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4-1.Complex numbers
normal
If $z$ is a complex number such that $|z - \bar{z}| = 2$ and $|z + \bar{z}| = 4 $, then which of the following is always incorrect -
A
$Amp(z)\in(-\frac{\pi}{6},0)$
B
$Amp(z)\in(\frac{5\pi}{6},\pi)$
C
$Amp(z)\in(0,\frac{\pi}{6})$
D
$Amp(z)\in(\frac{\pi}{6},\frac{\pi}{4})$
Solution
Let $z=x+i y \Rightarrow|z+\bar{z}|=4 \Rightarrow x=\pm 2$
$|z-\bar{z}|=2 \Rightarrow y=\pm 1$
If $\mathrm{Amp}(\mathrm{z})=\theta$ then $\tan \theta=\pm \frac{1}{2}$
$\Rightarrow \theta \in\left(0, \frac{\pi}{6}\right),\left(-\frac{\pi}{6}, 0\right),\left(\frac{5 \pi}{6}, \pi\right),\left(-\pi, \frac{-5 \pi}{6}\right)$
Standard 11
Mathematics
