If `cos(x+pi/3)+cos x=a` has real solutions, then number of integral values of `a` are 3 sum of number of integral values of a is 0 when `a=1` , number of solutions for `x in [0,2pi]` are 3 when `a=1,` number of solutions for `x in [0,2pi]` are 2

Options

A. number of integral values of a are 3
B. sum of number of integral values of a is 0
C. when `a=1`, number of solution for `x in [0, 2pi]` are 3
D. when `a=1`, number of solutions for `x in [0, 2pi]` are 2

Explanation

To solve the equation \( \cos(x + \frac{\pi}{3}) + \cos x = a \) and analyze the conditions given in the problem, we can follow these steps:

### Step 1: Rewrite the equation using the cosine addition formula
We start with the equation:
\[ \cos(x + \frac{\pi}{3}) + \cos x = a \]
Using the cosine addition formula, we can express \( \cos(x + \frac{\pi}{3}) \):
\[ \cos(x + \frac{\pi}{3}) = \cos x \cos \frac{\pi}{3} - \sin x \sin \frac{\pi}{3} \]
Substituting the values of \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \):
\[ \cos(x + \frac{\pi}{3}) = \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x \]
Now substituting this back into the original equation:
\[ \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x + \cos x = a \]
Combining like terms gives:
\[ \frac{3}{2} \cos x - \frac{\sqrt{3}}{2} \sin x = a \]

### Step 2: Rearranging the equation
Rearranging the equation, we have:
\[ \frac{3}{2} \cos x - \frac{\sqrt{3}}{2} \sin x = a \]

### Step 3: Express in terms of a single cosine function
This equation can be expressed in the form \( R \cos(x + \phi) = a \) where:
- \( R = \sqrt{\left(\frac{3}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} \)
- \( \phi \) is the phase shift.

Calculating \( R \):
\[ R = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} \]

### Step 4: Determine the range of \( a \)
The maximum and minimum values of \( R \cos(x + \phi) \) are \( R \) and \( -R \):
\[ -\sqrt{3} \leq a \leq \sqrt{3} \]

### Step 5: Find integral values of \( a \)
The integral values of \( a \) within this range are:
\[ a = -1, 0, 1 \]
Thus, there are 3 integral values of \( a \).

### Step 6: Sum of integral values of \( a \)
Calculating the sum:
\[ -1 + 0 + 1 = 0 \]

### Step 7: Determine the number of solutions for specific values of \( a \)
When \( a = 1 \):
\[ \frac{3}{2} \cos x - \frac{\sqrt{3}}{2} \sin x = 1 \]
This can be rewritten as:
\[ \sqrt{3} \cos(x + \phi) = 1 \]
The number of solutions in the interval \( [0, 2\pi] \) can be determined. Since \( \cos(x + \phi) = \frac{1}{\sqrt{3}} \) has two solutions in the interval \( [0, 2\pi] \).

### Final Summary
- The number of integral values of \( a \) is 3.
- The sum of the integral values of \( a \) is 0.
- When \( a = 1 \), there are 2 solutions for \( x \) in the interval \( [0, 2\pi] \).