For the following questions answer them individually
Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - x + 2 = 0$$ with $$\text{Im}(\alpha) > \text{Im}(\beta)$$. Then $$\alpha^6 + \alpha^4 + \beta^4 - 5\alpha^2$$ is equal to _______
All the letters of the word $$GTWENTY$$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $$GTWENTY$$ is _______
If $$\frac{^{11}C_1}{2} + \frac{^{11}C_2}{3} + \ldots + \frac{^{11}C_9}{10} = \frac{n}{m}$$ with $$\gcd(n, m) = 1$$, then $$n + m$$ is equal to _______
Equations of two diameters of a circle are $$2x - 3y = 5$$ and $$3x - 4y = 7$$. The line joining the points $$(-\frac{22}{7}, -4)$$ and $$(-\frac{1}{7}, 3)$$ intersects the circle at only one point $$P(\alpha, \beta)$$. Then $$17\beta - \alpha$$ is equal to _______
If the points of intersection of two distinct conics $$x^2 + y^2 = 4b$$ and $$\frac{x^2}{16} + \frac{y^2}{b^2} = 1$$ lie on the curve $$y^2 = 3x^2$$, then $$3\sqrt{3}$$ times the area of the rectangle formed by the intersection points is _______.
If the mean and variance of the data $$65, 68, 58, 44, 48, 45, 60, \alpha, \beta, 60$$ where $$\alpha > \beta$$ are $$56$$ and $$66.2$$ respectively, then $$\alpha^2 + \beta^2$$ is equal to _______
Let $$f(x) = 2^x - x^2$$, $$x \in R$$. If $$m$$ and $$n$$ are respectively the number of points at which the curves $$y = f(x)$$ and $$y = f'(x)$$ intersects the $$x$$-axis, then the value of $$m + n$$ is _______
The area (in sq. units) of the part of circle $$x^2 + y^2 = 169$$ which is below the line $$5x - y = 13$$ is $$\frac{\pi\alpha}{2\beta} - \frac{65}{2} + \frac{\alpha}{\beta}\sin^{-1}(\frac{12}{13})$$ where $$\alpha, \beta$$ are coprime numbers. Then $$\alpha + \beta$$ is equal to _______
If the solution curve $$y = y(x)$$ of the differential equation $$(1 + y^2)(1 + \log_e x)dx + xdy = 0$$, $$x > 0$$ passes through the point $$(1, 1)$$ and $$y(e) = \frac{\alpha - \tan(\frac{3}{2})}{\beta + \tan(\frac{3}{2})}$$, then $$\alpha + 2\beta$$ is _______
A line with direction ratio $$2, 1, 2$$ meets the lines $$x = y + 2 = z$$ and $$x + 2 = 2y = 2z$$ respectively at the point $$P$$ and $$Q$$. If the length of the perpendicular from the point $$(1, 2, 12)$$ to the line $$PQ$$ is $$l$$, then $$l^2$$ is _______
