KVPY SX SB Exam 20141 Question Paper

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Question : 81 of 120

Marks: +1, -0

PART II

MATHEMATICS

Let n > 3 be an integer. For a permutation

σ = (a_1,a_2,...,a_ n) \of\ ( 1 , 2,...,n)

we let

f _σ(x) = a_nx ^{n-1}+ a_{n-1}x^{n-2} +.....+ a_2x + a_1

. Let

S_σ

be the sum o f the roots of

f_σ (x) = 0

and let S denote the sum over all permutations σ of (1 ,2 ,...,n) of the numbers

S_σ.

Then

$S <-n!$
$ -n!
$0< S <-n!$
$n!< S$

Solution:

S =-[{λ-a_{n-1}}/a_{n-1}+{λ-a_n}/a_n+......{λ-a_1}/a_1]

∀ λ = 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 )

from A.M. ≥ H.M.

(a_1 + a_2 + ....... + a_n)(1/a_1 + 1/a_2 + ....... + 1/a )≥n^2

S ≤ -n(n -1)

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