If the arithmetic mean of the roots of the equation `4cos^(3)x-4cos^(2)x-cos(pi+x)-1=0` in the interval `[0,315]` is equal to `kpi`, then the value of `k` is

Options

A. `10`
B. `20`
C. `50`
D. `80`

Answer

To solve the equation \( 4\cos^3 x - 4\cos^2 x - \cos(\pi + x) - 1 = 0 \) and find the value of \( k \) such that the arithmetic mean of the roots in the interval \([0, 315]\) is equal to \( k\pi \), we can follow these steps:

### Step 1: Simplify the equation
We start with the given equation:
\[ 4\cos^3 x - 4\cos^2 x - \cos(\pi + x) - 1 = 0 \]
Using the identity \( \cos(\pi + x) = -\cos x \), we can rewrite the equation as:
\[ 4\cos^3 x - 4\cos^2 x + \cos x - 1 = 0 \]

### Step 2: Factor the equation
Next, we can factor the equation. We take \( 4\cos^2 x \) common from the first two terms:
\[ 4\cos^2 x(\cos x - 1) + \cos x - 1 = 0 \]
This can be factored further as:
\[ (\cos x - 1)(1 + 4\cos^2 x) = 0 \]

### Step 3: Solve for the roots
From the factored equation, we have two cases:
1. \( \cos x - 1 = 0 \) which gives \( \cos x = 1 \)
2. \( 1 + 4\cos^2 x = 0 \) which leads to \( \cos^2 x = -\frac{1}{4} \) (not possible for real numbers)

Thus, we only consider the first case:
\[ \cos x = 1 \]
The solutions for \( \cos x = 1 \) are:
\[ x = 2n\pi \quad (n \in \mathbb{Z}) \]

### Step 4: Find the roots in the interval \([0, 315]\)
Next, we need to find the values of \( n \) such that \( 0 \leq 2n\pi \leq 315 \):
\[ 0 \leq n \leq \frac{315}{2\pi} \]
Calculating \( \frac{315}{2\pi} \):
\[ \frac{315}{2\pi} \approx \frac{315}{6.28} \approx 50.2 \]
Thus, \( n \) can take integer values from \( 0 \) to \( 50 \), giving us \( 51 \) roots:
\[ x = 0, 2\pi, 4\pi, \ldots, 100\pi \]

### Step 5: Calculate the arithmetic mean of the roots
The roots are:
\[ 0, 2\pi, 4\pi, \ldots, 100\pi \]
The arithmetic mean (AM) is given by:
\[ \text{AM} = \frac{0 + 2\pi + 4\pi + \ldots + 100\pi}{51} \]
Factoring out \( 2\pi \):
\[ \text{AM} = \frac{2\pi(0 + 1 + 2 + \ldots + 50)}{51} \]
The sum of the first \( n \) integers is given by \( \frac{n(n+1)}{2} \):
\[ 0 + 1 + 2 + \ldots + 50 = \frac{50 \cdot 51}{2} = 1275 \]
Thus,
\[ \text{AM} = \frac{2\pi \cdot 1275}{51} = \frac{2550\pi}{51} = 50\pi \]

### Step 6: Find \( k \)
We know from the problem statement that the arithmetic mean is equal to \( k\pi \):
\[ 50\pi = k\pi \]
Comparing both sides, we find:
\[ k = 50 \]

### Final Answer
The value of \( k \) is:
\[ \boxed{50} \]