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Euler's Representation

If -1 n2-10...

Question

If ${cot}^{- 1} (\frac{n^{2} - 10 n + 21.6}{π}) > \frac{π}{6}, n \in N,$ then $n$ can be

A

3

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B

2

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C

4

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D

8

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Solution

The correct options are
A 3
D 4
Taking cot on both the sides, the inequality reverses.
$n^{2} - 10 n + 21.6 < \sqrt{3} π$
$(n - 5)^{2} - 25 + 21.6 < \sqrt{3} π$
$(n - 5)^{2} < 3.4 + \sqrt{3} π$
$- \sqrt{3.4 + \sqrt{3} π} < n - 5 < \sqrt{3.4 + \sqrt{3} π}$
$5 - \sqrt{3.4 + \sqrt{3} π} < n < 5 + \sqrt{3.4 + \sqrt{3} π}$
$2.02 < n < 7.97$
However $n ϵ N$
Hence the possible values of n are 3,4,5,6,7.

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{cot}^{- 1} \frac{n^{2} - 10 n + 21.6}{π} > \frac{π}{6}, n \in N

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