Question

Consider a system of equation,
$a+2b+3c=13a-b+2c=p2a+b+qc=2.$
Then which of the following is/are correct ?

• A. The system of equations has unique solution if $q\ne 3,p\in R$
• B. The system of equations has an infinite number of solutions if $q=3,p=3$
• C. The system of equations is inconsistent if $q=3,p\ne 3$
• D. The system of equations has an infinite number of solutions if $q\ne 3,p=3$

Answer 1

Answer: (A), (B), (C)

The system of equations is inconsistent if $q=3,p\ne 3$

$\Delta =\left|\begin{array}{ccc}1& 2& 3\\ 3& -1& 2\\ 2& 1& q\end{array}\right|$
$=-7q+21$

For unique solution, $\Delta \ne 0\Rightarrow q\ne 3$

${\Delta }_3=\left|\begin{array}{ccc}1& 2& 1\\ 3& -1& p\\ 2& 1& 2\end{array}\right|$
$=3p-9$

For infinite number of solutions, $\Delta =0,{\Delta }_1,{\Delta }_2,{\Delta }_3=0\Rightarrow q=3,p=3$

For inconsistant, $\Delta =0$ and at least one of ${\Delta }_1,{\Delta }_2$ or ${\Delta }_3$ is non-zero.
$\Rightarrow \Delta =0,{\Delta }_3\ne 0\Rightarrow q=3,p\ne 3$