Complex Numbers And Quadratic Equations question 647
Question: If $ z=\frac{\pi }{4}{{(1+i)}^{4}}( \frac{1-\sqrt{\pi }i}{\sqrt{\pi }+i}+\frac{\sqrt{\pi }-i}{1+\sqrt{\pi }i} ), $ then $ ( \frac{|z|}{am{p^{(z)}}} ) $ equals
Options:
A) 1
B) $ \pi $
C) $ 3\pi $
D) 4
Show Answer
Answer:
Correct Answer: D
Solution:
$ z=\frac{\pi }{4}{{(1+i)}^{4}}( \frac{1-\sqrt{\pi }i}{\sqrt{\pi }+i}+\frac{\sqrt{\pi }-i}{1+\sqrt{\pi }i} ) $ $ =\frac{\pi }{4}{{(1+i)}^{4}}[ \frac{1+\pi +\pi +1}{(\sqrt{\pi }+i)(1+\sqrt{\pi }i)} ]=\frac{\pi }{4}{{(1+i)}^{4}}\frac{2}{i} $ $ =\frac{\pi }{4}{{(2i)}^{2}}\frac{2}{i}=2\pi i\therefore ( \frac{|z|}{amp(z)} )=\frac{2\pi }{\frac{\pi }{2}}=4 $
BETA
