The system of linear equation
` 3x - 2y - kz =10`
` 2x - 4y - 2z =6`
` x +2y -z =5m`
is in-consistent if :

Options

A. `k=3, m = 4/5`
B. `k ne 3, m in R`
C. `k ne 3, m ne (4)/(5)`
D. `k=3 , m ne (4)/(5)`

Answer

To determine the values of \( k \) and \( m \) for which the given system of linear equations is inconsistent, we will analyze the coefficient matrix and the conditions for inconsistency.

### Given System of Equations:
1. \( 3x - 2y - kz = 10 \) (Equation 1)
2. \( 2x - 4y - 2z = 6 \) (Equation 2)
3. \( x + 2y - z = 5m \) (Equation 3)

### Step 1: Form the Coefficient Matrix
The coefficient matrix \( A \) of the system is given by:
\[ A = \begin{bmatrix} 3 & -2 & -k \\ 2 & -4 & -2 \\ 1 & 2 & -1 \end{bmatrix} \]

### Step 2: 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.

The determinant \( \Delta \) can be calculated using the formula:
\[ \Delta = \begin{vmatrix} 3 & -2 & -k \\ 2 & -4 & -2 \\ 1 & 2 & -1 \end{vmatrix} \]

Expanding the determinant along the first row:
\[ \Delta = 3 \begin{vmatrix} -4 & -2 \\ 2 & -1 \end{vmatrix} - (-2) \begin{vmatrix} 2 & -2 \\ 1 & -1 \end{vmatrix} - k \begin{vmatrix} 2 & -4 \\ 1 & 2 \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \( \begin{vmatrix} -4 & -2 \\ 2 & -1 \end{vmatrix} = (-4)(-1) - (-2)(2) = 4 + 4 = 8 \)
2. \( \begin{vmatrix} 2 & -2 \\ 1 & -1 \end{vmatrix} = (2)(-1) - (-2)(1) = -2 + 2 = 0 \)
3. \( \begin{vmatrix} 2 & -4 \\ 1 & 2 \end{vmatrix} = (2)(2) - (-4)(1) = 4 + 4 = 8 \)

Substituting back into the determinant:
\[ \Delta = 3(8) + 0 - k(8) = 24 - 8k \]

### Step 3: Set the Determinant to Zero
For the system to be inconsistent, we set the determinant to zero:
\[ 24 - 8k = 0 \]
Solving for \( k \):
\[ 8k = 24 \implies k = 3 \]

### Step 4: Analyze the Condition for \( m \)
Next, we need to find the condition for \( m \). We will form the augmented matrix and calculate the determinant of the modified matrix \( A_1 \) where the first column is replaced by the constants from the equations.

The augmented matrix \( A_1 \) is:
\[ A_1 = \begin{bmatrix} 10 & -2 & -k \\ 6 & -4 & -2 \\ 5m & 2 & -1 \end{bmatrix} \]

Calculating the determinant of \( A_1 \):
\[ \Delta_1 = \begin{vmatrix} 10 & -2 & -k \\ 6 & -4 & -2 \\ 5m & 2 & -1 \end{vmatrix} \]

Expanding this determinant similarly, we will focus on the conditions under which it equals zero. The calculations will show that for inconsistency, we will find a condition on \( m \).

### Final Condition for Inconsistency
After performing the necessary calculations, we find that the system is inconsistent if:
- \( k = 3 \)
- \( m \neq \frac{4}{5} \)

### Conclusion
Thus, the system of equations is inconsistent if:
- \( k = 3 \)
- \( m \) is not equal to \( \frac{4}{5} \)