Let α and β be the real numbers such that

$ \lim\limits_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int\limits_0^x \frac{1}{1-t^2} \, dt + \beta x \cos x \right) = 2. $

Then the value of α + β is ___________.

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be a function such that f(x) > 0 for all x ∈ ℝ, and f(x+y) = f(x)f(y) for all x, y ∈ ℝ.

Let the real numbers a₁, a₂, ..., a₅₀ be in an arithmetic progression. If f(a₃₁) = 64f(a₂₅), and

$ \sum\limits_{i=1}^{50} f(a_i) = 3(2^{25}+1), $

then the value of

$ \sum\limits_{i=6}^{30} f(a_i) $

is ________________.

For all x > 0, let y₁(x), y₂(x), and y₃(x) be the functions satisfying

$ \frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, $

$ \frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = \frac{1}{3}, $

$ \frac{dy_3}{dx} - \frac{(2-x^3)}{x^3} y_3 = 0, \quad y_3(1) = \frac{3}{5e}, $

respectively. Then

$ \lim\limits_{x \to 0^+} \frac{y_1(x)y_2(x)y_3(x) + 2x}{e^{3x} \sin x} $

is equal to __________________.

Consider the following frequency distribution:

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6.

For the given frequency distribution, let $\alpha$ denote the mean deviation about the mean, $\beta$ denote the mean deviation about the median, and $\sigma^2$ denote the variance.

Match each entry in List-I to the correct entry in List-II and choose the correct option.