JEE Mathematics 2024 Question with Solution

Given: The lines are $2x - y + 3 = 0$, $6x + 3y + 1 = 0$, and $ax + 2y - 2 = 0$.

Find: The values of $\alpha$ for which these three lines do not form a triangle.

For three lines to not form a triangle, either two of them must be parallel or all three must be concurrent.

From the solution, checking parallelism:

gives

and

gives

So the values obtained from the working are $-4$ and $4$.

The solution is inconsistent with the listed options, and the answer key points to option B. Since the extracted solution does not support any listed option value, the correct option cannot be resolved reliably from the supplied content.

The solution appears to belong to a different version of the question. It discusses the sum of squares of values of $\alpha$ and concludes with $32$, whereas the present question asks for the values of $\alpha$ themselves and provides ordered-pair options.

Also, the solution uses both $a$ and $\alpha$ inconsistently. Because of this mismatch, the final option cannot be mapped with confidence. Hence the answer is marked AMBIGUOUS.