To solve the problem, we need to evaluate the truth values of the statements \( p \), \( q \), and \( r \) and then analyze the given options based on these truth values.
### Step 1: Evaluate the statements
1. **Statement \( p \)**: "4 is an even prime number."
- The number 4 is even, but it is not a prime number (the only even prime number is 2). Therefore, \( p \) is **false**.
2. **Statement \( q \)**: "6 is a divisor of 12."
- Since 12 divided by 6 equals 2, 6 is indeed a divisor of 12. Therefore, \( q \) is **true**.
3. **Statement \( r \)**: "The HCF of 4 and 6 is 2."
- The factors of 4 are 1, 2, and 4; the factors of 6 are 1, 2, 3, and 6. The highest common factor (HCF) is 2. Therefore, \( r \) is **true**.
### Step 2: Analyze the options
Now we have:
- \( p \): false
- \( q \): true
- \( r \): true
We will analyze the truth values of the given options based on logical operations.
1. **Option 1**: \( p \land q \)
- \( p \) is false and \( q \) is true.
- \( p \land q \) (false AND true) = **false**.
2. **Option 2**: \( p \lor q \land \neg r \)
- \( p \) is false, \( q \) is true, and \( r \) is true (thus \( \neg r \) is false).
- \( p \lor q \) = false OR true = **true**.
- Now, \( true \land false \) = **false**.
3. **Option 3**: \( \neg p \land r \)
- \( \neg p \) is true (since \( p \) is false) and \( r \) is true.
- \( \neg p \land r \) = true AND true = **true**.
4. **Option 4**: \( \neg p \lor (q \land r) \)
- \( \neg p \) is true, \( q \) is true, and \( r \) is true.
- \( q \land r \) = true AND true = **true**.
- Now, \( \neg p \lor (q \land r) \) = true OR true = **true**.
### Conclusion
- From the analysis:
- Option 1: false
- Option 2: false
- Option 3: true
- Option 4: true
Since both Option 3 and Option 4 are true, we conclude that at least one of the options is true.
### Final Answer
The true statements are Options 3 and 4.
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