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 $\left(\frac{|Z|}{amp\left(Z\right)}\right)$ 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)

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