Complex Numbers And Quadratic Equations question 264
Question: If $ n $ is a positive integer greater than unity and $ z $ is a complex number satisfying the equation $ z^{n}={{(z+1)}^{n}} $ , then
Options:
A) $ Re(z)<0 $
B) $ Re(z)>0 $
C) $ Re(z)=0 $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
We have $ z^{n}={{(1+z)}^{n}}\Rightarrow {{( \frac{z}{z+1} )}^{n}}=1 $
Þ $ \frac{z}{z+1}={1^{1/n}} $
Þ $ \frac{z}{z+1} $ is a $ n $ th root of unity
Þ $ | \frac{z}{z+1} |=1 $
Þ $ \frac{|z|}{|z+1|}=1 $
Þ $ |z|=|z+1| $
Þ $ x+\frac{1}{2}=0 $
Þ $ x=\frac{-1}{2} $
Þ $ Re(z)<0 $ .
BETA
