Question

Consider the system of equations $x+y+z=6,x+2y+3z=10,x+2y+\lambda z=\mu$

The system has no solution if:

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

Answer 1

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

Given system of equations $AX=B$
where $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}6\\ 10\\ \mu \end{array}\right]$
For no solution,
$D=0$
$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right|=0$
$\Rightarrow 2(\lambda -3)-(\lambda -3)=0$
$\Rightarrow \lambda =3$
At least one of $D_1,D_2,D_3$ is non-zero
Also, $D_1=\left|\begin{array}{ccc}6& 1& 1\\ 10& 2& 3\\ \mu & 2& 3\end{array}\right|\ne 0$
$\Rightarrow \mu \ne 10$
$D_2=\left|\begin{array}{ccc}1& 6& 1\\ 1& 10& 3\\ 1& \mu & 3\end{array}\right|\ne 0$
$\Rightarrow \mu \ne 10$
$D_3=\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& \mu \end{array}\right|\ne 0$
$\Rightarrow \mu \ne 10$