Let $f(x) = \lim\limits_{\theta \to 0} \left( \frac{\cos \pi x - x^\left( \frac{2}{\theta} \right) \sin(x-1)}{1 + x^\left( \frac{2}{\theta} \right) (x-1)} \right),\ x \in \mathbb{R}$. Consider the following two statements :

(I) $f(x)$ is discontinuous at $x=1$.

(II) $f(x)$ is continuous at $x = -1$.

Then,

The value of

$$ \lim\limits_{x \rightarrow 0} \frac{\log _e\left(\sec (e x) \cdot \sec \left(e^2 x\right) \cdot \ldots \cdot \sec \left(e^{10} x\right)\right)}{e^2-e^{2 \cos x}} $$

is equal to

Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$, and $\lim\limits_{t \rightarrow x}\left(\frac{t^2 y(x)-x^2 y(t)}{x-t}\right)=3$ for each $x > 0$. Then $2 y(2)$ is equal to :

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function

$$ f(x)=\left\{\begin{array}{cl} b^2 \sin \left(\frac{\pi}{2}\left[\frac{\pi}{2}(\cos x+\sin x) \cos x\right]\right), & x<0 \\ \frac{\sin x-\frac{1}{2} \sin 2 x}{x^3} & , x>0 \\ a & , x=0 \end{array}\right. $$

is continuous at $x=0$, then $a^2+b^2$ is equal to :