Choose the correct answer from the following :
The value of `|{:(0,2,0),(1,3,5),(-1,0,4):}|is:`

Options

A. -6
B. 18
C. -18
D. None of these

Answer

To find the value of the determinant \( | \begin{pmatrix} 0 & 2 & 0 \\ 1 & 3 & 5 \\ -1 & 0 & 4 \end{pmatrix} | \), we will use the formula for the determinant of a 3x3 matrix.

### Step-by-step Solution:

1. **Write the Determinant**:
\[ D = \begin{vmatrix} 0 & 2 & 0 \\ 1 & 3 & 5 \\ -1 & 0 & 4 \end{vmatrix} \]

2. **Apply the Determinant Formula**:
The formula for the determinant of a 3x3 matrix is:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]
For our matrix:
- \( a = 0, b = 2, c = 0 \)
- \( d = 1, e = 3, f = 5 \)
- \( g = -1, h = 0, i = 4 \)

3. **Calculate Each Part**:
- First term: \( a(ei - fh) = 0(3 \cdot 4 - 5 \cdot 0) = 0 \)
- Second term: \( -b(di - fg) = -2(1 \cdot 4 - 5 \cdot -1) = -2(4 + 5) = -2 \cdot 9 = -18 \)
- Third term: \( c(dh - eg) = 0(1 \cdot 0 - 3 \cdot -1) = 0 \)

4. **Combine the Results**:
\[ D = 0 - 18 + 0 = -18 \]

5. **Final Answer**:
The value of the determinant is:
\[ | \begin{pmatrix} 0 & 2 & 0 \\ 1 & 3 & 5 \\ -1 & 0 & 4 \end{pmatrix} | = -18 \]

### Conclusion:
The correct answer is \(-18\).