Home
Class 12
MATHS
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)`
Promotional Banner

Topper's Solved these Questions

  • KVPY

    KVPY PREVIOUS YEAR|Exercise Part A - Mathematics|20 Videos

  • KVPY

    KVPY PREVIOUS YEAR|Exercise Part B- Mathematics|10 Videos

  • KVPY

    KVPY PREVIOUS YEAR|Exercise PART -I MATHEMATICS|20 Videos

  • KVPY 2021

    KVPY PREVIOUS YEAR|Exercise PART II MATHEMATICS|4 Videos

Similar Questions

Explore conceptually related problems

(1+x)^(n)=a_(0)+a_(1)x+a_(2)x^(2) +......+a_(n)x^(n) then Find the sum of the series a_(0) +a_(2)+a_(4) +……

Let alpha_(1),alpha_(2),......alpha_(n) are roots of a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)+....+a_(n)=0 then s_(3)=? ( s_n = sum of roots taken n at a time)

Differentiate |x|+a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-1)+...+a_(n-1)x+a_(n)

Let f(x)=a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)+......+a_(n),(a_(0)!=0) if a_(0)+a_(1)+a+_(2)+......+a_(n)=0 then the root of f(x) is

Let a_(1),a_(2),a_(3),...a_(n) be an AP.Prove that: (1)/(a_(1)a_(n))+(1)/(a_(2)a_(n-1))+(1)/(a_(3)a_(n-2))+......+(1)/(a_(n)a_(1))=

Let n be a positive integer and (1+x+x^(2))^(n)=a_(0)+a_(1)x+...+a^(2n)x^(2n) Show that a_(0)^(2)-a_(1)^(2)+a_(2)^(2)+...+a_(2n)^(2)=a_(n)

(1+x)^(n)=a_(0)+a_(1)x+a_(2)*x^(2)+......+a_(n)x^(n) then prove that

Given that (1+x+x^(2))^(n)=a_(0)+a_(1)x+a_(2)x^(2)+.......+a_(2n)x^(2n) find the value a_(0)+a_(1)+a_(2)+......+a_(2)n

If a_(0)=x , a_(n+1)=f*(a_(n)) , n=01,2,3,…., find a_(n) when f(x)=(1)/(1-x)

let f(x)=a_(0)+a_(1)x^(2)+a_(2)x^(4)+.........a_(n)x^(2n) where 0