Assertion (A)
The system of equations $3 x - 2 y + 1 = 0 and 6 x - 4 y + 3 = 0$ is inconsistent.
Reason (R)
The system of equations $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ has a unique solution when $\frac{a_{1}}{b_{2}} \neq \frac{b_{1}}{b_{2}} .$
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

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Solution

Option (b) is correct.
Reason (R) is clearly true.
For the given system to be inconsistent, we must have,
$\frac{a}{b} = \frac{c}{d} \neq \frac{e}{f}$
Also, $\frac{3}{6} = \frac{- 2}{- 4} \neq \frac{1}{3} \Rightarrow k = 6$
So, Assertion (A) is true, but Reason (R) is not a correct explanation of Assertion (A).

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