Complex Numbers And Quadratic Equations question 272
Question: If $ \alpha $ and $ \beta $ are imaginary cube roots of unity, then the value of $ {{\alpha }^{4}}+{{\beta }^{28}}+\frac{1}{\alpha \beta } $ ,is [MP PET 1998]
Options:
A) 1
B) $ -1 $
C) 0
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Since $ \alpha $ and $ \beta $ are complex roots of unity, we may write $ \alpha =\omega ,\beta ={{\omega }^{2}} $ Hence, $ {{\alpha }^{4}}+{{\beta }^{28}}+\frac{1}{\alpha \beta }={{\omega }^{4}}+{{({{\omega }^{2}})}^{28}}+\frac{1}{\omega .{{\omega }^{2}}} $ $ =\omega +{{\omega }^{56}}+1=\omega +{{\omega }^{2}}+1=0 $
BETA
