If `[]` denotes the greatest integer less than or equal to the ral number under consideration, and `-1lt=x<0,0lt=y<1,1lt=a<2,` then the value of the determinant `|[x]+1[y][z][x][y]+1[z][x][y][z]+1|` is `[x]` b. `[y]` c. `[z]` d. none of these

Options

A. `[x]`
B. `[y]`
C. `[z]`
D. none of these

Explanation

To solve the determinant given in the problem, we will follow these steps:

### Step 1: Understand the Given Conditions
We have:
- \( -1 < x < 0 \)
- \( 0 < y < 1 \)
- \( 1 < a < 2 \)

From these conditions, we can determine the greatest integer values:
- \( [x] = -1 \) (since \( x \) is between \(-1\) and \(0\))
- \( [y] = 0 \) (since \( y \) is between \(0\) and \(1\))
- \( [z] = 1 \) (since \( z \) is between \(1\) and \(2\))

### Step 2: Write the Determinant
The determinant we need to evaluate is:
\[ D = \begin{vmatrix} [x] + 1 & [y] & [z] \\ [x] & [y] + 1 & [z] \\ [x] & [y] & [z] + 1 \end{vmatrix} \]

Substituting the values we found:
\[ D = \begin{vmatrix} -1 + 1 & 0 & 1 \\ -1 & 0 + 1 & 1 \\ -1 & 0 & 1 + 1 \end{vmatrix} \]
This simplifies to:
\[ D = \begin{vmatrix} 0 & 0 & 1 \\ -1 & 1 & 1 \\ -1 & 0 & 2 \end{vmatrix} \]

### Step 3: Calculate the Determinant
Now we will calculate the determinant using the formula for a \(3 \times 3\) matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Where:
- \( a = 0, b = 0, c = 1 \)
- \( d = -1, e = 1, f = 1 \)
- \( g = -1, h = 0, i = 2 \)

Calculating each term:
- \( ei - fh = 1 \cdot 2 - 1 \cdot 0 = 2 \)
- \( di - fg = -1 \cdot 2 - 1 \cdot -1 = -2 + 1 = -1 \)
- \( dh - eg = -1 \cdot 0 - 1 \cdot -1 = 0 + 1 = 1 \)

Substituting back into the determinant formula:
\[ D = 0 \cdot 2 - 0 \cdot (-1) + 1 \cdot 1 = 0 + 0 + 1 = 1 \]

### Step 4: Conclusion
The value of the determinant \( D \) is \( 1 \).

### Final Answer
Since \( [z] = 1 \), the answer corresponds to option (c) \([z]\).