Complex Numbers And Quadratic Equations question 233
Question: If $ iz^{4}+1=0 $ , then $ z $ can take the value [UPSEAT 2004]
Options:
A) $ \frac{1+i}{\sqrt{2}} $
B) $ \cos \frac{\pi }{8}+i\sin \frac{\pi }{8} $
C) $ \frac{1}{4i} $
D) i
Show Answer
Answer:
Correct Answer: B
Solution:
$ iz^{4}=-1 $ $ z^{4}=\frac{-1}{i}\Rightarrow z^{4}=i\Rightarrow z={{(i)}^{1/4}} $ $ z={{(0+i)}^{1/4}} $ $ z={{( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} )}^{1/4}} $ $ z=\cos \frac{\pi }{8}+i\sin \frac{\pi }{8} $ (using De Moivre?s theorem)
BETA
