Which of the following compound statements is/are true?
p : All fraction numbers are rational or all rational numbers are fraction
q : Number of prime factors of 12 is 3 or number . of total factors is 6
r : Square of an integer is always positive or cube · of an integer may be negative or positive

Options

A. p only ·
B. q only
C. q and r
D. p, q and r

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**: "All fraction numbers are rational or all rational numbers are fractions."

- Let \( A \): "All fraction numbers are rational."
- Let \( B \): "All rational numbers are fractions."

**Analysis**:
- A fraction is defined as a number that can be expressed in the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \). By definition, all fractions are rational numbers.
- Therefore, \( A \) is **true**.

- A rational number can be expressed as \( \frac{p}{q} \) where \( p \) and \( q \) are integers. For example, the rational number \( 1 \) can be expressed as \( \frac{1}{1} \), but it is not considered a fraction in the traditional sense. Thus, \( B \) is **false**.

Since \( A \) is true and \( B \) is false, the compound statement \( p \) (A OR B) is **true**.

### Step 2: Analyze Statement \( q \)

**Statement**: "Number of prime factors of 12 is 3 or number of total factors is 6."

- Let \( A \): "Number of prime factors of 12 is 3."
- Let \( B \): "Number of total factors of 12 is 6."

**Analysis**:
- The prime factorization of 12 is \( 2^2 \times 3^1 \). The prime factors are \( 2 \) and \( 3 \), so the number of distinct prime factors is **2**. Therefore, \( A \) is **false**.

- The total factors of 12 are \( 1, 2, 3, 4, 6, 12 \), which gives us a total of **6 factors**. Therefore, \( B \) is **true**.

Since \( A \) is false and \( B \) is true, the compound statement \( q \) (A OR B) is **true**.

### Step 3: Analyze Statement \( r \)

**Statement**: "Square of an integer is always positive or cube of an integer may be negative or positive."

- Let \( A \): "Square of an integer is always positive."
- Let \( B \): "Cube of an integer may be negative or positive."

**Analysis**:
- The square of any integer \( n \) (whether positive or negative) is \( n^2 \), which is always non-negative. Therefore, \( A \) is **true**.

- The cube of an integer \( n \) is \( n^3 \). If \( n \) is positive, \( n^3 \) is positive; if \( n \) is negative, \( n^3 \) is negative. Therefore, \( B \) is also **true**.

Since both \( A \) and \( B \) are true, the compound statement \( r \) (A OR B) is **true**.

### Conclusion

All three statements \( p \), \( q \), and \( r \) are true.

**Final Answer**: All statements \( p \), \( q \), and \( r \) are true.

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