Assertion(A): A,B,C are positive angles such that A+B+C=7π, then maximum value of cotAcotBcotC=13√3 Reason(R): A.M.≥G.M.
A
A is true, R is false
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B
Both A and R are false
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C
A is false, R is true
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D
Both A and R are true
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Solution
The correct option is C Both A and R are true Assertion (A) : tan(A+B+C)=tanA+tanB+tanC−tanAtanBtanC1−tanAtanB−tanAtanC−tanBtanC tan(7π)=0⇒tan(A+B+C)=0Given(A+B+C=7π) tanA+tanB+tanC=tanAtanBtanC ... (i) Taking 3 positive numbers tanA,tanB,tanC A.M.≥G.M. ⇒tanA+tanB+tanC3≥(tanAtanBtanC)1/3 tanAtanBtanC3≥(tanAtanBtanC)1/3 [From (i)] (tanAtanBtanC)2/3≥3 ⇒cotAcotBcotC≤13√3 --------True. Reason (R) A.M. of three positive numbers ≥ their G.M. ------------- True. Therefore, both are True. Hence, option 'D' is correct.