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Question

If the triangle PQR varies, then the minimum value of cos(P+Q)+cos(Q+R)+cos(R+P) is

A
32
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B
53
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C
32
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D
53
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Solution

The correct option is A 32
Given PQR is a triangle.
Hence, P+Q+R=π
P+Q=πR,Q+R=πP,P+R=πQ

E=cos(P+Q)+cos(Q+R)+cos(P+R)
=cos(πR)+cos(πP)+cos(πQ)=cosRcosPcosQ=(cosP+cosQ+cosR)

To minimize the given expression (which is under negative sign), we need to maximize cosP+cosQ+cosR.
We know that cosP+cosQ+cosR will be maximum when cosP=cosQ=cosR
P=Q=R=π3 (i.e, PQR is an equilateral triangle)
Emin=3cos(π3)=32
Hence, option A is correct.

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