Complex Numbers And Quadratic Equations question 308
Question: Let $ x=\alpha +\beta ,y=\alpha \omega +\beta {{\omega }^{2}},z=\alpha {{\omega }^{2}}+\beta \omega ,\omega $ is an imaginary cube root of unity. Product of xyz is [Orissa JEE 2005]
Options:
A) $ {{\alpha }^{2}}+{{\beta }^{2}} $
B) $ {{\alpha }^{2}}-{{\beta }^{2}} $
C) $ {{\alpha }^{3}}+{{\beta }^{3}} $
D) $ {{\alpha }^{3}}-{{\beta }^{3}} $
Show Answer
Answer:
Correct Answer: D
Solution:
$ x=\alpha +\beta ,y=\alpha \omega +\beta {{\omega }^{2}},z=\alpha {{\omega }^{2}}+\beta \omega $
$ \therefore $ $ xyz=(\alpha +\beta )(\alpha \omega +\beta {{\omega }^{2}})(\alpha {{\omega }^{2}}+\beta \omega ) $ = $ (\alpha +\beta )[{{\alpha }^{2}}+\alpha \beta (\omega +{{\omega }^{2}})+{{\beta }^{2}}] $ = $ (\alpha +\beta )({{\alpha }^{2}}-\alpha \beta +{{\beta }^{2}})={{\alpha }^{3}}+{{\beta }^{3}} $ .
BETA
