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Consider the following two propositions:
\( P_1 : \sim (p\to \sim q) \)
\( P_2 : (p \wedge \sim q) \wedge ((\sim p) \vee q) \)
If the proposition \( p \to ((\sim p) \vee q) \) is evaluated as FALSE, then:

Consider the following statements:
\( P: \) Ramu is intelligent.
\( Q: \) Ramu is rich.
\( R: \) Ramu is not honest.
The negation of the statement “Ramu is intelligent and honest if and only if Ramu is not rich” can be expressed as:

\( \text{The Boolean expression } \big(\sim (p \land q)\ \lor\ q\big) \text{ is equivalent to:} \)

If the truth value pf the statement \( (P \land (\sim R)) \rightarrow ((\sim R) \land Q) \) is \( F \), then the truth value of which of the following is \( F \)?

The statement \( (p \land q) \Rightarrow (p \land r) \) is equivalent to:

Which of the following statement is a tautology?

The maximum number of compound propositions, out $p \vee r \vee s, p \vee r \vee \sim s, p \vee \sim q \vee s, \sim p \vee \sim r \vee s, \sim p \vee \sim r \vee \sim s, \sim p \vee q \vee \sim s, q \vee r \vee \sim s, q \vee \sim r \vee \sim s, \sim p \vee \sim q \vee \sim s$ that can be made simultaneously true by an assignment of the truth values to $p, q, r$ and $s$, equal to _________________.

Let \( \Delta \in \{\land, \lor, \Rightarrow, \Leftrightarrow\} \) be such that \( (p \land q)\ \Delta\ \big((p \lor q) \Rightarrow q\big) \) is a tautology. Then \( \Delta \) is equal to:

Negation of the Boolean expression \( p \Leftrightarrow (q \Rightarrow p) \) is:

Let the operation \(\ast, \odot \in \{\land, \lor\}\). If \( (p * q)\\odot\ (p \odot \sim q) \) is a tautology, then the ordered pair \( (*,\ \odot) \) is:

Let
\( p \): Ramesh listens to music
\( q \): Ramesh is out of his village
\( r \): It is Sunday
\( s \): It is Saturday
Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday” can be expressed as:

\( (p \lor r) \Leftrightarrow \big(p \land (\sim q)\big) \) is equivalent to \( (\sim p) \) when \( r \) is:

Consider the following statements:
\( A \) : Rishi is a judge.
\( B \) : Rishi is honest.
\( C \) : Rishi is not arrogant.
The negation of the statement “if Rishi is a judge and he is not arrogant, then he is honest” is:

The negation of the Boolean expression \( \big((\sim q)\land p\big)\Rightarrow \big((\sim p)\lor q\big) \) is logically equivalent to:

Negation of the Boolean statement \( (p \vee q) \Rightarrow ((\sim r) \vee p) \) is equivalent to:

Let \( \Delta, \nabla \in \{\wedge, \vee\} \) be such that \( p \nabla q \Rightarrow ((p \Delta q)\nabla r) \) is a tautology. Then \( (p \nabla q)\Delta r \) is logically equivalent to:

Which of the following statements is a tautology?

The number of choices for $\Delta \in \{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$, such that $(p \Delta q) \Rightarrow ((p \Delta \sim q) \vee ((\sim p) \Delta q))$ is a tautology, is:

The mean and variance of the data \( 4, 5, 6, 6, 7, 8, x, y \), where \( x < y \), are \( 6 \) and \( \frac{9}{4} \) respectively. Then \( x^4 + y^2 \) is equal to

Let $r \in \{p, q, \sim p, \sim q\}$ be such that the logical statement $r \vee (\sim p) \Rightarrow (p \wedge q) \vee r$ is a tautology. Then $r$ is equal to:

The statement \( \big(\sim(p \Leftrightarrow\sim q)\big)\land q \) is:

Let \( p,q,r \) be three logical statements. Consider the compound statements
$S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)$ and $S_2 : p \rightarrow (q \vee r)$
Then, which of the following is NOT true?

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JEE Main 2022 28th July Morning Shift

Let the operation \(\ast, \odot \in \{\land, \lor\}\). If \( (p * q)\\odot\ (p \odot \sim q) \) is a tautology, then the ordered pair \( (*,\ \odot) \) is: