Question

The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6,3x+5y+5z=26,x+2y+\lambda z=\mu$ has no solution, are:

• A. $\lambda \ne 2,\mu =10$
• B. $\lambda =3,\mu \ne 10$
• C. $\lambda =3,\mu =5$
• D. $\lambda =2,\mu \ne 10$

Answer 1

The correct option is D $\lambda =2,\mu \ne 10$

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 3& 5& 5\\ 1& 2& \lambda \end{array}\right|=0$
$\Rightarrow \lambda =2$ and ${\Delta }_1=0$ for any value of $\mu .$
For $\mu =10,{\Delta }_1={\Delta }_2={\Delta }_3=0$ which corresponds to the case of infinite solutions.
$\therefore \mu \ne 10$ for the system to be inconsistent.