Question

The system of equations :
$x+y+z=6$
$x+2y+3z=10$
$x+2y+\lambda z=\mu$
has no solution for

• A. $\lambda \ne 3,\mu =10$
• B. $\lambda =3,\mu \ne 10$
• C. $\lambda \ne 3,\mu \ne 10$
• D. none of these

Answer 1

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

Transformation matrix is $\left[\begin{array}{cccc}1& 1& 1& 6\\ 1& 2& 3& 10\\ 1& 2& \lambda & \mu \end{array}\right]$
applying $R_3\to R_3-R_2$
$\left[\begin{array}{cccc}1& 1& 1& 6\\ 1& 2& 3& 10\\ 0& 0& \lambda -3& \mu -10\end{array}\right]$
when $\lambda =3$ and $\mu \ne 10$ rank of coefficient matrix is less than transformation matrix.
i.e system of equations has no solution
$\therefore$ $\lambda =3$ and $\mu \ne 10$