Complex Numbers And Quadratic Equations question 737
Question: Let $ \alpha $ and $ \beta $ be the roots of the equation $ x^{2}+x+1=0 $ The equation whose roots are $ {{\alpha }^{19}},{{\beta }^{7}} $ is [IIT Screening 1994]
Options:
A) $ x^{2}-x-1=0 $
B) $ x^{2}-x+1=0 $
C) $ x^{2}+x-1=0 $
D) $ x^{2}+x+1=0 $
Show Answer
Answer:
Correct Answer: D
Solution:
Given $ x^{2}+x+1=0 $ \ $ x=\frac{1}{2}[-1\pm i\sqrt{3}]=\frac{1}{2}(-1+i\sqrt{3}),\frac{1}{2}(-1-i\sqrt{3}) $ $ =\omega ,{{\omega }^{2}} $ But $ {{\alpha }^{19}}={{\omega }^{19}}=\omega $ and $ {{\beta }^{7}}={{\omega }^{14}}={{\omega }^{2}}. $ Hence the equation will be same.
BETA
