For the following questions answer them individually
The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ________
If $$\left(\frac{1}{\alpha+1} + \frac{1}{\alpha+2} + \ldots + \frac{1}{\alpha+1012}\right) - \left(\frac{1}{2 \cdot 1} + \frac{1}{4 \cdot 3} + \frac{1}{6 \cdot 5} + \ldots + \frac{1}{2024 \cdot 2023}\right) = \frac{1}{2024}$$, then $$\alpha$$ is equal to ________
Let $$A, B$$ and $$C$$ be three points on the parabola $$y^2 = 6x$$ and let the line segment $$AB$$ 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 $$\left(\frac{AM \cdot BN}{CD}\right)^2$$ is equal to ________
Consider the circle $$C : x^2 + y^2 = 4$$ and the parabola $$P : y^2 = 8x$$. 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 $$(2q - p)^2$$ is equal to ________
Consider the matrices : $$A = \begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}, B = \begin{bmatrix} 20 \\ m \end{bmatrix}$$ and $$X = \begin{bmatrix} x \\ y \end{bmatrix}$$. Let the set of all $$m$$, for which the system of equations $$AX = B$$ has a negative solution (i.e., $$x < 0$$ and $$y < 0$$), be the interval $$(a, b)$$. Then $$8\int_a^b |A| \, dm$$ is equal to ________
Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $$2\sin^{-1} x + 3\cos^{-1} x = \frac{2\pi}{5}$$, is ________
Let $$A = \{(x, y) : 2x + 3y = 23, x, y \in \mathbb{N}\}$$ and $$B = \{x : (x, y) \in A\}$$. Then the number of one-one functions from $$A$$ to $$B$$ is equal to ________
For a differentiable function $$f : \mathbb{R} \to \mathbb{R}$$, suppose $$f'(x) = 3f(x) + \alpha$$, where $$\alpha \in \mathbb{R}$$, $$f(0) = 1$$ and $$\lim_{x \to -\infty} f(x) = 7$$. Then $$9f(-\log_e 3)$$ is equal to ________
Let the set of all values of $$p$$, for which $$f(x) = (p^2 - 6p + 8)(\sin^2 2x - \cos^2 2x) + 2(2 - p)x + 7$$ does not have any critical point, be the interval $$(a, b)$$. Then $$16ab$$ is equal to ________
The square of the distance of the image of the point $$(6, 1, 5)$$ in the line $$\frac{x-1}{3} = \frac{y}{2} = \frac{z-2}{4}$$, from the origin is ________
