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,

$ \frac{6}{3^{26}} + \frac{10 \cdot 1}{3^{25}} + \frac{10 \cdot 2}{3^{24}} + \frac{10 \cdot 2^2}{3^{23}} + \ldots + \frac{10 \cdot 2^{24}}{3} $ is equal to :

Let [.] denote the greatest integer function. Then

$$ \int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{12(3+[x])}{3+\left[\sin x\right]+\left[\cos x\right]} \right) dx $$
is equal to :

Let the ellipse $E: \frac{x^2}{144} + \frac{y^2}{169} = 1$ and the hyperbola $H: \frac{x^2}{16} - \frac{y^2}{\lambda^2} = -1$ have the same foci. If $e$ and $L$

respectively denote the eccentricity and the length of the latus rectum of $H$, then the value of $24(e+L)$ is :