Suppose $X$ and $Y$ are random variables. The conditional expectation of $X$ given $Y$ is denoted by $E[X \mid Y]$. Then $E[E[X \mid Y]]$ equals

The sum of the elements in each row of $A \in \mathbb{R}^{n \times n}$ is 1 . If $B=A^3-2 A^2+A$, which one of the following statements is correct (for $x \in \mathbb{R}^n$ )?

Let $f(x)=\frac{e^x-e^{-x}}{2}, x \in R$. Let $f^{(k)}(a)$ denote the $k^{\text {th }}$ derivative of $f$ evaluated at $a$. What is the value of $f^{(10)}(0)$ ?(Note: ! denotes factorial)

Let $p$ and $q$ be any two propositions. Consider the following propositional statements.

$$ \begin{aligned} & S_1: p \rightarrow q, \quad S_2: \neg p \wedge q, \quad S_3: \neg p \vee q, \\ & S_4: \neg p \vee \neg q, \end{aligned} $$

Where $\wedge$ denotes conjunction (AND operation), $\vee$ denotes disjunction (OR operation), and $\neg$ denotes negation

(NOT operation). Which one of the following options is correct?

(Note: $\equiv$ denotes logical equivalence)