Verified Solution • 2025–26 Session moderate

QJEEM23Ma15ae: JEE Main 2023 Mathematics - Class 11 JEEM23Ma15ae Solution: The number of values of for which is a tautology, is :

The number of values of r∈{p,q,∼p,∼q}\mathrm{r} \in\{\mathrm{p}, \mathrm{q}, \sim \mathrm{p}, \sim \mathrm{q}\} for which ((p∧q)⇒(r∨q))∧((p∧r)⇒q)((\mathrm{p} \wedge \mathrm{q}) \Rightarrow(\mathrm{r} \vee \mathrm{q})) \wedge((\mathrm{p} \wedge \mathrm{r}) \Rightarrow \mathrm{q}) is a tautology, is :

Multiple Choice Options

  • A 2
  • B 1
  • C 4
  • D 3

Step-by-Step Verified Solution

((p∧q)⇒(r∨q))∧((p∧r)⇒q)≡T((p \wedge q) \Rightarrow(r \vee q)) \wedge((p \wedge r) \Rightarrow q) \equiv T (given)

≡((∼p∨∼q)∨(r∨q))∧(∼p∨∼r∨q)\equiv((\sim p \vee \sim q) \vee(r \vee q)) \wedge(\sim p \vee \sim r \vee q)

≡((∼p∨r)∨(∼q∨q))∧(∼p∨∼r∨q)\equiv((\sim p \vee r) \vee(\sim q \vee q)) \wedge(\sim p \vee \sim r \vee q)

≡∼p∨∼r∨q\equiv \sim p \vee \sim r \vee q

For above statement to be tautology

rr can be ∼p\sim p or qq

∴\therefore Two values of rr are possible.

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