Mathematics Question from JEE Main 2024 · 6 April, Shift 1

Let $\vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}$, $\vec{b} = -\hat{i} + \hat{k}$, $\vec{c} = \beta \hat{j} - \hat{k}$, where $\alpha$ and $\beta$ are integers and $\alpha\beta = -6$. Let the values of the ordered pair $(\alpha, \beta)$ for which the area of the parallelogram of diagonals $\vec{a} + \vec{b}$ and $\vec{b} + \vec{c}$ is $\frac{\sqrt{21}}{2}$, be $(\alpha_1, \beta_1)$ and $(\alpha_2, \beta_2)$. Then $\alpha_1^2 + \beta_1^2 - \alpha_2\beta_2$ is equal to: