Complex Numbers And Quadratic Equations question 818
If a, b and c are the roots of equation $ x^{3}-3x^{2}+x+5=0 $ then $ y=\sum {{\alpha }^{2}}+\alpha \beta \gamma $ satisfies the equation [J & K 2005]
Options:
A) $ y^{3}+y+2=0 $
B) $ y^{3}-y^{2}-y-2=0 $
C) $ y^{3}+3y^{2}-y-3=0 $
D) $ y^{3}+4y^{2}+5y+20=0 $
Show Answer
Answer:
Correct Answer: B
Solution:
Given equation $ x^{3}-3x^{2}+x+5=0 $ . Then $ \alpha +\beta +\gamma =3 $ , $ \alpha \beta +\beta \gamma +\gamma \alpha =1 $ , $ \alpha \beta \gamma =-5 $ $ y=\Sigma {{\alpha }^{2}}+\alpha \beta \gamma ={{(\alpha +\beta +\gamma )}^{2}}-2(\alpha \beta +\beta \gamma +\gamma \alpha )+\alpha \beta \gamma $ = $ 9-2-5=2 $
$ \therefore $ $ y=2 $ It satisfies the equation $ y^{3}-y^{2}-y-2=0 $ .
BETA
