The arithmetic mean of the roots of the equation `4cos^3x-4cos^2x-cos(315pi+x)=1` in the interval `(0,315pi)` is equal to (A)`50pi` (B) `51pi` (C)`100pi` (D) `315pi`

Options

A. `50pi`
B. `51pi`
C. `100pi`
D. `315pi`

Answer

To solve the equation \( 4\cos^3 x - 4\cos^2 x - \cos(315\pi + x) = 1 \) in the interval \( (0, 315\pi) \) and find the arithmetic mean of the roots, follow these steps:

### Step 1: Simplify the Equation
Start by rewriting the equation:
\[ 4\cos^3 x - 4\cos^2 x - \cos(315\pi + x) = 1 \]
Using the cosine addition formula, we know that:
\[ \cos(315\pi + x) = \cos(315\pi)\cos(x) - \sin(315\pi)\sin(x) \]
Since \( \cos(315\pi) = \cos(135\pi) = -\frac{1}{\sqrt{2}} \) and \( \sin(315\pi) = \sin(135\pi) = \frac{1}{\sqrt{2}} \), we have:
\[ \cos(315\pi + x) = -\frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x \]
Substituting this back into the equation gives:
\[ 4\cos^3 x - 4\cos^2 x + \frac{1}{\sqrt{2}}\cos x + \frac{1}{\sqrt{2}}\sin x - 1 = 0 \]

### Step 2: Rearranging the Equation
Rearranging gives:
\[ 4\cos^3 x - 4\cos^2 x + \frac{1}{\sqrt{2}}\cos x + \frac{1}{\sqrt{2}}\sin x - 1 = 0 \]

### Step 3: Finding Roots
To find the roots, we can factor or use numerical methods. However, we can also observe that:
- The equation can be solved for \( \cos x \) values.

### Step 4: Determine the Values of \( x \)
The roots of the equation will be of the form \( x = 2k\pi \) where \( k \) is an integer. The values of \( k \) will be determined by the interval \( (0, 315\pi) \).

### Step 5: Count the Roots
The possible values of \( k \) are from \( 0 \) to \( 100 \) (since \( 2k\pi \) must be less than \( 315\pi \)). Thus, the roots are:
\[ x = 0, 2\pi, 4\pi, \ldots, 200\pi \]
This gives us \( 101 \) roots.

### Step 6: Calculate the Sum of Roots
The sum of the roots can be calculated as:
\[ \text{Sum} = 0 + 2\pi + 4\pi + \ldots + 200\pi = 2\pi(0 + 1 + 2 + \ldots + 100) \]
Using the formula for the sum of the first \( n \) integers:
\[ \sum_{i=0}^{n} i = \frac{n(n + 1)}{2} \]
For \( n = 100 \):
\[ \sum_{i=0}^{100} i = \frac{100 \cdot 101}{2} = 5050 \]
Thus, the sum of the roots becomes:
\[ \text{Sum} = 2\pi \cdot 5050 = 10100\pi \]

### Step 7: Calculate the Arithmetic Mean
The arithmetic mean of the roots is given by:
\[ \text{Arithmetic Mean} = \frac{\text{Sum of Roots}}{\text{Number of Roots}} = \frac{10100\pi}{101} = 100\pi \]

### Conclusion
Thus, the arithmetic mean of the roots of the equation in the interval \( (0, 315\pi) \) is:
\[ \boxed{100\pi} \]