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Question

# If a,b,c are three distinct positive real numbers which are in H.P then 3a+2b2aâˆ’b+3c+2b2câˆ’b is

A
Greater than or equal to 10
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B
Less than or equal to 10
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C
Only equal to 20
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D
None of these
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Solution

## The correct option is A Greater than or equal to 10a,b,c in H.P, b=2aca+c...(1)3a+2b2a−b+3c+2b2c−b=6ac+4bc−3ab−2b2+6ac+4ab−3bc−2b24ac−2ab−2bc+b2=12ac+b(a+c)−4b24ac−2b(a+c)+b2Replace b(a+c)=2ac by relation (1)=12ac+2ac−4b24ac−4ac+b2=14ac−4b2b2=14acb2−4Replace 2acb=a+c,7(a+c)b−4As a,b,c in H.P ∴2b=1a+1c⇒7(a+c)b−4=7(a+c)12(1a+1c)−4=72(1+ac+ca+1)−4=72(2+ac+ca)−4=3+72(ac+ca)By Arithmetic mean relation on ac,ca⇒(ac+ca)2≥√ac×caac+ca≥2∴3+72(ac+ca)≥3+72×2=10Will be greater than or equal to 10.

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