InterviewSolution
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InterviewSolution
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If `alpha,beta and gamma` are roots of equation `x^(3)-3x^(2)+1=0`, then the value of `((alpha)/(1+alpha))^(3)+((beta)/(1+beta))^(3)+((gamma)/(1+gamma))^(3)` is __________. |
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Answer» Correct Answer - -1 Let `alpha_(1)=(alpha)/(1+alpha),beta_(1)=(beta)/(1+beta),andgamma_(1)=(gamma)/(1+gamma)`, Also, let `y=alpha_(1),beta_(1) and gamma_(1)` be roots of an equation is y. Then `y=(x)/(1+x)` `therefore x =(y)/(1-y)` Now,`x^(3)-3x^(2)+1=0` `rArr((y)/(1-y))^(3)-3((y)/(1-y))^(2)+1=0` `rArr3y^(3)-3y+1=0` `rArralpha_(1)+beta_(1)+gamma_(1)=0 and alpha_(1)beta_(1)gamma_(1)=-(1)/(3)` `therefore alpha_(1)^(3)+beta_(1)^(3)+gamma_(1)^(3)=3alpha_(1)beta_(1)gamma_(1)=3(-(1)/(3))=-1` |
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