Which of the following statements is true ?

Options

A. Product of two irrational numbers is always irrational
B. Product of a rational and an irrational number is always irrational
C. Sum of two irrational numbers can never be irrational
D. Sum of an integer and a rational number can never be an integer

Answer

To determine which of the given statements is true, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Statement 1**: The product of two irrational numbers is always irrational.
- Let's consider two irrational numbers: √3 and √7.
- The product is √3 * √7 = √(3 * 7) = √21, which is indeed irrational.
- However, consider another pair: √2 and √2.
- The product is √2 * √2 = √4 = 2, which is rational.
- Therefore, this statement is **false**.

2. **Statement 2**: The product of a rational number and an irrational number is always irrational.
- Let’s take a rational number, say 2, and an irrational number, say √3.
- The product is 2 * √3, which is irrational.
- In general, multiplying a non-zero rational number by an irrational number will always yield an irrational number.
- Thus, this statement is **true**.

3. **Statement 3**: The sum of two irrational numbers can never be irrational.
- Consider two irrational numbers: √2 and -√2.
- Their sum is √2 + (-√2) = 0, which is rational.
- Therefore, this statement is **false**.

4. **Statement 4**: The sum of two integers and a rational number can never be an integer.
- Let’s take two integers, say 3 and 2, and a rational number, say 0.5.
- Their sum is 3 + 2 + 0.5 = 5.5, which is not an integer.
- However, if we take two integers, say 3 and 2, and a rational number that is also an integer, say 1, then 3 + 2 + 1 = 6, which is an integer.
- Therefore, this statement is also **false**.

### Conclusion:
The only true statement among the given options is **Statement 2**: The product of a rational number and an irrational number is always irrational.