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Let nge3 be an integer. For a permutaion...

Let `nge3` be an integer. For a permutaion `sigma=(a_(1),a_(2),.....,a_(n))` of (1,2,.....,n) we let `f_(sigma)(x)=a_(n)X^(n-1)+a_(n-1)X^(x-2)+....+a_(2)x+a_(1)`. Let `S_(sigma)` be the sum of the roots of `f_(sigma)(x)=0` and let S denote the sum over all permutations `sigma` of (1,2,.....,n) of the numbers `S_(sigma)`. Then-

A

`Slt0n!`

B

`-n!ltSlt0`

C

`0lt Slt n!`

D

`n!leS`

Text Solution

Verified by Experts

The correct Answer is:
B
`S=-[(lambda-a_(n))/(a_(jn))+(lambda-a_(n-1))/(a_(n-1))+.....+(lambda-a_(1))/(a_(1))]`
`AAlambda=a_(1)+a_(2)+....+a_(n)`
`S=-[(a_(1)+a_(2)+....+a_(n))((1)/(a_(1))+(1)/(a_(2))+.....+(1)/(a_(n)))-n]`
`S=n-(a_(1)+a_(2)+....+a_(n))((1)/(a_(1))+(1)/(a_(2))+.....+(1)/(a_(n)))`
Fom `A.M.geH.M`
`(a_(1)+a_(2)+....+a_(n))((1)/(a_(1))+(1)/(a_(2))+.....+(1)/(a_(n)))gen^(2)`
`Sle-n(n-1)`
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