One root of the equation `|(3x-8, 3, 3),(3,3x-8, 3),(3,3,3x-8)|=0 is (A) `8/3` (B) `2/3` (C) `1/3 ` (D) `16/3`

Options

A. `8//3`
B. `2//3`
C. `1//3`
D. `16/3`

Answer

To solve the equation given by the determinant \( |(3x-8, 3, 3), (3, 3x-8, 3), (3, 3, 3x-8)| = 0 \), we will follow these steps:

### Step 1: Set up the determinant
We denote the determinant as \( \Delta \):
\[ \Delta = \begin{vmatrix} 3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8 \end{vmatrix} \]

### Step 2: Apply column operations
We can simplify the determinant by adding all three columns:
\[ \Delta = \begin{vmatrix} (3x - 8 + 3 + 3) & 3 & 3 \\ 3 & (3x - 8 + 3 + 3) & 3 \\ 3 & 3 & (3x - 8 + 3 + 3) \end{vmatrix} \]
This simplifies to:
\[ \Delta = \begin{vmatrix} 3x - 2 & 3 & 3 \\ 3 & 3x - 2 & 3 \\ 3 & 3 & 3x - 2 \end{vmatrix} \]

### Step 3: Apply row operations
Next, we can simplify the determinant further by subtracting the first row from the second and third rows:
\[ \Delta = \begin{vmatrix} 3x - 2 & 3 & 3 \\ 0 & (3x - 2 - 3) & 0 \\ 0 & 0 & (3x - 2 - 3) \end{vmatrix} \]
This gives us:
\[ \Delta = \begin{vmatrix} 3x - 2 & 3 & 3 \\ 0 & 3x - 5 & 0 \\ 0 & 0 & 3x - 5 \end{vmatrix} \]

### Step 4: Calculate the determinant
The determinant can now be calculated as:
\[ \Delta = (3x - 2) \cdot (3x - 5) \cdot (3x - 5) \]
This simplifies to:
\[ \Delta = (3x - 2)(3x - 5)^2 \]

### Step 5: Set the determinant to zero
To find the roots, we set the determinant equal to zero:
\[ (3x - 2)(3x - 5)^2 = 0 \]

### Step 6: Solve for \( x \)
This gives us two equations to solve:
1. \( 3x - 2 = 0 \)
2. \( (3x - 5)^2 = 0 \)

From the first equation:
\[ 3x - 2 = 0 \implies x = \frac{2}{3} \]

From the second equation:
\[ 3x - 5 = 0 \implies 3x = 5 \implies x = \frac{5}{3} \]

### Conclusion
The roots of the equation are \( x = \frac{2}{3} \) and \( x = \frac{5}{3} \). Since we need to find one root, the answer is:
\[ \boxed{\frac{2}{3}} \]