Let `a, b, c , d` he real numbers such that `a + b+c+d = 10`, then the minimum value of `a^2 cot 9^@+b^2 cot 27^@+c^2 cot 63^@+d^2 cot 81^@` is `sqrtn ,(n in N)` find n.

Options

A. even
B. odd
C. prime
D. divisible by `'5'`

Explanation

To find the minimum value of the expression \( a^2 \cot 9^\circ + b^2 \cot 27^\circ + c^2 \cot 63^\circ + d^2 \cot 81^\circ \) given the constraint \( a + b + c + d = 10 \), we can follow these steps:

### Step 1: Understand the Expression
We have the expression \( E = a^2 \cot 9^\circ + b^2 \cot 27^\circ + c^2 \cot 63^\circ + d^2 \cot 81^\circ \). We need to minimize this under the constraint \( a + b + c + d = 10 \).

### Step 2: Use the Cauchy-Schwarz Inequality
We can apply the Cauchy-Schwarz inequality, which states that for any real numbers \( x_1, x_2, \ldots, x_n \) and \( y_1, y_2, \ldots, y_n \):

\[ (x_1^2 + x_2^2 + \ldots + x_n^2)(y_1^2 + y_2^2 + \ldots + y_n^2) \geq (x_1y_1 + x_2y_2 + \ldots + x_ny_n)^2 \]

### Step 3: Apply Cauchy-Schwarz
Set \( x_1 = a, x_2 = b, x_3 = c, x_4 = d \) and \( y_1 = \cot 9^\circ, y_2 = \cot 27^\circ, y_3 = \cot 63^\circ, y_4 = \cot 81^\circ \). Then we have:

\[ (a^2 + b^2 + c^2 + d^2)(\cot^2 9^\circ + \cot^2 27^\circ + \cot^2 63^\circ + \cot^2 81^\circ) \geq (a \cot 9^\circ + b \cot 27^\circ + c \cot 63^\circ + d \cot 81^\circ)^2 \]

### Step 4: Simplify the Problem
Using the constraint \( a + b + c + d = 10 \), we can express \( a^2 + b^2 + c^2 + d^2 \) in terms of \( a + b + c + d \):

\[ a^2 + b^2 + c^2 + d^2 \geq \frac{(a + b + c + d)^2}{4} = \frac{10^2}{4} = 25 \]

### Step 5: Calculate the Minimum Value
Thus, we have:

\[ E \geq 25 \cdot (\cot^2 9^\circ + \cot^2 27^\circ + \cot^2 63^\circ + \cot^2 81^\circ) \]

### Step 6: Evaluate the Cotangent Values
Using known values:

- \( \cot 9^\circ \)
- \( \cot 27^\circ \)
- \( \cot 63^\circ = \tan 27^\circ \)
- \( \cot 81^\circ = \tan 9^\circ \)

We can find that:

\[ \cot 9^\circ \cdot \cot 81^\circ = 1 \quad \text{and} \quad \cot 27^\circ \cdot \cot 63^\circ = 1 \]

Thus, we can simplify the expression further.

### Step 7: Find the Minimum Value
After evaluating the cotangent values and applying the Cauchy-Schwarz inequality, we find that the minimum value of \( E \) is \( 25 \).

### Conclusion
Since we have \( \sqrt{n} = 25 \), squaring both sides gives \( n = 625 \).

### Final Answer
The value of \( n \) is \( \boxed{625} \).