Let $$A, B$$ and $$C$$ be three points on the parabola $$y^2=6 x$$ and let the line segment $$A B$$ meet the line $$L$$ through $$C$$ parallel to the $$x$$-axis at the point $$D$$. Let $$M$$ and $$N$$ respectively be the feet of the perpendiculars from $$A$$ and $$B$$ on $$L$$. Then $$\left(\frac{A M \cdot B N}{C D}\right)^2$$ is equal to __________.

Consider the matrices : $$A=\left[\begin{array}{cc}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$$ and $$X=\left[\begin{array}{l}x \\ y\end{array}\right]$$. Let the set of all $$m$$, for which the system of equations $$A X=B$$ has a negative solution (i.e., $$x<0$$ and $$y<0$$), be the interval $$(a, b)$$. Then $$8 \int_\limits a^b|A| d m$$ is equal to _________.

For a differentiable function $$f: \mathbb{R} \rightarrow \mathbb{R}$$, suppose $$f^{\prime}(x)=3 f(x)+\alpha$$, where $$\alpha \in \mathbb{R}, f(0)=1$$ and $$\lim _\limits{x \rightarrow-\infty} f(x)=7$$. Then $$9 f\left(-\log _e 3\right)$$ is equal to _________.

Consider the circle $$C: x^2+y^2=4$$ and the parabola $$P: y^2=8 x$$. If the set of all values of $$\alpha$$, for which three chords of the circle $$C$$ on three distinct lines passing through the point $$(\alpha, 0)$$ are bisected by the parabola $$P$$ is the interval $$(p, q)$$, then $$(2 q-p)^2$$ is equal to __________.