Let a, , R. Consider the system of linear equations a x+2 y= 3 x-… - Vidyadip

MCQ

Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations

$a x+2 y=\lambda$

$3 x-2 y=\mu$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 $a \neq-3$, 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 $\lambda+\mu \neq 0$, then the system has no solution for $a=-3$

A
$A,C$
B
$B,C$
✓
$B,C,D$
D
$B,C,A$

✓

Answer

Correct option: C.

$B,C,D$

c
$\alpha x+2 y=\lambda$

$3 x-2 y=\mu$

$\Delta=\left|\begin{array}{ll}\alpha & 2 \\ 3 & -2\end{array}\right|=-2 \alpha-6$

$\Delta=0, \therefore, \alpha=-3$

$\Delta_1=\left|\begin{array}{ll}\lambda & 2 \\ \mu & -2\end{array}\right|=-2 \lambda-2 \mu=-2(\lambda+\mu)$

$\Delta_2=\left|\begin{array}{ll}-3 & \lambda \\ 3 & \mu\end{array}\right|=-3 \mu-3 \lambda=-3(\lambda-\mu)$

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