Question

The values of $\lambda ,\mu$ for which the system of equations $x+y+z=6,x+2y+3z=10,x+2y+\lambda z=\mu$ has an infinite number of solutions, is

• A. $\lambda =4,\mu =11$
• B. $\lambda =3,\mu =10$
• C. $\lambda =2,\mu =8$
• D. $\lambda =1,\mu =11$

Answer 1

The correct option is B $\lambda =3,\mu =10$

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right|=1(2\lambda -6)-1(\lambda -3)+1(2-2)$
$=2\lambda -6-\lambda +3+0=\lambda -3$

${\Delta }_1=\left|\begin{array}{ccc}6& 1& 1\\ 10& 2& 3\\ \mu & 2& \lambda \end{array}\right|=6(2\lambda -6)-1(10\lambda -3\mu )+1(20-2\mu )$

$12\lambda -36-10\lambda +3\mu +20-2\mu =2\lambda +\mu -16$

For infinite solutions $\Delta ={\Delta }_1=0\Rightarrow \lambda =3;\mu =10$