Question

If $z=\frac{\pi }{4}(1+i{)}^4(\frac{1-\pi i}{\pi +i}+\frac{\pi -i}{1+\pi i})$, then $\left(\frac{|z|}{arg\left(z\right)}\right)$ equals

• A. $1$
• B. $\pi$
• C. 4
• D. Noe of these

Answer 1

Answer: (C)

4

Given:

$z=\frac{\pi }{4}(1+i{)}^4(\frac{1-\pi i}{\pi +i}+\frac{\pi -i}{1+\pi i})$

Let $P=(\frac{1-\pi i}{\pi +i}+\frac{\pi -i}{1+\pi i})=\frac{1-{\pi }^2{i}^2+{\pi }^2-i^2}{(\pi +i)(1+\pi i)}$

$\Rightarrow P=\frac{2+2{\pi }^2}{\pi +{\pi }^{2}i+i+\pi i^2}=\frac{2(1+{\pi }^2)}{i(1+{\pi }^2)}=\frac{2}{i}=-2i$

$\Rightarrow z=\frac{\pi }{4}{(1+i)}^4(-2i)$

$\Rightarrow {(1+i)}^2=(1+i)(1+i)=1+i^2+2i=1-1+2i=2i$

$\Rightarrow {(1+i)}^4={\left(2i\right)}^2=4i^2=-4$

$\Rightarrow z=\frac{\pi }{4}\times -4\times -2i=2\pi i$

$\Rightarrow \left|z\right|=2\pi$

$\Rightarrow arg\left(z\right)={\tan }^{-1}\left(\frac{2\pi }{0}\right)=\frac{\pi }{2}\Rightarrow \frac{\left|z\right|}{amp\left(z\right)}=\frac{2\pi }{\pi /2}=4$