NIMCET 2026 — Mathematics PYQ

To find the number of ordered tuples $(p,q,r)$ that make the statement $(\neg p\vee q)\Rightarrow r$ true, we analyze the truth values of the implication.

An implication $A\Rightarrow B$ is false only when the antecedent $A$ is True and the consequent $B$ is False. In all other cases, it is true.

Let $A=(\neg p\vee q)$ and $B=r$. We want to find cases where $A\Rightarrow r$ is True. It is easier to find cases where it is false and subtract from the total number of combinations.

1. Total combinations: Since there are 3 variables ($p,q,r$), the total number of possible tuples is $2^3=8$.

2. Condition for False: The statement is false if:

   • $(\neg p\vee q)$ is True

   • $r$ is False

3. Analyzing $(\neg p\vee q)=T$ when $r=F$:

   • $(\neg p\vee q)$ is False only when both $\neg p$ is False and $q$ is False.

   • $\neg p=F⟹p=T$

   • $q=F$

   • So, $(\neg p\vee q)$ is False only for the tuple $(p,q)=(T,F)$.

   • Therefore, $(\neg p\vee q)$ is True for the other 3 combinations of $(p,q)$: $(T,T),(F,T),(F,F)$.

4. Identifying False Tuples:

   The statement is False for $(p,q,r)\in \{(T,T,F),(F,T,F),(F,F,F)\}$.

   There are exactly 3 cases where the statement is False.

5. Final Calculation:

   Number of True cases = (Total cases) - (False cases)

   Number of True cases = $8-3=5$

Correct Option: (d) 5