Mathematical Reasoning
2023
Q1
JEE Mains
Numerical
14 Mar 2026
The number of ordered triplets of the truth values of $p, q$ and $r$ such that the truth value of the statement $(p \vee q) \wedge(p \vee r) \Rightarrow(q \vee r)$ is True, is equal to ___________.
Correct Answer: 7
Explanation:
$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline \boldsymbol{p} & \boldsymbol{q} & \boldsymbol{r} & \boldsymbol{p} \vee \boldsymbol{q} & \boldsymbol{p} \vee \boldsymbol{r} & \begin{array}{c}
(\boldsymbol{p} \vee \boldsymbol{q}) \wedge \\
(\boldsymbol{p} \vee \boldsymbol{r})
\end{array} & \boldsymbol{q} \vee \boldsymbol{r} & \begin{array}{c}
(\boldsymbol{p} \vee \boldsymbol{q}) \wedge(\boldsymbol{p} \vee \boldsymbol{r}) \\
\Rightarrow(\boldsymbol{q} \vee \boldsymbol{r})
\end{array} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline
\end{array}
$
Hence, the total number of ordered triplets are 7.
Hence, the total number of ordered triplets are 7.
2022
Q2
JEE Mains
Numerical
14 Mar 2026
The maximum number of compound propositions, out of p$\vee$r$\vee$s, p$\vee$r$\vee$$\sim$s, p$\vee$$\sim$q$\vee$s, $\sim$p$\vee$$\sim$r$\vee$s, $\sim$p$\vee$$\sim$r$\vee$$\sim$s, $\sim$p$\vee$q$\vee$$\sim$s, q$\vee$r$\vee$$\sim$s, q$\vee$$\sim$r$\vee$$\sim$s, $\sim$p$\vee$$\sim$q$\vee$$\sim$s that can be made simultaneously true by an assignment of the truth values to p, q, r and s, is equal to __________.
Correct Answer: 9
Explanation:
There are total 9 compound propositions, out of which 6 contain $\sim s$. So if we assign $s$ as false, these 6 propositions will be true.
In remaining 3 compound propositions, two contain $p$ and the third contains $\sim r$. So if we assign $p$ and $r$ as true and false respectively, these 3 propositions will also be true.
Hence maximum number of propositions that can be true are 9.
In remaining 3 compound propositions, two contain $p$ and the third contains $\sim r$. So if we assign $p$ and $r$ as true and false respectively, these 3 propositions will also be true.
Hence maximum number of propositions that can be true are 9.