If the system of linear equation `x+y+z=6,x+2y+3c=14 ,a n d2x+5y+lambdaz=mu(lambda,mu R)` has a unique solution, then

Options

A. `lambda ne 8`
B. `lambda =8,mu ne 36`
C. `lambda=8 , mu =36`
D. none of these

Explanation

To determine the conditions under which the given system of linear equations has a unique solution, we can express the equations in matrix form and analyze the determinant of the coefficient matrix.

The given equations are:
1. \( x + y + z = 6 \)
2. \( x + 2y + 3z = 14 \)
3. \( 2x + 5y + \lambda z = \mu \)

We can write this system in matrix form \( A \mathbf{x} = \mathbf{b} \), where:

\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 2 & 5 & \lambda \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 6 \\ 14 \\ \mu \end{bmatrix} \]

For the system to have a unique solution, the determinant of matrix \( A \) must be non-zero:

\[ \text{det}(A) \neq 0 \]

Now, we will calculate the determinant of matrix \( A \):

\[ \text{det}(A) = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 2 & 5 & \lambda \end{vmatrix} \]

We can expand this determinant using the first row:

\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 2 & 3 \\ 5 & \lambda \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 3 \\ 2 & \lambda \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 2 \\ 2 & 5 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} 2 & 3 \\ 5 & \lambda \end{vmatrix} = 2\lambda - 15 \)

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

3. \( \begin{vmatrix} 1 & 2 \\ 2 & 5 \end{vmatrix} = 5 - 4 = 1 \)

Substituting these back into the determinant expression:

\[ \text{det}(A) = 1(2\lambda - 15) - 1(\lambda - 6) + 1(1) \]

Simplifying this:

\[ \text{det}(A) = 2\lambda - 15 - \lambda + 6 + 1 \]
\[ = \lambda - 8 \]

For the system to have a unique solution, we require:

\[ \lambda - 8 \neq 0 \]

Thus, we find:

\[ \lambda \neq 8 \]

### Final Answer:
The condition for the system of equations to have a unique solution is \( \lambda \neq 8 \).