If $cot^{-1}\frac{n}{\pi}>\frac{\pi}{6}$, n ∈ N, then the maximum value of n, is

5

We have,

 $cot^{-1}\left(\frac{n}{\pi}\right)>\frac{\pi}{6}$

$⇒ cot \begin{Bmatrix}cot^{-1}\left(\frac{n}{\pi}\right)\end{Bmatrix} < cot \frac{\pi}{6}$ [∵ cot θ is a decreasing function on (0, θ)]

$⇒ \frac{n}{\pi} < \sqrt{3} ⇒ n < \sqrt{3} \pi ≈ 5.5 $

So, the maximum value of n is 5.