The system of equations `x+y+z=6, x +2y + 3z = 10, x+2y + lambda z = k` is inconsistent if `lambda =.........,k!=.........`

Options

A. `lambda=1`
B. `lambda=2`
C. `lambda=-2`
D. `lambda=3`

Explanation

To determine the values of \( \lambda \) and \( k \) for which the system of equations is inconsistent, we will follow these steps:

### Step 1: Write the system of equations
The given system of equations is:
1. \( x + y + z = 6 \)
2. \( x + 2y + 3z = 10 \)
3. \( x + 2y + \lambda z = k \)

### Step 2: Formulate the coefficient matrix
The coefficients of the variables \( x, y, z \) can be arranged in a matrix:
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \end{bmatrix} \]

### Step 3: Calculate the determinant of the coefficient matrix
To find when the system is inconsistent, we need to calculate the determinant of matrix \( A \) and set it to zero:
\[ \text{det}(A) = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \end{vmatrix} \]

Using the determinant formula for a \( 3 \times 3 \) matrix, we have:
\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 2 & 3 \\ 2 & \lambda \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 3 \\ 1 & \lambda \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 2 \\ 1 & 2 \end{vmatrix} \]

Calculating the minors:
1. \( \begin{vmatrix} 2 & 3 \\ 2 & \lambda \end{vmatrix} = 2\lambda - 6 \)
2. \( \begin{vmatrix} 1 & 3 \\ 1 & \lambda \end{vmatrix} = \lambda - 3 \)
3. \( \begin{vmatrix} 1 & 2 \\ 1 & 2 \end{vmatrix} = 0 \)

Putting it all together:
\[ \text{det}(A) = 1(2\lambda - 6) - 1(\lambda - 3) + 0 = 2\lambda - 6 - \lambda + 3 \]
\[ = \lambda - 3 \]

### Step 4: Set the determinant to zero
For the system to be inconsistent:
\[ \lambda - 3 = 0 \]
Thus, we find:
\[ \lambda = 3 \]

### Step 5: Substitute \( \lambda \) back into the equations
Now we substitute \( \lambda = 3 \) into the third equation:
\[ x + 2y + 3z = k \]

The first two equations become:
1. \( x + y + z = 6 \)
2. \( x + 2y + 3z = 10 \)

### Step 6: Analyze the third equation
If \( k = 10 \), the third equation becomes identical to the second equation:
\[ x + 2y + 3z = 10 \]
This would mean that the system has infinitely many solutions (the equations are dependent).

### Step 7: Determine the condition for \( k \)
For the system to remain inconsistent, \( k \) must not equal 10:
\[ k \neq 10 \]

### Final Answer
Thus, the values are:
\[ \lambda = 3, \quad k \neq 10 \]