Question

# The least value of 6tan2θ+54cot2θ+18 is I. 54 when A.M.≥G.M. is applicable for 6tan2θ,54cot2θ,18 II. 54 when A.M.≥G.M. is applicable for 6tan2θ,54cot2θ;18 is added further III. 78 when tan2θ=cot2θ

A
I is correct
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B
I and II are correct
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C
III is correct
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D
I is false, II is correct
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Solution

## The correct options are A I is correct B I and II are correct I. 6tan2θ≥0,54cot2θ≥0 Using A.M.≥G.M. 6tan2θ+54cot2θ+183≥[6×54×18]1/3⇒6tan2θ+54cot2θ+18≥54 II. Using A.M.≥G.M. 6tan2θ+54cot2θ2≥[6×54]1/2⇒6tan2θ+54cot2θ≥36⇒6tan2θ+54cot2θ+18≥54 III. tan2θ=cot2θ⇒tan4θ=1⇒tan2θ=1=cot2θ 6tan2θ+54cot2θ+18=6+54+18=78 Which is constant. So, only statement I and II are correct.

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