# The value of $\lambda\, \, and \, \, \mu$  for which the system of equations $x+y+z=6$, $x+2y+3z=10$  and  $x+2y+\lambda z=\mu$  have no solution are Option 1) $\lambda = 3,\mu =10$ Option 2) $\lambda = 3,\mu \ne 10$ Option 3) $\lambda \neq 3,\mu =10$ Option 4) $\lambda \neq 3,\mu \ne 10$

As learnt in

Inconsistent system of linear equation -

If the system of equations has no solutions

-

For no solution $\Delta =0$

$\begin{vmatrix} 1 & 1 & 1\\ 1 & 2 & 3\\ 1 & 2 & \lambda \end{vmatrix}=0$

$1(2 \lambda-6)-1(\lambda-3)+1 \times 0=0$

$2 \lambda-6-\lambda+3=0$

$\lambda-3=0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\lambda=3$

Also,

$\Delta_{3}\neq0\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{vmatrix} 1 & 1 & 6\\ 1 & 2 & 10\\ 1 & 2 & \mu \end{vmatrix}\neq 0$

$1(2 \lambda -20)-1(2 \mu -10)+6 \times 0 \neq 0$

$\Delta_{1} \neq 0 \begin{vmatrix}6 & 1 & 1\\ 10 & 2 & 3\\ \mu & 2 & 3\end{vmatrix}=6 \times 0-1 (3_{0}-3 \mu)+1(2_{0}-2\mu)\neq 0$

$\mu -10 \neq 0\:\:\:\; \Rightarrow\mu \neq10$

Option 1)

$\lambda = 3,\mu =10$

This option is incorrect.

Option 2)

$\lambda = 3,\mu \ne 10$

This option is correct.

Option 3)

$\lambda \neq 3,\mu =10$

This option is incorrect.

Option 4)

$\lambda \neq 3,\mu \ne 10$

This option is incorrect.

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