Which one of following is correct ?
The three planes `2x+3y-z-2=0, 3x+3y+z-4=0,x-y+2z-5=0` intersect

Options

A. at a point
B. at two points
C. at three points
D. in a line

Explanation

To determine whether the three given planes intersect and to find the nature of their intersection, we can follow these steps:

### Step 1: Write down the equations of the planes.
The equations of the planes are:
1. \( 2x + 3y - z - 2 = 0 \) (Plane 1)
2. \( 3x + 3y + z - 4 = 0 \) (Plane 2)
3. \( x - y + 2z - 5 = 0 \) (Plane 3)

### Step 2: Convert the equations into a matrix form.
We can represent the system of equations in matrix form \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix.

\[ A = \begin{bmatrix} 2 & 3 & -1 \\ 3 & 3 & 1 \\ 1 & -1 & 2 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 2 \\ 4 \\ 5 \end{bmatrix} \]

### Step 3: Calculate the determinant of the coefficient matrix \( A \).
To check whether the planes intersect at a point, we need to calculate the determinant of matrix \( A \).

\[ \text{det}(A) = \begin{vmatrix} 2 & 3 & -1 \\ 3 & 3 & 1 \\ 1 & -1 & 2 \end{vmatrix} \]

Calculating the determinant using the rule of Sarrus or cofactor expansion:

\[ = 2 \begin{vmatrix} 3 & 1 \\ -1 & 2 \end{vmatrix} - 3 \begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix} - 1 \begin{vmatrix} 3 & 3 \\ 1 & -1 \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \( \begin{vmatrix} 3 & 1 \\ -1 & 2 \end{vmatrix} = (3)(2) - (1)(-1) = 6 + 1 = 7 \)

2. \( \begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix} = (3)(2) - (1)(1) = 6 - 1 = 5 \)

3. \( \begin{vmatrix} 3 & 3 \\ 1 & -1 \end{vmatrix} = (3)(-1) - (3)(1) = -3 - 3 = -6 \)

Now substituting back into the determinant calculation:

\[ \text{det}(A) = 2(7) - 3(5) - 1(-6) = 14 - 15 + 6 = 5 \]

### Step 4: Analyze the determinant.
Since \( \text{det}(A) \neq 0 \), the three planes intersect at a unique point.

### Conclusion:
The three planes intersect at a single point, which means they do not intersect in a line. Therefore, the correct statement is that the three planes intersect at a point.