Which of the following is equivalent to the Boolean expression $p \wedge \sim q$?

The statement pattern $(p \vee q) \to \sim r$ is logically 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 $\sqrt{2}+\sqrt{3}=\sqrt{5}$.

The contrapositive statement of the statement "If $x$ is a prime number,then $x$ is odd" is

If $p:$ Seema is fat. $q:$ She is happy,then the logical equivalent statement of 'If Seema is fat,then she is happy' is

The logical statement $[\sim(\sim p \vee q) \vee (p \wedge r) \wedge (\sim q \wedge r)]$ is equivalent to