Complex Numbers And Quadratic Equations question 461
Question: Sum of common roots the equations $ z^{3}+2z^{2}+2z+1=0 $ and $ z^{1985}+z^{100}+1=0 $
Options:
A) -1
B) 1
C) 0
D) 1
Show Answer
Answer:
Correct Answer: A
Solution:
[a] we have, $ z^{3}+2z^{2}+2z+1=0 $
$ \Rightarrow (z^{3}+1)+2z(z+1)=0 $
$ \Rightarrow (z+1)(z^{2}+z+1)=0 $
$ \Rightarrow z=-1, $ $ \omega $ , $ {{\omega }^{2}} $ Since z=-1 does not satisfy $ z^{1985}+z^{100}+1=0 $ while z= $ \omega $ , $ {{\omega }^{2}} $ satisfy it; hence, sum is $ \omega +{{\omega }^{2}}=-1 $ .
BETA
