Question

JEE Mains Maths question is attached

Answer 1

Since $p\to q\equiv\lnot p\lor q,$

$\longrightarrow (p\land q)\to(\lnot q\lor r)\equiv \lnot(p\land q)\lor(\lnot q\lor r)$

Since $\lnot(p\land q)\equiv\lnot p\lor\lnot q,$

$\longrightarrow (p\land q)\to(\lnot q\lor r)\equiv (\lnot p\lor \lnot q)\lor(\lnot q\lor r)$

The brackets can be ignored now.

$\longrightarrow (p\land q)\to(\lnot q\lor r)\equiv\lnot p\lor \lnot q\lor\lnot q\lor r$

$\longrightarrow (p\land q)\to(\lnot q\lor r)\equiv\lnot p\lor( \lnot q\lor\lnot q)\lor r$

Since $p\lor p\equiv p,$

$\longrightarrow (p\land q)\to(\lnot q\lor r)\equiv\lnot p\lor \lnot q\lor r$

Given that the truth value of this statement is false.

Then the three statements $\lnot p,\ \lnot q$ and $r$ should also be false, since they're connected by OR.

$\longrightarrow \lnot p=F\quad\implies\quad\underline{\underline{p=T}}$

$\longrightarrow \lnot q=F\quad\implies\quad\underline{\underline{q=T}}$

$\longrightarrow\underline{\underline{r=F}}$

Thus the truth values of $p,\ q$ and $r$ are $\bf{T,\ T,\ F}$ respectively.

Hence (1) is the answer.

Answer 2

$p,q,r \: are \: statements \: with \: truth \: values \: T,F,T$

respectively

consider

(~p ∨ q) ^ ~ r $\implies$ p

= ( F ∨ F) ^ F $\implies$ T

= (F ^ F) $\implies$ T

= F $\implies$ T = ~(F) ∨ T = T ∨ T = T

Clearly, option A is answer.