Question

Let $a,\lambda ,\mu \in R$. Consider the system of linear equation
$ax+2y=\lambda$
$3x-2y=\mu$
which of the following statement(s) is(are) correct?

• A. If $a=-3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$
• B. If $a\ne -3$, then the system has a unique solution for all values of $\lambda$ and $\mu$
• C. If $\lambda +\mu =0$, then the system has infinitely many solutions for $a=-3$
• D. If $\lambda +\mu \ne 0$, the the system has no solution for $a=-3$

Answer 1

Answer: (B), (C), (D)

If $a\ne -3$, then the system has a unique solution for all values of $\lambda$ and $\mu$
If $\lambda +\mu =0$, then the system has infinitely many solutions for $a=-3$
If $\lambda +\mu \ne 0$, the the system has no solution for $a=-3$

$ax+2y=\lambda$
$3x-2y=\mu$
(A) $a=-3$ gives
$\lambda =\mu$
or $\lambda +\mu =0$ not for all $\lambda ,\mu$
(B) $a\ne -3\Rightarrow \Delta \ne 0$ where $\Delta =\left|\begin{array}{cc}a& 2\\ 3& -2\end{array}\right|=-2a-6$
$\therefore$ (B) is correct
(C) correct
(D) if $\lambda +\mu \ne 0$
$3x+2y=\lambda$ ........(1)
& $3x-2y=\mu$ ........(2)
inconsistent $\Rightarrow$ (D) correct