If the chord joining the points $\mathrm{P}_1\left(x_1, y_1\right)$ and $\mathrm{P}_2\left(x_2, y_2\right)$ on the parabola $y^2=12 x$ subtends a right angle at the vertex of the parabola, then $x_1 x_2-y_1 y_2$ is equal to

If $\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 3 & 5\end{array}\right]$, then the determinant of the matrix $\left(\mathrm{A}^{2025}-3 \mathrm{~A}^{2024}+\mathrm{A}^{2023}\right)$ is

The value of $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x$, where $[\cdot]$ denotes the greatest integer function, is

Let $\overrightarrow{\mathrm{AB}}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\overrightarrow{\mathrm{AD}}=\hat{i}+2 \hat{j}+\lambda \hat{k}, \lambda \in \mathbb{R}$. Let the projection of the vector $\vec{v}=\hat{i}+\hat{j}+\hat{k}$ on the diagonal $\overrightarrow{\mathrm{AC}}$ of the parallelogram ABCD be of length one unit. If $\alpha, \beta$, where $\alpha>\beta$, be the roots of the equation $\lambda^2 x^2-6 \lambda x+5=0$, then $2 \alpha-\beta$ is equal to