Which of the following compound statements is/are true?
p : `sqrt7` is an irrational number and `sqrt(12)` is a rational number
q : Cow gives us milk and ox is used for ploughing
r : 105 is multiple of 3, 5, 7 and 35.

Options

A. p only
B. q and r
C. p and r
D. All of these

Answer

To determine which of the compound statements \( p \), \( q \), and \( r \) are true, we will analyze each statement step by step.

### Step 1: Analyze Statement \( p \)
Statement \( p \): "The square root of 7 is an irrational number and the square root of 12 is a rational number."

1. **Evaluate \( \sqrt{7} \)**:
- The square root of 7 is indeed an irrational number. This part of the statement is **true**.

2. **Evaluate \( \sqrt{12} \)**:
- We can simplify \( \sqrt{12} \) as follows:
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
- Since \( \sqrt{3} \) is irrational, \( 2\sqrt{3} \) is also irrational. Therefore, this part of the statement is **false**.

3. **Combine the evaluations**:
- The statement \( p \) uses the conjunction "and". Since one part is true and the other is false, the entire statement \( p \) is **false**.

### Step 2: Analyze Statement \( q \)
Statement \( q \): "A cow gives us milk and an ox is used for ploughing."

1. **Evaluate the first part**:
- "A cow gives us milk" is **true**.

2. **Evaluate the second part**:
- "An ox is used for ploughing" is also **true**.

3. **Combine the evaluations**:
- Both parts are true, and since they are connected by "and", the entire statement \( q \) is **true**.

### Step 3: Analyze Statement \( r \)
Statement \( r \): "105 is a multiple of 3, 5, 7, and 35."

1. **Evaluate each part**:
- **Multiple of 3**:
\[ 105 \div 3 = 35 \quad (\text{True}) \]
- **Multiple of 5**:
\[ 105 \div 5 = 21 \quad (\text{True}) \]
- **Multiple of 7**:
\[ 105 \div 7 = 15 \quad (\text{True}) \]
- **Multiple of 35**:
\[ 105 \div 35 = 3 \quad (\text{True}) \]

2. **Combine the evaluations**:
- All parts are true, and since they are connected by "and", the entire statement \( r \) is **true**.

### Conclusion
- Statement \( p \) is **false**.
- Statement \( q \) is **true**.
- Statement \( r \) is **true**.

Thus, the true compound statements are \( q \) and \( r \).

### Final Answer
The true statements are \( q \) and \( r \).

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