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Given $z_{1} = x^{2020} + i y^{2020}, z_{2} = x^{2019} + i y^{2019}$ where
$x, y \in R, x \neq 0, y \neq 0, x \neq y$ . Then which of the following must be true for every x, y in the given domain, is/are ……… (Arg(z) means principal argument of ‘z’, where
Arg (z) $\in$ , $(- π, π],$ are (z) means argument of z)

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$A r g (- {\bar{z}}_{1})$ is always given by $π - A r g (z_{1})$

$A r g (- z_{2})$ is always given by $- π + A r g (z_{2})$

$A r g (z_{1} z_{2}) = A r g (z_{2}) + A r g (z_{1}) + 2 | k | π, k \in I - {0}$

$A r g ({\bar{z}}_{2}) + A r g ({\bar{z}}_{1}) = - (A r g (z_{1}) + A r g (z_{2}))$

answer is A, D.

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Detailed Solution

$A \to a l w a y s t r e u B \to f a i l s (e x : x < 0, y < 0 i n z_{2}) C \to s i m p l y f a i l s (i f x < 0, y < 0) D \to a l w a y s t r u e$

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