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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 option is B I and II are correct
I.
6tan2θ0,54cot2θ0
Using A.M.G.M.
6tan2θ+54cot2θ+183[6×54×18]1/36tan2θ+54cot2θ+1854

II.
Using A.M.G.M.
6tan2θ+54cot2θ2[6×54]1/26tan2θ+54cot2θ366tan2θ+54cot2θ+1854

III.
tan2θ=cot2θtan4θ=1tan2θ=1=cot2θ
6tan2θ+54cot2θ+18=6+54+18=78
Which is constant.

So, only statement I and II are correct.

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