Class 11

Math

Algebra

Sequences and Series

The least value of $6tan_{2}ϕ+54cot_{2}ϕ$ is

(I) $54$ when $A.M≥G.M$ is applicable for $6tan_{2}ϕ,54cot_{2}ϕ,18$

(II) $54$ when $A.M≥G.M$ is applicable for $6tan_{2}ϕ,54cot_{2}ϕ,18$ is added further

(III) $78$ when $tan_{2}ϕ=cot_{2}ϕ$

- (I) is correct, (II) is false
- (I) and (II) are correct
- (III) are correct
- none of the above are correct

by applying $A.M≥G.M$ in $6tan_{2}ϕ+54cot_{2}ϕ+18$ we get

$36tan_{2}ϕ+54cot_{2}ϕ+18 ≥(6×54×18)_{1/3}≥18$

Now equality holds when

$6tan_{2}ϕ=54cot_{2}ϕ=18⇒tan_{2}ϕ=3;cot_{2}ϕ=31 $

Hence, the statements I and II are correct.