`Delta=|(cos""(theta)/(2),1,1),(1,cos""(theta)/(2),-cos""(theta)/(2)),(-cos""(theta)/(2),1,1)|` lies in the interval

Options

A. `[2, 4]`
B. `[0, 4]`
C. `[1, 3]`
D. `[-2, 2]`

Answer

To solve the problem, we need to find the value of the determinant \( \Delta \) given by:

\[ \Delta = \begin{vmatrix} \frac{\cos(\theta)}{2} & 1 & 1 \\ 1 & \frac{\cos(\theta)}{2} & -\frac{\cos(\theta)}{2} \\ -\frac{\cos(\theta)}{2} & 1 & 1 \end{vmatrix} \]

### Step 1: Simplifying the Determinant
We can perform column operations to simplify the determinant. Let's subtract the second column from the third column:

\[ \Delta = \begin{vmatrix} \frac{\cos(\theta)}{2} & 1 & 0 \\ 1 & \frac{\cos(\theta)}{2} & -\cos(\theta) \\ -\frac{\cos(\theta)}{2} & 1 & 0 \end{vmatrix} \]

### Step 2: Expanding the Determinant
Now we can expand the determinant along the third column:

\[ \Delta = 0 \cdot \begin{vmatrix} 1 & 1 \\ 1 & \frac{\cos(\theta)}{2} \end{vmatrix} - (-\cos(\theta)) \cdot \begin{vmatrix} \frac{\cos(\theta)}{2} & 1 \\ -\frac{\cos(\theta)}{2} & 1 \end{vmatrix} \]

### Step 3: Calculating the 2x2 Determinant
Now we calculate the 2x2 determinant:

\[ \begin{vmatrix} \frac{\cos(\theta)}{2} & 1 \\ -\frac{\cos(\theta)}{2} & 1 \end{vmatrix} = \left(\frac{\cos(\theta)}{2} \cdot 1\right) - \left(1 \cdot -\frac{\cos(\theta)}{2}\right) = \frac{\cos(\theta)}{2} + \frac{\cos(\theta)}{2} = \cos(\theta) \]

### Step 4: Substituting Back
Substituting back into the determinant expression:

\[ \Delta = \cos(\theta) \cdot \cos(\theta) = \cos^2(\theta) \]

### Step 5: Finding the Range of \( \Delta \)
Since \( \cos^2(\theta) \) varies from 0 to 1 as \( \theta \) varies, we can multiply this by 4 (as per the operations we did earlier):

\[ \Delta = 4 \cos^2(\theta) \]

### Step 6: Final Range of \( \Delta \)
Thus, the range of \( \Delta \) is:

\[ \Delta \in [0, 4] \]

### Conclusion
The determinant \( \Delta \) lies in the interval \( [0, 4] \).