If `Delta = |[1, 3costheta, 1], [sintheta, 1, 3costheta], [1, sintheta, 1]|`, the maximum value of `Delta` is

Options

A. -10
B. `-sqrt(10)`
C. `sqrt(10)`
D. 10

Answer

To find the maximum value of the determinant \( \Delta = \begin{vmatrix} 1 & 3 \cos \theta & 1 \\ \sin \theta & 1 & 3 \cos \theta \\ 1 & \sin \theta & 1 \end{vmatrix} \), we will follow these steps:

### Step 1: Write the determinant
We start with the determinant as given:
\[ \Delta = \begin{vmatrix} 1 & 3 \cos \theta & 1 \\ \sin \theta & 1 & 3 \cos \theta \\ 1 & \sin \theta & 1 \end{vmatrix} \]

### Step 2: Apply row operations
We can simplify the determinant by performing row operations. We will replace the third row \( R_3 \) with \( R_3 - R_1 \):
\[ R_3 \rightarrow R_3 - R_1 \]
This gives us:
\[ \Delta = \begin{vmatrix} 1 & 3 \cos \theta & 1 \\ \sin \theta & 1 & 3 \cos \theta \\ 0 & \sin \theta - 3 \cos \theta & 0 \end{vmatrix} \]

### Step 3: Expand the determinant
Now we can expand the determinant along the third row:
\[ \Delta = 0 + 0 + (0)(\sin \theta - 3 \cos \theta) = 0 \]
However, we need to calculate it properly considering the non-zero elements. Let's expand it correctly:
\[ \Delta = 1 \cdot \begin{vmatrix} 1 & 3 \cos \theta \\ \sin \theta & 3 \cos \theta \end{vmatrix} - (3 \cos \theta) \cdot \begin{vmatrix} \sin \theta & 3 \cos \theta \\ 0 & \sin \theta - 3 \cos \theta \end{vmatrix} \]

### Step 4: Calculate the 2x2 determinants
Calculating the first determinant:
\[ \begin{vmatrix} 1 & 3 \cos \theta \\ \sin \theta & 3 \cos \theta \end{vmatrix} = 1 \cdot (3 \cos \theta) - (3 \cos \theta)(\sin \theta) = 3 \cos \theta - 3 \cos \theta \sin \theta \]
Calculating the second determinant:
\[ \begin{vmatrix} \sin \theta & 3 \cos \theta \\ 0 & \sin \theta - 3 \cos \theta \end{vmatrix} = \sin \theta (\sin \theta - 3 \cos \theta) \]

### Step 5: Substitute back into the determinant
Substituting back, we have:
\[ \Delta = 3 \cos \theta (1 - \sin \theta) - (3 \cos \theta) \sin \theta (\sin \theta - 3 \cos \theta) \]

### Step 6: Simplify the expression
Now, simplifying this expression:
\[ \Delta = 3 \cos \theta - 3 \cos \theta \sin \theta - 3 \cos \theta \sin^2 \theta + 9 \cos^2 \theta \sin \theta \]

### Step 7: Find the maximum value
To find the maximum value of \( \Delta \), we can express it in terms of \( A = 3 \cos \theta - \sin \theta \). The maximum value of \( A \) can be found using the formula:
\[ \text{Maximum of } A = \sqrt{(3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]
Thus, the maximum value of \( \Delta \) is:
\[ \Delta_{\text{max}} = \left(\sqrt{10}\right)^2 = 10 \]

### Final Answer
The maximum value of \( \Delta \) is \( 10 \).

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