Question

How many values of $c$, for which the system of equations
$6x+3y=c-3,12x+cy=c$ has infinitely many solution?

• A. $1$
• B. $2$
• C. $3$
• D. Infinite

Answer 1

The correct option is A $1$

$\to$ From Cramers' rule, each of the following determinants should be $O$ for infinte solution.

$D=\left|\begin{array}{cc}6& 3\\ 12& C\end{array}\right|=0\Rightarrow 6C-36\Rightarrow C=6D_1=\left|\begin{array}{cc}C-3& 3\\ C& C\end{array}\right|=0\Rightarrow C(C-3)-x\Rightarrow C^2-6C=0\Rightarrow C=0orC=6\Rightarrow D_2=\left|\begin{array}{cc}6& C-3\\ 12& C\end{array}\right|=0\Rightarrow 6C-12C+36=0\Rightarrow 6C=36\Rightarrow C=6$

$\Rightarrow$ for $C=6$ all three are zero $\Rightarrow \left(A\right)$