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InterviewSolution
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If `alpha` is complex fifth root of unity and `(1+alpha +alpha^(2)+ alpha^(3))^(2005) = p + qalpha + ralpha^(2) + salpha^(3)` (where p,q,r,s are real), then find the value of `p+ q+r+s`. |
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Answer» Correct Answer - `-1` `alpha ` is a complex fifth roots of unity. `therefore 1+ alpha +alpha^(2) + alpha^(3)+ alpha^(4) = 0` Now, given that `(1+ alpha + alpha^(3) + alpha^(3))^(2005) = p + qalpha + ralpha^(2) + salpha^(3)` `rArr (-alpha^(4)) ^(2005) = p + qalpha + ralpha^(2) + salpha^(3)` `rArr - alpha^(8020) = p+ qalpha + ralpha^(2)+ salpha^(3)` `rArr -(alpha^(5))^(1640) = p +qalpha+ = ralpha^(2) + salpha^(3)` `rArr -1 = p + qalpha + ralpha^(2) + salpha^(3)` `rArr p = - 1, q = r = s = 0` `therefore p + q+ r+s = -1` |
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