Question

The least value of $6{\tan }^2\theta +54{\cot }^2\theta +18$ is
$\text{I.}$ $54$ when $A.M.\ge G.M.$ is applicable for $6{\tan }^2\theta ,54{\cot }^2\theta ,18$
$\text{II.}$ $54$ when $A.M.\ge G.M.$ is applicable for $6{\tan }^2\theta ,54{\cot }^2\theta ;18$ is added further
$\text{III.}$ $78$ when ${\tan }^2\theta ={\cot }^2\theta$

• A. $\text{I}$ is correct
• B. $\text{I}$ and $\text{II}$ are correct
• C. $\text{III}$ is correct
• D. $\text{I}$ is false, $\text{II}$ is correct

Answer 1

Answer: (A), (B)

$\text{I}$ and $\text{II}$ are correct

$\text{I}.$
$6{\tan }^2\theta \ge 0,54{\cot }^2\theta \ge 0$
Using $A.M.\ge G.M.$
$\frac{6{\tan }^2\theta +54{\cot }^2\theta +18}{3}\ge [6\times 54\times 18{]}^{1/3}\Rightarrow 6{\tan }^2\theta +54{\cot }^2\theta +18\ge 54$

$\text{II}.$
Using $A.M.\ge G.M.$
$\frac{6{\tan }^2\theta +54{\cot }^2\theta }{2}\ge [6\times 54{]}^{1/2}\Rightarrow 6{\tan }^2\theta +54{\cot }^2\theta \ge 36\Rightarrow 6{\tan }^2\theta +54{\cot }^2\theta +18\ge 54$

$\text{III}.$
${\tan }^2\theta ={\cot }^2\theta \Rightarrow {\tan }^4\theta =1\Rightarrow {\tan }^2\theta =1={\cot }^2\theta$
$6{\tan }^2\theta +54{\cot }^2\theta +18=6+54+18=78$
Which is constant.

So, only statement $\text{I}$ and $\text{II}$ are correct.