If the function $f(x) = \frac{K\sin x + 2\cos x}{\sin x + \cos x}$ is increasing for all values of $x,$ then

Consider the following statements $S$ and $R$:
$S: \sin x$ and $\cos x$ are both decreasing functions in the interval $\left( \frac{\pi}{2}, \pi \right)$.
$R:$ If a function $f$ is decreasing in an interval $(a, b)$,then its derivative $f'(x)$ is also decreasing in the interval $(a, b)$.
Which of the following is true?

Consider two statements $S_1$ and $S_2$.
$S_1$: If $f(x)$ is a differentiable function with $f'(x) > 0$ in $(a, b)$ and $f(x)$ is increasing in $(a, b)$,then $\frac{f(x)}{f'(x)}$ is also increasing in $(a, b)$.
$S_2$: Both $\sin x$ and $\tan x$ are increasing functions in $(0, \frac{\pi}{2})$.
Which of the following is true?

The function $f(x) = \frac{4x^3 - 3x^2}{6} - 2 \sin x + (2x - 1) \cos x$:

Let $n$ be a natural number and let $a$ be a real number. The number of zeroes of $x^{2n+1} - (2n+1)x + a = 0$ in the interval $[-1, 1]$ is:

For what values of $a$ is the function $f$ given by $f(x) = x^2 + ax + 1$ increasing on the interval $[1, 2]$?