Complex Numbers And Quadratic Equations question 66
Question: Let $ z $ be a complex number, then the equation $ z^{4}+z+2=0 $ cannot have a root, such that
Options:
A) $ |z|<1 $
B) $ |z|=1 $
C) $ |z|>1 $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Suppose there exists a complex number $ z $ which satisfies the given equation and is such that $ |z|<1 $ . Then $ z^{4}+z+2=0 $
Þ $ -2=z^{4}+z $
Þ $ |-2|=|z^{4}+z| $
Þ $ 2\le |z^{4}|+|z| $
Þ $ 2<2, $ because $ |z|<1 $ But $ 2<2 $ is not possible. Hence given equation cannot have a root $ z $ such that $ |z|<1 $
BETA
