Let $n>1$. Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$. Let the vector ( 0,1 , $0,0, \ldots, 0) \in \mathbb{R}^n$ be in the null space of $M$.

Which of the following options is/are always correct?

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as follows:

$$ f(x)=\left\{\begin{array}{cc} c_1 e^x-c_2 \log _e\left(\frac{1}{x}\right), & \text { if } x>0 \\ 3, & \text { otherwise } \end{array}\right. $$

where $c_1, c_2 \in \mathbb{R}$.

If $f$ is continuous at $x=0$, then $c_1+c_2=$ $\_\_\_\_$ . (answer in integer)

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 $v \in S$ ?