If the system of equations `3x+y+z=1, 6x+3y+2z=1` and `mux+lambday+3z=1` is inconsistent, then

Options

A. `mu ne 9, lambda ne 5`
B. `mu ne 9, lambda=5`
C. `mu=9, lambda=5`
D. `mu=9, lambda ne 5`

Answer

To solve the problem, we need to determine the conditions under which the given system of equations is inconsistent. The equations are:

1. \(3x + y + z = 1\) (Equation 1)
2. \(6x + 3y + 2z = 1\) (Equation 2)
3. \(\mu x + \lambda y + 3z = 1\) (Equation 3)

### Step 1: Write the system in matrix form
We can express the system of equations in the form of a matrix:

\[ \begin{bmatrix} 3 & 1 & 1 \\ 6 & 3 & 2 \\ \mu & \lambda & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \]

### Step 2: Determine the condition for inconsistency
For the system of equations to be inconsistent, the determinant of the coefficient matrix must be zero (i.e., \(\Delta = 0\)) and the determinant of the augmented matrix must not be zero (i.e., \(\Delta_1 \neq 0\)).

### Step 3: Calculate the determinant \(\Delta\)
The determinant \(\Delta\) is given by:

\[ \Delta = \begin{vmatrix} 3 & 1 & 1 \\ 6 & 3 & 2 \\ \mu & \lambda & 3 \end{vmatrix} \]

Calculating this determinant using the first row:

\[ \Delta = 3 \begin{vmatrix} 3 & 2 \\ \lambda & 3 \end{vmatrix} - 1 \begin{vmatrix} 6 & 2 \\ \mu & 3 \end{vmatrix} + 1 \begin{vmatrix} 6 & 3 \\ \mu & \lambda \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \(\begin{vmatrix} 3 & 2 \\ \lambda & 3 \end{vmatrix} = 9 - 2\lambda\)
2. \(\begin{vmatrix} 6 & 2 \\ \mu & 3 \end{vmatrix} = 18 - 2\mu\)
3. \(\begin{vmatrix} 6 & 3 \\ \mu & \lambda \end{vmatrix} = 6\lambda - 3\mu\)

Substituting back into the determinant:

\[ \Delta = 3(9 - 2\lambda) - (18 - 2\mu) + (6\lambda - 3\mu) \]
\[ = 27 - 6\lambda - 18 + 2\mu + 6\lambda - 3\mu \]
\[ = 9 - \mu \]

Setting \(\Delta = 0\) for inconsistency:

\[ 9 - \mu = 0 \implies \mu = 9 \]

### Step 4: Calculate the determinant \(\Delta_1\)
Now we need to check the condition for \(\Delta_1\):

\[ \Delta_1 = \begin{vmatrix} 1 & 1 & 1 \\ 3 & 2 & 3 \\ \lambda & 3 & 3 \end{vmatrix} \]

Calculating this determinant:

\[ \Delta_1 = 1 \begin{vmatrix} 2 & 3 \\ 3 & 3 \end{vmatrix} - 1 \begin{vmatrix} 3 & 3 \\ \lambda & 3 \end{vmatrix} + 1 \begin{vmatrix} 3 & 2 \\ \lambda & 3 \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \(\begin{vmatrix} 2 & 3 \\ 3 & 3 \end{vmatrix} = 6 - 9 = -3\)
2. \(\begin{vmatrix} 3 & 3 \\ \lambda & 3 \end{vmatrix} = 9 - 3\lambda\)
3. \(\begin{vmatrix} 3 & 2 \\ \lambda & 3 \end{vmatrix} = 9 - 2\lambda\)

Substituting back into \(\Delta_1\):

\[ \Delta_1 = -3 - (9 - 3\lambda) + (9 - 2\lambda) \]
\[ = -3 - 9 + 3\lambda + 9 - 2\lambda \]
\[ = \lambda - 3 \]

Setting \(\Delta_1 \neq 0\):

\[ \lambda - 3 \neq 0 \implies \lambda \neq 3 \]

### Conclusion
The system of equations is inconsistent if:

- \(\mu = 9\)
- \(\lambda \neq 3\)

### Final Answer
Thus, the values that make the system inconsistent are:

\(\mu = 9\) and \(\lambda \neq 3\).