If the system of equations,
$x+ y + z= 2 $
$2x+ 4y -z= 6 $
$3x+ 2y + \lambda z= \mu $
has infinitely many solutions, then
Answer & explanation
Correct answer: option 3
The correct answer is option (3) : $2\lambda + \mu = 14$
If the given system of equations has infinitely many solutions, then $D= D_1= D_2= D_3= 0.$
Now, $D=0⇒\begin{vmatrix}1 & 1 & 1\\2 & 4 & -1\\3 & 2 & \lambda \end{vmatrix}=0 ⇒\lambda =\frac{9}{2}$
and $D_3= 0 ⇒\begin{vmatrix}1 & 1 & 2\\2 & 4 & 6\\3 & 2 & \mu \end{vmatrix}=0 ⇒\mu = 5 $
Hence, $2\lambda + \mu = 14.$