Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as follows:

$$ f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right) $$

Which of the following statements is/are true?

Let $G(V, E)$ be a simple, undirected graph. A vertex cover of $G$ is a subset $V^{\prime} \subseteq V$ such that for every $(u, v) \in E, u \in V^{\prime \prime}$ or $v \in V^{\prime}$. Let the size of the smallest vertex cover in $G$ be $k$. Let $S$ be any vertex cover of size $k$.

For a vertex $v \in V$, which of the following constraints will always ensure that $v \in S$ ?

Let $G$ be an undirected graph, which is a path on 8 vertices. The number of matchings in $G$ is $\_\_\_\_$ (answer in integer)

Let $X$ be a random variable which takes values in the set $\{1,2,3,4,5,6,7,8\}$.

Further, $\operatorname{Pr}(X=1)=\operatorname{Pr}(X=2)=\operatorname{Pr}(X=5)=\operatorname{Pr}(X=7)=\frac{1}{6}$ and $\operatorname{Pr}(X=3)=\operatorname{Pr}(X=4) =\operatorname{Pr}(X=6)=\operatorname{Pr}(X=8)=\frac{1}{12}$.

The expected value of $X$, denoted by $E[X]$, is equal to $\_\_\_\_$ . (rounded off to two decimal places)