JEE Mathematics 2024 Question with Solution

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

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.

First, check whether the first two lines intersect:

So, the lines $2x - y + 3 = 0$ and $6x + 3y + 1 = 0$ are not parallel.

Now check when the third line is parallel to the first line:

This gives

The solution is inconsistent with the question. It discusses the sum of squares of values of $\alpha$ and concludes $32$, which corresponds to a different numerical-value question, not this MCQ asking for the values themselves.

Using the mathematically relevant condition mentioned in the working, the third line is parallel to the second line when

which gives

Hence the values obtained from the parallel-line condition are $-4$ and $4$.

These values are not present in the listed options. Since the solution's itself marks option (2) as correct, and the output answer must map to one of the given options, the most defensible extracted answer is B with a discrepancy noted between the options and the solution content.