The value of the determinant `|{:(1+x,2,3,4),(1,2+x,3,4),(1,2,3+x,4),(1,2,3,4+x):}|`is

Options

A. `x^(2)(x +10)`
B. `x^(3) (x + 10)`
C. `x^(4) (x+10)`
D. None ot these

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} 1+x & 2 & 3 & 4 \\ 1 & 2+x & 3 & 4 \\ 1 & 2 & 3+x & 4 \\ 1 & 2 & 3 & 4+x \end{vmatrix} \]

we will perform row operations and simplify the determinant step by step.

### Step 1: Row Operation
We can simplify the determinant by performing the row operation \( R_1 \rightarrow R_1 - R_2 \). This will help us eliminate some elements in the first column.

\[ D = \begin{vmatrix} (1+x) - 1 & 2 - (2+x) & 3 - 3 & 4 - 4 \\ 1 & 2+x & 3 & 4 \\ 1 & 2 & 3+x & 4 \\ 1 & 2 & 3 & 4+x \end{vmatrix} \]

This simplifies to:

\[ D = \begin{vmatrix} x & -x & 0 & 0 \\ 1 & 2+x & 3 & 4 \\ 1 & 2 & 3+x & 4 \\ 1 & 2 & 3 & 4+x \end{vmatrix} \]

### Step 2: Factor Out x
Now, we can factor out \( x \) from the first row:

\[ D = x \begin{vmatrix} 1 & -1 & 0 & 0 \\ 1 & 2+x & 3 & 4 \\ 1 & 2 & 3+x & 4 \\ 1 & 2 & 3 & 4+x \end{vmatrix} \]

### Step 3: Further Row Operations
Next, we can perform \( R_2 \rightarrow R_2 - R_1 \), \( R_3 \rightarrow R_3 - R_1 \), and \( R_4 \rightarrow R_4 - R_1 \):

\[ D = x \begin{vmatrix} 1 & -1 & 0 & 0 \\ 0 & 3 & 3 & 4 \\ 0 & 3 & 3+x & 4 \\ 0 & 3 & 3 & 4+x \end{vmatrix} \]

### Step 4: Expand the Determinant
Now we can expand the determinant. Notice that the first column has zeros, which simplifies our calculations:

\[ D = x \cdot 1 \cdot \begin{vmatrix} 3 & 3 & 4 \\ 3 & 3+x & 4 \\ 3 & 3 & 4+x \end{vmatrix} \]

### Step 5: Calculate the 3x3 Determinant
Next, we can simplify this 3x3 determinant:

\[ D = x \cdot \begin{vmatrix} 3 & 3 & 4 \\ 3 & 3+x & 4 \\ 3 & 3 & 4+x \end{vmatrix} \]

We can perform \( R_2 \rightarrow R_2 - R_1 \) and \( R_3 \rightarrow R_3 - R_1 \):

\[ D = x \begin{vmatrix} 3 & 3 & 4 \\ 0 & x & 0 \\ 0 & 0 & x \end{vmatrix} \]

### Step 6: Final Calculation
The determinant simplifies to:

\[ D = x \cdot 3 \cdot x \cdot x = 3x^3 \]

### Final Answer
Thus, the value of the determinant is:

\[ D = 3x^3 \]