If `cot^(-1)(n/(pi))>(pi)/6, n in N`, then the maximum value of n is :

Options

A. 6
B. 7
C. 5
D. none of these

Explanation

To solve the inequality \( \cot^{-1}\left(\frac{n}{\pi}\right) > \frac{\pi}{6} \) where \( n \) is a natural number, we can follow these steps:

### Step 1: Rewrite the inequality
Given the inequality:
\[ \cot^{-1}\left(\frac{n}{\pi}\right) > \frac{\pi}{6} \]

### Step 2: Apply the cotangent function
Since \( \cot^{-1}(x) \) is a decreasing function, we can take the cotangent of both sides. This reverses the inequality:
\[ \frac{n}{\pi} < \cot\left(\frac{\pi}{6}\right) \]

### Step 3: Calculate \( \cot\left(\frac{\pi}{6}\right) \)
We know that:
\[ \cot\left(\frac{\pi}{6}\right) = \frac{1}{\tan\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

### Step 4: Substitute the value of cotangent
Now, substituting this value back into the inequality:
\[ \frac{n}{\pi} < \sqrt{3} \]

### Step 5: Multiply both sides by \( \pi \)
To eliminate the fraction, multiply both sides by \( \pi \) (since \( \pi > 0 \), the inequality remains unchanged):
\[ n < \pi \sqrt{3} \]

### Step 6: Calculate \( \pi \sqrt{3} \)
Using the approximate value of \( \pi \approx 3.14 \) and \( \sqrt{3} \approx 1.732 \):
\[ \pi \sqrt{3} \approx 3.14 \times 1.732 \approx 5.441 \]

### Step 7: Determine the maximum natural number \( n \)
Since \( n \) must be a natural number, the largest integer less than \( 5.441 \) is \( 5 \).

### Conclusion
Thus, the maximum value of \( n \) is:
\[ \boxed{5} \]