Assertion (A)
The system of equations $x + 3 y - 5 = 0 and 2 x - 6 y + 8 = 0$ has infinitely many solutions.
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 infinitely many solutions when $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} .$
(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 (d) is correct.
Here, Reason (R) is clearly true.
We have a₁ = 1, b₁ = 3, c₁ = −5 and a₂ = 2, b₂ = −6, c₂ = 8
Thus, we have:
$\frac{a_{1}}{a_{2}} = \frac{1}{2}, \frac{b_{1}}{b_{2}} = \frac{3}{- 6} = \frac{- 1}{2} and \frac{c_{1}}{c_{2}} = \frac{- 5}{8}$
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ is not true and so, Assertion (A) is false.
Thus, Assertion (A) is false and Reason (R) is true.

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