The value of the determinant `|{:(1^(2),2^(2),3^(2),4^(2)),(2^(2),3^(2),4^(2),5^(2)),(3^(2),4^(2),5^(2),6^(2)),(4^(2),5^(2),6^(2),7^(2)):}|` is

Options

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

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} 1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2 \end{vmatrix} \]

we can simplify it step by step.

### Step 1: Write the determinant
First, we write the determinant explicitly with the squared values:

\[ D = \begin{vmatrix} 1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 9 & 16 & 25 & 36 \\ 16 & 25 & 36 & 49 \end{vmatrix} \]

### Step 2: Perform row operations
We will perform row operations to simplify the determinant. We will subtract the previous row from the current row:

1. \( R_2 \leftarrow R_2 - R_1 \)
2. \( R_3 \leftarrow R_3 - R_2 \)
3. \( R_4 \leftarrow R_4 - R_3 \)

After performing these operations, the determinant becomes:

\[ D = \begin{vmatrix} 1 & 4 & 9 & 16 \\ 3 & 5 & 7 & 9 \\ 5 & 7 & 9 & 11 \\ 7 & 9 & 11 & 13 \end{vmatrix} \]

### Step 3: Continue simplifying
Now we perform the same row operations again:

1. \( R_2 \leftarrow R_2 - R_1 \)
2. \( R_3 \leftarrow R_3 - R_2 \)
3. \( R_4 \leftarrow R_4 - R_3 \)

After these operations, the determinant becomes:

\[ D = \begin{vmatrix} 1 & 4 & 9 & 16 \\ 2 & 1 & -2 & -5 \\ 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 \end{vmatrix} \]

### Step 4: Identify identical rows
Notice that rows 3 and 4 are identical. When two rows of a determinant are identical, the value of the determinant is zero.

### Conclusion
Thus, the value of the determinant is:

\[ D = 0 \]

### Final Answer
The value of the determinant is **0**.