Which of the following logical statements is a tautology?

If $\sim p \to q$ is false and $q \leftrightarrow r$ is false, then the truth value of $(p, q, r)$ is...

$(p \wedge r) \Leftrightarrow (p \wedge (\sim q))$ is equivalent to $(\sim p)$ when $r$ is.

Consider the following statements:
$p$: the switch $S_1$ is closed.
$q$: the switch $S_2$ is closed.
$r$: the switch $S_3$ is closed.
Then the switching circuit represented by the statement $(p \wedge q) \vee (\sim p \wedge (\sim q \vee p \vee r))$ is

The statement $(p \wedge q) \Rightarrow (p \wedge r)$ is equivalent to:

Which of the following statements has the truth value $T$?
$A$: Cube roots of unity are in Geometric Progression and their sum is $0$.
$B$: $4+7 > 10$ iff $2+8 < 10$.
$C$: $\exists x \in N$ such that $x^2-3x+2=0$ and $\exists n \in N$ such that $n$ is an odd number.
$D$: $3+i$ is a complex number or $\sqrt{2}+\sqrt{3}=\sqrt{5}$.