Consider the following three statements for the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=\left|\log _e x\right|-|x-1|$ :

(I) $f$ is differentiable at all $x>0$.

(II) $f$ is increasing in $(0,1)$.

(III) $f$ is decreasing in $(1, \infty)$.

Then.

Let $P=\left[p_{i j}\right]$ and $Q=\left[q_{i j}\right]$ be two square matrices of order 3 such that $q_{\mathrm{ij}}=2^{(\mathrm{i}+\mathrm{j}-1)} \mathrm{p}_{\mathrm{ij}}$ and $\operatorname{det}(\mathrm{Q})=2^{10}$. Then the value of $\operatorname{det}(\operatorname{adj}(\operatorname{adj} \mathrm{P}))$ is:

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 :

The sum of all values of $\alpha$, for which the shortest distance between the lines $\frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha}$ and $\frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2 \alpha}$ is $\sqrt{2}$, is