Let
`f(theta)=|(1,cos theta, -1),(-sin theta, 1, -cos theta),(-1,sin theta, 1)|` Suppose A and B are respectively maximum and minimum value of `f(theta)`. Then (A,B) is equal to

Options

A. (2,1)
B. (2,0)
C. (`sqrt(2),0)`
D. `(2,1/(sqrt(2)))`

Answer

To solve the problem, we need to evaluate the determinant given by the function \( f(\theta) \) and find its maximum and minimum values. Let's go through the steps systematically.

### Step 1: Write the determinant
The function is defined as:
\[ f(\theta) = \left| \begin{array}{ccc} 1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1 \end{array} \right| \]

### Step 2: Calculate the determinant
We can calculate the determinant using the formula for a \( 3 \times 3 \) matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix:
- \( a = 1, b = \cos \theta, c = -1 \)
- \( d = -\sin \theta, e = 1, f = -\cos \theta \)
- \( g = -1, h = \sin \theta, i = 1 \)

Calculating the determinant:
\[ f(\theta) = 1 \cdot (1 \cdot 1 - (-\cos \theta) \cdot \sin \theta) - \cos \theta \cdot (-\sin \theta \cdot 1 - (-\cos \theta)(-1)) + (-1) \cdot (-\sin \theta \cdot \sin \theta - 1 \cdot (-\cos \theta)) \]

This simplifies to:
\[ = 1 \cdot (1 + \cos \theta \sin \theta) - \cos \theta \cdot (-\sin \theta - \cos \theta) + (-1) \cdot (\sin^2 \theta + \cos \theta) \]

### Step 3: Simplify the expression
Now, we simplify:
\[ = 1 + \cos \theta \sin \theta + \cos \theta \sin \theta + \cos^2 \theta - \sin^2 \theta - \cos \theta \]
\[ = 1 + 2 \cos \theta \sin \theta + \cos^2 \theta - \sin^2 \theta - \cos \theta \]
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ = 1 + 2 \cos \theta \sin \theta - \cos \theta \]
\[ = 1 - \cos \theta + \sin 2\theta \]

### Step 4: Find maximum and minimum values
Now, we need to find the maximum and minimum values of:
\[ f(\theta) = 1 - \cos \theta + \sin 2\theta \]
The maximum value of \( \sin 2\theta \) is \( 1 \) and the minimum value is \( -1 \). Thus:
- Maximum of \( f(\theta) \):
\[ f_{\text{max}} = 1 - (-1) + 1 = 3 \]
- Minimum of \( f(\theta) \):
\[ f_{\text{min}} = 1 - 1 - 1 = -1 \]

### Step 5: Conclusion
Thus, the maximum value \( A \) is \( 3 \) and the minimum value \( B \) is \( -1 \). Therefore, the pair \( (A, B) \) is:
\[ (A, B) = (3, -1) \]