Choose the correct answer from the following :
If a, b, c are in arithmetic progression, then `|{:(x+1,x+4,x+a),(x+2,x+5,x+b),(x+3,x+6,x+c):}|=`

Options

A. 2x
B. 1
C. 0
D. None of these

Answer

To solve the given determinant problem, we need to evaluate the determinant:

\[ D = \begin{vmatrix} x + 1 & x + 4 & x + a \\ x + 2 & x + 5 & x + b \\ x + 3 & x + 6 & x + c \end{vmatrix} \]

where \(a\), \(b\), and \(c\) are in arithmetic progression.

### Step 1: Understanding Arithmetic Progression
Since \(a\), \(b\), and \(c\) are in arithmetic progression, we have the relation:
\[ 2b = a + c \]

### Step 2: Apply Row Operations
We can simplify the determinant using row operations. We will perform the operation \(R_1 \rightarrow R_1 + R_3 - 2R_2\).

Calculating \(R_1\):
\[ R_1 = (x + 1) + (x + 3) - 2(x + 2) \]
This simplifies to:
\[ R_1 = (x + 1 + x + 3 - 2x - 4) = 0 \]

### Step 3: Update the Determinant
Now, the first row of the determinant becomes:
\[ \begin{vmatrix} 0 & \cdots & \cdots \\ x + 2 & x + 5 & x + b \\ x + 3 & x + 6 & x + c \end{vmatrix} \]

### Step 4: Evaluate the Determinant
Since the first row has become all zeros, the value of the determinant \(D\) is:
\[ D = 0 \]

### Conclusion
Thus, the value of the determinant is zero, which means the correct answer is:

\[ \boxed{0} \]