If -π

Question

If -π < arg (z) < -(π/2) then arg barz - arg (- barz) is (A) π  (B) -π  (C) (π/2)  (D) -(π/2)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/5a65779522e11c425b272537c5588038-.png" /> if arg (z) =-π+θ ⇒ arg ( barz) =π-θ arg (- barz) =-θ arg ( barz)- arg (- barz) = π-θ-(-θ) = π-θ+θ=π

Q. If $- π < a r g (z) < - \frac{π}{2}$ then arg $\overset{z}{ˉ} - a r g (- \overset{z}{ˉ})$ is

$π$

$- π$

$\frac{π}{2}$

$- \frac{π}{2}$

if $ar g (z) = - π + θ$
$\Rightarrow ar g (\overset{z}{ˉ}) = π - θ$
$ar g (- \overset{z}{ˉ}) = - θ$
$ar g (\overset{z}{ˉ}) - ar g (- \overset{z}{ˉ})$
$= π - θ - (- θ)$
$= π - θ + θ = π$

Q. If −π<arg(z)<−2π​ then arg zˉ−arg(−zˉ) is

π

−π

2π​

−2π​

if arg(z)=−π+θ ⇒arg(zˉ)=π−θ arg(−zˉ)=−θ arg(zˉ)−arg(−zˉ) =π−θ−(−θ) =π−θ+θ=π

Q. If $- π < a r g (z) < - \frac{π}{2}$ then arg $\overset{z}{ˉ} - a r g (- \overset{z}{ˉ})$ is

$π$

$- π$

$\frac{π}{2}$

$- \frac{π}{2}$

if $ar g (z) = - π + θ$
$\Rightarrow ar g (\overset{z}{ˉ}) = π - θ$
$ar g (- \overset{z}{ˉ}) = - θ$
$ar g (\overset{z}{ˉ}) - ar g (- \overset{z}{ˉ})$
$= π - θ - (- θ)$
$= π - θ + θ = π$