The number of integral values of a for which the equation `cos2x+a sin x=2a-7` possessess a solution.

Options

A. `0`
B. `1`
C. `3`
D. `5`

Answer

To find the number of integral values of \( a \) for which the equation \( \cos 2x + a \sin x = 2a - 7 \) possesses a solution, we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ \cos 2x + a \sin x = 2a - 7 \]
We know that \( \cos 2x = 1 - 2 \sin^2 x \). Substituting this into the equation gives:
\[ 1 - 2 \sin^2 x + a \sin x = 2a - 7 \]
Rearranging this, we have:
\[ -2 \sin^2 x + a \sin x + (1 - 2a + 7) = 0 \]
This simplifies to:
\[ -2 \sin^2 x + a \sin x + 8 - 2a = 0 \]

### Step 2: Multiply by -1
To make the leading coefficient positive, we multiply the entire equation by -1:
\[ 2 \sin^2 x - a \sin x + (2a - 8) = 0 \]

### Step 3: Identify coefficients
This is a quadratic equation in terms of \( \sin x \):
\[ 2 \sin^2 x - a \sin x + (2a - 8) = 0 \]
Here, \( A = 2 \), \( B = -a \), and \( C = 2a - 8 \).

### Step 4: Use the quadratic formula
For the quadratic equation \( A y^2 + B y + C = 0 \), the solutions for \( y \) (where \( y = \sin x \)) are given by:
\[ y = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Substituting our coefficients:
\[ \sin x = \frac{a \pm \sqrt{(-a)^2 - 4 \cdot 2 \cdot (2a - 8)}}{2 \cdot 2} \]
This simplifies to:
\[ \sin x = \frac{a \pm \sqrt{a^2 - 8a + 64}}{4} \]

### Step 5: Ensure the discriminant is non-negative
For \( \sin x \) to have real solutions, the discriminant must be non-negative:
\[ a^2 - 8a + 64 \geq 0 \]
Factoring gives:
\[ (a - 8)^2 \geq 0 \]
This is always true, so we proceed to find the range of \( \sin x \).

### Step 6: Set bounds for \( \sin x \)
Since \( \sin x \) must lie between -1 and 1, we set up the inequalities:
\[ -1 \leq \frac{a - \sqrt{(a - 8)^2}}{4} \quad \text{and} \quad \frac{a + \sqrt{(a - 8)^2}}{4} \leq 1 \]

### Step 7: Solve the inequalities
1. For the lower bound:
\[ -4 \leq a - (a - 8) \implies -4 \leq 8 \implies \text{always true} \]

2. For the upper bound:
\[ a + (a - 8) \leq 4 \implies 2a - 8 \leq 4 \implies 2a \leq 12 \implies a \leq 6 \]

### Step 8: Combine results
From the analysis, we have:
\[ 2 \leq a \leq 6 \]
The integral values of \( a \) in this range are \( 2, 3, 4, 5, 6 \).

### Step 9: Count the integral values
The integral values of \( a \) are:
- 2
- 3
- 4
- 5
- 6

Thus, the total number of integral values of \( a \) is 5.

### Final Answer
The number of integral values of \( a \) is \( \boxed{5} \).