Question

The values of $\lambda$ and $\mu$ for which the system of linear equations
$x+y+z=2$
$x+2y+3z=5$
$x+3y+\lambda z=\mu$
has infinitely many solutions are, respectively

• A. $6$ and $8$
• B. $5$ and $8$
• C. $5$ and $7$
• D. $4$ and $9$

Answer 1

The correct option is B $5$ and $8$

$x+y+z=2$
$x+2y+3z=5$
$x+3y+\lambda z=\mu$
has infinitely many solutions.
$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& \lambda \end{array}\right|=0$
$R_2\to R_2-R_1$
$R_3\to R_3-R_1$
$\left|\begin{array}{ccc}1& 1& 1\\ 0& 1& 2\\ 0& 2& \lambda -1\end{array}\right|=0$
$(\lambda -1-4)=0$
$\Rightarrow \lambda =5$
${\Delta }_3=\left|\begin{array}{ccc}1& 1& 2\\ 1& 2& 5\\ 1& 3& \mu \end{array}\right|=0$
$R_2\to R_2-R_1$
$R_3\to R_3-R_1$
$\left|\begin{array}{ccc}1& 1& 2\\ 0& 1& 3\\ 0& 2& \mu -2\end{array}\right|=0$
$(\mu -2-6)=0$
$\Rightarrow \mu =8$
$\lambda =5,\mu =8$