For the following questions answer them individually
Let $$\alpha, \beta$$ be roots of $$x^2 + \sqrt{2}x - 8 = 0$$. If $$U_n = \alpha^n + \beta^n$$, then $$\frac{U_{10} + \sqrt{2}U_9}{2U_8}$$ is equal to ___________
If $$S(x) = (1+x) + 2(1+x)^2 + 3(1+x)^3 + \cdots + 60(1+x)^{60}$$, $$x \neq 0$$, and $$(60)^2 S(60) = a(b)^b + b$$, where $$a, b \in N$$, then $$(a + b)$$ equal to ___________
The length of the latus rectum and directrices of a hyperbola with eccentricity $$e$$ are 9 and $$x = \pm \frac{4}{\sqrt{13}}$$, respectively. Let the line $$y - \sqrt{3}x + \sqrt{3} = 0$$ touch this hyperbola at $$(x_0, y_0)$$. If $$m$$ is the product of the focal distances of the point $$(x_0, y_0)$$, then $$4e^2 + m$$ is equal to ___________
In a triangle $$ABC$$, $$BC = 7$$, $$AC = 8$$, $$AB = \alpha \in \mathbb{N}$$ and $$\cos A = \frac{2}{3}$$. If $$49\cos(3C) + 42 = \frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to ___________
If the system of equations $$2x + 7y + \lambda z = 3$$, $$3x + 2y + 5z = 4$$, $$x + \mu y + 32z = -1$$ has infinitely many solutions, then $$(\lambda - \mu)$$ is equal to ___________
Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Let $$f : [0, \infty) \rightarrow \mathbb{R}$$ be a function defined by $$f(x) = \left[\frac{x}{2} + 3\right] - [\sqrt{x}]$$. Let $$S$$ be the set of all points in the interval $$[0, 8]$$ at which $$f$$ is not continuous. Then $$\sum_{a \in S} a$$ is equal to ___________
Let $$[t]$$ denote the largest integer less than or equal to $$t$$. If $$\int_0^3 \left([x^2] + \left[\frac{x^2}{2}\right]\right) dx = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7}$$, where $$a, b, c \in \mathbb{Z}$$, then $$a + b + c$$ is equal to ___________
If the solution $$y(x)$$ of the given differential equation $$(e^y + 1)\cos x \, dx + e^y \sin x \, dy = 0$$ passes through the point $$\left(\frac{\pi}{2}, 0\right)$$, then the value of $$e^{y\left(\frac{\pi}{6}\right)}$$ is equal to ___________
If the shortest distance between the lines $$\frac{x - \lambda}{3} = \frac{y - 2}{-1} = \frac{z - 1}{1}$$ and $$\frac{x + 2}{-3} = \frac{y + 5}{2} = \frac{z - 4}{4}$$ is $$\frac{44}{\sqrt{30}}$$, then the largest possible value of $$|\lambda|$$ is equal to ___________
From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $$X$$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $$X$$ is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$n - m$$ is equal to ___________
