If the system of simultaneous linear equations $\mathrm{x}+\mathrm{y}+\mathrm{z}=\lambda, 5…
If the system of simultaneous linear equations $\mathrm{x}+\mathrm{y}+\mathrm{z}=\lambda, 5 \mathrm{x}-\mathrm{y}+\mu \mathrm{z}=10$ and $2 \mathrm{x}+3 \mathrm{y}-\mathrm{z}=6$ has unique solution, then
$\mu=23$ and $\lambda \in \mathrm{R}$
$\mu \in \mathrm{R}$ and $\lambda \neq 23$
$\mu \neq 23$ and $\lambda \in \mathrm{R}$
$\mu=23$ and $\lambda=16$
Solution
Given system of equations
$x+y+z=\lambda$...(i)
$5 x-y+\propto z=10$...(ii)
$2 x+3 y-z=6$...(iii)
has unique solution
$\therefore\left|\begin{array}{rrc}1 & 1 & 1 \\ 5 & -1 & \mu \\ 2 & 3 & -1\end{array}\right| \neq 0$
$\begin{aligned} & 1(1-3 \propto)-1(-5-2 \propto)+1(15+2) \uparrow 0 \\ & 1-3 \propto+5+2 \propto+17 \uparrow\end{aligned}$
$\propto \uparrow 23$ for all $\lambda \in \mathrm{R}$ system.
of solution has unique solution