JEE Advanced 2024 · Paper 1, Mathematics question 12

Let $p = \frac1\alpha$ and $q = \frac1\beta$, so $\overrightarrow{OP} = (1-p, 1, 1)$, $\overrightarrow{OQ} = (1, 1-q, 1)$ and $\overrightarrow{OR} = \left(1, 1, \frac12\right)$.

Scalar triple product:
$$\begin{vmatrix} 1-p & 1 & 1 \\ 1 & 1-q & 1 \\ 1 & 1 & \frac12 \end{vmatrix} = \frac{(1-p)(1-q)}{2} - (1-p) - (1-q) + \frac32 = \frac{pq + p + q}{2} = 0.$$

So $\frac{1}{\alpha\beta} + \frac1\alpha + \frac1\beta = 0$, i.e. $1 + \beta + \alpha = 0$, so $\alpha + \beta = -1$.

$(\alpha, \beta, 2)$ on the plane: $3(\alpha+\beta) - 2 + l = 0 \implies -3 - 2 + l = 0 \implies l = 5$.