# If the system of equations

Question:

If the system of equations

$x+y+z=2 \quad 2 x+4 y-z=63 x+2 y+\lambda z=\mu$

has infinitely many solutions, then :

1. (1) $\lambda+2 \mu=14$

2. (2) $2 \lambda-\mu=5$

3. (3) $\lambda-2 \mu=-5$

4. (4) $2 \lambda+\mu=14$

Correct Option: , 4

Solution:

$\left|\begin{array}{ccc}1 & 1 & 1 \\ 2 & 4 & -1 \\ 3 & 2 & \lambda\end{array}\right|=0 \quad[\because$ Equation has many solutions $]$

$\Rightarrow-15+6+2 \lambda=0 \Rightarrow \lambda=\frac{9}{2}$

$\therefore D_{Z}=\left|\begin{array}{llc}1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 2 \mu\end{array}\right|=0 \Rightarrow \mu=5$

$\therefore 2 \lambda+\mu=14$