Question

For a complex number $Z$ , if $Z=\frac{\pi }{4}\left(1 + i\right)^{4}\left(\frac{1 - \sqrt{\pi } i}{\sqrt{\pi } + i} + \frac{\sqrt{\pi } - i}{1 + \sqrt{\pi } i}\right),$ then the value of $\left(\frac{\left|Z\right|}{amp \left(Z\right)}\right)$ is equal to (where $amp\left(Z\right)\in \left(- \pi , \pi \right]$ )

• A. $\pi $
• B. $4$
• C. $1$
• D. $3\pi $

Answer 1

Answer: (B)

$Z=\frac{\pi }{4}\left(\left(1 + i\right)^{2}\right)^{2}\left(\frac{\left(1 - \sqrt{\pi } i\right) \left(1 + \sqrt{\pi } i\right) + \left(\sqrt{\pi } - i\right) \left(\sqrt{\pi } + i\right)}{\sqrt{\pi } + \pi i + i - \sqrt{\pi }}\right)$
$\Rightarrow Z=\frac{\pi }{4}\left(1 - 1 + 2 i\right)^{2}\left(\frac{1 + \pi + \pi + 1}{\left(\pi + 1\right) i}\right)$
$\Rightarrow Z=\frac{\pi }{4}4\left(- 1\right)\left(\frac{2}{i}\right)=-\frac{2 \pi }{i}=2\pi i$
$\Rightarrow \left|Z\right|=2\pi $ and $amp\left(Z\right)=\frac{\pi }{2}$
$\Rightarrow \frac{\left|Z\right|}{amp \left(Z\right)}=4$