JEE Mathematics 2024 Question with Solution

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

Find: The greatest integer less than or equal to $p$, where $p$ is the sum of squares of all real values of $\alpha$ for which these lines do not form a triangle.

Three lines do not form a triangle if at least two of them are parallel. The solution analyzes the parallelism condition using

For lines $2x - y + 3 = 0$ and $6x + 3y + 1 = 0$,

so these two lines are not parallel.

For lines $2x - y + 3 = 0$ and $\alpha x + 2y - 2 = 0$,

Hence,

For lines $6x + 3y + 1 = 0$ and $\alpha x + 2y - 2 = 0$,

Hence,

Therefore, the real values are $\alpha = -4$ and $\alpha = 4$. So,

The greatest integer less than or equal to $p$ is $32$. Therefore, the correct option is A.

The second extracted approach mentions an additional value $\alpha = \frac{4}{5}$ from concurrency, but it also states

which is numerically inconsistent. Since the first approach gives a coherent derivation and concludes $p = 32$, the defensible answer is $32$.