JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Mathematical Reasoning *

First simplify $\sim (p\iff \sim q)$. Since $p\iff \sim q$ is true when $p$ and $\sim q$ have the same truth value, its negation is true when they differ, i.e., when $p$ and $q$ have the same truth value. Thus $\sim (p\iff \sim q)\equiv (p\iff q)$. Now $(p\iff q)\wedge q$ is true when both $p\iff q$ and $q$ are true. If $q$ is true and $p\iff q$ is true, then $p$ must be true. Thus $(p\iff q)\wedge q\equiv p\wedge q$. Option (D) states $(p\Rightarrow q)\wedge p$, which is also equivalent to $p\wedge q$ (since if $p$ is true and $p\Rightarrow q$ is true, then $q$ must be true). Therefore the answer is (D).