A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $\frac{m}{n}$, $\gcd(m, n) = 1$, then $n^2 - m^2$ is equal to :

If the range of the function $ f(x) = \frac{5-x}{x^2 - 3x + 2} , \ x \neq 1, 2, $ is $ (-\infty , \alpha] \cup [\beta, \infty) $, then $ \alpha^2 + \beta^2 $ is equal to :

The number of real roots of the equation $x |x - 2| + 3|x - 3| + 1 = 0$ is :

Let the length of a latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be 10. If its eccentricity is the minimum value of the function $f(t) = t^2 + t + \frac{11}{12}$, $t \in \mathbb{R}$, then $a^2 + b^2$ is equal to :