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Q.

The descending order of the principal values of arguments of the complex numbers $z_{1} = 1 + i$ , $z_{2} = - \sqrt{3} + i$ , $z_{3} = 1 + \sqrt{3} i$ , $z_{4} = 1 - i$ is

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a

$z_{2}, z_{3}, z_{1}, z_{4}$

b

$z_{4}, z_{1}, z_{3}, z_{2}$

c

$z_{3}, z_{1}, z_{2}, z_{4}$

d

$z_{3}, z_{2}, z_{1}, z_{4}$

answer is B.

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Detailed Solution

$z_{1} = 1 + i$	$Arg (z_{1}) = {Tan}^{- 1} 1 = \frac{π}{4}$
$z_{2} = - \sqrt{3} + i$	$Arg (z_{2}) = π - {Tan}^{- 1} \frac{1}{\sqrt{3}} = π - \frac{π}{6} = \frac{5 π}{6}$
$z_{3} = 1 + i \sqrt{3}$	$Arg z_{3} = {Tan}^{- 1} (\sqrt{3}) = \frac{π}{3}$
$z_{4} = 1 - i$	$Arg z_{4} = {Tan}^{- 1} (- 1) = \frac{- π}{4}$

$Descending order z_{2}, z_{3}, z_{1}, z_{4}$

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