22cos3θ2cosθ2 =tan−1tanθ2 =θ2=12argz

z is a unimodular complex number Statement 1: arg(z2+z¯)=argz Statement 2: z¯=cos(argz)−isin(argz)

|z|=1⇒z=cosθ+isinθ where θ=argz z¯=cosθ−isinθStatement -2 is truearg(z2+z¯)=arg{cos2θ+isin2θ+cosθ−isinθ}=tan−1sin2θ−sinθcos2θ+cosθ=tan−12cos3θ2sinθ22cos3θ2cosθ2 =tan−1tanθ2 =θ2=12argz

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z is a unimodular complex number
Statement 1: $\arg (z^{2} + \bar{z}) = \arg z$

Statement 2: $\bar{z} = \cos (\arg z) - i \sin (\arg z)$

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Statement -1 is false, statement-2 is true.

Statement -1 is true, statement-2 is true and statement-2 is correct explanation for statement-1

Statement-1 is true, statement-2 is false

Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1

answer is D.

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Detailed Solution

$| z | = 1 \Rightarrow z = \cos θ + i \sin θ$ where $θ = \arg z$

$\bar{z} = \cos θ - i \sin θ$

Statement -2 is true

$\arg (z^{2} + \bar{z}) = \arg {\cos 2 θ + i \sin 2 θ + \cos θ - i \sin θ}$

$= \tan^{- 1} \frac{\sin 2 θ - \sin θ}{\cos 2 θ + \cos θ}$

$= \tan^{- 1} \frac{2 \cos \frac{3 θ}{2} \sin \frac{θ}{2}}{2 \cos \frac{3 θ}{2} \cos \frac{θ}{2}} = \tan^{- 1} \tan \frac{θ}{2} = \frac{θ}{2} = \frac{1}{2} \arg z$