KVPY SA Exam 05-Nov-2017 Question Paper

Question : 13 of 80

Marks: +1, -0

Let A, G and H be the arithmetic mean, geometric mean and harmonic mean, respetively of two distinct positive real numbers. If α is the smallest of the two roots of the equation

A(G -H)x^2 + G (H -A) x + H(A -G) = 0

, then

$-2 < α< -1$
$0 < α< 1$
$-1 < α< 0$
$1 < α< 2$

Validate

Solution:

x = 1 is a root as sum of coefficient = 0
Now

αβ={H(A-G)}/{A(G+H)

Put

β = 1

α={HA -HG}/{AG- AH}={G^2- HG}/{AG- AH}

={G(G -H)}/{A(G -H)}=G/A<1

[as A.M. > G.M.]

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