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Question

Let A,G and H be the arithmetic mean, geometric mean and harmonic mean, respectively of two distinct positive real numbers. If α is the smallest of the two roots of the equation A(GH)x2+G(HA)x+H(AG)=0, then

A
2<α<1
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B
0<α<1
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C
1<α<0
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D
1<α<2
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Solution

The correct option is A 0<α<1
Consider two positive real numbers a and b.
Then, A=a+b2, G=ab and H=2aba+b.
From that we can conclude that AH=G2
Now consider the given equation,
One of the root is 1.
Given the other roots is α.
Product of roots = α = H(AG)A(GH).
Now substitute H=G2A in the equation of α.
α=G(AG)A(AG)
α=GA
We know that A>G>H (Because the numbers are distinct and positive)
0<GA<1
0<α<1
Therefore, option B is correct.

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