Let $y = y(x)$ be the solution curve of the differential equation

$(1 + \sin x)\dfrac{dy}{dx} + (y + 1)\cos x = 0,\ y(0) = 0.$ If the curve $y = y(x)$ passes through the point $\left( \alpha , \dfrac{-1}{2} \right)$,

then a value of $\alpha$ is:

If the domain of the function

$f(x) = \sqrt{\log_{(0.6)} (\left| \frac{2x-5}{x^2-4} \right|)}$ is $(-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)$, then the value of $a + b + c + d + e$ is ________.

If $\sum\limits_{k=1}^{n} a_k = 6 n^3$, then $\sum\limits_{k=1}^{6} \left( \frac{a_{k+1} - a_k}{36} \right)^2$ is equal to ________.

Let a, b, c ∈ {1, 2, 3, 4}. If the probability, that $a x^2 + 2\sqrt{2} bx + c > 0$ for all $x \in \mathbb{R}$, is $\frac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to ________.