JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Mathematical Reasoning *

$p\to q$ is false only when $p$ is true and $q$ is false. In that single case $\sim q$ is true and $p$ is true, so $\sim q\to p$ is true; in every other case $p\to q$ is itself true. The disjunction $(p\to q)\vee (\sim q\to p)$ is therefore true for all truth values of $p$ and $q$, so S1 is a tautology.
For S2, $\sim p\vee q$ is logically equivalent to $p\to q$, while $p\wedge \sim q$ is equivalent to $\sim (p\to q)$. Their conjunction is a statement conjoined with its own negation, which is false for all truth values, so S2 is a fallacy. Both S1 and S2 are true.