Question

Consider the system of equations
$x+y+z=6,$
$x+2y+3z=10$ and
$x+2y+\lambda z=\mu$
Statement $1$: If the system has infinite number of solutions, then $\mu =10.$
Statement $2$: The value of $\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& \mu \end{array}\right|=0\text{for}\mu =10.$

• A. Both the statements are true and Statement $2$ is the correct explanation of Statement $1$.
• B. Both the statements are true but Statement $2$ is not the correct explanation of Statement $1$.
• C. Statement $1$ is true and Statement $2$ is false.
• D. Statement $1$ is false and Statement $2$ is true.

Answer 1

The correct option is B Both the statements are true but Statement $2$ is not the correct explanation of Statement $1$.

Let $\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right|=0\Rightarrow \lambda =3.$
Now,
${\Delta }_1=\left|\begin{array}{ccc}6& 1& 1\\ 10& 2& 3\\ \mu & 2& \lambda \end{array}\right|$
$=\left|\begin{array}{ccc}6& 1& 1\\ 10& 2& 3\\ \mu & 2& \lambda \end{array}\right|=\mu -10(\because \lambda =3)$

${\Delta }_2=\left|\begin{array}{ccc}1& 6& 1\\ 1& 10& 3\\ 1& \mu & \lambda \end{array}\right|$
$=\left|\begin{array}{ccc}1& 6& 1\\ 1& 10& 3\\ 1& \mu & 3\end{array}\right|=20-2\mu$

${\Delta }_3=\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& \mu \end{array}\right|=\mu -10$

Clearly, for $\mu =10,$ all of ${\Delta }_1,{\Delta }_2,{\Delta }_3$ are zero.