If $q$ is false and $p\, \wedge \,q\, \leftrightarrow \,r$ is true, then which one of the following statements is a tautology?

The negation of the compound statement $^ \sim p \vee \left( {p \vee \left( {^ \sim q} \right)} \right)$ is

For the statements $p$ and $q$, consider the following compound statements :

$(a)$ $(\sim q \wedge( p \rightarrow q )) \rightarrow \sim p$

$(b)$ $((p \vee q) \wedge \sim p) \rightarrow q$

Then which of the following statements is correct?

The expression $ \sim ( \sim p\, \to \,q)$ is logically equivalent to

$\sim (p \vee q) \vee (\sim p \wedge q)$ is logically equivalent to

The contrapositive of statement 'If Jaipur is capital of Rajasthan, then Jaipur is in India' is