The set of values of 'a' for which the equation
`sqrt(a) "cos" x -2 "sin" x = sqrt(2) + sqrt(2-a)` has a solution is

Options

A. `(0, 2)`
B. `[0, 2]`
C. `(sqrt(5)-1, 2)`
D. `[sqrt(5)-1, 2]`

Explanation

To solve the equation \( \sqrt{a} \cos x - 2 \sin x = \sqrt{2} + \sqrt{2-a} \) for the set of values of \( a \) for which it has a solution, we will follow these steps:

### Step 1: Analyze the equation
The equation can be rewritten as:
\[ \sqrt{a} \cos x - 2 \sin x = \sqrt{2} + \sqrt{2-a} \]
This implies that the left-hand side must be able to take values that are equal to the right-hand side.

### Step 2: Determine the range of the left-hand side
The left-hand side \( \sqrt{a} \cos x - 2 \sin x \) can be expressed in terms of its maximum and minimum values. The maximum value occurs when \( \cos x = 1 \) and \( \sin x = 0 \), which gives:
\[ \text{Max} = \sqrt{a} \]
The minimum value occurs when \( \cos x = 0 \) and \( \sin x = 1 \), which gives:
\[ \text{Min} = -2 \]
Thus, the left-hand side can range from \(-2\) to \(\sqrt{a}\).

### Step 3: Determine the range of the right-hand side
The right-hand side \( \sqrt{2} + \sqrt{2-a} \) must also be analyzed. For this to be defined, we need:
\[ 2 - a \geq 0 \implies a \leq 2 \]
Thus, the right-hand side is defined for \( a \in [0, 2] \).

### Step 4: Find the maximum value of the right-hand side
The maximum value of the right-hand side occurs when \( a = 0 \):
\[ \sqrt{2} + \sqrt{2-0} = \sqrt{2} + \sqrt{2} = 2\sqrt{2} \]
When \( a = 2 \):
\[ \sqrt{2} + \sqrt{2-2} = \sqrt{2} + 0 = \sqrt{2} \]
Thus, the right-hand side varies from \(\sqrt{2}\) to \(2\sqrt{2}\) as \( a \) varies from \(0\) to \(2\).

### Step 5: Set the conditions for the existence of solutions
For the equation to have a solution, the maximum of the left-hand side must be at least the minimum of the right-hand side:
\[ \sqrt{a} \geq \sqrt{2} \]
This implies:
\[ a \geq 2 \]
However, since \( a \) must also satisfy \( a \leq 2 \), we conclude:
\[ a = 2 \]

### Step 6: Conclusion
The set of values of \( a \) for which the equation has a solution is:
\[ \{2\} \]