Answer: B — does not exist
Rewriting: $kx - y - 2 = 0$ and $6x - 2y - 3 = 0$. For infinitely many solutions: $\dfrac{k}{6} = \dfrac{-1}{-2} = \dfrac{-2}{-3}$, i.e., $\dfrac{k}{6} = \dfrac{1}{2}$ gives $k = 3$, but $\dfrac{1}{2} \neq \dfrac{2}{3}$. So no value of $k$ satisfies all three ratios simultaneously; the answer does not exist.
For infinitely many solutions, all three ratios must be equal: $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$. Here $\dfrac{b_1}{b_2} = \dfrac{1}{2}$ but $\dfrac{c_1}{c_2} = \dfrac{2}{3}$; these two are never equal regardless of $k$, so the condition can never be fully satisfied. Always check all three ratios, not just two.