GATE CS 1993 Q8.2 — Propositional and First Order Logic

For two different persons $x$ and $y$, the predicate $M(x, y)$ denotes that $x$ knows $y$. Consider the following statement. There is a person who does not know anyone else, but that person is known by everyone else. Which one of the following expressions represents the above statement?

Which of the following predicate logic formulae/formula is/are CORRECT representation(s) of the statement: "Everyone has exactly one mother"? The meanings of the predicates used are: mother $(y, x): y$ is the mother of $x$ noteq $(x, y): x$ and $y$ are not equal

​​​Let $P(x)$ be an arbitrary predicate over the domain of natural numbers. Which ONE of the following statements is TRUE?

​​​​Let $p$ and $q$ be the following propositions: $p$ : Fail grade can be given. $q$ : Student scores more than $50 \%$ marks. Consider the statement: "Fail grade cannot be given when student scores more than $50 \%$ marks." Which one of the following is the CORRECT representation of the above statement in propositional logic?

Geetha has a conjecture about integers, which is of the form $$\forall x(P(x) \Longrightarrow \exists y Q(x, y)),$$ where $P$ is a statement about integers, and $Q$ is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha's conjecture?

Let $p$ and $q$ be two propositions. Consider the following two formulae in propositional logic. $S_1: (\neg p\wedge(p\vee q))\rightarrow q$ $S_2: q\rightarrow(\neg p\wedge(p\vee q))$ Which one of the following choices is correct?